Large deformation analysis of orthodontic appliances

The deformations of orthodontic appliances used for space closure are large so that any mathematical analysis will require a nonlinear approach. Existing incremental finite element and finite difference numerical methods suffer from excessive computational effort when analyzing these problems. An accurate segmental technique is proposed to handle these difficulties in an extremely efficient fashion. The segmental technique starts by assuming that an orthodontic appliance is composed of a number of smaller segments, the ends of which undergo small relative rotation. With an appropriate choice of local coordinate system the equilibrium equations for each segment are linearized and solved in a straightforward manner. The segments are then assembled using geometric and force compatibility relations similar to the transfer matrix method. Consequently, the original nonlinear boundary value problem is solved as a sequence of linear initial value problems which converge to the required boundary conditions. As only one segment need be considered at a time, the computations can be performed accurately and efficiently on a PC type computer. Although an iterative solution is used to match the boundary conditions, the time required to solve a given problem ranges from a few seconds to a couple of minutes depending on the initial geometric complexity. The accuracy of the segmental technique is verified by comparison with an exact solution for an initially curved cantilever beam with an end load. In addition, comparisons are made with existing experimental and numerical results as well as with a new set of experimental data. In all cases the segmental technique is in excellent agreement with the results of these other studies.     

A novel method for the three-dimensional (3-D) analysis of orthodontic tooth movement-calculation of rotation about and translation along the finite helical axis.

The purpose of this study was to establish a novel method for evaluating orthodontic tooth movement in three-dimensional (3-D) space. The present system consisted of the following procedures at a given treatment period: (1) 3-D tooth positions were measured with a 3-D surface-scanning system using a slit laser beam; (2) the 3-D shape data were registered automatically at the maxillary first molars, and the coordinate systems were normalized; (3) the rotation matrix and translation vector were calculated from the automatic registration of the two position data for a given tooth; (4) the finite helical axes of teeth were calculated as the locus of zero rotational displacement; and (5) tooth movement was presented as rotation about and translation along the finite helical axis. To test this system, a male patient (age 22 yr 2 months) with Angle Class III malocclusion and moderate crowding of the anterior teeth, who had been treated using a standard multi-bracket appliance, was used as a model case in this study. Impressions for a dental cast model were taken at five phases; immediately before and after application of the appliance, and 10 days, 1 month and 2 months after beginning treatment. The results demonstrated that the present analytical method can more simply describe the movement of a given tooth by rotation about and translation along the finite helical axis, and provides quantitative visual 3-D information on complicated tooth movement during orthodontic treatment.     

Finite element method analysis of the tooth movement induced by orthodontic forces

The finite element method is a useful technique for measuring structural stress and for movement analyses. The objective of this investigation was to get a more accurate estimation of tooth movement depending on application point when a tipping orthodontic force is applied. The three-dimensional model of un upper canine, consisting of 4,000 hexahedron elements with 2,367 nodes was obtained. Horizontal, orally directed 1N tipping orthodontic force was applied to the model on five different levels of the tooth crown. The three-dimensional mathematical finite element model is useful in analyzing the tooth movement in response to orthodontic forces. The tipping tooth movement is greater if the force is applied closer to its neck, or more gingivally.     

Simulation of bone remodelling in orthodontic treatment

Orthodontic treatments not only displace irregular teeth but also induce responses in surrounding bone tissues. Bone remodelling is regarded as the regulatory mechanism triggered by mechanical loading. This study was aimed at investigating the effect of orthodontic loading on both tooth movement and neighbouring bone density distribution. A set of computational algorithms incorporating both external and internal remodelling mechanisms was implemented into a patient-specific 3D finite element (FE) model to investigate and analyse orthodontic treatment under four typical modes of orthodontic loading. The consequence of orthodontic treatment was reproduced numerically by using this FE-based technique. The results indicated that the diverse modes of orthodontic loading would result in different magnitudes of tooth movement and particular morphology of bone density distribution. It is illuminated that the newly developed algorithms may replicate the clinical situation more closely compared with the previous proposed method.     

Cellular response to orthodontically-induced short-term hypoxia in dental pulp cells

Orthodontic force application is well known to induce sterile inflammation, which is initially caused by the compression of blood vessels in tooth-supporting apparatus. The reaction of periodontal ligament cells to mechanical loading has been thoroughly investigated, whereas knowledge on tissue reactions of the dental pulp is rather limited. The aim of the present trial is to analyze the effect of orthodontic treatment on the induction and cellular regulation of intra-pulpal hypoxia. To investigate the effect of orthodontic force on dental pulp cells, which results in circulatory disturbances within the dental pulp, we used a rat model for the immunohistochemical analysis of the accumulation of hypoxia-inducible factor-1α in the initial phase of orthodontic tooth movement. To further examine the regulatory role of circulatory disturbances and hypoxic conditions, we analyze isolated dental pulp cells from human teeth with regard to their specific reaction under hypoxic conditions by means of flow cytometry, immunoblot, ELISA and real-time PCR on markers (Hif-1α, VEGF, Cox-2, IL-6, IL-8, ROS, p65). In vivo experiments showed the induction of hypoxia in dental pulp after orthodontic tooth movement. The induction of oxidative stress in human dental pulp cells showed up-regulation of the pro-inflammatory and angiogenic genes Cox-2, VEGF, IL-6 and IL-8. The present data suggest that orthodontic tooth movement affects dental pulp circulation by hypoxia, which leads to an inflammatory response inside treated teeth. Therefore, pulp tissue may be expected to undergo a remodeling process after tooth movement.     

Distribution of 3H-proline within transseptal fibers of the rat following release of orthodontic forces

Maxillary right first molar teeth of rats were tipped mesially with an orthodontic appliance for 2 weeks (experimental group), 3H-proline was injected, and orthodontic forces were removed 6 hr later (time 0). The contralateral molar teeth of treated (internal control group) and age- and weight-matched untreated animals (external control group) were also studied. Diastemata were created between the molar teeth by the orthodontic appliance, and transseptal fibers between first and second (P less than 0.001) and second and third molars (P less than 0.005) were significantly lengthened as compared to external and internal controls at time 0. Diastemata between molar teeth were closed 5 days after removal of orthodontic force. Transseptal fibers adjacent to the source of the orthodontic force (mesial region) had the highest mean number of 3H-proline-labeled proteins at time 0 and at all times following removal of the force (P less than 0.001), and had the highest rate of labeled protein removal (P less than 0.001). Half-lives for removal of 3H-proline-labeled transseptal fiber proteins were significantly greater in mesial and distal regions and significantly less in middle regions of experimentals than in corresponding regions of external controls (P less than 0.001). These data suggest the following: 1) transseptal fibers adjust their length by rapid remodeling in regions experiencing a tensile force; 2) collagenous protein turnover within the middle third of the transseptal fibers is more rapid subsequent to release of orthodontic force than during normal physiologic drift, suggesting that this region adapts rapidly to changes in adjacent tooth position and that these fibers do not play a significant role in relapse of orthodontically relocated teeth; and 3) significant differences in turnover rates of 3H-proline-labeled transseptal ligament proteins of external and internal control quadrants suggest that tooth movement produces both local and systemic effects on collagenous protein metabolism.     

Comparison of methods to determine the centre of resistance of teeth

In orthodontic treatment, the locations of the centre of resistance (CR) of individual teeth and the applied load system are the major determinants for the type of tooth movement achieved. Currently, CR locations have only been specified for a relatively small number of tooth specimen for research purposes. Analysing cone beam computed tomography data samples from three upper central incisors, this study explores whether the effort to establish accurate CR estimates can be reduced by (i) morphing a pre-existing simplified finite element (FE) mesh to fit to the segmented 3D tooth-bone model, and (ii) individualizing a mean CR location according to a small parameter set characterising the morphology of the tooth and its embedding. The FE morphing approach and the semi-analytical approach led to CR estimates that differ in average only 0.04 and 0.12 mm respectively from those determined by very time-consuming individual FE modelling (standard method). Both approaches may help to estimate the movement of individual teeth during orthodontic treatment and, thus, increase the therapeutic efficacy.     

A visco-elastic model for the prediction of orthodontic tooth movement

This study presents a biomechanical model of orthodontic tooth movement. Although such models have already been presented in the literature, most of them incorporate computationally expensive finite elements (FE) methods to determine the strain distribution in the periodontal ligament (PDL). In contrast, the biomechanical model presented in this work avoids the use of FE methods. The elastic deformation of the PDL is modelled using an analytical approach, which does not require setting up a 3D model of the tooth. The duration of the lag phase is estimated using the calculated hydrostatic stresses, and bone remodelling is predicted by modelling the alveolar bone as a viscous material. To evaluate the model, some typically used motion patterns were simulated and a sensitivity analysis was carried out on the parameters. Results show that despite some shortcomings, the model is able to describe commonly used motion patterns in orthodontic tooth movement, in both single- and multi-rooted teeth.     

Stress distribution and displacement analysis during an intermaxillary disjunction--a three-dimensional FEM study of a human skull

The goal of this study was to contribute to an understanding of how much expansion force is needed during a maxillary expansion (ME) and where bony reaction takes place. A finite element (FE) model of a dry human male skull was generated from CT scans. The FE model, which consists of cortical and cancellous bone and teeth, was loaded with the same force magnitudes, directions and working points as in rapid maxillary expansion (RME). A three-dimensional finite element stress analysis (FESA) of the forces and displacement was performed. The highest stress was observed in the maxilla in the region where the forces were applied, and spreads more or less throughout almost the whole frontal skull structures. The displacement distribution which causes stress in the skull is highly dependant on the thickness of the bone and its structure. All areas with high compressive and tensile stress are exactly the regions which determine the maximal amount of force to be used during the maxillary expansion and should be examined in case of any complication during a patient's treatment. Regions with significant compressive and tensile stress are the regions observed to have an increase in cellular activity. Further simulations with a given displacement (0.5mm) showed that displacement simulations need extra caution otherwise they will lead to very high forces which are not realistic in an orthodontic treatment.     

Quantitative approach for the prediction of tooth movement during orthodontic treatment

The orthodontic treatment is aimed to displace and/or rotate the teeth to obtain the functionally correct occlusion and the best aesthetics and consists in applying forces and/or couples to tooth crowns. The applied loads are generated by the elastic recovery of metallic wires linked to the tooth crowns by brackets. These loads generate a stress state into the periodontal ligament and hence, in the alveolar bone, causing the bone remodeling responsible for the tooth movement. The orthodontic appliance is usually designed on the basis of the clinical experience of the orthodontist. In this work, a quantitative approach for the prediction of the tooth movement is presented that has been developed as a first step to build up a computer tool to aid the orthodontist in designing the orthodontic appliance. The model calculates the tooth movement through time with respect to a fixed Cartesian frame located in the middle of the dental arch. The user interface panel has been designed to allow the orthodontist to manage the standard geometrical references and parameters usually adopted to design the treatment. Simulations of specific cases are reported for which the parameters of the model are selected in order to reproduce forecasts of tooth movement matching data published in experimental works.     

Distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method

In this paper, we present a distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method. The dynamic model comprises system of three coupled reaction-diffusion models, involving the tumor cells, the normal tissues and the drug concentration. An optimal control problem is formulated with the goal of minimizing the tumor cell density and reducing the side effects of the drug. A distributed parameters method based on the application of variational calculus is used on an integral-Hamiltonian, which is then used to obtain an optimal coupled system of forward state equations and backward co-state equations. The Galerkin finite element method is used to realistically represent the brain structure as well as to facilitate computation. Finally a three-dimensional test case is considered and partitioned into a set of spherical finite elements, using tri-linear basis functions, except for the elements affected by singularities of polar and azimuthal angles, as well as the origin.     

Determination of stress levels and profiles in the periodontal ligament by means of an improved three-dimensional finite element model for various types of orthodontic and natural force systems

This study was undertaken to investigate the stress-strain levels and distribution within the periodontal ligament for various types of physiological and orthodontic force systems, assuming that the bone resorption process, leading to tooth movements, is partly controlled by those conditions. Two finite element models were developed, simulating a full and partial mandibular morphology, respectively. Both models were based on morphology and physical parameters of human autopsy material. The effect of changing material parameters and structure, type of boundary conditions, calculation method and fineness of the model on the stress levels and profiles in the periodontal ligament was evaluated by a series of tests. A structure optimization technique was used to investigate the load bearing characteristics of the mandible and the influence of the anisotropic material properties of both the mandible and the segment. A ‘multiple modelling’ technique based on both the mandible and the segment was developed to test various types of boundary conditions in the analysis of the segment. Results presented as ‘stress profiles’ showing the correlation between the applied force system and the stress distribution in the periodontal ligament, based on the improved finite element models, were established.     

A biomechanical analysis of deformation and strain on lower jaw model

Forces, occurring as result of orthodontic appliances, present a guiding system which controls force activity during the growth and development of lower jaw. By qualitative photoelasticimetric analysis author decided to evaluate the influence of extension apparatus to the lower jaw. The concentration and stress distribution, intensity and force direction on contacts with insert (extensor device) were analysed on araldyte lower jaw models. Three different loadings 41.0, 57.4 and 73.8 N under four different loading conditions with and without the correction appliance were used. The results of investigation indicated that the force created by the orthodontic appliance is correct at the beginning. The change from surface contact to linear and punctual contact increased the static moment and local stress, resulting in new regional conditions.     

Numerical analysis of orthodontic treatment elements of pseudo-elastic NiTi alloys]

In orthodontic treatment malpositions of teeth are often corrected by fixed appliances, consisting, in part, of loops made by the orthodontist. The most important alloys in use are steel, cobalt-chromium, or titanium-molybdenium alloys. The static force systems of fixed appliances made of these materials are well known from experimental and numerical studies, but as they may change during tooth movement, we are often confronted with problems in therapy. The introduction of pseudoelastic nickel titanium alloys (NiTi) into orthodontic treatment, offers the chance of improving the effectiveness and reliability of orthodontic devices. In the present paper a plane finite element (FE) for the analysis of orthodontic loops is presented. It enables the determination of the nonlinear behaviour of pseudoelastic NiTi-alloys and is capable of simulating large structural displacements and rotations accompanied by moderate strains. A comparative numerical and experimental study shows the efficiency of this element. The associated results reflect pseudoelastic effects on certain loop designs, and reveal the benefits for the orthodontist and his patients.     

A quantitative study on magnesium alloy stent biodegradation

Insufficient scaffolding time in the process of rapid corrosion is the main problem of magnesium alloy stent (MAS). Finite element method had been used to investigate corrosion of MAS. However, related researches mostly described all elements suffered corrosion in view of one-dimensional corrosion. Multi-dimensional corrosions significantly influence mechanical integrity of MAS structures such as edges and corners. In this study, the effects of multi-dimensional corrosion were studied using experiment quantitatively, then a phenomenological corrosion model was developed to consider these effects. We implemented immersion test with magnesium alloy (AZ31B) cubes, which had different numbers of exposed surfaces to analyze differences of dimension. It was indicated that corrosion rates of cubes are almost proportional to their exposed-surface numbers, especially when pitting corrosions are not marked. The cubes also represented the hexahedron elements in simulation. In conclusion, corrosion rate of every element accelerates by increasing corrosion-surface numbers in multi-dimensional corrosion. The damage ratios among elements with the same size are proportional to the ratios of corrosion-surface numbers under uniform corrosion. The finite element simulation using proposed model provided more details of changes of morphology and mechanics in scaffolding time by removing 25.7% of elements of MAS. The proposed corrosion model reflected the effects of multi-dimension on corrosions. It would be used to predict degradation process of MAS quantitatively.     

Quantitative Approach for the Prediction of Tooth Movement During Orthodontic Treatment

The orthodontic treatment is aimed to displace and/or rotate the teeth to obtain the functionally correct occlusion and the best aesthetics and consists in applying forces and/or couples to tooth crowns. The applied loads are generated by the elastic recovery of metallic wires linked to the tooth crowns by brackets. These loads generate a stress state into the periodontal ligament and hence, in the alveolar bone, causing the bone remodeling responsible for the tooth movement. The orthodontic appliance is usually designed on the basis of the clinical experience of the orthodontist. In this work, a quantitative approach for the prediction of the tooth movement is presented that has been developed as a first step to build up a computer tool to aid the orthodontist in designing the orthodontic appliance. The model calculates the tooth movement through time with respect to a fixed Cartesian frame located in the middle of the dental arch. The user interface panel has been designed to allow the orthodontist to manage the standard geometrical references and parameters usually adopted to design the treatment. Simulations of specific cases are reported for which the parameters of the model are selected in order to reproduce forecasts of tooth movement matching data published in experimental works.     

Structural response of relaxed and constricted arterioles

Analysis of viscoelastic properties of blood vessel walls presents difficult analytical problems in view of their non-homogeneity, anisotropy, and non-linear viscoelastic characteristics. The analytical technique used in this study is the numerical method of direct stiffness which has been successfully applied in preliminary studies. The direct stiffness method constitutes a finite element analysis, where the wall structure in an electron micrograph is represented by an assembly of triangular elements. Different material properties can be assigned to each triangular element; the method is thus ideally suited for computer analysis of non-homogeneous biological structures. Anatomical components considered in the analysis were two types of collagen fibers and smooth muscle cells. Typical circumferential stress-time and radius-time histories for selected points have been obtained in relaxed and constricted arterioles under static pressure loads. Plots of the stress distribution in cross sections of the vessel wall have been obtained in relaxed and constricted arterioles.     

Finite element modeling of superelastic nickel–titanium orthodontic wires

Thanks to its good corrosion resistance and biocompatibility, superelastic Ni–Ti wire alloys have been successfully used in orthodontic treatment. Therefore, it is important to quantify and evaluate the level of orthodontic force applied to the bracket and teeth in order to achieve tooth movement. In this study, three dimensional finite element models with a Gibbs-potential-based-formulation and thermodynamic principles were used. The aim was to evaluate the influence of possible intraoral temperature differences on the forces exerted by NiTi orthodontic arch wires with different cross sectional shapes and sizes. The prediction made by this phenomenological model, for superelastic tensile and bending tests, shows good agreement with the experimental data. A bending test is simulated to study the force variation of an orthodontic NiTi arch wire when it loaded up to the deflection of 3 mm, for this task one half of the arch wire and the 3 adjacent brackets were modeled. The results showed that the stress required for the martensite transformation increases with the increase of cross-sectional dimensions and temperature. Associated with this increase in stress, the plateau of this transformation becomes steeper. In addition, the area of the mechanical hysteresis, measured as the difference between the forces of the upper and lower plateau, increases.     

The effect of unerupted permanent tooth crowns on the distribution of masticatory stress in children

Human mothers wean their children from breast milk at an earlier developmental stage than do ape mothers, resulting in human children chewing solid and semi-solid foods using the deciduous dentition. Mechanical forces generated by chewing solid foods during the post-weaning period travel through not only the deciduous teeth, but also the enamel caps of the developing permanent teeth within the maxilla and mandible, which are not present in the adult face. The effects of mechanical stress propagating through these very stiff structures have yet to be examined. Based on a heuristic model, we predicted that the enamel of the embedded developing teeth would act to reduce stresses in the surrounding bony elements of the juvenile face. We tested this hypothesis by simulating occlusal loading in a finite element (FE) model of a child's cranium with a complete set of deciduous teeth and the first permanent molars embedded in the bony crypt in the maxilla. We modeled bone and enamel with appropriate material properties and assessed the effect of embedding high-stiffness enamel structures on stress distribution in the juvenile face. Against expectation, the presence of unerupted enamel caps does not affect the magnitude or location of stresses in the juvenile face. Our results do not support the hypothesis that the unerupted secondary teeth act to moderate stresses in the juvenile face.