Spreading speeds for a two-species competition-diffusion system 

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General study of a heterogeneous in vitro tumour model

We use phenotypical observations of lung cancer cells A549, sensitive to Epothilene, and their resistant version, A549 Epo40. These lineages are used by M. Carré, researcher at the CRO2 1 in experiments on cancerous cells interactions. We also only consider two months in vitro experiments, where only the two types of cells are present, and nutrients are distributed in abundance.
The cell population is divided into two compartments: the sensitive cells, denoted s, and the resistant cells, denoted r. We neglect any mutation or metabolism change that would make sensitive cells become resistant during the experiment: such events are rare, and would not contribute a lot to the global resistance over two months of treatment. Moreover, the resistant ligneage of cells is resistant to very high concentrations of Epothilene (40 nMol), while dosages in this kind of experiment is limited to tolerable levels for an animal organism (5 nMol). Hence the selective pressure is not sufficient to make A549 cells mutate into A549 Epo40 cells. We thus decided to focus on other dynamics that explain the emergence of a globally resistant tumour. When cultivated alone, the cells behave as in a logistic model, which we will use for the rest of the study. This model is expressed for a generic population x as: dx dt (t) = x(t)(1 − x(t) K ) where is the initial growth rate of the population, and K the carrying capacity of the environment. The term x K represents the total space already occupied in the environment. In our case, the two kinds of cells consume space, so this term should be s+r K . But we should take into account the difference of size between the two lineages: resistant cells are bigger than sensitive cells in M. Carré’s experimental setting. For that purpose, the term representing the used space is s(t)+mr(t) K , with m > 1. The growth rates of s and r are denoted, respectively, s and r. ( ds dt (t) = ss(1 − s+mr K ), dr dt (t) = rr(1 − s+mr K ).

Optimal control of the tumour size at the end of the experiment

Our model is able to reproduce the phenomena we desired. It now should be used to determine treatments that achieve medical purposes. The choice of an objective, mathematically, will greatly influence the form of the corresponding optimal treatment. Thus, we investigate different kinds of objectives to present their advantages and drawbacks. We present here a first example, to reduce the tumour size at the end of the experiment. Optimal Control Problem 2. Given an initial condition (s0, r0) 2 T, a maximal concentration of the treatment Cmax and a duration T of experiment, find a Lebesgue measurable function C : [0, T] ! [0,Cmax] that minimizes the global quantity of effective cancerous cells: (s, r) := s(T) + mr(T).

Treatment protocols with different aims

Since the optimal treatment defined in 2.2.3 is not very well suited for medical protocols, we define two new problems. The first one based solely on the study of trajectories in 2.2.2 gives a way to stabilize the tumour at low, constant state. The second one is a new optimal control problem, which takes into account the size of the tumour during the whole experiment.

Adaptive stabilization protocol

A consequence of Theorem 5 is that the total tumour s + r never goes extinct. For medical reasons, it is interesting to reach a constant state, with mostly sensitive cells, and at the lowest possible level. If we define a treatment so that the tumour reaches such a state, the cancer will no longer be considered a single-event disease, but a chronic one, that requires constant care but is under medical control. In this outlook, we want to stabilize the system at the fixed point with only sensitive cells Es(C), with this point as close as possible to O = (0, 0), with the constraint of Es(C) being stable. The fixed point Es(C) is stable for 0 < C < Cmetro, so our objective is the point Es,l := ( K +K , 0), that is the position of Es(C) for C = Cmetro.

Biological interpretation: a model of cancer growth

Model (3.1) has been used to study heterogeneous tumour growth. Solid tumours are subject to non uniform phenomena, and as such, can develop heterogeneous behaviours. Especially, the use of chemotherapy to cure cancers often triggers the emergence of resistant lineages, that will not be affected by the drug, by selecting them against more sensitive cells. When the tumour does not reply to the treatment any more, medical doctors then have to find a different drug, if it exists, to tackle this new kind of cancerous cells, which can be more harmful for the patient.
In any case, appearance of a resistance to the chemotherapy is a cause of treatment failure, and should be avoided. Moreover, cytotoxic drugs, which are widely used in classical protocols, cause unwanted toxic side effects, thus their dosage should be carefully designed. These problems can be addressed by in vivo or in vitro experimentation, and also by mathematical modelling of the different phenomena inside tumours. Therapy optimization in the case of heterogeneous tumours, for example, has been studied among others with the use of cellular automata [RTGGA15, GSGF09], with the construction of complex systems and their numerical studies [ABB+13], and also with the construction of simple models and their analytical studies, using, for example, Pontryagin Maximum Principle [dLMS09, LS14]. We refer to [CLC16] for a review of mathematical modelling of resistance apparition in solid tumours.
In a previous article [1], the author studied the effect of chemotherapy on heterogeneous tumours. A series of biological experimentations on in vitro tumours, composed of sensitive and resistant cells, showed that large doses of chemotherapy would kill all sensitive cells and let resistant cells grow to a maximum population. A mathematical ODE model adapted to these experimentations was designed, and different adaptive treatment protocols to reduce risks of resistance to chemotherapy were constructed, notably with optimal control theory. The main idea is to choose the maximal drug dosage such that, in the model (3.4), a population with only sensitive cells is stable and locally attractive, and to bring the system in this bassin of attraction.
In order to better understand the experiments, it appeared crucial that spatial diffusion should be taken into account, which is why system (3.1) was constructed. Spatial heterogeneity has a great influence on cancer virulence and evolution, as illustrated in [GG96]. In [LLC+15], the authors construct a model of solid tumours taking into account spatial diffusion of nutrients, treatment and cells, as well as the resistance of cells as a continuous trait. Diverse treatment protocols can be tested and compared in this framework, like combination of constant infusion and alternative maximum-minimum delivery of cytotoxic drugs. Game theory is used in [BHD08] to explain how a more invasive lineage can be selected against a proliferative one. It uses cellular automata, and support the use of therapies that would increase the cost of motility, in order to maintain the tumour at a benign state. We are developing in our article a PDE model that enhances this idea, that cytotoxic drugs will be efficient as long as it does not favour a more motile lineage.
In model (3.1), s represents the population of cells that are sensitive to a certain drug, and r a lineage of resistant cells. They divide, spread and compete at different rates. We do not investigate how the resistant trait emerge, but only how they spread with the tumour once they have appeared. Thus, we do not take into account mutations of sensitive cells into resistant cells, or resistant cells into sensitive cells. Using the hypothesis of mutations, [GR16] showed the existence of a travelling wave connecting (0, 0) to a coexistence state. Finally, the action of chemotherapy is taken into account through parameter 0, the growth rate of s. If no treatment is applied, 0 is at a maximal value, and as treatment dosage increases, 0 decreases. The bistability criterium (3.7) states that if 0 < 1/0, then the sensitive population (s, r) = (0, 0) is no longer stable: this property gives us a limit value for treatment dosage. Since we are interested in biological coherent solutions, we only consider bounded solutions of (3.1).

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Table of contents :

Introduction 
0.1 Biological definitions and experiments
0.2 Optimization of chemotherapy via optimal control theory
0.3 Spreading of a system of competition-diffusion equations
0.4 Dynamic programming
1 Modélisation pour le traitement des cancers 
1.1 Cellules cancéreuses : quelques caractéristiques
1.2 Du laboratoire au patient
1.3 Traitements des cancers : quelle place pour la modélisation mathématique ?
1.3.1 Traitements localisés
1.3.2 Traitements systémiques
1.4 Modélisation mathématique de la résistance aux anti-cancéreux
1.4.1 Échec de la distribution du médicament et arrêt forcé du traitement
1.4.2 Les mécanismes de résistance spécifique
1.4.3 Hétérogénéité intratumorale
1.5 Amélioration des protocoles
1.5.1 Personnalisation des traitements
1.5.2 Le schéma de traitement métronomique
1.6 Données biologiques expérimentales
2 Optimization of an in vitro chemotherapy to avoid resistant tumours 
2.1 Introduction
2.2 General study of a heterogeneous in vitro tumour model
2.2.1 Mathematical modelling
2.2.2 Phase plan analysis for a constant treatment
2.2.3 Optimal control of the tumour size at the end of the experiment
2.3 Treatment protocols with different aims
2.3.1 Adaptive stabilization protocol
2.3.2 Optimal control of the tumour size during the whole experiment
2.4 Comparison of numerical results
2.5 Discussion
3 Spreading speeds for a two-species competition-diffusion system 
3.1 Introduction
3.2 Biological interpretation: a model of cancer growth
3.3 Outside of competition
3.3.1 Comparison lemma
3.3.2 Limitation of speeds in both directions
3.3.3 Invasion of the empty space by the fastest species
3.4 Competition between species
3.4.1 Left side of the interaction zone
3.4.2 Right side of the interaction zone
4 Dynamic programming 
4.1 Introduction
4.2 Problem presentation
4.2.1 Three models for heterogeneous tumour growth
4.2.2 Objectives and constraints
4.3 Stability and reachability
4.3.1 Definition of the stability kernels
4.3.2 Value function of the viability problem
4.3.3 Time minimum function
4.4 Numerical implementation
4.4.1 Viability problem
4.4.2 Reachability and time minimum
4.4.3 Trajectories reconstruction
4.5 Uncertainties
4.6 Numerical Simulations
4.6.1 Models without immune system
4.6.2 Model with immune system
Bibliography 

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