Numerical simulations in 3-dimensions of reaction–diffusion models for brain tumour growth

Jaroudi, Rym; Åström, Freddie; Johansson, Björn; Baravdish, George

doi:10.1080/00207160.2019.1613526

Cited by 12 publications

(4 citation statements)

References 38 publications

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“…Jaroudi et al [13] proposed a novel nonlinear Landweber approach to tackle the inverse problem of pinpointing the source of brain tumors, leveraging established reactiondiffusion models for brain tumor progression. Subsequently, Jaroudi [12] delved into 3D simulations of brain tumor development using a reaction-diffusion framework. They applied standard finite difference discretization with first-order accuracy in time and space to analyze two types of imaging data.…”

Section: Introductionmentioning

confidence: 99%

Numerical approach for brain tumor growth model using higher order compact finite difference scheme

Das,

Sen

2024

Preprint

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In this article, we focus on analyzing the reaction-diffusion model applied to brain gliomas, incorporating two distinct types of growth functions. To gain a comprehensive understanding, we extend our analysis to different types of tissue environments. We explore scenarios where the diffusion coefficient remains constant, reflecting a homogeneous tissue environment. Additionally, we investigate cases where the diffusion coefficient becomes spatially variable, introducing heterogeneity into the model. This spatial variability accounts for the varying properties of different regions within the brain. Subsequently, we extend our investigation to heterogeneous tissue environments in $2$-dimensions. Our analysis of simulation results reveals differences in growth patterns based on the parameters utilized.

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Section: Introductionmentioning

confidence: 99%

Numerical approach for brain tumor growth model using higher order compact finite difference scheme

Das,

Sen

2024

Preprint

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“…Rihan and Rahman [ 34 ] examined the interactions between a malignant tumor and the immune system of healthy effector cells under human immunodeficiency virus by employing ordinary and delay differential equations. Jaroudi et al [ 35 ] studied a brain tumor growth model with reaction–diffusion equations and two three-dimensional different numerical simulations. Laib et al [ 36 ] developed a numerical model for general form of a system of nonlinear Volterra delay integro-differential equations and applied this model to novel coronavirus (COVID-19) epidemic in China, Spain, and Italy.…”

Section: Introductionmentioning

confidence: 99%

Modeling and Solution of Reaction–Diffusion Equations by Using the Quadrature and Singular Convolution Methods

et al. 2022

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In the present work, polynomial, discrete singular convolution and sinc quadrature techniques are employed as the new techniques to derive accurate and efficient numerical solutions for the reaction–diffusion equations. Three models, Fitzhugh-Nagumo, Newell–Whitehead–Segel, and tumor growth models, were presented. The equations of three models are reduced to nonlinear ordinary differential equations by using different quadrature schemes. Then, Runge–Kutta fourth-order method is employed to solve nonlinear ordinary differential equations. In addition, the MATLAB program is used to solve these problems. Comparisons between the new methods and the existing ones are included, demonstrating the ease of implementation and efficiency. Also, the calculated results are supported by four various statistical errors. It is found that the rate of error reaches ≤ 10–6 in discrete singular convolution depending on regularized Shannon kernel which is better than others. Further, a parametric analysis is presented to discuss the influence of diffusion and reaction parameters on the solution.

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“…While there has been much theoretical study of this model as part of a general theory of reaction diffusion equations [37], it is typically very rare to solve models that capture the gross behaviour of glioma tumors in heterogeneous brain tissue based on data imaging. A number of different numerical approaches for the description of glioma tumors' heterogeneous rate of invasion and the dynamics of their highly diffusive nature (mostly without full convergence analysis) have been employed, for example, see [6,9,22,24,25,27,34,38,44]. However, few studies have so far been paid to the convergence analysis of non conforming methods for the reaction diffusion equation and its corresponding models.…”

Section: Introductionmentioning

confidence: 99%

A gradient Discretisation Method For Anisotropic Reaction Diffusion Models with applications to the dynamics of brain tumours

Alnashri,

Alzubaidi

2020

Preprint

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A gradient discretisation method (GDM) is an abstract setting that designs the unified convergence analysis of several numerical methods for partial differential equations and their corresponding models. In this paper, we study the GDM for anisotropic reaction diffusion problems, based on a general reaction term, with Neumann and Dirichlet boundary conditions. With natural regularity assumptions on the exact solution, the framework enables us to provide proof of the existence of weak solutions for the problem, and to obtain a uniform-in-time convergence for the discrete solution and a strong convergence for its discrete gradient. It also allows us to apply non conforming numerical schemes to the model on a generic grid; (the Crouzeix-Raviart scheme and the hybrid mixed mimetic (HMM) methods). Numerical experiments using the HMM method are performed to study the growth of glioma tumours in heterogeneous brain environment. The dynamics of their highly diffusive nature is also measured using the fraction anisotropic measure. The validity of the HMM is examined further using four different mesh types. The results indicate that the dynamics of the brain tumour is still captured by the HMM scheme, even in the event of a highly heterogeneous anisotropic case performed on the mesh with extreme distortions.

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Numerical simulations in 3-dimensions of reaction–diffusion models for brain tumour growth

Cited by 12 publications

References 38 publications

Numerical approach for brain tumor growth model using higher order compact finite difference scheme

Numerical approach for brain tumor growth model using higher order compact finite difference scheme

Modeling and Solution of Reaction–Diffusion Equations by Using the Quadrature and Singular Convolution Methods

A gradient Discretisation Method For Anisotropic Reaction Diffusion Models with applications to the dynamics of brain tumours

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