Radiative transfer equation and diffusion theory for photon transport in biological tissue
Encyclopedia
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.
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 
denotes time (Figure 1).
Several other important physical quantities are based on the definition of 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 μa, scattering coefficient μs, and scattering anisotropy 
are taken as time-invariant but may vary spatially. Scattering is assumed to be elastic.
The RTE (Boltzmann equation
) is thus written as:
where
, 
, and 
from 
, polar angle
and azimuthal angle 
from 
, and 
). 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:
It should be noted that both of these assumptions require a high-albedo
(predominantly scattering) medium.
n, m. In diffusion theory, radiance is taken to be largely isotropic, so only the isotropic and first-order anisotropic terms are used:
where
n, 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
solid angle. For the vector form, the RTE is multiplied by direction 
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
.
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
μs is the reduced scattering coefficient.
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
. 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.
as the situation demands.
, where 
is the position at which fluence rate is measured and 
is the position of the source. The pulse peaks at time 
. 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.
:
where
is normal to and pointing away from the boundary. The diffusion approximation gives an expression for radiance 
in terms of fluence rate 
and current density 
. Evaluating the above integrals after substitution gives:
Substituting Fick's law (
) gives, at a distance from the boundary z=0,
at a physical boundary is, in general, not zero. An extrapolated boundary, at 
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
. Thus, by definition, 
b must be 
z as defined above. Notably, when the index of refraction is the same on both sides of the boundary, 
F is zero and the extrapolated boundary is at 
b
.
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
is the distance from observation point 
to source location 
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
d
via Fick's law:
is the distance from the observation point 
to the source at 
and 
is the distance from the observation point to the image source at 
b
.
. 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.
Radiative transfer
Radiative transfer is the physical phenomenon of energy transfer in the form of electromagnetic radiation. The propagation of radiation through a medium is affected by absorption, emission and scattering processes. The equation of radiative transfer describes these interactions mathematically...
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
Radiance
Radiance and spectral radiance are radiometric measures that describe the amount of radiation such as light or radiant heat that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction. They are used to characterize both emission from...
. Radiance is defined as energy flow per unit normal area per unit solid angleSolid angle
The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...
per unit time. Here,
denotes position, denotes unit direction vector and denotes time (Figure 1).Several other important physical quantities are based on the definition of radiance:
- Fluence rate or intensity
- FluenceFluenceIn physics, fluence is the flux integrated over time. For particles, it is defined as the total number of particles that intersect a unit area in a specific time interval of interest, and has units of m–2...
- Current densityCurrent densityCurrent density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...
(energy fluxFluxIn the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
)
. 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 energyConservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
. Briefly, the RTE states that a beam of light loses energy through divergence and extinction (including both absorption
Absorption (electromagnetic radiation)
In physics, absorption of electromagnetic radiation is the way by which the energy of a photon is taken up by matter, typically the electrons of an atom. Thus, the electromagnetic energy is transformed to other forms of energy for example, to heat. The absorption of light during wave propagation is...
and scattering
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
away from the beam) and gains energy from light sources in the medium and scattering directed towards the beam. Coherence
Coherence (physics)
In physics, coherence is a property of waves that enables stationary interference. More generally, coherence describes all properties of the correlation between physical quantities of a wave....
, polarization and non-linearity are neglected. Optical properties such as refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
, absorption coefficient μa, scattering coefficient μs, and scattering anisotropy are taken as time-invariant but may vary spatially. Scattering is assumed to be elastic.The RTE (Boltzmann equation
Boltzmann equation
The Boltzmann equation, also often known as the Boltzmann transport equation, devised by Ludwig Boltzmann, describes the statistical distribution of one particle in rarefied gas...
) is thus written as:
where
is the speed of light in the tissue, as determined by the relative refractive index- μt
μa+μs is the extinction coefficient 
is the phase function, representing the probability of light with propagation direction 
being scattered into solid angle 
around 
. In most cases, the phase function depends only on the angle between the scattered 
and incident 
directions, i.e. 
. The scattering anisotropy can be expressed as 
describes the light source.
Assumptions
In the RTE, six different independent variables define the radiance at any spatial and temporal point (, , and from , polar anglePolar angle
In geometry, the polar angle may be* one of the two coordinates of a two-dimensional polar coordinate system;* one of the three coordinates of a three-dimensional spherical coordinate system; in this case it is also called the zenith....
and azimuthal angle from , and ). 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 theoryPhoton diffusion
Photon diffusion is a situation where photons travel through a material without being absorbed, but rather undergoing repeated scattering events which change the direction of their path. The path of any given photon is then effectively a random walk...
(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 mean free path, the fractional change in current density is much less than unity. This property is sometimes called temporal broadening.
It should be noted that both of these assumptions require a high-albedo
Albedo
Albedo , or reflection coefficient, is the diffuse reflectivity or reflecting power of a surface. It is defined as the ratio of reflected radiation from the surface to incident radiation upon it...
(predominantly scattering) medium.
The RTE in the diffusion approximation
Radiance can be expanded on a basis set of spherical harmonicsSpherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...
n, m. In diffusion theory, radiance is taken to be largely isotropic, so only the isotropic and first-order anisotropic terms are used:where
n, 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
solid angle. For the vector form, the RTE is multiplied by direction 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
Mean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...
.
The diffusion equation
Using the second assumption of diffusion theory, we note that the fractional change in current density over one transport mean free pathMean free path
In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...
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'sμs is the reduced scattering coefficient.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
. 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 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 . The diffusion equation is solved for fluence rate to yieldThe 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 coefficientAttenuation coefficient
The attenuation coefficient is a quantity that characterizes how easily a material or medium can be penetrated by a beam of light, sound, particles, or other energy or matter. A large attenuation coefficient means that the beam is quickly "attenuated" as it passes through the medium, and a small...
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 (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 reflectanceFresnel equations
The Fresnel equations , deduced by Augustin-Jean Fresnel , describe the behaviour of light when moving between media of differing refractive indices...
:where
is normal to and pointing away from the boundary. The diffusion approximation gives an expression for radiance in terms of fluence rate 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 at a physical boundary is, in general, not zero. An extrapolated boundary, at b for which fluence rate is zero, can be determined to establish image sources. Using a first order Taylor seriesTaylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
approximation,
which evaluates to zero since
. Thus, by definition, b must be z as defined above. Notably, when the index of refraction is the same on both sides of the boundary, F is zero and the extrapolated boundary is at b.Pencil beam normally incident on a semi-infinite medium
Using boundary conditions, one may approximately characterize diffuse reflectance for a pencil beamPencil beam
In optics, a pencil or pencil of rays is a geometric construct used to describe a beam or portion of a beam of electromagnetic radiation or charged particles, typically in the form of a narrow cone or cylinder....
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
2
for the scattering medium and set the new scattering coefficient μs2 to the original μs1 multiplied by 
1
, where 
1 is the original scattering anisotropy. - Convert the pencil beam into an isotropic point source at depth of one transport mean free path
' below the surface and power = 
'. - Implement the extrapolated boundary condition by adding an image source of opposite sign above the surface at
'
b.
The two point sources can be characterized as point sources in an infinite medium via
is the distance from observation point to source location in cylindrical coordinates. The linear combination of the fluence rate contributions from the two image sources isThis can be used to get diffuse reflectance
d via Fick's law:
is the distance from the observation point to the source at and is the distance from the observation point to the image source at b.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.