ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

A NON-LINEAR BIPHASIC MODEL FOR THE PERIODONTAL
LIGAMENT: MODELLING AND SIMULATION
1. ABSTRACT
M. Favino 1 , M. Drolshagen 2 , C. Bourauel 2 and R. Krause 1
We present a new model for the Periodontal Ligament (PDL). Our approach is based on
a biphasic model including a Neo-Hookean stress-strain relation. Due to the porous
microstructure of the PDL, this model allows for a better description of the elastic
properties of the PDL than standard purely elastic models. The non-linear saddle-pointlike
system arising from the temporal discretisation of the time-dependent PDE system
is solved by means of Newton’s method. Numerical simulations are performed on a
realistic geometry of a porcine tooth and the model is validated against data obtained by
in-vitro measurements.
2. INTRODUCTION
Mathematical models for biphasic materials are well-known in engineering related
applications such as ground water flow and reservoirs. In the last two decades
multiphase model was also employed in the context of biological materials. For
example [1] and [2] used it to model the mechanical behaviour of cartilage.
The PDL is a thin layer of dense soft connective tissue located between the tooth root
and the jaw bone; it consists of a solid phase formed by collagen fibres and an
interstitial fluid filling up the tissue. Although the biphasic nature of the PDL is wellknown,
existing models for the PDL take into account the influence of the interstitial
liquid only by means of macroscopical material parameters.
In this work, we propose the idea of using a multiphase model based on the theory of
the porous media (TPM) for the mathematical modelling and numerical simulation of
the PDL. According to the TPM, we do not have a geometrical resolution of the two
phases but they can be seen as two superimposed continua. At every spatial point the
two materials coexist and the local distributions are described by means of macroscopic
variables: the volume fractions. Beside this, we will also assume the incompressible
nature of the two phases. Following the work of [1], we extend our model [3] removing
the hypotheses of small displacements and employing a Neo-Hookean material law for
the solid tissue.
We will present a coupled non-linear elastic-biphasic model for the description of a
tooth-PDL system. The numerical solution strategy is based on the Newton’s method
and low-order finite elements. The model will be tested on a realistic geometry of a
porcine tooth.
1 Institute of Computational Science, University of Lugano, Lugano 6900, Switzerland
2 Endowed Chair of Oral Technology, University of Bonn, Bonn, Germany
This work was in part supported by German Research Foundation (DFG)