Research Article - Biomedical Research (2017) Volume 28, Issue 20

## Analysis of mechanical properties at different levels of the periodontal ligament

**Yu Yang and Wencheng Tang ^{*}**

School of Mechanical Engineering, Southeast University, Nanjing, PR China

- *Corresponding Author:
- Wencheng Tang

School of Mechanical Engineering

Southeast University, PR China

**Email:**[email protected]

**Accepted on** October 27, 2017

**Visit for more related articles at**Biomedical Research

### Abstract

**Background/Purpose:** As a soft connective tissue, the Periodontal Ligament (PDL) exhibits material properties which are not the same at different levels. Study concerning the mechanical properties of human PDL was not reported in nanoindentation method with a flat punch.

**Materials and Methods:** A quantitative and a qualitative analysis for cervical margin, midroot and apex were made. According to the Oliver-Pharr theory, the mean value of the elastic modulus was calculated for each level. A generalized Kelvin model was used to fit the experimental load-displacement curve of the loading part at different levels, the model parameters were obtained by least-square method. The difference of stress distribution of the PDL for linear and viscoelastic model was compared with finite element method.

**Result:** In midroot section, the average elastic modulus ranges from 0.11 to 0.23 MPa, the range of the cervical margin and apex changed from 0.21 to 0.53 MPa, and 0.44 to 0.62 MPa, respectively. Apex section shows a higher creep compliance than that on the other two sections.

**Conclusion:** Experimental results indicate that average elastic modulus in midroot was lower than that at the cervical margin and the apex. The results derived with generalized Kelvin model agrees well with the test data, the stress in the PDL derived with viscoelastic model is lower than that with elastic model.

## Keywords

Periodontal ligament, Elastic modulus, Viscoelastic model, Nanoindentation.

## Introduction

PDL is a soft tissue that connects tooth root and alveolar bone, it transmits orthodontic forces to alveolar bone resulting in tooth mobility during orthodontic treatment [1,2]. According to literature, clinical orthodontic treatment is a long-term and iterative process, even accompanied with possible side effects. Furthermore, the magnitude of forces and moments is considered to be indeterminate and the biologic response to orthodontic forces is unknown [3,4]. Study of the material properties of PDL is important to determine the role of PDL in absorbing treatment load and increase the understanding of tooth movement under orthodontic loading.

The Finite Element Method (FEM) is an important tool to analyse orthodontic movement and distribution of stress and strain within teeth and the periodontium. In order to make the FEM analysis more reliable, accurate material property parameters are needed. Since the 1960s, a large number of researchers have measured the elastic modulus of the PDL through a series of experiments, such as uniaxial tensional [5,6], intrusion and extrusive test [7]. Other studies tried to investigate the distribution of stress and strain within the tooth and periodontium, with laser holography [8], optoelectronical set-ups [9], photoelastic models [10] and Electronic Speckle Pattern Interferometry (ESPI) [11]. In spite of the effort, the measured value of elastic modulus varies dramatically, of which the minimum value is 0.01 MPa, while the maximum value is 1750 MPa [12], which differed by a factor of 105. The differences may be attributed to the complexity of PDL's structure, differences of modeling assumptions for mechanical behavior and the impact of experimental conditions. The property model is needed for element analysis on PDL, many researchers assume it as a linear [13], bilinear [14] and nonlinear model [15] in finite element analysis. The viscoelastic constitutive models of time-dependent were shown to describe the creep and stress-relaxation behavior on various species. In the early stage, the most acceptable theory is the Quasilinear Viscoelasticity (QLV) proposed by Feng [16] in which soft tissue relaxation behavior is described very well [17]. Schapery's nonlinear viscoelastic theory considering the temperature effect was applied for ligament tissue. Recently a modified superposition method was used to predict creep and stress relaxation behavior of animals’ PDL at different levels. Compared to Burgers four parameter and five parameter models, the Modified Superposition (MST) models has been proved better to describe the PDL viscoelastic behavior though a series of tests [18]. Nevertheless, the material properties among different levels of the PDL need to evaluate in quantitative and qualitative analysis.

With the development of science and technology, nanoindentation technique is widely applied to thin film material, new functional material and biological tissue material. Based on the load-displacement data of indentations on a nanometer scale, the selected material properties can be obtained by test the surface of the materials. Unlike other traditional experimental methods for estimation of material properties, local precision mechanical properties of biomaterials structures can be obtained by nanoindentation technology. As two important performance parameters of material properties, elastic modulus and hardness can be directly extracted from the unloading curve of indentations by utilizing Oliver and Pharr’s method [19]. The ability of microstructure of the PDL to resist deformation caused by external force can be understood by nanoindentation techniques, the method investigate the microcosmic mechanical properties and the intrinsic parameters of the material, which providing the theoretical base for developing the thin material of artificial biology.

Nanoindentation experiment is performed with different types of tip geometries. Currently, the measuring indenter involves pyramids, spheres, punches, and wedges. Since each indenter is provided with its own advantages and disadvantages, it is important to choose a suitable one for a given application. As a type of pyramid, the Berkovich indenter is extensively used for numerous applications. The reason is that the sharp geometry offers a high degree of spatial resolution and induces plasticity at a shallow depth. However, a tiny error would exist because of the calculation of the changing contact area. In terms of cell soft tissue, the indenter tip pierces the cell surface so that activity losses and the measured elastic modulus value are higher. Huang et al. [20] verified the applicability of the V-W exponential hyperelastic model to describe the instantaneous elastic of the PDL in nanoindentation method with a spherical indenter. For the flat punch, the contact area can be directly measured and it is not affected by transient behavior or thermal drift.

The objective of this paper is to determine the elastic modulus of the PDL at different levels and to fit the loading curve by generalized Kelvin viscoelastic model based on loaddisplacement date in nanoindentation with a flat punch, as well as predictions of the stress distribution in the PDL were determined under a constant tipping force.

## Materials and Methods

*Formula derivation for elastic modulus*

According to the Oliver-Pharr scheme [19] the reduced elastic modulus can be obtained from the initial slope of the unloading curve which can be fit by a power function:

(1)

Where P is the indentation load, C and m are fitting
parameters, h is the indenter displacement and ^{h}R is the
residual indentation depth. Taking the derivative of h for
Equation 1, unloading stiffness S is obtained by

(2)

The contact depth between the indenter and the sample can be determined by

(3)

Where P_{max} denotes peak load, ε is a constant depending on
the indenter geometry. According to the geometry and contact
depth of the indenter the contact area can be determined.
Reduced elastic modulus is calculated by

(4)

β is a constant related to the indentergeometry (β=1.0 for a flat punch), E* denotes the reduced elastic modulus. When considering the elastic deformation of the indenter, the reduced elastic modulus can be denoted by Equation 5.

(5)

where E and v are the elastic moduli and Poisson's ratios of the
measured material respectively. E_{i} and v_{i} are corresponding
elastic moduli and Poisson's ratios (for the diamond flat punch
indenter, E_{i}=1141 GPa and v_{i}=0.07), respectively. It is
important to note that due to very high stiffness of the indenter,
E* can be employed to represent the sample stiffness when
comparing with the PDL, which can be derived from the
unloading curve.

For a cylindrical flat punch indenter with a section radius of R and contact area constant for the cross-sectional area the following formula is given:

(6)

However, the biological soft tissue of the PDL performance
viscoelastic property, the unloading part curve is more convex
than that of elastic-plastic materials, which cause greater
contact stiffness. If the unloading rate is too low, a dotted curve like a nose in **Figure 1** may become evident lead to negative
contact stiffness. An error will occur when the elastic modulus
is calculated by using the measured contact stiffness S_{e}. Feng
et al. [21],have proposed that calculation of the true contact
stiffness S_{e} use the following equation:

Where is the creep rate (7)

(dh/dt) before unloading, is the unloading rate (dP/dt) at the
end of holding. Replacing S in Equations 2-4 and 6 with S_{e}, the
reduced modulus can be accurately determined.

*Formula derivation for creep compliance*

The flat punch consideration of a rigid indenter into a halfspace constitutes a linearly elastic homogeneous material, the relationship between indentation displacement and load based on Sneddon [21] is showed below:

P=4Gh/πR (1-v) → (8)

In this case, unlike other indenter shapes, the contact area between the flat punch and selected material is constant, G is the shear modulus and v is Poisson's ratio.

For the linearly viscoelastic materials under a prescribed arbitrary indentation loading history P (t). The relation of timedependent indentation displacement and load for flat punch was deduced by application of hereditary integral in Equation 8.

Where J (t) and ξ are the creep compliance in shear at time t and a dummy time variable of integration, respectively.

While the ramp force at a constant loading rate v_{0}, P (t)=v_{0}t,
substitute P (t) into Equation 9, it can be expressed as

(10)

Differentiating Equation 10 with respect to t, the J (t) is expressed as

(11) On the other hand, in terms of linear viscoelastic material, the creep function based on the generalized Kelvin model is given as

Where J_{0}-J_{i} are the compliance numbers, and τ is the
retardation time.

For the flat indenter, substituting P (t)=v_{0}t and Equation 12 into
Equation 9 leads to

(13)

We can use Equation 6 to fit the load-displacement curve
obtained from nanoindentation experiments by least square
correlation. All compliance constants, J_{0}, ...J_{i} and time
constants parameters τ_{0},...τ_{i} (i=1,..., N) can be obtained.
Substituting these parameters into Equation 5 determine the
creep compliance J (t) for a flat indenter.

*Experiments and details*

The experiments used four cases of human cadaveric maxillary
that were provided by Stomatological Hospital of Jiangsu
Province, All procedures were approved from the Research
Ethics Committee of Affiliated Hospital of Nanjing Medical
University. The canine samples including tooth, PDL and
alveolar bone were removed from the maxillary. With the help
of a slow cutting machine, samples were cut into 3 slices with
a thickness of approximately 2 mm, which were perpendicular
to the longitudinal axis of the tooth (**Figure 2**). For each PDL
level, four slices were obtained, giving a total of 12 slices from
the four canine specimens. All slice specimens were stored in
saline solution at 10°C until the beginning of the experiments
(<2 d).

The testing was conducted using a Nanotest machine (Agilent
Technologies Nano Indenter G200) with a flat punch in a room
with constant temperature of 27°C. First, the sample was
pasted on the petri dish and fixed on the working table (**Figure
3**). Then, using a low microscopic magnification to find a flat
test area, subsequently changing to a high magnification to
confirm the test position (**Figure 4B**). A total of 12 specimens
were used to test, each specimen has 8 test points around the
PDL (**Figure 4A**). During testing, the PDL region was coated
with saline solution every 1/2 hour to guarantee tissue activity.
A typical indentation data representing the average value for
each level indentation was chosen to calculation after the
experiments. In a total of 3 typical loading segment load displacement curve determine the creep function J (t) based on
least squares fit method, respectively (**Figure 5**).

During the test, a flat punch with radius of 20 μm was used to test PDL specimens. An indentation force of 1 mN was reached at the loading rate of 0.1 mN/s. The dwell time lasted for 20 s and finally the PDL was unloaded at the same rate. The load-displacement data in the whole testing process were recorded to investigate material properties of the PDL.

*Finite element analysis*

Before preparing the sample, the mandible canine sample was
scanned by a micro-CT, obtained 92 CT image at intervals of
0.1 mm. A 3D outline shapes of the canine and alveolar bone
was developed based on image-processing software (Mimics
10.1). To simplify, the PDL tissue was modeled as thickness of
0.2 mm between the root of the tooth and alveolar bone and
divided to three sections contained cervical margin, midroot
and apex (**Figure 6**), material properties of linear and
viscoelastic were assigned to the PDL respectively, based on
the experimental date which include calculation of the average
elastic modulus and creep equation for each section. Due to the
elastic modulus of tooth and bone are 1000-3000 times of the
PDL, the variation of materials properties settings for them
have a minimal effect on the FE result of the PDL. Therefore,
the tooth and alveolar bone were assumed to be linear elastic
and isotropic, the elastic modulus and Poisson's ratio were set
at 20 GPa and 0.3, 2 GPa and 0.3, respectively [22]. To find a
proper element size for each component, a mesh convergence
study for each element is completed and the results are shown
in **Table 1**. This produced the FE model consisting of 919627
tetrahedron elements in which the PDL had 402009 elements.
The contact between bone and PDL as well as between tooth
and PDL were defined as "tie" connection. Constraint in all
direction applied to the nodes located at the base of the
alveolar bone. A ramp load to the maximum of 1N was applied
at buccal-lingual direction to the midroot of crown and held
constant for 20 s.

Components | Element size |
---|---|

Tooth | 0.2 mm |

Alveolar bone | 0.2 mm |

PDL | 0.1 mm |

**Table 1.** The results of mesh convergence.

## Results

In the process of dealing with the data, some curves which fail
to fit the unloading part and defect curves were abandoned (i.e.
the PDL was glued to the indenter lead to the loading and
unloading become to a straight line). Due to the difference
between specimens and anisotropy of the PDL, it can be seen
that, obviously difference exist among load-depth curves when
testing the different samples at different levels with a same
loading velocity (**Figure 7**). However, in the same region of
different specimens, the variation of load-depth curves
performance is small relatively compared with the different
specimens (**Figure 8**). The mean value and standard deviations
of elastic modulus for the PDL at different levels from
specimens 1-4 were summarized in **Table 2**.

Cervical margin M ± SD | Midroot M ± SD | Apex M ± SD | |
---|---|---|---|

Specimen 1 | 0.53 ± 0.0758 | 0.11 ± 0.0274 | 0.48 ± 0.1065 |

Specimen 2 | 0.42 ± 0.0592 | 0.23 ± 0.0604 | 0.62 ± 0.3610 |

Specimen 3 | 0.28 ± 0.1032 | 0.19 ± 0.0187 | 0.54 ± 0.1342 |

Specimen 4 | 0.21 ± 0.0809 | 0.15 ± 0.0474 | 0.44 ± 0.1470 |

Average value | 0.36 ± 0.1542 | 0.17 ± 0.0853 | 0.52 ± 0.2425 |

**Table 2.** The average elastic modulus and standard deviations of PDL
for different levels along longitudinal direction (Unit: MPa).

A paired t-test was adopted to compare different levels.
Consequently, the elastic modulus showed a significant
difference along longitudinal direction. Moreover, a highly
significant difference was observed among the midroot,
cervical margin and apex (**Table 3**).

Analysis | P |
---|---|

Cervical margin vs. midroot |
0.0004 |

Cervical margin vs. apex |
0.028 |

Midroot vs. apex |
0.0003 |

**Table 3.** Results of statistical analyses of the PDL on different section
date. P<0.05: a significant difference; P<0.01: a highly significant
difference; NS: No Significant difference.

The elastic modulus in midroot (ranges from 0.11 to 0.23 MPa) was lower than cervical margin (ranges from 0.21 to 0.53 MPa) and apex (ranges from 0.44 to 0.62 MPa), among which the maximum value is 1.23 Mpa at apex on distal aspect, while the minimum value is 0.07 MPa at midroot on mesial aspect.

The creep and retardation time parameter were obtained from
good fitted curve (the squared correlation coefficients>0.99),
substituting these parameters into Equation 12, the J (t)
equations for different levels are shown. Variability of creep
compliance is apparent among the three levels (**Figure 9**). The
results show that creep compliance increase followed by apex,
cervical margin and midroot.

Cervical margin (t)=3.31+1.12 (1-e^{-t/6.51})+1.41 (1-e^{-t/
21.5})+0.483 (1-e^{-t/709.1})

Midroot J (t)=3.65+2.59 (1-e^{-t/4.36})+0.357 (1-e^{-t/44.53})+0.159
(1-e^{-t/355.62})

Apex J (t)=3.06+0.944 (1-e^{-t/6.99})+0.531 (1-e^{-t/30.61})+0.924 (1-
e^{-t/490.4})

According to the classic "pressure-tension" theory [23], the
major cause of tooth movement is compression and tension
within the periodontal tissue generated by external force, thus,
the maximum principle stress (P_{1}) is recorded for the two EF
models. Positive represents the tensile stress and negative
represents the compressive stress. The EF results cross the
PDL for linear and viscoelastic under a rapid growth load of 1
N are showed in **Figure 10**. It reveals that the stress distribution
of P_{1} exist distinction at cervical margin and apex for the two
material properties, especially in the maximum value areas. For
midroot, the stress distribution patterns are similar in trend but
different in magnitudes. At all location, the highest value of P_{1} for linear model was observed at the cervical margin on the lingual aspect and tapered more at apex than at the buccal
margin, this agrees with previous results [24]. Similar situation
can be observed with the viscoelastic model. However, the
zone of maximum value areas was smaller than linear model at
cervical margin and apex. The compressive stress magnitude is
smaller than the tensile stress on the opposite side. This is
corresponds with previous publication [25].

For the stage of constant force, three elements are selected
from each section (**Figure 10**), the strain variation of them
versus time is presented (**Figure 11**). It shows the different
strain variation at different stress level, the strain range of
element 1, element 2 and element 3 during creep segment is
about 0.049026-0.0490971 for a stress of 0.059 MPa, about
0.0125112-0.0125152 for a stress of 0.0039 MPa, about
0.0260272-0.026422 for a stress of 0.0365 MPa, respectively.

## Discussion

The above measurement details are influenced by different
factors that are difficult to judge quantitatively, in order to guarantee the measurements accuracy, the sample's storage and
experiment conditions were standardized. The order of
magnitude for measurement is very small, the experiment
result is strongly affected by surface conditions, therefore, to
make the test surface relatively smooth, the canine samples
were embedded in paraffin, which is convenient for fixing
positions and guaranteeing sample accuracy. During testing,
the PDL position is found through moving a low power micro
lens (10 times of magnification, **Figure 4B**), and then a high
power micro lens (50 times of magnification) is used to focus
and test after determining the position. If the sample's surface
roughness of the testing zone in the focalization process is less
than 5 μm, the focalization can be carried out to ensure the
reliability of the results to some extent.

A novel technology was used to evaluate the elastic modulus and creep compliance of the PDL at different levels. During testing, the dwell time prior to unloading was useful in predicting the flaw resistance of the viscoelastic materials [26]. In this analysis, the PDL was assumed to be a linear viscoelastic material at different levels for the PDL. The selected curve represents a typical indentation curve for flat punch, and the indentation depth denotes the majority testing depth at certain level. The results show that a generalized Kelvin model agrees well with the loading data for each level through nanoindentation method with a flat punch.

The PDL is a soft connective tissue which mainly consists of collagen fibers accounting for 50% of total mass. Additionally, it contains cells, blood vessels, nerves and ground substance matrix [27]. The principal fibers are composed of a large fiber bundle of collagen, its functions and sequence direction are not the same at different levels. The orientation of collagen fibers is more oblique along the midroot than one at cervical margin and apex [28], therefore, the elastic modulus and J (t) equations are different at different levels. Moreover, it is indicated that lower elastic modulus leads to higher creep compliance [29].

The present study demonstrates the difference that the indentation curves of the load-depth in different regions were determined by inherent property of the PDL. In this study, the elastic modulus of upper (cervical margin) and lower (apex) regions are higher than that of intermediate regions (midroot). This means that there has change stiffness along the axial direction for the PDL [7]. The reason may be that the percentage of blood vessels, nerves and ground substance is high in the midroot area compared to the other two areas [30].

The finite element method has been applied to dental biomechanical for a long time. In early stages, it was confined to the study of axisymmetric and two-dimensional problems [16]. Technology development is particularly attractive to a number of orthodontic investigators, since they can establish elaborative three-dimensional models and accurately simulate tooth movement when subjected to various orthodontic forces. Most of the previous computational models considered the PDL as a linear elastic, bilinear or nonlinear hyperelastic material [8,11,29]. However, teeth are always subjected to a long-term and lasting force during orthodontic treatment. Therefore, a viscoelastic model of time-dependent is more suitable to describe the PDL's mechanical properties for orthodontic treatment. Van et al. [31] reported that linear elastic model overestimate the PDL's stress, this is consistent with the FE results in this study. In reality, the PDL is anisotropic and non-uniform, we assumed the PDL as a uniform thickness of 0.2 mm that represent the PDL average value. Tom [24] verified that the stress distribution of the PDL with uniform and nonuniform are different in FE analysis.

In orthodontics, a low and continuous forces apply to the crown of a tooth is recommended that can cause steady tooth movement, whereas, heavy forces induce abrupt starts and stops of migration synchronous with collapse of necrosed bone and cementum [32,33]. Penedo et al. [34] study that the stress magnitude in the PDL exceeding capillary blood stress 2.6 KPa would cause onset of bone remodeling. Aoyama [35] suggested that bone resorption occurred on the compressive side was resulted from 4.9 KPa stress. However, the quantitative relationship between the PDL's stress and the bone resorption remains an open question.

The difference between linear elastic and viscoelastic for stress
distribution of the PDL at different levels through FE software
is compared in the rapid growth stage for the load. With the
constant force, the change of strain for the selected elements
from each section are exemplified, showing that higher loading
level lead to lager strain range, namely stronger creep effect.
This can be validated by the creep test, the creep curve of the
PDL at three different load levels is plotted in **Figure 12**. It is
found that the creep depth increases with the increase of force.
Moreover, the creep effect of the PDL is related not only with
inherent material property, but also with the applied stress.

## Conclusion

In this investigation, a novel method is used to evaluate the mechanical properties of the PDL at different levels based on a nanoindentation technique with a flat punch, the elastic modulus and the creep compliance for the PDL are obtained by loading and unloading curve, respectively. It is revealed that the minimum value of elastic modulus was in midroot and the maximum was in apex. A good fit was obtained between the generalized Kelvin model and test data. In addition, an accurate FE model is established to predict the stress distribution of the PDL for linear and viscoelastic. Under the constant force, the strain variation of the elements from each level is performed. The limitation of the FE model is without considering the non-uniform thickness for the PDL and the mechanical inter actions of adjacent teeth. Furthermore, a precise and complete model of the PDL needs different types of experimental data. Further research that determining a more accurate viscoelastic model is required based on creep and stress-relaxation test.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant # 51305208).

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