# Methodology for comparing absorbed dose rate calculation algorithms in molecular radiotherapy dosimetry

José-Alejandro Fragoso-Negrín, Lore Santoro, Kevin Hébert, Marilyne Kafrouni, Sébastien Vauclin, Pierre-Olivier Kotzki, Jean-Pierre Pouget, Emmanuel Deshayes, Manuel Bardiès

► To cite this version:

José-Alejandro Fragoso-Negrín, Lore Santoro, Kevin Hébert, Marilyne Kafrouni, Sébastien Vauclin, et al. Methodology for comparing absorbed dose rate calculation algorithms in molecular radiotherapy dosimetry. Physica Medica European Journal of Medical Physics, 2025, 133, pp.104965. 10.1016/j.ejmp.2025.104965. hal-05301986

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Original paper

# Methodology for comparing absorbed dose rate calculation algorithms in molecular radiotherapy dosimetry

<figure><img src="image_3.png" /></figure>

José-Alejandro Fragoso-Negrín<sup>a,b,c</sup>, Lore Santoro<sup>a,b</sup>, Kevin Hébert<sup>b</sup>, Marilyne Kafrouni<sup>c</sup>, Sébastien Vauclin<sup>c</sup>, Pierre-Olivier Kotzki<sup>a,b</sup>, Jean-Pierre Pouget<sup>a</sup>, Emmanuel Deshayes<sup>a,b,c</sup>, Manuel Bardiès<sup>a,b,*</sup>

## ARTICLE INFO

### Keywords:

- Dosimetry algorithms

- Absorbed dose rate

- Molecular radiotherapy

- OpenDose3D

- Density correction

## ABSTRACT

**Introduction:** This study compared the results obtained using three of the most frequently proposed algorithms, Local Energy Deposition (LED), Convolution of Voxel S Values (CONV) and Monte Carlo modeling (MC).

**Methods:** OpenDose3D, a free, open-source clinical dosimetry software was used to perform this comparison. The assessment focused on absorbed dose rate (ADR), thereby increasing the dataset size and avoiding potential bias associated with the time integration step. From patients treated with <sup>90</sup>Y-microspheres, [<sup>177</sup>Lu]Lu-DOTATATE or Na [<sup>131</sup>I], 52 datasets were processed. Voxel-based ADR maps were computed in homogeneous (water) and heterogeneous (CT-derived) media, for a total of 312 datasets. In a heterogeneous medium, density correction was applied at the voxel level. ADR values were averaged over VOIs, and relative differences (rd) were calculated using the MC results as the reference.

**Results:** In the homogeneous medium, LED underestimated the ADR by up to 10 % in soft tissues, particularly when cross-irradiation cannot be neglected. Conversely, the ADR obtained with CONV presented excellent agreement with the MC simulations (rd ≤ 1 %).

In the heterogeneous medium, density correction was crucial. For example in <sup>131</sup>I, LED underestimated the ADR values by up to 50 % in lungs and bone marrow. CONV showed excellent agreement with MC in soft tissues (rd ~ 1 %) and good agreement in organs/tissues where self-irradiation was predominant. The implemented density correction was much less efficient for organs/tissues that experience mostly cross-irradiation.

**Conclusions:** Our work proposes a procedure for evaluating ADR algorithms. It also underscores the impact of the medium heterogeneity and cross-irradiation on dosimetry calculations.

## 1. Introduction

Molecular radiotherapy (MRT) encompasses all therapeutic applications in nuclear medicine based on radionuclides that emit most often beta and possibly gamma radiation. Electrons deliver high irradiation over a short range (i.e., the intended objective of selective irradiation). Gamma rays contribute to undesired cross-irradiation, but can provide quantitative images of the radioactive drug in the patient body over time (pharmacokinetics) and for dosimetry.

Dosimetry allows determining the irradiation delivered and may help to develop personalized treatment plans. Clinical dosimetry is most often based on the Medical Internal Radiation Dose (MIRD) formalism

[1]. The activity is obtained from quantitative scintigraphy images. The absorbed dose can be calculated using two broad approaches, depending on the objective (e.g. diagnostic or therapy): model-based or patient-specific absorbed dose calculations [2]. Model-based approaches use reference dosimetric models [3,4], whereas patient-specific approaches rely on 3D geometry definition, most often from SPECT/CT or PET/CT depending on the radionuclide.

The clinical dosimetry workflow (CDW) is the chain of steps that lead from gamma camera calibration to absorbed dose calculation and reporting [5]. Recent articles have considered the sources of variation in clinical dosimetry [6–8] observed by using different software or users. Assessing the accuracy/precision of clinical dosimetry must consider the

<footnote><sup>a</sup> Institut de Recherche en Cancérologie de Montpellier (IRCM), Équipe Labellisée Ligue Contre le Cancer, INSERM U1194, Université de Montpellier, Institut régional du Cancer de Montpellier (ICM), Montpellier, France</footnote>

<footnote><sup>b</sup> Nuclear Medicine Department, Institut régional du Cancer de Montpellier (ICM), Montpellier, France</footnote>

<footnote><sup>c</sup> DOSIsoft, Cachan, France</footnote>

<footnote>* Corresponding author.</footnote>
whole (end-to-end) procedure [9], and how they are implemented in different software. However, since every CDW step influences the final results it is mandatory to study each step independently. Quantification (with partial volume effect correction), registration, segmentation, absorbed dose (rate) calculation, time-dependent variable integration are examples of the steps that must be studied.

The absorbed dose (rate) calculation step is an essential component in the available clinical dosimetry software tools. Various algorithms have been proposed to calculate the absorbed dose in MRT [10]. These include local energy deposition (LED), convolution (CONV) and Monte Carlo simulations (MC). LED and CONV are the most commonly used algorithms in clinical routine. LED assumes that particle kinetic energy is deposited locally, even at the voxel level [11]. Essentially, LED considers only beta and electron emissions (and possibly alpha particles or Auger electrons). CONV uses kernels, voxel S-values (VSV) or absorbed dose voxel kernels, pre-calculated in a homogeneous medium with analytical methods [12], from point to voxel source transformation [13], or directly from voxel sources at a fine scale generated via MC simulations [14]. MC simulations are considered the gold standard for absorbed dose rate (ADR) calculations due to their high accuracy and are a valuable benchmark for validating other algorithms. MC simulations are adapted to model radiation transport in heterogeneous media. However, their implementation in clinical software tools is often limited by the important simulation time, although ongoing advancements in GPU compatibility may help overcome these limitations.

Specific challenges are related to the absorbed dose calculation. Specifically, the accuracy of the different algorithms must be compared, and put into perspective with the time required for generating results. LED is a very attractive (and fast) algorithm, but its validity must be assessed by studying the relationship between radiation range in the medium and spatial sampling of the image. This means that depending on the radionuclide radiation range and desired scale of the calculation, LED may be adopted or not. CONV has been essentially designed for homogeneous media, although approaches have been proposed to compensate for heterogeneity [15]. Indeed, density heterogeneity, especially at a small scale (possibly below the spatial resolution of the CT) must be carefully addressed and MC approaches are considered the most appropriate to take into account such heterogeneities [16].

Moreover, the variety of VSV kernels proposed in the literature led researchers to compare the available kernels [17]. At the European Federation of Organisations for Medical Physics (EFOMP), the Focus Group 4 of the Special Interest Group for Radionuclide Internal Dosimetry (SIG_FRID) is currently working on generating and benchmarking VSV kernels.<fnref n="1" /> Lastly, as the different clinical dosimetry software tools may implement different algorithms, their comparison can be challenging. Means to compare the different algorithms and understand the sources of variation must be developed.

The aim of this work was to assess and compare absorbed dose calculation algorithms used in MRT. It was initially motivated by the need to evaluate the goodness of the algorithms implemented in OpenDose3D (OD3D) [18]. During this process, several aspects of the absorbed dose calculation procedures were studied, thereby allowing us to propose a global methodology adapted to dosimetry algorithm evaluation.

# 2. Materials and methods

## 2.1. OpenDose3D

Absorbed dose calculation algorithms were compared using OpenDose3D, a freely available open-source software designed for research and benchmarking purposes. OpenDose3D is integrated into the 3D Slicer software [19] in the Radiotherapy extension category.

OpenDose3D includes 3D Slicer advanced tools for 3D image import/export, display, registration and segmentation. Specific MRT dosimetry modules were developed within the OpenDose collaboration [20] and include time-dependent variable fitting and integration, and algorithms for radiation transport and energy deposition algorithms.

An interesting feature of OpenDose3D is the possibility to select between two main CDW: Activity and Absorbed Dose Rate. The CDW implemented in OpenDose3D are described in the User's manual.<fnref n="2" /> In the Activity workflow, time-activity curves are integrated to give the time integrated activity before performing the absorbed dose calculation. In the Absorbed Dose Rate workflow, ADR calculations are done first, at each time point, and then integrated to give the absorbed dose. This second CDW is more suitable to evaluate radiation transport and energy deposition algorithms because the comparison between algorithms can be based on the ADR computed at each time point.

## 2.2. Radionuclides

Three radionuclides widely used in MRT ($^{90}$Y, $^{177}$Lu, and $^{131}$I) were selected to compare different radiation types and energies. $^{90}$Y is a beta particle emitter, with a maximum energy ($E_{\beta max}$) of 2.28 MeV [21], which corresponds to a maximum penetration range of 11 mm in water. $^{177}$Lu emits both gamma rays and beta particles with an $E_{\beta max}$ of 497.8 keV [21], resulting in a maximum penetration range of ~2 mm in water. Approximately 82 % of the emitted energy comes from electron emissions, while the remaining 18 % is due to gamma rays. $^{131}$I emits gamma rays and beta particles, with an $E_{\beta max}$ of 806.9 keV [21] and a maximum penetration range of ~2 mm in water. In this case, gamma rays contribute to 66 % of the total energy emitted, and beta particles to the remaining 34 %.

The radiation range must be considered relative to the usual spatial sampling observed in clinical dosimetry (e.g. voxels of ~4.4 mm for a conventional 128 × 128 SPECT matrix). For the chosen radionuclides ($^{90}$Y, $^{177}$Lu, $^{131}$I), the energy deposited within a water sphere from a point source placed at the center of the sphere is represented in Fig. 1. This gives a first idea of the energy deposition as a function of the distance from the emission point.

## 2.3. Patient data

Images of patients who underwent selective internal radiotherapy with $^{90}$Y-loaded microspheres were obtained from the University of Michigan Deep Blue data repository [23]. This dataset includes a set of images from six patients: pre-therapy [$^{99m}$Tc]Tc-macroaggregated albumin SPECT/CT, post-therapy $^{90}$Y Bremsstrahlung SPECT/CT, and $^{90}$Y PET/CT. The administered activity varied among patients based on the pre-therapy assessment. Post-treatment $^{90}$Y PET/CT images were selected for this analysis. The initial non-cubic PET images (4.07 × 4.07 × 2 mm³) were resampled to generate 4.07 mm cubic voxels.

Images of patients treated with $^{[177}$Lu]Lu-DOTATATE were obtained from the Montpellier Cancer Institute (ICM), France, where dosimetry assessment following treatment is routinely performed. $^{177}$Lu SPECT images were from a subset of five patients who completed the initial two therapy cycles with a fixed administered activity of 7.4 GBq per cycle. The voxel size for SPECT images was 4.42 mm³ for a 128 × 128 matrix size. Acquisitions were performed at 4, 24, 72 and 192 h [24].

Data from two patients from ICM who received Na[$^{131}$I]I (administered activities of 3.7 and 5.6 GBq, respectively) for remnant thyroid cancer, without and with bone metastases respectively, were also included in the study. The voxel size for SPECT images was 4.42 mm³ for a 128 × 128 matrix size. For the patient with bone metastases, two fields of view were required to cover all lesions, thereby leading to a 128 ×

<footnote n="1">¹ https://www.efomp.org/index.php?r=pages&id=sig-frid.</footnote>

<footnote n="2">² https://gitlab.com/opendose/opendose3d/-/blob/develop/User_Manual.md.</footnote>
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Energy deposited per decay (keV.Bq⁻¹.s⁻¹) within a water sphere in function of its radius (mm), for a point source (⁹⁰Y, ¹⁷⁷Lu, ¹³¹I) placed at the center of the sphere. Data from the OpenDoseDVKData repository [22].</figcaption></figure>

128 × 256 image dataset.

Imaging acquisitions at ICM were performed with a SPECT/CT Discovery NM/CT 670 (General Electric [GE] Healthcare), including a BrightSpeed 16 CT scanner and a 3/8-inch NaI(Tl) crystal. [¹⁷⁷Lu]Lu-DOTATATE and Na[¹³¹I]I nuclear medicine images were acquired using a medium and a high-energy general purpose parallel-hole collimator, respectively. A 20 % energy window centered on the photopeak (208 and 364 keV, respectively) with scatter correction windows were applied. Acquisitions were performed using a body contour option, rotation of 180° per detector, a total of 60 projections and 45 s or 40 s each, respectively. For attenuation correction, low dose CT images were acquired with standard reconstruction. The application “Volumetrix” of Xeleris software (GE Healthcare), was used for SPECT/CT image reconstructions. The Ordered Subset Expectation Maximization iterative reconstruction algorithm was used with attenuation, scatter, recovery resolution corrections.

In summary, 52 patient datasets were processed: 6 for ⁹⁰Y-microspheres (6 patients, one time point), 40 for [¹⁷⁷Lu]Lu-DOTATATE, (5 patients, 2 cycles, 4 time points), 6 for Na[¹³¹I]I (2 patients, 3 time points).

## 2.4. Workflow

After importing the patient images, OpenDose3D resampled the CT images to match the voxel size of the SPECT or PET images. For [¹⁷⁷Lu]Lu-DOTATATE and Na[¹³¹I]I, SPECT images were then converted from counts to activity maps using a camera sensitivity factor of 6.62 and 27.32 cps/MBq, respectively, whereas ⁹⁰Y-microspheres PET images were already quantitative in Bq/mL. CT images were converted to density maps (kg/m³) using a CT-to-density calibration function based on Schneider [25].

## 2.5. Absorbed dose rate calculation algorithms

In this work, ADR were calculated using the three algorithms available in OpenDose3D: LED, CONV and MC.

### Local Energy Deposition

LED is the most straightforward (and fast) algorithm used in MRT dosimetry. To transform activity into voxel-based ADR maps (in Gy.s⁻¹), the algorithm multiplies the activity (Bq) by the emitted energy per decay (keV.Bq⁻¹.s⁻¹) and divides it by the voxel mass in kg (assuming a

water medium or the voxel mass derived from the CT for density correction). In OpenDose3D, emission values are taken from the ICRP Publication 107 [21], and for LED, only the average kinetic energy of the emitted beta particles, Auger and internal conversion electrons is considered. The emission values used for ⁹⁰Y, ¹⁷⁷Lu, and ¹³¹I were 931.81, 148.23, and 191.97 keV.Bq⁻¹.s⁻¹ respectively.

### Convolution

The CONV algorithm produces ADR images by convolving an activity image with a VSVs kernel. The Fast Fourier Transform (FFT) ensures fast computing times, but requires that both the activity image and the VSV kernel have identical voxel sizes. By default, OpenDose3D uses the OpenDoseDVKData repository [22] that proposes a wide range of 99 × 99 × 99 VSVs, with cubic voxel sizes from 0.1 to 6.0 mm.

In this work, VSVs were primarily sourced from the OpenDoseDVKData repository. OpenDose3D also allows incorporating VSV kernels from external sources. For example, for ¹⁷⁷Lu, VSV from Lanconelli et al. [26] with a sampling of 11 × 11 × 11 were also used to highlight the impact of VSV selection.

Also, for ¹³¹I calculations, VSVs directly generated by MC simulations with a sampling of 301 × 301 × 301 were used to illustrate the impact of kernel size for high energy gamma emissions in a context of image datasets composed of multiple field of views.

Water was chosen as the default medium for the calculations. When density correction was applied, it was done by multiplying the target voxel by the ratio of the kernel density (water) to the CT-derived density.

### Monte Carlo simulations

OpenDose3D can generate input macro files for GATE [27] to implement complete MC radiation transport and energy deposition simulations. We previously demonstrated how GATE can be used for nuclear medicine dosimetry [28]. ADR maps are created from SPECT-derived activity and CT-derived density maps. The resulting GATE ADR maps are then imported and processed in OpenDose3D.

GATE also generates statistical uncertainty images, which can be used to calculate the global uncertainty in each volume of interest (VOI). Here, the simulation completion criterion was set to ensure that relative statistical uncertainties were <1 %, ensuring a precision on the mean ADR values within each VOI up to two decimal places. Additional details about the GATE simulations are available in the Supplemental data.

To compare the results obtained with the other two tools in a
homogeneous medium (LED without density correction and CONV in water), specific MC simulations were performed considering a homogeneous water medium. Then additional simulations were generated using the real, CT-derived density maps to assess the impact of density heterogeneity on the ADR results.

A summary of the characteristics of the algorithms implemented in OpenDose3D is presented in Table 1 to highlight the aspects of radiation transport and energy deposition taken into account.

A voxel-based ADR map was computed for each algorithm (LED, CONV and MC), considering homogeneous (water) and heterogeneous (CT-derived) media, thus providing 312 ADR maps.

## 2.6. Volumes of interest

For each patient, VOIs were defined mostly in organs at risk for each treatment. Although tumors were in general not considered, the delineated VOI ensured a range of volumes and density adapted to the purpose.

For $^{90}$Y-microspheres, VOIs were delineated in the liver, right kidney, and lungs. For $^{[177}$Lu]Lu-DOTATATE, the chosen VOIs were in liver, spleen, right and left kidney and bone marrow (based on the L2, L3, L4 vertebrae), following the dosimetry protocol routinely performed at ICM [24]. For Na$^{[131]}$I, the segmented VOIs included salivary glands, lungs, bone marrow, and metastases.

The segmentation of organs at risk was performed at each time point using TotalSegmentator [29] version 2. Segmentation was initially performed on the original CT images and then resampled to align with the SPECT images, ensuring consistency with the voxel size of the ADR maps. For $^{131}$I images, metastases were segmented based on activity thresholding (40 % of maximum) on the SPECT images. The ADR computed at each time point at the voxel level with the different algorithms or settings (with or without density correction) were averaged for each VOI, and compared.

## 2.7. Relative differences

The ADR were averaged over VOIs. The relative differences between ADR values were calculated by considering the MC values as reference and using Eq. (1):

$$ \text{Relative Difference (\%)} = \left( \frac{ADR_{\text{calculated}} - ADR_{\text{MC}}}{ADR_{\text{MC}}} \right) \times 100 \quad (1) $$

# 3. Results

## 3.1. Homogeneous medium

ADR values were calculated in a homogeneous water medium to eliminate the effect of density heterogeneity. Fig. 2 presents the relative differences between LED and CONV vs. MC, taken as reference, for the three radionuclides considered in this study. LED showed deviations of up to -10 % in the liver, kidneys, spleen, and tumors, with higher deviations observed in the bone marrow and lungs. For $^{90}$Y in the liver, an excellent agreement was obtained, with deviations of less than 1 %.

<table><caption>**Table 1**

Characteristics of the algorithms for ADR calculation.</caption><thead><tr><th></th><th>Particle emissions (beta, electrons)</th><th>Gamma emissions (if any)</th><th>Propagation beyond the source voxel</th><th>Voxel-level density correction</th></tr></thead><tbody><tr><td>LED</td><td>x</td><td>-</td><td>-</td><td>x</td></tr><tr><td>CONV</td><td>x</td><td>x</td><td>x (in homogeneous medium)</td><td>x</td></tr><tr><td>MC</td><td>x</td><td>x</td><td>x</td><td>Not relevant</td></tr></tbody></table>

CONV, using the OpenDoseDVKData kernel (99 × 99 × 99), showed excellent agreement for both $^{90}$Y and $^{177}$Lu. However, deviations exceeding 1 % were observed for $^{131}$I.

## 3.2. Impact of kernel amplitude

Fig. 3 presents the results obtained using a large 301 × 301 × 301 kernel in the context of $^{131}$I therapy for thyroid cancer. The CONV calculations were performed on two patients, including one patient with bone metastases, with image acquisition using two fields of view, matrix size of 128 × 128 × 256. The CONV method demonstrated excellent agreement, with deviations of less than 1 %.

## 3.3. Impact of VSV selection

Fig. 4 presents the results obtained with convolution in a homogeneous medium using OpenDose3D and Lanconelli VSV compared with LED. Results are expressed as relative deviation to Monte Carlo, taken as the reference. The relative differences were calculated for pooled data from five OARs (both kidneys, spleen, liver, and bone marrow) at 4 time points for a patient treated with $^{[177}$Lu]Lu-DOTATATE. Details regarding the raw data used for the comparison are available in the Supplemental data.

Results obtained with OpenDose3D VSV are perfectly aligned with Monte Carlo simulations. However, results obtained with Lanconelli's VSV are in general inferior to those obtained assuming local energy deposition.

## 3.4. Heterogeneous medium

ADR calculations were performed in a heterogeneous medium to compare the density corrections implemented in the different algorithms. Fig. 5 presents the relative differences of the LED and CONV (with density correction) results vs. MC taken as reference, for the three radionuclides considered in this study. LED displayed deviations of up to -10 % in soft tissue, with a more significant underestimation of median ADR values reaching up to 50 % in the lungs and bone marrow. CONV demonstrated excellent agreement in soft tissue. However, notable deviations in the median ADR values were observed in bone marrow (up to -8%) and lungs (up to 26 %) for $^{177}$Lu and $^{131}$I therapies, respectively.

Fig. 6 presents the raw data used for the comparison of the algorithms in heterogeneous media. The LED and CONV results were obtained with density correction. Source organs/tissues are usually receiving a higher absorbed dose rate than targets, and ADRs are varying in time except for $^{90}$Y-microspheres (only one time point).

# 4. Discussion

In this work, the ADR values obtained with the LED and CONV algorithms were compared to those obtained with MC taken as the gold standard. This verification (the term validation must be used only when comparing results with measurements) was done in two steps by calculating the ADR first in a homogeneous medium and then in a heterogeneous medium. The results confirmed the reliability of the Open-Dose3D absorbed dose (rate) calculation algorithms.

## 4.1. Homogeneous medium

The ADR with $^{90}$Y-microspheres in the liver calculated with the LED algorithm were in good agreement with those obtained with MC simulations (<1% relative difference). However, LED underestimated the ADR for right kidney and lungs. For “target-only” regions, i.e. VOI with no activity, this can be explained by the fact that the ADR value is strictly zero with LED. Conversely, there is always a (very low) cross-ADR in the VOI when computed by MC, thereby explaining the important relative deviation.
<figure><img src="image_5.png" /><figcaption>**Fig. 2.** ADR relative differences (%) between the ADR calculated with the LED (top row, blue) and CONV (bottom row, red) algorithms, and the ADR calculated by Monte Carlo (MC) simulations in the homogeneous medium. The center line in each box is the sample median, and each box indicates the interquartile range. The whiskers indicate the range of non-outliers. Each individual data point is shown in the box plot.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Relative differences (%) between the ADR calculated with the CONV algorithm and MC for ¹³¹I in the homogeneous water medium. CONV data were acquired using a 301 × 301 × 301 voxel kernel (voxel size: 4.42 mm), covering a maximum orthogonal distance of 66.3 cm from the central-source voxel.</figcaption></figure>

Similarly, LED underestimated the ADR values with [¹⁷⁷Lu]Lu-DOTATATE in liver, kidneys and spleen (relative differences from -2% to -10%) and even more in bone marrow where the relative difference reached -30%.

With Na[¹³¹I]I, the underestimation by LED was much higher, with relative differences ranging from -10% to -80%, particularly in bone marrow and lungs. Like for ¹⁷⁷Lu, this cannot be attributed to differences in density because all calculations were made in water. Therefore, they could only be explained by the non-consideration of cross-irradiation by the LED algorithm, which is more evident for target-only VOIs. This is demonstrated by the results obtained with the CONV algorithm. The relative differences were consistently close to 0% for all three radio-nuclides and regions. The only exception was the ADR with Na[¹³¹I]I in the lung that was underestimated by CONV (compared with MC), probably due to the limited kernel size, which triggered the calculations reported in Fig. 3.

## 4.2. Impact of kernel amplitude

As mentioned previously, the default OpenDose3D kernels have a fixed size of 99 × 99 × 99, possibly not optimal for high energy penetrating photons, especially for large datasets that result from 2 fields of view (128 × 128 × 256 images).

Using the kernel of 301 × 301 × 301, the relative differences between CONV and MC for Na[¹³¹I]I in the homogeneous medium were reduced from 3.16% (Fig. 2) to <1% (Fig. 3).

## 4.3. Impact of VSV selection

As discussed in the article from Danieli et al. [17] the impact of VSV quality on dosimetric results may be important. To illustrate this point,
<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Relative differences (%) of the ADR calculated with CONV with 2 VSV (CONV-Lanconelli and CONV-OD3D) and with LED compared with MC. The relative differences calculated for pooled data from five OARs (both kidneys, spleen, liver, and bone marrow) at 4 time points for a patient treated with [¹⁷⁷Lu]Lu-DOTATATE.</figcaption></figure>

data from one patient treated with [¹⁷⁷Lu]Lu-DOTATATE were used to compare VSV kernels [26] pulling various VOIs (liver, kidneys, spleen and bone marrow) and four time-points, see Fig. 4. All calculations (LED, CONV and MC) were performed in a homogeneous (soft tissue) medium. The results obtained with OpenDose3D kernels were in line with those of MC. However, when Lanconelli kernels were used, results for CONV were inferior than with LED, which is not logical as CONV considers not only electrons (self-irradiation), but also gamma ray irradiation (cross-irradiation). A tentative explanation can be the absence of mono-energetic electron emissions in the central voxel [12,17] and the limited kernel size (11 × 11 × 11 for 4.42 mm voxels). Indeed the physical settings for the kernels presented in [26] were a compromise with calculation time and computing performances of more than a decade ago.

### 4.4. Heterogeneous media

First, for soft tissues, results in the heterogeneous medium were similar to those obtained in the homogeneous (water) medium. This is normal because the density differences between water and soft tissue are low. It is also in line with the results obtained by Götz et al. for ¹⁷⁷Lu, showing a deviation around 2 % for both kidneys and spleen for a kernel size of 81 × 81 × 81 (see Table 1 of [15], in the “Average deviation per organ” section). This is in agreement with our results (1 %) for a slightly larger kernel (99 × 99 × 99).

For ⁹⁰Y, the results obtained with LED and CONV were in agreement with those obtained by MC for liver (<1%). CONV gave excellent results also for kidneys, whereas the difference observed with LED in the right kidney was already observed in the homogeneous medium (Fig. 2).

Likewise, for ¹⁷⁷Lu and ¹³¹I, the LED and CONV results in soft tissues were similar to those obtained in water, with an excellent agreement between CONV and MC.

<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Relative differences (%) of the ADR calculated with the LED (top row, blue) and CONV (bottom row, red) algorithms compared with Monte Carlo (MC) simulations in the heterogeneous medium. CT-derived density correction was applied for both the LED and CONV algorithms.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 6. ADR values calculated from patient image dataset using the LED, CONV and MC algorithms in the heterogeneous medium.</figcaption></figure>

The LED and CONV ADR values for bone and lung tended to deviate from those obtained using MC, even when the density correction was applied. In the lung regions for ⁹⁰Y, the maximum deviation observed was −15 %, even after applying voxel-level density corrections. A similar trend was noted by d'Andreas [30], who reported relative differences of approximately −28 %. For ⁹⁰Y, the results for lungs tended to be better with CONV than with LED, but some differences compared with MC remained. For ¹⁷⁷Lu, the same behavior was observed for bone marrow, where the differences observed with LED were not fully compensated by implementing density-corrected CONV. For ¹³¹I, the ADR for lungs was overestimated with CONV and underestimated with LED.

These results must be considered in function of the irradiation delivered in these organs/tissues. Comparison of the ADR results (mGy/h) obtained with the three algorithms in the three clinical situations considered (Fig. 6) indicated that the high relative differences observed among algorithms in the heterogeneous medium concerned mostly regions where the ADR was low. However, for bone metastases that present a non-negligible irradiation (¹³¹I), the density correction seemed to perform rather well.

The proposed methodology allowed better understanding of the differences induced by the use of different ADR algorithms in MRT dosimetry. Specifically, in homogeneous media, LED tends to underestimate the ADR because only non-penetrating radiations are taken into account (no cross irradiation). This behavior is more pronounced for radionuclides that emit significant gamma radiation (i.e. more apparent for ¹³¹I than for ¹⁷⁷Lu). Moreover, when cross-irradiation is the major contributor to irradiation (i.e. target-only organs/tissues), the relative deviation can become important (although the absolute values are small).

In a homogeneous medium, CONV and MC should yield identical results. Here, CONV and MC ADR values differed by < 1 %, which

represents a good verification of the VSV and also of the whole CONV algorithm. Strikingly, the relative differences observed between LED and MC in the homogeneous medium (lungs/⁹⁰Y, bone marrow/¹⁷⁷Lu, various metastases/¹³¹I) were almost fully compensated by CONV. However, differences between CONV and MC were observed when cross irradiation became important and if the amplitude of the convolution kernel was not sufficient (high energy gamma emissions). This may concern acquisitions performed with more than one field of view, or diagnostic radionuclides where (low) irradiation is mostly due to gamma contribution.

The preliminary study only in the homogeneous medium was crucial for understanding the results obtained in the heterogeneous medium. Although it was shown that density correction improves the accuracy for soft tissue and abdominal regions [31], this correction was insufficient to improve accuracy in bone marrow and lungs. This may question the results obtained using the LED and CONV algorithms. In such cases, the recommended algorithm for bone marrow and lung dosimetry is MC. Obviously there may be a penalty in terms of computation times, but fast MC solutions are now appearing on the market [32]. For example, fast MC (GPU-based) can provide results within minutes [33,34], representing a marked improvement compared to our simulations, which required approximately 17 days to complete the calculations for one patient. In addition, approaches that take into account the density effect beyond voxel size are available [35,36], along with the application of multiple VSV approach [37].

The required accuracy in absorbed dose (rate) calculation will ultimately depend on the clinical application: for reference dosimetry, the calculation of the S values requires high accuracy, but time is usually not a constraint, whereas for clinical dosimetry in MRT, the decision-making process often requires to perform calculations within a reasonable time frame. Therefore, the calculation framework will ultimately depend on the clinical requirements.
Our results were based on the mean ADR, obtained at the voxel level and averaged on VOIs. This was sufficient to draw conclusions in our study; however, there are situations where voxel-based results should be compared directly using specific dosimetric variables (absorbed dose-volume histograms, and also statistical parameters obtained from ADR variations in a given volume). The comparison of 3D ADR maps could also be performed using indices derived from external beam radiotherapy (gamma index). Fig. 7 presents some examples of what can be generated to go further in algorithm comparison. For a patient treated with $^{131}$I, the CONV and MC results in the heterogeneous medium were compared at 24 h (first time point). ADR volume histograms were generated for the lungs. The local gamma index was calculated using the criteria of a 3 % absorbed dose difference and 4 mm distance-to-agreement applied to all whole-body voxels in the field of view. About 30 % of the voxels passed the criteria.

This study considered radionuclides and spatial sampling adapted to largely used applications in clinical MRT but of course it cannot represent all MRT situations. Equivalent approaches performed at different scales (small animals or even cell dosimetry) may yield different conclusions.

A range of factors may influence ADR calculations, such as the CT-to-density calibration function, radionuclide emission spectra, kernel size and density. The implementation of density correction or of FFT-based convolution also plays a role. Therefore, a detailed report of these parameters is necessary to improve dosimetry traceability and to understand the observed deviations.

## 5. Conclusions

The initial objective of this work was to benchmark the algorithms implemented in OpenDose3D. In addition, this study allowed us to discuss the methodology required to compare algorithms commonly used to compute absorbed doses at the voxel level in MRT.

The initial comparison in a homogeneous medium allows performing a first assessment of the algorithms. The direct comparison between LED and CONV can also be used for an initial evaluation, although MC represents the gold standard to evaluate the goodness of absorbed dose calculation algorithms.

In soft tissue, and for the radionuclides and spatial samplings currently used in clinical situations, all algorithms tend to generate

acceptable results, although LED is more indicated in situations where self-irradiation is preponderant. CONV is a fast and accurate tool to derive absorbed dose rates in homogeneous media. Density correction may be implemented differently in various software tools, and should be benchmarked carefully, acknowledging the fact that in some situations only MC will provide accurate results.

A summary of the recommended methodology to assess absorbed dose rate algorithms is presented below:

1. Calculation in a homogeneous medium is essential for verifying CONV calculations.

* CONV calculations should yield higher values than LED due to cross-irradiation impact. CONV/LED ratio should serve as a cross-irradiation indicator.

* CONV & MC must be in excellent agreement (less than 1 %).

* If those criterions are not met, and assuming that CONV is implemented correctly, VSVs should be checked so that:

* all particle emissions are included.

* kernel sizes are large enough to account for all deposited energy.

2. Heterogeneous medium calculation.

* CONV & MC must agree in soft-tissue medium using density correction.

* Density correction may not be sufficient in heterogeneous media. Bone marrow and lungs may require MC radiation transport modeling, especially since they are often OARs that deserve particular attention.

* It is advisable to conduct a comparison using 3D ADR maps to highlight detailed differences. Approaches such as the gamma index and/or DVH analysis can be used.

## Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: MK, SV are employees of DOSIsoft, Cachan, France.

JAFN is sponsored by DOSIsoft within the PhD contract CIFRE n°201273A10.

MB and LS supervise a PhD Student (JAFN) sponsored by DOSIsoft within the PhD contract CIFRE n°201273A10.

<figure><img src="image_10.png" /><figcaption>Fig. 7. Absorbed dose rate – volume histogram (right) and gamma index image (left): comparison of the CONV and MC results in heterogeneous medium for a patient treated with Na[$^{131}$I]I. The table presents the dosimetric indices extracted from the absorbed dose volume histogram.</figcaption></figure>
MB is acting as consultant advisor for Bain Capital and ITM.

ED received honoraria from Janssen and AAA/Novartis and travel expenses from AAA/Novartis and GE Healthcare.

KH received fees from AAA/Novartis and Astellas.

## Acknowledgements

This work was partly performed within the framework of the PhD CIFRE contract n°201273A10.

The ICRM team (JAFN, LS, POK, JPP, ED and MB) is an “Équipe Labellisée Ligue Contre le Cancer”.

## Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.ejmp.2025.104965.

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