Method Of Approximate Particular Solutions Research Articles

In-silico research on the effects of variable voltage pulse protocols on the cell survival and permeabilization fractions in electroporated cancerous tissues

Reversible electroporation is used to increase the cell membrane permeability by the application of short duration, high voltage electric pulses with electrodes strategically positioned to induce localized drug concentration in the tumor tissue. However, inadequate selection of pulses’ characteristics can lead to relevant loss of cell viability or lack of cell permeabilization, affecting aversively the results of the therapy. Accordingly, the balance between cell membrane electro-permeabilization and cell death is crucial, and the recursive modification of electroporation parameters can favor this balance to increase the drug uptake into viable cells. Particularly, the controlled diminution of the voltage level as the number of pulses increases can reduce the cell damage without compromising the membrane electro-permeabilization. In this work, a previously developed in-silico tool, based on the Global Method of Approximate Particular Solutions (GMAPS), is calibrated, validated, and used to study the differences between constant voltage and variable voltage pulse protocols (CVP and VVP) in terms of the magnitude and distribution of cell survival and permeabilization fractions along the tissue domain, and the internalization efficiency of the therapy. In VVP protocols, the reduction of the applied voltage with the number of pulses is numerically estimated to reach a target volume-averaged survival fraction in the most sensitive zone of the tissue. Numerical results indicate that differences between the pulse protocol types (CVP and VVP) depend on the initial voltage level ∅0 and number of pulses K, with the VVP protocol bringing about superior results for the larger values of ∅0 and K considered here.

High-resolution strategy for localized method of approximate particular solutions to solve unsteady Navier–Stokes problems

In this paper, a high resolution strategy for the localized method of approximate particular solutions (LMAPS) is proposed. In order to avoid solving large and ill-conditioned equations, in the original LMAPS, low-order linear equations are solved by choosing the support region containing five nearest points. In order to improve the accuracy and stability of numerical calculation, for the LMAPS, a high-resolution strategy with limiters is proposed by choosing upwind interpolation templates. Numerical results demonstrate that the proposed high-resolution LMAPS is effective and accurate. This method is suitable for solving the Navier–Stokes equations with high Reynolds number.

Influence of electric field, blood velocity, and pharmacokinetics on electrochemotherapy efficiency

The convective delivery of chemotherapeutic drugs in cancerous tissues is directly proportional to the blood perfusion rate, which in turns can be transiently reduced by the application of high-voltage and short-duration electric pulses due to vessel vasoconstriction. However, electric pulses can also increase vessel wall and cell membrane permeabilities, boosting the extravasation and cell internalization of drug. These opposite effects, as well as possible adverse impacts on the viability of tissues and endothelial cells, suggest the importance of conducting in silico studies about the influence of physical parameters involved in electric-mediated drug transport. In the present work, the global method of approximate particular solutions for axisymmetric domains, together with two solution schemes (Gauss-Seidel iterative and linearization+successive over-relaxation), is applied for the simulation of drug transport in electroporated cancer tissues, using a continuum tumor cord approach and considering both the electropermeabilization and vasoconstriction phenomena. The developed global method of approximate particular solutions algorithm is validated with numerical and experimental results previously published, obtaining a satisfactory accuracy and convergence. Then, a parametric study about the influence of electric field magnitude and inlet blood velocity on the internalization efficacy, drug distribution uniformity, and cell-kill capacity of the treatment, as expressed by the number of internalized moles into viable cells, homogeneity of exposure to bound intracellular drug, and cell survival fraction, respectively, is analyzed for three pharmacokinetic profiles, namely one-short tri-exponential, mono-exponential, and uniform. According to numerical results, the trade-off between vasoconstriction and electropermeabilization effects and, consequently, the influence of electric field magnitude and inlet blood velocity on the assessment parameters considered here (efficacy, uniformity, and cell-kill capacity) is different for each pharmacokinetic profile deemed.

In-silico study about the influence of electroporation parameters on the chemotherapeutic drug transport in cancer tissues using the meshless method of approximate particular solutions

Electroporation has become an important technique in cancer treatment to induce the drug internalization into cells by applying controlled electric pulses. In the present work, the simulation of transport of two chemotherapeutic drugs (doxorubicin and cisplatin) in electroporated cancerous tissues is carried out using the Global Method of Approximate Particular Solutions (Global MAPS), which is a meshless method having several computational advantages over finite element and finite volume methods commonly used in this kind of problems. Numerical parameters involved in Global MAPS simulations are calibrated by means of validation with results previously published in literature, points-number and time-step dependency analyses, and convergence analysis on a relaxation factor, obtaining satisfactory results. Voltage levels of 300 V, 350 V and 400 V, and pulse spacing of 10 s, 100 s and single pulse are considered as input electroporation parameters, whose influence on the following variables is analyzed: magnitude and distribution of electric field in the pulse stage; extracellular and intracellular concentrations of medicament in the pre-pulse and post-pulse diffusion stages, identifying different transport mechanism (internalization and extracellular diffusion) and the concentration equilibrium condition when reached; and efficiency and aggressiveness of the electroporation protocol. In general, numerical results point out that this influence highly depends on the drug type used during the electrochemotherapy treatment. • Transport simulation of drugs in electroporated cancer tissues using meshless technique. • Influence of voltage and pulse spacing on electric field, extracellular and intracellular concentrations. • Influence of voltage and pulse spacing on efficiency and aggressiveness of electroporation protocol. • Differences between cisplatin and doxorubicin behavior in pre-pulse and post-pulse diffusion stages. • Study of main transport mechanisms and concentration equilibrium condition for both drugs.

The localized method of approximate particular solutions for solving an optimal control problem

In this paper, we consider the localized method of approximate particular solutions (LMAPS) for solving a two-dimensional distributive optimal control problem governed by elliptic partial differential equations. Both radial basis functions and polynomial basis functions (RBFs) are used in the LMAPS discretization, while the leave-one-out cross-validation is adopted for the selection of the shape parameter appeared in RBFs. Numerical experiments are presented to demonstrate the accuracy and efficiency of the proposed method.

Hermite Method of Approximate Particular Solutions for Solving Time-Dependent Convection-Diffusion-Reaction Problems

This article describes the development of the Hermite method of approximate particular solutions (MAPS) to solve time-dependent convection-diffusion-reaction problems. Using the Crank-Nicholson or the Adams-Moulton method, the time-dependent convection-diffusion-reaction problem is converted into time-independent convection-diffusion-reaction problems for consequent time steps. At each time step, the source term of the time-independent convection-diffusion-reaction problem is approximated by the multiquadric (MQ) particular solution of the biharmonic operator. This is inspired by the Hermite radial basis function collocation method (RBFCM) and traditional MAPS. Therefore, the resultant system matrix is symmetric. Comparisons are made for the solutions of the traditional/Hermite MAPS and RBFCM. The results demonstrate that the Hermite MAPS is the most accurate and stable one for the shape parameter. Finally, the proposed method is applied for solving a nonlinear time-dependent convection-diffusion-reaction problem.

Upwind Strategy for Localized Method of Approximate Particular Solutions with Applications to Convection Dominated Diffusion Problems

Upwind Strategy for Localized Method of Approximate Particular Solutions with Applications to Convection Dominated Diffusion Problems

Numerical examination of the effect of different boundary conditions on the method of approximate particular solutions for scalar and vector problems

This work presents a numerical study of the influence of different boundary conditions (BC) on the precision of the method of approximate particular solutions (MAPS) with radial basis functions in two dimensions. The state of the art merely mentions that the MAPS accuracy decreases near the boundaries. Therefore, the influence that the choice of boundary condition has on this method should be studied with more detail. The present analysis with numerical experiments was applied to the solution of homogeneous Stokes flow equations, backward facing step flow, Slip flow, convective flow between parallel vertical plates and to elliptic differential equations with variable coefficients. The results show that care must be taken in choosing the boundary conditions for this type of solution methods because this selection affects the precision of the results. It is shown that the error associated with Neumann boundary condition, can be reduced by using a refined nodal distribution towards the inlet and outlet zones. Conversely, mixed boundary conditions require careful combination of traction and velocity to achieve better accuracy with MAPS for some flow problems.

Extending meshless method of approximate particular solutions (MAPS) to two-dimensional convection heat transfer problems.

Natural, forced and mixed convection heat transfer problems are solved by the meshless Method of Approximate Particular Solutions (MAPS). Particular solutions of Poisson and Stokes equations are employed to approximate temperature and velocity, respectively. The latter is used to obtained a closed expression for pressure particular solution. In both cases, the source terms are multiquadric radial basis functions which allow to obtain analytical expressions for these auxiliary problems. In order to couple momentum and energy equations, a relaxation strategy is implemented to avoid convergence problems due to the difference between successive temperature and velocity changes when solving the steady problem from an initial guess. The developed and validated numerical scheme is used to study flow and heat transfer in two two-dimensional problems: natural convection in concentric annulus between a square and a circular cylinder and non-isothermal flow past a staggered tube bundle. Numerical solutions obtained by MAPS are comparable in accuracy to solutions reported by authors who uses denser nodal distributions, showing the capability of the present method to accurately solve heat convection problems with temperature and heat flux boundary conditions as well as curve geometries and internal flow situations.

Symmetric method of approximate particular solutions for solving certain partial differential equations

In this study, a symmetric method of approximate particular solutions (MAPS) is proposed for solving certain partial differential equations (PDEs). Inspired by the unsymmetric MAPS and symmetric radial basis function collocation method (RBFCM), the symmetric MAPS is developed by using the bi-particular solutions of the multiquadrics (MQ). Similar to the unsymmetric MAPS, the right-hand-side of the governing equation is mainly approximated by the MQ in the proposed method. In addition, the system matrix of the prescribed method is symmetric. Numerical examples are solved by the unsymmetric & symmetric RBFCM and MAPS for different problems with different types of governing equations and boundary conditions. Numerical results with different shape parameters are analyzed to show that the symmetric methods are more stable. In addition, the accuracy improvement of the symmetric MAPS is studied. Finally, the stability performance of the symmetric MAPS is further studied for convection-diffusion problems at high Péclet numbers.

A localized MAPS using polynomial basis functions for the fourth-order complex-shape plate bending problems

In this paper, the localized method of approximate particular solutions using polynomial basis functions is proposed to solve plate bending problems with complex domains. The closed-form particular solutions of fourth-order differential equations can be reduced to the linear combination of the particular solutions of Helmholtz and modified Helmholtz equations. To alleviate the difficulty of solving overdetermined fourth-order plate bending problems using the localized collocation method, additional ghost points outside the computational domain are introduced to improve the stability and accuracy. Three examples are illustrated to validate the feasibility of the proposed localized MAPS.

Evaluation of Localization Strategies with Method of Approximate Particular Solutions Without Mesh

Evaluation of Localization Strategies with Method of Approximate Particular Solutions Without Mesh

Simulation of the influence of voltage level and pulse spacing on the efficiency, aggressiveness and uniformity of the electroporation process in tissues using meshless techniques.

Electroporation is a widely used method consisting of application of high-voltage, short-duration electric pulses to increase cell membrane permeability, allowing cellular internalization of medications. In this work, the influence of two primary parameters, voltage level (V) and pulse spacing (N), on electroporation efficiency, uniformity and aggressiveness, as quantified by the total mass transport to viable cells, intracellular concentration gradients and an aggressiveness factor introduced here, is studied by means of numerical simulations of drug transport in electroporated tissues. The global method of approximate particular solutions (Global MAPS) is used to solve the governing equations, together with domain scaling, singular value decomposition and smoothing algorithms, to address the ill-conditioning of the final system and suppress small scale oscillations. The accuracy of Global MAPS is evaluated by comparing the initial extracellular concentration, Ce , and final intracellular concentration, Ci , with previous finite volume method results, obtaining similar behavior of Ce and Ci along the tissue domain, with some differences for Ci in high-gradient zones. According to the Global MAPS results, the influence of V and N on Ci is only significant over a certain range, within which the largest drug transport to viable cells occurs. In general, both electroporation efficiency and aggressiveness change in nonuniform manner with V and decrease with N, whereas the electroporation uniformity decreases as V increases and N decreases. The contour plots obtained here can be considered useful tools to compare electroporation-based treatments in terms of their efficiency, aggressiveness and uniformity, assisting in the selection of a suitable treatment plan for cancer.

The method of approximate particular solutions to simulate an anomalous mobile‐immobile transport process

AbstractIn the current work, a generalized mathematical model based on the Coimbra time fractional derivative of variable order, which describes an anomalous mobile‐immobile transport process in complex systems is investigated numerically. A robust numerical technique based on the meshfree strong form method combined with an efficient time‐stepping scheme is performed to compute the approximate solution of the problem with high accuracy. For this purpose, firstly, an effective implicit time discretization approach is used for discretizing the variable‐order time fractional problem in the time direction. Then a global meshless technique based on the method of approximate particular solutions is performed to fully discretize the model in the spatial domain. The validity and performance of the procedure to numerically simulate the proposed generalized solute transport model on regular and irregular domains are demonstrated through some numerical examples.

Method of Approximate Particular Solutions of Electro-Kinetic Fluids in Micro and Nano channels

Method of Approximate Particular Solutions of Electro-Kinetic Fluids in Micro and Nano channels

The MAPS with polynomial basis functions for solving axisymmetric time-fractional equations

This study presents a parallel meshless approach for time-fractional equations in the axisymmetric geometry. In the present parallel meshless approach, the method of approximate particular solutions (MAPS), in conjunction with Laplace transform, extended precision arithmetic (EPA) and multiple scale technique (MST) is implemented. In this application, the Laplace transform is employed to eliminate the dependence on time. The MAPS is used to obtain the solution of time-independent equation in Laplace space. Furthermore, the solution of time dependent equation is restored by Stehfest’s algorithm. Here the multiple scale technique (MST) and extended precision arithmetic (EPA) are applied to alleviate the effect of an ill-conditioned matrix on a numerical inverse Laplace transform. Then multi-cores parallel computing can improve effectively computing speed of the proposed meshless approach. In the numerical implementation, both the 2D axisymmetric equations and the original 3D equations are considered and compared.

An efficient method of approximate particular solutions using polynomial basis functions

The most challenging task of the method of approximate particular solutions (MAPS) is the generation of the closed-form particular solutions with respect to the given differential operator using various basis functions. These particular solutions have to be generated prior to the solution process of the partial differential equations. In this paper, we propose a different approach without the tedious and inefficient solution procedure using symbolic computation to produce the closed-form particular solutions. The proposed approach is introduced and extended to solve a large class of elliptic partial differential equations (PDEs) based on the method of approximate particular solutions (MAPS).Numerical results show the proposed approach is simple, efficient, accurate, and stable. Five different numerical examples are presented to demonstrate the effectiveness of the proposed method.

Implicit surface reconstruction with radial basis functions via PDEs

We propose a partial differential equations-based algorithm for the 3D implicit surface reconstruction from a set of scattered cloud data. In the solution process, the method of approximate particular solutions with the IMQ radial basis function is employed for solving the modified Helmholtz or Poisson equation with a constant Dirichlet boundary condition. We also consider to repair the surfaces when a certain region of cloud data points are missing. The selection of several parameters is studied for the optimal recovery of the surfaces. Four examples are presented to validate the effectiveness of the proposed method.

Numerical solution of two-dimensional nonlinear sine-Gordon equation using localized method of approximate particular solutions

Sine-Gordon equation is one of the most famous nonlinear hyperbolic partial differential equations, it arises in many science and engineering fields. In the present work, we consider the numerical solution of two dimensional sine-Gordon equation using localized method of approximate particular solutions (LMAPS), in this technique, the method of approximate particular solutions (MAPS) occurs on some local domains, that greatly reduces the size of the collection matrix, and by combining the conditional positive radial basis function (RBF) generalized thin plate splines (GTPS) with additional low-order polynomial basis to avoid selecting shape parameters during localization. This method is effective compared with other existing methods and since this method is really meshless, it can be used to solve the nonlinear model with complicated computational domains. Several numerical examples are given to demonstrate the ability and accuracy of the present approach for solving nonlinear sine-Gordon equation.

An efficient MAPS for solving fourth order partial differential equations using trigonometric functions

In this paper, we apply the method of approximate particular solutions (MAPS) based on the trigonometric functions to solve the fourth order partial differential equations (PDEs) through the use of particular solutions of two second order differential equations. The derivation of the closed-form particular solutions for higher order PDEs is a challenge and could be tedious. Such task can be achieved by a simple algebraic procedure to alleviate the difficulty of the derivation of particular solutions for fourth order PDEs. Since the closed-form particular solutions for the second order PDEs with constant coefficients are known, the proposed solution procedure is simple and direct. Five numerical examples are illustrated to demonstrate the feasibility and effectiveness of the proposed method.