Introduction
Diffusion theory is the modeling of photon transport due to photon movement down concentration gradients. Diffusion theory is appropriate in medium dominated by scattering rather than absorption so that each photon undergoes many scattering events before being terminated by an absorption event. The photon has a relatively long residence time which allows the photon to engage in a random walk within the medium.
_{Diffusion theory = Photons falling down a gradient of photon concentration.} _{
} The instantaneous fluence rate, F(r,t) [J s^{1} cm^{2}] or [W cm^{2}], is proportional to the concentration of optical energy, C(r,t) [J cm^{3}] and the speed of light, c [cm/s], in the medium:
The units of energy concentration [J cm^{3}] times the units of velocity [cm/s] yield the units of fluence rate [J s^{1} cm^{2}]. This relationship between F and C allows us to discuss the timeresolved spatial distribution of photon concentration, C(r,t), in a lightscattering medium using the same math of diffusion which applies to many things such as solutes in a solution or heat in a material.
F(r,t) = cC(r,t)
Fick's 1st law of diffusion
Diffusion occurs in response to a concentration gradient expressed as the change in concentration due to a change in position, . The local rule for movement or flux J is given by Fick's 1st law of diffusion:
in which the flux J [cm^{2} s^{1}] is proportional to the diffusivity [cm^{2}/s] and the negative gradient of concentration, [cm^{3} cm^{1}] or [cm^{4}]. The negative sign indicates that J is positive when movement is down the gradient, i.e., the negative sign cancels the negative gradient along the direction of positive flux.
The flux J is driven by the negative gradient in the direction of increasing x.
For light, the diffusivity is proportional to the diffusion length D [cm] and the speed of light c:
where D = 1/(3 µ_{s}(1g)). The units of velocity [cm/s] times the units of length [cm] yield the units of diffusivity [cm^{2}/s]. The following example describes the local diffusion of red light in milk.
For optical diffusion, Fick's 1st law is expressed as the energy flux J [W cm^{2}] proportional to the diffusion constant D [cm] and the negative fluence gradient dF/dx:
which was obtained by substituting cD for and substituting F/c for C. The factors c and 1/c cancel to yield the above equation.
The flux J is driven by the negative gradient in the direction of increasing x.
Examplearbitrary units  Consider a local concentration of 10^{6} units per cm^{3} which drops by 10% over a distance of 1 cm. Then the gradient is 10^{5} [units cm^{4}]. Assume the diffusivity is 10^{3}[cm^{2}/s]. Then the flux J equals: 
For light, the diffusivity is proportional to the diffusion length D [cm] and the speed of light c:
= cD
Exampleoptical energy  Consider a local fluence rate F of 1 W/cm^{2} in milk (n = 1.33, µ_{s}(1g) = 10 cm^{1}).

For optical diffusion, Fick's 1st law is expressed as the energy flux J [W cm^{2}] proportional to the diffusion constant D [cm] and the negative fluence gradient dF/dx:
Fick's 2nd law of diffusion
Consider diffusion at the front and rear surfaces of an incremental planar volume. Fick's 2nd law of diffusion describes the rate of accumulation (or depletion) of concentration within the volume as proportional to the local curvature of the concentration gradient. The local rule for accumulation is given by Fick's 2nd law of diffusion:
in which the accumulation, dC/dt [cm^{3} s^{1}], is proportional to the diffusivity [cm^{2}/s] and the 2nd derivative (or curvature) of the concentration, [cm^{3} cm^{2}] or [cm^{5}]. The accumulation is positive when the curvature is positive, i.e., when the concentration gradient is more negative on the front end of the planar volume and less negative on the rear end so that more flux is driven into the volume at the front end than is driven out of the volume at the rear end.
Incremental planar volume accumulates concentration because the front gradient at x _{1} drives more flux J_{1} into the volume than the flux J_{2} driven out of the volume by the rear gradient at x_{2}.
The differential equation for optical diffusion is simply Fick's 2nd law with the substitution of the product cD for the diffusivity and substitution of F/c for concentration C, although the 1/c factors introduced on both sides of the equation cancel:
Incremental planar volume accumulates concentration because the front gradient at x _{1} drives more flux J_{1} into the volume than the flux J_{2} driven out of the volume by the rear gradient at x_{2}.
The differential equation for optical diffusion is simply Fick's 2nd law with the substitution of the product cD for the diffusivity and substitution of F/c for concentration C, although the 1/c factors introduced on both sides of the equation cancel:
Connecting optical transport theory and Fick's 1st law of diffusion.
The early work on neutron scattering theory in nuclear reactors first developed the connection between transport theory and Fick's 1st law of diffusion. The resulting statement of diffusion is also applied to optical transport. Consider the following problem:
For photon transport, the term µ_{s}' = µ_{s}(1  g) is substituted the above µ_{s} which described isotropic scattering used in the above treatment.
In summary, the connection between transport theory and Fick's 1st law of diffusion is based on the linearization of F(r) around the value F(0) and on the approximation of D by the value 1/(3 µ_{s}').
Light scattered at position r passes through a small aperture with area A contributing to the flux J_{+}. 
 given a homogeneous isotropically scattering medium with scattering coefficient µ_{s}
 given a small aperture with area A at the origin (r = 0)
 given a fluence rate F(0) at the origin
 assume that the fluence rate F(r) at a position r near to the aperture is approximated by
 calculate the net flux through the small aperture with area A due to scattering from all the surrounding volume:
 the scattered flux at r is µ_{s}F(r) = µ_{s}( F(0) + r(dF(r)/dr_{at r=0}) ) where r is the distance from r to the origin.
 the fraction of scattered flux from r that survives a pathlength r without being scattered or absorbed is exp(µ_{t}r), where µ_{t} = µ_{s} + µ_{a}. Consider the case of negligible absorption compared to scattering so µ_{t} = µ_{s}.
 the fraction of surviving scattered flux from r that passes through A is Acos/(4r^{2}).
 integrate the flux from r over all possible and r for the hemisphere above the aperture:
 a similar integration for the scattering from the hemisphere below the aperture yields a positive value J_{} for flux passing through A in the other direction opposite to J_{+}.
 the net flux J = J_{+}  J_{}
 The final expression for the net flux J has the form of Fick's 1st law of diffusion if the diffusion constant D has the following value:
, where
For photon transport, the term µ_{s}' = µ_{s}(1  g) is substituted the above µ_{s} which described isotropic scattering used in the above treatment.
In summary, the connection between transport theory and Fick's 1st law of diffusion is based on the linearization of F(r) around the value F(0) and on the approximation of D by the value 1/(3 µ_{s}').
Limits of diffusion theory
There are limits to the applicability of diffusion theory.
It is important to remember that diffusion theory simply attempts to mimic Fick's 1st law of diffusion by a proper choice of D to relate J and dF/dr. The approximation of F(r) as simply a linear perturbation of the value F(0) neglects higher order terms that depend on d^{2}F/dr^{2}. Sometimes the gradient has significant curvature within the spherical regime from which exp(µ_{s}' r) allows photons to pass through the aperture. In such regions, the linear approximation to F(r) is inadequate and diffusion theory is inaccurate. Two cases of curvature deserve emphasis:
However, we have conducted Monte Carlo simulations of light diffusion from an isotropic point source to yield the spatial distribution for F(r) which suggests a different value for D:
If µ_{a} << 3µ_{s}', the effect of absorption can be neglected.
It is important to remember that diffusion theory simply attempts to mimic Fick's 1st law of diffusion by a proper choice of D to relate J and dF/dr. The approximation of F(r) as simply a linear perturbation of the value F(0) neglects higher order terms that depend on d^{2}F/dr^{2}. Sometimes the gradient has significant curvature within the spherical regime from which exp(µ_{s}' r) allows photons to pass through the aperture. In such regions, the linear approximation to F(r) is inadequate and diffusion theory is inaccurate. Two cases of curvature deserve emphasis:
Near a source
The gradients are very steep near a point source or collection of point sources with both the exponential term exp(r/(4cDt)) and the term 1/r causing significant curvature in the gradients. Distant from sources, gradients become gradual and diffusion theory is more accurate.Absorption
Strong absorption prevents photons from engaging in an extended random walk. The approximation µ_{t} = µ_{s}' becomes inadequate. In common use is the approximation:Timeresolved Diffusion Theory
The solution to Fick's 2nd Law of diffusion for the case of a point source of energy, U_{o} [units], deposited at zero time (t = 0) at the origin (r = 0) is C(r,t) [units cm^{3}]:
where r [cm] is the distance from the source to the point of observation, t [s] is the time of observation, and [cm^{2}/s] is the diffusivity. In the denominator, the product t has units of [cm^{2}] which is taken to the 3/2 power to yield units of [cm^{3}]. The exponential in the numerator is dimensionless. The units of U_{o} can be the units of any extensive variable which is able to diffuse throughout a volume. Hence, C(r,t) has the proper dimensions for concentration, [units cm^{3}]. The above expression for C(r,t) is a solution to Fick's 2nd law of diffusion and describes spherically symmetric diffusion from a point impulse source in a homogeneous medium with no boundaries.
For the case of optical diffusion, let F = cC and = cD. Let the point source be an impulse of energy U_{o} [J]. The expression for fluence rate F(r,t) [W cm^{2}] becomes:
The above equation is very useful and the student of tissue optics should know this equation very well.
For the case of optical diffusion, let F = cC and = cD. Let the point source be an impulse of energy U_{o} [J]. The expression for fluence rate F(r,t) [W cm^{2}] becomes:
Timeresolved diffusion Theory:
Examples
The following figures illustrate the timeresolved transport of light from an impulse isotropic point source of energy within a homogeneous unbounded medium with absorption and scattering properties.
The C program that generated the data in the above figures is listed here.
The C program that generated the data in the above figures is listed here.
Steadystate Diffusion Theory
The timeresolved fluence rate F(r,t) [W/cm^{2}] in response to an impulse of energy U_{o} [J] and the steadystate fluence rate F_{ss}(r) [W/cm^{2}] in response to an isotropic point source of continuous power P_{o} [W] are summarized:
where the transport factors T(r,t) [cm^{2} s^{1}] and T_{ss}(r) [cm^{2}] have been introduced to more carefully distinguish the source, the transport, and the fluence rate. Note on notation: In this class, we use F_{ss}(r) rather than F(r) to especially emphasize the steadystate fluence rate from the timeresolved fluence rate. However, F(r) should be used outside this class.
The above expression for T_{ss}(r) can be obtained by integrating T(r,t)exp(µ_{a}ct) over all time to yield the total accumulated amount of photon transport to each position r. The factor exp(µ_{a}ct) accounts for photon absorption (recall that ct = pathlength so this expression is simply Beer's law for photon survival) and causes photon concentration to approach zero as time goes to infinity. The expression for T_{ss}(r) is derived:
Note that the final expression above has made the substitutions:
which removes the diffusion length D and introduces the optical penetration depth which is the incremental distance from the source that causes F_{ss}(r) to decrease to 1/e its initial value. The penetration depth is a parameter which is very easily understood in experimental measurements and consequently has more intuitive value to some people than D which is important from the perspective of the local step size of the diffusion process. In this class we will often use the following expression for F_{ss}(r) when we prefer to emphasize the roles of µ_{a}and :
where the transport factors T(r,t) [cm^{2} s^{1}] and T_{ss}(r) [cm^{2}] have been introduced to more carefully distinguish the source, the transport, and the fluence rate. Note on notation: In this class, we use F_{ss}(r) rather than F(r) to especially emphasize the steadystate fluence rate from the timeresolved fluence rate. However, F(r) should be used outside this class.
The above expression for T_{ss}(r) can be obtained by integrating T(r,t)exp(µ_{a}ct) over all time to yield the total accumulated amount of photon transport to each position r. The factor exp(µ_{a}ct) accounts for photon absorption (recall that ct = pathlength so this expression is simply Beer's law for photon survival) and causes photon concentration to approach zero as time goes to infinity. The expression for T_{ss}(r) is derived:
Steadystate diffusion Theory:
Examples
The following figures illustrate the transport of light from a steadystate isotropic point source of power within a homogeneous unbounded medium with absorption and scattering properties.
The C program that generated the data in the above figures is listed here.
The C program that generated the data in the above figures is listed here.
Frequencydomain Diffusion Theory:
Isotropic point source in infinite medium
A light source may be modulated sinusoidally to yield a sinusoidally varying fluence rate distribution at a distant observation point within a medium. Such a modulated concentration will propagate in the medium and is often called a photon density wave. Consider a modulated isotropic point source of light within a homogenous turbid medium with no boundaries.
The point source S has a steadystate power S_{o} [W] which is modulated sinusoidally by a modulation factor M_{o}sin(t), where 0 < M_{o} < 1:
where = 2f [radians/s] is the angular frequency of modulation, and the modulation frequency f is in hertz [cycles/s]. The position of observation r is located a distance r from the source. The above equation shows two ways to express S(t), one using a sine function and the other using the equivalent and wellknown convention of an exponential with an imaginary exponent.
In response to this modulated source, the modulated fluence rate F(r,t) at r is described:
where T_{ss}(r) is the steadystate transport, k" is the imaginary wavenumber that describes the attenuation of the photon density wave, and k' is the real wavenumber that describes the phase lag of the observed photon density wave.
The expressions for T_{ss}(r), k" and k' are:
where
The behavior of k' and k" are shown in the following figure which plots k' and k" as functions of /(µ_{a}c):
Click figure to enlarge
The above expressions are equivalent to the expressions published by Schmitt et al. 1992 which in turn are equivalent to the expressions published by Fishkin et al. 1991,1993. Link to references..
At the observation point, the persistence of the source modulation is called the modulation, M, and is often described in the literature as (AC_{out}/DC_{out})/(AC_{in}/DC_{in}) which equals:
At the observation point, the phase of the signal lags the phase of the source by an angle called the phase, [radians], which equals:
The ratio /(µ_{a}c) describes the number of radians of modulation cycle that occur in one mean photon lifetime. Only 1/e or 37% of photons survive after a time period of 1/(µ_{a}c) [s]. is an angle specified by the ratio /(µ_{a}c), and approaches zero for low modulation frequencies and approaches 90° at the highest modulation frequencies, >> µ_{a}c.
The point source S has a steadystate power S_{o} [W] which is modulated sinusoidally by a modulation factor M_{o}sin(t), where 0 < M_{o} < 1:
In response to this modulated source, the modulated fluence rate F(r,t) at r is described:
The expressions for T_{ss}(r), k" and k' are:
where
The behavior of k' and k" are shown in the following figure which plots k' and k" as functions of /(µ_{a}c):
The above expressions are equivalent to the expressions published by Schmitt et al. 1992 which in turn are equivalent to the expressions published by Fishkin et al. 1991,1993. Link to references..
At the observation point, the persistence of the source modulation is called the modulation, M, and is often described in the literature as (AC_{out}/DC_{out})/(AC_{in}/DC_{in}) which equals:
The ratio /(µ_{a}c) describes the number of radians of modulation cycle that occur in one mean photon lifetime. Only 1/e or 37% of photons survive after a time period of 1/(µ_{a}c) [s]. is an angle specified by the ratio /(µ_{a}c), and approaches zero for low modulation frequencies and approaches 90° at the highest modulation frequencies, >> µ_{a}c.
 At very low modulation frequencies, << µ_{a}c, there is little modulation during the lifetime of a photon. Consequently, photon migration has little impact on the transport of the modulation. The value of k" approaches one so k" approaches 1/, therefore M approaches unity. The value of k' approaches zero, therefore approaches zero. The observed modulation of F_{ss}(r,t) mimics the source modulation with no loss of modulation and no phase lag. F(r,t) approaches the behavior S_{o}T_{ss}(r)(1 + M_{o}sin(t)).
 At high modulation frequencies, as the modulation frequency approaches the value of µ_{a}c or greater, there is significant modulation during the lifetime of the photon. Consequently, the photon can diffuse during its lifetime and thereby smear the spatial resolution of the modulation. F(r,t) exhibits significant loss of modulation (M < 1) and an increased phase lag ( > 0): F(r,t) = S_{o}T_{ss}(r)(1 + M_{o}Msin(t  k'r))
Frequencydomain diffusion theory:
Examples
The following figure illustrates the timeresolved signal F(r,t) in blue and the source S(t) in red. Note the greatly decreased F_{ss}, the slightly decreased modulation M, and the phase lag at the observation point.
The C program that generated the data in the above figure is listed here.
The following figures illustrates the modulation of the photon density wave as it propagates into the medium.
The C program that generated the data in the above figures is listed here.
source: f = 400 MHz, S_{o} = 1 W, M_{o} = 1.0 medium: µ_{a} = 1.0 cm^{1}, µ_{s}(1  g) = 10.0 cm^{1}, n_{t} = 1.33  Click on figure to enlarge 
The following figures illustrates the modulation of the photon density wave as it propagates into the medium.
The C program that generated the data in the above figures is listed here.
Source:http://omlc.ogi.edu/classroom/ece532/class5/fdexamples.html
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