Abstract results, linear elastic, hyperfoam and Ogden models were

 

Abstract

Tooth
movement in the alveolar bone is due to the periodontal ligament (PDL) response
to forces applied to the tooth in different situations during orthodontics and physiological
conditions. This response affects the tooth, the PDL and bone morphology. The
aim of this work was to simulate PDL behavior considering porous conditions. In this research, CT images were used to
construct a 3D model of the human incisor and its surrounding alveolar bone.
The finite element model was created by constructing real geometry of each
tissue, and hydro-mechanical coupling of this model was investigated. To
compare the results, linear elastic, hyperfoam and Ogden models were assigned
to solid phase of PDL. Results of tooth movement in short time force applied to
tooth for different constitutive models for solid phase of PDL is reported.
Results show fluid phase of PDL plays an important role in behavior of tooth
movement. PDL interstitial fluid behavior showed a good agreement with
physiological concept of fluid movement in the PDL. Considering interstitial fluid of each component of the
periodentium can give the researchers a proper understanding as to how the PDL behaves when
forces are applied to the tooth.

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Keywords:
periodontal ligament, finite element
method, hydro- mechanical coupling, interstitial fluid, tooth movement

1          
Introduction

The periodontal ligament (PDL) is a soft tissue which
connects the tooth to the surrounding alveolar bone 1, 2. In addition to the tooth support, the PDL plays a
significant role in distribution of physiological and supraphysiological forces
on the alveolar bone and precludes stress concentration during mastication and
orthodontic treatment 3, 4. The PDL is
also responsible for bone remodeling during orthodontic tooth movement 5, 6. Following structural three elements
within the PDL govern mentioned mechanical functions: (1) the collagen and
elastic fibers, (2) the ground substance which consists of 30% glycoproteins
and proteoglycans, and 70% bound water, and (3) the vasculature 7. Indeed, the PDL consists of a solid phase,
formed by collagen fibers, and a fluid phase, filling up the tissue with
interstitial fluid 2, such that solid
phase resist tensile loads and fluid phase tolerates compressive loads 8, 9. Thus, the mechanical response of the
PDL to tensile and compressive loads is different. As a result, the accurate
constitutive model that might be used for numerical simulation of the PDL should
take the biphasic nature of this material into account. 

Definition of a proper constitutive model, considering
structural configuration and experimental results, denotes a reliable and
hopeful approach to the biomechanics of the tissues. However, material models
of the PDL are not realistic. A lot of constitutive models of the PDL such as
linear elastic 10, 11, nonlinear
elastic 12-14 and viscoelastic 15, 16
have been reported in literature. The elastic models can only describe the
PDL’s behavior at the absence of viscous phenomena, whereas viscoelastic model
is a class of phenomenological model which describes overall tensile response
of the PDL 17. Due to the existence of
complex interaction between solid phase and fluid phase of the PDL, elastic and
viscoelastic constitutive models are unable to well describe the microscopic
behavior of the soft tissue.

Recently, biphasic material formulation has been used for
describing the mechanical behavior of the soft tissue based on the porous media
theory 18, 19. In the biphasic models,
a framework able to describe hydro-mechanical coupling between a porous elastic
solid matrix and a pore filling fluid has been chosen. So that, under the
mechanical loading, the soft tissue’s matrix deformed, letting the fluid
content flow through its pores 17, 20-22. van Driel et al 21 showed that assigning of poroelastic
material properties to the PDL and to the surrounding alveolar bone is
appropriate for describing of creep response of the tooth. Natali et al 22 assumed biphasic material formulation and
small strains framework for the PDL and also, imposed prevention of fluid
exchange between the PDL and surrounding bone and cementum. Their proposed
model proved to be quite accurate in the simulations of experimental tests
reported. Bergomi et al 20 considered a
porohyperelstic finite element model to interpret behavior of the PDL under
harmonic tension-compression loading. Wei et al 17
compared the effect of different constitutive models of solid phase of the PDL
(the linear elastic model, the hyperfoam model, and the Ogden model) and showed
that the Ogden model is the most appropriate one among other models, based on
the in vivo experimental test.

In order to better understand the behavior of PDL in
tooth movement, the present study was designed to investigate tooth
displacement in response to orthodontic loading. For this purpose, a 3D finite
element model of incisor tooth, PDL and bone was built. Then, to compare the
effect of the mechanical behavior of the model components, elastic and biphasic
material formulations were assigned to each them. It should be noted that in
the simulation with the biphasic material formulation, unlike most previous
studies, was considered fluid exchange between the PDL and surrounding bone and
the tooth.

2          
Materials and methods

2.1          
Geometry

The construction of a 3D model, was done by means of the
CT image scanning from cross sections of the sample. This work was carried out
by the Sky-Scan 172 high-resolution micro CT scanner (Sky-Scan, Kontich,
Belgium) with 1.7 micrometer of each section. The images of the sample were
imported into MIMICS 10 (Materialise, Leuven, Belgium). After segmentation of
each layer of the periodontium, the 3D models that contained STL meshes were
obtained. Solid model was obtained by importing STL meshes into CATIA V5
(Dassault Systèmes, Vélizy-Villacoublay, France). After preparing the 3D
models, finite element simulation was done by ABAQUS 6.12 (Dassault Systèmes,
Vélizy-Villacoublay, France).

2.2          
Mass and momentum balance laws

The multi-phase mixture is composed of a solid matrix and
fluid phase, therefore the interaction between phases is given with coupled
equations. Ignoring mass exchanges between the two phases, balance of mass for
solid and fluid phase can be written as follows 23:

(1)

(2)

Where  denotes a material
time derivative,  is porosity which
relates to void ratio as ,  and are the intrinsic Cauchy pressures of the solid and fluid
phases, respectively.  is solid velocity,
 and  are bulk modules
of solid and fluid phases, respectively and  is Darcy’s flux.
By summing these two equations, the total mass balance is obtained by:

(3)

The total Cauchy stress tensor  is obtained from
the sum of   and . By the theory of multi-phase mixture, the balance of
momentum is written as:

(4)

(5)

Where  is gravity,  and  are body forces
per unit current volume of the solid matrix exerted on the solid and fluid
phase, respectively. Adding (4) and (?5) for two phases and noticing that , we obtain balance of momentum for the entire mixture:

(6)

2.3          
Material definition

With the use of multi-phase theory, a micro model was
presented which is capable of simulating the hydro-mechanical coupling of each
component. The considered phases are related to solid skeleton (collagen and
elastin fibers) and fluid phase. In fact, behavior of the studied tissue is an
outcome of the presence of different phases 24.
The multi-phase model considers the interaction of different phases and
exhibits a more acceptable behavior.

2.3.1    Solid phase

Fundamental equations of solid matrix based on Hook’s law
were applied for bone and tooth, and to compare mechanical behavior of PDL
linear elastic and hyperelastic material was considered for the PDL.

2.3.1.1  
Linear elastic material
model

Tooth and bone is modeled using homogeneous and linear
elastic model. Each of tooth and bone considered single material and
inhomogeneous character of trabecular and cortical layer for bone and dentine,
enamel pulp, etc for tooth is not necessary for the purpose of this work.

2.3.1.2  
Hyperelastic materials
method

Hyperelastic materials account for both nonlinear
material behavior and large deformations. The constitutive law for an isotropic
hyperelastic material is defined by an equation which relates the strain energy
density of the material to the deformation gradient or, for an isotropic solid
to the three principal stretch ratios 25:

(7)

Where ,  and  are the principal
stretch ratios.

As a type of hyperelastic model, hyperelastic foam
(Hyperfoam) strain energy function was used as following 20:

(8)

Where ,  and  are material
parameters,  is Poisson’s
ratio,  is the Jacobian and the  is order of the
strain energy potential which first order potential  considered in this
study.

The other hyperelastic model used is the Ogden model,
which is formulated in terms of the principle stretch ratios as follows 25, 26:

(9)

The second order Ogden model with material parameters of  and  which were derived
from data fitting from experimental test in the literature 12. Solid phase
properties constants are shown in Table 1.

2.3.2       
Fluid phase

It is assumed that fluid flux follows up Darcy’s law. To
achieve this purpose, Darcy’s law was used to describe fluid movements in solid
porous matrix. The total discharge rate, , at an area of  and a thickness of
 which is subjected
to a pressure difference of  with a
permeability value of  is obtained from
the Darcy’s law as below:

(10)

Permeability  is related to
fluid and solid properties.  The Kozney-
Carman equation 27, for permeability is defined as:

 

(11)

Where  is the specific
area per unit volume of matrix, ,  and   are unit weight,
viscosity and Kozney- Carman constant, respectively. According to the Kozney-
Carman equation, permeability is affected by porosity, thus porosity changes
cause permeability changes. Compared to PDL, bone and tooth sustain low
strains. Therefore, their porosity can be assumed constant during loading. The permeability of PDL can be defined as Argoubi and Shirazi-
Adl 28, equation:

(12)

Where  and  are permeability
and void ratio in zero strain, respectively,  is the void ratio
and  is material’s
constant.

 The permeabilities of the
bone and the tooth were considered 5.27× and 3.87× , respectively. The permeability of PDL is given by
equation (?12) 29. Fluid phase properties of tooth, PDL
and bone are shown in Table 2.

2.4      Boundary conditions and loading

A force with constant magnitude of 0.05N is applied to
the tooth in mesial-distal direction as shown in Figure 1 until 2.5 seconds and then removed suddenly.
Both distal sides of bone are assumed to be fixed. The pore pressure at the
outer surface was set to 0.0 MPa 20, 21. Based on physiological finding we
assume that the flow is conserved and there is no pressure drop at the
interface of each part. The boundary conditions, loading and meshing are shown
in Figure 1.

2.5          
Finite element analysis

The constructed 3D solid models were imported to Abaqus
software for finite element simulation. C3D4P mesh was chosen for poroelastic
solution of each component. Total number of elements after convergence test
with controlling the maximum von-Mises stress variations of less than 5% was
347,302 of which 143,024 were dedicated to the tooth, 82,864 to the PDL and
121,414 to the bone. Numerical solutions were applied with SOILS analysis
(couples pore fluid pressure and solid displacement) of Abaqus software.

3          
Results

Three different materials were assigned to solid matrix
of PDL as linear elastic, Ogden hyperelastic and hyperfoam. Orthodontic force
with 0.05N magnitude applied in mesial distal direction to the tooth during 2.5
seconds and suddenly removed. Figure 2 shows the curve of force- displacement
for three different material which chosen for solid matrix of PDL.
Displacement- time curve for the point which load applied plotted in Figure 2, after unloading tooth tends to return to its
initial state. The hyperfoam model shows higher displacement while the linear
elastic model shows lower displacement. Although the linear elastic model is
assigned to the solid matrix, but due to the presence of interstitial fluid,
nonlinear time-dependent behavior is evident in both loading and unloading steps.

The fluid velocity curve versus time for the point where
the maximum velocity was observed in that area which located in the cervical
third area is shown in Figure 3. Maximum magnitude of 13.58 ?m/s observed in
initial time at distal side. As liquid squeezes out from PDL, the volume of
fluid content decreases and fluid velocity deceases too. With the removal of
force, the direction of fluid velocity changes and the fluid returns toward the
PDL. At this time pore pressure is at its highest negative value as depicted in
Figure 4. As stated by Darcy’s low, the pressure gradient is a motivation for
fluid flow in porous media, and the negative sign in equation (?10) shows the transport of fluid from high to low
pressures.

Figure 6 and Figure 7 show pore pressure and equivalent stress
before and after unloading of three models. There is no significant difference
in distribution of pore pressure and stress in three models, but in magnitude,
hyperfoam shows higher while linear elastic shows lower pore pressure and
stress.

Fluid velocity with direction and deformation for entire
model with cut view are shown in Figure 5 and Figure 8 respectively. Maximum displacement is in the
crown of the tooth is observed just before unloading. The contour after
unloading represents the tooth return to its original state as PDL recover its
fluid content.

4          
Discussion

Biomechanical analysis of the periodontal ligament tissue
has always been accompanied by a lot of simplifications. These simplifications
were mostly due to problems encountered in obtaining empirical data and
numerical formulation. For example, considering the
periodontal ligament as an isotropic and linear elastic material is such
simplifications. While one of the most
characteristic features of the periodontal ligament is nonlinear behavior and
also the dependence of the behavior of this substance on time. The precise
behavior of solid matrix and interstitial fluid of PDL with considering all physiological
conditions cannot easily be taken into account. With the simplifications
considered in the finite element, it can be expected to somewhat predict the
behavior of this tissue. The results obtained from the simulation should be in
good agreement with the behavior of this tissue within the body. The proposed
constitutive model should represent the behavior of fibers and interstitial
fluid. In this study multi-phase method was used for PDL to
simulate tooth movement, this method can directly model the behavior of fluid
and solid phase of tissues.

Tooth movement depends on PDL behavior in the adjacent
tooth and alveolar bone. Since the measurement of physical parameters in the
periodontium has limitations, study of these parameters is done with the use of
finite element analysis. In this study, with obtaining real geometry described
above, the periodentium hydro-mechanical coupling was simulated. Three
dimensional finite element model simulations with three different materials for
PDL solid matrix have been discussed throughout this paper to study the
behavior of PDL in loading and unloading conditions. Similar results are
obtained in the previous studies with lateral loading on the tooth with
considering hydromechanical coupling 22,
viscoelastic model 30 and experimental 31.

Studies 21, 32
show that the viscoelastic mechanical behavior of PDL is strongly influenced by
the movement of interstitial fluid into the vascular reservoir of the bone
marrow through multiple holes in the alveolar wall. Bien 32, thoroughly
analyzes the PDL fluid dynamics in relation to tooth movement and identifies
three systems (cellular, vascular, and interstitial) for transmitting and
dampening of the forces acting on the teeth. The interstitial fluid is
restricted in the ground matrix and acts as a thixotropic gel. It is gel-like
when it’s not moving and it flows easily under pressure. When loaded, this
fluid squeezed out of compression areas and pulled into tensile areas 33 (see Figure 5). In the initial stage
of loading, since the fibers are slack, the tooth moves inside the bone cavity
and the interstitial fluid squeezes out. Over time, the ordinarily slack fibers
tightened 32, and the tooth movement
rate drops. The effect of this phenomena can be seen in Figure 2 which tooth displacement reaches to constant
value.

As shown in Figure 3 at the beginning of the procedure, fluid
within the PDL space rapidly squeezes out and the fluid velocity increases to
its peak. PDL compression and fluid squeezing lead to tooth movement in the
bone cavity. As Davidovitch 33 stated this fluid flow is a crucial step in
the physiochemical behavior of PDL. With the removal of force, the teeth tend
to return to its previous state. This process, which termed “relapse” 34,
causes the fluid to be returned to the solid matrix due to the collapse of
collagen fibers and the creation of negative pore pressure (Figure 6).

With pore pressure rising, more fluid squeezing is
observed (Figure 3 and Figure 4). Comparison of stress and pore pressure
contours in different times shows the impact of these two. The
results show that during the initial application and removal of the load, the
pore pressure is more effective than stress. This indicates that the fluid
first reacts and then the PDL fibers enter into action. Not considering the
role of fluid in tissue behavior will not deliver acceptable results. This
hydromechanical coupling between liquid phase and the solid matrix leads to the
deformation of collagen fibers that cause tooth movement.

Immediately after force, the tooth moves into the bone
socket. Collagen fibers that bind tooth to the bone, stretch which leads to
tensional deformation of alveolar bone while on the other side collagen fibers
compress 35. This mechanical stimulus
(tension and compression) lead to bone remodeling and consequently tooth moves
to a new position. Bone absorption is observed in the area under pressure,
while bone is formed in the tensile region 36.
An important theory describing bone reaction to stress and strain is the
flow-induced shear phenomenon 37-39,
which is based on the presence of osteocytes trapped in the lacunae inside the
bone 37. The strain in the bone causes
fluid flow inside the canaliculi, which causes shear stress on the osteocytes.
With the transfer of fluid from the canaliculi and flow decrease, osteoclasts
activate, leading to bone loss in that area 41.
As shown in Figure 5, the fluid tends to leave the pressurized
area in the bone socket by applying the force and movement of the fluid in the
PDL and bone. On the other side, in the stretching area, the activation of
osteoblast will cause bone formation due to fluid intake in that area.

5          
Conclusion

The ultimate goal of this study is computer aided
simulation of tooth movement. With proper simulation, better prediction of
tooth movement in the bone cavity and how to interact with the tissues of the
teeth, PDL and bones can be obtained. The effect of the
fluid inside the tissue on the time-dependent behavior is quite evident. To
properly simulate the behavior of the PDL, taking into account the
hydromechanical coupling between the components will provide more acceptable
results on how the tissue behaves.

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