Biomedical Engineering Reference
In-Depth Information
where
Δα
is the update of optical parameters and ϕ(
and
φ
(
α
) are the measured
and calculated values for photon density function, respectively. If the
Δα
s are small,
the higher order differential components in equation 9.28 can be disregarded. In dis-
crete form, it can be written (as averaged values in each imaging voxel) as
N
∂
∂
ϕ
α
∑
()
i
∆
ϕϕ ϕα
=
()
α
−
=
∆
α
,
i Mj
=… =…
1
,
,
,
1
,
,
N
(9.29)
i
i
i
j
j
=
1
j
where
M
is the total number of measurements and
N
is the total number of imaging
voxels.
equation 9.29 can be converted to matrix equation as
∆
φ
= W
∆
α
(9.30)
where
W
is the weight (Jacobian) matrix and represents the sensitivity of photon
density function, corresponding to each source-detector pair, to optical parameters
of different voxels (
W
ij
= ∂
φ
i
/∂
α
j
).
For a point source and detector, photon density function can be written as
0
Ω
′
⊗⊗
′
′
ϕ(
rrt SGrrt Ot Grrtdr
ds
,,
)
=
(
,,
)
()
(
,,
)
(9.31)
em d
ex
s
where
G
(
r
′,
r
″,
t
) is the Green function of photon propagation to position
r
′ in
response to a delta source located at initial point
r
″. Subscripts “
ex
” and “
em
” repre-
sent the calculated or measured variables at the excitation and emission wavelengths,
respectively.
Ot
(
)
η
−
t
τ τ
is the pulse response of fluorescence target with
quantum yield of η and lifetime of τ. Ω represents the target region where fluorescent
probes are located.
For FD imaging systems, equation 9.31 can be written as
()
=
exp
S
DD i
ηε
ω
0
∫
′
′
′
ϕω
π
(
rr
,,
)
=
Grr Grr Nr d
(
,,
ω
)
(
,,
ω
)
(
)
Ω
(9.32)
ds
em d
ex
s
4
1
−
τ
Ω
ex
em
where
ω
,
η
, and
ε
are the modulation frequency, quantum yield, and extinction coef-
ficient of fluorophore, respectively.
N
(
r
) is the concentration of fluorophores. FD
equation can be used in CW systems by inserting
ω
= 0.
A comprehensive discussion on Green functions for different boundary conditions
and imaging geometries is provided in refs. [69-71]. Green functions for diffusion
equation in the FD and an infinite medium can be written as
(
)
exp
ik rr
cD
− ′
,
′
=
Grr A
(
)
(9.33)
4π
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