Optical Imaging - Correction Techniques in Emission Tomography - page 247

The radiance L(r; s;t) [W cm 2 st 1 ], see Figure 12.4, is defined as the
flux per unit projected area per unit solid angle leaving a source or a reference
surface: L = dP=dsdA proj , with P [W] the power or rate at which energy is
transferred from one region to another by the radiation field, dA proj = dA cos
is the projected area, and is the angle between the outward surface normal
of the area element dA and the direction of observation s. The ux density
vector j(r;t) equals the rst moment of the radiance:

Z Z

j(r;t) = L(r; s;t)sds (12.4)

4

The irradiance I [W cm 2 ] is defined as the flux per unit area received by
a real or imaginary surface, I = dP=dA. The ux density vector is also related
to the irradiance by:

I(r;t) = j(r;t) · n: (12.5)

The principle of conservation of energy can now be formulated in terms of
the above-mentioned quantities as:

Z Z Z

1
c

@(r;t)
@t

dV = j(r;t) · ndS + ( a (r;t) + S(r;t)dV (12.6)

V @V V

| {z } | {z } | {z }

change in V flow production

where c [cm s 1 ] is the speed of light in the medium, (r;t) [W cm 2 ] is the
uence rate, j(r;t) [W cm 2 ] is the ux density vector, n is the normal on
the surface, a [cm 1 ] is the absorption coecient and S(r;t) [W cm 3 ] is
the power per unit volume produced by the sources. After applying Gauss'
theorem to the surface integral, Equation 12.6 reduces for an arbitrary volume
to:

1
c

@(r;t)
@t

= rj(r;t) a (r;t) + S(r;t): (12.7)

To solve Equation 12.7, another relationship between the fluence rate and
the flux density vector is necessary. For highly scattering media, the photons
follow random path trajectories. After a few scattering events the photons
have a diffusive behavior. An accurate description of the diffusion process is
provided by Fick's law [38]:

j(r;t) Dr(r;t); (12.8)

with D the diffusion coecient. The diffusion coecient D depends on the
absorption and scattering coecients through [7]:

1
3[ 0 s + a ] ;

D = (12.9)