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Report SAM-TR.77-38
LASER-INDUCED THERMAL DAMAGE OF SKIN
SA. N. Takata, M.S.L. ZaneveWd6 D.V.M., Ph.D.W. Richter, M.S., D.V.M.lIT Research Institutow10 West 35th Street.C hicago, Illinois 60616Oi
S.", December 1977
Final Report for Period September 1976-April 1977
S[Approvod for public release; distribution unlirnitd.d
Prepared for'
USAF rCHOc OF AEROSPACE MEsDICIN(1Aerospace Medical Division -(AF$C)Brooks Air Force -as., Texas 7$235
NOT!CES
This final report was submitted by UIT Resear,-h Institute(IITRI), urnder contract F33615-76-C-0608, job order 7757-02-5,.with the USAF School of Aerospace Medicine, Aerospace Medical , *41Division, AFSC, Brooks Air Force Base, Texas. Captain Alan R. . ,.Mertz and Captain Kenneth L. Schepler (SAM/RZL) were the Lab-oratory Project Scientists-in-Charge.
When U.S. Government drawings, specifications, or a.herdata are used for any purpose other than a definitely r:late,Government procurement operation, the Government thereby incursno responsibility nor any obligation whatsoever; and the fact thatthe Government may have formiulated, furnished, or in awv w"y suppliedthe said drawings, specifications, or other data is not te be re-garded by implication or otherwise, as In any manner licensing th(4
holder or any other person or corporation, or conveying any rights or Apermission to manufacture, use, or sell any patented invention thatmay in any way be related thereto.
The animals involved in this study were procured, maintainee, atidused in accordance with the Animal Welfare Act of 1970 and the ý'Guide 4,for the Care and Use of Laboratory Animals" prepared by the Instituteof Laboratory Animal Rnsources - National Research Council.
This report has been reviewed by the Information Of-ice (OT) andi Is releasable to the National Technical Information Service (N%'7S).At. NTIS, it will be available to the general public, includinc, fore'yn "ore
nations.
This technical report has been reviewed and is approved forpublication.
ALAN R. MERTZ, Captain, USAF Major, !%AFProject Scientist I Su srvis;r
KENNETH L. SCHEPM aptatn USAFProject Scientist
ROBERT G. MWAVERBrigadier General, USAF, XCComiander
Editor: ANGELINA DAVIS Supervisory Editor: MARION ., fIEN , A
SEC LRTV StSIF1CATION OF THIS PAGE (When Date Bntorr~d)____________________
A., -.:TLE (and Sub.100 S.Tie)ODCVEE
,?AMGE OF,,;KIN ,/Sep9 3@76 - Apri 1077
L./Zaneveld .V.M.~, Ph.D. (Univ of III Med Ctr) _
W./Richter M.S., D.V.M. (Univ of Chicago)
IIT Research Institutei RA&WR j!NMEý
10 West 35th Street 6?$ý
Ciaerospc, eia Divisoi on 6066 FSC)5
11. CONITRORLING OGENCE NAME AN ADDRESS(I dtfr~tioCnrltjO I2c) 5.SEURIYCASS.(o ttl r-7J
USAF School of Aerospace Medicine (RZL) / Uclasified177Aerospace Medical Division (AFSC) _________________OF__PAGES
BrosA~ir Force BasepTxs725_9d~ ~~I. ELSIFCTO/ONRDNNG. AGTR EINC STAEMEN (o, AVhie S Repnrolnoffc)I.rEUITtLS,)o hs opi
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17. 0 &3TRIBUTION STATEMENT (of the abstract entered in Block 20, It different trom Report)
"It
IS. SUPPLEMENTARY NOTES
.~..*19. KEY WORDS (Continue on reverse eide it necessary and identify by block number)
Skin BurnsSkin ModelLaser Effects
\Thermal Predictions
20 ABSTRACT (Continue on reverse side it necessary and identify by block number)
A computerized model was developed for predicting thermal damage of skin bylaser exposures. Thermal, optical, and physiological data are presented forthe model. Model predictions of extent of irreversible damage were comparedwith histologic determinations of the extent of damage produced in pig skin
by 7,ad ruby lasers.L
DD FJA, 1473 EDITION OF' I NOV 65 15 OUSOLTICEUCASFEt SECURITY CLASSIFICATION 0F THIS PAGE (When Data Entered)Ale ~~7
TABLE OF CONTENISPage
INTRODUCTION . . . . . . . . . . 5
DATA NEEDED BY THE SKIN MODEL * * 7IITRI Measurements of the Optical Properties
of Pig Skin ..... . . . . 8Optical Properties of Human Skin and Water 17Thermal Properties of Skin . . . . . . . 18
Specific Heat. , . . . . . . . . 22Thermal Conductivity . . . . . . . . 22Density. . . . . . . . . . . . 23
Blood Flow . . . . . . . . . . . . . 23Heat-Transfer Coefficients . . . . . . 24
Surface of Skin . . . . . . . . . 24Steam Blisters . . . . . . . . . 25
Damage/Burn Predictions . . . . . . . . . 26Extent of Thermal Damage . . . . . . . 26Degree of Burn . . . . . . . . . 26
SKIN MODEL . . . . . . . . . . 27Laser Exposures. . . . . . . . . . . . 28Grid . . . . . . . . . . . . 28Assignment of Properties and Energy Deposition
Rates . . . . . . . . . . . 30Heat Transfer . . . . . . . . . . . 30Thermal Damage Predictions . . . . . . 33
PIG EXPERIMENTS . . . . . . . . . . . . . 33Optical Setups and Measurements . . . . . . 34
CW CO Laser 34Ruby Laser 37
Animal Procedure, Autopsy, Histological Procedure,and Results . . . . . . . . . . . 37
Histology Thicknesses/Depths of Skinand Other Media . . . . . . . 42
Qualitative Effects of Excessive Heat 43Histology Procedures and Observationa 43Dimensional Changes of Skin Specimens 44
Burn Results . . . . . . . . . . . . 45
Criteria for First-, Second-, and Third-Degree Burns . . . . . . . . . . 51
Criterion for Fourth-Degree Burns (SteamBlisters) . . . . . . . . . . . 52
Criterion for Fifth-Degree Burns . . 52Comparison of Predicted and Experimental Radii/
Depths of Irreversible Damage . . . . . . 52C02 Laser Exposures . . . . . . 52Ruby Laser Exposu-es . . . . . . .. 53
SUMMARY AND CONCLUSIONS . . . . . . . . . . 55
REFERENCES . . . . . . . . . . . . 57
APPENDIXES
A. FINITE-DIFFERENCE METHOD USED TO CALCULATE TEMPERATURES 59
B. APPROXIMATE TECHNIQUE FOR REDUCING COMPUTATIONAL TIMESOF LONG PULSE TRAINS . . . . . . . . . 75
C. SKIN MODEL . . . . . . . . . . . . . . . 85
LIST OF ILLUSTRATIOI1 SFigure Page
1 Schematic of Integrating-Sphere Reflectometer 92 Optical Data for Undamaged Pig Skik . . . 103 Optical Data for Thermally Dar.aged Pig Skin . 114 Optical Data for Pig Fat . . . . . . . . . 125 Linear Absorption Coefficients of Fair Skinned Humans
from in Vivo Measurements . . . . . . . . 156 Spectral Reflectance of White and Negro Skin . 177 Transmission Spectrum of Human Epidermis , . 188 Radial and Axial Grid Poinkts and Increments 299 CO2 Laser Setup and Resulcant Burns 35
10 Optical System for Controlling Animal Exposure 3611 Profiles of Unexpanded "02 Laser Beam . . . 3812 Profiles of Expanded C02 Laser Beam . . . . 3913 Optical System for Ruty Laser . . . . . 4014 Profile of Normal Ruby Laser Beam . . . . . . 4115 Radii and Depths of T amage Producld by C02 Laser
(1.53 watts, beam radius at l/e1 point - 0.710 cm) 4816 Radii and Depths of Damage Produced by CO2 Laser
(1.97 watts, bean radius at i/ez point - 0.383 cm) 4917 Radii and Depths of Damage Producid by C02 Laser
(1.55 watts, beam radius at l/e1 point - 0.383 cm) 5018 Temperature Predictions Due to Ruby Laser Pulse. 54
A-1 Radial and Axia'. Grid Points and Increments . . . 61B-1 Scheme for Grot-ping Large Numbers of Laser Pulses . 76B-2 Structuring PuLse Train (40 pulses) Using Mean Power
to Conserve Execution Time . . . . . . . . 83C-1 Flow Diagram of Skin Model . . . . . . . . 87
2
LIST OF TABLES
Table Page
1 Experimentally Determined Absorption Coefficientsof Various Pig Tissues . . . 14
2 Summary of Optical Properties for Various PigTissues . . . . . . . . . . . 16
3 Optical Properties of Human Skin . . . . . 194 Absorption Constants of Water . . ... 205 Histology Results from CO exposures (1.53 + 0.04
watts, beam radius at 17e• points - 0.710-cm) 466 Histology Results from CO9 Exposures (1.97 + 0,07
watts, beam radius at 1We2 points - 0.383-cm) 467 Histology Results from CO2 •xposures (1.55 + 0.07
watts, beam radius at I1ei points 0.383-cm) 478 Radii of Red, White Burn Areas Prior to Excising
Specimens (CO2 Exposures) .. . . . . 519 Comparison of Predicted and Experimental Damage
Radii and Depths (CO Laser) ... . 53B-1 Ratios of Approximate io Exact Surface Temperature
Rises (T/TO - 0.1) , . .I. . .G . . . 79B-2 Ratios of Approximate to Exact Surface Temperature
Rises (T/TO = 0.001) . . . . . . . . 80B-3 Ratios of Approximate to Exact Surface Temperature
Rises (T/TR - 0.0001) . . ... 81B-4 Programming Mean Powers" into Noncoded Pulse
Trains . . . . . . . . . . 82
ACCESSION forNTIS White Section
DOC Buff SectiontUNANNOUNCED DJUSTIFICA'ION ............
... ...... I .................................
BYByITRBTfIUN/AYAI A01IJfY •UES
3
M
LASER-INDUCED THERMAL DAMAGE OF SKIN
INTRODUCT ION
Numerous experimental investigations have been conductedto determine the degree of burns produced in skin by a vari-ety of heat sources such as flames, hot water, and radiation(16,18,21,22).
Studies by Henriques (11), Stoll and Greene (21), andTakata (22) have used such data to develop burn criteriabased upon transient temperatures. Moreover, a number ofskin models exist (15,21) for computing skin damage causedby simple heating conditions--one-dimensional, surface ab-sorption, etc. Unfortunately such models are not capableof handling a wide variety of phenomena. This is particu-larly true with variable spacial and temporal heating pro-duced by lasers. What is needed is a model capable ofaccounting-for two-dimensional heat deposition/transfer,blood flow, spacial/temporal variations of tissue proper-ties, hair follicles, steam blisters, and evaporation ofwater. In this regard the Corneal Model (23), developedfor the USAF School of Aerospace Medicine, Brooks Air ForceBase, provides an excellent basis from which to build sucha model. It is with this goal that this study is directed.
Specific objectives of the program are listed below:
a develop a comprehensive computerized modelcapable of predicting the extent and degreeof skin burns produced by lasers
9 identify data required by the code throughliterature searches and experiments
* conduct laser exposures of pigs and compareresultant damage with model predictions
The SKIN MODEL was designed to predict transient tem-peratures and thermal damage produced in skin by any radiallysymmetric laser beam of normal incidence.. Basic to the modelis the use of an implicit-explicit finite difference tech-nique for computing transient temperatures. This techniquewas originally presented by Peaceman and Rachford (17).Since then it has been applied to cylindrical coordinatesby Mainster et al . (14), and used by Takata et al. (23) topredict transient temperatures produced in eyes by laserirradiation.
5 -
Ia
Additional features providad in the SKIN MODEL aretemporal changes in optical/thermal properties and thermalbarriers created by steam blisters. Provisions have alsobeen made for hot spots created by the interception ofradiation by hair follicles. As with the eye model, ther-mal damage is predicted using Henriques damage integral(11). This criterion involves integrating temperature-dependent rates of damage with respect to time. Irrever-sible damage is predicted when the integral equals orexceeds a given value. The region of irreversible damageis predicted by evaluating damage at various depths andradii.
Data for the computerized model were obtained fromthe literature as well as from IITRI experiments. Thesedata include thermal and optical properties of skin tis-sues, tissue densities and water content, blood flow rates,heat-transfer coefficients, and criteria for predictingthe degree of burn and blister formation. Specific ex-periments conducted by IITRI are presented below:
"* measurement of optical properties of excisedpig skin
"* measurement of heat-transfer coefficientsassociated with heat losses from moist andfrom dry skin to the surrounding air
"* exposure of white skin pigs to CW CO2 laser(nominal 5 watts) and to a pulsed ruby laser(nominal 25 joule pulses of 500-usec dura-tion and nominal 10 joule pulses of 50-jisecduration)
The pig experiments served to validate the model and toacquire additional data for the model. These data includedcriteria for the onset of blister formation auA the degreeof burn. Medical aspects of the pig expc't nts were super-vised by Dr. Larry Zaneveld of thec Univerr I, llinoisMedical Center. Extent of irreversible ¶ measuredhistologically by Dr. Ward Richter of -•y ofChicago.
Two-thirds of the predicted del ir-reversible damage were within one ) )f thehistological measurements of dama e 13 ro-duced by the C02 laser. The CO2 . xpL - veddifferent beam diameters, las-" pc p es. 1
1 4•
! • n . ',•
No damage was detected as a consequence of exposingpigs to single ruby laser pulses of 50 and 500 usec. Thisresult conflicts with model predictions and suggests ap-preciable energy was lost by some means not accounted forin the model. One possibility is attenuation of the laserbeam by materials evolved from the skin surface. Thisappears likely in that a pronoumced cracking sound accom-panied each exposure. Wisps of smoke or vapor were observedimmediately following several exposures.
DATA NEEDED BY THE SKIN MODEL
Prior to model development, a literature search -'.sconducted. Of primary concern were property data withwhicb to characterize skin and subcutaneous tissues such as
"* reflectance"* absorption coefficients9 thermal conductivitya specific heat* density
* water content* blood flow rates
Sburn criteria
This endeavor involved computer searches of the follow-ing sources
1) SSIE -- Smithsonian Science InformationExchange
2) Biosis -- Biological Abstracts
3) Scisearch -- Current awareness of primaryjournals
4) Compendex -- Engineering Index
5) DDC-- Defense Documentation Center
6) NTIS -- National Technical InformationService
In this section we shall present pertinent data foundin the literature along with IITRI measurements of the opti-cal properties of pig skin. In addition heat-transfercoefficients are given for predicting the rates of heattransfer from skin surfaces to the surrounding air andacross steam blisters.t I
a - :]i R~~q i l I iH II •ip'b
IITRI Measurements of the Optical Properties of Pig Skin
Diffuse reflectance and absorption coefficients ofvariouR skin specimers were determined by exposing excisedsamples of pig skin to radiation of varying wavelengths andmeasuring thel reflected and transmitted radiant energy.This determination involved use of the integrating sphereshown in Figure 1. Here measurements of the radiation fromthe sample or skin specimen are compared to that producedby thie reference beam.
To separate the reflected radiation from the trans.-mitted radiation, two power measurement3 were made witheach specimen. One measurement was made with a black filmon the unexposed side of the specimen to prevent escape ofradiation transmitted through the specimen. The othermeasurement involved the specimen without the black backing.
Three types of tissues were used, namely
"e normal epidermal/dermal specimens (5 differentthicknesses)
"* irreversibly thermal damaged epidermal/dermalspecimens (3 different thicknesses)
"e normal fat tissues (3 different thicknesses)
Skin was irreversibly damaged by placing a 700C aluminumdisk upon live skin. Contact was maintained for 1 minute.
All specimens were hairless and stretched in a sampleholder by approximately 12% in one direction to ensure aflat surface. On the live animal the skin is stretched intwo directions by approximately the same amount. Thus, thespecimens were approximately 12% thicker than when on thelive animal,
Percentages of the radiant energy diffusely reflectedand transmitted through individual specimens are shown bythe dashed curves of Figures 2, 31, and 4. The solid curverepresents the sum of these curves. Absorption coefficientswere obtained using Beer's Law namely
q - exp(-cxz)()
where q - frac:tion of absorbed radiant intensitytransmitted through the specimen
a - absorption coefficient -
z - specimen thicknessj
8
SPHERE
SAMPLE SAMPLE HOLDER MOUNT\ HOLDER•_•,
DETECTOR
PORT j REFERENCE BEAM
SAMPLE BEAM
Figure 1. Schematic of integrating-sphere reflectometer.
C%4
+
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CIO
.4J 4. -0
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Ito
iI I r4
-j -
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* 0.
-, 9 1
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LL
z 44-4
CNO!I!UY~ 1N31~N 10 ~1Y±33A
12>
Solving equation I for a yields
a - -ln(q)/z (2)
Absorption coefficients obtained from equation 2 are shownin Table 1 for 10 skin specimens over wavelengths rangingfrom 0.33 to 2.6 pm. It may be observed that the coeffi-cients for the two fat specimens are in good agreement witheach other while the coefficients for normal skin vary con-siderably. Particularly noteworthy are the higher absorp-tion coefficients with the thinner specimens of skin whichsuggest greater absorption (percentagewise) by the epidermaland shallow dermal tissues than by the deeper dermal tissues.Such absorption is less evident with irreversibly damagedskin.
The coefficients for the deeper dermal tissues weredetermined by the following analysis using the absorptioncoefficients for the thinner specimens. The mean value aoof the coefficients for the two thinner specimens shallbe considered to apply to some unknown depth z , which re-mains to be determined. Beyond the depth zo t~e coefficientwill be represented by a. Using Beer's Law, the fractionof the nonreflected radiant flux passing through a depthz greater than zo is given by
q = exp-(aoz 0+al(z-zo)) (3)
To determine the coefficient a for the deeper thermal tis-sues, we shall use the transmilsion data for each of thethree thicker specimens presented in Table 1. Each speci-men will yield a somewhat different a1 value according tothe value selected for zo. Basic to this analysis is thechoice of zo yielding the most consistent a, values for thethree thickest specimens. For each trial z, value, thedeviations of the individual al values from their mean valuewere squared and summed over each of the wavelengths con-sidered. This procedure was performed using a variety ofz values until the sum of the squares of the deviation wasminimal. By this means the most consistent a, values werefound to occur with a zo value of 0.054 cm.
Results of this endeavor for normal skin are preseptedin Table 2. Included in Table 2 are the mean values of co-efficients measured with irreversibly thermal damaged skinand with normal fat. In addition, mean reflection data arepresented for the thicker specimens.
13
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Use of two absorption coefficients for the skin is con-sistent with the measurements involving fair-skinned humanbeings shown in Figure 5 for wavelengths from 0.3 to 1.0 Pm.Beyond 1.0 Pm, Figure 5 presents only a single absorptioncoefficient and is in variance with data presented in Table 2.On comparing the results of Figure 5 with the coefficientsof Table 2 for the shorter wavelengths and the mean coeffi-cients of Table I for the longer wavelengths, observe tiatour values are lower across the entire spectrum. Thisobservation suggests that the skin of young pigs transmits
* radiation better than fair-skinned human beings.
300
v-4, 200
4. Y)1100-,-I4.
4 60
o 00 -
ro-
"V 20 " '9-4 60 60 - - " ,
S0 0.2 0.6 1.0 1.4 1.8 2.2 2.6Wavelength, um
iFigure 5. Linear absorption coefficients of fair-skinnedS• humans from in vivo measurements (ref. 6).
a 71 curve for superficial skin layers.- y2 curve-for deeper tissues
Before concluding it is important to recognize that theskin specimens were 12% thinner when on the live animal.Hence the critical depth of 0.054 cm presented in Table 2should be reduced by 12%. to 0.048 cm. Coefficients remainthe same.
15
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4-16
CIO
Optical Properties of Human Skin and Water
Figures 5 through 7 present optical properties of humanskin found in the literature. Figure 6 shows the reflectanceof white and black skin over wavelengths from 0.4 to 40.0 pm(10). Notice that skin pigmentation is important for wave-
lengths below about 2 pm. Beyond 2 pm pigmentation is notimportant.
WHITEIa I
50 ------ -;S~IL5
30 lr • -
S20 VNEGRO
1 0 - i - .-- - - --10VI SIBLE il L li .0.2 0.4 0.6 0.8 1.0 2.0 4.C 6.0 8.0 10.0 20.0 40.0
WAVELENGTH, Um
Figure 6. Spectral reflectance of white and negro skin(from ref. 10)
Figure 7 presents transmittance data (8) for a 0.003-cm
thick layer of wet and dry epidermis. IITRI calculationsof the absorption coefficient are presented at tie right of
the figure. Over wavelengths from 1.0 to 2.4 om, it may be
observed that the coefficients for thin epide•:mal layers
are much larger than the coefficients for the entire skin
presented in Figure 5. This observation suggests there
should be twd absorption coefficients for the skin--one for
the outer epidermal layer and one for the remainder of the
skin as was found for pig skin. Table 3 presents our best
estimates of the reflectance and absorption coefficients
of human skin based upon the literature, and IITRI measure-
"ments using pig skin.
[, 17
60 170
50 5.A.Dry
~40 -I -- 3305Li.
~30 40
kn- 0'M 20
Ie
537~~
10 Wet 0:
-768 Lj
0 -1 2 3 456 7 8 9 10 11
WAVELENGTH, Uim
Figure 7. Transmission spectrum of human epidermis(0.003 cm thick), Wet and dry (from ref. 8).
Absorption coefficients could not be found for skintissues deeper than 0.003 cm for wavelengths beyond 2.4 pim.In this regard we recommend using the values for water shownin Table 4 (7). This recommendation is made for two reasons.First, the fraction of the radiant intensity passing throughthe thin (0.003 cm) outer layer of epidermis is usuall~ysmall at the longer wavelengths. Secondly, water presentin the tissues has very large absorption values at wave-lengths beyond 2.4 Uim. The '-esult is very shallow penetra-tion of the radiant energy beyond the relatively thin epi-dermal layer. Epidermal layers are of the order of 0.01 cmthick as contrasted with values of the order of 0.2 cm fordermal layers.
Thermal Properties of Skin
The most important parameter affecting the thermal pro-perties is the amount of water in the tissues. Water isimportant in that it is an excellent heat sink as well as arelatively good conductor of heat. By comparison, the ef-fec ts of temperature upon thermal properties are- of secondaryimportance and hence will not be considered.j
18
TABLE 3. OPrICAL PROPERTIES OF HUMAN SKIN
Absorption coefficients AbsorptionWavelength, Reflectiona (normal tissuesa. 1/cm coefficientse,
(am)) M Outer V0.003 cm Renalnpng (irreversiblyWhite Negro Sweatb of epidermisc skin Fate damaged tissues),
1/cm
0.33 (35) - 0 200 40 22 53
0.40 22 9 0 110 21 20 39
0.50 38 10 0 55 13 13 19
0.60 38 12 0 29 12 12 15
0.65 37 15 0 25 8 11 12
0.70 68 22 0 40 9 10 12
0.80 67 37 0 16 10 10 10
0.90 57 39 0 13 12 10 10
1.00 5ý 38 0 238 13 10 10
1.10 58 40 0 231 13 10 10
1.20 .58 40 i 233 12 12 12
1.42 35 27 1 U17 31 24 34
1.70 13 11 7 231 20 27 25
1.85 11 9 23 205 40 28 35
2.20 3 3 17 202 30 23 43
2.40 3 3 50 244 50 28 54
3.00 2 2 11394 1448 --
4.00 2 2 145 380 - -
4.70 2 2 420 572 - -
5.00 2 2 312 401 - -
6.00 2 2 2241 1230 - -
7.00 1 1 574 999 - -
8.00 1 1 539 921 - -
9.00 1 1 537 821 - -
10.00 1 1 638 727 - -
11.00 1 1 1106 911 - -
aValue. fror. Figure 6. reflection at 0.33 im from Table 2.
hResults foi sweat taken from Table 4 for water.
cValues beiow 1 pm from Figure 5; other values from Figure 7.dValues from Figure 5.
'Values from Table 2.
19
TABLE 4. ABSORPTION CONSTANTS OF WATER (ref. 7)
_____,_)_.... ____, __, x(, )o••, x( ,u.)_• e,1cm"1 )
0.200 6.¶V 15 .lO"2 0.900 6.7858.10"2 3.400 7.2072.102
0.225 2.7367.10-2 0.925 1.4400.10"1 3.450 4.8080'102
0.250 1.6839-10-2 0.950 3.8757T10"1 3.500 3.3750.102
0.275 1.0739'10"2 0.975 4.4852"10"01 3.600 1.7977.102
-0.300 6.7021-'I0" 3 1.OOC 3.6317.10'1 3.700 1.2227.102
"0.325 4.1759"10" 3 1.200 1.0357 3.800 1.1244.102
0.350 2.3338-10- 3 1.400 1.2387.10+1 3.900 1.2244 .102
0.375 1.1729-10"3 1.600 6.7152 4.000 1.4451.102
0.400 5.8434.10-4 1.800 8.0285 4.100 1.7225.102
0.425 3.8438-10-4 2.000 6.9115.10+1 4.200 2.0585.102
0.450 2.8484,10 -4 2.200 1.6508-.10+1 4.300 2.4694.102
0.475 2.4736.10-4 2.400 5.0056.10+1 4.400 2.9417.102
0.500 2.5133.1&-4 2.600 1.5321.10+2 4.500 3.74+20.102
0.525 3.1595-10-4 2.650 3.1772.102 4.600 4.0158.102
0.550 4.4782.10-4 2.700 8.8430.102 4.700 4.1977.102
0.575 7.8676"10-4 2.750 2.6961.103 4.800 3.9270.102
0.600 2.2829.10-3 2.930 5.1612.103 4,900 3.5135,102
0.625 2.7948.10-3 2.850 8.1571.103 5.000 3.1165.102
0.650 3.1706.10-3 2.900 1.1613.104 5.100 2.7350.102
0.675 4.1516.10-3 2.950' 1.2694.104 5.200 2.4408.102
0.700 6.0139.10-3 3.000 1.1394.104 5.300 2.3236.102
0.725 i.5f60'10" 2 3.050 9.bR83-10 3 5.400 2.3969.102
O.75rf 2.6138.10"2 3.100 7.7830,103 5.500 2.6504-.102
0.775 2.3998 .10-2 3.150 5.3856 .103 5.600 3.1865.102
Q.O00 1.9635.10"2 3.200 3.6285.103 5.700 4.4754.102
0.825 2.7722.10-2 3.250 1.3586.103 5.800 7.1493.1020.850 4.3317.10"2 3.30C 1.4013.103 5.900 1.3248.103
C.875 5.6154.10-2 3.350 9.7905.102 6.000 2.241o.103
20
TABLE 4 (Continued)11,,,) -x (cm-~._ 1(, 1 (Ln") 1 um c•(m"1
6.100 2.6987'103 9.800 6.1421.102 27.000 1.6011.103
6.200 1.7836.103 10.000 5.3837.102 28.000 1.5169"103 2 3
6.300 '.1370"102 10.500) 7.9228.102 29.000 1.4430'103
6.400 8.8161.102 11.000 1.1058'103 30.000 1.3739.103
6.500 7.5785.102 11.500 1.5517.103 32.000 1.2724-103
6.600 6.7782- 102 12.000 2.0839"103 34.000 1.2160-103
6.700 6.3207.102 12.500 2.6038-103 36.000 1.1973.103
6.800 6.0430"102 13.000 2.9483.103 38.000 1.1938-103
6.900 5.8643-102 13.500 3.1928-103 40.000 1.2095.103
7.000 5.7446.102 14.000 3.,211.103 42.000 1.2237"103
7.100 5.6637-102 14.500 3.3626"103 44.000 1.2452-103
7.200 5.6025.102 15.000 3.3678"103 46.000 1.2621.10 3
7.300 5.5430."02 15.500 3.3564-103 48.000 1.2776.103
7.400 5.5020.102 16.000 3.3144-103 50.000 1.2918"103
7.500 5.4622.102 16.500 3.2596"103 60.000 1.2294.103
7.600 5.4234'.102 17.000 3.1712.103 70.000 1.0340,103
7.700 5.4019.102 17.500 3.0806.-103 80.000 8.5923.1 02
7.800 5.3971.102 18.000 2.9740-10 3 90.000 7.4840.102
7.900 5.3924.102 18.500 2.8597"10 3 100.000 6.6853.102
8.000 5.3878.102 19.000 2.7382.103 110.000 6.0661.102
8.200 5.3790.102 19.500 2.6035'103 120.000 5.5083.102
8.400 5.4006-102 20.000 2.4693"103 130.000 4.9686.102
8.600 5.4357.102 21.000 2.2859"103 140.000 4.4880.102
8.800 5.4978.102 22. 000 2. 1306 "103. 150,000 4.1469.102
9.000 5.5711"102 23.000 2.0052 .103 160.000 3.8956"102
9.200 5.6685.102 24.000 1.8902 .103 180.000 3.4837.102
9,400 5.7886.102 25.000 1.7895-103 190.000 3.3136.102
9.600 5.9429 102 26.000 1.6916'103 200.000 3.1667.102
21
I.. m illl IIil~
In the discussion to follow, water content shall beexpressed in terms of g/cm3 and represented by w. Thisparameter varies with the type of tissue, and of course.with water losses at elevated temperatures. Values of ware reported by Spells (20) as 0.20 and 0.75 g/cm3 for fatand muscle, respectively. Our measurements for dermal tis-sues indicate a w value of 0.80 g/cm3 . Values for theepidermis will vary widely according to the relative hu-midity, temperature of the skin, and most importantly withwhether or not sweat is present. If sweat is absent, weshall assume that w increases linearly with depth startingwith a value of 0.2 g/cm3 at thý surface of the skin andending with a value of 0.8 g/cm at the epidermal/dermalinterface. If sweit is present, we shall consider a wvalue of 1.05 g/cm. at the suriace of epidermis whichdecreases linearly to 0.8 g/cmr at the interface with thedermis.
Specific Heat--Awbery and Griffiths (1) have measuredthe specific heat of meat containing various amounts of water.Figure 2 of their paper indicates that the specific heat Cvaries linearly with water content w divided by skin densityp as follows
C - 0.37 + 0.63(w/p) (4)
When w and p are set equal to one, equation 4 approximatesthe specific heat of water.
In equation 4, the value 0.37 represents the specificheat of the non-water constituents. To check equation 4pig fat was oven dried and its specific heat measured as0.39 + 0.03 cal/g-*C. Within the estimated experimentalerror- this measurement agrees with the value 0.37 given byequation 4.
Thermal Conductivity--Spells (20) has plotted thethermal conductivities ot a wide variety of biologicalmedia as a function of water content. This includes muscle,fat, kidney tissues, blood, liver, milk, eggs, etc. Theresult shows that a linear relationship exists betweenthermal conductivity K and water content w as describedbelow.
K - (0.133 + 1.36 w/p)10 3 cal/cm-sec-OC. (5)
If one substitutes a value of 1 for w/p, equation 5 predictsa conductivity of 1.49.10- cal/cm-sec-*C which iq in sub-stantial agreement with the conductivity 1.53.10-4 for pure Iwater at normal body temperature.
22
For a typical w/: value of 0.8 fcr skiri equation 5indicates a thermal conductivity of 1.22.10" cal/cm-sec-C,This value -ay be compared with values ranging from 0.85,10-3to 1.60-10- found in the literature (5,25). The relativelylarge variation of conductivity reported above could be dueto differences of water content.
Density--Densities of skin, fat, and muscle were cal-culated using the weight and volume measurements of Cooperet al. (4) for each of three media. Each media involvedfive sets of measurements and yielded the mean densitiespresented below
density of skin - 1.03 g/cm3 (6)
density of-fat - 0.98 g/cm (7)
density of muscle - 1.01 g/cm3 (8)
"Here one must recognize that the density for skin rep-resents a mean value for the entire skin which consistspredominantly of dermal tissues. The density givenibyetuation 6 should not be applied to the epidermis unlessit is saturated with moisture.
The above brings us to the problem of computing densi-ties in terms of varying water content. Here we shallassume the volume remains invariant with changes in watercontent. Proceeding upon this basis the density P of skin,fat, or muscle is given by
P - s + w (9)
where s represents the mass of all nonaqueous ingredientsper cm3 .
In equation 9, w represents the primary variable inthat organic materials require temperatures well aboveboiling water for distillation or decomposition. The vari-able s may be estimated for various tissues using the den-sities presented by equations 6 through 8 along with thew values presented at the beginning of this section.
Blood Flow
Of all the parameters, blood flow is the most difficultto quantify in that it varies from individual to individualas well as with skin location and temperature. With inclinedresting human beings, Cooper et al. (41 found that bloodflow can range from 0 to 0.0118 cm3/cm3 of muscle/sec at
23
-I -_m l~ ~ o'
normal environmental temperatures. At an environmentaltemperature of 35*C, the results of Hardy and SolderstOm(9) and Behnke and Willman (2) indicat blQod flows in theforearm skin of 0.00025 and 0.00020 cmi/cmi of skin/secrespectively. In spite of the elevated environmentaltemperature, these flows are well within the range of valuesfound by Cooper et al. (4).
Senay et al. (19) report increases of blood flow broughtabout by temperature changes. In raising the forearm skinfrom 32*C to 370C, blood flows were found to increase by0.000027 cm/cm,3 of forearm skin/sec, and increase by0.000100 cm3/cm3 of calf skin/sec. They indicate that vaso-dilation occurred at 31.40C and 33.7*C for skin at the calfand forearm, respectively. While these blood flow changesare significant with respect to the flow rates determinedfrom the results of Hardy and Solderstom (9) and Behnke andWillman (2), they are very small compared to the range ofpossible blood flows cited previously from Cooper et al. (4).In this regard no data could be found describing the effectsof local heating upon increased blood flow.
Until such data are available, we suggest use of thehighest values found for skin and muscle by Cooper et al.(4). Such values will not appreciably affect skin temper-atures unless the exposures are of the order of 1 sec orlonger in duration.
Heat-Transfer Coefficients
Surface of Skin--This section is concerned with ratesof heat loss from skin surfaces. Such fluxes equal theproduct of a heat-transfer coefficient and the differencebetween the temperatures of the surface of the skin and thesurrounding air. it depends upon
* air motion
9 surface orientation, moisture, and temperature* relative humidity of surrounding air.
Here we have experimentally determined the heat-transfercoefficient associated with wet and dry vertical surfacesexposed to still air at 22*C with a relative humidity of50%. These conditions are considered typical of thoselikely to be encountered. Dependence of the heat-transfercoefficient on temperature is of minor importance and hencewill be neglected.
24
In the experiments, an electrical heater was placed ina 10-cm diameter, 2-cm-thick aluminum disk whose sides andback face were insulated with 2 cm of styrofoam. The testface was left bare to simulate dry skin or covered withwet tissue paper to simulate wet skin. Temperatureswithin the aluminum disk and styrofoam wete measured bymeans of thermocouples.
The rate of heating was adjusted until the aluminumdisk reached 70'C. Heat fluxes leaving the test face weredetermined by first calculating the relatively small heatlosses through the styrofoam. These losses were then sub-tracted from the measured rate of heating, and divided bythe product of face area and the temperature differencebetween the disk and surrounding air. Resultant heat-transfercoefficients he for dry and wet surfaces are as follows:
-4 2_r2,10 cal/cm -sec-°C, dry surfaceshe = 1(10),
-7 1 0- cal/cm -sec-*C, wet surfaces
The above vale for dly surfaces contrasts well with thevalue 1.5.10 ' cal/cm -sec--C found in ref. 26.
Because the epidermis is usually quite dry, we recom-mend use of the coefficient for dry surfaces unless sweatis present.
If water commences to boil, tissues will rapidly losetheir water. Rates at which heat is expended in transform-ing water into steam were roughly 100 times larger than heatfluxes due to natural convection/reradiation. Thus verylittle is gained by altering the coefficient as the skinsurface dries.
Steam Blisters--Heat is transferred across steam blis-ters by all modes of heat transfer, namely conduction,radiation, and convection. Rates of heat transfer will bedescribed by the product of a heat-transfer coefficient andthe temperature difference across blisters. Of all theparameters, the coefficient is the most difficult to quan-tify in view of variations in blisters.
The section "criterion for Fifth-degree3Burns" sti-mates the heat-transfer coefficient as 6.10 cal/cm4-sec-*C,This cc .ýfficient will be considered constant independent oftemperature, and it is the factor by which temperature
`'54-4
differences across the blisters are multiplied to determ3L.ieheat fluxes crossing the water vapor within the blister.Much of this heat transfer is by thermal conduction and toa lesser extent by thermal radiation.
Damage/Burn Predictions
Extent of Thermal Damage--Extent of irreversible damageis predicted using the same expression used by the eyemodels (23). In this regard thermal damage a is computedas follows:
fA I Aexp-A 2 /va(t) dt (11)
0where AI = step function of Val 1/sec
A2 = step function of Va, K
v = temperature in *K
t = time in sec.
This expression was originally proposed by Henriques (11)and subsequently used by a number of other investigatorssuch as Stoll and Greene (21). Values for A1 and A2 arepresented below (22):
A = 4.322-106 4 /sec 317 < va S 323°K (12)
A2 = 50,000*K } 31 -(S9.389I101041/se 323 < v < 333°K (13)
A1 =/e
A2 = 80,000OK 2 - -
Below 317*K AI is zero. In this regard we were not ableto evaluate Al and A2 values beyond 333 0 K due to the-absenceof high temDeratures at burn thresholds. Higher temperatureswill 'e achieved only with very short pulses--of the orderof I.1- 5 sec or less.
Degree of Burn--Burns range from first degree to fifthdegree. Vis-•ally these burns are characterized as follows:
first-degree burn --- red
second-degree burn --- red with white spots
third-degree burn --- entirely white
fourth-degree burn --- steam blister
fifth-degree burn --- char.
26
Characterizing burns in the above manner provides a roughestimate of the damage. However, it is not indicative ofthe depths to which tissues are irreversibly damaged. Forexample, very intense short-duration heating can char thesurface with only shallow dermal damage. On the otherhand, low intensity long-duration heating can entirelydestroy the dermis with no charring.
Criteria for each of the five degrees of burn were ob-tained by examining computed peak temperatures and a valuesat particular burn radii. These criteria are presentedbelow:
first-degree burns --- a = 0.1 (assumed)
second-degree burns --- = 1.0third-degree burns --- = 10,000
fourth-degree burns --- 131*C at base of epidermis
fifth-degree burns --- 400*C at exterior surfaceof epidermis
SKIN MODEL
As noted earlier, the skin model utilizes an implicit-explicit finite-difference method (14,23) to predict transienttemperatures produced in skin by lasers. Thermal conductionis treated using polar coordinates.
In this program the method cited above was upgraded toaccount .for:
"* coded pulses (variable pulse durations and power)"* temporal property changes (thermal and optical)"* steam blisters
"* heat losses due to the generation of steam. interception of radiation by hair follicles* variable blood flow rates with respect to depth.
In addition, criteria were introduced for predicting thedegree of burn. Means for predicting the extent of irrever-sible thenral damage remain the same as that developed forskin (22) and used for the eye (23).
27U _
This section is concerned with describing essentialfeatures of the model. Details are presented in AppendixesA, B, and C. Appendix A presents the finite-differencerepresentation of the heat conduction problem along withits solution; Appendix B describes means by which longpulse trains are treated to conserve computational time;and Appendix C discusses functional details of the code.
Laser Exposures
The model is capable of handling any 'Laser exposureinvolving radially symmetric beams that are perpendicularto the surface of the skin. Beam profiles may have anysymmetric shape while the laser may have any wavelengthand be continuous or pulsed. Laser powers are consideredconstant during each pulse.
Two types of pulse trains may be treated by the code.The first is noncoded pulses in which all the pulses areidentical. The second is coded pulses in which the pulsesvary in power and duration. Times between successivecoded pulses may vary from 0 to any value desired. Usingthis provision one can consider laser exposures with time-dependent power of any description.
Grid
A nonuniform grid is used to conserve computation timeWithin regions of pronounced heat deposition, grid pointsare spaced at regular small intervals as shown in Figure 8.Beyond this region, the grid spaces are progressivelyincreased as follows:
Az i = C1 Zi-l (14)
Ar . C 2r. (15
The constants C1 and C are chosen large enough so that re-mote grid points , such as rN~l, z *F,d5o not sense any heat-ing whatsoever.
The only exception to the grid shown in Figure 8 iswhen a hair follicle is to be included. The follicle islocated on the axis and occupies the first radial incrementArl. Subsequent radial increments are uniform and thenexpanded as per Figure 8. Axial increments are selected sothat the follicle occupies one of the Az increments. Toachieve this end the axial grid is expanded for several
28
increments as shown in Figure 8 and then cont.racted to thediameter of the follicle. Beyond the follicle the expan-sion is resumed. Further details are presentedin Appendix C.
Surface of skin Z-
z2z 3
"" L
1* I
rI rN1+1 rN 1
Radius r
Figure 8. Radial and axial grid points and increments.
29IF
m
Assignment of Properties and Energy Deposition Rates
Thermal properties, absorption coefficients, densities,water contenlL, and blood flow rates are varied with depthin a stepwise fasn.Lon according to the thicknesses of theepidermis, dermis, and any sweat layer. The steps may bemade as small as the data allow.
Energy deposition rates are calculated by first assess-ing the beam intensities at each of the grid points illus-trated in Figure 8. These intensities are computed usingBeer's Law at depths zi beneath ZLl1 as follows:
q(rj,zi) = (l-H)(l-C)qo(rj)exp-(aoZo+al(Zl-Zo)
+ ... aL(Zi-ZL L)) (16)
where q(rj,zi) - intensity at grid point rj, Zi,cal/cm2 -sec
H = fraction of radiant energy inter-cepted by hairs, dimensionless
= reflectance, dimensionless
qo(r4 ) = incident b~am intensity at radiusr., cal/cm -sec
o,i ... a L = absorption coefficients of tissuesbetween depths 0 to Zo, Zo to Zl,... , ZLl to ZL, respectively, 1/cm
Rates of energy deposition q(rj zi) are computed as follows
qrZrjzi-l) - q(r, i+l) (17)q~rj~zi -i+lZi-l
When a hair follicle is present at rjzi then
q(rj zi) = q(rjizi)/(zi-zi+j) (18)
Heat Transfer
Heat transfer within the skin and subcutaneous tissuesis computed using finite-difference approximations of theheat conduction equation (modified) for heat deposition andblood flow. This equation is presented below:
30
PC+ 2-+L(K-) + z b q (19)
where p - tissue density, g/cm3
C - specific heat of tissues, cal/g-*C
v - temperature, 0C
t = time, sec3 S- rate of heat deposition , cal/cm -sec
K - thermal conductivity, cal/cm-sec*C
r - radius, cmz - depth, cm
- rate of blood flow, g/cm3-sec-- specific heat of blood, cal/g-*C
- rate of heat loss in transforming water intosteam, cal/cm3 -sec
Initially, the thermal properties and blood flow rates varyonly with depth. Thereafter, the densities p, specificheats C and thermal conductivities K are altered with re-spect to z'adius and depth as water is transformed into steam.Expressions are presented in the section "Thermal Propertiesof skin" for the thermal conductivity and specific heatsas a function of water content w divided by density p.
In equation 19, the blood flow W is considered toenter capillaries through large blood vessels at the initialskin temperature. Thereafter the blood assumes the temper-ature of the tissues before leaving through other majorblood vessels. In this respect the blood acts solely as aheat sink. Heat transport from one location to another isneglected in that it is of minor importance compared toother heat losses, namely by thermal conduction and byblood acting as a heat sink.
Initially the skin gnd subcutaneous tissues are assumedto be at a uniform temperature. Biological heating isneglected. To simplify the temperature calculations, alltissue temperatures are given as temperature differenceswith respect to the temperature v of the surrounding air.Ambient air trmperatures may be hlgher or lower than theinitial temperature of the skin.
Rates of heat loss from the surface of the skin aredescribed by
-K4r-hev, z-0, t.> 0 (20)
31.
-i sm.-I
Due to radial symmetry heat does not flow across the axis.Thus
KavS= 0, r-0, t> 0 (21)
At distances far away from the regions of energy deposition,temperatures must approach their initial value vo-ve. Thisis expressed by
V ÷ vo - ve as r and/orz ÷ (22)
Once a steam blister forms, heat fluxes to the under-lying tissues are reduced by the film of water vapor gener-ated within the blister. Neglecting the heat capacity ofthe water vapor within the blister, the heat fluxes leavingthe blister are identical to those entering the underlyingtissue. Mathematically this is described as follows:
z"i = h,(vI - vi ) (23)115zZw-6 KtlZZw+6 hW v Zw-6 - IZw+6Zw-w
where Zw = depth at which blister forms (namely thatof the epidermal-dermal interface), cm
6 = an incrementally small displacement fromZwS, cm
hw - heat-t ansfer coefficient across blister,cal/cm4-sec-_C
Water within tissues beneath the blister is assumed tobe contained following blister formation. This, of course,raised the possibility of superheated water along withelevated pressures within tissues beneath the blister.No provisions have been made for blister rupture.
Heat losses due to the generation of steam are accountedfor by introducing positive values for q in equation 19.Such values are introduced for grid points within the epi-dermis wherein temperatures reach that of boiling waterduring periods while water is present. Other grid pointshave I values of zero.
Values for q are adjusted by trial and error until thetemperatures at the appropriate grid points approximate thatof boiling water. Usually four trials are required. Fol-lowing each trial, the q values are revised based uponcalculated temperatures using the previous i values. After
32
the last trial, the water content withtn the tissues isadjusted according to the loss of water. Once tissues loseall their water, temperatures are allowed to increasewithout bounds.
Thermal Damage Predictions
Tissues are considered irreversibly damaged when thevalue Ql of equation 11 equals or exceeds 1. In this equa-tion rates of damage increase exponentially with respectto temperature. In the model rates of damage are computedduring selected time steps. Cumulative damage is predictedby multiplying these rates by the associated time steps andsumming the damage over time.
Degrees of burn are predicted as described in the sec-tion "Damage /aurn Predictions." Here peak Q values areused to describe the occurrence of first-, second-, andthird-degree burns. Fourth-degree burns are predictedwhen a steam blister forms and fifth-degree burns when thepeak temperature is adequate to char tissues.
PIG EXPERIMENTS
This section d~escribes exposure of 20 pigs (30 burnsites per pig) to a o. and ruby laser. Included are dis-cussions of:
* optical and animal-handling procedures
9 histological procedures and results
* criteria describing the
- formation of steam blisters., and- degree of burns.
e comparison of radial and axial extent ofirreversible damage with model predictions.
The CO laser was continuous with a nominal power of 5 watts.The rady laser provided single pulses of 50- or 500-uisecduration with a maximum nominal energy of 25 joules. Bothlasers functioned somewhat below their nominal, power/energylevels. Beam powers were measured by means of a power meterat various times during the experiments. In each set ofexperiments, the power remained reasonably stable as willbe shown later in this section.
33
Immnediately before each exposure, skin temperatureswere stabilized at 17*C; This was achieved by contactingthe skin with a warm aluminum disk held at 37*C for one-half minute.
Various aspects of the CO2 experiments are shown inFigure 9. Included are the test setup as well as pic-tures of the resultant burns. The scale in the photographat the lower left is inumm; it shows a typical fifth-degreeburn as evidenced by black char at the center of the burnsite. A third-degree burn is shown at the lower right asevidenced by the relatively large white circular area.Around this and all other white areas is a red ring.
The photograph at the upper right shows burns createdby exposures to the CO9 las eF with a power of 1.55 wattsand a beam radius at the 1/e-points of 0.710 cm. Eachcolumn represents a different exposure time. Burns incolumns A, B, C, D, and E were produced by 10-, 15-, 20-,7.5- and 5-sec duration exposures to the C02 laser. -Burnsin columns A, B, and C were white surrounded by a red ringwhile those in column E were entirely red. Burns in columnD were predominantly red with a barely perceptible whitespot at the center.
In -the remainder of this section we shall discuss theexperimental procedures, compare model predictions withthose found experimentally, and interpret the data in re-gard to heat-transfer coefficients and burn criteria.
Optical Setups and Measurements
CW C02 Laser--To provide reproducible exposures carewas taken so that all burn sites were located at a fixeddistance of 150 cm. from the laser. The normal beamdivergence producedi a spot size of approximately 0.73 cmmeasured at the l/e2 points. The optical arrangement isshown in Figure 10. Two folding mirrors are used. Thefirst mirror is used to change the beam direction in thehorizontal plane, and the second mirror is used to directthe beam downward through an apeture onto the animal.The aperture is an adjustable iris so that all or any por-tion of the beam may be used. Further increases in beam-diameter may be made by replacing the first mirror with aconvex mirror of long focal length. Exposure times arecontrolled by an electromechanical shutter driven by anelectronic interval timer.
34
EU
coW cor
0x
Cod
ý4 Co
00Cx aL
a) *1C
00.
CN W 4
p. Co
Cd
444
44r
5-44
35
mechanicalshutter
C02 laer
Apertureplate
•- - -- 92 cm -
Flat or
Flat mirror, 450 convex mirror
Top view
Flat mirror, 45* convex mirror------ - • -• • ----- • • 7--
.CO2 1aspr
25 cm
_perture plate,•---(adjustable iris)
~ Animal
Side view
Figure 10. Optical system for controlling animalexposure (C02 laser).
36
The entire optical system described above, along withlaser power supply, is mounted on a portable cart to enablethe system to be moved to the animal facility.
Beam profiles for our CW CO2 laser are presented inFigures 11 and 12. Figure 11 presents the profiles of thebeam as it leaves the laser, while Figure 12 presents theprofiles after the beam is expanded by a convex mirror.The two profiles represent the profiles in two perpendicu-lar directions. These profiles were obtained by traversinga 0.030-cm pinhole across the beam in steps of 0.064-cm,and measuring the power passing through the pinhole. Themean deviation of the power measurements from the curvesof Figures 11 and 12 was 0.05 milliwatts.
All profiles shown in Figures 11 and 12 are essentiallygaussian in shape. Notice, however, that the beams are notquite circular--deviating in radius by about 5% in the twodirections. In this regard we shall use the mean value ofthe two radii cited in each of the figures.
Ruby Laser--The ruby laser operates in two modes at awavelength of 0.6943 pm. The first mode provides 25 joulesof energy over a pulse duration of 500 usec. The secondmode provides a variable output of 0 to 10 joules with apulse duration of 50 psec. Repetition rates may be variedprovided there is a minimum of 60 sec between pulses.
The setup used to expose pigs to the ruby laser isshown in Figure 13. In essence the beam passed through aprism and a Galilean telescope before being diverted down-ward through a second prism. Measurements of the beamprofile were made using the same procedure as that describedfor the CO2 laser beam. Results are shown in Figure 14.
Animal Procedure, Autopsy, Histological Procedure, and Results
Twenty pigs, each weighing approximately 27 kg were usedin the experiments. They arrived at the animal facility atleast 1 to 2 days before being used in the experiments.Food and water were withheld from the animals the nightbefore exposing them to the lasers.
Three to four pigs were exposed each day. The pigs werefirst tranquilized with Sernylan (0.5 - 1.0 mg/kg), makingthem groggy and allowing administration of pentobarbitalthrough a catheter placed in their ear vein. Pentobarbitalwas infused as needed--based on the response of the animalto external stimuli. Although it was originally planned to
37
Jc 0
44.
a% .L af
t4 4
- 41
ago4
4 04o 1.44.
U r4- e4
'-4
44,
14 0
140
'0SUVNfl1IN E31OHNId w*60O IAOMW.L 9NISSVd VUN~d V3SV1
38
CL CC
0:9 0
10 4 1-44
4)
10-
L&A
=J v-4
*0 N
0.
La .
0 0
V-4
ob
.-W4
SILVI-11W '3W.Nl N3-0'0H9(= L 9NSSt.UAW 3S:
39-
1,L'aser
I Afocal optical system,Galilean telescope
Prism
Top view
Prisms
I Aperture plate
End view
Figure 13. Optical system for ruby laser.
n.0
.07
_.06 .0
"C>
05
U6A 04
,,•z 03 "
o.04
o01
0
DISTANCE ACROSS RUBY LASER BEAN, CM
Figure 14. Profile of normal ruby laser beam.Curve is "eyeball" fit to the data.
use Sernylan for the entire procedure, it was difficult tokeep the animals quiet with only thi-s agent. i,-ntobarbitalproved to be an excellent anesthetic and none of the ani-reals died during its use. Care had to be taken, however,since it is easy to overdose the animals with pentobarbital.
: ~One animal showed respiratory arrest, but imm, liate adminis-S~tration of artificial respiration as well as the respira-S~tory 'stimul-.t Dopram revived the animal.
S~Once the pig was sufficiently anesthetized, the animal'sflank was shaved with an electric razor. Extreme care was
• ~taken since this can produce some erythema or even abrasions.S~The hair remover (Nair) was subsequently applied for approxi-
mately 10-15 minutes after which it was rinsed off with warmi water. Two applications of Nair were usually necessary toI remove all the hair and to produce a smooth, abrasion-freei skin.
I I-
41
Idl
Fifteen areas were marked and numbered for referencepurposes with indelible ink on each side of the animal.All burn sites were completely white. These areas werevaried in size, depending on the laser exposure, to assistin detecting barely visible burns at autopsy.
Deep anesthesia was required when the animals wereactually exposed to the lasers since they have a tendencyto flinch their skin. Just before exposure, pentobarbitalwas administered until the second or third (surgical) stageof anesthesia was obtained. After exposure to the laserbeams, the pigs were kept overnight in individual cages toprevent the exposure sites from getting dirty and otherpigs from biting and licking at the burn sites.
Twenty-four hours after the laser treatment, the pigswere euthanized. First they were tranquilized with Sernylanand then received an intravenous administration of Euthobarb.The pigs were immediately autopsied. Deep cuts were madesome distance from the edges of the burn sites and went allthe way through the fat to the abdominal muscle layers.The tissues were placed in plastic bags containing Bouin'sfixative and delivered to the pathologist for sectioning andobservation.
Once these techniques were developed, the entire pro-cedure became very routine and no difficulties were encoUn-tered.
Histology Thicknesses/Depths of Skin and Other Media--From the outsidie inwaras, Ene skin consIsts or a thin epidermis that is composed of several layers of cells and athicker dermis that is composed of connective tissue. Under-neath the dermis lies a layer of fat. Glands, hair follicles,and small blood vessels are also present in the skin.
Thicknesses of significant normal structures of pigskin are listed below:
Epithelium or epidermis: 0.0081 - 0.0161 cm(mean - 0.0121 cm)
Dermis: 0.1078 - 0.2479 cm(mean 0.1779 cm)
Capillaries were located at depths ranging from 0.0135 to0.0216 cm while glands or hair follicles ranged in depthsfrom 0.1617 to 0.2695 cm (mean - 0.2156 cm). All of theabove measurements were corrected for tissue shrinkage.
42
Qualitative Effects of Excessive Heat--When the skin isdamaged by lasers, cells will die. Cell death or damage maybe indicated by a complete absence of the cell or more subtlechanges in the appearance or staining reaction of the cellnucleus or cytoplasm. Concomitant with cell death is normallyan influx of white blood cells attempting to remove the deadand damaged cells. The extent of damage caused by the laserbeam is therefore indicated by the depth and radius in theskin that dead cells can be found as well as the presence ofwhite blood cells.
Depending on the characteri3tics of the laser beam, thedamage varies in depth but occasionally goes all the wayinto the fat layer. Some cells are more sensitive to laserdamage than others. For instance, cells lining blood vesselsand glands are particularly susceptible, whereas fat cellsare less susceptible than other cells of the dermis. Thus,it is possible to find areas, particularly in the fat andalso in the dermis, where the blood vessels and glandularcells show damage but the surrounding tissue cells do not.A sharp line of demarcation between damaged and undamagedtissue is therefore not present. In the measurements, thefurthest point was taken at which any cell damage could beobserved.
Histology Procedures and Observations--A total of 582of the 600 burn sites were submitted for histological examin-ation. The skin samples were received fixed in Bouin'sfixative. They remained in fixative 1-3 days and were thenwashed in three changes of 507 alcohol to remove the fixa-tive. The tissue was trimmed for processing by cuttingthrough the center of the burn site, embedding one halfwith the lesion down to insure sectioning through the cen-ter of the lesions for accurate measurement. The otherhalf was stored for future reference. The tissue was thendehydrated in alcohol, embedded in paraffin, sectioned at 6microns and stained with hematoxylin and eosin. Four to sixsections were cut from the face of the paraffin block toobtain a section through the lesion center.
The tissues were then evaluated by histologic examina-tion and the radius and depth of the thermal damage weredetermined. The injured tissue is described as follows.There was coagulative necrosis of varying depth at the-burnsite with loss of epithelium covering some of the moresevere burns. The pattern of necrosis was roughly theshape of a flattened cone. Epithelial cells had pyknotic,or shrunken dense nuclei or occasionally fragmented nuclei.The cytoplasm lost its basophilia and was more eosinophilicthan adjacent normal epithelium. Similar changes were seen
43
IJ
in cells lining the hair shafts. Necrotic fibroblasts weredetected by the presence of pyknotic nuclei; cytoplasm wastoo sparse to be visible. The upper layers of collagen wereblue in color and the filamentous structure was obscured.Deeper in the lesion, the collagen retained its structurebut was altered in color. Deeper there was an indistinctregion of questionable change although cellular elementsat this level were necrotic. There was mild and variablepolymorphonuclear infiltration at the deep edge of the lesion.
Some cellular elements were more sensitive to injury thanothers. They were necrotic at a greater depth than resistanttissues. Thus there was no sharp edge to the burn lesion.The blood vessel endothelium and supporting tissue are mostsensitive as well as the glandular epithelium. Fat tissueand fibroblasts are less sensitive than the above elements.Glands and blood vessels well within the panniculus adiposuswere necrotic in some animals but the surrounding fat wasnormal.
The depth of the lesion was measured to the deepestpoint of observed damage. The radius was very precise inits outline as the surface epithelium presented a sharpjunction of normal and injured cells. When the epitheliumwas lost from the surface, the depth measurement includedan estimate of its thickness.
Dimensional Changes of Skin Specimens--Skin specimenscontract upon being excised from the animals. Further con-traation was observed in preparing the specimens for histo-logic measurements. No dimensional changes occurred betweenexposing and sacrificing the animals.
Skin contraction caused by excising was quantified bycomparing photographs of the burn sites before and imme-diately after excising. From these photographs the follow-ing results were obtained:
* Normal undamaged skin contracts linearly alongthe surface by 12.0+3.2%. This figure appliesto each of two direEtions along the surface ofche skin.
e Burn sites contract much less than undamagedskin, i.e., only by 2.2+4.2%.
Based upon the above results, burn radii measured usingexcised skin should be multiplied by 1/0.978. To arrive atthe correction factor for burn depth, we shall assume thatskin volume remains constant. Wis means that burn depths:hould be multiplied by (0.978)4.
44
To correct for shrinkage caused by subsequent treatmentof the specimens, scaled photographs of cross sections ofexcised skin were compared with photographs of the samecross sections after completion of all histologic prepara-tions. These photographs showed that the radii were reducedby 10.3+4.2%. while depths were reduced by 11.3+5.5%..
The above results indicate the following factors forcorrecting histologic measurements
(1/0.978)(l/ '0.897) -1.140 for burn radii
(0.978)2(1/0.887) -1.078 for burn depths.
Burn Results
A total of 19 pigs were exposed to the CW C02 laser and1 pig to the ruby laser. Only one pig was used for the rubylaser in that it was not sufficiently powerful to cause anyburn damage.
Tables 5, 6, and 7 present the histologic determinationsof the depths and radii of damage for three different C02beams, namely
Power C02 Beam Radius at l/e2 Point
1.53 + 0.04 watts 0.710 cm
1.97 + 0.07 VI 0.383 V
1.55 + 0.07 "0.383"
The difference in power of the smaller' beam (0.383 cm) wascaused by a drift in power during the period between the twosets of experiments.
These tables include approximately 757. of the burn sites.Remaining burn sites were, omitted because of one or more ofthe following reasons
* sites were too small to be evident
* sites were elliptical in shape
* specimens curled or were lost in transit.
Exposure times-were controlled electronically via a shuttersystem with an estimated accuracy of a few hundredths of asecond.
45
TABLE 5. HISTOLOGY RESULTS FROM C0 2 EXPOSURES(1.53 + 0.04 watts, beam radius at 11e' points - 0.710 cm)
Pulse duration Number of Damage Depths of damageb
(sec) measurementsa radiib (6m)(cm)
03.0 23(24) 0.0997 + 0.0432 0.0211 + 0.0101
04.0 22(26) 0.1747 4- 0.0603 0.0357 + 0.0170
05.0 52(52) 0.2237 + 0.0570 0.0567 + 0.0423
06.0 10(10) 0.2835 + 0.0537 0.0438 + 0.0345
10.0 22(22) 0.3708 + 0.0582 0.1294 + 0.0483
15.0 21(19) 0.4326 + 0.0670 0.1419 + 0.0410
20.0 22(20) 0.5100 + 0.0956 0.2594 + 0.0957
30.0 17(12) 0.5632 + 0.0531 0.2430 + 0.0396
40.0 19 (4) 0.6344 + 0.0667 0.3799 + 0.2095
aRadial measurements plus depth measurements given in parentheses.
bDimensions corrected for tissue shrinkage.
TABLE 6. HISTOLOGY RESULTS FROM CO2 EXPOSURES'1.97 + 0.07 watts, beam radius at 1/e 2 points - 0.383 cm)
.?ulse duration Number of Damage Depths of damageb(sec) measurementsa radiib (cm)
0.3 4 (4) 0.0800 + 0.0330 0.0167 4- 0.00630.4 19(19) 0.1282 + 0.0316 0.0160 + 0.0055
0.6 28(28) .0.1940 + 0.0333 0.0351 + 0.0145
1.0 25(25) 0.2531 + 0.0418 0.0290 + 0.01272.0 6 (6) 0.2655 + 0.0405 0.1004 + 0.0246
3.0 0 (5) - 0.1177 + 0.0140
5 11(14) 0.3627 + 0.0605 0.1878 + 0.04467.5 10 (9) 0.3910 + 0.0712 0.1476 + 0.0527
10.0 14 (9) 0.4328 + 0.0616 0.1749 + 0.0566
15.0 13(11) 0.4806 + 0.0631 0.1979 + 0.0619
aRadial mes ,_aent plus depth measurements in parentheses.
bDimensions corrected for tissue shrinkage.
46
TABLE 7. HISTOLOGY RESULTS FROM C02 EXPOSURES(1.55 + 0.07 watts, beam radius at 1/e 2 points - 0.383 cm)
Pulse duration Number of Damage Depths of damageb(sec) measurementsa radiib (cm)
(cm)0.3 10(10) 0.0326 + 0.0257 0.0079 + 0.0022
0.5 10(10) 0.1018 + 0.0369 0.0132 + 0.00602.0 10 (9) 0.2246 + 0.0800 0.0432 + 0.01965.0 14(14) 0.3082 + 0.0218 0.1183 + 0.01907.5 11(14) 0.3652 + 0.0366 0.1784 + 0.0197
10.0 14(14) 0.4006 + 0.0307 0.1430 + 0.050220.0 25(25) 0.4431 + 0.0575 0.2738 + 0.0412
aRadial measurements plus depth measurements given in parentheses.
bDimensions corrected for tissue shrinkage,
On a percentage basis, the mean standard deviationsassociated with the damage radii of Tables 5, 6, and 7 are21.3, 19.7, and 26.97, respectively. Mean standard'devia-tions associated with depths of damage are 47.1, 31.6, and28.07, respectively. In each case the radial measurementsare more precise than the depth measurements due in largepart to variations in skin composition and contours withdepth.
Plots of the tabular results are presented in Figures15 through 17. Again one can see the greater variabilityin the depth measurements.
Skin burns are evident as a consequence of discolora-tion. With marginal burns, the sites were a pinkish redand remained so during the 24 hours prior to sacrificingthe animal. More severe burns exhibit a white area sur-rounded by a well-defined red ring. Radii of these circu-lar areas were measured by means of microcalipers (finestdial reading - 0.003 cm). These measurements were madewithin 1 hour following exposure and immediately beforesacrificing the animals. No pronounced differences werenoticed in the measurements. Results are presented inTable 8.
By comparing the radii given in Table 8 with thosegiven in Tables 5 and 6, it may be observed that the radiiof the red areas correspond reasonably well with the radiiof irreversible damage for each of the exposures.
4St 47
0.65
0.60
0.50
CIo.
0.30
0.20 O. 20 .•Depths
0.10
0 10 20 30 40
PULSE DURATION. sec
Figure 15. Radii and depths of damage produced byCO2 laser (1.53 watts, beam radius ati/e 2 point - 0.710 cm).
48
0.50
0.40
0 0.30
00
0.20
• Depths
0.10
00 5 10 15
PULSE DURATION, sec
JFigure 16. Radii and depths of damage produced by C02
laser (1.97 watts, beam radius at 1/ehpoint - 0.383 cm).
jI 49
0.50
0.40
Radii
0. 10
0 3
0 5 10 15 20
PULSE DURATION, sec
Figure 17. Radii and depths of damage produced bi CO2laser (1.55 watts, beam radius at 1/eC
point - 0.383 cm).
50
I • -' '' -" r" • , a - qm ~ •-- " • " mm m m~l~m m m~m m •• , • l1 b
TABLE 8. RADII OF RED, WHITE BURN AREAS PRIORTO EXCISING SPECIMENS (CO2 EXPOSURES)
Incident Beam radius Pulse Radii of Radii ofbeam at 1/e2 points duration red areas white areasapower (cm) (sec) (cm) (cm)(watts)
1.53 0.710 5.0 0.287 + 0.011 0.177 + 0.014it It 10.0 0.402 + 0.029 0.267 + 0.023" " 15.0 0.434 + 0.036 0.316 + 0.036" " 20.0 0.485 + 0.018 0.391 + 0.032
"30.0 0.550 + 0.031 0.444 + 0.032
"" 40.0 0.606 + 0.041 0.504 + 0.032
1.97 0.383 7.5 0.368 + 0.027 0.342 + 0.026it to 10.0 0.400 + 0.034 0.366 + 0.016
15.0 0.426 + 0.024 0.392 + 0.020
a"White" areas produced by 1.53-watt exposures were lessdistinct than those produced by 1.97-watt exposures.
Three different ruby laser exposures were used, namely
Number of Ruby biam radiusexposures Pulse energy, time at 1/es point
12 3.48 joules (500 psec) 0.17 cm
3 11.8 joules (500 usec) 0.77 cm5 5.9 joules (50 usec) 0.77 cm
Histologic examinations of skin sites exposed to the rubylaser revealed no damage whatsoever.
Criteria for First-, Second-, and Third-Degree Burns--Third-degree burns are completely white without blisters orcharring. Radii of white areas shown in Table 8 are largerthan that of any blisters. The mean valite of the damage Aintegral at these radii was (1.0+1.0) 101. The large stan-dard deviation is attributable t6 the fact that small error.--,jin temperature (or location) produce very substantial cheoges:in Q.
Third-degree burns may be estimated using the above •value. Second-degree burns should have a a value of 1based upon the agreement between the damage radii presentIedin Tables 5 and 6, and the red burn radii of Table 8. FLrit-degree burns should have a ' value of the order of 0.1 orless. More precise definition is not possible in that sucnburns were not evident during the course of the pig experiments.
51
Criterion for Fourth-Degree Burns (Steam Blisters)--Blisters form as a consequence of pressures developed inter-nally by trapped steam. We have found that the skin sepa-rates at the epidermal/dermal interface. Measurements ofthe blister radius produced by 20-sec 902 pulses (power -1.55 watts, beam radius - 0.383 at l/ez point) yieldedvalues of 0.220+0.073 cm. At this blister radius, themodel predicted-a peak temperature of 1310 C. This temper-ature represents a simple first-order criterion for pre-dicting the formation and growth of steam blisters. Atthis temperature, superheated water would develop pressuresof two atmospheres.
Criterion for Fifth-Degree Burns--Fifth-degree burnsarise-when tissues char. Char areas are always surroundedby white and red rings. Steam blisters will usually formbefore the start of charring.
Charring represents a complex temperature/time depen-dent process (24). Tissues will char upon exposure totemperatures of a few hundred degrees centigrade and higher,depending upon the exposure times. Based upon our experi-ences with other organic materials we have assumed a tem-perature of 400 0 C in view of the short exposure times tosuch temperatures.
With the CO2 laser power of 1.55 watts and the beamradius of 0.383 cm (1/e2 points), black spots formed onblisters in 10 of 25 20-second exposures. No char was ob-served with 10-second exposures. To achieve epidermaltemperatures of 400*C, the code indicated a heat transfercoefficient (associated with heat t ansfer across thevapor space) of approximately 6.10- cal/cmL-sec-*C. Heatfluxes crossing the blister space are equal to the productof the above value and the temperature difference acrossthe blister. While these fluxes can be appreciable, theyare small by comparison to conductive fluxes just prior toblister formation.
Comparison of Predicted and ExperimentalRadii/Depths of Irreversible Damage
CO Laser Exposures--Table 9 compares predicted radiiand depths of irreversible damage with their experimentalcounterparts. Experimental data are from Tables 5, 6, and7. Overall it may be observed that the agreement is good.The one exception is the depths of damage produced by thelong-duration exposures to 1.97 watts. This discrepancyis probably due to the presence of fat beneath the skin.
52
TABLE 9. COMPARISON OF PREDICTED AIG; EXPERIMENTAL WAGERADII AND DEPTHS (CC 2 LASER)
Pnwer. A Beaq radius Pulse dur., Radius of awrage (ca) Depth of damage (Cu)
(watts) at1Ie' point (cm) atior, (sec) Eip Calc Exp Calc
1.55 0.383 0.3 0.0326 0 0.0079 0
1.55 0.383 0.5 0.1018 0.0G07 0.0132 0.0147
1 55 0.383 2.0 0.2246 0.2766 0.0432 0.0654
i.55 0.383 5.0 0.3082 0.3510 0.1183 0-1385
1.55 0.383 7.5 0.3652 0.3770 0.1784 0.2579
1.55 0.383 10.0 0.4006 0.4013 0.1430 0.2985
1.55 0.383 20.0 0.431 0.4498 0.2738 0.3011
1.53 0.71r. 3.0 0.0997 0.0364 0.0211 0.0137
1,53 0.710 4.0 0.1747 0.2078 0.0357 0.0272
1.53 0.710 5.0 0.2237 0.281,4 0.0567 0.0407
1.53 0.710 6.0 0.2835 0.3254 0.0438 0.0542
1.53 0.71.0 10.0 0.3700 0.4Z27 0.1294 0.1229
.1.53 O.710 15.0 0.4326 0.4882 0.1419 P.1433
1.53 0.710 20.0 0.5100 0.5277 0.2594 0.2890
1.53 0.710 30.0 0.5632 0.6043 0.2430 0.2739
1.53 0.710 40.0 0.6344 0.6446 0.3799 0.3251
1.97 0.710 0.3 0.0800 0 ....
1.97 0.710 0.4 0.1282 0.0955 0.0160 0.0169
1.97 0.710 0.6 0,1940 0.1792 0.0351 0.0777
1.97 0.710 1.0 0.2531 0.2444 0.0"() 0..0&511
1,97 0.710 2.C 0.2655 0.3065 0.1004 O.C695
1.97 0.710 3.0 -- 0.3420 n. .).60
1.97 0.710 5.0 0.3627 0.3754 0.1878 0.2467
1.97 0.710 7.5 0.3910 0.4081 0.1476 0.2376
1.97 0.710 10.0 0.4328 0.4290 0.1749 0.3200
1.97 0.710 15.0 0.4806 0.4531 0.1979
Such fat is much more difficult to damage than skin tissues,while media such as glands, end blood vessels are not.As a consequence depths of damage beneath the skin werehighly dependent upon the presence of glands or blood ves-sels In regions of pronounced heating.
Ruby Laser Exposures---No irreversible damage was pre-dicted fo= th 5.9 and TI. 8 joule exposures involving a beamradius of 0.77 cm described beneath Table S. This, of course,is what we found experimentally. However, damage was pre-dicted as a consequence of the 3.48 joule exposure with abeam radius of 0.17 cm. The damage extended over a radiusof 0,1541 cm and a depth of 0.1067 cm. This finding con-flicts with the fact that no damage was found histologically.Only one explanation could be advanced for this discrepancy--namely attenuation of the laser beam by vapors/particulatesreleased by beam impact.
53
- .- ~d4
Experimental results are consistent with observationsof Kuhns et al. (12) and Lawrence (13). Kuhns indicatesthat 42 joules/cm2 are needed to produce a slight lesion inwhite pig skin. If all the energy of our ruby laser werec;mtained within 0.17 cm, the energy density would be 38.3joules/cm2 .
To gain a better insight into the magnitude of anyattenuation, a compute rula was executed with the incidentreduce,. by half. The predicted radius and depth of damagewere still significant, namely, 0.1142 and 0.0558 cm, re-specti',;i,'ly. Figure 18 shows that the temperatures are veryapprec~iablv even when only half of the incident energy isconsiderec. These results suggest that attenuation is verysubstantii.
100
Radius = 0 (Beam axis)
Radius = 0.068 cm
S~Radius = 0.136 cm
o Radius - 0.204 caCoU,e
--
9 Initial temperature
ka•" Note: Temperature wavesdue to variations of
absorption coefficientswith depth.
50
0 011 0.2 0.24
DEPT BELOW SURFACE OF SKIN, cm
Figure 18. Temperature predictio,.Is due to ruby laser pulse(pulsN energy - 1.74 joules, pulse duratwona5-10i 8 sec; beam radius du0.17 t m at l/er points).
54
SUHMARY AND CONCLUSIONS
The skin model provides a wide variety of options tothe user. These include
*laser exposures involving
-single pulses
-multiple pulses which are identical; or hav~edifferent powers, durations.. and repetitionrates
-any wavelength
Beam profiles may have any shape provided they are radiallysymmetric. Incident beams are considered normal to thesurface of the skin.
The model allows for
e differences of thermal, optical, physiologicalproperties with depth
o temporal changes in optical/thermal propertiesand in water content
e blood flow variations with depth
* formaticn/growth of steam blisters
e heat losses to the environmente heat losses in transforming water into steam
* interception of radiation by a hair follicle.
The model predicts transient temperatures, extent of irre-versible damage, and degree of burn..
Basic techniques used to develop the model are describedin the section "Skin Model" and in Appendixes A and B. Opera-tional details are described in Appendix C. Input data forthe code are given in the section "Data Needed by the SkinModel," and sample computer runs are presented in Appendix C.
Depths and radii of irreversible damage predicted bythe code were generally in good agreement with those deter-.mined histologically from CO2 laser exposures of pigs, butthere was poor agreement with exposures of pigs to our rubylaser. In this case, the model predicted appreciable damagewhile none was evident from histologic observations. Wesuspect that the discrepancy is due to the attenuation ofthe beam by vapors and particulates rather than due to data
55
errors. Material discharge was observed on impacting theskin by the intense ruby pulses. Similar discharge was notproduced by the CO2 beam in that the CO2 laser power isseveral orders of magnitude less than tee ruby laser.
Beam attenuation, if such is the cause of the abovediscrepancy, is very appreciable. Model runs indicate thatbeam intensities must be reduced by more than half beforedamage becomes insignificant. This aspect of the problemremains to be resolved. For this reason, we recommend thatanalytical/experimental studies be undertaken to determinethe conditions under which attenuation becomes importantand devise means to account for such in the model. Unlessthis is done, large uncertainties will exist in the modelpredictions involving rapid delivery of energy.
56
REFERENCES
1. Awbery, J. H., and E. Griffiths. Thermal properties ofmeat. J Soc Chain, Inc., 52, 1933.
2. Behnke, A. R., and T. L. Willman. Cutaneous diffusionof helium in relation to peripheral blood flow andabsorption of atmospheric nitrogen through skin.Am J Physiol 131:627-632 (1941).
3. Carslaw, H. S., and J. C. Jaeger. Conduction of heatin solids. Oxford: Clarendon Press, 1947.
4. Cooper, K. E., et al. The blood flow in skin and muscleof the human forearm. J Physiol 128:258-267 (1955).
5. Dussan, V. Study of burn hazard in human tissues andits implication on consumer product design. ASMEPublication 71-4A/HT-39, 2-7 (1971).
6. Goldman, L., and J. R. RockwellJr. Lasers in medicine.New York: Gordan Press, 1971.
7. Hale, G. M., and M. R. Querry.. Optical constants ofwater in the 200 nm to 200 pm wavelength region.Appl Optics 12 (3):555-564 (1973).
8. Hardy, J. D., et al. The radiation of heat from thehuman body. IV. The emission, reflection and trans-mission of infra-red radiation by the human skin.J Clin Invest 13:817 (1934).
9. Hardy, J. D., and G. F. Solderstom. The heat loss fromnude body and peripheral heat flows at temperaturesof 22-35 C. J Nutrition 16:493-510 (1938).
10. Hendler, E. R., J. Crosbie, and J. D. Hardy. Measurementof skin heating during exposure to infrared heating.Project NM 17-01-13-2 of Naval Air Materiel Center,Philadelphia, Pa., March 1957.
11. Henriques, F. C. Studies of thermal injury. J ArchPathol 43:489 (1947).
12. Kuhns, J. G., et al. Laser injury in skin. Lab Invest17(1):1-13 (1967).
13. Lawrence, J. C. In vitro studies of skin after irradia-tion by a ruby laEer. Brit J Plastic Surg 20:257-262(1967).
57
14. Mainster, M. A., et al. Transient thermal behaviorin biological systems. Bull Math Biophysics 32:303-314 (1970).
15. Mehta, A. K., and F. Wong. Measurement of flammabilityand burn potential of fabrics. MIT Report Nat Sci Fdon Grant GI-31881, 1972.
16. Moritz, A. R. Studies of thermal injury III. The pathol-ogy and pathogenesis of cutaneous burns--An experi-mental study. Am J Pathol, Nov. 1947.
17. Peaceman, D. W., and H. H. Rachford. The numericalsolution nf parabolic and elliptic differentialequations. J Soc Indust Appl Math 3:28-41 (1955).
18. Perkins, J. B., et al. Studies of flash burns: Relationof the time and intensity of applied thermal energyto the severity of burns, University of RochesterReport UR-217, Dec. 1952.
19. Senay, L., et al. Cutaneous blood flows in calf,forearm, cheek and ear during changing ambient tem-perature. WADD Technical Report 61-190, Wright-Patterson Air Force Base, 1961.
20. Spells, K. E. The thermal conductivities of some bio-logical fluids. Med Biol5:139-153 (1960).
21. Stoll, A. M., and L. G. Greene. Relationship betweenpain and tissue damage due to thermal radiation.J Appl Physiol, Vol 14, No 3, May 1959.
22. Takata, A. N. Developmnt of criteria for skin burns.Aerosp Med, June 1974.
23. Takata, A. N., et al. Thermal model of laser inducedeye damage, sponsored by USAF School of AerospaceMedicine on Contract No. F41609-74-C-0005, Oct 1974.
24. Takata, A. N., and W. Wulff. Study of heat transferthrough thermally decomposing linear materials, DASAReport 1554, June 1964.
25. The intracutaneous variation of temperature due toexternal temperature stimuli and intracutaneoustemperature sensation and heat transfer. Archiv Fiurdie Gesamte Physiologic 252:146 (1950).
26. Thermal problems in biotechnology, published by theAmerican Society of Mechnical Engineers, Dec 1968.
58t
APPENDIX A
FINITE-DIFFERENCE METHOD USED TO CALCULATE TEMPERATURES
INTRODUCTION
In this appendix we shall describe the finite-differencetechnique for predicting transient temperatures in biologicaltissues. The technique provides for awede variety of thermalconditions and was originally developed by Peaceman andRachford (17). Subsequently it was applied to problems in-volving cylindrical coordinates by Mainster (14) and then byTakata (23) to predict laser-induced thermal damage to eyes.
In essence the method is based upon finite-differencesin which thermal gradients associated with one of the coor-dinates are treated implicitly while the gradients in theother coordinate are treated explicitly. This procedure isreversed following each set of calculations. The result isstable, accurate temperature predictions using time steporders of magnitude larger than those required by standardexplicit finite-difference techniques.
Here we have upgraded the technique to accommodate
o heat losses to the environment
* thermal effects of variable blood flow with deptho steam blisterso heat losses due to the generation of escaping
steam* temporal as well as spacial variations of thermal
properties
FINITE-DIFFERENCE TECHNIQUE
Three assumptions are made in developing the finite-difference equations. These are
* Initially the biological media is at a uniformtemperature. Biological heating is neglected.
o Blood flow acts as a heat sink in which bloodenters incremental small tissue volumes at theinitial'body Lemperature of the tissues and
59.
leaves at the tissue temperature. Changes inthe blood flow with temperature are neglected.
e The heat-transfer coefficient associated withenvironmental heat exchange is considered con-stant independent of temperature.
These assumptions do not reflect upon the capabilities ofthe method and can easily be upgraded or eliminated according.to how well one is able to quantify each of the phenomena forspecific individuals and environmental conditions.
Spacial Grid
To conserve on computational time, an expanding gridis used as illustrated in Figure A-1. Within regions ofgreatest heat deposition, small uniform Ar, Az spacial in-crements are selected. Here the region of most pronouncedheating is considered located just below the surface aswould be the case for skin burns. In this figure, thefirst Nl radial increments, i.e. Ar1 , Ar 2 ... ArNl.are equalas are the first M1 axial increments AzI, Az2 ... AZMI.Beyond the uniform grid, the-radial and axial incremaentsare progressively increased as follows:
&rj+I CI Ar., j = Nl to N (A-1)
Azi+l C2 Azi, i = M] to M (A-2)
One important condition needs to be satisfied in selectingthe grid--that is that the expanded grid must extend intoregions of negligible temperature rise.
Problem Definition
In the equations to follow, all tissue temperatures vare measured from the environmental temperature v. to sim-plify calculations of environmental heat transfer.Heat Conduction Equation:
v - K Dv (A-3)S~-t- q-S+ 9(Ky+ + "X e A3
Initial Condition:
v- vo e (A-4)
60
Ar1Ar Surface. of Skin
- .2ZM1+1
S- - -ZM+ri r N1+l r N+I
Figure A-i. Radial and axial grid points and increments.
61
,A.
Boundary Conditions:
K = 0 at r - 0, all t (A-5)
-K-" -hve at z = 0, t > 0 (A-6)
v Vo - ve as r Z, z4 (A-7)
Steam Blisters:
all=ii~ hw(vI v A8
zw zw +6 zw- z+
Finite-Difference Approximations
The following finite-difference approximations are usedto represent the various partial derivatives of equation A-3,subject to the gradients at various boundaries expressed byequations A-5, A-6, and A-8 for time index k.K~v K i-i v i.IJ+l,k -Vi.J -Ik
'"Lv) Iu E i+ivjl l j -2, ... N, all i
= 0, j - 1, all i (A-9)
Si Vj
2Ki [viJ+l,k - vi,jk i.J.k vij,-l~k
rj+I-rjI rj - rj r.-r "
K KijJ+l+l - Ki--l • vi.+-ka - vi-j-lak
+ r l , J =2, ... N, all i
2 K-= (vi 2 k - vi l jk) 1, all i (A-10)
1r1
62
.1 . ., -- . . . • • -•' -• • • '.. - =
ij
2Ki I i+l~j,k 'i~jk - i i k -vi-ji k
i+l- zi-I Zi+l- zi z1- zi-i
SKi+lj Ki-l [Vi+lk - vi-i , i-2, M, all j+ Zi+l - Zi-il .. Zi+l -zi-I
_Kk+2"k+ v1 )AZ2 j,k ,j,k) ,jk ,j,k+1/2)'
i = 1,.all j (A-Il)
Steam blisters are allowed to form or spread radiallywhenever a given temperature is achieved at the depth zj.For values of I greater than 2, the following two equationsreplace the particular expressions of equation A-l1 with i,j values associated with the blister.
a 9 [ zv.j VI-l)j k)7z ý z'I-l1ij I Z 1-2 1 ,Jk- v-~~
- KIlI [v-l,J,k - VI-2u,k] (A-12)1-1,zI-1 z 1-2a 2 _[ Kj [vI+l~ijk - v1 '•'k
. ,z'aVI z+,..I+1,j - zij
- hw(vljk - vi.ljk)] (A-13)
For half time steps, one merely replaces k by k+1/2. For anI value of 2 one uses the following expressions.
*(K'v)
A Z-- " (V1 ,J.,k +vl,,k+l/.2) ,hw(vljk ,2j,k)]"(A-14)
63
KA
a. (Kv 2z-'W (K 2.-' + K3"j)
2j)I - "3 (Vl,j,k - V2j,k) + 2(zS _2)2,J L
"(v3,j,k " v2,jk)] (A-15)
Finally the expression for rates of temperature change issimply described as follows:
S 2 Pi jCi'i(v - vi ,k) all i, j kDC•tl i j,k+1/2 Atk (A j kl2 ''16
where Atk tk+- tk (A-17)
Finite-Difference Equations
Here half-time steps Atk/2 are considered. During thefirst half-time step, radial gradients are treated explicitlywhile axial gradients are treated implicitly. This set ofequations is termed ROW. During the second half-time step,the radial gradients are treated implicitly while the axialgradients are treated explicitly. This set of equations istermed COLUMN.
Substituting the finite-difference approximations of equa4tions A-9, A-10, A-11 and A-16 into equation A-3 yields the fol-lowing set of equations:
ROW:
- Al(ii)Vi-l,jk+l/ 2 + (A2 (iJ) + 2 PijCij/tk)Vi,jk+l/2
- A3 (iJ)vi+l,J,k+l/2 - Bl(iJ)vi,j-l,k - B2(iJ)vi,j,k
+ B3 (ilJ)vi,j+lk + 2 PijCi,jvij,k/Atk + ;i,j,k+1/2
Sijj,k+l/2 (A-18)
COLUMN:
- B1 (iJ)vi,j-l,k+l + (B2 (i,J) + 2 Pi,jCij/ Atk)vi,j,k+l
- B3 (iiJ)vij+,k+1 " Al(iiJ)vi-l,j,k+1/2 - A2 (iJ)vij,k+i/2
•" 64
~p
+ A3 (itj)vi+l,j,k+i/ 2 + 2PijCij Vi,jk+l/2/Atk + qij,k+i/2
-qij ,k+1/2 (A-19)
Expressions for the coefficients Al, A2 , A3, Bi, B2 , and B3are presented below
Al(i,j) = Ki,jZ,(i) + (KE+l,j - Ki_l j)Z 2 (i)
A2 (ij) = Ki jZ 3 (i) + NiCb/2 i12, ... M, allj
A3 (ij) = Ki,JZ4 (i) - (Ki+ij - Ki_l j)Z 2 (i) (A-20)
Al(ij) = 0
A2ij K I.- + K 2j + heh+F~/ 1 l
K +he
A2ij ( z+)2+ B1 2i- i, all ]
A3 (ij) = + K? (A-21)(AZI)2
BI(ij) -Ki'jRI(J) + (Kij+I - Ki,j_I)R2 (J)]
B2 (ij) = KijR 3(j) + WiCb/2 j j-2, ... N, all i
3 (i,j) = Ki,jR4(j) - (,j+ Kij_)R2 (J (A-22)
i,2iL 1 + 01
SKi 2 + Kile2,B (i'l) = 2( +l 1 't •i~b/2 + he•" -= , all i
S(i,1),= ,2 il (A-23)
Whenever a steam blister forms or grows, one should re-
place those Al, A2,and A3 coefficients having i values of II and I-i. When I > 2 the coefficients are:
_____65
A2 (1-l"J) KT~l~j.Z(I-1)+7BIlCb/2 + hWZ5 (I-i)
A2 (I'J) -KI JZ4(I) + lIb + bwZ'5 (I)
A 3 "IJ) - KI JZ4 (I) (A-24)
If I - 2 one should replace those Al, Aoa and A3j coefficientshaving i values of 1 and 2 by
A2(lJ) - (he + 2h)z + W~/
A3(l,j) - 2hw/z1
Al (2,j) - 2h /z3
A2 (2,J) - 2hw/z3 + (K2 ,j + K ,)(3z-2)+ Fb/
A 3 (2.j) -(K 2.j + K .)(3z-2)(A-25)
In equations A-20 through A-25 the functions, Rl, .... 4Zl~..Z 9 depend only upon the grid variables rj, zi as
described7 below
Z 1 (i) = -( 2 (A-26)iz+l - Zij...iZi 1 )
Z Mi = -1 -.-2 (A-27)2(zi~
z3(i) - i+l - 2 ____+ l
ilzi-l ) -i-+l- Zi) (Zi+l ZQzij) Z Z-l
2Z 4 (i) (z~ i.~ i- z1) (A-29)
66
Z 5 (0) _- 2 (A-30)zi+I - z_
_1 +2
R(J) = rj(r Il-r +) (rj+, - rj_)(r _ r2-) (A-31)
R2 (j) = 1 1 r1 (A-32)
2 2 (A-33)R3(J) -(rj+l - r j-l) (rj+l ": r.)+(rj+l - rj l(rj - rjI
R 4 (j) rj(rj+- 1 - 2,_r+ - rj) (A-34)~ 31 -rjQ (rjl r.<(rj+l
In matrix form, equation A-18 and equation A-19 are oftridiagonal form with non-zero elements along the main dia-gonal and on either side of the diagonal. Means for sc vingsuch matrices are described by Takata et al. (23) and involvesimple algebraic solution of each set of equations for ROWand COLUMN. To illustrate the procedure consider the firsttwo expressions of equation A-18 wherein i = I and 2.These expressions are given below.
(A2 (I'j• + 2PljClj/A•tk)vljk+I/2-A 3 (l"j)v2 ,j~k+l/2
= (A-35)
- Al(2ij)vl,j,1l+1/ 2 + (A 2 (2,j) + 2P 2 ,jC 2 .j/Atk)v2,j k+l/2
- A3 ( 2 "j)v 3 jk+i/ 2 = G2 ,j (A-36)
where G1I and G2 i represent the values of the right-handside of uation A-18.
First equation A-35 is solved for vlj k+1J/2 and usedto eliminate vI j k+l/1 from equation A-36. The resultantequation is the-A olved for v2, ,k+l/2 and used to eliminatev2i1.k+l,, from the next expression I - 3) of equation A-18.By' epeating the process for all M+I expressions of equationA-18, one arrives at the following set of equations.
6
67
-A 3 (13j)S,j ,k+l2 F1 V2,j ,k+1/2 1
- 3 (2 , j)v =D
v2jk+l/2 F2 v3,j,k+1/2 D2
(A-37)
VM-li,j,k+1/2 - A3 (M-l'j)vM.Jk+1 2 =,1-1
7M,j,k+1/2 = DM
for i equal to 1:
F1 A2 (1,j) + 2 PIjCIf/Atk
E1 -A 3 (lj)/Fl
D 1- G1,j /FI
for i greater than 1:(A-38)
Fi A2 (ij) + 2i,jCi,j /Atk + A<i.J)Ei_
E i =-A3(i,j)/Fi
Di (Gij + A( + (i,j)D_)/Fi
To solve equation A-37 for a specific value of J, onefirst evaluates each of the terms Di, Ei, and Fi of equationA-38 for i values ranging from 1 to M. Then the temperaturesvi,j k+112 are computed starting with i-M and ending with i=lfor ,articular values of j. The result is described below:
VMJ'k+1/2 j DM
VM-l,j,k+l/2 ' M- + M- vMJ,k+1/2 (A-39)
Vljk+l/2 D F + -iv2 ,Jk+1/2
68
Ar
The procedure for solving equation A-19 is the same asthat described above for ROW. In doing so the arrays A1 ,A2 , and A3 of equations A-37 and A-38 are replaced by B1 ,B2 , and B3 ; the Gi j array assumes the values of the right-hand side of equation A-19; and i is held fixed while j isallowed to vary from 1 to N. Order of the ROW and COLUMNcalculations is not important.
In solving equations A-18 and A-19, all terms on theright-hand side of these equations are known apriori ex-cept for the array !T- k-i/- This array describes therate of heat losses IA ransforming water into escapingsteam and is zero unless steam is being generated. At thestart of each full-time step Atk, all elements of thearray c&k 1k are set equal to zero. Then the ROW and
1OCA 6taltjons are performed. If none of the pre-dicted temperatures exceeds the temperature Xw of boilingwater, the computations are completed. Otherwise estimatesare introduced for I at the locations of excessive tem-perature. Following each set of temperature calculationsthe trial i values are revised according to temperaturedeviations from that of boiling water, and a new set oftemperature calculations is conducted. Each set of cal-culations utilizes temperatures computed from the previoustime step. Usually 4 trials are needed to arrive at theappropriate ! values.
Once water is expended from as element, the particularqiijk+i/2 values are set equal to zero. This, of course,a Iws temperatures to rise above the temperature of boil-ing water.
Two techniques are used to determine .i i l/ Eachtechnique computes additive changes to the pviob A ,'ij k+1/2values. The first technique provides fine refinements and ispresented below.
S= Z-(vi,j k+1 - xC)pi,jCi/Atk (A-40)
The magnitude'of the above correction is controlled via thedimensionless factor Zw. Unless one encounters a problem ofconvergence, this factor is set equal to 1. When ZW - 1,the numerator of equation A-40 represents the excess uensi-ble heat per unit volume.
The second technique yields larger f corrections, andis used when much of the heat absorbed from the laser es-capes to other grid elements. In such instances, largercorrections are needed to achieve i values of the proper
69
magnitude with the fewest trials. Choice between the twotechniques is made by comparing the absorbed heat with therise in sensible heat when all qj k+l/2 values are zero.i corrections are estimated by th6 second technique as fol-lows:
Z(X vijk+l).
q = (vi,j,k+l -- vsi.,jk+l) qi.jk+l/2 (A-41)
where vsi .1k+l and vi 4 k+l represent the temperatures be-fore and roilowing the ji k+l12 change. The denominatorrepresents the temperature' ang* produced by the previousAi; while Xw - vi, k+l represents the desired temperaturechange needed to yield, the temperature of boiling water at1 atmosphere of pressure.
The expressions given by equations A-40 and A-41 areonly used If the resultant I values do not abstract moreheat than allowed by the available water. If the availablewater is too small, then i is set equal to the followingexpression
'ti,j,k+1/2 - Wij Q/Atk (A-42)
The numerator represents the amount of heat necessary totransform the remaining water into steam. Onice the wateris expended, temperatures will rise above that for boiling.
Generally 4 sets of temperature calculations are re-quired to closely approximate the 4i, ,k+l/2 values neededto achieve temperatures of boiling water.
Immediately thereafter, the mass of eacaping steam issubtracted from the amount of water per cmi surroundingeach of the affected grid points. Once the water has beenexpended from the media surrounding a particular grid point,all future T are zero for the grid point. This, of course,allows temperatures to rise without bounds.
Errors
Errors associated wit" the method are discussed byMainster (14). In his paper Mainster indicated that hecould maintain excellent computational accuracies in spiteof sequential Increases of thg time steps. This was trewith KAt/(PC~z ) or Kht/(pCArz) values as large as 3,104.We concur with this finding provided the temporal/spacialderivatives of the thermal gradients are not excessivelylarge. It is not true immediately following abrupt
70
pronounced changes of heating such as at the start or endof a laser pulse. At such times it is necessary to usesmall time steps approximating those required by standardexplicit finite-difference. Subsequent time steps may beincreased. In this regard we have found that the timesteps may be expanded sequentially by a constant factorwithout adversely degrading accuracy as follows
Atk+l - C3Atk (A-43)
To maintain accuracies of the order of a few percent or less,C3 should not exceed approximately 1.4. Accuracy will beimproved, of course, with smaller C3 values.
Nomenclature
AlA2.,A 3 coefficients of temperatures
1 blood flow per unit volume of tissue, g/cm -sec
-BB 2,B 3 coefficients of temperatures
C specific heat, cal/g-*C
CIC21C 3 factors used to expand radial increments,axial increments and time steps, respectively,dimensionless
h heat-transfer coefficient, cal/cm2 -sec-*C
i index associated with axial grid incrementsor nodal points
I i index associated with all axial grid pointsat depth of blister formation
J index associated with radial grid incrementsor nodal points
k index of time steps
K thermal conductivity, cal/cm-sec-*C
*M total number of axial increments
M14 number of uniform axial increments
N total number of radial increments
N1 number of uniform radial increments
71i a.
Q heat of evaporation for water, cal/g
q(r,z) radiant intensity per unit area, cal/cm2-sec
rate of heat diposition into unit volume oftissue, cal/cm -sec-
rate of heat loss per unit volume of tissuedue to the generation and escape of steam,cal/cm3 -sec
r radius, cm
t elapsed time from start laser exposure, sec
v temperature, 0C
w mass of water per unit volume of tissue, g/cm3
Xw temperature of boiling water (1 atm. of pres-sure), °C
z depth, cm
Zw dimensionless factor used for purposes ofconvergence, usually equals 1
zw depth of steam blister, cm
6 incrementally small distance approaching zero,cm
p density, g/cm
Subscripts
b blood
e environment
i,j see above
k see above
M,Ml see above
N,Nl see above
72
o initial or incident value
w steam blister
iI
73
APPENDIX B
APPROXIMALTE TECHNIQUE-FOR REDUCINGCOMPUTATIONAL TIMES OF LONG PULSE'TRAINS
(NON-CODED PULSES - CONSTANT LASER POWERS/TIMES)
INTRODUCTION
Long pulse trains involve numerous fl=. or power changes.These numerous changes require many time steps because timesteps can only be expanded geometrically during periods ofconstant flux such as during and between pulses. Immediatelyfollowing the start or end of each pulse, it is necessary toreturn to small time steps to preserve accuracy.
Two options may be used to minimize the number of timesteps. The first applies to situations in which temporalchanges in the thermal ptoperties are of negligible impor-tance, arid all boundary conditions are independent or linearlydependent upon temperature.. Under such conditions, the tem-perature rises produced by each pulse are independent of eachother. This allows one to calculate the temperature riaesproduced by a single pulse, and simply sum the contributionsfrom each pulse all.owing for time differences. This tech-nique was used for the eye models (23).
Figure B-1 illustrates the technique-for treati~ng mul-tiple pulses allowinrg for temporal changes in the thermalor optical properties. In essence the method eliminatesmany flux changes by replacing several consecutive pulsesby a single "ipulse"l having the mean flux needed to conserveenergy from the several pulses. In doing so, selected pulsesare preserved to afford accurate temperature/damage predic-tions during representative pulses of the train. Incrementaldamage predicted during the selected pulses is then weightedto account for all pulses.
ACCURACY
This section assesses the consequence of v'sing mean flux.upon temperatures produced during succeeding pulses. Thetotal number of pulses shall be designated by nl+n2, with thefirst n1 pulses replaced by the mean f 'lux. The remaining n2pulses shall be preserved and shown at the center of the firstgroup of pulses shown in Figure B-1. Accuracy will be deter-mined by comparing temperatures predicted during the last,. ofthe n1+n2 pulses with exact temperatures in which all pulses
'.4
4-44
oo '-4
""o4 14
0 0W4) 04
,0
0,
w
04-4
17 A
-4-
76
are preserved. For this purpose we shall consider a semi-infinite body in which the flux is uniformly distributedover the surface. The energy will be considered entirelyabsorbed by the surface.
To arrive at the exact surface temperature rises weshall apply Duhamel's Principle to the solution (3) forthe surface temperature rises produced by a constant flux.The temperature rise Ts at the beginning of the last of thenl+n2 pulses is given by
2q, n 1+n 2 -lTs Z ( 1`70 - VTW-r_ ) (B-1)
n-1
while-the temperature rise at the end of the last pulse is
given by
nl1+n2- 12qo
Te ' -,V-- +T, - Vin) (B-2)
n-0
where
q - flux during each pulse, cal/cm 2-sec
K - thermal conductivity, cal/cm-sec- 0 C
p - density, g/cm3
C - specific heat, cal/g-*C
To - reciprocal of repetition rate, sec
T - pulse width, sec
Using the mean laser flux of Tq/To, yields the approxi-mate values Tsand Te for T. and Tet respectively, presentedbelow:
Y, 2TS (- -ýn o-s Tj°l• ( j(nl+n2 "l)r 0o- n")o
n 2 -1+ 2q n- (B-3)•
n-l
77
Te = 2Tq (" (nl+n2 -1)to'-&- • (n 2 -1)to+T)Te 2 o+T
+ 21 (no+ T B-4)
n-0
In examining eauations B-i through B-4 it may be noticedthat the ratios T5 /Ts and Te/Te are independent of the fluxq and thermal properties K, p and C. They do however dependupon T/,ro and upon the values for n1 and n 2 . Results ofevaluating these ratios on a computer are presented in TablesB-1 through B-3 as a function of nq and n 2 for three T/Tovalues. It may be observed that the agreement improves withthe number of pulses nI and n2 for each of the three T/TOvalues used, namely 0.1, 0.001 and 0.00001. The best agree-ment is achieved with the smallest T/TO value, namely 0.00001.This value is more typical of the values associated withpulsed lasers than the other two values employed.
Of primary concern is the minimum number of individualpulses n2 to achieve accurate temperatures. In examiningTables B-i through B-3, it may be observed that 2 or 3pulses will limit the errors in the peak temperatures toless than about 27. While errors in the temperatures at thestart of the last pulse are slightly larger, they are ofminor importance insofar as damage is concerned. Moreovererrors associated with indepth temperatures and two-dimensionalheat transfer will be less than the values indicated above.
To follow transient changes in the damage, about 10groups should be adequate to represent 100 or more pulses.Less groups should be used with shorter pulse trains. Re-maining pulses should be placed at the end of the pulsetrain and treated individually.
PROGRAMMING
This method requires accurate temperature predictionsduring and immediately following selected pulses. Theselected pulses should be spaced at regular intervals along 4the length of the pulse train. At least one pulse shouldbe preserved prior to each of the selected pulse or pulses.All other pulses are represented by the mean laser powerdistributed uniformly with respect to time from one pulseto the next.
78
7-4 0) -T 120' '- Ch m %D 4 %* ýl %Q T .4to In 00a -"0000 -40000
0 44.4 r4 0000a0 0 0000 000000 0 :3 . . . . . .-14 CL -4 -4 -4 1 4 - -4 -
0
415 4.)
04.4 W - "'.t4-0 M)-4 000, C1400000 "v4 00000 00000 00000
41 O. .- 4 -4 4 14.4 - 4--- 4 - -
E-.4
'44
444- Ul
co4o 000000000 00000 : -O 00000 !?0000 00000 r4
4- - .4 0 cy4. 0l1~% 0 0 Q.
W00
E4-4
0 4.5
fU % Cn L C, G rfl4 P-4. <7 . 4aOV0 00 1-4 C-4 .-4 V4 -4 -4N 44, -4 -4 -4- -4- -4-)- U)
4.4- 04 r-
P4) 'd >4 0
1.4 Q- 0 I r4.0 4--4 C400.000.400
.41
41 4-0E-4 CS 4
0 4 Wj e 414"4 W)Cn C")4 Ifý C4ý4- -'4U W .0
1.4C40 0 40 0 40 + 1401
.0 - 000 0 MU~If~fl~~ 0 000 -a - ('
0 -'"-4 ~('~ ~ 0 04E-4 04inO ) 4
79 4
r--40 'T M N' 4Cr en en eqN .- 4 -a en C
0U
00
o. c44 Lne , (%rUD v)a at- '4.4 M .NC4- 4 ený4000 NC40 00F 0~ 00 00000 000800
E414.
It-- 1 n UL 4C4C)Uj 40 TL
'4-
0)
E44 1 4. 1
0A -80000 -0800 "8800wP04 0000 0C)0 0 0 00 0 ) W-
00
0 '-440000 0W00 4~ LU -40
l C4 1 004 93 -4C4C 4C)T& 4( 0 0p
4.11 U (
LL
00
.0 04 00000 tM '1Un~ 00000 W-4 N43 --1-4 N N4N N in 0f 0l% 3
80
H) C.4... coo C4.-4'0 Ce.4.-4 -A 000000 0000 0000
_ ~4.4-400000 00000 00000
01 1
0 IA
00n w
444) 0
~~co 0' coo- 0~.400 .0000000000~4 00 000 0
0 -W*1.4 I
00
T0~ 00 0 0000 to0
4100000000000
cct 114.' ~1.4 t
P4f 45.. ' 0
P~ r4 in 4 . .
E-4 41 .4~ 4 14 1 4 4 -4 -:-4 14 -4~ 14 144
0 0
co 0 .~4
41 %4.4
N 0
4.4
0nnfL~ 0000.4C1.4 -+ -4 v- I4N 0 , n k n LiLi1
Sc
To illustrate the method we shall consider a train of40 pulses. Six series of consecutive pulses shall be re-placed by the mean power as illustrated by the squat areasof Figure B-2. Number of pulses eliminated are presentedover the shaded areas. The beam power associated with in-dividual pulses obviously exceeds the mean power as indi-cated by the figure. Pulses yielding accurate temperaturesare blackened. At least one pulse should precede each ofthese pulses to fully develop temperature transients.
Table B-4 indicates the valueu needed for the array&NPG(L) and NPR(L) to structure the train as shown inFigure B-2.
TABLE B-4. PROGRAMHING 'IMAMW P(AERS" IETO DONCODED PULSE TRIjS
L NPG(L) NIR(L) L XPG(L) UPK(L)
1 7 0 12 1 72 1 0 13 5 03 1 7 14 1 0
4 5 0 15 1 75 3. 0 16 3 06 1 7 17 1 07 5 0 18 1 2
8 1 0 19 1 19 1 7 20 1 1
10 5 0 21 1 1II 1 0 40 40
The index L indicates the order in which individualpulses or mean powers are to be considered. The arrayAPG(L) represents both the number of pulses and how theyare to be ereated. For normal treatment of pulses, NP%(L)is set equal to 1. NPG(L) values greater than 1 will re-srlt in replacement of the pulses by the riean laser poweras shown in Figure B-2.
The array NPA(L) should be set equal to zero exceptfor the , acted pulseas. For the salected pulses, NPR(L)is equal the number of pulse, represented by the pulse.For example, the first "sel'.cted" pulse of Figure B-shown as black, his an L value of 3. It represents 7oulses, three preceding it and three in the series of 5p,.•lses following it. This is also true for the selectedpulses with L values of 6, 9, 12, and 15.
82
46 "4
cn
a 4)
___ ___ ___ __4
o4 0)
N4 VI
~,N~ 0.0
4,4
IIl
14W
Z4W
.4 5.4aMR9
a. 483
APPENDIX C
SKIN MODEL
INTRODUCTION
The skin model consists of a main program plus the 8subroutines listed below:
* TIME that computes the time steps and associatedlaser powers
* GRID that computes the grid increments (radialand axial)
"* PROF that computes the laser intensities at eachof the radial grid points determined by GRID
"* CWATER that computes the thermal conductivitiesand volumetric heat capacities at each of thegrid points in terms of variable water contentin the tissues
"* BA that computes the matrix elements needed toassess the temperature rises
"* HTDEP that computes the rates of energy deposi-tion per unit volume of tissue at each of thegrid points
"* TEMP that uses the alternating explicit-implicitfinite-difference technique of the CORNEAL MODELto compute temperatures allowing f:.r
- heat losses to the environment
- heat losses due to blood flow- heat losses due to transforming water into
steam
- steam blisters
"£ DAMAGE that computes the cumulative thermal damageat each of the grid points as well as inteimalpressures created by superheated water
85i
Jr_
Code listings, nomenclature, and sample runs are pre-sented at the rear of this appendix.
A flow diagram of the major code routines is shown inFigure C-I. All routines are used during the first timestep. Thereafter only the main program and subroutinesTEMP and DAMAGE are used unless water losses or irreversibledamage have not occurred. If water losses occur during theprevious time step, subroutines CWATER and BA are reenteredto reevaluate thermal properties and matrix elements. Con-trol is exercised via the parameter IMAX which indicatesthe maximum depth Z(IMAX) over which water has been lostduring the previous time step. During the first time stepIMAX is set equal to M (the number of axial intervals) toensure subroutine CWATER is entered.
When irreversible damage occurs, subroutine HTDEP isreentered to reevaluate the rates of heat deposition usingthe absorption coefficients of damaged tissue. Here theparameter LMI is used for purposes of entry. During thefirst time step it equals N. Thereafter LI indicates themaximum depth (Z(LMI)) of irreversible damage.
In the remainder of this appendix we shall describe thecode according to the listing presented at the rear ofthis appendix. Statement numbers are in sequence in eachroutine. We shall use these statement numbers for refer-ence purposes. For example, 5.2 would be used to indicatethe second line following statement 5. Detailed informa-tion required to operate the code are presented in theUser's Manual.
Nomenclature describing the variables to be discussedis presented at the rear of this appendix in alphabeticalorder. Variables followed by dote are read into the codeas input data.
DESCRIPTION OF COMPUTER PROGRAM
Main Program
This portion of the code performs the following functions
e reads input data
* calculates the DR and DZ increments along withthe maximmn radius RN and the maximum depth ZM
9 controls course of calculations
86
SMAIN k m 1 T M R DP O
NoPROGRAM (first time s iep) o e1 .k k> 1
wt Yes
NoCWATER
i sY e s
T E M N o t e : k -i n d e x o f t i m e c t e p - 1 ,2 . ..K T
IWA - the maximum depth index at which] water has been lost from tissue.
LMI - the maximum depth index at whichirreversible damage occurs.
DAMAGE 1 - depth index where skin tissue isalive.
Figure C-1. Flow diagram of skin model.
87.
"* prints input data, temperatures, damage, andpeak pressure caused by superheated water
"* interpolates extent of irreversible damage(radial and axial)
"* computes degree of burn
Data cards are read in sequence according to the numbersshown in columns 73-80 Of the listing. Numbers followed byletters indicate the particular cards needed. For examplewith uniform beam profiles one should use card (12, UNIF)and not cards (12, GAUSS) and (12, IRREG). The latter cardsare for gaussian and irregular profiles, respectively.
With single-pulse exposures card 14,NCI is needed.Card 14,NC,X is needed for pulse trains.
If the pulse train consists of uniform pulses (non-coded) and is to be treated on a pulse-by-pulse basis, noadditional cards 14, ... are needed. Otherwise cards (14,NC,GP) are required to group the noncoded pulses.
Coded pulse trains require cards labeled 14,CODED.
Statements 7.2-7.4--Here the heat-transfer coefficientH is set equal to that for dry surfaces unless a sweat layeris present. The value 0.0001 cm represents an arbitrarilysmall sweat thickness below which sweat is neglected.
Statements 7.3-9.3--These statements are used only ifsweat is consi ered. The arrays TH, ABS, TB, and BI. areadjusted to allow for the depth of sweat TSWEAT.
Statements 11.1-18.4--This portion of the Code com-putes the radial increment DR and the radial extent XO ofuniform-, gaussian-, and irregular-beam profiles. At,statements 12.1 and 12.2, the DR increment is chosen sothat the uniform profile ends halfway between the Nl-th and(NI+l) increments. This is dcne so that heat is depositedequally on either side of rNi. Statement 16.2 computesthe radius XO at which gaussian profiles become insignifi-cant. Intensities q of normalized gaussian profiles aredescribed by
q w exp-(2r 2 /(SIGMA) 2 (C-i)
where r - radius in cm, q - 1 at r - 0, and SIGMA indicatesthe extent of the profile. Substituting CUT for q and RIMfor r and solving for SIGMA yields
88
SIGMA - RIM V'-2/ln(CUT) (C-2)
Thus equation C-i becomes
q = exp(r 2 ln(CUT)/(RIM) 2) (C-3)
As defined, q - CUT at r = RIM. This portion of the codereplaces q with an arbitrarily small number exp-5 and solvesfor a radius r at which the intensities are insignificant.
Statements 22.5, 24.2, and 42.0--These statements com-pute the duration XX of the laser exposure.
Statements 40.0-41.3--These statements estimate thedepth X4 at which the beam intensity is insignificant.This estimate involves choosing a depth at which the inten-sity is a very small fraction (i.e, exp-8) of that at thesurface.
Statements 44.0 and 44.1--These statements estimatethe distance X3 over which heat will be conducted duringthe duration of the expcsure, namely XX.
Statement 44.4--Here.ZM is set equal to 3 times thesum of the depths X4 and X3 associated with negligible beamintensity and heat conduction, respectively. The factor 3is greater than 1 to ensure that ZM is beyond the maximumdepth sensing heat.
Statement 44.6--Here we have chosen to fit uniformDZ increments within the sweat plus epidermis by settingMi-i. The number 1 is arbitrary. More than 2 incrementswill appreciably increase execution times with only mar-ginal improvements in accuracy.
Statements 44.7-44.10--These statements adjust theuniform radial grid to accommodate a hair follicle withinthe first radial increment. The remaining portion of theuniform grid is subdivided into equal increments DR ifthe depth DHF of the follicle is set equal to any numbergreater than ZM.
Statement 46.0--Here RN is set equal to 1.5 times thesum of the radii xo and X3 associated with negligible beamintensity and heat conduction, respectively. The factor1.5 is greater than 1 to ensure that the maximum radiusRN extends beyond the region heated.
89
Statements 86.3-86.5--The indices IMAX and LMI are setequal to M so that all subroutines are callea prior to thefirst time step. JMAX is set equal to N to ensure all Jvalues are used in subroutines CWATER and BA.
Statements 86.6-88.4--These statements
"* establish the first grid point z11 within "livetissues"'
"* establish the first grid point zIp within theregion in which water may become superheated
"* initialize ZB5, RGV, IB, RB, V, VO, KK and indexof time steps k.
Statements 90.0-90.2--Here the time step DT and laserpower POW are set equal to the values computed in subroutineTIME and the elapsed time TTIME is adjusted accordingly.
Statements 90.3-90.5--Subroutines CWATER, BA, HTDEPare called prior to the first time step and following timesteps during which water has been lost (IMAX> 0) or damagehas occurred (LMI> II). Subroutines CWATER and BA revisethe thermal propefties and matrix elements for water losses,respectively. Subroutine HTDEP alters deposition rateswhen irreversible damage occurs. The latter is accomplishedby replacing the absorption coefficients at the particulargrid points with the coefficients for damaged tissue.
Statements 106.4-119.0--These statements have to dowith printing temperature rises. Temperatures are printedat intermediate time steps only when the following threeconditions are satisfied
"* XP(K) equals 1.1. Otherwise the temperatures aretoo approximate for printing in that either amean power is being used to represent a group ofpulses or the pulse following the mean power.
"* time TTIME is between a specified pair of timesTIMEl(KK) and TIME2(KK). This provision is pro-vided to control periods over which printoutsare desired. If one wished printouts at alltimes, set KX-1; TIME1(l) -0; and TIME2(l) to avalue several times larger than the duration ofthe exposure. Extremely large values will causeno problem.
90
* ITYPEX must be greater or equal to the specifiedvalue ITYPE. Here printouts will be provided atevery time step if ITYPE is set equal to 1, atevery other time step of ITYPE equals 2, etc.
Temperatures and damage are always printed following thelast time step.
Statement 119.1--This statement stores the peak tem-perature at(rl, zl) in ZB5. This temperature is used topredict fifth-degree burns following completion of thetemperature/damage calculations.
Statements 119.2-119.3--Here the time index k is in-creased by 1, and the calculations are continued providedk does not exceed the total number of time steps KT. Inthis regard, the KT time steps include NTX time stepsfollowing completion of the exposure. This provision ismade to account for the fact that damage continues afterthe last pulse. For this purpose, NTX should be assigneda reasonably large number, but not so large that KT exceedsthe maximum dimensibn MK allowed in dimensioning arrays.A value such as 20 thould suffice, Once damage becomesinsignificant, subroutine DAMAGE aborts additional timesteps.
Statements 119.4-124.0--At the end of the calculationsthe cumulative damage is printed for each of the grid points.
Statements 124.1-132.0--Here the peak temperature RBof any superheated water is used to calculate the maximuminternal pressure. The resultant pressure represents anupper limit in that it does not allow for tissue deforma-tion, i.e., water remains as water without expansion.
Statements 133.1-154.0--Here the radial and axialextent of irreversible damage is determined by interpola-tion using the largest radius rjD. and depth zIDZ ofirreversible damage found for any of the grid points.The maximum depth is assumed to be on the axis while themaximum radius is assumed to be. at the shallowest depthzIU of. "live" tissue. Only under very unusual heatingconditions would these assumptionh not be true, i.e., beamprofiles with their greatest intensities far removed fromthe axis. If one encounters such conditions, one considersall i, j values.
In that the same scheme is used to interpolate thres-hold damage for both coordinates., we shall illustrate theprocedure only for r. Damage D is interpolated assumingit decreases exponentially'with r in fhe following fashion.
91
U = DjDR exp-(Xl'(X 2 - rJDR)) (C-4)
Substituting DjDR+l for D and rJDR+l for r, and solving forthe constant X1 yields
Xl = -ln(DJDR+l/DJDR)/(rJDR+I-rJDR) (C-5)
Setting D equal to 1 in equation C-4 and solving for thedesired radius X2 yields
X2 rjDR + ln(DjDR)/Xl (C-6)
Statements 160.0-162.0--These statements determine thedegree of burn LDB. First-, second-, and third-degree burnsare based upon the level of damage Q (see equation 11) atrI, z:-. Fourth-degree burns are indicated if a steam blis-ter forms (JW>l). JW indicates radial extent rJw of a blisterand is evaluated at statement 14.1 of subroutine DAMAGE.Fifth-degree burns are predicted if the peak temperature at(rI, zl) exceeds DB5.
Subroutine TIME
This subroutine comutes and stores the time stepsDTX(k). Relatively small time steps are used immediatelyfollowing the start and end of each pulse. Subsequent timesteps are sequentially increased as long as the laser powerremains constant (including zero). These periods corres-pond to the interpulse and intrapulse times.
In addition the code evaluates the array XP(k) whichaccounts for pulses replaced by mean power. It representsa weight factor which is other than zero only for timesteps corresponding to individual pulses yielding accuratetemperature predictions. When all the pulses are treatedindividually, XP(k) - 1 for all time steps, i.e., k - 1 -oKT. When pulses have been eliminated, XP(k) will assumevalues greater than 1 to account for the number of eliminatedpulses. XP(k) values are greater than zero only for thosetime steps DTX(k) yielding accurate temperature predictions.
Initially the arrays POWER(k) and XP(k) are set equalto their most common values, i.e., 0 and 1, respectively.Subsequently, some of these values are revised in the sub-routine.2
S~92
Statements 2.1-2.7--This portion of the code assessesthe minimum time step DTO needed for accurate pree'ctionsof temperature. The criterion is of the same form as usedwith explicit-finite differences. The factor ZR is aninput parameter and should be approximately 0.5.
Statements 2.12-30.1--This portion of the code computesthe time steps TX(k) and associated arrays POWER(k) andXP(k) for single-pulse exposures.
To ensure accurate damage calculations, a minimum ofNTP time steps are used to embrace the pulse duration DPULSE.Statements 2.14-2.17 use a series of uniform time steps DTlif DT1 is less than or equal to DTO. Otherwise the follow-ing series of increasing time step is considered.
DTO(I+XC+(XC) 2+ (XC)Lll) - DTO X-l1 (c-7)
where the number of terms Ll is chosen so that
DTO(XC)Ll"lS DTl (C-8)
This series is used provided that it does not exceed DPULSE.Any residual time is subdivided into equal time steps lessthan DTO. If the series exceeds DPULSE then the factor R3is determined such that
DTO(1+Pt3+(R3)2 ... (R3 )Lll) - DPULSE (C-9)
This, is accomplished by rewriting equation C-9 a's follows
R3 - exp(In(DPULSE(R3-!) + l)/Ll) (C-10)SDTO
Trial values (Xl) are then substituted for R3 on the right-hand side of equation C-10 and solved for R3. If R3 doesnot agree with Xl, Xl is set equal to R3 and the processrepeated until they do agree. Usually only a few trialsare needed.
During each time step within DPULSE, POWER(k) is setequal to the actual laser power POWX.'
Statements 29.0-30.1 compute the NTX time steps follow-ing the pulse. he first of these time steps is DTO followedby (XC)DTO, (XC)YDTO, etc. Then the total number of timesteps k is stored KT.
93
Statements 31.0-50.0--This portion of the subroutinecomputes the arrays DTX(k), POWER(k), and XP(k) for non-coded pulse trains. Statements 31.9 through 38.2 deal withindividual pulses while statements 38.3 through 48.1 dealwith the mean laser power XX3 used to represent variousgroups of pulses. The array NPG(L) equals 1 for individualpulses. When a group of pulses is replaced by the meanpower, NPG(L) equals the number of pulses in the group.
Following each pulse or group of pulses, the time stepDTO is increased by the specified factor ZZ. Usually oneshould set ZZ - 1. To conserve on computation time, tryZZ values several percent lar er than 1. Reduce ZZ closerto 1 if any temperature oscillations occur.
Statements 31.9 and 31.10 compute the number of timesteps NP needed to embrace the pulse. The number of timesteps is never allowed to fall below 2. Immediately there-after the time steps DTX(k) are set equal to the constantvalue XXM, and POWER(k) and XP(k) set equal to POWX andNPR(L), respectively. The array NPR(L) equals the numberof pulses represented by individual pulses, and equals 1when all pulses are treated individually.
Statements 32.3 through 38.0 compute time steps betweenpulses. Here XX4 represents the first of two expanded timesteps, XX4 and XX4(XC), needed to embrace the interpulr .time XX2. These time steps are used if XX4 is less thanor equal to DTO. Otherwise, the number Ll of expanded timesteps needed to exceed XX2 is evaluated at statement 36.0.This is achieved by solving equation C-11 for Ll and drop-ping the decimal.
(XCLi-I_ l)DT0 - XX2 (C-11)XC-I
Then the smallest time step DT2 is evaluated by replacingDTO of equation C-11 with DT2 and solving for DT2. Theresult is the following Ll time steps:
DT2, DT2(XC), DT2(XC) 2 "- DT2(XC)Lll1 (C-12)
The array DTX(k) is set equal to these time steps betweenstatements 36.2 and 38.0.
Statements from 38.1 through 48.1 deal with groups ofpulses. The duration X2 of the group of pulses is evaluatedat statement 40.0. If X2 is less than DTO, one time stepis used. Otherwise, the time steps are computed betweenstatements 46.0 and 48.0 using the same procedure of state-ments 36.0 and 38.0.
94
Statements 50.1-54.1--This portion of the code computesNTX time steps following the end of the pulse train .'Thesesteps start with DTO followed by DTO(XC), DTO(XC)I, etc.
Statements 70.0-100.0--This portion of the subroutinecomputes the time steps for coded pulses. The procedure isidentical to that used for single pulses between statements2.12 and 28.0. The only difference is in the pulse durationsand number of pulses.
Statements 100.1, ll0,1--Here NTX time steps are addedonto the end of the coded pulses in the same fashion usedwith noncoded pulses.
Statements 120.0-121.1--If the total number of timesteps exceeds the number MK allocated in dimensioning ofarrays, the subroutine will indicate this fact and stopfurther computations. MK must be <200.
Subroutine GRID
This subroutine locates the grid points based upon the
"* values selected for Nl, N, and M
"* Ml, DR. DZ, RN, and ZM values computed in themain program
9 diameter SHF and depth DHF of hair follicle
A hair follicle is considered only if it lies at a depth lessthan the maximum depth ZM (DHF-cZM). A typical grid systemwithout a hair follicle is shown in Figure A-1 presented onpage 61.
Statements to 10.3--Following the last uniform DR, theradial increments are expanded by the factor R2 chosen soCK increments exist between (RN-CP) and RN. This requiresthat
DR , _- - CP (C-13)
R2 is determined by rewriting equation C-13 as follows
R2 - exp(ln(LP )DR + 1)/CK) (C-14)
This equation is solved for R2 in the same fashion describedin solving equation C-10.
95
Statements 13.0-15.0--Here the first N1+I - N4 gridpoints R(J) are determined for the uniform portion of theradial grid. When a hair follicle is present, the firstincrement equals half the diameter SHF of the follicle.The next N1-1 increments DR are uniform.
Statements 15.1-16.0--Here the factor R2 is used to-evaluate the grid points R(j) over the expanded portion ofthe radial grid.
Statements 16.1-18.0--Here the first M1+1 - M4 gridpoints Z(i) are computed over the uniform portion of theaxial grid.
Statements 18.5-26.0--These statements assume no hairfolliEcle.They evaluate the grid points Z(i) over theexpanded portion of the axial grid. The technique is thesame as that used for the radial grid.
Statement 30.0--To accommodate a hair follicle, it isnecessary to commence to contract the axial grid pointsearly in the exparsion phase to achieve an increment havingdimensions identical to the follicle. Thereafter theincrements are expanded to ZP4.
Statement 30.0 partitions the M-M1 increments for theexpansion phases.
Statements 30.1-36.0--Here the first LX incrementsfollowing the uniform grid are eryanded to a depth roughlymidway between the base of the epidermis and the follicle.
Statements 36.1-46.0--The next LX-l increments arecontracted to the follicle, with the last increment equalto the follicle diameter SHF. The follicle is centeredat Z(L2). One additional increment, Az - SHF, is providedbeneath the follicle.
Statements 47.0-62.0--Finally the remaining M-L2-1increments are expanded to ZM.
Statements 70.0-72.0--This portion of the subroutineassesses the index 1W associated with Z(IW) nearest ZBL.ZBL represents the depth at which blisters form and shouldbe set equal to the thickness of the epidermis in inputingdata.
i
Subroutine PROF
This subroutine computes the profile HR(J) of the laserbeam in terms of the intensity of the incoming beam. Threeradially symmetric profiles are considered, namely
e Irregular profiles--whose shape is described on apoint-by-point basis using the arrays PX(L) andRX(L)
* Gaussian profiles--whose shape is described interms of intensity CUT for normalized profilesat radius RIM
* Uniform profiles--which have a constant internsityover a radius RUNIF and zero intensity thereafter
The radius R11 of the grid work is selected in the main pro-gram so that it extends well beyond the profile.
Input data describe the shape of the profiles. Inten-sities are adjusted to yield 1 watt of beam power using thefactor QP. Actual beam powers are introduced in subroutineTEMP.
Statements 11.6-18.1--This portion of the subroutinecomputes the factor QP for irregular profiles. The profileis linearly interpolated between points using
p(r) - RX(L- (PX(L)-PX(L-I)) (C-15)p~r)--P(L-) +RX(L)-- RX(L-I)
where p(r) represents the value of PX at r.
The term X5 in the listing represents the integral ofp(r) with respect to radial area over the entire profile.Y5 is determined by integrating between successive pairs ofpoints RX(L-l) and RX(L) as follows
RX(L)
f(r) 27rdr (C-16)
RX(L-!)
Results ahre stored in X5, QP is then computed allowing forenergy losses due to hairs and reflect:ion.
97
"4'
Statements 18.2-23.0--Here the irreg!11ar profile isinterpolated at radial increments RiNT starting at r - 0using equation C-15. Results are stored in FX(L),
Statements 23.1-34.0--This portion of the subroutinecomputes the radial area JFA(L) and power FP(L) betweenr = 0 and r = (L-.5)RINT. Small values are used for RINT,consistent with accurate integration.
Statements 34.1-35.0--Here the values of FA and FPare determined at radii midway between successive radialgrid points R(j). Increments of FP are then divided byincrements of FA and multiplied by QP to arrive at HR(j)for irregular profiles.
Statements 44.0-47;0--Gaussian profiles have the fol-lowing shape
exp-(2r 2 /(SIGMA) 2 (C-17)
This expression is normalized sQ that it equals 1 at r = 0.Input data indicate its value, CUT, at the specified radiusRIM. If one chooses a value of l/e2 for CUT then r = SIGMA,
Setting the above expression equal to CUT and replac-ing r with RIM yields
SIGMA = RIM 'V-'-2/ln(CUT) (C-18)
Multiplying equation C-17 by 27rr and integrating overall r yields ff(SI@GA)4/2. This result is then used tocompute the QP value given by statement 46.1.
Statements 59.0-72.0--Uniform profiles are consideredto be of unit magnitude, and to extend to the specifiedradius RUNIF. The integral of such profiles with respectto radial area is sirmply 7r(RUNIF)L and yields the QP valuecited at statement 59.0.
In that r - RUNIF lies midway between R(NI) and R(Nl+l),only the first NI HR(j) values are set equal to QP. Allotber HR(j) values, with j = N+I, ... N3 are equal to 0.
Subroutine CWATER
This subroutine assigns values for the blood floas,thermal conductivities and volumetric heat capacitie tothe grid points. Blood flows remain constant with respectto time. Thermal properties are expressed in terms ofwater content and hence change with water loss.
98
Statements 6.0-8.2--Blood flows BL(LI) vary with depthby assigninig constant values over tissue layers of thick-nesses of TB(Ll). Blood flows are assigned at each of thedepths Z(i) according to the layer in which Z(i) resides.
Statements 8.3-26.0--This section assigns values forthe water content WW(i), density DD(i), non-water consti-tuents SS(i), and specific heat CH(i) at the various depthsZ(i).
Sweat is considered only if it has a depth greater thanthe arbitrarily small value of 0.0001 cm. Water contentand density are considered constant across the dermis andwithin subcutaneous tissues. If sweat is absent, watercontent and density vary linearly across the epidermisstarting with values W1 and Dl, respectively, at its outersurface, and ending with values W2 and D2 at its base.When sweat is present the values Wl and Dl are replaced bythose for sweat, namely WO and DO.
Equations used to calculate specific heat and thethermal conductivities are presented in the section "ThermalProperties of Skin."
Statements 26.3-33.0--This portion of the subroutineassigns values for water content, heat capacity, and thermalconductivity to each of the grid points.
Statements 40.0-44.0--Whenever water losses occur,the water content WATER(i,j) is adjusted in subroutine TEMP.Then this portion of the subroutine is used to revise theheat capacities and thermal conductivities at the affectedgrid points, i- 1, IMAX and j- 1, JMAX.
Subroutine BA
Subroutine BA computes the matrix elements.Al(i,j),A2(ij), A3(ij), Bl(ij)., B2(ij) and B3(ij). Thesematrix elements are described in Appendix A.
Statements to 11.0--This section computes the arrayspresented by equations A-26 to A-29 and A-31 to A-34.
Statements 12.,0-26.0--The matrix elements are evaluatcdfor a 1-gria ý5"f'ts except for those incurring no heating,points with i.M3 or j -N3. Thereafter, the matrix elementsare revised imnmdiately after any water is lost, Theserevisions pertain to elements associated with grid pointscontained within R(JMAX) and Z(IMAX), The indices JMAXand IMAX ai:e determined in subroutine TEMP.
99
_ _ _ ... .• ,..... -- '' ...-- . . .. • r .. . . . ' I • • . .. • ... . . .. . .j • i . . '
Subroutine HTDEP
Subroutine HTDEP computes the rates of energy depositionS(ij) per unit volume of tissue at each of the grid points(Z(i), R(j)). This is accomplished by computing the radiantintensities at depths ZH(i) midway between the grid pointsand using the intensity differences to determine the ratesof energy deposition.
Statements 5.1-20.0--First, determinations are made ofthe radiant intensities at each of the depths ZE(L) at whichthe absorption coefficients undergo change. Here we startwith a unit intensity (IE(l) =1) at the surface (ZE(l) -0)and compute the intensities IE(L) at each of the depthsZE(L). The depths ZE(L) of interest are
L=I
ZE(L) = TH(k) (C-19)
The intensities at these depths are
IE(l) = 1
IE(2) = *IE(l)exp(-ABS(i)"TH(I))
IE(LZ+!) = IE(LZ)exp(-ABS(LZ)'TH(LZ)) (C-20)
These computations are conducted between statements 5.1 and12.0.
The next step is to use the above intensities IE(L) todetermine the intensities IZ(i) at the depth ZH(i). At thesurface, IZ(1) equals 1. Otherwise
IZ(i+l) = IE(L)exp(-ABS(L) (ZH(i)-ZE(L)) (C-21)
where the index L is such that
ZE(L) < ZH(i) < ZE(L+l) (C-22)
Heat deposition rates are computed between statements16.3 and 20.0. This computation is accomplished by dividingthe difference of successive pairs of IZ(i) by the distancebetween the corresponding depths ZH(i), and multiplying byHR(J).
100
Statements 20.2-22.1--This portion of the subroutineis entered only if a hair follicle is considered. Thefollicle is at depth 7.(IHF) between radii r = 0 and r = R(2),and is considered op~aque.
Statements 32.0-42.0--Whenever irreversible damageoccurs, the code adjusts the rates of heat deposition asso-ciated with grid points within the damage radius R(LNJ).
Here LXX represents the largest index to be considered,It equals the maximum i index associated with heat depositionand damage. First, all S(ij) of concern are initialized tozero.
Of interest are grid points within the maximum depthZ(LMI) of damage. In that some of the tissues may be un-damaged, it is necessary to allow for both damaged andundamaged tissues. From statements 34.1 to 36.0 the radiantintensity is computed for each depth- ZH(i) by multiplyingthe intensity at the prior depth ZH(i-l) by one of the fol-lowing factors:
damaged tissues: exp-[(ABSC)(ZH(i)-ZH(i-l))] (C-23)
undamaged tissues: IZ(i)/IZ(i-l) (C-24)
The remaining computations are similar to that described forstatement 20.0.
Statements 36.1 through 37.0 concern undamagedtissues and follow the same procedure described above.
Statements 42.1-44.1--Here the energy is entirely ab-sorbed by the hair follicle, so no energy is depositeddirectly below the follicle.
Subroutine TEMP
This subroutine performs the following functions
e computes transient temperatures allowing forsteam generation
e adjusts water content WATER(i,j) for any waterloss
e computes maximum temperatures at (R(l), Z(1)),of hair follicle, and of all grid points
101
If mean laser powers are used and boiling commences, thissubroutine treats remaining pulses individually by reeval-uating the arrays DTX(k), POWER(k), and XP(k) for futuretime steps k.
Statements to 2.3--On each entry of the subroutine,IMAX is set equal to zero to account for the fact thatboiling may cease between pulses and certainly after theexposure. Remaining statements in this section initializevarious arrays and parameters.
Statements 3.0-4.1--The previous temperatures V(ij)are preserved VO(ij) for reuse in successively approximat-ing the rates of heat loss in transforming water into steam.After each trial the resultant V values are returned toVO. Here KW (number of trials) is initialized to 0.
Statements 9.0-48.7, 50.0--This portion of the sub-routine computes the temperatures using half time stepsDT/2 for COLUMN and half time steps DT/2 for ROW. Exceptfor indices, the COLUMN computations are of identical formas those for ROW discussed in Appendix A. Statements 9.1-44.0 for COLUMN and 45.3-48.0 for ROW evaluate the termsD, E, F, and G of Appendix A. In this subroutine theterms DXC and DXR correspond with D; CXC and CXR with E;FXC and FXR with F; and SUM with G. Rates of heat deposi-tion are described by POWcS(ij) and rates of heat losses'due to steam formation by SW(i,j).-
Statements 44.1-45.0 for COLUMN, and statements 48.0-48.5 plus 50.0 for ROW evaluate the temperatures using theabove arrays. The equations for ROW are those of equationA-39 of Appendix A.
Statements 48.6-4811, 50.1, 50.2--This portion of thecode evaluates the i and j indices IMAX and JM(i) t"-scribingthe region over which water is lost. These evaluations aremade with all SW(ij) values set equal to zero. Tempera-ture computations are completed if none of the grid points(i = 1 to IP) with available water exceeds the temperatureof boiling water. Otherwise, KW is increased to 1 and atrial and error method is used to determine the SW(ij)values needed to reduce the temperatures to that of boiling.
Statements 50.3-57.0--Two techniques are used to deter-mine SW(ij) by successive approximations. These techniqueswill be labeled 1 and 2. Technique 1 perturbs the previousSW(i,j) values as follows
SW(i,J) ÷SW(i,j) +ZW(KW)(V(i,J) - XW)VSH(i,j)/DT. (C-25)
102
Here the array ZW(KW) is inputed by the user and should havevalues of about 1 to 1.5. The most recent temperaturesare V while the previous temperatures are VS. This techniqueworks best when
1) most of the absorbed heat remains in the parti-cular grid increments during the time step
2) SW(i,J) corrections are relatively small
Statement 50.8 determines whether or not condition 1 is true.If it is true, JSW(i,j) is set equal to 1. Otherwise, itis equal to 0. Condition 2 is judged by comparing the mostrecent temperature changes with the arbitrarily small tem-perature difference of 1IC. Technique 1 always is called forthe first of the NW trials and for trials KW = IKW to NW.
The second technique alters previous SW(ij) values bylarge amounts and is important in rapidly achieving SWvalues of the correct magnitudebut it is not aa good astechnique 1 in making small refinements. Technique.2 in-volves using the previous SW(i,j) correction divided by theresulting V(i,j) changes. The ratio is multiplied by thedesired temperature change XW-V(i,j) and the factor ZW(KW)to arrive at subsequent changes of SW(i,j). If the result-ing SW correction is less than that described by equationC-25, equation C-25 is used in that it provides more accurateestimates whenever the revisions are small.
Statements 57.1-57.2--These statements prevent SW valuesfrom becoming negative or exceeding values allowed by theamount of available water.
Statements 57.3-62.1--Here the change in SW(i,J) isstored in ZSW(1,j) and the most recent temperatures V(i,j)are stored in VS(i,j).
Statements 60.4-62.1 reset the V(ij) values back totheir original values before directing the code to statement9.0 for the next of the NW :rials.
Statements 63.0-70.1--This portion of the subroutineis entered after completing the NW trials. It performs thefollowing functions
"* subtracts water losses from WATER(ij)
"* reinitializes the arrays SW(i~j), JSW(ij),and JM(i)
"* determines the maximum radius R(JMAX) overwhich water has been lost.
103
The next section is entered if some of the pulses have beengrouped together to conserve on computational time.
Statements 70.2-84.2--It is not valid to treat pulsesusing a mean power once steam commences to be generated.In such situations it is necessary to treat the remainingpulses individually. To accomplish this, the subroutinerecomputes the arrays DTX(k), POWER(k), and XP(k) forindividual pulses. The method is identical to that describedin subroutine TIME for noncoded pulse trains.
Statements 70.3-74.0 determined the
* number of pulses that have been treated, Ll
e number of pulses remaining
o minimum time step DTO
@ initialize POWER(k) and XP(k) for remainingpulses
e evaluate the parameters XX2, XX4, XX5, and XX6used in subroutine TIME
* sets the variable time index LK equal to k
* directs the subroutine to statement 82 once allKP remaining pulses have been treated
Statements 74.1-78.1 compute the time steps betweenpulses. These statements are identical to statements 32.1-38.2 of subroutine TIME except for omitting the array XP(k),and the use of KO and LK for k and L3, respectively.
Statements 79.0-80.1 evaluate time steps during pulses.The same procedure is used as described for statements 31.6-32.1 of subroutine TIME. The array NPG(L) is )mitted inthat all remaining pulses are treated indi!',,ally..
Statement 80.2 reduces the remainir s KP to betreated by 1. Then the subroutine (hec if suffi-cient storage is available for the nc ere LKrepresents the number of time steps ' repre-sents the number of time steps nee' oulse,and NTX equals the number of time -lowingthe last pulse. If the number of. thanthe allotted storage MK, the subt . • r thenext pulse. Otherwise it will. ir intstorage is not available, abort :t andindicate the number of pulses ) )rt,
104
* -4 ~ * '~ ~~** ~ ~ 0
Statements 82.0-84.1 compute the time steps followingthe last pulse. The procedure is identical to that per-formed in time. NGX is set to zero to alert the code to thefact that the pulses are being treated individually and toprevent reentry of this portion of the subroutine.
Statements 90.0-92.1--This section is used followingeach time step. It seEtsRGV to the highest temperatureexceeding RGV. Also it searches the region in which super-heated water may result for peak temperature. The peaktemperature and associated depth index i are stored in RBand IB, respectively. Finally, the peak temperature ofany hair follicle is stored in VHF.
Subroutine DAMAGE
This subroutine performs the following funct 'ons:
* evaluates thermal damage D(i,j) at each of thegrid points as well as the radial R(LNJ) indaxial Z(LMI) extent of irreversible damage
* revises matrix elements Al, A2, and A3 if ablister forms
* aborts future time steps when damage becomesinsignificant
Statements to 4.0--KTO represents the index k asso-ciated with the last time step associated with the lastpulse and it is used to abort time steps following thelast pulse once further damage becomes insignificant.
Statements 4.7-6.4--Damage is computed only if thetemperature V exceeds XD, where XD is the lowest tempera-ture at which damage is significant. Damage calculationsare also aborted if XP(k) is not an integer value greaterthan 0. The index LL equals 1 when pulsee remain to betreated. This index prevents damage calculaticns frombeing aborted until the laser exposure is completed.
Statements 6.6-12.0--Here the indices ID and JD areevalua•ed. ThiseInaces correspond to the depths Z(ID)and R(JD) over which damage is occurrinag.
Statements 12.1-14.1--This portion of the subroutinecompares the temperature of grid points, ner. rest to thedepth ZBL at which blisters form, with the temperatureDTENP associated with blister formation. Matrix elementsAl, A2, and A3 are revised, if DTENP is achieved. This in-volves use of equations A-24 and A-25 presented in Appendix A.
105
The index JW is used to avoid recalculating the naatrixelements.
Statements 14.2-22.0--Here damage is calculated onlyif the grid points ie within the region of "live tissue,"at Z(II) or below. Damage is calculated using stepwiseapproximations of equation 11. Incremental damage X3 ismultiplied by the number of pulses XP(k) being represented.
Statements 20.2 and 20.3 determine the i and j indices(LMI and LNJ) of the radius R(LNJ) and depth Z(LMI) ofirreversible damage. These indices are used to alter theabsorption coefficient in subroutine HTDEP whenever irre-versible damage occurs.
Statement 22,1--This statement aborts additional cal-culations when damage becomes insignificant. Five timesteps are allowed at the end of the last pulse beforethis condition is exercised. This provision is providedin that the incremental damage X3 of statement 14.13 maybe small as a consequence of the small time steps usedimmediately following a pulse.
CODE DOCUMENTATION
This section provides compute printouts of the follow-ing:
*nomenclature used in the code
a code-listing
e three sample computer runs with input datacards: a single pulse, multiple noncoded pulsestreated in groups, and coded pulses.
Property data used in the sample runs are identical to thatused in the section "Pig Experiments".
106
BES AVAILABLE COPyC *USUUSUSoUSSmS*6o * 'm~esa.
C *oo PANAHETERS FOLLOWED 4v *OT [ REPRESENT INPUT OATAC AGICL). ANGOAPTIVITY OF SNIM ASSOCIATtD WIT04 LAVER TN(L~,1/CmC AMSCe AhbOr$tIv:TY OP !RmgvEmIOSmv DAMAOGEO SINOI/CMc Aeon, A11,OOPTIVITY OF WAT1R,'.CMc ALPP-A iiRMAL DIFFUGIVITY.CmJIiECC AI.Al9A3 MATRIX ELEMENTS AiSOCIATED WITH Z POP TEMPERATUREC CALCULATION89CALICM-isECo-C ILOOD(IJ) BLOOD FLOW AT 1)oRCI),SMICMI[CC SUAL), BLOOD LOm IN LsTM LAYER OF TNICNNEiS Ti(L),MI4CM3m-|CC 1ie96u8 MATRIX ELEMENTS ASSOCIATED MITH R FOR TEMPERATUREc CfLCUATIONitCAL0CM3-1fCwCc Cl, SPECIFIC MEAT OF SLOODOCA4•OGaCC C(1) SPECIFIC MW or SWEAT OR TISSUES AT OEPTh Z(I).C CALISM-CC CK NUNSIE OF RI0 INCREMENTS PROM LAST OF UNIFORM INCRE.C M[NTI TO FINAL INCREMENTC COhI,9CONI. CONITANTS FOR CALCULATING THERMAL CONDUCTIVITIES AsC FUNCTION OF WATER CONTENTCALICH06E9CCC CON(I.J) THERMAL CONDUCTIVITIES OF SMEAT OR TISSUES AT CI),fR(J)oC CAL4CA-OSECcC CP OI81ANCE FPOM BEGINNING OF LAST UNIFORM GRID INCREMENTC TO END OF FINAL 11CREMENToCOC CUTe CONSTANT INDICATING INTENSITY OF GAUISIAN PROFILEC AT RADIUS RIM CONsIOERING UNIT INTENIITY AT AXIS*C DIMENIIONLEIIC OD(ZIJ) CUMMULATIVE THERMAL DAMASG AT Z(I)vR(J)sDAMAGEC REVERSIBLE "NEN OCIJ) LESS THAN 1IoR[RVIRSIS6L WHENC OCIIJ) GREATER THAN 10OIME1lIONLElC DAML9•I)v NATURAL LOSS OF COEFFICIENTS OP EXPREBIION FOR RATE OFC THERMAL OAMASEmTEPERATUREI BELOW SO C FOR 6L8I TEMPER0C ATURKS ABOVE SO C FOR LualoiLNC(8C)C OAW 93(Ll) ARGUMENTS OF EXPONENTIAL$ OF EXPRESSION FOR RATE OFC TNER1AL DAMASETIMPERATURES BELOW 90 C FOR L@I, TEIMPERmC ATURES ABOVE 90 C FOR LGIMREI 9C Dole AMOUNT OF DAMAGE NEEDED TO CAUSE FIRST DEGREE iURN,C DIMENSIONLEISC Dole AMOUNT OF DAMASK NIEE TO CAUSE 4ECONO 0ESREE $URN*C DININ&IONLIllC 095i TEMPERATUPE ABOVE MNICM TISIUES CHAR.CC Doti) DENSITY AT Z(dI)GM/CM3C OEPID, THICKNESS OF ODEAD* EPIDERMAL LAYERCMC OmFe 09PTH OF HAIR FOLLICLE LOCATED ON BEAM AXIS.CM, WILLC CONSIDER HAIR FOLLICLE ONLY IF OMF IS LESS THAN ZM,C TMEREFORE TO OMIT MAIR FOLLICLE SET OMP TO VERY LARGEc NUMBER SUCH As 1000.C DPULIE, PULSE WIDTH OF NON-COOEO PULSEI CALL UNIFORM)q89C
DPULIC(L), PULIE WIDTHS OF CODED PULSES, Lot FOR FIRST PULtE, LoaC FOR SECOND PULSE9ETC*' HERE OPULSE8S MAY HAVE ANYc POWER FROM 0 AND UPiECC DR MINIMUM RADIAL INCRtMENTCMC OT TIME SUTP WHICH VARIES WITH INDEX MelECC DTLMP, TEMPERATUHE OF SUPERWEATtO WATER AT MMICH TISSUES ARtC ASSUMED TO SEPARATECC DTO MINIMUM TIME ITEPSE9CC OTX(K) TINE INTERVALS AIGOCIATIO.WIT LAMdER POWERS POMER(K)vC SECC 0l MINIMUM AXIAL INCOIMENTCMC PO..DO,9Dl.o DENSITY OP lWEATEPIDIRMAL IURFACEOERMISIAND IUB-
107
C Die CUTANEOUS TISSU28SSN/CM3C PAEL) TOTAL RADIAL AREA FROM R60 TO (Lool)ORINtsCillC FP(L) INTEGRAL OR Mks~) WITH RESPECT TO RADIAL ARIA PROM 480
C TO (L-*S)ORINYCNMSC P(L) INYERPOLATED VALUE OF PX'AT (L"W~RINTOIMNION8hiLill
C 14 MEATOTRANIPER COEPPICIINT AT $KIM GURPACECALoCRelfecCC Hit MEATuTRANSPER COEFFICIENT AT SURPACI(DRY)vCAlsiCMlwl&C-tC "as MCATeTRANIFIR COEPPICIENT AT SURPACKCWET3,CALoCMlwSECeCC "AIRS FRACTION OF RADIATION INTAPCMITD BY 'AR.IESO.Bc MRCJ) 4SAEGR SAM PROFILE A& FUNCTION OP RADIUS R(J) VOQC LASER POwER OP I WAtTyCALiCMS.RICwWATTC "We MEAT TRANSFER COEFFICIENT ACROSS STAM BLISTER.eC CAL#CMl.UEfCwCC I INDEX OF AXIAL INCREMENT$C Io I INDEX OP DEPTH 2(I) AT WHICH PEAK PRESSURES OCCURC ID 1 INDEX OF MAXIMUM DEPTH ?(I$ AT WHICH TWERMAL6 DAMAGEC IS OCCURRING, DAMAGE MAY OR MAY NOT 6E REVERSIBLEC 101. MINIMUM I VALUE AT WHICH ARRAYS V(I*J)90(I.J)oS(IJ~tC ARE PRINTEDC Iotle MAXIMUM IVALUE AT WHICH ARRAYSVIJ)D.JB(J.C ARE PRINTEDC 11(L) PRACTION OP @RAM INTENSITY AT 0EPTH 1ECL)vWMEREC 2911261 AT SURFACE CONRESPONOIWS, TO Z(I)mO.,DI4ENlION-C LESSC IMF I INDEX OP DEPTM CtIMP) OP MAIR FOLLICLEC II 1 :NDEE INDICATING DEPTH OF OLIVE# TISSUleS TISSUES LIVEC PROM I(II) AND BEYONDC 111641116 M4INIMUM AND MAXIMUM I INDICES FOR 1 OR 300 PLMtC II). I INDEX TO SE MARKED ON 1 OR 500 PLOTC IMAN MAXIMUM I INDICI AT WHICH WATER NAI BEEN LOSTC IFP I INDEX INDICATING SMALLOWEtl DEPTH Z(IP) AT WHICHC STEAM IS CONFINE*C IPROO, INDEX INDICATING SMAPE OP LASER PROPIL2999 UNIFORM$C 81 BAUIBIAN. BE IRREGULARC ITYP9, INDEX CONTROLLING PRESUgNCY OP PRINTING TEMPWRTURK89C FOR ITYPEIe TEMPERATURE WILL It PRINTED FOR EACH TIM9C STEP BETWEEN TIME INTERVAL PROM 710WE1(%K) TO YIMI&~(MX)C FOR ITYPEUE EVERY OTMER TIME SylEptyceC ITYPEX INDEX USED TO CONTROL FREQUENCY OP ,PRINTING TEMPERATURE$C IN I INDEX OF DEPTH 1(1) NEARSTl DEPTH ROL AT WHICH TISSUE$C SEPARATEC 12(1) PRACTION OP SEAM INTENSITY AT VARIOUS DEPTNS IM(I.I)oC WHERE IZCI)sI. AT IURPACEDIMINSIONLESSC J INDEX OP RADIAL INCREMENTS,C ~jD J INDEX OF MAXIMUM RADIUS R(J) AT WHICH THERMAL DAMASKC If OCCURRING, DAMAGE MAY OR MAY NOT BE REVERSIBLEC JOI. MINIMUM J VALUE At WHIC04 ARRAYS V(j9J)vDCI9J)9S(IvJlsC ARE PRINTEDC Mel MAXIMUM J VALUE AT MHICM ARRAYS V(IqJ)qO(IqJ)qlCIj)#C ARE PRINTEDC JJIOIJJRO MINIMUM AND MAXIMUM J INDICES FOR a OR 3.0 PLOTSC JM(I) MAXIMUM J INDEX At WHICH WATER It LOST FOR I INDIC9SC LESS THAN IPC JMAX MAXIMUM J INDEX AT WHICM WATER NAB SEEN LOSTC J"RADIAL INDEX ASSOCIATED WITH BLISTER PORMATION, 81C IF GLISTER NAS NOT FORMED* OTHERWISE OLISTER EXTENTSC TO MADIAL DISTANCE R(JW)C K INDEX OP TIME STEPS@ RANGES PROM I TO XTC KK INDEX OF SPECIFIC TIME INTERVAL TIMEI(XK) TO tIMEINKN)C TO ASSES WHETMER OR NOT TEMPERATURES ARE TO ME PRINTED
108
c KO INDEd OF TIME STEPC KP INDEX OF NUMBER OF PULSES REMAINING TO Of TREATEDC ItT TOTAL tdUNSER OF TIME STEP$ TO SE CON41DERED IN CALCUOC LATIONSC RTYPE, IN.DEX INDICATING WHETHER OR NOT TO PRjT CARDS FOR a ORC 3-0 PL0T39KTvPffwI FOR CAROSPKTYPE80 FOR NO CARDSC, KM. NUMBER OF TIME INTERVALS TIMEICKK) TO TIMEZ(KK) DURINJGC NMICM TEMPERATURES ARE TO BE PRINTED.K9I1 TO KiC LASER. INDEX INDICATING WHETHER OR NOT PULSE TRAIN IS CODED*C al IF NOT CODED. 82 If CODEDC LB. NUMBER OF LAVERS AT WHICH4 OLCOD FLOW 1S DESCRIBEDC LOb HIGHEST DEGREE Of BURN PRODUCED90IMENSIONLESSC LI NUMBER OF RADIAL INCREMENTS WITH WHICH TO SUB'PIVIOEC RADIAL DISTANCE FROM R86D TO RXCLR)C LNI@LNJ MAXIMUM I AND J INDICES AT WHICH IRREVERSIBLE DAMAGEC OCCURS.INITIALLY LW! SET EOUAL TO N TO ENTER SUbROUT':NEC CNIATERC LR. NUMBER OF INTENSITIES SPECIFIED IN IRREGULAR PROFILEC LN INDEX CONTROLLING WHETHER OR NOT TO NAINTAIN RATE* OFC HEAT LOSS SW FOR NEXT TIME STEPC LZ9 NUMBER OF SKIN LATER$ HAVING DIFFERENT OPTICAL ASSORPwC TIVITIESC LisL~vL~s.. DUMMY PARAMETERSC P, TOTAL NUMBER OF Z GRID INCREMENTSC "No MAXIMUM NUMBER OF TIME STEPS TO RE CONSIDERED, SHOULDC NOT EXCEED DIMENSION OF ARRAYS POWER9XPv AND OTEC NI. NUMBER OF UN IFORM OZ INCREMENTSC M3 TOTAL NUMBER OF AXIAL.-GRID POINTSC N' TOTAL NUMBER OF RADIAL'GRID INCREMENTSC %I* NUMBER OF UNIFORM DR INCREMENTSc N3 TOTAL MUNEIR OF RADIAL GRIC' POINTSC Mae MUMOER 0F-GROUPS OF PULSES USEO F0'p- NON-COOED PULSEC CALCULATIONSC MGX' INDEX INDICATIING WMETMER OR NOT MULTIPLE PULSES AREC TREATED INOIVIDUALLYCNOX9O), OR IM SROUPICNOX61)C hP NUMBER OF TINE STEPS DURING PULSEC NPUCL), NUMBER oF PULSES IN SUCCESSIVE GROUPSC NPF4CL) MUMBER OF PULSES REP .RESENTED BY PARTICULAR PULSE,C SET s0 EXCEPT FOR LAST OF A SERIES OF INDIVIDUAL PULSESC NPULSE. NUMBER OF*LASER PULSES CONSIDEREDC NTP. MINIMUM NUMBER Of TINE STEP USED TO EMBRACE PULSEC WIDTM ASSOCIATED WITH SINGLE PULSE EXPOSURESC NTZ. NUMMER OF TINE STEPS DESIRED FOLLOWING EXPOSURE;C SHOULD SE SET EQUAL TO TO RELATIVELY LARGE NUMBER TOC ENSURE ALL DAMAGE "AS OCCURRED, CODE MILL END CVMP.C UTATIONS, IF DAMAGE WAI CtASED. NTX VALUES OF ABOUT 20C SHOULD SUFFICE. ALEL^, USED TO CWECK IF SUFFICIENT TIMEC STEPS HAVE SEEN ALLOCATED IN ARRAYS INVOLVING K. SOC DO NOT USE EXTREMELY LARGE NUMBERS OF ORDER OF MgC No, NUMBER OF'TINES TEMPERATURE CALCULATIONS ARE RFCYCLLDC IN ESTIMATING RATES OF NEAT LOSS DUE TO MTAM GENERATIONC PON LASER POWERWATTSC POWEg(K) LASER POWER Ar K-TW TIME STEP9WATTSC PO~cmC(L)e LASER POWERS Of SUCCESSIVE CODED FULSEBYWATTSC. polo. LASER POWER Of NON-CODED-PULIES CALL EGUAL3,WATTSC Px*.i. MAGNITUDE OF IRREGULAR LASER PROFILE AT RADIALC DISTANCES *XCL). VALUES NEED ONLY TO BE RELATIVE* DI~MN-C SIONLESSC PC PREVIOUS LASER POWER POWMATTIC P1 MOST RECENT LASER POWER PONONATTU
109
BEST AVAI ABLE MC uP FACTOR ADJUSYING MAGNITUDE OP PROFILE FOR POWER OfC I wATTY CAL#Chl-SECaWA!TC R(j) RAUIAL DISTA¢CEG STARTZNG AT R(j•wO. AND ENDING ATc RCN3)BRNNCMC Re PEAK TEMPERATURE IN C AT DEPTH$ AT WHICH STEAM CANNOTC ESCAPEC RaF. REFLECTIVITY Of SLIAPACE OF $KIN FOR RADIATION OF GIVENC HAVELENGTH90IMENSIONLE&IC REPRYE. REPETITION RATE FCR NON-CODEO PULSEa.PUL5ESE8#8CC Rgv MAXIMUM TEMPERATURE FOR 2 OR 1-0 PLOTloCC RIM. PADIUS AT WHICH NORMALIZED GAUSSIAN INTENSITY EQUALSC CUTe CMC HINT INCREMENTALLY SMALL RADIAL DISTANCE USED TO INTEGRATEC IRREGULAR PRGFILESO CHC RN MAXIMUM RADIAL OISTANCE.CMC RUNIF, RADIAL EXTENT OF UNIFORM PROFILE*oCC RX(L). RADIAL DISTANCES AT WHICH INTENSITIES PX(L) ARE SPEC-C IFIEDVCMC RIORZOR) FACTORS SY WHICH AXIAL INCREMENTSRADIAL INCREMENT8,ANDC TIME STEPS ARE EXPAND0D9IMENSIONLESSC MCIOJ) RATE OF MEAT DEPOSITION PER UNIT VOLUME OF TISSUE ATC Z(I),R(J)9CAL/CM3-SEC-WATTC 80F6 DIAMETER OF "AIR FOLLtCLEeCHC "00. SPECIFIC MEAT OF NON-WATER CONSTITUENTS IN SWEAT.C CAL/GM-CC SMI.6sIRe CONSTANTS FOR CALCULATING SPECIFIC MEAT OP TISSUESC AS FUKCTION OF MATER CONYENT#CALGMOCC SB(I) GRAMS OF NON-MATER CONSTITUENTS PER CM¢ AT ZCI)C SW(lqJ) ESTIMATED FLUX EXPENDED IN EVAPORATING WATER9CAL.¢CMI.ECC TSCL), THICKNE8E OF SUCCESSIVE LAYERS USED TO DESCRIBE RLOOD4 PLO* BLCL)@CMC TSL(L), TEMPERATURE IN C AT WHICH PRESSURE EQUALS L ATMOSPHERESC TOERM. THICKNESS OF DERMISOCMC TE. ENVIRONMENTAL TEMPEPATUREOCC TEPID, THICKNESS OF EPIODRMIsCMC THE). THICKNESSES OF SUCCESSIVE SKIN LAYERS USED TO DESCRIBEC ABS(L)yCMC TTIME TOTAL ELAPSED TIM•E•SECC TIMEI(KK)., TIMES SETWEEN WHICH TEMPERATURES WILL BE PRINTEDOvECC TIMEC(KK).C TO. INITIAL SKIN TEMPERATUREeCC TOE INITIAL TEMPERATURE OF SKIN ABOVE THAT OF ENVIRONMENTtCC TSuEAT, THICKNESS OF SWEAT LAYER#CMC VOVO NEW AND PREVIOUS TEMPERATURES ABOVE ENVIRONMENTALC TEMPERATURE TE9CC VmF P[AK TEMPERATURE OF HAIR FOLLICLEvCC VSMcIqJ) VOLUMETRIC HEAT CAPACITY OF SKIN AT ZCI)@R(J)*CAL/CM3NCC NAVEL, thAVELENGTN OF LASERS RADIATION*NICRONSC ow(I) GRAMS OF MATER PER CH3 AT DEPTH Z(I)C M0o.W1o9w.q9 GRAMS OF WATER PER C€3 OF SWEAT@JUST BELOW EPIDERMALC w3. SURFACEDERNIS, AND SUBCUTANEOUS TISSUES. RESPECTIVELYC XC, FACTOR BY WHICH TIME STEPS ARE PROGRESSIVELY INCREASED.C IF TEMPERATURES COMMENCE TO OSCILLATE JUST PRIOR TOC PULSES* REDUCE XC CLOSER TO IC XP(L) NUMSER OF PULSES REPRESENTED BY PARTICULAR PULSE IkC DAMAGE CALCULATlONS. ALWAYS I EXCEPT WHEN PULSES AREC GROUPED TO CONSERVE COMPUTATIONAL TIMEC XleXZ.X3.*9 DUMMY PARAMETERSC ZSL DEPTH AT WHICH TISSUES MAY SEPARATE TO FORM SLIITECRCMC *MUST BE EQUAL OR GREATER THAN DEPTH ZOEP
110
!&NINE I
C ZOEP DEPTH S[NEATH WHICH WATER DOES NOT ESCAPE.CmC ZE(L) DEPIM EQUAL TO TF(I),TMC?).,eTM(L.1) FOR L GREATERC THAN IvOTHERWISEto .CMC ZH(I) DEPTHS BELOW SKIN SURFACE. EQUALS CZCI)+ZCI*I))/2.C CMC ZM MAXIMUM AXIAL DIbTANCE.CW
C ZR. CONSTANT CONTRCLLING SIZE OF MININUH TIME STEP DTO.C SHOULD BE 0.! OR SLIGNTLY GREATERoTEMPERATUPE$ N:LLC OSCILLATE IF ZR IS TOO LARGE, DIMENSIONLESSC ZZ. FACTOR BY wHICH SMALLEST TIME STEP IS PROGRESSIVELYC INCREASED IN TREATING SUCEEDING PULSES. USED To CON-C SERVE COMPUTATIONAL TIME WHEN GREATER THAN I. IF TEMP-C ERATURES COMMENCE TO GSCILLATE AFTER SEVERAL PULSESC REDUCE 1Z CLOSER TO !(NEVE9 LOWER)o DIMENSIONLESSC ZW(L). NUMBERS USED TO E6TI AT N HEAT LOSSES DUE TO STEAMC GENERATION* DIMENSION4ES8C 0*9 DIMENSION!MG OF ARRAYS
C ARHAYS :11TH OZKENSION K7e-I"TXPOWER, ANO XPC ARNAYS WITH DIMENSION KXwo-. TIMEIAND TIME2C ARRAYS WITH DIMENSION LB....SLAND TbC ARRAYS WITH DIMENSION LR..*uP•oAND RXC ARRAYS WITH DIMENSION LZew.-ASSvIZTHoAND ZEC ARRAYS WITH DIMENSION 03a --- CW CXCECX.tDCDXRFXCFZ4RJMZZEtwWoC XRIEXR2,XRJXRO.ERSIEZ19XZ12XZ1,XZi.AND XZSC ARNAY$ WITH 0IHtNSIONS W3.N3.uueAlgAltA3,IBB2QSSCONDDostSS.iC VIV$HVXO*VXXaND WATERC ARRAYS WITH DIMENSION NG..--NPGAND NPRAC ARRAYS WITH DIMENSION NPULSE,--OPULSCANO POMERCC ARRAY WITH DIMENSION NW....ZW
C ARkAYS WITH DIMENSION N3*---HRAND RCC INFOR4ATION FOR INPUTING DATA INTO CODE
C NI GREATER THAN OR EQUAL TO.2 v MR LESS THAN TIME INTERVAL DIMENSIONC NIH. RUNIF, AND Rv(LR) GREATER THAN SMPC ZSL GREATER THAN OR EQUAL TO ZDEPC ALL DEPTH INPUT DATA RMFEPRED TO SKIN SURFACEm--NOT FROM SURFACEC OF ANY SWEAT LAYER. THIS INCLUDES DEPlOvOMF9TS(L).TEPIO'THCL).C ZOL9 AND ZOEPC IF TEMPERAIURE OSCILLATIONS ARE EXPERIENCED EXAMINE WHEN THEY OCCUR.C IF IMMEDIATELY FOLLOWING THE START OR END OF THE FIRST FEW PULSES;C REDUCE ZR. IF BETWEEN THE FIRST TWO PULSES REDUCE XC. IF AFTER SEVERALC PULSES REDUCE ZZ..9
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"41 XaX2+A"#(L,.TeL,
IPCK2oLTeS..ANO.L*LF*LZOo TO AtRPADCS~s1)M. IIFCLASEROFOOI)GO fp tad
44ALPNAUCCONI*CON2*wI/DI)/(SHI*DI*gH2*w,,
ZM83,0*(XS+X3)
O?m(TsWFA?.TzPynI),o~lIFC94pT~eNGoTn 46
46 RNNIS*(X0,xI)
IT'rPE~uTTYPE
RfAO(5,S)(TT94E2CL)9Lsl .KX) toRFAOt54P)NW, (ZW(L)*Lwal Nwi
112
WRITE(b,469)0B1 .*t'2.0S3.won
69 pO0mhTC1Nf4O~Ivm,4E1O.4)CALL TIME CALL ICALL GRID CALL ICALL PROF CALL IIF(KTYPEaEGQ*0G~J TO SS
C 0*0 DATA/CARDS VO 2 OR 3-D PLOTfREA0(5. 110jzl#Ia~on!SJJ1,JJk atWRI7E(1.5O)IiIII29II39MOMJ Mips
50 FONM#AT(SI7)WRITEC SoSUN),303i'RITECI.S23 (R(.J).,JmlN3)
52 FOIRIATCIOF?.44)
ISMI9Sma.TDERMoTETEP:0,TO.tSWEATtwotlwIowa.3,WAVELoXC.ZDEPZRoZZii FORMAT(IN
43MTEa.F5,lEX6HTIPIDueF?.4taX,3MTOaFS*o~ag.IyMSWEATutp,.4vtx,53MMOUop~.3,2X,3HwIU.P?~.3,3No3Mw2*.Fb.3,2X.3NW3E6p~.3.IXv~eWAVELgv&F~ie3/I)s.3MXCuqP6q3,IX9gM2DgPUopb.3,3X.3IZRm.FS.1,IX.3HZZu.eF7.I)wR1TECA,19) ?PROPLASERLSL!,NNI,44.NI NPULSE.NCXNTX
99 PORMATCIM ,MPOm I,~4HA~U 1a.M~,aar3~u?,EIaMNE9:?,aX#3MNlUI2,aK,2MMU.Za,,X3S4Mlaol2/IX,?,lNPUL5EGI4.1Ee6,4GN2Kg, Il92W94HNTKU, IiWRITE(696O)ZIL,14wDTEMPOPSHFoNse
60 p0IRMAYCIN ,L4MZSLu.ES.3.lX,3MHWaIS.3,IXqb4DTEMPgF6.0,lX,44DMOMamlEa. 311K OMS4Hu.S*F&3)
64i FORMAT(IN 96IRUS4IFQ~9E3)17(IPR'WEG.1)WRITEC6,&5)CUTRZMew
6S FORMATCIN ,dIMCUTuFA.AI.2Xd4WRIMvqFg6*)ZFCIPROV.R9,1)WRZTE(6...)LR.CRXCL),LE1.LR) b
46 FORMATC1M #lHL~mv13#3wKWalX.l0O9e3))IF(?'PftOPsEQl)WRITEC4467) (PXCL) ,LUI LR)ow
67 FOHN-TC1N ,3MFXm/(lX.l04*3))I!(LAlfX.EQ~a)Go TO 72WRITE(6964)OPULSE*NTPiP0WX was
64 FORMAYCIN ,7IMDPUL5Es.t9.3.?u,4NNTp..1?,lx,5NmPowK.,Iq.3)IP(NPULUf*GT.1)WRYE(6.5e) CNPSCL),La1.NG)mo
69 FORMAT(IM 04AMNP~u/CIE.IOI))IFCNPULIE.9T.1)WN!TE(6,70) CNPRCL) ,LSING)No
70 PIJRMATCIN q4w4P*S/(IKqlOI7))IP(NPULSaS.T.l)WRITEce,71)AfPET swow
71 FORMATCIe4 96MRIPETIagi.4)40 TO TO
71 WRITECS.?e)COPULcC(L),LuloNPUL.I)Be744 FORMATCIM 9THDPULlCm,,(Xl1OSe3))
WPTIE(44,?5) POwEICtL) .LuI tNPULSE)on7% FORMAYCIM *7MP0W9RCU/C1K.IoES.'3))
8'WRITECA6o9)1(TIMEICl),LmIJDg9J,,CTII E woo~sNK
SO FORMAYCIN STI4.ut~lE.)
113
~SEAVAIAB~ COPYWRITE (6.5 *) CTN(L) sLul LZ)se
#I 7OIRMAT(114 3HMT199#1X,1O7Y4)) m
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WRITE6o,83) (TBCL) eLoI*LS)agS3 FORMAT(M t314T5m1(1XIOP7.4))
WRITE(6#84) C*LCL)vm9S wooIIS FO14MAT(11 v39MOAMCIX91)0. ,N9OA( l-,*0.X9))C.~g .
,iRZTECA6g8)DAM(1,13,DAMCIlo),OAMC2,l),OAM(l,8)so17A*jq2Xq9MD140C29il~uqF6q0)WRITE(6vaa)OR*RNvOz9ZM9NG un
86 FORtMATC 14 o314DRgoF6.o~o2X394RNU,76.s4oRK,3MDZoofw6*491IX93HZNB.FS.
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POVETOE
NIE TOE
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C START 0f TEMPERATURE AND DAMAGE CALCULATIONS
(K01
90 DTBDTXCI()TTIMEsTTIME*OTPOWuPOWER I(K)ALIFCIMAX.GToO)CALL CWATER CLIF(IMAX*GToO)CALL SA CALL IIF(L,4196teII)CALL MTDEP CALL xI7(g(,BT.1)B TO 105WRZTEC69100) owe
100 FORMATCIM #28iNMEAT DEPOSITION AEIue.u.)WRIT9(69101) CRCJ) .JJOI ,JO1)ag
101 FORMAT(IH IElR,7.1M3M.uu~~.u..~.uu..00 104 181019102WRITIC6.108)Z(I). $CaciJ) .SJu1JOI.l) s
102 FORMAYCIM v2MZU.*S4dIvlX99ffaI)106 CONTINUE105 WRITEC6.106)DTTT'IME.POW.XPCK).:MAX.JMAX e106 FOkMATCIMO,3NDTUEI.3.2tlKImYIMEUE9.4,INobMPOWEP~gE9.3,EKIM4XPU.
175.09714 IMAXN#12,?H JMAX8911)CALL TEMP CALLCALL DAMAGE CALLI7CXPCK)vLTov409)GO TO 119IF(ITYPEXN.LT.ITYPE.0R.NKX4T.KX)SO TO 119IF(TTIME.LToTIMElCI(K).OR.TT1ME.ST,¶1MEl(KK))S0 TO 119DO 107 !'ID1'!O2
00 107 JUJoIjo?
114
107 VTCI*J)ovtZJ)-TOEWRlTECO. !O1, C(Rj) 'JwjoI .0)00 110 !uDID!2WRITEC..loa)Z(T),evTC1,s),JuJD1,JD2I
106 FORMAT(IN 92MZs*F6.d4.1X99FG.1)110 CONTINUE
ITYPEXo0IFCKTYPEsEQe0)GO TO 11600 112 16111#112
* DO 112 JGJJI#JJ2112 VT(I*J)$V(19J3.TOE
DO 116 181311912WXITE(I.111)(VTCIqJ) .J.JJI ,JJa) wee
*115 FORMATtIOFT.1)116 CONTINUE
WRITEC19I 5)RGYwa116 ITVPEXuITYPEX,1119 1FCTTimte.T.TIMEaCKK).AN0.KK.LT.IX)KxocuIK,
IFCV(1 .1) .T*165)Z8SSUVC 1,)
IFC'(.LE.I(T)G0 TO 90WRITEO,0122)
SRI FOR'IATCIM 97MOAMAGEu,1014....ammmuw)WAZTE(6v1OI)CR(RJ) .JJDI@JOI) wasDO 124 ImII#ID2WRITEC6.123)ZCI). (0C2.J),JuJDl.JD2)wa
13a3 PORMATCIM vlHZw9PI.0,lX99ffl.2)130 CONTINUE
IPCRS.LT*IOO.)GO TO 130L82
136 ZF(RBLTsTSLCL))B0 TO IRSL~sL+1IF(LoLE.1T)GO TO 126
SIG X1UCP~uTSL(L.1l))'(TSLCL)-TSLCLs1))X49L.1 ..XI
130 WRITEC&,131)XotZCIB) .
131 FORMAT(10M 914MPEAX PRISSUP989PT.191'4 ATm.93X96MVEPTNU.PS.*,3Ht'M)[email protected],,F)WRITEC, 133)VHMwo
133 pO9RMATC1M t3TMLOCA'TED AT HAIR FOLLZCLEMTMPERATURE89!1.0,31M C))JDHBO00 140 JdI1N
140 CONTINUEZFCJDOR.EQ,0)60 TO 160XIUALOGCO(I1.IJ0U)JOCIIJ0R,1l))/CRCJOA.1).R(J0R))XIN0(JDR)+ALOBCO( !I.JDA) )/XIWAITECC. 144)XIUw
100 FORMATCIMO917MftADZUI OF OAMAGE., P?.4@3M CM)00 110 I8II9M
ISO CONTINUEX1UALO6(COID1.1)jDCIDZ,1,1))OCZC10Z,1).!(ID2))X28Z(IDZ)+ALOBCDCIDZtI))MK
114 FORMATCINO#16MDEPTM OP DAMAuE89 F?.0,3M CM)160. LONNO'
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161 FOI4MAT(lMO,15I4OEGREE or *URNSW11jST!OP
116
C *.TIME COM~PUTES LASER POWPRS POWIR(K2 FOR tACO TIME STEP 0TXCK)COMMON Sl1 ASL4AA(S3 ,l3.S,(3.1 S(S3,
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C 0** NONOCOOED USS~~mabum~um~ua..mmmuue.m
IP(NPULSE.GTs1)GO TO 31C *** EXPOSURES INVOLVIN4G A SINGLE PULPE
DTIBDPULIE/NTPIFCOTI.GTDTO)GO TO SDO 4 Kwj@NTPDTXCK)uoTl
4 POWER(K)uPOWXL38NTP6O TO 29
*Ll§ALOGIOTl0DTOl/XX6*1XIODTO*CXC*SLloI.)/4XXSIP(X1.~e.Oi4ULS9)GO TO a0LlG(OPULlE-WW)DTI*1OTIN(OPULSK-XI) IP00 10 9819LIDTX (K) oDTTP0 MERKN C po wE
10 OTTmXC*OT'LINLIOLELIGLI.1DO 12 MuLl .LlOTX(K)vOTl
12 POWER(K)NPOWXL39L2so TO aq
4-0 xluaeaa R3UEXPCALOG(CDPULSC*X1DOPULSE)IDTO,.)ILlI)
IFC13#XI.GT..9q9999AND.RS'Xi.Lt.1.00001)0O TO 14
6O TO atAS WR1TE(6916)R3us86 FORMAT(jIN 91144349S~o)
00 26 KUe*LIOTE (K) uDTTPOMEIR(K)@PDWX
AS OTTwRSSDTT
117
_______ _-~
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L.2*L3#MVxDO. 30 X*LILi
so to 110C ** EXPOI"'ES INVOLVJINS MUMtPLE PULSES
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00 38 KeLleL3
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POEKR (K) aEE345 aXPCK*OlKOL3
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DTOUOTO/ZZDO 100 L5INPULS1OTQBZZ*OTOOTTOOTOOYINOPULlC CL)M~TPPCOTI.BT.,OTO)0O TO 76
0O 74 KG.Io1Li
14 PONERCK)sPOWERCCL)KlLa6O TO 100
?6 LIBALOCO(T1#OTO)#EE&9X1SDTO*CXC*$LI.1.) /EESIPCE1,G[*OPUL5CCL))OO TO 90Llu(DPULcC(L)wX1) IDTI*lDTlI(OPULlCCL)mXI).0LlLlsK,1LOGK*LI00 SO KSL39LIIDYE CK)aDTTPONEO(K)OPOW&RCCLI
00 *TTwXCSOiTLIBL4+1
00 82 KvL5vLbDTE (K) soI
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119
100 CONTINUEL.'Iu*Nyx00 Ito KGL39LG
110 DORNuc*DTo
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114 F0OMAYCIM1N .ePOWfR§/tlxqg*,l))
IbFOROAT(im 4)MXpal(IEISF44o))RETURNEN4D
120
101
C 0*0 CALOCULATES RADIA 4R6AOIAI *h WA PAESESS()AD1t3 R(OnoCleSCAPA,13,I,13.SA(I3)S(I3)
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60 TO 32
00 3.6 MIULaLZ(I)al(Iui).-Ki
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6070 TO44 LINLROLM
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II4F§LlIU(LM.LE92)6O TO 47
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67 CMU"30LI
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s0 TO is*0 MIEiSmplo
00 62 ImLI."3
IptXlSXl*GT*MI)GO To 72
?A CONTINUEwRZT9(*.46S) ((J) .Jw.N3)wa
44 FORKAT(IWNm(i.e6)WRITE(*,9@)(CRC?) ,?uiWwasu
90 poN"MAT" qil~u.'(xqlpO.6))RET URNEND
122
*.wew~inemmUPROUT INE 0OeeU U
C *S RO0OMUE INTINSITIEs or LASER SEAN AT GIVEN RADII rOm LAWE
C 000 PONER of I WATT-.INTINS!TIES IN UNITS OF CALICOR-SECCOMMON ABSCII),AB$c.ALPW4AlAC35.35),A2C35.35).43C3
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It FX(L)mo.IFCIPROP.EQ*1'9O TO 44tFCIPROF.EQo0)GO TO S9
C 004 IRREGULAR WEAK PROFILERIkTwRKcLR).4(LI-1)
C *INTEGRATE PROFILE OVER ALL RADII TO DETERMINE 00x5.0sD0 18 L62OLNX2mCPX(L)-PXCLm1))l(RXCL)-RXCLel))Xl*PX(Lwt)inX*RXCL-1)X33E1*CRX(L,)*RKCL)-RXCLwl)*PECL-1))/l.Xsaxl*(RK(L)SRX(L)ORX(L).RXCL.I)*RX(L.&I)RX(L-1) )#3.
to Sm s5UK6e263I3*CX3#4)GPU.23904SC1.-REF)*C I ..NA10IRXi
C * INTERPOLATE PROFILE AT IN6TERVALS OF RIN?L322RIsC.JO 23 LwubLI
20 IF(RX(L2)oGTaKK)6O TO It
IFCLR.LEoLR)GO TO 20GO To 23
PwXCL)sPKCLauI)*X2*(PVCL2)-PX(Li-i))
C CALCULATE TOTAL AREA@PA(LJ@ AND INTEGRAL OF FX(L) WITWd RESPECT TOc RADIAL AREA9FPCL)*FROM RmC TO VARIOUS RADIAL OISTAkC~S (LO*S)*RZNT
00 34 LPZ*LlXI8CLaeS)*RINiTX20CL-1 .5)ORINTFPCL)ufP(L-1).PX(L)S3.141b*(XISXI.El*Xl)
34 FACL)uFACL.I),3.I416*CKI*YIwE2SXl2C 4 CALCULATE PROFILE 04R(J) rOR ALL R(J)
RISo,
0O 35 Jo1,N* X3uCRCJ).R(J*l1))(2.*RINT)4.SOOO@OI
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B ESTAVAILABLE COPY
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weCj)ooPo(Egmxl )lnx-xa,
IS CONTitwue
00 To 70C 040 GAUSSIAN BEAM PRCFILf
MRC J) UOP' NP (wX3)47 CONTINUE
GO TO ToC 9$0 UNIFORM SCAN PROFILE
S9 QPsea390be(1.OEaF)sC1.-NArR)/(3.1416asCNIP*mUN:,,)00 60 JUI.o4I
60 MRCJ)wGP70 wRITE(e.72)(tiR(J),jg~,qN)O72 FORMAT(114 o3kNRv/(lxSE;.3))
RETURNEND
124
C 000 COATE* COMPUTES CONDUCTTVITtESMEAT CAPACITIES VERSUS WATER CONTENTC $00 AND BLOOD FLOWS
COMMON ABSOCII) .ASSC.ALPMA.AI (35.35) .AiI(9353) A3(315#3) .51(353S).SlC23 5.35),B3(3S.35).6Lcl1¶.RLOOOt3S.)sC~oCN1,CONl.CON(35,3S).CUT,
lO(&S,35) iDANCReI) .OWF.DRoT.OTXCZOO2.DPULSEDPULSCCSO),OTEMPDZ.300.DIDZ.03.M.NAIR*1wR(3S) .MWl~ISMjPItYMAX,1~PPRO~,IW.J.AX.KKt,4LAFER.LS.LMILNJLR.LZo~eiMMMIM3.NNI.N3.NG.N6XNPGC3O.) NPRC3O),
?TO&RMTETEPIO.T~ttII).TOE.*TSWEATTT!MEV(31,35) .VMF.V0135.3S).*V514C35.35) WATgR(35.35) ,WO.W1Oow2,W3.EcXP8CIO) 2(35) .ZSLsZDEP.
DIM4ENSION CMC3S)90D(39),65C35)9"WC3g)IPCK*GTot)f;O TO 40
C 0*0 BLOOD FLOWS AT VARIOUS DEPTH$ 2(I)00 6 IuIOM3
6 GLOOD(I)NO,LIGIxl@Tb(I)Tel
I7 IFCZCI).LText)SO TO SLI*Ll.IIFCL.sGT.L8)BO TO 9
so TO 7*SLOODCI)GSLCLI)
*IF(I.LE.N)QG TO 7C *SDENSITIES 009 WATER CONTENT Mw9 NONswATER CONTENT is AND SPECIFICC H* EATS C14 WITH 0R WITWOUT SWEAT
9IFCTSWEAT,.LT..ooo1,bo To 20D18DOwiswo
80 00 a6 IvI.M3IFUZ.GTsI.OR.T9WEATGrYEPID)GO To 21OD(I)s( +@D)TPI*El+6*loa0)TPDT$ET(t~)
ITSIwEAT*TSWEAT) IC4.*TEPID*CTSWEATTEPIO))MMCI )uC CI*3s*ul)STEPID*TEPID*Ce.*Wlua.*Wa)*TEPIDeTgWfAt+COiIuwa,.1TSNEATSTSWEAT) #C400TEPtD*CSWEAT.TCPID)I6O To 23
at IF(ZCI),GT.TSMEAT..o0oI350 TO 22"MM(1)sWoDDCI)wDO
CH(1)wcM(S$0SS( I)#wo)/Oo9O TO 26
a8 IFC(CI).BTTSWEAT#TEP1D)SO TO 24Xi0CZ(I)wTQMEATWjEPID
DOCI)umDI.EIS~Dl.D)
SO TO 8684 IP(Z(I).GToTSWEAT.TEPt!D*TO2Rm)aO TO if
DDCI)uO2
SS(I)uD*'.l+ M "2 aso TO 26 BE.WAAI1LABLE COPY
125
BTEST-AVAIL[ABLE CON
96 CONTZIkulC*** DETERMINE ZkATIAt YTHRMAL CONDUCTIvITJeg AND VOLUMETRIC HEAT CAP*C $99 AC21tjs
DO 33 ISI.M3
00 33 J829%j3
CONCI.j)SCONCII)J3 vSmcI9j)FIv&NcI.I;
Lb RlftIT(6,36(CONCI. 1) .XuI"3)
W~lTEt6936S-3 CS(I, I) .jvi '3)no36 ORMAT(jiq 9AHVlHs4clxqf~q0 4))RETURN
C**ADJUST CONOUCTZt IES AND MEAT CA'~cITes VON WATER LOSS40 DO 44 1mlIMAXDO 44 J*Ioj"AXDDXAPWATER( I J) +$I( I
vfcMXS ~C5M *OszAy~zJ),x
WOND
126
C ; WA O4P"U"TES wMATQIX ELE'M!N73 NEEDED FOR TEPMPERAYURE CUI4PUTATION3
DILMEWLSLM RI.LNJLALZ.M.MKMI e3i.N.NI .3,NG*X E.P~oJ~lCSg3 ,NP(3O, Ilx~cs
IFCKoGT,I)G0 TO 1200 10 J429N
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1a L If1.4 A X*IL24JNAX,1DO 13 [email protected](J.LT.JW.AND.!w.ZQ.ZiGC TO 19
A2C1,JINCCGN(1.J).CONC2,Jl))iZ*OZ),.4107,SLOODCI)SCe4V2*
19 00 23 I42#LIZPCI.LT.JW.ANO.III.EGIW-l)G0 TO 23IF(J.V?.Jw,ANO.Iw,90.Iw)00 TO 23
A3(1,J)uCON(Ij)*2ZPZ3~ )*SLOOD(IiJ)SCfl. lo)*Xil
a3 CONTINUEDO 25 16190.
DO a5 JvlvLZb1(IqJ)C0NCIeJ)*XR1(j),(C0N(1#fj*12scoNCI.j1))*XnR3JIIalCI,4)ICONC1,J, *MA3CJ).SLOODCZ)*CS,2.S3C19J)SC0N(1,mi)SXR4(J)w(CoN~ij,1)ucoN(I.j.1))*Np2cj)
as CONTINUED0 26 JsisLa
86 Sl(IJ)vd2(tj)#i4,oZl
END
BEST- AVAILABLE COPY
127
BESLAVAILABLE -COPY-C H~ TOEP COMPUTES RATES OF ENERGY DEPOSITION AT fACM GRID POINT PERC **UNIT VOLUME OF TISSUE
COMMON ASi),8CAPAa 3~S.23.5.33,5 SCI3)1b2(35.35),W3C35.35).BL(ll).5L000C35),CICONlC0N2,CON(31,SS),CUT92D(35,35) ,0AMC1o2),OPHONoDT.TXC200),DPULSEDOULSCCSO),OTEMPoz,3D0,D1,D2,03,MMAIRNR(35),MWIS.IMPlIM,1AXIIP9PPOPIM,44MAxKKT.IILASE9,LIS.LMILNJLR.LZ.MMKM' ,M3,N.NIN3,NGNGKNPSC3O),NPR(30).SNPULSENTPNTX.NM.POMPOMERC200) ,POMERCIsO) POwX.PXC5O) OR(35) ,Rs,bREP.P6EPETRIMRGV.PNRUNIFR(CSO),U(35,35) .SNF.UNOMlSMIIIT(l1),TTDE'6MTETEPIDTMCII).TOETSWEATTT!MEVCSS,35).VNFoVO(35,35),8VSHt3S,3S),wATERC35.35~.wO~bI19w2,w3,KCWPt2@O).?C3S) ,RSL.ZOEP,9ZM,.!NZWCIQ) ,ZZ*DRS*JwDIMENSION IECII)9IZ(36) .ZFCII) .Z.(36)REAL IE*IZIF(KsGTe1)GO TO 32
IZCI)sO.
00 G IuIOM
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LIULZ~lDO 12 Ls2fLIIECL).IECL.1)SEXPC.AUS(LwI)STMCL.1))IFtIE(L) qLT.I.E10) IECL)vO.
12 ZE(L)wZE(Lw1)+TI4CL-1)
LotLX81.
146 IFCZHCLX)oLT*ZE(L+I))GO TO 16
GO TO 1416 IZ(LX+I)uIECL)*EXP(.A9SCLI*(ZM(LX)m7E(L)))
LWELX,1IPCLX.LE.M.At4o.IZCLX).GT.1.EulO)GO TO 14DO 20 I1,LE6
IPvz(I.T.)XZ(Iste
X3uC IZCT)oZZCI.1 3IXDO 20 JUI.N
80 &CZ9J)vx3*MRtj)JFClNF,EQ*O)9O TO SaDO 22 101MP914
C ADJUST S(ItJ) FOR CHANCES IN ABSOAPTIVITV18 LXX8AMAV1(LX#LMI)
0O 4a2 JulvLNJ
128
00 S0 !uII9LXX
ooWzZ( I I)xxot.00 36 IuI!.LMIIF(X0,LT*.ttEe1)0) TO 36X202"(C1)
ZPCDCIj) .GE.1,3X1UK0*EXP(UAS5C*Xl3
xxo.tiiiz(I1)
36 CONTINUELlmLMI*1IF(LI.GT.LYX)G0 TO 48200 37 ImLI.LXXIP(EO*LT.1.E-S0)GC) TO 3?
IF(X1.LTsl* E-IO3XtmO,
37 CONTINUE483 CONTINUE
IPCINPFG,OQ)6O TO 54DO 414 J.1g47,M
44 8CI1*3u00ICTMP,1)mX3
S4 RkYURN
BEST AVAILABLE-LOPY
129
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1J8MC35,35),UW(35,3S),VS(35.35),VXXC3S,35).ZSWC3S93SI)IMAX80JMAEUOIF(%sGTsl)GO TO 3
00 2 !IsMJMCI).ODO 2,JuINJG*(IqJ~Eo
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3 00 '4 !ulMDO '4 JoloN
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DO '4S L8I.N
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130
IDINC I) SU04/FAW( 1)
458 CONTINI.Jt
00 SO LUI.T m
VXmuXp(T)0CxR(I)$VXIF (V9.LT.1O0.JV'AvtlFIFtl.GE.IP.0N.K*I,G1.0)Gfl TO SO
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L9 *1 M A XDO S1 1u1.LXLIBJMCI)DO $I JueL.l
'31 C ON 71 wE131 IF(K.G.P)i To 6.3
DO 60 IU1,LKLIBJM(IiUO 6V J81*LljFC4ATjECIJ)eLY,.0OI)G0 TO bO
XUS39.*W.ATEX(IsJ)jnl'X385wC19J)
1FCJSiC TJJ ~TO SS
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C ~ CO~kkT FOk LFFE~CTS OF %qATER LOSS6) DO bb IuleLX
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6% JM4AX8AMAXItJMAJ~sLl)jF(NGX.eU,0.UR.LA8ER.,iIa)G0 TO 90IP(POwWtJKC+1)iG7,1I.Ft)Gfl TO 90
c * IEiET TO SIN46LE hflNeC0ODEI PULSES TO HANDlLE WAUkR LOSS
131 BSIVIAI
BESL-AVAI[ABLE -COPA I a AM INI(0dOZ W )
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L3ELK+LI~00? LKOLitL3
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IFCWd.GTV(19J))G0 TO 9211441
132
MaUv C I2j'fi CONTINUt
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133
C see 61AG ePMPUTEG CUMMULAYIVI TMcpR4ALoAmA6f A la c onto Pgg OINTCOMMON S(1)ASaLM.13.5.tfe~ aC.3)S(II.
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10 CONTINUE
C ** EVSEMATIXELMETS P LItE 13M4a AIOS*J
NORM
IP(I0.LT.1l)GO TO 10C 000 EVALUATE TINMEAL WAGES AT FACM S410 POINT OF LIVE TIMSES
Do at 141191000 IN SU1.JO
Iv(Xl.LT~e4,.)O TO It
ZP(LL.ESI)SO TO 20
IC0CI.J~ .LTe1..AN~oX.NS.*ST.)LLUIto 0C1.J)wOCIqJ)*N3
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44 P0IkMAT( IN ILII32.LNU1,X31mI3l3Nb1)is RETURN
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135
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DEPTH OF DAMAGES 0%~58 CM
DEGNEE OF SURN83
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