Biomedical Optics and Lasers - Introduction to Biomedical Engineering - page 1125

ds

I ( r , s ) I + d I

s

s '

FIGURE 17.7 Light intensity change in the transport theory approach including scattered light from other

directions.

of scattering and absorption, m t ¼ m a þ m s , is defined as the attenuation coefficient. The

s 0 Þ

function is called the “phase function” and is related to the scattering amplitude

of a particle, a scaled form of the probability distribution of scattering angles. Note that
by ignoring scattering and assuming a collimated light source, Eq. (17.34) reduces to
Eq. (17.29), which was derived in the last subsection. S

p ð s ,

is the source term that could
be irradiation on the surface, fluorescence generated inside the tissue, or an internal vol-
umetric light source.
The equation of transfer is an integro-differential equation for which a general solution
does not exist. However, several approximate solutions have been found, such as the
two-flux and multiflux models, the discrete ordinate finite element method, the spherical
harmonic method, the diffusion approximation, and the Monte Carlo method. Each of these
is subject to certain limitations and assumptions. In this chapter the focus is on the diffusion
approximation, which is one solution.

ð~ r, s Þ

^

Diffusion Approximation
The diffusion approximation is a second-order differential equation that can be derived
from the radiative transfer equation (17.34) under the assumption that the scattering is
“large” compared with absorption. The solution to this equation provides a useful and
powerful tool for the analysis of light distribution in turbid media. The governing differen-
tial equation for the diffusion approximation is

m s
D

2 f

3m a m t

d

f

d ¼

f ð 17 : 35 Þ

—

c

where f d is the diffuse fluence rate and the parameters of the equation are m' ¼ m s (1 - ),

g

m' ¼ m a þ m s 'and

D ¼

1/3 m' , in which is defined as the anisotropy of the medium.

g

[W/cm 2 ], is the sum of the collimated part,

The total light fluence rate, f f c ,andthe

diffuse part, . The total fluence rate, as given by the following equation, is a key

parameter in laser-tissue interaction. It can be thought of as the total light received at
a point in space, or within the medium, through a small sphere divided by the area of
that sphere.

f d

f ð r , z Þ¼ f

c ð r

, z Þþ f

d ð r

, z Þ: ð 17 : 36 Þ

The collimated fluence rate is given by

m t z Þ f

c ¼ I o ð r Þð

1 r sp Þ exp ð , ð 17 : 37 Þ