Impact of beam hardening on proton stopping-power ratio estimation using photon-counting, dual-energy, and single-energy computed tomography

Abstract

Background and purpose

Accurate stopping-power ratio (SPR) estimation is essential for treatment planning of proton therapy. However, SPR accuracy is influenced by object size and position due to beam hardening. This study compared proton SPR estimation using photon-counting computed tomography (PCCT), dual-energy CT (DECT), and single-energy CT (SECT) across varying phantom sizes and insert positions.

Materials and methods

Two phantoms with tissue-equivalent inserts were scanned using PCCT, DECT, and SECT scanners. Based on the PCCT and DECT scans, virtual monoenergetic images (VMIs) were created. Two SPR estimation methods were used: a DECT-based method and a SECT-based Hounsfield look-up table (HLUT). SPRs were calibrated on the Gammex Advanced Electron Density phantom and evaluated on a cylindrical phantom with four diameters. To assess size-dependency of the estimated SPR, tissue-equivalent bone inserts were placed centrally in the evaluation phantom, for each of the four diameters. To assess position-dependent SPR uncertainty, the inserts were placed at different distances from the centre of the evaluation phantom with fixed diameters.

Results

For smaller phantom diameters, SPR estimation accuracy was comparable across PCCT, DECT, and SECT. For larger phantom diameters (30 and 40 cm), SPRs estimated based on SECT and DECT showed position-dependency. PCCT demonstrated the most stable SPR estimations across positions, with deviations between 0.9% and 2.1%. SPR estimations were more stable using a HLUT than the DECT-based SPR method.

Conclusions

SPR estimated based on PCCT-VMI with a HLUT showed reduced impact of beam hardening effects related to size and position.

Graphical abstract

Highlights

• •Three different computed tomography (CT) modalities were compared.
• •Two different stopping-power ratio estimation methods were compared.
• •The most stable stopping-power ratio estimations were seen for photon-counting CT.
• •Stopping-power ratio deviations for photon-counting CT were between 0.9% and 2.1%.

1. Introduction

Protons deposit most of their energy at the end of their range, enabling highly conformal dose distributions and sparing of healthy tissue. Treatment planning in proton therapy typically relies on a computed tomography (CT) scan of the patient to estimate stopping-power ratios (SPRs) of tissues relative to water [1], [2], as SPR is needed to calculate the proton range and dose [3].

Single-energy CT (SECT) is commonly used for treatment planning in proton therapy [3], [4]. However, only a heuristic conversion from CT numbers to SPRs can be established [4], which introduces uncertainty in the SPR estimation. CT numbers for the same tissue can vary with patient size and location inside the patient due to beam hardening caused by the higher X-ray attenuation at lower photon energies [5], [6]. Beam hardening is one of the major sources to uncertainties in SPR estimation [7], [8]. A total range uncertainty of 3.0%–3.5% is therefore typically accounted for in treatment planning [9], [10].

By acquiring CT scans with two different energy spectra, dual-energy CT (DECT) exploits the energy dependence of photon interactions to provide additional information on the tissue composition [11]. This enables the estimation of relative electron density (RED) and effective atomic number (EAN), which can be used for SPR estimation [[12], [13], [14]]. Applying DECT, range uncertainties can potentially be reduced to 1.7%–2.0% [[8], [15], [16], [17], [18], [19]]. Further, DECT also enables generation of virtual monoenergetic images (VMIs) [20], [21] which can reduce the impact of beam hardening [5], [22].

In photon-counting CT (PCCT), solid-state detectors enable detection of individual photons and their energies [23]. This allows for better spectral separation compared to most DECT techniques [24]. PCCT has a SPR accuracy comparable to DECT [25], [26], [27], [28], at low imaging doses even superior to DECT [29], [30]. PCCT can be used for treatment planning in both photon [31], [32] and proton therapy [25]. Furthermore, PCCT can reduce CT number variations in different phantom configurations [33], [34], [35], [36], [37] and therefore potentially impact SPR estimations. The aim of this study was therefore to assess the impact of beam hardening on SPR estimation accuracy based on PCCT, DECT, and SECT, including both phantom size and insert position.

2. Materials and methods

2.1. Phantoms

The Gammex Advanced Electron Density (AED) phantom (Sun Nuclear – A Mirion Medical Company, Middleton, WI, USA), and a custom-made Modular Evaluation (ME) phantom, also from Sun Nuclear (Fig. 1) were used. Both phantoms were made from the same solid water material. Twelve cylindrical tissue-equivalent phantom inserts with a diameter of 2.8 cm and a length of 16.5 cm from the AED phantom were used for both phantoms. The AED phantom had a small round inner part with a diameter of 20 cm, and a large oval part with the dimensions 40 cm × 30 cm, simulating head and abdomen of an adult. The ME phantom consisted of an inner cylinder and three cylindrical shells with respective diameters of 10, 20, 30, and 40 cm.

Fig. 1.

Schematic workflow of the beam hardening assessment. The AED phantom and ME phantom were scanned with PCCT, DECT, and SECT. For the PCCT and DECT scans, VMIs were created, while the low (L) and high (H) energy scans were also used. The AED phantom was used for calibration of the SPR estimation, applying two different methods – a DECT-based method proposed by Saito and Sagara [14], and SECT-based Hounsfield look-up tables (HLUTs) [4]. The ME phantom was used for evaluation. For both phantoms, the grey numbered areas indicate the insert positions, and the dark blue circles indicate the different phantom diameters. The insert configurations for the AED phantom followed the consensus guide [4], and the inserts configurations for the ME phantom are described in Supplementary Material A. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The AED phantom was used for calibration of the SPR estimation methods, while the ME phantom was used for evaluation, as it consisted of several phantom sizes. The calibrations based on the small AED phantom were applied to the ME phantom diameters of 10 and 20 cm, while the calibrations based on the large AED phantom were applied to the ME phantom diameters of 30 and 40 cm. Insert configurations for the AED scans followed the consensus guide for Hounsfield look-up table (HLUT) generation [4]. For the ME phantom scans, the non-bone (i.e. lung, adipose, and soft tissue) inserts were placed close to the centre (Supplementary Material A). Additionally, for the ME phantom with 30 and 40 cm diameters, non-bone inserts were placed further away from the centre. The impact of beam hardening was evaluated using two bone inserts, ‘CaCO₃ 50%’ and ‘HE Cortical Bone’. The two bone inserts were scanned individually at all four phantom diameters, at positions 1, 4, 11, and 17, corresponding to increasing distances from the centre (Fig. 1).

2.2. Image acquisitions

CT scans were acquired using two different CT scanners: a NAEOTOM Alpha PCCT scanner (Siemens Healthineers, Forchheim, Germany), and a SOMATOM Definition Edge CT scanner (Siemens Healthineers), the latter operated in both DECT and SECT mode. PCCT scans were acquired at 140 kVp, DECT scans were acquired in dual-spiral mode at 80/140 kVp, and SECT scans were acquired at 120 kVp. The field-of-view of each scan was set to the phantom diameter plus five centimetres. PCCT scans were iteratively reconstructed with the QIR algorithm strength 3, Qr40 kernel, and iBHC Quantum beam hardening correction. DECT and SECT scans were iteratively reconstructed with the ADMIRE algorithm strength 3, Qr40 kernel, and iBHC bone beam hardening correction. All images were reconstructed with a matrix size of 512 × 512 pixels, and 2 mm slice thickness, resulting in voxel sizes between 0.29 × 0.29 × 2.00 mm³ and 0.88 × 0.88 × 2.00 mm³. For PCCT, a standard abdominal protocol was used. For DECT and SECT, two imaging dose levels were used for the larger phantom sizes. The applied imaging doses are stated in Table 1.

Table 1.

CT dose levels for the different scans. Abbreviations: AED: Gammex Advanced Electron Density phantom, ME: Modular evaluation phantom, CTDI_vol: CT dose index, NA: not applicable.

CT modality	Phantom	Phantom size	Dose level	CTDIvol (mGy)
PCCT	AED	Small	NA	CTDIvol,16cm = 42.4
PCCT	AED	Large	NA	CTDIvol,32cm = 20.7
DECT	AED	Small	Low	CTDIvol,16cm = 39.3
DECT	AED	Large	Low	CTDIvol,32cm = 20.0
DECT	AED	Large	High	CTDIvol,32cm = 35.0
SECT	AED	Small	Low	CTDIvol,16cm = 40.0
SECT	AED	Large	Low	CTDIvol,32cm = 19.7
SECT	AED	Large	High	CTDIvol,32cm = 35.1
PCCT	ME	All diameters	NA	CTDIvol,32cm = 10.2
DECT	ME	All diameters	Low	CTDIvol,32cm = 20.1
DECT	ME	30 cm, 40 cm	High	CTDIvol,32cm = 35.0
SECT	ME	All diameters	Low	CTDIvol,32cm = 19.7
SECT	ME	30 cm, 40 cm	High	CTDIvol,32cm = 35.1

For PCCT and DECT, 16 VMIs were created at energies ranging from 40 to 190 keV in 10 keV increments. Additionally, the low- and high-energy bin images for PCCT and the low- and high-energy spectra images for DECT were also used, as the original Saito method was based on low- and high-energy DECT scans [14]. CT numbers were extracted from cylindrical volumes-of-interest (VOIs) with a diameter of 1.9 cm and a length of 2.0 cm placed centrally in each insert.

2.3. Calibration of SPR estimation methods

SPR calibration was based on the mean CT numbers of the inserts in the AED phantom. Two SPR estimation methods were used: a DECT-based method following Saito and Sagara (denoted Saito method) [[14], [38]], and a SECT-based HLUT approach following the consensus guide (denoted HLUT method) [4].

For the Saito method, RED (${\rho }_{\mathrm{e}}$) was estimated as a weighted difference of CT numbers, $\mathcal{H}$, in scans acquired at different energies

$${\rho }_{\mathsf{e}}=a\frac{\left(1+c_0\right)\bullet {\mathcal{H}}_{\mathsf{H}}-c_0\bullet {\mathcal{H}}_{\mathsf{L}}}{1000}+b$$ (1)

where $H$ and $L$ denote the high and low energies, and $a$, $b,$ and $c_0$ are fit parameters.

The EAN ($Z_{\mathrm{eff}})$ was estimated as

$${\left(\frac{Z_{\mathsf{eff}}}{Z_{\mathsf{eff},\mathsf{w}}}\right)}^n-1={\gamma }_{\mathsf{L}}\left(\frac{\frac{{\mathcal{H}}_{\mathsf{L}}}{1000}+1}{{\rho }_{\mathsf{e}}}-1\right)+{\gamma }_0$$ (2)

where subscript $w$ denotes water, $n=3.3$ [38], and ${\gamma }_{\mathrm{L}}$ and ${\gamma }_0$ are fit parameters [[14], [26], [39], [40]], with ${\gamma }_0$ accounting for non-perfect measurements of the CT number of water [26], [39].

The mean excitation energy ($I$) was estimated as

$$\ln \frac{I}{I_{\mathsf{w}}}=d_1\left({\left(\frac{Z_{\mathsf{eff}}}{Z_{\mathsf{eff},\mathsf{w}}}\right)}^n-1\right)-d_0$$ (3)

where $d_0$, $d_1$ are fit parameters. Separate fits for soft tissue and bone were performed, with a threshold of $Z_{\mathrm{eff}}=8.8$ [14]. The theoretical mean excitation energies were calculated applying the Bragg additivity rule [41] based on Table 2.8 and 2.11 in ICRU report 49 [42], resulting in $I_{\mathrm{w}}$ = 75.3 eV. The parameters $a$, $b,$ $c_0,$ ${\gamma }_{\mathrm{L}},$and ${\gamma }_0$ were fitted based on the AED phantom CT scans, while $d_0$ and $d_1$ were fitted based on 71 tabulated human tissues collected by Woodard and White [43].

The SPR was estimated using the Bethe formula [3]:

$$\mathsf{SPR}={\rho }_{\mathsf{e}}\frac{\ln \frac{2m_{\mathsf{e}}{c}^2{\beta }^2}{I\left(1-{\beta }^2\right)}-{\beta }^2}{\ln \frac{2m_{\mathsf{e}}{c}^2{\beta }^2}{I_{\mathsf{w}}\left(1-{\beta }^2\right)}-{\beta }^2}$$ (4)

where $m_{\mathrm{e}}$ is the electron mass, $c$ is the speed of light, and $\beta$ is the relative proton speed for 100 MeV protons.

When fitting Eq. (1) and (2), lung inserts were excluded due to the heterogeneity of these inserts [44]. Fits were created for PCCT and DECT, based on either VMI pairs or the low- and high-energy bins/spectra, resulting in 121 individual fits. In the evaluation, voxels with negative, i.e. unphysical, RED, EAN, and SPR values could occur. These were replaced with ${\rho }_{\mathrm{e}}=0.001$, $Z_{\mathrm{eff}}=0,$ and $\mathrm{SPR}=0$. The chosen RED value approximated that of air [45] to avoid a division by zero in Eq. (2).

For the HLUT method, piecewise linear conversion curves were created applying the consensus guide open-source code [4]. The HLUTs were scanner specific, and they were fitted based on theoretical SPRs for tabulated human tissues and AED phantom inserts, calculated with Eq. (4). Conversion curves were created for PCCT, DECT, and SECT. For PCCT and DECT, one conversion curve was created per VMI energy, and low- and high-energy bin/spectre, resulting in 18 conversion curves. All conversion curves were only used for research purposes.

2.4. VMI energy selection

For the evaluations performed on the ME phantom, a single VMI energy / energy pair was used for each CT modality / dose level. These VMI energies / energy pairs were selected as those minimising the root-mean-square deviation (RMSD) of the estimated SPRs in the AED phantom. SPR deviations between estimated and theoretical SPR (Supplementary Material B) were calculated relative to the SPR of water (SPR_water = 1) as [26], [39]:

$${\mathsf{SPR}}_{\mathsf{deviation}}=\left(\mathsf{SP}{\mathsf{R}}_{\mathsf{estimated}}-{\mathsf{SPR}}_{\text{theoretical}}\right)/{\mathrm{SPR}}_{\text{water}}·100\%$$ (5)

For each insert, the SPR deviations were calculated per voxel in the VOI, and the averaged value was used for RMSD calculations:

$$\mathsf{RMSD}=\sqrt{\frac{1}{N}{\sum }_{i\in \text{inserts}}^N\mathsf{SP}{\mathsf{R}}_{\mathsf{deviation},i}^{\mathsf{2}}}$$ (6)

where N is the number of included inserts. Only inserts placed in the centre of the AED phantom were included, i.e. N = 12 (eight non-bone and four bone inserts) for the HLUT method, but N = 10 for the Saito method, as lung inserts were excluded. RMSDs were calculated for all energy pairs, but pairs were excluded if more than 1.0% of the voxels for an insert had negative RED, EAN, or SPR values. For the HLUT method, RMSDs were calculated for all energies (here, no negative values could occur).

2.5. SPR accuracy and beam hardening evaluation

The SPR accuracy was assessed for each insert and ME phantom diameter with both SPR estimation methods. The SPR deviations (Eq. (5)) were calculated voxel-wise in the VOIs, and the mean and standard deviation were extracted. Size-dependent beam hardening was assessed by comparing SPR deviations of centrally placed bone inserts across different ME phantom diameters. Position-dependent beam hardening was assessed by comparing SPR deviations for the bone inserts placed at different distances from the centre in the ME phantom with constant diameter. For both bone inserts and each configuration, a linear model was fitted for the estimated SPRs as a function of diameter or position, indicating the impact of beam hardening affecting SPR estimation.

3. Results

The chosen VMI energy for the HLUT method for PCCT was 150 keV (RMSD = 0.5%) for the small phantom and 170 keV (RMSD = 0.5%) for the large phantom (Fig. 2). For DECT, it was 170 keV (RMSD = 0.3%) for the small phantom, 190 keV (RMSD = 0.6%) for the large phantom in the low-dose scans, and 120 keV (RMSD = 0.7%) for the large phantom in the high-dose scans. For the Saito method, the minimum RMSD among the VMI energy pairs for PCCT was obtained at 150/160 keV (RMSD = 0.4%) for the small phantom and at 80/90 keV (RMSD = 0.6%) for the large phantom. The low/high-energy bin was excluded due to negative EANs in 1.9% of the voxels in the adipose insert, despite the low RMSD of 0.5%. For DECT, the minimum RMSDs were found at 40/50 keV (RMSD = 0.7%) for the small phantom, 40/120 keV (RMSD = 0.6%) for the large phantom at low dose, and 40/100 keV (RMSD = 0.8%) for the large phantom at high dose. For both SPR methods, RMSDs for at least 60% of the energies/energy pairs stayed within 0.2 percentage points of the chosen energy/energy pair.

Fig. 2.

Root-mean-square deviations (RMSDs) of the estimated SPRs based on the AED phantom for all VMI energies / energy pairs as well as for the low- and high-energy spectra/bins (indicated by L and H, placed in grey boxes). The triangular heatmaps (a-e) show RMSDs following the Saito method applying two energies, and the lines (f) show RMSDs following the HLUT method applying a single energy. The energy / energy pair selected for further evaluations is indicated with a blue box. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

SPR deviations for non-bone inserts for PCCT following the HLUT method were within ±1.0% for all ME phantom diameters, except for the HE Liver insert in the 10 cm phantom, with a deviation of −1.1%. For DECT and SECT scans, following the HLUT method, SPR deviations for centrally located non-bone inserts remained within ±1.0%, however, deviations increased to a maximum of −2.0% for inserts located in peripheral phantom positions (Fig. 3 and Supplementary Material C). For the Saito method, both PCCT and DECT showed SPR deviations above −2.5% for some non-bone inserts in the larger phantom diameters. Large deviations occurred for lung inserts, which were not used in the calibration.

Fig. 3.

SPR deviations for centrally placed non-bone inserts in the 40 cm ME phantom. The mean SPR deviation (marker) and its standard deviation over all voxels (bar) are shown. The names on the x-axis correspond to the phantom insert names. Insert name abbreviations: LN300: Lung LN300, LN450: Lung LN450, Adi: HE Adipose, Breast: HE Breast 50:50, LW: Liquid Water, SW: HE Solid Water. The results for the remaining diameters can be found in Supplementary Material C.

For the different ME phantom sizes, SPR deviations for the CaCO₃ 50% insert, using the HLUT method, were lowest for PCCT (below 0.5%), and highest for low-dose DECT with 3.5% (Fig. 4). Increasing the dose for DECT, reduced SPR deviations to 1.6%. For SECT, deviations were below 1.2% for both dose levels. For HE Cortical Bone, deviations were lowest for PCCT following the HLUT method, with 0.9%, and highest for high-dose SECT, up to 2.8%. In general, SPR deviations were smaller when following the HLUT method than for the Saito method. Across all phantom diameters, no correlation between the estimated SPR and the phantom size was observed (Supplementary Table S2).

Fig. 4.

Size-dependent beam hardening evaluation. SPR deviations for the CaCO₃ 50% insert (a) and the HE Cortical Bone insert (b). The inserts were placed centrally in the ME phantom with different diameters. The mean SPR deviation (marker) and its standard deviation over all voxels (bar) are shown.

For the bone inserts placed in different positions in the 20 cm phantom, SPR deviations were within 2.3% for all three CT modalities (Fig. 5), with the largest deviations for the HE Cortical Bone at position 4 (periphery) using PCCT following the Saito method. For the 30 cm phantom, SPR deviations for PCCT, high-dose DECT, and SECT with both dose levels remained within 1.7% for the HLUT method. SPR estimations based on the Saito method showed larger positional dependence, with the largest deviations seen for low-dose DECT following the Saito method, reaching up to −2.7%. For the 40 cm phantom, the strongest positional impact was observed. While SPR deviations were similar for the centrally placed inserts between the CT modalities and SPR estimation methods, deviations increased towards the periphery. PCCT combined with the HLUT method had the lowest peripheral deviations, with up to +2.1% for HE Cortical Bone. Deviations for inserts in the periphery were largest for DECT and SECT, reaching −5.5% for both high-dose SECT and low-dose DECT using the Saito method. Across the 30 cm and 40 cm phantoms, the positional impact on SPR could be described by linear fits (Supplementary Material D). The strongest positional impact was observed for PCCT following the Saito method, with an increase in SPR of 0.18 percentage point/cm (R² = 0.99).

Fig. 5.

Position-dependent beam hardening evaluation. SPR deviations for the CaCO₃ 50% insert (a, c, e) and HE Cortical Bone insert (b, d, f). The inserts were placed at different distances from the centre in the ME phantom with a diameter of 20 cm (a, b), 30 cm (c, d), and 40 cm (e, f). The numbers on the x-axes correspond to the insert positions as shown in Fig. 1 and Supplementary Fig. S1. The mean difference and its standard deviation over all voxels are shown.

4. Discussion

This study compared the influence of size- and position-dependent beam hardening on SPR estimation accuracy for PCCT, DECT, and SECT scans, using a DECT-based approach and a SECT-based HLUT approach. For smaller phantom sizes, SPR deviations were comparable across the three CT modalities. For larger phantom sizes, PCCT combined with the HLUT method provided the most accurate and stable SPR estimates. Compared to the Saito method, SPR estimation following the HLUT method was more accurate and showed reduced variations with phantom size and insert position.

The SPR deviations were comparable to prior studies reporting SPR deviations below 1.0% for PCCT. Hu et al. [46] achieved this using a 60/180 keV VMI pair. Huijskens et al. [25] reported similar deviations using a commercial SPR algorithm. Lustermans et al. [26] compared DECT with PCCT using the Saito [14] and Hünemohr [12] SPR estimation methods and observed no significant differences between the two SPR methods, both with deviations below 1.0%. In an anthropomorphic head phantom, Zimmermann and Poludniowski [27] found RMSD of 2.2% for PCCT.

For DECT, Saito and Sagara [14] reported SPR deviations up to −0.9% for bone, and −1.7% for non-bone inserts. Chacko et al. [7] reported SPR deviations of 2.0% for bones in different phantom sizes and reported a linear relationship between phantom diameter and SPR deviation. In contrast, we found no clear relationship between the phantom diameter and the estimated SPR, possibly due to individual SPR calibrations for small and large phantoms.

The optimal VMI energy pairs found in this study deviated from those in previous studies which were typically well-separated in energy [26], [39], [46]. However, the influence of the VMI pair seemed negligible. Using the 70/180 keV pair (used by Lustermans et al. [26]), RMSDs only differed by 0.1 percentage points (Fig. 2), and SPR estimates for HE Cortical Bone in the 40 cm phantom across all positions deviating up to 2.7%, −4.9%, and −4.9% for PCCT, low-, and high-dose DECT (compared to 2.8%, −4.9%, −4.9% in Fig. 5). This might reflect that spectral information is already incorporated during the VMI creation. Still, higher energy pairs yielded lower RMSDs for bone inserts, while well-separated energy pairs yielded lower RMSDs for non-bone and slightly reduced the impact of uncertainty in the fit parameters for the Saito method (Supplementary Material E).

The large SPR deviations for bone inserts in the periphery of the ME phantom (Fig. 5) could be due to large, systematic CT number fluctuations that were observed along the scan direction, and were more pronounced for high-dose scans (Supplementary Material F). In addition, artefacts for bone inserts in the periphery in low-dose DECT and SECT likely contributed to CT number inconsistencies and reduced SPR accuracy. Lustermans et al. [29], [30] showed that the imaging dose influences the CT number consistency and SPR accuracy more strongly for DECT than PCCT, which was seen for DECT and SECT for some insert and phantom configurations (Fig. 5). A similar behaviour of low- and high-dose SECT to high-dose DECT was observed (Fig. 4-Fig. 5), indicating that SECT was not sensitive to the chosen imaging dose. Accuracy differences between low- and high-dose DECT (Fig. 4) could be caused by VMI dependent CT number differences between the AED and ME phantoms. Overall, noise of the SPR estimations between the different CT modalities and SPR estimation methods was mostly impacted by the imaging dose, SPR estimation method, and position within the phantom (Supplementary Material F).

A strength of this study was the use of different phantom sizes and geometries to calibrate and evaluate estimated SPRs. However, the phantoms had the same composition, and the same inserts were used for calibration and evaluation. The phantom itself had a simple structure, which did not allow for the evaluation of beam hardening effects in more realistic anthropomorphic geometries. Furthermore, only one SECT, one dual-spiral CT, and one PCCT scanner were included, which are not necessarily representative for all available CT scanners [[15], [47], [48], [49]].

In conclusion, this study assessed the impact of size- and position-dependent beam hardening on SPR estimation across PCCT, DECT, and SECT scans, using two different SPR estimation methods. When SPRs were estimated based on VMIs following the HLUT method, PCCT showed less impact of beam hardening compared to DECT and SECT, resulting in a more stable SPR estimation across phantom sizes and insert positions.

CRediT authorship contribution statement

Lasse Bassermann: Writing – review & editing, Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis, Data curation. Jens Morgenthaler Edmund: Writing – review & editing, Supervision, Resources. Weronika Elżbieta Olech: Writing – review & editing, Investigation. Stine Elleberg Petersen: Writing – review & editing, Supervision. Maria Fuglsang Jensen: Writing – review & editing, Methodology. Ludvig Paul Muren: Writing – review & editing, Supervision. Vicki Trier Taasti: Writing – review & editing, Supervision, Project administration, Methodology, Funding acquisition, Conceptualization.

Declaration of competing interest

Given the role as Associate Editor in Physics and Imaging in Radiation Oncology, Vicki Trier Taasti had no involvement in the peer-review of this article and had no access to information regarding its peer-review. Full responsibility for the editorial process for this article was delegated to another journal editor.

Appendix A. Supplementary data

Supplementary material