Physiologically accurate 3D models of a mandibular second molar and a maxillary central incisor were created in Solidworks (Dassault Systèmes) based on computer tomography (CT) data provided by eHuman, Inc. The CT data contained a point-cloud that traced surfaces of enamel, dentin and pulp chamber, as well as the interfaces between enamel and dentin. 3D CAD models were created by connecting the relevant points together and generating surfaces that passed through the said points and surfaces. The tooth models then include enamel and dentin components with a pulp chamber in the dentin. The present tooth geometries and periodontal ligament are shown in Fig. 1.
The solid models of these teeth were meshed for finite element analysis (FEA) in MSC Apex. The pre-processing, processing and post-processing steps were performed in MSC Marc/Mentat software. The finite element mesh that we developed for modeling a mandibular second molar is illustrated in Fig. 2. A percussion rod was also included in the present 3D models to simulate QPD accurately and for predicting the force measured by a sensor in the rod as a function of time. The present FEA meshes have a total number of about one million elements each. The location and angle of the percussion rod to each tooth was based on typical clinical use of the Periometer hand piece. For molars, the rod percussion is generally positioned at the buccal-mesial cusp since this is the easiest cusp on these teeth to access. This rationale was also used to apply similar percussion rod positioning for in vivo studies conducted previously [26,27,28,29].
We used second order isoparametric three-dimensional ten-node tetrahedron elements for the PDL, eight-node, isoparametric, arbitrary hexahedral elements for the percussion probe, and linear isoparametric three-dimensional tetrahedron elements for the remaining structures. Boundary conditions were defined to prevent free body motion in that the elements on the outer surfaces of the supporting bone were constrained. This constraint is reasonable when one considers the mass of the human mandible and maxilla compared to the relatively low forces induced by QPD. The percussion rod tip and at the tooth enamel were specified as touching contact bodies in MSC Marc [30] in to order implement the percussion impact. They are initially separated from each other prior to contact and then are in touching contact during the percussion response. The tooth enamel, dentin, PDL and bone were specified as “glued” contact bodies in MSC Marc [30].
Mesh size sensitivity of the models with various PDL thicknesses was studied. The results showed that increasing the mesh size causes the maximum force to increase slightly and the time duration of the response to decrease slightly. This is expected as reducing the number of elements (increasing the mesh size) in a model generally makes the model stiffer. Regardless, the curves remained bell-shaped which matches experimental results. The present mesh size was then selected to achieve a reasonable combination of accuracy and FEA run time (Fig. 2).
At least two layers of elements were used in the PDL for each FEA model [7]. To validate the use of two layers, results were compared for two otherwise identical models that had either two or three layers of elements in the PDL. The number of elements in the model with three layers was approximately 50% greater than that in the model with two layers in the PDL. A comparison of the results for two models, one with two layers of elements in the PDL and one with three layers of elements in the PDL, is presented in Fig. 3. The difference in force between the two sets of results is less than about 4% over the duration of the percussion response. Thus, it was concluded that two layers of elements in the PDL are sufficient for reasonably accurate results.
The percussion rod was free to move in only its axial direction with a mass of 9 g and an initial velocity of approximately 60 mm/s, which are consistent with the operation of the Periometer hand piece [31]. The models were run with a time increment of 2 µs. A direct integration method was used to obtain the solution to the equations of motion for our models.
Proportional viscous damping is commonly employed to apply modal analysis of undamped systems to damped systems. A proportional damping model expresses the damping matrix as a linear combination of the mass and stiffness matrices, that is.
$$D=\sum _{i=1}^{n}\left\{{\alpha }_{i}{{M}}_{i}+{\beta }_{i}{{K}}_{i}\right\}$$
(1)
where \({D}\) is the global damping matrix, \({{M}}_{i}\) is the mass matrix, and \({{K}}_{i}\) is the stiffness matrix, \({\alpha }_{i}\) is the mass matrix multiplier damping coefficient, and \({\beta }_{i}\) is the stiffness matrix multiplier damping coefficient for the \(i\)th element [30]. This damping model is also known as ‘Rayleigh damping’. Modes of Rayleigh damped systems preserve the simplicity of the real normal modes as in the undamped case [32]. Thus, \({\alpha }_{i}\) and \({\beta }_{i}\) can be readily determined from a modal analysis of the undamped structure. In addition, MSC Marc used in the present work employs tangent stiffness which avoids the possibility of any artificial Rayliegh damping [30, 32]. Accordingly, Rayleigh damping was applied in the present models to simulate the viscoelastic behavior of the PDL. The damping matrix for the present models is defined as a linear combination of the mass and stiffness matrices of the system and damping coefficients are specified on an element-by-element basis.
Orthogonality and relations between the modal equations were used to calculate the \(\alpha\) and \(\beta\) coefficients based on a modal damping factor and the undamped circular natural frequency [20, 33]. Accordingly, α and \(\beta\) are given by
$$\alpha =\frac{2\left(\sum _{j=1}^{n}{\omega }_{i}^{2}\sum _{j=1}^{n}\frac{{\xi }_{j}}{{\omega }_{j}}-n\sum _{j=1}^{n}{\omega }_{j}{\xi }_{j}\right)}{\left(\sum _{j=1}^{n}{\omega }_{j}^{2}\sum _{j=1}^{n}\frac{1}{{\omega }_{j}^{2}}\right)-{n}^{2}}$$
(2a)
$$\beta =\frac{2\left(\sum _{j=1}^{n}{\omega }_{j}{\xi }_{j}\sum _{j=1}^{n}\frac{1}{{\omega }_{j}}-n\sum _{j=1}^{n}\frac{{\xi }_{j}}{{\omega }_{j}}\right)}{\left(\sum _{j=1}^{n}{\omega }_{j}^{2}\sum _{j=1}^{n}\frac{1}{{\omega }_{j}^{2}}\right)-{n}^{2}}$$
(2b)
where \({\xi }_{j}\) and \({\omega }_{j}\) are the modal damping ratio and the undamped circular natural frequency for the jth mode, respectively.
The damping ratio is a dimensionless parameter associated with the decay in oscillations for a system subjected to an external force and disturbed from its static equilibrium position. Huang and coworkers [34] measured the damping ratios for a maxillary central incisor in vivo, which were determined to be in the range of 0.09 to 0.24. Following the procedure described by Liao and colleagues [20], we assigned these damping ratios to the first and 20th modes, respectively. Interpolation was used to estimate the damping ratios for the second to 19th modes as described in [20, 33]. We obtained the first 20 undamped natural frequencies, i.e., \({{\omega }}_{{i}}\) for i = 1 to 20 by performing a modal analysis for the maxillary central incisor model in MSC Marc. We assumed that no significant damping occurs in the tooth enamel and in the stainless steel percussion rod. In addition, the total strain energy of the entire structure and the damping energy dissipated in the entire structure were determined for our FEA maxillary central incisor model. The maximum value of the total strain energy, which is the peak restored energy, was then recorded. The damping ratio was calculated from the present FEA results for the entire tooth complex using the equation:
$$\xi = \frac{{d}}{{4}\pi{P }}$$
(3)
where P is the peak strain energy (stored) and d is the corresponding value of the damping energy (dissipated).
Equation (2b) was used to calculate α and \(\beta\) for damping ratios determined for several PDL elastic moduli reported by others. The resulting α and \(\beta\)properties apply to the entire tooth-ligament-bone complex although each of the different material components of the finite element model should have its own damping properties. Accordingly, a simple Reuss composite model [35] was adapted to estimate the partition of overall α and \(\beta\) between these components since they are arranged in series with respect to the loading direction. The Reuss model gives the following relationship for the effective elastic modulus of the tooth structure, Eeff:
$$\frac{{1}}{{{E}}_{{eff}}}=\frac{{{V}}_{{E}}}{{{E}}_{{E}}}+\frac{{{V}}_{{D}}}{{{E}}_{{D}}}+\frac{{{V}}_{{PDL}}}{{{E}}_{{PDL}}}+\frac{{{V}}_{{B}}}{{{E}}_{{B}}}$$
(4)
where EE and VE are the elastic modulus and volume fraction of the enamel, ED and VD are the elastic modulus and volume fraction of the dentin, EPDL and VPDL are the elastic modulus and volume fraction of the PDL, and EB and VB are the elastic modulus and volume fraction of the bone. We note that the modulus of the enamel is several times greater than the other tissues in the model while it also has a relatively small volume fraction. Therefore, the first term in Eq. (4) can be ignored. In addition, the moduli for the dentin and cortical bone are roughly the same so that the second and last terms in Eq. (4) may be combined for a single elastic modulus. To a first approximation, Ashby showed that the damping ratio for soft and hard polymers is inversely related to the elastic modulus according to:
$${E }\cong \frac{{C}}{\xi }$$
(5)
where constant C has roughly the same value for a range of polymeric materials including biological materials [36]. Assuming that ligament and dentin tissue also follow this relation, Eq. (5) can be substituted into Eq. (4) to give an approximation of the effective damping ratio:
$${\xi }_{{eff}} \cong {{V}}_{{PDL}}{ \xi }_{{PDL}}+({{{V}}_{{D}}+{{V}}_{{B}})\xi }_{{D}}$$
(6)
where ξPDL and ξD are the damping ratios for the PDL and dentin, respectively. In addition, the values of ξD is approximately equal to 10− 3ξPDL based on Eq. (5) and the ratio of the elastic modulus of the PDL and that for either dentin or bone. For the incisor tooth geometry, the volume of the dentin and adjacent bone is approximately 19 to 20 times greater than that for the PDL. Substituting the corresponding volume fractions into Eq. (6) followed by substitution of 10− 3ξPDL for ξD gives
$${\xi }_{{eff}}\cong {0.05}{ \xi }_{{PDL}}+{{0.95} \xi }_{{D}}\cong {0.05}{ \xi }_{{PDL}}+ {{9.5\times}{{10}}^{{-4}} \xi }_{{PDL}}\approx {0.05}{ \xi }_{{PDL}}.$$
(7)
Thus, the damping ratio for the PDL, ξPDL, should be about 20 times greater than ξeff determined for the tooth and PDL together. Since α and \(\beta\) are both directly proportional to the corresponding value of ξ, it follows that their values for the PDL should also be about 20 times greater than those determined for the entire tooth complex. Based on the ratio of PDL elastic modulus over dentin elastic modulus and Eq. (4), it follows that the α and \(\beta\)values for the dentin and bone should be about 10− 3 times those for the PDL and about 10− 2 times those determined for the tooth complex. Accordingly, 20 α and 20 \(\beta\) were used in each model for the PDL while 0.001 α and 0.001 \(\beta\) were used for the bone and dentin as reasonable damping parameter estimates. Since these partitioned values can be considered material properties, they can also be applied in the molar models even though they were determined using damping data and tissue volume fractions for a central incisor.
The mechanical properties of the hard dental tissues (bone, dentin and enamel) were assumed to be linear-elastic and isotropic. While it is well known that the properties of these hard tissues are somewhat anisotropic, it is generally assumed unnecessary to implement anisotropic properties [4, 37, 38]. For the present model, this assumption appears reasonable considering the single loading direction imposed by the percussion rod. These properties as well as the density and constitutive model for each dental material are summarized in Table 1.
Table 1 Mechanical properties of hard tissues and stainless steel [4, 35,36,38] A large range of elastic moduli for the PDL that have been proposed previously were implemented in the present study. Specifically, we exercised models with a PDL having a typical thickness of 180 μm and each of the different reported linear elastic moduli, which range from 0.01 to 6.9 MPa, as well as Poisson’s ratios of 0.45 and 0.49.
Modeling PDL Thicknesses
The percussion behaviors of the present teeth were simulated with a PDL thickness of 180 μm and one of three different PDL elastic moduli: 0.01, 0.1, and 6.9 MPa. Thicknesses in the range of 180 μm have been commonly reported in the literature for healthy teeth [2].