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A Stochastic Interacting Particle-Field Algorithm for a Haptotaxis Advection-Diffusion System Modeling Cancer Cell Invasion

Boyi Hu, Zhongjian Wang, Jack Xin, Zhiwen Zhang

TL;DR

This work develops a stochastic interacting particle-field (SIPF) algorithm to simulate cancer cell invasion within a haptotaxis advection-diffusion (HAD) system, using a particle representation for cell density and a spectrally discretized ECM field solved by FFT-based methods. The method updates the MDE field implicitly via a Green’s-function kernel, the ECM field explicitly in Fourier space, and the cell density through Euler–Maruyama dynamics driven by the ECM gradient, yielding a mesh-free, low-cost, and self-adaptive solver that handles nearly singular free-boundary phenomena in 3D. The authors establish global well-posedness results for the HAD system and derive integral identities that facilitate validation of numerical approximations. Numerical experiments show SIPF outperforms finite-difference and spectral methods in 3D, especially for small diffusion, and remain stable under challenging conditions such as multiple clusters and reduced diffusion, highlighting its potential for realistic tumor invasion modeling and future extensions toward oxygen-coupled and multi-species haptotaxis models.

Abstract

The investigation of tumor invasion and metastasis dynamics is crucial for advancements in cancer biology and treatment. Many mathematical models have been developed to study the invasion of host tissue by tumor cells. In this paper, we develop a novel stochastic interacting particle-field (SIPF) algorithm that accurately simulates the cancer cell invasion process within the haptotaxis advection-diffusion (HAD) system. Our approach approximates solutions using empirical measures of particle interactions, combined with a smoother field variable - the extracellular matrix concentration (ECM) - computed by the spectral method. We derive a one-step time recursion for both the positions of stochastic particles and the field variable using the implicit Euler discretization, which is based on the explicit Green's function of an elliptic operator characterized by the Laplacian minus a positive constant. Our numerical experiments demonstrate the superior performance of the proposed algorithm, especially in computing cancer cell growth with thin free boundaries in three-dimensional (3D) space. Numerical results show that the SIPF algorithm is mesh-free, self-adaptive, and low-cost. Moreover, it is more accurate and efficient than traditional numerical techniques such as the finite difference method (FDM) and spectral methods.

A Stochastic Interacting Particle-Field Algorithm for a Haptotaxis Advection-Diffusion System Modeling Cancer Cell Invasion

TL;DR

This work develops a stochastic interacting particle-field (SIPF) algorithm to simulate cancer cell invasion within a haptotaxis advection-diffusion (HAD) system, using a particle representation for cell density and a spectrally discretized ECM field solved by FFT-based methods. The method updates the MDE field implicitly via a Green’s-function kernel, the ECM field explicitly in Fourier space, and the cell density through Euler–Maruyama dynamics driven by the ECM gradient, yielding a mesh-free, low-cost, and self-adaptive solver that handles nearly singular free-boundary phenomena in 3D. The authors establish global well-posedness results for the HAD system and derive integral identities that facilitate validation of numerical approximations. Numerical experiments show SIPF outperforms finite-difference and spectral methods in 3D, especially for small diffusion, and remain stable under challenging conditions such as multiple clusters and reduced diffusion, highlighting its potential for realistic tumor invasion modeling and future extensions toward oxygen-coupled and multi-species haptotaxis models.

Abstract

The investigation of tumor invasion and metastasis dynamics is crucial for advancements in cancer biology and treatment. Many mathematical models have been developed to study the invasion of host tissue by tumor cells. In this paper, we develop a novel stochastic interacting particle-field (SIPF) algorithm that accurately simulates the cancer cell invasion process within the haptotaxis advection-diffusion (HAD) system. Our approach approximates solutions using empirical measures of particle interactions, combined with a smoother field variable - the extracellular matrix concentration (ECM) - computed by the spectral method. We derive a one-step time recursion for both the positions of stochastic particles and the field variable using the implicit Euler discretization, which is based on the explicit Green's function of an elliptic operator characterized by the Laplacian minus a positive constant. Our numerical experiments demonstrate the superior performance of the proposed algorithm, especially in computing cancer cell growth with thin free boundaries in three-dimensional (3D) space. Numerical results show that the SIPF algorithm is mesh-free, self-adaptive, and low-cost. Moreover, it is more accurate and efficient than traditional numerical techniques such as the finite difference method (FDM) and spectral methods.
Paper Structure (11 sections, 2 theorems, 57 equations, 6 figures, 3 tables, 4 algorithms)

This paper contains 11 sections, 2 theorems, 57 equations, 6 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Cancer Cell Invasion HAD System
  3. Global Well-Possedness of Nonnegative Solutions
  4. Integral Identities
  5. SIPF Algorithm for Cancer Cell Invasion HAD System
  6. Numerical Experiments
  7. Comparing radial/FDM/SIPF methods in 3D
  8. Regime of Small Diffusion Coefficient
  9. Two Clusters of Cancer Cells Evolution
  10. Comparing FDM and SIPF based on Integral Identities
  11. Conclusion and Future Work

Key Result

Proposition 1

Let $\Omega \subset \mathbb{R}^N$, $N \geq 1$ be a domain with $C^2$ boundary and $p > N$. Given the non-negative initial value $(\rho_0, f_0, m_0) \in W^{1,p}(\Omega) \times W^{1,\infty}(\Omega) \times X_{p}^{\theta}$, $\theta \in \left(\frac{N+p}{2p}, 1\right)$, there exists $T > 0$ (depending onl Moreover, the solution depends continuously on the initial data.

Figures (6)

  • Figure 1: A numerical simulation of the system \ref{['cancer system']}, with constant tumor cell diffusion, reveals the spatio-temporal dynamics of the tumor invasion process. The figure shows the emergence of a ring of cells that breaks away from the primary tumor mass and invades deeper into the ECM.
  • Figure 2: 3D numerical solutions of the system \ref{['cancer system']} with constant tumor cell diffusion showing the cell density computed by radial, FDM, and SIPF.
  • Figure 3: 3D relative $L^2$ errors of $m$ in SIPF (radial solution being the reference).
  • Figure 4: Comparing radial solutions, FDM (red) and SIPF (green) at $d_n=0.0002$, shows under-shoot (violating positivity) and inaccurate peak locations in FDM.
  • Figure 5: Two clusters of cancer cells merge into a larger single cluster and spread further.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Proposition 1: Theorem 3.3, lictcanu2010asymptotic
  • Proposition 2