Simulation of multi-stage nonlinear bone remodeling induced by fixed partial dentures of different configurations: a comparative clinical and numerical study

Abstract

This paper aimed to develop a clinically validated bone remodeling algorithm by integrating bone’s dynamic properties in a multi-stage fashion based on a four-year clinical follow-up of implant treatment. The configurational effects of fixed partial dentures (FPDs) were explored using a multi-stage remodeling rule. Three-dimensional real-time occlusal loads during maximum voluntary clenching were measured with a piezoelectric force transducer and were incorporated into a computerized tomography-based finite element mandibular model. Virtual X-ray images were generated based on simulation and statistically correlated with clinical data using linear regressions. The strain energy density-driven remodeling parameters were regulated over the time frame considered. A linear single-stage bone remodeling algorithm, with a single set of constant remodeling parameters, was found to poorly fit with clinical data through linear regression (low $R^{2}$ and R), whereas a time-dependent multi-stage algorithm better simulated the remodeling process (high $R^{2}$ and R) against the clinical results. The three-implant-supported and distally cantilevered FPDs presented noticeable and continuous bone apposition, mainly adjacent to the cervical and apical regions. The bridged and mesially cantilevered FPDs showed bone resorption or no visible bone formation in some areas. Time-dependent variation of bone remodeling parameters is recommended to better correlate remodeling simulation with clinical follow-up. The position of FPD pontics plays a critical role in mechanobiological functionality and bone remodeling. Caution should be exercised when selecting the cantilever FPD due to the risk of overloading bone resorption.

Appendix

The anisotropic properties of the cortical bone can be identified in terms of nine orthotropic parameters in the elasticity tensor, i.e., $E_{1}$, $E_{2}$, $E_{3}$, $G_{12}$, $G_{23}$, $G_{31}$, $\nu$ $_{12}$, $\nu$ $_{23}$, $\nu$ $_{31 }$ (Eq. (16)), where radial, circumferential and axial directions are denoted as subscripts 1, 2 and 3, respectively, while radial–circumferential, circumferential–axial and axial–radial planes are denoted as subscripts 12, 23 and 31, respectively. Nonlinear correlations between the elasticity parameters and elemental densities were established using curve fitting (Eqs. ( 2–7), based on the data reported by SchwartzDabney and Dechow (2003) (Fig. 10a–c). However, $R^{2}$ values for the fitted equations of orthotropic Poisson’s ratios against density are significantly low (ranging from 0.023 to 0.15), using several different curve fitting strategies. The correlations between orthotropic Poisson’s ratios and HUs reported by Hellmich et al. (2008) were therefore adopted (Eqs. 8–10 and Fig. 10d).

$$\begin{aligned} \left\{ {\begin{array}{l} \varepsilon _{11} \\ \varepsilon _{22} \\ \varepsilon _{33} \\ \gamma _{12} \\ \gamma _{23} \\ \gamma _{31} \\ \end{array}} \right\} =\left[ {{\begin{array}{llllll} {1/{E_1 }}&{} {{-\nu _{21} }/{E_1 }}&{} {{-\nu _{31} }/{E_3 }}&{} 0&{} 0&{} 0 \\ {{-\nu _{12} }/{E_1 }}&{} {1/{E_2 }}&{} {{-\nu _{32} }/{E_3 }}&{} 0&{} 0&{} 0 \\ {{-\nu _{13} }/{E_1 }}&{} {{-\nu _{23} }/{E_2 }}&{} {1/{E_3 }}&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} {1/{G_{12} }}&{} 0&{} 0 \\ 0&{} 0&{} 0&{} 0&{} {1/{G_{23} }}&{} 0 \\ 0&{} 0&{} 0&{} 0&{} 0&{} {1/{G_{31} }} \\ \end{array} }} \right] \left\{ {\begin{array}{l} \sigma _{11} \\ \sigma _{22} \\ \sigma _{33} \\ \sigma _{12} \\ \sigma _{23} \\ \sigma _{31} \\ \end{array}} \right\} \end{aligned}$$

(16)

The local material orientations were identified by generating multiple sections accommodating the morphology of mandible so that each divided section resembles a cylinder and hence features an orthogonal coordinate system (SchwartzDabney and Dechow 2003). Two field variables were set up in Abaqus subroutine, i.e., density and HU for the calculation of all nine engineering constants defining orthotropy, which are nine solution-dependent variables. The upper threshold for cancellous density is 1.53 g/cm$^{3}$ (Misch et al. 1999), and the lower threshold for cortical density is 1.70 g/cm$^{3 }$(SchwartzDabney and Dechow 2003) for differentiating them.