Interdisciplinary center for Mathematical modelling in Biomedicine
Interdisciplinary center for Mathematical modelling in Biomedicine




S.M. Nikol’skii Mathematical Institute


Vitalii Volpert

Doctor of Sciences, professor, participant

Structural unit: S.M. Nikol’skii Mathematical Institute.

Mathematical modelling in biomedicine is one of rapidly developing scientific disciplines motivated by the fundamental research and by the applications to public health. It requires close collaboration between different disciplines and includes the development of mathematical models of complex physiological processes, mathematical analysis of these models and their computer simulations. Scientific Center on mathematical modelling in biomedicine is recently created in order to promote scientific research in this area and to form a young generation of scientists working in this field.

Main scientific directions
  • Mathematical modeling of cardiovascular diseases.
  • Mathematical immunology.
  • Mathematical oncology.
Achievements All achievements

Development of viral infection in the tissues such as lymph nodes or spleen is studied depending on virus multiplication in the host cells, their transport and on the immune response. The properties of the cells of the immune system and the initial viral load determine the spatiotemporal regimes of infection dynamics. It is shown that infection can be completely eliminated or it can persist at some level together with a certain chronic immune response in a spatially uniform or oscillatory mode. Finally, the immune cells can be completely exhausted leading to a high viral load persistence in the tissue. Our study shows that both the motility of immune cells and the virus infection propagation represented by the diffusion rate coefficients are relevant control parameters determining the fate of virus-host interaction. 
Reaction-diffusion equation with delay arising in modeling the immune response is investigated. We prove the existence of traveling waves in the bistable case using the Leray– Schauder method. Differently from the previous works, we do not assume here quasi-monotonicity of the delayed reaction term.

Following a stroke, cortical networks in the penumbra area become fragmented and partly deactivated. We develop a model to study the propagation of waves of electric potential in the cortical tissue with integrodifferential equations arising in neural field models. The wave speed is characterized by the tissue excitability and connectivity determined through parameters of the model. Post-stroke tissue damage in the penumbra area creates a hypoconnectivity and decreases the speed of wave propagation. It is proposed that external stimulation could restore the wave speed in the penumbra area under certain conditions of the parameters. Model guided cortical stimulation could be used to improve the functioning of cortical networks.

Formation of blood clot in response to the vessel damage is triggered by the complex network of biochemical reactions of the coagulation cascade. The process of clot growth can be modeled as a traveling wave solution of the bistable reaction–diffusion system. The critical value of the initial condition which leads to convergence of the solution to the traveling wave corresponds to the pulse solution of the corresponding stationary problem. In the current study we prove the existence of the pulse solution for the stationary problem in the model of the main reactions of the blood coagulation cascade using the Leray–Schauder method.
The mechanics of platelet initial adhesion due to interactions between GPIb receptor with von Willebrand factor (vWf) multimers is essential for thrombus growth and the regulation of this process. Multimeric structure of vWf is known to make adhesion sensitive to the hydrodynamic conditions, providing intensive platelet aggregation in bulk fluid for high shear rates. But it is still unclear how it affects the dynamics of platelet motion near vessel walls and efficiency of their adhesion to surfaces. Our goal is to resolve the principal issues in the mechanics of platelet initial attachment via GPIb-vWf bonds in near-wall flow conditions: when the platelet tends to roll or slide and how this dynamics depends on the size, conformation and adhesive properties of the vWf multimers. We employ a 3D computer model based on a combination of the Lattice Boltzmann method with mesoscopic particle dynamics for explicit simulation of vWf-mediated blood platelet adhesion in shear flow. Our results reveal the link between the mechanics of platelet initial adhesion and the physico-chemical properties of vWf multimers. This has implications in further theoretical investigation of thrombus growth dynamics, as well as the interpretation of in vitro experimental data.