Volume 27, Issue 5

- Mathematical models and numerical algorithms of substance transport over a closed graph of a cardiovascular system must satisfy the property of conservativity. A conservative model and a numerical algorithm for calculation of substance transport through a working heart are proposed in the paper. The efficiency of the algorithm considered here is confirmed by test calculations. A method for simulation of substance transport in tissues surrounding vessels is also proposed.

A technology for high-resolution efficient numerical modelling of bioimpedance measurements is considered that includes 3D image segmentation, adaptive unstructured tetrahedral mesh generation, finite-element discretization, and analysis of simulation data. The first-order convergence of the proposed numerical methods on a series of unmatched meshes and roughly second-order convergence on a series of nested meshes are shown. The current, potential, and sensitivity field distributions are computed for conventional schemes of bioimpedance measurements using segmented geometrical torso model of the Visible Human Project (VHP) man. Use of the adaptive tetrahedral meshes reduces significantly the number of mesh elements and, hence, the associated computational cost compared to rectangular meshes while keeping the model accuracy.

Visualization of the Pareto frontier is a new approach in multi-criteria decision making, i.e. searching for efficient decisions taking multiple decision criteria into account. In this paper, visualization of the Pareto frontier is used in the process of multi-criteria search for efficient strategies of HIV therapy. By using a non-linear mathematical model of virus dynamics, which takes into account the phenomenon of virus variability under the influence of a drug, we show that in the framework of the model it is possible to construct a therapy strategy that balances the conflicting requirements imposed on the therapy process.

A discrete stochastic model of HIV infection spread within a heterogeneous population is presented. A system of high-dimensional stochastic difference equations is used in the model. A system of upper estimates for mathematical expectations of the sizes of groups of individuals in the population is constructed. Sufficient conditions for the extinction of the infection process are obtained in terms of M-matrices. Based on the Monte Carlo method, a numerical algorithm and a simulation code are developed, which allows one to study probabilistic characteristics of the observed variables of the model. Results of numerical experiments are presented.

A complex method of study is proposed, which includes a complete set of operations combining formal and informal methods, simulation and analytic approaches. This method has been used in construction and analysis of a set of models of tundra cenosis, i.e., ‘vegetation-lemmingspolar foxes’ (VLPF) model, an individual-oriented model (IOM) of a lemming population, and a simplified model in the form of a difference equation. Calculation formulas relating the parameters of the VLPF model and the difference equation are obtained, hypotheses concerning the principal mechanisms determining fluctuations in animal populations are formulated. Analysis of the properties of the difference equation and their manifestations in the VLPF and IOM models is performed.

We have simulated a chain of sinoatrial pacemaker cells connected via gap junctions. To study the dynamics we introduce a phase of oscillations and show that Burgers equation is adequate to describe the phase dynamics in the sinoatrial node. We show that the propagating action potential bears properties of either trigger waves, or phase waves, which depends upon its wave number. We propose a definition of the safety factor that is applicable to the oscillatory tissue of the sinoatrial node and show that the maximum of the safety factor relates to the boundary case separating phase waves and trigger waves.

Platelet aggregation at the site of the vascular injury leads to the formation of a hemostatic plug covering the injury site, or a thrombus in the pathological case. The mechanisms that control clot growth or lead to growth arrest are not yet completely understood. In order to study these mechanisms theoretically, we use the Dissipative Particle Dynamics method, which allows us to model individual platelets in the flow and in the clot. The model takes into account different stages of the platelet adhesion process. First, a platelet is captured reversibly by the aggregate, then it is activated and adheres firmly, becoming a part of its core.We suggest that the core of the clot is composed of platelets unable to attach new platelets from the flow due to their activation by thrombin and/or wrapping by the fibrin mesh. The simulations are in a good agreement with the experimental results [9]. Modelling shows that stopping the growth of a hemostatic plug (and thrombus) may result from its exterior part being removed by the flow and exposed its non-adhesive core to the flow.