An expansion to the Nägerl’s theory of proportionality in reduced alveolar bone height models: a 3D finite element analysis

INTRODUCTION: Orthodontic tooth movement is a basic theme in an orthodontic treatment. According to Nägerl, A nearer force application to the center of resistance will bring a farther center of rotation to the center of resistance. The main goal of this study was to evaluate this theory of proportionality by Finite element method (FEM) and find out its applicability in a bone loss situation. MATERIALS AND METHODS: Three 3D FEM of an upper central incisor were designed in Solidworks 2016, the first one with a healthy bone height (model 1), with 3 mm of bone loss (model 2) and with 4.5 mm of bone loss (model 3). An 0.5 N force was applied in different predefined locations on the labial surface of the crown in model 2. This was 0.2 N for the model 3. The exact location of the center of resistance (Cres), center of rotation (Crot) for each force application point was calculated using a C++ code specially design for this study in all models. RESULTS: An apical shift of the Cres positions were shown in gradual steps of bone loss from 7.9708 mm incisal from the apex to 6.6292 mm in model 2 and 5.6105 mm in model 3. Modification of the location of the Crot in different force magnitudes and points of force applications were shown whit a constant a*b. CONCLUSION: In healthy teeth and teeth with alveolar bone loss, Cres located in the apical third of the root. The product of the distance between the point of force application and Cres (“a”) and the Cres and Crot (“b”) is constant, thus; Nägerl theory is applicable in both healthy and reduced bone height. In this way, applying a single force nearer to the cervical point will result in a more apical location of the Crot, reducing the angle change in the long axis of the tooth. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1186/s12903-022-02476-9.