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.
Contents
- Definitions
- Radiative transfer equation
- Assumptions
- The RTE in the diffusion approximation
- The diffusion equation
- Solutions to the diffusion equation
- Point sources in infinite homogeneous media
- Fluence rate at a boundary
- The extrapolated boundary
- Pencil beam normally incident on a semi infinite medium
- Diffusion theory solutions vs Monte Carlo simulations
- References
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
Several other important physical quantities are based on the definition of radiance:
Radiative transfer equation
The RTE is a differential equation describing radiance
where
Assumptions
In the RTE, six different independent variables define the radiance at any spatial and temporal point (
It should be noted that 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
Hence we can approximate radiance as
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
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
Using the second assumption of diffusion theory, we note that the fractional change in current density
Notably, there is no explicit dependence on the scattering coefficient in the diffusion equation. Instead, only the reduced scattering coefficient appears in the expression for
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 and defining the source term
Point sources in infinite homogeneous media
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. The source term in the diffusion equation becomes
The term
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
where
Substituting Fick's law (
The extrapolated boundary
It is desirable to identify a zero-fluence boundary. However, the fluence rate
which evaluates to zero since
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):
- Set scattering anisotropy
g 2= 0 for the scattering medium and set the new scattering coefficient μs2 to the original μs1 multiplied by( 1 − g 1) , whereg 1 is the original scattering anisotropy. - Convert the pencil beam into an isotropic point source at a depth of one transport mean free path
l ' below the surface and power =a '. - Implement the extrapolated boundary condition by adding an image source of opposite sign above the surface at
l '+ 2 z b.
The two point sources can be characterized as point sources in an infinite medium via
This can be used to get diffuse reflectance
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 μa increases and the scattering coefficient μs 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.