Chapter 10: Monte Carlo Photon Transport

Real X-ray photons are absorbed or scattered off in stray directions as they pass through matter. Scattered photons contaminate the detector and cause artifacts (Chapter 12), and absorbed photons become patient dose (Chapter 9). Following this elementary physics one photon at a time and solving it statistically is the Monte Carlo method, the computation that production tools like Geant4 carry out. Every forward projection so far has been a line integral (the Beer–Lambert law), the idealization that an X-ray weakens by $\exp(-\int \mu\,dl)$ as it passes through matter, which was enough to build the mathematics of reconstruction but captures only part of the physics of real CT.

The three interactions of a photon with matter

In the diagnostic range (about 20–150 keV), X-rays interact with matter mainly in three ways.

• Photoelectric absorption: the photon is absorbed whole by an atom and ejects an electron. The photon vanishes and deposits all its energy locally (dose). It is more likely at low energy and in high-atomic-number materials (roughly $Z^3/E^3$).
• Compton scatter: the photon collides with an electron, changes direction, and loses some energy. The scattered energy is $E' = E/(1 + (E/511)(1-\cos\theta))$ (keV), and the lost part $E-E'$ becomes local dose. It is the leading process in soft tissue over the diagnostic range.
• Rayleigh scatter: the photon scatters coherently off the whole atom, changing direction slightly with no energy loss. Its contribution is small, but it adds near-forward scatter.

How much each process occurs depends on energy. At low energy photoelectric absorption dominates, photons are readily absorbed, and dose is high. Raising the energy makes Compton scatter dominate and increases scattered radiation. The total attenuation $\mu$ is also larger at low energy, and this is the starting point of the beam hardening of Chapter 12 (low-energy components are absorbed first, hardening the spectrum).

The Monte Carlo method: tracking by probability

Where a line integral expresses the whole beam in one equation, Monte Carlo follows photons one at a time, probabilistically. The procedure is roughly this.

1. Sample the distance to the next interaction from the total attenuation $\mu$: $s = -\ln(\xi)/\mu$ (with $\xi$ a uniform random number).
2. At that point, draw which interaction happened, weighted by the cross sections.
3. For photoelectric, the photon vanishes and leaves its energy locally. For Compton, draw a scattering angle from the Klein–Nishina distribution and update direction and energy. For Rayleigh, change direction only.
4. Repeat 1–3 until the photon reaches the detector, vanishes, or leaves the field of view.

The fate of one photon is set by random numbers, but averaging over many photons converges to the expected distribution (the law of large numbers). The Poisson statistics of Chapter 9 apply directly here. The more photons $N$ you collect, the smaller the relative spread of the estimate, falling as $1/\sqrt{N}$. Few photons give a grainy result; more photons smooth it out. In practice this slow convergence (halving the error needs four times the photons) is the pain point, and variance-reduction techniques are used to fight it.

Simulation: seeing scatter and dose from the photons

A water cylinder with a dense bone insert is exposed to a parallel beam from the top, and photons are tracked one at a time. The middle panel is the deposited-energy (dose) distribution; the right panel is the primary and scatter profiles that reach the detector. Raising the energy lets photons penetrate more, so the dose fraction falls clearly. The scatter-to-primary ratio (SPR) does not simply increase with energy: at low energy the heavy absorption leaves few primary photons at the detector, so the relative effect of scatter (SPR) is actually larger there. What raises SPR cleanly is more material in the path (a thicker or denser object). Raising the photon count reduces the Monte Carlo noise in the dose map and the profiles. Notice the gentle pedestal of scattered radiation sitting under the sharp shadow of the primary beam. That is the scatter that breaks the quantitative accuracy of the projection.

Geant4 and production Monte Carlo

This simulation is a teaching model: two-dimensional, with simplified cross sections and in-plane scatter only. Real research and device design use more rigorous Monte Carlo tools. Geant4 (GEometry ANd Tracking) is a general-purpose particle-transport toolkit developed at CERN, used from particle physics to medical physics. For medical imaging there is GATE built on Geant4, TOPAS used for proton therapy and more, and codes such as MCNP and EGSnrc. These handle arbitrary 3D geometry, continuous energy spectra, accurate cross-section data, and detector response (down to scintillator light and photomultiplication), and are used to design CT scatter correction, evaluate detector response for PCCT (Chapter 14), and compute dose. The toy model of this chapter reproduces, minimally and in the browser, the elementary processes those tools run inside.

Key points

Forward projection by line integral is an idealization; real photons undergo photoelectric absorption, Compton scatter, and Rayleigh scatter. Photoelectric absorption produces dose, and scatter changes direction and contaminates the detector. Which process dominates depends on energy: at low energy absorption and dose, at high energy scatter. The Monte Carlo method follows this elementary physics one photon at a time and converges, as an average over many, to the true distribution, with error inversely proportional to the square root of the photon count. It is the reference computation behind scatter correction (Chapter 12) and dose evaluation (Chapter 9), and production tools like Geant4 carry it out rigorously.

References

• Agostinelli S et al. Geant4—a simulation toolkit. Nuclear Instruments and Methods in Physics Research A 506, 250–303 (2003) — the original Geant4 paper.
• Jan S et al. GATE: a simulation toolkit for PET and SPECT. Physics in Medicine and Biology 49, 4543–4561 (2004).
• Klein O, Nishina Y. Über die Streuung von Strahlung durch freie Elektronen…. Zeitschrift für Physik 52, 853–868 (1929) — the Compton scattering cross section.
• Rogers DWO. Fifty years of Monte Carlo simulations for medical physics. Physics in Medicine and Biology 51, R287–R301 (2006).