title: Mass concentration in a nonlocal model of clonal selection.
abstract: Self-renewal is a constitutive property of stem cells. Testing the cancer stem cell hypothesis requires investigation of the impact of self-renewal on cancer expansion. To better understand this impact, we propose a mathematical model describing the dynamics of a continuum of cell clones structured by the self-renewal potential. The model is an extension of the finite multi-compartment models of interactions between normal and cancer cells in acute leukemias. It takes a form of a system of integro-differential equations with a nonlinear and nonlocal coupling which describes regulatory feedback loops of cell proliferation and differentiation. We show that this coupling leads to mass concentration in points corresponding to the maxima of the self-renewal potential and the solutions of the model tend asymptotically to Dirac measures multiplied by positive constants. Furthermore, using a Lyapunov function constructed for the finite dimensional counterpart of the model, we prove that the total mass of the solution converges to a globally stable equilibrium. Additionally, we show stability of the model in the space of positive Radon measures equipped with the flat metric (bounded Lipschitz distance). Analytical results are illustrated by numerical simulations.
authors.affiliation: Institute of Applied Mathematics, BIOQUANT, University of Heidelberg, Im Neuenheimer Feld 294, 69120, Heidelberg, Germany.
authors.forename: J-E
authors.initials: JE
authors.lastname: Busse
authors.affiliation: Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097, Warsaw, Poland.|Institute of Mathematics, Polish Academy of Science, Śniadeckich 8, 00-656, Warszawa, Poland.
authors.forename: P
authors.initials: P
authors.lastname: Gwiazda
authors.affiliation: Institute of Applied Mathematics, BIOQUANT, University of Heidelberg, Im Neuenheimer Feld 294, 69120, Heidelberg, Germany. Anna.Marciniak@iwr.uni-heidelberg.de.|Interdisciplinary Center of Scientific Computing (IWR), University of Heidelberg, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany. Anna.Marciniak@iwr.uni-heidelberg.de.|Bioquant, University of Heidelberg, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany. Anna.Marciniak@iwr.uni-heidelberg.de.
authors.forename: A
authors.initials: A
authors.lastname: Marciniak-Czochra
mesh_terms: D002454:Cell Differentiation; D049109:Cell Proliferation; D002999:Clone Cells; D003198:Computer Simulation; D006801:Humans; D008954:Models, Biological; D009369:Neoplasms; D013234:Stem Cells
keywords: Bounded Lipschitz distance; Cell differentiation model; Clonal evolution; Integro-differential equations; Lyapunov function; Mass concentration; Selection process