Abstract
We consider a biphasic continuum model for avascular tumour growth in two spatial dimensions, in which a cell phase and a fluid phase follow conservation of mass and momentum. A limiting nutrient that follows a diffusion process controls the birth and death rate of the tumour cells. The cell volume fraction, cell velocity–fluid pressure system, and nutrient concentration are the model variables. A coupled system of a hyperbolic conservation law, a viscous fluid model, and a parabolic diffusion equation governs the dynamics of the model variables. The tumour boundary moves with the normal velocity of the outermost layer of cells, and this time-dependence is a challenge in designing and implementing a stable and fast numerical scheme. We recast the model into a form where the hyperbolic equation is defined on a fixed extended domain and retrieve the tumour boundary as the interface at which the cell volume fraction decreases below a threshold value. This procedure eliminates the need to track the tumour boundary explicitly and the computationally expensive re-meshing of the time-dependent domains. A numerical scheme based on finite volume methods for the hyperbolic conservation law, Lagrange \(\mathbb {P}_2 - \mathbb {P}_1\) Taylor-Hood finite element method for the viscous system, and mass-lumped finite element method for the parabolic equations is implemented in two spatial dimensions, and several cases are studied. We demonstrate the versatility of the numerical scheme in catering for irregular and asymmetric initial tumour geometries. When the nutrient diffusion equation is defined only in the tumour region, the model depicts growth in free suspension. On the contrary, when the nutrient diffusion equation is defined in a larger fixed domain, the model depicts tumour growth in a polymeric gel. We present numerical simulations for both cases and the results are consistent with theoretical and heuristic expectations such as early linear growth rate and preservation of radial symmetry when the boundary conditions are symmetric. The work presented here could be extended to include the effect of drug treatment of growing tumours.
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Data Availability Statement
The datasets—specifically, MATLAB code for NUM simulations—generated during and/or analysed during the current study are available in the GitHub repository, https://github.com/gopikrishnancr/2D_tumour_growth_FEM_FVM.
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Acknowledgements
The authors are grateful to Prof. Neela Nataraj, Indian Institute of Technology Bombay, India for the valuable suggestions and help. The authors are grateful to Dr. Laura Bray, Queensland University of Technology, Australia and Ms Berline Murekatete, Queensland University of Technology, Australia for helpful discussions and providing image data for the irregular tumour depicted in Fig. 9a.
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Appendix
Appendix
1.1 A Some Classical Definitions and Results
We recall two classical results used in this article.
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a.
Theorem (Korn’s second inequality) [10, Theorem 3.78]. If \(\Omega \subset \mathbb {R}^d\), where \(d =2,3\) is a domain, then there exists a positive constant \(\mathscr {C}_K\) such that, for every \(\varvec{v} \in \mathbf {H}^1_d(\Omega )\),
$$\begin{aligned} \mathscr {C}_K ||{\varvec{v}}||_{1,\Omega } \le ||\nabla _s{\varvec{v}}||_{0,\Omega } + ||{\varvec{v}}||_{0,\Omega *}. \end{aligned}$$ -
b.
Lemma (Petree-Tartar) [10, Lemma A.38]. If \(X,\,Y,\,\) and Z are Banach spaces, \(A : X \rightarrow Y\) is an injective operator, \(T : X \rightarrow Z\) is a compact operator, and there exists a positive constant \(\mathscr {C}_{1}\) such that \(\mathscr {C}_{1} ||x||_X \le ||Ax||_Y + ||Tx||_Z\), then there exists a positive constant \(\mathscr {C}_{PT}\) such that \(\mathscr {C}_{PT} ||x||_X \le ||Ax||_Y\).
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c.
Definition (Bounded variation) By the space BV(A), where \(A \subset \mathbb {R}^d\) is an open set we mean the collection of all functions \(u : A \rightarrow \mathbb {R}\) such that \(||u||_{BV} < \infty \), where
$$\begin{aligned} ||u||_{BV} := \sup \left\{ \int _{A} u\,\mathrm {div}(\varphi )\,\mathrm {d}{\varvec{x}}: \varphi \in \mathscr {C}_c^1(A;\mathbb {R}^d), ||\varphi ||_{L^\infty (A)} \le 1 \right\} . \end{aligned}$$
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Droniou, J., Flegg, J.A. & Remesan, G.C. Numerical Solution of a Two Dimensional Tumour Growth Model with Moving Boundary. J Sci Comput 85, 22 (2020). https://doi.org/10.1007/s10915-020-01326-6
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DOI: https://doi.org/10.1007/s10915-020-01326-6