Variability Estimation in a Nonlinear Crack Growth Simulation Model With Controlled Parameters Using Designed Experiments Testing

Abstract. Uncertainties in an output of interest that depends on the solution of a complex system (e.g., of partial differential equations with random inputs) are often, if not nearly ubiquitously, determined in practice using Monte Carlo (MC) estimation. While simple to implement, MC estimation fails to provide reliable information about statistical quantities (such as the expected value of the output of interest) in application settings such as climate modeling, for which obtaining a single realization of the output of interest is a costly endeavor. Specifically, the dilemma encountered is that many samples of the output of interest have to be collected in order to obtain an MC estimator that has sufficient accuracy – so many, in fact, that the available computational budget is not large enough to effect the number of samples needed. To circumvent this dilemma, we consider using multifidelity Monte Carlo (MFMC) estimation which leverages the use of less costly and less accurate surrogate models (such as coarser grids, reduced-order models, simplified physics, and/or interpolants) to achieve, for the same computational budget, higher accuracy compared to that obtained by an MC estimator – or, looking at it another way, an MFMC estimator obtains the same accuracy as the MC estimator at lower computational cost. The key to the efficacy of MFMC estimation is the fact that most of the required computational budget is loaded onto the less costly surrogate models so that very few samples are taken of the more expensive model of interest. We first provide a more detailed discussion about the need to consider an alternative to MC estimation for uncertainty quantification. Subsequently, we present a review, in an abstract setting, of the MFMC approach along with its application to three climate-related benchmark problems as a proof-of-concept exercise.

Thermal-pulse method is a powerful tool for measuring space charge distributions in polymer films. The data analysis for thermal-pulse method involves the Fredholm integral equation of the first kind, which requires an appropriate numerical procedure to obtain a solution. Various numerical techniques, including scale transformation and regulation method, are proposed. Of those numerical methods, the scale transformation (ST) is the simplest and the most widely used method. However, it presents a high spatial resolution only near the sample surface. Monte Carlo (MC) method is one of the recently proposed ways to solve the equation numerically and has been successfully applied to the analysis of laser intensity modulation method data, which also involves the Fredholm integral equation of the first kind. In this paper we attempt to analyze thermal-pulse data in frequency domain with the MC method and discuss its effectiveness based on some numerical simulations. The simulation results indicate that the electric field profiles can be effectively extracted by the MC method. The computed profiles by the MC method consist well with the supposed distributions in the entire thickness of the sample, while the profiles reconstructed by the ST method fit very well to the supposed one at the vicinity of the target surface and distort sharply along the direction of the thermal pulse propagation in the sample bulk. On the other hand, the oscillations in the computed results by the MC method could deteriorate its accuracy in this study. The influence of noise level on the analysis based on the MC method is also tested by the use of the simulated data. The results show that the computed profiles would become more fluctuant as the noise level increases. This problem can be solved by selecting a larger value of tolerance during the singular value decomposition procedure. Thus, the value of tolerance is considered to be one of the key parameters in this algorithm, which is actually hard to determine. Additionally, the experimental data obtained from a polypropylene film under applied electric field are analyzed to illustrate the feasibility of MC method to be applied to the thermal-pulse experimental data. The results also show that the spatial accuracy by the MC method in the entire sample thickness is higher than by the ST method, which verifies that the MC method is more suitable for detecting the electric field distribution in the deep bulk of the sample. Owing to noise and error, the accuracy of MC calculation depends on the chosen tolerance value, which is now considered to be an obstacle in applying this method to the practical thermal-pulse measurement.

Uncertainties in an output of interest that depends on the solution of a complex system (e.g., of partial differential equations with random inputs) are often, if not nearly ubiquitously, determined in practice using Monte Carlo (MC) estimation. While simple to implement, MC estimation fails to provide reliable information about statistical quantities (such as the expected value of the output of interest) in application settings such as climate modeling for which obtaining a single realization of the output of interest is a costly endeavor. Specifically, the dilemma encountered is that many samples of the output of interest have to be collected in order to obtain an MC estimator having sufficient accuracy; so many, in fact, that the available computational budget is not large enough to effect the number of samples needed. To circumvent this dilemma, we consider using multifidelity Monte Carlo (MFMC) estimation which leverages the use of less costly and less accurate surrogate models (such as coarser grids, reduced-order models, simplified physics, interpolants, etc.) to achieve, for the same computational budget, higher accuracy compared to that obtained by an MC estimator or, looking at it another way, an MFMC estimator obtains the same accuracy as the MC estimator at lower computational cost. The key to the efficacy of MFMC estimation is the fact that most of the required computational budget is loaded onto the less costly surrogate models, so that very few samples are taken of the more expensive model of interest. We first provide a more detailed discussion about the need to consider an alternate to MC estimation for uncertainty quantification. Subsequently, we present a review, in an abstract setting, of the MFMC approach along with its application to three climate-related benchmark problems as a proof-of-concept exercise.

It is very time-consuming to obtain a high-precision Monte Carlo (MC) estimate of the fuel temperature reactivity coefficient (FTC) through direct subtraction of two reactivity values from MC calculations at two different fuel temperatures. As an alternative to the direct subtraction MC estimate of the FTC, this paper presents a new method based on the adjoint-weighted correlated sampling technique. The new method translates the change in fuel temperature as the corresponding changes in both the microscopic cross sections and the transfer probabilities in scattering kernels described by the free gas model. The effectiveness of the new method is examined through continuous-energy MC neutronics calculations for pressurized water reactor pin cell and CANDU pressurized heavy water reactor lattice problems. The isotope-wise and reaction-type–wise contributions to the FTCs in the two problems are examined for two free gas models: the constant-cross-section and the resonance-cross-section models. It is demonstrated that the new MC method can predict the reactivity change due to fuel temperature variation as accurately as the conventional, more time-consuming direct subtraction MC method.

Background Small megavoltage photon fields are increasingly used in modern radiotherapy techniques such as stereotactic radiotherapy. Therefore, it is important to study the reliability of dosimetry in the small-field conditions. The IBA Razor Nano Chamber (Nano chamber) ionization chamber is particularly intended for small-field measurements. In this work, properties of the Nano chamber were studied with both measurements and Monte Carlo (MC) simulations. Material and methods The measurements and MC simulations were performed with 6 MV, 6 MV FFF and 10 MV FFF photon beams from the Varian TrueBeam linear accelerator. The source-to-surface distance was fixed at 100 cm. The measurements and MC simulations included profiles, percentage depth doses (PDD), and output factors (OF) in square jaw-collimated fields. The MC simulations were performed with the EGSnrc software system in a large water phantom. Results The measured profiles and PDDs obtained with the Nano chamber were compared against IBA Razor Diode, PTW microDiamond and the PTW Semiflex ionization chamber. These results indicate that the Nano chamber is a high-resolution detector and thus suitable for small field profile measurements down to field sizes 2 × 2 cm2 and appropriate for the PDD measurements. The field output correction factors and field OFs were determined according to TRS-483 protocol In the 6 MV FF and FFF beams, the determined correction factors were within 1.2% for the field sizes of 1 × 1 cm2–3 × 3 cm2 and the experimental and MC defined field output factors showed good agreement. Conclusion The Nano chamber with its small cavity volume is a potential detector for the small-field dosimetry. In this study, the properties of this detector were characterized with measurements and MC simulations. The determined correction factors are novel results for the NC in the TrueBeam fields.

In these days, Monte Carlo (MC) simulation is a method that can calculate the radiation dose that occurs in an environment in the most accurate way. The correct measurement of the dose occurring on the patient’s surface is of great importance to estimate the reactions that may occur on the patient’s skin. This importance encouraged us to do this study. The aim of this study is to determine buildup region and surface doses using MC simulation and to compare them with results of the parallel plane ion chamber and Treatment Planning System (TPS) measurements for 6-MV photon beams. Surface doses normalized to the maximum dose for the parallel plane ion chamber, MC simulation, fast photon (FP) algorithm, and collapsed cone convolution superposition (CC) algorithm are 13.6%, 30.28%, 0%, and 27.33%, respectively. The CC algorithm and parallel plane ion chamber measurements are compatible with MC simulation but the FP algorithm has calculated the dose less to a depth of 0.8 cm. Measuring the surface dose and the doses in the buildup region is of great importance in terms of accurately predicting the complications that may occur in the patient’s skin and taking precautions early. Using some methods and correction factors, the surface dose and the doses that may occur in the buildup region can be accurately calculated. It is recommended not to use the FP algorithm for stereotactic body radiation therapy and intensity-modulated radiation therapy treatments, as it cannot calculate doses correctly in the buildup region and surface.

A novel cone-beam CT (CBCT) system has been developed with promising capabilities for musculoskeletal imaging (e.g., weight-bearing extremities and combined radiographic / volumetric imaging). The prototype system demonstrates diagnostic-quality imaging performance, while the compact geometry and short scan orbit raise new considerations for scatter management and dose characterization that challenge conventional methods. The compact geometry leads to elevated, heterogeneous x-ray scatter distributions - even for small anatomical sites (e.g., knee or wrist), and the short scan orbit results in a non-uniform dose distribution. These complex dose and scatter distributions were investigated via experimental measurements and GPU-accelerated Monte Carlo (MC) simulation. The combination provided a powerful basis for characterizing dose distributions in patient-specific anatomy, investigating the benefits of an antiscatter grid, and examining distinct contributions of coherent and incoherent scatter in artifact correction. Measurements with a 16 cm CTDI phantom show that the dose from the short-scan orbit (0.09 mGy/mAs at isocenter) varies from 0.16 to 0.05 mGy/mAs at various locations on the periphery (all obtained at 80 kVp). MC estimation agreed with dose measurements within 10-15%. Dose distribution in patient-specific anatomy was computed with MC, confirming such heterogeneity and highlighting the elevated energy deposition in bone (factor of ~5-10) compared to soft-tissue. Scatter-to-primary ratio (SPR) up to ~1.5-2 was evident in some regions of the knee. A 10:1 antiscatter grid was found earlier to result in significant improvement in soft-tissue imaging performance without increase in dose. The results of MC simulations elucidated the mechanism behind scatter reduction in the presence of a grid. A ~3-fold reduction in average SPR was found in the MC simulations; however, a linear grid was found to impart additional heterogeneity in the scatter distribution, mainly due to the increase in the contribution of coherent scatter with increased spatial variation. Scatter correction using MC-generated scatter distributions demonstrated significant improvement in cupping and streaks. Physical experimentation combined with GPU-accelerated MC simulation provided a sophisticated, yet practical approach in identifying low-dose acquisition techniques, optimizing scatter correction methods, and evaluating patientspecific dose.

As energy demand increases and global warming progresses, expectations for fuel cells, which generate electricity through chemical reactions between hydrogen and oxygen, are rising. There are various types of fuel cells, which are classified according to the electrolyte. Polymer electrolyte fuel cells (PEFCs) are expected to be used in stationary power sources for home use and fuel cell vehicles because of their low operating temperature, short start-up time, and ease of miniaturization. The cost has been an issue in the widespread use of fuel cells. The amount of platinum used in the catalyst layer (CL) must be reduced to reduce costs. However, it increases the current density per unit area of platinum. There are three factors that contribute to voltage drop: ohmic polarization, activation polarization, and diffusion polarization. At high current densities, the voltage drop due to diffusion polarization is dominant. The main cause of diffusion polarization is the transport resistance of oxygen molecules in the CL, and improving the oxygen transport properties in the CL will lead to lower cost PEFCs. In this study, we analyzed the transport properties of oxygen molecules in the CLs of PEFCs using the Monte Carlo (MC) method. The effect of oxygen scattering on the overall oxygen transport properties was investigated by considering the behavior on the surface of the ionomer film (surface diffusion) into the MC method, which simulates the overall transport in the PEFC CL. The objective of this study is to reproduce accurate oxygen transport by comparing the results of MC and molecular dynamics (MD) methods. First, we used the MC method to analyze oxygen scattering phenomena. In this study, we used the 3D structure data of CL in the MC simulations. The three-dimensional structure of the sample was reconstructed from sequential slice images. Molecules were assumed to reflect specularly on the boundaries of the simulation domain. Oxygen molecules were randomly placed in the simulation system. The time step was set at 5 ps and the simulation time was set at 4 μs. In order to examine the effect of surface diffusion on the transport properties of oxygen molecules, the simulation was performed using various conditions of surface diffusion. We calculated the effective diffusion coefficient of oxygen in the CL as the transport property of oxygen molecules in the MC method. As the surface diffusion coefficient increases, the effective diffusion coefficient of oxygen increases. When the surface diffusion coefficient is small, the effective diffusion coefficient of oxygen increases due to a decrease in the time constant, which determines the probability distribution of the surface residence time. However, when the surface diffusion coefficient is large, a different trend emerges. As the time constant increases, the effective diffusion coefficient has a peak value and then decreases. Next, to verify the model of surface diffusion in the MC method, we analyzed the scattering phenomenon of oxygen molecules in a simulation system that simulates the pores in the CL using the MD method. We modeled the nano pores in the CL as a slit between the ionomer walls in the MD method. Nafion with the equivalent weight (EW) of 1146 was used as polymer model. The number of Nafion chains is set at 4, and the water content λ is set at 7. Three-dimensional periodic boundary conditions were applied to the simulation domain. After the system was equilibrated, 1 ns of NVT simulation was performed for production. The temperature was set at 297 K. The behavior of oxygen molecules impacting on the ionomer wall was analyzed. We analyzed the behavior of oxygen molecules colliding with the ionomer membrane using the MD method. The average residence time of surface diffusion on the ionomer thin film was of the order of 10-10s. When the calculation results were compared with the MC results, it was found that the average residence time obtained by the MD method was the value where the surface diffusion coefficient affects much on the effective diffusion coefficient. Comparing the results of the MD method with the surface diffusion model used in the MC method, there are some discrepancies between the results and the model. For example, the MC method used a model in which oxygen molecules reflect diffusively on the ionomer surface. However, in the MD method, it was found that oxygen molecules tend to reflect in the direction of travel (Fig. 1). Therefore, a more accurate model of the MC method needs to be developed in the future. Figure 1

Purpose: To create a Monte Carlo (MC) model of an ocular proton therapy device, or ocular nozzle, that can predict relative doses to within 3% or 0.5 mm, D/MU values, and be used to develop and commission an analytical algorithm to predict doses and D/MU values to within similar criteria. Methods and Materials: Depth dose profiles and D/MU values were measured in eight ocular proton therapy fields including four unmodulated and four modulated beams using a parallel‐plane ionization chamber in water. Crossfield profiles were measured at three or more depths for each of the modulated beams using film. A MC model of the ocular nozzle was constructed using the MCNPX radiation transport code. After benchmark tests against the measurements, the MC model was used to provide dosimetric data needed to develop a broad‐beam algorithm capable of predicting relative and absolute doses and to test the algorithm's predictions more comprehensively than could be afforded with measured data. Specifically, MC and analytical predictions were calculated and compared in two‐dimensions in a water phantom and in an anthropomorphic model of the eye. Results: Relative depth dose profiles between MC and measurements agreed to within 3% or 0.5 mm. MC predicted D/MU values agreed to within 1% of the measured values. Similarly, the analytical algorithm accurately predicted dose distributions where the predictions were generally within 3% or 0.5 mm of measured and simulated values at water depth > 6 mm. At shallower depths, the analytical model underestimated the dose by approximately 3% to 6%, which MC simulations revealed was due to edge‐scattered protons from the nozzle's collimators. Conclusion: The MC model accurately predicted dose distributions and D/MU values in ocular proton therapy beams. Furthermore, the present work demonstrates the value of the MC method for developing and testing contemporary analytical dose algorithms.

We combine molecular dynamics (MD) and Monte Carlo (MC) simulations to study the properties of gas molecules confined between two hard walls of a microchannel or nanochannel. The coupling between MD and MC simulations is introduced by performing MD near the boundaries for accuracy and MC in the bulk because of the low computational cost. We characterize the influence of different densities and molecule sizes on the equilibrium properties of the gas in the microchannel. The effect of the particle size on the simulation results is very small in the case of a dilute gas and increases with the density. The hybrid MD-MC simulation method is validated by comparing the results for density and temperature profiles with those of pure MD and pure MC simulations. These results compare well for pure MD and pure MC, as well as hybrid MD-MC, both in the bulk and near the boundaries, when hard-sphere interactions are used. When Lennard-Jones potentials are used to accurately model the interactions between the gas and wall molecules instead, the results of pure MD simulations differ significantly from the pure MC simulations near the boundaries, but the results of the hybrid method compare well with the pure MD results near the wall, and with the pure MC and pure MD results in the middle of the channel. The hybrid method also very accurately simulates the interface between the MD and MC simulation domains. Comparisons between MD, MC, and hybrid MD-MC computational costs are outlined. The speedup when using 50% of the domain for MD simulations and 50% for MC simulations is very small compared to pure MD simulations times, but this speedup increases drastically for more realistic situations where the region near the wall is small compared to the bulk region.

Purpose: For microdosimetric calculation event-by-event Monte Carlo (MC) simulation is considered the most accurate, but it is very time-consuming. In this work we present an event-by-event MC simulation of low energy electron and proton for accelerated microdosimetric MC simulations on a graphic processing unit (GPU). Methods: The MC simulation of particle was implemented in C and executed on a multi-core CPU, and a commercially available general purpose GPU using the compute unified device architecture (CUDA). Additionally, a hybrid implementation scheme was realized by employing OpenMP and CUDA so that both GPU and multi-core CPU were utilized simultaneously. The two implementation schemes have been tested and compared with the sequential single threaded MC simulation on the CPU. A calculation time comparison was established on the speed-up for a set of benchmarking cases of electron and proton. Results: Dosimetric results were obtained with both the parallel and serial MC codes. A GPU over CPU speed-up of 67.2 and 19.2 times was achieved for 300 eV and 2 keV electron tracks respectively. For proton tracks, the GPU-based code was approximately 5 times faster than the CPU-single-thread code. By incorporating a multi-core CPU and running the MC code simultaneously on the GPU and CPU, an increase of 2%-7% and 20% in the speedup was noticed for electrons and protons tracks respectively. A good dosimetric agreement between the CPU- and GPU-based MC methods was observed for both electrons and proton. Conclusions: A GPU-based MC method for microdosimetric calculations of low energy electron and proton has been presented. The results indicate the capability of our GPU-based implementation for accelerated MC simulations of both electron and proton without loss of accuracy. Lastly, the potential of a hybrid approach by utilizing simultaneously a GPU and a multi-core CPU for further acceleration of MC microdosimetric calculations has been demonstrated.

Objective. Monte Carlo (MC) simulations are the benchmark for accurate radiotherapy dose calculations, notably in patient-specific high dose rate brachytherapy (HDR BT), in cases where considering tissue heterogeneities is critical. However, the lengthy computational time limits the practical application of MC simulations. Prior research used deep learning (DL) for dose prediction as an alternative to MC simulations. While accurate dose predictions akin to MC were attained, graphics processing unit limitations constrained these predictions to large voxels of 3 mm × 3 mm × 3 mm. This study aimed to enable dose predictions as accurate as MC simulations in 1 mm × 1 mm × 1 mm voxels within a clinically acceptable timeframe. Approach. Computed tomography scans of 98 breast cancer patients treated with Iridium-192-based HDR BT were used: 70 for training, 14 for validation, and 14 for testing. A new cropping strategy based on the distance to the seed was devised to reduce the volume size, enabling efficient training of 3D DL models using 1 mm × 1 mm × 1 mm dose grids. Additionally, novel DL architecture with layer-level fusion were proposed to predict MC simulated dose to medium-in-medium (D m,m ). These architectures fuse information from TG-43 dose to water-in-water (D w,w ) with patient tissue composition at the layer-level. Different inputs describing patient body composition were investigated. Main results. The proposed approach demonstrated state-of-the-art performance, on par with the MC D m,m maps, but 300 times faster. The mean absolute percent error for dosimetric indices between the MC and DL-predicted complete treatment plans was 0.17% ± 0.15% for the planning target volume V 100, 0.30% ± 0.32% for the skin D 2cc , 0.82% ± 0.79% for the lung D 2cc , 0.34% ± 0.29% for the chest wall D 2cc and 1.08% ± 0.98% for the heart D 2cc . Significance. Unlike the time-consuming MC simulations, the proposed novel strategy efficiently converts TG-43 D w,w maps into precise D m,m maps at high resolution, enabling clinical integration.

The Monte Carlo (MC) method has been employed through the years as a gold-standard method to reconstruct numerical solutions of the RTE. The MC method provides a physical simulation of light propagation; i.e., photon trajectories are generated by using the same statistical rules that govern propagation in random media. In practice, the MC method estimates the expected characteristics of the photon population as statistical averages over a large number of case histories of photon life that are simulated by a computer. These average statistical characteristics of the photon population are exactly what the RTE describes, and, intuitively, this is why the MC method allows one to estimate numerically solutions of the RTE. In fact, comparisons of MC-based results with exact analytical solutions of the RTE demonstrate that for the limit of an infinite number of photons used, the simulated data converge rigorously to the analytical results. The existing literature shows that the MC method can be implemented by using different approaches. In this chapter, the approaches usually employed are briefly reviewed by illustrating their main peculiarities. These approaches are also the basis of more complex MC procedures, such as perturbation MC methods, voxel-based MC methods, mesh-based MC methods, or GPU-accelerated MC codes.

Purpose: To investigate dose distributions in the prostate and surrounding tissues for I–125 brachytherapy using Monte Carlo (MC) calculations with patient‐specific phantoms derived from CT images. We explore the effectiveness of different techniques to mitigate streaking artifacts due to brachytherapy sources in post‐implant CT images. Methods: Nine patients (45–87 sources) are considered. Streaking artifacts in post‐implant CT images are mitigated using various metallic artifact reduction (MAR) techniques: raw sinogram, fan beam virtual sinogram and 3D median filter. Segmented structures (CTV, rectum and bladder) guide the assignment of tissues (air, muscle, prostate, calcification, average tissue and bone) to develop patient‐specific MC phantoms. The EGSnrc user‐code Brachydose and GEANT4 user‐code ALGEBRA are employed for MC dose calculations using patient‐specific phantoms derived from CT images. Dose distributions generated with patient models derived using different MAR techniques and different calculation methods (MC, TG‐43) are compared directly, and using recommended dose metrics. Results: Application of each MAR technique to patient CT data results in comparable mitigation of streaking artifacts within the treatment volume in CT images. MC calculations based on uncorrected CT data Result in high dose spikes (>200%) in the treatment volume and lower doses (40–70%) in surrounding tissues compared to MC calculations with corrected phantoms. Dose distributions within the prostate from MC simulations using phantoms generated with different MAR techniques are comparable to each other but differ from TG‐43 calculations with significant (10%+) local differences. Conclusion: Mitigation of streaking artifacts in CT images is necessary for patient‐specific MC dose calculations; however, dose distributions and clinical metrics in the target and OAR are insensitive to the particular MAR technique applied. Differences between MC and TG‐43 dose distributions within the target volume and surrounding organs for I‐125 prostate brachytherapy underline the importance of patient‐specific model‐based dose calculations for treatment planning and evaluation. NSERC, CCSRI, The Canada Research Chairs Program

Purpose: Investigate the heterogeneity effect due to air cavity and lead sheets in intro‐operative High‐Dose‐Rate (IOHDR) brachytherapy dose calculation using Monte Carlo (MC) simulation and GAFCHROMIC EBT film measurement. Method and Materials: The organs‐at‐risk (OARs) in IOHDR treatment are usually pushed away from applicators and covered by lead sheets. The dose estimations to OARs are affected by local heterogeneities like air gap and lead sheets and difficult to calculate by conventional treatment planning systems, where infinitely homogeneous water environment is assumed. This study used both MC simulation and Gafchromic film measurement to access the heterogeneity effect of dose calculations in a phantom setting. Nine 5×5cm 2 sloid water slabs, each 2mm thick, were put on a 30×30×20cm 3 solid water phantom. Gafchromic EBT films were place at 0, 4, 8 12 mm depth between the slabs. The Ir‐192HDR source was place 19 mm above the slabs in the air. 0–3 lead sheets, each 0.9mm thick, were placed directly on top of the slabs. MC Photo Transport code (PTRAN) was used to simulate doses at each film positions and compared with film measurements. Results: The MC simulation agrees with film measurements within 5% except data points directly under lead sheets where dose enhancement by secondary electrons was not simulated in PTRAN. The heterogeneity correction factor (HCF), defined as dose ratio of with and without lead sheets at surface of the slabs is 147%, 121% and 96%, for 1–3 lead sheets respectively. At 4–12mm depth in water slabs, the HCFs are approximately 81%, 65% and 53% for 1–3 lead sheets. Conclusion: Significant dose enhancement was found directly under lead sheets. HCF data obtained in this study can be used to estimate doses to OARs in IOHDR treatment. The MC simulation was proved to be useful to calculate doses to OARs at complex clinical settings.