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

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

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

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

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

The stochastic dynamics of chemical reactions can be accurately described by chemical master equations. An approximated time-evolution equation of the Langevin type has been proposed by Gillespie based on two explicit dynamical conditions. However, when numerically solve these chemical Langevin equations, we often have a small stopping time–a time point of having an unphysical solution–in the case of low molecular numbers. This paper proposes an approach to simulate stochasticities in chemical reactions with deterministic delay differential equations. We introduce a deterministic Brownian motion described by delay differential equations, and replace the Gaussian noise in the chemical Langevin equations by the solutions of these deterministic equations. This modification can largely increase the stopping time in simulations and regain the accuracy as in the chemical Langevin equations. The novel aspect of the present study is to apply the deterministic Brownian motion to chemical reactions. It suggests a possible direction of developing a hybrid method of simulating dynamic behaviours of complex gene regulation networks.

A general framework for age-structured predator-prey systems is introduced. Individuals are distinguished into two classes, juveniles and adults, and several possible interactions are considered. The initial system of partial differential equations is reduced 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 are provided. Previous models are considered and analysed in view of the above results. A Rosenzweig-MacArthur model with delay is presented as an example.

A mathematical model, coupling the dynamics of short-term stem-like cells and mature leukocytes in leukemia with that of the immune system, is investigated. The model is described by a system of seven delay differential equations with seven delays. Three equilibrium points E0, E1, E2 are highlighted. The stability and the existence of the Hopf bifurcation for the equilibrium points are investigated. In the analysis of the model, the rate of asymmetric division and the rate of symmetric division are very important.

Mathematical and computational neuroscience have contributed to the brain sciences by the study of the dynamics of individual neurons and more recently the study of the dynamics of electrophysiological networks. Often these studies treat individual neurons as points or the nodes in networks and the biochemistry of the brain appears, if at all, as some intermediate variables by which the neurons communicate with each other. In fact, many neurons change brain function not by communicating in one-to-one fashion with other neurons, but instead by projecting changes in biochemistry over long distances. This biochemical network is of crucial importance for brain function and it influences and is influenced by the more traditional electrophysiological networks. Understanding how biochemical networks interact with electrophysiological networks to produce brain function both in health and disease poses new challenges for mathematical neuroscience.

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