Sergei Bosiakov

Viscoelasticity of periodontal ligament: an analytical model

BackgroundUnderstanding of viscoelastic behaviour of a periodontal membrane under physiological conditions is important for many orthodontic problems. A new analytic model of a nearly incompressible viscoelastic periodontal ligament is suggested, employing symmetrical paraboloids to describe its internal and external surfaces.MethodsIn the model, a tooth root is assumed to be a rigid body, with perfect bonding between its external surface and an internal surface of the ligament. An assumption of almost incompressible material is used to formulate kinematic relationships for a periodontal ligament; a viscoelastic constitutive equation with a fractional exponential kernel is suggested for its description.ResultsTranslational and rotational equations of motion are derived for ligament’s points and special cases of translational displacements of the tooth root are analysed. Material parameters of the fractional viscoelastic function are assessed on the basis of experimental data for response of the periodontal ligament to tooth translation. A character of distribution of hydrostatic stresses in the ligament caused by vertical and horizontal translations of the tooth root is defined.ConclusionsThe proposed model allows generalization of the known analytical models of the viscoelastic periodontal ligament by introduction of instantaneous and relaxed elastic moduli, as well as the fractional parameter. The latter makes it possible to take into account different behaviours of the periodontal tissue under short- and long-term loads. The obtained results can be used to determine loads required for orthodontic tooth movements corresponding to optimal stresses, as well as to simulate bone remodelling on the basis of changes in stresses and strains in the periodontal ligament caused by such movements.

Mathematical Model for Analysis of Translational Displacements of Tooth Root

AbstractAnalytical modeling of stress-strain state of a periodontal ligament in the case of the translational displacement of a tooth root was carried out. The tooth root was assumed as a rigid body. The boundary conditions corresponding to the translational displacement of the root and fixed external surface of the periodontal ligament in the dental alveolus were considered. The system of differential equations describing the periodontal ligaments plane-strain state induced by the translational motion of the tooth were used as the governing equations. An analytical solution was found for the governing equations in the explicit form. Comparative analysis of the concentrated force generated by the prescribed translational motion of the tooth root was performed using the obtained analytical solution and the model of an incompressible periodontal ligament in the form of a circular paraboloid and hyperboloid. The mathematical model developed in this paper can be used to analyze stresses and strains in the pe...

Viscoelastic Behavior of Periodontal Ligament: Stresses Relaxation at Translational Displacement of a Tooth Root

Understanding of viscoelastic response of a periodontal membrane under the action of short-term and long-term loadings is important for many orthodontic problems. A new analytic model describing behavior of the viscoelastic periodontal ligament after the tooth root translational displacement based on Maxwell approach is suggested. In the model, a tooth root and alveolar bone are assumed to be a rigid bodies. The system of differential equations for the plane-strain state of the viscoelastic periodontal ligament is used as the governing one. The boundary conditions corresponding to the initial small displacement of the root and fixed outer surface of the periodontal ligament in the dental alveolus are utilized. A solution is found numerically for fractional viscoelasticity model assuming that the stress relaxation in the periodontal ligament after the continuing displacement of the tooth root occurs approximately within five hours. The character of stress distribution in the ligament over time caused by the tooth root translational displacement is evaluated. Effect of Poisson’s ratio on the stresses in the viscoelastic periodontal ligament is considered. The obtained results can be used for simulation of the bone remodelling process during orthodontic treatment and for assessment of optimal conditions of the orthodontic load application.

Analytical Modelling of the Tooth Translational Motions: Comparative Analysis for Different Shapes of Root

The orthodontic treatment planning may be carried out based on the finite element and analytical models of periodontal ligament (PDL). For analytical modelling of the PDL behavior the shape of the tooth root mainly was approximated by circular or elliptical paraboloid. Another shape of the tooth root is the elliptical two-sheeted hyperboloid. Semi-axes of the tooth root cross-section in the shape of the elliptical paraboloid and two-sheeted hyperboloid on the alveolar crest level are the same, but the shape of a two-sheeted hyperboloid allows employing the additional parameter for describing the root apex rounding. The aim of this study is the comparative analysis of the hydrostatic stresses patterns during the tooth root translational displacements in the almost incompressible PDL for the root in the shape of the elliptical paraboloid and two-sheeted hyperboloid. As a result, patterns of the hydrostatic stresses in the PDL during translational displacement are nearly identical for the tooth root in the shape of a paraboloid and the tooth root in the shape of a two-sheeted hyperboloid with the rounded apex of the tooth root. The translational movement of the tooth root with a pointed apex leads to the higher hydrostatic stresses in the PDL compared with the tooth root with a rounded apex. The obtained results indicated that the rounding of the tooth root should be considered during planning of orthodontic treatment.

CRANIOFACIAL STRESS PATTERNS AND DISPLACEMENTS AFTER ACTIVATION OF HYRAX DEVICE: FINITE ELEMENT MODELLING

Rapid maxillary expansion is employed for the treatment of cross-bite and deficiency of transversal dimension of the maxilla in patients with and without cleft of palate and lip. For this procedure, generally, different orthodontic appliances and devices generating significant transversal forces are used. The aim of this study is the finite-element analysis of stresses and displacements of the skull without palate cleft and the skull with unilateral and bilateral cleft after activation of the Hyrax orthodontic device. Two different constructions of the orthodontic device Hyrax with different positions of the screw relative palate are considered. In the first case, the screw is in the occlusal horizontal plane, and in the other, the screw is located near the palate. Activation of the orthodontic device corresponds to the rotation of the screw on one-quarter turn. It is established that the screw position significantly affects the distributions of stresses in skull and displacements of the cranium without palate cleft and with unilateral or bilateral palate cleft. Stresses in the bone structures of the craniums without cleft and with cleft are transferred from the maxilla to the pterygoid plate and pharyngeal tubercle if the screw displaces from the occlusal plane to the palate. Depending on the construction of the orthodontic appliance, the maxilla halves in the transversal plane are unfolded or the whole skull is entirely rotated in the sagittal plane. The stresses patterns and displacements of the skull with bilateral palate cleft are almost unchanged after activation of the orthodontic devices with different positions of the screw, only magnitudes of stresses and displacements are changed. The obtained results can be used for design of orthodontic appliances with the Hyrax screw, as well as for planning of osteotomies during the surgical assistance of the rapid maxillary expansion.

Analytical Modeling of the Viscoelastic Behavior of Periodontal Ligament with Using Rabotnov’s Fractional Exponential Function

The mathematical modeling of a stress-strain state of the viscoelastic periodontal membrane is carried out. Internal and external surfaces of the periodontal ligament are described by a symmetrical two-sheeted hyperboloid. Tooth root is assumed to be a rigid body. Displacements of points on the internal surface of the periodontal ligament coincide with the displacements of the corresponding points of the external surface of the tooth root. The relationships between the displacements and strains for periodontal ligaments are formulated as an assumption that the periodontal tissue approaches to incompressible materials. Viscoelastic constitutive law with a fractional exponential kernel for periodontal ligament was used. The equations of motion for the periodontal ligament relative to translational displacements and rotation angles of its points are derived. In the particular case the vertical translational motion of the tooth root, as well as corresponding displacements are analyzed. Constants of the fractional viscoelastic function were assessed on the basis of the experimental data about the behavior of the periodontal ligament. The obtained results can be used to determine a load for orthodontic tooth movement corresponding to the optimal stresses, as well as to simulate bone remodeling on the basis of changes of stresses and strains in the periodontal ligament during orthodontic movement.

Biomechanical Effects of Maxillary Expansion in Cross-Bite Patients During Orthodontic Treatment with Hyrax Screw

The aim of this study was finite element analysis of stress-strain state of the human maxillary complex with and without cleft palate. Loading the skull is carried out by activating orthodontic device HYRAX. Model of the skull and supporting teeth of upper jaw obtained on the basis of tomographic data for dry intact skull of an adult. Design of orthodontic device differ position of screws and rods relative to the palate. Equivalent stresses in the bones of the craniofacial complex are assessed. It is shown that large stresses occur in the maxillary complex, if the screw and rods of orthodontic devices are located in a horizontal plane for skull with and without cleft. Also in the intact skull big stresses appear in the bone of the upper jaw with location of the screw and rods of orthodontic device in a horizontal plane. In the rest of the skull bones stresses are insignificant. By moving the device screw to the palate the values of maximum stresses are reduced, but the region of big stresses displaced to the pterygoid plate and pharyngeal tubercle. In the skull with cleft for different positions of screws and rods orthodontic device the upper jaw is loaded fragmentary. High stresses are observed in the region of the maxilla near the zygomatic arches and along the edges of eye-sockets. When placing screw of orthodontic device close to palate the stresses decreases, but are observed in most part of the zygomatic arches.

Damage Prediction of the Femur with Postresection Defect

The aim of this study is to develop an approach to assessing the strength of the femur after sectoral resection in cases of benign bone tumors, tumor-like and metastatic lesions. The proposed approach is based on the finite element calculation of dangerous volumes in the area of bone defect. Load is static and equivalent to average human weight. Model of the femur is based on tomographic data. Postresection defect is localized in the middle third of the lateral side of the femur. As a conditions for the selection of dangerous volume fracture criterion Coulomb–Mohr is used. The analysis of damage near the concentrators of the bone defect is carried out for different loads. The domain of the bone defect with the largest damage is determined. For concentrators of the postresection hole the emergence and growth of crack is considered as a change of the dangerous volume with taking account the removal of the damaged finite elements. The ranges of the load corresponding to the various cases of damage development are determined. Three options to compensate for bone strength and the prevention the pathological bone fracture after sectoral resection are suggested.

Assessment of Eigenfrequencies of the Middle Ear Oscillating System: Effect of the Cartilage Transplant

Finite-element models of the intact middle ear and the diseased one with the eardrum subjected to retraction in the posterosuperior quadrant are proposed. Because the natural frequencies of the middle ear oscillating system are the principle dynamic characteristics, finite-element calculations of eigenfrequencies for both the normal middle ear and the middle ear with pathological changes of the tympanic membrane in the posterosuperior quadrant are carried out. The geometrical model of the middle ear consisting of the eardrum, malleus, incus, and stapes was obtained on basis of the tomographic data. When the eardrum has a retraction pocket, a cartilage transplant superimposed on the posterosuperior quadrant is considered as a reinforcing element. The optimal thicknesses of the cartilage transplant are defined in such a manner that the natural frequencies of the reconstructed middle ear were as close to those of the normal middle ear as possible. The obtained results can be used to predict the thickness of the cartilage transplant needed to restore the functions of the human middle ear.