Multimodal Diffuse Optical Imaging - Biomedical Optical Imaging Technologies: Design and Applications - page 358

cerebrospinal fluid, synovial fluid in joints) [ 54 , 55 ]. In other cases, approximations
to the RTE are introduced to provide more computationally efficient methods.
In the case of predominant scattering, the diffusion equation that is an approxima-
tion of the radiative transport equation is preferred due to its simpler mathematical
expression. Its mathematical expression both in the frequency domain and the time
domain can be easily solved analytically for planar geometries [ 56 , 57 ] (the CW case
corresponds to a null frequency). In the case of heterogeneous media, the diffusion
equation can be also solved analytically by using perturbative approaches such as the
Born or the Rytov approximation [ 58 ]. However, for complex boundary conditions,
numerical methods or more refined analytical algorithms [ 59 ] are necessary to
model accurately the light propagation, leading to increase in the computational
burden. Even though the diffusion equation is less accurate than MC methods or the
RTE, it is the most commonly used forward model in dynamically diffuse optical
tomography. It has been employed successfully to provide 3D functional maps of
tissue activities in the brain [ 60 , 61 ], breast [ 62 - 64 ], and muscle [ 65 , 66 ].

10.4.2 Use of A Priori Information

Even though considerable advances have been performed in modeling and recon-
struction techniques, due to the diffuse nature of the light collected, diffuse optical
tomography remains an ill-posed, poorly conditioned inverse problem that can
suffer from nonuniqueness and low spatial resolution. One way to improve the
performance of optical tomography is to incorporate a priori information in the
image formation process. Lately, the incorporation of two different kinds of priors
has been demonstrated to significantly improve the performance of functional
optical properties: spectral and spatial priors.
Incorporating spectral priors in the optical inverse formulation consists in
utilizing all the multispectral information collected in one inverse problem to
estimate directly the functional and structural parameters of interest conversely
to classical methods that derive these parameters from optical reconstructions
of absorption and scattering performed at each individual wavelength. In this
formulation, the absorption coefficient is modeled as the linear sum of all the
individual chromophores in the tissue:
a j D X
i

" j C i

(10.1)

where ". j / is the wavelength-dependent extinction coefficient, known a priori, of
the i th chromophore with a concentration C i (Hb, HbO 2 ,H 2 O, and lipids in the
case of the breast). The scattering coefficient is modeled with a spectral power law
derived from a simplified Mie scattering theory [ 44 , 45 ]:
s 0 j D A b (10.2)

j