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The adequate complexity three-dimensional finite element model of biomechanical system with space, shell and beam-type elements was built. The model includes the Ilizarov fixator and tibial bone’s simulator with the regenerating tissue at the fracture location. The proposed model allows us to specify the orthotropic elastic properties of tibial bone model in cortical and trabecular zones. It is also possible to change the basic geometrical and mechanical characteristics of biomechanical system, change the finite element mash density and define the different external loads, such as pressure on the bone and compression or distraction between the repositioned rings of Ilizarov device.

By using special APDL ANSYS program macros the mode of deformation was calculated in the fracture zone for various static loads on the simulator bone, for compression or distraction between the repositioned rings and for various mechanical properties during different stages of the bone regenerate formation (gelatinous, cartilaginous, trabecular and cortical bone remodeling). The obtained results allow us to estimate the permissible values of the external pressure on the bone and of the displacements of the Ilizarov fixator rings for different stages of the bone regeneration, based on the admittance criterion for the maximum of the stresses in the callus. The presented data can be used in a clinical condition for planning, realization and monitoring of the power modes for transosseous osteosynthesis with the external Ilizarov fixator.

This paper presents the development and verification of a finite element algorithm for modeling nonlinear deformation in the functional spinal unit (FSU) of the human lumbar spine. The study is motivated by the increasing prevalence of musculoskeletal disorders and the need for improved predictive methods for spinal motion segment behavior under various loading conditions and pathologies.

Current modeling approaches often employ simplified geometry and linear deformation laws (Hooke’s law) for all spinal elements, including the intervertebral disc. However, these models inadequately represent human anatomy and the physical properties of the intervertebral disc, which comprises the annulus fibrosus and nucleus pulposus exhibiting pronounced nonlinear and viscoelastic characteristics.

To address these limitations, the authors propose a mathematical model wherein vertebrae are treated as linearly elastic, heterogeneous bodies, while the intervertebral disc is modeled according to nonlinear deformation theory. An anatomically accurate 3D reconstruction of the lumbar spine was generated from computed tomography (CT) images. Volumetric tetrahedral finite elements were employed for discretization. The intervertebral disc behavior is described using thin plate bending theory based on Kirchhoff-Love assumptions.

The algorithm’s key feature is a two-step solution procedure: an initial linear elastic solution is obtained, followed by the calculation of additional nodal forces for disc elements, enabling efficient nonlinear analysis without stiffness matrix modification.

Model verification was performed using two geometric representations of the lumbar spine: a simplified version comprising parallelepipeds and cylinders, and a patient-specific realistic model derived from CT data. Numerical simulation results demonstrate that incorporating nonlinear properties of the intervertebral disc substantially alters both deformation patterns and magnitudes compared to purely elastic models, producing nonlinear segments on stress-strain curves. The discrepancy between results obtained with the proposed algorithm and those from Comsol software did not exceed 10%, confirming the model’s validity. This algorithm shows promise for investigating lumbar spine biomechanics under normal conditions, degenerative changes, and post-surgical states.

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.