A PBPK model for PRRT with [177Lu]Lu-DOTA-TATE: Comparison of model implementations in SAAM II and MATLAB/SimBiology.

Physiologically based pharmacokinetic (PBPK) models offer the ability to simulate and predict the biodistribution of radiopharmaceuticals and have the potential to enable individualised treatment planning in molecular radiotherapy. The objective of this study was to develop and implement a whole-body compartmental PBPK model for peptide receptor radionuclide therapy (PRRT) with [177Lu]Lu-DOTA-TATE in SimBiology to allow for more complex analyses. The correctness of the model implementation was ensured by comparing its outputs, such as the time-integrated activity (TIA), with those of a PBPK model implemented in SAAM II software. METHODS: A combined PBPK model for [68Ga]Ga-DOTA-TATE and [177Lu]Lu-DOTA-TATE was developed and implemented in both SAAM II and SimBiology. A retrospective analysis of 12 patients with metastatic neuroendocrine tumours (NETs) was conducted. First, time-activity curves (TACs) and TIAs from the two software were calculated and compared for identical parameter values. Second, pharmacokinetic parameters were fitted to activity concentrations, analysed and compared. RESULTS: The PBPK model implemented in SimBiology produced TIA results comparable to those generated by the model implemented in SAAM II, with a relative deviation of less than 0.5% when using the same input parameters. The relative deviation of the fitted TIAs was less than 5% when model parameter values were fitted to the measured activity concentrations. CONCLUSION: The proposed PBPK model implemented in SimBiology can be used for dosimetry in radioligand therapy and TIA prediction. Its outputs are similar to those generated by the PBPK model implemented in SAAM II, confirming the correctness of the model implementation in SimBiology.

Implementation of dosimetry for molecular radiotherapy; results from a European survey.

PURPOSE: The use of molecular radiotherapy (MRT) has been rapidly evolving over the last years. The aim of this study was to assess the current implementation of dosimetry for MRTs in Europe. METHODS: A web-based questionnaire was open for treating centres between April and June 2022, and focused on 2020-2022. Questions addressed the application of 16 different MRTs, the availability and involvement of medical physicists, software used, quality assurance, as well as the target regions for dosimetry, whether treatment planning and/or verification were performed, and the dosimetric methods used. RESULTS: A total of 173 responses suitable for analysis was received from centres performing MRT, geographically distributed over 27 European countries. Of these, 146 centres (84 %) indicated to perform some form of dosimetry, and 97 % of these centres had a medical physicist available and almost always involved in dosimetry. The most common MRTs were 131I-based treatments for thyroid diseases and thyroid cancer, and [223Ra]RaCl2 for bone metastases. The implementation of dosimetry varied widely between therapies, from almost all centres performing dosimetry-based planning for microsphere treatments to none for some of the less common treatments (like 32P sodium-phosphate for myeloproliferative disease and [89Sr]SrCl2 for bone metastases). CONCLUSIONS: Over the last years, implementation of dosimetry, both for pre-therapeutic treatment planning and post-therapy absorbed dose verification, increased for several treatments, especially for microsphere treatments. For other treatments that have moved from research to clinical routine, the use of dosimetry decreased in recent years. However, there are still large differences both across and within countries.

Traceable calibration with 177Lu and comparison of activity meters at hospitals in Norway and Sweden.

INTRODUCTION: The activity meter is used to determine the activity of delivered radiopharmaceuticals, administered activity to patients and reference activity when gamma-cameras are calibrated prior to imaged-based dosimetry. The aim is to describe a procedure for cross-calibration of activity meters at different clinical sites, and report on 177Lu activity results when using factory-set calibration factors compared to when calibration is performed with traceability to a primary standard. METHODS: Thirty activity meters placed at seven hospitals in Norway and Sweden from four manufacturers: Capintec, Commecer, NuviaTech and Veenstra were included. A stock solution with 177Lu was prepared at the local sites and measured in each activity meter with factory settings. The solution was shipped to the reference site at Lund University for measurements in a secondary standard activity meter. Deviations between local and reference activity measurements were determined for three geometries: 25-mL vial, 10-mL syringe and 1-mL syringe. RESULTS: The median of the deviations was 6.4 % for the 25 mL vial, 5.9 % for the 10 mL syringe and 6.8 % for the 1 mL syringe. The median of the deviations for the 25 mL vial, was 1.5 % for activity meters from Capintec, 7.0 % for Comecer, 11.0 % for NuviaTech and 2.4 % for Veenstra. The majority of the deviations were positive and the maximum deviation was 14.5 %. CONCLUSION: The activity of 177Lu measured in an activity meter with factory-set dial settings may yield deviations up to 14.5%, compared to activities measured with traceability to a primary standard. This would imply an undertreatment of patients.

Phase II trial demonstrates the efficacy and safety of individualized, dosimetry-based 177Lu-DOTATATE treatment of NET patients.

PURPOSE: Radionuclide therapy with 177Lu-DOTATATE is well established for patients with advanced somatostatin receptor-positive neuroendocrine tumors with a standard schedule of 7.4 GBq at four occasions. However, this approach does not consider individual variability affecting the tumor radiation dose or dose to organs at risk. Therefore, it is important to assess more personalized strategies. The aim of this phase II trial was to evaluate individualized 177Lu-DOTATATE for which the number of cycles varied based on renal dosimetry. METHODS: Patients were eligible if they had a progressive, somatostatin receptor-positive neuroendocrine tumor with a Ki 67 labeling index < 20%. They received cycles of 7.4 GBq of 177Lu-DOTATATE at 10 ± 2-week intervals until a predefined radiation dose to the kidneys was reached. The primary endpoint was objective tumor response (RECIST v 1.1). Secondary endpoints included progression-free survival (PFS), overall survival (OS), and toxicity (CTCAE v. 4.0). RESULTS: Ninety-six patients who had received a median of 5 cycles (range 1-9) were evaluable for efficacy. The objective tumor response was 16% partial response, 66% stable disease, and 19% progressive disease. The median PFS and OS were 29 months and 47 months, respectively, and were significantly associated with kidney dose, performance status, and Ki 67 levels but not with tumor origin. The overall toxicity was mild, and the most common events were grade 1-2 anemia, thrombocytopenia, fatigue, nausea, and diarrhea. Grade 3-4 toxicity occurred in < 10% of patients and was mostly hematological, with no grade 3-4 renal toxicity. CONCLUSION: Individualized treatment with 177Lu-DOTATATE based on renal dosimetry is clearly feasible with low toxicity and promising efficacy, showing the potential to further improve outcome beyond the standard approach, and should be further assessed in randomized trials. TRIAL REGISTRATION: EudraCT 2011-000,240-16. NCT01456078. https://clinicaltrials.gov/ct2/show/NCT01456078.

Dosimetric Quantities in Neuroendocrine Tumors over Treatment Cycles with 177Lu-DOTATATE.

Tumor dosimetry was performed for 177Lu-DOTATATE with the aims of better understanding the range and variation of the tumor-absorbed doses (ADs), how different dosimetric quantities evolve over the treatment cycles, and whether this evolution differs depending on the tumor grade. Such information is important for radiobiologic interpretation and may inform the design of alternative administration schemes. Methods: The data came from 41 patients with neuroendocrine tumors (NETs) of grade 1 (n = 23) or 2 (n = 18) who had received between 2 and 9 treatment cycles. Dosimetry was performed for 182 individual lesions, giving a total of 880 individual AD assessments across all cycles. Hybrid planar-SPECT/CT imaging was used, including quantitative SPECT reconstruction, voxel-based absorbed-dose-rate calculation, semiautomatic image segmentation, and partial-volume correction. Linear mixed-effect models were used to analyze changes in tumor ADs over cycles, absorbed-dose rates and activity concentrations on day 1, effective half-times, and tumor volumes. Tumors smaller than 8 cm3 were excluded from analyses. Results: Tumor ADs ranged between 2 and 77 Gy per cycle. On average, the AD decreased over the cycles, with significantly different rates (P < 0.05) of 6% and 14% per cycle for grade 1 and 2 NETs, respectively. The absorbed-dose rates and activity concentrations on day 1 decreased by similar amounts. The effective half-times were less variable but shorter for grade 2 than for grade 1 (P < 0.001). For grade 2 NETs, the tumor volumes decreased, with a similar tendency in grade 1. Conclusion: The tumor AD, absorbed-dose rate, and activity uptake decrease, in parallel with tumor volumes, between 177Lu-DOTATATE treatment cycles, particularly for grade 2 NETs. The effective half-times vary less but are lower for grade 2 than grade 1 NETs. These results may indicate the development of radiation-induced fibrosis and could have implications for the design of future treatment and dosimetry protocols.

A multicentre and multi-national evaluation of the accuracy of quantitative Lu-177 SPECT/CT imaging performed within the MRTDosimetry project.

PURPOSE: Patient-specific dosimetry is required to ensure the safety of molecular radiotherapy and to predict response. Dosimetry involves several steps, the first of which is the determination of the activity of the radiopharmaceutical taken up by an organ/lesion over time. As uncertainties propagate along each of the subsequent steps (integration of the time-activity curve, absorbed dose calculation), establishing a reliable activity quantification is essential. The MRTDosimetry project was a European initiative to bring together expertise in metrology and nuclear medicine research, with one main goal of standardizing quantitative 177Lu SPECT/CT imaging based on a calibration protocol developed and tested in a multicentre inter-comparison. This study presents the setup and results of this comparison exercise. METHODS: The inter-comparison included nine SPECT/CT systems. Each site performed a set of three measurements with the same setup (system, acquisition and reconstruction): (1) Determination of an image calibration for conversion from counts to activity concentration (large cylinder phantom), (2) determination of recovery coefficients for partial volume correction (IEC NEMA PET body phantom with sphere inserts), (3) validation of the established quantitative imaging setup using a 3D printed two-organ phantom (ICRP110-based kidney and spleen). In contrast to previous efforts, traceability of the activity measurement was required for each participant, and all participants were asked to calculate uncertainties for their SPECT-based activities. RESULTS: Similar combinations of imaging system and reconstruction lead to similar image calibration factors. The activity ratio results of the anthropomorphic phantom validation demonstrate significant harmonization of quantitative imaging performance between the sites with all sites falling within one standard deviation of the mean values for all inserts. Activity recovery was underestimated for total kidney, spleen, and kidney cortex, while it was overestimated for the medulla. CONCLUSION: This international comparison exercise demonstrates that harmonization of quantitative SPECT/CT is feasible when following very specific instructions of a dedicated calibration protocol, as developed within the MRTDosimetry project. While quantitative imaging performance demonstrates significant harmonization, an over- and underestimation of the activity recovery highlights the limitations of any partial volume correction in the presence of spill-in and spill-out between two adjacent volumes of interests.

Bone Marrow Absorbed Doses and Correlations with Hematologic Response During 177Lu-DOTATATE Treatments Are Influenced by Image-Based Dosimetry Method and Presence of Skeletal Metastases.

This study aimed to compare different image-based methods for bone marrow dosimetry and study the dose-response relationship during treatment with 177Lu-DOTATATE in patients with and without skeletal metastases. Methods: This study included 46 patients with advanced neuroendocrine tumors treated with at least 2 fractions of 177Lu-DOTATATE at Sahlgrenska University Hospital. High- and low-uptake compartments were automatically outlined in planar images collected at 2, 24, 48, and 168 h after injection. The bone marrow absorbed doses were calculated from the cross doses of the high- and low-uptake compartments and the self-dose, using the time-activity concentration curve for the low-uptake compartment. This time-activity concentration curve was adjusted using a fixed constant of 1.8 for the planar dosimetry method and using the activity concentrations in vertebral bodies in SPECT images at 24 h after injection of 177Lu-DOTATATE in 4 hybrid methods: L4-SPECT used the activity concentration in the L4 vertebra, whereas V-SPECT, L-SPECT, and T-SPECT used the median activity concentration in all visible vertebrae, lumbar vertebrae, and thoracic vertebrae, respectively. Results: Using the planar method, L4-SPECT, V-SPECT, L-SPECT, and T-SPECT, the estimated median bone marrow absorbed doses were 0.19, 0.36, 0.40, 0.39, and 0.46 Gy/7.4 GBq, respectively, with respective ranges of 0.12-0.33, 0.15-1.44, 0.19-1.71, 0.21-1.60, and 0.18-2.12 Gy/7.4 GBq. For all methods, the bone marrow absorbed dose significantly correlated with decreased platelet counts. This correlation increased after treatment fraction 2: the Spearman correlation (rs) were -0.49 for the planar method, -0.61 for L4-SPECT, -0.63 for V-SPECT, -0.63 for L-SPECT, and -0.57 for T-SPECT. A separate analysis revealed an increased correlation for patients without skeletal metastases using the planar method (rs = -0.67). In contrast, hybrid methods had poor correlations for patients without metastases and stronger correlations for patients with skeletal metastases (rs = -0.61 to -0.74). The mean bone marrow absorbed doses were 3%-69% higher for patients with skeletal metastases than for patients without. Conclusion: The estimated bone marrow absorbed doses by image-based techniques and the correlation with platelets are influenced by the choice of measured vertebrae and the presence of skeletal metastases.

EANM practical guidance on uncertainty analysis for molecular radiotherapy absorbed dose calculations.

A framework is proposed for modelling the uncertainty in the measurement processes constituting the dosimetry chain that are involved in internal absorbed dose calculations. The starting point is the basic model for absorbed dose in a site of interest as the product of the cumulated activity and a dose factor. In turn, the cumulated activity is given by the area under a time-activity curve derived from a time sequence of activity values. Each activity value is obtained in terms of a count rate, a calibration factor and a recovery coefficient (a correction for partial volume effects). The method to determine the recovery coefficient and the dose factor, both of which are dependent on the size of the volume of interest (VOI), are described. Consideration is given to propagating estimates of the quantities concerned and their associated uncertainties through the dosimetry chain to obtain an estimate of mean absorbed dose in the VOI and its associated uncertainty. This approach is demonstrated in a clinical example.

Dosimetry-based treatment planning for molecular radiotherapy: a summary of the 2017 report from the Internal Dosimetry Task Force.

BACKGROUND: The European directive on basic safety standards (Council directive 2013/59 Euratom) mandates dosimetry-based treatment planning for radiopharmaceutical therapies. The directive comes into operation February 2018, and the aim of a report produced by the Internal Dosimetry Task Force of the European Association of Nuclear Medicine is to address this aspect of the directive. A summary of the report is presented. RESULTS: A brief review of five of the most common therapy procedures is included in the current text, focused on the potential to perform patient-specific dosimetry. In the full report, 11 different therapeutic procedures are included, allowing additional considerations of effectiveness, references to specific literature on quantitative imaging and dosimetry, and existing evidence for absorbed dose-effect correlations for each treatment. Individualized treatment planning with tracer diagnostics and verification of the absorbed doses delivered following therapy is found to be scientifically feasible for almost all procedures investigated, using quantitative imaging and/or external monitoring. Translation of this directive into clinical practice will have significant implications for resource requirements. CONCLUSIONS: Molecular radiotherapy is undergoing a significant expansion, and the groundwork for dosimetry-based treatment planning is already in place. The mandated individualization is likely to improve the effectiveness of the treatments, although must be adequately resourced.

Biologically effective dose in fractionated molecular radiotherapy--application to treatment of neuroblastoma with (131)I-mIBG.

In this work, the biologically effective dose (BED) is investigated for fractionated molecular radiotherapy (MRT). A formula for the Lea-Catcheside G-factor is derived which takes the possibility of combinations of sub-lethal damage due to radiation from different administrations of activity into account. In contrast to the previous formula, the new G-factor has an explicit dependence on the time interval between administrations. The BED of tumour and liver is analysed in MRT of neuroblastoma with (131)I-mIBG, following a common two-administration protocol with a mass-based activity prescription. A BED analysis is also made for modified schedules, when due to local regulations there is a maximum permitted activity for each administration. Modifications include both the simplistic approach of delivering this maximum permitted activity in each of the two administrations, and also the introduction of additional administrations while maintaining the protocol-prescribed total activity. For the cases studied with additional (i.e. more than two) administrations, BED of tumour and liver decreases at most 12% and 29%, respectively. The decrease in BED of the tumour is however modest compared to the two-administration schedule using the maximum permitted activity, where the decrease compared to the original schedule is 47%.

Uncertainty propagation for SPECT/CT-based renal dosimetry in (177)Lu peptide receptor radionuclide therapy.

A computer model of a patient-specific clinical (177)Lu-DOTATATE therapy dosimetry system is constructed and used for investigating the variability of renal absorbed dose and biologically effective dose (BED) estimates. As patient models, three anthropomorphic computer phantoms coupled to a pharmacokinetic model of (177)Lu-DOTATATE are used. Aspects included in the dosimetry-process model are the gamma-camera calibration via measurement of the system sensitivity, selection of imaging time points, generation of mass-density maps from CT, SPECT imaging, volume-of-interest delineation, calculation of absorbed-dose rate via a combination of local energy deposition for electrons and Monte Carlo simulations of photons, curve fitting and integration to absorbed dose and BED. By introducing variabilities in these steps the combined uncertainty in the output quantity is determined. The importance of different sources of uncertainty is assessed by observing the decrease in standard deviation when removing a particular source. The obtained absorbed dose and BED standard deviations are approximately 6% and slightly higher if considering the root mean square error. The most important sources of variability are the compensation for partial volume effects via a recovery coefficient and the gamma-camera calibration via the system sensitivity.

Pharmacokinetic digital phantoms for accuracy assessment of image-based dosimetry in (177)Lu-DOTATATE peptide receptor radionuclide therapy.

Patient-specific image-based dosimetry is considered to be a useful tool to limit toxicity associated with peptide receptor radionuclide therapy (PRRT). To facilitate the establishment and reliability of absorbed-dose response relationships, it is essential to assess the accuracy of dosimetry in clinically realistic scenarios. To this end, we developed pharmacokinetic digital phantoms corresponding to patients treated with (177)Lu-DOTATATE. Three individual voxel phantoms from the XCAT population were generated and assigned a dynamic activity distribution based on a compartment model for (177)Lu-DOTATATE, designed specifically for this purpose. The compartment model was fitted to time-activity data from 10 patients, primarily acquired using quantitative scintillation camera imaging. S values for all phantom source-target combinations were calculated based on Monte-Carlo simulations. Combining the S values and time-activity curves, reference values of the absorbed dose to the phantom kidneys, liver, spleen, tumours and whole-body were calculated. The phantoms were used in a virtual dosimetry study, using Monte-Carlo simulated gamma-camera images and conventional methods for absorbed-dose calculations. The characteristics of the SPECT and WB planar images were found to well represent those of real patient images, capturing the difficulties present in image-based dosimetry. The phantoms are expected to be useful for further studies and optimisation of clinical dosimetry in (177)Lu PRRT.

Long-Term Retention of 177Lu/177mLu-DOTATATE in Patients Investigated by γ-Spectrometry and γ-Camera Imaging.

UNLABELLED: Dosimetry in peptide receptor radionuclide therapy using (177)Lu-DOTATATE is based on patient imaging during the first week after administration and determination of the activity retention as a function of time for different tissues. For calculation of the absorbed dose, it is generally assumed that the long-term activity retention follows the pattern determined from the first week. This work aimed to investigate the validity of this assumption by performing additional patient measurements between 5 and 10 wk after administration. A further aim was to investigate to what extent absorbed dose values were affected when including these measurements, also taking into account the radionuclide impurity of (177m)Lu and build-up of secondary (177)Lu from the (177m)Lu decay. METHODS: A combination of methods was used: planar γ-camera imaging as part of the clinical dosimetry protocol, determination of the whole-body activity between 5 and 9 wk after injection using spectrometric NaI(Tl) and HPGe detectors, and imaging between 5 and 10 wk after injection for assessment of the activity distribution. From these measurements the long-term retention of activity was determined and the relative influence on absorbed doses calculated. RESULTS: The most important finding was a clearly visualized tumor uptake in images from between 5 and 7 wk after injection and in 1 patient also kidney and spleen uptake in images acquired on day 33. As a consequence, the total-body time-activity curve had a tail, which was not completely captured by imaging during the first week. The absorbed doses to total body and tumors obtained when including these late time points were on average 5%-6% higher than those obtained when using data acquired during the first week. The contributions to the absorbed dose from (177m)Lu and secondary (177)Lu were negligible. CONCLUSION: At approximately 5-7 wk after injection, there was a measureable amount of (177)Lu-DOTATATE in patients, which is mainly governed by retention in tumors. For tumor dosimetry, imaging at a later time than the routinely used 7 d may be warranted. The contribution to the absorbed dose from the radionuclide impurity of (177m)Lu was negligible.

Hybrid Imaging for Patient-Specific Dosimetry in Radionuclide Therapy.

Radionuclide therapy aims to treat malignant diseases by systemic administration of radiopharmaceuticals, often using carrier molecules such as peptides and antibodies. The radionuclides used emit electrons or alpha particles as a consequence of radioactive decay, thus leading to local energy deposition. Administration to individual patients can be tailored with regards to the risk of toxicity in normal organs by using absorbed dose planning. The scintillation camera, employed in planar imaging or single-photon emission computed tomography (SPECT), generates images of the spatially and temporally varying activity distribution. Recent commercially available combined SPECT and computed tomography (CT) systems have dramatically increased the possibility of performing accurate dose planning by using the CT information in several steps of the dose-planning calculation chain. This paper discusses the dosimetry chain used for individual absorbed-dose planning and highlights the areas where hybrid imaging makes significant contributions.

The evidence base for the use of internal dosimetry in the clinical practice of molecular radiotherapy.

Molecular radiotherapy (MRT) has demonstrated unique therapeutic advantages in the treatment of an increasing number of cancers. As with other treatment modalities, there is related toxicity to a number of organs at risk. Despite the large number of clinical trials over the past several decades, considerable uncertainties still remain regarding the optimization of this therapeutic approach and one of the vital issues to be answered is whether an absorbed radiation dose-response exists that could be used to guide personalized treatment. There are only limited and sporadic data investigating MRT dosimetry. The determination of dose-effect relationships for MRT has yet to be the explicit aim of a clinical trial. The aim of this article was to collate and discuss the available evidence for an absorbed radiation dose-effect relationships in MRT through a review of published data. Based on a PubMed search, 92 papers were found. Out of 79 studies investigating dosimetry, an absorbed dose-effect correlation was found in 48. The application of radiobiological modelling to clinical data is of increasing importance and the limited published data on absorbed dose-effect relationships based on these models are also reviewed. Based on National Cancer Institute guideline definition, the studies had a moderate or low rate of clinical relevance due to the limited number of studies investigating overall survival and absorbed dose. Nevertheless, the evidence strongly implies a correlation between the absorbed doses delivered and the response and toxicity, indicating that dosimetry-based personalized treatments would improve outcome and increase survival.

Dynamic (99m)Tc-MAG3 renography: images for quality control obtained by combining pharmacokinetic modelling, an anthropomorphic computer phantom and Monte Carlo simulated scintillation camera imaging.

In dynamic renal scintigraphy, the main interest is the radiopharmaceutical redistribution as a function of time. Quality control (QC) of renal procedures often relies on phantom experiments to compare image-based results with the measurement setup. A phantom with a realistic anatomy and time-varying activity distribution is therefore desirable. This work describes a pharmacokinetic (PK) compartment model for (99m)Tc-MAG3, used for defining a dynamic whole-body activity distribution within a digital phantom (XCAT) for accurate Monte Carlo (MC)-based images for QC. Each phantom structure is assigned a time-activity curve provided by the PK model, employing parameter values consistent with MAG3 pharmacokinetics. This approach ensures that the total amount of tracer in the phantom is preserved between time points, and it allows for modifications of the pharmacokinetics in a controlled fashion. By adjusting parameter values in the PK model, different clinically realistic scenarios can be mimicked, regarding, e.g., the relative renal uptake and renal transit time. Using the MC code SIMIND, a complete set of renography images including effects of photon attenuation, scattering, limited spatial resolution and noise, are simulated. The obtained image data can be used to evaluate quantitative techniques and computer software in clinical renography.

On the biologically effective dose (BED)-using convolution for calculating the effects of repair: I. Analytical considerations.

This work presents a new mathematical formulation of biologically effective dose (BED) for radiation therapy where effects of repair need to be considered. The formulation is based on the observation that the effects of repair, both during protracted irradiation and of incomplete repair between fractions, can be written using a convolution, i.e. BED(T)=∫(∞)(-∞)R(T)(t)dt+1/(α/ß) ∫(∞)(-∞)[R(T)(t)xI(t)]dt where T is the total irradiation time, R(T)(t) is the absorbed dose rate as a function of time t and I(t) is the function describing repair. To validate this formulation, the previously published expressions for instant and protracted irradiation are first summarized. Then, by analytical derivation, it is shown that the new formulation gives identical results. The calculation of BED can thus be treated within one single mathematical framework, applicable in external beam therapy, brachytherapy, radionuclide therapy, or a combination of these treatment modalities. Moreover, the new formulation allows for a straightforward incorporation of different repair models and has the advantage of being numerically applicable.