Light and temperature distribution in laser irradiatedtissue: the influence of anisotropic scattering andrefractive index

Massoud Motamedi, Sohi Rastegar, Gerald LeCarpentier, and Ashley J. Welch

The rigorous method of discrete ordinates was used to evaluate the effects of anisotropic scattering andoptical discontinuity at the boundaries on light and temperature distribution in tissue. The influence ofoptical parameters of tissue on its thermal response was examined by using a finite element solution of theheat conduction equation. Calculations were performed for wide ranges of scattering albedo, the anisotropyfactor, as well as interface reflectivities. This study shows that the presence of an optical discontinuity due toan air-tissue interface forces the maximum peak intensity to move from subsurface to the surface for tissuewith high scattering albedo, which leads to a higher fluence rate in the near surface region. Temperature fieldcalculations show a higher subsurface temperature for a highly scattering medium during tissue coagulation.Neglecting the anisotropic properties of tissue as well as the optical discontinuity at the boundaries wouldresult in considerable error in the calculated temperature rises. Additionally the accuracy of the photondiffusion theory for predicting light and temperature distribution near the tissue surface is examined.

1. Introduction

The advent of various new lasers, the introduction ofthe wideband tunable free electron laser, and the de-velopment of new photochemical and photothermalsensitizers have broadened the horizons in the field ofphotomedicine in terms of both diagnostic and thera-peutic applications. Most of the current applicationsof lasers in medicine are designed for tissue coagula-tion or precise removal of tissue. Fundamental tothese applications of lasers is the determination oflight distribution in tissue, rate of heat generation,heat conduction, temperature dependent rate reac-tions, optothermal properties of tissue, and the dy-namics of tissue ablation.

One of the most common approaches used to formu-late light propagation in turbid media has been theradiative transfer theory. A complete treatment forradiative transport theory was given by Chandrasek-

Massoud Motamedi is with Wayne State University, Departmentof Medicine, Detroit, Michigan 48201; S. Rastegar is with TexasA&M University, Bioengineering Program, College Station, Texas77843; and the other authors are with University of Texas at Austin,Biomedical Engineering Program, Austin, Texas 78712.

Received 22 December 1988.0003-6935/89/122230-08$02.00/0.© 1989 Optical Society of America.

har' who derived the integrodifferential equationwhich describes the rate of change in the radiance of apencil beam of light as a function of the optical proper-ties of the participating medium

s VL(r,s) = -g4L(r,s) + 1 1t/4,r J P(s,s')L(r,s')dw' (1)

whereL(r,s) = radiance at a position r in direction s(W/cm2/sr),

At = Ma + A, total attenuation coefficient(cm-'),

A = scattering coefficient (cm-'),Ma = absorption coefficient (cm-'),s = direction of propagation of the beam,s' = direction of scatter of the beam,

p(s,s') = phase function, which defines the prob-ability that the light is scattered from s'into s, and

= solid angle.The quantity L(r,s) is the monochromatic radiance

at location r in the direction s, which is the amount ofenergy transported at point r in the direction s per unitarea perpendicular to this direction per unit solid an-gle. The quantity on the left-hand side of Eq. (1)represents the gradient of the radiance at point r in thedirection of propagation s. The first term on the right-hand side of Eq. (1) accounts for the decrease in theradiance due to absorption and scattering. The lastterm represents the increase in radiance due to energyscattered into the direction of interest from all other

2230 APPLIED OPTICS / Vol. 28, No. 12 / 15 June 1989

directions. The phase function p represents the prob-ability density function that a photon moving in thedirection s' will be scattered into direction s.

Since a general solution for Eq. (1) is not available,various approximate solutions have been used to studylight transport in tissue. Analyses which have ac-counted for scattering in tissue have been primarilybased on the isotropic assumption. van Gemert et al.2

and Wan et al.3 have published a computational meth-od for light distribution in tissue. This method isbased on the heuristic 1-D approach of Kubelka-Munk,4 which is a two-flux isotropic approximation tothe radiative transfer theory. More recently, photondiffusion theory has become a popular model for esti-mating light distribution in tissue for isotropic andanisotropic scattering and index matched conditions.5-7

However, this approximation is only valid deep withina medium, and its accuracy near the surface has notbeen examined. Three-dimensional models have alsobeen developed which account for the finite size of thelaser beam.8 9 These models have generally been de-veloped based on the photon diffusion theory8 or adiscrete anisotropic phase function9 and matchedboundaries.

Despite all the above approximate methods, whichin certain cases yield satisfactory results, careful andaccurate examination of the effect of anisotropic scat-tering and boundary conditions on the light and tem-perature distribution in laser irradiated tissue is need-ed. The discrete-ordinate method for radiativetransport theory, introduced originally by Chandra-sekhar,' is a suitable method for serving such a pur-pose. An advantage of this method is its ability toaccount easily for various boundary conditions as wellas optical heterogeneity. Moreover, the 1-D modelcan be easily extended to several dimensions with cy-lindrical and spherical geometries,'0 and implementa-tion of multilayer calculations is straightforward."This method has been successfully utilized in manydisciplines including meteorology'2 and oceanogra-phy.' 3

In this study a 1-D discrete ordinate approximationto the radiative transfer equation and a finite elementsolution to the bioheat equation are used to evaluatethe effect of tissue anisotropicity and optical boundaryconditions on light and temperature distribution, re-spectively. The predictions of the optical field nearthe surface as calculated by the photon diffusion the-ory and the discrete ordinate method are also com-pared.

A. Light Distribution in Tissue

For the purpose of this study, the tissue is modeledas a semi-infinite slab. The incident radiation is as-sumed to be collimated and normal to the interface,and the positive z direction corresponds to the tissuedepth. The first step in modeling the propagation ofcollimated incident light in tissue is to separate thepropagating radiation fields into collimated and non-collimated components. The collimated radiance canbe modeled as

dL,(z)/dz = -L,(z)- (2)

To estimate the scattered light, a discrete ordinatemethod utilizing Lobatto quadrature and phase func-tion renormalization developed by Houf and Incro-pera'3 is used. This method divides the radiation fieldat any optical depth (r = IAtz) into a number of discretestreams. Consider the transport equation in its 1-Dform shown as Eq. (3). By using the theorem of spher-ical harmonics,'4 the phase function is then expandedinto a series of N Legendre polynomials about M (thecosine of scattering angle) in Eq. (4):

(3),u[dL(ru)Id7I = -L(T,u) + wo/2 J P(',)L(7,A')du';

+1,u[dL(Tu)/dr] = -L(T,,u) + w0/2 L(r,u')

-1

N

X > WkPk(u')Pk(u)du'.K=O

Finally Lobatto quadrature formula15 is used to ap-proximate the integral in Eq. (4). This analysis resultsin a system of M coupled linear nonhomogeneous dif-ferential equations of the form

M

gi[dLi(T,si)IdT] = -Li(T,Ai) + wo/2 E aLi(r,gi)ji=1

N

X W WkPk(ASi)Pk(Ak) 1 • i < M, (5)k=O

where Li represents the noncollimated radiation field,A1i is the ith discrete ordinate whose value depends onthe order of quadrature, ai is the ith weighting factor ofLobatto integration, Gk is the kth coefficient for Legen-dre polynomials expansion of the phase function, Pk isthe kth Legendre polynomial, and wo = Ms/t is thesingle scattering albedo.

For the purpose of this study at the air-tissue inter-face, a Fresnel boundary condition is applied, whereasfor the lower interface the boundary is assumed to bediffusely reflecting.'3

Due to the heterogeneous properties of tissue and theabsence of information about the individual scatteringparticles in biological media, attempts have been madeto develop simple analytical methods to represent theangular distribution of scattered light within the tur-bid medium. Recent studies have suggested that theHenyey-Greenstein phase function,16 which has oneparameter and satisfies a closed form relationship,

p(A) = 0 (l - g2)(1 + g2 _ 2g- )-1.5 (6)

provides a reasonable estimate for the scattering phasefunction of biological media such as dermis and vascu-lar tissue in the visible portion of the spectrum.9 "17

Here g is an adjustable parameter that completelycharacterizes the phase function and represents themean cosine of the scattering angle. It can range from-1 for purely backward scattering through 0 for isotro-pic scattering to +1 for purely forward scattering. Inthe present analysis the following relation between thephase function represented by Legendre polynomials

15 June 1989 / Vol. 28, No. 12 / APPLIED OPTICS 2231

(4)

and the mean cosine of the scattering angle g is used'8:N

P(O) = (2k + l)g&P,(cos0), (7)k=O

where a value of N = 150 suggested by Incropera andHouf'9 is used to better represent a highly forwardscattering phase function.

In this study the total fluence rate is defined as thesum of the scattered and collimated fluxes and is givenby an expression

ML(z) = E aLj(r,,)Aj + Lj(z). (8)

ill

B. Thermal Response of Tissue

The thermodynamics of laser interaction with tissueconsists of two distinct threshold dependent phases.The first phase is the coagulation of tissue due to thedenaturation of proteins and enzymes which eventual-ly leads to destruction of live cells. The second phaseof tissue response is the ablation phase, which is aprocess that can involve a number of endothermic andexothermic reactions and tissue vaporization resultingin explosion, disintegration, and carbonization. Theissue of the influence of the light field on the thermalfield during the preablation phase is addressed in thisstudy.

Heat generation in tissue is a function of the productof the absorption coefficient and the total light fluencerate:

QL(rz) = ia(r,z) I L(r,z)dw. (9)

The temperature rise in tissue due to photon absorp-tion is calculated using the heat conduction equation

pj(aT/Ot) = V- (kVY) + QL, (10)

where T is the temperature (K), p is the density (kg/In3 ), c is the specific heat of the material (J/kg K), k isthe thermal conductivity (W/m K), and QL is the laser-generated heat source (W/m3).

In the past, different approximate solutions to thebioheat equation have been used to predict the ther-mal response of laser irradiated tissue.2 0'2' These so-lutions have been based on the assumption that thelight distribution in tissue could be modeled by esti-mating the total fluence rate in tissue using Beer's lawand matched boundary conditions. To account for theeffects of light scattering on tissue thermal response,van Gemert et al.2 2 used a heat source based on a two-flux approximation to describe the light distribution intissue. Other investigators7 have applied the diffu-sion approximation to the radiative transport equationwhich accounts for the anisotropic nature of tissue andallows for collimated incident light but does not con-sider the mismatched boundary conditions.

To examine the influence of the optical parameterson the thermal response of tissue, Eq. (10) was reducedto 1-D geometry and solved using a finite elementmodel (see the Appendix). The heat source term QL

was based on a 1-D reduction of Eqs. (9) and (8). For

this solution, ninety-eight linear elements were usedover a length of 3 mm. Near the surface, where thehighest temperature gradients were located, the ele-ments were more densely distributed. A Crank-Nicol-son time stepping scheme with a fixed step size of 5 Mswas used.23 The calculations were performed based onthe reported thermal properties of the vessel wall.24

For the purpose of this study, the following uniforminitial condition was used:

T(z,0) = T = 370C. (11)

The boundary condition of no heat flux for a finite slabwas used at z = 0 and z = d:

-k(dT/dz) = 0. (12)

The effect of blood perfusion and heat generation dueto metabolic reactions were neglected.

II. Results and Discussion

A. Parametric Study

The following parametric study using the discreteordinate approximation to the transport theory wasconducted to evaluate the influence of anisotropicityand optical boundary conditions on the light distribu-tion in tissue. In these calculations the incident irra-diance of 100 W/cm2 and the recommended value of Pd= 0.4727 for the diffuse internal reflection for tissuewere used. Figures 1 and 2 show the effect of varyingthe phase function asymmetry factor g on the totalfluence rate. The results show that if the medium hasa low single scattering albedo, the propagating beamwill assume almost the same angular distribution afterpenetrating one optical depth. However, within thefirst optical depth as g increases, transmission (for-ward scattering) increases, and consequently the localabsorption rate near the surface decreases.

For a highly scattering medium, the influence ofanisotropic scattering shifts the maximum of the scat-tered light intensity to greater optical depths, and thetotal fluence rate decays more slowly with higher gcompared with a more isotropic medium (Fig. 2).

B. Matched vs Mismatched for Front Surface

The effect of front surface boundary conditions onthe total fluence rate was studied as a function of singlescattering albedo coo, phase function asymmetry factorg, and index of refraction n. First, the effect of aniso-tropic scattering was evaluated. The distribution ofthe total fluence rate for low single scattering albedo(wo = 0.5) as a function of phase function anisotropicity(g = 0.8 or 0.0) for matched vs mismatched conditionsis displayed in Fig. 3. For this study, the index ofrefraction of tissue was assumed to be the same as thatof water, n = 1.33. As the phase function becomesmore isotropic the difference between the value of thetotal fluence rate at the surface between matched andmismatched conditions increases. This is mainly dueto the increased backscattered photons which reachthe front boundary and are bounced back due to totalinternal reflection at the tissue-air interface. Thiseffect is more apparent in a medium with high albedo.

2232 APPLIED OPTICS / Vol. 28, No. 12 / 15 June 1989

150

125

75

50

25

Albedo =0.50

E

6

Ici

L- a --- g=0.80- g=0.50

- g=0.00

0 +0 1 2 3 4 5 6

Optical Depth

Fig.1. Computed total fluence rate as a function of anisotropyg fora moderately absorbing medium. Incident irradiance of 100 W/cm2.

Index matched at surface.

300

Albedo =.98

200 . - _ =° 8U -.--- ~~~~~~~~~~~~~~~g=05

o 100

0-0 3 6 9 12

Optical Depth

Fig. 2. Computed total fluence rate as a function of anisotropy

factor g for a highly scattering medium. Incident irradiance of 100W/cm2. Index matched at surface.

This is illustrated in Fig. 4 where the total fluence rateis calculated for high albedo (wo = 0.98) for an isotropic(g = 0.0) and a highly anisotropic medium (g = 0.8).The total internal reflection tends to smooth thefluence profile. Hence the local absorption profile ismore uniform, and the highest rate for absorption ofenergy occurs at the surface.

Since there are no known values for the index ofrefraction of biological media, it is of interest to deter-mine theoretically the effect of different indices ofrefraction for tissue on the total fluence rate. This isimportant for cases where the water content of tissuechanges. The sensitivity of the total fluence rate tothe value for the index of refraction (varied from 1.0 to1.46) was examined for two different conditions. For alow albedo (00 = 0.50, g = 0.50), relative small differ-

100

75

50

Albedo = 0.50

g=0.80, n=1.00-- g=0.80, n=1.33-Q---D- g=0.00, n=1.00

- g=0.00, n=1.33

25:1

0 1 2 3 4 5 6

Optical Depth

Fig. 3. Total fluence rate as a function of boundary conditions andanisotropy factor. Incident irradiance of 100 W/cm2 .

500

400

300

200

0

1-

100

0

Albedo=0.98

g=0.80, n=1.00

- g=0.80, n=1.33

-u--- g=0.00 , n=1.00

v g=0.00, n=1.33

0 2 4 6 8 10 12

Optical Depth

Fig. 4. Total fluence rate as a function of anisotropy factor g formatched and mismatched boundaries. Incident irradiance of 100

W/cm 2 .

ences (<10% at the surface) between the total fluencerate predicted for matched boundary vs that predictedfor mismatched boundaries are depicted in Fig. 5.Moreover, the predicted fluence rate values for thedifferent indices converge at about one optical depth.However, for high albedo (0 = 0.91, g = 0.50) thedifferences between matched boundary and mis-matched boundaries as a function of refractive indexare considerably higher (Fig. 6).

C. Reflection at the Distal Surface

Many tissues are multilayered in nature; therefore,it is important to quantify the effect of reflection ateach interface. For a relatively low absorbing andstrongly forward scattering medium, the interface be-tween the layers can make significant contributions tothe light distribution within the top layer. Figure 7shows the comparison between a nonreflective distal

15 June 1989 / Vol. 28, No. 12 / APPLIED OPTICS 2233

2

0

-

125

Albedo=0.50, g=0.50

- n=1.00

n=1.20

-a--- n=1.33

------ n=1.46

250

200

150

_ It

ac 250

C 200

a 150

100

50

0

+ DO: g=0.5- Dif:g=0.5

1 2 3 4 5 6

+ DO: g=0.8- Dif : g=0.8

0 1 2 3 4 5 6

Optical Depth

0.0 0.3 0.6 0.9 1.2 1.5 Fig. 8. Comparison of total fluence rates computed using discreteOptical Depth ordinate and diffusion approximation.

Fig. 5. Total fluence rate as a function of index of refraction for lowalbedo. Incident irradiance of 100 W/cm2.

300

61 200

1-u

7a 100

0

Albedo=0.91, g0.50

a, n=1.00

- n=1.20

-- n=1.33- n=1.46

0 2 4 6 8 10 12

Optical Depth

Fig. 6. Total fluence rate as a function of index of refraction forhigh albedo. Incident irradiance of 100 W/cm 2 .

300 -

Albedo=0.98

- g- - g0. 8 0, R=0.47

6-00--- ga0.80, R-0.00

-1 200 -id ~-s- g=°0.00, R=0.47

-,-. -g0.00, R=0.00

1-o

0

interface and a diffusely reflective distal interface. Inthe case of a highly scattering medium with an acutelyforward phase function (g = 0.8, coo = 0.98), there is lessradiation absorbed near the upper surface, whichmeans more photons will reach the distal interface,and consequently more photons are diffusely reflectedback from the reflective surface. This effect tends toincrease the effective absorption deep inside the tis-sue, and thus the local rate of photon absorption ismore uniform. For the isotropic case (g = 0), with highsingle scattering albedo (e.g., 0 = 0.98), the reflectionat the distal interface plays no significant role. Forthe case of low albedos (such as c0 = 0.50), the lightdistribution is dominated primarily by absorption, andthe role of diffuse reflection from the bottom interfaceis reduced significantly.

D. Comparison of Photon Diffusion Theory and DiscreteOrdinate Method

Recently many investigators have used the photondiffusion theory to model light propagation in tissue.Since the diffusion approximation is assumed to bemost accurate in the region far from the sources andboundaries in highly scattering media, it is importantto examine its accuracy near the surface against thediscrete ordinate method, which will be used as abench mode.

An analytical solution to photon diffusion theory6

was used to estimate the total fluence rate and com-pare its predictions with those made by the discreteordinate method. The calculated total fluence rate asa function of asymmetry factorg is shown in Fig. 8. Upto a moderate degree of anisotropicity (g = 0.5), thedifference between these techniques is insignificant.These differences are due to the nature of the analyti-cal solution used for the diffusion model and the nu-merical solution used in the discrete ordinate method.However, the difference between these models be-comes significant when a highly anisotropic phasefunction is used (g = 0.8). In this case the calculationof total fluence rate based on the diffusion model over-estimates the predictions of discrete ordinate in the

2234 APPLIED OPTICS / Vol. 28, No. 12 / 15 June 1989

125

Ue 1003

0 75

175

0 3 6 9Optical Depth

Fig. 7. Total fluence rate as a function of bottom reflection.

. . . . . .

Isotropic Scattering (g=0)

a-- Albedo=.98

- Albedo=.91_-. Albedo=.50

0.30.0 0.1 0.2

Thickness (mm)

Fig. 9. Relative volumetric power density as a function of scatter-ing albedo for isotropic scattering.

0.0

Anisotropic Scattering (g=0.8)

- Albedo=.98- Albedo=.91

Albedo=.S0

0.1 0.2

Thickness (mm)

Fig. 10. Relative volumetric power density as a function of scatter-

ing albedo for anisotropic scattering.

transition zone between the multiple scattering regionand diffusion zone where the number of scatteringevents has not reached the required level for formationof the in-depth condition for the diffusion approxima-tion to be valid. After passing the transition region(about four optical depths), the two predictions showexcellent agreement (Fig. 8). This indicates that thediffusion model is not valid in the first few opticaldepths, which is the critical region for most biomedicalapplications. This difference is primarily due to howthe phase function is implemented in each technique.The diffusion approximation considers only the firsttwo terms of the Legendre polynomials, whereas thediscrete ordinate model accounts for the first 150terms of Legendre polynomials. With only two termsto represent the phase function, the diffusion modeloverestimates the isotropic diffuse flux.

GI&

E

0.00 0.05 0.10 0.15 0.20 0.25

Depth (cm)

Fig. 11. Predicted temperature rise in tissue as a function of tissue

anisotropicity. Incident power density of 65 kW/cm2 .

E. Temperature Distribution in Laser Irradiated Tissue

Parametric analysis was performed to evaluate theinfluence of the optical properties of tissue as well asthe optical boundary conditions on the thermal re-sponse of tissue. In this parametric study both opticaland thermal properties of tissue have been assumedconstant. The irradiation condition used in this studywas based on the assumption of using a single pulse oflaser light with fluence of 65 kW/cm2 and pulse dura-tion of 100 ,s.

The effect of the absorption coefficient on the heatsource term is shown in Figs. 9 and 10 for isotropic (g =0.0) and anisotropic scattering medium (g = 0.80).This analysis indicates that for highly absorbing medi-um, the maximum intensity of the heat source is at thesurface. This is in contrast to the case of high scatter-ing albedo for which the maximum intensity of theheat source is below the surface (Fig. 10). Because theheat source is directly proportional to the light distri-bution in tissue [Eq. (7)], the heat source term assumesthe same profile as the light distribution in tissue.

Figure 11 shows the effect of the phase function onthe temperature field. Higher surface temperatures.are calculated for the isotropic medium, because thereis more backscattered light deposited near the surface,whereas for the anisotropic case, the predicted tem-perature deep within the tissue is higher due to itshighly forward scattering behavior (Fig. 11).

The influence of the surface boundary condition onthe temperature rise in laser irradiated tissue was pre-dicted and is shown in Figs. 12 and 13. For a mediumwith a slightly anisotropic scattering behavior (g =0.5), the difference between the surface temperaturerise in a matched condition (n, = n2) and mismatchedcondition (n1 sz, n2 ) was -35%. The same effect waspredicted for highly anisotropic medium, but the dif-ference in the surface temperatures had decreased be-cause there were fewer photons participating in theprocess of total internal reflection at the interface. Itshould be recognized that high scattering properties

15 June 1989 / Vol. 28, No. 12 / APPLIED OPTICS 2235

3000

E

2 2000

.I_

00

4) 1000E

a2

3000

2000-

1000

E

-a3'5:ER

0

110

100

90

80

70

60

501-

401

30 1-0.00

Mismatched

75

653

0.05 0.10 0.15 0.20 0.25 0.30

55 -

45 -0.00 0.05 0.10 0.15 0.20 0.25

Depth (cm)

Fig. 12. Predicted temperature rise in tissue with matched andmismatched boundary conditions (isotropic scattering). Incident

power density of 65 kW/cm2 .

95

85

&

I-

75

65

55

45 Al

35 1--0.00

Fig. 13. Predicmismatched boi

tend to estaltant temperaIn the case ofseen; howeveof the scatterof total internoted that focoagulation a

Depth (cm)

Fig. 14. Predicted temperature rise in tissue on discrete ordinateand diffusion models. Incident power density of 65 kW/cm2 .

the surface. This means that for high temperaturerise in tissue, the explosive vaporization process maybe delayed mainly because the maximum temperature

\ smatched is positioned at the surface during the preablationphase.

Figure 14 shows the predicted temperature risesbased on photon diffusion theory and discrete ordinatemethod for a highly asymmetric phase function. Thisanalysis shows that the diffusion model overestimatesthe predicted temperature near the surface consider-ably.

Ill. Conclusion

bedo =.98, p 5=1/cm, g=0.50 An analysis has been presented for modeling of lightdistribution in tissue. The influence of the refractiveindex, scattering phase function, and scattering albedo

0.05 0.10 0.15 0.20 0.25 0.30 on light and temperature distribution in laser irradiat-Depth (cm) ed tissue has been demonstrated. The accurate meth-

cted temperature rise in tissue with matched and od of discrete ordinates and heat conduction equationindary conditions (anisotropic scattering). Inci- were used for predicting light and temperature distri-dent power density of 65 kW/cm2. bution in tissue. As a result of this study, the following

conclusions can be drawn: (1) For highly scatteringtissue, the maximum light intensity and temperature

dlish a maximum fluence rate and resul- rise occur below the surface. Furthermore, the posi-ture rise below the surface of the medium. tions of these peaks shift more into the tissue as thematched boundaries, this effect is easily single scattering albedo is increased. (2) The presence

r, in a mismatched condition, the results of optical discontinuity at the interface results in aning properties are obscured by the process increase in the total fluence rate and the temperaturenal reflection (Fig. 13). It must also be near the surface. (3) The diffusion approximation isir laser applications which involve tissue insufficient for describing light distribution near thend ablation, the surface boundary condi- surface for a highly anisotropic medium.

tion plays a critical role in determining the extent ofsurface damage in the coagulation phase and the mech-anism of the initiation of tissue destruction in theablation phase (explosive ablation vs surface ablation).Thus the role of the surface boundary condition mustbe considered in interpreting the results of in vitro andin vivo experiments. For example, the presence of amismatched boundary condition forces the peak tem-perature rise to move from the subsurface position to

Appendix: Numerical Solution for a Heat ConductionEquation

A finite element discretization of the heat conduc-tion equation can be done as follows. An approximatesolution U(z,t) to T(z,t) is sought of the form

U(zt) = EUj(t)j(z),j

2236 APPLIED OPTICS / Vol. 28, No. 12 / 15 June 1989

E

EE2

85

where Oj are finite element basis functions and Uj arenodal finite element solutions to be sought. A weakformulation of the differential Eq. (8) can be written as

(pcaU/Ot,,Pj) = (LU,0j) + (QL,'5i), i = 1,2,...,m,

where (., .) denotes an L2 inner product of the form(4j, ij) = fD(z)A(z)dz in a domain D. The operatorin this case in L is ka2/az2 . The system of ordinarydifferential equations to be solved, in proper initialconditions, for this formulation is

AdU/dt = BU + c,

where Aij = (pcpj,<p)Bij= kdj/dzxafjldz),Ci = QL,oi ),

Uj = U(zj,t).For time discretization a generalized Crank-Nicol-

son time stepping scheme is employed which is writtenas

A(Un - Un-)/At = P[BUn + cn] + (1 - P)[BUn-1 + cn-1]

In this formulation v = 0 results in a forward differ-ence scheme, v = 1 results in a backward differencescheme, and v = 1/2 results in a Crank-Nicolson-typescheme. For the present analyses v = 1/2 was used.

The authors wish to thank Wai-Fung Cheong of theUniversity of Texas at Austin for helpful discussions.

This work was supported in part by the Office ofNaval Research under contract N14-86-K-0875.

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