Monte Carlo-based Methods Research Articles

Efficient participating media rendering with differentiable regularization

Highly scattering media, such as milk, skin, and clouds, are common in the real world. Rendering participating media is challenging, especially for high-order scattering dominant media, because the light may undergo a large number of scattering events before leaving the surface. Monte Carlo-based methods typically require a long time to produce noise-free results. Based on the observation that low-albedo media contain less noise than high-albedo media, we propose reducing the variance of the rendered results using differentiable regularization. We first render an image with low-albedo participating media together with the gradient with respect to the albedo, and then predict the final rendered image with a low-albedo image and gradient image via a novel prediction function. To achieve high quality, we also consider the gradients of neighboring frames to provide a noise-free gradient image. Ultimately, our method can produce results with much less overall error than equal-time path tracing methods.

GATE Monte Carlo approach to heterogeneity dose distribution in small fields used in radiation therapy

The Accurate dosage prediction in Radiation Therapy is challenging, prompting a need for precision beyond conventional clinical Treatment Planning Systems (TPS). Monte Carlo-based methods are sought for their superior accuracy. The aim of this study is to compare dose distributions between the ACUROS algorithm and the GATE platform in various tissue densities and field sizes, focusing on smaller fields. This study was initiated with a homogeneous validation of the TrueBeam STX system, using measurements obtained from the Centre Hospitalier Interregional Edith Cavell (CHIREC) in Brussels. The validation compared dosimetric functions (Percentage Depth Dose (PDD), Dose profile (DP) and Collimator scatter fraction (CSF)) employing the GAMMA index with a 2% / 2 mm criterion tolerance. Following this, heterogeneous studies examined dose distributions between the ACUROS algorithm and the GATE platform in various tissue densities and field sizes, with a specific focus on smaller fields. Simulations were conducted using both platforms on chest phantoms with heterogeneous slabs representing bone, lung, and heart, each housing a central tumor. The impact of electronic equilibrium on tumors for different small field sizes was evaluated. Results showed a remarkable 99% agreement between measurements and GATE calculations in the homogeneous validation of the TrueBeam STX system. However, in heterogeneous studies, ACUROS consistently overestimated lung doses by up to 8% compared to GATE simulation, especially evident with a flattening filter and smaller beam sizes at density interfaces. This highlights significant dose estimation discrepancies between ACUROS and GATE, emphasizing the need for precise calculations. The findings support exploring Monte Carlo-based methods for enhanced accuracy in Radiation Therapy treatment planning.

A comparative study of methods for estimating model-agnostic Shapley value explanations

Shapley values originated in cooperative game theory but are extensively used today as a model-agnostic explanation framework to explain predictions made by complex machine learning models in the industry and academia. There are several algorithmic approaches for computing different versions of Shapley value explanations. Here, we consider Shapley values incorporating feature dependencies, referred to as conditional Shapley values, for predictive models fitted to tabular data. Estimating precise conditional Shapley values is difficult as they require the estimation of non-trivial conditional expectations. In this article, we develop new methods, extend earlier proposed approaches, and systematize the new refined and existing methods into different method classes for comparison and evaluation. The method classes use either Monte Carlo integration or regression to model the conditional expectations. We conduct extensive simulation studies to evaluate how precisely the different method classes estimate the conditional expectations, and thereby the conditional Shapley values, for different setups. We also apply the methods to several real-world data experiments and provide recommendations for when to use the different method classes and approaches. Roughly speaking, we recommend using parametric methods when we can specify the data distribution almost correctly, as they generally produce the most accurate Shapley value explanations. When the distribution is unknown, both generative methods and regression models with a similar form as the underlying predictive model are good and stable options. Regression-based methods are often slow to train but quickly produce the Shapley value explanations once trained. The vice versa is true for Monte Carlo-based methods, making the different methods appropriate in different practical situations.

Numerical impact of variable volumes of Monte Carlo simulations of heterogeneous conductivity fields in groundwater flow models

The knowledge of aquifer systems, their geological setting, their structure, and subsequent modeling is highly uncertain and is usually faced through Monte Carlo-based methods in hydrogeology. One of the most important uncertainty sources for groundwater models is represented by input hydraulic conductivities, related to the aquifer’s structure. There are no specific rules when simulating hydraulic conductivity fields within Monte Carlo frameworks to instruct numerical models, and information about employed conductivity fields and their numerical convergence is often not given. This technical work aims to fill this gap by investigating the impact of employing conductivity information upon different volumes of Monte Carlo simulations applied to a real case study. Thus, this work estimates the minimum volumes of Monte Carlo hydraulic conductivity fields to be employed in groundwater flow models for achieving numerically stable (i) boundary conditions, (ii) global model performances, and (iii) local ones such as simulated hydraulic heads. The present results aim to be indicative of similar hydrogeological settings and will serve as a basis for more complex ones and for investigating transport problems.

Optimal constrained design of control charts using stochastic approximations

In statistical process monitoring, control charts typically depend on a set of tuning parameters besides its control limit(s). Proper selection of these tuning parameters is crucial to their performance. In a specific application, a control chart is often designed for detecting a target process distributional shift. In such cases, the tuning parameters should be chosen such that some characteristic of the out-of-control (OC) run length of the chart, such as its average, is minimized for detecting the target shift, while the control limit is set to maintain a desired in-control (IC) performance. However, explicit solutions for such a design are unavailable for most control charts, and thus numerical optimization methods are needed. In such cases, Monte Carlo-based methods are often a viable alternative for finding suitable design constants. The computational cost associated with such scenarios is often substantial, and thus computational efficiency is a key requirement. To address this problem, a two-step design based on stochastic approximations is presented in this paper, which is shown to be much more computationally efficient than some representative existing methods. A detailed discussion about the new algorithm’s implementation along with some examples are provided to demonstrate the broad applicability of the proposed methodology for the optimal design of univariate and multivariate control charts. Computer codes in the Julia programming language are also provided in the supplemental material.

Monte Carlo Sampling of Inverse Problems Based on a Squeeze-and-Excitation Convolutional Neural Network Applied to Ground-Penetrating Radar Crosshole Traveltime: A Numerical Simulation Study

Monte Carlo-based sampling methods (MCMC) can be used to solve inverse problems affecting ground penetrating radar (GPR) data. However, due to their high computational complexity, they have not been widely used in practical applications. This article uses neural network methods to replace the computationally complex forward problem of Monte Carlo methods. However, the neural network method is an approximation of the accurate formula method, and this may introduce model errors. In order to reduce the impact of model errors, in this study, we incorporate the Squeeze-and-Excitation (SE) attention mechanism into Convolutional Neural Networks (CNN) to further improve the accuracy of the network. Moreover, with the statistical advantages of the MCMC method, model errors can be explained during the inversion process, further reducing their impact. We apply the proposed method to solve the inversion problem of crosshole ground-penetrating radar travel time data. Compared with commonly used approximate forward models, the method proposed in this paper has better accuracy. The results of data experiments indicate that this method can effectively invert the velocity of underground media.

THE COMPARISON OF 2D DOSE PATIENT-SPECIFIC QUALITY ASSURANCE BETWEEN MONTE CARLO-CONVOLUTION AND MODIFIED CLARKSON INTEGRATION ALGORITHM

A sophisticated machine of radiotherapy treatment process follows the complexity of the quality assurance (QA) measurement. Non-measurement QA becomes one of the solutions to reduce the medical physicists’ workload. However, this method has not been clinically established. This study compared two non-measurement methods of patient-specific quality assurance (PSQA) to find the feasible algorithm for the adaptive radiotherapy process. Monte Carlo-based (MC) PSQA used a phase space file of the medical linear accelerator (Linac) to obtain the photon energy fluence and forward projected to the isoplane. In contrast, Modified Clarkson Integration-based (MCI) used a non-uniform fluence map in the isoplane. For the modulated intensity, we used a pair of the dynamic log files of the multileaf-collimator (MLC) and then employed them in the algorithms. The dose distributions of MC and MCI methods were compared to the treatment planning system (TPS) using gamma index analysis. We found that the gamma pass rates (GPR) for MC-TPS and MCI-TPS were 99.54% and 99.57%, respectively. Further, the dose distribution in the off-axis region for the MCI method showed lesser accuracy due to the higher secondary dose contribution. The linac log file information can be used and calculated into a 2D dose distribution using both MC and MCI methods, providing high-accuracy results.

Greedy separation algorithm finding community for a stochastic block model

We propose a greedy separation algorithm that finds the most fitted candidate among stochastic block models for a network, based on three known approaches. The first approach tests whether the network has one or more than two communities based on the distribution of the largest eigenvalue of the adjacency matrix. The second uses the algorithm to infer the classed label of each node in the network. The third approach ascertains the optimal number of clusters using an information criterion based on the Bayesian information criterion. The algorithm combined with the above approaches can find a suitable candidate from successively generated stochastic block models. However, in the second approach, the estimated labels heavily depend on the initial labels. The collection of the hub node and its neighbors is expected to construct one class with the same label. We find the hub nodes and enhance the initial labels by the PageRank method. We also conduct experiments with real data to evaluate the accuracy of the proposed method by comparing it with Markov chain Monte Carlo methods. The greedy separation algorithm with the PageRank method is preferable to the Monte Carlo-based methods.

An efficient procedure on the evaluation of the probability distribution function of the cumulative seismic loss

The existing loss assessment frameworks in the literature are mainly limited to the evaluation of the expected lifetime seismic losses, and do not assess the probability distribution of the long-term losses. The long-term loss is defined as the cumulative seismic loss of the structures over the service life. However, the probability distribution of the long-term losses can also be informative for decision-making under uncertain seismic consequences. This paper proposes an efficient procedure to calculate the probability distribution of the long-term losses. The procedure is based on the discretization of continuous random variables and performing two sets of recursive convolutions. The proposed method is compatible with PEER loss assessment framework, such that it can be readily implemented in engineering practice. The main contribution of the developed method is its simplicity compared to the existing analytical procedures and its computational efficiency compared to the Monte Carlo-based simulation methods. An illustrative example is provided to demonstrate the details of the procedure. A precise consistency is observed between the results of the proposed procedure and the scenario-based sampling method as the benchmark. The results show that among the common distribution functions, the log-logistic distribution has a better relative fit with the probability distribution of lifetime repair cost. It the studied cases, the difference between the median and expected lifetime loss turn out to be significant. Therefore, when dealing with similar scenarios, relying on the expected lifetime loss calculated using the conventional methods may result in an uneconomical design.

Distributed Stochastic Model Predictive Control With Taguchi’s Robustness for Vehicle Platooning

Vehicle platooning for highway driving has many benefits, such as lowering fuel consumption, improving traffic safety, and reducing traffic congestion. However, its performance could be undermined due to uncertainty. This work proposes a new control method that combines distributed stochastic model predictive control with Taguchi’s robustness (TR-DSMPC) for vehicle platooning. The proposed method inherits the advantages of both Taguchi’s robustness (maximizing the mean performance and minimizing the performance variation due to uncertainty) and stochastic model predictive control (ensuring a specific reliability level). Taguchi’s robustness is achieved by introducing a variation term in the control objective to bring a trade-off between mean performance and its variation. TR-DSMPC propagates uncertainty via an approximation method: First-Order Second Moment, which is far more efficient than Monte Carlo-based methods. The uncertainty is considered from two perspectives, time-independent uncertainty by random variables and time-dependent uncertainty by stochastic processes. We compare the proposed method with two other MPC-based methods in terms of safety (spacing error) and efficiency (relative velocity). The results indicate that our proposed method can effectively reduce the performance variation and maintain the mean performance.

Statistic selection and MCMC for differentially private Bayesian estimation

This paper concerns differentially private Bayesian estimation of the parameters of a population distribution, when a noisy statistic of a sample from that population is shared to provide differential privacy. This work mainly addresses two problems. (1) What statistics of the sample should be shared privately? For this question, we promote using the Fisher information. We find out that the statistic that is most informative in a non-privacy setting may not be the optimal choice under the privacy restrictions. We provide several examples to support that point. We consider several types of data sharing settings and propose several Monte Carlo-based numerical estimation methods for calculating the Fisher information for those settings. The second question concerns inference: (2) Based on the shared statistics, how could we perform effective Bayesian inference? We propose several Markov chain Monte Carlo (MCMC) algorithms for sampling from the posterior distribution of the parameter given the noisy statistic. The proposed MCMC algorithms can be preferred over one another depending on the problem. For example, when the shared statistic is additive and added Gaussian noise, a simple Metropolis-Hasting algorithm that utilises the central limit theorem is a decent choice. We propose more advanced MCMC algorithms for several other cases of practical relevance. Our numerical examples involve comparing several candidate statistics to be shared privately. For each statistic, we perform Bayesian estimation based on the posterior distribution conditional on the privatised version of that statistic. We demonstrate that the relative performance of a statistic, in terms of the mean squared error of the Bayesian estimator based on the corresponding privatised statistic, is adequately predicted by the Fisher information of the privatised statistic.

Quantifying performance of passive systems in an integrated small modular reactor under uncertainties using multilevel flow modelling and stochastic collocation method

Many newly designed advanced nuclear reactors deploy several passive systems to increase the inherent safety. However, uncertainties often exist in these new designs because of lacking sufficient operation experience and experimental data, which leads to the possible deviation on the performance of systems from their prospective values and increases the risk of the plant. Thus, methods should be developed to quantify the influence of uncertainties on the performance of the passive systems properly to improve our understanding of these new designs and provide support for decision-making.Traditionally, the system performance analysis under uncertainties is performed by Monte-Carlo based methods (MCS), which is extremely time-consuming. Thus, to circumvent the limitation of traditional Monte-Carlo based method, a novel hybrid method that combines multilevel flow modelling and stochastic collocation (MFM-SC) is proposed in this paper. The MFM-SC method utilizes the MFM to model the functions of collaborative working passive systems and inference the relevant uncertain parameters according to causal reasoning rules. Then, the stochastic collocation method which combines the strength of Monte Carlo methods and stochastic Galerkin methods is used to create an orthogonal polynomial based surrogate model for the thermal hydraulic model. With the efficiency of the surrogate model in computation, the quantification of the performance of passive systems under uncertainties which requires thousands of repetitive computations can be easily conducted.An example test case is also presented in this paper to analyze the performance of core makeup tanks and the steam drum in an integrated small modular reactor during a station blackout accident under multiple uncertainties using MFM-SC method. The statistical results of the deviation range of the systems’ performance under uncertainties are given according to thousands of experiments using a surrogate model, and the response surfaces are also calculated based on the MFM-SC method to reveal the relation between the uncertain inputs and the systems output separately. The result shows the significant improvement in the computational efficiency of the suggested MFM-SC method, and it can provide essential information for decision-making.

Stochastic simulation with informed rotations of Gaussian quadratures

Given the fast growth of available computational capacities and the increasing complexity of simulation models addressing agro-environmental issues, uncertainty analysis using stochastic techniques has become a standard modeling practice. However, conventional uncertainty/sensitivity analysis methods are either computationally demanding (Monte Carlo-based methods) or produce results with varying quality (Gaussian quadratures). In this article, we present a computationally inexpensive and reliable uncertainty analysis method for simulation models called informed rotations of Gaussian quadratures (IRGQ). We also provide an R script that generates IRGQ points based on the required input data. The results demonstrate that this method is able to produce approximations that are close to the estimated benchmarks at low computational costs. The method is tested in three different simulation models using different input data in order to demonstrate the independence of the proposed method on specific model types and data structures. This is a methodological paper for practitioners rather than theorists.

Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation

Pricing of financial derivatives, in particular early exercisable options such as Bermudan options, is an important but heavy numerical task in financial institutions, and its speed-up will provide a large business impact. Recently, applications of quantum computing to financial problems have been started to be investigated. In this paper, we first propose a quantum algorithm for Bermudan option pricing. This method performs the approximation of the continuation value, which is a crucial part of Bermudan option pricing, by Chebyshev interpolation, using the values at interpolation nodes estimated by quantum amplitude estimation. In this method, the number of calls to the oracle to generate underlying asset price paths scales as widetilde{O}(epsilon ^{-1}), where ϵ is the error tolerance of the option price. This means the quadratic speed-up compared with classical Monte Carlo-based methods such as least-squares Monte Carlo, in which the oracle call number is widetilde{O}(epsilon ^{-2}).

A generalized computationally efficient copula-polynomial chaos framework for probabilistic power flow considering nonlinear correlations of PV injections

This paper develops a general computationally efficient copula-polynomial chaos expansion (copula-PCE) framework for power system probabilistic power flow that can handle both linear and nonlinear correlations of uncertain power injections, such as wind and PVs. A data-driven copula statistical model is used to capture both nonlinear and linear correlations of uncertain power injections. This allows us to resort to the Rosenblatt transformation to transform correlated variables into independent ones while preserving the dependence structure. It paves the way for leveraging PCE for surrogate modeling and uncertainty quantification of power flow results. To further improve the computational efficiency of the proposed method without sacrificing accuracy, two strategies are developed and unified together. Specifically, the branch flow sensitivity analysis (BFSA) is used to select an appropriate number of outputs that need PCE surrogate modeling while the analysis of variance (ANOVA) scheme enables us to reduce the number of basis functions and coefficients evaluations for each PCE surrogate. Simulations carried out on the IEEE 57-bus and 118-bus and PEGASE 1354-bus systems show that the proposed framework can obtain better accuracy and computational efficiency than the existing PCE-based methods with linear correlation assumption and other Monte Carlo-based methods. The sensitivities to different copula types and parameters are also investigated.

Risk Assessment of Water Treatment Plants Using Fuzzy Fault Tree Analysis and Monte Carlo Simulation

Urban water supply, transmission, and distribution systems are considered as the basic infrastructures of each community. Deficiency of water resources in system leads to lack of some subscribers’ access to safe water at any time. The water treatment units have a high potential for massive crisis and importance of sustainable performance in water treatment plants, due to their special role in high quantity and quality water supply even in critical situations, requires a comprehensive approach in assessing and increasing the reliability. A comprehensive approach in assessing the performance of water treatment plants and reducing their vulnerability can lead to cost reduction of inappropriate performance in critical situations and also a focus on the most important vulnerable parts of water treatment plants to decrease their vulnerability, as well as cost-effective budgeting in recovery programs. In this research, due to the uncertainties in the input information, two methods of fuzzy fault tree analysis (FFTA) and fault tree analysis (FTA) based on Monte Carlo simulation were evaluated in Jalaliyeh water treatment plant in Tehran, and results of both approaches were compared in terms of uncertainty analysis. The results indicate the low to moderate failure risk in the treatment plant in both approaches and a small difference between results in both cases. Failure probability of top event for FFTA and Monte Carlo-based FTA methods is 0.194 and 0.27, respectively. In the basic event rating, it was found that threats such as inappropriate reservoir design, power equipment failure, transfer pipe failure, and inappropriate maintenance of the pumps play the major role in treatment plant failure.

Monte Carlo-Based Reliability Estimation Methods for Power Devices in Power Electronics Systems

Monte Carlo simulation has been widely used for reliability assessment of power electronic systems. In this approach, multiple simulations are carried out during the lifetime estimation of the components in power converter, e.g., power devices, where the parameter variations are considered. In the previous mission-profile based reliability assessment methods, the dynamic thermal stress profiles are usually converted into a set of static parameters. However, this simplification may introduce a certain uncertainty during the reliability assessment, since the static parameters may not be able to accurately represent the thermal stress under highly dynamic conditions. Moreover, the previous research did not take into account the correlation between the method of introducing the parameter variation and the required number of Monte Carlo simulations. This can significantly affect both the accuracy and computation burden of the Monte Carlo simulation. To address this issue, an in-depth analysis of Monte Carlo simulation applied to reliability assessment of power devices in power electronic systems is provided in this paper. Two additional Monte Carlo simulation approaches based on semi-dynamic and dynamic parameters are proposed, and their reliability evaluation results are compared with the traditional static parameter method. It is demonstrated in a case study of photovoltaic (PV) inverter application that the reliability of power converter can be overestimated up to 30% when using the static parameters.

Multiple rotations of Gaussian quadratures: An efficient method for uncertainty analyses in large-scale simulation models

Concerns regarding the impact of climate change, food price volatility, and weather uncertainty have motivated users of simulation models to consider uncertainty in their simulations. One way to do this is to integrate uncertainty components in the model equations, thus turning the model into a problem of numerical integration. Most of these problems do not have analytical solutions, and researchers, therefore, apply numerical approximation methods. This article presents a novel approach to conducting an uncertainty analysis as an alternative to the computationally burdensome Monte Carlo-based (MC) methods. The developed method is based on the degree three Gaussian quadrature (GQ) formulae and is tested using three large-scale simulation models. While the standard single GQ method often produces low-quality approximations, the results of this study demonstrate that the proposed approach reduces the approximation errors by a factor of nine using only 3.4% of the computational effort required by the MC-based methods in the most computationally demanding model.

The interacting nature of dwarf galaxies hosting superluminous supernovae

Context. Type I superluminous supernovae (SLSNe I) are rare, powerful explosions whose mechanism and progenitors remain elusive. Several studies have shown a preference for SLSNe I to occur in low-metallicity, actively star-forming dwarf galaxies. Aims. We investigate whether the host galaxies of SLSNe I show increased evidence for interaction. Galaxy interaction can trigger star formation and provide favourable conditions for these exceptional explosions to take place. Methods. Based on SLSN host galaxy images obtained with the Hubble Space Telescope (HST), we narrowed down a sample of 42 images obtained in the rest-frame ultraviolet over the redshift range between 0 < z < 2. The number of host galaxy companions was measured by counting the number of objects detected within a given projected radius from the host. As a comparison, we used two different Monte Carlo-based methods to estimate the expected average number of companion objects in the same HST images, as well as a sample of 32 dwarf galaxies that have hosted long gamma-ray bursts (GRBs). Results. About 50% of SLSN I host galaxies have at least one major companion (within a flux ratio of 1:4) within 5 kpc. The average number of major companions per SLSN I host galaxy is 0.70−0.14+0.19. Our two Monte Carlo comparison methods yield a lower number of companions for random objects of similar brightness in the same image or for the SLSN host after randomly redistributing the sources in the same image. The Anderson-Darling test shows that this difference is statistically significant (p-value < 10−3) independent of the redshift range. The same is true for the projected distance distribution of the companions. The SLSN I hosts are, thus, found in areas of their images, where the object number density is greater than average. The SLSN I hosts have more companions than GRB hosts (0.44−0.13+0.25 companions per host distributed over 25% of the hosts) but the difference is not statistically significant. The difference between their separations is, however, marginally significant with SLSN companions being closer, on average, than those of GRBs. Conclusions. The dwarf galaxies hosting SLSNe I are often part of interacting systems. This suggests that SLSNe I progenitors are formed after a recent burst of star formation. Low metallicity alone cannot explain this tendency.

Absorbed dose distributions from beta-decaying radionuclides: Experimental validation of Monte Carlo tools for radiopharmaceutical dosimetry.

This study aims to experimentally validate the Monte Carlo generated absorbed doses from the beta particles emitted by 90 Y and 177 Lu using radiochromic EBT3 film-based dosimetry. Line sources of 90 Y and 177 Lu were inserted longitudinally through blocks of low-density polyethylene and tissue-equivalent slabs of cortical bone and lung equivalent plastics. Radiochromic film (Gafchromic EBT3) was laser cut to accommodate orthogonal line sources of radioactivity, and the film was sandwiched intimately between the rectangular blocks to achieve charged particle equilibrium. Line sources consisted of plastic capillary tube of length (13±0.1)cm, with 0.42-mm inner diameter and a wall thickness of 0.21mm. 90 Y line sources were prepared from a solution of dissolved 90 Y resin microspheres. 177 Lu line sources were prepared from an aliquot of 177 Lu-DOTATATE. Film exposures were conducted for durations ranging from 10min to 38h. Radiochromic film calibration was performed by irradiation with 6-MV-bremsstrahlung x rays from a calibrated linear accelerator, in accordance with literature recommendations. Experimental geometries were precisely simulated within the GATE Monte Carlo toolkit, which has previously been used for the generation of dose point kernels. The mean percentage difference between measured and simulated absorbed doses were 5.04% and 7.21% for 90 Y and 177 Lu beta absorbed dose in the range of (0.1-10)Gy. Additionally, 1D gamma analysis using a local 10%/1mm gamma criterion was performed to compare the absorbed dose distributions. The percentage of measurement points passing the gamma criterion, averaged over all tests, was 93.5%. We report the experimental validation of Monte Carlo derived beta absorbed dose distributions for 90 Y and 177 Lu, solidifying the validity of using Monte Carlo-based methods for estimating absorbed dose from beta emitters. Overall, excellent agreement was observed between the experimental beta absorbed doses in the linear region of the radiochromic film and the GATE Monte Carlo simulations demonstrating that radiochromic film dosimetry has sufficient sensitivity and spatial resolution to be used as a tool for measuring beta decay absorbed dose distributions.