M. Burger, A. Lorz, M.-T. Wolfram, Balanced growth path solutions of a Boltzmann mean field game model for knowledge growth - submitted - preprint
    In this paper we study balanced growth path solutions of a Boltzmann mean field game model proposed by Lucas et al to model knowledge growth in an economy. Agents can either increase their knowledge level by exchanging ideas in learning events or by producing goods with the knowledge they already have. The existence of balanced growth path solutions implies exponential growth of the overall production in time. We proof existence of balanced growth path solutions if the initial distribution of individuals with respect to their knowledge level satisfies a Pareto-tail condition. Furthermore we give first insights into the existence of such solutions if in addition to production and knowledge exchange the knowledge level evolves by geometric Brownian motion.
    T. Lorenzi, A. Lorz, B. Perthame, On interfaces between two cell populations with different mobilities and proliferation rates - submitted
    Partial differential equations describing the dynamics of cell population densities from a fluid mechanical perspective can model the growth of avascular tumours. In this framework, we consider a system of equations that describes the interaction between a population of dividing cells and a population of non-dividing cells. The two cell populations are characterised by different mobilities. We present the results of numerical simulations displaying two-dimensional spherical waves with sharp interfaces between dividing and non-dividing cells. Furthermore, we numerically observe how different ratios between the mobilities change the morphology of the interfaces, and lead to the emergence of finger-like patterns of invasion above a threshold. Motivated by these simulations, we study the existence of one-dimensional travelling wave solutions.
    M. Burger, A. Lorz, M.-T. Wolfram, On a Boltzmann mean field model for knowledge growth - submitted - preprint
    In this paper we analyze a Boltzmann type mean field game model for knowledge growth, which was proposed by Lucas and Moll. We discuss the underlying mathematical model, which consists of a coupled system of a Boltzmann type equation for the agent density and a Hamilton-Jacobi-Bellman equation for the optimal strategy. We study the analytic features of each equation separately and show local in time existence and uniqueness for the fully coupled system. Furthermore we focus on the construction and existence of special solutions, which relate to exponential growth in time - so called balanced growth path solutions. Finally we illustrate the behavior of solutions for the full system and the balanced growth path equations with numerical simulations.
    A. Lorz, B. Perthame, C. Taing, Dirac concentrations in a chemostat model of adaptive evolution - Accepted in Chinese annals of mathematics
    We consider parabolic systems of Lotka-Volterra type that describe the evolution of phenotypically structured populations. Nonlinearities appear in these systems to model interactions and competition phenomena leading to selection. In this paper, the equation on the structured population is coupled with a differential equation on the nutrient concentration that changes as the total population varies. We review different methods aimed at showing the convergence of the solutions to a moving Dirac mass. Setting first two frameworks based on weak or strong regularity assumptions in which we study the concentration of the solution, we state BV estimates in time on appropriate quantities and derive a contrained Hamilton-Jacobi equation to identify the Dirac locations.
    R.H. Chisholm, T. Lorenzi, A. Lorz, Effects of an advection term in nonlocal Lotka-Volterra equations - Accepted in Communications in Mathematical Sciences
    Nonlocal Lotka-Volterra equations have the property that solutions concentrate as Dirac masses in the limit of small diffusion. In this paper, we show how the presence of an advection term changes the location of the concentration points in the limit of small diffusion and slow drift. The mathematical interest lies in the formalism of constrained Hamilton-Jacobi equations. Our motivations come from previous models of evolutionary dynamics in phenotype-structured populations [R.H. Chisholm, T. Lorenzi, A. Lorz, et al., Cancer Res., 75, 930-939, 2015], where the diffusion operator models the effects of heritable variations in gene expression, while the advection term models the effect of stress-induced adaptation.
    T. Lorenzi, R.H. Chisholm, M. Melensi, A. Lorz, M. Delitala, Mathematical model reveals how regulating the three phases of T-cell response could counteract immune evasion - Immunology, Vol. 146, pp. 271–280, 2015 - published version
    T cells are key players in immune action against the invasion of target cells expressing non-self antigens. During an immune response, antigen-specific T cells dynamically sculpt the antigenic distribution of target cells, and target cells concurrently shape the host's repertoire of antigen-specific T cells. The succession of these reciprocal selective sweeps can result in "chase-and-escape" dynamics and lead to immune evasion. It has been proposed that immune evasion can be countered by immunotherapy strategies aimed at regulating the three phases of the immune response orchestrated by antigen-specific T cells: expansion, contraction and memory. Here, we test this hypothesis with a mathematical model that considers the immune response as a selection contest between T cells and target cells. The outcomes of our model suggest that shortening the duration of the contraction phase and stabilising as many T cells as possible inside the long-lived memory reservoir, using dual immunotherapies based on the cytokines IL-7 and/or IL-15 in combination with molecular factors that can keep the immunomodulatory action of these interleukins under control, should be an important focus of future immunotherapy research.
    R.H. Chisholm, T. Lorenzi, A. Lorz, A. Larsen, L. Almeida, A. Escargueil, J. Clairambault, Emergence of drug tolerance in cancer cell populations: an evolutionary outcome of selection, non-genetic instability and stress-induced adaptation - Cancer Res., 75, pp. 930-939, 2015 - published version
    In recent experiments on isogenetic cancer cell lines, it was observed that exposure to high doses of anti-cancer drugs can induce the emergence of a subpopulation of less-proliferative, drug-tolerant cells, that display markers associated with cancer stem cells. After a period of time, some of the surviving stem-like cells were observed to change their phenotype to resume normal proliferation, and eventually repopulate the sample. Interestingly, these evolutionary dynamics were shown to be the result of epigenetic modifications, rather than genetic mutations. We propose a theoretical mechanism for the transient emergence of such drug tolerance, that might be a mandatory intermediate state towards established drug resistance. We formulate both an individual-based model and an integro-differential equation model of phenotypic evolution in a cell population exposed to cytotoxic drugs. The outcomes of both models suggest that, on intermediate time scales, selection, non-genetic instability, stress-induced adaptation and the interplay between these mechanisms can push an actively proliferating cell population to transition into a less-proliferative and drug-tolerant state. Hence, the cell population experiences much less stress in the presence of the drugs and, in the long run, reacquires a proliferative phenotype, due to selection pressure and phenotypic fluctuations. These results provide a possible evolutionary explanation for the experimental results presented by Sharma and co-workers. Furthermore, our results highlight how the transient appearance of the less-proliferative and drug-tolerant cells is strictly related to the usage of high doses of cytotoxic drugs. Finally, we show how stem-like characteristics can act to stabilise the transient, less proliferative and drug-tolerant subpopulation for a longer period of time.
    A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil, B. Perthame, Modeling the effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors - Bulletin of Mathematical Biology, 77, pp. 1-22, 2015 - preprint - published version
    Histopathological evidence supports the idea that the emergence of phenotypic heterogeneity and resistance to cytotoxic drugs can be considered as a process of adaptation, or evolution, in tumor cell populations. In this framework, can we explain intra-tumor heterogeneity in terms of cell adaptation to local conditions? How do anti-cancer therapies affect the outcome of cell competition for nutrients within solid tumors? Can we overcome the emergence of resistance and favor the eradication of cancer cells by using combination therapies? Bearing these questions in mind, we develop a model describing cell dynamics inside a tumor spheroid under the effects of cytotoxic and cytostatic drugs. Cancer cells are assumed to be structured as a population by two real variables standing for space position and the expression level of a cytotoxic resistant phenotype. The model takes explicitly into account the dynamics of resources and anti-cancer drugs as well as their interactions with the cell population under treatment. We analyze the effects of space structure and combination therapies on phenotypic heterogeneity and chemotherapeutic resistance. Furthermore, we study the efficacy of combined therapy protocols based on constant infusion and/or bang-bang delivery of cytotoxic and cytostatic drugs.
    A. Lorz, and B. Perthame, Long-term behaviour of phenotypically structured models - Proceedings of the Royal Society A, Vol. 470, no. 2167, 2014 - published version - preprint
    Phenotypically structured equations arise in population biology to describe the interaction of species with their environment that brings the nutrients. This interaction usually leads to the selection of the fittest individuals. Models used in this area are highly nonlinear, and the question of long term behaviour is usually not solved. However, there is a particular class of models for which convergence to an Evolutionary Stable Distribution is proved, namely when the quasi-static assumption is made. This means that the environment, and thus the nutrient supply, reacts immediately to the population dynamics. One possible proof is based on a Total Variation bound for the appropriate quantity. We extend this proof to cases where the nutrient is regenerated gradually. A simple example is the chemostat with a rendering factor, then our result does not use any smallness assumption. For a more general setting, we can treat the case with a fast equilibration of the nutrient concentration.
    A. Lorz, P. Markowich, B. Perthame, Bernoulli variational problem and beyond - Archive for Rational Mechanics and Analysis, Vol. 212, 2014, pp. 415-443 - published version - preprint
    The question of 'cutting the tail' of the solution of an elliptic equation arises naturally in several contexts and leads to a singular perturbation problem under the form of a strong cut-off. We consider both the PDE with a drift and the symmetric case where a variational problem can be stated. It is known that, in both cases, the same critical scale arises for the size of the singular perturbation. More interesting is that in both cases another critical parameter (of order one) arises that decides when the limiting behaviour is non-degenerate. We study both theoretically and numerically the values of this critical parameter and, in the symmetric case, ask if the variational solution leads to the same value as for the maximal solution of the PDE. Finally we propose a weak formulation of the limiting Bernoulli problem which incorporates both Dirichlet and Neumann boundary condition.
    T. Lorenzi, A. Lorz, G. Restori, Asymptotic Dynamics in Populations Structured by Sensitivity to Global Warming and Habitat Shrinking - Acta Applicandae Mathematicae, Vol. 131, 2014, pp. 49-67 - published version
    How to recast effects of habitat shrinking and global warming on evolutionary dynamics into continuous mutation/selection models? Bearing this question in mind, we consider differential equations for structured populations, which include mutation, proliferation and competition for resources. Since mutations are assumed to be small and to occur on a longer time scale than proliferation, a parameter $\varepsilon$ is introduced to model the ratio between these time scales as well as the average size of mutations. A well-posedness result is proposed and the asymptotic behavior of the density of individuals is studied in the limit $\varepsilon \rightarrow 0$. In particular, we prove the weak convergence of the density to a sum of Dirac masses and characterize the related concentration points. Moreover, we provide numerical simulations illustrating the theorems and showing an interesting sample of solutions depending on parameters and initial data.
    F. Thomas, D. Fisher, P. Fort, J.-P. Marie, S. Daoust, B. Roche, C. Grunau, C. Cosseau, G. Mitta, S. Baghdiguian, F. Rousset, P. Lassus, E. Assenat, D. Grégoire, D. Missé, A. Lorz, F. Billy, W. Vainchenker, F. Delhommeau, S. Koscielny, R. Itzykson, R. Tang, F. Fava, A. Ballesta, T. Lepoutre, L. Krasinska, V. Dulic, P. Raynaud, P. Blache, C. Quittau-Prevostel, E. Vignal, H. Trauchessec, B. Perthame, J. Clairambault, V. Volpert, E. Solary, U. Hibner, M. E. Hochberg, Applying ecological and evolutionary theory to cancer: a long and winding road - Evolutionary Applications, Vol. 6, 2013, pp. 1-10 - published version
    Since the mid 1970s, cancer has been described as a process of Darwinian evolution, with somatic cellular selection and evolution being the fundamental processes leading to malignancy and its many manifestations (neoangiogenesis, evasion of the immune system, metastasis, and resistance to therapies). Historically, little attention has been placed on applications of evolutionary biology to understanding and controlling neoplastic progression and to prevent therapeutic failures. This is now beginning to change, and there is a growing international interest in the interface between cancer and evolutionary biology. The objective of this introduction is first to describe the basic ideas and concepts linking evolutionary biology to cancer. We then present four major fronts where the evolutionary perspective is most developed, namely laboratory and clinical models, mathematical models, databases, and techniques and assays. Finally, we discuss several of the most promising challenges and future prospects in this interdisciplinary research direction in the war against cancer.
    A. Lorz, T. Lorenzi, M. E. Hochberg, J. Clairambault, B. Perthame, Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies - ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 47, 2013, pp. 377-399 - published version - preprint
    Resistance to chemotherapies, particularly to anticancer treatments, is an increasing medical concern. Among the many mechanisms at work in cancers, one of the most important is the selection of tumor cells expressing resistance genes or phenotypes. Motivated by the theory of mutation-selection in adaptive evolution, we propose a model based on a continuous variable that represents the expression level of a resistance gene (or genes, yielding a phenotype) influencing in healthy and tumor cells birth/death rates, effects of chemotherapies (both cytotoxic and cytostatic) and mutations. We extend previous work by demonstrating how qualitatively different actions of chemotherapeutic and cytostatic treatments may induce different levels of resistance. The mathematical interest of our study is in the formalism of constrained Hamilton-Jacobi equations in the framework of viscosity solutions. We derive the long-term temporal dynamics of the fittest traits in the regime of small mutations. In the context of adaptive cancer management, we also analyse whether an optimal drug level is better than the maximal tolerated dose.
    A. Chertock, K. Fellner, A. Kurganov, A. Lorz, P. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach - Journal of Fluid Mechanics, Vol. 694, 2012, pp. 155-190 - published version - preprint
    Aquatic bacteria like Bacillus subtilis are heavier than water yet they are able to swim up an oxygen gradient and concentrate in a layer below the water surface, which will undergo Rayleigh-Taylor type instabilities for sufficiently high concentrations. In several articles, a simplified chemotaxis-fluid system has been proposed as a model for modestly diluted cell-suspensions. It couples a convective chemotaxis system for the oxygen-consuming and oxytactic bacteria with the incompressible Navier-Stokes equations subject to a gravitational force proportional to the relative surplus of the cell-density compared to the water-density. In this paper, we derive a high-resolution vorticity-based hybrid finite-volume finite-difference scheme, which allows us to investigate the nonlinear dynamics of a two-dimensional chemotaxis fluid system with boundary conditions matching an experiment. We present selected numerical examples, which illustrate (i) the formation of sinking plumes, (ii) the possible merging of neighbouring plumes and (iii) the convergence towards numerically stable stationary plumes. The examples with stable stationary plumes show how the surface-directed oxytaxis continuously feeds cells into a high-concentration layer near the surface, from where the fluid-flow (recurring upwards in the space between the plumes) transports the cells into the plumes, where then gravity makes the cells sink and constitutes the driving force in maintaining the fluid-convection and, thus, in shaping the plumes into (numerically) stable stationary states.
  18. A. Lorz, S. Mirrahimi, B. Perthame, Dirac mass dynamics in a multidimensional nonlocal parabolic equation - Communications in Partial Differential Equations, Vol. 36, 2011, pp. 1071-1098 - published version - preprint
    Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Is it possible to describe the dynamics of the limiting concentration points and of the weights of the Dirac masses? What is the long time asymptotics of these Dirac masses? Can several Dirac masses co-exist? We will explain how these questions relate to the so-called ''constrained Hamilton-Jacobi equation'' and how a form of canonical equation can be established. This equation has been established assuming smoothness. Here we build a framework where smooth solutions exist and thus the full theory can be developed rigorously. We also show that our form of canonical equation comes with a kind of Lyapunov functional. Numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation. Our motivation comes from population adaptive evolution a branch of mathematical ecology which models darwinian evolution.
    M. Di Francesco, A. Lorz, P. Markowich, Chemotaxis fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior - Discrete and Continuous Dynamical Systems, Series A, Vol. 28, No. 4, 2010, pp. 1437-1453 - published version - preprint
    We study a system arising in the modelling of the motion of swimming bacteria under the effect of diffusion, oxygen-taxis and transport through an incompressible fluid. The novelty with respect to previous papers in the literature lies in the presence of nonlinear porous--medium--like diffusion in the equation for the density $n$ of the bacteria, motivated by a finite size effect. We prove that, under the constraint $m\in(3/2, 2]$ for the adiabatic exponent, such system features global in time solutions in two space dimensions for large data. Moreover, in the case $m=2$ we prove that solutions converge to constant states in the large--time limit. The proofs rely on standard energy methods and on a basic entropy estimate which cannot be achieved in the case $m=1$. The case $m=2$ is very special as we can provide a Lyapounov functional. We generalize our results to the three--dimensional case and obtain a smaller range of exponents $m\in(m^*,2]$ with $m^*>3/2$, due to the use of classical Sobolev inequalities.
    R.-J. Duan, A. Lorz, P. Markowich, Global Solutions to the coupled chemotaxis-fluid equations - Communications in Partial Differential Equations, Vol. 35, No. 9, 2010, pp. 1635-1673 - published version - preprint
    In this paper, we are concerned with a model arising from biology, which is a coupled system of the chemotaxis equations and the viscous incompressible fluid equations through transport and external forcing. The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the Chemotaxis-Navier-Stokes system over three space dimensions, we obtain global existence and rates of convergence on classical solutions near constant states. When the fluid motion is described by the simpler Stokes equations, we prove global existence of weak solutions in two space dimensions for cell density with finite mass, first-order spatial moment and entropy provided that the external forcing is weak or the substrate concentration is small.
    A. Lorz, Coupled Chemotaxis Fluid Model - Mathematical Models and Methods in Applied Sciences, Vol. 20, No. 6, 2010, pp. 987-1004 - published version - preprint
    We consider a model system for the collective behaviour of oxygen-driven swimming bacteria in an aquatic fluid. In certain parameter regimes such suspensions of bacteria feature large-scale convection patterns as a result of the hydrodynamic interaction between bacteria. The presented model consist of a parabolic-parabolic chemotaxis system for the oxygen concentration and the bacteria density coupled to an incompressible Stokes equation for the fluid driven by a gravitational force of the heavier bacteria. We show local existence of weak solutions in a bounded domain in $R ^d$, $d=2,3$ with no-flux boundary condition and in $R^2$ in the case of inhomogeneous Dirichlet conditions for the oxygen.
  22. Proceedings

    F. Billy, J. Clairambault, O. Fercoq, T. Lorenzi, A. Lorz, B. Perthame, Modelling targets for anticancer drug control optimization in physiologically structured cell population models - AIP Conference Proceedings, Vol. 1479, 2012, pp. 1323-1326 - published version
    The main two pitfalls of therapeutics in clinical oncology, that limit increasing drug doses, are unwanted toxic side effects on healthy cell populations and occurrence of resistance to drugs in cancer cell populations. Depending on the constraint considered in the control problem at stake, toxicity or drug resistance, we present two different ways to model the evolution of proliferating cell populations, healthy and cancer, under the control of anti-cancer drugs. In the first case, we use a McKendrick age-structured model of the cell cycle, whereas in the second case, we use a model of evolutionary dynamics, physiologically structured according to a continuous phenotype standing for drug resistance. In both cases, we mention how drug targets may be chosen so as to accurately represent the effects of cytotoxic and of cytostatic drugs, separately, and how one may consider the problem of optimisation of combined therapies.
  24. A. Decoene, A. Lorz, S. Martin. B. Maury, M. Tang, Simulation of self-propelled chemotactic bacteria in a Stokes flow - ESAIM: Proceedings, Vol. 30, 2010, pp. 105-124 - published version - preprint
    We present a method to simulate the motion of self-propelled rigid particles in a two-dimensional Stokesian fluid, taking into account chemotactic behaviour. Self-propulsion is modelled as a point force associated to each particle, placed at a certain distance from its gravity centre. The method for solving the fluid flow and the motion of the bacteria is based on a variational formulation on the whole domain, including fluid and particles: rigid motion is enforced by penalizing the strain rate tensor on the rigid domain, while incompressibility is treated by duality. This leads to a minimization problem over unconstrained functional spaces which can be easily implemented from any finite element Stokes solver. In order to ensure robustness, a projection algorithm is used to deal with contacts between particles. The particles are meant to represent bacteria of the \emph{Escherichia coli} type, which interact with their chemical environment through consumption of nutrients and orientation in some favorable direction. Our model takes into account the interaction with oxygen. An advection-diffusion equation on the oxygen concentration is solved in the fluid domain, with a source term accounting for oxygen consumption by the bacteria. In addition, self-propulsion is deactivated for those particles which cannot consume enough oxygen. Finally, the model includes random changes in the orientation of the individual bacteria, with a frequency that depends on the surrounding oxygen concentration, in order to favor the direction of the concentration gradient and thus to reproduce chemotactic behaviour. Numerical simulations implemented with FreeFem++ are presented.