Theoretical numerical modeling of the oxygen diffusion effects within the periodontal ligament for orthodontic tooth displacement

Journal of Cellular Immunotherapy 4 (2018) 44–47

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Theoretical numerical modeling of the oxygen diffusion effects within the periodontal ligament for orthodontic tooth displacement Camille Spingarna, Delphine Wagnerb, Yves Rémonda, Daniel Georgea,∗ a b

ICUBE Laboratory, University of Strasbourg, CNRS, 67000, Strasbourg, France Dental Faculty, 8, rue Sainte Elisabeth, 67000, Strasbourg, France

ARTICLE INFO

ABSTRACT

Keywords: Oxygen concentration Osteoblast Orthodontics Bone remodelling Tooth movement

Orthodontic tooth movement results from alveolar bone remodelling around the tooth. With applied mechanical forces, the periodontal ligament and its vascularization are deformed, the blood flow changes and so the amount of oxygen import, provoking, through cell density variation, bone remodelling and tooth movement. A numerical finite element (FE) model is presented in which the amount of oxygen and the cells densities of osteoclasts and osteoblasts are deduced from the oxygen variation inside the deformed periodontal ligament, leading to bone remodelling. Predicting this evolution will help the orthodontists in their decision making for patient dependent treatments.

1. Introduction Bone is a living material in continual renewal. Trying to model its evolution has been going on for a long time at continuous and microstructural levels [1,2]. Although lately sophisticated theoretical numerical models were developed [3–13], precise understandings of the coupled multiphysical multiscale underlying physics are still difficult to comprehend [14–17]. During orthodontic treatment, the bone adapts to the applied mechanical forces [18,19] on the teeth. While in orthopaedics, mechanical unloading is usually related to boneresorption and mechanical loading with bone remodelling, in orthodontics, a compressed bone shows boneresorption in front of the tooth, while tension shows bone reconstruction at the back, allowing the displacement of the tooth [20]. The exact process of bone remodelling for orthodontic tooth movement is still unknown but it is usually accepted that the periodontal ligament (PDL) plays a crucial role as it is highly vascularized (about 10%) and impacts directly on the cell differentiation processes through the oxygen concentration variation. The important blood supply involves a significant turnover of cells [21]. Under low amplitude forces, the ligament's vascularization is partially deformed resulting in a cell density change and differentiation, and bone remodelling leading to the tooth motion. When the PDL is completely compressed (above a given force level) and no longer vascularized, necrosis of the surrounding tissue occurs [22]. Deformation of the ligament is therefore the key factor for a good orthodontic movement

[23]. It is known that angiogenesis plays a key role in osteogenesis, but it also influences the multiplication of osteoblasts (responsible of bone construction) and osteoclasts (responsible of bone resorption). Two studies [24,25] extracted osteoblastic and osteoclastic cells from rats and analysed their evolutions at different level of oxygen. They concluded that osteoclasts multiply in hypoxia [24] and osteoblasts multiply with increase of oxygen [25]. This can be linked to the oxygen concentration variation in the vascularized PDL and play a role in the cells differentiation leading to the bone remodelling process in orthodontics. Here, a theoretical numerical model is developed to predict cell multiplication in the PDL and predict bone remodelling initiation for orthodontic application. 2. Material and methods The developed model is implemented within the software ABAQUS [26] through a UMAT subroutine encoded in FORTRAN. The bone, the teeth and the ligament are considered homogeneous and isotropic with a linear elastic mechanical behaviour. Their material properties are extracted from the literature [27,28]. The vascularization of the ligament is considered homogeneously distributed throughout the volume leading to an oxygen concentration variation equal to the relative volume variation of the ligament as:

cO 2 = f

Vvasc (t ) Vinit

= f {Tr [ (t )]}

Peer review under responsibilfity of Shanghai Hengrun Biomedical Technology Research Institute. ∗ Corresponding author. E-mail address: [email protected] (D. George). https://doi.org/10.1016/j.jocit.2018.09.011

Available online 12 September 2018 2352-1775/ © 2018 Shanghai Hengrun Biomedical Technology Research Institute. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

(1)

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Table 1 Variation of cells percentage over oxygen concentration variation according to oxygen concentration calculated from experiments [24,25]. Oxygen concentration c O2 (%)

Variation of percentage of cells over variation of oxygen concentration osteoclasts variation (%) oxygen concentration variation

osteoblasts variation (%) oxygen concentration variation

0.2 to 1 1 to 2 2 to 5 5 to 12 12 to 20

0.075 0.40 −0.30 −0.31 −0.38

4.65 −2.16 3.32 0.21 0.88

0

0.2

sr

dosteoblasts (t0)

dosteoclasts (t0)

1.15

4.6

1.8/100 μm2

0.45/100 μm2

th tens

0.5

Tr ( ) th tens

th comp th comp

Tr ( ) <

th comp

if

Tr ( ) >

th tens

else

(2)

The threshold and are hypothesised with being the threshold compressive strain of the PDL below which no more oxygen variation is set, with a limit of oxygen concentration of 0%, and th tens being the threshold tensile strain of the PDL above which no more oxygen variation is set, and with a limit oxygen concentration of 20% [24,25]. In addition, we hypothesise that the oxygen concentration, and the cells concentration, varies proportionally with blood volume variation through the variable Tr ( ) . The cell density variations as a function of oxygen concentration are calculated from available experimental data [24,25]. These variations are presented in Table 1. Also, as defined by Arnett et al. and Utting et al. [24,25], assumption is made in the current work that there is equilibrium between osteoblasts and osteoclasts activity at the average oxygen concentration of 10% in the blood (normal physiological level). The bone remodelling process leading to the tooth displacement occurs at the interface between the bone and the PDL. Hence, only the cells located at this interface are considered active. An algorithm is developed (similar to the one developed in Ref. [29] for detecting the bone trabecular surface) to localize the cells close to the bone/PDL interface that enables to track its moving in space. To carry out the bone density evolution analysis, input data are required. Histology experiments were carried out on pig's PDL teeth to quantify the initial cells density distribution. An average density of 1.8 ( ± 3) osteoblasts per 100 μm2 was determined into the ligament. As it is defined that bone kinetic resorption is about four times faster than bone kinetic reconstruction [30], we assumed that the osteoclasts density is four times smaller than the osteoblasts one at 10% oxygen concentration. This leads to an average density of 0.45 osteoclasts per 100 μm2. Finally, the percentage of bone density variation at the interface is calculated from the given cells density and associated kinetics as: th comp (≤0)

Table 2 Definition of the parameters used in the current model. ss

0.2

cO 2 =

if

th comp

−0.5

d Fig. 1. Simplified 3D model of tooth within PDL and bone.

bone

dt

= ss . dosteoblasts

th tens

sr . dosteoclasts

th comp

(3)

where dosteoblasts and dosteoclasts are the number of active cells, ss and sr are respectively the velocity of forming and resorbing the bone density percentage per unit cell. The data used in the current model are resumed in Table 2. Tooth movement in orthodontics requires the application of both forces and moments to get the proper tooth displacement. This is a rather complex scenario to implement. Hence, a simpler analysis is developed with only a uniaxial force leading to a simple rotation of the tooth. To highlight the effects of oxygen impact on the bone remodelling process, a simplified 3D geometry model is developed (see Fig. 1)

Where cO2 is the oxygen concentration into the PDL, Vvasc is the variation of the volume of the PDL, Vinit its initial volume and Tr ( ) , the trace of the strain tensor. The function f is assumed to be with a linear variation and defined such that no deformation is set for an oxygen concentration of 10% (as the normal average concentration in the blood). The function f provides the oxygen concentration with:

Fig. 2. (a) PDL strain distribution, (b) variation of oxygen concentration, (c) variation of active osteoblasts density, and (d) variation of active osteoclasts density into the PDL with an applied force of 1.5 N on the teeth. 45

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C. Spingarn et al.

Fig. 3. Time evolution of bone density around the teeth with an orthodontic force of 1.5N (x100%).

characterize the oxygen variation within the PDL, hence the bone cells multiplication/activation process. From these, bone density evolution could be predicted for a simplified geometry through coupled mechanobiological parameters. These could, at term, be applied on real patients’ cases, to help the orthodontist procedures.

including bone, tooth and PDL. The thickness of the PDL is set as 0.2 mm as reported in the literature [31]. The orthodontic force is set at 1.5 N [18–20] as a surface tension. The external surface of the bone is fixed. The model is meshed with 8 noded linear bricks (C3D8 element of Abaqus). The model includes 9856 hexahedral elements and 10566 nodes.

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3. Results and discussion

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Fig. 2 shows, in the ligament and for an applied force of 1.5N, the developed strain, the oxygen concentration, active osteoclasts density, and active osteoblasts density. As expected, the largest variations are located at places where the PDL deformations are highest (compression and tension sides). Due to the low force applied and the defined hypotheses, the oxygen concentration shows a small variation, which is followed by corresponding variations of cell densities. It is possible that the oxygen concentration variation due to small strains may be more important in particular through the porosity of the vascular network. This needs to be addressed experimentally in future works. Nevertheless, the variations of cell densities generate, through the applied mechanical force, a variation in the cell activation process leading to the bone remodelling. Following equation [3], the results for the bone density evolution are presented in Fig. 3. As described previously, the bone density is resorbed in the compressed area (at the front). On the contrary, the bone is constructed in the area under tension (at the back). Although the experimental quantification of the model parameters need be extracted, results show the bone density variations with very small applied mechanical forces through oxygen variations. Here, the bone remodelling simulation is based not on strain energy density as in the usual bone remodelling numerical models [1–8], but only on biological quantities being determined via the applied mechanical forces. These could help in more detailed definition of multiphysics coupled mechanobiological theoretical-numerical models as for example in Ref. [11], and integrating more realistic load cases to help the orthodontists in the prediction of patient specific tooth movement. 4. Conclusion Orthodontists require predictive models to decide on optimum procedure to correct patients' malocclusions. The teeth realignment being oxygen PDL dependent, a theoretical numerical model was developed and implemented within a finite element numerical model to 46

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