Mathematical modeling in biomedicine

Research and Innovation

Research Projects

Mathematics and Telecommunications Differential Equations

Year 2019

Department S.M. Nikol’skii Mathematical Institute

Tags mathematical modeling heart-vascular system oncological disease immune response viral infection

Development and research of blood clotting models and description of thrombin production in normal and pathological (hemophilia) cases; comparison with experimental data. The study of spatial models of blood clotting based on reaction-diffusion equations. The study of the rate of thrombosis, considered as a reaction-diffusion wave. The study of blood clotting in the stream (veins, arteries), determination conditions for normal growth of the clot and excessive growth leading to the development of thrombosis.

The study of mathematical models of cancer growth, taking into account angiogenesis; determination of optimal protocols for drug administration, taking into account the interaction of chemotherapy and angiogenesis. The study of hematological cancers, including multiple myeloma. The study of mutations of malignant cells and the emergence of resistant clones. The study of the interaction of cancer with the body's immune system and the determination of various modes of tumor growth.

Development and study of mathematical models of the immune response to viral infection taking into account mutations of viruses. Determination of the conditions and dynamics of the evolution of viruses. Construction and calibration of mathematical models of various extent of specification for a compact description of key processes of regulation of the immune response, taking into account the structure of lymphoid organs. Investigation of integrative mathematical models of the immune system response to HIV infection on the criterion of controllability and the structure of reachable sets.

List of key publications on the project:

G. Bocharov, V. Volpert, B. Ludewig, A. Meyerhans. Mathematical Immunology of Virus Infections. Springer, 2018;
Bocharov, G., Meyerhans, A., Bessonov, N., Trofimchuk, S., Volpert, V. Interplay between reaction and diffusion processes in governing the dynamics of virus infections. Journal of Theoretical Biology, 2018;
Beuter, A., Balossier, A., Trofimchuk, S., Volpert, V. Modeling of post-stroke stimulation of cortical tissue. Mathematical Biosciences. 2018;
Belyaev, A.V., Dunster, J.L., Gibbins, J.M., Panteleev, M.A., Volpert, V. Modeling thrombosis in silico: Frontiers, challenges, unresolved problems and milestones. Physics of Life Reviews. 2018;
Galochkina, T., Marion, M., Volpert, V. Initiation of reaction–diffusion waves of blood coagulation. Physica D: Nonlinear Phenomena. 2018.

Project goals

Mathematical modelling in biology and medicine in three priority directions: cardiovascular system, oncological diseases, the immune response and infectious diseases.

Project leader All participants

Gennady Bocharov

Project results

Blood coagulation - analysis

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.

Blood coagulation – modelling

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.

Application area

The results obtained in the course of the project have applications in oncology, immunology, and the field of cardiovascular diseases.