Biomedical Engineering Reference
In-Depth Information
1
1
3( ()
lifetime at location
r
.
D
and
D
are the
=
=
x
m
3(µ
()
r
+ ′
µ
())
r
µ
r
+ ′
µ
())
r
ax
sx
am
sm
diffusion coefficients of the media at excitation and emission wavelengths. In
equation 9.19, symbol ⊗ is the temporal convolution operator,
U
(
t
) is the unit-step
function, and
c
is the speed of light in the media.
In the FD, by taking the Fourier transform of equations 9.18 and 9.19, the diffusion
formula can be written as
1
∇
()
∇−
()
+
()
−
(
)
=−
(
)
.
Dr
µ
r
µ
r
c
j
ωϕ ω
r r
,,
S r
,
ω
(9.20)
x
ax
af
x
s
s
)
=−
() ()
+
ω ϕ ω
ηµ
ω
r
r
1
∇
()
∇−
()
−
(
(
)
.
Dr
µ
r
c
j
r r
,,
af
ϕ
x
rr
,,
ω
(9.21)
m
am
m
d
1
j
τ
The above formula can be simplified for CW imaging by applying
ω
= 0:
(
)
∇
()
∇−
()
+
()
(
)
=−
()
.
Dr
µ
r
µ
r
ϕ
rr Sr
,
(9.22)
x
ax
af
x
s
s
∇
()
∇−
()
(
)
=−
() () (
)
.
Dr
µ
r
ϕ ηµ ϕ
r r
,
r
r
r r
,
(9.23)
m
am
m
d
af
x
s
In all cases, equation 9.16 in section “Boundary Conditions” can be used to model
the boundary condition.
9.5.4
simulation algorithms of Forward Models
In the last three decades, many computational algorithms based on analytical,
numerical, and statistical models have been developed. These algorithms can be
divided into three different categories. The first category includes the fastest algo-
rithms and is based on analytical approximation of diffusion equations, such as Born
or rytov approximations [43, 48, 49]. The algorithms in the second category are
based on the numerical models of diffusion approximation or radiative transport
equation, using finite element method [50-54], boundary element method [55], or
finite difference method [56]. These solutions are more computational intensive but
are capable of modeling any arbitrary geometry and boundary conditions.
The third category includes algorithms based on statistical modeling, for example,
Monte Carlo simulations and random walk theory [57, 58]. These algorithms track
and record the path of individual photons in turbid media. Their results are very close
to the radiative transfer equation; however, since they are based on statistical models,
they require tracking a significant number of photons, a process that can be compu-
tationally intensive and time-consuming. They can be used to model any arbitrary
geometry with different spatial and temporal resolutions. In these methods, a specific
weight is attributed to each photon (initialized as one). As the photon moves in the
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