In the previous section we have seen how radiation interacts with matter at an atomic level. In this section we will examine the bulk "macroscopic" aspects of the interaction of radiation with matter, with particular reference to positron emission and detection.

Calculations of photon interactions are given in terms of atomic cross sections (o) with units of cm2/atom. An alternative unit, often employed, is to quote the cross section for interaction in barns/atom (b/atom) where 1 barn = 10-24cm2. The total atomic cross section is given by the sum of the cross sections for all of the individual processes [2], i.e.,

Otot = 0pe + Oincoh + 0 coh + 0pair + 0 tripl + 0nph (24)

where the cross sections are for the photoelectric effect (pe), incoherent Compton scattering (incoh), coherent (Rayleigh) scattering (coh), pair production (pair), triplet production (tripl), and nuclear photoabsorption (nph). Values for attenuation coefficient are often given as mass attenuation coefficients (u/p) with units of cm2.g-1. The reason for this is that this value can be converted into a linear attenuation coefficient (u) for any material simply by multiplying by the density (p) of the material:

Figure 2.13. Total atomic cross-section as a function of photon energy for lead. The scattering cross-sections (o) are given for coherent (COH), incoherent (INCOH) or Compton scattering, photonuclear absorption (PH.N.), atomic photoelectric effect (t), nuclear field pair production (Kn), electron field pair production (triplet) (Ke), and the overall total cross section (TOT). (Reproduced with permission of the Institute of Physics Publishing from: Hubbell JH. Review of photon interaction cross section data in the medical and biological context. Phys Med Biol 1999;44(1):R1-22).

Figure 2.13. Total atomic cross-section as a function of photon energy for lead. The scattering cross-sections (o) are given for coherent (COH), incoherent (INCOH) or Compton scattering, photonuclear absorption (PH.N.), atomic photoelectric effect (t), nuclear field pair production (Kn), electron field pair production (triplet) (Ke), and the overall total cross section (TOT). (Reproduced with permission of the Institute of Physics Publishing from: Hubbell JH. Review of photon interaction cross section data in the medical and biological context. Phys Med Biol 1999;44(1):R1-22).

The mass attenuation coefficient is related to the total cross section by

where u(g) = 1.661 x 1024g is the atomic mass unit (1/Na where NA is Avogadro's number) defined as 1/12th of the mass of an atom of 12C, and A is the relative atomic mass of the target element [2].

An example of the total cross section as a function of energy is shown in Fig. 2.13.

We have seen that the primary mechanism for photon interaction with matter at energies around 0.5 MeV is by a Compton interaction. The result of this form of interaction is that the primary photon changes direction (i.e., is "scattered") and loses energy. In addition, the atom where the interaction occurred is ionized.

For a well-collimated source of photons and detector, attenuation takes the form of a mono-exponential function, i.e.,

Material |
Density (p) [g.cm-3] |
[cm-1] | |

Adipose tissue* |
0.95 |
0.142 |
0.090 |

Water |
1.0 |
0.150 |
0.095 |

Lung* |
1.05' |
~0.04-0.065 |
~0.025-0.045 |

Smooth muscle |
1.05 |
0.155 |
0.101 |

Perspex (lucite) |
1.19 |
0.173 |
0.112 |

Cortical bone* |
1.92 |
0.284 |
0.178 |

Pyrex glass |
2.23 |
0.307 |
0.194 |

NaI(Tl) |
3.67 |
2.23 |
0.34 |

Bismuth germanate |
7.13 |
~5.5 |
0.95 |

(BGO) | |||

Lead |
11.35 |
40.8 |
1.75 |

(Tabulated from Hubbell [3] and *ICRU Report 44 [4]). 1This is the density of non-inflated lung. §Measured experimentally. |

where I represents the photon beam intensity, the subscripts "0" and "x" refer respectively to the unat-tenuated beam intensity and the intensity measured through a thickness of material of thickness x, and m refers to the attenuation coefficient of the material (units: cm1). Attenuation is a function of the photon energy and the electron density (Z number) of the attenuator. The attenuation coefficient is a measure of the probability that a photon will be attenuated by a unit length of the medium. The situation of a well-collimated source and detector are referred to as narrow-beam conditions. The narrow-beam linear attenuation coefficients for some common materials at 140 keV and 511 keV are shown in Table 2.4 and Fig. 2.14.

However, when dealing with in vivo imaging we do not have a well-collimated source, but rather a source

Detector emitting photons in all directions. Under these uncolli-mated, broad-beam conditions, photons whose original emission direction would have taken them out of the acceptance angle of the detector may be scattered such that they are counted. The geometry of narrow and broad beam detection are illustrated in Fig. 2.15.

In the broad-beam case, an uncollimated source emitting photons in all directions contributes both un-scattered and scattered events to the measurement by the detector. In this case the detector "sees" more photons than would be expected if unscattered events were excluded, and thus the transmission rate is higher than anticipated (or, conversely, attenuation appears lower). In the narrow-beam case, scattered photons are precluded from the measurement and thus the transmission measured reflects the bulk attenuating properties of the object alone.

Figure 2.15. Broad-beam geometry (left) combines an uncollimated source of photons and an uncollimated detector, allowing scattered photons to be detected. The narrow-beam case (right) first constrains the photon flux to the direction towards the detector, and second, excludes scattered photons by collimation of the detector.

Detector

Figure 2.15. Broad-beam geometry (left) combines an uncollimated source of photons and an uncollimated detector, allowing scattered photons to be detected. The narrow-beam case (right) first constrains the photon flux to the direction towards the detector, and second, excludes scattered photons by collimation of the detector.

Figure 2.16. Scattered photons in SPECT and PET are shown. In SPECT, the recorded scatter is constrained within the object boundaries as there is low probability for scattering in air. In PET, as two photons are utilized, the line of response connecting the detectors may not intersect the object at all. This fact can be used to infer the underlying scatter distribution within the object by interpolation of the projections (see Ch. 6).

Figure 2.16. Scattered photons in SPECT and PET are shown. In SPECT, the recorded scatter is constrained within the object boundaries as there is low probability for scattering in air. In PET, as two photons are utilized, the line of response connecting the detectors may not intersect the object at all. This fact can be used to infer the underlying scatter distribution within the object by interpolation of the projections (see Ch. 6).

The geometry of scattered events is very different for PET and single photon emission computed tomography (SPECT). As PET uses coincidence detection, the line-of-sight ascribed to an event is determined by the paths taken by both annihilation photons. In this case, events can be assigned to lines of response outside of the object. This is not true in the single-photon case where, assuming negligible scattering in air, the events scattered within the object will be contained within the object boundaries. The difference in illustrated in Fig. 2.16.

Positron emission possesses an important distinction from single-photon measurements in terms of attenuation. Consider the count rate from a single photon emitting point source of radioactivity at a depth, a, in an attenuating medium of total thickness, D (see Fig. 2.17). The count rate C observed by an external detector A would be:

where C0 represents the unattenuatted count rate from the source, and | is the attenuation coefficient of the medium (assumed to be a constant here). Clearly the count rate changes with the depth a. If measurements were made of the source from the 180° opposed direction the count rate observed by detector B would be:

where the depth b is given by (D - a). The count rate observed by the detectors will be equivalent when a = b.

Now consider the same case for a positron-emitting source, where detectors A and B are measuring coincident photons. The count rate is given by the product of the probability of counting both photons and will be:

Figure 2.17. Detectors A and B record attenuated count rates arising from the source (•) located a distance a from detector A and b from detector B. For each positron annihilation, the probability of detecting both photons is the product of the individual photon detection probabilities. Therefore, the combined count rate observed is independent of the position of the source emitter along the line of response. The total attenuation id determined by the total thickness (D) alone.

b a which shows that the count rate observed in an object only depends on the total thickness of the object, D; i.e., the count rate observed is independent of the position of the source in the object. Therefore, to correct for attenuation of coincidence detection from annihilation radiation one measurement, the total attenuation path length (-^D), is all that is required. In singlephoton measurements the depth of the source in the object, in principle, must be known as well.

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