Monte-Carlo - Self Consistent Field Method in the Polyelectrolyte Theory

Abstract

A new time saving numerical method for calculation of equilibrium potential and density distribution of mobile ions around the polyion in a polyelectrolyte system is proposed: the region around the polyion is being divided into two zones - internal and external; in the internal zone all the ions are accounted explicity with the aid of Monte-Carlo procedure; in the external zone the combined Monte-Carlo - self consistent field method proposed earlier is applied, an exchange of ions between regions is being implied. For 1:1 electrolyte the optimal choice of the boundary between the zones has been demonstrated. As an example of a more complicated system calculation for 2:2:1:1 electrolyte was carried out.     

Monte-Carlo-self consistent field method in the polyelectrolyte theory

A new time saving numerical method for calculation of equilibrium potential and density distribution of mobile ions around the polyion in a polyelectrolyte system is proposed: the region around the polyion is being divided into two zones-internal and external; in the internal zone all the ions are accounted explicity with the aid of Monte-Carlo procedure; in the external zone the combined Monte-Carlo-self consistent field method proposed earlier is applied, an exchange of ions between regions is being implied. For 1:1 electrolyte the optimal choice of the boundary between the zones has been demonstrated. As an example of a more complicated system calculation for 2:2:1:1 electrolyte was carried out.     

A Monte Carlo analysis of breast screening randomized trials

PurposeTo analyze breast screening randomized trials with a Monte Carlo simulation tool.MethodsA simulation tool previously developed to simulate breast screening programmes was adapted for that purpose. The history of women participating in the trials was simulated, including a model for survival after local treatment of invasive cancers. Distributions of time gained due to screening detection against symptomatic detection and the overall screening sensitivity were used as inputs. Several randomized controlled trials were simulated. Except for the age range of women involved, all simulations used the same population characteristics and this permitted to analyze their external validity. The relative risks obtained were compared to those quoted for the trials, whose internal validity was addressed by further investigating the reasons of the disagreements observed.ResultsThe Monte Carlo simulations produce results that are in good agreement with most of the randomized trials analyzed, thus indicating their methodological quality and external validity. A reduction of the breast cancer mortality around 20% appears to be a reasonable value according to the results of the trials that are methodologically correct. Discrepancies observed with Canada I and II trials may be attributed to a low mammography quality and some methodological problems. Kopparberg trial appears to show a low methodological quality.ConclusionMonte Carlo simulations are a powerful tool to investigate breast screening controlled randomized trials, helping to establish those whose results are reliable enough to be extrapolated to other populations and to design the trial strategies and, eventually, adapting them during their development.     

Clinical implementation of a Monte Carlo based treatment plan QA platform for validation of Cyberknife and Tomotherapy treatments

PurposeThe main focus of the current paper is the clinical implementation of a Monte Carlo based platform for treatment plan validation for Tomotherapy and Cyberknife, without adding additional tasks to the dosimetry department.MethodsThe Monte Carlo platform consists of C++ classes for the actual functionality and a web based GUI that allows accessing the system using a web browser. Calculations are based on BEAMnrc/DOSXYZnrc and/or GATE and are performed automatically after exporting the dicom data from the treatment planning system. For Cyberknife treatments of moving targets, the log files saved during the treatment (position of robot, internal fiducials and external markers) can be used in combination with the 4D planning CT to reconstruct the actually delivered dose. The Monte Carlo platform is also used for calculation on MRI images, using pseudo-CT conversion.ResultsFor Tomotherapy treatments we obtain an excellent agreement (within 2%) for almost all cases. However, we have been able to detect a problem regarding the CT Hounsfield units definition of the Toshiba Large Bore CT when using a large reconstruction diameter. For Cyberknife treatments we obtain an excellent agreement with the Monte Carlo algorithm of the treatment planning system. For some extreme cases, when treating small lung lesions in low density lung tissue, small differences are obtained due to the different cut-off energy of the secondary electrons.ConclusionsA Monte Carlo based treatment plan validation tool has successfully been implemented in clinical routine and is used to systematically validate all Cyberknife and Tomotherapy plans.     

Monte Carlo transport of swift protons and light ions in water: The influence of excitation cross sections,relativistic effects,and Auger electron emission in w-values

PurposeTo develop a particle transport code to compute w-values and stopping power of swift ions in liquid water and gases of interest for reference dosimetry in hadrontherapy. To analyze the relevance of inelastic and post-collisional processes considered.MethodsThe Monte Carlo code MDM was extended to the case of swift ion impact on liquid water (MDM-Ion). Relativistic corrections in the inelastic cross sections and the post-collisional Auger emission were considered. The effects of introducing different electronic excitation cross sections were also studied.ResultsThe stopping power of swift ions on liquid water, calculated with MDM-Ion, are in excellent agreement with recommended data. The w-values show a strong dependence on the electronic excitation cross sections and on the Auger electron emission. Comparisons with other Monte Carlo codes show the relevance of both the processes considered and of the cross sections employed. W and w-values for swift electron, proton, and carbon ions calculated with the MDM and MDM-Ion codes are in very close agreement with each other and with the 20.8 eV experimental value.ConclusionWe found that w-values in liquid water are independent of ion charge and energy, as assumed in reference dosimetry for hadrontherapy from sparse experimental results for electron and ion impact on gases. Excitation cross sections and Auger emission included in Monte Carlo codes are critical in w-values calculations. The computation of this physical parameter should be used as a benchmark for micro-dosimetry investigations, to assess the reliability of the cross sections employed.     

Nonadditive Monte Carlo Simulation of Liquid Hydrogen Chloride Difference Algorithm and Parallel Implementation

Abstract

This paper continues our Monte Carlo simulation study of liquid hydrogen chloride [1]. The importance of non-additive interactions is carefully analyzed. Computed atom pair correlation functions are compared to neutron scattering experiments [2]. A difference algorithm (“Δ—algorithm”) is developed, which makes non-additive Monte Carlo simulations practicable. We also report an implementation of this algorithm on a transputer network, taking advantage of the inherent parallelism of the Δ — algorithm.     

Nature of the Stacking of Nucleic Acid Bases in Water: a Monte Carlo Simulation

Abstract

The results of a Monte Carlo simulation of the hydration of uracil and thymine molecules, their stacked dimers and hydrogen-bonded base pairs are presented. Simulations have been performed in a cluster approximation. The semiempirical atom-atom potential functions have been used (cluster consisting of 200 water molecules). It has been shown that the stacking interactions of uracil and thymine molecules in water arise mainly due to the increase in the water-water interaction during the transition from monomers to dimer. It has been found out that stacked base associates are more preferable than base pairs in water. This preference is mainly due to the energetically more favourable structure of water around the stack.     

A semi-analytical model for calculating the penetration depth of a high energy electron beam in a water phantom with a magnetic field

PurposeAs an electron beam is incident on a uniform water phantom in the presence of a lateral magnetic field, the depth-dose distribution of the electron beam changes significantly and forms the well-known ‘Bragg peak’, with a depth-dose distribution similar to that of heavy ions. This phenomenon has pioneered a new field in the clinical application of electron beams. For such clinical applications, evaluating the penetration depth of electron beams quickly and accurately is the critical problem.MethodsThis paper describes a model for calculating the penetration depth of an electron beam rapidly and correctly in a water phantom under the influence of a magnetic field. The model was used to calculate the penetration depths under different conditions: the energies of electron beams of 6, 8, 12 and 15 MeV and the magnetic induction intensities of 0.75, 1.0, 1.5, 2.0 and 3.0 T. In addition, the calculation results were compared with the results of a Monte Carlo simulation.ResultsThe comparison results indicate that the difference between the two calculation methods was less than 0.5 cm. Moreover, the computing time of the calculation model was less than a second.ConclusionsThe semi-analytical model proposed in the present study enables the penetration depth of the electron beam in the presence of a magnetic field to be obtained with a computational efficiency higher than that of the Monte Carlo approach; thus, the proposed model has high potential for application.     

Calculations of the spatial distribution of charge density in the environment of DNA

A technique for modeling the structured environmental charge distribution about isolated polyions of arbitrary geometry is presented and applied to B-DNA. It describes the three-dimensional variation of the continuous space charge and allows estimation of local electrostatic potentials and fields that the electrolytic environment induces at nuclei of the polyion. Calculations involve an iterative solution to the set of equations coupling electrostatic potential and average charge density in space. By dividing the region around a DNA segment into finite volume elements, sets of numerically stable atmospheric charge densities have been obtained over a range of concentrations of added monovalent salt. Results are in good agreement with those of Poisson-Boltzmann calculations on comparable systems and are consistent with findings from Monte Carlo simulations of DNA.     

Simulations on the Primitive Electrolyte Environment of a High Charge-density Polyelectrolyte. A Sampling Problem and its Solution

Abstract

We show that the classical Metropolis Monte Carlo (MMC) algorithm converges very slowly when applied to the primitive electrolyte environment for a high charge-density polyelectrolyte. This slowness of convergence, which is due to the large density inhomogeneity around the polyelectrolyte, produces noticeable errors in the ion distribution functions for MMC runs of 1.3 × 10⁶ trial steps started from nonequilibrium distributions. We report that an algorithm which we call DSMC (for density-scaled Monte Carlo) overcomes this problem and provides relatively rapid convergence in this application. We suggest that DSMC should be well-suited for other Monte Carlo simulations on physical systems where large density inhomogeneities occur.     

Using matrix summation method for three dimensional dose calculation in brachytherapy

AimThe purpose of this study is to calculate radiation dose around a brachytherapy source in a water phantom for different seed locations or rotation the sources by the matrix summation method.BackgroundMonte Carlo based codes like MCNP are widely used for performing radiation transport calculations and dose evaluation in brachytherapy. But for complicated situations, like using more than one source, moving or rotating the source, the routine Monte Carlo method for dose calculation needs a long time running.Materials and methodsThe MCNPX code has been used to calculate radiation dose around a ¹⁹²Ir brachytherapy source and saved in a 3D matrix. Then, we used this matrix to evaluate the absorbed dose in any point due to some sources or a source which shifted or rotated in some places by the matrix summation method.ResultsThree dimensional (3D) dose results and isodose curves were presented for ¹⁹²Ir source in a water cube phantom shifted for 10 steps and rotated for 45 and 90° based on the matrix summation method. Also, we applied this method for some arrays of sources.ConclusionThe matrix summation method can be used for 3D dose calculations for any brachytherapy source which has moved or rotated. This simple method is very fast compared to routine Monte Carlo based methods. In addition, it can be applied for dose optimization study.     

Structure of Lennard–Jones fluid near large spherical particles: further test of the “universality” of adjustable parameter in perturbation density functional theory

Grand canonical Monte Carlo simulation is used to study the density profiles of Lennard–Jones (LJ) fluid next to a large hard sphere (mimicking a colloidal particle) of various sizes. The LJ fluid in the inhomogeneous system thus maintains equilibrium with the bulk LJ fluid. The chosen density and potential parameters for the bulk fluid correspond to the conditions situated at “dangerous” regions of the phase diagram, i.e. near the critical temperature or close to the gas–liquid coexistence curve. The aim of present extensive simulations is to provide exact data for the broad range of the bulk parameters against which the “universality” of adjustable parameter associated with a perturbation density functional approximation (DFA) can be tested. Here the term “universality” means independence of this parameter on the particular external field responsible for the generation of a non-uniform density profile of the fluid. It is shown that the “universality” of this parameter associated with a third order+second order perturbation DFA holds also in the present case of a large spherical particle as a source of external potential, similarly as established in previous studies dealing with other interaction potential and other external fields [J. Chem. Phys., 122, 064503 (2005); J. Chem. Phys., 123 124708 (2005)]. This DFA can be used as input into a recently proposed framework for the calculation of interparticle potential of mean force (PMF).     

Comparison of the RayStation photon Monte Carlo dose calculation algorithm against measured data under homogeneous and heterogeneous irradiation geometries

PurposeThis work compares Monte Carlo dose calculations performed using the RayStation treatment planning system against data measured on a Varian Truebeam linear accelerator with 6 MV and 10 MV FFF photon beams.MethodsThe dosimetric performance of the RayStation Monte Carlo calculations was evaluated in a variety of irradiation geometries employing homogeneous and heterogeneous phantoms. Profile and depth dose comparisons against measurement were carried out in relative mode using the gamma index as a quantitative measure of similarity within the central high dose regions.ResultsThe results demonstrate that the treatment planning system dose calculation engine agrees with measurement to within 2%/1 mm for more than 95% of the data points in the high dose regions for all test cases. A systematic underestimation was observed at the tail of the profile penumbra and out of field, with mean differences generally <0.5 mm or 1% of curve dose maximum respectively. Out of field agreement varied between evaluated beam models.ConclusionsThe RayStation implementation of photon Monte Carlo dose calculations show good agreement with measured data for the range of scenarios considered in this work and is deemed sufficiently accurate for introduction into clinical use.     

Gibbs-Ensemble Molecular Dynamics: Liquid-Gas Equilibria for Lennard-Jones Spheres and n-Hexane

Abstract

We present a novel method to simulate phase equilibria in atomic and molecular systems. The method is a Molecular Dynamics version of the Gibbs-Ensemble Monte Carlo technique, which has been developed some years ago for the direct simulation of phase equilibria in fluid systems. The idea is to have two separate simulation boxes, which can exchange particles (or molecules) in a thermodynamically consistent fashion. Here we pres the derivation of the generalized equations of motion and discuss the relation of the resulting trajectory averages to the relevant ensemble. We test this Gibbs-Ensemble Molecular Dynamics algorithm by applying it to an atomic and a molecular system, i.e. to the liquid-gas coexistence in a Lennard-Jones fluid and in n-hexane. In both cases our results are in good accord with previous mean field and Gibbs-Ensemble Monte Carlo results as well as with the experimental data in the case of hexane. We also show that our Gibbs-Ensemble Molecular Dynamics algorithm like other Molecular Dynamics techniques can be used to study the dynamics of the system. Self-diffusion coefficients calculated with this method are in agreement with the result of conventional constant temperature Molecular Dynamics.     

Determination of initial electron parameters by means of Monte Carlo simulations for the Siemens Artiste Linac 6 MV photon beam

AimIn this study, we investigated initial electron parameters of Siemens Artiste Linac with 6 MV photon beam using the Monte Carlo method.BackgroundIt is essential to define all the characteristics of initial electrons hitting the target, i.e. mean energy and full width of half maximum (FWHM) of the spatial distribution intensity, which is needed to run Monte Carlo simulations. The Monte Carlo is the most accurate method for simulation of radiotherapy treatments.Materials and methodsLinac head geometry was modeled using the BEAMnrc code. The phase space files were used as input file to DOSXYZnrc simulation to determine the dose distribution in a water phantom. We obtained percent depth dose curves and the lateral dose profile. All the results were obtained at 100 cm of SSD and for a 10 × 10 cm² field.ResultsWe concluded that there existed a good conformity between Monte Carlo simulation and measurement data when we used electron mean energy of 6.3 MeV and 0.30 cm FWHM value as initial parameters. We observed that FWHM values had very little effect on PDD and we found that the electron mean energy and FWHM values affected the lateral dose profile. However, these effects are between tolerance values.ConclusionsThe initial parameters especially depend on components of a linac head. The phase space file which was obtained from Monte Carlo Simulation for a linac can be used as calculation of scattering, MLC leakage, to compare dose distribution on patients and in various studies.     

Comparative dose levels between CT-scanner and slot-scanning device (EOS system) in pregnant women pelvimetry

PurposeTo estimate fetal absorbed doses for pregnant women pelvimetry, a comparative study between EOS imaging system and low-dose spiral CT-scanner was carried out. For this purpose three different studies were investigated: in vivo, in vitro and Monte Carlo calculations.MethodsIn vivo dosimetry was performed, using OSL NanoDot dosimeters, to determine the dose to the skin of twenty pregnant women. In vitro studies were established by using a cubic phantom of water, in order to estimate the out of field doses. In the latter study, OSLDs were placed at depths corresponding to the lowest, average and highest position of the uterus. Monte Carlo calculations of effective doses to high radio-sensitive organs were established, using PCXMC and CTExpo software suites for EOS imaging system and CT-scanner, respectively.ResultsThe EOS imaging system reduces radiation exposure 4 to 8 times compared to the CT-scanner. The entrance skin doses were 74% (p-values <0.01) higher with the CT-scanner than with the EOS system. In the out of field region, the measured doses of the EOS system were reduced by 80% (p-values <0.02).Monte Carlo calculations confirmed that effective doses to organs are less accentuated for EOS than for CT pelvimetry.ConclusionsThe EOS system is less irradiating than the CT exam. The out-of-field dose which is significant, is lower in the EOS than in the CT-scanner and could be reduced even further by optimizing the time used for image acquisition.     

Monte Carlo Simulation of the β-Sheet–Random Coil Transition of a Homopolypeptide. I. Equilibrium Study

Abstract

Monte Carlo simulation of the β-sheet – random coil conversion of a homopolypeptide chain was carried out on the basis of a model where successive two amino acid residues were assumed to change their states simultaneously and hence constituted a basic unit. Only three states were considered for each unit: extended, turn and coil. The conversion exhibited a transition between two states, random coil (C) and the β-sheet (B). In the transition region, two population maxima were always found, each corresponded to the local minimum of the free energy and there was an energy barrier between them. This behavior is characteristic of the all-or-none type transition. We have found that the nature of the first-order transition is retained in the case of a small system consisting of 100 units. The size of the cooperative unit was evaluated. According to the analytical theory of Kanô, a transition curve was obtained which was very close to the present one. This consistent result has suggested that equilibrium properties of the β-sheet-random coil transition are well evaluated with the mean field approximation. The matrix method of Mattice is also discussed.     

A comparison of approximation techniques for variance-based sensitivity analysis of biochemical reaction systems

Background

Sensitivity analysis is an indispensable tool for the analysis of complex systems. In a recent paper, we have introduced a thermodynamically consistent variance-based sensitivity analysis approach for studying the robustness and fragility properties of biochemical reaction systems under uncertainty in the standard chemical potentials of the activated complexes of the reactions and the standard chemical potentials of the molecular species. In that approach, key sensitivity indices were estimated by Monte Carlo sampling, which is computationally very demanding and impractical for large biochemical reaction systems. Computationally efficient algorithms are needed to make variance-based sensitivity analysis applicable to realistic cellular networks, modeled by biochemical reaction systems that consist of a large number of reactions and molecular species.

Results

We present four techniques, derivative approximation (DA), polynomial approximation (PA), Gauss-Hermite integration (GHI), and orthonormal Hermite approximation (OHA), for analytically approximating the variance-based sensitivity indices associated with a biochemical reaction system. By using a well-known model of the mitogen-activated protein kinase signaling cascade as a case study, we numerically compare the approximation quality of these techniques against traditional Monte Carlo sampling. Our results indicate that, although DA is computationally the most attractive technique, special care should be exercised when using it for sensitivity analysis, since it may only be accurate at low levels of uncertainty. On the other hand, PA, GHI, and OHA are computationally more demanding than DA but can work well at high levels of uncertainty. GHI results in a slightly better accuracy than PA, but it is more difficult to implement. OHA produces the most accurate approximation results and can be implemented in a straightforward manner. It turns out that the computational cost of the four approximation techniques considered in this paper is orders of magnitude smaller than traditional Monte Carlo estimation. Software, coded in MATLAB^®, which implements all sensitivity analysis techniques discussed in this paper, is available free of charge.

Conclusions

Estimating variance-based sensitivity indices of a large biochemical reaction system is a computationally challenging task that can only be addressed via approximations. Among the methods presented in this paper, a technique based on orthonormal Hermite polynomials seems to be an acceptable candidate for the job, producing very good approximation results for a wide range of uncertainty levels in a fraction of the time required by traditional Monte Carlo sampling.     

Monte Carlo evaluation of the dose calculation algorithm of TomoTherapy for clinical cases in dynamic jaws mode

PurposeFor the TomoTherapy^® system, longitudinal conformation can be improved by selecting a smaller field width but at the expense of longer treatment time. Recently, the TomoEdge^® feature has been released with the possibility to move dynamically the jaws at the edges of the target volume, improving longitudinal penumbra and enabling faster treatments. Such delivery scheme requires additional modeling of treatment delivery. Using a previously validated Monte Carlo model (TomoPen), we evaluated the accuracy of the implementation of TomoEdge in the new dose engine of TomoTherapy for 15 clinical cases.MethodsTomoPen is based on PENELOPE. Particle tracking in the treatment head is performed almost instantaneously by 1) reading a particle from a phase-space file corresponding to the largest field and 2) correcting the weight of the particle depending on the actual jaw and MLC configurations using Monte Carlo pre-generated data. 15 clinical plans (5 head-and-neck, 5 lung and 5 prostate tumors) planned with TomoEdge and with the last release of the treatment planning system (VoLO^®) were re-computed with TomoPen. The resulting dose-volume histograms were compared.ResultsGood agreement was achieved overall, with deviations for the target volumes typically within 2% (D₉₅), excepted for small lung tumors (17 cm³) where a maximum deviation of 4.4% was observed for D₉₅. The results were consistent with previously reported values for static field widths.ConclusionsFor the clinical cases considered in the present study, the introduction of TomoEdge did not impact significantly the accuracy of the computed dose distributions.     

Structure of Mixed Solvent Electrolyte Solutions via Gibbs Ensemble Monte Carlo Simulation

Abstract

Previously reported Gibbs ensemble Monte Carlo simulations of vapor-liquid equilibrium in methanol-water and methanol-water-NaCl mixtures are extended to permit study of the microscopic structure of the liquid phases of these systems. The salt effect in a prototypical mixed solvent electrolyte solution (water-methanol-NaCl) is microscopically interpreted in terms of the structural changes undergone by the solvation shells of the ions in the liquid phase of water-methanol-NaCl systems in vapor-liquid equilibrium at constant pressure.