Approximation and existence of a viscoelastic phase-field model for tumour growth in two and three dimensions

Harald Garcke; Dennis Trautwein

doi:10.3934/dcdss.2023181

Article Contents

2024, Volume 17, Issue 1: 221-284. Doi: 10.3934/dcdss.2023181

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Approximation and existence of a viscoelastic phase-field model for tumour growth in two and three dimensions

Harald Garcke^1, and
Dennis Trautwein^1, ,

1.
University of Regensburg, Germany

^*Corresponding author: Dennis Trautwein
^*Corresponding author: Dennis Trautwein

Received: May 2023

Revised: September 2023

Early access: September 2023

Published: January 2024

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Abstract

In this work, we present a phase-field model for tumour growth, where a diffuse interface separates a tumour from the surrounding host tissue. In our model, we consider transport processes by an internal, non-solenoidal velocity field. We include viscoelastic effects with the help of a generalized Oldroyd-B type description with relaxation and possible stress generation by growth. The elastic energy density is coupled to the phase-field variable which allows to model invasive growth towards areas with less mechanical resistance. The main analytical result is the existence of weak solutions in two and three space dimensions in the case of additional stress diffusion. The idea behind the proof is to use a numerical approximation with a fully-practical, stable and (subsequence) converging finite element scheme. The physical properties of the model are preserved with the help of a regularization technique, uniform estimates and a limit passage on the fully-discrete level. Finally, we illustrate the practicability of the discrete scheme with the help of numerical simulations in two and three dimensions.

Keywords:

Mathematics Subject Classification: Primary: 65M12; Secondary: 65M60, 76A10, 35Q92, 35K35.

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Figure 1. Typical setting for phase-field models, where a smooth interface with width related to $ \varepsilon>0 $ separates the pure phases

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Figure 2. Framework of multiple configurations

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Figure 3. Snapshots of the first example with $ \kappa_t = 0 $. First row: $ \varphi_h^n $ at $ t = 0, 5, 10, 14 $, where $ \varphi_h^n = 1 $ (red) in the tumour tissue and $ \varphi_h^n = -1 $ (blue) in the host tissue. Second row: the nutrient $ \sigma_h^n $, $ \left\lvert{{ {\mathbf{v}}_h^n}}\right\rvert $ (with the velocity field $ {\mathbf{v}}_h^n $), and both eigenvalues of $ {\mathbb{B}}_h^n $ at $ t = 14 $

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Figure 4. Snapshots of the first example with $ \kappa_t = 0.5 $. First row: $ \varphi_h^n $ at $ t = 0, 5, 10, 14 $, where $ \varphi_h^n = 1 $ (red) in the tumour tissue and $ \varphi_h^n = -1 $ (blue) in the host tissue. Second row: the nutrient $ \sigma_h^n $, $ \left\lvert{{ {\mathbf{v}}_h^n}}\right\rvert $ (with the velocity field $ {\mathbf{v}}_h^n $), and both eigenvalues of $ {\mathbb{B}}_h^n $ at $ t = 14 $

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Figure 5. Snapshots of the first example with $ \kappa_t = -0.5 $. First row: $ \varphi_h^n $ at $ t = 0, 5, 10, 14 $, where $ \varphi_h^n = 1 $ (red) in the tumour tissue and $ \varphi_h^n = -1 $ (blue) in the host tissue. Second row: the nutrient $ \sigma_h^n $, $ \left\lvert{{ {\mathbf{v}}_h^n}}\right\rvert $ (with the velocity field $ {\mathbf{v}}_h^n $), and both eigenvalues of $ {\mathbb{B}}_h^n $ at $ t = 14 $

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Figure 6. Snapshots of the second example with $ \kappa_t = -2 $. First row: $ \varphi_h^n $ at $ t = 0, 5, 10, 13.5 $, where $ \varphi_h^n = 1 $ (red) in the tumour tissue and $ \varphi_h^n = -1 $ (blue) in the host tissue. Second row: the nutrient $ \sigma_h^n $, $ \left\lvert{{ {\mathbf{v}}_h^n}}\right\rvert $ (with the velocity field $ {\mathbf{v}}_h^n $), and both eigenvalues of $ {\mathbb{B}}_h^n $ at $ t = 13.5 $

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Figure 7. Snapshots of the second example with $ \kappa_t = -1 $. First row: $ \varphi_h^n $ at $ t = 0, 5, 10, 13.5 $, where $ \varphi_h^n = 1 $ (red) in the tumour tissue and $ \varphi_h^n = -1 $ (blue) in the host tissue. Second row: the nutrient $ \sigma_h^n $, $ \left\lvert{{ {\mathbf{v}}_h^n}}\right\rvert $ (with the velocity field $ {\mathbf{v}}_h^n $), and both eigenvalues of $ {\mathbb{B}}_h^n $ at $ t = 13.5 $

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Figure 8. Snapshots of the second example with $ \kappa_t = 1 $. First row: $ \varphi_h^n $ at $ t = 0, 5, 10, 13.5 $, where $ \varphi_h^n = 1 $ (red) in the tumour tissue and $ \varphi_h^n = -1 $ (blue) in the host tissue. Second row: the nutrient $ \sigma_h^n $, $ \left\lvert{{ {\mathbf{v}}_h^n}}\right\rvert $ (with the velocity field $ {\mathbf{v}}_h^n $), and both eigenvalues of $ {\mathbb{B}}_h^n $ at $ t = 13.5 $

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Figure 9. Snapshots of the example in 3d with $ \kappa_t = 0 $. First row: time evolution of the interface of $ \varphi_h^n $ at $ t = 0, 1, 2, 3 $, as well as $ \varphi_h^n $ with the corresponding mesh at $ t = 3 $. Second row: the nutrient $ \sigma_h^n $, the velocity magnitude $ \left\lvert{{ {\mathbf{v}}_h^n}}\right\rvert $ (with the velocity field $ {\mathbf{v}}_h^n $), and the three eigenvalues of $ {\mathbb{B}}_h^n $ at $ t = 3 $. A cut was taken along the plane with normal $ (1, 1, 0)^\top $ and origin $ (0, 0, 0)^\top $

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Figure 10. Snapshots of the example in 3d with $ \kappa_t = 4 $. First row: time evolution of the interface of $ \varphi_h^n $ at $ t = 0, 1, 2, 3 $, as well as $ \varphi_h^n $ with the corresponding mesh at $ t = 3 $. Second row: the nutrient $ \sigma_h^n $, the velocity magnitude $ \left\lvert{{ {\mathbf{v}}_h^n}}\right\rvert $ (with the velocity field $ {\mathbf{v}}_h^n $), and the three eigenvalues of $ {\mathbb{B}}_h^n $ at $ t = 3 $. A cut was taken along the plane with normal $ (1, 1, 0)^\top $ and origin $ (0, 0, 0)^\top $

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Figure 11. Snapshots of the example in 3d with $ \kappa_t = -5 $. First row: time evolution of the interface of $ \varphi_h^n $ at $ t = 0, 1, 2, 3 $, as well as $ \varphi_h^n $ with the corresponding mesh at $ t = 3 $. Second row: the nutrient $ \sigma_h^n $, the velocity magnitude $ \left\lvert{{ {\mathbf{v}}_h^n}}\right\rvert $ (with the velocity field $ {\mathbf{v}}_h^n $), and the three eigenvalues of $ {\mathbb{B}}_h^n $ at $ t = 3 $. A cut was taken along the plane with normal $ (1, 1, 0)^\top $ and origin $ (0, 0, 0)^\top $

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