An intramembranous ossification model for the in-silico analysis of bone tissue formation in tooth extraction sites

TL;DR

The paper develops a dedicated mathematical model of intramembranous ossification in tooth extraction sockets, built on a bioregulatory PDE framework and implemented with FEM. By adapting and extending prior fracture-healing models, it couples cell populations, ECM densities, and growth factors to capture diffusion, chemotaxis, haptotaxis, proliferation, differentiation, and apoptosis under depth-dependent vascularization. Validation against canine in-vivo data yields a mean absolute error of about 3.04%, supporting the model’s ability to reproduce spatiotemporal tissue evolution and ossification fronts. The work offers a tool for planning dental procedures, evaluating treatment strategies, and guiding future in-silico studies on osseointegration and mechanobiology, while outlining necessary enhancements such as clot dynamics, 3D geometry, and mechanobiological coupling.

Abstract

The accurate modeling of biological processes allows to predict the spatio-temporal behavior of living tissues by computer-aided (in-silico) testing, a useful tool for the development of medical strategies, avoiding the expenses and potential ethical implications of in-vivo experimentation. A model for bone healing in mouth would be useful for selecting proper surgical techniques in dental procedures. In this paper, the formulation and implementation of a model for Intramembranous Ossification is presented aiming to describe the complex process of bone tissue formation in tooth extraction sites. The model consists in a mathematical description of the mechanisms in which different types of cells interact, synthesize and degrade extra-cellular matrices under the influence of biochemical factors. Special attention is given to angiogenesis, oxygen-dependent effects and growth factor-induced apoptosis of fibroblasts. Furthermore, considering the depth-dependent vascularization of mandibular bone and its influence on bone healing, a functional description of the cell distribution on the severed periodontal ligament (PDL) is proposed. The developed model was implemented using the finite element method (FEM) and successfully validated by simulating an animal in-vivo experiment on dogs reported in the literature. A good fit between model outcome and experimental data was obtained with a mean absolute error of 3.04%. The mathematical framework presented here may represent an important tool for the design of future in-vitro and in-vivo tests, as well as a precedent for future in-silico studies on osseointegration and mechanobiology.

Figures (11)

• Figure 1: Graphical representation of the mathematical model. The inner blue boxes indicate the relevant modes of motility simulated for each cell: $D_H$ stands for haptokinetic diffusion, CT for chemotaxis (chemo-attractant in brackets) and HT for haptotaxis (substrate in brackets). Dashed arrows indicate production of growth factors, double line arrows indicate differentiation and continuous line arrows indicate matrix synthesis. Blue dashed arrows refer to the H-D production of angiogenic growth factor ($g_v$). Arrows ending in a perpendicular line ($\Downarrow$) indicate: decay or consumption of growth factors (dashed arrows); apoptosis (double arrows); tissue resorption or degradation (continuous arrows). Inspired on the schematic representation by Geris et al.geris2008angiogenesis.
• Figure 2: Assumed functional relationship between oxygen tension and $\tilde{m}_v$. The processes are listed in the table according to the data reported by Carlier et al.carlier2015oxygen
• Figure 3: Comparison between the activation functions for H-D fibroblast apoptosis using a 6$^{th}$ order Hill function ($\boldsymbol{---}$), bFGF-dependent fibroblast apoptosis using a 6$^{th}$ order Hill function ($\boldsymbol{\cdots}$) and both thresholds modeled with a Heaviside constant function ($\boldsymbol{-}$)
• Figure 4: Geometry for the simulations.
• Figure 5: Model conditions: graphical representation. Continuous and dashed lines represents boundary and domain conditions respectively. Blue lines indicate non-uniform (NU) conditions, which depend on wound depth (Figure \ref{['fig:NumCell']}), while red lines are defined homogeneously. Arrows ending in a horizontal line mean that the condition is released at that precise moment.