Radiative transfer equation and diffusion theory for photon transport in biological tissue
Photon transport in biological tissue can be equivalently modeled numerically with Monte Carlo simulations or analytically by the radiative transfer equation (RTE).
Overall, solutions to the diffusion equation for photon transport are more computationally efficient, but less accurate than Monte Carlo simulations.
[1] The RTE can mathematically model the transfer of energy as photons move inside a tissue.
By making appropriate assumptions about the behavior of photons in a scattering medium, the number of independent variables can be reduced.
[1] Radiance can be expanded on a basis set of spherical harmonics
Using properties of spherical harmonics and the definitions of fluence rate
, the isotropic and anisotropic terms can respectively be expressed as follows: Hence, we can approximate radiance as[1] Substituting the above expression for radiance, the RTE can be respectively rewritten in scalar and vector forms as follows (The scattering term of the RTE is integrated over the complete
):[1] The diffusion approximation is limited to systems where reduced scattering coefficients are much larger than their absorption coefficients and having a minimum layer thickness of the order of a few transport mean free path.
Using the second assumption of diffusion theory, we note that the fractional change in current density
The vector representation of the diffusion theory RTE reduces to Fick's law
, which defines current density in terms of the gradient of fluence rate.
Substituting Fick's law into the scalar representation of the RTE gives the diffusion equation:[1]
Notably, there is no explicit dependence on the scattering coefficient in the diffusion equation.
[1] For various configurations of boundaries (e.g. layers of tissue) and light sources, the diffusion equation may be solved by applying appropriate boundary conditions and defining the source term
A solution to the diffusion equation for the simple case of a short-pulsed point source in an infinite homogeneous medium is presented in this section.
represents the exponential decay in fluence rate due to absorption in accordance with Beer's law.
Taking time variation out of the diffusion equation gives the following for a time-independent point source
is the effective attenuation coefficient and indicates the rate of spatial decay in fluence.
[1] Consideration of boundary conditions permits use of the diffusion equation to characterize light propagation in media of limited size (where interfaces between the medium and the ambient environment must be considered).
The direction-integrated radiance at the boundary and directed into the medium is equal to the direction-integrated radiance at the boundary and directed out of the medium multiplied by reflectance
b for which fluence rate is zero, can be determined to establish image sources.
Using a first order Taylor series approximation, which evaluates to zero since
[3] Using boundary conditions, one may approximately characterize diffuse reflectance for a pencil beam normally incident on a semi-infinite medium.
The linear combination of the fluence rate contributions from the two image sources is This can be used to get diffuse reflectance
be the Green function solution to the diffusion equation for a homogeneous medium of optical properties
, then the Green function solution for a homogeneous medium which differs from the former only by optical properties
The usefulness of the property resides in taking the results obtained for a given geometry and set of optical properties, typical of a lab scale setting, rescaling them and extending them to contexts in which it would be complicated to perform measurements due to the sheer extension or inaccessibility.
be the Green function solution to the diffusion equation for a non-absorbing homogeneous medium.
The assumptions involved in characterizing photon behavior with the diffusion equation generate inaccuracies.
[7][8] For a photon beam incident on a medium of limited depth, error due to the diffusion approximation is most prominent within one transport mean free path of the location of photon incidence (where radiance is not yet isotropic) (Figure 3).
