Optical Imaging - Correction Techniques in Emission Tomography - page 246

1= a is the average distance covered by photons before being absorbed, and
1= s is the average distance between scattering events [7, 11, 12, 48, 52, 54].
In the following sections, we describe some key equations used in the to-
mographic problem following the diffusion approach [33, 3]. Their purpose
is to provide the reader a basic outline of formulating a fluorescence optical
tomography problem.

12.2.1.2 The diffusion approximation

The diffusion model can be derived from the general principle of conser-
vation of energy and Fick's law. The principle of conservation of energy for
a diffusive medium applied over a finite volume can be stated as follows: the
change of the energy density in the volume over time equals: power density
leaving the volume power density absorbed in the volume + power density
produced by sources present.
The energy density w(r;t) [J cm 3 ] is the energy per unit volume of the
radiation eld, w(r;t) = dE=dV , where V is the volume. The energy density
is related to the uence rate (r;t) by:

w(r;t) = (r;t)=c; (12.2)

where c is the speed of light in the medium. The uence rate (r;t) [W
cm 2 ], also called spectral irradiance, is defined as the power incident on a
small sphere at a given point r in space, divided by the cross-sectional area of
that sphere. It can be written as the integral over all directions of the radiance
L(r; s;t):

Z Z

(r;t) = L(r; s;t)ds: (12.3)

4

FIGURE 12.4: The radiance L is defined as the flux per unit projected area
per unit solid angle leaving a source or a reference surface: L = dP=dsdA proj .
(From [33].)