Monte Carlo treatment planning for photon and electron beams

During the last few decades, accuracy in photon and electron radiotherapy has increased substantially. This is partly due to enhanced linear accelerator technology, providing more flexibility in field definition (e.g. the usage of computer-controlled dynamic multileaf collimators), which led to intensity modulated radiotherapy (IMRT). Important improvements have also been made in the treatment planning process, more specifically in the dose calculations. Originally, dose calculations relied heavily on analytic, semi-analytic and empirical algorithms. The more accurate convolution/superposition codes use pre-calculated Monte Carlo dose “kernels” partly accounting for tissue density heterogeneities. It is generally recognized that the Monte Carlo method is able to increase accuracy even further. Since the second half of the 1990s, several Monte Carlo dose engines for radiotherapy treatment planning have been introduced. To enable the use of a Monte Carlo treatment planning (MCTP) dose engine in clinical circumstances, approximations have been introduced to limit the calculation time. In this paper, the literature on MCTP is reviewed, focussing on patient modeling, approximations in linear accelerator modeling and variance reduction techniques. An overview of published comparisons between MC dose engines and conventional dose calculations is provided for phantom studies and clinical examples, evaluating the added value of MCTP in the clinic. An overview of existing Monte Carlo dose engines and commercial MCTP systems is presented and some specific issues concerning the commissioning of a MCTP system are discussed.     

Super-Monte Carlo: A photon/electron dose calculation algorithm for radiotherapy

Super-Monte Carlo (SMC) is a method of dose calculation for radiotherapy which combines both analytical calculations and Monte Carlo electron transport. Analytical calculations are used where possible, such as the determination of photon interaction density, to decrease computation time. A Monte Carlo method is used for the electron transport in order to obtain high accuracy of results. To further speed computation, Monte Carlo is used once only, to form an electron track kernel (etk). The etk is a dataset containing the lengths and energy deposition of each step of a number of electron tracks. The etk is transported from each incident particle interaction site, from which the dose is calculated. Dose distributions calculated in heterogeneous media show SMC results similar to those of Monte Carlo. For the same statistical uncertainty, SMC takes an order of magnitude less computation time than a full Monte Carlo simulation. SMC has only been implemented for photons and electrons, however the same basic method could be used for the transport of other particles. Current development includes the optimisation of the etks and the code in order to decrease computation time, and also the inclusion of SMC onto a clinical planning system.     

Finding a suitable shield for mixed neutron and photon fields based on an Am–Be source

This study aimed to investigate a shielding design against neutron and photon rays from neutron irradiators based on Am–Be sources, using the Monte Carlo simulation. Different shielding materials were studied, including borated polyethylene, DaGa concrete, and epoxy resin with colemanite. The Monte Carlo N-particle code (MCNP) was used to design shielding. A new type of neutron and photon shielding material based on 40 % galena, 55 % polyethylene, and 5 % boric acid is proposed. The results show that the total effective dose of radiation is significantly reduced by the optimum radius of this shielding system.     

Correlation method for variance reduction of Monte Carlo integration in RS-HDMR

The High Dimensional Model Representation (HDMR) technique is a procedure for efficiently representing high-dimensional functions. A practical form of the technique, RS-HDMR, is based on randomly sampling the overall function and utilizing orthonormal polynomial expansions. The determination of expansion coefficients employs Monte Carlo integration, which controls the accuracy of RS-HDMR expansions. In this article, a correlation method is used to reduce the Monte Carlo integration error. The determination of the expansion coefficients becomes an iteration procedure, and the resultant RS-HDMR expansion has much better accuracy than that achieved by direct Monte Carlo integration. For an illustration in four dimensions a few hundred random samples are sufficient to construct an RS-HDMR expansion by the correlation method with an accuracy comparable to that obtained by direct Monte Carlo integration with thousands of samples.     

Gamma distribution model to provide a direct assessment of the overall quality of quantum Monte Carlo-generated electron distributions

Our objective is to assess the accuracy of simulated quantum Monte Carlo electron distributions of atoms and molecules. Our approach is first to model the exact electron distribution by a linear combination of gamma distribution functions, with parameters chosen to exactly reproduce highly accurate literature values for a number of selected moments for the system of interest. In application to the ground-state electron distributions of helium and dihydrogen, a high level of accuracy of the model was confirmed upon comparing its predicted moments, not used in the model's parametrization, to those calculated from high-level theory. Next, we generated electron-electron and electron-nucleus distributions for dihydrogen from electron positions outputted from a variety of quantum Monte Carlo algorithms. Upon juxtaposition of the simulated distributions with the putatively exact one that we derived from the model, we quantified the error in simulated distributions. The most accurate distributions were obtained from no-compromise reptation quantum Monte Carlo, a recently developed algorithm designed to ameliorate the distributions' time-step bias. Marginally less accurate distributions were generated from fixed-node diffusion Monte Carlo with descendant counting and detailed balance.     

Radial dose profiles for pencil electron beams

The relationship between the Boltzmann and Fermi-Eyges-Yang equations governing electron transport is examined. Radial dose profiles for a pencil beam obtained by numerical solution of the Boltzmann equation in the small angle approximation are compared with both the Gaussian approximation and with Monte Carlo simulations for a carbon medium. For energies ranging from 5 to 20 MeV and penetration depths up to 75% of the range the numerical results are within 10% of the Monte Carlo results for the radial distance encompassing 63% of the energy deposition.     

剩余函数量子Monte Carlo方法

为量子Ｍｏｎｔｅ Ｃａｒｌｏ方法提出一条新途径－剩余函数法,引入了Ｓｃｈｒｏｅｄｉｎｇｅｒ方程剩余函数的概念,利用剩余函数将一种新的有明显物理意义的试探函数应用到量子Ｍｏｎｔｅ Ｃａｒｌｏ过程中,这种试探函数是通过一种迭进式的方式确定的,它不需要在Ｍｏｎｔｅ Ｃａｒｌｏ过程中优化参数。文中我们将给出这种试探函数的具体形式,证明由这种试探函数求出的能量期望值收敛于体系真实的能量值;文中还给出这种试探     

Monte Carlo sampling algorithm for searching a scale-transformed energy space of polypeptides

A Monte Carlo sampling algorithm for searching a scale-transformed conformational energy space of polypeptides is presented. This algorithm is based on the assumption that energy barriers can be overcome by a uniform sampling of the logarithmically transformed energy space. This algorithm is tested with Met-enkephalin. For comparison, the entropy sampling Monte Carlo (ESMC) simulation is performed. First, the global minimum is easily found by the optimization of a scale-transformed energy space. With a new Monte Carlo sampling, energy barriers of 3000 kcal/mol are frequently overcome, and low-energy conformations are sampled more efficiently than with ESMC simulations. Several thermodynamic quantities are calculated with good accuracy.     

Approaching chemical accuracy with quantum Monte Carlo

A quantum Monte Carlo study of the atomization energies for the G2 set of molecules is presented. Basis size dependence of diffusion Monte Carlo atomization energies is studied with a single determinant Slater-Jastrow trial wavefunction formed from Hartree-Fock orbitals. With the largest basis set, the mean absolute deviation from experimental atomization energies for the G2 set is 3.0 kcal/mol. Optimizing the orbitals within variational Monte Carlo improves the agreement between diffusion Monte Carlo and experiment, reducing the mean absolute deviation to 2.1 kcal/mol. Moving beyond a single determinant Slater-Jastrow trial wavefunction, diffusion Monte Carlo with a small complete active space Slater-Jastrow trial wavefunction results in near chemical accuracy. In this case, the mean absolute deviation from experimental atomization energies is 1.2 kcal/mol. It is shown from calculations on systems containing phosphorus that the accuracy can be further improved by employing a larger active space.     

Monte Carlo approach to the decay rate of a metastable system with an arbitrarily shaped barrier

A path integral Monte Carlo method based on the fast-Fourier transform technique combined with the important sampling method is proposed to calculate the decay rate of a metastable quantum system with an arbitrary shape of a potential barrier. The contribution of all fluctuation actions is included which can be used to check the accuracy of the usual steepest-descent approximation, namely, the perturbation expansion of potential. The analytical approximation is found to produce the decay rate of a particle in a cubic potential being about 20% larger than the Monte Carlo data at the crossover temperature. This disagreement increases with increasing complexity of the potential shape. We also demonstrate via Langevin simulation that the postsaddle potential influences strongly upon the classical escape rate.     

Prompt-gamma neutron activation analysis system design: Effects of D-T versus D-D neutron generator source selection

Prompt-gamma neutron activation (PGNA) analysis is used for the non-invasive measurement of human body composition. Advancements in portable, compact neutron generator design have made those devices attractive as neutron sources. Two distinct generators are available: D-D with 2.5 MeV and D-T with 14.2 MeV neutrons. To compare the performance of these two units in our present PGNA system, we performed Monte Carlo simulations (MCNP-5; Los Alamos National Laboratory) evaluating the nitrogen reactions produced in tissue-equivalent phantoms and the effects of background interference on the gamma-detectors. Monte Carlo response curves showed increased gamma production per unit dose when using the D-D generator, suggesting that it is the more suitable choice for smaller sized subjects. The increased penetration by higher energy neutrons produced by the D-T generator supports its utility when examining larger, especially obese, subjects. A clinical PGNA analysis design incorporating both neutron generator options may be the best choice for a system required to measure a wide range of subject phenotypes.     

Comments on the primary and scatter dose-spread kernels used for convolution methods

The basic primary and scatter dose-spread kernels used for convolution methods are usually produced by Monte Carlo simulations with the interaction point forced to the center of a large water phantom. However, it is still not clear whether such Monte Carlo based kernels allow accurate dose calculations with a wide range of field sizes and depths, especially in thorax phantoms. Using the differential primary and scatter concept, this paper proposes another type of basic kernel, with which perfectly accurate primary and scatter absorbed dose calculations can be performed under conditions that the beam is parallel, the incident beam intensity is uniform within and zero outside the field, and the primary beam attenuation coefficient along raylines is not a function of depth and off-axis distance.     

X-ray treatment at 5 MeV and above

There is agreement in the scientific community that X-ray treatment of food at 7.5 MeV can be safe. Possible process improvements for treating at higher than 5 MeV X-rays have been re-visited. Monte Carlo methods have been applied to simulate the X-ray conversion process and to calculate dose distributions in homogeneous phantoms. Experimental data obtained using X-rays produced with a Rhodotron TT200 at 5 and 10 MeV verifies a representative set of data which is calculated with the presented method.

With this qualified Monte Carlo tool, calculations at 7.5 MeV incident electron energy were performed. The analysis gives special attention to higher photon yield, improved product penetration, as well as surface and edge effects.     

Ratio control variate method for efficiently determining high-dimensional model representations

The High-Dimensional Model Representation (HDMR) technique is a family of approaches to efficiently interpolate high-dimensional functions. RS(Random Sampling)-HDMR is a practical form of HDMR based on randomly sampling the overall function, and utilizing orthonormal polynomial expansions to approximate the RS-HDMR component functions. The determination of the expansion coefficients for the component functions employs Monte Carlo integration, which controls the accuracy of the RS-HDMR interpolation. The control variate method is an established approach to improve the accuracy of Monte Carlo integration. However, this method is often not practical for an arbitrary function f(x) because there is no general way to find the analytical control variate function h(x), which needs to be very similar to f(x). In this article, we show that truncated RS-HDMR expansions can be used as h(x) for arbitrary f(x), and a more stable iterative ratio control variate method was developed for the determination of the expansion coefficients for the RS-HDMR component functions. As the asymptotic error (standard deviation) of the estimator given by the ratio control variate method is proportional to 1/N(sample size), it is more efficient than the direct Monte Carlo integration, whose error is proportional to 1/square root(N). In an illustration of a four-dimensional atmospheric model a few hundred random samples are sufficient to construct an RS-HDMR expansion by the ratio control variate method with an accuracy comparable to that obtained by direct Monte Carlo integration with thousands of samples.     

Electron pair localization function: a practical tool to visualize electron localization in molecules from quantum Monte Carlo data

In this work we introduce an electron localization function describing the pairing of electrons in a molecular system. This function, called "electron pair localization function," is constructed to be particularly simple to evaluate within a quantum Monte Carlo framework. Two major advantages of this function are the following: (i) the simplicity and generality of its definition; and (ii) the possibility of calculating it with quantum Monte Carlo at various levels of accuracy (Hartree-Fock, multiconfigurational wave functions, valence bond, density functional theory, variational Monte Carlo with explicitly correlated trial wave functions, fixed-node diffusion Monte Carlo, etc). A number of applications of the electron pair localization function to simple atomic and molecular systems are presented and systematic comparisons with the more standard electron localization function of Becke and Edgecombe are done. Results illustrate that the electron pair localization function is a simple and practical tool for visualizing electronic localization in molecular systems.     

Numerical study of the accuracy and efficiency of various approaches for Monte Carlo surface hopping calculations

A one-dimensional, two-state model problem with two well-separated avoided crossing points is employed to test the efficiency and accuracy of a semiclassical surface hopping technique. The use of a one-dimensional model allows for the accurate numerical evaluation of both fully quantum-mechanical and semiclassical transition probabilities. The calculations demonstrate that the surface hopping procedure employed accounts for the interference between different hopping trajectories very well and provides highly accurate transition probabilities. It is, in general, not computationally feasible to completely sum over all hopping trajectories in the semiclassical calculations for multidimensional problems. In this case, a Monte Carlo procedure for selecting important trajectories can be employed. However, the cancellation due to the different phases associated with different trajectories limits the accuracy and efficiency of the Monte Carlo procedure. Various approaches for improving the accuracy and efficiency of Monte Carlo surface hopping procedures are investigated. These methods are found to significantly reduce the statistical sampling errors in the calculations, thereby increasing the accuracy of the transition probabilities obtained with a fixed number of trajectories sampled.     

Quantum mechanical single molecule partition function from path integral Monte Carlo simulations

An algorithm for calculating the partition function of a molecule with the path integral Monte Carlo method is presented. Staged thermodynamic perturbation with respect to a reference harmonic potential is utilized to evaluate the ratio of partition functions. Parallel tempering and a new Monte Carlo estimator for the ratio of partition functions are implemented here to achieve well converged simulations that give an accuracy of 0.04 kcal/mol in the reported free energies. The method is applied to various test systems, including a catalytic system composed of 18 atoms. Absolute free energies calculated by this method lead to corrections as large as 2.6 kcal/mol at 300 K for some of the examples presented.     

Advantages of mesh tallying in MCNPX for 3D dose calculations in radiotherapy

The energy deposition mesh tally option of MCNPX Monte Carlo code is very useful for 3-Dimentional (3D) dose calculations. In this study, the 3D dose calculation was done for CT-based Monte Carlo treatment planning in which the energy deposition mesh tally were superimposed on merged voxel model. The results were compared with those of obtained from the common energy deposition (*F8) tally method for all cells of non-merged voxel model. The results of these two tallies and their respective computational times are compared, and the advantages of the proposed method are discussed. For this purpose, a graphical user interface (GUI) application was developed for reading CT slice data of patient, creating voxelized model of patient, optionally merging adjacent cells with the same material to reduce the total number of cells, reading beam configuration from commercial treatment planning system transferred in DICOM-RT format, and showing the isodose distribution on the CT images. To compare the results of Monte Carlo calculated and TiGRT planning system (LinaTech LLC, USA), treatment head of the Siemens ONCOR Impression accelerator was also simulated and the phase-space data on the scoring plane just above the Y-jaws was created and used. The results for a real prostate intensity-modulated radiation therapy (IMRT) plan showed that the proposed method was fivefold faster while the precision was almost the same.     

Prediction of sound velocity and compressibility via molecular simulation at fixed entropy

A molecular-level simulation route is proposed to compute the isentropic thermodynamic properties in a fluid system by Monte Carlo simulation at fixed entropy. The method involves computation of the pressure response of a system to an infinitesimal change in system density by introduction of a single molecule, while retaining the system volume as well as the absolute molar entropy. The probability for accepting a change in temperature during the Monte Carlo moves was weighted against the argument proposed by Smith et al. [W.R. Smith, M. Lísal, I. Nezbeda, Chem. Phys. Lett. 426 (2006) 436–440]. Application to fluid argon has confirmed superior accuracy for the technique within the gas state to yield results within 1.2% of the measured values for the range of thermodynamic conditions investigated.