Photon transport in biological tissue can be equivalently modeled numerically with
Monte Carlo simulations
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
or analytically by the
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 trans ...
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
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's la ...
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
In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiati ...
. Radiance is defined as energy flow per unit normal area per unit
solid angle
In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point.
The poi ...
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
*
Fluence
*
Current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional a ...
(energy
flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...
)
. 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
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means tha ...
. Briefly, the RTE states that a beam of light loses energy through divergence and
extinction
Extinction is the termination of a kind of organism or of a group of kinds (taxon), usually a species. The moment of extinction is generally considered to be the death of the Endling, last individual of the species, although the Functional ext ...
(including both
absorption
Absorption may refer to:
Chemistry and biology
*Absorption (biology), digestion
**Absorption (small intestine)
*Absorption (chemistry), diffusion of particles of gas or liquid into liquid or solid materials
*Absorption (skin), a route by which s ...
and
scattering
Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
away from the beam) and gains energy from light sources in the medium and scattering directed towards the beam.
Coherence
Coherence, coherency, or coherent may refer to the following:
Physics
* Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference
* Coherence (units of measurement), a deriv ...
,
polarization and non-linearity are neglected. Optical properties such as
refractive index
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, ...
,
absorption coefficient
The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient valu ...
μ
a, scattering coefficient μ
s, and scattering anisotropy
are taken as time-invariant but may vary spatially. Scattering is assumed to be elastic.
The RTE (
Boltzmann equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
) is thus written as:
:
where
*
is the speed of light in the tissue, as determined by the relative refractive index
*μ
tμ
a+μ
s 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.
Diffusion theory
Assumptions
In the RTE, six different independent variables define the radiance at any spatial and temporal point (
,
, 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 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 ...
(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.
Both of these assumptions require a high-
albedo
Albedo (; ) is the measure of the diffuse reflection of solar radiation out of the total solar radiation and measured on a scale from 0, corresponding to a black body that absorbs all incident radiation, to 1, corresponding to a body that refle ...
(predominantly scattering) medium.
The RTE in the diffusion approximation
Radiance can be expanded on a basis set of
spherical harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form ...
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
In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as ...
.
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
In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as ...
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
Diffusivity, mass diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species (or the driving force for diffusion). Diffusivity is enc ...
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
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
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 yield
: