Alexander Lorz

 

 A. Lorz, T. Lorenzi, A. Escargueil, J. Clairambault, and B. Perthame, An explanation for the heterogeneity in solid tumours  - in preparation   
 

 

 J. Chung, A. Lorz, and B. Perthame, On  non-local Lotka-Volterra equations coupled with a  differential equation for the non-local quantity  - in preparation   
 

 

 L. Boudin, C. Grandmont, A. Lorz, A. Moussa, Modeling and numerics for respiratory aerosols  - in preparation   
 
We write a model for an aerosol (air/particle mixture) in the respiratory system for a fixed or moving domain. It consists of the  incompressible Navier-Stokes equations for the air and the Vlasov equation for the particles, coupled through a drag force. We propose a discretisation of the model and
 investigate fundamential numerical properties of the C++ code which has been developed. Moreover, we focus on the influence of the aerosol particles on the airflow (retroaction force).
   

 

 T. Lorenzi, A. Lorz, Polymorphism in cancer cell populations evolving under drug pressures  - in preparation   
 
On phenotype-structured equations modeling the dynamics of cancer cells, we ask the question:
 Can these equations reproduce the evolution of a monomorphic population into a polymorphic one, due to the effects of therapeutic pressures? Bearing this question in mind, we consider a class of differential equations describing the dynamics of a cell population structured by two continuous real variables modeling the resistance levels to two different types of cytotoxic drugs. The asymptotic long-time behavior of the cell density is studied. In particular, we provide general assumptions on the parameter functions allowing us to prove weak convergence of the density to one Dirac mass (monomorphism) or several Dirac masses (polymorphism).
   

 

 A. Lorz, P. Markowich, B. Perthame, Bernoulli variational problem and beyond  - Submitted   
 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  - Submitted   
 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. 
   

 

A. Lorz, A coupled Keller–Segel–Stokes model: global existence for small initial data and blow-up delay  - Communications in Mathematical Sciences, Vol. 10, 2012, pp. 555–574 - published version  - preprint 
 We study a system consisting of the elliptic-parabolic Keller-Segel equations coupled to Stokes equations by transport and gravitational forcing. We show global-in-time existence of solutions for small initial mass in 2D. In 3D we establish global existence assuming that the initial $L^{3/2}$-norm is small.
Moreover, we give numerical evidence that for this extension of the Keller-Segel system in 2D, solutions exist with mass above $8\pi$, which is the  critical mass for the system without fluid. 
   

 

J.-G. Liu, A. Lorz, A Coupled Chemotaxis-Fluid Model: Global Existence  - Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Vol. 28, 2011, pp. 643-652
- published version - preprint
 We consider a model arising from biology,
consisting of chemotaxis equations coupled to  
viscous incompressible fluid equations through transport and
external forcing. Global existence of solutions to the Cauchy
problem is investigated under certain conditions. Precisely, for the
Chemotaxis-Navier-Stokes system in two space dimensions, we
obtain global existence for large data. In three space dimensions, we prove global existence of weak
solutions  for the Chemotaxis-Stokes system with nonlinear diffusion for the cell density. 
    

 

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.
   

 

 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.