Application of Maxwell models for bone remodeling under orthodontic loading

Orthodontic treatments apply controlled displacement or mechanical forces (quasi-static but persistent) to teeth via dental braces. These forces are transmitted through the periodontal ligament (PDL) to the surrounding alveolar bone, initiating a bone remodeling process that allows the teeth to gradually shift position. The PDL transmits load to the alveolar bone and initial stress peaks relax with time as bone resorbs in regions of compression and deposits in regions of tension. Bone tissue, although mostly elastic on short time scales, exhibit therefore adaptive behaviors over longer durations due to cellular remodeling. In this paper we use viscoelastic Maxwell models to capture such an adaptive behavior. Two models are considered. The first is a two-degrees-of-freedom (2DOF) model. The second investigates the continuum two-dimensional case. We derive them, first time in this context, with the use of a Hamilton–Rayleigh principle. For both cases, we show that the load free configuration changes according to the experienced load and the models quantify these concept. The models are general and can be applied to any load history. Besides, the second 2D model can also be applied to any geometric configuration. The examples are nevertheless sufficiently simple to be solved analytically. The target is to help to predict optimal loading schedules to avoid excessive stress or damage.