In Vivo Molecular Fluorescence Imaging - Nanotechnology for Biomedical Imaging and Diagnostics - page 272

9.5.2 Boundary conditions

Based on the Fresnel reflection theory, if the reflection indices of the strongly
scattering medium and the boundary medium are different, a fraction of the radiation
energy will be reflected back into the scattering medium. If we set the irradiance at
the boundary equal to the integral of the reflected radiance, the extrapolated boundary
conditions [46] in terms of the fluence rate and its normal derivative will be equal to

ˆ .

0

(9.16)

nD rt qrt ∇+ φ ( , ) φ ( , ) =

where n is the outward normal unit vector at the boundary and q is a positive number
related to the internal reflection at the boundary and can be defined by

05 1
1

−
+

R
R

eff

q = . (9.17)

eff

R eff is the effective reflection coefficient and depends on the refractive index of two
media. In a semi-infinite absorption boundary condition, all the photons are absorbed
by the boundary and nothing is reflected back to the media, R eff = 0 and q = 0.5.

9.5.3 Forward Model for Fluorescence light propagation in tissue

The forward model of a fluorescence light propagation in tissue consists of two stages:
(i) excitation of fluorophores inside tissue (by absorbing the excitation photons),
followed by (ii) emission of fluorescence light. Therefore, two coupled Boltzmann
or diffusion equations are needed to describe the propagation of (i) excitation photons
and (ii) emission photons in the tissue [47]:

1 ∂

∂

∇ − . () Dr (µ ( ) r + µ ( )) r + ϕ ( rr t Srt ,, ) () =− , (9.18)

x ax af x s s

ct

1 ∂

∂

ηµ
τ

() ()
()

r r − t Ut

∇ − . () Dr µ ( ) r − ϕ ( rrt ,, ) =−

af

ϕ ( r , rrt

s ,) exp

⊗ ( )

m am m d x

ct r τ

(9.19)

where the subscripts x and m indicate the variables measured at excitation λ x and
emission λ f wavelengths, respectively. φ x ( r , r s , t ) is photon density function at time t
and position r in response to the excitation source S ( r s , t ) with wavelength λ x at loca-
tion r s . φ m ( r d , r , t ) is photon density detected at location r d and time t in response to
fluorescence signal with wavelength λ f emitted from location r. μ ax , μ am , and μ af are
the absorption coefficients of media at excitation and emission wavelengths and of
fluorophore, respectively. η ( r ) is the quantum efficiency and τ ( r ) is the fluorescence