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Most biomechanical tasks of interest to clinicians can be solved only using personalized mathematical models. Such models allow to formalize and relate key pathophysiological processes, basing on clinically available data evaluate non-measurable parameters that are important for the diagnosis of diseases, predict the result of a therapeutic or surgical intervention. The use of models in clinical practice imposes additional restrictions: clinicians require model validation on clinical cases, the speed and automation of the entire calculated technological chain, from processing input data to obtaining a result. Limitations on the simulation time, determined by the time of making a medical decision (of the order of several minutes), imply the use of reduction methods that correctly describe the processes under study within the framework of reduced models or machine learning tools.

Personalization of models requires patient-oriented parameters, personalized geometry of a computational domain and generation of a computational mesh. Model parameters are estimated by direct measurements, or methods of solving inverse problems, or methods of machine learning. The requirement of personalization imposes severe restrictions on the number of fitted parameters that can be measured under standard clinical conditions. In addition to parameters, the model operates with boundary conditions that must take into account the patient’s characteristics. Methods for setting personalized boundary conditions significantly depend on the clinical setting of the problem and clinical data. Building a personalized computational domain through segmentation of medical images and generation of the computational grid, as a rule, takes a lot of time and effort due to manual or semi-automatic operations. Development of automated methods for setting personalized boundary conditions and segmentation of medical images with the subsequent construction of a computational grid is the key to the widespread use of mathematical modeling in clinical practice.

The aim of this work is to review our solutions for personalization of mathematical models within the framework of three tasks of clinical cardiology: virtual assessment of hemodynamic significance of coronary artery stenosis, calculation of global blood flow after hemodynamic correction of complex heart defects, calculating characteristics of coaptation of reconstructed aortic valve.

The paper present a dynamic macro model of world dynamics. The world is divided into 19 geographic regions in the model. The internal development of the regions is described by regression equations for demographic and economic indicators (Population, Gross Domestic Product, Gross Capital Formation). The bilateral trade flows from region to region describes interregional interactions and represented the trade submodel. Time, the gross product of the exporter and the gross product of the importer were used as regressors. Four types were considered: time pair regression — dependence of trade flow on time, export function — dependence of the share of trade flow in the gross product of the exporter on the gross product of the importer, import function — dependence of the share of trade flow in the gross product of the importer on the gross product of the exporter, multiple regression — dependence of trade flow on the gross products of the exporter and importer. Two types of functional dependence were used for each type: linear and log-linear, in total eight variants of the trading equation were studied. The quality of regression models is compared by the coefficient of determination. By calculations the model satisfactorily approximates the dynamics of monotonically changing indicators. The dynamics of non-monotonic trade flows is analyzed, three types of functional dependence on time are proposed for their approximation. It is shown that the number of foreign trade series can be approximated by the space of seven main components with a 10% error. The forecast of regional development and global dynamics up to 2040 is constructed.

Efficiency and completeness of the numerical simulation in oceanography and hydrometeorology are entirely determined by algorithmic features of the construction of an interactive computer simulations in the scale of the oceans with adaptive coated closed seas and coastal waters refined mathematical models, with the possibility of specifying software parallelization calculations near the concrete — the protected areas of the sea coast. An important component of the research is continuous graphical visualization techniques in the course of calculations, including those undertaken in parallel processes with shared RAM or test points on the external media. The results of computational experiments are used in the description of hydrodynamic processes near the coast, which is important in keeping the organization of sea control services and forecasting marine hazards.