FUN OFRO
UR 553
At 7:22
UR-553
AEC RESEARCH AND
DEVELOPMENT REPORT
O
AUG
3
1960
RARY
A THEORETICAL AND EXPERIMENTAL INVESTIGATION OF THE TEMPERATURE
RESPONSE OF PIG SKIN EXPOSED TO THERMAL RADIATION
by
Thomas P. Davis
THE UNIVERSITY OF ROCHESTER
ATOMIC ENERGY PROJECT
MELIORA
ROCHESTER, NEW YORK
OTVAS
LEGAL NOTICE
This report was prepared as an account of Government
sponsored work. Neither the United States, nor the Commission,
nor any person acting on behalf of the Commission:
A. Makes any warranty or representation, expressed or
implied, with respect to the accuracy, completeness, or
usefulness of the information contained in this report, or
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disclosed in this report may not infringe privately owned rights;
or
B. Assumes any liabilities with respect to the use of,
or for damages resulting from the use of any information,
apparatus, method, or process disclosed in this report.
As used in the above, "person acting on behalf of the
Commission" includes ariy employee or contractor of the Commission,
or employee of such contractor, to the extent that such employee
or contractor of the Commission, or employee of such contractor
prepares, disseminates, or provides access to, any information
pursuant to his employment or contract with the Commission, or
his employment with such contractor.
UN CLASSIFIED
UR-553
Biology and Medicine
TID-4500, (15 th Ed.)
THE UNIVERSITY OF ROCHESTER
Atomic Energy Project
P. 0. Box 287, Station 3
Rochester 20, New York
* * *
Contract W-7401-eng -49 between the U. S. Atomic
Energy Commission and the University of Rochester,
administered by the Department of Radiation Biology
of the School of Medicine and Dentistry.
A THEORETICAL AND EXPERIMENTAL INVESTIGATION OF THE TEMPERATURE
RESPONSE OF PIG SKIN EXPOSED TO THERMAL RADIATION
by
Thomas P. Davis
Division: Special Programs
Division Head:
H. A. Blair
Section:
Flash Burns
Section Head:
J. R. Hinshaw

Date Completed: 6/10/59
Date of Issue: 7/8/59
U N CLASS IF I E D
A THEORETICAL AND EXPERIMENTAL INVESTIGATION OF THE TEMPERA TURE
RESPONSE OF PIG SKIN EXPOSED TO THERMAL RADIATION
ABSTRACT
This study was undertaken as a part of the problem of predicting the
irreversible thermal injury of skin.
Of first importance in this problem is
the determination of the temperature-time-depth history of skin subjected to
an arbitrary thermal insult. Although this paper deals with radiant energy
heating of skin, it is intended that the results may be applied to other
modes of energy input, as well.
The theoretical analysis of heat flow in irradiated skin is considered
first, and the predicted responses of two models are presented.
Methods are
developed for the comparison of these theoretical predictions with experimen-
tal results; these comparison schemes define the experimental procedures and
materials which are next described in detail. Finally, the experimental
results are presented, together with an evaluation of the constants of skin,
whenever possible.
While the experimental results failed to supply all the information
necessary to compute the temperature response of skin to any arbitrary energy
input, several important conclusions were reached. A most interesting result
of the theoretical analysis is that the absorption pattern of radiation in

skin may be determined directly by temperature measurements, and the data may
be tested rigorously for their validity. Also, it is now possible to plan
future experiments which will provide the necessary data which this study
failed to supply.
Most important is the indication that the desired pre-
diction of thermal injury of skin may be achieved.
iii
TABLE OF CONTENTSTES
Page
Abstract
ii
List of Figures and Tables VT EERSTATED
viii
CHAPTER I. Introduction
1.1
Genesis of the Problem to be sent
1
1.2
Review of Previous Work
3
1.3
The Method of Approach
5
1.4
Original Contributions
6
1.5
General Conclusions
7
Footnotes
8
CHAPTER II. General Aspects of the Theoretical Analysis
2.1
The Method of Analysis
9
2.2
Derivation of the Heat Conduction Equation in a
Substance at Rest
9
2.3
Further consideration of the Heat Conduction Equation
12
2.4
The Model to be Analyzed--General Assumptions
16
2.5
Summary
19
Footnotes
20
CHAPTER III. Analysis of the Opaque Solidato

3.1
Definition of the Model
21
3.2
The Solution of the Heat Conduction Equation for
the Opaque Solid
21
3.3
Specialization of the Solution for Rectangular
Irradiance Function
24
3.4
The Fitting of Experimental Results to the
Normalized Solution
26
3.5
The Opaque Composite Solid
32
3.6
The Opaque Solid Subjected to a Trapezoidal
Irradiance Pulse
38
iv
3.7
Summary of Predictions from Opaque Solid Theory
44
3.8
Utility of the Opaque Solid Model
49
Footnotes
50
CHAPTER IV. Analysis of the Diathermanous Solid
4.1
Definition of the Model
51
4.2
General Consideration of the Absorption of Radiation
in Skin
52
4.3
Experimental Determinations of the Absorption of
Radiation in Skin
57
4.44
The Solution of the Heat Conduction Equation for
the Diathermanous Solid with Double Exponential
Absorption
63
4.5
Experimental Evaluation of the Constants of the
Dia thermanous Solid
78
4.6
Direct Experimental Determination of the Pattern of
Absorption of Radiation in Skin
80
4.7
Summarys
85
Footnotes
85
CHAPTER V. The Influence of the Quality of Radiation on the
Diathermanous Solid Response
5.1
Development of the Problem
87
5.2
Superposition of Solutions for the Diathermanous Solid
87
5.3
Superposition of the Normalized Solutions for Double
Exponential Absorption
90

5.4
Values of the Constants of Skin Obtained from the
Literature
91
5.5
The Influence of the Quality of Radiation on the
Initial Time Rate of Change of Temperature
101
5.6
Direct Determination of the Absorption Pattern of
Heterochromatic Radiation in Skin
104
5.7
Correction of Initial Slopes for Finite Irradiance
Rise Time
106
5.8
Summary
107
Footnotes
108
V
CHAPTER VI. Experimental Materials and Methods
6.1
Introduction
110
6.2.
Biological
110
6.3
Physical--General Arrangement
112
6.4
Radiation Source
112
6.4.1 Shutter System
115
6.5
Animal Holder
117
118
6.6
Temperature Sensing Elements
BOEL
6.6.1 Wollaston Wire Thermocouples
122
6.6.2 Butt-welded Thermocouples
123
6.6.3 Plated Thermocouples
124
6.6.4 Soldered Thermocouples
125
6.6.5 Calibration of Thermocouples
126
6.6.6 Placement of Thermocouples
127
6.6.7 Electrical Connections to the Thermocouples
128
6.7
Depth Measuring Device
129
6.8
Reference Elements
132
6.8.1 Temperature Regulated Water Bath
133
6.9
Calibration Circuits
133
6.10 Preamplifiers
134
6.ll Recording Assembly and System Response
137

6.12 Pulse Measuring Assembly
142
6.13 Timing Marker Generator
143
6.14 Exposure Procedure
145
6.15 Reduction of Data
1446
6.16 Summary
147
Footnotes
148
vi
CHAPTER VII. Experimental Results; Opaque Skin
7.1
Scope of the Experimental Work
150
7.2
Animal No. 1389
150
7.3
Animal No. 1418
152
7.4 Animal No. 1428
156
7.5 Animals No. 14:45 and 1455
159
7.6
Summary of Results; Average Thermal Constants of
Pig Skin
163
7.7
Discussion of Results
165
7.8
Summary
169
Footnotes
169
CHAPTER VIII. Experimental Results; Bare Skin
8.1
Scope of the Experimental Work
171
8.2
Reduction of Data
171
8.3
Summary of Temperature Response Measurements
173
8.4
Summary of Initial Slope Measurements
180

8.5
Comparison of Experimental Results with
Theoretical Predictions
180
8.6
The Source of Error in the Temperature Measurements
185
8.6.] Variation of Heat Capacity per Unit Volume with
Depth
186
8.6.2 Location of Zero Time and Extrapolation of
Responses to Zero Time
188
8.6.3 Direct Action of Radiation on the Thermocouples
189
8.7
Further Consideration of the Initial Slope
Measurements
190
8.8
Summary
194
8.9
Location of Data
196
Footnotes
196
vii
CHAPTER IX. Summary of the Study
9.1
Discussion of the Theoretical Analysis
197
9.2
Discussion of the Experimental Results
203
9.3
Consideration of the Direct Measurement of the
Absorption Pattern of Radiation in Skin
206
9.4
Conclusion
208
Footnotes
209
Bibliography
210
APPENDIX I. Nomenclature
214
APPENDIX II. The Application of Heat Flow Theory and Chemical
Kinetics to the Prediction of Irreversible
Thermal Injury of Skin
216
APPENDIX III. The "Opaque Solid Function," R(x)
229
APPENDIX IV. Solution of the Composite Opaque Solid Model
231
APPENDIX V. Solution of the Diathermanous Solid Model,
Double Exponential Absorption
240
ACKNOWLEDGEMENTS
260

viii
LIST OF FIGURES AND TABLES
Page
Figure 3-1.
Predicted Response of the Opaque Solid
27
Figure 3-2.
Predicted Response of the Opaque Solid
28
Figure 3-3.
Factor for Correcting Trapezoidal Pulse Response
to Ideal Step--Function Response, for Opaque
Solid Surface
45
Figure 4-1.
Predicted Response of the Diathermanous Solid,
Single Exponential Absorption. 12 = 1
71
Figure 4-2a. Predicted Response of the Dia thermanous Solid,
Double Exponential Absorption. 12 3, de 0.1
SAR
72
Figure 4-2b. Predicted Response of the Diathermanous Solid,
Double Exponential Absorption. 12 3, , 0.5
73
74
Figure 4-2c. Predicted Response of the Diathermanous Solid,
Double Exponential Absorption. 12 3, di = 1.0
Figure 4-3a. Predicted Response of the Diathermanous Solid,
Double Exponential Absorption. 72 = 5, 7, = 0.1
75
76
Figure 4-3b. Predicted Response of the Diathermanous Solid,
Double Exponential Absorption. 12 = 5, 1,
5, , = 0.5
Figure 4-3c. Predicted Response of the Diathermanous Solid,
Double Exponential Absorption. 12 = 5, 1, 1.0
Figure 5-1. Relative Spectral Energy Absorption, (1-R ). Ja
Chester White Pig Exposed to Carbon Arc Image
Furnace

77
95
Figure 5-2.
Linear Absorption Coefficients for Shallow ( 8 )
and Deep (82) Tissue
97
Table 5-1.
Numerical Values for Solution of the Normalized
Response Equation
99
Figure 5-3.
Predicted Temperature Response of Bare Pig Skin
Exposed to Carbon Arc Image Furnace
100
Figure 5-4.
Predicted Initial Time Rate of Change of Tempera-
ture Response of Bare Pig Skin Exposed to Carbon
Arc Image Furnace
103
Figure 6-1.
Block Diagram of the Physical Equipment
113
Figure 6-2.
The Carbon Arc Image Furnace
114,
ix
Figure 6-3.
Animal Holder, Open
119
Figure 6-4.
Animal Holder, Closed
120
Figure 6-5.
Animal Holder, in Exposure Position
121
Figure 6-6.
Amplifier-Recorder System
135
Figure 7-1.
Measured Response, Animal No. 1418 (Opaque Skin)
154
Figure 7-2.
Measured Response, Animal No. 1418 (Opaque Skin)
155
Figure 7-3.
Measured Response, Animal No. 1428 (Opaque Skin)
158
Figure 7-4.
Measured Response, Animals No. 1445 and 1455
(Opaque Skin)
160
Figure 7-5.
Measured Response, Animals No. 1445 and 1455
(Opaque Skin)
162
Figure 7-6.
Normalized Measured Response, Animals No. 144445
and 1455 (Opaque Skin)
164
Table 8-1.
Scope of the Experimental Work on Bare Skin
172
Table 8-2.
Summary of Bare Skin Temperature Responses
174
Figure 8-la. Measured Response, Bare Skin Exposures
176
Figure 8-lb. Measured Response, Bare Skin Exposures
177
Figure 8-1c. Measured Response, Bare Skin Exposures
178
Figure 8-2.
Average Measured Response, Bare Skin Exposures
179
Figure 8-3.
Measured Initial Time Rate of Change of Tempera-
ture Response, Bare Skin Exposureş
181
Figure 8-4.
Comparison of Predicted and Measured Bare Skin
Response
183

1
A THEORETICAL AND EXPERIMENTAL INVESTIGATION OF THE TEMPERATURE
RESPONSE OF PIG SKIN EXPOSED TO THERMAL RADIATION*
CHAPTER I.
INTRODUCTION
1.1
Genesis of the Problem
The expanding utilization of nuclear energy has posed many problems,
not the least of which are in the field of medicine. In particular the
existence of nuclear weapons as possible instruments of war demands planning
on an unprecedented scale for treatment of mass casualties; yet these casual-
ties may differ not only in number but in kind from anything known heretofore.
There has been, therefore, considerable acceleration in research in many
branches of the biological sciences aimed at defining the basic traumata in-
volved, and thus providing a reasonable basis for the necessary planning.
In this laboratory the flash burn has been under investigation for
over ten years (1, 2). This is the lesion resulting from the exposure of
an individual to a brief pulse of visual and infrared radiation as released
by an atomic detonation.
The importance of this problem is indicated by the
estimate of Leroy (3) that of the atomic bomb casualties in Japan, 65 to 85
percent were burned; over 90 percent of the lesions were of the primary, or
flash-burn, type. Modern high-yield weapons have intensified this problem.
Estimates from The Effects of Nuclear Weapons (4) indicate that lº burns on

bare skin will be sustained 30 miles from a 10 megaton blast; this radius
enclosed an area of almost 3000 square miles.
These estimates have been
partly derived from, and are completely supported by work in this labora-
tory (5). When it is recalled that the hospitals of Boston and the sur-
rounding area were severely taxed by the relative handful of casualties
*The material in this paper was submitted to the University of
Rochester in partial fulfillment of the requirements for the
Doctor of Philosophy degree in Biophysics.
2
from the Cocoanut Grove fire (6), the magnitude of the problem may be
appreciated.
While the reason for existence of this laboratory is thus perfectly
clear, it is also obvious that mere routine testing under specified conditions
can never supply the information needed in the planning for treatment of mass
casualties.
New and untested situations continually arise for which no data
are available.
Further it would be highly desirable to inter-compare re-
sults obtained with various modes of energy input; this would make results
from the older literature useful, and would aid in integrating the findings
of various laboratories.
Therefore, this laboratory has attempted to focus
major attention on the more basic problems of the flash burn, in the hope
of developing principles rather than specific results.
With the continual development and refining of experimental materials
and methods it has become increasingly clear that highly precise tools are
now available for the study, not merely of the atomic bomb burn, nor even
of the more general radiant energy burn, but of the burn, per se.
That is,
it has now become feasible and most desirable to attempt to elucidate the
fundamental mechanisms of irreversible tissue damage produced by a hyper-
thermic episode.
This problem may be termed "the prediction of irreversible

thermal injury of skin"; the desideratum is the ability to define "damage"
mathematically in terms of the characteristics of the thermal insult.
This over-all problem has been treated by the writer in a paper pre-
pared several years ago.
This report (from which some of the foregoing
material has been taken) is included as Appendix II of this paper; while
many of the ideas have since been discarded, the general approach therein
described is still valid.
In brief, the prediction problem is specialized
to exclude photochemical reactions; it then follows that "damage" is a
function of the temperature of the tissue involved, while the temperature,
3
in turn, is dependent upon the characteristics of the energy input.
In
the terminology of Appendix II:
s
♡ (7)
where
ㅜ ​T
T (9" x t no
(All symbols are defined in Appendix I.)
Thus, the prediction of thermal injury of skin is now broken down into
two sub-problems: first the prediction of the temperature response of skin
for an arbitrary energy input, and second the prediction of irreversible
tissue injury based on this temperature response.
It is with the first prob-
lem, the determination of the temperature response of skin that this study
is concerned.
It has been the purpose of the above remarks to define the rationale
of this study. It is, clearly, but a small part in the total problem of burn
prediction; however extension of the results to the latter will not be
attempted here, but must await further research.
1.2 Review of Previous Work

An extensive historical review of the literature will not be given
here; rather, citations of pertinent reports will be made as appropriate
throughout this paper. However, two particularly interesting studies will
be mentioned, since they have served as guides for much of the research
here presented.
Apparently, the first worker in this field to attempt a mathematical
formulation of thermal injury was F. C. Henriques, Jr., working with A. R.
Moritz and others.
His work is described briefly in Appendix II, and in full
in a series of papers, two of which (7, 8) are of particular interest here.
The procedure was as outlined above :
first the temperature response of skin
(to long-time application of hot water) was investigated theoretically and
4
experimentally (7). These data were then used in predicting certain burn
thresholds by means of a "punishment integral," which was based on reaction
kinetics considerations (8). The agreement between theory and experiment is
extraordinarily good; too good, one might suspect, to be completely
fortuitous. However, application of Henriques' punishment integral to radiant
energy burns leads to predictions (9, 10), which are seriously at variance
with recent data from this laboratory. It is possible that the difficulty
lies in the method used in computing tissue temperatures (see Appendix II).
The second study is that of Buettner, who utilized Henriques' punish-
ment integral in a slightly altered form, and estimated skin temperatures
by a numerical analysis (11). This very thorough analysis culminates in a
table giving the predicted depth of thermal injury for various radiant energy
exposures (12). The postulated experimental conditions were such that they
could be fulfilled quite precisely by the equipment of this laboratory. An
experiment was therefore carried out to test these predictions (13); in all
cases, the actual lesions were considerably less severe than predicted. Here
again, one may question the accuracy of the temperature predictions.
These two studies would appear to be the only ones in the literature
which attempt to predict injury from radiant energy exposures.
Since neither

has been fully in agreement with existent experimental data, it is clear that
further work is necessary.
As indicated in the above section, the proper
starting point is the determination of the correct temperature response of
skin. Indeed, it may be that when this information is obtained, the dis-
crepancies in Henriques' or Buettner's work may disappear, and the ultimate
goal of the prediction of irreversible thermal injury will have been
achieved.
5
1.3 The Method of Approach
The problem of predicting the temperature response of skin is completely
solved when one has specified the correct differential equation describing
the heat flow in skin, and determined the numerical values of the constants
of the system. If the equation is completely general, the predictions will
hold for any general conditions; if the equation is restricted to certain
cases, the predictions are likewise restricted.
General solutions of the classical heat flow equation are not obtain-
able, as pointed out in the next chapter; it is necessary to specialize this
equation if solutions are to be obtained.
The method adopted here, then, is
the so-called model approach. First a simplified model of skin and the
tissue-energy interaction is proposed which will allow the general heat flow
equation to be so specialized as to be amenable to formal mathematical
attack. Analytical solutions of this specialized equation are next obtained,
which solutions are then normalized, or expressed in dimensionless variables,
so as to be applicable to any system satisfying the basic requirements of the

model.
The normalizing factors will, in general, involve the unknown con-
stants of the system.
These normalized solutions (which may be presented in numerical form)
must now be examined for possible me thods of comparison with experimental
data.
When such comparison schemes are developed, they will define the
appropriate experimental procedures to be followed.
When these experiments
have been completed and the necessary data obtained, the comparisons are
made, and if the agreement be satisfactory, then the normalizing factors--
i.e., the unknown constants of the system--may be evaluated, and the nor-
malized solutions specialized to the particular system under investigation;
namely, skin.
If the agreement be unsatisfactory, then one must either alter
the original model or the experimental methods, whichever is at fault.
6
In a later chapter it will be shown that a most interesting result
of the theoretical analysis permits one to interpret the experimental data
independently of any choice of a model; indeed the data themselves can define
the model.
This is one of the most important results of this study, and will
be treated in some detail.
1.4 Original Contributions
The derivation of the heat conduction equation presented in the next
chapter is obviously not original, and has been given only for comparison
with the more detailed derivation of Section 2.3, which, so far as the author
is aware, has not been given before, and is of interest primarily for its
consideration of the enthalpy function. Also, as indicated by the references
cited, the solution presented in Chapter III for the opaque solid model is
not original, nor is the method selected for evaluation of the thermal inertia.
However, the development of a method for the evaluation of thermal diffusivity
is an original contribution, as is the treatment of the surface response of
the composite opaque solid, together with the correction for finite rise time
of the irradiance pulse.
The diathermanous solid model with double exponential absorption, as

defined in Chapter IV, was first proposed by Hardy, but the analytical
solution for this model has apparently not been given before. Likewise, it
is believed that the extensive treatment of the initial time rate of change
of temperature response of the diathermanous solid, and the utilization of
this factor in the analysis of experimental results is original with this
study, and is a contribution of some value.
The extension of the solution
to the case of heterochromatic radiation depends directly upon superposition
of solutions, which is so obvious that it would be embarrassing to claim
originality for this development, although it seems not to have been treated
7
explicitly in the literature.
In any event, the resultant curves of Figures
5.-3 and 5-4 are original with this study, and, it is hoped, represent the
best predictions to date of the response of bare pig (and human?) skin to
solar-like radiation.
While the experimental materials and methods used in this study were
quite straightforward, the simultaneous recording of the irradiance pulse and
the temperature response is an interesting contribution of some merit.
An
extensive discussion of the experimental results will be found in later
chapters, but it might be mentioned here that the clear demonstration of errors
in the measurements, and the lengthy investigation of the source of these
errors is a unique feature of this study.
1.5 General Conclusions
For the measurement of the so-called "thermal constants" of skin,

the theoretical predictions and experimental results were in good agreement,
which permitted these constants to be evaluated with considerable confidence.
However, for the determination of the "optical" constants, or more specifi-
cally, the absorption pattern of radiation in the skin, the experimental
measurements were seriously in error.
Thus, the completely general prediction
of the temperature response of skin to an arbitrary thermal insult has not
been achieved by this study; on the other hand, the results are of great
value in preparing specific plans for the work which remains to be done.
Aside from the demonstration that the combination of the theoretical
and experimental approaches is not only reasonable but necessary, a most
important contribution of this study is the indication that the future re-
search is practicable, and the goal of the prediction of irreversible thermal
injury of skin may be achieved.
8
FOOTNOTES
(1) Pearse, H. E., J. T. Payne, and L. Hogg, The Experimental Study of
Flash-Burns, Ann. of Surg. 130, 774-789 (1949).
(2) Pearse, H. E. and H. D. Kingsley, Thermal Burns from the Atomic Bomb,
Surg. Gyn. and Obst. 98, 385-394 (1954).
(3) Leroy, G. V., Medical Sequelae of the Atomic Bomb Explosion, J. Am.
Med. Assoc. 134, 1143 (1947).
(4) Glasstone, S., ed., The Effects of Nuclear Weapons, 299, Fig. 7.47
(U. S. Government Printing Office, Washington, D.C., 1957).
(5) Lerman, B. and J. R. Hinshaw, Relationships Between Burn Severity and
the Simulated Thermal Pulses of Various Nuclear Weapons, University
of Rochester Atomic Energy Project Report UR-533 (1958).
(6) Pearse, H. E., Thermal Injuries of Nuclear Warfare, Military Medicine
118, 274 (1956).
(7) Henriques, F. C., Jr. and A. R. Moritz, Studies of Thermal Injury I.
The Conduction of Heat to and Through Skin and the Temperatures
Attained Therein. A Theoretical and Experimental Investigation,
Am. J. Path. 23, 531-549 (1947).
(8) Henriques, F. C., Jr., Studies of Thermal Injury V. The Predictability
and the Significance of Thermally Induced Rate Processes Leading to
Irreversible Epidermal Injury, Arch. Path. 43, 489-502 (1947).
(9) Henriques, F. C., Jr. and R. A. Maxwell, The Applicability of the Skin
Damage Integral to the Prediction of Flash Burn Injury Thresholds,
Including those Caused by Atomic Detonations, Technical Operations,
Inc., Report TOI 54-19 (1956). (SECRET).
(10) The Thermal Data Handbook, Armed Forces Special Weapons Publication
AFSWP 700 (1954). (SECRET).
(11) Buettner, K., Effects of Extreme Heat and Cold on Human Skin III.
Numerical Analysis and Pilot Experiments on Penetrating Flash Radiation
Effects, J. Appl. Physiol. 5, 207-220 (1952).

(12) Ibid. 216.
(13) Davis, T. P., Unpublished research (1956).
CHAPTER II FORUM
GENERAL ASPECTS OF THE THEORETICAL ANALYSIS
2.1
The Method of Analysis
The general procedure in the theoretical analysis of the conduction
of heat in skin is determined by the choice of a model approach as described
previously.
Of first importance in this approach is the obtaining of a for-
mal solution, in dimensionless terms, of the system. Obviously, then, one
cannot employ numerical methods; indeed, the constants necessary for such
methods are precisely the unknowns of the problem.
The only suitable pro-
cedure is a formal, analytical attack on the classical differential equation
of heat conduction.
2.2 Derivation of the Heat Conduction Equation in a Substance at Rest
The derivation of this equation is based upon the experimentally de-
duced equation of Fourier (1) which states that the rate of heat conduction,
q, between two plates separated by some specific material is directly pro-
portional to the area of the path connecting the plates, A, and their tem-
perature difference, AT; and inversely proportional to the separation of
the plates, Ax.
The constant of proportionality is termed the heat con-
ductivity, k, and may be considered a basic property of the material
separating the plates. Explicitly:

9: kA
А
AT
(2-1)
AX
This equation may be written in differential form by allowing Ax to
decrease to the infinitesimal length dx, and considering the process only
during the infinitesimal time interval dt.
The differential heat conducted
in the x direction in this time is then
dQ=- kA dI dt
(2-2)
10
where the term dt/dx is called the temperature gradient, and the minus sign
is introduced to maintain the positive sense of heat flow in the positive x
direction.
(Heat flows from a region of higher temperature to one of lower
temperature; i.e., in the direction of a negative temperature gradient.)
Now, following the standard procedure employed in the derivation of
the equation of heat conduction (2), one selects as a system a differential
volume element av = dx dy dz in the interior of a body through which heat is
flowing, and considers the components of heat flow in the directions x, y,
and z.
While the presence of distributed heat sources or sinks within the
body is permitted, no movement of parts of the body, as flow of fluid, for
example, is allowed. Temperature is considered a function of space coordinates
and time; i.e., T = T(x, y, z, t).


z
1
dz
dQ,,,
- dQz, x
4
Idy
(*4,2).
dx
x
In the time interval dt, the heat entering this system from the
positive x direction, dQ],x, will be, from equation (2-2):
ƏT(x, 4.2,t) dt,
kx (dy.dz)
dQ1, x = kx (dy.dz)
where kx is the thermal conductivity in the x direction at the point X.
Әх

Now, the value of kx will, in general, depend upon x and T, independently,
or ky = kx(x,r). But the latter is a function of the former, so that one
may consider kx as a function of x alone, that is:
=
2 kx
dx
2x
not eeh
dkx =
akx
2x

11
The heat flow out of the differential volume in the positive x-direction,
dQ2,x, at x + dx, is given by
d Q 2, (kx + Olen dx)(dy.dz) ((*, 4, 2, 6)
2T (x, y, z, t) dx 7
+
Х
dx
X
Hence, the net heat flow into the system from the x direction is
at (x, y, z tl at
2x
2x
+
d Qi, x- dQ2x = -Kx (dy dz)
+ Kx (dy.dz)
2T
at (xy, z, t) at
+ Kx (dx dy.dz) Q2Tlxu zal de
akx
dx
(dx.dy.dz) 07/2,4,3,7)
dt
+ 2 (4x4 đề
22T (x, y, ztl dx.dt
2
2x²
(dx.dy.dz) [K. (x,,2,0) ]dt
x = T(x, y, z
axt
2x
عالی به موارد ، و ف ا ا ا ا ا ا ا ) -
Exactly similar expressions may be written for the component of heat
flow in the y and z directions, giving the net heat flow into the system:
+
dQ.-dQ2 = (dxody.dz. dt) [Ez (k« BJ) = o(ky )
22 (ka
21 )]
ake a2I dx +
aky a ² I dy
ako 2²T
+
(2-3)

Continuing with the usual procedure, a heat storage factor dQ3 is de-
fined as the product of the total system heat capacity at constant pressure,
Now, Cp = mcp)
Cp, and the average temperature rise, dT:
dQz = Cp dT.
mcp, where m is the mass of the system and cp the specific heat
capacity, while m =
pdV, where
Р P
is the density of the material. Further
dT
әТ
dt, hence
at
ОТ
dt
dQz = p Cp (dx.dy.de)
(2-4)
at
12
Finally, the influence of distributed heat sources or sinks is considered
by defining the heat production in the system, dQ4, as
d Q4
(2-5)
where q'il is the rate of heat production per unit volume.
(The problem of
a point source, which in essence implies infinite temperature, will not be
considered here.)
Now, taking a Wheat balance," one states that heat storage dQ3
(equation 2-4), must be equal to the net heat input dQ] - dQ2 (equation 2-3)
plus the internal heat generation dQL (equation 2-5),
or
pop Idx dy dz dt) 1 = (dx.dy.dede) fille
, FL) + f (ley BJ)
2
oky 22 I dy
(kce 37) : du
+
ay
take a ? I da
qu?
az
If the "space-time" volume element db = dx dy dz dt is cancelled on each side
of the above equation, then in the limit, as dx, dy and dz all approach
zero, one obtains the classic heat conduction equation:

por meno leve trong (key ) + gom (ke ) + 9.".
(2-6)
2.3 Further Consideration of the Heat Conduction Equation
With the basic equation of heat conduction now established (equation
2-6) one may proceed to the specification of suitable geometrical configura-
tions and boundary conditions which will allow analytical solutions of the
equation to be obtained.
Before doing so, however, it is of interest to
consider the derivation in some detail.
At the outset, it should be noted that Fourier's papers go back to
at least 1807 (although publication was delayed until 1822), while the crucial
13
experiments of Joule were not performed until 1840. Thus, Fourier's work
antedated by some 30 years the firm establishment of the mechanical theory of
heat and the exact expression of the First Law of Thermodynamics.
It should not be too surprising, therefore, to note that all statements
in the previous section are quite correct if one substitutes everywhere the
word "caloric" for "heat," and if one is willing to renounce the First Law
in favor of the old theory, assuming that caloric is a massless fluid.
(It
will be shown later that it would be convenient to endow caloric with inertia,
however.) Actually, this should not be too distressing, since the science
of heat transfer is quite distinct from thermodynamics, and it is a real
simplification for workers in the former field to consider the thermo-
dynamicist's Wheat" as a fluid, strictly analogous to the "electric fluid"
popular with early workers in electricity.
Obviously, it is not impossible, nor particularly difficult, to
phrase the derivation of equation (2-6) in agreement with proper thermo-
dynamic concepts.
The first change necessary is to note that Q is not a
function of thermodynamic coordinates, but depends upon the path traversed
by a particular system; hence an infinitesimal amount of heat is not an
exact differential, dQ, but an inexact differential, here represented by
SQ. With this rather trivial change in nomenclature, equation (2-3). But
follows precisely as derived above, with the left-hand side reading te
8 Q1 - 8Q2.
The next factor considered, the "heat storage," requires somewhat

more extensive consideration. Quoting from Zemansky (3) "Heat is energy in
transit.... When the flow of heat has ceased, there is no longer any occasion
to use the word heat." The term "heat storage," then, is simply devoid of
meaning.
14
Now the system being considered is of differential volume dx dy dz,
and processes occur in a differential time interval dt. Recognizing that
these differentials may independently be made as small as desired, one may
be assured that the system will always be infinitesimally near equilibrium
states (i.e., the system undergoes infinitesimal quasistatic processes);
hence thermodynamic coordinates may be written for the system as a whole.
It is then permissible to speak of the enthalpy, H, of the system.
The
enthalpy, in turn, may be considered to be a function of any two of the three
thermodynamic coordinates of this closed system, temperature, T, pressure, P,
and volume, V. Selecting the first two:
HE HIT, P).
Hence
dH=(), dT+
CH), dp,
(ОН)
OPT
and for an isobaric process (dP
0)
dHz (OH), dt,
or
DHE Cp dT
(2-7)
from the definition of C
Cp
Now, the change in temperature, dt, in equation (2-7) is a change in
0

time only, since spatial dimensions are not thermodynamic coordinates.
(The system is always infinitesimally near equilibrium states.) Hence,
equation (2-7) may be rewritten as
I dt
d H= Cp at
or, as in equation (2-4):
d H= PC (dx dy dz) aI.dt
at
(2-8)
Finally, the "distributed heat source" factor (equation 2-5) must be
considered.
The terminology again is poor, for what is actually being des-
cribed is the performance of work, SW, on the system. This is clear,
15
since this process can change the state of the system by other than a heat
flow process.
This is not ordinary "P DV" work, but at the molecular level,
represents increased molecular kinetic energy due to dissipative processes.
Hence, one may write
- SW:9" (dx.dy da dt)
(2-9)
in direct analogy with equation (2-5). The minus sign is used to indicate
that work is performed on the system.
The heat flow equation is now obtained by substitution in the dif-
ferential expression of the First Law:
dE = d Q - dw',
where the SW' term includes all work elements including P dV, and E is the
internal energy of the system. Letting SW' = P AV + d W, where S W repre-
sents the "non-P dV" work elements, one obtains
dE=SQ - Pov-dw.
From the definition of the enthalpy function
H = Et Py
one may write
de dH-PdV-VIP

which upon substitution in the expression for the First Law gives
dH-Pdv-vdp. SQ- pdv-sw
or
d H= SQ-dW+Vdp.

Now, for an isobaric process the V DP term is zero, and substitution
of equation (2-8) for dh, equation (2-3) for SQ = SQ] - 8Q2, and
equation (2-9) for 8 w, yields, in the limit:
pce a Parks PHD) - okna (ky ) Balke 3 - que
at
16
which is in exact agreement with equation (2-6).
Thus, the heat conduction equation may be derived without recourse to
such barbarous terms as "heat storage" or "heat generation," nor to such a
questionable concept as a "heat balance” which implies some sort of con-
servation of "heat fluid."
This is admittedly pedantic, but it is of real
interest to note that a careful derivation of this equation has introduced
no new limitations on its validity beyond the stated requirement that no
movement of parts of the body is allowed, and the restriction to a constant
pressure process implied in writing equation (2-4). (For a constant volume
process, one need only replace cp by Cyo This follows from the definition
of Cv = () v, whence de Cv dT for dV = 0.) Also noteworthy is the
point, brought out in this derivation, that equation (2-6) is valid in the
neighborhood of a discontinuity, but not "on" the discontinuity, since here
one cannot define the thermodynamic coordinates of a differential volume
element.
2.4 The Model to be Analyzed
General Assumptions
A general solution of the heat conduction equation does not exist.
In order to proceed, one must set up a model within the framework of which
the equation can be so specialized as to be amenable to formal or numerical
methods of attack.
This model must specify the geometrical configuration of

the system, and a sufficient number of boundary conditions (including
initial conditions).
The ability of equation (2-6) to predict temperatures in agreement
with those which actually occur in a real, physical system is completely
dependent upon how faithfully the model describes the true situation. In
even rather ideal conditions, the solution of this equation is not simple,
particularly in the non-steady state (i.e. aT/ at €0). Here, however,
17
it is desired to predict the temperature rise in skin exposed to brief pulses
of radiant energy. If one were to require that the model reflect with high
precision the detailed anatomy of the skin and the peculiarly difficult
processes of radiant energy absorption therein, an overwhelming complex
system would result. Clearly, for any progress, the model must embody rather
severely simplifying assumptions.
First, the skin will be replaced by a so-called semi-infinite solid.
This is a solid which extends infinitely far in both positive and negative
y and z directions, and in the positive x direction. (As in the above
sections, x, y, and z are mutually perpendicular Cartesian coordinates.)
By definition, the surface of the solid coincides with the y-z plane, or
the plane x = 0; this choice has been made simply as a matter of convenience.
Next, it will be assumed that the thermal conductivity in the x-direction,
kx, the density, p , and the specific heat capacity, Cp, are all independent
of y and z. Initially the solid is to be at uniform temperature throughout,
and at zero time the surface is exposed uniformly to an arbitrary pulse of
radiant energy. The absorption pattern of this radiation, whatever it may
be, is likewise assumed independent of y and z.
With these assumptions only, considerable simplification of equation
(2-6) has already been achieved.
Of first importance is the result that the
isothermals will be plane surfaces parallel to the surface of the solid;

i.e., the temperature will be independent of y and z.
The heat conduction
equation thus reduces to the so-called "unidimensional" form
IT(x, t)
()
РСР
9" (x, t),
at
(2-10)
**O, t?0.
Actually, this is an equation in two dimensions, x and t, and the functional
dependence has been explicitly indicated.
The equation is defined in the
18
region x > 0 and t = 0, which has also been indicated explicitly.
In addition, two boundary conditions have been established. First,
since the solid is initially at uniform temperature throughout, one may
write:
Tlx,0) =
To
Secondly, it is physically reasonable that as x tends to infinity, the tempera-
ture will be unaffected by any energy input at the surface of the solid; i.e.
Lim T(xt) To
This latter condition will hold so long as the radiant energy input,
per unit area of surface of the solid, is finite.
If one considers a column
of this solid, of unit cross-sectional area, and extending in the positive
x-direction, the total heat capacity of this column will tend to infinity as
the length of the column tends to infinity. For finite energy input to this
column, the average temperature rise, over all x, must then be
zero,
or

toimen + S.[T(8,4) - To] ds.0.
It follows then that [T (x,t) - T. ) can be different from zero for a finite
range of values of x, only, which establishes the validity of the condition
given above.
As a matter of convenience, a new temperature scale will be defined
by
U = T - TO
so that the symbol U means temperature rise (in Kelvin or Centigrade degrees)
above initial temperature.
The symbol T is reserved for absolute temperature
(in degrees Kelvin). Also, the terms p and Cp will always appear as the
product (pop); hence the symbol v will be defined by V = pep. Equation (2-10)
and the boundary conditions thus far established now become:
19
v
OU (x, t)
& kxlx
pe [kx(x) ] + 91" (x, t), *20,720
(2-11)
W(x,0) =
(2-12)
W (0,t)
(2-13)
using the abbreviated notation U(00,t) for Lim U(x,t).
One additional assumption will now be made.
In the derivation of the
heat conduction equation, the final expression was obtained by substitution
in the differential form of the First Law:
dH = -
SQ - SW + V dP. The
left-hand side is given by equation (2-8) as
d H= PCp (dx.dy de dt) at
at
Thus far, no assumptions concerning the constancy of p and
Cp
have been
made since only a differential temperature change (dT = a I at) is being
considered. In the final expression the differential space-time volume
element (dx dy dz dt) is cancelled out, leaving one apparently free to con-
sider any temperature rise.
Such treatment is allowed only if suitable
analytical expressions for p and Cp as functions of temperature are used.
In the following sections, the assumption will be made that the product
le cp) = v is a constant.
One may take a slightly different view of this assumption by recog-

nizing that v is, in fact, one of the unknowns of the system. Hence, the
value which will be determined by experiment is an average over the tempera-
ture range utilized.
2.5 Summary
While the various assumptions made above have resulted in considerable
simplification of the basic equation of heat conduction, the resultant
20
equation (2-11) is still not sufficiently specialized to permit a formal
solution to be obtained. The functional forms of both ky(x) and q'''(x,t)
must be specified, and one additional boundary condition given. In the
following chapters, two specific models will be proposed which will provide
this needed additional information. Normalized solutions of the heat con-
duction equation will then be obtained, and methods for the experimental
evaluation of the various constants of the models will be developed. The
way will then be prepared for a logical presentation of the experimental
materials and methods which were employed in this investigation.
FOOTNOTES
(1) Jakob, M., Heat Transfer, Vol. I, 2 (John Wiley and Sons, Inc., New
York, 1949).
(2) Ibid., 9.
(3) Zemansky, M. W., Heat and Thermodynamics, Third Edition, 63 (McGraw-Hill
Book Company, Inc., New York, 1951).

21
CHAPTER III
ANALYSIS OF THE OPAQUE SOLID
3.1 Definition of the Model
A particularly simple and quite useful model is the isotropic semi-
infinite opaque solid. Since the solid is opaque, the impinging radiation is
either absorbed at the mathematical surface (x = 0), or is reflected, the
reflectance being considered a constant.
The conductivity is assumed to be
independent of temperature and position (ky(x) = k), and it is also assumed
that there are no distributed sources or sinks (q'''(x,t) = 0). Additionally,
the surface (x = 0) is considered to be insulated against all heat losses,
so that the only heat flow across the surface is due to the absorbed radiant
energy. From the equation of Fourier, as expressed by equation (2-2), this
last statement implies:
L/ขาว Oulx,)
x >07
+ 11-R) HTT),
2x
k
where the irradiance, H(t), is the rate of radiant energy incident per unit
area, and R is the reflectance of the surface. Since one always considers
the absorbed irradiance, or (1-R)H(t), the symbol Ha(t)
(1-R)H(t) will be
used for convenience.
3.2 The Solution of the Heat Conduction Equation for the Opaque Solid (1)
Under the assumptions above concerning k(x) and q'''(x,t), the right

hand side of equation (2-11) becomes simply
k
2²U (x,t)
,
8x²
Dz[k bulut)] + 0 = K O* U13,7)
Using the definition of thermal diffusivity, a,
-
k
pcr
H 습
​22
the heat conduction equation reduces for this model, to:
dulx, t)
ozu (xt), *20, 720
Әx 2
(3-1)
W(x,0)
=0, x=0
(3-2)
U 100,) = 0, tzo
(3-3)
dow 10,t)
ou(0,t).-€ Halt), +20.
(3-4)
5x
The solution of this boundary value problem is readily obtained by
application of the Laplace transformation (2). This standard method reduces
the partial differential equation (3-1) in x and t to an ordinary differen-
tial equation in x.
Denoting the Laplace transformation with respect to t
of U(x,t) by u(x,s) and that of Ha(t) by ha(s); i.e.:
L {W (x, t)} = u(x, )
L {Halt)} hals),
then taking the transform of both sides of (3-1) and using condition (3-2),
one obtains:
d²ulx,s)
Sulxs) = 0
dx²
while (3-3) and (3-4) become
(3-5)
uloos) =
(3-6)
and
du (x,s) -
I hals).
)3-7)
k
dx

Rearranging (3-5) slightly
overená) U1x,s) - 0
O,
and the solution of this simple differential equation is
Ulx,5) = A et is
+Betrs
where A and B are the two constants of integration. In view of condition
23
(3-6), B must be identically zero, while, from (3-7):
- hals)
=
VELA
,
or
A: vhals)
K
Now, by definition o
k/v , hence
va
I ㅗ
​k к
k.
A new constant, the "thermal inertia," (3) u , is now defined as
Duk
k: PCp ,
M = k v
whence
hals) =
A =
VES
rū
Vs
and the solution for u(x,s) becomes :
hals)
Ulx,5) =
ع . (
(3-8)
Using the convolution integral and tabulated transforms (4), one may im-
mediately obtain the inverse transform of u(x,s) as
t
TiTu Halton)
dd,
(3-9)
मरत्र
e
v
where 1 is the dummy variable of integration.
This form allows the calcula-
tion of the temperature rise, u(x,t), for any arbitrary irradiance pulse
H(t), subject only to the restriction that H(t) must be zero for time less

than zero.
(This is equivalent to the statement that the input pulse com-
mences at zero time.)
The important point in the development so far is that equation (3-9)
contains but two constants, u and a, which may be called the "thermal
constants of the solid.
Thus, by rendering any solid opaque, say by simply
covering the surface with a thin opaque layer, these thermal constants of
24
the solid may be determined by measuring the temperature response as a function
of depth and time. (The surface reflectance must be determined by separate
means, as with a spectrophotometer fitted with a diffuse reflectance head.)
3.3 Specialization of the Solution for Rectangular Irradiance Function
To reduce the temperature response equation (3-9) to a somewhat simpler
form, a specific function, the step function, will be selected for H(t).
Thus:
Hasteln (t) = otco,
Ha, tzo,
where Ha is a constant. The integration indicated in equation (3-9) may now
be performed, or the transform of the above-defined functional form for
Hq (t) may be substituted in equation (3-8) and the inverse transform taken
directly. Following the latter course:
Step
L { Hastee (t)} = ha
/s):
Ha
Ha
S
j
hence,
ustee (x5)= Ha
() og
the is
s 372
and (5),
Uster (x, t) = 2 [VF
**
e
weerfel ]
(3-10)

where erfc(x) is the complementary error function defined by
erfe(x) = l- erf (w) - Pahina la medida,
(Here, erf(x) is the error function:
erf (x) = #S,*--a'da).
Now, the step function form for H(t) requires that the irradiance
must continue indefinitely at a constant level, while any actual pulse must
25
be terminated at some time.
It will be convenient to consider a rectangular
pulse:
Ha
1+) =
Ha, osten
o, to ni
i.e. a pulse of constant amplitude over the exposure time, M. Clearly,
this form may be synthesized from two step functions as defined above,
Ha reet (t) = Hastep (t)- Ha Step It-n).
.
The linearity of the original differential equation insures that
superposition of solutions is allowed, hence
ureet (x,+)
= w Step (x,+)
ten
SHOP (2,t) - w Step (x,t-n), t>m,
or
creet (wit) = 240 [VF e* - gerte (int)], 0575m
2H01W - Terte last
Verme Toptan va ertelemben ) ] notes nap3
t>n

It is of some convenience to generalize these results by defining
the dimensionless variables:
Y = Tran
and
-
By slight rearrangement, equation (3-10) then becomes:
u rect (vaan &, ma) = 2 Han
A ve {va [te
F 쓸
​- Pertel)]}, 05251
2 Haun FR (F), Ostel,
JIM
26
where
2.
R(x) = p [
xerfe (x)
е e
The function R(x) is defined so that R(0) is equal to unity. Thus, when
You is zero (i.e. on the surface of the solid) the temperature rise is simply:
Urect 10, me) = ? Ha van
ott El
(3-11)
which is maximal at 7 = 1, or
2 Havn
u reet (09)
(3-12)
V TATTOO
The temperature response equation may now be further simplified by
defining a normalized response, T
T(4, 7) = wreet (14x ny, na)
u reet lon)
whence the complete expression becomes :
T (1,7) = V RO,0$751
=TRI) - UT-T RIFE) > 1.
(3-13)

Equation (3-13) presents the complete temperature-time-depth history of the
opaque solid subjected to a rectangular heat input, in compact form.
A
brief table of the values of the function R(x) is given in Appendix III.
The accompanying curves display T (V,?) as a function of dimensionless
time, ? , with dimensionless depth, y , as a parameter (Figure 3-1), and
as a function of x with r as a parameter (Figure 3-2).

3.4 The Fitting of Experimental Results to the Normalized Solution
While equation (3-13) possesses a measure of mathematically satis-
fying elegance, the directly obvious physical variables have been somewhat
lost in the normalization. Now the normalizing constants are essentially
four: (1) the absorbed irradiance, Ha, (2) the exposure time, m, (3) the
thermal inertia, M, and (4) the thermal diffusivity, a. In any experimental
27

1.0
mm
T
דיייי
v=0
W=0.05
0.10
Dimensionless Temperature Rise, T
p=0.1
z'o = the
Y=0.4
?
= 0.8
है
9.1 =
0.01
.003
.003
0.01
10
50
0.10
1.0
Dimensionless Time, I
Figure 3-1. Predicted Response of the Opaque Solid
28

1.0
1
T=1.0
T=0.8
T=0.4
T=2.0
T=0.2
T:40
Dimensionless Temperature Rise, T
Tool
0.1
.05
.02
1.0
0.1
Dimensionless Depth, Y
Figure 3-2. Predicted Response of the Opaque Solid
29
measurements, the first two, the absorbed irradiance and the exposure time,
will depend upon the arrangement of the exposure source (and the independently
measured surface reflectance), and must be known with as high an accuracy
as possible. The latter two, the thermal inertia and thermal diffusivity,
are the "unknowns" of the material; it is precisely the purpose of the experi-
ment to yield numerical values for these constants. A logical question,
then, is how one might best use the theoretical predictions in conjunction
with experimental data to determine these values.
It is of interest, although probably not of profound theoretical sig-
nificance, that the thermal inertia and the thermal diffusivity can be de-
termined quite independently of each other.
The evaluation of the former
follows quite directly from equation (3-11), which gives the expected surface
temperature rise as a function of dimensionless time, T.
The only unknown
involved is Mo
It is convenient to reduce the experimental results to
temperature rise per unit absorbed irradiance, or U(x,t)/Ha; for brevity,
this quantity is denoted by the symbol U*(x,t). Thus, if the measured
values of U*(x,t) are divided by rm , and plotted against ve/^ , then a
straight line should result, passing through the origin, with a slope of

2/VTM ; i.e., from equation (3-11):
Urect (0,t)
2
vil
(3-14)
This will hold, of course, only for times such that 0 = t/41. The mameri-
cal value of u can thus be found from this slope.
For the evaluation of the diffusivity, a, two methods present
themselves. First, one may plot the logarithm of experimental values of
T; i.e. U*(x,t)/U*(0, m ) against log of x/17 with t/rn as a parameter.
(The actual determinations of x and m will be described in detail in a later
chapter.) The resultant family of curves should be identical to Figure 3-2,
This employs the convenient
except for a simple translation of the abscissa.
30
property of a logarithmic scale that a multiplicative constant causes a
TE simple linear shift, only. Thus, if the experimental curves are shifted
over the theoretical curves until the "best" fit is obtained, the multi-
plicative constant can be readily determined. Here, this constant is simply
1/ V44 ; hence O may be evaluated directly.
PER
The second method employs a somewhat different approach. Inspection
of Figure 3-l reveals that for any non-zero value of dimensionless depth,
Y у
the dimensionless temperature reaches a "smooth" maximum (i.e. the
first derivative of T with respect to 7 becomes zero ) at a value of
(always greater than unity) which increases with increasing y
Letting
Tmax
be the value of the normalized time at which such a maximum response
occurs, the functional dependence of y on I max will now be deduced.
Returning to the original form of urect(x,t) it will be recalled that
this solution was obtained by superposition of two step function responses as
Urect (xit) = Usten lx, t) - U Step (x, tom) eta
Hence, it follows that
a w rect (x, t)
dt 아
​,
de
& [u ste® (x, 4) – Uster (*, tom)]
ou step (xit)
(xst) au step
au step (x,tom)
at
The derivatives on the right are readily found from equation (3-8), utiliz-

ing the useful property of the Laplace transform (6) that
au(xt)
Le "{su1x,s) - U1x.0)};
hence
2
au step(x,+)
x
Ha
TU

31
Now the extrema are at values such that: andre
pins
rect
O
at
t = tmar
or
Ha
& *
e² 르
​lite
4x tmax
e 4x(tmaxon)
1 -
o
TM 페
​tmax
Vtmaxon
Rearranging slightly, and introducing y and I max
as previously defined
SI
het v?
r? lamany
ma
Tmax-1
Tmaxl
Tmar
Tmar
е e
1
or
x ?
- 116 br Tmax (Tmax - 1)
Tag 11
Tmax
Tmax-1
and finally:
Tmax (Tmax-1)
7 = 1
Tmaa
In
(3-15)
2
Tmax-1
Equation (3-15) gives the expected value of You (= x/ 44 ) for which a
Tmux
temperature maximum will occur at a specified value of
(= tmax/ m ).
The application of this relationship to the experimental determination of
a is straightforward. For each subsurface temperature measurement, one
Then
notes the time, tmax' at which the maximum temperature was reached.
plotting x/ ñ (again, these values must be known) against
Itman
2
tmas /tmax-1)
2
In
Itmax
n
should yield a straight line, passing through the origin, with slope of
4a
. This follows from the trivial rearrangement of equation (3-15)

to
tmax ltmax_l)
tmax
: 140
ñ
n
?
In
V
2
tmax
(3-16)
n
Since the subsurface temperature maxima always occur at times such that
tmax
l, there is no problem with negative or zero values in the
n
32
denominator of the logarithm, or with negative values under the radical.
In reviewing the above methods for the experimental evaluation of the
thermal constants, u and a , it may again be pointed out that they are de-
termined quite independently of each other, with surface temperature measure-
ments only being used for it and subsurface measurements for a
In fact,
it is clear that surface measurements, which are certainly the simplest to
make, can give information concerning til, only, and do not permit an evalua-
tion of a to be made.
For the latter constant, subsurface measurements,
complete with accurate values of the depth of placement of the temperature
sensing element, must be made.
It might also be mentioned that the thermal inertia, M
and dif-
fusivity, a
, may be considered "derived" constants, defined in terms of
the "fundamental" constants, thermal conductivity, k, and heat capacity per
unit volume, v , by
Mkov
a= k/v.
Obviously, the latter two constants may be found from the former by
ku ma
v=rula.
hence, it would seem to be a matter of complete indifference which a
set-u and a or k and V --be considered "fundamental," and which

9
"derived."

3.5 The Opaque Composite Solid
The foregoing prescriptions for determining the thermal constants
from experimental data will, of course, be valid only if the experimental
material satisfies the assumptions made in deriving the various equations
involved.
It is reasonable now to ask what changes in the experimental
results might be expected if certain of these assumptions are not satisfied,
E 33
in particular the constancy of V and k. The model selected for this investi-
gation is the opaque composite semi-infinite solid. Here, the various assump-
tions in Chapter II and Section 3.1 are still applicable except that the
thermal conductivity is assumed to be a constant value ky for the values of
x from zero to a value b, and a constant value k2 for x greater than b.
Likewise the heat capacity per unit volume is taken as v, for x from zero
to b and V2 for x greater than b. Equations (3-1) through (3-4) are now
replaced by
aulxit) -
V
ki
0² U (x,t)
2x²
o Excb, tzo
(3-17)
ve out) - k ozunt), *>b, tzo
(3-18)
Ulx,0) =
D) T (3-19)
,0
U700, 7) =
O
(3-20)
ou10,r)
I Halt), tzo
(3-21)
ki
(6-0,+) = 16+0, +)
(3-22)
ki
au lb-ot)
aulb+o,t)
ke ou
(3-23)
-
2x
Equations (3-17) and (3-18) follow directly from the definition of the com-
posite solid; equations (3-19) through (3-21) are identical to (3-2) through
(3-4) which were discussed previously; equations (3-22) and (3-23) are

"continuity" conditions.
The first, which is more correctly written:
Lim u(xt) =
Lim ulxit)
tzo
*>be
*6+
simply states that the temperature response function suffers no jump at the
break point b.
Such a jump, in view of the absence of internal sources or
sinks, is a physical impossibility. Equation (3-23) expresses the physically
34
necessary condition that heat flow into the break point from the left must
be equal to the flow out to the right. Here again the notation is an ab-
breviation for
Lim [k. Ulet)
Lim [k. dux, t)
2U lx, t) 1 tao.
x →6
and az
Letting a, =
ky/v. a₂ = k2/02 and using condition (3-19),
the Laplace transforms of (3-17) and (3-18) give two ordinary differential
equations analogous to (3-5), which are readily solved to yield:
U(x,s) = A e tais & Bears
о<x<ь,
and
us
U (x,s) = Ce tars & De
*>b,
where four constants of integration, A, B, C, and D must be evaluated. From
the transform of (3-20), ulce ,s) = 0, it follows that D must be identically
zero.
Further, from the transform of condition (3-21) it follows that
- hals) :
I A + B B.
or
mi to
hals)
B: A -
: A-
Hence the solutions in u(x,s) become
U14,5) = Ale ons te ters) - hals)
e
Jis
V5
OLX cb,
ulxs), Centrs
xəb.
The constants A and C must be evaluated from the transforms of (3-22)

and (3-23), namely:abs
4/6-0,5) = u(6+0,5)
u(6+0,5) broa)
and
ki
drulb-os)
dx
ke dulb+o,s)
dx
The evaluation of these constants is quite straightforward, and is presented
in detail in Appendix IV. Upon substitution of the complete expressions
for A and C, the complete solution in u(x,s) becomes:
35
е
214,5) =
hals)
vu.
et (1 E) te thot 13 (1-1)
rs le mars (1+ v)-ethers (1-1]
osxcb
Š
(3-24)
Sobotenente alph
povs
0(4,5) = hemen
x>b.
Ir5 let 5 (1 + vente - (-]
(3-25)
2 e
In Appendix IV, these expressions are reduced to a somewhat more
manageable form, and the inverse transforms taken to give general formulas
for the temperature response U(x,t). Here, however, it will be sufficient
to consider only the surface response to a step function. It is convenient
to make the substitution x = O in the transformed function u(x,s), rather
than in the general equation for U(x,t), since the problem of taking the
inverse transform is greatly simplified. As stated previously,
(=
for the step function form of H (t), hence equation (3-24) becomes
ha(s)= Ha
S
a
u(0,5) =
- Van
He,
eta 15(1+) + e ters (1-1)
sü [eters (1+) - (1-1)]

The dimensionless constant 1 is now defined as:
1 =
1- M2/Mi
1+ M2/M.
from which it is clear that 12 must always be less than unity.
The ex-
pression above for u(0,s) may now be written as:
36
Se
2 en
Ha
It de
Tu 53/2 T-xe this
or
2bn is
u(0,s) = Ha
the star 2 3 1
7
s 3/2
(3-26)
Equation (3-26) is in such form that the inverse Laplace transform can be
taken. However, it will be simpler to consider what form a plot of U(0,t)
(0,t)
against t will take, i.e. to derive the function
From
art
equation (3-11) it is seen that for the isotropic opaque solid
aucoot)
IVE
2 Ha
VITM
and the question here is what deviation from this simple relationship will
occur in the case of the composite solid.
Now,
ZVE
2 ulo,t)
OUF
while as mentioned previously
aucot)
at

Hence

- 5
2bn.
auco.t)
2.
at
Hoe
Ha
vui
"{ts
+ 2 21"
2Ž^" e
15
2
at
Se on Core 22
in e
31
UTE
and, from above
du 10,t)
alt
veten [1+ 28 ^**]
(3-27)
While the form of equation (3-27) is not immediately obvious, the limiting
3 37
values are quite simple. Letting t approach zero, the exponential will like-
wise approach zero, hence
OU 10,0)
2 Ha se to ne
a VF VTM
The initial slope of the temperature response plotted against the square
root of time thus should be 2H/1Tui
from which mi
can be obtained
directly.
Now, letting t increase without bound (i.e. tof) the exponen-
tial term will approach unity, and thus
2 Ha
VIMI
auto24
Praha Li + 2 Š^"]
[-1+
[-1 + ]
- [2]
2 Ha
2
2
1-2
Vali

But
I +
1- M21 Mi
It VM2 TM
.
Wu"
M2
WA
1-2
l-VM2Tui
/
1+VM2/MI
Hence
du (0,0)
a VF
2 Ha
TM2
or the plot of u(0,t) vs vt should tend to reach a straight line with
slope 2Ha/ VTT ME after a sufficient length of time. From this limiting
slope, the value of M2 may be obtained. If the experimental data indicate
2
that the slope of the temperature response vs the square root of time is
constant for all values of time, then the implication is that me
M2 :
This does not "prove" that the solid is isotropic, since compensating
variations in k and V could result in this constancy of M ; however it
38
will be shown later that in the case of skin such compensation is highly
unlikely. Thus, the constancy of the slope of y(0,t) vs
Nt may be con-
sidered sufficient justification for the assumption of constancy of k and V.
3.6 The Opaque Solid Subjected to a Trapezoidal Irradiance. Pulse
The considerations of the previous section focus attention on the
initial temperature response of the opaque solid,
In any actual experiment,
considerable care must be exercised in order to obtain reliable data at
these very early times, since several factors other than variation of u with
depth may affect the results. Certainly, some means must be provided to
indicate, on the actual temperature recordings, the precise moment of initia-
tion of the exposure. In addition, the transient response of the thermal
sensing element-amplifier-recorder system must be known in order to establish
the form of the true temperature response from the recorded response.
Both
of these matters will be dealt with in following chapters. There is another
factor, rather easily overlooked, which should be considered now; this con-
cerns the actual irradiance pulse shape delivered to the test specimen.
It is, unfortunately, a complete physical impossibility to deliver
a true mathematical step function in any real situation. While the rise
time of the irradiance pulse may be made very small, it can never become
precisely zero, as required for a true step function.
For any given situation,

it is probably always possible to design a shuttering mechanism which will
provide a rise time so short that it may be considered a true step function
with negligible error.
With an irradiance source such as a carbon arc furnace,
however, this approach poses several unique problems. The shutter must control
extremely high values of radiant power, must be completely opaque when closed,
and should be completely transparent when open for wavelengths from the near
ultraviolet out to about 2.6 H in the infrared.
Thus both light-weight,
39
fast-acting mechanical shutters and the vastly more rapid Kerr cell types
are not particularly suitable, although admittedly they probably could be
adapted for this use.
A simpler approach would seem to be to use a robust mechanical shutter,
with the fastest practicable opening time, and then apply mathematical cor-
rections to the resultant data to account for the finite rise time. This is
the procedure selected for this investigation.
Investigation revealed that the opening characteristic of the shutter
employed (which will be described fully in a later chapter) could be ap-
proximated quite closely by a linearly rising segment (ramp function) fol-
lowed by a "flat-topped" segment.
The closing characteristic is similar,
yielding a trapezoidal rather than a rectangular pulse. Graphically this
is represented as follows:

Ha
li
Locate
^
Halt)
1
1
1
8
n
n+8
ty
The absorbed irradiance pulse is thus given by
Tn Ha trap (t) = 0,
(t) = 0, tso
(
na ) t, ostsd

Ha,
sat en
- Ha -66) t, nstsner
= 0, t2n+d
40
This pulse may also be synthesized from four ramp functions, as follows:
trap It) = Haramp (t)- Ha
Ha
rump (tos)
ramp
- Haramo (ton)
+ Ha
(t-mos),
where
Ha
rump (t) = otco
(*)t, t20
Since the temperature response to the trapezoidal pulse form can be obtained
by superposition of solutions, it is only necessary to derive the temperature
Thus
trup
response to a ramp function.
utro
(xit) U
rame (xit), osted
= cramp (xit)- tramp (x, t-6), Jeton
= Urampixit)- wramp (x,t-0)- w ramp (x, ton),
nets
9
S
For the isotropic opaque solid, the ramp function solution can be
found directly from equation (3-9), the general solution for this model.

It is extraordinarily simple, however, to attack this problem in a more
general way, after which the special case of the isotropic opaque solid will
be considered.
Turning to the transformed equations in u(x,s), it will be noted that
for both the isotropic and the composite opaque solid, u(x,s) is directly
proportional to ha (s), the transform of the irradiance function.
It will
be seen in the next chapter that this same relationship holds true for the
diathermanous solid as well.
In fact, this must hold for any model, so long
as ) and k are independent of temperature; i.e. the governing differential
equation is linear. Certainly, for any linear model subjected to an ir-
radiance step function, if the absorbed irradiance be doubled, then the
temperature response will likewise be doubled; hence one can state that
Sa 41
Ustep (xt) = Ha • F(x,t),
where F(x,t) depends only upon the particular model involved. Taking the
Laplace transform of this expression gives
uster (x,5)
H Ha L {F(x, t)}
He sa {Flx,t)}.ru mobiston to genera
=
S
Clearly, the term Ha/s is simply the specialization of the transformed general
irradiance function, ha(s), for the particular case of a step function; since
this transformed response equation must be of the same form for any (trans-
formed) irradiance function, it follows that in general
bantu
Ulx,s) = ha(s). s { F 1x,t)}
France
(H)it,
Now consider the ramp function: La mostat molt
Haramp (t) = oteo estolbeskawa
Ha
to.
d
The Laplace transform is given by
ha (s) = {{ Ha
(t)}
Ha I ㅗ
​S 5²
and the transform of the temperature response by
te bylo
Ha
L
뽕
​S
ramp
rump ?

This last expression may be rearranged slightly to give any more
e rame (x,5) = . $ [et als op { F1x. t)}]
where the term in brackets is recognized as simply ustep(x,s), or
eramp(x, 5) = $.ts ustep (x,s).
Now, using the property of the Laplace transform (7) that if f(s) is the
transform of F(t) then
42
L-1 { & f(s)}
{+#
S* F(t) de
the inverse of the above expression may be written immediately as
Cromo (4,4)= S. Coste (3, 3) dt
Using the notation U*(x, t) as the response for unit absorbed irradiance, one
may write finally:
t
to u ramp
U rame (x, t) =
(x, t) = (**
(*) 1.
* step (x,7) d?,
(3-28)
where Hals is the slope of the ramp function, and U*step(x,t) may be recog-
nized as the indicial transfer function. Equation (3-28) is a general re-
lationship, valid for any linear model. (It is, in fact, a special case of
the superposition integral, commonly used in communications and network
analysis. The term "indicial transfer function" comes from this latter
field.)
One can now return to the question of the influence of the finite
shutter opening time on the initial temperature response of the opaque solid.

The surface temperature rise per unit absorbed irradiance of the isotropic
solid is given by
uster
(0,4)= VW VF
from equation (3-11). Hence the ramp function response is
cromo (0,4)= ahogy svedt,
and considering only the early phases of the previously defined trapezoidal
pulse,
43
Ut** (0,t) = over one s Fi de, osts 5
2 H Sonde - Sevedre], seten
건
​The latter expression may be simplified by noting that
S* 460) dr - S* $er) dt = S, tee) de
hence
w trae (0,t) = 24. Fedt
Siede ,osted
Best rem, jsts a
trap (0,t) =
Performing the indicated integrations gives
4 Ha (+)2, ostes
U
3 S VTM
4 Ha [+ 3/2_ (t-1)/2], Setsn.
38 VIM
A correction function ftrap, is now defined, which when multiplied by the

trapezoidal pulse surface response will give the "corrected" step function
surface response; i.e.
W ㅂ
​step (0,0)
Ftrar (t) =
trap
U
(0,7)
2 (4) ostes
(
1- (1-
)2
jstsn.
23/
=
)
The last expression, for time greater than $ , may be cleared up by ex-
panding the denominator as
1
1
1-(1-4)
1- [1-{(4)+ ({})(4) * ()+(+X+/() + ({}+... ]
=
3
(€)[1-5
() - GX45 ()?-6)(+)*)*)()-...]
2!
3($)
žo (1) 5G)- ...
1
1 - ź
The correction function may now be expressed as
Etrop (t) = {(), ostss,
,s$t$n.
|- * (*) - ģ('-42 ()-...
A plot of ftrap vs the dimensionless time ratio (t/d ) is given in
Figure 3-3, from which it can be seen that the correction factor approaches
unity rapidly as (t/S ) increases. With directly measured values of S,

the irradiance rise time, and using the curve of Figure 3-3, it is now a
simple matter to correct a measured temperature response to a true step
function response.
The measurement of d will be described in a subsequent
chapter.

3.7 Summary of Predictions from Opaque Solid Theory
The various relations now established enable one to examine in some
detail the temperature response of the opaque semi-infinite solid. For
convenience, the assumptions involved in the analysis and the predictions
based on these assumptions will now be briefly summarized.
45

100
1.5
80
60
40
1.4
Ha
30
201
f
Halt)
1.5
10
S
t-
8
1.2
dengt
4
1.1
2
1.0
0.1
0.2
0.3 0.4
0.6 0.8 1
2
4 6 8 10 20 30 40 60 80 100
(t/8)
Figure 3-3. Factor for Correcting Trapezoidal Pulse Response to Ideal Step-Function Response,
for Opaque Solid Surface
46
a. General assumptions
1. The receiver is a semi-infinite solid at rest, initially
at uniform temperature, To, throughout.
2. The heat capacity per unit volume, V , and the thermal
conductivity, k, are independent of lateral position
and time (or temperature). They may vary with depth, x,
only.
3. At zero time, the surface of the solid, in the plane
0, is to be exposed uniformly to a pulse of
radiation.
4. The receiver is opaque; i.e. radiation is absorbed at
the surface, x = 0.
The reflectance, R, is constant
in lateral position and time.
b. Specific model I
6.1. The solid is isotropic; i.e. k and v are constant.
2. The irradiance pulse is rectangular, of duration
m seconds.
3. Statement of the problem:
Oulx.t)
It
x
2²0(x,t)
, xzo
x=0, t20
2 x2
&
Ulx,0) = 0, $20
wc, t) = 0
to
aulo,t)
o, tco, tan
Ох
- the Ha, osten

47
4. Normalized solution:
rect
TIM, T) -
u rect1on)
,
Izl)
where =
= VERO) - VET RITA)
Van
th 틀
​R(x) = Va The
X2
- xerfe (x)]
and
5. Determination of thermal constants:
Thermal inertia, u , is evaluated from
U*rect (0,7)
2
os
tal
I
IR
n
Thermal diffusivity, a , is evaluated from
tmax
n
max ( tmax - 1)
1
In
tmax
n
tmax
n 7
2
where x is the depth for which the response urect(x,t)
reaches a maximum at time tmax.
c. Specific model II
2
1. The solid is composite; i.e.
V = V., osx cb
- Vỏ , x> 5
k=ki, o şx<b,
= K2,
)

x > b,
2. The irradiance pulse is a step function in
3. Statement of the problem:
48
2 U1x, t)
at
OLURD), o sxc6, +20
2 U (x,t)
at
22W(xt)
<> 5, + 2 O
Wlx,0) = 0
U100, t) =0 t20
aucot)
= 0 tro
- - Ha, tzo
Wlb-o,t) = u(b+o, t) +20
kowy
-
2W(6+0, t)
k,
t zo
Ох
4. Slope of surface response vs
vt:
$* ep
du
2 Ha
VITHI
[1+2 3 "e
+
]
n.1
Sos
grense 5. Evaluation of thermal inertias:
M, evaluated by
Lim
Ou* step (0,t)
tot
LES ES
2
VITHI
Me evaluated by
Lim
a u* Step (ot)
too
art
2
VIT M2

d. Correction for finite rise time of irradiance pulse
1. Define trapezoidal pulse as
Ha trap (t) =
O, to
(на
osted
auf beidara = 6t,
Ha, deten
= Ha- (Ha)t, a stento
tanto
BE
49
2. In general, for any model:
U trap (x,t)
(Не
* step
W*
(x, 2) dł ostes
16
Ha
д 8
S
Wstep(x, z) d?
(x, 2) da, dsten
E
3. For surface response of isotropic opaque solid
SHOP
F trap (t) =
3
U (0,t)
W trap (ot)
ostas
1
7- () - (+)-(*)'-...
Jeton.
3.8 Utility of the Opaque Solid Model
It requires no more than a casual observation to convince one that
normal living skin is far from opaque, but rather is remarkably translucenta
This is true even for rather heavily pigmented skins, as attested to by
transmission measurements in the visual and infrared spectral regions (8).
It is logical to ask, then, what possible utility the foregoing analysis
of the opaque solid may have in elucidating the temperature response of
normal skin.
First, it should be pointed out that simply by coating skin with an
extremely thin layer of an opaque material, it can be made to approximate

an opaque material quite closely. Thus there is no difficulty in altering
the experimental material to fit the theory, but this does not explain the
reasons for such a procedure.
This question could best be answered after the development, in the
next chapter, of the diathermanous solid theory. For the present, it will
only be pointed out that the solution for the temperature response of this
model is quite complex, in contradistinction to the relatively simple ex-
pressions derived for the opaque solid. In particular, the "thermal" and
50
"optical" constants are so entangled, that their experimental evaluation
becomes extraordinarily difficult. Now, a surface treatment to render the
skin opaque will not alter the thermal constants (k and D ) of the skin;
hence these may be determined by the methods developed previously, where-
upon they now become knowns, not unknowns, in the diathermanous solid
solution.
The evaluation of the remaining unknowns then becomes not only
possible, but quite straightforward.
Before leaving this matter, some consideration should be given to
the assertion that the surface treatment to render the skin opaque will not
alter the thermal constants of the material.
While this would certainly
seem to be true in theory, in actual practice it may not have been so, as
will be discussed in some detail in later chapters. The possible alteration
of the thermal constants by surface treatment remains one of the unresolved
problems of the present investigation.
FOOTNOTES
(1) Coulbert, C. D., W. F. MacInnes, T. Ishimoto, et al., Temperature Response
of Infinite Flat Plates and Slabs to Heat Inputs of Short Duration at One
Surface, University of California at Los Angeles, Department of Engineering
Report (1951).
(2) Churchill, R. V., Modern Operational Mathematics in Engineering,
Chapters IV and VII (McGraw-Hill Book Company, Inc., New York 1944).
(3) The term "thermal inertia for surface heating" was coined by J. D. Hardy.
See Section 7.7, and literature citations therein.

(4) Churchill, R. V., Op. cit., 37, and 299, Transform No. 84
(5) Ibid., 299, Transform No. 85
(6) Ibid., 294, Operation No. 3
(7) Ibid., 294, Operation No. 5
(8) Hardy, J. D., H. T. Hammel, and D. Murgatroyd, Spectral Transmittance
and Reflectance of Excised Human Skin, J. Appl. Physiol. 9, 259-264 (1956),
51
CHAPTER IV
ANALYSIS OF THE DIATHERMANOUS SOLID
4.1 Definition of the Model
The semi-infinite diathermanous solid model attempts to quantitate
the influence of the penetration of radiation into the skin.
The choice of
this term is rather unfortunate, since it implies the ability to transmit
heat (literally "heat through"), and the term "diathermic" is so used in
thermodynamics; hence the opaque solid is diathermanous, in this sense,
Complying with more or less accepted usage, however, the term will be employed
here in the restricted sense of describing a material which can be penetrated
by radiant energy.
In defining this model, the basic assumptions of Chapter II will be
followed, and in addition k and V will be considered constant.
The heat
conduction equation thus reduces to
au (xit)
at
0261x, t) + + 9" (xit), *20, t20
(4-1)
dx?
with boundary conditions
U(x,0) = 0, X20
(4-2)
1
(4-3)
U (0,t) = 0, t20
The model will then be completely defined by specifying the function
q!!!(x,t) and providing one additional boundary condition.

The latter problem will be considered first.
As before, the surface
of the solid is assumed to be insulated against heat losses, so that the
only energy crossing this boundary is the incoming radiation. Here, however,
absorption occurs in depth; i.e. the radiation must traverse a finite thick-
ness of material to suffer a finite absorption and conversion into internal
energy. Equation (3-4) for the opaque solid is therefore to be replaced by
52
the homogeneous boundary condition
aucot
o, tzo
dx
(This is the same condition which applies in the analagous problem of internal
Joulean heating of an electrical conductor, with insulated surface.)
The only remaining matter is the specification of the function qi(x,t).
This is equivalent to requiring that the absorption pattern of the radiation
in depth be defined, and is one of the major problems of this investigation.
This will now be considered in some detail.
4.2 General Consideration of the Absorption of Radiation in Skin
Before dealing with the precise form of q!!!(x, t), certain relations
which the function must satisfy will be developed. It is important to note,
first, that skin is a highly scattering medium, even a thin layer producing
almost complete diffusion of an initially collimated beam. Therefore, the
usual concepts of transmission and reflection do not apply, and one must
speak of total forward and back scatter. If a beam of radiation, of ir-
radiance Ho, is incident on this scattering medium, a certain fraction,
RH., will be back scattered into the total back hemisphere), where the
symbol R may be used in analogy with the surface reflectance of the opaque
solid. Similarly, the irradiance absorbed in the material must be (1-R)H.,
which again will be represented by Ha. Note that H. (and H2) may be a function

of time, but, as before R will be assumed constant.
Now, if a nonscattering diathermanous material (colored glass, for
example) is placed in a beam of collimated radiation, it is quite clear that
any diminution in the radiation with depth must be due solely to absorption.
If the irradiance at the depth x is H(x,t), then the absorption of radiation
at this depth in a volume element of unit cross section and thickness dx must
be simply H(x, t) - H(x+dx,t). This, however, will be precisely the same as
53
the "heat generation" per unit volume, qr(x,t), times the volume of the
element, dx. Hence
BRUGER "
q"(x,t) dx = H(,t) - H(** dx, +)
- H(xit) – [H7x, t) + 3 H1x, t) dx ]
2 Hlxit)
or
2HIxit)
q'"'(xit)
HEROES
It is also clear that the absorption (or "heat generation") per unit volume
at the depth x must be proportional to the irradiance at that depth, or
q"! (x,t) = 81x) Hlx.+),
where the proportionality constant (the linear absorption coefficient), 8,
may be a function of x, as noted. The assumption that 8 is independent of x
leads immediately to the familiar exponential attenuation pattern
H(x,+) • Holt) e-**
from which it follows that
9" (x,t) = 8 Holthe
In the more general case of
8
8(x) integration of the equations above
leads to
- S. *r(s) de
Hlxit) = Holthe
Sama
and
9:"18,t) = x(x) Holt) e-S"819)df

(In all of the above, the small loss by reflection at the interface has been
neglected.) Here, then, the determination of the linear absorption coef-
ficient as a function of depth completely specifies the function q''(x,t).
In the case of the scattering diathermanous material, the situation
is, unfortunately, more complex. Plane parallel radiation incident on such
a body will soon become quite diffused due to multiple internal scattering.
54
It is quite possible to define, for any position and time, a net irradiance
in the positive x direction, say H. (x,t), in analogy to the term H(x,t) for
the nonscattering case, above. However the absorption per unit volume
q''(x,t) will not depend directly on this net forward irradiance, since a
given volume of the material can absorb radiation from any direction. At
each level, one must consider the total irradiance, Hr(x,t), available for
absorption; given a plane surface of unit cross-section, and at the depth x,
the irradiance available for absorption at this depth is the arithmetic sum
of all the radiant flux which passes through the plane from any direction.
Thus at the surface this total irradiance will be equal to the sum of the
incident irradiance, Ho(t), and the total scattered back out of the material,
which from the definition of R above, is simply RH.(t). Explicitly:
Hq 10,) = (1+R) Holt).
Now, can one still claim that any reduction in Hp(x, t) with depth is due
solely to absorption, and hence
9" (xt) =
2 Hr(x,t)
?
2x

It is not difficult to show that the answer must be no.
Consider a scattering,
but perfectly non-absorbing semi-infinite solid.
Since there will be no
temperature rise in the material, the back scattered radiation must be equal
to the incident radiation, which means that R must be unity. It follows
from the discussion above, that, in this case Hp (0,t) = 2H.(t). If a de-
crease in Hr(x, t) with depth were due to absorption only, then since there
is no absorption in this ideal material, Hy(x, t) must equal 2H. (t) for all
values of x.
However, at least some part of the back scattered radiation
comes from a differential surface layer, hence in the immediately subad-
jacent layer there cannot be the same total amount of radiation. Repeating
this argument for successively deeper layers leads to the conclusion that
55
HT(x, t) must tend to zero as x increases without bound, whether absorption
occurs or not; i.e.
Ho 100, t) = 0. listy
Thus, the reduction in Hr(x,t) is not dependent upon absorption, only, and
9" lx, t)
2 Hlx, t)
OH
dx
It is convenient at this point to recognize that the time dependence
of q'''(x,t) is simply that of the absorbed irradiance pulse Ha, which de-
pends only upon the time, t, while the variation of q!!(x,t) with depth is
independent of the irradiance pulse shape, and hence depends only upon some
function of x, say F(x). Clearly, qpil(x,t) may be expressed as the product
bio je q"! (xit): Halt). F(x).
Now let the function V(x) be such that
dv(x)
dy
dx
Flx), *20.
V(x) is thus defined to within an additive constant; for convenience, let
V ( 0 ) = 0. Substitution of this relation in the expression above for
q!''(x, t) gives :
9lxit) = - Halt).avi
dx
or
9*"(xt): - [Halt). Vix)],
(4-5)
from which V(x) may be recognized as the absorption pattern of radiation in

the material, and Ha(t) • V(x) might be thought of as the net "absorbable"
irradiance at given values of t and x.
It will be useful now to develop a
necessary condition which will provide a limiting value of V(x).
For the uniformly irradiated semi-infinite scattering solid, the
integral of all absorption per unit volume in a column of unit cross section
and extending indefinitely far in depth must equal the radiation absorbed
56
for this unit cross section, or
Soq".(x, t) dx = (1+r) H.(t) =
Halt)
(4-6)
This relation holds, since any lateral scatter out of this column will be
exactly compensated for by scatter into the column from adjacent regions.
Equation (4-6) is simply a statement of conservation of energy. Substituting
equation (4-5) in (4-6), and noting that V( ) = 0 leads to the desired
relation.
So
Halth
om [ Halt). Vix)] dx
Halt)
= Halt) [VIO) - Vloo)]
V(O) = 1
or
(4-7)
Now, in analogy with the non-scattering diathermanous solid, one can
define, with complete generality, a linear absorption coefficient, 7 (x),
such that:
q"''(x,t) = 81x) [Halt). Vix) ]
From equation (4-5), then
dV+)
dx
- YIY) VIX)
or

In V(x)
VE - S.*r(e) ds)
vro)
whence from the value of V(0) as found above
V
-
е e
Substitution back in equation (4-5) gives, finally:
-Soyle) de
q"(xt) = 8(x) Halt) e
(4-8)
It must be realized that nothing of great theoretical significance has been
obtained in going from equation (4-5) to' (4-8) and, in fact, the latter
relation could have been obtained at the outset simply by defining the depth
8057
dependence of q''(x,t), F(x), as as
- So *r($) ds
F(x)=81x) e
although such a definition might have been a trifle startling. It is per-
tinent to ask, however, whether this development has resulted in defining
a depth function which can be evaluated experimentally, since the form of
qill(x,t) must be determined by some sort of experimental procedure. This
last statement must be qualified by noting that the most elegant method of
determining qar(x, t) would be to solve the transport equation, applying all
the known characteristics of skin.
This would be inordinately complex, and
probably not enough is known about the detailed tissue-radiant energy inter-
action to justify the procedure. Hence, at the present time at least,
q!!!(x, t) must be determined experimentally.
4.3 Experimental Determinations of the Absorption of Radiation in Skin
In the previous section three expressions for q'(x, t) were given
as:
91" (x,t): Halt).F(x)
q" (0,4) = -o [ Halt). Vor)]
9" (xit) = Yix) Halt) e-Sys) de
While these relationships are in reality completely interchangeable, they
serve to typify the different approaches to the problem of specifying

q''(x,t)
The first equation,
9"(x,t) = Halt). F(x)
would seem to be too noncommittal to permit direct experimental determination.
One of the most important results of this study is that such direct determina-
tion is possible. This will be considered in some detail in a later section
58
of this chapter; here it will merely be stated that one can, in theory at
least, determine the precise form of F(x) (which, of course, is all that is
necessary, since H, (t) is under the control of the experimenter) directly
from temperature measurements.
Another me thod of approach involves the concept of a net "absorbable"
irradiance, with the function q'''(x, t) determined according to the second
equation above.
The pattern of this absorbable radiation in depth--i.e. the
form of v(x)--may simply be assumed, or it may be based on experimental
evidence,
An interesting example of the former is a linear absorption
pattern (1), proposed apparently to secure a simple form for q'''(x,t).
Here,
Vex)= (1-E), osxs 4
0, x >L,
from which
9."(x, t) = Ź Halt), 05 X6L
0,X>L
It is simple to show that for this particular pattern, the linear absorption
coefficient, Ý (x), must be given by

1
x(x) =
O EXCL
L11-²
, X>L
Thus, the absorption coefficient increases steadily from a minimum value of
1/1 at the surface (x = 0), and becomes infinite at the depth x = L.
This
particular variation of = (x) with x is almost surely precisely the opposite
of the true situation, since the best measurements to date (which will be
discussed below) indicate a decrease in the linear absorption coefficient
with depth.
Another interesting example of the employment of the concept of
"absorbable" irradiance is contained in a paper by Buettner (2), with a
59
curve of V(x) deduced from various pieces of experimental evidence gleaned
from a thorough literature review. The slope of this curve (apparently
obtained graphically) gives q''(x,t), which is then used in a numerical
analysis of heat flow.
Of interest in Buettner's analysis is the treatment
of heavily pigmented Negro skin ("limited penetration") with a resultant
sharp "spike" in q''(x,t) at the melanin layer.
In the third expression above for q'''(x,t) the dependence of this
function on depth is described in terms of a linear absorption coefficient,
(x). If, for a scattering material, this coefficient bears some close
resemblance to that of a non-scattering medium, where its physical significance
is quite clear, then one should be able to determine the form of q'''(x, t) by
transmittance and reflectance (i.e., forward and back scatter) measurements
on skin sections.
It is, in the writer's opinion, extremely difficult either
to confirm or deny the propriety of such optical measurements on theoretical
grounds. Essentially, the problem is to make an intelligent appraisal of
the effect of the necessary process of removing sections of the material to
determine variation in forward and back scatter with thickness; the scatter
from the material thus removed can now no longer contribute to absorption in
that remaining.
While there may be some question as to how (or whether) measurements
of scatter may be employed in specifying the volume absorption of radiation

in intact skin, at least there are now no doubts as to the proper experi-
mental methods which must be followed in order that the measurements them-
selves be valid.
The work of Hardy and his collaborators, particularly in
the last decade, has furnished the best information available at present on
the scatter of visible and infrared radiation by skin, and also has established
quite clearly the procedures necessary for the obtaining of such reliable and
consistent data (3, 4, 5, 6). Briefly summarized, the requirements are as
60
follows:
1. Forward scatter ("transmission"), back scatter ("reflection"),
and absorption measurements must all be made on the same sample. (See also
Buettner (2), bottom of p.
207).
2. These measurements must be made on an absolute basis, that is, not
based on comparison with some questionable reflectance standard.
3. The scattered and absorbed components must be measured over the
desired spectral range with monochromatic radiation of moderate purity.
Because of the pronounced influence of wavelength on absorption coefficient,
the use of broad spectral bands--isolated, say, by filters--is inadvisable.
4. It is highly desirable, although probably not necessary, to es-
tablish not only the total scatter into each hemisphere, but also the polar
distribution of this scatter, again as a function of wavelength. Such data
yield important clues as to the details of the tissue-energy interaction.
5. The thickness of the sample under investigation should be known
accurately.

6. In addition, it would seem desirable to follow the procedure
established by Hardy and vary the thickness from one sample to another by
removing the deeper tissues, leaving always the intact surface to be exposed
to the incident monochromatic radiation.
The instrument used by Hardy, a goniometer spectrophotometer (7),
satisfies most elegantly the first four requirements above. For example,
,
with respect to the second of these, the summation of scattered radiation
for a non-absorbing material will be within a fraction of one percent of the
measured incident radiation.
A very slight amount of absorption can thus
be measured with good accuracy.
The tissues analyzed in this instrument were obtained from surgical
specimens or from autopsy, and kept in saline or a moist chamber at all
61
times. Samples were cut to thicknesses of from 0.3 to 2.1 mm, after which
each specimen of suitable thickness was mounted between microscope slides
and sealed with stop-cock grease.
From the thickness of this assembly,
measured with micrometer calipers, the thickness of the two microscope slides
was deducted to give the tissue thickness accurately. While Hardy himself
stresses the artificiality of these conditions, he points out that this
mode of preparation is necessary to obtain consistent results (8). Further,
while one must assume that the scattering properties of these preparations
have been altered, it cannot be claimed a priori that they are drastically
dissimilar from those of living, intact skin. Hence, it seems reasonable
to accept the results of Hardy et al. as the best current estimates of the
absorption of radiation in skin.
A preliminary observation might first be noted. Measurements on stacks
of ground glass plates led to the result that the increase in absorption of
radiation with stack thickness (i.e. number of plates) followed an exponential
form in exact accordance with what one would expect for a non-scattering
medium (9). This may provide some justification for expressing the absorption
pattern in terms of a linear absorption coefficient. With skin samples, in
the spectral region from 1.0 u out to the limit studied, 2.4 u , exactly
the same situation prevailed; i.e. the absorption of radiation could be
characterized by a single absorption coefficient, independent of depth,

although dependent on wavelength. For a short wavelength span, then,
equation (4-8) may be written
-PX
q" (x,x) = 8 Halt) er
For wavelengths shorter than 1.0M , however, the data were best fit by a
"double exponential" form with one absorption coefficient,
applicable
9
for the superficial layers, and another, rz, applying for the deeper tissue.
62
Again, both X, and rz depended upon wavelength, and in all cases, o
was
greater than 82 ; i.e. the absorption was more pronounced near the surface.
Now, equation (4-8) must be expressed (for a narrow wavelength interval) as:
91" (x, t) = 8. Halt) e-rix
04x<b
e-ralx-b), *>b,
= 82 Halthe-rib
(4-9)
where b is the depth of the "break" in the absorption pattern. Clearly,
the first equation above, involving only a single coefficient, r, is a
special case of the second set, where
V2. Thus, one may take equa-
tion (4-9) as the general expression for q!!!(x,t), allowing X, and 82 to
assume any (positive) values, including equality. This pattern of q!!!(x,t),
for brevity termed the double exponential pattern, is the one selected for
the theoretical analysis of the diathermanous solid, which will follow in
the next section.
The objective of this and the previous section was to establish a
functional form of q'''(x, t) for substitution in the governing differential
equation (4-1). This objective has been achieved with the statement of
equation (4-9). These sections have been quite extensive for the purpose
of offering some justification for the double exponential pattern; however,
it is definitely not claimed that this form has now been rigorously established
as being "true"; i.e. representing the actual state of affairs in skin. Rather,

equation (4-9) having now been presented, it should simply be regarded as
another assumption of the diathermanous solid model, which assumption can be
validated only by experiment. With this qualification, the main task of
establishing the temperature response of the diathermanous solid model will
now be resumed.
63
4.4
The Solution of the Heat Conduction Equation for the Dia thermanous Solid
with Double Exponential Absorption
Collecting the various equations presented previously, one may now
state the problem in full as:
DIE EERSTE DRESSES
ô z W (x, t)
a U (x, t)
ot
+
& Halterix
10
OLXab, tao
2x²
(4-10)
tot du(xit)
26th I Hollje ribe -rele-6)
*>b, txo (4-11)
Ulx, o)=0
(4-2)
Uls, t) = 0,
to
(4-3)
In orb (4-3)
20(0,t) = 0
tzo
2x
(4-4)
U (6-0,t) = U (6+0,t) tzo
an
(4-12)
ou lb-o, t) - dulb +o, t)
ox
tso
ox к
(4-13)
Equations (4-12) and (4-13) are continuity conditions analagous to equations
(3-22) and (3-23) of Chapter III.
As stated in the previous section, both
di and Y2 are dependent upon wavelength, hence the above set of equations
is applicable only in a wavelength interval sufficiently short that the
coefficients are approximately constant. Extension of the results to a

broad spectral range will be considered later.
Once again, the problem will be attacked by the method of the Laplace
transform, with the consequent reduction of equations (4-10) and (4-11) to
second order ordinary differential equations. Thus, taking the transform of
these equations, and utilizing condition (4-2), gives
d² uxs).
& ux,s) = - i halse
o<x<b
dx²
64
and ਓ ਚ ਤ ਕ ਰਤੇ ਨੇ ਡAB et
d²u(x,) sulx,5) =
е e
x>b,
dx²
(4-15)
where, as before, u(x,s) and ha(s) are the transforms of U(x,t) and Ha(t),
respectively.
The remaining boundary and continuity conditions become :
U10, S) = 0
(4-16)
dulo,s)
s
(4-17)
u(6-0,5) = u(btos)
(4-18)
dulb-os)
dulbtos)
dx
and
(4-19)
x
It is interesting to note that in this problem, the transform of the irradiance
function, ha(s), enters in the non-homogeneous differential equation, while
the boundary values are homogeneous, in contradistinction to the opaque solid
problem where the differential equation is homogeneous, and ha(s) enters via

a non-homogeneous boundary condition.
The homogeneous solution of equation (4-14), say up(x,s), is
+ rs
+ Be
О<x<ь,
UNITS) = A e te v
precisely as in the opaque solid case.
To this solution must be added the
particular integral, up(x,t) which may be assumed to have the form
Up(x,s)= Me-rix.
Substituting this expression in (4-14) gives
(v.2- a) me *
& hals) e-
xw
۔
or
2 흡
​ha is)
M =
(S-8²x)
The condition (4-17) requires that
65
Samet) e amo en el patients are all
At
B-
8,2 hals)
15-ria)
e sad
from which the constant of integration B may be evaluated in terms of A to
give the complete solution of (4-14) as:
)
tiers
& hals)
V 5-8,20
os xa b
ISSON
Similar considerations lead to the complete solution of (4-15) as:
rib
Ulx,5) = ce vers
+
& halsle
e dz/x-6)
x>b
Ger
where the limiting value (4-16) has been used to evaluate one of the con-
stants of integration. The two constants of integration remaining in the
above equations must now be evaluated by the continuity conditions (4-18)
and (4-19). The procedure is given in Appendix V, where the detailed solution
for the diathermanous solid model is presented. With the appropriate ex-
pressions for these two constants substituted in the equations above, one
obtains finally the transformed response for this model as
ulx,s) = hals) Irie
rie-
S-
us (s-8,²x)
8,212 out us
(4-20)

-rib
82
vs
rs
va
+
e los
tava) VENÉ Hribas)le
:)]
(15/15 + 8 252) T5 (U5 + 8.5x)
O & xeb,
and
Vs.
{
-rib-82(x-1)
е e
s-82²x
-
8,2 se
15(5-8,2x)
setur
е e
x+b.rs
va
(4-21)
+
2
* Creatments - vistas) e*
- lustis-rava] - 15 105-OVK
s-oras) ***]?
(
*>b
66
Letting the bracketed terms in equations (4-20) and (4-21) be f (x,s)
and fz(x,s) with inverse transforms F1(x,t) and F2(x,t), respectively, one
may write the solution for U(x, t) by use of the convolution as
OLX LG
>
Ulx,t) = Cialt-7) * F. (7) &T
-
SHA
Halt-a). I F2 (7) de
x>b,
for any general irradiance pulse form.
It is more useful, however, to con-
sider again the step function:
Halt) = otco
Ha, t20,
whereupon
Ha
hals) =
S
The determination of the inverse transforms of (4-20) and (4-21) is straight-
forward but quite lengthy, and has been relegated to Appendix V. As is shown
there, the final expression for the temperature response, ustep(x, t) can be

normalized by defining the dimensionless terms
>,= rib
72 = 81/82
Audi?at,
and
I (8,0) = kr. U step (8/8, 6/8%)
Ha

67
whereupon the normalized response becomes
y (5,0) = cavo e * -gerfeli)
{ [ererte (Vo + zie) + e Perfe (VB - Zo)]
Se allerfe (22) erte ()) (---)
+
Lemos é erfe(@ + ) + de este to enter ke + 2y
dits
210
die eerst
cenfeertc (18 + 2) de estos e fertel et )],
020, of ELA
(4-22)
and
co
+
2
FAvo e
va e***-serte (na) -e [tole * -1) +1]
' [e'erfe (18+ zabe) + Perfe (V6 - ze)]
Alerte (2018) + erte dz
eerte (tot get the case worteller
et ce erfelia + ) de esta emerference bet),
+
е e
2
di-
210

dots
217
-é
(4-23)
68
This function thus depends upon the dimensionless variables & ando,
proportional to depth and time, respectively, and the dimensionless para-
meters 1, and 12.
These two expressions are seen to be identical except that the term
e-s in (4-22) is replaced by - -^ [azie hi-1) +1] in (4-23). This,
incidentally makes it quite simple to check the continuity conditions (14-18)
and (4-19). Thus
and
Lim (-es) = - et
Limf-ed[nole -1) + ]]= -endi;
Lim os (-es).
→
further Liin
es
while
Lim, 8-*izle
of {-e'[-1)+]} -ende.
Hence the continuity equations must be satisfied.
It should also be noted
that if
12 = 1 (i.e. 8i = 12 ) both equations reduce to
115,0)= ahavo e * - $erfe czto) - eis
to le erfe (104 km) te fertelvo-zo)]
+

which is the expression for "single exponential" absorption. Finally, the
first two terms in the equations may be expressed as
aloe *. serfeli) v va Ramo),
where the function R(x) was defined in Chapter III (see Appendix III). This
does not imply that equation (4-22) or (4-23) will reduce to the solution
for the opaque solid for some limiting situation; this is clearly impossible
since the homogeneous boundary condition (4-4) can never reduce to the
69
non-homogeneous expression (3-4), which latter condition is a unique feature
of opaque solid theory.
Other than the few comments above, it is difficult to make any general
statements about equations (4-22) and (4-23). Their complexity makes it
virtually impossible to gain any intuitive "feel" of the form of the solu-
tions, and one is forced to employ numerical computations and point-by-
point curve drawing. Even this procedure is not simple; using a desk cal-
culator, it requires from 30 minutes to one hour to determine and check a
single value of Flę, ) for one set of values of F, 0, 1, , and
da . Accordingly, the solutions were programmed for numerical computation
on the University Computing Center's IBM 650 digital computer, and values
of 115,4) determined for rather wide ranges of the above-mentioned vari-
ables and parameters. (It is interesting to note that the computation time
per point was about 30 seconds, which is long for machine time, but an im-
provement by about a factor of 100 over average desk calculator time.)
The
programming, debugging, and actual machine operation were all done by
Dr. A. M. Dutton, of the Departments of Radiation Biology and Mathematics;
his labor of love is here acknowledged with profound gratitude.
The numerical values obtained upon print-out of the computer answer
cards may be thought of an representing points on a hyper-surface in five

dimensions, Ķeç,, di
and 1. To be useful, these must be re-
9
duced to two-dimensional plots, with the other variates represented as
parameters.
While the data have been plotted in various ways, the most
immediately appealing scheme is to present y, the normalized response, as
a function of 0 , the normalized time; this secures a close resemblance to
experimental curves, where temperature was recorded against time.
Accord-
ingly, this is the procedure followed in the following sets of curves.
In
Figure 4-1, the normalized temperature rise is plotted against the
70
normalized time, with normalized depth as a parameter; the ratio of linear
absorption coefficients, 12, is unity, hence this plot represents the
results for "single exponential" absorption. As noted above, the results in
this case are independent of di, the normalized depth of the break in the
absorption pattern. Figures 4-2a, b, and c present the same information
with 2 equal to 3, and ), equal to 0.1, 0.5, and 1.0.
In Figures 4-3a,
b, and c, z is 5, and, is 0.1, 0.5, and 1.0.
It will be noted in
several of the figures that values for equal to 10 do not appear. These
points were not calculated, since it seemed likely that this represented a
depth far below any which would actually be observed.
A feature of some interest in these curves is that for the smaller
values of
at least, the initial slope (i.e. Lim
OV (5)
) is finite;
it will be shown below that this holds for all values of St. This is in
marked contrast to the solutions for the opaque solid.
In this latter case,

the initial time rate of change of the temperature response is infinite on
the surface (x = 0), and zero for all subsurface positions (x > 0). An
immediate consequence of this last fact is that when one obtains the response
of the opaque solid for a rectangular pulse by superposition there will be
no immediate downward break in the subsurface response curves, and the tempera-
ture will continue to rise to a "smooth" maximum at some time following the

termination of the exposure.
Consider now this same superposition to obtain the rectangular pulse
response of the diathermanous solid.
Because of the initially finite slope
of temperature response for all depths, it follows that there will be a
break in all curves at the termination of the exposure. Inspection of the
accompanying figures suggests that for some depths (values of § up to
2.0, at least), the slope of the temperature response curves will actually
be negative immediately following the termination of the pulse, which is
71

10
9
22=1
8
S=0
Dimensionless Temperature Rise, T(F)
5=10
1
0
0
10
20
50
60
70
80
30
40
Dimensionless Time,
o
Figure 4-1. Predicted Response of the Diathermanous Solid, Single Exponential Absorption
72

10
9
22= 3
2,-0.1
8
0
Dimensionless Temperature Rise, IN (5, 4)
5=2
5=5
Ş=10
1
0
0
10
20
50
60
70
80
30
40
Dimensionless Time,
Figure 4-2a. Predicted Response of the Diathermanous Solid, Double Exponential Absorption
73

10
9
12:3
2,-0.5
8
Dimensionless Temperature Rise, I(50)
8=0
&=1
€ = 2
&=5
2.
S=10
1
0
0
10
20
50
60
70
80
30
40
Dimensionless Time,
Figure 4-2b. Predicted Response of the Diathermanous Solid, Double Exponential Absorption
74

10
9
^2=3
2,=1.0
8
6
6
Dimensionless Temperature Rise, I/5,9)
8=2
4
5=5
2
1
0
30
0
10
20
30
40
50
Dimensionless Time, o
60
70
80
Figure 4-2c. Predicted Response of the Diathermanous Solid, Double Exponential Absorption
75.

10
9
22=5
7,=0.1
8
Dimensionless Temperature Rise, I (54)
- 2
&=5
N
G=10
1
0
0
10
20
30
40
Dimensionless Time,
o
50
60
70
80
Figure 4-3a. Predicted Response of the Diathermanous Solid, Double Exponential Absorption
76

10
12=5
9
1,=0.5
6
Dimensionless Temperature Rise, F(50)
9:0
8 = 2
le
2
3:10
과
​0
0
10
20
50
60
70
80
30
40
Dimensionless Time, o
Figure 4-3b. Predicted Response of the Diathermanous Solid, Double Exponential Absorption
77

10
da = 5
2,=1.0
st
Dimensionless Temperature Rise, I(S,)
See
:1
E = 2
4
3=5
2.
1
10
20
50
60
70
80
30
40
Dimensionless Time, o
Figure 4-3c. Predicted Response of the Diathermanous Solid, Double Exponential Absorption
78
simply another way of stating that the temperature reaches its maximum value
at the end of the exposure, and thereafter declines back to its initial
value. This is a very characteristic feature of the diathermanous solid
response; not only on the surface, but also for some distance into the
material, the temperature decreases abruptly upon termination of a rec-
tangular pulse.
It might be mentioned here in passing, that the solutions for the
diathermanous solid response, equations (4--22) and (44-23) have been pre-
sented for an irradiance step function, only. For a rectangular pulse of
duration in seconds, one may write, for the normalized response,
rrect (8,0) = y step 15, o), oso srilan
= qonster (8,0) - Jax step 15, 9-0,40m), osti?«m.
If one writes these out in full, performing the indicated substitutions in
(4-22) and (4-23), one quickly becomes convinced that nothing of a general
nature can be obtained from the lengthy and unwieldy expressions.
4.5 Experimental Evaluation of the Constants of the Diathermanous Solid
While equations (4-22) and (4-23) represent an interesting exercise in
the solution of a particular partial differential equation, they will be of
value in the present study only if some means can be found for experimental
evaluation of the various unknowns of the equations.
Two of the constants
which appear are thermal conductivity, k, and thermal diffusivity, a
9

which occur in the normalizing factors for temperature rise and time,
respectively. These, however, are "thermal" constants, which may be eval-
uated directly from opaque solid studies, and hence may be considered as
known quantities in this present case.
(As explained in Chapter III, it
is precisely for this reason that one would study "opaqued" skin response.)
The normalizing factors thus contain but one unknown "optical" constant,
79
8 ४।
the linear absorption coefficient for the superficial layers. The
other unknowns which must be determined, then, are 82, the linear absorp-
tion coefficient for the deeper material, and b, the depth of the break
between di and 2.
If one compares the curves of Figures 4-1, 4-2, and 4-3, it is im-
mediately obvious that these sets of curves are all quite similar.
Thus, it
is virtually impossible to determine the unknown constants simply by com-
paring experimental curves with these predicted ones. There are simply no
outstanding characteristics to serve as a guide in making such comparisons,
and further, all variables--temperature, depth, and time--have been stretched
by unknown factors.
Considerable progress can be made if one turns to an alternative pro-
cedure developed from consideration of the initial slopes of the Y (6,9)

VS
A curves. As noted before, these values all appear to be finite, which
is a distinctive characteristic of the diathermanous solid. It is of in-
terest, then, to develop an analytical expression for these slopes.
It will be convenient at this point to revert to the solutions for
U(x, t) before normalization.
In Appendix V, the general expression for
2 u(x,t)/2 t is presented, and when one takes the limit as t approaches
zero, the following is obtained:
one oulx,o)
I Ha enix
o=xab
at

Ha e-rib
- 82 (x-1)
e
e
xab.
b, u
>
(4-24)
Now, these extraordinarily simple equations suggest a most obvious method
for determining 8, and 82 , and hopefully also b. Consider first the
initial time rate of change of the surface temperature. This will be simply
du 10,0)
8, Ha
D
80
or, using the previously defined U* as the response per unit absorbed
irradiance:
aw*(0,0)
at
sla
With the thermal constant v known, 6, is immediately given by simply
measuring this initial slope from the recorded data.
Now consider the
logarithmic form of equation (4-24):
In 2 U* (x, o). In rix,
os xab
at
18-82 db
In he
82x,
x>b.
Thus, plotting the logarithm of the measured initial slopes of tempera-
ture at various depths against depth, x, one should obtain two straight
lines, one with a slope of -8, for 0 < x < b, and the other with a
slope of - 82 for x > b.
This procedure, subject to the practical dif-
ficulties of achieving accurate measurements of the initial slope, offers
a straightforward means of determining the unknown constants ri, rz,
and b.
4.6 Direct Experimental Determination of the Pattern of Absorption of
Radiation in Skin
Equation (4-24) has been derived for the special case of an ir-
ta
radiance step function, i.e.
Halt) = otco
= Ha, tro
Let this functional form for H (t) now be substituted in the expression
for q''(x,t) selected for this analysis (equation 4-9):
q'"(x, t) = 0, T20,*20
8. Ha e-ma, tzo, Ostab
= 82 Hae e rib
e-P2 (x-b), t=0, x=b,

81
and determine, in particular, the value of q'''(x,t) at t = 0. Obviously,
this is:
9111(x, o)
8. Ha e
osxab
-rib
82 Hae
- Pz(x-b)
*>b.
Comparison of this last equation with equation (4-24) leads to the sur-
prising result that in this special case of double exponential absorption
and step-function input:
au (x, o)
at
q" (x, 0)
" x)
(4--25)
Since double exponential absorption contains single exponential as a special
case, for this latter pattern the relation (4-25) will be satisfied as well.
It is natural to ask whether this relationship is valid only for an
exponential-type form of q'''(x,t). From the solutions for the linear ab-
sorption pattern, briefly mentioned in Section 4.3, it may be shown that for
the step function form of Ha(t), equation (4-25) is satisfied. (These

solutions may be found in Appendix II; since this form of absorption is
known to be incorrect, the derivation of the solutions has not been pre-
sented.)
In addition, this relation was found to hold for several textbook
problems.
If one is willing to adopt the definition of q'''(x, t) for the
opaque solid as
9" («,0)
,
x > 0
then (4-25) is valid for this model, as well.
May equation (4-25) be accepted as a general relationship, valid not
only for various forms of the absorption pattern, but also for arbitrary
irradiance input functions? If so, it follows that one has at hand not merely
a method of checking some particular assumed absorption pattern, but rather a
means of actually establishing the pattern itself directly from experimental
82
data.
In Section 4.2, it was pointed out that q'''(x,t) may be expressed
with complete generality as a product
9" (xt) = Ha(t) F(x))
where the depth dependence of q'''(x,t), F(x), may be considered as the
general absorption pattern of the radiation. Substituting this expression
in equation (4-25) leads to
au (x, o)
I Halo). F(x)
at
or, if Halo) is non-zero:
(4-26)
F(x) = cos outro).
at
All terms on the right-hand side of this expression may be determined by
experiment; hence this relation establishes the previously mentioned claim
that one can define the absorption pattern of radiation in skin, F(x),
directly from temperature measurements. Equation (4-26) represents one of
the most important results of this study; it is, so far as the writer can
determine, an original contribution to the problem of determining the inter-
action of radiation with a diathermanous material.
In theory, at least, this
procedure could be useful in studies of microwaves, or, more generally, any
system governed by the diffusion (heat conduction) equation.
Now, all of this development clearly depends upon the general valid-
ity, as yet unestablished, of equation (4-25). It might be noted here that
substitution of equation (4-25) in equation (4-1) yields the equivalent

statement:
OU(x,0)
o, xzo
(4-27)
2x2 2
A review of several texts on heat transfer and mathematics failed to yield
an analytical proof of either equation (4-25) or (4-27); also the writer
was unable after some time and effort to establish such a proof, and hence
83
was forced to fall back on a physical argument which indicated the probable
generality of these expressions. This argument may be phrased as follows:
In the first instant after the irradiation commences, the temperature
rise at any position will be governed solely by the absorption of radiant
energy at that point, and will be uninfluenced by the temperature of sur.
rounding regions; from this, equation (4-25) follows. Now, the term
2?u(x,t)/2 x2 in the heat conduction equation represents the heat flow
tern; equation (4-27), then, follows from the argument that no heat will flow
until some infinitesimal temperature gradient is established.
This is equi."
valent to the claim above that initially, the temperature at any given point
is independent of that of surrounding regions.
While this argument may be intuitively appealing, it is quite un-
acceptable as proof of the general validity of equation (4-25) or (4-27).
First, it provides no clue as to the class of systems wherein these re-

lations are valid. Secondly, it employs, only thinly disguised, the concept
of heat as a fluid (caloric); indeed, this fluid, while still massless, must
be endowed with inertia. What is needed is a rigorous proof of these equa-
tions, whereupon the interesting concepts in the paragraph above become
consequences, not "proofs" of the relations.
Accordingly, attention must again be turned to the problem of con-
structing an analytical proof of either (4-25) or (4-27). Consider the ease
of unidirectional heat flow in a semi-infinite isotropic body initially at
uniform temperature throughout, and with insulated surface.
Two boundary
conditions may be given immediately:
U (x,0) = 0, xzo
(4-2)
and
aucot
o, tzo
84
Now, define the Laplace transform of U(x,t) with respect to x as
us;
equation (4-2) above thus becomes
uls, o) = o.
(4-28)
From the appropriate property of the Laplace transform (10) one can write
2² U(x, t)
(t
aulo,t)
2x2
The last term in this equation is zero, by condition (4-4). Now evaluate
this transform for zero time, i.e. t = 0:
Les 16.0)}. sºu15,0) - SW10,0).
But, from (4-28) and (4-2), both terms on the right are zero, hence
= 0
La formal x, 0))
Since ["{0}
d'u
0, it follows that
dood? Ulx,0)
dx² 2
which establishes equation (4-27), and hence (4-25) also.
This proof may
be unduly restrictive; i.e. these relations may be valid in a much broader
class of systems than here defined. (In fact, the proof has been presented
here in detail in the hope that someone with more adequate mathematical
background than that of the writer will develop a more general statement.)
However, equation (4-25) will certainly hold for the model defined by
equations (4-1) through (4-4), which is all that is necessary for the
present study.

One additional relation of considerable importance will now be
derived.
If equation (4-25) is substituted in equation (4-6), one obtains
S. 241,0) dx =
Hacol
V
(4-29)
85
This is a necessary condition, independent of any assumptions regarding the
form of the absorption pattern; one then has a valuable check on the ac-
curacy with which the initial slopes have been determined, since the terms
on the right hand side will be known, and the integral may be evaluated
graphically.
4.7 Summary
The foregoing treatment of the diathermanous solid has been rather
lengthy; probably unreasonably so.
However, many of the problems considered
have not been adequately treated in the literature, nor is it claimed that
they have been adequately treated here.
It is only hoped that these extensive
discussions will suggest the proper questions which must be answered before
the over-all problem of the interaction of radiation with tissue may be
considered solved.
Turning to the solutions for the double exponential absorption pattern
(equations 4-22 and 4-23), one important problem remains. As previously

stated, these solutions apply only over a wavelength interval sufficiently
narrow that the linear absorption coefficients may be regarded as constants.
The superposition of solutions to account for variation of these coefficients
will be considered in the following chapter.
The theoretical development
will then be sufficiently complete so that attention may be turned to the
experimental phase of this study.
FOOTNOTES
(1) A Study of the Physical Basis of Burn Production with Applications to the
Defensive Reactions to an Atomic Bomb Air Burst, Medical Research and
Development Board, Office of the Surgeon General, 34 (Author and date
of publication unknown.).
(2) Buettner, K., Effects of Extreme Heat and Cold on Human Skin III. Numeri-
cal Analysis and Pilot Experiments on Penetrating Flash Radiation Effects,
J. Appl. Physiol., 5, 209 (1952).
86
(3) Hardy, J. D. and C. Muschenheim, Radiation of Heat from the Human Body
IV. The Emission, Reflection, and Transmission of Infrared Radiation by
the Human Skin, J. Clin. Invest. 13, 817-831 (1934).
(4) Hardy, J. D. and C. Muschenheim, Radiation of Heat from the Human Body
V. The Transmission of Infrared Radiation Through Skin, J. Clin.
Invest. 15, 1-9 (1936).
(5) Hardy, J. D. Annual Progress Report for 1951 (to Chief of Naval Research,
Attn. Biology Branch).
(6) Hardy, J. D., H. T. Hammel, and D. Murgatroyd, Spectral Transmittance
and Reflectance of Excised Human Skin. J. Appl. Physiol. 9, 257-264
(1956).
(7) Clark, C., R. Vinegar, and J. D. Hardy, Goniometric Spectrometer for the
Measurement of Diffuse Reflectance and Transmittance of Skin in the
Infrared Spectral Region, J. Opt. Soc. Am. 43, 993-998 (1953).
(8) Hardy, J. D., H. T. Hammel, and D. Murgatroyd, op. cit., 257.
(9) Ibid., 260.
(10) Churchill, R. V., Modern Operational Mathematics in Engineering, 294,
Operation No. 4 (McGraw-Hill Book Company, Inc., New York, 1944).

3 87
CHAPTER V
THE INFLUENCE OF THE QUALITY OF RADIATION ON THE
DIA THERMANOUS SOLID RESPONSE
5.1 Development of the Problem
In the analysis of the opaque solid, the incident radiation was con-
sidered only as so much energy or power, and no consideration of the quality
(wavelength distribution) was necessary. Indeed, the general solution of the
opaque solid is equally applicable to the case of contact heating with the
absorbed irradiance, Ha(t), replaced by the more general power input per
unit area, q'!(t), which involves a suitable heat transfer coefficient.
In contrast to this situation, the formal solution for the diathermanous
solid is valid only so long as the linear absorption coefficients are con-
stant.
It has already been pointed out that these coefficients are strongly

dependent upon wavelength; hence the diathermanous solid response depends
upon both the quantity and the quality of the incident radiation.
It will
be the purpose of this chapter to develop methods for extending the previously
established relations to allow for this variation of the "optical" constants
with wavelength.
Specifically, the problem is to establish the validity of super-
position of the formal solutions for the diathermanous solid.
Rules will
then be developed for superposition of the normalized solutions, and the
results of machine computations, based on constants taken from the litera-
ture, will be given. Some consideration will also be given to the influence
of quality of radiation on the initial time rate of change of the tempera-
ture response.
5.2 Superposition of Solutions for the Diathermanous Solid
Let the relative spectral energy distribution of the incident
88
irradiance be divided into n contiguous segments, within each of which the
linear absorption coefficients, 81 and rz, and the backscatter factor, R,
may be considered essentially constant. Now, imagine a diathermanous solid
exposed to substantially monochromatic radiation, of wavelength correspond-
ing to the ith wavelength interval. At any position and time, the "internal
heat generation" or absorption per unit volume will be given by q'''}(x,t).
Similarly, the absorption per unit volume due to monochromatic radiation
corresponding to the jth interval will be q'''z(x,t). If both of these
monochromatic beams are now directed on the solid simultaneously, the ab-
sorption per unit volume at x and t will be
q'"'z(x,t) + q''';(x,t)
provided that the receiver is passive. (Note that this relation will hold
even if the time dependence of q'''}(x,t) is different from that of q'''
q'''z(x,t).)
By obvious extension of this argument, it follows that the total absorption
per unit volume at any position and time, q''(x,t), due to the simultaneous
action of all n wavelength intervals of incident irradiance is
q"(x,+) = E q'";(x,c).
(5-1)
The temperature response, U(x, t), due to this total volume absorption,
q'''(x,t), will be given by the solution of the differential equation (4-1):
au (xt) 2² W lith+ + 9" (+,+) , xzo, tzo.
ax ²
Rearranging slightly, and substituting (5-1), this relation becomes
ราคา

(be - med Ult)
1) 018)= 9"; (457)
q!"; (x,t), xz0, +20.
in
to (5-2)
Also U(x, t) must satisfy the boundary conditions
U (x, o)=0, x=0
(5-3)
Ulot) = 0, tzo
(5-4)

ou 10,t) = 0, tzo
and
(5-5)
89
Now consider the temperature response, Uı (x,t), of the diathermanous
solid exposed to monochromatic radiation of wavelength as
There will be
n such functions, corresponding to n wavelength intervals of the incident
1
heterochromatic radiation, each of which must satisfy the following set of
equations:
les
(BE-) wilx+) = q"; (x, t), x>0, +20, :1, 2. ..."
(5-6)
Vi (x,0) = 0, *20, 1 = 1,2, ..n
(5-7)
Uiloot) = 0, tzo, i = 1, 2,...n
(5-8)
DU; (0,5) = 0,620, i=1, 2, .***
(5-9)
If « is a constant, strictly independent of x and t, then the termos
2 22
2
at
may be recognized as a linear operator. As a consequence of this, if one
takes the sum over all n of both sides of equation (5-6);
Wilx, t)]= 2 = 9!";( x,t),
(87
9.-) Wilx,
then the order of summation and differentiation may be interchanged to give
(c) Wilrie È 9 ";(x,t).
(5-10)
Further, it is obvious from the homogeneity of conditions (5-7) through
(5-9) that

och (5-11)
Ź Wilx,0) = 0, 420
Ź , )
È Wilo,t) = o vi(0,t)=0, +20.
Wiloo,t) =0, tzo
(5-12)
Wi Wa
and
(5-13)
90
Comparing equations (5-2) through (5-5) with (5-10) through (5-13), it
follows that
Ulrit) - Ž Wilxit), *20, +20,
,
(5-14)
and the validity of superposition of solutions is established.
For the special case of double exponential absorption and step-function
irradiance input, as treated in the previous chapter, the expression for
q'
'i(x,t) becomes :
91; lx,t) = 0,
t co
rrix
=8i Hai e
tzo, ostab,
arzilx-6)
= 82, Hai e-rib
e
t2u, > b)
i=1,2,...n.
(5-15)
Note that the depth of the break in the absorption curve, b, is assumed in-
dependent of wavelength. Substitution of this relation for q'''}(x,t) in
(5-6) will yield n sets of equations, each set identical to (4-10) through
(4-13) of the previous chapter, with the trivial addition of the subscript
"i" (where i
1, 2.•.•n) to U(x, t), X,
and 82
The formal (non-

normalized) solution of each set will give a value of U. (x,t), and the
response for the total wavelength distribution, may then be found by super-
position, as demonstrated by (5-14), above.
5.3 Superposition of the Normalized Solutions for Double Exponential
Absorption
The procedure outlined above for determining the temperature response
of the dia thermanous solid to heterochromatic radiation is quite straight-
forward. For the special case under consideration, double exponential ab-
sorption, the solutions have been obtained in normalized form, F (5,9)
91
for convenience in computation, and here it is not true that too.
FIBA) = Ê F; (€, ).
Simple superposition does not hold for these normalized solutions because the
normalizing factors for , and all involve X , and this latter
coefficient is wavelength dependent.
In order to add these normalized solutions, then, it is necessary to
transform each back to real temperature rise, and add terms corresponding
to the same values of real depth and time.
From the definitions of ,
§ , and given in Chapter IV; i.e.:
Su Yi X
&= 8,2xt
Hotararea
I = kri u
en Haw
SES
it follows that the total temperature response U(x,t), is given by
Data
U(xt) =
I 8 x 8,.
(5-16)
krii
The implication of equation (5-16) is rather discouraging: it is,
in general, impossible to obtain normalized solutions for the diathermanous
solid exposed to any arbitrary heterochromatic radiation.
In order to com-
pare theoretical and experimental responses, it is necessary first to have
at hand numerical values of the constants of the diathermanous solid; it is

these very values, however, which the experiment is supposed to provide.
Thus the procedures developed in Section 4.5 of the previous chapter are
valid only if monochromatic radiation is employed, and have questionable
utility for heterochromatic irradiation.
5.4 Values of the Constants of Skin Obtained from the Literature
The most direct way out of this dilemma is to select a particular
92
set of conditions for the experimental study; this will include a choice of
radiant energy source, and an experimental animal. Now by direct measure-
ment, where possible, and literature review, the constants and parameters
appropriate to these conditions will be estimated, and numerical solutions
of equation (5-16) obtained for suitable ranges of depth and time.
This
predicted response may now be compared with the experimental results; if
these are in reasonable agreement, then at least one can claim that the form
of the equations and the constants from the literature are not contradicted
by experiment.
If the agreement is poor, then the model, the constants,
and/or the experimental results may be incorrect, and a different experimental
approach would be necessary to decide which is at fault.
The experimental conditions will be described in detail in a later
chapter, but for present purposes it will be sufficient to specify the
following:
1) Source
-- a carbon arc image furnace (1, 2). The relative spectral
energy distribution (3) and the total irradiance output in absolute units (4)
have been accurately determined.
2) Exposure pulse
CD
a rectangular pulse (i.e. trapezoidal with very
rapid rise time) of 0.5 second duration.
For time less than 0.5 second,
this is equivalent to a step function, and it will be so considered,

hereafter.
3) Experimental animals
young Chester White pigs.
The similarity
in structure of the skin of this animal and human skin has been described (5).
The spectral reflectance of the skin of the Chester White and fair human skin
is also quite similar (6, 7). Finally, closely comparable lesions are pro-
duced in pig and fair human skin by very nearly the same radiant exposures
(8, 9). These points provide some justification for the necessity of using
some constants derived from measurements on human skin.
93
The simplest term to evaluate (and the one least likely to be in error)
is the absorbed irradiance in the ith wavelength interval, Hai(t). Since the
incident irradiance, H (t), is a step function, and the spectral distribution
of the source is constant in time, it follows that Hi(t) is likewise a step
ai
function:
Hailt) = otco
Hai to.
Now, if R, is the spectral reflectance (or back-scatter factor) of the
skin (whence (1-- Ra ) is the spectral absorptance), and one defines
-1
as the lower and upper limits, respectively, of the ith
and a;
wave-
length interval, then H.
Hai
is given by
Hai = " (1-Ra) Hox dd
" (1-Ra) (Hox) dx,
Det
= Ho
dial
CATE
where Hoa is the absolute spectral distribution of the incident irradiance,
and H. is the total incident irradiance. Now the relative spectral energy
distribution, Ja is proportional to the ratio (Ho a /H.), i.e.
Hoa
Hol
Ja = Al Ho
where the proportionality constant is simply the total area under the relative

distribution curve, or
na obce
A = S. Ja da.
Now, from the above expression, the total absorbed irradiance, Ha,
is
Het to S.51-2) (E)
da
solo che tutto
94
Hence
Ho
bad Hai
Ha
SM (1-Ra) (Ho) da
Hofi-2016)
(1-R:) Ja dd
)
[(1-Ra) Jo di
or
(5-17)
Hai Haki
where
(5-18)
dial
Ki =
" (-Rs) Jo da
S(1-R.) Ja dd
Values of (1 - Ra )Jhave been determined from the measurements of RA
for the Chester White pig (10), and Ja
on the carbon arc furnace (11), and
are shown in Figure 5-l as a function of wavelength. For any choice of wave-

length intervals, values of Kį may be obtained from this curve by numerical
integration or planime try.
The next step is the evaluation of the linear absorption coefficients.
No information is available for pig skin, and it is here that one must turn
to values for human skin; some justification for this procedure has been given
above.
While considerable information on this latter tissue is available,
the task of selecting accurate values for the absorption coefficients is not
simple, since values cited in the literature vary widely (12), and no single
report covers the desired wavelength span.
As mentioned previously, the recent report of Hardy, et al. (13) con-
tains the most reliable data available.
Careful evaluation of and X2
are presented only at four wavelengths; for interpolation, only rather rough
transmission curves are given covering the spectrum from 0.7 to 2.44 • For
95
1.0
0.91
0.81
0.7
A

0.6H
ep.( Yu-t)
0.5
0.4
0.3
0.2
0.1
2.4
2.6
0
0.2 0.4
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
Wavelength, u
Figure 5-1. Relative Spectral Energy Absorption, (1-RX).Ja
White Pig Exposed to Carbon Arc Image Furnace
Chester
96
shorter wavelengths, the values obtained by Hansen (14) on human autopsy skin
were selected, as these seemed to converge toward reasonable agreement with
the data of Hardy Figure 5-2 presents the smoothed curves of the absorption
coefficients vs wavelength.
It must be stressed that this figure is com-
pounded of about equal parts of freehand art and literature research; it is
hoped, however, that it represents the best estimates available as to the
true values of these coefficients.
It will be noted that the absorption coefficients vary quite rapidly
with wavelength. This would suggest that, for accuracy, the wavelength inter-
vals must be made small.
In Fig. 5-2, the eight intervals selected are shown,
and average values of d, and 82 indicated for each interval by a horizontal
bar.
These wavelength segments are not short, and the variation of absorption
coefficients in several is large.
The increased precision which would be
gained by selecting a larger number of shorter intervals is hardly justified
in view of the dubious accuracy of the curves, and the resultant increase
in computation time.
Reference to equation (5-15) reveals that only one additional constant
must be evaluated; namely the depth of the break in the absorption pattern, b.
Hardy has suggested a value of 0.04 cm (15), based, however, not on anatomical

considerations, but rather on the thinnest section which could be conveniently
handled in his goniometer spectrophotometer.
It would seem more reasonable
to expect that a break in the absorption pattern would be associated with
some more or less abrupt change in the structure of skin, the epidermal-dermal
junction being the most likely candidate.
This would set the value of b
equal to 0.01 cm, which is a rough average of epidermal thickness in the
flank areas of young swine.
If this value be correct, then the values of
Ygiven by Hardy are low to the extent that his superficial tissue samples
contained some material with a lower absorption coefficient ( 82).
97
400
200
I-4
100
ca
80
60
40
Linear Absorption Coefficient, cm
8,
t
20
8,902
M
10
8
6
0.2 0.4 0.6 0.8
1.2
1.0
1.4 1.6
1.6
1.8 2.0 2.2 2.4 2.6
Wavelength, u
Figure 5-2. Linear Absorption coefficients for Shallow (%) and Deep (82) Tissue

98
Now, by employing equation (5-17) for Hai, equation (5-16) may be
written in terms of the temperature response per unit absorbed irradiance,
U*(x,t) as
Σ
Ki
;
(5-19)
krii
The numerical values necessary for the solution of Ti in each interval, and
the "weighting factor" Kz/k80i , are tabulated below. The numerical values
employed for the thermal constants, k and a
are
k = 9.8 x 10-4 cal cm-] deg-1 sec-1
a = 8.2 x 10-4 cm2 sec-l
which were determined from "opaqued" pig skin.
As noted previously, the weighted values of Yi must be added at values
of ſ and corresponding to the same real depth and time.
In real dimen-
sions, the following values were selected:
x = 0, 0.02, 0.04, and 0.08 cm;
t = 0.1, 0.2, 0.3, 0.5, 0.7, 1.0, and 1.5 sec
(There are thus 4 x 7 = 28 points in each interval, or a total of
8 x 28 = 224 points to be determined.) Numerical values of the normalized
coordinates corresponding to the above real variables were calculated, and,
together with the appropriate values of the parameters 2, and 12 , inserted
into the previously mentioned program for the IBM 650 computor. The re-

sultant values of normalized response were properly weighted and summed,
with the final values obtained being presented as a set of curves of
U*(x, t) vs t in Figure 5-3. These curves, then, represent the predicted
temperature response of skin based on available constants. Comparison with
experimental results will be given in a later chapter.
99
Bd 320 () ਤੋਂ ਬਚਨ 36 tਰ ਰ ਮਹ
. ਪਰ Bada ਚ ਬg 03 ਉੱਚਤਮ ਚ 1 32 3
BE
-
Table 5-1
Numerical Values for Solution of the Normalized Response Equation
Inter-
val
Wavelength
wu)
K
ri 82
(cm-I) (cm-1)
K
kri
2
12
8,²a
1
0-0.46
0.240
150
25
1.50 6.0 18.45
1.626
2
0.46-0.54
0.117
60
10
0.60 6.0
2.952 1.982
3
0.54-0.70
0.167
25
10
0.25 2.5
0.512
6.789
4
0.70-1.00
0.142
15
10
0.15 1.5
0.184
9.621
5
1.00-1.34
0.112
12
12
0.12
1.0
0.118
9.485
6.
1.34-1.86
0.117
25
25
0.25 1.0
0.512
4.756
7
1.86-1.98
0.015
80
80
0.80 1.0
5.248
0.191
8
1.98-00
0.090
40
ܘܢܐ
0.40 1.0
1.312
2.287

100

10
9
8
7
6
X- O cin
Temperature Rise per Unit Absorbed Irradiance, U*(x, t)
deg cal-l cm2 sec
SE
5
0.02cm
3F G
1
2
0.04 cm
1
0.08cm
0
0
0.1
0.4
0.5
0.2
0.3
Time, t, sec
Figure 5-3. Predicted Temperature Response of Bare Pig Skin
Exposed to Carbon Arc Image Furnace

101
5.5 The Influence of the Quality of Radiation on the Initial Time Rate of
Change of Temperature
The subjective comparison of theoretical and experimental results, as
outlined in the previous section, is clearly most unsatisfactory, and repre-
sents a rather low order of research.
It is difficult to decide a priori
just what will constitute "good" agreement. More important, if the agreement
be judged "poor," there is no suggestion as to the cause or causes of this
failure.
If one wishes to examine critically the literature values tabu-
lated above, or suggest other values as determined by experiment, a more
rigorous scheme must be devised.
Considerable aid in this direction is obtained if one turns to the
interesting relations concerning the initial time rate of change of tempera-
ture, as developed in Chapter IV.
It is only necessary to note that the
systems of equations defining the "interval" response, Ui(x,t), and the
total response, u(x, t) are included in the class of systems, as defined in
the previous chapter, for which
dulx, o)
at
- 9" (x, o)
(5-20)
and also

√ 9"; (x,0)
2 Ui(x,0)
(5-21)
at
Now with the superposition of (non-normalized) solutions established by
equation (5-14), it is an obvious step to note that
2
e Ŝ Ui (x, t).
du(x, t)
2 t
at
Interchange of differentiation and summation for this finite sum of well-
behaved functions gives
du(x, t)
at
dui lxit)
at
102
and in particular, in the limit as t approaches zero from above:
ou (x, o) - a wilx, o)
(5-22)
at
(This equation is obtained in more direct fashion by substitution of equations
(5-20) and (5-21) in the initial relation (5-1).) Superposition thus holds
also for the initial time rates of change.
If one now accepts the form of qilli(t) as defined by (5-15), then,
using expression (5-17) for Hair
du * (x,0)
riikie
o=xab
11
§ 3 Vaikietiiba-ti (x-6) som
x>6
(5-23)
e
where, again, the response per unit absorbed irradiance, U*, has been used.
The sum of exponential decay curves is not itself an exponential curve, hence
the plot of the logarithm of the initial slopes against depth, x, will not
consist of straight line segments.
The expected form is shown in Figure 5-4,
where the product of the heat capacity per unit volume, w, and the initial
slope @ U* (x,0)/2 t, has been plotted on a logarithmic scale against linear
depth in centimeters.
Numerical values for substitution in equation (5-23)
were taken from Table 5-1.
Now consider the comparison of the theoretical predictions embodied

in Figure 5-4 with experimental results.
While it is still difficult to
decide a priori just what will constitute "good" agreement, at least con-
siderable improvement has been made in the problem of selecting the probable
causes of manifestly "poor" agreement. First it will be noted that equation
(5-23) is expressed in real dimensions, and there are no thermal constants of
questionable accuracy "buried" in normalizing or weighting factors; the only
thermal constant which appears,
enters as a multiplicative factor, only.
103

100
80
60
20
8
Product of leat Capacity per Unit Volume by
Initial Slope per Unit Absorbed Irradiance, y
1
0.6
1
02
OL.06
LO
16
18
.20
.08
12
Depth, x, cm
Figure 5-4. Predicted Initial Time Rate of Change of Temperature
Response of Bare Pig Skin Exposed to Carbon Are Image Furnace
104
Secondly, at the depth of the break in the absorption pattern, b, there appears
a finite discontinuity in the predicted curve; if such a break appears, but
at a different depth, in the experimental curve, the error in the selection
of the numerical value of b is obvious, and can be immediately corrected.
The most important point, however, is that in following this method,
the experimental data are in a sense self-checking. It was established pre-
viously that a necessary condition on q!!!(x,t), independent of any assump-
tions regarding the depth dependence of this function, is:
9" (x, t) dx = Halt)
It follows from equation (5-20) that for the irradiance step function here
being considered,
Sv.
əw* (x,0) dx = 1.
(5-24)
Hence, the area under the experimentally determined curve of the initial
slopes multiplied by V plotted against depth, x, must be equal to unity, and
if this is not so, errors must exist in the measurements, themselves.
It
may be noted in passing that equation (5-23) satisfies this condition since:
Sou ow* (x, 0) dx =
Sariki e-rix dx
00
-rib -r2ilx-b)
е e
zil
dx
Ki

=
1.
5.6 Direct Determination of the Absorption Pattern of Heterochromatic
Radiation in Skin
The previous sections have treated the problems of comparing experi-
mental results with predictions from the pre-selected model; namely the
Bio
105
diathermanous solid with double exponential absorption.
Indeed it is this
comparison which is the crux of the model approach, as discussed in the first
chapter.
As was pointed out in the last chapter, however, the relation
Co roulx,0)
$ 9'''(x,0)
at
provides a means of establishing the desired depth dependence of the volume
absorption function, q!!!(x,t), independently of any hypothesized model.
Since this relation still holds with heterochromatic radiation, it is
in theory possible, for any particular spectral distribution, to obtain a
graphical, if not analytical, expression for qi'(x,t), but this expression
will hold only for the specific spectral distribution. Thus, if one wishes
to predict the response to, say, tungsten lamp illumination, then the form of
q!''(x,t) deduced for solar illumination will be of little or no value.
While it is not appropriate here to indulge in speculation, it might
be mentioned that this method of direct determination of the absorption pattern
suggests an intelligent application of "subtractive" experiments, where sharp
cut-off optical filters are used to absorb wavelengths shorter than the cut-
off wavelength, while the remainder are transmitted to the test animal (16).
Extensive series of such filters are available commercially, with rather
closely spaced cut-off wavelengths. Noting that (5-20) can be written as

V
2 W (x, o)
at
it follows that one can determine first
Ž q"; (x,0)
¿ q'*!/(x,t), then by removal of,
say, the first interval, 3 q'"'(x,t). The difference is simply
q'"'y (x,t). Piltering out the second band will give § q'"'}(x,t), from
which one can determine qi"12(x,t), and so on.
The demands on the accuracy
of such a procedure are extreme, but there would appear to be no theoretical
objections to the method.
106
5.7 Correction of Initial Slopes for Finite Irradiance Rise Time
It was pointed out in Chapter III, that no real device can produce
an ideal irradiance step function, and the actual irradiance pulse is more
closely approximated by superposition of ramp functions, or a trapezoidal
pulse. A general solution for a trapezoidal pulse response was developed as
t
U trap/xit) = (a ) c*ster (*. c) do, ostso
(
6) * step (x,c) da)
t
w 1
(x, 2) da, ista
n
where
Ha trap (+) = otro
=(4)t, osts]
= Ha, s stem
L
For the surface response of the opaque solid, a factor, ftrap, was then
derived, which allowed for correction of trapezoidal pulse response to ideal
step-function response. A brief examination of the solutions for the dia-
thermanous solid response shows that this analytical method becomes hope-
lessly complex; indeed, for heterochromatic radiation, no general correction
factor can be given.
Numerical integration of predicted response curves indicates that the
trapezoidal pulse response quickly becomes indistinguishable from the step-

function response, for the shutter opening times employed in this study.
The procedure followed, then, was that of simply extrapolating the measured
response curves back to the very carefully determined zero time. Since the
extrapolations were in all cases quite short, this procedure is subject to
little error.
107
5.8 Summary
At this point, the theoretical analysis of the temperature response of
skin to radiant energy may be considered to be sufficiently complete for
present purposes. Predicted responses for bare (diathermanous) and opaque
skin have been presented, and wherever possible, methods of comparing ex-
periment with theory have been developed. It is now appropriate to consider
the experimental methods and results of this study; these will be treated in
the following chapters.
It is important to realize that many severely simplifying assumptions
have been embodied in the models proposed, assumptions which in some cases
have been only briefly mentioned or even simply implied. As an example, the
"necessary" relation (5-26) is valid only for the uniformly irradiated semi-
infinite solid, as here stated, and it will be seen that the radiation was
actually delivered to a sharply circumscribed area of a definitely finite
animal.
Whenever possible, these assumptions will be justified by experi-
mental evidence, but (as stated in Appendix II) it must be admitted that the

complex process of a radiant energy exposure of living skin is extraordinarily
resistant to formal mathematical attack.
In spite of this, the theoretical
analysis is essential, if experimental findings are to be interpreted
intelligently.
The experiments described in the succeeding chapters are by no means
complete; they have posed almost as many questions as they have answered.
It is hoped, however, that the foregoing theoretical analysis is sufficiently
thorough to serve as a guide for future experimentation for some time to come.
It is for this reason that this analysis has been placed first not only in
presentation, but in over-all importance in this study.
108
FOOTNOTES
(1) Davis, T. P., L. J. Krolak, R. M. Blakney, and H. E. Pearse, Modifica-
tion of the Carbon Arc Searchlight for Producing Experimental Flash-
burns, J. Op. Soc. Am. 44, 766-769 (1954).
(2) Davis, T. P., "The Carbon Arc Image Furnace" in Proceedings of the
Symposium on High Temperature A Tool for the Future, 10-15
(Stanford Research Institute, Menlo Park, California, 1956).
(3) Krolak, L. J. and T. P. Davis, A Universal Spectrophotometer for the
Measurement of the Relative Spectral Distribution of the Carbon Arc
Source, University of Rochester Atomic Energy Project Report UR-367
(1954).
(4) Davis, T. P., A Standard Radiation Calorimeter, M.S. Thesis, Department
of Physics, The University of Rochester, 51, et seq. (1954).
(5) Moritz, A. R. and F. C. Henriques, Jr., Studies of Thermal Injury II.
The Relative Importance of Time and Surface Temperature in the Causation
of Cutaneous Burns, Am. J. Path. 23, 698 et seq. (1947).
(6) Kuppenheim, H. F., J. M. Dimitroff, P. M. Melotti, I. C. Graham, and
D. W. Swanson, Spectral Reflectance of the Skin of Chester White Pigs
in the Ranges 235-700 mu and 0.707-2.6 lg J. App. Physiol. 9, 75-78
(1956).
(7) Krolak, L. J., The Measurement of Diffuse Reflectance of Pig Skin,
Titanium Dioxide Paint, and India Ink; the Transmittance of Titanium
Dioxide and India Ink, University of Rochester Atomic Energy Project
Report UR-439 (1956).
(8) Perkins, J. B., H. E. Pearse, and H. D. Kingsley, Studies on Flash
Burns: The Relation of Time and Intensity of Applied Thermal Energy to
the Severity of Burns, University of Rochester Atomic Energy Project
Report UR-217, 21-25 (1952).
(9) Hinshaw, J. R. and T. P. Davis, Unpublished research to be incorporated
in a paper on the prediction of burn severity from the thermal pulse of
nuclear weapons.
(10) Krolak, L. J., op. cit., 13.
(11) Krolak, L. J. and T. P. Davis, op. cit., 33.

(12) Hardy, J. D., Annual Progress Report for 1951 (to Chief of Naval
Research, Attn. Biology Branch).
(13) Hardy, J. D., H. T. Hammel, and D. Murgatroyd, Spectral Transmittance
and Reflectance of Excised Human Skin. J. Appl. Physiol. 9, 262 (1956).
(14) Hansen, K. G., On the Transmission through Skin of Visible and Ultraviolet
Radiation, Ph.D. Thesis, The University of Copenhagen (Published as
Supplementum LXXI, Acta Radiologica), 96-98 (1948).
El 109
(15) Hardy, J. D., H. T. Hammel, and D. Murgatroyd, op. cit., 261.
The
(16) Berkley, K. M., T. P. Davis, and H. E. Pearse, Study of Flash Burns :
Effect of Spectral Distribution on the Production of Cutaneous Burns,
University of Rochester Atomic Energy Project Report UR-336 (1954).


110
!
CHAPTER VI
EXPERIMENTAL MATERIALS AND METHODS
6.1 Introduction
In the previous chapters, extensive consideration has been given to
the problem of comparing theoretical and experimental results. The comparison
schemes developed define the general types of experimental information which
must be obtained; procedures were thus selected so as to provide these
necessary data in proper form and with as high accuracy as possible. In
the following sections both the experimental materials and the methods of
their use will be described.
In addition, wherever possible the necessity
of including a particular item of equipment will be explained on the basis
of the demands imposed by the theoretical analysis.
6.2 Biological
As mentioned in the last chapter, the experimental animals employed
were young Chester and Yorkshire White pigs. Most animals were of the former
breed, and only a few of the latter were used. No differences in burn res-
ponse between these breeds have ever been noted in this laboratory, and no
differences in temperature response were found in this study. For these
purposes the breeds seem to be identical.
The animals available were weanlings with body weights between 9 and

12 kg. Prior to an experiment, a specimen was selected which was of suitable
size and without obvious abnormalities in skin appearance. Both males and
females were used. Eighteen to twenty-four hours before preparation, food
and water were withdrawn from the animal selected. Preparation consisted of
anesthetization with Dial in Urethane (Ciba) administered intraperitoneally
in a dosage of about 65 mg per kg of fasted body weight. Except for the
slightly lower dosage, the procedure was as developed by Kingsley, et al. (1).
111
In addition, 1.5 mg per kg of chlorpromazine (Thorazine, Smith, Kline and
French) was administered intramuscularly to suppress shivering (2). The hair
on both flanks was clipped and the remaining stubble removed with an ordinary
electric razor (Norelco or Schick). The entire animal was thoroughly hosed
and the shaved flanks washed thoroughly with detergent and water.
A pig thus prepared is not able to maintain normal body temperature
in a cool environment.
To avoid any difficulties with severe hypothermia,
and to achieve fairly constant skin temperatures (3), the temperature of the
room in which all experimental work was done was held at about 30°C (86°F).
Each animal was usually prepared at about 8:00 A .M., and satisfactory anes-
thesia and immobility were maintained in most cases until about 4:00 P.M.,
at which time the shiver reflex had returned, and the animal began to exhibit
"walking" motions. Experimental work was then terminated and the animal
returned to his cage. By the next morning, twenty-four hours after prepara-
tion, most pigs were able to gain their feet, but their gait was most uncertain.
After forty-eight hours, recovery was apparently complete. None of the fifteen
animals used were lost.
On a few animals, the lesions were biopsied twenty-four hours after
burning.
(For this brief procedure, anesthetization was by veterinary sodium

pentobarbital, with three to five ml administered intravenously.) Biopsies
were taken across the burned areas so that normal tissue was included at each
end of the sample. The tissues were fixed in 10% formalin and embedded in
paraffin. Duplicate sections were stained with hematoxylin and eosin, and by
a modification of Verhoeff's elastic tissue stain as developed by Hinshaw (4).
The excellent workmanship of Mrs. Fredette and Mrs. VanWinkle, of the Histology
laboratory of this Project, in the preparation of these sections is gratefully
acknowledged.
The average reflectance of the skin of the Chester White pig to the
112
carbon arc furnace employed in these studies is 0.40, as determined in this
laboratory (5). For the exposures on "opaque" skin, the skin was painted
with India ink, to which was added a few drops of a liquid detergent.
This
covering has also been investigated in this laboratory (6); the reflectance
to the carbon arc is 0.10, and the transmittance of a thin film of this
material is no greater than 0.ll.
Thus very little radiation can penetrate
into the skin.
In this study, absolutely no evidence of a "diathermanous-
type" response was ever observed with the India ink painted skin.
The thick.
ness of a typical ink film was determined by covering half of a sheet of
cellophane with the material, and measuring the thickness of the cellophane,
and the cellophane-plus-ink with micrometer calipers. The value obtained by
difference in these readings was 0.0005", or about 13 microns.
The film was
probably thinner on the skin, since the ink penetrated into the tissue somewhat.

6.3 Physical
General Arrangement
The physical equipment used in this study may be divided into several
major component groups, as indicated schematically in Figure 6-1.
The
various items were all assembled in the temperature-controlled room mentioned
above.
In the following sections the pertinent details of the component
groups will be presented, approximately in the order indicated in Figure 6-1.
6.4 Radiation Source
The radiation source used for all exposures was an arc imaging furnace,
constructed around a surplus 24 inch Corps of Engineers searchlight, Model
1942 (7) (Fig. 6-2). The primary source of radiation in this furnace is a
high current carbon arc (8), with a 10 mm positive carbon (National Carbon
Co. "Ultrex") operated at about 145 amperes (current density of 185 amps
cm?). Operating power for the unit is provided by a rotary converter
per
consisting of a 50 HP three-phase induction motor directly coupled to a 30 kw
113

Radiation
Source
Pulse
Measuring
Assembly
Animal
Holder
Temperature
Sensing
Elements
Reference
Elements
Calibration
Circuits
Preamp-
lifiers
Recording
Assembly
Depth
Reading
Device
Temperature
Regulated
Water Bath
Timing
Marker
Generator
Figure 6-1. Block Diagram of the Physical Equipment

01.
YO
Figure 6-2. The Carbon Arc Image Furnace
115
(300 amperes at 100 volts) de generator.
The standard steel parabolic mirror of the searchlight has been re-
placed by glass first-surfaced aluminized ellipsoidal reflector, manufactured
by the Bausch & Lomb Optical Co., with first and second focal lengths of ll
and 527 inches, respectively. The positive carbon crater of the arc is placed
at the first focal point of this mirror, whereupon it is imaged, with a magni-
fication of about five, at the second focus; the actual exposure plane is
located slightly inside this second focal point.
The spatial distribution of irradiance in the exposure plane dis-
plays approximate radial symmetry about the optic axis of the unit, and
follows quite closely a Gaussian function:
-nar2
H(r)
H(0) e
where H(r) is the irradiance at a radial distance r from the optic axis,
and H(0) is the irradiance on this axis (9). The maximum "full open" value
of H(o) for this furnace is 35 cal cm-2sec-1; while a = 0.13 cm-2. At a
radial distance of 0.564 cm the irradiance has dropped to about 88% of this

central maximum, while at 0.9 cm it has decreased to about 70% of maximum.
The irradiances cited in the remainder of this paper are always average values
2
over the central circular area of 1.0 cm (i.e. the area within a circle of
0.564 cm radius); as noted above, over this area the irradiance may be con-
sidered reasonably constant.
This spatial average irradiance may be varied
by a diaphragm and screens from a full open value of 33 cal cm-2 sec-l down
to slightly below 0.1 cal cm-2 sec-l. During the period covered by this
study, the calibration of the furnace was checked frequently to insure the
stability of the output.
6.4.1 Shutter System
The shutter normally used on this furnace employs two light aluminum
116
vanes, one for opening, the other for closing, actuated by rotary electrical
solenoids.
Each solenoid is powered through a thyratron; a synchronous motor
driven cam controls the thyratron grids, and provides precise timing. Ex-
posure times of from 0.1 second to as long as desired may be obtained.
For the present study, this shutter presented two difficulties.
The
opening and closing times are about 0.015 second, which is quite satisfactory
for routine work. However, as pointed out previously, it is desirable to
obtain the shortest possible irradiance rise times, in order to secure a
reasonable approximation to a true step function, and to avoid large cor-
rections of the initial temperature response.
A much more serious dif-
ficulty was the electrical pulse generated in the low-level temperature
measuring circuits each time a rotary solenoid was actuated. Although the

shutter had been expressly designed to keep these unwanted pulses to a mini-
mum, they could not be eliminated entirely; further, thiş noise occurred at
the initiation of the exposure, precisely where the maximum accuracy was
demanded by theoretical considerations.
Accordingly, an alternative arrangement was devised, using the pulse-
shaping wheel seen prominently in Figure 6-2 (10). This wheel was altered
so that over half of its periphery an opaque aluminum-foil screen extended
into and completely obscured the converging beam of radiation from the source;
over the remaining half of the wheel's periphery, the radiation could pass
unobstructed.
Between the pulse-shaping wheel and the exposure plane, and
as close to the former as possible, an opaque metal screen was mounted, with
a vertically disposed, 14 cm width slit through the center of the screen.
This wheel-and-slit system was used in conjunction with the previously des-
cribed vane-type shutter in the following way.
The wheel was rotated at one
second per revolution, with this speed accurately measured by an electronic
tachometer of special design. When it was desired to make an exposure, a
117
hand operated asbestos-board screen was swung out of the beam, thus energiz-
ing the shutter control circuits. As soon as the wheel rotated so that the
leading edge of the opaque half had swept over the slit, the opening solenoid
on the vane shutter was actuated. Approximately one-half second later, well
after any electrical disturbance due to the solenoid had damped out, the
trailing edge of the opaque section of the wheel swept past the slit, thus
initiating the exposure. After one-half second, the leading edge of the
opaque section again swept over the slit, terminating the exposure,
and

shortly thereafter the second rotary solenoid was fired, closing the vane
shutter. A single half-second pulse was thus delivered to the test animal;
measured rise time was about 0.005 second, or some three times faster than
the vane shutter.
Further, at the crucial opening point, no electrical noise

was injected into the measuring circuits. Synchronization between wheel and
vane shutter was automatic, and the system performed without malfunction
through the entire study. While only a single exposure time of 0.5 second
was used, this was no disadvantage for this investigation.
6.5 Animal Holder
In the usual investigation in this laboratory, the experimental animal
is simply hand-held behind a fixed water-cooled aperture plate which delimits
the exposure area (11). The circular aperture is centered on the optic axis
of the furnace, and is about 1.8 cm in diameter. The water circulated through
the aperture plate is held at about 35°C.
For the present study, this simple arrangement was not adeguate. In
order to make an exposure, it is necessary to have the exposed spiace in
the vertical exposure plane, and only a few inches behind the rene sitter;
in this position it is impossible to insert fine temperature sensing els-
ments into the burn site. To circumvent this problem, a hinged animal holder
118
was constructed as shown in Figures 6-3 (open), 6-4 (closed), and 6-5 (in
exposure position).
The animal was first placed in this holder, flank up, and fixed with
blocks, wedges, and straps.
The hinged aperture plate was swung down over
the animal's flank, and the limits of the intended burn site marked with ink,
This plate was then swung out of the way, and the "thermocouple pattern"
inked on the skin, as shown in Figure 6-3. After installation of the thermo-
elements (to be described below), the aperture plate was again closed over
the animal, and the entire holder rotated to bring the selected site into a
vertical plane.
The holder was so fixed to the furnace exposure table that
the aperture would now be in the exposure plane and precisely centered about

the optic axis of the source.
6.6 Temperature Sensing Elements
The search for a satisfactory temperature sensing element was the
first project undertaken in this investigation.
This search was initiated
in 1952, and has by no means ended.
It would not be appropriate here to
present a detailed history of this program, but in view of the somewhat
unusual materials employed, a brief review is in order.
The primary considerations were small size, coupled with best pos-
sible sensitivity, fast response time, adequate strength, and lack of tissue
reaction.
It was decided quite early that thermistors would not be suitable,
since even the unmounted bead types were larger than desirable.
In addition,
there were problems of insertion, insulation of lead wires, and the not par-
ticularly impressive response time constants available. Attention was there-
fore directed to thermocouples, which in spite of their meager output voltages
have the advantages of rapid response and ease of insertion (with proper
construction). Further, the only lower limit on their size is set by
119

B
Figure 6-3. Animal Holder, Open
120

Figure 6-4. Animal Holder, Closed
121

2.)
Figure 6-5. Animal Holder, in Exposure Position
122
considerations of strength and feasibility of construction.
6.6.1 Wollaston Wire Thermocouples
This last consideration, that of ease of construction of the thermo-
elements, was by no means of minor importance, since it was anticipated that
a large number of elements would be needed for this investigation.
An elegant
solution to this problem seemed to be the use of Wollaston process wire in
the production of fine thermocouples.
This wire is a composite material,
formed by placing a sleeve of one metal around a rod of another; the as-
sembly is then drawn through dies to reduce its diameter as though it were
a single wire (12). Normally, the Wollaston process is used to obtain wires
of very small diameter, by simply dissolving away the unwanted outer jacket
in a suitable reagent, thus freeing the fine inner core.
For thermocouple
production, this procedure is revised slightly.
A suitable length of the

composite wire is selected, and the jacket dissolved away for only half the
length of the wire.
If the resistances of jacket and core are selected
properly, the shunting effect of the latter in the former is negligible,
and one has easily formed a thermocouple at the jacket-core junction.
After extensive discussions with a leading supplier of Wollaston
process wire (13), one suitable pair of metals was selected:
a jacket of
fine silver over a pure palladium core.
Although the thermoelectric power
of this combination was given by tables as only 10 microvolts per degree,
the metals would be expected to be quite benign in tissue, which was im-
portant in order to avoid large circulating currents due to electrochemical
action.
Calculations indicated that a jacket to core diameter ratio of four
to one would yield insignificant shunting of the palladium within the silver:
hence an order for material with a jacket diameter of 0.002 inch, and a core
diameter of 0.0005 inch was placed.
On advice of the supplier, an order for
123
silver and palladium extension lead wires and foils for connecting clamps
was placed at the same time, with all these materials to be taken from the
same melts of the respective metals.
This precaution was deemed, necessary
to avoid unwanted thermal emf's generated at the junction of two supposedly
identical metals from different melts.
At the same time, extensive develop-
ment of an amplifying-recording system matching the indicated thermoelectric
power was begun.
Unfortunately, the supplier was unable to furnish the Wollaston wire
in the desired size; the smallest ratio of diameters which could be achieved
was ten to one.
The wire obtained, then, had a 0.003 inch diameter silver
jacket over a 0.0003 inch diameter palladium core.
Although the jacket was

somewhat larger, and the core much smaller than desired, about two dozen
thermocouples were constructed of this material.
This construction was fully as simple as had been hoped. First, an
eight inch length of 0.005 inch diameter spring steel wire was sharpened on
one end, and butt-welded (using a capacitor discharge) on the other to a
four inch length of the Ag-Pd, Wollaston wire.
This steel wire served as a
leader to introduce the thermocouple into the skin.
The terminal two inches
of the Wollaston wire were dipped into nitric acid, which quickly removed
the silver jacket. The completed assembly was slipped into a fine metal tube
for protection until use.
The palladium proved so fragile, however, that not
one of the elements constructed was ever successfully inserted into an animal
and connected to the electric circuit.
6.6.2 Butt-welded Thermocouples
Undoubtedly, in other hands or for other purposes, the Wollaston wire
thermocouples described above could be most satisfactory. However, after it
appeared that breakage would never be much below 100%, attention was turned
124
other means of fabrication.
In view of the extensive investment of time and
money in a system compatible with silver-palladium junctions, the decision
was made to continue using these metals.
(Immersion of thermocouples in
Ringer's solution had shown that electrochemical effects were completely
negligible.)
The first alternative procedure was the butt-welding, by capacitor
discharge, of two inch lengths of silver and palladium wires, each with a
diameter of 0.001 inch.
The Job-like patience in carrying out this method
of Mrs. Marjory Pecora Pawley, formerly of this laboratory, is gratefully
acknowledged. After three months, about ten feet of each of these wires
were consumed in the net production of six assemblies, complete with steel
leaders, as described above.
This procedure was then abandoned.
6.6.3 Plated Thermocouples
The ease of thermocouple construction from Wollaston wire suggested
the opposite procedure:
rather than dissolve the silver jacket from the
composite wire, one could plate silver on a fine palladium wire.
This

project was turned over to Peter Hudson, formerly of this laboratory; his
labor is gratefully acknowledged.
The procedure developed was as follows:
a four inch length of 0.001 inch diameter palladium wire was suspended ver-
tically under slight tension in a plastic holder.
The lower two inches of
this wire was dipped into a preliminary strike bath for twenty seconds;
this same section of wire was then immersed in a standard silver cyanide
bath for plating (14). The plating current was adjusted initially to 0.20
milliampere, corresponding to a current density of about 5 milliamperes per
cm²; this current was gradually increased to a maximum value of 1.0 milli-
ampere as the plating increased in thickness.
In a total plating time of
two hours, a silver coating of about 0.004 inch could be obtained.
Eight
125
plating cells were connected in series, so that eight thermocouples were
formed at the same time.
From electrical conductivity measurements, the silver coating was
judged to be uniform and dense, although under low power magnification, the
surface appeared rather grainy. The measured temperature-emf relation of
these couples was almost identical to those of the Wollaston wire and butt-
welded elements described above.
In general, these were quite satisfactory
thermocouples except for the large diameter of the silver jacket necessary
to eliminate shunting effects of the palladium core,
Further work on this
method was halted with the successful development of the procedure which
will be described next.

6.6.4 Soldered Thermocouples
The writer is indebted to Dr. J. B. Hursh, of this department, for
recommending that soldered thermo junctions be investigated. The procedure
which was developed yielded completely satisfactory elements of surprising
strength.
Construction was carried out under a steromicroscope, with the wires
held in a special manipulator constructed by J. A. Basso, of this laboratory.
Again, silver and palladium wires were used, each of two inch length, but of
0.002 inch diameter. First, one end of each wire was flattened slightly
over about 1 mm, and these two flattened portions carefully brought side by
side, with a film of a mild rosin flux on the mating surfaces.
A stick of
ordinary soft solder was wiped over the clean tip of a fine soldering pencil,
and the trace of solder adherent to the tip transferred to the junction; the
molten solder was immediately pulled up between the flattened wires by
capillary action.
The finished junction was but slightly larger than the
wires themselves, and in most cases, it could not be located by the unaided
126
eye.
Assembly was completed by soldering an eight inch length of spring
steel leader (both 0.005 and 0.010 inch diameters were used) to the free
end of the silver wire; the entire unit was inserted in a length of fine
plastic tubing for protection until use.
The writer was able to fabricate
about five such assemblies per hour.
The majority of the thermocouples used in this study were constructed
by Miss Marilyn Aldrich, of this laboratory,
Her patience and extraordinary
skill are deeply appreciated.
6.6.5 Calibration of Thermocouples
The temperature-emf relations of several of the soldered thermo-
couples were determined, and were found to be so nearly identical that it
was felt that individual calibration of each couple would be quite unnecessary.
In all, a total of five complete calibration runs were made, covering the tem-
co

perature range of 35°C to 135° C.
Each thermocouple was placed in a well-stirred mineral oil bath, pro-
vided with an external tape heater.
The temperature was raised slowly to
its highest value, and then returned slowly to the starting point. Tempera-
tures were read on an A. H. Thomas specification, etched stem, mercury-in-
glass thermometer, graduated from -5 to 250°C by 0.5°C.
This instrument was
pointed for total immersion, and emergent stem corrections were applied to
all readings.
Thermo-emf was measured with the amplifier-recorder system to
be described below.
This system, rather than a potentiometer, was used so
that any loading by the amplifier on the couple would be automatically
accounted for.
The resultant emf-temperature curves were essentially linear from
35° C to 100°C.
From the least squares fit over this range, the calibration
factor for these thermocouples (the inverse of the thermoelectric power)
127
was found to be 0.0875°C per microvolt, with a standard deviation of about
0.002°C per microvolt.
6.6.6 Placement of Thermocouples
The placement of thermocouples for subsurface temperature measurements
was quite simple.
The sharpened end of the steel wire leader was inserted
into the skin a few millimeters from the edge of the exposure site (which
was marked as noted in Section 6.5). When the tip of the leader was at
what was judged to be the desired depth, it was then guided laterally through
the skin, and directly under the center of this marked site.
Radial lines
inked on the skin, and visible in Figure 6-3, were helpful in maintaining
proper alignment. From the surface, the progress of the tip of the leader
could easily be felt, and when it had passed completely under the exposure

area,
it was urged back out to the surface by folding the skin over the
sharp point.
The leader was then pulled completely through, thus bringing
the attached thermocouple into position; the leader was then clipped off
and discarded.
The 0.010 inch diameter leaders were used for deep place-
ment, and the lighter 0.005 inch for shallow.
It was now necessary to adjust the thermocouple so that the junction
itself was precisely on the center line of the exposure site.
During con-
struction of each thermocouple, a small drop of red paint was placed on the
palladium wire a known distance (12.5 mm) from the junction. A simple paste-
board scale was made with two marks separated by this distance.
With the
couple now inserted in the animal, the scale was placed with one mark on the
center of the exposure site, and the thermocouple pulled back and forth until
the red "tag" was aligned with the other mark.
Since the steel leader had
been carefully centered, one was now assured that the. thermo junction was
exactly on the center line of the exposure area.
The thermocouple was fixed
128
in this position with tiny tabs of cellophane tape, while subsequent couples
were installed.
The measurement of the depth of the junctions will be con-
sidered in the next section.
The emplacement of surface thermocouples was, surprisingly, more dif-
ficult than the subsurface elements, the major difficulty being the maintain-
ing of good contact between skin and wire.
After considerable effort, a
satisfactory method was developed.
In order to secure and maintain good
contact it was found to be necessary to align the thermocouple with the fold
lines of the skin.
With the "grain" direction of the skin determined, the
tip of the leader was pushed into and immediately back out of the skin a few

millimeters from the edge of the exposure site.
This process was repeated on
the opposite side of the site, and, as before, the thermocouple pulled into
approximate position and the leader clipped off. Now, using a magnifying
glass, the junction was centered accurately, and with the skin slightly
compressed with one hand, each end of the thermocouple was tied in a simple

knot around the tab of skin through which that end passed.
When the skin
was released, the thermocouple was placed under gentle tension due to the
elasticity of the tissue.
With care, thermocouple breakage was held to about
one in five in this procedure. (In contrast, only rarely did breakage occur

in the placement of subsurface elements.)
In the early phase of this study, four thermocouples--one surface and
three subsurface--were installed in each exposure site.
This was later re-
duced to three--one surface and two subsurface.
This will be discussed in
more detail subsequently.
6.6.7 Electrical Connections to the Thermocouples
Screw clamps with silver and palladium foil clamping surfaces were
used to secure simple, rapid, and reliable electrical connections to the
129
thermocouples. Four pairs of clamps were mounted on (but electrically in-
sulated from) a metal ring, each pair consisting of one silver and one
palladium clamp on opposite sides of the ring.
This ring was mounted on
one end of a supporting tube, which was composed of a steel pipe (for
magnetic shielding) securely pinned within a copper pipe (for electrical
shielding). Silver and palladium extension lead wires were soldered to the
clamping foils, and these wires then passed through the supporting tube to
a reference junction assembly.
The lead wires, of 0.010 inch diameter,
were insulated with fine polyethylene tubing.
6.7 Depth Measuring Device
For determination of the thermal diffusivity and the absorption
pattern of skin, it is necessary that one know the real depth, x, at which

a temperature measurement is made.
Of the various schemes considered for
measuring the depth of placement of a thermocouple, the one finally selected
was patterned quite closely after that developed by Schilling, working with
Ross and Moritz at Western Reserve (15). The procedure, as modified for this
study, utilized the ferro-magnetic properties of the spring steel leader used
to insert the thermocouples, with a tape recorder erase head employed as the
sensing element.
The electrical circuitry was quite straightforward.
The erase head
was shunted by a variable capacitor, and this parallel L-C circuit placed
in one leg of a unity ratio resistive Wheatstone bridge, with the adjacent
resistance leg also variable.
The bridge was powered by a General Radio
Company 1000 cps tuning fork oscillator through a shielded bridge trans-
former, and the bridge output measured with a DuMont type 403 cathode ray
oscilloscope and a Heathkit ac vacuum tube voltmeter.
The bridge output
was also loaded by a General Radio Company narrow pass filter tuned to
130
1000 cps, to suppress the second harmonic content in the output voltage.
Life With no magnetic material near the erase head pickup, the resistor
and capacitor could be adjusted to secure minimum bridge output; these ad-
justments were made while observing the CRO presentation. Now, when any
magnetic material was brought near the gap in the erase head, the inductance
of the latter was increased, the bridge was therefore unbalanced, and the
unbalanced voltage could be read on the vacuum tube voltmeter.
The magni-
tude of this output was dependent upon the separation between the erase head
and the magnetic material.
The instrument was calibrated by placing slips
of paper of various thicknesses between the pickup head and a sample of either
the 0.005 or 0.010 inch diameter wire used for the leaders.
The head was
carefully positioned over the wire to secure maximum output voltage which
was then plotted against the paper thickness as measured by micrometer
calipers.
To measure the depth of a subsurface thermocouple, the circuit was
first balanced by an assistant while the erase head was well isolated from
any magnetic material.
The assistant then noted the bridge output voltage
for zero separation, by holding the head directly on a sample of the appro-
priate steel leader; this one-point calibration check proved to be a highly
reliable indication of any drifts in the system. During this checking pro-

cedure, the experimenter was inserting a thermocouple in an anesthetized
animal as described above, proceeding only to the point where the steel
leader had been passed completely under the exposure site, with the tip
protruding back out through the skin surface.
If the calibration check
was satisfactory, the erase head was passed to the experimenter, who posi-
tioned the pickup precisely over the leader, using a simple plastic centering
jig. Using very light pressure to avoid compressing the skin or flexing the
leader, the head was "rocked" for maximum bridge output voltage.
This
131
procedure was repeated until exact agreement was achieved in two successive
trials, and this value of output voltage recorded on the master data sheet.
These voltage values were later translated into depths from the calibration
curve, taking care to add the radius of the leader so as to obtain the depth
to the center of the leader.
The leader was then pulled on through, bringing the attached thermo-
couple into position, as has been previously described.
(It was assumed,
;
and must certainly be true, that the thermocouple would be at the same depth
as the leader.)
The silver and palladium wires being non-magnetic, the depth
of another leader in the same site could now be measured without interference
from the first thermocouple.
It will be noted that the medium between the erase head and the steel

wire was paper for the calibration, and skin (or skin plus minute amounts of
silver and palladium) in actual use.
The magnetic susceptibilities of these
materials are so small that the error introduced by this procedure is com-
pletely negligible.
The over-all error in these depth measurements is difficult to assess.
The calibration data are accurate to about 0.05 mm, but it is doubtful if the
thermocouple depths were measured to that accuracy. An estimate of 0.1 mm
is probably not unreasonable. Actually, since the temperature gradients
through the skin are very steep, an error in depth measurement of this mag-
nitude can be quite serious. Up to the present time, however, no better
measuring system has been devised, so one must hope that by taking many
measurements, these errors will be smoothed out.
This is hardly a satis-
factory situation, and certainly more consideration must be given to this
problem in the future.
132
6.8 Reference Elements
As mentioned in subsection 6.6.7, the extension lead wires from the
thermocouple clamping ring were brought through the composite iron and copper
support tube to a reference (or cold) junction assembly.
This consisted of
an open bottom steel box, securely fastened to the distal end of the support
tube, with two, four-terminal tie-point strips mounted within the box.
The
silver and palladium leads from each thermocouple were brought to adjacent
tie points, and there fastened securely to the tinned copper leads of a two
conductor cable.
This assembly is visible in Figure 6-5, with the four

shielded cables coming from the side of the box.
When the animal holder was rotated into exposure position, the open
side of the steel box was down. A brass vessel filled with water was slipped
up inside this box so as to immerse completely the reference junctions. Water
from a constant temperature bath was circulated rapidly around the junctions.
In theory, the entire amplifying-recording system could now be considered an

Min-between" metal, so that only the potential difference of Ag-Pd junctions
at the reference and measured temperatures would be recorded.
In practice,
all parasitic thermal emf's could not be eliminated, although in several
determinations the resultant zero off-set never exceeded 2°C equivalent
temperature difference.
In fact, from the consistent values of initial
temperatures observed throughout the experiment, it was inferred that the
zero off-set probably did not exceed 0.5°C.
It will be recalled that in the theoretical development of this pro-
gram, only the temperature rise was considered, with the initial temperature
throughout the material assumed constant.
This assumption is not strictly
true for skin, since a slight initial temperature gradient does exist through
this tissue. However, this may be corrected for fairly accurately by measuring
only the temperature rise above the initial temperature at each depth. Hence,
133
one is not particularly concerned with accurate measurements of the initial
temperatures, and zero off-set due to parasitic emf's is not serious.
6.8.1 Temperature Regulated Water Bath
The bath used to supply constant temperature water for the reference
junctions consisted of a 10 inch by 10 inch cylindrical Pyrex jar containing
an E. H. Sargent Company circulating and heating tower.
This device pro-
vides completely turbulent flow throughout the entire volume of the bath,
and is one of the most satisfactory units commercially available.
The
temperature was controlled by a Princo "Magneset" thermoregulator, operating

a 200 watt heater through a sensitive electronic relay (16). The bath tem-
perature was held at 35.0 + 0.02°C.
Water was pulled from this bath by a
small centrifugal pump, and forced at rather high velocity across the
reference junctions.
This water returned to the bath by gravity flow.
6.9 Calibration Circuits
BE
A voltage calibrating circuit was placed in each thermocouple
channel, ahead of the initial preamplification.
Insertion of these cir-
cuits in this position contributed to the input noise of the system, but
this increase in noise was justified by the high accuracy obtained by intro-
ducing the calibrating voltages directly in series with the thermocouples.
The four calibrating circuits were essentially identical.
A 10 ohm
+1% wire wound resistor was placed in one preamplifier input lead, with an
adjustable current driven through this resistor from a Wheatstone bridge
type circuit.
This current was measured directly by a 100-0-100 microampere,
center zero panel meter.
Since the input resistance of the following pre-
amplifier was about 7000 ohms, the shunting effect on the 10 ohm resistor
amounted to only about 1/7 of 1%, which is negligible. Hence the calibration
voltage was given simply by Ohm's law as ten times the measured current.
Each
134
calibrating circuit was powered individually by a Mallory No. RM-4R mercury
cell.
The circuits were placed on the same chassis which housed the
preamplifiers.
In use, a calibration signal of suitable magnitude was recorded in
each channel just prior to an exposure, and a second signal of the same
magnitude inserted at the end of the recording.
The average of these two cali-

bration deflections was divided into the calibration voltage to give the
voltage sensitivity, in microvolts per millimeter, of that channel for that
particular exposure.
This value was then multiplied by the known calibra-
tion factor of the Ag-Pd thermocouples to give the temperature sensitivity
of the channel in centigrade degrees per millimeter of deflection.
It will
be recalled that in the theoretical analysis, the temperature response per
unit absorbed irradiance was the quantity of primary concern; hence, the
temperature sensitivity was divided by the value of absorbed irradiance per-
taining to that particular exposure to give a "scale factor" as:
degrees temperature rise per unit absorbed irradiance
millimeters of chart deflection
This scale factor multiplied by the deflections recorded during the exposure
gave the temperature response in the desired form.
The scale factor was
determined for each thermocouple channel for every exposure made in this study.
6.10 Preamplifiers
The preamplifiers used in each thermocouple channel were low-level,
dc instruments designed and constructed by the writer.
Four identical ampli-
fiers were placed on a common chassis, with operating voltages obtained from
common power supplies.
The complete assembly is shown in Figure 6-6.
The amplifiers were of the chopper-input chopper-output type, with
heavy dc feedback around the complete circuits.
The low-level input signal
was first chopped at 60 cps by a Stevens-Arnold A-ll chopper, and the square-
BRAR
13
***
mo
430
135

11-01
Sok
220
x 296
baros
Figure 6-6. Amplifier-Recorder System
136
wave ac signal passed to an input transformer.
The latter was a special unit
made by Southwestern Industrial Electronics, with a turns ratio of 1:265 and
a reflected primary impedance of 40 ohms center tap to grid.
This trans-
former drove two conventional stages of R-C coupled, balanced amplification,
utilizing 12AX7 twin triodes.
All tube heaters were powered by well-filtered
dc.
The second stage was followed by a balanced cathode follower which pro-
vided a low impedance drive for the rectifying output chopper, also a Stevens-
Arnold A-ll type. The pulsating dc output was smoothed by a low-pass R-C
filter.
Current feedback around the entire circuit was provided by 5000 ohm
and 10 ohm resistors connected in series to ground, with the low side of the
preamplifier input (the center tap of the input transformer primary) con-
nected to the junction between these elements.
Since the closed-loop gain of each amplifier was determined almost
entirely by the ratio of these feedback resistors, low noise, low tempera-

ture coefficient, metal film types were used. Although the four preampli-
fiers were supposedly identical, their performance characteristics differed
slightly from one to another. Typical specifications are:
voltage gain,
500; input resistance, 7000 ohms; equivalent input noise with 10 ohms input
resistance, 3 microvolts peak-to-peak, predominantly 120 cps ripple. This
last characteristic was the one most subject to variation from one channel
to another.
The peak-to-peak equivalent input noise of the best channel
was less than 1 microvolt, while that of the worst was 5 microvolts. Since
this noise was for the most part the 120 cps ripple remaining in the pul-
sating dc output after filtering, its magnitude increased with increasing
output voltage. Thus, for the noisiest channel mentioned above, the equi-
valent input noise increased to about 12 microvolts peak-to-peak at an output
of 200 millivolts.
Since this corresponds to a 400 microvolt input signal,
the signal-to-noise ratio here is 33, which is quite acceptable.
137
The frequency response of a typical unit was fairly clean from de
to 40 cps, and within + 3 db to about 85 cps, with, however, considerable
heterodyning around 60 cps, the chopping frequency. The distortion at fre-
quencies above 40 cps was severe; at 200 cps, where there was still a
measurable response, the output waveform was approximately triangular. Thus,
the bandwidth here specified cannot be employed blindly in calculating the
amplifier response to an arbitrary input signal. Actually, these response
data are not particularly meaningful, the important measurements being those
of the complete thermocouple-preamplifier-recorder system.
These will be
discussed in the next section.
6.ll Recording Assembly and System Response

A Sanborn Model 150 four-channel recorder was used, with Model
150-1300 dc coupling preamplifiers in all four channels, providing a basic
sensitivity in each channel of 50 millivolts per cm of stylus deflection.
The channels used for temperature measurements were, of course, driven by
the output of the chopper-input preamplifiers described above.
This
additional gain of 500 increased the system gain to 10 microvolts per mm,
as measured from the input terminals of the preamplifiers. (These latter
should probably be termed "pre-preamplifiers," since the Sanborn assembly
contains "preamplifiers.")
No attempt was made to adjust the gain of each channel to secure pre-
cisely this sensitivity; rather, as noted in Section 6.9, the calibration
circuits preceding the chopper-input preamplifiers were used to establish
an exact scale factor for each channel on every exposure. However, the
sensitivities were always very nearly 10 microvolts per mm, so one can
define an average temperature sensitivity from the calibration factor for the
Ag-Pd couples; this factor, as stated previously, was found to be 0.0875° C
138
per microvolt output.
The temperature sensitivity, then, was 0.875°C per
mm of stylus deflection, or, inversely, 1.144 mm per °C. Since only the
central 40 mm section of the Sanborn chart was used, full scale response for
full recorder gain was 35ºC rise, or 70°C maximum.
With the Sanborn pre-
amplifier gain attenuated by a factor of 2 (X-2 setting), the full scale
temperature rise was 70°C, or a maximum temperature of 105°C. As mentioned
in subsection 6.6.5, the linearity of the Ag-Pd thermocouples was satis-
factory up to 100°C, and it is only over the range 35°C to 100°C that the
stated calibration factor is valid. Thus, by never utilizing attenuations
greater than 2, one automatically avoided operating beyond the linear range
of the thermocouples. If, as occasionally happened in the early exposures,
a stylus was driven off scale, the data were simply lost, and the furnace
output was reduced for the next exposure.
This means, of course, that at
no time in this investigation were skin temperatures in excess of 100°C
ever studied.
In the previous chapters, considerable discussion was devoted to the
measurement of temperature response at, or immediately after, the initia-
tion of the exposure. This, in turn focusses attention on the time response

of the thermocouple-amplifier-recorder system to arbitrary temperature input
functions, and specifically to a step function.
Three procedures were used
to investigate this system response; two employed electrical input signals
to the chopper-input preamplifier, while the third utilized a "temperature
input" to the entire system.
These procedures will now be described briefly.
The first test consisted of applying voltages to the amplifier-recorder
system which increased exponentially with time.
The form of the predicted
response curves of the diathermanous solid is vaguely suggestive of an ex-
ponential form; hence it was felt that an exponential voltage input function
would give some indication of the ability of the system to follow the expected
139
GO
response form.
These test voltages were generated in a simple resistance-
capacitance circuit with initial time rates of change of stylus deflection
selected to be approximately equal to those expected on the basis of the
predicted temperature response.
The output voltage of this circuit was de-
veloped across a simple, resistive, 500 to l voltage divider; full output
voltage (with maximum value of about 200 millivolts) was applied directly
to one channel of the Sanborn recorder, while a signal with precisely the
same form, but attenuated by a factor of 500 was applied to an adjacent re-
corder channel through one of the chopper-input preamplifiers.
Since the
latter had a gain of 500, the two recorded signals should have been precisely
the same, and any differences could be ascribed to distortion introduced by
the preamplifier. Further, if either or both recordings did not follow an
exponential form, this would suggest that the system probably could not
faithfully follow the predicted response form.

The results of this test were most encouraging.
Not only did the
recordings follow an exponential form with considerable accuracy, but the
two channels showed almost superimposable responses, showing that the chopper
ON
input preamplifiers could indeed handle this type of signal with satisfactory
fidelity.
The second test consisted of applying a voltage step function, of
about 200 microvolt amplitude, to a preamplifier input, and recording the
resultant response on one channel of the Sanborn recorder.
While there was
some variability in results, the typical response was that of a slightly
underdamped system; the recorder stylus, after an initial rapid upscale
deflection, oscillated briefly before coming to its new rest position.
The
first upward excursion overshot the final deflection by about 10%, with the
peak reached about 7 milliseconds after applying the step function input.
Using these values, one may express the response to a unit step function
140
input as
330 t
Alt) = s[i-
(cos 450€ + 0.73 sin 450+)
]
(6-1)
where A(t) (with units of, say, millimeters of deflection per microvolt input)
is by definition the indicial transfer function, and S is the system sensi-
tivity (with the same units as A(t)). Now, by the super position integral
(17) the recorder deflection, say Y(t), due to any arbitrary input voltage,
E(t), may be expressed as
Y1H) = E (0) Alt) + S JEAN) Alt-1)d).
(6-2)
da
Now, in principle, if one has a particular form of y(t), one can de-
termine the system input voltage, E(t), which gave this recording, although
the inversion of equation (6-2) to obtain E(t) in terms of y(t) is by no
means simple, and analogue computation is definitely indicated.
There is,
however, an even more fundamental objection to the use of this equation.
One
is, after all, interested in correcting the recordings to allow for any de-
ficiencies in the system response to an arbitrary temperature change;
equation (6-2) allows one to correct the thermocouple output voltage, Eſt),
but it is possible, and even likely, that the principal lag in the system
would be in the thermocouple itself.

The third method of testing the system response was designed to pro-
vide an over-all indicial transfer function, in terms of recorder stylus
deflection per unit temperature change in the immediate invironment of a
thermocouple. The procedure was quite simple.
A soldered Ag-Pd thermocouple,
as described in subsection 6.6.4, was mounted in the thermocouple clamping
ring (6.6.7), and this latter assembly then plunged rapidly into an oil bath
heated about 10°C above ambient.
This provided a temperature step function
input to the system.
141
Unfortunately, the results of this test were so variable that no
general indicial transfer function could be obtained.
In about half the
tests, the response was quite smooth, corresponding to a simple "resistance-
capacitance" type system.
In the remainder of the tests, some oscillation in
stylus position occurred, although the stylus did not always overshot its
final position. Possibly these oscillations --which were much slower than
those observed with the voltage step function input--were due to temperature
gradients in the oil bath.
One might also suspect movement artifacts,"
since the thermocouple was subjected to considerable stress in entering the
bath.
The one factor which was quite reproducible from one test to another

was the time required for the stylus to reach 67% (1 - e-l) of its final
value.
This rise time was about 10 milliseconds, which is satisfactorily
short.
If the over-all system were indeed analogous to a resistance-
capacitance network, this rise time would suggest a bandwidth of about 16 cps;
it would be most interesting if a "temperature sine-wave generator" could be
devised which could check this prediction.
To date, the highest "generator"
frequency the writer has been able to achieve is about 1 cps, using an in-
tentionally unstable feed-back control sys tem.
Further work in this area
would be desirable.
In view of this failure to derive a rigorous scheme for correcting
the initial recorded response to the "true" temperature response, it was
necessary to employ the classic method of extrapolation to zero time.
Since
any transients (or lag) introduced by the thermocouple-amplifier-recorder
system would be damped out within 30 or 40 milliseconds, at most, this extra-
polation actually could introduce very little error, providing the location
of initial time on each recording was known with high accuracy.
Two pieces
of equipment were used to achieve this, a pulse measuring assembly and a
142
timing marker generator; these will be described in the following sections.
6.12 Pulse Measuring Assembly
The pulse measuring assembly was designed to place, on one channel of
the four channel recorder, a recording of the exact irradiance pulse form
delivered to the experimental animal for each exposure.
The pickup element
was a small plane mirror (visible in Figure 6-4) located just in front of
the water-cooled aperture plate of the animal holder.
When this holder was
rotated into exposure position the mirror was below and slightly to one side
of the optic axis of the source, and inclined at 45° to this axis.
A lens
and type 929 vacuum photo tube were mounted on an optical bench placed on a
table beside the carbon arc furnace (Figure 6-6). During an exposure, some
of the radiation which was back-scattered from the animal's skin was picked
up by the plane mirror and directed to the lens which then focussed it onto
a tiny aperture in an aluminum foil shield around the phototube envelope.

This simple photo tube photometer thus measured, with essentially no delay,
the exact radiation pulse delivered to the animal.
The phototube was operated
well below saturation, to assure linear response.
To provide a low-impedance
drive for the recorder, the phototube output was coupled directly to a
12AU7 cathode follower, located immediately beneath the phototube socket.
(The other triode unit of the 12AU? was used to provide quiescent voltage
balancing, and to insert timing marks on the recording as will be described
below.)
The total voltage swing impressed on the recorder for most exposures
was about 2 volts (considerably less than this for India ink covered skin);
hence this particular recorder channel was operated at attenuations of 10 or
20, giving sensitivities of 0.5 or 1.0 volt per centimeter, respectively.
By placing a recording of the irradiance pulse directly on one channel
of the four-channel recorder, highly accurate determinations of zero time
143
and irradiance rise time were made possible, but at the expense of losing one
channel for temperature recording. Thus, temperature information was traded
for time information.
6.13 Timing Marker Generator
The pulse measuring assembly allowed one to determine the location of
zero time quite accurately only on the pulse recording channel.
It was
ascertained that there could be relative shifts between the four styli of
the recorder of as much as 0.3 mm along the printed time scale of the chart;
these shifts seemed to be due to slight errors in chart paper tracking, or

possibly to slight errors in printing the paper.
In the Sanborn instrument,
the styli write on the chart as the latter passes over a sharp edge of metal;
possibly this edge was not precisely normal to the direction of chart motion.
In any event, it was necessary to determine, for each recording, the precise
values of these relative displacements in order to locate the zero time
position on the three temperature recording channels.
This was accomplished by inserting marker signals on all four styli
simultaneously, and at regular intervals.
Thus, if for some exposure, it
was found that the scale distance from one marker to the initiation of the
pulse (on the pulse recording channel) was, say, 34.6 mm, then the distance
from this same marker to the zero time point on the other three channels was
likewise 34.6 mm
The essential point is that the marker signals were intro-
duced on all four styli simultaneously; hence, one did not need to depend
upon the accuracy of alignment of the printed time lines on the chart paper.
In addition, since the marker signals were applied at accurately known in-
tervals, one could determine the true chart speed for each recording.
The time base for the unit was the power line frequency; this is held
within a fraction of one percent of 60 cps, which is quite accurate enough
144
for this purpose.
This alternating current was used to drive a Leeds and
Northrup chopper, with the resultant 60 contacts-per-second signal fed to a
divide-by-30 unit.
The latter was a Model 160A Potter predetermined elec-
tronic counter, set for a count of 30; on reaching full count, a Western
Electric SPDT mercury relay, incorporated in the counter, was switched from
one pole to the other.
With this relay on one side, a 2 uifd capacitor was
charged from a lă volt dry cell, and when the relay was switched to the other
pole, the capacitor discharged rapidly through a 5000 ohm potentiometer.
This cycle was repeated once per second, thus giving one short voltage pulse
every second across the potentiometer.
The dc coupling preamplifiers of the Sanborn recorder feature balanced,
or differential, input, while the outputs of the chopper-type preamplifiers
which drove the three temperature measuring channels (A, B, and D) were single-
ended. Thus, an additional, low-side input was available on each of these
channels.
These three low-side inputs were placed in parallel and driven
by a voltage picked off the 5000 ohm potentiometer of the timing marker
circuit.
This voltage was adjusted to give styli deflections of 1 or 2

millimeters, depending upon recorder attenuator setting (X-2 or X-1, re-
spectively). The irradiance pulse recording channel (C) was operated at
considerably lower gain, as noted above; hence it was necessary to use a
larger marker voltage for this channel, which was obtained across the full
5000 ohm potentiometer.
This signal was applied to the grid of the cathode
follower providing quiescent voltage balancing to the low-side input of
channel C.
Timing marker pips were thus recorded simultaneously on each channel,
as desired.
Each pip had an abrupt downward-going leading edge followed by
a rapid return to initial position, and occupied about 10 milliseconds, or
1% of the total time between markers.
Thus, there was only about one chance
145
in a hundred that a pip would occur at the beginning of an exposure, and hence
obliterate the all-important initial response. Naturally, this occurred with
somewhat higher frequency than predicted, although it was usually possible
to smooth out the recordings by eye, and thus salvage the data.se
6.14 Exposure Procedure
For each exposure, the thermocouples were installed, with their depths
BE
measured, as described previously. The electrical connections were then
made, and circuit continuity checked; often it was necessary to eliminate
accidental shorts or grounds by insulating the offending sections of thermo-
couple wire with tiny tabs of cellophane tape.
The animal holder was then

rotated into exposure position, the reference bath raised into place, and the
location of the animal behind the water-cooled aperture and the contact of
the surface thermocouple given a final check.
While an asistant struck and adjusted the carbon arc furnace, the
recorder was started at a slow chart speed with all channels switched to off
position; this placed an initial baseline on each channel.
The recorder
inputs were then switched on, and calibration signals, from the circuits
described in Section 6.9, recorded.
These signals were usually either 200
or 400 microvolts, depending upon recorder attenuation (X-l or X-2,
respectively).
The recorder chart speed selector was then set to the highest chart
speed available, 100 mm per second, and the timing marker generator set into
operation.
The chart drive was turned on and one second later, the assistant
initiated the exposure cycle described in subsection 6.4.1,
The chart drive
was left on high speed for about 10 seconds, then shifted to a low speed,
1.0 mm per second, and the cooling phase followed out to about 120 seconds,
Since time markers were lost during this shift, the timing of this final
146
110 second section of chart was obtained with sufficient accuracy from a
stop watch which was started by hand at approximately zero time. During
this time, the animal was left undisturbed in exposure position.
By 120 seconds after the initiation of the pulse, the styli had all
returned to very nearly their initial positions. The recorder inputs were
again switched off to obtain a record of the final styli baseline positions;
then, with the inputs returned to the on position, final calibration signals
were recorded.
The animal was then rotated back out of exposure position, and the
thermocouples carefully removed.
The subsurface couples were completely
unharmed and could be re-used after soldering them to another length of steel
leader; some of these elements were used many times.
Because of the knots
tied in the surface thermocouples, the wires were badly fatigued and liable
to breakage upon re-use; they were therefore discarded.
This procedure was repeated through the day until the decrease in the
depth of anesthesia of the animal made further thermocouple emplacement

impossible. On one of the last animals used in this study, seven exposures
were made in one day, each involving one surface and two subsurface thermo-
couples, and of these 21 thermocouples, only one was broken.
Undoubtedly,
with further experience, this record can be improved.
6.15 Reduction of Data
All recorder charts were read under a low power microscope, with
tracing positions estimated to 0.1 mm.
The records were first read by
Miss Marilyn Aldrich, of this laboratory, and then re-read by the writer;
the two estimates never differed by more than 0.1 mm.
This precision is
quite remarkable when it is recalled that in some cases the noise (120 cps
ripple) was 1.2 mm, peak-to-peak.
147
The chart time scale positions of the one second timing marks were
first noted, and from these, the actual chart speed determined.
This factor
was used to reduce all time scale readings to true time.
The zero time
position on each channel was reckoned as outlined in Section 6.13. Next, the
initial and final calibration signals on each of the three temperature re-
cording channels were averaged, and these averages, together with the known
value of absorbed irradiance for the particular exposure, used to compute a
scale factor for each channel, as defined in Section 6.9. Finally, all
chart deflections were corrected for any shift in recorder baseline, assuming
a linear drift with time.
Chart readings were then converted to temperature

rise per unit absorbed irradiance, at true time after initiation of the pulse.
Since differences in tracing positions were always involved, the precision
of these differences was about 0.2 mm.
At a chart speed of 100 mm per second,
this corresponds to a precision in zero time location of about 2 milliseconds.
This latter figure also represents the accuracy of the time measurements,
since the time base (the power line frequency) was considerably more accurate
than this.
The over-all system temperature sensitivity at full recorder gain was
given previously as 0.875°C per mm of stylus deflection.
This would lead to
a precision in measuring the temperature rise of about 0.2°C.
This is not
a good estimate of the accuracy of the determinations, since thermocouple
calibration errors, differences between thermocouples, calibration circuit
errors, and other unknown errors could affect the accuracy.
It is suggested
that ten times this figure, or 2°C, is a safely pessimistic estimate of
the accuracy.
6.16 Summary
In the foregoing sections, the more important items of equipment used
148
in this study have been described, and their functions briefly sketched.
The general disposition of several of these items can be seen in the ac-
companying photographs. The experimental procedures which have been outlined
above evolved only after considerable experience.
In particular, in the early
exposures, all four channels were used for temperature recording, and no
attempt was made to obtain accurate placement on the recordings of the initia-
tion of the irradiance pulse. Also, after the pulse measuring assembly was
added, several exposures were made before the necessity of employing a
cathode follower coupling stage was appreciated.
In the experimental results presented in the following chapters,
these earlier measurements have been, for the most part, discarded.
As a
result, of the 15 animals employed in this study, the results from only 10
have been used.
It should be reemphasized that the experimental work is far
from complete, and much remains to be done.
FOOTNOTES
(1) Kingsley, H. D., L. Hogg, Jr., J. T. Payne, and J. H. Morton, Studies
in Prolonged Anesthesia in Swine and Dogs, University of Rochester Atomic
Energy Project Report UR-152, 65-75 (1951).

(2) Berkley, K. M., H. E. Pearse, and T. P. Davis, Studies of Flash Burns:
The Influence of Skin Temperature in the Production of Cutaneous Burns
in Swine, University of Rochester Atomic Energy Project Report
UR-338, 9 (1954).
(3) Ibid., 10.
(4) Hinshaw, J. R. and H. E. Pearse, Histologic Techniques for the Differen-
tial Staining of Burned and Normal Tissues, Surg. Gyn. and Obs. 103,
726-730 (1956).
(5) Krolak, L. J., The Measurement of Diffuse Reflectance of Pig Skin, Ti-
tanium Dioxide Paint, and India Ink; the Transmittance of Titanium
Dioxide and India Ink, University of Rochester Atomic Energy Project
Report UR-439, 12 (1956).
(6) Ibid., 12.
(7) Davis, T. P. and H. E. Pearse, The Use of the Carbon Arc and Burning Mag-
nesium as Thermal Sources for Experimental Burns, Ann. Surg. 144, 68 (1959).
FEE
149
(8) Finkelnburg, W., The High Current Carbon Arc, FIAT Final Report
No. 1052 (1946).
(9) Swope, G. A. and F. C. Henriques, Thermal Pulse Properties of Five
High Intensity Laboratory Sources, Technical Operations, Inc.,
Report No. TOI 54-8, 6 (1954).
(10) Mixter, G., Jr. and T. P. Davis, A Method of Shaping Thermal Energy
Pulses from a Carbon Arc Source, University of Rochester Atomic Energy
Project Report UR-387 (1955).
(11) Mixter, G., Jr. and L. J. Krolak, Critical Energy of Fabric as Indicated
by its Persistence Time Under Thermal Radiation, University of Rochester
Atomic Energy Project Report UR-260, 34-36 (1953).
(12) Strong, J., Procedures in Experimental Physics, 542 (Prentice-Hall,
Inc., New York, 1938).
(13) Baker & Co., Inc., Newark, N.J. In these discussions, George Mixter,
Jr., M.D., formerly of this laboratory, made many valuable con-
tributions. His interest and assistance is gratefully acknowledged.
(14) Personal communication from Mr. Edwin Wallin, of Stromberg-Carlson Co.

(15) Ross, 0. A., A. E. Axelrod, and A. R. Moritz, Progress Report of Project
No. G-3645 to Morphology and Genetics Study Section, Division of Re-
search Grants, Department of Health, Education, and Welfare, for
July 16, 1952 to August 31, 1954, p. 8. Also personal visit by the
author to this laboratory.
(16) Davis, T. P. An Electronic Relay, University of Rochester Atomic Energy
Project Report UR-348, 9-14 (1954).
(17) Arguimbau, L. A., Vacuum-Tube Circuits, 153-161 (John Wiley & Sons,
Inc., New York, 1948).
150
CHAPTER VII
EXPERIMENTAL RESULTS; OPAQUE SKIN
7.1 Scope of the Experimental Work
A total of five animals were used for this phase of the experimental
work, with 36 exposures made and 63 thermocouples emplaced.
of these totals,
the results of 42 thermocouples from 24 exposures on four animals were used.
Of these 42 thermocouples, 24 were surface (for evaluation of the thermal
inertia, u ), and 18 were subsurface (for evaluation of the thermal dif-
fusivity, oc ). Since the procedures varied from one animal to another, the
exposures and results for each pig will be described briefly in the fol-
lowing sections.
7.2 Animal No. 1389
Six exposures were made on this 12.4 kg pig on February 11, 1958.
This was the first experiment on opaque skin; flat black Krylon spray
enamel was used, rather than India ink.
This produced film thicknesses
ranging from 50 u to over 100 M
.
One might suspect that such thick films
would be quite unsatisfactory, and the experimental results showed that this
was the case.
Further, these exposures employed the vane-type shutter, rather

than the pulse wheel, and were made before installation of the pulse measuring
assembly or the timing marker generator. For these reasons, the data are not
included in the final averages.
It is interesting to consider the effect of the thick film of black
enamel.
For the first two exposures, the exposure site was painted before
installation of the thermocouples, so the surface couple was "over" the
enamel.
On the third exposure, the thermocouples were installed first, and
then the paint was applied; here the surface element was "under" the Krylon.
For each of the remaining three exposures, two surface couples were used, one
151
under and one over the paint.
On the average, the temperature rise for the
five elements over the enamel was higher than that for the four couples under
from
the film. Approximate values of thermal inertia were found to be:
the thermocouples over the paint, 11 x 10-4 cal2 cm-4 deg-2 sec-1, and from
those under the paint, 19 x 10-4 cal2 cm
deg sec-l. The average for all
seven elements was 114 x 10-4, with a truly enormous spread from a minimum
of 6.4 x 10-4 to a maximum of 27 x 10-4, all in the same units as above.
-2
cm-4
Part of this spread in values of u was due to the difficulty in locating
true zero time on the records, and it was this set of exposures which led to
the design and installation of the pulse measuring photometer and the timing
marker circuit.
Only three good subsurface temperature records were obtained from

these exposures, but they are of dubious value because of the large and
unknown thickness of enamel added after the depth readings were made.
NO
attempt was made, therefore, to obtain values of thermal diffusivity, a
from these measurements. All in all, the principal contribution of these
exposures was to illustrate the deficiencies in the experimental techniques
employed.
7.3 Animal No. 1418
Eight exposures were made on this 10.2 kg animal on March 20, 1958.
The pulse measuring assembly and the timing marker generator were both uti-
lized throughout. This series of exposures was designed to determine whether
the thermal constants of the skin were independent of depth, or whether the
skin should be treated as a composite solid.
It will be recalled that in
Section 3.5 of Chapter III, it was demonstrated that the linearity of the
surface temperature rise vs square root of time is the crucial test of the
constancy, with depth, of the thermal inertia, u
Hence, on this animal,
152
only surface temperatures were measured, and no subsurface elements were
emplaced.
The skin was rendered opaque by application of India ink plus
detergent. The thermocouples were tied into position and oriented with the
fold lines of the skin, as described in the previous chapter.
1996 The superficial layer of skin is often described as "dessicated,"
and while this term is probably incorrect, it is certainly true that the water
content is well below that of the dermis, and even the deeper strata of the
epidermis. One might suspect, then, that the heat capacity per unit volume,
of the superficial layer would be less than that of the deeper tissue.
It would seem reasonable that the thermal conductivity, k, would likewise
be smaller for the surface material.
Thus, if the thermal inertia u
( и
une
k v ) were, indeed, depth dependent, it would be logical that the
value for the superficial layer would be smaller than that for the deeper
material, or, in the terms of Section 3.5, M, CM2• Assuming that the skin
can be considered as the simple composite solid treated in the above-

mentioned section, it follows that the slope of the curve of surface tem-
perature rise vs square root of time would initially be proportional to
(Mi )-2, while after some time had elapsed, this slope would become pro-
portional to ( M2 )-ı; this initial slope would then be larger than that at
a later time.
Since the surface layer is quite thin, the transition between
these two slopes would be expected to occur very soon after the initiation
of the pulse; hence it was necessary in the exposures here described to
investigate carefully the temperature response in the first few hundredths
of a second.
In the first five exposures made on this animal, the pulse wheel and
vane shutter combination described in subsection 6.4.1 was used; the pulse
duration was therefore about i second, and the irradiance rise time (from zero
to maximum value) about 5 milliseconds.
In order to obtain large recorder
153
deflections and hence improve the precision of the measurements, a rather
high value of incident irradiance was used, 5.3 cal cm-2 sec-l; this leads
to a value of 4.8 cal cm -1 for the absorbed irradiance. . Because of
ca
sec
this, the surface temperatures reached 100°C in about second, and hence
the recorder stylus was driven off scale; in the first & second, however,
accurate determinations of temperature response could be made.
The last three exposures on this animal were made at a greatly reduced
furnace output, with the absorbed irradiance being only about 460 millical
-2 sec-l. The vane-type shutter only was used, giving irradiance rise
cm
times of about 15 milliseconds, and the exposures were of some 7 seconds
duration.
Electrical noise from the vane shutter obscured the early phases
of the recorded responses, and these exposures were primarily useful to
determine if there was a transition between cili
)
2 response

do
)
and Cha
dependence at times much later than expected.
The data from all eight exposures for times out to about à second
(v time = 0.5) are plotted in Figure 7-1, and the data for the three long-
time exposures in Figure 7-2.
In all cases, the surface temperature rise
per unit absorbed irradiance, u*(0,t), has been corrected for the irradiance
rise time, as explained in Section 3.6.
Of all the response curves, only
that for exposure No. 6 displays the change in curvature expected if the
thermal inertia were a function of depth, while that for exposure No. 8
has precisely the opposite curvature.
For the five exposures with accurate
temperature measurements at short times, no suggestion of this predicted
curvature is seen.
From this, one may state that within the precision of
these experiments, the thermal inertia of the skin is independent of depth.
It is highly unlikely that compensating errors in the thermal conductivity
and the heat capacity per unit volume could produce this constancy of thermal
inertia; hence one may claim with some confidence that these thermal constants
154

15
14
A
+xo
13
(7'0),
(
12
11
Х
f
10
Х
0
Sec
Xx
z
wo
8
G
8
Surface Temperature Rise per Unit Absorbed Irradiance,
7
deg cal-1
X
6
5
to
4
DO
3
0 - Exp No.1
Exp. No. 2
A
Exp. No 3
V - Exp. 16.4
+ - Exp No.5
X- Exp, Na 6
o. Exp. No. 7
Expo No. 8
2
1
A
OD
0
0.1
0.2
0.3
0.4
0.5
Square Root of Time, VE, secă
Figure 7-1. Measured Response, Animal No. 1418 (Opaque Skin)
155

100
90
80
70
60
D
Surface Temperature Rise per Unit Absorbed Irradiance, U*(0,t)
deg cal-l cm2 sec
50
***
017
30
xxxxxxxxxx
X- Exp. No. 6
0 - Exp. No.7
Q - Exp. No.8
20
10
2.8
0.44
0.8
1.2
1.6
2.0
2.4
Square Root of Time, Vt, secz
Figure 7-2. Measured Response, Animal No. 1418 (Opaque Skin)
156
ho ,
are likewise independent of depth, or at least that the data do not support
the suggestion that they are depth dependent.
ayola One disturbing feature of these results is that the best-fitting
straight line drawn in Figure 7-] does not pass through the origin, as
would be expected from theory. There would seem to be an error in the lo-
cation of true zero time of almost 3 milliseconds, which error is quite
consistent from one exposure to another.
This consistency strongly suggests
that this zero time error arose from a time delay in the thermocouple-
se amplifier-recorder system.
There exists some evidence that this delay is
in the chopper-input preamplifier, although this conclusion is not yet
definitely established. In any event, in future work, the timing marker
signals will be introduced at the inputs of these preamplifiers, so that
any delay introduced by them will be corrected for automatically.
From the expression derived in Chapter III for the surface tempera-
ture response of the opaque solid to a step function input,

U*(0,1) - PA VF
it follows that the slope of the u*(0,t) vs r t plot of Figure 7-1 is
equal to 2 (Tu )-2. The value obtained from the least squares fit is
M M 10.0 x 10-4 cal2 cm-4 deg-2 sec-1.
7.4 Animal No. 1428
Six exposures were made on this 11.6 kg pig on April 10, 1958. The
experimental details were precisely as described above for pig No. 1418,
with the exception that a commercial low-level de amplifier (Allegany
Instrument Co. Model 220) was substituted for the chopper-type preamplifier
in the thermocouple channel.
The commercial instrument has a band width
extending from de to 20 kc, which is broader by a factor of about 500 than
that of the shop-made unit, although the zero stability of the latter is
157
much better.
The principal aim of these exposures was to see if the use of a
broad band preamplifier would eliminate the zero point error noted in the
data from pig No. 1418. Again, only surface thermocouples were employed.
Calibration signals were obtained by bridging from the calibration circuit
preceding channel D of the chopper-type preamplifiers.
In Figure 7-3, the data from this animal are presented as the surface
temperature rise per unit absorbed irradiance, U*(0,t), vs the square root
of time after initiation of the irradiance pulse. As in Figures 7-1 and
7-2, the data have been corrected for the finite irradiance rise time to give
true step function response, as described in Chapter III.
The regression
line, obtained by the method of least squares, now crosses the square root of
time axis at a value of 0.029 sec., which leads to a zero time error of
secz, which leads to a zero time error of
0.0009 second, or a little less than 1 millisecond.
This is certainly
small, but the consistency from one exposure to another suggests that the

error is real.
Comparison of the zero point error for these two sets of exposures
(on animals 1418 and 1428) would point rather clearly to a small time lag in
y mouse
the thermocouples themselves, and a much larger time lag in the chopper--
input preamplifiers. However, in another set of exposures, to be described
below, a zero point error considerably smaller than 1 millisecond was ob-
tained, using one of these same chopper-type preamplifiers. In short, this
matter is still unresolved.
It is highly unlikely, however, that this zero
point error, whatever its origin, affects the accuracy of the determinations
of the thermal inertia, u , since, except for this slight time shift, the
temperature responses do follow the predicted square root of time functional
form most satisfactorily.
Before considering the next sets of exposures, one additional aspect
of the data from animal No. 1428 should be mentioned.
Twice during the
158

15
14
+
13
A
0
12
11
A
10
+
V
9
8
+
Surface Temperature Rise per Unit Absorbed Irradiance, U*(0,t)
deg cal-l cm2 sec
6
А
56
+
4
3.
O-Eep. No, 1
A-ExpNa 2
V- Exp. No. 3
- Exp, No.za
t- Exp. No.4
0-eup, NS
2
O
A
ter
1 1
E/
VA
Х
D/A
0
0
0.1
0.2
0.3
0.4
0.5
Square Root of Time, VE, secz
Figure 7-3. Measured Response, Animal No. 128 (Opaque Skin)
159
course of these exposures, mistakes in procedure were made which rendered
the exposure data valueless. Rather than remove the thermocouples and pre-
pare a new site immediately, the animal was left in position for about five
minutes, and a second exposure made on the same site.
The data from these
"double burns" are included in Figure 7-3 as exposures No. 2 and No. 3a;
the points are clearly in no way atypical, and in fact are quite close to
the best-fitting line.
This result is in agreement with the findings of
Lipkin and Hardy (1), who showed, however, that two prior exposures can
increase the value of u by a factor of four, apparently by causing local
vasodilatation. Further investigation of this phenomenon has high priority
in plans for future work in this laboratory.
From the slope of the best-fitting straight line in Figure 7-3, the
value of thermal inertia was found to be
M = 12.5 x 10-4 cal? cm-4 deg-2 sec-1
7.5 Animals No. 1445 and 1455

Identical procedures were used with these two animals, burned on
July 14, and September 8, 1958, respectively.
respectively. Eleven surface and subsurface
temperature measurements were made on pig No. 14445 in four exposures; and
17 such measurements in six exposures on No. 1455. The chart reading proce-
dure was altered slightly so as to obtain directly the time in the normalized
form suggested by the analysis of Chapter III.
This normalized time, T, is
defined as the ratio of real time, t, to the exposure time, M.
The latter
value was obtained directly from the recorded pulse form.
The surface
temperature responses for each exposure were read at specified values of
τ
and then averaged for each animal.
These averaged data are presented
in Figure 7-4 as "reduced" temperature response, u(0,t)/Ham
VS the square
$
SEMUA
160

341
32
30
28
26
244
70)
WAH
22
O
20
18
No. 1445
No. 1455
Reduced Surface Temperature Rise,
deg cal-] cm2 secz
16
14
12
101
8
8
6
2
o
0 0.1 0.2 0.3 0.4
0.5 0.6 0.7 0.8 0.9 1.0
Square Root of Dimensionless Time, 1t/m
Figure 7-4. Measured Response, Animals No. 1445 and 1455
(Opaque Skin)
161
root of normalized time, V. From opaque solid theory,
2 Ha .it
U (0,t)
/ VTM
hence
U (0,t) 2
7 T
Havn
VTM
and the slopes of the best-fitting straight lines shown in Figure 7-4 will
be equal to 2(71
)-.
)- The values of thermal inertia, M , thus determined
are:
for pig No. 14445
M 시
​= 10.8 x 10-4 cal2 cm-4 deg-2 sec-1,
and for pig No. 1455

M 시
​= 13.9 x 10-4 cal?
-4
cm
deg
Sec-1.
Note that the zero time error for the data from the former animal is about
1 millisecond (in real time), while from the latter it is slightly less
than 2 milliseconds.
Thus, these errors are comparable to that obtained
when using the broad-band preamplifier described previously, which argues
against the introduction of an appreciable time delay by the chopper-input
preamplifiers.
For each subsurface thermocouple the value of normalized time at
which the temperature response reached its maximum, ?
max?
was determined,
as well as the depth, x, to the center of the thermocouple as measured by
the depth reading device described in the previous chapter. These pairs of
values are presented in Figure 7-5 as x/r^ plotted against
V
Tmax (Tmax-1)
In
Emax
Tmax-1
2
From the development of Section 3.4,
ñ - 1401 Iman (Imax - 1)
In Imax
2
j
Imax-
-T
hence the slopes of the best-fitting straight lines in Figure 7-4 are equal
162
0.14
-

0.12
를
​0.10
cm sec-
0.08
0.06
Reduced Depth, x/rm's
No. 1455
No14450
0.04
0.02
1
0
0
0.4
0.8
1.2
1.6
2.0
2.4
Tmax(Tmax-1) in
2
Tmax.
Tmanel
Figure 7-5. Measured Response, Animals No. 1445 and 1455
(Opaque Skin)
163
for pig No. 1445
to vua. The values of thermal diffusivity obtained are:
8.65 x 10-4 cm2 sec-l
a
and for pig No. 1455:
a = 8.25 x 10-4 cm? sec-1
The scatter in these data is surprisingly small, and in general, they
seem to follow the predicted linear relationship most satisfactorily.
Here
again, the best-fitting straight lines do not pass through the origin, as
required by theory, but the errors are small, and it is hardly possible to
claim on the basis of these two curves that the error is consistent, rather
than random.

Considerable variation from one exposure to another was noted in the
measured values of temperature rise, so that the "curve matching" procedure
developed in Section 3.4 for the determination of was found to be quite
impractical. This scatter is seen clearly in Figure 7-6 which presents a
profile of normalized temperature response, T, as defined in Chapter III,
vs "reduced" depth, x/vn for a single value of normalized time, 1.0.
Som

The curve which has been drawn in was computed on the basis of the average
value of thermal diffusivity, of, as given below, and is not necessarily
a best-fitting curve through these particular plotted points.
7.6 Şummary of Results: Average Thermal Constants of Pig Skin
To obtain a single set of estimates of the thermal constants of pig
skin, the various values above were averaged, with each weighted according
to the number of exposures involved in its determination. Thus, for the
thermal inertia, u
M 시
​(8 x 10.0) + (6 x 12.5) + (4 x 10.8) + (6 x 13.9)
21
10-4,
while
(7 x 8.65) + (11 x 8.25)
18
20-4.
a
164

1.0F
0.84
0.6%
T=1.0
0.4
0.2
6
0.1-
Dimensionless Temperature Rise, T (72)
08L
1
06F
04
***8.4x10-4
cm² sec1
02
0-1445
0-1455
O
بانه
.011
01
.02
06
08 0.1
0.2
Reduced Depth, x/vm
cm sec
Figure 7-6. Normalized Measured Response, Animals No. 1445 and 1455
(Opaque Skin)
165
By definition
M = ku,
and
K/v;
hence the thermal conductivity, k, is
k: : 3
while the heat capacity per unit volume, V
>
is
v=Vula
Performing the indicated operations gives
M = 11.7 x 10-4 cal2 cm-4 deg-2 sec-1
a = 8.4 x 10-4 cm2 sec-1,
and
k = 9.9 x 10-4 cal cm-7 deg-1 sec-1,
V = 1.18 ca] cm-3 deg
g-1
.
7.7 Discussion of Results
One of the most important results of the measurements described above

is that apparently one may consider the thermal constants of skin as true
constants, independent of depth. It has been pointed out that the constancy
of u with depth, which follows from the close adherence to a simple square
root of time relationship of the surface temperature rise, is a strong in-
dication that all the thermal constants are likewise independent of depth.
The excellent fit to a linear relation of the data presented in Figure 7-5
gives independent confirmation of this claim, since this simple linear re-
lationship will hold only if a is independent of depth, and if both u and
are true constants, then k and must be, also.
This conclusion must be qualified to this extent:
it is true if and
only if the surface treatment (painting with India ink) in no way altered
166
the thermal constants of the skin.
Now, it was pointed out in Section 7.3
that because of the relative dessication of the superficial layer of the
epidermis, the value of the heat capacity per unit volume, v
, might well
be considerably lower for this region than for the deeper, more hydrated
tissues.
It follows that painting the surface with India ink, which is an
aqueous suspension, might so increase the water content of the superficial
layer that the heat capacity per unit volume of this region would be increased
to a value more nearly equal to that obtaining in the deeper layers.
The
conclusion that is independent of depth would thus be true only if the
skin surface were artificially wetted.
While this may seem to be a trivial matter, it actually is of con-
siderable importance in the interpretation of certain measurements on bare,
or diathermanous, skin.
This will be considered further in the next chapter,
where it will be shown that the value of ✓ for non-wetted superficial tissue
could be as low as 0.85 cal cm-3 deg-2, although this is a most approximate

estimate, and probably represents a lower limit.
All in all, however, it is
possible that the painting with India ink does alter slightly the thermal
constants of the surface layers of the skin, simply by increasing the water
content of these tissues.
Turning now to the numerical values of the thermal constants of pig
skin, it would be of some interest to compare the estimates given in the
previous section with those presented by others.
One of these constants
which has been studied extensively is the thermal inertia,
M M
; Hardy (who
coined the term "thermal inertia for surface heating") and his associates
have presented several papers discussing the measurement of this constant
(2, 3, 4). This authority has developed quite elegant procedures for de-
termining surface temperature by means of an infrared photometer.
The latest
and undoubtedly best measurements reported (4) were made on human skin, using
167
non-penetrating infrared radiation to heat the skin; this eliminated the
necessity of using India ink to render the skin opaque. The average value
of u obtained in this study was 10.8 x 10-4 cal? cm -4 deg-2 sec-1, with a
standard deviation of + 0.8 x 10-4. It is tempting to claim that this close
agreement confirms the accuracy of the measurements of the present study,
but the fact that different species are involved in this comparison raises
the possibility of compensating errors yielding accidental agreement.
The
writer, however, is inclined to believe that the demonstrated similarities of
pig and human skin may well extend to similarity in thermal constants; hence
this agreement is not fortuitous, but does, in fact, provide confirmation of
the accuracy of the surface temperature measurements here reported.
In order to evaluate any one of the other thermal constants of skin,
it is necessary to make subsurface temperature measurements, as in this study,
or to employ completely different procedures than measurement of the tempera-
ture response of the skin to radiant energy exposures.
So far as the writer
can determine, this study is the only one in which a direct experimental de-
termination of thermal diffusivity, a , was attempted; more commonly, the
thermal conductivity, k, or heat capacity per unit volume, v , has been the

constant determined, frequently in vitro. Buettner has presented a rather
extensive discussion of various measurements of k (5) and has suggested, as
a compromise of the available data, the expression
k = 7 (1 + 3x) x 10-4 cal sec
-] deg
-] cm-1
(for x in cm) during the heating phase (with penetrating radiation) and
-1 deg-1 cm-]
10-3 cal sec
sec-1
during the cooling phase. To the writer, this procedure is not entirely
clear. In any event, these values are in excellent agreement with the value,
9.9 x 10-4 cal sec-] deg-1 cm-7, reported here.
The value of heat capacity per unit volume, v , given in the last
168
section, 1.18 cal cm-3 deg-2, is quite interesting, being one of the highest
estimates the writer has seen.
This constant can be evaluated quite directly
in vitro by employing the classic method of mixtures to measure the specific
heat capacity, cp, and determining the density, P , by mass and volume
measurements; the value of is then, by definition, the product of c.
Cp
and p.
Henriques has reported such measurements, with the value obtained
being 0.69 cal cm-3 deg-1, for the epidermis only; the density of the epi-
dermis was assumed to be 0.8 gm cm-3 (6). In contrast to this rather low
value, Buettner (7) and Lawson, Thomas, and Simms (8) have assumed a value
of unity for this constant.
Chen and Jensen (9) have selected a value of
0.8 for "standard" skin, as read from Figure 3 of the referenced report.
On this same figure, Chen and Jensen have quoted data from Hardy for "moist
Sec-2
deg-2 cm-1
skin in summer" as
k
10-3 cal sec
M = 12.5 x 10-4 cal? cm-4 deg-2 sec-1.
These would lead to a value of v of 1.25 cal cm-3 deg-1, which is the only

and
value the writer has found which exceeds that obtained in this study.
In general, it would seem that most authors have simply assumed a
value of V
and there appears to be a remarkable paucity of careful in vivo
9
determinations of this constant. Hence, the writer tends to accept the value
here determined, 1.18 cal cm-3 deg-1, as more nearly representative of living
pig skin than the more commonly cited values of 0.8 to 1.0 cal cm-3 deg-1.
In any event, the values of thermal inertia, u , and thermal conductivity,
k, as determined in this study seem to be in excellent agreement with careful
determinations reported in the literature, and if these two values be accepted
then the values of thermal diffusivity, a
and heat capacity per unit
volume, v
, are, of course, fixed by the defining equations for these
constants.
169
7.8 Summary
Bo Bar In view of the above considerations, it may be suggested that the
values of the thermal constants of pig skin presented in Section 7.6 are
probably accurate to about # 10%; the accuracy may be better than this for
M, and poorer for a
since the latter involves the difficult and not
overly precise measurement of depth. Actually, a precise statement of ac-
curacy is not necessary for present purposes, inasmuch as these thermal
constants are only needed for the analysis of the more important case of
diathermanous skin.
In this connection, it should be pointed out that, due
to a slight error in the initial analysis of the opaque skin data, the values
of k and a used in Section 5.4 for the numerical solution of the diathermanous
solid response are respectively 1% and 2% below the final values presented in
Section 7.6.
It will become clear in the presentation of the following chapter
that errors of this magnitude, and even errors of 10%, are quite insignificant
when compared to the large discrepancies which were found between predicted
and measured responses for diathermanous skin.
FOOTNOTES
(1) Lipkin, M. and J. D. Hardy, Measurement of Some Thermal Properties of
Human Tissues, J. Appl. Physiol. 7, 216 (1954).

(2) Hardy, J. D., Method for the Rapid Measurement of Skin Temperature During
Exposure to Intense Thermal Radiation, J. Appl. Physiol. 5, 559-566 (1953).
(3) Lipkin, M. and J. D. Hardy, op. cit., 212-217.
(4) Hendler, E., R. Crosbie, and J. D. Hardy, Measurements of Skin Heating
During Exposure to Infrared Radiation, J. Appl. Physiol. 12, 177-185 (1958).
)
(5) Buettner, K., Effects of Extreme Heat and Cold on Human Skin III. Numerical
Analysis and Pilot Experiments on Penetrating Flash Radiation Effects,
J. Appl. Physiol. 5, 210-211 (1952).
(6) Henriques, F. C., Jr. and A. R. Moritz, Studies of Thermal Injury I. The
Conduction of Heat to and through Skin and the Temperatures Attained
Therein. A Theoretical and an Experimental Investigation, Am. J. Path.
23, 538-539 and 546 (1947).
170
(7) Buettner, K., op. cit., 212.
(8) Lawson, D. I., P. H. Thomas, and D. L. Simms, The Thermal Properties of
Skin, Fire Research Station (Boreham Wood, Herts., England) F. R. Note
224 / 1955, 7 (1955).
(9) Chen, N. Y. and W. P. Jensen, Skin Simulants with Depth Magnification,
Massachusetts Institute of Technology Fuels Research Laboratory
Technical Report No. 5, A-ll Figure 3 (1957).

S
171
CHAPTER VIII
EXPERIMENTAL RESULTS, BARE SKIN
8.1 Scope of the Experimental Work
For the investigation of the response of bare, or dia thermanous
skin, ten animals were employed, with 39 exposures made and 103 thermo-
couples emplaced.
Of these totals, the results from six animals, 25 ex-
posures, and 68 thermocouples were usable; of the last, 21 were on the
surface and 47 were subsurface.
The data from the first four animals employed
were discarded since techniques were poorly developed, and the pulse measur-
ing photometer and timing generator had not yet been installed,
Table 8-1

summarizes the scope of this phase of the experiment.
It should be noted that for the exposures on animals 1402 and 1403
the pulse measuring assembly was not yet in the form described in Chapter VI,
the phototube being coupled directly to the Sanborn recorder, without the
intervening cathode follower.
As a result, the photome ter-recorder system
had a long time constant, and measurements of irradiance pulse rise time
were completely inaccurate. However, the point of initiation of the pulse
was still recorded accurately, so the data from these exposures may be

included.
Four of the five exposures on animal No. 1446 were made with the skin
surface wetted with water plus a few drops of detergent.
The reason for this
"sham painting" will be described below. Except for these four exposures,
the experimental techniques were exactly as described in Chapter VI for the
six animals 1402 through 1450.
8.2 Reduction of Data
The recorded responses were reduced as previously explained, with the
temperature rise per unit absorbed irradiance, U*(x,t), determined at various
172
Table 8-1
Scope of the Experimental Work on Bare Skin
Pig
Date
Irradiance
Expo- Thermo- Pulse
Sures couples Recorded Timing Incident Absorbed
cal
cal
cm-sec cm-sec
1370 1-20-58
4
7
No
No 3.9
2.4
1373 1-23-58
4
7
No
No
3.9
2.2
1375 1-28-58
4
14
No No 5.0 3.0
1392 2-13-58
27
No
No 5.0 3.0
14022-25-58
25
Yes(a)
Yes
5.0 3.0
1403 2-27-58
5
13
Yes(a)
Yes
5.0 3,0
1408 3-4-58
4
10
Yes
Yes
5.3 3.2

1444 6-5-58
5.
15 Yes
Yes
5.2 3.1
1446 7-16-58
5(b)
13 Yes
Yes
5.2 3.1
1450 9-18-58
4
12
Yes
Yes
5.2
3.1
KA
10
39
103
Total
6
25
68
Usable
(a) No cathode follower
(b) The surface was wetted on four exposures
173
values of real time.
Measured depths were corrected for leader wire radius
to obtain the true depth to the center line of each thermocouple.
The data
from each exposure were plotted simply as U*(x, t) vs time, with one curve
Ah
for each thermocouple employed for that particular exposure.
Inasmuch as
the primary concern was comparison with the theoretical step-function re-
sponse, the records were analyzed only through the heating phase--i.e., during
the exposure pulse. As noted previously, it was not practical to derive an

analytical expression for correcting the recorded response to true step-
function response; hence the initial portion of each curve was obtained by
smooth extrapolation to zero time.
Because of the uncertainty in the source
and average magnitude of the time lag noted in the opaque skin results, no
attempt was made to correct for this lag, and the zero time position on each
SE
Brecording was located as described in Section 6.13.
8.3 Summary of Temperature Response Measurements
From these replotted response curves, values of v*(x, t) were read at
times of 0.05, 0.10, 0.15, 0.20, 0.30, 0.40, and 0.50 second after initia-
tion of the irradiance pulse. These data, together with the corresponding
depths, are given in Table 8-2, which is thus a complete summary of all the
usable bare skin exposures made in this study.
These tabulated values are
also presented graphically in Figures 8-la, 8-1b, and 8-1c as profiles of
U*(x, t) vs depth, x, for the seven values of time mentioned above. Note
that for x = o, only the arithmetic means of the 21 surface temperature measure-
ments were plotted for each time.
Finally, values of U*(x, t) were read from these seven smoothed curves
at values of depth of 0, 0.02, 0.04, 0.08, and 0.16 cm, and again plotted,
reverting to the U*(x,t) vs time presentation. This final set of doubly-
smoothed curves, Figure 8-2, is then a graphical representation of the
174
E
Table 8-2
Bust Summary of Bare Skin Temperature Responses
NEGRU
U*(x,t)
Exp. Corrected Ju*(x,0)
No. Depth, cm
SS
No.
ot
0.05 0.10 0.15 0.20 0.30 0.40 0.50
sec sec sec
(5)
(7) (8) (9) (10) (11)
sec
Sec
sec
sec
(1) (2)
(3)

1 4-1408
23-1408 0
3 1-1450 0
4 2-1408 0
5 4-1444 0
6 5-1444 0
73-1402 0
8 4-1402 0
94-1446 0
10 2-1403 0
ll 3-1450 0
12 5-1403 0
13 3-1403 0
14 4-1403 0
15 1-1403 0
16 2-1450
17 4-1450 0
18 3-14446*
19 6-1446* 0
20 5-1446* 0
21 2-1446* 0
22 1-1444 0.015
23 1-1450 0.016
24 2-1444 0.019
25 2-1450 0.020
26 1-1403 0.021
27 4-1450 0.021
28 5-1444 0.021
29 3-1402 0.023
30 4-1444 0.024
31 3-1444 0.025
32 3-1403 0.026
33 2-11146* 0.027
34 2-1408 0.029
35 5-1446* 0.029
36 3-1408 0.030
37 4-1408 0.031
38 2-1403 0.031
39 3-1446* 0.032
40 1-1408 0.035
264
232
218
207
206
197
190
183
175
172
167
163
160
158
155
151
150
145
144
124
120
150
62.0
105
80.0
69.6
64.7
60.3
63.6
72.2
104
112
75.2
80.5
59.1
81.2
90.9
73.7
90.3
85.2
8.51 10.34 11.58 12.70 14.39 15.68 16.74
5.50 6.88 7.94 8.74 10.13 11.37 12.47
7.50 9.86 11.59 12.92 11.83 16.36 17.64
4.16 5.33 6.01 6.71 7.83 8.78
3.62 4.85 5.88 6.73 8.17 9.43 10.60
4.70 6.34 7.49 8.39 10.00 11.59 13.14
5.35 7.87 9.20 10.36 12.27 13.78 15.16
5.90 8.73 10.18 11.50 13.47 15.20 16.86
5.73 7.72 8.83 9.87 11.78 13.51 15.13
4.27 5.85 6.97 7.86 9.45 10.96 12.33
5.03 6.65 7.73 8.72 10.34 11.82 12.97
5.40 7.32 8.62 9.60 11.43 12.73 14.07
3.92 5.60 6.88 7.96 9.63 11.43 12.96
4.85 6.48 7.60 8.40 9.92 11.23 12.48
4.80 6.65 7.87 8.83 10.34 11.68 12.91
6.27 9.48 11.50 12.88 14.73 16.14 17.32
5.88 8.29 9.62 10.81 12.54 13.81 14.85
3.53 5.14 6.38 7.53 9.43 11.14 12.72
4.10 5.67 6.82 7.78 9.31 10.72 11.89
2.62 4.16 5.62 6.81 8.88 10.62 12.18
2.76 4.44 5.80 7.13 9.26 11.25 13.02
3.93 6.33 8.10 9.50 11.94 13.92
2.37 3.77 5.05 6.06 8.02 9.59 11.04
2.56 3.84 4.83 5.83 7.48 8.944
2.48 3.77 4.93 5.86 7.66 9.16 10.45
2.07 3.25 4.24 5.01 6.51 8.03 9.22
2.00 3.13 4.08 4.84 6.24 7.53 8.52
1.72 2.76 3.75 4.60 6.24 7.76 9.06
2.00 3.44 4.48 5.50 7.34 8.97 10.43
1.59 2.46 3.18 3.86 5.22 6.57
2.32 3.47 4.53 5.58 7.48 9.18
2.37 3.82 5.05 6.05 7.91. 9.45 11.00
2.32 3.62 4.73 5.57 7.22 8.41
9.83
2.22 3.19 3.84 4.42 5.52 6.61
1.82 3.00 4.01 5.02 6.83 8.39 9.76
1.43 2.15 2.85 3.48 4.76 5.91 7.02
1.83 2.59 3.19 3.77 4.93 6.08 7.18
2.22 3.37 4.24 5.03 6.36 7.73 8.94
2.43 3.65 4.63 5.53 7.02 8.32 9.40
1.73 2.42 3.02 3.58 4.72 5.83 6.95
175
Table 8-2
Continued
(1) (2)
(3)
(4)
(5)
(6)
(7)
(8)
(9) (10) (11)
41 4-1446 0.039
42 3-1450 0.040
43 3-14444 0.045
44 H-1444 0.056
45 4-1403 0.061
46 1-1444 0.070
47 2-1444 0.076
48 5-1444 0.081
49 1-1444 0.083
50 2-1446* 0.098
51 3-142444 0.098
52 2-1450 0.100
53 1-1450 0.101
54 1-1403 0.113
55 3-1450 0.118
56 3-1408 0.120
57 5-1446* 0.122
58 4-1446 0.122
59 3-1403 0.122
60 4-1450 0.124
61 2-1403 0.127
62 4-1408 0.132
63 3-1446* 0.137
64 2-1444 0.137
65 l-1408 0.143
66 4-1402 0.143
67 4-1403 0.155
68 3-1402 0.169
81.0
68.2
90.9
45.4
68.2
71.0
77.2
42.0
54.7
53.4
34.7
52.2
48.1
38.2
30.5
44.3
53.2
50.2
40.8
26.3
38.2
26.7
35.6
32.0
40.0
32.4
37.7
28.7
2.50 3.77 4.86 5.86 7.68 9.40 11.02
2.58 3.79 4.76 5.57 6.98 8.14 9.34
1.86 2.79 3.63 4.32 5.71 6.99 8.20
1.06 1.47 1.84 2.16 2.73 3.30 3.82
1.83 2.61 3.10 3.56 4.36 5.09 5.82
1.80 2.43 2.78 3.11 3.77 4.29
2.07 2.81 3.30 3.74 4.48 5.04
0.95 1.39 1.70 1.99 2.46 2.87 3.24
1.75 2.67 3.32 3.83 4.86 5.71
1.57 2.23 2.72
2.72 3.11 3.79 4.37 4.89
0.69 0.97 1.15 1.33 1.60 1.84 2.12
1.60 2.43 2.94 3.37 4.00 4.54 5.02
1.56 2.37 2.90 3.40 4.08 4.70 5.23
1.44 2.14 2.56 2.88 3.41 3.80 4.16
1.23 1.90 2.40 2.80 3.444 3.91 4.37
0.82 1.12 1.41 1.62 2.03 2.35 2.64
1.52 2.19 2.77 3.17 3.91 4.47 5.03
1.63 2.57 3.13 3.68 4.55 5.29 5.99
1.02 1.48 1.84 2.17 2.63 3.00 3.41
0.79 1.13 1.38 1.60 1.94 2.29 2.51
0.83 1.13 1.34 1.45 1.73 1.92 2.13
0.82 1.21 1.52 1.73 2.13 2.41 2.68
1.21 1.77 2.16 2.46 3.16 3.53 4.00
0.89 1.24 1.49 1.68 1.94 2.20
1.10 1.51 1.74 1.95 2.34 2.67 2.93
0.87 1.27 1.52 1.73 2.07 2.34 2.58
0.85 1.22 1.47 1.67 2.05 2.34 2.60
0.98 1.53 1.85 2.16 2.56 2.82 3.12

*Surface wetted with water
176

14
U
13
12
11
1
10
A
9
AA
DPD
A
8
Temperature Rise per Unit Absorbed Irradiance, U* (x, t)
deg cal-1 cm2 sec
7
t=0.5 sec
00
o od
4
ADA
t-0.2 sec
7
0
A
0
o
ह
O OA
A
A
2
0
00
D
a
8
NO
1
o t=0.05seco
DOOD
T
0
0
20"
70°
..06
.12
.14
.16
08
.10
Depth, x, cm
Figure 8-la. Measured Response, Bare Skin Exposures
177

T
13 12
0
11
OT
9
8
O
O
7
Temperature Rise per Unit Absorbed Irradiance, U* (x, t)
deg cal-] cm2 sec
9
O
O
5
DO
t-o.3 sec
O
7
CO
0 0
GO
O
ليا
JE
D
C
O
N
t = 0.1 sec
2
1
O
0
0
.02
770°
.06
.12
.14
.16
08
.10
Depth, x, cm
Figure 8-lb. Measured Response, Bare Skin Exposures
178

13
1
12
11
10
OOO
8
00
7
Temperature Rise per Unit Absorbed Irradiance, U*(x,t)
deg cal-l cm2 sec
9
Ic
t=0.4 sec
O
5
O
.
O
77
OD 2
2
000
Q
زيا
200
2
t=0.15 sec
1
0
20
.OL
.06
.08
12
.
.10
Depth, x, cm
오뚜
​Figure 8-1c. Measured Response, Bare Skin Exposures
179

14
13
Sec
cm
12
I-
11
10F
X=0cmo
9.
$81
x=
X=0.02cm
.
7
가
​Temperature Rise per Unit Absorbed Irradiance, U*(x,t), deg cal
6
X=0.04cm
54
A
X= 0.0 Bcm
✓
3
Ô
x=0.16 cm
2
1
A
0
0
0.1
0.4
0.5
0.2
0.3
Time, t, sec
Figure 8-2. Average Measured Response, Bare Skin Exposures
180
"average" bare skin response, as determined by this experiment.
8.4 Summary of Initial Slope Measurements
For each of the replotted response curves, described in Section 8.2,
the initial slope, Cd U*(x,0)/2 t), was estimated by eye, simply by ad-
justing a straightedge to be tangent to the initial portion of the curve
(1). These values are also tabulated in Table 8-2, and are shown graphi-
cally in Figure 8-3 on a logarithmic scale against a linear depth scale.
The
values for x = 0 (surface measurements) are plotted as points except for the
four determinations with wetted surface, which are shown by x's.
The in-
dicated average value for the surface initial slope, 185 deg cal-l cm?,
was calculated as the arithmetic mean of the 17 "normal" values, and does
not include these four wetted-skin measurements.
The line drawn in Figure 8-3 was determined by the method of least
squares as the best-fitting straight line through the values for x greater
than zero; i.e. for subsurface measurements.
This line, which may be ex-
pressed as
U*(x,0)
ət
96.8 e-7.22x deg cal-1 cm?

(where x is in centimeters), is thus valid only for non-zero values of x,
and is not intended to fit the surface (x = 0) values of initial slope.
8.5 Comparison of Experimental Results with Theoretical Predictions
The most notable feature of Table 8-2 is the remarkable variability
in the data.
This scatter is even more noticeable in Figures 8-la, b, and c,
and, in fact, the smooth profile curves shown in these figures are based more
on free-hand art work than one would consider desirable.
The zero slope of
these curves at x = 0 was, of course, dictated by the boundary condition 4.4
of the diathermanous solid model.
In order to decide experimentally whether
181

2 U*(x,0)
300
7 e
200
100
0
80
60
Initial Slope per Unit Absorbed Irradiance,
deg cal-l cm2
8
40
0
O
201
10
0
02
.04
.06
.12
elle
.16
.18
O
08 .10
Depth, x, cm
Figure 8-3. Measured Initial Time Rate of Change of Tem-
perature Response, Bare Skin Exposures
182
this slope actually is zero or non-zero, it would be necessary to introduce
temperature sensing elements, of probably less than 10 u diameter, at depths
of about 10 to 20 . This would be a difficult task and for present pur-
poses, at least, not justified.) The regularity of the points defining the
"average" response curves of Figure 8-2 masks, but obviously has not elim-
inated, the variability in the original data. Thus, these curves must be
regarded as indicating the central tendency of the experimental data, and
not as a precise presentation of highly reproducible results. Undoubtedly,
the curves are better than order of magnitude estimates, but it would probably
be optimistic to claim that they represent the "true" average response with
an accuracy better than a factor of two.
This same sort of variability in experimental results was encountered
in the opaque skin studies described in Chapter VII, but in that case rather
elegant schemes were available for comparing experiment with theory; hence,
it was possible to compute the unknown constants of the theoretical model
with a fairly high level of confidence.
As pointed out in Chapter V, no such

comparison schemes are possible for bare, or diathermanous, skin exposed to
heterochromatic radiation, and one is forced to fall back on the manifestly
unsatisfactory procedure of gross comparison of theoretical predictions and
experimental results.
It would now seem that the scatter in the experimental
data would make such comparisons meaningless.
Such, however, is not the case.
If one compares the experimental
response curves of Figure 8-2 with the theoretical predictions shown in
Figure 5-3, one can see clearly that the agreement is poor, but it is not
at all clear whether the level of disagreement is such that one would re-
ject the hypothesis that the responses are, in fact, equal. Certainly,
there are no order of magnitude differences between these two sets of curves.
A much more revealing comparison, however, is shown in Figure 8-4 where the
183
ܢܐܐ

10
12
8
10
Sec
6
8
6
4
4
2
2
1
Temperature Rise per Unit Absorbed Irradiance, U*(x,t), deg cal-l cm
0.8
1
0.6
0.4
Measured
- Predicted
(Measured)
(Predicted)
0.2
0.1
0
0.1
0.4
0.5
0.2
0.3
Time, t, sec
Figure 8-4. Comparison of Predicted and Measured Bare Skin Response
184
responses are plotted on logarithmic scales against a linear time scale;
the ordinates of the two sets of curves--experimental and theoretical--have
been shifted arbitrarily so as to bring into coincidence the experimental
and predicted surface responses at 0.5 second.
Now it can be seen that un-
questionably theory and experiment do not agree, and the disagreement cannot
be removed simply by applying a multiplicative "correction factor" (which
corresponds to a relative shift of ordinates in Figure 8-4) to either set
of curves.
Not only do the two sets of curves disagree in form, but the

spacing between isodepth curves is completely different, and no amount of
free-hand redrawing of the profiles of Figures 8-la, b, and c, can reduce
this difference significantly.
Having thus concluded that the theoretical predictions and experi-
mental results do not agree, one is faced with the task of locating the
source, or sources, of this disagreement. At first thought, one might point

to the variability in the experimental data, and suggest that this raises
serious doubts as to the validity of the entire set of experimental results.
It must be recalled, however, that the so-called theoretical predictions are
based on literature values of linear absorption coefficients, some of which
values may be little better than order of magnitude estimates.
One is thus
forced to choose between two sets of not overly reliable data.
Real progress in this problem can be made by turning to the experi-
mentally determined values of the initial slope of the temperature response.
While again, there is considerable variability in these data, there can be
no doubt that the general trend shown in Figure 8-3 is real, and that the
indicated line fits the data reasonably well.
Now if one assumes, for the
moment, that this same line may be extended from x = 0 to x, then it
follows that
185
Su 7
2 u*(x,0) dx
at
= (1.18) (96.8)
- 7.22x
e
dx
x
(1.18) (96.8) MEELELA
7.22
16.
DE
(8-1)
Yet, it has been pointed out that this integral, as a consequence of the
conservation of energy, must necessarily be equal to unity!
Here there can
be no question; the measured values of initial slope are too high, on the
average, by a factor of about 16.
Even if one assumed that the curve of
initial slope dropped to zero for depths greater than 0.17 cm (the deepest
measurement made), the area under the curve would still be about ll.
The
actual area must be greater than this latter figure, since it is completely
unrealistic to assume that the initial slope of the temperature response will

be zero for depths greater than 0.17 cm (which is equivalent to claiming that
the tissue is opaque at these depths), and further the true curve must lie
above the simple exponential shown in Figure 8-3 for small values of x, since
the measured value of the initial slope at x = 0 is about 185 deg sec
sec-1
for
unit absorbed irradiance, and not 96.8.
Thus there is unquestionably a real
systematic error in the measured temperature responses, which leads to these
high values of initial slope. A major part of the experimental work done
in this study was devoted to the investigation of possible sources of this
error.
Several of the more pertinent results of this work will be des-
cribed in the next section.
8.6
The Source of Error in the Temperature Measurements
Several possible sources of error in the temperature measurements,
or specifically in the measurements of initial slope, were studied, including
variation in the value of the heat capacity per unit volume, V , with depth,
186
errors in zero time location or extrapolation of the response curves to zero
time, and direct interaction of the thermocouples with the radiation penetrating
through the skin.
These possibilities will be discussed and evaluated in the
following subsections.
8.6.1 Variation of Heat Capacity per Unit Volume with Depth
For the case of the diathermanous solid, with a double exponential
absorption pattern, and exposed to monochromatic radiation, it was established
in Chapter IV that
2 u*(0,0)
ot
di
욥
​where the left-hand side is the initial slope of the surface response, and
di is the linear absorption coefficient for the superficial layer of the
solid.
The analysis of Chapter V demonstrated that for heterochromatic
radiation, this expression becomes

aux 10.0)
WA
Poi
at
where di applies to the i
wavelength interval. From the literature
values of the linear absorption coefficients tabulated in Chapter V, one
predicts that
[ 24*(0,0)
58 deg cal-2 cm?
.
1 I theory
The very earliest measurements made in this study indicated that the value
of this initial slope of surface temperature response was about
200 deg cal-1 cm?; the final average value was found to be
[ ut 뼱
​6,0)
185 deg cal-1 cm?.
ou* (0,0)
Jexperimental
Focussing attention only on these values for the surface response it is clear
that theory and experiment would be in exact agreement if the value of the
heat capacity per unit volume, > , for the superficial layers were
187
0.32 cal cm
deg-1. It was this consideration which led to the investiga-
tion of the composite opaque solid as presented in Section 3.5, and the de-
velopment of experimental tests to determine if the value of ✓ could indeed
be this low for the surface layers.
These tests have been presented in the
previous chapter; the results showed that ✓ was independent of depth, with
a value of 1.18 cal cm -3 deg-1.
However, as pointed out in Section 7.7 of the previous chapter, there
is the possibility that wetting of the surface by the India ink coating
could have increased a low value of ✓ for the surface layers, to a value
more nearly equal to that obtaining in the highly hydrated deeper tissue.
This would, of course, result in the false conclusion that y was depth-
independent. Because of this possibility, the four sham-painting exposures
on animal No. 1446 were made; here, the skin was "painted" with tap water

with a few drops of detergent added.
If the India ink coating had increased
the value of ✓ for the superficial tissues from a true value of 0.32 to a
value of 1.18 cal cm-3 deg-], then the wetting of the surface with water
should have decreased the initial slope of the surface response from 185 to
50 deg cal-l cm². From the data for these four exposures as given in
Table 8-2, the average surface initial slope is about 133 deg cal-1 cm2.
cm-3
If this latter value is the correct one for a surface ✓ of 1.18 cal cm
deg-7, then the "dry skin" value of 185 deg cal-1 cm² for initial slope
would suggest a "true" value of V for the surface layers of about
0.85 cal cm- ,-3
deg which is still almost three times larger than the
.-1
value of 0.32 necessary to secure agreement between predicted and measured
values of initial slope of the surface response.
It is most important to note that in these four cases, the skin was
thoroughly wetted just prior to the exposure, and almost no drying time was
permitted.
This is in sharp contrast to the procedure when India ink was
188
applied to the skin; here, about two minutes elapsed between the painting and
the exposure.
Hence, the experimental conditions were quite different, and
deg-7)
the value of v for the surface layers of dry skin (namely, 0.85 cal cm-3
which was obtained by intercomparing results from these experiments, must be
regarded with considerable skepticism.
The lowered value of initial slope
of the surface response with wetted skin may have been caused simply by
additional energy being required to heat a finite film of water on the sur-
face, and not to an increase in tissue heat capacity. The essential point
is that this reduction in initial slope was far less than would be required
to secure agreement between the predicted and the experimental values.
It should also be made clear that this suggested variation of ✓ with
depth, even if true, would fail to explain the real problem: the area under
the curve of initial slope vs depth is greater by a factor of 16 than that
demanded by consideration of conservation of energy. Certainly, it is com-
pletely inadmissable to suggest that the value of ) , for all depths, must
be less by a factor of 1/16 than that measured.
This would lead to a value
of about 0.07 cal cm 2-3 deg-1, which is an order of magnitude below the lower
estimates in the literature.

8.6.2 Location of Zero Time and Extrapolation of Responses to Zero Time
In the previous chapter, considerable discussion was devoted to the
apparent existence of time lags in the thermocouple-amplifer-recorder
system which lead to a zero time error of several milliseconds.
This error
was particularly troublesome in analyzing the opaque skin data because of
the necessity of presenting temperature responses against a square root of
time scale; such a transformation expands the initial portion of the scale,
and hence "magnifies" the zero point error.
In contrast to this, examina-
tion of the curves of Figure 8-2 (where the response is plotted against a
189
linear time scale) reveals that a time lag of one or two milliseconds would
have only a very slight effect on the form of the curves near zero time.
Actually, the effect would be to increase the values of initial slopes slightly,
which is precisely the opposite effect from that necessary to reduce the error
in the measurements.
A much more important potential source of error is in the extrapola-
tion of the replotted response curves to zero time. Usually the first point
plotted was at about 10 to 20 milliseconds after zero time; obviously, by
rather heroic pencil manipulation, one could bring any response curve into
the origin with as small a slope as desired, including zero.
It is just as
obvious that such a procedure would result in a pronounced inflection in the
curve, and both intuition and theory (as shown in Figure 5-3) reject such an
inflection. Certainly, the necessity of employing this extrapolation is
most unfortunate; one would like to have an ideal, infinite band-width

temperature measuring and recording system (or, at least be able to correct
a non-ideal response to approximate closely the ideal) so as to obtain ac-
curate measurements arbitrarily close to zero time.
However an extrapolation
over the initial ten or twenty milliseconds can really be made with con-
siderable confidence, and it is hardly possible that consistent errors would
have been introduced by this process which would yield estimates of the
initial slopes an order of magnitude too high.
8.6.3 Direct Action of Radiation on the Thermocouples
In addition to the above possible sources of error, there is the
obvious question of error in the temperature or depth calibrations.
This
can be dismissed immediately, since the same instruments and techniques
used for these bare skin studies were also used for the opaque skin measure-
ments, and the latter are in good agreement with results in the literature.
190
Thus, it seems that one is forced to conclude that only one possibility
remains;' namely, the direct interaction of the thermocouples with the radia-
tion striking and penetrating the skin.
This is unquestionably the most
likely source of error; indeed, it will be shown shortly that to some extent
it is a necessary error.
At the same time, this is an extremely unfortunate situation, since
if the thermocouples were heated directly by the radiation, it follows that
the temperature of these elements was always higher, by an unknown amount,
than the temperature of the surrounding tissue.
It would seem then that one

is forced to admit that the bare skin measurements are of dubious value un-
less and until one makes some evaluation of the temperature difference be-
tween a thermoelement and its surroundings. Actually the situation is not
quite as desperate as this; considerable useful information can be extracted
from these measurements by employing once more the concept of initial slope,

which, interestingly enough, is the very concept which was used originally
to establish firmly the fact that the measurements are in error.
8.7 Further Consideration of the Initial Slope Measurements
From the analysis of Chapter IV, in any region at a depth x,
du(x, o)
ſ "
q" (x,0),
at
where q!!!(x,0) is the initial radiant power absorption per unit volume, Now
if one takes a thermocouple as the region under consideration, then this re-
lation should hold for the thermocouple itself irrespective of the composition
or temperature of the surrounding material.
One would expect, then, that the
initial response of a thermocouple in tissue should be exactly the same as
the initial response in any other medium, for example, air.
From this
reasoning follows the statement above that the initial slope of skin response
As a check on this line
as determined by the thermocouples, must be in error.
191
of reasoning, the following simple experiment was performed.
One of the standard 0.002 inch diameter Ag-Pd thermocouples was sus-
pended horizontally in air just behind and below the aperture in the water-
cooled aperture plate of the animal holder (Section 6.5). The thermocouple
was arranged to be within but not touching a pe tri dish, and a plane mirror
was placed just behind the aperture and inclined at an angle so that radia-
tion passing through the aperture would be reflected onto the thermocouple.
With the latter in air, several exposures were made and the carbon arc furnace
output adjusted to secure about mid-scale deflection of the recorder.
The
petri dish was now filled with water so that the thermo junction was completely
immersed; according to the argument above, the initial response of the couple
should still be the same as when the element was in free air. Upon making
repeated exposures, the thermoelement response was absolutely and uniqui-

vocally zero. (The length of exposure was not sufficient to produce any
measurable temperature rise in the water.) Some of the
Some of the water was now pipetted
out until the thermocouple lay approximately in the water surface; the re-
sponse upon exposure was again zero.
Finally, more water was removed until
only a very thin bridge of water was still held to the thermocouple wire by
surface tension. Now, a response approximately equal to 1/3 the noise level
was observed; it was too small to attempt to determine the initial slope.
Now these results apparently contradict the principle which at the
same time provides the only reasonable explanation of the known errors of
skin temperature measurements.
This is a most interesting situation, and is
not yet completely understood; however the most likely explanation of the
contradiction follows from a consideration of the "fine structure" of the
thermocouples themselves.
A thermocouple, being composed of metal, is an opaque solid.
This
means that, on the surface of the wire, the initial absorption per unit
192
volume, q''(x,0), will approach infinity, and thus the initial slope of the
thermocouple temperature might be expected to be infinite also, even with the
couple immersed in water.
But from the description of the thermoelements as
given in subsection 6.6.4, one notes that the actual thermo junction is a
surface in the interior of this opaque solid, and thus, the indicated thermo-
couple temperature will not be that of the surface of the wire.
Further, the
energy absorbed at the thermocouple surface can be very quickly dissipated
in a surrounding material such as water, since the heat transfer coefficient
at the metal-water interface will be very large.
The actual thermo junction
surface will then not be heated at all.
In tissue, however, where the heat

transfer coefficient will not be so high, the thermo junction will be
heated
by the energy absorbed at the wire surface, and the indicated initial response
will be that of a very shallow location in an opaque solid.
Note that the
subsurface response in an opaque solid has an initial slope of zero, and

begins to rise only after a finite delay.
It now appears that this indicated initial thermocouple response will
be a complex function of the heat transfer coefficient between the wire and
its surroundings, the thermoelement geometry, and the orientation of its
sensitive surface.
Since these factors were, for this study, unknown,
attempts to resolve this problem by an analytical approach have been un-
successful.
However, it is claimed that because of the extrapolation of the
records back to zero time, an initial slope is obtained which is not the
true value for either the thermo junction or the surrounding tissue, but which
is, nevertheless, directly proportional to the radiant flux striking the
thermocouple.
This important assertion has not been proved, and is probably
not susceptible of rigorous proof, but from a consideration of the linearity
of the system it seems that it must be so.
Now, assuming that this claim is true, it follows that Figure 8-3 does
193
not represent q!!!(x,0), as hoped, but instead indicates the depth distribu-
tion of the total "absorbable" irradiance; i.e., employing the noncommittal
symbol S.(x) for the measured initial slope at the depth x, for unit ab-
sorbed irradiance,
So (x) = K V(x),
(8-2)
where K is an unknown constant of proportionality, and V(x) is the absorption
pattern of radiation in the skin as defined in Section 4.2. Now, following
the development of this latter section:
(4-5)
hence
9" (xit) = - [Halt). V(x)];
9'" (x,t) = -2
on [ Halal Solx)]
9.'"(, t) : - Halt). d Solx)
],
(8-3)
or
(8-4)
K K

The constant of proportionality, K, can now be evaluated by the necessary
condition
91" (x,x) dx
Halt).
(4-6)
Thus:
Halt) = 59" (xit)
Hertes
[s. (o) – S.(m)].
11
Halt d So (x)
d So (x) dx
dx
Halt)
K K
Now, since V(x) must decline to zero as x increases without bound, it follows
that
SoC ) - 0,
and
K = S. (0)
194
The complete expression for q'''(x, t) now becomes
d Solx)
dx
9"(x, t) = Halt)
Socol
(8-5)
Х
where S.(x) is the function graphed in Figure 8-3, and S. (0) is the intercept
of this curve with the ordinate.
8.8 Summary
Unfortunately, the numerical value of this intercept cannot now be
determined.
The measured initial slopes are in some way dependent upon the

heat transfer coefficient between thermocouple and surrounding tissue, and
certainly this coefficient must have been considerably different for the
surface couples than that for the subsurface elements; hence the value of K,
the proportionality constant in equation (8-2), will likewise be different
for the surface than for the subsurface measurements.
Perhaps one should
select the value of so(0) as determined from the wetted-skin experiments,
since here the heat transfer coefficients for surface and subsurface thermo-
couples might be more similar.
However, the number of such exposures is
few, and the influence of the film of water on the surface response unknown;
thus it seems rather unreasonable to use these data.
Considering the positive results, the curve of Figure 8-3 suggests
that for depths greater than about 0.02 cm, at least, the absorption of
radiation from the carbon arc furnace can be approximated rather well by a
cm-],
simple exponential, with a linear absorption coefficient of about 7.2 cm
which is considerably lower than the values cited in Table 5-1. For the
superficial layers, the absorption is greater than this, as suggested by
Hardy (see Section 4.3), although the value of the linear absorption co-
efficient in this region cannot be determined from these measurements.
The
depth to the break between these absorption coefficients is not shown by
Figure 8-3, but it must lie above 0.02 cm, and is probably less than 0.015 cm.
195
This reinforces the supposition that this break coincides with the epidermal-
dermal junction, which is at a depth of about 0.008 to 0.01 cm for the young
Chester White pig.
Turning now to the comparison of predicted and measured temperature
response,
as shown in Figure 8-4, there is no doubt that the distinct dif-
ference in form between the two sets of curves is due to errors in measure-
ment.
Initially, the thermocouples were heated above the temperature of the
tissue, so the measured response curves are too high at early times.
It

seems unlikely that this gross temperature difference persisted throughout
the entire heating cycle, however, so that the measured temperatures may be
reasonably close to actual tissue temperatures after a few tenths of a
second.
In this connection it should be noted that in none of the biopsies
taken was there ever any evidence of abnormally high temperatures around a
thermocouple site.
In fact, no evidence of the presence of a thermocouple
was ever found at all, even though the sections were carefully studied in
regions where it was known a couple had been located.
Thus, at least the
thermocouples were not heated to tissue-damaging temperature.
Considering the difference in spacing between the predicted and
measured curves, for the various values of depth, the theoretical analysis
indicates that this spacing is dependent on the value of linear absorption
coefficient.
Thus the evidence of this study indicates that here the pre-
dicted curves are in error, since the measured coefficient, for subsurface
tissue, is considerably lower than predicted. Admittedly, the procedure used
to evaluate this absorption coefficient is open to question, but at least
this represents an in vivo determination, and the tissue was not in the
highly artificial condition as in the in vitro studies upon which the pre-
dictions are based.
196
Finally, it is most clear that a great deal more work remains to be
done in the measurement of bare skin response.
In the writer's opinion,
the results presented in this chapter definitely do not indicate that ac-
curate temperature measurements in bare skin are impossible, but only that
the procedures used here were not particularly suitable.
In the next chapter,
several schemes will be described briefly which, it is hoped, will lead to
the desired end of accurate temperature determinations in a diathermanous
material such as skin.

8.9 Location of Data
All original data pertaining to this study are on file in the physics
laboratory of the Flash Burn Section of the University of Rochester Atomic
Energy Project.
FOOTNOTES
(1) We have recently investigated an ingenious "grating tangentimeter" des-
cribed by T. E. Thompson, J. L. Oncley, and H. Svensson in Rev. Sci.
Inst. 29, 977 (1958), with which angular measurements may be made
with a precision of £0.1 degree, under favorable conditions. A
simplified model of such an instrument, capable of a precision of
+0.5 degree, has been set up in this laboratory, and used to check
several of these eye estimates of initial slopes. The values from these
two methods agree to within 20%. In the future, the grating tangenti-
me ter will be used for all initial slope determinations.
1
197
CHAPTER IX
SUMMARY OF THE STUDY
9.1 Discussion of the Theoretical Analysis
It was stated earlier that of the two phases of this study, theore-
tical and experimental, the former is considered the more important.
Con-
siderable effort has been devoted to making the analyses as complete and
general as possible, so as to provide a basis not only for the experimental
work presented here, but also for additional research for some time to
come.
In spite of this effort, the analyses will be valid only if the basic

assumptions employed are valid; therefore it will now be worthwhile to re-
examine several of these assumptions, particularly in the light of certain
of the experimental results.
Consider first the assumption that the skin could be considered a
uniformly irradiated solid.
It was pointed out in Chapter VI that the
exposure site was delimitted by a 1.8 cm diameter aperture in a water-
cooled plate, and that at the edges of this aperture the irradiance was only
about 70% of that at the center.
While this would seem to be a gross viola-
tion of the assumption, the real question is whether sufficient lateral heat

conduction occurred to negate the unidimensional form of the heat conduc-
tion equation.
The results of the opaque skin experiments confirm the fact
that skin is a rather poor conductor; hence if one considers only the central
circular area of one square centimeter, where the irradiance is uniform within
about 10%, the isothermals will be very nearly planes parallel to the skin
surface.
The tissue around this central column serves as a thermal guard,
and prevents excessive lateral conduction into the skin behind the water-
cooled shield.
Hence this assumption is probably fairly well satisfied.
There remains the additional question of whether, in the case of bare
198
skin exposures, considerable radiation may not have been scattered out of
this central, uniformly irradiated column.
This would have two effects.
First, the absorption of radiation in depth would appear to be more rapid
than is actually the case, since part of the loss of radiation would be by
scatter rather than true absorption. Second, the much-used condition that
the integral over all depths of the radiant power absorption per unit volume
must necessarily be equal to the absorbed irradiance would no longer hold,
since this assumes that any scatter out of a column of tissue will be com-
pensated for by scatter into the column from adjacent regions. If, in fact,
the outgoing scatter is greater than the incoming scatter, then
$9" (*. t) dx < Halt).
This is a difficult matter to evaluate, although if one assumes a

linear absorption coefficient of 7.2 cm-1, then radiation proceeding
laterally from the axis of the exposed column of skin to a point 0.564 cm
off the axis (the edge of a circle of 1 cm² area) would be attenuated by
over 98%. Further, scatter out of the central, uniformly irradiated column
would be in part compensated by scatter into this column from the tissue around
it but within the 1.8 cm diameter exposure site. Hence, it may be concluded
that loss of radiation by lateral scatter was probably not serious.
Next consider the assumption that the skin could be considered in-
finitely thick.
Since the skin is actually only about 0.18 cm thick, this
would also appear to be a very poor assumption. Here, the crucial point is
contained in the limiting condition based on this assumption; namely
Lim u(x, t) = 0.
xoo
Thus, when a depth has been reached at which the temperature remains approxi-
mately unaffected by an exposure on the surface, then, by definition, this
depth is "infinity."
For the opaque skin exposures, it was found that a
199
thermocouple emplaced at the dermal-fat junction displayed almost no tempera-
ture rise until many seconds after the exposure, and the very slight rise
seemed to be the result of local vasodilatation, not heat conduction.
Thus
opaque skin may be considered a very good approximation to a semi-infinite
solid, with the reservation that the exposure times must be short, as in
this study. The case of long exposures has been treated quite elegantly
by Hendler, Crosbie, and Hardy (1).
Again, it is necessary to consider separately the bare skin ex-
posures. Here, even for the deepest couples employed (dermal-fat interface),
prompt and significant temperature responses were noted.
This would be of
little consequence if one could assume that the thermal and optical constants
of the subadjacent fat, and the deeper fascia and muscle were similar to
those of the dermis.
Since this is probably incorrect, it is necessary to
consider just what effect these deep layers, with their different thermal
and optical properties, would have on the temperature response of the dermis.
The straight analytical attack on this question is quite complex, and no

useful results have been obtained as yet.
One might suspect, however, that
the insulating quality of the fatty tissue will produce higher deep dermal
temperatures than would be predicted by the simple theory.
This is an area
in which the theoretical analysis must be extended.
Another basic assumption, which was originally thought to be ex-
ceedingly poor, is that the thermal constants of skin may be considered
to be independent of depth.
The excellent agreement between theory and ex-
periment for the case of opaque skin indicates that the skin may indeed be
considered to be a uniform, isotropic material.
In view of the complex
structure of skin, with stratification both laterally and in depth, this is
a surprising conclusion.
It must again be pointed out that this uniformity of skin may be more
200
apparent than real, and be the result of the surface treatment employed to
render the skin opaque.
This has been discussed at length in Chapters VII
and VIII.
In this connection, it is interesting to note that in the measure-
ments of surface temperature response made by Hendler and his collaborators
(2) the skin was heated by non-pene trating infrared radiation which obviated
the necessity of covering the surface with an opaque film.
Since the tempera-
ture responses measured by these workers followed quite closely the simple
square root of time relation predicted for a uniform solid, one might conclude
that the noted uniformity of skin is not the result of surface treatment, but
is, in fact, real.
However, the response time of the infrared photometer
used by Hendler was probably quite long; hence, inflections in the response
curve characteristic of depth variations of the thermal constants could

probably not have been observed in these experiments.
It must be concluded
that the possibility exists that there is some variation of thermal constants
of skin with depth, but this variation, if it exists, is small.
Two additional basic assumptions will now be considered: first, that
the skin is initially at uniform temperature throughout, and second, that the
surface of the skin is insulated--that is, there are no conduction, convec-
tion, or radiation losses at this surface.
Now the skin actually should be
considered as a more or less thin lamina between a constant temperature core
and the environment.
Under normal conditions, heat is conducted from the
core through the skin and dissipated at the surface to the cooler sur-
roundings. Thus, there does exist an initial temperature gradient through
the skin, and the surface is not perfectly insulated.
While it is not dif-
ficult to set up the boundary value problem incorporating these facts, the
solution, particularly for diathermanous skin, becomes rather complex. The
above-mentioned analysis of Hendler, Crosbie, and Hardy treats this situation
for opaque skin, and these authors conclude that the assumptions of uniform
201
initial temperature and insulated surface are justified for exposures of a
few seconds duration, or less.
The measurements on opaque skin reported in
Chapter VII of this study confirm this conclusion, since the responses fol-
lowed the theoretical predictions quite closely.
It is reasonable that if
these assumptions be true for opaque skin, then they must also hold for bare,
or diathermanous skin, as well. Note, however, that if one considers the
long cooling phase after a short exposure, the assumption that the surface
is insulated becomes quite poor, since undoubtedly much of the energy absorbed
in the skin is eventually dissipated to the environment rather than being
lost solely by conduction into depth.
Code

Finally, the assumption that the skin may be considered a passive solid
must be investigated.
It is necessary first to recall the position which
the present study occupies in the over-all problem of the prediction of
irreversible thermal injury of skin, as defined in Chapter I.
It was there

claimed that damage is a function of temperature, while the temperature in
turn is a function of the characteristics of the thermal insult.
The over-
all problem was thus broken down into two problems: the determination of
tissue temperatures, and the correlation of tissue damage with these
temperatures.
It follows, then, that the assumption here being considered leads to
the paradoxical situation of computing the temperatures of a passive material,
and using these temperatures to predict the active degradation of the material.
Thus, if this study is to have any utility, this assumption must be false,
by definition.
The degree of failure of the assumption will depend upon such
factors as the activation energy of the damage process, or processes, which
factors are precisely those which require accurate temperature data for their
evaluation.
Now, in the present study, all exposures were designed to be so small
202
that no extensive tissue damage would occur; hence, it is probable that the
measured temperatures are representative of passive skin.
(However, one
might question whether the high value of heat capacity per unit volume of
skin may not reflect some energy "loss" in degrading dermal collagen.) The
important question is how accurately these experimental results and theoreti-
cal predictions can be extended to exposures such that tissue damage does
occur, whereupon the system becomes, by necessity, non-linear.
Further consideration of this question leads to two rather in-
teresting conclusions. First, the division of the over-all problem of burn
prediction into the two subproblems of tissue temperature determination, and
temperature-damage correlation is probably improper and, strictly speaking,
impossible,
If one wishes to avoid the paradox of employing "passive" tissue

temperature to predict "active" tissue damage, then temperatures and damage
must be considered together.
The second conclusion is that the very failure
of this assumption of a passive--hence linear--receiver may provide a most
fruitful method for the quantitation of the thermal damage process.
Thus,
consider the heat flow equation rephrased to include a distributed heat sink
term which would account properly for the endothermic tissue degradation.
The solution of the equation would predict non-linear (i.e. exposure-
dependent) temperature response, where the non-linearity would be due to
tissue damage.
But this solution, with all unknowns evaluated by measuring
temperatures for tissue-damaging exposures, would constitute the damage
prediction equation; that is, it would contain the information necessary
to predict tissue degradation as a function of the characteristics of the
energy input.
No attempt has yet been made in this laboratory to follow up this
line of reasoning, although it would seem to be most important to do so.
It might be noted that this problem can be attacked using opaque skin,
203
since this case is vastly easier to handle both theoretically and experi-
mentally. Eventually, of course, the study must be extended to bare skin,
since this is the case of maximum interest. However, present plans call for
a thorough study of the depth of damage of India ink painted skin, as re-
lated to duration and magnitude of radiant exposure, with concomitant
temperature measurements.
At the same time, the theoretical analysis of
the non-linear, actively responding system will be attacked, employing the
simple, opaque solid model. Hopefully, this approach may result in real
progress in the predicting of irreversible tissue damage.

9.2 Discussion of the Experimental Results
The measurements of temperature response of opaque skin were,
in
general, most satisfactory.
In the absence of penetrating radiation, it is
highly probable that the thermocouples were following the true tissue tem-

peratures quite faithfully, as suggested by the excellent agreement between
theory and experiment. A disturbing feature of these measurements, however,
was the significant variation in temperature response from one exposure to
another.
It is interesting to note that other workers in this laboratory
have reported large variations in depth of damage (in bare skin) produced
by identical exposures (3). Thus, it may be that the variability in tem-
perature response is real, and due to the biologist's neutrino, "biological
variation.
The studies described in the paragraph above may help clarify
this problem.
In contrast to the results with opaque skin, the bare skin measure-
ments were unsatisfactory, so far as the accurate determination of the optical
properties of living skin is concerned.
This does not mean, however, that
this phase of the experimental program was a total failure.
The major dif-
ficulty in making these measurements, the direct action of the penetrating
204
radiation on the thermocouples, was by no means unexpected; it is obvious
that this interaction may occur.
So far as the writer is aware, however,
this is the first study which proved conclusively that this effect does
occur, which is some accomplishment, if a rather negative one.
Actually, one can point out several positive accomplishments of this
work.
First, by application of the results of the theoretical analysis, it
was possible to quantitate the error caused by direct heating of the thermo-
couples, insofar as this affected the initial slope of the temperature
response. From these considerations, it was possible to derive a value of
the linear absorption coefficient for the dermis, by employing fairly reason-
able assumptions. While this value, 7.2 cm-1, is not precise, it is one

which must be disproved before it can be discarded.
Further, the results of
this study indicate that the linear absorption coefficient for superficial
tissue is higher than that for the deeper material, which is fully in agree-
ment with the measurements of Hardy and his collaborators (4).
The depth
of the break in the absorption pattern, however, seems to be considerably
less than that suggested by these authors, and apparently is near to or
coincides with the epidermal-dermal interface.
Now, it is reasonable to ask whether accurate temperature measure-
ments in radiated bare skin can ever be achieved.
The answer to this
question is almost surely in the affirmative.
From the failures of the
present study, one can see what steps must be taken to overcome these
failures; indeed this is a most important accomplishment of this study.
First, one would like to have experimental confirmation of the im-
portant theoretical result expressed in equation (4-24):
au (x,o)
at
습
​q!" (x,0).
205
This will be done by immersing thermocouples in a strongly colored aqueous
od solution; the thermal and optical constants of such a diathermanous, non-
scattering medium can be measured independently with considerable accuracy,
so it will be possible to verify the above relation directly. From the in-
Gitteresting observations reported in Section 8.7 concerning the zero thermocouple
response in clear water, it is inferred that these thermoelements can follow
Smithe temperature of an aqueous solution quite accurately.
There will be no
problem with convection currents in the solution, since only the initial
response is of interest.
Secondly, it is abundantly clear that the skin surface temperature
must be measured with an infrared radiometer, as recommended by Hardy (5).

From the results of Hendler, et al. (6), it appears possible to measure the
surface temperature of bare skin exposed to carbon arc radiation, simply by
sampling those wavelengths of the re-emitted radiation for which the skin is
very nearly opaque.
From preliminary studies, it appears that the only problem

is the practical one of obtaining an adequate signal-to-noise ratio in a
obroad band-width system.
Such a photometer will permit accurate surface
temperature measurements uncomplicated by direct interaction of the incident
irradiation with the temperature sensing system.
This development alone will
permit accurate in vivo measurements of the linear absorption coefficient of
the superficial tissue, 3, , as indicated by the expression developed in
Chapter IV:
au*(0,0)
at
I 숍
​Finally, it will be necessary to reconsider the problem of measuring
subsurface temperatures. Probably the most reasonable approach to this problem
is to attempt to derive an analytical expression for the response of an opaque
temperature sensing element imbedded in a scattering dia thermanous medium such
206
as skin.
If the dependence of the thermoelement temperature on that of the
surrounding tissue can be found, then it will be a straightforward matter to
correct the indicated responses to get the true tissue temperatures.
This analytical approach will be practical only if the temperature
sensing element has such ideal geometry that the mathematical boundary value
problem does not become unmanageably complex. Actually, such elements are
already at hand.
If one takes a suitable length of the Ag-Pd Wollaston wire
described in subsection 6.6.1, and dissolves away the silver jacket over only
a few millimeters in the center of this length, the remaining fine palladium
filament connecting the two halves forms an elegantly small and sensitive
resistance thermometer.
Not only is the geometry of this element quite
simple, but the heat transfer coefficient between thermometer and tissue
can be determined in vivo by passing a known current through the palladium

filament and measuring its equilibrium temperature rise.
These elements
are extremely fragile, so that placement will be difficult, although not
impossible.
Other methods of obtaining accurate subsurface tissue measurements are
also under consideration.
For example, one may compare the results from
thermocouples of various diameters--hence various surface-to-volume
ratios--and extrapolate to the ideal zero surface-to-volume ratio element.
Since such a thermoelement has an infinite diameter, this may be a rather
questionable extrapolation. However, employing different diameter elements
would be quite useful in checking theoretical predictions concerning this
parameter.
9.3 Consideration of the Direct Measurement of the Absorption Pattern of
Radiation in Skin
From the analysis of Chapter V it is easy to show that for a dia-
207
thermanous solid exposed to heterochromatic radiation the following relation
holds:
du (x, o)
-
q!"; (x,0),
at
where q'''}(x,0) is the initial rate of energy absorption per unit volume
for the ith wavelength interval of the incident radiation. Expressing
q!!!:(x,0) as the product function
91; 1x,0) = Hairo)-Fi(x),
babam
where Hai(0) is the initial absorbed irradiance function for the i
th
Wave.
length interval, and Fi(x) is the absorption pattern for this radiation, it
follows that one can, in principle, determine this latter function by direct
temperature measurement. By its very nature, this procedure offers the most
direct and unambiguous method for the evaluation of the heating effect of
penetrating radiation, which effect must be known for temperature prediction

purposes.
This underlines the importance of achieving accurate subsurface
temperature measurements in bare skin.
Actually, the functions Fi(x) must be known for each small wavelength
interval over the spectral range of interest, if one is to be able to predict
temperature responses to any arbitrary wavelength distribution.
Thus, even
accurate subsurface temperature measurements employing, say, carbon arc
radiation are not enough; one must employ either monochromatic radiation
or subtractive, sharp cut-off filters in order to evaluate the absorption
pattern as a function of wavelength. Both of these methods present real
experimental difficulties. Even with a fast monochromator, with a broad
spectral band-pass, the output irradiance is low.
It follows that tempera-
ture responses will be quite small, hence difficult to measure with accuracy.
On the other hand, the use of sharp cut-off filters, discussed in Section 5.6,
208
implies that one must determine each Fi(x) as
Fi(x) =
9"; (x0
(x,0) - į 9"; (x,0)
Hairol
1%, 0)]
d=i+1
In general, each F(x) will be a small difference between two relative large
i
numbers, which latter are subject to large experimental variability. Hence,
with this method also, it will be difficult to achieve good accuracy.
Assuming that these practical problems can be solved, one is im-
mediately faced by a much more difficult fundamental problem. An accurate
statement of the absorption pattern of radiation in skin, as a function of
wavelength, is primarily of use in predicting the temperature response of a

linear passive system. If one wishes to predict tissue injury, and still avoid
the "passive-active tissue" paradox discussed above, then the theory must be
extended to include the effects of such injury.
It would seem that the
factor q'?'(x,t), in the heat flow equation, must now contain both a
wavelength-dependent energy absorption term and a (presumably) wavelength-
independent term which accounts for the endothermic tissue degradation
process.
If one demands that the damage predicting scheme be capable of
handling an exposure to radiation of any arbitrary wavelength distribution,
then it is clear that these two terms must be evaluated separately. Will
the experimental procedures described above be of any utility in untangling
these terms?
If not, what procedures must be used?
The writer is unable to
answer these questions, and in fact does not hope to be able to do so until
some progress has been made in the opaque skin studies described in Section
9.1.
9.4. Conclusion
The specific results given in this chapter and the preceding ones
are not as extensive as one would desire.
In all honesty, it must be ad-
mitted that the completely general prediction of the temperature response
209
of skin to any arbitrary thermal insult remains an unsolved problem. Still,
this study does constitute a highly satisfactory demonstration of the utility
and the necessity of combining the theoretical and experimental approaches
to this problem.
If the above remarks seem to have dwelt more on plans for the future
than on accomplishments of this research, this is not unreasonable,
It is
obvious that much work remains to be done, but the real contribution of this
study is the conclusion that this future research is possible and practicable,
and that the ultimate goal of predicting the irreversible thermal injury of
skin may be achieved.

FOOTNOTES
(1) Hendler, E., R. Crosbie, and J. D. Hardy, Measurement of Skin Heating
During Exposure to Infrared Radiation, J. Appl. Physiol. 12, 177-185
(1958).
(2) Ibid.,
180.
(3) Hinshaw, J. R., The Irradiance Dependency of Exposure Time as a Factor
in Determining the Severity of Radiant Thermal Burns, University of
on Rochester Atomic Energy Project Report UR-451, 6 (1956).
(4) Hardy, J. D., H. T. Hammel, and D. Murgatroyd, Spectral Transmittance
and Reflectance of Excised Human Skin, J. Appl. Physiol. 9, 262 (1956).
(5) Stoll, A. M. and J. D. Hardy, Direct Experimental Comparison of Several
Surface Temperature Measuring Devices, Rev. Sci. Inst. 20, 678-686 (1949).
(6) Hendler, E., R. Crosbie, and J. D. Hardy, op. cit. It is interesting to
note that the instrumentation described in this report is now capable of
to measuring surface temperatures for bare skin exposed to penetrating
radiation. The system undoubtedly has a narrow band-width, but if the
indicial transfer function is determined so that measured responses may
be corrected, then these authors may immediately determine the linear
s absorption coefficient for the surface layer of skin for a variety of
spectral distributions.
doo
to the
210
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Exposure to Intense Thermal Radiation, J. Appl. Physiol. 5, 559-566 (1953).
19. Hardy, J. D., H. T. Hammel, and D. Murgatroyd, Spectral Transmittance and
Reflectance of Excised Human Skin, J. Appl. Physiol. 9, 257-264 (1956).
20. Hardy, J. D. and C. Muschenheim, Radiation of Heat from the Human Body
IV. The Emission, Reflection, and Transmission of Infrared Radiation by
be the Human Skin, J. Clin. Invest. 13, 817-831 (1934).
21. Hardy, J. D. and C. Muschenheim, Radiation of Heat from the Human Body
V. The Transmission of Infrared Radiation Through Skin, J. Clin. Invest.
15, 1-9 (1936).
22. Hendler, E., R. Crosbie, and J. D. Hardy, Measurement of Skin Heating
During Exposure to Infrared Radiation, J. Appl. Physiol. 12, 177-185
(1958).

23. Henriques, F. C., Jr., Studies of Thermal Injury V. The Predictability
and the Significance of Thermally Induced Rate Processes Leading to
Irreversible Epidermal Injury, Arch. Path. 43, 489-502 (1947).
24. Henriques, F. C., Jr. and R. A. Maxwell, The Applicability of the Skin
Damage Integral to the Prediction of Flash Burn Injury Thresholds,
Including those Caused by Atomic Detonations, Technical Operations, Inc.,
Report TOI 54-19 (1956) (SECRET).
25. Henriques, F. C., Jr. and A. R. Moritz, Studies of Thermal Injury I.
The Conduction of Heat to and Through Skin and the Temperatures Attained
Therein, A Theoretical and Experimental Investigation, Am. J. Path.
a 23, 531-549 (1947).
26. Hinshaw, J. R., The Irradiance Dependency of Exposure Time as a Factor in
Determining the Severity of Radiant Thermal Burns, University of Rochester
Atomic Energy Project Report UR-451 (1956).
27. Hinshaw, J. R. and H. E. Pearse, Histologic Techniques for the Differential
Staining of Burned and Normal Tissues, Surg. Gyn. and Obs. 103, 726-730
(1956).
28. Jakob, M., Heat Transfer, Vol. I (John Wiley and Sons, Inc., New York,
1949).
212
29. Kingsley, H. D., L. Hogg, Jr., J. T. Payne, and J. H. Morton, Studies in
Prolonged Anesthesia in Swine and Dogs, University of Rochester Atomic
Energy Project Report UR-152 (1951).
30. Krolak, L. J., The Measurement of Diffuse Reflectance of Pig Skin, Titanium
Dioxide Paint, and India Ink; the Transmittance of Titanium Dioxide and
India Ink, University of Rochester Atomic Energy Project Report UR-439
(1956).
31. Krolak, L. J. and T. P. Davis, A Universal Spectrophotometer for the
Measurement of the Relative Spectral Distribution of the Carbon Arc Source,
University of Rochester Atomic Energy Project Report UR-367 (1954).
32. Kuppenheim, H. F., J. M. Dimitroff, P. M. Melotti, I. C. Graham, and
D. W. Swanson, Spectral Reflectance of the Skin of Chester White Pigs in
the Ranges 235-700 mu and 0.707-2.6 H, J. App. Physiol. 9, 75-78 (1956).

33. Lawson, D. I., P. H. Thomas, and D. L. Simms, The Thermal Properties of
Skin, Fire Research Station (Boreham Wood, Herts., England) F.R. Note
224/1955 (1955).
34. Lerman, B. and J. R. Hinshaw, Relationships Between Burn Severity and the
Simulated Thermal Pulses of Various Nuclear Weapons, University of Rochester
Atomic Energy Project Report UR-533 (1958).
35. Leroy, G. V., Medical Sequelae of the Atomic Bomb Explosion, J. Am. Med.
Assoc. 134, 1143 (1947).
36. Lipkin, M. and J. D. Hardy, Measurement of Some Thermal Properties of
Human Tissues, J. Appl. Physiol. 7, 212-217 (1954).
37. Mixter, G., Jr. and T. P. Davis, A Method of Shaping Thermal Energy
Pulses from a Carbon Arc Source, University of Rochester Atomic Energy
Project Report UR-387 (1955).
38. Mixter, G., Jr. and L. J. Krolak, Critical Energy of Fabric as Indicated
by its Persistence Time Under Thermal Radiation, University of Rochester
Atomic Energy Project Report UR-260 (1953).
39. Moritz, A. R. and F. C. Henriques, Jr., Studies of Thermal Injury II.
The Relative Importance of Time and Surface Temperature in the Causation
of Cutaneous Burns, Am. J. Path. 23, 695-720 (1947).
40. Pearse, H. E., Thermal Injuries of Nuclear Warfare, Military Medicine
118, 274-278 (1956).
41. Pearse, H. E. and H. D. Kingsley, Thermal Burns from the Atomic Bomb,
Surg. Gyn. and Obst. 98, 385-394 (1954).
42. Pearse, H. E., J. T. Payne, and L. Hogg, The Experimental Study of
Flash-Burns, Ann. of Surg. 130, 774-789 (1949).
213
43. Perkins, J. B., H. E. Pearse, and H. D. Kingsley, Studies on Flash Burns:
The Relation of Time and Intensity of Applied Thermal Energy to the
Severity of Burns, University of Rochester Atomic Energy Project Report
UR-217 (1952).
444. Ross, 0. A., A. E. Axelrod, and A. R. Moritz, Progress Report of Project
No. G-3645 to Morphology and Genetics Study Section, Division of Research
Grants, Department of Health, Education, and Welfare, for July 16, 1952
to August 31, 1954.
45. Stoll, A. M. and J. D. Hardy, Direct Experimental Comparison of Several
Surface Temperature Measuring Devices, Rev. Sci. Inst. 20, 678-686 (1949).
46. Strong, J., Procedures in Experimental Physics (Prentice-Hall, Inc.,
New York, 1938).
47. Swope, G. A. and F. C. Henriques, Thermal Pulse Properties of Five High
Intensity Laboratory Sources, Technical Operations, Inc., Report No.
TOI 54-8 (1954).
48. Zemansky, M. W., Heat and Thermodynamics, Third Edition (McGraw-Hill
Book Company, Inc., New York, 1951).
49. A Study of the Physical Basis of Burn Production with Applications to the
Defensive Reactions to an Atomic Bomb Air Burst, Medical Research and
Development Board, Office of the Surgeon General (Author and date of
publication unknown).
50. The Thermal Data Handbook, Armed Forces Special Weapons Publication
AFSWP 700 71954) (SECRET).

214
APPENDIX I SORTE
Det ecce to Nomenclature
Only the more important symbols are included in this list, and trivial
constants, self-evident terms, and symbols adequately defined in context
have been omitted.
a
As subscript
Subs
Olta
"absorbed."
Cp
· Specific heat capacity (cal gm-] deg-1).
h
k
q
DO
- Laplace transform of irradiance function, H.
Thermal conductivity (cal sec-1 cm-1 deg-1).
Rate of heat flow (cal sec
sec-1).
q!! Rate of heat flow per unit area (cal cm-2 sec-l).
q!!! - Rate of heat generation (absorption) per unit volume
(cal cm-3 sec-1).
t
Time (sec).
ORO

- Laplace transform of temperature rise, U.
- Depth (cm).
Cp
Total heat capacity (cal deg-l).
H
Irradiance, radiant power per unit area (cal cm-2 sec-l).
Ja
- Relative spectral energy.
Q
do
Heat (cal), or Radiant exposure, the time integral of irradiance
(cal cm-2).
R
Reflectance, or back-scatter factor.
T
- Absolute temperature (°K).
U
Temperature rise (centigrade degrees).
a
Damage symbol.
8
Thermal diffusivity (cm? sec-1).
- Linear absorption coefficient (cm-1).
215
S
Trapezoidal irradiance pulse rise time (sec).
m
Rectangular irradiance pulse duration, exposure time (sec).
a
Dimensionless time, diathermanous solid model.
M
Thermal inertia kiv (cal2 cm-4 deg-2 sec-1).
Heat capacity per unit volume (cal cm-3 deg-1).
V
les
Dimensionless depth, diathermanous solid model.
p - Density (gm cm-3).
· Dimensionless time, opaque solid model.
Detta
r
Dimensionless depth, opaque solid model.
T Dimensionless temperature rise, opaque solid model.
F - Dimensionless temperature rise, diathermanous solid model.
betes

Badges
Inato
216
APPENDIX II
The Application of Heat Flow Theory and Chemical Kinetics to the
Prediction of Irreversible Thermal Injury of Skin
Thomas P. Davis
Biophysics Seminar
November 13, 1956
I. INTRODUCTION
During the course of our study of radiant energy "flash" burns it
has become increasingly evident that it would be highly desirable to be
able to intercompare results obtained with different time-irradiance input
Save
pulses, with various spectral distributions, or even with different modes
of energy input; i.e., conduction or convection. Similarly we have become

increasingly aware of the necessity of extending laboratory findings to new
unexplored situations; this latter problem is acute in the case of those
attempting to predict military and civilian casualties in the event of a
nuclear attack.
The desideratum for these and related problems is the
ability to express damage in terms of the characteristics of the thermal
insult; that is, = O(q'', n,
O(q!', ^ , *, t ...), where D is some undefined
measure of irreversible thermal damage to skin.
Some empirical expressions do exist (see, for example, Davis,
Hinshaw, and Pearse, A Comparison of the Effects on Bare Porcine Skin of
Radiant Energy Delivered in the Forms of Square and Simulated Field Pulses,
UR-418 (1955).), but they are only convenient summaries of existing labora-
tory data, and, having no theoretical significance, cannot be extended to
new situations. Obviously to obtain the desired result, a generally appli-
cable prediction formulation, a somewhat more fundamental approach is
indicated.
217
The approach adopted here is as follows:
First, the study is restricted to exclude photochemical reactions;
i.e., reactions of the type X + h)
x*
= P.
With this restriction it
is then claimed that damage is solely a function of temperature of the
concerned tissue, or
D - DCT)
where
T: T (9", n. x, t, ...).
Thus, the problem is now broken down into two subproblems: First the
prediction of the temperature-time-space history of the skin for any arbi-
trary energy input, and second, the prediction of irreversible tissue injury
based on this temperature response.
This rather formal presentation of what
is probably an immediately obvious approach has been adopted for purposes of
unifying the remaining presentation, which often seems far afield from the

central matter. Et
The "damage symbol," D , still remains undefined.
The problem of
definition is actually an integral part of the problem of prediction, and
further discussion of this matter will be deferred until after treatment of
the problem of determining the temperature response.
II. (,
T = T (q'', n , x, ty ••o)
The basic equation of heat conduction in a substance at rest is
pCp ôt
21 - Elke
Blk ) + (ky o T) + (ko #7) + q.
where the k's may be functions of position and time.
General solutions of
this equation are not obtainable, and one must proceed to set up a model
within the framework of which the equation can be so specialized as to
obtain a solution.
In proceeding to develop such a model the following
assumptions will be made:
CE
218
1. The receiver is semi-infinite, with the surface in the plane
0, and extending in the positive x direction.
2. The heat input to the system is uniform over the surface.
3. The receiver is isotropic and passive.
4. The surface is perfectly insulated (including no reradiation
losses).
5. The receiver is initially at uniform temperature throughout.
These assumptions are presented in more detail, and fully discussed with
respect to the actual system, skin, in reference (43)*.
Under these assumptions, isothermals will be planes parallel to the
surface, and the heat flow equation becomes

2²T(x,t)
ar2
OT/x+)
at
+
91(x, t)
РСР
The problem will now be specialized further by considering only radiant
energy input to the system, and the following three models will be proposed:
1. Opaque solid.
2. Diathermanous solid, exponential-type absorption.
3. Diathermanous solid, linear-type absorption.
Solutions of the above specialized heat flow equation may now be obtained
readily.
These will be presented below:
Model 1
opaque solid
Statement of the problem:
1. T¢"')(x,+) = a Txx"(x1+)
2. T" 1x,0) = To
3. T")(0,t) - To
4. T10,4)= - † (1-R) H1+)
*References throughout this appendix refer to the Bibliography preceding
Appendix I.
219
Solution:
2
a
e nd?
+ To
T"(t) = Votere (ci-e) H(t-1)
VI
Model 2
diathermanous solid, exponential-type absorption
Statement of the problem:
1. Thelxit) = x T 2 x lxst) + per C-R) Hit) e-rx
2. 7') (XO) - TO
3. T12) (x, t)
(t= To
4. Tx"10,7) = 0
Solution:
7="?(24) -Tozeca SCOR)Ht-)e****[2cosh xx
temperflakkee PVat) -e erflavka
+8rar
37)]dz.

Model 3 - Diathermanous solid, linear-type absorption
o=xLL
=
>
Statement of the problem:
(1-R) H(A)
1. T';(x,y) = a T Ixit) +
PCpL
XT 2 (xit)
>> L
2. T"(7,0) T.
3. T" (0,+) - To
4. T*(0,0) = 0
5. T(4-0,7) - T')(1+0, +)
6. T (L-0, +) = T(470, +)
Solution:
T13)(xt) - To
2ent Fru<2)H4t- +][ert (m): erf (**)]dt,
x>0
It may be noted that the solution for model I also describes the
temperature response for Newtonian (contact) heating, with replacement of
220
H(t) by a suitable heat input function q''(t), involving the heat transfer
coefficient.
The problem of selecting the "best" model to fit skin, and the de-
termination of constants would seem to be a formidable experimental task.
Some improvement may be obtained by further simplifying the heat input
function, and introducing certain dimensionless parameters.
For the input,
We will consider the following function:
HI+) = Ho, osten
otco, tan

The term (1 - R) H(t - Z ) may now be brought out from under the integral
sign in the foregoing expressions, and solutions obtained immediately for
the temperature
9
limits from o tom • For times greater than m
equals TM(x, t) – T '(x, t -m] by superposition.
While this is some
improvement, the expressions are still not easy to fit to experimental data.
However, one may note the following relatively simple expressions:
TT (8t) =
2(1-2)HO VE + To
Takpce
2. T10,0) =
1-R Ho
3. T10,0) =
1
10,0) = (1-12) H..
8 ४
pcp
РСР
L
Now,
by making the skin opaque, say with India ink or black Krylon, one may
obtain, from (1) above, the product k p cp from surface temperature measure-
ments.
The pop product is about 0.9, hence one may obtain k, and from
Next, according to the above equations (2) and (3), from
this a =
k/e cp
the initial slope of the surface temperature rise, one may obtain an ex-
perimental value, say o cm-1; which will be equal to either 8 or 1-1, de-
.
pending upon which model, 2 or 3, is "correct." Now, let us define two
dimensionless variables, in terms of this experimental constant o:
5=ox
A = orat.
221
Now solutions for models 2 and 3 may be written as DEBE
(1-R) H.
kom a TV (916) - hva e*- Serte (Ita) - e-
&[e'erte (v + zfa) + e forte (12 - zbo)]
and
ko
(1-R) HO
AT 13)
180): flert (474) ert (4)] d*
These are now being computed by Dr. Dutton on the IBM 650 for rather wide
ranges of & and ☺
These analytical expressions may be compared with
experimental measurements, permitting a choice of the better model.

It may be noted that this approach is somewhat different from that of
others (refs. 4, 49), who have obtained values of the optical and thermal
constants of skin from in vitro measurements, and applied these to the direct
solution of the heat flow equation, either by analytical or numerical methods.
It is believed that the method here presented is preferable, but only in case
a satisfactory model has been initially selected. Preliminary calculations,
for { = 0 and ę = 1, however, indicate that the temperature responses under
models 2 and 3 are remarkably similar, and it may be that the system is not
markedly sensitive to the absorption pattern.
It should also be stressed that both models 2 and 3 are valid only
for very white skin.
The presence of a heavily pigmented layer at the base
of the epidermis will undoubtedly alter the temperature-depth profiles
considerably. It is not planned, at the present time, to extend these in-
vestigations to cover this important situation.
The experimental measurements will be made by introducing thermocouples
into the skin of Chester White pigs, these being the experimental animals
used for virtually all work by this group.
The thermocouples are silver-
palladium, made by dissolving away the jacket of silver-palladium Wollaston
222
wire.
The silver is of 0.003 inch diameter, and the palladium 0.0003 inch
diameter.
For introduction, the silver end of the couple is butt-welded to
a length of 0.005 inch diameter spring steel wire, with the leading end of
this wire sharpened to permit easy entry into the skin.
The depth of the
couple is measured by utilizing the ferro-magnetic properties of the steel
leader, employing a tape recorder erase head as the sensing element.
Silver
and palladium extension wires lead from the couple to a reference junction
maintained near normal skin temperature. The thermal emf is amplified and
fed to a Sanborn recorder; the entire amplifier-recorder system has an upper
frequency limit of about 40 cps, insuring adequate time response.
Calibra-
tion circuits are also included ahead of the first amplifier, so that the
over-all system response may be determined.
The recorder available has
four input channels, so that four couples at various depths may be used for
each exposure.
At this point, it would be well to review the presentation to this
point. With certain simplifying assumptions, the general heat flow equation
has been specialized so as to be amenable to formal solution.
Certain models
for the interaction of radiation with skin have been proposed, and normalized
temperature-time-depth functions derived.
The solutions of these functions,
presented as families of curves, will then permit selection of the "best"

model, and determination of the constants involved.
With this accomplished,
Joone may then compute the temperature response of skin for any arbitrary heat
input, including that by conduction and convection, where suitable heat trans-
fer coefficients can be obtained.
We are now prepared to consider the next
phase of the problem, that of determining the relation of the temperature to
irreversible tissue damage.
handbou
223
III.
O(T).
As mentioned previously the definition of damage is a central part
of this problem, and hence will be considered first.
In the Flash Burn
Section, we have set up a scale of burn severity, ranging from 0 to 5+,
based on gross observation, or "surface" appearance.
This scale is defined
as follows:
ON
no burn
1+
red burn
2+
patchy white burn
3+ - uniform white burn
4+
blebbed white burn

5+
carbonized burn
Subdivisions of mild, moderate, and severe within each major group yield a
scale of 16 levels of severity. Experimental burn results are handled as
quantal data to establish median effective exposures for a given level of
severity, and are presented typically as iso-response curves on a plot of
radiant exposure vs exposure time, for example.
Thus, it might seem at first sight that the desirable prediction scheme
would be one which would yield median effective exposures for such iso- N+
curves; i.e., iso-response curves for the Nth level of grossly observable
severity. The principal objection to this scheme is the paradoxical statement
frequently made by workers in this laboratory that "one 2+ burn (say) is not
the same as another 2+ burn. 11
While such a statement is meaningless under
the definition of the surface appearance scale, it nevertheless represents
the undoubtedly correct subjective feeling as to the true severity of a
particular lesion.
It has long been agreed that a more meaningful criterion of damage
is the depth of tissue destruction.
In spite of such agreement, the criterion
224
has been little used in the past because of difficulties of depth assessment.
Recently, however, the development of certain staining methods by Hinshaw
(27) has greatly eased the problem of differentiating damaged from normal
tissue, and while certain problems still remain, notably those of shrinkage
and swelling of the injured tissue, it would seem that accurate depth of
to damage determinations are now readily obtainable.
Now, using this depth of damage as the criterion of burn severity, one
may design experiments to determine the median effective exposure for a given
depth (quantal data), or the mean depth for a given exposure.
While the ex-
perimental design would be quite different for these two approaches, it is
claimed that the end results are the same providing that for a given radiant
exposure, the distribution of depth of damage is symmetric; i.e., the mean
equals the median.
Hence the approach here employed is that of predicting the
mean depth of damage (as displayed by staining techniques) produced by a given
thermal insult.
Having now selected a definite quantity to be predicted, one must in-

vestigate means of accomplishing this.
The first method which might be
suggested is that of the "maximum temperature attainment" criterion.
Here,
one would postulate that irreversible tissue damage will result upon the
attainment of some fixed critical temperature.
This approach has the charm
of maximum simplicity, but the disadvantage of probable failure. Experience
with animals (23) and inanimate "skin simulants" (5) has indicated that the
maximum temperature attainment criterion is inadequate to explain experi-
mental results.
A more reasonable approach is that of Henriques and Moritz and sum-
marized in the "punishment integral":
RTITI
dr
Roofie Hai
225
While this is a frankly empirical approach, it does possess a reasonable
basis, and further has been the method most frequently employed for prediction
purposes.
It would seem worthwhile, then, to digress briefly to discuss this
formulation, a thorough exposition of which will be found in the references.
The punishment integral is based on work by Henriques and Moritz using
hot water burns on pigs. The experiment consisted in the determination, for
various water temperatures, of the minimum application times necessary to
produce trans-epidermal necrosis (threshold A), and maximum application times
which could be tolerated without irreversible epidermal damage (threshold B)
(25)
The quantity r was then defined as "an arbitrary function of epi-
dermal injury as determined by histological examination" (23), and it was

postulated that this "injury" follows a relation
Pé
RT
dt
where the similarity to chemical kinetics is obvious.
The temperature, T, is
to be taken as that at the dermal-epidermal junction. In the case of long-
time burns (time)
60 seconds) this temperature will be approximately equal
to that of the applied hot water, T
Ts, a constant, so that one may write
AE
Pte
2
Since P is an arbitrary constant, Henriques selected a value of n
= 1 for
48
threshold A, and by curve fitting obtained the values
P=3.1810
sec-
AE = 150,ooo cal molo -
Then, the expression
48
75 000
T(T)
/= 3.7 X10
е e
dr
fit the experimental data with high precision, not only in the steady-state-
region, but also in the non-steady-state region down to application times of
a few seconds. Further, with these same constants, a value of 12 0.53
UN
226
similarly followed the threshold B data.
Henriques has subsequently applied this punishment integral to radiant
energy burn data from this laboratory, using the following procedure: the
temperature at the dermal-epidermal junction is computed on opaque solid
theory, using the radiant exposure for a median effective exposure for an
N+ burn at, say, I second exposure time (square-wave pulse).
The resulting
value of n then obtained is used to predict the N+ median effective ex-
posure for other exposure times, or other input pulse forms (24, 50).
When compared with recent data from this laboratory, these predictions

are not notably good, except in the case of "opaqued" skin at the 2+ level,
and normal skin at the 4+ and 5+ levels.
In the former case, one might ex-
pect reasonable agreement since 2+ at all times considered corresponds rather
well with trans-epidermal necrosis, and the model used for computing
T
es
T(x,t) is not grossly dissimilar to the actual system.
In the latter
case, no reasonable explanation is apparent, other than compensating errors.
Buettner (4) has followed what would appear to be a somewhat better
procedure. He has used Henriques' formulation without change, but has pro-
posed that 2
= l be the criterion for irreversible injury and has then
determined the depth at which this holds, for a given energy input.
In his
computation of temperature response, he has also used a diathermanous solid
model, with constants obtained from a literature review and some direct
determinations.
The most interesting feature of his extraordinarily thorough
analysis is a set of predictions giving the expected depth of irreversible
injury for a variety of radiant exposures, all of two second duration square-
wave form.
Since his work is based on white skin, and uses the punishment
integral which, in turn, is based on the Chester White pig as the experimental
animal, we are in an excellent position to check directly his predictions.
(It may be noted that Buettner presents only pilot experiment results, and
227
not a direct check.) Such an experiment has been performed recently, and
while the results are not yet available, our prior work at exposure times
near two seconds indicates that these predictions are in error.
The approach which the writer has developed follows quite closely
those of Buettner and Henriques. First a mythical reaction Y > 2 in the
skin is hypothesized which follows a first-order reaction; i.e.,
day boy
.
dt
But, from Eyring's theory of absolute rates
h =
RT
Noh
е e Р e
Thus,
A sa
ZlHa
RT
R

day =
R
4 Ha
RT
e
dt
T e
Сү
Noh
or
A Ha
in lepi - Nah
450
R
RTF) da
е e
T(2) e
.
Now, assume that irreversible thermal injury, as defined by the above men-
tioned staining techniques, corresponds to a certain critical value of
cylcy Then, if a constant S is defined as proportional to, but not
Sa
R
necessarily equal to
e
Noh
then one may arbitrarily select S such
عليه
R
that at that value of x equal to the depth of irreversible injury, in lum) - /
Thus, the supposition is that
A Ha
RT (8,4)
| = 5$, 7(x, t)e
dt
defines the value of x equal to the depth of irreversible tissue injury for
a given hyperthermic episode. Note that the integration must be carried to
infinity--i.e., to the time when the temperature has returned to normal--
since one must consider the hypothetical reaction during the entire period of
elevated temperatures, including the relaxation phase.
Since this formulation must be capable of predicting not only our
data but also that of Henriques and Moritz, the values of the constants would
228
be expected to be approximately the same.
However, we have seen that Buettner's
careful analysis does not yield reliable depth estimates.
Therefore, it will
be necessary to investigate not only the question of the best possible tempera-
ture response calculations, but also the damage predicting scheme, per se.
For instance, the possibility of reaction order other than first, competing
reactions, or consecutive reactions should be considered.
IV. CONCLUSION
It will be realized that many severely simplifying assumptions have
been made in this development. Insofar as possible, these assumptions must
be checked by experiment, although it must be admitted that the extreme com-
plexity of the burn process in living skin is extraordinarily resistant to
formal mathematical attack.
It is to be hoped particularly that the proposed
hypothetical reaction Y - Z may eventually be translated into a meaningful
statement involving demonstrable species, for then the entire damage prediction
scheme could be based on solid experimental results. And, indeed, if the
present investigation proves successful, the information obtained as to heats
of activation may provide a clue as to the probable species involved, and thus
aid in its own theoretical justification.

229
APPENDIX III
The "Opaque Solid Function," R(x)
The "opaque solid function," R(x), was defined in Chapter III as
) e x
Numerical values of this function can be obtained conveniently and rapidly
using an ordinary desk calculator and the extensive Tables of Probability
Functions, Vol. I (A. N. Lowan, Technical Director), prepared by the
Federal Works Agency, Work Projects Administration for the City of New

York, Sponsored by the National Bureau of Standards (1941). This table
x2
presents values of patie
and erf(x) (the former is the first derivative
of the latter) at closely spaced values of the argument. For computation, it
is convenient to rearrange the above expression to
+ x
]
The following brief table of values of R(x) is of convenience in
calculating the (normalized) temperature response of the opaque, isotropic
solid, as outlined in Section 3.3.
230

X2
R(x) = viltte - x.erfe(x)]
X
0
1
2
3
44
5
6
7
8
9
0.0
1.00000 0.98237 0.96495 0.94773 0.93070 0.91388 0.89725 0.88082 0.86460 0.84857
0.1
0.83274 0.81711 0.80167 0.78643 0.77139 0.75655 0.74190 0.72744 0.71318 0.69912
0.2
0.68524 0.67156 0.65807 0.64477 0.63166 0.61874 0.60601 0.59346 0.58110 0.56893
0.3
0.55694 0.54513 0.53350 0.52206 0.51079 0.49970 0.48879 0.47805 0.46749 0.45710
0.4
0.44688 0.43685 0.42696 0.41725 0.40771 0.39833 0.38911 0.38006 0.37117 0.36243
0.5
0.35385 0.34543 0.33717 0.32905 0.32109 0.31327 0.30561 0.29809 0.29071 0.28348
0.6
0.27638 0.26943 0.26262 0.25594 0.24940 0.24299 0.23671 0.23056 0.22454 0.21864
0.7
0.21287 0.20722_0.20169 0.19628 0.19099 0.18581 0.18075 0.17580 0.17096 0.16623
0.8
0.16160 0.15708 0.15267 0.14835 0.14414 0.14003 0.13601 0.13209 0.12826 0.12453
0.9
0.12088 0.11733 0.11386 0.11048 0.10718 0.10396 0.10083 0.09778 0.09480 0.09190
231
APPENDIX IV
Solution of the Composite Opaque Solid Model
Statement of the problem, from Section 3.5:

ki, o,
ke, vi
Mi = kili M2 = K2 Ve
ka
I
Le
1
ANDT
1) Ue (x) = xx (x, t), o exab, tzo
Holt)
2) We(xt) = 42 Uxxlx, t), xb,
3) U (x,0) = 0
4) U (0o,t) = 0
Holt
5) Ux (0,4)=-
tzo
6) U (6-0, t): Ulbro, t)
7) ki Ux (6-0, t) = K2 Ux (bro, t),
ka Ux (6+0,t), tzo.
tzo
b ь
(1-12) H. (t)
ki
x
using (3)
3,
Transforming by
Le { } and using
1) , s)
d² ulx,s)
2'). s Ulx, s)
d?ulx,s)
X2
dx2

)
*>
A) u (0,5) = 0
s) dulos
)
-- halsh
Eestyle
ki
=
u lbtos)
61) ulb-0,5)
7) k, dulb-o,s)
k

-
ka dulb to,s)
AC
Solving (11) and (21)
2':
8) ulx.s) = A en VI
+
Betten is
Otxab,
+
ce the
a) u 18,5) = ce
+ De
x > b.
Differentiating (8), and applying 159)
232
ha(s) = - VA VB
ha(s)
k.
bars)
Vs
A hars)
he
Applying (41) to (9)
Thus (8) and (a) become:
+ tutor
10) Ulx,s) = Ale
) A strettivo)- beton
098b
11) u (x,s) = ce
(x,s
xxb.

The constants A and c
are
Mom
evaluated by
to (10) and (11). Thus i
applying (61) and (21)
tan
ers
12) Alette Hom)-celshowe
k, VT, Al-e te #")+ki ve cei te ti haces
2.V.
and
+
vs
or,
since
ki /2 = Tui,
12) Availe ters te tere) + Cutie
et te
hals)
Laula
Solving by determinants, let
le tera
e fairs)
val-eta te us )
te
+
A =
ale
een
st
233
or
Az étrs [et (1.- Vari) te tera Viet van)].
Then
to us
A.A = hals) e
is
sta
ī
a(s)
ters
e
Vī,
to ( 17 th a
A,
hals) e
Vs[ettvo (vart vare-
(vart view) - (vam - vac).]
urhile
utan
4.0 =
(e-4 vse tairs)
vail-e-toiuste toive
hals) e
or

2 hals)
C =
vse&s[e #v(Vers+ Vau) - e-t. Vi var - Viva)]
baxis
VOL
Ulx,S) =
Perius (1-04
Substituting in (10)
hais) {{1+1/2) (e
step by uns
Veious le tre / 1 + 2) - e tus )
(1+%e*(-Venta de la v
et 17 (1+ venti)-e-Hvoli - letgo
Vs
}
or
e
-
13/14 1 te Eus th 2
ettes (1+Vmi) - e *(1-)
14) U14s): hats)
E) +€ ***(1-vesti
0*b
234
Substitusting the expression for C in (11):
-
2 e
15) W1*,) = bers)
) U)
}
>b.
Haris (1+
(14) -ehv (1-
Now, define
2
dimensionless
constant a
1-15
n =
1+ ve
Note
thet since (i - zvezde<(1+2 Vita),
or (1 - Veh) ² < (1 + 2)? then it follows that

22 <1.
Utilizing this
constant, equation (14)
may
be
re-written
s.
haps)
bers
e
21(x,s) =
- DV
+ 2e
te hou
ter vs ci-de
)
hals)
Menu
fe
- This
26 *VS
ta e
| -de
2hrs
-}
The development to this point is very similar
to that presented by R. V. Churchill in operational
Mathematics Engineering,
pp. 122-124,
slight changes
in two of the boundary
conditions. The next step in the development
in
235
the
follows from observation that since lalal
and /s/a. So, then
2 bn ro
Šthe
[1- de er ) "?
2
Now, the exponential terms here involve the square
root of a complex number, s, so this formal ex-
pansion must be verified. This is best done by
verification of the formal solution in Wla,t), ob-
tained from the inverse of the expression in relx,5).
See R.V. Churchill, Phil. Mag., Ser. 7, Vol. 31,
pp. 81-87 (1941). Here, this expansion will be
verified by demonstrating that the solution
in W(x,t) reduces to that for the isotropic
opaque solid when a=0 (r.e. when Hi= M2).
Note that the
expansion is valid for
do under the usual
usual definition o'z 1.
Equation (14) may now be written :
20btx vs
Ënnt - 2560p) - XV
U (xs) =
- }

abore
hacs) Še - Born
+
or
O$**b.
(16) M(x,s) - Van
her {**+ * * [
emate ]}
C
Similarly, equation (15) becomes
236
•P3
Ver
RE
ta
26
uix,s) - bass i TV4) - he )
VS
24hrs
8
*>b.
2 ha(s)
vi(1+1) 75
3Vo
{
e
But
2
1+ 14
it
(1 + Val Mi) + (1 - Mad )
1 + MINT
hence
ha(s).
(2n+1)]Vs
17) W1*,5) =
- Internet
VI
* {lia) & 1
008 },
IS
$,x>..

Equations (16) and (17) now involve only the
form [(s)- e-AV5] (820), the inverse
transform of which
found immediately
(R. V. Churchill, Operational Mathematics in
Engineering, P. 299, No. 84)
h²
may be
VITE
convolution, the inverse transform of (16)
Thus, using the
becomes :
Ulvit) om St Haft - 4) { exp[-t.) + & A" (expl- fandt 424)
+ expl-(2n6 = 2*)]} dt, osxab,
237
while that of (19) becomes:
ce
U (2,4)= 7. S. Helt * {1+2) ŠX explain the 201 } do,
de.
X>6.
These unwieldy expressions could be handled by
machine methods, but for present purposes it is
preferable to specialize the problem by considering an
irradiance step function :
Halt) = otro
= Ha, t2o.
Then ha(s) = Hals, and the terms in (16) and
(17) take
-AVS
the
[
of this transform i's
e
S/
20exp[-*) - Bertelse)
- VE RITA)
Cena
R(A)= va [te-**-xerfe(x)])
in Section 3.3. (See Appendix III.)

where
defined

Thus equations (16) and (17) become:
ie) costo (5,y)= 2 de Fafe (ia) + Žac Presentantes
+ Rahman
Relevant to try} , osxeb
Canvas
,
238
and
19
) custer (2,t) - Bohem VE (1+2) Ž 1 Review the end
16
*>b.
SELLER
Note that if a-o- in which case
in which case a, 542 za
equations (18) and (19) both reduce to
Uster (x,t)/..- VF Rovan),
lachich is the correct expression for the isotropic
opaque solid. Also, it bos, equation (p)
reduces to
Us**(**) 2 /
He Ve Rllante), x70

*xtloc
xzo,
VTM
since
R(oo) O.
Consider now
the case where xso. From (IP)
20) UStr(0,8) - everything we [1+ 2 XR (M)].
As stated above
But
VER ( 1 ) = 17 Lt {
(A'*}
[VER (CAF)] = L"{s. She is so ease }
van
VITE
while [VE R(A)] 2VFFE [VER(e-
239
Hence,
21) du step (e,t)
ave [1 + 2 Šte**
which is identical to equation 3-27 of Chapter III,
there derived
The two limitting cases
du sree (oo) 2 Ha
art
Vipi
and
du step (0,00

2 Ha
VTM
seem so eminently reasonable, that the formal ver-
ification of equation's (18) and (19) was not made.
This, in the favorite text-book phrase, will be left as
exercise for the reader.

240
APPENDIX V
Solution of the Diathermanous Solid Model,
Double Exponential Absorption
:
Statement of the problem:
dp >6, +30
1) U+ (x,x) = x Uas (x,t) +
&
2) Wt (x,t) = of Vwx (x,x) + HACY) embe -volt-o)
3) u (x,0) = 0 , *20
4) U700,+):0,
1
5) Ux10,+) = otro
6) W(6-0, t) = Ulbro, e), tao
7) Ux(b-o,t) = Walbrot), t20

H 서
​Taking Le { }, and using (8):
() sulx,s) - a very reason helme *
& hals) e or otxeb
2') s U 11.5) = a c'hlas herede
& hals) erbe orlead), +06
4') N 100 s) = 0
s') dulos)
dx
6') w16-0,s): x16 +0,5)
7.) dulo-os) : 036670,)
The homogeneous solution of (1) is
UH(2,5) = A e-
+ BE
Bet
201
M:
to which must be added a particular integral. Assume
this has
the form
2* (x,$)ME
d²uclos)
Then
det
Via means and substituting
in
* Me
halso e
forma
ca
20
S Merit
ON
a hals)
The complete solution of (10) is thus :
8) Ulx,s)
A e
& haes) e
78 - 726)

es
ਹਰ
+ Be**
sa
otxeh
For equation (21) the particular integral
will have the form
Uplx,s) = Ne -P2 (told
hence
SNe -role-A) - 12)
**
and
N = 12 hals) e-rib
(s-rex)
28?a Ne Plood
It halede balk-6)

The
complete solution of (21) is, then :
9) 21x,s) = cek ve + De verhelshembento (wa)
v (s-86²)
*>b.
Differentiating (8) and applyang (54):
0:-VA + B - 2 )
8,3 hals)
P(
582)
hats)
B = A+
vis-8,2x)
*
242
Substituting in (8):
w1x= e+ shars)
V(S-1)
sxe ab
.
Now, from (4') it follows that in equation (9), Dro.
Thus
) wers) = ceva rohas) e-riben(6)
x>b.
(s-na)
The two constants A and Care
now to be
evaluated by applying the continuity conditions (6)
and (7). From (6)
6:
- $
12) Ale #1

hals F-
teorib
re
-]
and from (71) :
is) A(-e-Motte tors) + ce-tors
haces, memometiment
-)
Sorex
s-r*
serole
let
Solving by determinants
lehetet -1
(-e this te for
3)
%
to
e
+1
=
2 2.
213
Then,

[
是
​A: 12
als
2
急
​nb
1.
$
+
“”
)。
[5 sale
: sear Servernal an]。”
2,
:

T(6 - re), (E-rss)。
oris 急性
​stser tw) (5 - new)
an
“
and
對
​计
​Sofw
G= ha(s) a
AV
ret) 修​。
( 4 *)she
-b
ne
sor

hars)
(S-YR), (S-W)
St-tim) WS (S-r-or)
到​身-
​中
​2》
小
​)
+/ Metring) - 8/Fire)
vas-re
Mrs.tw),
- - 12:
me 2Fn.
7
Sent

Substituting for A in 110):
v(pc): hats) as-ruki(S-
S(ser *t) S(ser, k),
“nia
](
2
设​e)。
+256(s)=?
- 2 Esse
*”}
+ nein
AP
ܢܐܢܐ2
14) U(x) =
ha(s)
da,
S-x²x
?
V5 (S-rew)
tale
***)(ek het ravan
2
Vs(is travel is (verit
o sx ab.
Similarly, substituting for a in
VE
15) 2(x,s) = hals) forebroch-6)
V

Vs(s)
Xb
(antes
2
(15+ r)
valvs+VIVA
ma) a *****]}
'ava)
SIE-VB) V(V-VI)
river)
>>b.
ame
Equation's (14) and (15)
the complete solutions
in U(x,s).
To
approach the inverse transform, we
turn linmediately to
the special
where Halt)
is
stee function
case
a
Halt) =o, teo
- Ha, tao.
Thus hacs) = Hals, and noting that
en
- VS
Ire
15 (5-8,²)
V5 (V5 + 0,00 2 J5 (vs-rive)
8,2 Vale -
245
then (14) and
(15) ay
ay be
written
as;
Ha
16) 21*,5) =
en
27, epid
Sls-02)
S v3 (167)
SECUS rin)
erib
оскар
காம
erib retro
SVE(15 #riva)
sv3 (ề +86)
* nebo
erib
S13 (+TVA)
-
STS (VB *r)
oskab,

Fibo a Pala-6)
Ha freeribe
20 / 5ls - r, *)
17) U1*,5) 3
-
29.09
Arie
retag
SV5/13 +882) SVE(vs-1.4)
-eribri e
SVE(V3+, V)
e-ribne
SV (13-)
VO
3
Voc
tee
(सको हार
游​5
Sus(ve & P Q )
e-ribne
SV (15 - FL V
-
5}
x>b.
term
by termi
Now, we attack (16)
1st Term :
{
(sor.id)
}= 20, e-r* L { to satie?
2re**[******
eriale

= 2ries
da
2
next
e-PF
Sus(vs+a).
The remaining terms all involve the form
A partial fraction expansion gives
216
AVE
fe
SVS (ista)
az 15 (sta)
a ²5
a s 3/2
Thus, for the
2nd
term
eri
عطع- مم
Sus (Us + river)
}
er fe Winte trovare )- erte (DART)
ric
+
3 14 1
-
Žerte (2)
30 termi
vf
L's de
SP (F3 - rita)
]
Si
erte (rivat) erte (257)
*

VE
SONID
7
* 2 ertel;
avat
4th termi
bet ne vs
ribre
Sus (Vs +8, 82)
r. (b) 8,2+
L"{en
e
ria
e
erte (the hard to 7)
orib
-btch?
opib
ae
4
e
erte
2127
+
e
e
Tib (b +x) erfe (bat)
5th termi
bers
3
orib ri(b-x) rihat
e
е e
rid
si (b-x)
be
2107
toivat
svs (Vs + riva
2
-hib
edib
here the
4
erte (bota
+
grb (bx)
erte
219
247
ave
low
hogy
- terte
TT
217
erte ( 2
GRAD
en
te heraldr), Part
The 6th and
7th/ terims
obvious from
the
4th and
5th/ above. The
inverse of (16) can
be written out
as
U (xt)
Ha 2
erteles Tori vart
)
200
*
erte lentet) + 4 -- 2x erte (
rivert) + 1 certe (zutant )
27
ont rifbtw)
erte (+ n Vap)
erfo (at) - 2e-mope
tembra) erte (***ter ertelements
Pilb) 2,20€
% 륭
​émertel om trivat)
- 2 er vete + " >
(6x) erte (62
2007
and solbro) neut
te e butterte le net revetemente che non
+ 2e ** yote perih (lote) erte (+7
exte(burtant to Jaz) - certe (en ang
&

METS
-66-x)?

- 2
2V
(62) ²
+ 2e"#e
2 Vatt
Osxab.
248
This
an be
ins
X
cleared up extensively
to give :
18) Ulxit)" # $2V ertel (a ment) er
erte (river + 2) + mertefriter-
CORO
28,
+ e
2
erte
( 12
merfel (
ertelementi V77)
SI
el**)
ertelt ravar)
enfe (her than & kitap)
ertel 2 to 20
rust)

82
OsxLb,
where the identity
erfe(x) = 2 - erte (-x)
has been employed.
The inverse of (17) may now
be written
out by
inspection
W(wit) = f ze moeten
-r6 r1(2-0)
-06 -2 (4-0)
2
2e e - 1 er te laten
end
+
erte G + Oi Vat) + 4/5
8
nar
- 2 x
erte ( 2 maart
.
ertelenrivet)
+ 1 ente (e mart) -
riba (+b) neut
eerte
(en'+)
249
661²
tenerte om te - 2embrie
FE
(*+b) erte le certe list
tib r2(a+h) nhat
e
ertel xth or toe vat).
to 2 e
***
(*+b) erte (
x+12
2 UMT
-hb
pribo
orbriz-b) visit
e
e ertel
ertel van ons vot) -ente (open)
:-2 em
e
+ e-role-b) erte larta)
-rb-82(x+4) sikt
ertelette Bivat) te serte for the
72
2
+2e"hbet e
erio (x-blerte (rotor) }
*>b.
be done
Again, considerable canceling and regrouping cam
to give :
-(8-72)6 -ra
19) Ulrit) =
=
é*
-(t-tlet

hid
ra
+
+
fa
* lemerte (avat tutt) + emerte (orvat htt)]
& K )
erte (tent+ erte laten)) (+-ta)
* (**) riet
erte (2 + 2, 007)
e
(com')
250
terzibre), but
erte (bot + 8 2 007)
ertel
2 var
éricbatteriert
van + rivede)
che lo perdist ertele nu pavar)}
x>b.
>
Now, we define the dimensionless
define the dimensionless variables
à, = sib
o reat x₂ = 81/82,
PF) = U
kriu (8/8, 9/8,20),

and
may
where upon (16) and (19) immediately be re-written as:
20) 119,0) = 15e-to' - Serte (fe) - es
+
+
[e'erte (18+ sfo) te fertelvo --S)]
'[(erte( t) + erteller
2)
(1-22)
mit e 'erte ( ** ** evtelen
et-se ertelle + autor) + dze er
207
a-
er
ossada
and
251
e
21) 715,0) =
es
şerte (fr) -en-[nile
**(21e **-1)+1]
+ [ este (Vat fe) te fortelvo - stal]
exte ( 21 ) + erte
+ erte
site evertel vo + 11 + r) tize
the fierte(* ***)
eertelen met de inte ferteldar stand
(1-22)
2
е e
>di.
Equations (20) and (21), the complete normalized solutions
for this model,
to be identical to equations
(4-22 and
(4-23) of Chapter X. For further disa
ave
seen
see
p. 77
cussion
these relations,
et seq.
The next task is to derive the expressions
for du(x,t) /at. Noting that
deleted "{s u(x,s)}

it follows from (16) and (17) that for the step-
function
form of Halth :
- tur
dulritha Ha Laser
20
sora
US(vs rival
of
tetap
re
(1 + rv)
Us (US +852)
VS
be beter as
ne breite v
+
5 :]}
Osxeb;
is (rs + rata)
US (US +822)
252
triptichanderin't ente (rivar & tun)
- 2r,e "e+ ententerte (rivat - 2)
hence,
Ou(x,th Ha
2re
erte (rivat + at
2 vot
at
2
II
+ rie
rerit
*
2 var
tedio
+ r. (6) deat
* -r et com a
exte (rivat +
betet
2 vit
** 10-x) rint
erte(river & bon
+ e***(**) *u* erteltava + potente
26-mediat erfc (rotoct + *)]]
trze tr2(6x)
or

22) sulat) {riety ** [em" erte(r wat * *}* e***erte (vat-px;)]
-em[riem com um tot erte (rivat + niet ***
-reseta)ematertel vivait )
-roetelom sveterte (rukt to be t)]}
,
Oxb;
and,
253
23) əwixitha
Ha
27
{2010
riberalk-b) 2²at
tripvixentererte(FIVAT
+
X
21&t
-rie-rine
erte (arivat tatant )
-en[riemero
zlato) 8, 2018 erte (Givort &
x+4)
VO
-bile-6)
nie
77
erte (river that
emitertel osvort +*
-rela)
2107
-+*(x-1)
482
ertel-vztat + ) ] },
q>b.
Now, note
that
Lim erte (x) = 2,
X 00
and
Lim
ertelx) = 0.
X400
MO
be
evaluated in the
as
zero
Equations (22) and (23) may
limit
t approaches
Qulx,)
0Ha
gt
2V
from above. Thus!

Hare-rix
osxeb,
and
- r6 -72(x-6)
25) en
QU(x, 0+) = tem { 201
Hq
Hartenbe (*)
.
*>b.
>
iden tical +. (4-24)
of
These two equations are
Chapter I.
254
the
as.
-re not erte(22 URT :
I+ is interesting to note
following rather
startling situation. By employing again the fact that
erfax) = 2- erte (-x), equation (23) may be
re-written
au (x,t) He -ribor (b-x) diret
tree" erte (rivatp + +
at
20
. 28,0 t + dietin mut erte (bivortata)
rifbor) rinat
e* (p.exboms erte (binar & btt
+ *)
rico-x) at
tre erte (rivit +
-
*2 lbt) ***t erte (revent &
at bu)
+
ario rib-x) Frat
+24e e
}
Que
box
2168
rze

Sement
282ebre(boot) rout
the frie** [e***erte (rivat usein) *e**erte (rin
-erib [or, en (bral o et ente (oitett
)
bt
2187
rillo-x) ixt
tre
82(67%) sihat
-re е e
erte (tivet + bare
erfe (sabet + b )
-remolemeleri Pat erfe(WW2F +7)]}
*>b,
which is
the
identical to equation (22)! Thus
ou (x,+)/ot is
the
form
for
same
the
255
two regions, osxel, and xob, although the limitting
values (as to from above) are different. Further,
reference to equations (24) and (25) above shows
that
Lim Dux, 0+)
*
Lim ow (x,0+)
Xbt
In fact, it
fact, it can
in either (18) or (ia)
shown by substitution of x=b
and direct differentiation wor.t.
t
that
awlb, 0+)
Lim awrrot)
-js
Lim au(x, ot)
+
)
]
ál
Heems ()
that
this
sub-
Note
expression cannot be
derived be
stitution of x=6 in (22) (or equivalent (23)),
since is faced with the indeterminate form erfs ().
the
One important task remains. The solution of a partial
differential equation by the method of the Laplace trans-
form must always be justified by demonstrating that the
solution satifies the original d.e. and the boundary
values. In the present case, with
the above - demonstrated
discontinuity in the first partial woont. time, this is particu
larly important. This "brute force" verification is actually
considerably more lengthy and involved than the establishing
of the solution itself, which is usually the case.
Hence,
here, it will only be sketched in outline. The complete

256
verification is
on tile in this laboratory.
It is convenient
to revert now
to the normalised
Solution I in
Note
that
du
뾽
​content
ha
kri
2
08ht
sota
nitk. Ha
RS
Vi har et
,
and
2²u
2
- Bone
]
alu (49)
을
​2
A
K
The original de
de and boundary conditions
may
be written as:

OSSEN,
26) Solsel: G? TES)
27) ), >
28) (8,0) = 0
29) 7 (0,0) = 0,20
30) 20,0)
30)
, = 0,
020
A> O
/
31) 7/2,-0, 0), report (to, )
32) Orlar-eol = 02 (2+0,6)
ܘܨܣ
That the form of a given by (20) and (21) satisfies
the last two continuity
conditions
demonstrated
in Chapter 7. We
turn to (26) above
and note that (20) becomes
may then
257
V (8,0)
e-s+
iles.2] = 0, 05&a,
gives
while (21) gives
115,0) = - @[dzle **-1) +
- es tenidze
to
este (1-1)
VODOTTI
dze dne
O,
>a,.
Hence (28) is satished. The verification of c2a) is
in ore difficult, since one must first demonstrate
that Lim & exte(x) = 0, and also
Lim e erte (x)=o.
a trifle
X
Thus
the son vertelu): in fiende
afo
Le bonne
ne-
and Lim exerte (x)= Limba enda
ola
Lim
@m?
22
Then, from (21)
(0,0): -e- (1-22) + em (10m) = 0
-) ,
and (29) is satisfied.
SO)
Next we
This is a
tedious process, at best and only
the final
result will be given here. .
OT (5,0). ef-este () + [eberte (vat hal-e értelva-)]
tiene un centre este luotetieteet ontelimon )
tetteberte (va t***-esit e tiendelet )

25
+ܝܐ
of&cdi
258
hence
>$
OT 10,0): 1-1+ [erte (va) -ertelvo)]
+ [-e'eerlaattia) te te fortelle eine
tenerente (va+fte)-etenimentele ]
=o,
and (30) is satisfied
Finally, the second partial wor.t.
found
to be
autem
5* 196.0.-0'+{fle erfelbo+Alte Pertel-6)
tema [-e**bertel van
ertelen av de
-edmeertelvet domingo
ertellent

+
210
pentru
22
920, osad
and
OTISH). - e
***. , "=
{G(5,0)} 0?0, 8>2,
Og
where G(5,0) is
is the
the expression in brackets in
the
expression above for of fade. Now, inspection
of equation (22) the equivalent (23) reveals
that
(90) - G (5,4) $70, #20,
where G(5,0) is as defined above. Thus, substituting
259
in
(2) anal (20)
G(5,6). Les+G(9,-) tes
e
and
for the
+G(50) + entre
Q.E.D. !
that
an even
more
It might be noted in passines
interesting problem is that of
the diather manaus
composite solid with double exponential absorption,
where
8:00
fo
otxeb and
v=V
V:n
kiko
***
x>b.
leka

This has so far resisted a few not overly vigorous
attempts at solution. There is also the rather
rather in
pleasantly messy but otherwise straight forward problem
of tripls exponential absorption. However, as noted in
Chapter I, the first probleon which must be next
attacked is that of an opaque thermo element imbedded
in a diathermanous material
Siste viator !
260
ACKNOWLEDGEMENTS
The author would first like to express his profound gratitude for
the lively interest and continual encouragement of Dr. H. E. Pearse,
Department of Surgery, and Dr. William F. Bale, Department of Radiation
Biology, in the prosecution of this study.
The time-worn phrase, "Without
their assistance this paper could never have been written,
has become so
trite as to be almost devoid of meaning: in this case, however, it is quite
literally true.
For their support, the author is deeply grateful.
Of equal value has been the direct assistance and moral support of
the writer's many associates in the Flash Burn Section of The University of
O Rochester Atomic Energy Project; to each of these associates, past and
present, the author acknowledges his indebtedness.
In particular, mention
should be made of the early proposal of Dr. H. D. Kingsley, formerly Chief
of this section, that a study of temperature response of radiated skin should
be undertaken.
It is hoped that this paper, coming some nine years after
this proposal, will indicate to him that we are at least beginning to imple-
ment his suggestion.
The researches of Dr. J. R. Hinshaw, present Chief of this section,

have not only paved the way for this study, but also have served as models
of precise and thorough experimentation in the field of radiant energy burns.
Indeed, the entire problem of the prediction of radiant energy burns, of which
this study is a small part, is based almost entirely upon the work of Dr.
Hinshaw.
It is a pleasure to acknowledge the advice, assistance, and en-
couragement of this surgeon and scholar.
*This paper is based on work performed under contract with the United States
Atomic Energy Commission at The University of Rochester Atomic Energy
Project, Rochester, New York.
261
It would be a grave error to omit the name of Dr. George Mixter, Jr.,
from this listing of those who have assisted in this study. Formerly as-
sociated with this section, Dr. Mixter pioneered in the application of
analytical procedures to the extensive experimental results he obtained.
It
was largely through the close but too brief association with this ebullient
and astounding intellect that the writer was first attracted to the inter-
disciplinary field of biophysics.
The immediate co-workers of the writer, in the physics laboratory of
the Flash Burn Section, have contributed extensively to this study, Dr. R. M.
Blakney, formerly in charge of the laboratory, and L. J. Krolak were largely
responsible for the initial development of the carbon arc source employed for

all the experimental work reported herein.
Mr. Krolak also performed the
careful spectrophotometric measurements necessary for the proper interpre-
tation of the experimental results.
The author's present associates,
J. A. Basso and Mrs. Marilyn Aldrich Guck, have been of constant and in-
valuable assistance throughout the entire program.
Much of the specialized
equipment used in this study was constructed by Mr. Basso, who also is
responsible for the calibration of the carbon arc image furnace used as the
radiation source.
Mrs. Guck assisted in all the experimental work, and also
carried out the initial data reduction.
Mr. Basso prepared the figures in
this paper, and with Mrs. Guck proofread the final copy. Truly it is im-
possible to acknowledge adequately their aid except to say once more that
without it this paper could never have been written.
It is pleasant also to acknowledge the invaluable help of Mrs.
Cornelia Winslow, who labored long and conscientiously in translating the
manuscript into readable typescript and prepared the final product. The
author is particularly grateful for her finding and correcting many errors
and inconsistencies in the original copy.
262
Finally, it is a pleasant but impossible task to attempt to acknow-
ledge properly the help of my wife.
Certainly, her patience has made it
possible for me to devote so much time to this work.
Even more important,
her interest and enthusiasm have made it all the more interesting to me.
Most important, her love has made it all eminently worthwhile.