arXiv subjects

Charuka D. Wickramasinghe

Publications and source records attributed to Charuka D. Wickramasinghe.

5 recordsLinked to original sources

arXiv2026

TumorPred: A Computational Framework Implemented via an R/Shiny Web Application for Parameter Estimation and Sensitivity Analysis in Compartmental Brain Modeling

It is difficult or infeasible to directly measure how much of a drug actually enters the human brain and a brain tumor, how long it remains there, and to estimate drug-specific or patient-specific parameters, as well as how changes in these parameters influence model outputs and pharmacokinetic characteristics. Compartmental modeling offers a powerful mathematical approach to describe drug distribution and elimination in the body using systems of differential equations. This study introduces TumorPred, an R/Shiny-based web application designed for model simulation, sensitivity analysis, and pharmacokinetic parameter calculation in a permeability-limited four-compartment brain model. The model closely mimics human brain functionality for drug delivery and aims to predict the pharmacokinetics of drugs in the brain blood, brain mass, and cranial and spinal cerebrospinal fluid (CSF) of the human brain. The app provides real-time output updates in response to input modifications and allows users to visualize and download simulated plots and data tables. The computational accuracy of TumorPred is validated against results from the Simcyp Simulator (Certara Inc.). TumorPred is freely accessible and serves as an invaluable computational tool and data-driven resource for advancing drug development and optimizing treatment strategies for more effective brain cancer therapy.

math.DS↗

arXiv2026

PBPK-iPINNs: Inverse Physics-Informed Neural Networks for Physiologically Based Pharmacokinetic Brain Models

Physics-Informed Neural Networks (PINNs) integrate machine learning with differential equations to solve forward and inverse problems while ensuring that predictions adhere to physical laws. Physiologically based pharmacokinetic (PBPK) modeling advances beyond classical compartmental approaches by employing a mechanistic, physiology-focused framework. Such models involve many unknown parameters that are difficult to measure directly in humans due to ethical and practical constraints. PBPK models are constructed as systems of ordinary differential equations (ODEs) and these parametric ODEs are often stiff, and traditional numerical and statistical methods frequently fail to converge. In this study, we consider a permeability-limited, four-compartment PBPK brain model that mimics human brain functionality in drug delivery. We introduce PBPK-iPINN, a method for estimating drug-specific or patient-specific parameters and drug concentration profiles using inverse PINNs. We also conducted parameter identifiability analysis to determines whether the parameters can be uniquely and reliably estimated from the available data. We demonstrate that, for the inverse problem to converge to the correct solution, the components of the loss function (data loss, initial condition loss, and residual loss) must be appropriately weighted, and the hyperparameters including the number of layers and neurons, activation functions, learning rate, optimizer, and collocation points must be carefully tuned. The performance of the PBPK-iPINN approach is then compared with established numerical and statistical methods. Accurate parameter estimation yields precise drug concentration-time profiles, which in turn enable the calculation of pharmacokinetic metrics. These metrics support drug developers and clinicians in designing and optimizing therapies for brain cancer.

stat.ML↗

arXiv2024

A Graded Mesh Refinement for 2D Poisson's Equation on Non Convex Polygonal Domains

This work delves into solving the two dimensional Poisson problem through the Finite Element Method which is relevant in various physical scenarios including heat conduction, electrostatics, gravity potential, and fluid dynamics. However, finding exact solutions to these problems can be complicated and challenging due to complexities in the domains such as re-entrant corners, cracks, and discontinuities of the solution along the boundaries, and due to the singular source function. Our focus in this work is to solve the Poisson equation in the presence of re entrant corners at the vertices of domain where some of the interior angles are greater than 180 degrees. When the domain features a re entrant corner, the numerical solution can display singular behavior near the corners. To address this, we propose a graded mesh algorithm that helps us to tackle the solution near singular points. We derive H1 and L2 error estimate results, and we use MATLAB to present numerical results that validate our theoretical findings. By exploring these concepts, we hope to provide new insights into the Poisson problem and inspire future research into the application of numerical methods to solve complex physical scenarios

math.NA↗

arXiv2023

A Mixed Finite Element Method for Singularly Perturbed Fourth Oder Convection-Reaction-Diffusion Problems on Shishkin Mesh

This paper introduces an approach to decoupling singularly perturbed boundary value problems for fourth-order ordinary differential equations that feature a small positive parameter $ε$ multiplying the highest derivative. We specifically examine Lidstone boundary conditions and demonstrate how to break down fourth-order differential equations into a system of second-order problems, with one lacking the parameter and the other featuring $ε$ multiplying the highest derivative. To solve this system, we propose a mixed finite element algorithm and incorporate the Shishkin mesh scheme to capture the solution near boundary layers. Our solver is both direct and of high accuracy, with computation time that scales linearly with the number of grid points. We present numerical results to validate the theoretical results and the accuracy of our method.

math.NA↗

arXiv2022

A $C^0$ finite element method for the biharmonic problem with Dirichlet boundary conditions in a polygonal domain

In this paper, we study the biharmonic equation with Dirichlet boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem into a system of two Poison equations and one Stokes equation, or a system of one Stokes equation and one Poisson equation. It is shown that the solution of each system is equivalent to that of the original fourth-order problem on both convex and non-convex polygonal domains. Two finite element algorithms are in turn proposed to solve the decoupled systems. In addition, we show the regularity of the solutions in each decoupled system in both the Sobolev space and the weighted Sobolev space, and we derive the optimal error estimates for the numerical solutions on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings.

math.NA↗