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Modeling tumor growth with variable mass and angiogenesis-driven perfusion through a 3D-1D coupled framework

Chiara Giverso, Denise Grappein, Stefano Scialò

Abstract

Tumor growth beyond a critical size relies on the development of a functional vascular network, which ensures adequate oxygen and nutrient supply. In this work, we present a modeling framework based on an optimization-based 3D-1D coupling strategy to simulate perfusion in a tumoral tissue with growing mass, interacting with a dynamically evolving capillary network. The tumor is described as a multiphase system including tumor cells and interstitial fluid, governed by a non-linear PDE system for cell volume fraction, pressure, oxygen, and VEGF, and discretized via finite elements. Capillary growth is tackled using a continuous-discrete hybrid tip-tracking approach. The vascular geometry is updated over time according to angiogenic signals, and coupled to the tissue model through a constrained optimization formulation that enforces fluid and nutrient exchange via interface variables. A sensitivity analysis using the Morris elementary effect method identifies key parameters influencing system behavior. Results highlight the critical role of vascular development in regulating tissue perfusion and tumor progression. Overall, the proposed numerical approach provides a versatile tool for investigating tumor-vascular interactions and can support further quantitative analysis of angiogenesis and tumor perfusion dynamics.

Modeling tumor growth with variable mass and angiogenesis-driven perfusion through a 3D-1D coupled framework

Abstract

Tumor growth beyond a critical size relies on the development of a functional vascular network, which ensures adequate oxygen and nutrient supply. In this work, we present a modeling framework based on an optimization-based 3D-1D coupling strategy to simulate perfusion in a tumoral tissue with growing mass, interacting with a dynamically evolving capillary network. The tumor is described as a multiphase system including tumor cells and interstitial fluid, governed by a non-linear PDE system for cell volume fraction, pressure, oxygen, and VEGF, and discretized via finite elements. Capillary growth is tackled using a continuous-discrete hybrid tip-tracking approach. The vascular geometry is updated over time according to angiogenic signals, and coupled to the tissue model through a constrained optimization formulation that enforces fluid and nutrient exchange via interface variables. A sensitivity analysis using the Morris elementary effect method identifies key parameters influencing system behavior. Results highlight the critical role of vascular development in regulating tissue perfusion and tumor progression. Overall, the proposed numerical approach provides a versatile tool for investigating tumor-vascular interactions and can support further quantitative analysis of angiogenesis and tumor perfusion dynamics.

Paper Structure

This paper contains 21 sections, 56 equations, 11 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Notation and main assumptions
  3. Mathematical model
  4. Cell aggregate balance equations
  5. The tumor cells problem
  6. The pressure problem
  7. The oxygen concentration model
  8. The chemotactic growth factor model
  9. The capillary growth model
  10. Summary of model equations
  11. Problem discretization and solving strategy
  12. Treatment of 3D-1D coupled problems
  13. Capillary network update
  14. Linearized time-discrete tumor cell fraction problem
  15. Time-discrete 3D-1D pressure problem
  16. ...and 6 more sections

Figures (11)

  • Figure 1: VEGF source term $\Gamma_g(\phi_c,c)$ for $G=1$, $c^\star=11.5~ \mathrm{mmHg}$ and $b=11.5$.
  • Figure 2: Example of branching probability function used in the simulations. In this case $P_{br} (g)= \frac{1}{1 + e^{-a(1 - d)}}$ with $a$ and $d$ such that $P_{br}(g_{br})=0.99$ and $P_{br}(g_{lim})=0.05$, for $g_{br}=1.5\cdot 10^{-13}$ mmHg and $g_{lim}=0.25 \cdot 10^{-13}$ mmHg.
  • Figure 3: Initial capillary network $\Lambda^0$ and meshed computational domain $\Omega$.
  • Figure 4: SA1: EE sensitivity tests at $t = 7, 14, 21$ days.
  • Figure 5: SA1: EE sensitivity tests at $t = 7, 14, 21$ days.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Remark 1