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A Three-dimensional tumor growth model and its boundary instability

Jian-Guo Liu, Thomas Witelski, Xiaoqian Xu, Jiaqi Zhang

TL;DR

This work extends Feng et al. 2023 by developing a 3D sharp-interface tumor growth model and analyzing boundary stability through perturbation theory with spherical harmonics, complemented by 2D ADI validation to match front propagation speeds. By deriving and analyzing evolution functions that govern δ^{-1} dδ/dt, the authors identify a consistent in vitro stability across dimensions and a dimension- and parameter-dependent transition to instability in vivo, including a proposed λ^*(ℓ) threshold. The combination of analytical constructions (including 3D spherically symmetric solutions and perturbations) with numerical ADI verification provides a robust framework for predicting front dynamics in 3D tumor growth and offers insights into how nutrient supply and consumption shape tumor morphology. The results have potential implications for understanding malignant boundary evolution and for developing computational tools to anticipate tumor boundary behavior in realistic three-dimensional geometries.

Abstract

In this paper, we investigate the tumor instability by employing both analytical and numerical techniques to validate previous results and extend the analytical findings presented in a prior study by Feng et al 2023. Building upon the insights derived from the analytical reconstruction of key results in the aforementioned work in one dimension (1D) and two dimensions (2D), we extend our analysis to three dimensions (3D). Specifically, we focus on the determination of boundary instability using perturbation and asymptotic analysis along with spherical harmonics. Additionally, we have validated our analytical results in a two-dimensional framework by implementing the Alternating Directional Implicit (ADI) method, as detailed in Witelski and Bowen (2003). Our primary focus has been on ensuring that the numerical simulation of the propagation speed aligns accurately with the analytical findings. Furthermore, we have matched the simulated boundary stability with the analytical predictions derived from the evolution function, which will be defined in subsequent sections of our paper. These alignment is essential for accurately determining the stability or instability of tumor boundaries.

A Three-dimensional tumor growth model and its boundary instability

TL;DR

This work extends Feng et al. 2023 by developing a 3D sharp-interface tumor growth model and analyzing boundary stability through perturbation theory with spherical harmonics, complemented by 2D ADI validation to match front propagation speeds. By deriving and analyzing evolution functions that govern δ^{-1} dδ/dt, the authors identify a consistent in vitro stability across dimensions and a dimension- and parameter-dependent transition to instability in vivo, including a proposed λ^*(ℓ) threshold. The combination of analytical constructions (including 3D spherically symmetric solutions and perturbations) with numerical ADI verification provides a robust framework for predicting front dynamics in 3D tumor growth and offers insights into how nutrient supply and consumption shape tumor morphology. The results have potential implications for understanding malignant boundary evolution and for developing computational tools to anticipate tumor boundary behavior in realistic three-dimensional geometries.

Abstract

In this paper, we investigate the tumor instability by employing both analytical and numerical techniques to validate previous results and extend the analytical findings presented in a prior study by Feng et al 2023. Building upon the insights derived from the analytical reconstruction of key results in the aforementioned work in one dimension (1D) and two dimensions (2D), we extend our analysis to three dimensions (3D). Specifically, we focus on the determination of boundary instability using perturbation and asymptotic analysis along with spherical harmonics. Additionally, we have validated our analytical results in a two-dimensional framework by implementing the Alternating Directional Implicit (ADI) method, as detailed in Witelski and Bowen (2003). Our primary focus has been on ensuring that the numerical simulation of the propagation speed aligns accurately with the analytical findings. Furthermore, we have matched the simulated boundary stability with the analytical predictions derived from the evolution function, which will be defined in subsequent sections of our paper. These alignment is essential for accurately determining the stability or instability of tumor boundaries.
Paper Structure (29 sections, 147 equations, 17 figures)

This paper contains 29 sections, 147 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Model formulation
  3. Two models for the nutrient distribution
  4. The in vitro model.
  5. The in vivo model.
  6. The Two-Dimensional Problem
  7. Overview of the steps
  8. Results of 2D tumor boundary stability/instability in feng2023tumor
  9. The Three-Dimensional Problem
  10. 3D spherically symmetric solution for the in vitro model
  11. 3D spherically symmetric solution for the in vivo model
  12. Stability of 3D spherically symmetric in the in vitro model
  13. Stability of 3D spherically symmetric in the in vivo model
  14. Results and Discussion
  15. Results for 3D tumor model
  16. ...and 14 more sections

Figures (17)

  • Figure 1: Schematic figure of the tumor and the surrounding exterior. The cell population density $\rho$ is defined in the red shaded tumoral region; $c^{(o)}$ and $c^{(i)}$ are the nutrient concentration defined outside and inside the red shaded tumoral region feng2023tumor.
  • Figure 2: 2D in vivo evolution function with $G_0=1, c_B=100$; top (left): $\lambda=100$ and $R \in [0,20]$; top (right): $\lambda=100$ and $R \in [0,1]$; bottom (left): $\lambda=1$ and $R\in[0.50]$; bottom (right): $\lambda=0.8$ and $R\in[0,50]$. Here the $l$ denotes as wave number. Figure from feng2023tumor.
  • Figure 3: 2D tumor growth boundary evolution with $\lambda=0.5$ (three time profiles evolving from left to right) for an initial perturbation with wave number $m=8$
  • Figure 5: 3D in vitro tumor evolution (\ref{['3d vitro speed']})
  • Figure 6: 3D in vivo tumor evolution (\ref{['3d speed in vivo']})
  • ...and 12 more figures