Dans cette partie de la thèse, un formalisme Bayésien rigoureux est introduit pour permettre de prédire le parcours le plus probable (MLP) de n’importe quel ion traversant un matériel et détecté à l’entrée et à la sortie. Le parcours est prédit en se basant sur une combinaison du diffusé moyen de la particule dans un matériel ainsi que sur les mesures de la position initiale/finale, de la direction initiale/finale et de l’énergie d’entrée et de sortie. La précision de l’estimé de parcours est comparée à des trajectoires extraites de simulations Monte Carlo. Chaque ion de l’hydrogène jusqu’au carbone est simulé dans deux scénarios, 1) le premier où la portée de la particule est fixée et 2) le second où la vélocité initiale de la particule est fixée. Dans le scénario où la portée de la particule est fixée, l’erreur quadratique moyenne
maximale entre l’estimé de parcours et la trajectoire Monte Carlo diminue significativement entre le proton (0.50 mm) et l’hélium (0.18 mm), puis diminue moins sévèrement jusqu’au carbone (0.09 mm). Cependant, ce scénario est identifié comme le scénario qui produit la dose maximale au patient tout en minimisant la résolution spatiale de l’image de sortie. Dans le second scénario où la vitesse initiale est fixée, l’erreur quadratique moyenne maximale entre l’estimé de parcours et la trajectoire Monte Carlo diminue significativement entre le proton (0.29 mm) et l’hélium (0.09 mm), mais augmente à nouveau pour les ions plus lourds jusqu’au carbone (0.12 mm). En conclusion, l’hélium est considéré comme la particule avec l’estimé de parcours le plus précis pour la dose la plus faible. Nous posons l’hypothèse que les images tomographiques en hélium auront la plus haute résolution spatiale.
4.2
Abstract
In this work, a generic rigorous Bayesian formalism is introduced to predict the most likely path of any ion crossing a medium between two detection points. The path is predicted based on a combination of the particle scattering in the material and measurements of its initial and final position, direction and energy. The path estimate’s precision is compared to the Monte Carlo simulated path. Every ion from hydrogen to carbon is simulated in two scenarios, 1) where the range is fixed and 2) where the initial velocity is fixed. In the scenario where the range is kept constant, the maximal root-mean-square error between the estimated path and the Monte Carlo path drops significantly between the proton path estimate (0.50 mm) and the helium path estimate (0.18 mm), and decrease steadily up to the carbon path estimate (0.09 mm). However, this constant-range scenario is identified as the configuration that maximizes the dose while minimizing the path resolution. In the scenario where the initial velocity fixed, the maximal root-mean-square error between the estimated path and the Monte Carlo path drops significantly between the proton path estimate (0.29 mm) and the helium path estimate (0.09 mm) but increases for heavier ions up to carbon (0.12 mm). As a result, helium is found to be the particle with the most accurate path estimate for the lowest dose, potentially leading to tomographic images of higher spatial resolution.
4.3
Introduction
Proton radiography (pRad) and proton computed tomography (pCT) were first proposed by Cormack et al. [93] and later experimentally proved by the same group [94]. However, photon tomography soon proved to be much more efficient and straightforward, and research in proton imaging halted.
Recently, this research emerged again with the advent of proton therapy. The proton therapy planning system requires knowledge of the proton stopping power within the patient, which can
be measured by proton tomography. As of now, this quantity is clinically obtained through a conversion from X-ray tomography Hounsfield units [78]. Such a process introduces significant uncertainties in planning and reduces the flexibility and advantages of proton treatment [18,80,
119,120]. It has been proven that single-event detection pCT could help lessen the uncertainty by directly measuring the proton stopping power in the patient [121]. Various reconstruction algorithm have been proposed to produce high spatial resolution proton tomographic images [39,114,122]. Moreover, proton imaging possesses several clinical and diagnostic qualities. It has a higher density resolution, a significantly lower noise level, and lower dose to the patient [91, 96] than in the conventional X-ray CT imaging. Finally, pCT suffers from different artifacts than X-ray CT [91]. However, one of the major problem encountered in pCT is the lower spatial resolution compared to X-ray CT.
The multiple deflections a proton suffers throughout its path, known as multiple Coulomb scattering (MCS), substantially reduce the spatial resolution of the images acquired. Conse- quently, the conventional X-ray tomographic algorithm struggles when using unaltered proton radiographies to reconstruct the pCT and the extracted images are of poor quality. Accurate proton path estimate methods have been proposed to solve the problem of MCS. Of those, the most likely path (MLP) algorithm is the most widely applied ([17, 98,100,123]). It is a method to calculate the proton path given position and direction information as well as the proton beam scattering formulated from the Fermi-Eyges [65,66] scattering equation. Starting from a different perspective, Collins-Fekete et al. [124] proposed a phenomenological approach to retrieve the proton path from a fit of the cubic spline direction magnitude that best repro- duces the Monte Carlo estimated path. Every approach relies on a sophisticated proton by proton detection system [38,50,125] to acquire precise entry and exit position/direction data and energy loss. Still, the MLP algorithm is limited by the inherent uncertainty associated with the MCS and can not resolve the proton path with high accuracy, leading to tomographic images of inferior spatial resolution than X-ray CT.
On the other hand, heavier ions suffer less from MCS due to smaller average angular devia- tions and are viable candidates to acquire high-quality tomographic images. However, more sophisticated accelerators are required to produce a beam of heavier particles with a minimal energy to cross a clinically relevant distance [14,126–128]. Moreover, as of now, no trajectory estimate has been proposed to extract the ions MLP.
This work first presents a generalized formalism that strictly follows the Bayesian theory to estimate the most likely trajectory of an ion. The proposed formalism is demonstrated to encompass the prior formalism by Schulte et al. [98], itself based on work by Schneider et al. [17] and Williams [100], and replicate the phenomenological cubic spline path (CSP) prediction
made by Collins-Fekete et al. [124]. Furthermore, the formalism strict Bayesian definition allows for future extensions with an example shown in this work. The new formalism is used to investigate the accuracy of the path estimate for heavier ions crossing a medium using the accuracy of the proton’s path estimate as a comparison. The MLP maximal root mean square (RMS) error to the Monte Carlo path is investigated for every ion up to carbon.