We are interested in optimizing the co-administration of two drugs for some acute myeloidleukemias (AML), and we are looking for in vitro protocols as a first step. This issue canbe formulated as an optimal control problem. The dynamics of leukemic cell populations inculture is given by age-structured partial differential equations, which can be reduced toa system of delay differential equations, and where the controls represent the action ofthe drugs. The objective function relies on eigenelements of the uncontrolled model and ongeneral relative entropy, with the idea to maximize the efficiency of the protocols. Theconstraints take into account the toxicity of the drugs. We present in this paper themodeling aspects, as well as theoretical and numerical results on the optimal controlproblem that we get.

In this paper we present a fluid-structure interaction model of neuron’s membranedeformation. The membrane-actin is considered as an elastic solid layer, while thecytoplasm is considered as a viscous fluid one. The membrane-actin layer is governed byelasticity equations while the cytoplasm is described by the Navier-Stokes equations. Atthe interface between the cytoplasm and the membrane we consider a match between the solidvelocity displacement and the fluid velocity as well as the mechanical equilibrium. Themembrane, which faces the extracellular medium, is free to move. This will change thegeometry in time. To take into account the deformation of the initial configuration, weuse the Arbitrary Lagrangian Eulerian method in order to take into account the meshdisplacement. The numerical simulations, show the emergence of a filopodium, a typicalstructure in cells undergoing deformation.

We analyse a reduced version of the Grodins et al. control model [14] of respiration involving only CO2, andshow that it can be dramatically simplified by the use of judicious approximations. Inparticular, we show that the conceptual basis of the popular model of Mackey and Glass[20] is at odds with the important transportprocesses of the Grodins model. Despite this, a realistic approximation of the Grodinsmodel yields a Mackey-Glass type model with almost the same criterion for the onset ofCheyne-Stokes breathing.

While the reduced Grodins model does apparently provide a realistic mechanism forinstability, consideration of the buffering effect of the blood-brain barrier appears tomake it unlikely. We conclude that a realistic physiological model of Grodins type toexplain Cheyne–Stokes breathing is not yet in place, and raise the question whether thebicarbonate buffering system has a rôle to play.

A two dimensional two-delays differential system modeling the dynamics of stem-like cellsand white-blood cells in Chronic Myelogenous Leukemia is considered. All three types ofstem cell division (asymmetric division, symmetric renewal and symmetric differentiation)are present in the model. Stability of equilibria is investigated and emergence ofperiodic solutions of limit cycle type, as a result of a Hopf bifurcation, is eventuallyshown. The stability of these limit cycles is studied using the first Lyapunovcoefficient.

The stochastic dynamics of chemical reactions can be accurately described by chemicalmaster equations. An approximated time-evolution equation of the Langevin type has beenproposed by Gillespie based on two explicit dynamical conditions. However, whennumerically solve these chemical Langevin equations, we often have a small stopping time–atime point of having an unphysical solution–in the case of low molecular numbers. Thispaper proposes an approach to simulate stochasticities in chemical reactions withdeterministic delay differential equations. We introduce a deterministic Brownian motiondescribed by delay differential equations, and replace the Gaussian noise in the chemicalLangevin equations by the solutions of these deterministic equations. This modificationcan largely increase the stopping time in simulations and regain the accuracy as in thechemical Langevin equations. The novel aspect of the present study is to apply thedeterministic Brownian motion to chemical reactions. It suggests a possible direction ofdeveloping a hybrid method of simulating dynamic behaviours of complex gene regulationnetworks.

A general framework for age-structured predator-prey systems is introduced. Individualsare distinguished into two classes, juveniles and adults, and several possibleinteractions are considered. The initial system of partial differential equations isreduced to a system of (neutral) delay differential equations with one or two delays.Thanks to this approach, physically correct models for predator-prey with delay areprovided. Previous models are considered and analysed in view of the above results. ARosenzweig-MacArthur model with delay is presented as an example.

A mathematical model, coupling the dynamics of short-term stem-like cells and matureleukocytes in leukemia with that of the immune system, is investigated. The model isdescribed by a system of seven delay differential equations with seven delays. Threeequilibrium points E0, E1,E2 are highlighted. The stability and the existence of theHopf bifurcation for the equilibrium points are investigated. In the analysis of themodel, the rate of asymmetric division and the rate of symmetric division are veryimportant.

Mathematical and computational neuroscience have contributed to the brain sciences by thestudy of the dynamics of individual neurons and more recently the study of the dynamics ofelectrophysiological networks. Often these studies treat individual neurons as points orthe nodes in networks and the biochemistry of the brain appears, if at all, as someintermediate variables by which the neurons communicate with each other. In fact, manyneurons change brain function not by communicating in one-to-one fashion with otherneurons, but instead by projecting changes in biochemistry over long distances. Thisbiochemical network is of crucial importance for brain function and it influences and isinfluenced by the more traditional electrophysiological networks. Understanding howbiochemical networks interact with electrophysiological networks to produce brain functionboth in health and disease poses new challenges for mathematical neuroscience.

We show that simple diffusion processes are weak limits of piecewise continuous processesconstructed within a totally deterministic framework. The proofs are based on thecontinuous mapping theorem and the functional central limit theorem.