Photon transport in biological tissue can be equivalently modeled numerically with Monte Carlo simulations or analytically by the radiative transfer equation (RTE). However, the RTE is difficult to solve without introducing approximations. A common approximation summarized here is the diffusion approximation. Overall, solutions to the diffusion equation for photon transport are more computationally efficient, but less accurate than Monte Carlo simulations.

Definitions

The RTE can mathematically model the transfer of energy as photons move inside a tissue. The flow of radiation energy through a small area element in the radiation field can be characterized by radiance with units Radiance is defined as energy flow per unit normal area per unit solid angle denotes time (Figure 1). Several other important physical quantities are based on the definition of radiance:

• Fluence rate or intensity measured in

• Fluence measured in

• Current density (energy flux measured in This is the vector counterpart of fluence rate pointing in the prevalent direction of energy flow.

Radiative transfer equation

The RTE is a differential equation describing radiance. It can be derived via conservation of energy. Briefly, the RTE states that a beam of light loses energy through divergence and extinction (including both absorption and scattering away from the beam) and gains energy from light sources in the medium and scattering directed towards the beam. Coherence, polarization and non-linearity are neglected. Optical properties such as refractive index, absorption coefficient , scattering coefficient are taken as time-invariant but may vary spatially. Scattering is assumed to be elastic. The RTE (Boltzmann equation) is thus written as: where

• is the speed of light in the tissue, as determined by the relative refractive index

• is the extinction coefficient

• describes the light source.

Diffusion theory

Assumptions

, polar angle). By making appropriate assumptions about the behavior of photons in a scattering medium, the number of independent variables can be reduced. These assumptions lead to the diffusion theory (and diffusion equation) for photon transport. Two assumptions permit the application of diffusion theory to the RTE:

• Relative to scattering events, there are very few absorption events. Likewise, after numerous scattering events, few absorption events will occur, and the radiance will become nearly isotropic. This assumption is sometimes called directional broadening.

• In a primarily scattering medium, the time for substantial current density change is much longer than the time to traverse one transport mean free path. Thus, over one transport means free path, the fractional change in current density is much less than unity. This property is sometimes called temporal broadening.

Both of these assumptions require a high-albedo (predominantly scattering) medium.

The RTE in the diffusion approximation

Radiance can be expanded on a basis set of spherical harmonics. In diffusion theory, radiance are the expansion coefficients. Radiance is expressed with 4 terms: one for n = 0 (the isotropic term) and 3 terms for , the isotropic and anisotropic terms can respectively be expressed as follows:

Hence, we can approximate radiance as before evaluation.):

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.

The diffusion equation

over one transport mean free path is negligible. 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: is the diffusion coefficient and is the reduced scattering coefficient. . This leads to an important relationship; diffusion is unaffected if the anisotropy of the scattering medium is changed while the reduced scattering coefficient stays constant.

Solutions to the diffusion equation

For various configurations of boundaries (e.g. layers of tissue) and light sources, the diffusion equation may be solved by applying appropriate boundary conditions as the situation demands.

Point sources in infinite homogeneous media

. The diffusion equation is solved for fluence rate to yield the Green function represents the exponential decay in fluence rate due to absorption in accordance with Beer's law is the effective attenuation coefficient and indicates the rate of spatial decay in fluence.

Boundary conditions

Fluence rate at a boundary

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). To begin to address a boundary, one can consider what happens when photons in the medium reach a boundary (i.e. a surface). 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. Evaluating the above integrals after substitution gives:

) gives, at a distance from the boundary z=0,

The extrapolated boundary

at a physical boundary is, in general, not zero. An extrapolated boundary, at for which fluence rate is zero, can be determined to establish image sources. Using a first order Taylor series. Thus, by definition, must be as defined above. Notably, when the index of refraction is the same on both sides of the boundary, is zero and the extrapolated boundary is at .

Pencil beam normally incident on a semi-infinite medium

Using boundary conditions, one may approximately characterize diffuse reflectance for a pencil beam normally incident on a semi-infinite medium. The beam will be represented as two point sources in an infinite medium as follows (Figure 2):

1. for the scattering medium and set the new scattering coefficient to the original multiplied by is the original scattering anisotropy.

2. .

3. .

This can be used to get diffuse reflectance is the distance from the observation point to the image source at

Properties of diffusion equation

Scaling

, can be obtained with the following rescaling:

.

Such property can also be extended to the radiance in the more general framework of the RTE, by substituting the transport coefficients with the extinction coefficients

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.

Dependence on absorption

can be obtained as:

Again, the same property also holds for radiance within the RTE.

Diffusion theory solutions vs. Monte Carlo simulations

Monte Carlo simulations of photon transport, though time consuming, will accurately predict photon behavior in a scattering medium. The assumptions involved in characterizing photon behavior with the diffusion equation generate inaccuracies. Generally, the diffusion approximation is less accurate as the absorption coefficient increases and the scattering coefficient decreases. 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). Among the steps in describing a pencil beam incident on a semi-infinite medium with the diffusion equation, converting the medium from anisotropic to isotropic (step 1) (Figure 4) and converting the beam to a source (step 2) (Figure 5) generate more error than converting from a single source to a pair of image sources (step 3) (Figure 6). Step 2 generates the most significant error.

See also

• Monte Carlo method for photon transport

• Radiative transfer

References

Further reading

• (2011)