Simulation of the relaxed-degenerate Cahn-Hilliard model and effect of the relaxation
To solve the computational cost issue of previous numerical methods for the Cahn-Hilliard equation with a biologically relevant choice of potential, we proposed a relaxation of the equation. scheme, we were able to show analytically that it retrieves the properties of the relaxed model. For the efficient linear scheme, the preservation of the structure of the model is observed during the numerical simulations. Still, we can not prove analytically that the discrete energy is dissipating. This property is essential to minimize the approximation error and is required to prove that the solution of the discrete scheme converges to one solution of the continuous model. We aim to adapt the Scalar Auxiliary Variable method to our relaxed-degenerate Cahn- Hilliard model to solve this problem. Indeed, the RDCH model can be formulated as a gradientflow of its associated energy, and the SAV method can thus be applied. We adapted the SAV method for the volume-filling Keller-Segel model, and showed that it allows to design an efficient linear scheme that preserves the dissipation of a modified energy. To make sure that the modified energy is close to the original one, we evaluated the error for a well-known nonlinear dispersive equation analytically and numerically. Our results for the Keller-Segel model and the Nonlinear Schrödinger equation show that the SAV method is robust, efficient, and if combined with an upwind stabilization, enhances the spatial accuracy. However, it remains unclear at the moment how to evaluate the error between the solutions of the RDCH model and of the DCH model due to the relaxation parameter . Indeed, understanding quantitatively the error introduced by the relaxation remains an open question. Furthermore, the comparison of our numerical scheme for the RDCH model with results of simulations of the original DCH model was only qualitative. To allow for a quantitative comparison, we need to identify a relevant quantity. We propose to compare the phase-ordering dynamics in two dimensions for simulations of the original model performed in  and the ones given in Chapter 3. We know from the literature that the coarsening domains follow a growth law of the form L(t) t.
This law is estimated from the inverse of the first moment of the spherically averaged structure factor  L(t) =< k >1= R RkS(k; t) dk S(k; t) dk .
Support a deeper understanding of key mechanisms in tumor progression
In this manuscript, we investigated the role of mechanical effects in the progression and organization of tumors. In particular, we proposed to give an explanation for two observed phenomena in the organization of tumor cells. On the one hand, to understand the shrinking of tumor cells due to a chemotherapeutic drug, we studied the assumption that tumor cells change their mechanical properties: from a solid to a semi-elastic body. Our work relies on the derivation and numerical simulations of a nonlinear volume-filling Keller-Segel model that takes into account the effect of the drug. On the other hand, to explain the formation of irregularities at the border of tumors during invasion processes, we proposed a mathematical model consistent with basic mechanics and thermodynamics. Our mathematical model is rather complicated but considers the effects of friction on the extracellular matrix, viscosity, attraction and repulsion between the cells, and proliferation. These two works proposed mathematical models that focus on physical effects as an explanation of the organization of tumor cells. In consequence, they are coarse approximations of the reality, and to get closer to the biological reality, it is necessary to take into account more effects.
Building upon the results of these two previous works, we are interested in investigating a particular scenario for in-vivo tumors that can help the development of a recent therapy. Indeed, recent researches in Medicine indicates that immunotherapy is a promising cure for malignant tumors. Different immunotherapy treatments exist: targeted antibodies, cancer vaccines, adoptive cell transfer, tumor-infecting viruses, checkpoint inhibitors, cytokines, and adjuvants. The response to this treatment depends on many factors, but one of the most important is the T lymphocytes’ infiltration inside the tumor before the treatment. The different types of infiltration of tumors by lymphocytes is an indicator of the prognosis. Galon et. al.  proposed a classification in four categories (see Figure 1.13). The hot tumors are inflamed and infiltrated with activated T cells even in the center of it. The category « altered-immunosuppressed » denotes tumors with a small amount of infiltrated T cells. Tumors that enter the « altered-excluded » category present different regions: their border is infiltrated by activated T cells while the center is deprived from lymphocytes. The last category is « cold » tumors and they are often correlated to a poor response to immunotherapy since no T cells are inside the tumor. However, very little is known about T cells’ mechanisms and their different regulators to obtain the different observed patterns.
Non-linear semi-implicit multi-dimensional upwind numerical scheme
As we have seen in the previous section, to preserve the non-negativity of the discrete solutions, a particular approximation of the mobility function is needed. Based upon the results obtained on finite volume schemes for nonlinear parabolic models, we propose an adaptation of the upwind method within the finite element method.
Upwind approximation of mobility. We approximate the continuous mobility b(uk+1 h ) by a piecewise continuous function ~B (nk+1 h ). This latter is constant on specific subdomains that we define for each element. We consider for each element K 2 T h, the decomposition of K in (d + 1) subdomains defined by ~D K ij = fx 2 Kji; j k; k 6= i; jg; for i = 1; 2; 3, j = 2; 3, and i 6= j. Setting k+1 i := ‘k+1 h + 0 +(nk+1 h ) (xi), we define on each of the subdomains ~D K ij (for each K 2 T h).
Table of contents :
Table des matières
1.2 Mathematical representation of living tissues
1.2.1 First steps of mathematical modelling
1.2.2 The Cahn-Hilliard model for modelling of tissues and tumours
1.2.3 The Keller-Segel model and the volume-filling approach
1.3 General assumptions and preliminaries
1.4 Numerical simulation: foundations
1.5 Summary of the thesis
1.5.1 Towards an efficient numerical scheme for the degenerate Cahn-Hilliard model for Biology
1.5.2 Modelling of specific scenarios in Biology
1.5.3 Structure-preserving numerical method for nonlinear models
1.6 Discussion and perspectives
1.6.1 Simulation of the relaxed-degenerate Cahn-Hilliard model and effect of the relaxation
1.6.2 Support a deeper understanding of key mechanisms in tumor progression .
I The Cahn-Hilliard equation for Biology
2 Relaxation of the Cahn-Hilliard equation for Biology
2.2 The regularized problem
2.2.1 Regularization procedure
2.2.2 Existence for the regularized problem
2.2.3 Energy, entropy and a priori estimates
2.3 Existence: convergence as ! 0
2.4 Convergence as ! 0
2.5 Long-time behavior
3 Structure-preserving numerical method for the relaxed-degenerate Cahn- Hilliard model
3.3 Definition of the regularized problem
3.4 Nonlinear semi-implicit scheme
3.4.1 Description of the nonlinear numerical scheme.
3.4.2 Well-posedness of the regularized problem and stability bounds
3.4.3 Well-posedness of the non regularized problem and stability
3.4.4 Convergence analysis
3.5 Non-linear semi-implicit multi-dimensional upwind numerical scheme
3.6 Linearized semi-implicit numerical scheme
3.7 Numerical simulations
3.7.1 Numerical results: test cases
3.7.2 Effect of the relaxation parameter
3.A Proof of M-matrix properties in the 1D and 2D cases
II Modification of existing nonlinear PDE models, numerical simulation, and application in Biology.
4 Treatment-induced shrinking of tumour aggregates: A nonlinear volumefilling chemotactic approach
4.2 Description of the experiments
4.3 Mathematical model
4.3.1 Volume-filling approach for chemotaxis: first part P1
4.3.2 PDE system including the treatment: Part P2
4.4 Linear stability analysis and pattern formation
4.4.1 Dimensionless model
4.4.2 First part: Formation of the aggregates
4.4.3 Second part: Treatment
4.5 Numerical simulations
4.5.1 Biological relevance of the model parameters
4.5.2 Numerical results for a one dimensional case
4.5.3 Numerical results for a two dimensional case
4.6 Discussion of results and perspectives
4.A Derivation of the general model
4.B Stability analysis
4.C Description of the numerics
4.D One dimensional numerical results
5 Compressible Navier-Stokes-Cahn-Hilliard model for the modelling of tumor invasion in healthy tissue.
5.2 Derivation of the model
5.2.1 Notation and definitions
5.2.2 Mass balance equations
Table des matières xvii
5.2.3 Balance of linear momentum
5.2.4 Energy balance
5.2.5 Entropy balance and Clausius-Duhem inequality
5.2.6 Constitutive assumptions and model equations
5.2.7 Summary of the model equations
5.3 General assumptions and biologically relevant choice of the model functions .
5.3.1 General forms and assumptions
5.3.2 Biologically consistent choice of functions
5.3.3 Non-dimensionalized model
5.4 Large friction hypothesis
5.5 Finite volume numerical scheme
III Structure-preserving numerical method for nonlinear PDEs
6 The Scalar Auxiliary Variable method for the volume-filling Keller-Segel model.
6.2 Numerical scheme
6.2.1 Finite element framework
6.2.2 Fully discrete scheme
6.2.3 Matrix formulation
6.2.4 Upwind stabilization
6.2.5 Solving Algorithm
6.3 Existence of a non-negative solution and stability bound
6.3.1 Existence of a discrete non-negative solution
6.3.2 Discrete energy a priori estimate
6.4 Numerical results
6.4.1 1D numerical results
7 Conservation properties and long time behavior of the Scalar Auxiliary Variable method for nonlinear dispersive equations.
7.2 Numerical scheme
7.2.1 Time and space discretisation of the SAV model
7.2.2 The fully discrete SAV scheme
7.3 Conservation properties and inequalities
7.4 Convergence analysis
7.4.2 Convergence theorem
7.5 Error analysis
7.6 Numerical experiments
7.6.1 First test case: cubic nonlinearity
7.6.2 Second test case: cubic nonlinearity with non-smooth initial condition .
7.6.3 Third test case: non-integer exponent
7.6.4 Computing ground states
7.A Gradient flow with discrete normalization for computing ground state