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Mechanobiological modeling can explain orthodontic tooth movement: Three case studies
Journal of Biomechanics 46 (2013) 470–477
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Mechanobiological modeling can explain orthodontic tooth movement: Three case studies A. Van Schepdael a,n, J. Vander Sloten a, L. Geris b,c a
Biomechanics Section, KU Leuven, Celestijnenlaan 300C, box 2419, 3001 Heverlee, Belgium Biomechanics Research Unit, U. Lie ge, Belgium c Prometheus division of Skeletal Tissue Engineering, KU Leuven, Belgium b
a r t i c l e i n f o
a b s t r a c t
Article history: Accepted 26 October 2012
Progress in medicine and higher expectation of quality of life has led to a higher demand for several dental and medical treatments. This increases the occurrence of situations in which orthodontic treatment is complicated by pathological conditions, medical therapies and drugs. Together with experiments, computer models might lead to a better understanding of the effect of pathologies and medical treatment on tooth movement. This study uses a previously presented mechanobiological model of orthodontic tooth displacement to investigate the effect of pathologies and (medical) therapies on the result of orthodontic treatment by means of three clinically relevant case studies looking at the effect of estrogen deficiency, the effect of OPG injections and the influence of fluoride intake. When less estrogen was available, the model predicted bone loss and a rise in the number of osteoclasts present at the compression side, and a faster bone resorption. These effects were also observed experimentally. Experiments disagreed on the effect of estrogen deficiency on bone formation, while the mechanobiological model predicted very little difference between the pathological and the non-pathological case at formation sites. The model predicted a decrease in tooth movement after OPG injections or fluoride intake, which was also observed in experiments. Although more experiments and model analysis is needed to quantitatively validate the mechanobiological model used in this study, its ability to conceptually describe several pathological conditions is an important measure for its validity. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Mechanobiology Mathematical model Tooth movement
1. Introduction Progress in medicine and higher expectation of quality of life have led to a higher demand for several dental and medical treatments (Rinchuse et al., 2007), making it more and more common for other medical conditions needing to be taken into account by the orthodontist when planning orthodontic treatment. The primary aim of orthodontic treatment is obtaining the correct occlusion, in order to improve chewing, esthetics and patient comfort. During treatment, tooth displacement is achieved by applying orthodontic forces to the tooth. Under the influence of these forces, the pressure side of the tooth root will experience bone resorption while bone formation will take place on the tension side. The coordination of these two processes through cellular communication results in permanent tooth displacement through the alveolar bone. Orthodontic treatment can
n
Corresponding author. Tel.: þ32 16 328997; fax: þ32 16 327994. E-mail address: [email protected] (A. Van Schepdael).
0021-9290/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2012.10.037
be complicated by different pathological conditions and medical treatments. Estrogen deficiency is a common cause of bone loss and an increased bone turnover rate, as demonstrated in a number of studies with rats, monkeys and humans (Cesnjaj et al., 1991; Parfitt et al., 1983; Whyte et al., 1982). Osteoporotic changes in the alveolar bone of rats were shown by Tanaka et al. (2002). The mechanism by which estrogen exerts its effect on bone remodeling is not entirely understood. However, a normal estrogen concentration may limit the size of the pre-osteoclast population by stimulating apoptosis (Rattanakul et al., 2003) and limit the number of osteoclasts further by decreasing the sensitivity of maturing osteoclasts to RANKL (Weitzmann and Pacifici, 2006). Since tooth movement is achieved through the resorption and formation of alveolar bone, pathologies affecting bone metabolism will most likely affect the outcome of orthodontic treatment. As OPG binds to RANKL, making it inert, it is known to prevent bone resorption. While the benefits of OPG injections in systemic conditions like osteoporosis are clear, there are also possible uses for RANKL inhibitors like OPG in orthodontic tooth movement. It is often necessary to minimize undesired movement of teeth
A. Van Schepdael et al. / Journal of Biomechanics 46 (2013) 470–477
which are for example to be used as an anchor unit. Sometimes, teeth are subjected to unnecessary, but unavoidable, forces generated by parts of the orthodontic appliances they come in contact with. Tooth movement should be avoided in those cases. The use of RANKL inhibitors may also be used to prevent teeth from returning to their original position after finishing orthodontic treatment (Kohno et al., 2005). The use of fluoride has already been a successful strategy for preventing tooth decay and caries; it is commonly added to tooth paste. In mineralized tissues, fluoride replaces the hydroxyl group of the hydroxy apatite crystals, forming fluoroapatite (Gonzales et al., 2011; Robinson and Kirkham, 1990). The structure of fluoroapatite is larger and less soluble then hydroxy apatite, making it more resistant to demineralization (Foo et al., 2007). Therefore, it is believed that systematic fluoride intake may prevent orthodontically induced root resorption (Gonzales et al., 2011; Foo et al., 2007), although it will probably decrease the rate of tooth movement. Together with experiments, computer models might lead to a better understanding of orthodontic treatment and the pathologies affecting the outcome. Most existing models (Mengoni and ˜ iz, 1998; Bourauel et al., 1999; Bourauel Ponthot, 2010; Alcan et al., 2000; Schneider et al., 2002; Soncini and Pietrabissa, 2002) describing tooth movement are based on an empirical bone remodeling function derived from experiments. Moreover, the biological activity in the PDL and the alveolar bone is not taken into account. Taking into account this biology makes the model more complex, but makes it possible to more accurately describe the process. The work presented in this article builds on the mechanobiological model of tooth movement previously presented by the authors (Van Schepdael et al., 2012) and investigates three case studies: the effect of estrogen deficiency, the effect of OPG injections and the influence of fluoride intake.
Table 1 Overview of the parameters of the mechanobiological model, their value, unit and origin. (1) Derived from Geris et al. (2008). (2) Derived from Pivonka et al. (2008). (3) Derived from Pfeilschifter et al. (1998). (4) Derived from Sandberg et al. (1988). Parameter
Value
Unit
Origin
Pms Qmd Pcs
3.42 3.6 2 13.55 2 10 0.54 2 0.18 3.27 10 0.3 3.06 551.6 0.7 0.25 1.06 10 1 0.11 6.13 6.03 0.1 1 100 3440 9.15 10.05 2.5 1.67 1 4.58 6.83 8.3 35 248.5 48.6 98 1 0.68
ml cells 1 day 1 ml cells 1 day 1 g cells 1 day 1 ml g 1 g cells 1 day 1 ml g 1 day 1 ml cells 1 day 1 cells ml 1 day 1 ng ml 1 [dimensionless] mm2 day 1 cells ng 1 day 1 day 1 mm2 day 1 ml g 1 day 1 [dimensionless] ml cells 1 day 1 mm2 day 1 ng cells 1 day 1 ml ng 1 ng cells 1 day 1 day 1 ng ml 1 day 1 ng cells 1 day 1 ml ng 1 day 1 ml ng 1 day 1 ng cells 1 day 1 mm2 day 1 ng cells 1 day 1 ml ng 1 day 1 [dimensionless] ml ng 1 MPa MPa MPa
From steady state conditions (1) (1) (1) (1) (1) (1) and stability analysis (1) and stability analysis (1) (1) (1) Estimated Estimated (2) Using life span of osteoclast (1) (1) Estimated (1) From steady state conditions (1) (1) (3), (4) Estimated (1) (2) (2) (2) (2) (2) Estimated Using molecular weight of OPG (2) (2) (2) From H2 (2)
kc Pcsf
kcf Ab0
ab db Y11 H11 mbt Cmh Y2 dl0 Df Af0 Afs
af df Dgb Ggb
ag Egb dgb Prs R1 dgr B1r B1o Egrf Dgo Pos
ko 2. Materials and methods 2.1. Model development The mechanobiological model consists of a set of nine coupled non-linear partial differential equations, of the taxis–diffusion–reaction (TDR) type. The equations describe the concentration of various cells, growth factors, cytokines and matrix-components. The periodontal ligament consists of collagen fibers (mc) and contains a large amount of fibroblasts (cf). The alveolar bone consists of mineralized collagen, with mm representing the degree of mineralization of the collagen. The bone has a small concentration of osteoblasts (cb) and osteoclasts (cl), constantly remodeling and renewing the bone. To coordinate bone remodeling, osteoclasts, osteoblasts and fibroblasts communicate through the RANKLRANK-OPG signaling pathway. In the model, RANKL (gr ¼grb þgrf) is produced by fibroblasts (grf) and osteoblasts (grb), while OPG (go) is produced by osteoblasts only. The osteogenic differentiation of mesenchymal stem cells into osteoblasts is regulated by active TGF-b (gb), also produced by osteoblasts and fibroblasts. Multinucleated osteoclasts are formed through the fusion of hematopoietic stem cells, which are present in the vascular matrix in the PDL and the bone. Fibroblasts are modeled to respond to mechanical stretching by producing the osteogenic growth factor TGF-b, along with other osteogenic factors of the TGF-b superfamily (Wescott et al., 2007; Kimoto et al., 1999; Marotti 2000; Pinkerton et al., 2008). The upregulation of the TGF-b production results in the appearance of a high number of osteoblasts in and around the PDL. This leads to bone formation in the tension zones. Fibroblasts respond to compression by upregulating the production of RANKL (Kanzaki et al., 2002; Nishijima et al., 2006; Yamaguchi et al., 2006; Krishnan and Davidovitch, 2009). This results in a higher number of osteoclasts, which start resorbing the alveolar bone, making it possible for the tooth to move. More information concerning the biological assumption made in this model can be found in Van Schepdael et al. Van Schepdael et al., (2012) as well as in the online Supplementary material, and a more comprehensive overview of the biology of tooth movement can be found in Garant (2003), Krishnan and Davidovitch (2009), Krishnan and Davidovitch (2006) and Henneman et al. (2008). The specific equations for all nine variables are represented below. More information on the parameters, equations and initial conditions can be found in Van Schepdael et al. (2012) and in the online Supplementary material. An overview of the origin and value of all parameters can be found in Table 1.Table 2
471
dgo D2 H2 A B C
Table 2 Initial values applied to the model domain.
mm mc cb cl cf gb grb grf go
PDL
Alveolar bone
Unit
0 1 0 0 1 0 0 0 0
0.9 0.075 3.2 2.3 0 2 2.9 0 6.2
[dimensionless] g ml 1 cells ml 1 cells ml 1 cells ml 1 ng ml 1 ng ml 1 ng ml 1 ng ml 1
shows the initial values of all variables in the PDL and the bone. @mc ¼ @t
P cs ½1kc mc cb |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
production by osteoblasts
@mm ¼ @t
Pms ½1mm cb |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
þ P csf ½1kcf mc cf |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð1Þ
production by fibroblasts
mineralisation by osteoblasts
Q md cl Hðmm Þ |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
@cb Y 11 g b ½1mm Hðmm mbt Þ þ Ab0 mm cb ½1ab cb db cb ¼ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |ffl{zffl} @t H11 þ g b |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} apoptosis proliferation differentiation from MSC
ð2Þ
demineralisation by osteoclasts
ð3Þ
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A. Van Schepdael et al. / Journal of Biomechanics 46 (2013) 470–477
@cl ¼ @t
Y2g |ffl{zffl}r
dl0 ½D2 þ H2 g b cl r ½C mh cl rmm |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
fusion from HSC’s
ð4Þ
attachment to bone matrix
apoptosis
@cf ¼ Af 0 ½1 þ Af s 9S9mc cf ½1af cf df mm cf þ r ½Df rcf F @t |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
ð5Þ
@g b ¼ @t
ð6Þ
apoptosis
proliferation
Ggb ½1ag g b cb |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
þ
production by osteoblasts
@g rb ¼ @t
Egb ½S HðSÞcf |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
production by fibroblasts
dgb g b þ r ½Dgb rg b |fflffl{zfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} denaturation
diffusion
dgr g rb B1r g rb g o |fflfflfflfflffl{zfflfflfflfflffl} |fflfflffl{zfflfflffl}
production by osteoblasts
ð7Þ
binding to OPG
denaturation
@g rf Df ¼ Egrf ½S ½HðSÞ1cf dgr g rf B1r g rf g o þ r g rf rcf df mm g rf @t c |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflffl{zfflffl} |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} f |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} apoptosis of fibroblasts production by fibroblasts denaturation binding to OPG moving with fibroblasts
ð8Þ @g o ¼ @t
P os ½1ko g o cb |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
production by osteoblasts
dgo g o |fflffl{zfflffl} denaturation
2.3. Numerical implementation
diffusion
g P rs 1 rb R1 cb |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
within will be stimulated in the same way. The parameter Sext was chosen to be Sext ¼ 0.3 MPa on the left side of the root, modeling bone formation, and Sext ¼ 2 MPa on the right side of the root, modeling bone resorption.
B1o g r g o |fflfflfflffl{zfflfflfflffl}
binding to RANKL
þ r ½Dgo rg o |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
ð9Þ
The nine variables in the model are non-negative and the numerical simulations must respect conservation of mass. The finite volumes technique was employed for its inherent mass conservation properties. The method of lines (MOL) was applied to separate the spatial and temporal discretization (Gerisch and Chaplain, 2006; Gerisch and Chaplain, 2008). For the time integration of the resulting system of ordinary differential equations, the ROWMAP time integrator (Weiner et al., 1997) was used. The model parameters and variables were nondimensionalized for the numerical calculations. A typical length scale during orthodontic tooth movement is the thickness of the periodontal ligament, L0 ¼ 0.2 mm and a typical time scale of T0 ¼ 1 day was chosen. A representative concentration of collagen content in the tissue is m0 ¼0.1 g/ml (Geris et al., 2008). Typical growth factor concentrations are in the order of magnitude of g0 ¼ 100 ng/ml, and a non-dimensionalization value of c0 ¼106 cells/ml was used for the cell densities (Geris et al., 2008). All results and parameter values mentioned in this article are presented in their undimensionalized value.
diffusion
The parameter S describes the mechanical stimulus to which the cells respond. A positive value of S implies tension, while a negative value occurs under compression. Furthermore, it is assumed that the cells in the PDL and the alveolar bone respond to the mechanical strains induced by the orthodontic appliance (Middleton et al., 1996; Kawarizadeh et al., 2004). Defining Sext as a variable related to the externally applied stress, the mechanical stimulus S becomes S ¼ Sext =E:
ð10Þ
where E is Young’s modulus, dependent upon the local matrix densities. The higher the degree of mineralization, the higher Young’s modulus of the bone (Eq. (11)). E ¼ Am2m þ Bmm þ C
ð11Þ
The parameters of the second order function is chosen in such a way that for mm ¼ 0 Young’s modulus equals EPDL ¼ 0.68 MPa (Provatidis 2001) and rises quickly once the mineralization surpasses mm ¼ 0.2 (see also Table 1).
2.4. Modeling of estrogen deficiency To examine the effect of estrogen on tooth movement, an estrogen deficiency was modeled and orthodontic tooth movement was simulated. The results were then compared to the non-pathological simulations. Both bone resorption and bone formation were studied. Estrogen limits the size of the pre-osteoclast population and decreases the sensitivity of osteoclasts to RANKL (Rattanakul et al., 2003; Weitzmann and Pacifici, 2006). In the mechanobiological model, these are both modeled by the parameter Y2, as can be seen from Eq. (4). Estrogen deficiency was thus modeled by increasing the parameter Y2. For the first 50 days of the simulation, no force was applied, allowing the system to reach its new osteoporotic equilibrium. At day 50, the parameter Sext was set to Sext ¼ 0.3 MPa for bone formation and Sext ¼ 2 MPa for bone resorption. The simulation was ended 30 days after load application.
2.2. Simulation details
2.5. Modeling of OPG injections
The numerical simulations were performed on a domain that consists of two rectangular parts that represent small sections of the tooth root as shown in Fig. 1A, and are located about halfway between the tooth crown and the tooth apex. The objective of the simulation is to model pure horizontal translation to the right and thus the part on the left side of the root will be used to model bone formation. The part on the right is a mirror image of the one on the left and is used to model bone resorption. At the start of the simulation, a distributed force is applied to the tooth root in a horizontal direction and to the right as shown in Fig. 1B. This results in a homogeneous stress distribution in the periodontal ligament and all the cells
To examine the effects of OPG on orthodontic tooth movement, OPG injections at the compression side were simulated and bone resorption was modeled. The results were then compared to a control case, in which no OPG injections were modeled. The OPG injection into the soft tissue adjacent to the resorption side, was modeled by applying an exponentially decreasing boundary condition (Seeherman et al., 2003). go ¼
7 X
d½tt i g inj Hðtt i Þ 0 e
ð12Þ
i¼1
u f
y
x
f
Alveolar Bone PDL
Model Domain
Tooth Root
Fig. 1. Schematic representation of the model domain. (A) The model domain consists of two rectangular parts that are located on the left and the right side of the tooth root. For the simulations, a horizontal translation u is applied to the root. The left part of the domain will thus experience bone formation, the right part experiences bone resorption. (B) Detail of the formation side of the model domain at the start of force application. The left side of the domain represents the alveolar bone, next to the alveolar bone is the PDL and the rest of the domain represents the tooth root. (C) Detail of the formation side of the model domain at the end of the simulation. Due to bone formation, more alveolar bone is now present, and the PDL has shifted to the right.
A. Van Schepdael et al. / Journal of Biomechanics 46 (2013) 470–477 Injections were administered at {ti} ¼{0,3,7,10,14,17,21} days and g inj 0 was given a value of g inj 0 ¼ 0:05 (non-dimensionalized value). An orthodontic force corresponding with Sext ¼ 2 MPa was applied for 50 days. 2.6. Modeling of fluoride intake To examine the effects of fluoride intake on tooth movement, a higher concentration of fluoroapatite in the alveolar bone was modeled and bone resorption was simulated. The simulation was run for several concentrations of fluoroapatite, and the rate of tooth movement was compared for all cases. In the mechanobiological model, the mineralization of the alveolar bone was modeled as shown by Eq. (2). In this equation, Pms denotes the rate at which osteoblasts can mineralize the tissue, while Qmd represents the rate of demineralization by osteoclasts. Since the replacement of hydroxy apatite by fluoride apatite makes the demineralization of the bone more difficult, an increase in fluoride intake was modeled by decreasing the parameter Qmd. An orthodontic force corresponding with Sext ¼ 2 MPa was applied for 21 days, and tooth movement was estimated for Qmd ¼ 36, Qmd ¼ 18, Qmd ¼ 8 and Qmd ¼ 3.6.
473
higher in the pathological case, before and during tooth movement (Fig. 3B). Due to this increased osteoclast concentration, the newly formed bone has a lower mineralization. The predicted speed of bone formation was not affected by the estrogen deficiency (Fig. 3A). 3.2. Predicted effect of OPG injections on tooth movement The model predicted a RANKL increase after force application in both cases, but the RANKL concentration is lower when OPG injections are modeled (Fig. 4B). As a consequence, the predicted increase in osteoclast density after force application is lower when OPG injections are modeled (Fig. 4A). Tooth movement was subsequently slowed down by OPG injections (Fig. 4C). 3.3. Predicted effect of fluoride intake on tooth movement
3. Results 3.1. Predicted effect of estrogen deficiency on tooth movement During the first 50 days of the simulation, the model predicted a general decrease in the mineralization of alveolar bone, as can be seen from Figs. 2A and 3A. At the resorption side, the osteoclast density rises after force application at day 50. This increase in osteoclast density is higher in the pathological case (Fig. 2B). Combined with the decreased bone mass, this leads to faster bone resorption. There is a slight increase in osteoblast density in the PDL at the compression side after force application in both the pathological and non-pathological case. The rise is slightly higher in the pathological case (Fig. 2C), but the predicted effect of estrogen deficiency on the osteoblast density at the resorption side is not as clear as the effect on osteoclasts at the resorption side. At the formation side, an increase in osteoblast density in the PDL was predicted after force application. No difference was predicted between the pathological and non-pathological case (Fig. 3C). The predicted osteoclast density at the tension side only rises slightly after force application at day 50 under normal conditions and stays constant in the pathological case. The total amount of osteoclasts predicted by the model is however always
The model predicted the amount of tooth movement recorded after 21 days to decrease from 0.3 mm to 0.18 mm when the demineralization rate was halved. Decreasing the demineralization rate tenfold (to Qmd ¼3.6) halted tooth movement completely (Table 3).
4. Discussion Although mechanobiological models have been used to simulate fracture healing, bone remodeling, tumor growth and soft tissue damage (see e.g. Geris et al., 2010a, 2010b; Roose et al., 2007 for a general overview), models of orthodontic tooth movement are primarily biomechanical, using empirical bone remodeling laws to estimate tooth movement. This study uses a previously presented mechanobiological model of orthodontic tooth displacement, incorporating cell densities and growth factor concentrations, to investigate the effect of pathologies and (medical) therapies on the result of orthodontic treatment by means of three case studies looking at the effect of estrogen deficiency, the effect of OPG injections and the influence of fluoride intake. For an extensive discussion on the assumptions and limitations of the mechanobiological model used in this study as well as the challenges
Fig. 2. Predicted effect of estrogen deficiency on bone resorption during tooth movement. (A) Predicted degree of mineralization on day 50 (before force application) and day 80 (30 days after force application) is shown for the simulation under normal conditions and under estrogen deficient conditions. Due to the nature of the boundary conditions and the applied force, the results of the simulation are essentially one-dimensional. For this reason the results are shown using a one-dimensional representation. The x-axis corresponds to the x-coordinate as defined in Fig. 1C. The sharp decrease shown in the figure corresponds to the boundary between alveolar bone (left) and the PDL (right). (B) Osteoclast density in the PDL during tooth movement at the compression side, under normal and estrogen deficient conditions. Note that the horizontal axis starts at day 40. (C) Osteoblast density in the PDL during tooth movement at the compression side under normal and estrogen deficient conditions. It should be noted that there is a large difference in scale between the vertical axes of (B) and (C).
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Fig. 3. Predicted effect of estrogen deficiency on bone formation during tooth movement. (A) Predicted degree of mineralization on day 50 (before force application) and day 80 (30 days after force application) is shown for the simulation under normal conditions and under estrogen deficient conditions. (B) Osteoclast density in the PDL during tooth movement at the tension side, under normal and estrogen deficient conditions. The oscillations present in the osteoclast density are a result of the chosen grid size and the post-processing, but did not influence the overall result. (C) Osteoblast density in the PDL during tooth movement at the tension side under normal and estrogen deficient conditions.
Fig. 4. Predicted effect of OPG injections on tooth movement. (A) Osteoclast density in the PDL during tooth movement at the compression side, under normal conditions and with OPG injections. (B) RANKL density in the PDL during tooth movement at the compression side, under normal conditions and with OPG injections. (C) Predicted rate of bone resorption under normal conditions and when OPG injections are modeled.
Table 3 Predicted effect of fluoride intake on orthodontic tooth movement. Predicted tooth displacement after 21 days is shown for different values of Qmd. Qmd represents the demineralization of the alveolar bone by osteoclasts, and high fluoride intake results in low values of Qmd. Qmd
Tooth movement (mm)
36 18 8 3.6
0.3 0.18 0.02 0
presented in developing such a model, the reader is referred to Van Schepdael et al. (2012). Estrogen deficiency was modeled by increasing the sensitivity of osteoclasts to RANKL and the size of the pre-osteoclast population. The model predicted a decrease in bone mass in an estrogen deficient situation (Fig. 2), as is noted generally among patients with estrogen deficiency (Cesnjaj et al., 1991; Parfitt et al., 1983; Whyte et al., 1982). The rate of tooth movement, determined by the rate of bone resorption (Masella and Meister 2006; Roberts et al., 2004), was predicted to increase, which was experimentally observed by both Yamashiro and Takano-Yamamoto (2001) and Arslan et al. (2007) (Fig. 5). They investigated the effect of estrogen
deficiency by monitoring tooth movement in ovariectomized rats. The rate of bone formation predicted by the model was unaffected by estrogen deficiency. At the compression side, the model predicted a higher concentration of osteoclasts and a slightly increased concentration of osteoblasts during resorption with respect to non-pathological conditions. At resorption sites, a significantly higher number of osteoclasts in the pathological case was noted by both Yamashiro and Takano-Yamamoto (2001) and Arslan et al. (2007). On the other hand, their observations on the number of osteoblasts disagreed, one showing an increase (Yamashiro and TakanoYamamoto, 2001), one observing a decrease (Arslan et al., 2007) in the pathological case with respect to the non-pathological case. At the tension side, the model predicted very little difference between the pathological and the non-pathological case. During tooth movement, the osteoblast concentration increased with the same amount in both cases. Although Yamashiro and TakanoYamamoto (2001) observed a non-significant increase in osteoblast density in ovariectomized rats compared to the nonpathological case, experiments conducted by Arslan et al. (2007) showed a decrease of the number of osteoblasts. The model presented in this study predicted an increase in osteoclast density in the PDL in homeostasis, which was confirmed experimentally (Weitzmann and Pacifici, 2006; Yamashiro and Takano-Yamamoto,
A. Van Schepdael et al. / Journal of Biomechanics 46 (2013) 470–477
475
Fig. 5. Comparison of model and experimental results concerning the effect of estrogen deficiency on tooth movement. (A) Model results for predicted rate of tooth movement comparing normal and estrogen deficient conditions. No tooth movement is seen in the first 50 days, before force application (not shown). Tooth movement starts about 3 days after force application at day 50. (B) Experimental results reproduced from Yamashiro and Takano-Yamamoto (2001), who investigated the effect of estrogen deficiency by monitoring tooth movement in ovariectomized rats. (C) Experimental results reproduced from Arslan et al. (2007), who investigated the effect of estrogen deficiency by monitoring tooth movement in ovariectomized rats.
Fig. 6. Comparison of model and experimental results concerning the effect of OPG injections on tooth movement. (A) Model results for predicted rate of tooth movement comparing tooth movement with and without OPG injections. (B) Experimental results reproduced from Dunn et al. (2007), showing experimentally observed tooth movement of rat molars. Average observations are shown for the control group, a group receiving injections of 0.5 mg/kg OPG and a group receiving injections of 5 mg/kg OPG.
2001). The predicted number of osteoclasts at the tension side increased slightly after force application in the non-pathological case, but stayed constant when estrogen deficiency was modeled. Although Yamashiro and Takano-Yamamoto (2001) noted an elevated osteoclast concentration at the formation side in ovariectomized rats, Arslan et al. (2007) did not confirm that observation. OPG injections were modeled by applying time-dependent boundary conditions for OPG. Dunn et al. (2007) experimentally investigated the effect of twice weekly OPG injections into the soft tissue adjacent to the resorption side of rat molars. They were able to inhibit molar movement by 78.7% with injections of 5.0 mg/kg OPG. Even with injections of 0.5 mg/kg, they observed a significant reduction in tooth movement (Fig. 6). By simulating similar conditions, this decrease was also predicted by the mechanobiological model. Compared to the control case, the model predicted a decrease in RANKL concentration and osteoclast density when OPG was injected. Dunn et al. (2007) also observed a decrease in osteoclast concentration at the compression side (Fig. 7). The increase was related to the dose of OPG administrated to the animals. High doses of OPG decreased the osteoclast concentration significantly. Fluoride intake was modeled by decreasing the rate at which osteoclasts can demineralize bone tissue. Gonzales et al. (2011) investigated the effect of fluoride intake from birth on
orthodontic tooth movement and root resorption. They recorded the amount of tooth movement occurring in rats after two weeks of force application. The rats were divided into four groups: a control group receiving no fluoride, and groups receiving daily fluoride doses for two, four and 12 weeks. Gonzales et al. (2011) concluded that there was an inverse relationship between the duration of the exposure to fluoride and the amount of tooth movement. A similar effect was also observed experimentally by Hellsing and Hammarstrom (1991). A similar relationship was predicted by the mechanobiological model (Fig. 8), since a low value of Qmd indicates the alveolar bone contains high concentrations of fluoroapatite and the duration of the exposure to fluoride is thus reflected by the value of the parameter Qmd. In this study three clinically relevant case studies of orthodontic tooth movement were studied. Estrogen deficiency, OPG injections and fluoride intake. The results obtained by the model are qualitatively in agreement with experimental observations, but a quantitative agreement could not be reached. This is due to several limitations and assumptions made during model development (Van Schepdael et al., 2012). The first of three major simplifications is the representation of the geometry of the tooth root. In this study, only a small part of the alveolar bone is modeled. However, the software used to implement the model allows more complex geometries, and the model domain can be
476
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Fig. 7. Comparison of model and experimental results concerning the effect of OPG injections on osteoclast density in the PDL during tooth movement. (A) Model results for predicted osteoclast densities (relative to the normal osteoclast density in bone) after 21 days, comparing tooth movement with and without OPG injections. (B) Experimental results reproduced from Dunn et al. (2007), showing observed osteoclast numbers. Average observations are shown for the control group, a group receiving injections of 0.5 mg/kg OPG and a group receiving injections of 5 mg/kg OPG.
Fig. 8. Comparison of model and experimental results concerning the effect of fluoride intake on tooth movement. (A) Model results for predicted tooth movement after 21 days of force application. Predicted tooth movement is shown for various values of Qmd. (B) Experimental results reproduced from Gonzales et al. (2011), showing experimentally observed tooth movement in rats. Results are shown for a control group, and groups receiving fluoride intake for two, four and 12 weeks.
expanded for further studies. Secondly, other factors such as NO, PGE2 and BMPs have an influence on orthodontic tooth movement. The modeled variables were selected for their relative importance in tooth movement as it is currently perceived (Wise and King, 2008). Finally, the model contains many parameters for which no values are available in literature. Often measuring those parameters is difficult or even impossible, and estimations have to be made, influencing the quantitative results (Fig. 9 in supplementary material). Although more experiments and model analysis is needed to quantitatively validate the mechanobiological model used in this study, its ability to conceptually describe several pathological conditions is an important measure for its validity. By changing the value of certain parameters or the nature of the boundary conditions, the model was able to qualitatively capture the effects of a number of pathological conditions and/or medical therapies on orthodontic tooth movement.
Conflict of interest statement All authors have no conflict of interest.
Acknowledgments An Van Schepdael is a research assistant of the Research Foundation Flanders (FWO-Vlaanderen). The authors gratefully acknowledge the technical assistance of Dr. Alf Gerisch.
Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jbiomech. 2012.10.037.
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Contents lists available at SciVerse ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Mechanobiological modeling can explain orthodontic tooth movement: Three case studies A. Van Schepdael a,n, J. Vander Sloten a, L. Geris b,c a
Biomechanics Section, KU Leuven, Celestijnenlaan 300C, box 2419, 3001 Heverlee, Belgium Biomechanics Research Unit, U. Lie ge, Belgium c Prometheus division of Skeletal Tissue Engineering, KU Leuven, Belgium b
a r t i c l e i n f o
a b s t r a c t
Article history: Accepted 26 October 2012
Progress in medicine and higher expectation of quality of life has led to a higher demand for several dental and medical treatments. This increases the occurrence of situations in which orthodontic treatment is complicated by pathological conditions, medical therapies and drugs. Together with experiments, computer models might lead to a better understanding of the effect of pathologies and medical treatment on tooth movement. This study uses a previously presented mechanobiological model of orthodontic tooth displacement to investigate the effect of pathologies and (medical) therapies on the result of orthodontic treatment by means of three clinically relevant case studies looking at the effect of estrogen deficiency, the effect of OPG injections and the influence of fluoride intake. When less estrogen was available, the model predicted bone loss and a rise in the number of osteoclasts present at the compression side, and a faster bone resorption. These effects were also observed experimentally. Experiments disagreed on the effect of estrogen deficiency on bone formation, while the mechanobiological model predicted very little difference between the pathological and the non-pathological case at formation sites. The model predicted a decrease in tooth movement after OPG injections or fluoride intake, which was also observed in experiments. Although more experiments and model analysis is needed to quantitatively validate the mechanobiological model used in this study, its ability to conceptually describe several pathological conditions is an important measure for its validity. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Mechanobiology Mathematical model Tooth movement
1. Introduction Progress in medicine and higher expectation of quality of life have led to a higher demand for several dental and medical treatments (Rinchuse et al., 2007), making it more and more common for other medical conditions needing to be taken into account by the orthodontist when planning orthodontic treatment. The primary aim of orthodontic treatment is obtaining the correct occlusion, in order to improve chewing, esthetics and patient comfort. During treatment, tooth displacement is achieved by applying orthodontic forces to the tooth. Under the influence of these forces, the pressure side of the tooth root will experience bone resorption while bone formation will take place on the tension side. The coordination of these two processes through cellular communication results in permanent tooth displacement through the alveolar bone. Orthodontic treatment can
n
Corresponding author. Tel.: þ32 16 328997; fax: þ32 16 327994. E-mail address: [email protected] (A. Van Schepdael).
0021-9290/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2012.10.037
be complicated by different pathological conditions and medical treatments. Estrogen deficiency is a common cause of bone loss and an increased bone turnover rate, as demonstrated in a number of studies with rats, monkeys and humans (Cesnjaj et al., 1991; Parfitt et al., 1983; Whyte et al., 1982). Osteoporotic changes in the alveolar bone of rats were shown by Tanaka et al. (2002). The mechanism by which estrogen exerts its effect on bone remodeling is not entirely understood. However, a normal estrogen concentration may limit the size of the pre-osteoclast population by stimulating apoptosis (Rattanakul et al., 2003) and limit the number of osteoclasts further by decreasing the sensitivity of maturing osteoclasts to RANKL (Weitzmann and Pacifici, 2006). Since tooth movement is achieved through the resorption and formation of alveolar bone, pathologies affecting bone metabolism will most likely affect the outcome of orthodontic treatment. As OPG binds to RANKL, making it inert, it is known to prevent bone resorption. While the benefits of OPG injections in systemic conditions like osteoporosis are clear, there are also possible uses for RANKL inhibitors like OPG in orthodontic tooth movement. It is often necessary to minimize undesired movement of teeth
A. Van Schepdael et al. / Journal of Biomechanics 46 (2013) 470–477
which are for example to be used as an anchor unit. Sometimes, teeth are subjected to unnecessary, but unavoidable, forces generated by parts of the orthodontic appliances they come in contact with. Tooth movement should be avoided in those cases. The use of RANKL inhibitors may also be used to prevent teeth from returning to their original position after finishing orthodontic treatment (Kohno et al., 2005). The use of fluoride has already been a successful strategy for preventing tooth decay and caries; it is commonly added to tooth paste. In mineralized tissues, fluoride replaces the hydroxyl group of the hydroxy apatite crystals, forming fluoroapatite (Gonzales et al., 2011; Robinson and Kirkham, 1990). The structure of fluoroapatite is larger and less soluble then hydroxy apatite, making it more resistant to demineralization (Foo et al., 2007). Therefore, it is believed that systematic fluoride intake may prevent orthodontically induced root resorption (Gonzales et al., 2011; Foo et al., 2007), although it will probably decrease the rate of tooth movement. Together with experiments, computer models might lead to a better understanding of orthodontic treatment and the pathologies affecting the outcome. Most existing models (Mengoni and ˜ iz, 1998; Bourauel et al., 1999; Bourauel Ponthot, 2010; Alcan et al., 2000; Schneider et al., 2002; Soncini and Pietrabissa, 2002) describing tooth movement are based on an empirical bone remodeling function derived from experiments. Moreover, the biological activity in the PDL and the alveolar bone is not taken into account. Taking into account this biology makes the model more complex, but makes it possible to more accurately describe the process. The work presented in this article builds on the mechanobiological model of tooth movement previously presented by the authors (Van Schepdael et al., 2012) and investigates three case studies: the effect of estrogen deficiency, the effect of OPG injections and the influence of fluoride intake.
Table 1 Overview of the parameters of the mechanobiological model, their value, unit and origin. (1) Derived from Geris et al. (2008). (2) Derived from Pivonka et al. (2008). (3) Derived from Pfeilschifter et al. (1998). (4) Derived from Sandberg et al. (1988). Parameter
Value
Unit
Origin
Pms Qmd Pcs
3.42 3.6 2 13.55 2 10 0.54 2 0.18 3.27 10 0.3 3.06 551.6 0.7 0.25 1.06 10 1 0.11 6.13 6.03 0.1 1 100 3440 9.15 10.05 2.5 1.67 1 4.58 6.83 8.3 35 248.5 48.6 98 1 0.68
ml cells 1 day 1 ml cells 1 day 1 g cells 1 day 1 ml g 1 g cells 1 day 1 ml g 1 day 1 ml cells 1 day 1 cells ml 1 day 1 ng ml 1 [dimensionless] mm2 day 1 cells ng 1 day 1 day 1 mm2 day 1 ml g 1 day 1 [dimensionless] ml cells 1 day 1 mm2 day 1 ng cells 1 day 1 ml ng 1 ng cells 1 day 1 day 1 ng ml 1 day 1 ng cells 1 day 1 ml ng 1 day 1 ml ng 1 day 1 ng cells 1 day 1 mm2 day 1 ng cells 1 day 1 ml ng 1 day 1 [dimensionless] ml ng 1 MPa MPa MPa
From steady state conditions (1) (1) (1) (1) (1) (1) and stability analysis (1) and stability analysis (1) (1) (1) Estimated Estimated (2) Using life span of osteoclast (1) (1) Estimated (1) From steady state conditions (1) (1) (3), (4) Estimated (1) (2) (2) (2) (2) (2) Estimated Using molecular weight of OPG (2) (2) (2) From H2 (2)
kc Pcsf
kcf Ab0
ab db Y11 H11 mbt Cmh Y2 dl0 Df Af0 Afs
af df Dgb Ggb
ag Egb dgb Prs R1 dgr B1r B1o Egrf Dgo Pos
ko 2. Materials and methods 2.1. Model development The mechanobiological model consists of a set of nine coupled non-linear partial differential equations, of the taxis–diffusion–reaction (TDR) type. The equations describe the concentration of various cells, growth factors, cytokines and matrix-components. The periodontal ligament consists of collagen fibers (mc) and contains a large amount of fibroblasts (cf). The alveolar bone consists of mineralized collagen, with mm representing the degree of mineralization of the collagen. The bone has a small concentration of osteoblasts (cb) and osteoclasts (cl), constantly remodeling and renewing the bone. To coordinate bone remodeling, osteoclasts, osteoblasts and fibroblasts communicate through the RANKLRANK-OPG signaling pathway. In the model, RANKL (gr ¼grb þgrf) is produced by fibroblasts (grf) and osteoblasts (grb), while OPG (go) is produced by osteoblasts only. The osteogenic differentiation of mesenchymal stem cells into osteoblasts is regulated by active TGF-b (gb), also produced by osteoblasts and fibroblasts. Multinucleated osteoclasts are formed through the fusion of hematopoietic stem cells, which are present in the vascular matrix in the PDL and the bone. Fibroblasts are modeled to respond to mechanical stretching by producing the osteogenic growth factor TGF-b, along with other osteogenic factors of the TGF-b superfamily (Wescott et al., 2007; Kimoto et al., 1999; Marotti 2000; Pinkerton et al., 2008). The upregulation of the TGF-b production results in the appearance of a high number of osteoblasts in and around the PDL. This leads to bone formation in the tension zones. Fibroblasts respond to compression by upregulating the production of RANKL (Kanzaki et al., 2002; Nishijima et al., 2006; Yamaguchi et al., 2006; Krishnan and Davidovitch, 2009). This results in a higher number of osteoclasts, which start resorbing the alveolar bone, making it possible for the tooth to move. More information concerning the biological assumption made in this model can be found in Van Schepdael et al. Van Schepdael et al., (2012) as well as in the online Supplementary material, and a more comprehensive overview of the biology of tooth movement can be found in Garant (2003), Krishnan and Davidovitch (2009), Krishnan and Davidovitch (2006) and Henneman et al. (2008). The specific equations for all nine variables are represented below. More information on the parameters, equations and initial conditions can be found in Van Schepdael et al. (2012) and in the online Supplementary material. An overview of the origin and value of all parameters can be found in Table 1.Table 2
471
dgo D2 H2 A B C
Table 2 Initial values applied to the model domain.
mm mc cb cl cf gb grb grf go
PDL
Alveolar bone
Unit
0 1 0 0 1 0 0 0 0
0.9 0.075 3.2 2.3 0 2 2.9 0 6.2
[dimensionless] g ml 1 cells ml 1 cells ml 1 cells ml 1 ng ml 1 ng ml 1 ng ml 1 ng ml 1
shows the initial values of all variables in the PDL and the bone. @mc ¼ @t
P cs ½1kc mc cb |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
production by osteoblasts
@mm ¼ @t
Pms ½1mm cb |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
þ P csf ½1kcf mc cf |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð1Þ
production by fibroblasts
mineralisation by osteoblasts
Q md cl Hðmm Þ |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
@cb Y 11 g b ½1mm Hðmm mbt Þ þ Ab0 mm cb ½1ab cb db cb ¼ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |ffl{zffl} @t H11 þ g b |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} apoptosis proliferation differentiation from MSC
ð2Þ
demineralisation by osteoclasts
ð3Þ
472
A. Van Schepdael et al. / Journal of Biomechanics 46 (2013) 470–477
@cl ¼ @t
Y2g |ffl{zffl}r
dl0 ½D2 þ H2 g b cl r ½C mh cl rmm |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
fusion from HSC’s
ð4Þ
attachment to bone matrix
apoptosis
@cf ¼ Af 0 ½1 þ Af s 9S9mc cf ½1af cf df mm cf þ r ½Df rcf F @t |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
ð5Þ
@g b ¼ @t
ð6Þ
apoptosis
proliferation
Ggb ½1ag g b cb |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
þ
production by osteoblasts
@g rb ¼ @t
Egb ½S HðSÞcf |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
production by fibroblasts
dgb g b þ r ½Dgb rg b |fflffl{zfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} denaturation
diffusion
dgr g rb B1r g rb g o |fflfflfflfflffl{zfflfflfflfflffl} |fflfflffl{zfflfflffl}
production by osteoblasts
ð7Þ
binding to OPG
denaturation
@g rf Df ¼ Egrf ½S ½HðSÞ1cf dgr g rf B1r g rf g o þ r g rf rcf df mm g rf @t c |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflffl{zfflffl} |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} f |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} apoptosis of fibroblasts production by fibroblasts denaturation binding to OPG moving with fibroblasts
ð8Þ @g o ¼ @t
P os ½1ko g o cb |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
production by osteoblasts
dgo g o |fflffl{zfflffl} denaturation
2.3. Numerical implementation
diffusion
g P rs 1 rb R1 cb |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
within will be stimulated in the same way. The parameter Sext was chosen to be Sext ¼ 0.3 MPa on the left side of the root, modeling bone formation, and Sext ¼ 2 MPa on the right side of the root, modeling bone resorption.
B1o g r g o |fflfflfflffl{zfflfflfflffl}
binding to RANKL
þ r ½Dgo rg o |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
ð9Þ
The nine variables in the model are non-negative and the numerical simulations must respect conservation of mass. The finite volumes technique was employed for its inherent mass conservation properties. The method of lines (MOL) was applied to separate the spatial and temporal discretization (Gerisch and Chaplain, 2006; Gerisch and Chaplain, 2008). For the time integration of the resulting system of ordinary differential equations, the ROWMAP time integrator (Weiner et al., 1997) was used. The model parameters and variables were nondimensionalized for the numerical calculations. A typical length scale during orthodontic tooth movement is the thickness of the periodontal ligament, L0 ¼ 0.2 mm and a typical time scale of T0 ¼ 1 day was chosen. A representative concentration of collagen content in the tissue is m0 ¼0.1 g/ml (Geris et al., 2008). Typical growth factor concentrations are in the order of magnitude of g0 ¼ 100 ng/ml, and a non-dimensionalization value of c0 ¼106 cells/ml was used for the cell densities (Geris et al., 2008). All results and parameter values mentioned in this article are presented in their undimensionalized value.
diffusion
The parameter S describes the mechanical stimulus to which the cells respond. A positive value of S implies tension, while a negative value occurs under compression. Furthermore, it is assumed that the cells in the PDL and the alveolar bone respond to the mechanical strains induced by the orthodontic appliance (Middleton et al., 1996; Kawarizadeh et al., 2004). Defining Sext as a variable related to the externally applied stress, the mechanical stimulus S becomes S ¼ Sext =E:
ð10Þ
where E is Young’s modulus, dependent upon the local matrix densities. The higher the degree of mineralization, the higher Young’s modulus of the bone (Eq. (11)). E ¼ Am2m þ Bmm þ C
ð11Þ
The parameters of the second order function is chosen in such a way that for mm ¼ 0 Young’s modulus equals EPDL ¼ 0.68 MPa (Provatidis 2001) and rises quickly once the mineralization surpasses mm ¼ 0.2 (see also Table 1).
2.4. Modeling of estrogen deficiency To examine the effect of estrogen on tooth movement, an estrogen deficiency was modeled and orthodontic tooth movement was simulated. The results were then compared to the non-pathological simulations. Both bone resorption and bone formation were studied. Estrogen limits the size of the pre-osteoclast population and decreases the sensitivity of osteoclasts to RANKL (Rattanakul et al., 2003; Weitzmann and Pacifici, 2006). In the mechanobiological model, these are both modeled by the parameter Y2, as can be seen from Eq. (4). Estrogen deficiency was thus modeled by increasing the parameter Y2. For the first 50 days of the simulation, no force was applied, allowing the system to reach its new osteoporotic equilibrium. At day 50, the parameter Sext was set to Sext ¼ 0.3 MPa for bone formation and Sext ¼ 2 MPa for bone resorption. The simulation was ended 30 days after load application.
2.2. Simulation details
2.5. Modeling of OPG injections
The numerical simulations were performed on a domain that consists of two rectangular parts that represent small sections of the tooth root as shown in Fig. 1A, and are located about halfway between the tooth crown and the tooth apex. The objective of the simulation is to model pure horizontal translation to the right and thus the part on the left side of the root will be used to model bone formation. The part on the right is a mirror image of the one on the left and is used to model bone resorption. At the start of the simulation, a distributed force is applied to the tooth root in a horizontal direction and to the right as shown in Fig. 1B. This results in a homogeneous stress distribution in the periodontal ligament and all the cells
To examine the effects of OPG on orthodontic tooth movement, OPG injections at the compression side were simulated and bone resorption was modeled. The results were then compared to a control case, in which no OPG injections were modeled. The OPG injection into the soft tissue adjacent to the resorption side, was modeled by applying an exponentially decreasing boundary condition (Seeherman et al., 2003). go ¼
7 X
d½tt i g inj Hðtt i Þ 0 e
ð12Þ
i¼1
u f
y
x
f
Alveolar Bone PDL
Model Domain
Tooth Root
Fig. 1. Schematic representation of the model domain. (A) The model domain consists of two rectangular parts that are located on the left and the right side of the tooth root. For the simulations, a horizontal translation u is applied to the root. The left part of the domain will thus experience bone formation, the right part experiences bone resorption. (B) Detail of the formation side of the model domain at the start of force application. The left side of the domain represents the alveolar bone, next to the alveolar bone is the PDL and the rest of the domain represents the tooth root. (C) Detail of the formation side of the model domain at the end of the simulation. Due to bone formation, more alveolar bone is now present, and the PDL has shifted to the right.
A. Van Schepdael et al. / Journal of Biomechanics 46 (2013) 470–477 Injections were administered at {ti} ¼{0,3,7,10,14,17,21} days and g inj 0 was given a value of g inj 0 ¼ 0:05 (non-dimensionalized value). An orthodontic force corresponding with Sext ¼ 2 MPa was applied for 50 days. 2.6. Modeling of fluoride intake To examine the effects of fluoride intake on tooth movement, a higher concentration of fluoroapatite in the alveolar bone was modeled and bone resorption was simulated. The simulation was run for several concentrations of fluoroapatite, and the rate of tooth movement was compared for all cases. In the mechanobiological model, the mineralization of the alveolar bone was modeled as shown by Eq. (2). In this equation, Pms denotes the rate at which osteoblasts can mineralize the tissue, while Qmd represents the rate of demineralization by osteoclasts. Since the replacement of hydroxy apatite by fluoride apatite makes the demineralization of the bone more difficult, an increase in fluoride intake was modeled by decreasing the parameter Qmd. An orthodontic force corresponding with Sext ¼ 2 MPa was applied for 21 days, and tooth movement was estimated for Qmd ¼ 36, Qmd ¼ 18, Qmd ¼ 8 and Qmd ¼ 3.6.
473
higher in the pathological case, before and during tooth movement (Fig. 3B). Due to this increased osteoclast concentration, the newly formed bone has a lower mineralization. The predicted speed of bone formation was not affected by the estrogen deficiency (Fig. 3A). 3.2. Predicted effect of OPG injections on tooth movement The model predicted a RANKL increase after force application in both cases, but the RANKL concentration is lower when OPG injections are modeled (Fig. 4B). As a consequence, the predicted increase in osteoclast density after force application is lower when OPG injections are modeled (Fig. 4A). Tooth movement was subsequently slowed down by OPG injections (Fig. 4C). 3.3. Predicted effect of fluoride intake on tooth movement
3. Results 3.1. Predicted effect of estrogen deficiency on tooth movement During the first 50 days of the simulation, the model predicted a general decrease in the mineralization of alveolar bone, as can be seen from Figs. 2A and 3A. At the resorption side, the osteoclast density rises after force application at day 50. This increase in osteoclast density is higher in the pathological case (Fig. 2B). Combined with the decreased bone mass, this leads to faster bone resorption. There is a slight increase in osteoblast density in the PDL at the compression side after force application in both the pathological and non-pathological case. The rise is slightly higher in the pathological case (Fig. 2C), but the predicted effect of estrogen deficiency on the osteoblast density at the resorption side is not as clear as the effect on osteoclasts at the resorption side. At the formation side, an increase in osteoblast density in the PDL was predicted after force application. No difference was predicted between the pathological and non-pathological case (Fig. 3C). The predicted osteoclast density at the tension side only rises slightly after force application at day 50 under normal conditions and stays constant in the pathological case. The total amount of osteoclasts predicted by the model is however always
The model predicted the amount of tooth movement recorded after 21 days to decrease from 0.3 mm to 0.18 mm when the demineralization rate was halved. Decreasing the demineralization rate tenfold (to Qmd ¼3.6) halted tooth movement completely (Table 3).
4. Discussion Although mechanobiological models have been used to simulate fracture healing, bone remodeling, tumor growth and soft tissue damage (see e.g. Geris et al., 2010a, 2010b; Roose et al., 2007 for a general overview), models of orthodontic tooth movement are primarily biomechanical, using empirical bone remodeling laws to estimate tooth movement. This study uses a previously presented mechanobiological model of orthodontic tooth displacement, incorporating cell densities and growth factor concentrations, to investigate the effect of pathologies and (medical) therapies on the result of orthodontic treatment by means of three case studies looking at the effect of estrogen deficiency, the effect of OPG injections and the influence of fluoride intake. For an extensive discussion on the assumptions and limitations of the mechanobiological model used in this study as well as the challenges
Fig. 2. Predicted effect of estrogen deficiency on bone resorption during tooth movement. (A) Predicted degree of mineralization on day 50 (before force application) and day 80 (30 days after force application) is shown for the simulation under normal conditions and under estrogen deficient conditions. Due to the nature of the boundary conditions and the applied force, the results of the simulation are essentially one-dimensional. For this reason the results are shown using a one-dimensional representation. The x-axis corresponds to the x-coordinate as defined in Fig. 1C. The sharp decrease shown in the figure corresponds to the boundary between alveolar bone (left) and the PDL (right). (B) Osteoclast density in the PDL during tooth movement at the compression side, under normal and estrogen deficient conditions. Note that the horizontal axis starts at day 40. (C) Osteoblast density in the PDL during tooth movement at the compression side under normal and estrogen deficient conditions. It should be noted that there is a large difference in scale between the vertical axes of (B) and (C).
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Fig. 3. Predicted effect of estrogen deficiency on bone formation during tooth movement. (A) Predicted degree of mineralization on day 50 (before force application) and day 80 (30 days after force application) is shown for the simulation under normal conditions and under estrogen deficient conditions. (B) Osteoclast density in the PDL during tooth movement at the tension side, under normal and estrogen deficient conditions. The oscillations present in the osteoclast density are a result of the chosen grid size and the post-processing, but did not influence the overall result. (C) Osteoblast density in the PDL during tooth movement at the tension side under normal and estrogen deficient conditions.
Fig. 4. Predicted effect of OPG injections on tooth movement. (A) Osteoclast density in the PDL during tooth movement at the compression side, under normal conditions and with OPG injections. (B) RANKL density in the PDL during tooth movement at the compression side, under normal conditions and with OPG injections. (C) Predicted rate of bone resorption under normal conditions and when OPG injections are modeled.
Table 3 Predicted effect of fluoride intake on orthodontic tooth movement. Predicted tooth displacement after 21 days is shown for different values of Qmd. Qmd represents the demineralization of the alveolar bone by osteoclasts, and high fluoride intake results in low values of Qmd. Qmd
Tooth movement (mm)
36 18 8 3.6
0.3 0.18 0.02 0
presented in developing such a model, the reader is referred to Van Schepdael et al. (2012). Estrogen deficiency was modeled by increasing the sensitivity of osteoclasts to RANKL and the size of the pre-osteoclast population. The model predicted a decrease in bone mass in an estrogen deficient situation (Fig. 2), as is noted generally among patients with estrogen deficiency (Cesnjaj et al., 1991; Parfitt et al., 1983; Whyte et al., 1982). The rate of tooth movement, determined by the rate of bone resorption (Masella and Meister 2006; Roberts et al., 2004), was predicted to increase, which was experimentally observed by both Yamashiro and Takano-Yamamoto (2001) and Arslan et al. (2007) (Fig. 5). They investigated the effect of estrogen
deficiency by monitoring tooth movement in ovariectomized rats. The rate of bone formation predicted by the model was unaffected by estrogen deficiency. At the compression side, the model predicted a higher concentration of osteoclasts and a slightly increased concentration of osteoblasts during resorption with respect to non-pathological conditions. At resorption sites, a significantly higher number of osteoclasts in the pathological case was noted by both Yamashiro and Takano-Yamamoto (2001) and Arslan et al. (2007). On the other hand, their observations on the number of osteoblasts disagreed, one showing an increase (Yamashiro and TakanoYamamoto, 2001), one observing a decrease (Arslan et al., 2007) in the pathological case with respect to the non-pathological case. At the tension side, the model predicted very little difference between the pathological and the non-pathological case. During tooth movement, the osteoblast concentration increased with the same amount in both cases. Although Yamashiro and TakanoYamamoto (2001) observed a non-significant increase in osteoblast density in ovariectomized rats compared to the nonpathological case, experiments conducted by Arslan et al. (2007) showed a decrease of the number of osteoblasts. The model presented in this study predicted an increase in osteoclast density in the PDL in homeostasis, which was confirmed experimentally (Weitzmann and Pacifici, 2006; Yamashiro and Takano-Yamamoto,
A. Van Schepdael et al. / Journal of Biomechanics 46 (2013) 470–477
475
Fig. 5. Comparison of model and experimental results concerning the effect of estrogen deficiency on tooth movement. (A) Model results for predicted rate of tooth movement comparing normal and estrogen deficient conditions. No tooth movement is seen in the first 50 days, before force application (not shown). Tooth movement starts about 3 days after force application at day 50. (B) Experimental results reproduced from Yamashiro and Takano-Yamamoto (2001), who investigated the effect of estrogen deficiency by monitoring tooth movement in ovariectomized rats. (C) Experimental results reproduced from Arslan et al. (2007), who investigated the effect of estrogen deficiency by monitoring tooth movement in ovariectomized rats.
Fig. 6. Comparison of model and experimental results concerning the effect of OPG injections on tooth movement. (A) Model results for predicted rate of tooth movement comparing tooth movement with and without OPG injections. (B) Experimental results reproduced from Dunn et al. (2007), showing experimentally observed tooth movement of rat molars. Average observations are shown for the control group, a group receiving injections of 0.5 mg/kg OPG and a group receiving injections of 5 mg/kg OPG.
2001). The predicted number of osteoclasts at the tension side increased slightly after force application in the non-pathological case, but stayed constant when estrogen deficiency was modeled. Although Yamashiro and Takano-Yamamoto (2001) noted an elevated osteoclast concentration at the formation side in ovariectomized rats, Arslan et al. (2007) did not confirm that observation. OPG injections were modeled by applying time-dependent boundary conditions for OPG. Dunn et al. (2007) experimentally investigated the effect of twice weekly OPG injections into the soft tissue adjacent to the resorption side of rat molars. They were able to inhibit molar movement by 78.7% with injections of 5.0 mg/kg OPG. Even with injections of 0.5 mg/kg, they observed a significant reduction in tooth movement (Fig. 6). By simulating similar conditions, this decrease was also predicted by the mechanobiological model. Compared to the control case, the model predicted a decrease in RANKL concentration and osteoclast density when OPG was injected. Dunn et al. (2007) also observed a decrease in osteoclast concentration at the compression side (Fig. 7). The increase was related to the dose of OPG administrated to the animals. High doses of OPG decreased the osteoclast concentration significantly. Fluoride intake was modeled by decreasing the rate at which osteoclasts can demineralize bone tissue. Gonzales et al. (2011) investigated the effect of fluoride intake from birth on
orthodontic tooth movement and root resorption. They recorded the amount of tooth movement occurring in rats after two weeks of force application. The rats were divided into four groups: a control group receiving no fluoride, and groups receiving daily fluoride doses for two, four and 12 weeks. Gonzales et al. (2011) concluded that there was an inverse relationship between the duration of the exposure to fluoride and the amount of tooth movement. A similar effect was also observed experimentally by Hellsing and Hammarstrom (1991). A similar relationship was predicted by the mechanobiological model (Fig. 8), since a low value of Qmd indicates the alveolar bone contains high concentrations of fluoroapatite and the duration of the exposure to fluoride is thus reflected by the value of the parameter Qmd. In this study three clinically relevant case studies of orthodontic tooth movement were studied. Estrogen deficiency, OPG injections and fluoride intake. The results obtained by the model are qualitatively in agreement with experimental observations, but a quantitative agreement could not be reached. This is due to several limitations and assumptions made during model development (Van Schepdael et al., 2012). The first of three major simplifications is the representation of the geometry of the tooth root. In this study, only a small part of the alveolar bone is modeled. However, the software used to implement the model allows more complex geometries, and the model domain can be
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Fig. 7. Comparison of model and experimental results concerning the effect of OPG injections on osteoclast density in the PDL during tooth movement. (A) Model results for predicted osteoclast densities (relative to the normal osteoclast density in bone) after 21 days, comparing tooth movement with and without OPG injections. (B) Experimental results reproduced from Dunn et al. (2007), showing observed osteoclast numbers. Average observations are shown for the control group, a group receiving injections of 0.5 mg/kg OPG and a group receiving injections of 5 mg/kg OPG.
Fig. 8. Comparison of model and experimental results concerning the effect of fluoride intake on tooth movement. (A) Model results for predicted tooth movement after 21 days of force application. Predicted tooth movement is shown for various values of Qmd. (B) Experimental results reproduced from Gonzales et al. (2011), showing experimentally observed tooth movement in rats. Results are shown for a control group, and groups receiving fluoride intake for two, four and 12 weeks.
expanded for further studies. Secondly, other factors such as NO, PGE2 and BMPs have an influence on orthodontic tooth movement. The modeled variables were selected for their relative importance in tooth movement as it is currently perceived (Wise and King, 2008). Finally, the model contains many parameters for which no values are available in literature. Often measuring those parameters is difficult or even impossible, and estimations have to be made, influencing the quantitative results (Fig. 9 in supplementary material). Although more experiments and model analysis is needed to quantitatively validate the mechanobiological model used in this study, its ability to conceptually describe several pathological conditions is an important measure for its validity. By changing the value of certain parameters or the nature of the boundary conditions, the model was able to qualitatively capture the effects of a number of pathological conditions and/or medical therapies on orthodontic tooth movement.
Conflict of interest statement All authors have no conflict of interest.
Acknowledgments An Van Schepdael is a research assistant of the Research Foundation Flanders (FWO-Vlaanderen). The authors gratefully acknowledge the technical assistance of Dr. Alf Gerisch.
Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jbiomech. 2012.10.037.
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