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Analysis of diffusion theory and

similarity relations for light reflectance

by turbid media

Lihong V. Wang, Steven L. Jacques

Lihong V. Wang, Steven L. Jacques, "Analysis of diffusion theory and

similarity relations for light reflectance by turbid media," Proc. SPIE 1888,

Photon Migration and Imaging in Random Media and Tissues, (14 September

1993); doi: 10.1117/12.154627

Event: OE/LASE'93: Optics, Electro-Optics, and Laser Applications in

Scienceand Engineering, 1993, Los Angeles, CA, United States

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Analysis of diffusion theory and similarity relations

for light reflectance by turbid media

Lihong Wang,

Steven

L. Jacques

Laser Biology Research Laboratory —

Box

The University of Texas M. D. Anderson Cancer Center

1515 Holcome Blvd., Houston, Texas 77030

ABSTRACT

Both diffusion theory and similarity relations for light reflectance by semi-infinite turbid

media have been analyzed by comparing their computational results with Monte Carlo

simulation results. Since a large number of photon packets are traced, the variance of the Monte

Carlo simulation results is small enough to reveal the detailed defects of diffusion theories and

similarity relations. We have demonstrated that both diffusion theory and similarity relations

provide very accurate results when the photon sources are isotropic and one transport mean free

path below the turbid medium surface or deeper. This analysis has led to a hybrid model of

Monte Carlo simulation and diffusion theory, which combines the accuracy advantage of Monte

Carlo simulation and the speed advantage of diffusion theory. The similarity relations are used

for the transition from the Monte Carlo simulation to the diffusion theory.

1. INTRODUCTION

In laser-tissue interaction, there is a growing demand for accurate and fast models to

theoretically predict the light distribution in turbid media, such as biological tissue, with given

optical

,23

and

to inversely deduce the optical properties from measurable

parameters.4''6 One of the measurable parameters is the diffuse reflectance as a function of the

distance between the observation and incident points of the laser beam on the medium surface,

where the diffuse reflectance in this paper is defined as the photon probability of escape from

inside a semi-infinite turbid medium per unit surface area regardless of whether the photon

source is in- or outside. Measurement of the reflectance can be used to determine the optical

properties of tissue non-invasively.7 Therefore, an efficient and accurate model is needed to

relate the reflectance and optical properties of a turbid medium.

Monte Carlo simulation8'9 offers a flexible and accurate approach toward photon

transport in turbid media. It can deal with complex geometries in a straightforward manner, and

can score multiple physical quantities simultaneously. The accuracy of Monte Carlo simulation

has been tested with experimental results.t0 In this paper, Monte Carlo simulation results are

used as standards to be compared with results from other theories. However, due to its statistical

nature, Monte Carlo simulation usually requires tracing a large number of photons to get

acceptable variance; hence, it is computationally expensive, especially when the absorption

coefficient is much less than the scattering coefficient of the media, in which photons may

propagate over a long distance before being absorbed.

Although diffusion theories5'7 offer a fast approach to approximate certain physical

quantities of light transport in turbid media, it is not valid near the photon source or boundary

where the photon intensity is strongly anisotropic, which violates the assumption of diffusion

theory. In therapeutic applications of lasers in medicine, the photon fluence near the source is

the site of the most intense laser-tissue interaction. This region is where diffusion theory is most

inaccurate. In diagnostic and dosimetric measurements, such as the local diffuse reflectance

Rd(r), the reflectance near the source is the strongest, and therefore, can be more easily and

accurately measured experimentally. Again, this region is where diffusion theory is most

inaccurate.

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We would like to build a hybrid model of Monte Carlo simulation and diffusion theory to

combine the advantages of both eoes1 The hybrid model will be used to compute the

diffuse reflectance of an infinitely narrow photon beam normally incident on a semi-infinite

homogeneous turbid medium with given optical properties. Although no photon beam is

infinitely narrow, its response can be convolved to compute the response of a finite size photon

beam.8 Although the real tissue can never be infmnitdy wide, it can be so treated on the

condition that it is much wider than the spatial extent of the photon distribution.

Before building the hybrid model, we need to consider two problems. First, where and

when can we use the diffusion theory to obtain acceptable accuracy? Second, how to connect

the Monte Carlo simulation and the diffusion theory? The studies in this paper will answer

these two questions.

DIFFUSION

THEORY AND SIMILARITY RELATIONS

The optical properties of a turbid medium can be described using four parameters:

relative refractive index, nra; absorption coefficient, jig; scattering coefficient, i; and anisotropy

factor, g. The refractive index re1 jS the ratio between the refractive index of the turbid medium

and the ambient medium. The absorption coefficient ji is defined as the probability of photon

absorption per unit infinitesimal pathlength, and the scattering coefficient jx is defined as the

probability of photon scattering per unit infinitesimal pathlength. The anisotropy factor g is the

average cosine of the scattered angle. The anisotropic scattering of tissue is well represents12 by

a Henyey-Greenstein scattering function.13 We will use the following optical parameters as an

example: re1

1, t2 =

0.1

cm1, lAs

100 cmt ,

0.9,

which are typical for tissues in the

visible and infrared wavelength.'4

A cylindrical coordinate system is set up for this problem. The origin of the coordinate

system is the point of photon incidence on the medium surface. The z =

plane is on the

surface of the medium, and the z-axis points downward into the medium. The radial coordinate

and azimuthal angle are denoted by r and 0, respectively.

In the diffusion approximation of the transport equation, the diffuse photon intensity is

assumed to be almost uniform in all directions.'5 The fluence rate response for a point source in

an infinite turbid medium can be easily solved analytically under the diffusion approximation.

The fluence rate response for a point source in a semi-infinite medium can be estimated by a

linear combination of the responses for the original point source and an added image point

source, where both sources are in an infinite medium, as described by Farrell et al.7 The added

image point source is positioned to satisfy the boundary condition so that the problem of the

semi-infinite medium can be converted into that of an infinite medium.

The real problem we want to solve is the response of an infinitely narrow photon beam

normally incident on a semi-infinite turbid medium with anisotropic scattering. In order to use

the established diffusion theory for an isotropic point source, the infinitely narrow photon beam

has to be converted into an isotropic point source, or into a spatial distribution of isotropic point

sources. Farrell et al.7 convert an infinitely narrow photon beam into an isotropic point source

which is 1 transport mean free path (mfp'), where mfp' =

i/(ji

+ ji(1—g)), down below the

medium surface, and the strength of the point source is the original photon strength multiplied

by the transport albedo a', where a' =

j.t(1—g)/(jx

+ j.z(1—g)). The diffuse reflectance Lj(r) as a

function of r, the distance between the photon incident point and the observation point,

estimated in this model is satisfactory only when the distance r is larger than 1 mfp' (1 mfp' is

about 0.1 cm for this simulation), but poor when the distance is small (Fig. 1).

In this diffusion theory, there are several steps of approximation (Fig. 2).

Initially,

similarity relations16'17 are invoked to convert the anisotropic scattering (Fig. 2(a)) of the medium

into isotropic scattering (Fig. 2(b)). Second, the infinitely narrow photon beam is converted into

an isotropic point source at a depth of 1 mfp' below the surface (Fig. 2(c)). Third, the surface

108/SPIE

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