Models of Discrete and Continuous Cell Differentiation in the Framework of Transport Equation

We introduce a class of structured population models describing cell differentiation that consists of discrete and continuous transitions. The model is defined in a framework of measure-valued solutions of a nonlinear transport equation with a growth term. To obtain ODE-type quasi-stationary node points we exploit the idea of non-Lipschitz zeroes in the velocity. This, in combination with the so-called measure-transmission conditions, allows us to prove the existence and uniqueness of solutions. Since the analysis has biological motivations, we provide examples of its application. (An erratum is attached.)

[1]  L. Ambrosio,et al.  Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields , 2009 .

[2]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[3]  Marie Doumic,et al.  A Structured Population Model of Cell Differentiation , 2010, SIAM J. Appl. Math..

[4]  Arthur D Lander,et al.  The 'stem cell' concept: is it holding us back? , 2009, Journal of biology.

[5]  Brian I Lord,et al.  13 – Biology of the haemopoietic stem cell , 1997 .

[6]  Agnieszka Ulikowska,et al.  Structured Population Models in Metric Spaces , 2013 .

[7]  Michael C Mackey,et al.  A mathematical model of hematopoiesis: II. Cyclical neutropenia. , 2005, Journal of theoretical biology.

[8]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[9]  Michael C Mackey,et al.  A mathematical model of hematopoiesis--I. Periodic chronic myelogenous leukemia. , 2005, Journal of theoretical biology.

[10]  C. M. Dafermos,et al.  Hyberbolic [i.e. Hyperbolic] conservation laws in continuum physics , 2005 .

[11]  Fabien Crauste,et al.  A Mathematical Study of the Hematopoiesis Process with Applications to Chronic Myelogenous Leukemia , 2009, SIAM J. Appl. Math..

[12]  Arnold Kriegstein,et al.  The glial nature of embryonic and adult neural stem cells. , 2009, Annual review of neuroscience.

[13]  P. Gwiazda,et al.  A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients , 2010 .

[14]  D. Willett.  Nonlinear vector integral equations as contraction mappings , 1964 .

[15]  Helmut Neunzert,et al.  An introduction to the nonlinear Boltzmann-Vlasov equation , 1984 .

[16]  Carlo Cercignani,et al.  Kinetic Theories and the Boltzmann Equation , 1984 .

[17]  Willi Jäger,et al.  Modeling of asymmetric cell division in hematopoietic stem cells--regulation of self-renewal is essential for efficient repopulation. , 2009, Stem cells and development.

[18]  Odo Diekmann,et al.  Boundedness, global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations , 2005 .

[19]  李幼升,et al.  Ph , 1989 .

[20]  Yukihiko Nakata,et al.  Stability analysis of multi-compartment models for cell production systems , 2012, Journal of biological dynamics.

[21]  Tomás Alarcón,et al.  A model for stem cell population dynamics with regulated maturation delay , 2011 .

[22]  M. Mackey,et al.  Age-structured and two-delay models for erythropoiesis. , 1995, Mathematical biosciences.

[23]  Q. Nie,et al.  Cell Lineages and the Logic of Proliferative Control , 2009, PLoS biology.

[24]  Qing Nie,et al.  Feedback regulation in multistage cell lineages. , 2008, Mathematical biosciences and engineering : MBE.

[25]  Anna K. Marciniak-Czochra,et al.  Characterization of stem cells using mathematical models of multistage cell lineages , 2011, Math. Comput. Model..

[26]  N. L. Carothers.  Real Analysis: Preface , 2000 .

[27]  Benoît Perthame,et al.  Stability Analysis of a Simplified Yet Complete Model for Chronic Myelogenous Leukemia , 2010, Bulletin of mathematical biology.