Photon diffusion equation is a second order

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

describing the time behavior of photon fluence rate distribution in a low-absorption high-scattering medium. Its mathematical form is as follows.

\nabla(D(r)\cdot\nabla)\Phi(\vec,t)-v\mu_a(\vec)\Phi(\vec,t)+vS(\vec,t)=\frac{\partial t}

where

\Phi

is photon

fluence rate In radiometry, radiant exposure or fluence is the radiant energy ''received'' by a ''surface'' per unit area, or equivalently the irradiance of a ''surface,'' integrated over time of irradiation, and spectral exposure is the radiant exposure per un ...

(W/cm²),

\nabla

del Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes ...

operator,

\mu_a

is absorption coefficient (cm⁻¹),

D

diffusion constant Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion equ ...

v

is the speed of light in the medium (m/s), and

S

is an isotropic source term (W/cm³). Its main difference with

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 law ...

in physics is that photon diffusion equation has an absorption term in it.

Application

Medical Imaging

The properties of photon diffusion as explained by the equation is used in

diffuse optical tomography Diffuse optical imaging (DOI) is a method of imaging using near-infrared spectroscopy (NIRS) or fluorescence-based methods. When used to create 3D volumetric models of the imaged material DOI is referred to as diffuse optical tomography, whereas ...

External links

Diffuse Optics Lab at University of Pennsylvania, Philadelphia, USA
Equations