Structural modeling of drug release from biodegradable porous matrices based on a combined diffusion/erosion process.

Biodegradable, porous microspheres exhibit a wide range of release profiles. We propose in this paper a unifying approach based on the dual action of diffusion and erosion to establish which mechanisms are responsible for the variety of release kinetics observed during in vitro experiments. Our modeling procedure leads to the partitioning of the matrix into multiple, identical elements, thus simplifying significantly the mathematical and numerical treatment of the problem. The model equations cannot be solved analytically, since the domain contains a moving interface, and must therefore be solved numerically, using specific methods designed for that purpose. Our model confirms the major role that the relative dominance between diffusion and erosion plays in the release kinetics. In particular, the velocity of erosion, the effective diffusion coefficient of the drug molecule in the wetted polymer, the average pore length, and the initial pore diameter are sensitive parameters, whereas the porosity and the effective diffusion coefficient of the drug in the solvent-filled pores is seen to have little influence, if any, on the release kinetics. The model is confirmed by using release data from biodegradable microspheres with different ratios of low and high molecular weight PLA. Excellent goodness of fit is achieved by varying two parameters for all types of experimental kinetics: from the typical square root of time profile to zero-order kinetics to concave release curves. We are also able to predict, by interpolation, release curves from microspheres made of intermediate, untested ratios of PLA by using a relation between two model parameters.

Variable maturation velocity and parameter sensitivity in a model of haematopoiesis.

We analyse an age-structured model for haematopoiesis, describing the development of specialized cells in the blood from undifferentiated stem cells and including the controlling effects of hormones. Variation in the length of time for maturing of precursor cells in this model has a stabilizing influence. When the maturing process does not vary, then the age-structured model reduces to a delay differential equation. Depending on the death process considered, either a differential equation with two time delays or a differential equation with a state-dependent delay is obtained. Each of these is analysed in turn, for its linear stability. A sensitivity analysis of the parameters in this model shows which biochemical processes in the negative feedback most strongly affect the solutions.

Numerical bifurcation analysis of delay differential equations arising from physiological modeling.

This paper has a dual purpose. First, we describe numerical methods for continuation and bifurcation analysis of steady state solutions and periodic solutions of systems of delay differential equations with an arbitrary number of fixed, discrete delays. Second, we demonstrate how these methods can be used to obtain insight into complex biological regulatory systems in which interactions occur with time delays: for this, we consider a system of two equations for the plasma glucose and insulin concentrations in a diabetic patient subject to a system of external assistance. The model has two delays: the technological delay of the external system, and the physiological delay of the patient's liver. We compute stability of the steady state solution as a function of two parameters, compare with analytical results and compute several branches of periodic solutions and their stability. These numerical results allow to infer two categories of diabetic patients for which the external system has different efficiency.

Regulation of platelet production: the normal response to perturbation and cyclical platelet disease.

An age-structured model for the regulation of platelet production is developed, and compared with both normal and pathological platelet production. We consider the role of thrombopoietin (TPO) in this process, how TPO affects the transition between megakaryocytes of various ploidy classes, and their individual contributions to platelet production. After the estimation of the relevant parameters of the model from both in vivo and in vitro data, we use the model to numerically reproduce the normal human response to a bolus injection of TPO. We further show that our model reproduces the dynamic characteristics of autoimmune cyclical thromobocytopenia if the rate of platelet destruction in the circulation is elevated to more than twice the normal value.

Hematopoietic model with moving boundary condition and state dependent delay: applications in erythropoiesis.

An age-structured model for erythropoiesis is extended to include the active destruction of the oldest mature cells and possible control by apoptosis. The former condition, which is applicable to other population models where the predator satiates, becomes a constant flux boundary condition and results in a moving boundary condition. The method of characteristics reduces the age-structured model to a system of threshold type differential delay equations. Under certain assumptions, this model can be reduced to a system of delay differential equations with a state dependent delay in an uncoupled differential equation for the moving boundary condition. Analysis of the characteristic equation for the linearized model demonstrates the existence of a Hopf bifurcation when the destruction rate of erythrocytes is modified. The parameters in the system are estimated from experimental data, and the model is simulated for a normal human subject following a loss of blood typical of a blood donation. Numerical studies for a rabbit with an induced auto-immune hemolytic anemia are performed and compared with experimental data.

Age-structured and two-delay models for erythropoiesis.

An age-structured model is developed for erythropoiesis and is reduced to a system of threshold-type differential delay equations using the method of characteristics. Under certain assumptions, this model can be reduced to a system of delay differential equations with two delays. The parameters in the system are estimated from experimental data, and the model is simulated for a normal human subject following a loss of blood. The characteristic equation of the two-delay equation is analyzed and shown to exhibit Hopf bifurcations when the destruction rate of erythrocytes is increased. A numerical study for a rabbit with autoimmune hemolytic anemia is performed and compared with experimental data.

Feedback and delays in neurological diseases: a modeling study using dynamical systems.

Numerous regulatory mechanisms in motor control involve the presence of time delays in the controlled behavior of the system. Experimentally, we have shown that an increase of the time delay in visual feedback induces different oscillations in control subjects and in patients with neurological diseases during the performance of a simple compensatory tracking task. A preliminary model is proposed to describe the oscillations observed in control subjects and in patients with neurological diseases. The influence of delays in two feedback loops are the main components of the motor control circuitry involved in this task and are studied from an analytical and physiological perspective. We analytically determine the influence in the model of each of these delays on the stability of the finger position. In addition, the influence of stochastic elements ("noise") in the modeling equation is seen to contribute qualitatively to a more accurate reproduction of experimental traces in patients with Parkinson's disease but not in patients with cerebellar disease.

Production of acetone and conversion of acetone to acetate in the perfused rat liver.

The utilization of millimolar concentrations of [2-14C]acetone and the production of acetone from acetoacetate were studied in perfused livers from 48-h starved rats. We devised a procedure for determining, in a perfused liver system, the first-order rate constant for the decarboxylation of acetoacetate (0.29 +/- 0.09 h-1, S.E., n = 8). After perfusion of livers with [2-14C]acetone, labeled acetate was isolated from the perfusion medium and characterized as [1-14C]acetate. No radioactivity was found in lactate or 3-hydroxybutyrate. After 90 min of perfusion with [2-14C]acetone, the specific activity of acetate was 30 +/- 4% (n = 13) of the initial specific activity of acetone. We conclude that, in perfused livers from 2-day starved rats, acetone metabolism occurs for the most part via free acetate.

Dynamics of pure parasystole.

A mathematical model is proposed for ventricular parasystole. In this model, there are two separate rhythms, a sinus rhythm and a ventricular ectopic rhythm. An ectopic beat will occur if the ectopic depolarization falls during a time interval when the ventricles are not refractory. Following an ectopic beat there is a compensatory pause. Analysis of this model, utilizing numerical simulation and techniques in number theory, demonstrates several new rules for parasystole. Specifically, for any set of fixed values for the sinus and ectopic frequencies and the ventricular refractory time, there are at most three different values for the number of sinus beats between ectopic beats. One and only one of these values is odd, and the sum of the two smaller values is one less than the larger value. The variation in the allowed values of the number of sinus beats between ectopic beats, as a function of the parameters of the model, is classified. Clinical cases found in the literature display certain aspects of the predictions of the theoretical model. Theoretical analysis of this kind provides new approaches to assessing the mechanism of complex ventricular arrhythmias.

Periodic pulsatile stimulation of a nonlinear oscillator.

We consider pulsatile periodic stimulation of an integrate-and-fire oscillator, and investigate the possible phase-locked patterns between the intrinsic rhythm and the forcing system. These stimulations are varied according to their period and intensity, corresponding to controllable experimental parameters. Phase transition curves are derived and analyzed under functional iteration. Two different perturbative mechanisms are considered, leading to significant differences in possible behaviors. Bistability can be obtained in one case. The analysis is reduced to the investigation of two-parameter families of discontinuous maps of the circle into itself.