Biomedical Engineering Reference
In-Depth Information
in the medium r
0
produced by the perturbation O(r
0
) can be written in terms
of the incident field
0
(r
0
) as [24, 37]
sc
(r
0
) = O
f
(r
0
)(r
0
; r
s
;!):
(12.14)
The measured field at the boundary of the scattering medium can be derived
by integrating Equation 12.14 over the complete medium and propagating the
fluence rate from the fluorochrome to the boundary:
Z
g(r;r
0
;!)O
f
(r
0
)(r
0
; r
s
;!)dr
0
sc
(r;r
s
;!) =
(12.15)
where g(r;r
s
;!) is the Green's function solution of the diusion equation for a
single delta function at r
0
. To solve Equation 12.15 bears the diculty that the
eld (r;r
0
) is generally a function of
sc
(r;r
s
), because this photon distribu-
tion depends on O(r). To solve Equation 12.15 analytically, a linearization is
performed using an approximation such as the Born or the Rytov approxima-
tion [24, 36]. In both approximations, the quantity (r) is essentially assumed
equal to the photon field
0
(r) that is established by the source at r
s
in a
similar medium without the heterogeneity present. As derived from the Born
approximation, Equation 12.15 can then be written as
Z
g(r;r
0
;!)O
f
(r
0
)
0
(r
0
; r
s
;!)dr
0
:
sc
(r;r
s
;!) =
(12.16)
The difference between this equation and Equation 12.15 is the replace-
ment of (r;r
0
;!)dr
0
by
0
(r;r
0
;!)dr
0
. We can further refine this expression
when considering that the light traveling toward the fluorochrome has the ex-
citation wavelength
ex
, and the light traveling from the fluorochrome to the
boundary travels at the emission wavelength of the fluorochrome
em
[9, 37]:
Z
g
em
(r
0
; r
s
;!)O
f
(r
0
;!)
ex
(r;r
0
;!)dr
0
:
fl
(r;r
s
;!) =
(12.17)
0
The distribution of fluorescence Of
f
is complex and depends on the mod-
ulation frequency of the source as well as on the quantum yield of the u-
orochrome, the extinction coecient of the uorochrome , the uorochrome
concentration F(r) and the uorochrome lifetime [31]:
O
f
(r)
[F(r
0
)]
1 i!
:
(12.18)
12.2.2 Reconstruction of the fluorochrome distribution
Fluorescence measurements can always be referenced to excitation meas-
urements obtained under identical experimental characteristics by use of dif-
ferent filter sets; see Figure 12.5. Advantage of this additional information
can be taken by normalizing the fluorescence measurement with the emission
Search WWH ::
Custom Search