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J. Clairambault, A. Lorz, B. Perthame,
Populational adaptive evolution and resistance to therapy - Submitted
Resistance to treatments, in particular anticancer therapy, is a raising concern in the medical community. Among the many mechanisms at work, a possible scenario is the selection of tumor cells expressing a resistance gene or phenotype. Motivated by the theory of mutation-selection in adaptive evolution, we propose a model based on a continuous variable that represents the level of expression of the resistance gene (or genes, yielding a phenotype), the birth/death rates of tumor cells, the effect of chemotherapy and mutations. Relying on previous works, we give several results expressing that for healthy cells a low level of gene expression is selected but chemotherapy is able to induce resistance.
The mathematical interest comes from the formalism which uses constrained Hamilton-Jacobi equations in the framework of viscosity solutions. We are able to derive the long time dynamics of the concentration points in the regime of small mutations. For the model at hand, we are also able to analyze whether an optimal drug level is better than the maximal tolerated dose.
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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 - Accepted in Journal of Fluid Mechanics -
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.
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- 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 - 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.
- 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 - 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.
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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 -
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.
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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 -
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.
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