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A BDF B-spline Method for a Nonlocal Tumor Growth Model

Bouhamidi Abderrahman, El Harraki Imad, Melouani Yassine

TL;DR

This work develops a nonlocal continuum model for tumor growth with nonlocal velocity effects and advection-dominated transport among proliferative cells, healthy cells, and nutrients. It provides a rigorous existence/uniqueness theory for the coupled nonlocal PDE system, first at the local level and then for the full nonlocal model, using semigroup and fixed-point methods on appropriate function spaces. The authors then exploit rotational invariance to derive a radial reformulation and introduce a cubic B-spline collocation in space together with a 6th-order implicit BDF time integration, forming a large, stiff ODE system solved by Newton iteration. Numerical experiments validate convergence against an analytical solution and demonstrate biologically meaningful tumor growth patterns, highlighting nonlocal adhesion and nutrient-driven proliferation in avascular scenarios, with potential applicability to optimal-control strategies for treatment planning.

Abstract

This paper presents a model for tumor growth using nonlocal velocity. We establish some results on the existence and uniqueness of the solution for a nonlocal tumor growth model. Many experiences show that tumor spheroid can be invariant by rotation and can guard the shape of the spheroid during the growth process in some particular cases. Here, we assume that the multiple components of the system are invariant by rotation. Then, we use the Backward Differentiation Formula (BDF) spline to solve the nonlocal system. To illustrate the effecienty of the proposed method, we performed numerical tests that simulate a tumor growth scenario. Such techniques may be used to provide informations on practical applications of the model.

A BDF B-spline Method for a Nonlocal Tumor Growth Model

TL;DR

This work develops a nonlocal continuum model for tumor growth with nonlocal velocity effects and advection-dominated transport among proliferative cells, healthy cells, and nutrients. It provides a rigorous existence/uniqueness theory for the coupled nonlocal PDE system, first at the local level and then for the full nonlocal model, using semigroup and fixed-point methods on appropriate function spaces. The authors then exploit rotational invariance to derive a radial reformulation and introduce a cubic B-spline collocation in space together with a 6th-order implicit BDF time integration, forming a large, stiff ODE system solved by Newton iteration. Numerical experiments validate convergence against an analytical solution and demonstrate biologically meaningful tumor growth patterns, highlighting nonlocal adhesion and nutrient-driven proliferation in avascular scenarios, with potential applicability to optimal-control strategies for treatment planning.

Abstract

This paper presents a model for tumor growth using nonlocal velocity. We establish some results on the existence and uniqueness of the solution for a nonlocal tumor growth model. Many experiences show that tumor spheroid can be invariant by rotation and can guard the shape of the spheroid during the growth process in some particular cases. Here, we assume that the multiple components of the system are invariant by rotation. Then, we use the Backward Differentiation Formula (BDF) spline to solve the nonlocal system. To illustrate the effecienty of the proposed method, we performed numerical tests that simulate a tumor growth scenario. Such techniques may be used to provide informations on practical applications of the model.

Paper Structure

This paper contains 9 sections, 3 theorems, 140 equations, 7 figures, 4 tables.

Table of Contents

  1. A nonlocal continuum model
  2. Existence and uniqueness of solution
  3. Existence and uniqueness of the solution of the local system
  4. Existence and uniqueness of the nonlocal system
  5. Radial model reformulation
  6. The BDF B-spline method
  7. Numerical experiments
  8. Convergence test with known analytical solution
  9. Numerical Test for Tumor Growth

Key Result

Theorem 1

Let the assumptions (A1)-(A6) hold. Then there exists a maximal time $T_{max}$ such that equation (localequation) has a unique positive solution in $C\left([0,T],L^\infty(\Omega_{R})^3\right)$. Furthermore, $U(t,\cdot)_{w}-U_{R}$ is a compactly supported function in $\Omega_{R}$ for all $t\leq T_{ma

Figures (7)

  • Figure 1: Approximate solution $\widetilde{U}_{h}$ and Exact solution $\widetilde{U}_{ex}$ for $t=0,\,\, t=25,\, \, t=50,\, \, t=75,\, \, t=100$.
  • Figure 2: Approximation of $\widetilde{P}$ on $[0,1]\times[0,100]$.
  • Figure 3: Approximation of $\widetilde{S}$ on $[0,1]\times[0,100]$.
  • Figure 4: Approximation of $\widetilde{M}$ on $[0,1]\times[0,100]$.
  • Figure 5: Radial tumor cells density $\widetilde{P}$, healthy cells density $\widetilde{S}$, and the concentration of nutriment and oxygen $\widetilde{M}$ for different time instances.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark
  • Proposition 1: The convolution of two radial functions
  • proof