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A time-fractional Fisher-KPP equation for tumor growth: Analysis and numerical simulation

Marvin Fritz, Nikos I. Kavallaris

TL;DR

This work addresses tumor-growth modeling via a time-fractional Fisher–KPP equation in which the Riemann–Liouville derivative acts on the diffusion term, capturing memory-driven subdiffusive transport. The authors develop a rigorous analytical framework, proving local well-posedness with a Galerkin method and extending to global well-posedness for small initial data using Henry–Gronwall–Bihari-type estimates, complemented by a nonuniform convolution-quadrature numerical scheme. Numerically, the physically consistent RL-based model exhibits slower, subdiffusive front propagation and monotone mass evolution compared to a Caputo-in-time formulation that can show overshoot and accelerated fronts. These results underscore the importance of correctly placing fractional derivatives in diffusion terms for subdiffusive reaction–diffusion models of tumor growth and offer a solid foundation for further study, including inverse problems and parameter estimation under memory effects.

Abstract

We study a time-fractional Fisher-KPP equation involving a Riemann-Liouville fractional derivative acting on the diffusion term, as derived by Angstmann and Henry (Entropy, 22:1035, 2020). The model captures memory effects in diffusive population dynamics and serves as a framework for tumor growth modeling. We first establish local well-posedness of weak solutions. The analysis combines a Galerkin approximation with a refined a priori estimate based on a Bihari-Henry-Gronwall inequality, addressing the nonlinear coupling between the fractional diffusion and the reaction term. For small initial data, we further prove global well-posedness and asymptotic stability. A numerical method based on a nonuniform convolution quadrature scheme is then proposed and validated. Simulations demonstrate distinct dynamical behaviors compared to conventional formulations, emphasizing the physical consistency of the present model in describing tumor progression.

A time-fractional Fisher-KPP equation for tumor growth: Analysis and numerical simulation

TL;DR

This work addresses tumor-growth modeling via a time-fractional Fisher–KPP equation in which the Riemann–Liouville derivative acts on the diffusion term, capturing memory-driven subdiffusive transport. The authors develop a rigorous analytical framework, proving local well-posedness with a Galerkin method and extending to global well-posedness for small initial data using Henry–Gronwall–Bihari-type estimates, complemented by a nonuniform convolution-quadrature numerical scheme. Numerically, the physically consistent RL-based model exhibits slower, subdiffusive front propagation and monotone mass evolution compared to a Caputo-in-time formulation that can show overshoot and accelerated fronts. These results underscore the importance of correctly placing fractional derivatives in diffusion terms for subdiffusive reaction–diffusion models of tumor growth and offer a solid foundation for further study, including inverse problems and parameter estimation under memory effects.

Abstract

We study a time-fractional Fisher-KPP equation involving a Riemann-Liouville fractional derivative acting on the diffusion term, as derived by Angstmann and Henry (Entropy, 22:1035, 2020). The model captures memory effects in diffusive population dynamics and serves as a framework for tumor growth modeling. We first establish local well-posedness of weak solutions. The analysis combines a Galerkin approximation with a refined a priori estimate based on a Bihari-Henry-Gronwall inequality, addressing the nonlinear coupling between the fractional diffusion and the reaction term. For small initial data, we further prove global well-posedness and asymptotic stability. A numerical method based on a nonuniform convolution quadrature scheme is then proposed and validated. Simulations demonstrate distinct dynamical behaviors compared to conventional formulations, emphasizing the physical consistency of the present model in describing tumor progression.

Paper Structure

This paper contains 16 sections, 5 theorems, 84 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Modeling
  3. Analytical Preliminaries
  4. Notation
  5. Sonine kernels
  6. Fractional derivatives
  7. Gronwall-type inequalities
  8. Well-Posedness Analysis
  9. Local well-posedness
  10. Global well-posedness for small data
  11. Numerical experiments
  12. Temporal and spatial discretization
  13. Fully discrete formulation
  14. Implementation and setup
  15. Numerical results
  16. ...and 1 more sections

Key Result

Lemma 1

Let $c > 0$, $u, f \in C^0([0, T];\mathbb{R}_{\geq 0})$ for some $T>0$, and $\psi \in C^0(\mathbb{R}_{\geq 0};\mathbb{R}_{\geq 0})$ be nondecreasing with $\psi(0) = 0$. If then where

Figures (4)

  • Figure 1: Initial conditions: (left) single circular patch, (middle) four separated circles, (right) irregular "blob" geometry.
  • Figure 2: Total mass evolution under both models: solid lines correspond to \ref{['Eq:Fisher']}, dashed lines to \ref{['Eq:WrongModel']}.
  • Figure 3: Comparison of consistent \ref{['Eq:Fisher']} and Caputo-in-time \ref{['Eq:WrongModel']} for $\alpha=\tfrac{1}{2}$ at different times.
  • Figure 4: Evolution of the complex initial condition for fractional orders $\alpha\in\{\tfrac{1}{4},\tfrac{1}{2},\tfrac{3}{4},1\}$.

Theorems & Definitions (11)

  • Lemma 1: Henry--Gronwall--Bihari, cf. ouaddah2021fractional
  • Lemma 2
  • proof
  • Theorem 1: Local well-posedness
  • proof
  • Lemma 3: Regularity of the history force
  • proof
  • Theorem 2: Global well-posedness for small data
  • proof
  • Remark 1
  • ...and 1 more