KAHIBARO
33.1. X-Ray Physics
Table of Contents
Photon interactions
In CT and general X ray imaging, you mainly deal with photons in the energy range from a few tens of keV up to a few hundred keV. In this range, several interaction processes can occur when an X ray photon crosses matter. GATE uses Geant4 physics models to sample these interactions one by one, using Monte Carlo methods.
The most important processes for CT simulation are the photoelectric effect and Compton scattering. At higher energies, pair production can appear, but it is usually negligible for diagnostic X ray imaging.
In the photoelectric effect, the photon is completely absorbed by an atom, and its energy is transferred to a bound electron, which is then ejected. The photon disappears, so no scattered photon continues. The probability for the photoelectric effect depends strongly on the atomic number $Z$ of the material and on photon energy $E$. Roughly, the photoelectric cross section behaves like $Z^n / E^3$, with $n$ between about 3 and 4 depending on the energy range. This means that high $Z$ materials such as iodine, barium, or bone show strong photoelectric absorption at CT energies. In images, this enhances contrast between bone and soft tissue and is also the basis of contrast agents.
In Compton scattering, the photon interacts with a quasi free electron. The photon loses part of its energy and changes direction, and the electron is ejected. The scattered photon continues to propagate with reduced energy. Compton scattering dominates in soft tissue at typical CT energies, especially above about 50 to 60 keV. It is mainly responsible for image noise and scatter artifacts, because scattered photons may still reach the detector but no longer carry correct information about the original path through the patient.
The Compton scattering angle and energy change follow the Compton formula. If the initial photon energy is $E$ and it scatters by an angle $\theta$, the scattered energy $E'$ is given by
$$
E' = \frac{E}{1 + \frac{E}{m_e c^2} (1 - \cos \theta)} ,
$$
where $m_e c^2 \approx 511\ \text{keV}$ is the electron rest energy.
Remember: in the photoelectric effect the photon is fully absorbed and no scattered photon remains. In Compton scattering, a lower energy photon continues in a new direction, which can still be detected and contributes to image scatter.
At much higher photon energies, above 1.022 MeV, pair production becomes possible. The photon converts into an electron positron pair in the field of a nucleus. For diagnostic CT this process is practically absent and is usually not a concern in CT simulations, but it becomes important for megavoltage beams used in radiation therapy.
In GATE, you do not have to code each of these interactions manually. Instead, you choose an appropriate electromagnetic physics list. The physics list defines which processes are active for photons, electrons, and other particles, and how their cross sections are modeled. For CT simulation, you select a list designed for low energy photons so that photoelectric and Compton interactions are modeled accurately in the 20 to 150 keV range of the X ray spectrum.
Attenuation
As an X ray beam passes through matter, some photons interact and are removed from the original beam by absorption or scattering. This loss of photons is described by attenuation. In CT, the measured signal in each detector pixel reflects how strongly the beam was attenuated along a given path through the patient or phantom.
In a simple homogeneous material, the number of uncollided photons $I(x)$ remaining after traveling a distance $x$ along a straight path follows the exponential attenuation law
$$
I(x) = I_0 \, e^{-\mu x} ,
$$
where $I_0$ is the initial intensity and $\mu$ is the linear attenuation coefficient of the material.
Key attenuation law: for a monoenergetic, narrow X ray beam in a homogeneous material,
$$I(x) = I_0 e^{-\mu x}.$$
The linear attenuation coefficient $\mu$ depends on photon energy and material composition.
The coefficient $\mu$ is the sum of contributions from all interaction processes that remove photons from the primary beam, primarily photoelectric absorption and Compton scattering in CT. It depends on photon energy, so a real polyenergetic X ray spectrum does not follow a single exponential with one fixed $\mu$. Instead, lower energy photons are attenuated more strongly, and the spectrum hardens as it passes through matter. This is called beam hardening.
To compare attenuation across materials independently of density, it is common to use the mass attenuation coefficient, $\mu / \rho$, with units of $\text{cm}^2/\text{g}$, where $\rho$ is material density. For a given photon energy,
$$
\mu = \left(\frac{\mu}{\rho}\right) \rho .
$$
In practice, patient anatomy consists of many different tissues, and the beam traverses a path that passes through multiple materials with varying densities. For a monoenergetic beam crossing a sequence of slabs $i$ with thickness $x_i$ and linear attenuation coefficient $\mu_i$, the transmitted intensity becomes
$$
I = I_0 \exp\left(-\sum_i \mu_i x_i \right).
$$
CT scanners work with broad polyenergetic spectra and complex geometries, so analytical formulas become approximations. Monte Carlo simulation with GATE tracks each photon independently through the full 3D geometry and evaluates interactions process by process. The effective attenuation is then obtained by counting how many photons reach each detector element and with what energy.
In CT reconstruction theory, it is common to work with line integrals of the attenuation coefficient along many paths. For a monoenergetic beam and a continuous, position dependent attenuation coefficient $\mu(\mathbf{r})$, the ideal transmitted intensity along a ray path $L$ is
$$
I = I_0 \exp\left( - \int_L \mu(\mathbf{r})\, \mathrm{d}l \right).
$$
CT projections approximate measurements of these integrals for many angles and positions. Reconstruction algorithms then estimate $\mu(\mathbf{r})$ throughout the volume. Monte Carlo simulation in GATE allows you to generate realistic projection data that include not only ideal attenuation but also scatter, detector response, and spectrum effects, which will be explored in later chapters on CT geometry and projection data.
In GATE, attenuation is not specified directly as a single parameter. Instead, you define materials with given composition and density, and you choose a suitable physics list. Geant4 then uses built in cross section tables to determine $\mu$ as a function of energy for each material. When you simulate an X ray source and a detector, the fraction of photons that survive to the detector follows from these microscopic interactions and automatically produces the correct macroscopic attenuation behavior.
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