Systems describing the dynamics of proliferative and quiescent cells are commonly used as computational models, for instance for tumor growth and hematopoiesis.Focusing on the very earliest stages of hematopoiesis, st...Systems describing the dynamics of proliferative and quiescent cells are commonly used as computational models, for instance for tumor growth and hematopoiesis.Focusing on the very earliest stages of hematopoiesis, stem cells and early progenitors, the authors introduce a new method, based on an energy/Lyapunov functional to analyze the long time behavior of solutions. Compared to existing works, the method in this paper has the advantage that it can be extended to more complex situations. The authors treat a system with space variable and diffusion, and then adapt the energy functional to models with three equations.展开更多
This paper deals with a non-local parabolic equation of Lotka-Volterra type that describes the evolution of phenotypically structured populations. Nonlinearities appear in these systems to model interactions and compe...This paper deals with a non-local parabolic equation of Lotka-Volterra type that describes 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.Different methods aimed at showing the convergence of the solutions to a moving Dirac mass are reviewed. Using either weak or strong regularity assumptions, the authors study the concentration of the solution. To this end, BV estimates in time on appropriate quantities are stated, and a constrained Hamilton-Jacobi equation to identify where the solutions concentrates as Dirac masses is derived.展开更多
In the recent biomechanical theory of cancer growth,solid tumors are considered as liquid-like materials comprising elastic components.In this fluid mechanical view,the expansion ability of a solid tumor into a host t...In the recent biomechanical theory of cancer growth,solid tumors are considered as liquid-like materials comprising elastic components.In this fluid mechanical view,the expansion ability of a solid tumor into a host tissue is mainly driven by either the cell diffusion constant or the cell division rate,with the latter depending on the local cell density(contact inhibition) or/and on the mechanical stress in the tumor.For the two by two degenerate parabolic/elliptic reaction-diffusion system that results from this modeling,the authors prove that there are always traveling waves above a minimal speed,and analyse their shapes.They appear to be complex with composite shapes and discontinuities.Several small parameters allow for analytical solutions,and in particular,the incompressible cells limit is very singular and related to the Hele-Shaw equation.These singular traveling waves are recovered numerically.展开更多
The propagation of unstable interfaces is at the origin of remarkable patterns that are observed in various areas of science as chemical reactions, phase transitions, and growth of bacterial colonies. Since a scalar e...The propagation of unstable interfaces is at the origin of remarkable patterns that are observed in various areas of science as chemical reactions, phase transitions, and growth of bacterial colonies. Since a scalar equation generates usually stable waves, the simplest mathematical description relies on two-by-two reaction-diffusion systems. The authors' interest is the extension of the Fisher/KPP equation to a two-species reaction which represents reactant concentration and temperature when used for flame propagation,and bacterial population and nutrient concentration when used in biology.The authors study circumstances in which instabilities can occur and in particular the effect of dimension. It is observed numerically that spherical waves can be unstable depending on the coefficients. A simpler mathematical framework is to study transversal instability, which means a one-dimensional wave propagating in two space dimensions.Then, explicit analytical formulas give explicitely the range of paramaters for instability.展开更多
基金supported by the European Research Council(ERC) under the European Union’s Horizon 2020 Research and Innovation Programme(No.740623)
文摘Systems describing the dynamics of proliferative and quiescent cells are commonly used as computational models, for instance for tumor growth and hematopoiesis.Focusing on the very earliest stages of hematopoiesis, stem cells and early progenitors, the authors introduce a new method, based on an energy/Lyapunov functional to analyze the long time behavior of solutions. Compared to existing works, the method in this paper has the advantage that it can be extended to more complex situations. The authors treat a system with space variable and diffusion, and then adapt the energy functional to models with three equations.
基金supported by ANR-13-BS01-0004 funded by the French Ministry of Research(ANR Kibord)
文摘This paper deals with a non-local parabolic equation of Lotka-Volterra type that describes 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.Different methods aimed at showing the convergence of the solutions to a moving Dirac mass are reviewed. Using either weak or strong regularity assumptions, the authors study the concentration of the solution. To this end, BV estimates in time on appropriate quantities are stated, and a constrained Hamilton-Jacobi equation to identify where the solutions concentrates as Dirac masses is derived.
基金Project supported by the ANR grant PhysiCancer and the BMBF grant LungSys
文摘In the recent biomechanical theory of cancer growth,solid tumors are considered as liquid-like materials comprising elastic components.In this fluid mechanical view,the expansion ability of a solid tumor into a host tissue is mainly driven by either the cell diffusion constant or the cell division rate,with the latter depending on the local cell density(contact inhibition) or/and on the mechanical stress in the tumor.For the two by two degenerate parabolic/elliptic reaction-diffusion system that results from this modeling,the authors prove that there are always traveling waves above a minimal speed,and analyse their shapes.They appear to be complex with composite shapes and discontinuities.Several small parameters allow for analytical solutions,and in particular,the incompressible cells limit is very singular and related to the Hele-Shaw equation.These singular traveling waves are recovered numerically.
基金supported by the FONDECYT Grant(No.1130126)the ECOS Project(No.C11E07)the Fondo Basal CMM and the French"ANR Blanche"Project Kibord(No.ANR-13-BS01-0004)
文摘The propagation of unstable interfaces is at the origin of remarkable patterns that are observed in various areas of science as chemical reactions, phase transitions, and growth of bacterial colonies. Since a scalar equation generates usually stable waves, the simplest mathematical description relies on two-by-two reaction-diffusion systems. The authors' interest is the extension of the Fisher/KPP equation to a two-species reaction which represents reactant concentration and temperature when used for flame propagation,and bacterial population and nutrient concentration when used in biology.The authors study circumstances in which instabilities can occur and in particular the effect of dimension. It is observed numerically that spherical waves can be unstable depending on the coefficients. A simpler mathematical framework is to study transversal instability, which means a one-dimensional wave propagating in two space dimensions.Then, explicit analytical formulas give explicitely the range of paramaters for instability.
