Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Maximum Bound Principle and Bound Preserving ETD schemes for a Phase-Field Model of Tumor Growth with Extracellular Matrix Degradation

Qiumei Huang, Zhonghua Qiao, Cheng Wang, Huiting Yang

TL;DR

The paper addresses phase-field modeling of tumor growth with ECM degradation by developing MBP- and bound-preserving, decoupled exponential time differencing (ETD) schemes. The approach yields explicit, linear, and easily solvable updates for a coupled system of phi_T, phi_N, phi_sigma, phi_M, and theta, with stabilization terms and a cut-off ensuring physical bounds. Key contributions include proving MBP/bound preservation for the numerical scheme, implementing FFT-accelerated ETD/ETDRK methods, and validating the model via 2D and 3D simulations that reveal how ECM degradation, MDE activity, and nutrient availability shape tumor dynamics. The work offers a reliable, efficient computational framework for exploring tumor-ECM interactions with potential implications for understanding progression and evaluating therapeutic strategies.

Abstract

In cancer research, the role of the extracellular matrix (ECM) and its associated matrix-degrading enzyme (MDE) has been a significant area of focus. This study presents a numerical algorithm designed to simulate a previously established tumor model that incorporates various biological factors, including tumor cells, viable cells, necrotic cells, and the dynamics of MDE and ECM. The model consists of a system that includes a phase field equation, two reaction-diffusion equations, and two ordinary differential equations. We employ the fast exponential time differencing Runge-Kutta (ETDRK) method with stabilizing terms to solve this system, resulting in a decoupled, explicit, linear numerical algorithm. The objective of this algorithm is to preserve the physical properties of the model variables, including the maximum bound principle (MBP) for nutrient concentration and MDE volume fraction, as well as bound preserving for ECM density and tumor volume fraction. We perform simulations of 2D and 3D tumor models {and discuss how different biological components impact growth dynamics. These simulations may help predict tumor evolution trends, offer insights for related biological and medical research,} potentially reduce the number and cost of experiments, and improve research efficiency.

Maximum Bound Principle and Bound Preserving ETD schemes for a Phase-Field Model of Tumor Growth with Extracellular Matrix Degradation

TL;DR

The paper addresses phase-field modeling of tumor growth with ECM degradation by developing MBP- and bound-preserving, decoupled exponential time differencing (ETD) schemes. The approach yields explicit, linear, and easily solvable updates for a coupled system of phi_T, phi_N, phi_sigma, phi_M, and theta, with stabilization terms and a cut-off ensuring physical bounds. Key contributions include proving MBP/bound preservation for the numerical scheme, implementing FFT-accelerated ETD/ETDRK methods, and validating the model via 2D and 3D simulations that reveal how ECM degradation, MDE activity, and nutrient availability shape tumor dynamics. The work offers a reliable, efficient computational framework for exploring tumor-ECM interactions with potential implications for understanding progression and evaluating therapeutic strategies.

Abstract

In cancer research, the role of the extracellular matrix (ECM) and its associated matrix-degrading enzyme (MDE) has been a significant area of focus. This study presents a numerical algorithm designed to simulate a previously established tumor model that incorporates various biological factors, including tumor cells, viable cells, necrotic cells, and the dynamics of MDE and ECM. The model consists of a system that includes a phase field equation, two reaction-diffusion equations, and two ordinary differential equations. We employ the fast exponential time differencing Runge-Kutta (ETDRK) method with stabilizing terms to solve this system, resulting in a decoupled, explicit, linear numerical algorithm. The objective of this algorithm is to preserve the physical properties of the model variables, including the maximum bound principle (MBP) for nutrient concentration and MDE volume fraction, as well as bound preserving for ECM density and tumor volume fraction. We perform simulations of 2D and 3D tumor models {and discuss how different biological components impact growth dynamics. These simulations may help predict tumor evolution trends, offer insights for related biological and medical research,} potentially reduce the number and cost of experiments, and improve research efficiency.

Paper Structure

This paper contains 25 sections, 7 theorems, 82 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model review
  3. Total tumor cells dynamics
  4. Necrotic tumor cell dynamics
  5. Critical nutrient dynamics
  6. MDE dynamics
  7. ECM dynamics
  8. The coupled tumor growth system
  9. The numerical method
  10. Two-dimensional spatial discretization
  11. Fully discrete numerical scheme for $\phi_T$
  12. Fully discrete numerical scheme for $\phi_N$
  13. Fully discrete numerical scheme for $\phi_{\sigma}$
  14. The fully discrete scheme
  15. Discrete MBP for $\psi_\sigma$
  16. ...and 10 more sections

Key Result

Theorem 3.1

(Discrete bound preservation for $\Phi_N$ of the trapezoidal rule scheme): If the initial values satisfy $0\leq\phi_{N_{i,j}}^{0} \leq \phi_{T_{i,j}}^{0}\leq 1$ hold for any $t_{n+1}$, then the trapezoidal rule scheme TR_phi_N maintains the discrete bound preservation. Specifically, for any time ste

Figures (12)

  • Figure 1: Organization of tumor cells and phase of tumor dynamics
  • Figure 2: Convergence test of the solution $\phi_T$
  • Figure 3: The volume fraction evolution of total tumor cells $\phi_T$, necrotic cells $\phi_N$, and viable cells $\phi_V$.
  • Figure 4: Evolution of ECM density and nutrient concentration in the tumor growth
  • Figure 5: MBP test of $\psi_\sigma,$ and $\psi_M$
  • ...and 7 more figures

Theorems & Definitions (13)

  • Theorem 3.1
  • proof
  • Lemma 3.1
  • Lemma 3.2: du2021maximum
  • Theorem 3.2
  • proof
  • Remark 3.1
  • Lemma 3.3
  • Theorem 3.3
  • proof
  • ...and 3 more