Biomedical Engineering Reference
In-Depth Information
Lrst
is the radiance,
s
is the unit vector pointing in the direction of
interest,
ω
is a solid angle, and
μ
s
and
μ
a
are the scattering and absorption coeffi-
cients, respectively. (
ˆˆ
)
ˆ
where
(,,)
f ss
′
is the normalized probability for a photon traveling in
the direction of
s
and being scattered in the direction
ˆ
s
, which satisfies
ˆ
Qrst
is the source and defines as the power injected from the
direction
s
into a unit solid angle centered at
r
[44].
By integrating the transport equation over all solid angles, equation 9.8 can be
simplified as
∫∫
ˆˆ
′
Ω=
.
(,,)
(
ss d
,
)
1
4
π
1
c
∂
ϕ
()
.( )
rt
t
,
+∇
jrt
,
=−
µϕ
(
r t Srt
,
)
+
(
,
)
(9.9)
a
∂
where the source
S
, photon density (fluence rate)
φ
(
r
,
t
), and the flux
j
are defined as
(9.10)
∫∫
ˆ
Srt
(,)
=
Qrstd
(,,)
Ω
4
π
(9.11)
∫∫
ˆ
φ
(,)
rt
=
Lrstd
(,,)
Ω
4
π
(9.12)
∫∫
(,,
ˆˆ
jrt
(,)
=
Lrstsd
Ω
4
π
For some geometries, the general closed-form solutions of the rTe are not available,
and the closed forms, based on diffusion approximation, are usually applied. If the
reduced scattering coefficient is much larger than the absorption coefficient (in other
words, if many scattering events happen before an absorption event, i.e.,
μ
a
<<
μ
s
′), the
transport equation can be simplified to the diffusion equation
1
∂
ϕ
()
()
rt
t
,
0
(9.13)
∇∇ −
.
Dr t
ϕ
(
,
)
µϕ
(
r t
,
)
+
+
Srt
,
=
a
c
∂
(
)
=
(
)
′
+
where
D
=
13 1
[(
−
g
)
µµ µµ
s
+
]
1 3
(
)
is the photon diffusion coefficient,
g
is the mean cosine of the scattering angle, and µ
s
is the reduced scattering coeffi-
cient. In the FD, the diffusion formula can be written as
a
s
a
j
c
ω
φω ω
−∇ ∇
.
Dr
φω µ
(
,
)
+
−
(
r
,
)
=
S r
(
,
)
(9.14)
a
where
ω
is the angular modulation frequency. For CW mode,
ω
= 0 and the diffusion
equation simplifies to
(9.15)
−∇ ∇+ =
.
Dr rSr
ϕµϕ
a
( )
. () ()
The diffusion approximation is not accurately very close to the source and/or
boundaries and very close source-detector pairs [45].
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