In the traditional Monte Carlo ray-trace (MCRT) method, a large number of energy bundles are traced from a source, through a participating medium, and onto a screen where they are absorbed. If a sufficiently large number of energy bundles are traced, their distribution on the screen may be accurately interpreted as an image. In the application at hand, the authors seek to simulate the image produced on the screen of a CT scanner due to the presence of a target mass, referred to here as a “phantom,” interposed between the X-ray source and the screen. The effort is motivated by the fact that, in traditional MCRT simulation environments, many if not most of the energy bundles fail to reach the screen. This leads to significant numerical inefficiencies and excessively long simulation times. The authors describe and demonstrate an application of the Metropolis-Hastings algorithm which assures that all energy bundles emitted by the X-ray source follow paths that terminate on the detector screen.

A sequence of candidate paths is created, with each path consisting of a chain connecting the fixed source point with a random point on the screen while passing through a small number of randomly positioned scattering centers within the phantom. The probability associated with each path is computed using standard scattering and attenuation models. While relevant to the CT scan community, the specific set of absorption and scattering rules used in the article is not its essential feature. Of far greater relevance is the fact that, once a path connecting the source with a point on the screen has been identified, its probability can be evaluated based on a suitable set of absorption and scattering rules. The probability 𝑝_accept(𝑥_𝑖−1 → 𝑦) that path 𝑦 is an acceptable mutation of the previous path 𝑥_𝑖−1 is computed using the Metropolis-Hastings algorithm, after which path 𝑦 is either accepted (𝑥_𝑖 = 𝑦) with probability 𝑝_accept or rejected (𝑥_𝑖 = 𝑥_𝑖−1) with probability 1−𝑝_accept. In this way, a sequence of statistically meaningful paths is thus created connecting the source to the screen.

The authors benchmark their algorithm against a traditional Monte Carlo ray-trace algorithm running on the same platform. They obtain good agreement between the two approaches while reporting a speed-up of between 20 and 40 times using their approach, depending on the nature of the phantom being imaged.

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