330 research outputs found
Eigenelements of a General Aggregation-Fragmentation Model
We consider a linear integro-differential equation which arises to describe
both aggregation-fragmentation processes and cell division. We prove the
existence of a solution (\lb,\U,\phi) to the related eigenproblem. Such
eigenelements are useful to study the long time asymptotic behaviour of
solutions as well as the steady states when the equation is coupled with an
ODE. Our study concerns a non-constant transport term that can vanish at
since it seems to be relevant to describe some biological processes like
proteins aggregation. Non lower-bounded transport terms bring difficulties to
find estimates. All the work of this paper is to solve this problem
using weighted-norms
Biomixing by chemotaxis and efficiency of biological reactions: the critical reaction case
Many phenomena in biology involve both reactions and chemotaxis. These
processes can clearly influence each other, and chemotaxis can play an
important role in sustaining and speeding up the reaction. In continuation of
our earlier work, we consider a model with a single density function involving
diffusion, advection, chemotaxis, and absorbing reaction. The model is
motivated, in particular, by the studies of coral broadcast spawning, where
experimental observations of the efficiency of fertilization rates
significantly exceed the data obtained from numerical models that do not take
chemotaxis (attraction of sperm gametes by a chemical secreted by egg gametes)
into account. We consider the case of the weakly coupled quadratic reaction
term, which is the most natural from the biological point of view and was left
open. The result is that similarly to higher power coupling, the chemotaxis
plays a crucial role in ensuring efficiency of reaction. However,
mathematically, the picture is quite different in the quadratic reaction case
and is more subtle. The reaction is now complete even in the absence of
chemotaxis, but the timescales are very different. Without chemotaxis, the
reaction is very slow, especially for the weak reaction coupling coefficient.
With chemotaxis, the timescale and efficiency of reaction are independent of
the coupling parameter.Comment: 10 pages. arXiv admin note: text overlap with arXiv:1101.244
On the Calibration of a Size-Structured Population Model from Experimental Data
The aim of this work is twofold. First, we survey the techniques developed in
(Perthame, Zubelli, 2007) and (Doumic, Perthame, Zubelli, 2008) to reconstruct
the division (birth) rate from the cell volume distribution data in certain
structured population models. Secondly, we implement such techniques on
experimental cell volume distributions available in the literature so as to
validate the theoretical and numerical results. As a proof of concept, we use
the data reported in the classical work of Kubitschek [3] concerning
Escherichia coli in vitro experiments measured by means of a Coulter
transducer-multichannel analyzer system (Coulter Electronics, Inc., Hialeah,
Fla, USA.) Despite the rather old measurement technology, the reconstructed
division rates still display potentially useful biological features
On interfaces between cell populations with different mobilities
Partial differential equations describing the dynamics of cell population densities from a fluid mechanical perspective can model the growth of avascular tumours. In this framework, we consider a system of equations that describes the interaction between a population of dividing cells and a population of non-dividing cells. The two cell populations are characterised by different mobilities. We present the results of numerical simulations displaying two-dimensional spherical waves with sharp interfaces between dividing and non-dividing cells. Furthermore, we numerically observe how different ratios between the mobilities change the morphology of the interfaces, and lead to the emergence of finger-like patterns of invasion above a threshold. Motivated by these simulations, we study the existence of one-dimensional travelling wave solutions
A General Inverse Problem for the Growth-Fragmentation Equation
The growth-fragmentation equation arises in many different contexts, ranging
from cell division, protein polymerization, biopolymers, neurosciences etc.
Direct observation of temporal dynamics being often difficult, it is of main
interest to develop theoretical and numerical methods to recover reaction rates
and parameters of the equation from indirect observation of the solution.
Following the work done in (Perthame, Zubelli, 2006) and (Doumic, Perthame,
Zubelli, 2009) for the specific case of the cell division equation, we address
here the general question of recovering the fragmentation rate of the equation
from the observation of the time-asymptotic solution, when the fragmentation
kernel and the growth rates are fully general. We give both theoretical results
and numerical methods, and discuss the remaining issues
Wave-like solutions for nonlocal reaction-diffusion equations: A toy model
Traveling waves for the nonlocal Fisher Equation can exhibit much more complex behaviour than for the usual Fisher equation. A striking numerical observation is that a traveling wave with minimal speed can connect a dynamically unstable steady state 0 to a Turing unstable steady state 1, see [12]. This is proved in [1, 6] in the case where the speed is far from minimal, where we expect the wave to be monotone. Here we introduce a simplified nonlocal Fisher equation for which we can build simple analytical traveling wave solutions that exhibit various behaviours. These traveling waves, with minimal speed or not, can (i) connect monotonically 0 and 1, (ii) connect these two states non-monotonically, and (iii) connect 0 to a wavetrain around 1. The latter exist in a regime where time dynamics converges to another object observed in [3, 8]: a wave that connects 0 to a pulsating wave around 1. © 2013 EDP Sciences
Kinetic models for epidemic dynamics with social heterogeneity
We introduce a mathematical description of the impact of sociality in the
spread of infectious diseases by integrating an epidemiological dynamics with a
kinetic modeling of population-based contacts. The kinetic description leads to
study the evolution over time of Boltzmann-type equations describing the number
densities of social contacts of susceptible, infected and recovered
individuals, whose proportions are driven by a classical SIR-type compartmental
model in epidemiology. Explicit calculations show that the spread of the
disease is closely related to moments of the contact distribution. Furthermore,
the kinetic model allows to clarify how a selective control can be assumed to
achieve a minimal lockdown strategy by only reducing individuals undergoing a
very large number of daily contacts. We conduct numerical simulations which
confirm the ability of the model to describe different phenomena characteristic
of the rapid spread of an epidemic. Motivated by the COVID-19 pandemic, a last
part is dedicated to fit numerical solutions of the proposed model with
infection data coming from different European countries
Modeling the Effects of Space Structure and Combination Therapies on Phenotypic Heterogeneity and Drug Resistance in Solid Tumors
Histopathological evidence supports the idea that the emergence of phenotypic heterogeneity and resistance to cytotoxic drugs can be considered as a process of selection in tumor cell populations. In this framework, can we explain intra-tumor heterogeneity in terms of selection driven by the local cell environment? Can we overcome the emergence of resistance and favor the eradication of cancer cells by using combination therapies? Bearing these questions in mind, we develop a model describing cell dynamics inside a tumor spheroid under the effects of cytotoxic and cytostatic drugs. Cancer cells are assumed to be structured as a population by two real variables standing for space position and the expression level of a phenotype of resistance to cytotoxic drugs. The model takes explicitly into account the dynamics of resources and anticancer drugs as well as their interactions with the cell population under treatment. We analyze the effects of space structure and combination therapies on phenotypic heterogeneity and chemotherapeutic resistance. Furthermore, we study the efficacy of combined therapy protocols based on constant infusion and bang–bang delivery of cytotoxic and cytostatic drugs
A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows
We consider the Saint-Venant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when completed with friction. Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle shocks and contact discontinuities. However, in general these schemes are known to be quite inaccurate for near steady states, as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions. This difficulty can be overcome by using the so-called well-balanced schemes. We describe a general strategy, based on a local hydrostatic reconstruction, that allows
us to derive a well-balanced scheme from any given numerical flux for the homogeneous problem.
Whenever the initial solver satisfies some classical stability properties, it yields a simple and fast
well-balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality
Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies
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. © EDP Sciences, SMAI, 2013
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