Prashant K. Jha CEAD Lab

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Mixed-dimensional modeling of tumor growth and angiogenesis

A mixed-dimensional model of tumor growth in which an evolving one-dimensional vascular network carries blood and nutrients through three-dimensional tissue.

past2019–2022 MD Anderson
Mechanics
  • Multiphase mixture theory
  • Transport in porous media
  • Growth and remodeling
Methods
  • Phase-field models
  • Mixed-dimensional 3D-1D coupling
  • Finite elements
  • Bayesian calibration

Coupling a 3D tissue model to a 1D vascular network

This was joint work with the late J. Tinsley Oden at the Oden Institute, UT Austin, and colleagues at the Technical University of Munich. The study analyzed a 3D-1D model of tumor growth that couples flow and nutrient transport in the vessel network to nutrient and tumor evolution in the tissue domain, and established well-posedness for the resulting coupled multidimensional nonlinear model. See the article.

Figure 1 shows the tumor advancing toward the nutrient-rich artery at right. The top row tracks the φ_T = 0.8 contour of the total tumor volume fraction, the sum of the proliferative, hypoxic and necrotic cell species. The bottom row separates the necrotic (φ_N = 0.4, black), hypoxic (φ_H = 0.45, orange) and proliferative (φ_P = 0.5, green) populations. Figure is taken from article.

Figure 1: tumor growth toward a nutrient-rich artery

Growing the vasculature with the tumor

The subsequent study added angiogenesis. Nutrient-starved tumor cells release tumor angiogenesis factors that stimulate nearby vessels to grow toward the tumor, altering the coupled supply of nutrients and the evolution of the tumor. The resulting paper is Fritz et al. (2021).

Figure 2 demonstrates the angiogenesis effect and simultaneous tumor growth. Figure 3 shows quantities of interest — total tumor φ_T and the proliferative φ_P, hypoxic φ_H and necrotic φ_N masses — with and without angiogenesis. Angiogenesis is stochastic, so the simulation was run repeatedly to capture the variation in those quantities. Figures 2 and 3 and the simulation videos are from Fritz et al. (2021).

Figure 2: Network growth for two-vessels and complex network settings
Figure 3: effect of angiogenesis on tumor-associated quantities of interest
Angiogenesis simulation, two-vessels setup (sample 1)
Angiogenesis simulation, two-vessels setup (sample 2)
Angiogenesis simulation, complex network setup (sample 1)
Angiogenesis simulation, complex network setup (sample 2)

What it established, and where it led

This project concluded at the end of the Oden Institute appointment. Its central computational contribution was a mixed-dimensional formulation coupling an evolving one-dimensional vascular network and dynamic flow to transport and growth in a three-dimensional tissue domain.

The predictive-science questions developed around that model later informed work on model-error estimation, Bayesian calibration, optimal experimental design and reliable scientific computing.

Publications