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This paper describes a method for generating a synthetic database of stenoses, consisting of 1620 entries. Each entry represents the results of a numerical experiment simulating the three-dimensional flow of a viscous incompressible fluid through a tube with a variable cross-section: pressure drop, mean flow rate, cross-sectionally averaged inlet blood flow velocity, maximum stenosis severity, stenosis length, stenosis asymmetry, tube radius, and Reynolds number. The database was validated by comparison with other models (with elastic walls) and bench experiments, showing a deviation in pressure drops of no more than 4%. The synthetic stenosis database was used to train a physics-informed neural network for the rapid estimation of pressure drop based on four key input parameters: Reynolds number, stenosis length, stenosis severity, and stenosis asymmetry coefficient. The physics-informed aspect was achieved by introducing penalties into the loss function for the absence of a positive pressure drop and for the lack of monotonicity of the pressure drop with respect to the input parameters. The physics-informed neural network demonstrated higher accuracy on hemodynamically significant stenoses when tested on a validation set and on new stenoses not represented in the database. The mean relative error for stenoses with a length of 8 healthy vessel radii was 6% for the physics-informed network and 13% for a classical neural network. The errors for short stenoses with a length of 4 radii were nearly identical: 9.5% for the physics-informed network and 10% for the classical neural network. The developed method for the functional assessment of the hemodynamic significance of stenoses can be used both as a standalone tool for clinical stenosis evaluation and as a component of network blood flow models. The approach becomes most relevant when modeling multi-vessel disease, which is predominant in clinical practice. The key advantage of the method lies in the physical correctness of the results and accuracy comparable to classical modeling, but with significantly lower computational costs.

The work is devoted to the description of the author’s formal statements of the clustering problem for a given number of clusters, algorithms for their solution, as well as the results of using this toolkit in medicine.

The solution of the formulated problems by exact algorithms of implementations of even relatively low dimensions before proving optimality is impossible in a finite time due to their belonging to the NP class.

In this regard, we have proposed a hybrid algorithm that combines the advantages of precise methods based on clustering in paired distances at the initial stage with the speed of methods for solving simplified problems of splitting by cluster centers at the final stage. In the development of this direction, a sequential hybrid clustering algorithm using random search in the paradigm of swarm intelligence has been developed. The article describes it and presents the results of calculations of applied clustering problems.

To determine the effectiveness of the developed tools for optimal clustering of multidimensional objects according to a variety of heterogeneous indicators, a number of computational experiments were performed using data sets including socio-demographic, clinical anamnestic, electroencephalographic and psychometric data on the cognitive status of patients of the cardiology clinic. An experimental proof of the effectiveness of using local search algorithms in the paradigm of swarm intelligence within the framework of a hybrid algorithm for solving optimal clustering problems has been obtained.

The results of the calculations indicate the actual resolution of the main problem of using the discrete optimization apparatus — limiting the available dimensions of task implementations. We have shown that this problem is eliminated while maintaining an acceptable proximity of the clustering results to the optimal ones. The applied significance of the obtained clustering results is also due to the fact that the developed optimal clustering toolkit is supplemented by an assessment of the stability of the formed clusters, which allows for known factors (the presence of stenosis or older age) to additionally identify those patients whose cognitive resources are insufficient to overcome the influence of surgical anesthesia, as a result of which there is a unidirectional effect of postoperative deterioration of complex visual-motor reaction, attention and memory. This effect indicates the possibility of differentiating the classification of patients using the proposed tools.

Aortic stenosis and other forms of aortic valve dysfunction are associated with impaired intracardiac hemodynamics, left ventricular overload, and an increased risk of cardiovascular complications. Assessment of valve function requires not only integral clinical indicators but also local mechanical and hemodynamic characteristics, which, as a rule, cannot be measured directly in vivo. Therefore, mathematical modeling is regarded as one of the main tools for the quantitative analysis of the aortic valve. Despite the widespread use of various deformable-solid models and coupled fluid-structure interaction formulations, FSI, for describing leaflet dynamics, the limits of applicability of simplified formulations relative to the fully coupled problem remain insufficiently defined. In this study, an idealized model of the aortic root with the sinuses of Valsalva and a tricuspid valve was considered. The leaflets were described using an anisotropic hyperelastic material model. Five computational scenarios were compared, including a fully coupled FSI formulation that accounts for both solid and fluid dynamics, as well as a deformable-solid model with four loading variants replacing the effect of blood flow, differing in the way pressure was represented and in the direction of load application to the leaflets. The comparison criteria included deformation, displacement, von Mises stress, leaflet oscillatory dynamics, and the geometric opening area of the valve. It was shown that the FSI model provides the most consistent description of valve function, including asymmetric leaflet opening, smoother opening dynamics, and the absence of pronounced nonphysiological flutter. Structural formulations with loads applied along the local normal to the leaflet surface lead to overestimation of deformation and stress, as well as to more pronounced oscillatory regimes. Scenarios with restricted load direction produce a more moderate response, but they also fail to reproduce the spatial load structure and the temporal organization of leaflet opening. It was concluded that, in aortic valve modeling, not only the magnitude of the pressure difference but also the way it is applied to the leaflets in space and time is of decisive importance. Structural deformable-solid models may be used for the qualitative assessment of selected mechanical trends, but they cannot serve as a full substitute for the FSI formulation in the analysis of leaflet kinematics, oscillatory regimes, stress-strain state, and valve opening dynamics.

The mechanisms of hydrodynamical activation of blood coagulation system are investigated in stenosed vessels for a wide range of Reynolds number values (from 10 up to 500). It is assumed that the vessel wall permeability for procoagulant factors rapidly increases when wall shear stress exceeds specific threshold value. A number of patterns of blood coagulation processes development are described. The influence of blood flow topology changes on activation of blood coagulation is explored. It is established that not only blood flow decrease, but also its increase may promote activation of blood coagulation. It was found that dependence of thrombogenic danger of stenosis on vessel lumen blockage ratio is non-monotonic. The relevance of obtained theoretical results for clinical practice is discussed.

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.