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<p>This article discusses a&#xa0;mathematical model based on the reaction-diffusion equation, which represents an interpretation of the classical biological definition of cancer as uncontrolled cell proliferation with the potential for local invasion. The behavior of individual gliomas can be determined by two dominant factors: the net proliferation rate and infiltration or diffusion. One of the strengths of this model is that the product of these two rates forms the basis of the well-known Fisher approximation for the predicted constant radial expansion rate of a&#xa0;visible tumor, allowing one to estimate the time required for a&#xa0;tumor to expand from its detectable size at diagnosis to its size at death. For many years, this model has been studied as the Fisher-Kolmogorov-Petrovskii-Piskunov equation. In this paper, we develop a&#xa0;new discrete one-dimensional model of the spatiotemporal evolution of glioma. A distinctive feature of the difference model is its computational efficiency, achieved by replacing the multivariate model with a&#xa0;one-dimensional one while maintaining its adequate predictive ability, as well as by using an efficient algorithm for the numerical solution of systems of linear algebraic equations. The applied difference scheme, having a&#xa0;second-order approximation in the spatial variable and a&#xa0;first-order in the time variable, made it possible to reduce the problem of finding a&#xa0;solution to a&#xa0;nonlinear equation to solving a&#xa0;system of linear algebraic equations using the tridiagonal matrix algorithm. For the proposed scheme, monotonicity is studied and a&#xa0;sufficient condition for stability is proven. The results of the numerical implementation of the scheme are presented on model examples with real data in the form of combinations: [low diffusion, low proliferation], [high diffusion, low proliferation], [low diffusion, high proliferation], [high diffusion, high proliferation] and [medium diffusion, medium proliferation].</p>

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A discrete model of glioma growth and spread

  • 期刊论文

  • 发表日期：

    2025年11月7日

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  • Olga P. Barabash,
  • Igor P. Polovinkin

摘要

This article discusses a mathematical model based on the reaction-diffusion equation, which represents an interpretation of the classical biological definition of cancer as uncontrolled cell proliferation with the potential for local invasion. The behavior of individual gliomas can be determined by two dominant factors: the net proliferation rate and infiltration or diffusion. One of the strengths of this model is that the product of these two rates forms the basis of the well-known Fisher approximation for the predicted constant radial expansion rate of a visible tumor, allowing one to estimate the time required for a tumor to expand from its detectable size at diagnosis to its size at death. For many years, this model has been studied as the Fisher-Kolmogorov-Petrovskii-Piskunov equation. In this paper, we develop a new discrete one-dimensional model of the spatiotemporal evolution of glioma. A distinctive feature of the difference model is its computational efficiency, achieved by replacing the multivariate model with a one-dimensional one while maintaining its adequate predictive ability, as well as by using an efficient algorithm for the numerical solution of systems of linear algebraic equations. The applied difference scheme, having a second-order approximation in the spatial variable and a first-order in the time variable, made it possible to reduce the problem of finding a solution to a nonlinear equation to solving a system of linear algebraic equations using the tridiagonal matrix algorithm. For the proposed scheme, monotonicity is studied and a sufficient condition for stability is proven. The results of the numerical implementation of the scheme are presented on model examples with real data in the form of combinations: [low diffusion, low proliferation], [high diffusion, low proliferation], [low diffusion, high proliferation], [high diffusion, high proliferation] and [medium diffusion, medium proliferation].