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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). 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.
Additional recommended knowledge
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 . Radiance is defined as energy flow per unit normal area per unit solid angle per unit time. Here, denotes position, denotes unit direction vector and t denotes time (Figure 1).
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 n, absorption coefficient μa, scattering coefficient μs, and scattering anisotropy g are taken as time-invariant but may vary spatially. Scattering is assumed to be elastic. The RTE (Boltzmann equation) is thus written as:
In the RTE, six different independent variables define the radiance at any spatial and temporal point (x, y, and z from , polar angle θ and azimuthal angle φ from , and t). 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:
The RTE in the diffusion approximation
Radiance can be expanded on a basis set of spherical harmonics Yn,m. In diffusion theory, radiance is taken to be largely isotropic, so only the isotropic and first-order anisotropic terms are used:
where Ln,m are the expansion coefficients. Radiance is expressed with 4 terms; one for n = 0 (the isotropic term) and 3 terms for n = 1 (the anisotropic terms). Using properties of spherical harmonics and the definitions of fluence rate and current density , the isotropic and anisotropic terms can respectively be expressed as follows:
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 4π solid angle. For the vector form, the RTE is multiplied by direction before evaluation.):
The diffusion equation
Using the second assumption of diffusion theory, we note that the fractional change in current density 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 μ's = (1 − g)μs is the reduced scattering coefficient.
Solutions to the diffusion equation
For various configurations of boundaries (eg layers of tissue) and light sources, the diffusion equation may be solved by applying appropriate boundary conditions and defining the source term as the situation demands.
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 , where is the position at which fluence rate is measured and is the position of the source. The pulse peaks at time t'. The diffusion equation is solved for fluence rate to yield
The term represents the exponential decay in fluence rate due to absorption in accordance with Beer's law. The other terms represent broadening due to scattering. Given the above solution, an arbitrary source can be characterized as a superposition of short-pulsed point sources. 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.
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 (ie 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 RF:
where is normal to and pointing away from the boundary. The diffusion approximation gives an expression for radiance L in terms of fluence rate Φ and current density . Evaluating the above integrals after substitution gives:
Substituting Fick's law () gives, at a distance from the boundary z=0,
The extrapolated boundary
It is desirable to identify a zero-fluence boundary. However, the fluence rate Φ(z = 0,t) at a physical boundary is, in general, not zero. An extrapolated boundary, at zb 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 . Thus, by definition, zb must be − Az as defined above. Notably, when the index of refraction is the same on both sides of the boundary, RF is zero and the extrapolated boundary is at zb = − 2D.
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):
The two point sources can be characterized as point sources in an infinite medium via
ρ is the distance from observation point (r,θ,z) to source location (r',θ',z') in cylindrical coordinates. The linear combination of the fluence rate contributions from the two image sources is
This can be used to get diffuse reflectance Rd(r) via Fick's law:
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).
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Radiative_transfer_equation_and_diffusion_theory_for_photon_transport_in_biological_tissue". A list of authors is available in Wikipedia.|