Modelling light transport in tissue

Before image reconstruction can be attempted, a model of photon transport in tissue is required. For diffuse optical imaging, the diffusion equation, in which the propagation of light is assumed to be isotropic, is commonly used. In this section, we discuss the derivation of the diffusion equation from the radiative transport equation. Arridge (1999) provides a more detailed review.

A full description of light propagation in tissue is provided by the radiative transport equation (RTE; Equation 1). This is a conservation equation which states that the radiance (the number of photons per unit volume), $$\phi(r, \hat s, t)$$, for photons travelling from point r in direction $$\hat s $$ at time t is equal to the sum of all the mechanisms which increase minus those effects which reduce it. Equation (1) shows the time-domain RTE. The RTE in the frequency-domain is obtained by replacing $${\partial}/{\partial t}$$ by $$i\omega$$.

$$\left( \frac{1}{c} \frac{\partial}{\partial t} + \hat s . \nabla + \mu_{tr}(r) \right) \phi(r, \hat s, t) = \mu_s(r) \int_{S^{n-1}} \Theta(\hat s, \hat s') \phi(r, \hat s, t) d \hat s' +q(r, \hat s,t)$$. (1)

In (1), $$\Theta(\hat s, \hat s')$$ is the scatter phase function, which gives the probability of a photon scattering from direction $$\hat s $$ to $$\hat s'$$, and $$q(r, \hat s,t)$$ is the light source at r at time t travelling in direction $$\hat s$$. For clarity, we use the same symbols as Arridge (1999).

The RTE is an approximation to Maxwell's equations and has been used successfully to model light transport in diffusive media, turbulence in the earths atmosphere, and neutron transport. However, it does not include wave effects, so the wavelength must be much smaller than the dimensions of the object under study. It also requires the refractive index to be constant in the medium, although extensions for spatially varying refractive index have been derived (Ferwerda 1999, Marti-Lopez et al. 2003, Tualle and Tinet 2003). Unfortunately, to solve for the properties of the light field at all points in a large 3D volume, as is required in optical tomography, the RTE is extremely computationally expensive and simpler models are generally sought. Kim and Keller (2003) recommend the use of a modified Fokker-Planck equation as an approximation to the RTE where scatter is strongly forward-biased.

Three variables in the RTE depend on direction $$\hat s$$: the specific intensity $$\phi$$, the phase function $$\Theta$$, and the source term q. If these are expanded into spherical harmonics, we obtain an infinite series of equations which approximate to the RTE. The $$P_N$$ approximation is obtained by taking the first N spherical harmonics, which gives $$(N+1)^2$$ coupled partial differential equations. As N increases, the $$P_N$$ approximation models the RTE more accurately, but with increasing computational requirements (Aydin et al. 2002).

If we now take the $$P_1$$ approximation and assume that the phase function $$\Theta(\hat s, \hat s')$$ is independent of the absolute angle, i.e. $$\Theta = \Theta(\hat s,\hat s')$$, that the photon flux changes slowly and that all sources are isotropic, we obtain the diffusion equation (2): $$-\nabla. \kappa(r) \nabla \Phi(r,t) + \mu_a \Phi(r,t) + \frac{1}{c} \frac{\partial \Phi(r,t)}{\partial t} = q_0(r,t)$$. (2)

where photon density $$\Phi(r,t) = \int_{S^{n-1}} \phi(r,\hat s,t) d \hat s'$$, and the diffusion coefficient $$\kappa=1/{3(\mu_a+\mu'_s)}$$.

The diffusion equation (2) has been widely and successfully used to model light transport in tissue, although it is necessary to assume that light propagates diffusively. This is generally the case in bulk tissue but the assumption breaks down near the source, near the surface, near internal boundaries, in anisotropic tissues, and in regions of either high absorption or low scatter. In these situations, higher order approximations to the RTE may be required.