1. Population balances in case of crossing characteristic curves: Application to T-cells immune response
- Author
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Ali Elkamel, Frédéric Gruy, Eric Touboul, Claude Lambert, Qasim Ali, Centre Ingénierie et Santé (CIS-ENSMSE), École des Mines de Saint-Étienne (Mines Saint-Étienne MSE), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT), Department of Applied Mathematics, University of Western Ontario (UWO), Department of Chemical Engineering [Waterloo], University of Waterloo [Waterloo], Laboratoire Georges Friedel (LGF-ENSMSE), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), Département PROcédés Poudres, Interfaces, Cristallisation et Ecoulements (PROPICE-ENSMSE), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-SPIN, Centre Sciences des Processus Industriels et Naturels (SPIN-ENSMSE), Institut Fédératif de Recherche en Sciences et Ingénierie de la Santé (IFRESIS-ENSMSE), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-IFR143, Département Décision en Entreprise : Modélisation, Optimisation (DEMO-ENSMSE), Institut Mines-Télécom [Paris] (IMT)-Institut Mines-Télécom [Paris] (IMT)-Institut Henri Fayol, Laboratoire d'Informatique, de Modélisation et d'optimisation des Systèmes (LIMOS), SIGMA Clermont (SIGMA Clermont)-Université d'Auvergne - Clermont-Ferrand I (UdA)-Ecole Nationale Supérieure des Mines de St Etienne-Centre National de la Recherche Scientifique (CNRS)-Université Blaise Pascal - Clermont-Ferrand 2 (UBP), Department of Applied Mathematics-University of Western Ontario-London-ON-Canada, Department of Chemical Engineering-University of Waterloo-Waterloo-ON-Canada, CHU-Laboratoire d'immunologie-Université Jean Monnet-Saint Etienne, ENSM.SE-Institut Fayol-Saint Etienne, Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-École des Mines de Saint-Étienne (Mines Saint-Étienne MSE), Institut Fédératif de Recherche en Sciences et Ingénierie de la Santé (IFRESIS), Université Jean Monnet - Saint-Étienne (UJM)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Ecole Nationale Supérieure des Mines de St Etienne (ENSM ST-ETIENNE)-Centre National de la Recherche Scientifique (CNRS), and Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Université d'Auvergne - Clermont-Ferrand I (UdA)-SIGMA Clermont (SIGMA Clermont)-Ecole Nationale Supérieure des Mines de St Etienne (ENSM ST-ETIENNE)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Work (thermodynamics) ,General Chemical Engineering ,Population ,Population balance equation ,Monotonic function ,02 engineering and technology ,hyperbolic conservation laws ,Quantitative Biology - Quantitative Methods ,020401 chemical engineering ,T-cell activation ,Cell Behavior (q-bio.CB) ,0204 chemical engineering ,[MATH]Mathematics [math] ,education ,Quantitative Methods (q-bio.QM) ,Mathematics ,characteristic curves ,Conservation law ,education.field_of_study ,Advection ,Mathematical analysis ,population balances ,Function (mathematics) ,021001 nanoscience & nanotechnology ,FOS: Biological sciences ,Quantitative Biology - Cell Behavior ,[SDV.IMM]Life Sciences [q-bio]/Immunology ,0210 nano-technology ,Hyperbolic partial differential equation - Abstract
The progression of a cell population where each individual is characterized by the value of an internal variable varying with time (e.g. size, weight, and protein concentration) is typically modeled by a Population Balance Equation, a first order linear hyperbolic partial differential equation. The characteristics described by internal variables usually vary monotonically with the passage of time. A particular difficulty appears when the characteristic curves exhibit different slopes from each other and therefore cross each other at certain times. In particular such crossing phenomenon occurs during T-cells immune response when the concentrations of protein expressions depend upon each other and also when some global protein (e.g. Interleukin signals) is also involved which is shared by all T-cells. At these crossing points, the linear advection equation is not possible by using the classical way of hyperbolic conservation laws. Therefore, a new Transport Method is introduced in this article which allowed us to find the population density function for such processes. The newly developed Transport method (TM) is shown to work in the case of crossing and to provide a smooth solution at the crossing points in contrast to the classical PDF techniques., Comment: 18 pages, 10 figures
- Published
- 2016
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