ECE532 Biomedical Optics|
Steven L. Jacques, Scott A. Prahl
Oregon Graduate Institute
Time-resolved Diffusion Theory
The solution to Fick's 2nd Law of diffusion for the case of a point source of energy, Uo [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 [cm2/s] is the diffusivity. In the denominator, the product t has units of [cm2] which is taken to the 3/2 power to yield units of [cm3]. The exponential in the numerator is dimensionless. The units of Uo 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 Uo [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.