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.

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.