Combination of direct methods and homotopy in numerical optimal control: application to the optimization of chemotherapy in cancer

We consider a state-constrained optimal control problem of a system of two non-local partial-differential equations, which is an extension of the one introduced in a previous work in mathematical oncology. The aim is to minimize the tumor size through chemotherapy while avoiding the emergence of resistance to the drugs. The numerical approach to solve the problem was the combination of direct methods and continuation on discretization parameters, which happen to be insufficient for the more complicated model, where diffusion is added to account for mutations. In the present paper, we propose an approach relying on changing the problem so that it can theoretically be solved thanks to a Pontryagin Maximum Principle in infinite dimension. This provides an excellent starting point for a much more reliable and efficient algorithm combining direct methods and continuations. The global idea is new and can be thought of as an alternative to other numerical optimal control techniques

Modelling interactions between tumour cells and supporting adipocytes in breast cancer

International audienceIn breast cancer, invasion of the micro-environment implies potential bidirectional communica- tion between cancer cells and the adipose tissue. Biological evidence suggests that adipocytes, in particular, are key-actors in tumorigenesis and invasion: they both enhance proliferation of the cancer cells and favor acquisition of a more invasive phenotype. To understand these effects, mathematical modelling (thanks to tools developed in theoretical ecology) is used to perform asymptotic analysis in number of cells and distribution of phenotypes. These models can be tuned through confrontation with experimental data coming from co-cultures of cancer cells with adipocytes.The first part of this report is devoted to presenting the biological background on breast cancer and its environment. We then present the main aspects of the mathematical modelling, and how it is expected to be validated experimentally. In the third part, we summarize and prove many important results that have been obtained on a single integro-differential equation rep- resenting the evolution of a population of individuals structured with a phenotypic trait. The next part consists of a first glance at possible generalizations of those results to a system of integro-differential equations coupled mutualistically. In a fifth and last part, we introduce how we intend to parametrize the models through explicit computations and numerical simulations

Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues

Multiphase mechanical models are now commonly used to describe living tissues including tumour growth. The specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous medium equation, including cross-reaction terms. We study the incompressible limit, when the pressure becomes stiff, which generates a free boundary problem. We establish the complementarity relation and also a segregation result. Several major mathematical difficulties arise in the two species case. Firstly, the system structure makes comparison principles fail. Secondly, segregation and internal layers limit the regularity available on some quantities to BV. Thirdly, the Aronson-B{\'e}nilan estimates cannot be established in our context. We are lead, as it is classical, to add correction terms. This procedure requires technical manipulations based on BV estimates only valid in one space dimension. Another novelty is to establish an L1 version in place of the standard upper bound

Linear inverse problems with nonnegativity constraints through the β\beta-divergences: sparsity of optimisers

We pass to continuum in optimisation problems associated to linear inverse problems $y = Ax$ with non-negativity constraint $x \geq 0$. We focus on the case where the noise model leads to maximum likelihood estimation through the so-called $\beta$-divergences, which cover several of the most common noise statistics such as Gaussian, Poisson and multiplicative Gamma. Considering~$x$ as a Radon measure over the domain on which the reconstruction is taking place, we show a general sparsity result. In the high noise regime corresponding to $y \notin \{{Ax}\mid{x \geq 0}\}$, optimisers are typically sparse in the form of sums of Dirac measures. We hence provide an explanation as to why any possible algorithm successfully solving the optimisation problem will lead to undesirably spiky-looking images when the image resolution gets finer, a phenomenon well documented in the literature. We illustrate these results with several numerical examples inspired by medical imaging

Modelling interactions between tumour cells and supporting adipocytes in breast cancer

International audienceIn breast cancer, invasion of the micro-environment implies potential bidirectional communica- tion between cancer cells and the adipose tissue. Biological evidence suggests that adipocytes, in particular, are key-actors in tumorigenesis and invasion: they both enhance proliferation of the cancer cells and favor acquisition of a more invasive phenotype. To understand these effects, mathematical modelling (thanks to tools developed in theoretical ecology) is used to perform asymptotic analysis in number of cells and distribution of phenotypes. These models can be tuned through confrontation with experimental data coming from co-cultures of cancer cells with adipocytes.The first part of this report is devoted to presenting the biological background on breast cancer and its environment. We then present the main aspects of the mathematical modelling, and how it is expected to be validated experimentally. In the third part, we summarize and prove many important results that have been obtained on a single integro-differential equation rep- resenting the evolution of a population of individuals structured with a phenotypic trait. The next part consists of a first glance at possible generalizations of those results to a system of integro-differential equations coupled mutualistically. In a fifth and last part, we introduce how we intend to parametrize the models through explicit computations and numerical simulations

A phenotype-structured model for the tumour-immune response

This paper presents a mathematical model for tumour-immune response interactions in the perspective of immunotherapy by immune checkpoint inhibitors (ICIs). The model is of the integrodifferential Lotka-Volterra type, in which heterogeneity of the cell populations is taken into account by structuring variables that are continuous internal traits (aka phenotypes) representing a lumped ''aggressiveness'', i.e., for tumour cells, ability to thrive in a viable state under attack by immune cells or drugs-which we propose to identify as a potential of de-differentiation-, and for immune cells, ability to kill tumour cells. We analyse the asymptotic behaviour of the model in the absence of treatment. By means of two theorems, we characterise the limits of the integro-differential system under an a priori convergence hypothesis. We illustrate our results with numerical simulations, which show that our model exemplifies the three Es of immunoediting: elimination, equilibrium, and escape

A survey of adaptive cell population dynamics models of emergence of drug resistance in cancer, and open questions about evolution and cancer

This article is a proceeding survey (deepening a talk given by the first author at the BioMath 2019 International Conference on Mathematical Models and Methods, held in Będlewo, Poland) of mathematical models of cancer and healthy cell population adaptive dynamics exposed to anticancer drugs, to describe how cancer cell populations evolve toward drug resistance. Such mathematical models consist of partial differential equations (PDEs) structured in continuous phenotypes coding for the expression of drug resistance genes; they involve different functions representing targets for different drugs, cytotoxic and cytostatic, with complementary effects in limiting tumour growth. These phenotypes evolve continuously under drug exposure, and their fate governs the evolution of the cell population under treatment. Methods of optimal control are used, taking inevitable emergence of drug resistance into account, to achieve the best strategies to contain the expansion of a tumour. This evolutionary point of view, which relies on biological observations and resulting modelling assumptions, naturally extends to questioning the very nature of cancer as evolutionary disease, seen not only at the short time scale of a human life, but also at the billion year-long time scale of Darwinian evolution, from unicellular organisms to evolved multicellular organs such as animals and man. Such questioning, not so recent, but recently revived, in cancer studies, may have consequences for understanding and treating cancer. Some open and challenging questions may thus be (non exhaustively) listed as:-May cancer be defined as a spatially localised loss of coherence between tissues in the same mul-ticellular organism, 'spatially localised' meaning initially starting from a given organ in the body, but also possibly due to flaws in an individual's rms of evolution towards drug resistance governed by the phenotypes which determine landscape such as imperfect epigenetic control of differentiation genes?-If one assumes that "The genes of cellular cooperation that evolved with multicellularity about a billion years ago are the same genes that malfunction in cancer." (Davies and Lineweaver, 2011), how can these genes be systematically investigated, looking for zones of fragility-that depend on individuals-in the 'tinkering' (F. Jacob, 1977) evolution is made of, tracking local defaults of coherence?-What is such coherence made of and to what extent is the immune system responsible for it (the self and differentiation within the self)? Related to this question of self, what parallelism can be established between the development of multicellularity in different species proceeding from the same origin and the development of the immune system in these different species