Page 296 - Biomedical Engineering and Design Handbook Volume 2, Applications

274 DIAGNOSTIC EQUIPMENT DESIGN

T
σ = σ pe + Z σ c (10.16)
where Z is the atomic number. Therefore, integrating Eq. (10.1) for the removal of photons from a
2
narrow x-ray beam of initial flux/m Φ , over a path of length x, gives
0
T
Φ = Φ exp( −σ nx ) (10.17)
0 a
This relation holds for monochromatic x-rays propagating through a homogeneous medium. The
T
quantity s n is known as the linear attenuation coefficient m, with photoelectric and Compton
a
components
μ pe = n a σ pe
(10.18)
c
μ = n e σ c
so that the overall attenuation coefficient is given by
T
=
μσ n a = μ pe + μ c (10.19)
According to Eq. (10.15), the linear attenuation coefficient for the Compton event can be separated
into that due to absorption and that due to scattered radiation, so that
c
c
μ = μ + μ s c (10.20)
e
This means that for a given material substance we have the linear attenuation coefficients grouped
as follows:
pe
• Total attenuation m + m c
pe
• Total absorption m + m c
e
• Photoelectric absorption m pe
• Compton absorption m c
e
• Compton scattering m c
s
The variation of these terms with x-ray energy is usefully demonstrated for water (Fig. 10.7). With
reference to the overall attenuation coefficient [Eq. (10.19)], and to the associated cross section
pe
c
[Eq. (10.16)], for each element there is a photon energy for which m = m . Values of these energies
can be used to indicate the region in which either process is dominant (Fig. 10.8).
The linear attenuation coefficient can be written as m = s N r/A, where Avagadro’s number N =
T
A
A
−1
23
6.02 × 10 mol , A is the atomic weight, and r is the density. Given that the interaction cross sec-
T
T
tion s is not a function of the density of the medium, a mass absorption coefficient m/r = s N /A
A
can be defined. This is related to the mass of a material required to attenuate an x-ray beam by a
given amount. It is the form most often quoted for x-ray attenuation in tables of physical constants.
According to Eq. (10.17), the beam photon flux Φ can be written as
Φ = Φ exp( −μx ) (10.21)
0
which is the familiar Beer-Lambert law for the attenuation of an x-ray beam passing through matter.
When the medium is nonhomogeneous, the attenuation must be integrated along the ray-path length l
according to Φ=Φ exp(−∫ m(x) dl). For the location r in two- or three-dimensional space, this becomes
0 r
Φ = Φ exp ( ∫ ( ) dl ) (10.22)
− μ
r
0
l
If the x-ray source is polychromatic, the attenuation must be integrated over the wavelength l as
well, to give