# Proton computed tomography reconstruction from sandwich time-of-flight using a lookup table approach

Aurélien Coussat, Nils Krah, Jean Michel Létang, Simon Rit

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Aurélien Coussat, Nils Krah, Jean Michel Létang, Simon Rit. Proton computed tomography reconstruction from sandwich time-of-flight using a lookup table approach. Fully 3D Image Reconstruction in Radiology and Nuclear Medicine, May 2025, Shanghai, China. pp.646-649. hal-05375602

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# Proton computed tomography reconstruction from sandwich time-of-flight using a lookup table approach

Aurélien Coussat¹, Nils Krah¹, Jean Michel Létang¹, and Simon Rit¹

¹INSA-Lyon, Université Claude Bernard Lyon 1, CNRS, Inserm, CREATIS UMR 5220, U1294, F-69373, Lyon, France

## Abstract

Proton computed tomography (pCT) is an imaging modality which offers numerous advantages for proton therapy, including more precise estimation of the proton stopping power map compared to its current estimation based on X-ray computed tomography (CT). List-mode pCT scanners measure the energy loss of each proton to estimate the integral of the proton stopping power along its path. However, there is currently no list-mode pCT scanner which is fast enough for clinical use. A time-of-flight (TOF) pCT scanner was recently proposed as a potential solution, referred to as *sandwich TOF*, where the stopping power integral must be estimated using the TOF of protons between two detectors located before and after the scanned object along the proton beam. The advantage of this kind of set-up is its compactness. Conversion from sandwich TOF to the line integral of the relative proton stopping power is not trivial as the proton speed and energy change differently depending on the sought stopping power of the traversed materials. In this work, we propose a simple reconstruction procedure using a lookup table approach. The method is assessed using Monte Carlo simulations of several objects and compared to images obtained with energy-based pCT.

## 1 Introduction

Proton therapy is a radiotherapy technique that leverages the localized dose deposition of protons in matter to target deep-seated volumes while minimizing damages to surrounding healthy tissues. The location of the end of the proton beam range, where it deposits the maximum energy and known as the Bragg peak, depends on the relative stopping power (RSP) of the traversed materials. As a consequence, treatment planning for proton therapy requires as input a map of the RSP distribution within the patient's body. This distribution is currently determined by converting Hounsfield units obtained from an X-ray computed tomography (CT) to RSP values. However, this conversion introduces uncertainties ranging from 1.6% to 5% [1], resulting in increased safety margins around the treated volume and higher normal tissue complication probability.

Proton computed tomography (pCT) is an imaging method that can be used to directly characterize the RSP distribution in the treatment room. In pCT, protons travel through the patient and lose energy depending on the traversed tissues. The residual energy of each proton is then measured using an energy detector located after the patient, and then converted into the corresponding water-equivalent path length (WEPL) which is a line integral of the sought RSP. The RSP distribution can then be recovered with a tomographic reconstruction algorithm.

Protons undergo multiple deflections when traversing the

patient due to multiple Coulomb scattering and nuclear interactions, and hence follow curved trajectories, which limits the spatial resolution. List-mode pCT scanner designs address this issue by tracking each proton individually using two detectors located before (upstream) and after (downstream) the scanned object. The measured positions and directions are then used to estimate the most-likely path of each proton individually. To keep the acquisition time reasonable, data should be acquired at a rate of at least a few MHz which has not been achieved so far.

A potential solution is to infer the energy loss indirectly from the times-of-flight (TOFs) of the protons between two detectors located after the object as the proton speed depends on the proton energy [2]. However, placing these two detectors with sufficient distance is impractical and an alternative solution [3], referred to as "sandwich TOF (STOF) pCT" in the following, is to place two 4D-tracking detectors that sandwich the object to estimate both the path (3D) and the TOF (+1D) of each proton. The TOF depends on the materials traversed by the proton, therefore the WEPL should be directly inferable from the TOF, removing the need for a residual energy detector. The 4D-tracking detectors could be implemented using low gain avalanche detectors as they are capable of sufficient timing and spatial resolutions [4]. One of the many challenges in implementing STOF pCT stems from the fact that protons traversing different materials with equivalent WEPLs result in equal energy losses, but significantly different TOFs, hence the conversion between WEPL and TOF is non-bijective. Consequently, the forward problem is non-linear and non-trivial to inverse, and the solution might be non-unique.

This work investigates the image quality, limitations, and artifacts of STOF when using a naive reconstruction method. The intention is to create a baseline for more advanced reconstruction techniques in the future. We are aware of the simplifying assumptions that restrict the set of objects for which the RSP distribution can be accurately reconstructed. More precisely, a Monte Carlo simulation of STOF pCT was implemented (Section 2.1), TOF and energy loss data were fitted to WEPL to convert measurements to WEPL (Section 2.2) and several phantoms were reconstructed (Section 2.3). The fits and the reconstructions are displayed in Section 3. The results and outcomes are discussed in Sections 4 and 5, respectively.
<figure><img src="image_2.png" /><figcaption>**Figure 1:** Schematic representation of a STOF pCT scanner, with the trajectory of a deflected proton shown as an example.</figcaption></figure>

# 2 Materials and Methods

## 2.1 Monte Carlo simulation

In this study, it is assumed that protons travel along the z axis. Protons are emitted in parallel in the z direction from position y = 0 mm and z = -1000 mm. The beam width is 40 cm in the x direction. The initial energy of the protons is set to 200 MeV. The upstream and downstream detectors are positioned at z = -110 mm and z = 110 mm, respectively. Detectors have a dimension of 40 cm in both x and y directions with no thickness. The simulated detectors perfectly measure the positions and directions of protons (to retrieve their most-likely path as per Ref. [5]), their time (to measure TOF) and their energy (in order to compare the proposed method with energy-based pCT). The scanned object is positioned between the two detectors and rotates around the y axis with a step size of 0.5°. At each step, 72 000 protons are shot, resulting in approximately 720 × 72 000 proton pairs measured by the two detectors over a 360° arc. The rest of the world is filled with air. The setup is illustrated by Figure 1.

Several scanned objects were considered in this study. First, a pure water cylinder with a radius of 10 cm and a length of 40 cm in order to assess the method in a simple case. Second, a water cylinder with a radius of 5 cm and centered at x = -5 cm, with the objective to evaluate the method in an asymmetrical configuration, as non-uniformity in the quality of the reconstructed image is anticipated. Third, a more complex phantom consisting of a water cylinder with multiple aluminum inserts arranged in a spiral [6] (referred to as the “spiral phantom”), to evaluate the resolution of the reconstruction of the inserts. Fourth, a water cylinder similar to the first object with two additional inserts: a cylindrical bone insert with a radius of 1.5 cm located at z = 5 cm, and a cylindrical air insert with a radius of 1.5 cm located at z = -5 cm (referred to as the “bone phantom”), with the goal to assess the impact of highly different RSPs on the reconstruction of the rest of the water phantom.

All simulations were prepared using GATE (the Geant4 Application for Tomographic Emission) [7] version 10.

## 2.2 TOF to WEPL conversion

TOF pCT reconstruction necessitates the conversion of the measured TOF to the equivalent WEPL. For this purpose, a second Monte Carlo simulation was prepared, employing the same geometry as outlined in Section 2.1 with a box of water in place of the phantom, placed at the isocenter of the detector, i.e. there was always the same amount of air before and after the object. The dimensions of the box were 40 × 40 × L cm, where L denotes the length of the box along the z axis. The Monte Carlo simulation was repeated for 100 distinct, evenly-spaced values of L ranging from 0 mm (no box, protons travel in air only) to 220 mm (the distance between the two detectors, i.e. in the last simulation all the space between the detectors was filled with water). In each simulation, 10⁴ protons were emitted along the z axis from a point-like source located at z = -120 mm.

For each proton, the WEPL corresponds to the width of the water box L, and the TOF was determined using the time registered by the upstream and downstream detectors. For each value of L, the median over all TOFs was stored. A polynomial curve from TOF to WEPL was then fitted along the list of stored medians. To avoid aberrant WEPL occasionally obtained for rare events with exceedingly high TOFs, TOFs above an empirically determined threshold were mapped to a maximum WEPL.

For comparison purposes, the exact same methodology was applied to fit the measured energy loss to the WEPL.

## 2.3 Tomographic reconstruction

The Monte Carlo simulation described in Section 2.1 yielded in 720 pairs of list-mode datasets for up- and downstream detectors. The reconstructions were then performed in four steps. (1) Data collected by the two detectors were paired by proton. The present study does not simulate realistic proton pairing and assumes that protons are perfectly paired. Once paired, the TOF was calculated then converted to WEPL using the method detailed in Section 2.2. (2) Energy and angle cuts (at 3 standard deviations) were applied in order to filter out protons that underwent nuclear interactions as suggested in [5]. (3) Proton pairs were binned following the procedure described in Ref. [6], yielding distance-driven projections of the scanned object. (4) Reconstruction was performed using distance-driven FDK, as described in Ref. [6]. For simplicity, only one central slice (1 mm thick around y = 0) was reconstructed, but the method is equally applicable to 3D reconstructions. The output image was reconstructed on a grid of 210 × 210 pixels of 1 mm³. For comparison, the reconstructions were also performed using the energy-loss-to-WEPL fit mentioned in Section 2.2. All reconstructions were executed with PCT<fnref n="1" />, a proton CT toolkit based on the Reconstruction Toolkit [8].

<footnote n="1">¹https://github.com/RTKConsortium/PCT</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 2:** Red: fitted curve from energy loss to WEPL. Blue: fitted curve from TOF to WEPL. The leftmost medians correspond to the case where *L* = 0 mm. The rightmost medians correspond to the case where *L* = 220 mm. The horizontal bars highlight the range between the first and the third quartile.</figcaption></figure>

## 3 Results

Figure 2 illustrates the data collected to calculate the two polynomial fits used for energy-based pCT reconstruction (red) and STOF pCT reconstruction (blue). Figure 3 shows reconstructions of the phantoms listed in Section 2.3 performed with energy pCT (on the left) and with STOF pCT (on the right). Specifically, Figure 3 displays the reconstruction of the central slice of (a) the centered uniform cylinder of water, (b) the same uniform cylinder of water shifted at $x = -5$ cm, (c) the spiral phantom, and (d) the bone phantom. Additionally, Figure 4 shows profiles through the central slice of the images shown in Figures 3c and 3d. Finally, the Monte Carlo simulation of the uniform cylinder (shown in Figure 3a) was repeated 100 times. The pixel-wise standard deviations across all STOF pCT images were calculated and are shown in Figure 5.

## 4 Discussion

As expected, and as illustrated in Figure 2, the WEPL is nonlinearly proportional to both the energy loss and the TOF. The uncertainty in the measured quantities increases with *L* due to energy straggling. The resulting fitted curves serve as a simple model for conversion to WEPL that permits image reconstruction, although sometimes with strong artifacts, as shown in Figure 3.

This approach has several limitations. The fitted curves are only valid for a given distance between detectors and a given initial energy, set to 220 mm and 200 MeV in this study, respectively. The method assumes that the proton beam is parallel, which might not be the case depending on the implementation of the pCT scanner. Additionally, several assumptions are made regarding the object: it is assumed to be positioned at the isocenter, composed of pure water, and convex.

When the object is not located at the isocenter, the RSP values are not correctly reconstructed, as illustrated by Figure 3b.

<figure><img src="image_4.png" /><figcaption>(a) Uniform water cylinder.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(b) Off-centered uniform cylinder.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(c) Spiral phantom.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(d) Bone phantom.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(a) Spiral phantom.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(b) Bone phantom.</figcaption></figure>

<caption>**Figure 4:** Profiles through the lines displayed in Figures 3c and 3d.</caption>
<figure><img src="image_10.png" /><figcaption>**Figure 5:** Top: pixel-wise standard deviation of 100 realizations of the reconstruction of the uniform water cylinder (see Figure 3a). Pixels outside the scanner field-of-view were ignored (value set to 0). Left is energy-based pCT and right is STOF pCT. The grayscale is [0,0.027]. Bottom: profiles through the standard deviations shown above. The root mean square (RMS) over the whole cylinder is of 0.011 for energy-based pCT, and 0.014 for STOF.</figcaption></figure>

This is explained by the following: when protons exit the phantom and reach the downstream detector, their TOF is dependent on the amount of air between the phantom and the detector, but not their energy. As a consequence, an off-centered phantom does not respect the assumptions of the TOF-to-WEPL model shown in Figure 2 (blue) and the RSP values of the STOF reconstruction are incorrect, but not those of the energy-based reconstruction. This is one of the limitation of the method.

Non-water inserts are incorrectly reconstructed, as can be seen in Figures 3c and 3d when looking at the profiles. RSP for the aluminum and bone inserts are overestimated in the STOF case compared to values obtained with energy-based pCT. On the opposite, RSP of the air insert (Figure 3d) is underestimated by STOF pCT. Since the TOF-to-WEPL polynomial is fitted using only a water box, it is not designed to properly reconstruct non-water elements. The spatial resolution of TOF pCT is not deteriorated compared to that of energy-based pCT (Figure 3c) due to the fact that perfect detectors are considered. The resolution of a realistic scanner is in practice dependent on the resolution of its detectors. Moreover, the reconstruction of the water background was not strongly affected with the small aluminum inserts (Figure 3c), but much more by the larger bone and air inserts (Figure 3d), because they induce larger discrepancies between the TOF-to-WEPL fit that only accounts for water and the actual TOF measurements.

Finally, even though perfect detectors are considered, the standard deviation of reconstructed RSP values is higher in the STOF case (root mean square (RMS) of 0.014) compared to the energy-based one (RMS of 0.011), as illustrated by Figure 5. This is because the variation of measured TOF

is larger than that of measured energy loss, especially for larger amount of traversed water, as illustrated by the quartile spread for the rightmost medians on Figure 2.

## 5 Conclusion

The proposed tomographic reconstruction method allows for reconstruction of the RSP map of the scanned object from STOF data, but with strong limitations when it comes to non-water or off-centered objects. Nevertheless, this method serves as an initial step towards STOF pCT image reconstruction and can be used as a baseline method for comparison against more advanced techniques, which will be considered in future works. These more advanced techniques may be based on anatomical segmentation, or direct, iterative inversion of the non-linear forward model.

## Acknowledgment

With financial support from IT Cancer of Inserm within the framework of the 2021-2030 Cancer Control Strategy, on funds administered by Inserm, within the framework of the SIRIC LYRICAN INCa-INSERM-DGOS-12563 and the LABEX PRIMES (ANR-11-LABX-0063) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the ANR.

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