Optical Imaging - Correction Techniques in Emission Tomography - page 248

where 0 s is the reduced scattering coecient defined by 0 s = s (1g), with
g the anisotropy factor. Using this expression, the diffusion of photons can be
approximated as an isotropic scattering process, while the actual anisotropy is
captured by g. A medium scattering mostly in the forward direction, for which
g 1, is assumed equivalent to a medium having a smaller reduced scattering
coecient 0 s . In the near-infrared range, 0 s >> a , and it is customary to
define the diffusion coecient by:

1
3[ 0 s ] :

D = (12.10)

Substituting Fick's law in Equation 12.7 leads to the following diusion equa-
tion:

1
c

@(r;t)
@t

= rDr(r;t) a (r;t) + S(r;t): (12.11)

For the time-independent case, the diffusion equation simplifies to

rD(r)rr;t) + a (r;t) = S(r;t): (12.12)

The general dependence of the photon field on optical properties and source
detector distance has the following form:

(r;t) e ikr

r :

(12.13)

Where r is the source-detector distance assuming a point source and a
point detector, and k = ( c a +i!

cD ) 1=2 is the propagation wavenumber of the
photon wave that depends on the absorption coecient a , the diffusion co-
ecient D, the speed of light c in tissue and the modulation frequency ! of
the photon beam that illuminates the tissue. For light of constant intensity,
! = 0. Equation 12.13 describes a generic dependence that does not account
for the effects of heterogeneities or of boundaries, but illuminates the complex
nature of photon attenuation in tissues.

12.2.1.3 Model for a fluorescence heterogeneity

Next to the general description of light traveling through scattering me-
dia, we need an expression for the light propagation when a fluorochrome is
present in the medium [31]. Fluorescence heterogeneities in the medium can
be treated similar to absorption or scattering heterogeneities in the medium.
Here we will derive the corresponding expressions using the Born approxima-
tion [24]. We consider the total eld (r) to be a sum of 0 (r) and sc (r),
where 0 (r) is the incident eld, dened as the eld without heterogeneities,
and sc (r) is the scattered eld, dened as the eld that can be attributed to
the heterogeneities. We assume a distribution O(r) of an optical property per-
turbation; this could be the distribution of the absorption coecient around
a homogeneous optical property value, or similarly the distribution of the flu-
orochromes inside the medium. For fluorescence, the fluence rate at one point