Linear Boltzmann equation solver for voxel-level dosimetry in radiopharmaceutical therapy: Comparison with Monte Carlo and kernel convolution

Abstract

Background:

With recent interest in patient-specific dosimetry for radiopharmaceutical therapy (RPT) and selective internal radiation therapy (SIRT), an increasing number of voxel-based algorithms are being evaluated. Monte Carlo (MC) radiation transport, generally considered to be the most accurate among different methods for voxel-level absorbed dose estimation, can be computationally inefficient for routine clinical use.

Purpose:

This work demonstrates a recently implemented grid-based linear Boltzmann transport equation (LBTE) solver for fast and accurate voxel-based dosimetry in RPT and SIRT and benchmarks it against MC.

Methods:

A deterministic LBTE solver (Acuros MRT) was implemented within a commercial RPT dosimetry package (Velocity 4.1). The LBTE is directly discretized using an adaptive mesh refined grid and then the coupled photon-electron radiation transport is iteratively solved inside specified volumes to estimate radiation doses from both photons and charged particles in heterogeneous media. To evaluate the performance of the LBTE solver for RPT and SIRT applications, ¹⁷⁷Lu SPECT/CT, ⁹⁰Y PET/CT, and ¹³¹I SPECT/CT images of phantoms and patients were used. Multiple lesions (2–1052 mL) and normal organs were delineated for each study. Voxel dosimetry was performed with the LBTE solver, dose voxel kernel (DVK) convolution with density correction, and a validated in-house MC code using the same time-integrated activity and density maps as input to the different dose engines.The resulting dose maps,difference maps,and dose-volume-histogram (DVH) metrics were compared,to assess the voxel-level agreement.Evaluation of mean absorbed dose included comparison with structure-level estimates from OLINDA.

Results:

In the phantom inserts/compartments,the LBTE solver versus MC and DVK convolution demonstrated good agreement with mean absorbed dose and DVH metrics agreeing to within 5% except for the D90 and D70 metrics of a very low activity concentration insert of ⁹⁰Y where the agreement was within 15%. In the patient studies (five patients imaged after ¹⁷⁷Lu DOTATATE RPT, five after ⁹⁰Y SIRT, and two after ¹³¹I radioimmunotherapy), in general, there was better agreement between the LBTE solver and MC than between LBTE solver and DVK convolution for mean absorbed dose and voxel-level evaluations. Across all patients for all three radionuclides, for soft tissue structures (kidney, liver, lesions), the mean absorbed dose estimates from the LBTE solver were in good agreement with those from MC (median difference < 1%, maximum 9%) and those from DVK (median difference < 5%, maximum 9%). The LBTE and OLINDA estimates for mean absorbed dose in kidneys and liver agreed to within 10%, but differences for lesions were larger with a maximum 14% for ¹⁷⁷Lu, 23% for ⁹⁰Y, and 26% for ¹³¹I. For bone regions, the agreement in mean absorbed doses between LBTE and both MC and DVK were similar (median < 11%, max 11%) while for lung the agreement between LBTE and MC (median < 1%, max 8%) was substantially better than between LBTE and DVK (median < 16%, max 33%). Voxel level estimates for soft tissue structures also showed good agreement between the LBTE solver and both MC and DVK with a median difference < 5% (maximum < 13%) for the DVH metrics with all three radionuclides. The largest difference in DVH metrics was for the D90 and D70 metric in lung and bone where the uptake was low. Here, the difference between LBTE and MC had a median value < 14% (maximum 23%) for bone and < 4% (maximum 37%) for lung, while the corresponding differences between LBTE and DVK were < 23% (maximum 31%) and < 67% (maximum 313%), respectively. For a typical patient with a matrix size of 166 × 166 × 129 (voxel size 3 × 3 × 3 mm³), voxel dosimetry using the LBTE solver was as fast as ~2 min on a desktop computer.

Conclusion:

Having established good agreement between the LBTE solver and MC for RPT and SIRT applications, the LBTE solver is a viable option for voxel dosimetry that can be faster than MC.Further analysis is being performed to encompass the broad range of radionuclides and conditions encountered clinically.

1 |. INTRODUCTION

Radiopharmaceutical therapy (RPT) for cancer treatments has evolved in recent years with more FDA-approved therapeutic agents being available for clinical use.¹ Often, these treatments rely on administering a fixed activity within certain intervals, such as four cycles of 177Lu-DOTATATE at 7.4 GBq per cycle with an 8-week interval between cycles,² disregarding a wide range of factors including disease severity, and physiological variability in patients that leads to variability in absorbed dose delivered to tumors and healthy organs. Patient-specific dosimetry guided RPT is desirable for optimizing tumor response while keeping normal-organ absorbed dose and toxicity at an acceptable level.³

Traditionally, absorbed doses in RPT have been estimated using the MIRD formalism and the standardized reference model-based S-values,⁴ which are integrated in software such as OLINDA/EXM⁵ and more recently in MIRDCalc.⁶ Although scaling for the difference in organ mass is possible with model-based dosimetry,this approach is not individualized for each patient, as it assumes reference geometries with uniform activity and homogeneous tissue within each organ. Voxel-based dosimetry addresses some of the limitations of the reference model-based approach by fully exploiting the availability of 3D emission (SPECT, PET) and anatomical (CT) images of patients to make the dosimetry highly patient-specific.⁷ Although the poor spatial resolution of SPECT and PET imaging limits the capabilities of voxel-level estimation,^8,9 non-uniform dose deposition at a scale that is larger than the spatial resolution can still be captured.^10,11 Voxel-level dosimetry enables calculation of dose-volume histogram metrics and radiobiologic quantities such as the equivalent uniform dose, which accounts for the biologic consequences of spatial nonuniformities in absorbed dose that has the potential to impact therapy outcome.^12,13

The commonly used methods to perform RPT dosimetry at voxel scale include assumption of local-energy deposition that does not involve any radiation transport, dose point kernel (DPK) convolution where a DPK represents the radial distance-dependent absorbed dose about a point source in a homogeneous medium (typically water or soft tissue),dose voxel kernel (DVK) convolution where S-values are defined for different voxel sizes, and direct Monte Carlo (MC)-based radiation transport.⁷ MC simulations have long been used as a standard method for determining stochastic solutions to the linear Boltzmann transport equation (LBTE) governing the interaction of radiation with matter. Direct MC simulation is generally considered to be the most accurate^7,13 for RPT dosimetry, especially in heterogeneous media where the kernel-based methods have limitations when a single kernel is used even if density correction is applied.^14–17 Some general-purpose MC codes available for voxel-based dosimetry include EGSnrc,¹⁸ MNCPX,¹⁹ Geant4,²⁰ and GATE,²¹ while other research software include 3D-ID/3D-RD,²² and DPM.²³ These codes allow for voxel-level charged particle and photon transport in heterogeneous media where they are tracked through various materials and energy deposition is recorded as a 3D absorbed dose matrix. Several approximations are made to enhance the speed of the MC simulations, for example, including energy or range cut off values to limit the generation of secondary particles or the use of variance reduction techniques depending on the application. However, since a large number of histories (typically in order of a 10 million to a billion) are required to avoid high statistical noise, the computation time is extensive, i.e., ranging from 1 to 30 h for clinically relevant voxel/matrix sizes even when using simplified assumptions, whereas the DVK convolution-based method could be as quick as a few seconds.²⁴

In addition to stochastic MC approaches for solving the LBTE, deterministic methods also exist that exclude statistical noise inherent to MC.²⁵ In deterministic methods, energy, position, and angle of particles are discretized. Both stochastic and deterministic approaches will converge to the same solution in the limit of an infinite number of histories for MC and an infinite number of discretization for deterministic solvers of the LBTE, provided that the energy cross-section values are the same. In the deterministic method that has been implemented in clinical software for absorbed dose calculations, the LBTE is directly discretized using an adaptive mesh refined grid and then the coupled photon-electron radiation transport is iteratively solved inside specified volumes to estimate radiation doses.²⁶ This approach has shown considerable potential as an alternative to MC techniques in applications such as absorbed dose estimation for treatment planning and modeling particle transport for ¹⁹²Ir sealed source brachytherapy,²⁷ external beam radiotherapy with megavoltage photon beams,²⁸ and external beam radiotherapy (EBRT) in magnetic fields.²⁹ In these applications,^27,30 deterministic solvers demonstrated accuracies equivalent to MC methods, but with faster computational speed that is similar to the speed of kernel convolution methods. In addition,the deterministic method has also been investigated in the radionuclide therapy regime but was limited to phantom studies with ⁹⁰Y and ¹³¹I.³¹

Motivated by the value of voxel-based dosimetry that is both accurate and fast for clinical applications,^11,13 the goal of this work is to evaluate a new implementation of the LBTE solver for RPT,(Acuros MRT) within the Velocity 4.1 platform (Varian Medical Systems,Palo Alto, California, USA). This new voxel-level dose engine is benchmarked against the Dose Planning Method (DPM) MC code that has been extensively validated with experimental measurements, including in the setting of RPT. Additionally, the LBTE solver is compared with DVK convolution, a widely used and computationally less intensive alternative to MC for voxel dosimetry and with model-based dosimetry using the MIRD formalism and reference S-values, which is the widely used approach for organ-level dosimetry. Since the goal is to compare dose engines, the same time-integrated activity maps (TIA) and density maps are used as the input for all dosimetry methods. Both phantom and patient studies with three widely used radionuclides in RPT and selective internal radiation therapy (SIRT) are considered in this work. Comparisons are made between the LBTE solver and the other dose engines for mean absorbed doses, dose maps, and dose volume histogram (DVH) metrics.

2 |. METHODS

2.1 |. Phantom setup

The following physical phantoms were used to mimic activity distributions in SPECT imaging after 177Lu-DOTATATE peptide receptor radionuclide therapy (PRRT), PET imaging after SIRT with ⁹⁰Y glass microspheres, and SPECT imaging after ¹³¹I radioimmunotherapy (RIT). The phantom compartments that included activity were filled with water.

2.1.1 |. Lung/liver torso phantom used for ¹⁷⁷Lu SPECT/CT and ⁹⁰Y PET/CT

The torso phantom (Data Spectrum) consisted of a fillable liver with “lesion” inserts, lung compartments filled with polystyrene beads to simulate lung tissue and a PTFE rod to simulate the spine. For the ¹⁷⁷Lu study, spherical (15 mL) and ovoid (29 mL) shaped “lesion” inserts were placed in the liver. In case of ⁹⁰Y, two spherical (8 , 16 mL) and one ovoid (29 mL) shaped inserts were placed in the liver along with spherical (14 mL) and ovoid (11 mL) shaped inserts of very low activity concentration in the ‘cold’ background compartment to mimic inadvertent extrahepatic microsphere deposition. The total ¹⁷⁷Lu activity in the phantom was 122.1 MBq with an insert-to-liver background ratio of 3.8:1, which is a realistic ratio when compared to the lesion-to-liver ratio in ¹⁷⁷Lu-DOTATATE patient studies. Since the insert activity concentration (0.3 MBq/mL) was ~ 10% of typical values encountered for lesions in imaging patients following ¹⁷⁷Lu-DOTATATE therapy,³² a prolonged acquisition time (275 min) was used for the phantom compared with the typical acquisition time for patients. Thus, the phantom study represented a clinically relevant count (noise) level. The total ⁹⁰Y activity in the phantom was 1850 MBq with insert-to-liver background ratios of 5:1 to 6:1. Since the insert activity concentration (6 to 7 MBq/mL) was similar to that reported for lesions when imaging following ⁹⁰Y SIRT,³³ the acquisition time used for the phantom (30 min) was similar to that used for patients to achieve a clinically relevant count-level.

2.1.2 |. Elliptical phantom used for ¹³¹I SPECT/CT

This study consisted of six spherical ‘lesion’ inserts in an elliptical phantom with the sphere volumes ranging from 2 to 82 mL.The total ¹³¹I activity was 88 MBq with a sphere-to-background ratio ranging from 5:1 to 17:1 and activity concentration in inserts ranging from 0.04 to 0.13 MBq/mL. Since the activity concentration was representative of lesions imaged after ¹³¹I RIT,³⁴ the acquisition time used for the phantom was similar to that used for patients to achieve a clinically relevant count level.

2.2 |. Patient studies

This work used post-therapy (day 0 or 1) ¹⁷⁷Lu-SPECT/CT images from five patients with neuroen-docrine tumors (NETs) treated with ¹⁷⁷Lu-DOTATATE PRRT (7 to 7.3 GBq), post-therapy (day 0) ⁹⁰Y PET/CT images from five patients with intrahepatic malignancies treated with ⁹⁰Y SIRT using glass microspheres (3.8–12.3 GBq) and post-therapy (day 2) ¹³¹I SPECT/CT images from two non-Hodgkin’s lymphoma patients treated with ¹³¹I-tositumomab RIT (5.3 GBq). These patient studies, described in detail previously,^33,35,36 were selected to encompass a wide range of tissue heterogeneities including soft tissue, bones, and lungs as well as activity non-uniformities so as to test the performance of LBTE solver under varying and clinically relevant conditions. All the patients underwent the respective therapies at the University of Michigan Medical Center and the research imaging were approved by the Institutional Review Board. The patients signed a written informed consent to participate in the imaging.

2.3 |. Imaging and activity quantification

¹⁷⁷Lu scans were performed on a Symbia Intevo Bold SPECT/CT camera equipped with a medium-energy collimator and a 15.8 mm crystal. The SPECT field-of-view (FOV) for patient studies included the kidneys, liver, and any nearby tumor. The images were reconstructed using the Siemens xSPECT Quant software that provides absolute activity quantification (image in Bq/mL units). ⁹⁰Y scans were acquired on a Siemens Biograph mCT PET/CT with time-of-flight and a field of view (FOV) encompassing the entire liver and a portion of the lung. The images were reconstructed with Siemens 3D-ordered-subset expectation maximization (OSEM) software and were available in activity units (in Bq/mL). ¹³¹I scans were performed on a Siemens Symbia SPECT/CT system using a high-energy parallel-hole collimator and a 15.8 mm crystal. The FOV was chosen to include the index lesions. These images were reconstructed using an in-house developed 3D-OSEM software and quantified using a calibration factor from phantom measurements as reported.³⁴ Partial volume corrections were not performed for either SPECT or PET as the goal of the current study was to compare different dose engines. Acquisition and reconstruction parameters for all three radionuclides are given in Table 1. The CT scans in both SPECT/CT and PET/CT were performed in low dose mode and the reconstructed matrix size was 512 × 512 (0.97 × 0.97 mm²).

TABLE 1.

Acquisition settings and reconstruction parameters for phantom and patient studies

Imaging	Acq. time (mins)	Energy windows (keV)	Recon	Corrections	SPECT or PET matrix size	SPECT or PET voxel size (mm3)	TIA/Dose map matrix size	TIAa/Dose map voxel size (mm3)
177Lu SPECT/CT	25 (Phantom: 275)	208 ± 10% 178 ± 5% 241 ± 5%	OSCGMM 48i, 1ss	CT-based AC, TEW SC, RR	256 × 256 × 199	1.95 × 1.95 × 1.95	166 × 166 × 129	3 × 3 × 3
90Y PET/CT	30	542 ± 20%	PSF+TOF 1i, 21ss	CT based AC, Model based SC, RR	200 × 200 × 135	4.07 × 4.07 × 3	200 × 200 × 135	4 × 4 × 3
131I SPECT/CT	20 (Phantom: 40)	364 ± 10% 318 ± 3% 413 ± 3%	3DOSEM 20i, 6ss	CT based AC, TEW SC, RR	128 × 128 × 80 (registered to CT of 512 × 512 × 197)	4.80 × 4.80 × 4.80 (CT: 0.97 × 0.97 × 2)	166 × 166 × 130	3 × 3 × 3

AC and SC, attenuation and scatter correction, respectively; PSF, point source function; RR, resolution recovery; TEW, triple energy window; TOF, time of flight.

^aActivity maps are resampled to a minimum voxel size of 3 × 3 × 3 mm³ within Velocity.

2.4 |. Lesion/organ segmentation

For the phantoms, inserts were segmented manually on CT. For patients, the relevant organs of interest within the FOV and up to five tumors were segmented as described previously.³⁶ The organs of interest were kidneys, liver, lungs, and bones for ¹⁷⁷Lu PRRT; total liver and lungs for ⁹⁰Y SIRT,and lumbar vertebrae representing regions of marrow uptake for ¹³¹I RIT.Normal organs were segmented on the CT using deep learning based auto-segmentation tools and verified by a radiologist, while the lumbar regions were segmented manually on the CT. The lesions were contoured manually by a radiologist on baseline diagnostic anatomical scans (CT or MRI) and then transferred to the SPECT/CT or PET/CT following co-registration. In case of any evident misregistration, manual fine tuning of lesion locations was performed by the radiologist.

2.5 |. Generation of time-integrated activity (TIA) and density maps

Each patient dataset (SPECT/CT or PET/CT) were imported into Velocity along with the corresponding contours (RTStruct). In this work, the Dosimetry navigator from the Theranostics category,one of the several available navigators (or workflow) was used. This workflow enables the calculation of absorbed doses from input CT and emission tomographic images using Acuros MRT.All activity maps are resampled in Velocity to a minimum of 3 × 3 × 3 mm³ voxel sizes.Since the goal was to compare dose engines,regardless of time-activity fitting regime,TIA maps were generated from single SPECT or PET images. A mono-exponential decay with an effective half-life of 60 h for ¹⁷⁷Lu DOTATATE and 50 h for ¹³¹I tositumomab was assumed based on typical values in patient studies, while for ⁹⁰Y SIRT, the physical half-life of 64 h was used as microspheres do not redistribute. The TIA maps, resampled CTs (to match the voxel size of the TIA map) and the respective contours were exported from Velocity in order to have the same input for the computation of absorbed doses in different dose transport engines (Figure 1). The matrix and voxel size of the TIA maps are indicated in Table 1.

FIGURE 1.

Comparison of voxel dose estimates using the same density and TIA maps as input to the different dose engines.

The density maps for radiation transport and for density scaling post DVK convolution were generated via a CT-to-density calibration curve, which was determined specifically for the SPECT or PET camera used. The Gammex 467 Tissue Characterization Phantom consisting of 16 rods of known material was imaged and the Hounsfield unit versus density data were plotted and fitted with a bi-linear function (i.e.,one from −1000 to 0 HU and another from 0 to 1000 HU),as is commonly used.³⁷ While this bilinear curve was used to generate the density map for DPM and DVK, Acuros MRT LBTE solver required the measured data points to be entered to the workflow where a piece-wise linear fit is performed and used as the calibration curve. To achieve a high level of concordance in the CT-to-density calibration curves used by DPM, DVK, and LBTE, the input data to the Acuros calibration was limited to 3 points to enforce the same bilinear fit.

2.6 |. Dose calculation methods

2.6.1 |. Acuros MRT

The deterministic solver, previously used in EBRT (Acuros XB),²⁸ was implemented for internal emitter therapies, including the solution of the LBTE for photons and the linear Boltzmann-Fokker-Planck transport equation (LBFPTE) for electrons. The new implementation starts with an internal, or fixed, isotropic volumetric photon or electron source, whereas the previous version starts with external focal photons and electrons (contaminate electrons) a meter or so away from the patient. The photon and electron energies for internal therapy applications are considerably lower than for EBRT applications.

All known beta decay modes, gamma, and x-ray lines are included for dose computation. Spectra for electron emission from beta decay is computed using the Fermi function and associated kinematics formula while the emission data was obtained from the LUND/LBNL Nuclear Data³⁸ (Table 2). Similar to the implementation for EBRT, the solver includes 37 electron energy groups and 15 photon energy groups from 1 keV to 3 MeV. Both the angular quadrature order (Sn) and scatter order expansion (Pn) are variable and adaptive. Given the isotropic nature of radionuclide emissions and distributed sources, as well as the lower (compared with EBRT) energies of the photons and electrons, it is expected that low order expansions should be sufficient in most instances. The energy cutoffs for terminating transport in space are 1 keV for photons and 200 keV for electrons while the overall energy threshold is set to 1 keV for both electrons and photons. Contribution of absorbed dose from Bremsstrahlung photons is negligible in RPT and SIRT, thus in Acuros MRT, electrons do not generate photons, and hence, any energy from bremsstrahlung is discarded.

TABLE 2.

Settings used for the different dose computations.

	Material composition	Material density	Decay data	Beta spectrum	Energy cutoff	Number of simulated particles
DVK	Water	1.0 g/cm3	NNDC48	RADAR49	1 keV	106 per emission type
MC	Water	Variable based on CT	NuDat 3.046	BetaShape45	59 keV (electrons), 4 keV (photons)	109
LBTE solver	ICRP: Air, Lung, Adipose Tissue, muscle, cartilage, bone	Variable based on CT	LUND/LBNL Nuclear Data38	Fermi function	200 keV (electrons), 1 keV (photons)	NA

Like the codes for brachytherapy and EBRT,^27,28 the solver incorporates a linear discontinuous finite element method for spatial discretization; the solution is allowed to linearly vary across an element with discontinuities possible between elements. The patient volume is discretized via adaptive mesh refinement to increase computational efficiency. The adaptive spatial mesh is based on the location of the volumetric source coming from the radionuclide source, whereas for the EBRT implementation, it is based upon the beam(s) entering and traversing the patient. In general, larger cells would occur in low activity gradient regions, while smaller cells would be generated within and near areas where the activity is high and changing rapidly.The use of the linear discontinuous finite-element spatial differencing helps to keep the overall number of cells to a minimal amount in each patient geometry. The absorbed dose is reported on the 3−5 mm dose grid. Acuros MRT requires multi-group energy cross sections for each material, which were generated using an in-house code ZERKON (Varian Medical Systems, Palo Alto, California, USA) that is based on CEPXS.³⁹ These materials include ICRP⁴⁰ lung, adipose tissue, muscle, cartilage, air, and bone. The CT to density curve was performed by providing a CT calibration curve for piecewise CT-to-density conversion. The internal mapping from HU numbers to mass densities and further from mass densities to material identification is hard-coded and the upper limit of density is set to 3 g/cm³. A density threshold of 0.01 g/cm³ is used to avoid extremely high absorbed doses in very low-density voxels. The voxels in TIA map corresponding to density below the threshold is set to zero before the dose calculation step.

All dosimetry results from the LBTE solver in this work are reported as dose-to-medium. Acuros MRT solves for the electron fluence for all energies throughout the calculation volume, so dose-to-water could also be reported. However, dose-to-water is not an option in the current implementation. Additional details of the LBTE, LBFPTE, and calculating absorbed dose as a post-processing step have been published elsewhere.^41,42

2.6.2 |. DPM Monte Carlo

A fast MC code—Dose Planning Method (DPM), initially developed for dosimetry in external beam radiotherapy⁴³ and then adapted in-house for internal emitter therapy²³ was used for this work. Details of the MC transport mechanics and physics data based on PENELOPE are given in these prior publications. The considerable speed-up of DPM over standard MC is the result of its unique transport mechanics, which allow for long transport steps over heterogeneous boundaries. DPM was adapted for nuclear medicine energy regimes especially for patient-specific dosimetry by sampling the decay locations internally. Following initial benchmarking of this adaptation of DPM for RPT, it has been used for clinical research with multiple radionuclides and therapies at University of Michigan, including ¹³¹I RIT,^{35 90}Y SIRT,³³ and ¹⁷⁷Lu-DOTATATE PRRT.³⁶ Recently,the code has also been validated against physical phantom measurements with radiochromic films and a good agreement (within 4%) in absorbed doses was obtained between experiments and DPM for multiple radionuclides.⁴⁴

In the current version of DPM,the beta energy spectra are from BetaShape software v1.0 (Laboratoire National Henri Becquerel),⁴⁵ while the photon, x-ray, auger electron, and electron capture spectrums are from NuDat 3.0⁴⁶ (Table 2). The low-energy electron and photon cutoffs used in DPM MC were 59 and 4 keV, respectively, while the number of histories were set to 1E+09 to achieve a statistical uncertainty < 1% (discussed in Results section). Similar to Acuros MRT, a density threshold of 0.01 g/cm³ was set in DPM.However,unlike Acuros, in DPM, this threshold is applied as a post-processing step by setting the voxels in absorbed dose maps below this threshold to zero.

2.6.3 |. DVK

The dose volume kernels used for this analysis were based on the publicly available DPKs generated by Graves et al.⁴⁷ in unit density water. The source of the beta spectrum, decay data, and cutoff energies are given in Table 2. The beta, gamma, and mono-energetic electron radial dose distributions were summed for each radionuclide to create a single radial distribution. With radial dose distributions, MC integration was performed to generate DVKs to match the voxel sizes (3 and 4 mm) used in this work. For each voxel in the DVK, a MC integration was performed by 1) randomly selecting positions in the source voxel, 2) randomly selecting positions in the target voxel, 3) calculating the distance between the source and target positions, 4) looking up the absorbed dose in the summed radial distribution for each distance, and 5) calculating the average absorbed dose to the voxel over the distances. The resulting DVK was individualized for each study,matching the shape of each voxel and covering a volume of 80 × 80 × 80 cm. DVK convolution with the TIA map was performed using a fast Fourier transformation (FFT) followed by a voxel-level density correction using the CT-derived density map in MATLAB 2022a. The density correction scales the voxel value by the ratio of density of the homogenous medium used to generate the kernel to the density of the voxel.

2.6.4 |. Organ-level (S-values)

OLINDA v1.1⁵ with the reference adult male/female phantom (with mass scaling to account for difference between patient and phantom organ mases) was used to compute organ-level mean absorbed doses. The unit density sphere model within OLINDA, which provides the self-dose component only, was used for the lesions.

3 |. RESULTS

3.1 |. Phantom studies

For the phantom studies, comparison of absorbed dose estimates from the LBTE solver against MC, DVK, and OLINDA are presented in Table 3 and dose maps and line profiles are given in Figure S-1. For all three radionuclides and all inserts/compartments,there was excellent agreement (within 5%) with both MC and DVK for mean absorbed dose. The agreement for DVH metrics were also within 5% except for the D90 and D70 values of the two extra-hepatic inserts with very low activity concentration in the ⁹⁰Y phantom where the difference was up to 15%. The comparison of mean absorbed dose estimates from LBTE solver with OLINDA showed higher differences (up to 21%).

TABLE 3.

Phantom data showing the median [min,max] percentage difference of mean absorbed doses and DVH metrics between LBTE solver and other dosimetry methods

Radionuclide	Structures	Mean absorbed doses			D90		D70		D50		D30		D10
		% diff LBTE vs. MC	% diff LBTE vs. DVK	% diff LBTE vs. OLINDA	% diff LBTE vs. MC	% diff LBTE vs. DVK	% diff LBTE vs. MC	% diff LBTE vs. DVK	% diff LBTE vs. MC	% diff LBTE vs. DVK	% diff LBTE vs. MC	% diff LBTE vs. DVK	% diff LBTE vs. MC	% diff LBTE vs. DVK
177 Lu	Liver	1.4	4.7	9.0	−0.5	1.9	0.5	4.5	0.7	3.9	2.6	3.6	0.2	3.4
	Lesion Inserts	0.2 [−0.1, 0.5]	3.6 [3.3, 3.9]	2.1 [1.6,2.7]	0.7 [0.5, 1.0]	4.4 [4.4, 4.4]	0.7 [0.4, 0.9]	3.9 [3.6, 4.1]	0.7 [0.7, 0.8]	4.0 [3.8, 4.1]	0.2 [−0.1, 0.5]	3.7 [3.5, 3.9]	0.4 [0.2, 0.6]	3.8 [3.5, 4.0]
90 Y	Liver	2.2	0.5	−2.5	−2.2	−1.3	−0.2	−0.5	0.0	−0.4	−0.2	−0.7	−0.5	−1.0
	Lesion Inserts	0.2 [−0.7, 2.4]	−1.2 [−3.4, −0.9]	18.8 [15.5, 20.7]	0.9 [−2.2, 15.4]	0.8 [−2.2, 15.4]	0.5 [−1.5, 7.8]	0.3 [−1.8, 5.6]	0.3 [−0.8, 2.7]	−0.1 [−1.3, 1.2]	0.2 [−1.2, 0.7]	−0.3 [−2.0, 0.6]	−1.2 [−2.9, 0.3]	−1.3 [−2.4, 0.0]
131 I	Lesion Inserts	−1.4 [−2.7, −0.5]	0.6 [−1.0, 2.1]	10.3 [7.4, 12.7]	−0.8 [−1.1, −0.1]	2.0 [1.4, 2.4]	−0.7 [−1.3, −0.4]	1.7 [0.5, 2.2]	−0.5 [−0.7, 0.3]	2.0 [1.0, 2.3]	−0.3 [−0.9, 0.0]	1.7 [1.0, 2.3]	−0.5 [−0.8, −0.2]	1.4 [1.0, 2.3]

Difference % = 100 * (LBTE–XX)/LBTE where XX = MC, DVK or OLINDA. Here liver indicates whole liver including the lesions

3.2 |. Patient studies

The patient evaluation covers a wide range of structures with different densities and uptake concentration. A total of 93 contours were evaluated including one to five lesions per patient along with normal organs—kidneys, liver, lungs, and bones for ¹⁷⁷Lu; liver and lungs for ⁹⁰Y and lumbar vertebrae (spongiosa) for ¹³¹I. All lesions were in soft tissue with volumes ranging from 2 to 1052 mL (¹⁷⁷Lu), 5–870 mL (⁹⁰Y), and 4–60 mL (¹³¹I).

All mean absorbed dose values and DVH metrics for individual patients are presented in Table S1–S3. Figure 2 shows the absorbed dose maps from Acuros LBTE solver, difference maps between dose engines, profiles, and DVHs, for example, patients in the ¹⁷⁷Lu-DOTATATE, ⁹⁰Y-SIRT, and ¹³¹I RIT cohorts. Profile locations were selected to encompass multiple tissue regions such as soft tissue, bone, and lung along with lesions. As evident from the difference maps and profiles,in general,there is better agreement between dose maps corresponding to LBTE and MC than between LBTE and DVK. This is more pronounced for the lung and the skeletal regions and at the interface of different tissue regions. Although the DVHs corresponding to the different dose engines look very similar, there is better agreement between LBTE and MC than between LBTE and DVK.

FIGURE 2.

Coronal slice of absorbed dose map from Acuros (LBTE) for a patient treated with ¹⁷⁷Lu-DOTATATE (row 1), ⁹⁰Y-SIRT (row 2), and ¹³¹I-RIT (row 3). Also shown are the relative difference maps (LBTE vs MC and LBTE vs DVK), line profiles across dose maps and DVHs (LBTE represented as solid, MC as dot and DVK as dash).

The voxel level percentage differences in absorbed doses derived from LBTE in comparison to MC,and DVK convolution are plotted in Figure 3 for all defined structures. The comparisons of DVH metrics are presented in Figure S-2 and Table S-4. In general, the voxel level estimates from LBTE show better agreement with MC than with DVK.Voxel level estimates for soft tissue structures show good agreement between the LBTE solver and both MC and DVK with a median difference < 5% (maximum < 13%) for the DVH metrics with all three radionuclides.The largest difference in DVH metrics was for the D90 and D70 metric in lung and bone. Here, the difference between LBTE and MC had a median value < 14% (maximum 23%) for bone and < 4% (maximum 37%) for lung, while the corresponding differences between LBTE and DVK were < 23% (maximum 31%) and < 67% (maximum 313%), respectively. It is worth noting that for the ¹⁷⁷Lu-DOTATATE therapy, the bone and lung regions had very low uptake, hence voxel-level estimates can be noisy.

FIGURE 3.

Relative percentage differences in voxel absorbed dose estimates computed with LBTE compared with MC and DVK convolution for lesions and normal organs. Difference % = (LBTE –XX)/LBTE XX = MC, DVK. The horizontal line in the box represents the median value, the top and bottom of the box shows the 3rd (Q3) and 1st (Q1) quartile, and the whiskers extend to the largest (1.5*(Q3 − Q1) + Q3) or smallest (Q1 − 1.5 * Q3 − Q1) value.

Since mean absorbed doses to lesions/organs is currently the most widely used dosimetric quantity in RPT, the mean values averaged over all voxels in the structures are also compared. In this case, comparison with the S-value based calculation from OLINDA is also possible. The percentage difference in mean absorbed doses derived from the LBTE solver in comparison to MC, DVK and OLINDA are plotted in Figure 4 and reported in Table S-4. As with the voxel level estimates, in general, mean absorbed doses in various structures from LBTE show better agreement with DPM MC than with DVK. For all three radionuclides, mean absorbed doses from LBTE were in good agreement with those from MC and DVK for soft tissue (median difference < 5%, maximum 9%) and for bones (median difference < 11%, maximum 11%). For lungs with LBTE, mean absorbed dose agreed well with MC (median difference < 1%, maximum 8%), but the difference was substantially higher when compared to DVK (median difference < 16%, maximum 33%).

FIGURE 4.

Relative percentage differences between mean absorbed dose estimates computed with LBTE compared with other methods. Difference % = (LBTE –XX)/LBTE XX = MC, DVK or OLINDA. The horizontal line in the box represents the median value, the top and bottom of the box shows the 3rd (Q3) and 1st (Q1) quartile, and the whiskers extend to the largest (1.5*(Q3 − Q1) + Q3) or smallest (Q1 − 1.5 * Q3 − Q1) value. Numerical values are given in Table S-4.

Absorbed doses from the LBTE solver agreed reasonably well (within 10%) with OLINDA for normal organs for all three radionuclides, but differences were more pronounced for lesions where OLINDA underestimated the absorbed doses by up to 14% for ¹⁷⁷Lu, up to 23% for ⁹⁰Y, and up to 26% for ¹³¹I. These differences can be attributed to the fact that OLINDA uses a spherical tumor model for its computation, while the voxel-level calculation (LBTE, MC, and DVK) utilizes the tumor shape delineated on CT and to the fact that OLINDA considers only self-absorbed doses for spheres while the other dose engines compute both the self and cross-dose components. Such variations between OLINDA and DPM MC has also been previously reported by Howard et al.⁵⁰ The non-self-dose contribution is particularly relevant for ¹³¹I, which has high-intensity gamma-ray contribution, while the gamma-ray intensity of ¹⁷⁷Lu is relatively low.

3.3 |. Computation time

The computation times for a typical voxel dosimetry calculation for a patient with matrix size of 166 × 166 × 129 (voxel size 3 × 3 × 3 mm³) are compared in Table 4. As the time for the MC calculation depends on the number of histories, the time for DPM is given for 10⁹ histories (used to generate the MC results presented in the current study) and for 10⁸ histories. The MC statistical noise decreases with the total number of histories and is indicated in the table both for organ and voxel-based calculation.These values were obtained from the uncertainty maps generated from the DPM runs, where the history-by-history method for estimating uncertainty is used.⁴³ Uncertainty values can be considered to select the number of MC histories as a compromise between computational time and noise.

TABLE 4.

Computation time for the different dose engines to generate the absorbed dose map for a typical patient (¹⁷⁷Lu-DOTATATE).

	DPM MC
	With 108 histories	With 109 histories (used in current study)	DVK FFT convolution	Acuros MRT
Time (min)	3.2	32	< 1	< 2
System and cores used	iMac i7, 3.80 GHz. 1 core	iMac i7, 3.80 GHz. 1 core	iMac i7, 3.80 GHz; 1 core	Xeon W-2235, 3.80 GHz.6 cores
Statistical Uncertainty expressed as Relative uncertainty in absorbed dose. Mean ± STD	0.10 ± 0.14% (organ/lesion level) 7.7 ± 6.0% (voxel-level: voxels within organ/lesion)	0.03 ± 0.04% (organ/lesion level) 3.1 ± 2.7% (voxel-level: voxels within organ/lesion)	N/Aa	N/A

^aMinimal as kernels were generated using a long MC run.

4 |. DISCUSSION

Deterministically solving the LBTE is relatively novel in the realm of RPT and SIRT dosimetry despite its availability for EBRT dosimetry. In this work, a fast LBTE solver was implemented within a commercial dosimetry platform, and benchmarked against MC, which is generally accepted to be the gold standard for voxel dosimetry in complex geometries/heterogeneities encountered in the human body. In clinically relevant phantom studies and RPT/SIRT patient studies,absorbed dose estimates from the LBTE solver were compared to those from a previously validated MC-dosimetry code, DVK convolution, and also with OLINDA, using the same TIA maps as the input to the different dose engines.The structures evaluated included lesions and organs most relevant for the therapy (e.g., kidneys and liver in the case of ¹⁷⁷Lu DOTATATE and liver in the case of ⁹⁰Y- SIRT) as the SPECT and PET field-of-view for post-therapy imaging was selected to encompass these organs.

Our findings showed very good agreement between dosimetry results from LBTE solver and DPM MC with all three radionuclides for mean absorbed dose in soft tissue, lung and bone (Figure 4, Table S-4). For mean absorbed dose, the median difference was < 1% (maximum 6%) in soft tissue, < 10% (maximum 11%) in bone, and < 1% (maximum 8%) in lung, and are in agreement with prior publications in EBRT and brachytherapy application where differences < 5% have been reported when benchmarking LBTE solvers against MC transport codes such as MCNPX, EGS4.⁴² There was also good agreement in LBTE and MC voxel level estimates in general with the largest difference observed for D90 in lung for ¹⁷⁷Lu-DOTATATE patients, where the images were noisy due to low uptake. Differences between the LBTE solver and MC-based dosimetry software could be attributed to differences in the energy cross section libraries, in the transport and additionally, in the case of RPT and SIRT, differences in the radionuclide decay data including beta and gamma-ray energy spectra.The current study was not designed to isolate these effects, but a previous study³¹ showed that much of the differences between LBTE and MC could be attributed to differences in energy cross section data used by different algorithms. We did perform some additional testing for an example patient to identify potential differences between performance of LBTE solver and MC: 1) In the current work the electron energy cut-off for terminating transport was 200 keV for the LBTE solver while DPM MC used 59 keV. However, re-running DPM with a threshold of 200 keV led to minimal difference in dosimetry results (< 1% difference in mean absorbed doses between the two DPM runs). 2) It was also established that the DPM simulations had converged by re-running select cases with a larger number of histories (re-run with 10¹⁰ instead of 10⁹ used in the current work) and observing minimal differences (< 1% in mean absorbed doses), which negated the possibility that statistical noise was a significant contribution. 3) Generation of Bremsstrahlung photons in the DPM MC transport physics, which is disregarded in the current implementation of the LBTE solver was investigated, however differences between DPM runs with and without inclusion of Bremsstrahlung photons was observed to be negligible (< 0.5% difference in mean absorbed doses).

In general, for both the voxel-level (Figure 3) and organ-level estimates (Figure 4) the absorbed doses from LBTE agreed better with MC than with DVK, which is also evident from the difference maps,line profiles and DVHs of Figure 2. This may be attributed to the presumption of homogeneous (single) media to generate the kernels for DVK convolution. Using DVKs derived for a homogenous media (water), even with density scaling applied as we did in the current study, does not fully reflect the impact of density variations on the dose deposition. Differences in bone can also be expected for the photon component given the higher effective Z relative to water and DVK reports dose to water, while Acuros MRT reports dose to medium. While some efforts have been made to use multiple kernels^15–17 of varied densities to take tissue heterogeneities into consideration, this was not evaluated in the current work. In addition to generation of multiple kernels, such an approach requires generation of medium-specific masks for the convolution step. Using digital phantoms and patient data from RPT and SIRT, prior studies have reported on close agreement between the conventional (single-kernel) DVK convolution and MC for soft tissue, but substantially larger differences for lung-liver interfaces and bone marrow.^14–16

A limitation of the study is that while benchmarking against MC was performed, no ground-truth dosimetry data were available for direct validation.However,the in-house MC code,DPM used in the current work has been extensively benchmarked and validated against experimental measurement in both EBRT^51,52 and RPT/SIRT applications.⁴⁴ DPM is substantially faster than general purpose MC for MRT dosimetry applications as it has been optimized for dose computation in voxelated geometries using simplified cross-sectional models that are accurate at energies relevant to radiation therapy. This has enabled its use for patient specific dosimetry in clinical research. The LBTE-solver achieved comparable results to DPM but was faster than this efficient implementation of MC, which will facilitate its use in routine clinical practice. The discretization of the phase space variables affects the computation speed of the LBTE solver; however, the speed-accuracy tradeoff associated with discretization was not investigated.

In addition to the dose calculation step evaluated in the current study, multiple other factors such as activity quantification, registration of multi-timepoint serial images to determine pharmacokinetics,and time-activity fitting impacts dosimetry results in RPT. The inaccuracies of voxel-level dosimetry due to the limited spatial resolution of SPECT and PET, even when resolution recovery is included in the iterative reconstruction model as in the current study, is well-known.⁹ When reporting mean absorbed doses, a widely used approach to mitigate spatial resolution effects and the associated partial volume effects is to apply volume-based recovery coefficients.^8,10 However,voxel-level partial volume correction is much more challenging,⁵³ and a well-validated and practical method is not yet available although recently a deep-learning based approach has been proposed to reduce the impact of poor SPECT resolution on absorbed dose maps.⁵⁴ Use of perfect resolution virtual data as the starting image for the gold standard has been recommended for evaluating voxel dosimetry as in that case, the impact of spatial resolution on the dose estimate will be included in the evaluation.⁵⁵ However, in the current study the SPECT-and PET-derived patient TIA maps were used as the input to all of the dose engines, including the reference MC. The current study benchmarked the LBTE-solver based absorbed dose estimation step of the Acuros MRT dosimetry workflow by comparison with a well-validated MC dosimetry code,⁴⁴ while impact of PET and SPECT spatial resolution on dosimetry was not investigated.

5 |. CONCLUSION

Within soft tissue and bone, the deterministic LBTE solver exhibited good agreement with the more computationally intensive MC radiation transport making it a potential alternative for voxel-level dosimetry in RPT and SIRT. In general, dose estimates from Acuros MRT agreed more closely with estimates from MC than with estimates from DVK convolution. Further analysis is being performed to encompass the wide range of radionuclides, geometries, and tissue densities observed in clinical settings. Additionally, the registration, segmentation, and time-activity fitting modules that are coupled to the dose engine of the Theranostics Navigator will be evaluated. Testing the dose engine on radiopharmaceutical therapies other than ¹⁷⁷Lu-DOTATATE, ⁹⁰Y-SIRT, and ¹³¹I-RIT and conditions not evaluated here, such as lesions in the bone spongiosa, is envisaged.