• R.M. Stupnytskyi Ivano-Frankivsk National Medical University.
  • V.R. Yarychkivskyi Danylo Halytskyi Lviv National Medical University
Keywords: 3D model, mathematical modeling, Young modulus, Poisson coefficient


In the modern world, dentistry has achieved a significant possibilities in aesthetic and functional restoration of teeth and dental arches thanks to modern computer technologies. Every day, the latest developments in the dental industry, such as CAD-CAM systems, clinical and laboratory scanners, 3D printers, microscopes, become indispensable attributes of dental institutions. Modern equipment expands the capabilities of dentists in the diagnosis of pathologies of the masticatory system, greatly facilitates the choice of an optimal plan of prosthetic treatment and allows to have a good quality of rehabilitation of patients. An individual place in the concept of treatment relates to variety of software that allows to make a treatment plan and predict its results.

Objective. To create a virtual model of the tooth with periodontal tissues and determine the theoretical aspects, conditions and parameters of its application in examination stresses and deformations that arise in different occlusion relationships.

Methods and materials. 3D modeling is a method of creating various forms and complexity of three-dimensional computer models of real or fantastic objects of the world with the use of various techniques and mechanisms.

Autodesk 3ds MAX (3D Studio MAX) and a polygonal modeling method were used to create 3D models. Mathematical simulation (mathematical modeling) is a method of studying processes or phenomena by creating their mathematical models and studying these models. The method is based on the identity of the form of equations and the uniqueness of the relations between the variables in the equations of the original and the model, that is, their analogy.

Mathematical modeling allows you to replace real objects with its virtual model and then study the last one. As with any simulation, the mathematical model is designed taking into account the physical characteristics of the original object.

Mathematical modeling is carried out in Comsol Multiphysics 4.2a software program ("Comsol AB" (Sweden).

For creation of the three-dimensional model, we chose the frontal area of the upper jaw. The model was created by the polygonal modeling method, taking into account the anatomical parameters of natural teeth and periodontal tissues (Autodesk 3Ds Max software). The size of the teeth, the thickness and shape of the bone tissue contours, the magnitude of the deflection of the tooth axis and the alveolar appendix and the thickness of the mucosa were modeled in accordance to the average parameters.

The created models of anatomical elements were later integrated into the program for computer mathematical modeling. During the study we used the finite element method and entered the following values: Young's modulus, Poisson's coefficient and body density.

Characteristics of the materials for calculating the stress-strain state were absolutely identical to the tissues of the tooth and bone. Each model applied forces in different planes according to occlusal movements: in sagittal plane - forward, force 100 N; in horizontal - transversal movements, force 120N; in the vertical - the force is 400 N. The calculation of the magnitude of force was carried out according to the average statistics of the cross-sectional area of masticatory muscles involved in the movements of the mandible. We also note that these values are critical and maximally possible.

Conclusion: The construction of three-dimensional models of teeth and tissues of periodontium helps to understand in detail the essence of processes occurring in the masticatory system during its functioning, to measure stresses, strains and deformations during occlusion relationships. Analysis of the data obtained with the help of mathematical modeling improves the capabilities of dentists at different stages of functional rehabilitation of patients, simplifies the choice of orthopedic design and has a significant predictive value.


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