Biomedical Engineering Reference
In-Depth Information
is the surface flux density [W/cm
2
] of the incident beam and r
sp
where
I
o
is the specular
reflectance. The optical boundary condition at the beam axis (
r
¼
0) is
r
¼0
¼
@
f
d
@
r
0
:
ð
17
:
38
Þ
The boundary condition elsewhere is
f
d
2AD
r
f
d
n
¼
0
ð
17
:
39
Þ
where A is the internal reflectance factor and
is the inward unit normal vector. The inter-
nal reflectance factor A can account for the effect of mismatch in the index of refraction
between the boundary and the surrounding medium and is given by
n
1
þ
r
i
A
¼
ð
17
:
40
Þ
1
r
i
where
r
i
is evaluated by an empirical formula
n
2
rel
n
1
rel
r
i
¼
1
:
440
þ
0
:
710
þ
0
:
688
þ
0
:
0636
n
rel
,
ð
17
:
41
Þ
and
n
rel
istheratiooftherefractiveindicesofthetissueandthemedium.Theinternal
reflectance factor
A
reduces to 1 in cases where the boundary is matched—that is, when
n
rel
¼
1.
Equations (17.35) through (17.41) provide the governing differential equations
and boundary condition for the diffusion approximation. A solution of these, either
by analytical or numerical methods, allows the calculation and analysis of fluence rates
within a scattering and absorbing media such as biological tissues. As an example,
for an isotropic point source inside an infinite medium, the solution for fluence rate
as measured by a detector embedded inside the medium at a “large” distance
r
from the fiber can be derived using the Green's functions solution of the preceding
equations to be
e
r
=d
r
f
o
4p
f
ð
r
Þ¼
ð
17
:
42
Þ
D
p
D
where
d ð¼
=m
a
Þ
is the penetration depth.
EXAMPLE PROBLEM 17.5
It is desired to measure the concentration of an absorber in a scattering medium with known
scattering coefficient. If the reduced scattering coefficient
0
s
m
is known and the relative intensity
at a distance
r
o
from an isotropic source can be measured, solve an algebraic equation for the
absorption coefficient based on the diffusion approximation given
r
o
is large enough for diffusion
approximation to be valid.
