**Feasible Simulation of Diseases Related to Bone Remodelling and of Their Treatment**

Václav Klika1,2 and František Maršík1

*1Institute of Thermomechanics, v.v.i., Academy of Sciences of Czech Republic 2Dept. of Mathematics, FNSPE, Czech Technical Univerzity in Prague Czech Republic* 

### **1. Introduction**

186 Theoretical Biomechanics

Yamaguchi, G., 2001. Dynamic Modeling of Musculoskeletal Motion: A Vectorized

Publishers, Boston.

Approach for Biomechanical Analysis in Three Dimensions, Kluwer Academic

A great advance in understanding mechanosensing and mechanotransduction of bone tissue has occurred in the past years. However, a clear answer is yet to come. There is plentiful evidence that most cells in human body are able to sense their mechanical environment, including osteoblasts and osteocytes. Li et al. found that marrow stromal cells change their proliferation rate and gene expression patterns in response to mechanical stimulation (Li et al, 2004). Ehrlich and Lanyon mention that osteocytes produce significantly higher levels of PGE2 (prostaglandin E2) and PGI2 (prostacylin) than osteoblasts and in vivo inhibition of prostaglandins prevents bone adaptation in response to mechanical strains (Ehrlich and Lanyon, 2002). Further, nitric oxide NO is also a mediator of mechanically induced bone formation. It is released, similarly, as prostanoids, in higher levels after exposure to physiological levels of mechanical loading. Moreover, Rubin discovered that dynamic loading down-regulates osteoclastic formation (Rubin et al, 2000) (more precisely, the strained bone cell downregulates its expression of RANKL (Rubin et al, 2004)), which suggests that mechanical forces play an important role in bone adaptation process.

Candidate mechanoreceptors within a cell are stretch-activated channels, integrins (membrane spanning proteins that couple the cell to its extracellular environment), connexins (membrane spanning proteins that form channels that allow the direct exchange of small molecules with adjacent cells - including intercellular communication via gap junctions), and membrane structure (Rubin, 2006). Nowadays, there is also a completely different possible explanation for transformation mechanical signal into a biochemical function. Valle et al. explain in their review that only a proper mechanical loading may lead to exposure of binding sites and thus enabling further biochemical processes to proceed (Valle, 2007). Most probably multiple mechanosensors are involved in receiving mechanical signals. Moreover, there are other studies showing the importance of many other mechanisms beside those mentioned above (Lemaire et al, 2004).

On tissue level, it is clear that mechanical loading is important in the bone remodelling process. Heřt described an interaction between the mechanical stimulation of local cells and the bone adaptation process in the 1970s (Heřt et al, 1972), Frost observed the same behaviour in his clinical praxis and summed it up in his "Utah paradigm" (Frost, 2004). More recently, the fundamental importance of dynamic loading was accepted. Comparison of the static versus the dynamic loading effects on bone remodelling is given in a nice and

Feasible Simulation of Diseases Related to Bone Remodelling and of Their Treatment 189

external biomechanical effects (stress changes). Our thermodynamic model enables to combine biological and biomechanical factors (Klika and Maršík, 2010). Such a model may also reflect changes in remodelling behaviour resulting from pathological changes to the bone metabolism or from hip joint replacement. However, it is a model and thus it is a great simplification of the complex process of bone remodelling. In this chapter, a more detailed description of biochemical control mechanisms will be added to the mentioned model (Klika and Maršík, 2010; Klika et al, 2010) which in turn leads to possibility to study several

In our previous work, the influence of mechanical stimulation on (chemical) interactions in general was studied and it was shown how to comprise this effect into a model of studied biochemical processes (Klika and Maršík, 2009; Klika, 2010). These findings were used to describe the bone remodelling phenomenon (Klika and Maršík, 2010; Klika et al, 2010; Maršík et al, 2010). In this chapter, an extension of the mentioned bone remodelling model (influences of concrete biochemical factors) will be presented where the essential

Firstly, fundamental control factors will be mentioned. As was mentioned in the introduction, the RANKL-RANK-OPG pathway is essential in the bone remodelling control. Osteoprotegerin (OPG) inhibits binding of ligand RANKL to receptor RANK and thus prevents osteoclastogenesis. Since osteoclasts are the only resorbing agents in bone, osteoprotegerin "protects bone" (osteo-protege). Further, one of the major problems connected to bone remodelling is a rapid bone loss after menopause that affects a significant portion of women after 50 years of age. Menopause is linked to a rapid decrease in estrogen levels. And because estrogen significantly affects bone density, it would be beneficial to be able to simulate the influence of estrogen levels on the bone remodelling process. Similarly, the parathyriod hormone PTH, tumour growth factor TGF-β1, and nitric oxide NO play a

PTH causes a release of calcium from the bone matrix and induces MNOC differentiation from precursor cells, estrogen has complex effects with final outcome in decreasing bone resorption by MNOC, calcitonin decreases levels of blood calcium by inhibiting MNOC function, and osteocalcin inhibits mineralisation (Sikavitsas et al, 2001). The discovery of the RANKL-RANK-OPG pathway enabled a more detailed study of the control mechanisms of bone remodelling. Robling et al. states that all PTH, PGE (prostaglandin), IL (interleukin), and vitamin D are "translated" by corresponding cells (osteoblasts) into RANKL levels (Robling et al, 2006). Further, nitric oxide NO is known to be a strong inhibitor of bone resorption and recently it has been known that it works in part by suppressing the expression of RANKL and, moreover, by promoting the expression of OPG (Robling et al, 2006). Both these effects eventually lead to a decrease of numbers of active osteoclasts MNOC, which in turn causes decrease of bone resorption. Kong and Penninger mention that the OPG expression is induced by estrogen (Kong and Penninger, 2000). Boyle et al. add that OPG production by osteoblasts is based on anabolic stimulation from TGF-β or estrogen (Boyle et al, 2003). Martin also deals with the question how hormones and cytokines influence contact-dependent regulation of MNOC by osteoblasts. He summaries results from the carried out experiments (mainly in vitro) that PTH, IL-11, and vitamin D (1.25(OH)2D3 more precisely) promotes RANKL

RANKL-RANK-OPG pathway mediates many of these above mention biochemical factors. Moreover, RANKL levels also reflect microcrack density. Hence, it is essential to incorporate

concrete bone related diseases using this model.

**1.2 Simulation of diseases and of their treatment** 

significance of dynamic loading will still be apparent.

significant role during the bone adaptation process.

formation, which in turn increases osteoclastogenesis (Martin, 2004).

inspectional review by Ehrlich and Lanyon (2002). Further, Robling provides experimental results that confirm the essential importance of the dynamic loading (Robling, 2006). It is worth mentioning that Heřt referred to this fact in his observations more than 35 years ago (Heřt et al, 1972).

From the phenomenological relations for the rate of chemical reaction based on classical irreversible thermodynamics including coupling with mechanical processes, it was shown that, although tissues are exposed to all variety of mechanical factors: straining, shear, pressure, and even dynamic electric fields, the volume variation rate is the most important mechanical stimulus driving the processes in them (Klika, 2010). However, we believe that shear rate might be important for triggering the process. (for details see (Klika, 2010; Klika and Maršík, 2009)). Thus, in the presented manuscript, the mechanical stimulation of bone remodelling was assumed to be proportional to rate of volume change. As it will be seen from the features and results of the model, it seems to capture the response of bone to changes of its mechanical environment on tissue level. Probably, the other ways of mechanosensing are controlling the triggering of bone remodelling process in a given loci. We realize that biochemical reactions are initiated and influenced primarily by genetic effects and then by external biomechanical effects (stress changes). The aim of the presented thermodynamic model is to combine biological and biomechanical factors whereas currently available models of bone remodelling focus only on one of these factors which is actually the reason why this model was developed. It should provide an estimate of the effects of increased physical activity on quality of bone even in several disease states. Such a model may also reflect changes in remodelling behaviour resulting from pathological changes to the bone metabolism or from hip joint replacement and also may help for better assessment of the risk of osteoporosis-related fractures (Lindsay, 2003). Preliminary version of the mentioned approach has been published by our team in the past but not until this considerably improved version was the model applicable to praxis (both qualitative and quantitative results) (Klika and Maršík, 2010; Klika et al., 2010) which is here being extended.

### **1.1 Available models of bone remodeling**

With the development of computer-aided strategies and based on the knowledge of bone geometry, applied forces, and elastic properties of the tissue, it may be possible to calculate the mechanical stress transfer inside the bone (Finite Elements analysis or FE analysis). The change of stresses is followed by a change in internal bone density distribution. This allows to formulate mathematical models that can be used to study functional adaptation quantitatively and furthermore, to create the bone density distribution patterns (Beaupré et al, 1990; Carter, 1987). Such mathematical models have been built in the past. Since they calculate just mechanical transmission inside the bone and not considering cell-biologic factors of bone physiology, they just partially correspond to the reality seen in living organisms. Basically, there are essentially two groups of models for bone remodelling. One assumes that the mechanical loading is the dominant effect, almost to the exclusion of other factors, and treatment of biochemical effects are included in parameter with no physical interpretation (e.g. Beaupré et al, 1990; Carter, 1987; Huiskes et al, 1987; Ruimerman et al, 2005; Turner et al, 1997). The results or predictions of these models yield the correct density distribution patterns in physiological cases. However, they have a limited ability to simulate disease. The second group, the biochemical models, consider control mechanisms of bone adaptation in great detail, but with limited possibilities for including mechanical effects that are known to be essential (Komarova et al, 2003; Lemaire et al, 2004). We realize that biochemical reactions are initiated and influenced primarily by genetic effects and then by external biomechanical effects (stress changes). Our thermodynamic model enables to combine biological and biomechanical factors (Klika and Maršík, 2010). Such a model may also reflect changes in remodelling behaviour resulting from pathological changes to the bone metabolism or from hip joint replacement. However, it is a model and thus it is a great simplification of the complex process of bone remodelling. In this chapter, a more detailed description of biochemical control mechanisms will be added to the mentioned model (Klika and Maršík, 2010; Klika et al, 2010) which in turn leads to possibility to study several concrete bone related diseases using this model.

### **1.2 Simulation of diseases and of their treatment**

188 Theoretical Biomechanics

inspectional review by Ehrlich and Lanyon (2002). Further, Robling provides experimental results that confirm the essential importance of the dynamic loading (Robling, 2006). It is worth mentioning that Heřt referred to this fact in his observations more than 35 years ago

From the phenomenological relations for the rate of chemical reaction based on classical irreversible thermodynamics including coupling with mechanical processes, it was shown that, although tissues are exposed to all variety of mechanical factors: straining, shear, pressure, and even dynamic electric fields, the volume variation rate is the most important mechanical stimulus driving the processes in them (Klika, 2010). However, we believe that shear rate might be important for triggering the process. (for details see (Klika, 2010; Klika and Maršík, 2009)). Thus, in the presented manuscript, the mechanical stimulation of bone remodelling was assumed to be proportional to rate of volume change. As it will be seen from the features and results of the model, it seems to capture the response of bone to changes of its mechanical environment on tissue level. Probably, the other ways of mechanosensing are controlling the triggering of bone remodelling process in a given loci. We realize that biochemical reactions are initiated and influenced primarily by genetic effects and then by external biomechanical effects (stress changes). The aim of the presented thermodynamic model is to combine biological and biomechanical factors whereas currently available models of bone remodelling focus only on one of these factors which is actually the reason why this model was developed. It should provide an estimate of the effects of increased physical activity on quality of bone even in several disease states. Such a model may also reflect changes in remodelling behaviour resulting from pathological changes to the bone metabolism or from hip joint replacement and also may help for better assessment of the risk of osteoporosis-related fractures (Lindsay, 2003). Preliminary version of the mentioned approach has been published by our team in the past but not until this considerably improved version was the model applicable to praxis (both qualitative and quantitative results) (Klika and

With the development of computer-aided strategies and based on the knowledge of bone geometry, applied forces, and elastic properties of the tissue, it may be possible to calculate the mechanical stress transfer inside the bone (Finite Elements analysis or FE analysis). The change of stresses is followed by a change in internal bone density distribution. This allows to formulate mathematical models that can be used to study functional adaptation quantitatively and furthermore, to create the bone density distribution patterns (Beaupré et al, 1990; Carter, 1987). Such mathematical models have been built in the past. Since they calculate just mechanical transmission inside the bone and not considering cell-biologic factors of bone physiology, they just partially correspond to the reality seen in living organisms. Basically, there are essentially two groups of models for bone remodelling. One assumes that the mechanical loading is the dominant effect, almost to the exclusion of other factors, and treatment of biochemical effects are included in parameter with no physical interpretation (e.g. Beaupré et al, 1990; Carter, 1987; Huiskes et al, 1987; Ruimerman et al, 2005; Turner et al, 1997). The results or predictions of these models yield the correct density distribution patterns in physiological cases. However, they have a limited ability to simulate disease. The second group, the biochemical models, consider control mechanisms of bone adaptation in great detail, but with limited possibilities for including mechanical effects that are known to be essential (Komarova et al, 2003; Lemaire et al, 2004). We realize that biochemical reactions are initiated and influenced primarily by genetic effects and then by

Maršík, 2010; Klika et al., 2010) which is here being extended.

**1.1 Available models of bone remodeling** 

(Heřt et al, 1972).

In our previous work, the influence of mechanical stimulation on (chemical) interactions in general was studied and it was shown how to comprise this effect into a model of studied biochemical processes (Klika and Maršík, 2009; Klika, 2010). These findings were used to describe the bone remodelling phenomenon (Klika and Maršík, 2010; Klika et al, 2010; Maršík et al, 2010). In this chapter, an extension of the mentioned bone remodelling model (influences of concrete biochemical factors) will be presented where the essential significance of dynamic loading will still be apparent.

Firstly, fundamental control factors will be mentioned. As was mentioned in the introduction, the RANKL-RANK-OPG pathway is essential in the bone remodelling control. Osteoprotegerin (OPG) inhibits binding of ligand RANKL to receptor RANK and thus prevents osteoclastogenesis. Since osteoclasts are the only resorbing agents in bone, osteoprotegerin "protects bone" (osteo-protege). Further, one of the major problems connected to bone remodelling is a rapid bone loss after menopause that affects a significant portion of women after 50 years of age. Menopause is linked to a rapid decrease in estrogen levels. And because estrogen significantly affects bone density, it would be beneficial to be able to simulate the influence of estrogen levels on the bone remodelling process. Similarly, the parathyriod hormone PTH, tumour growth factor TGF-β1, and nitric oxide NO play a significant role during the bone adaptation process.

PTH causes a release of calcium from the bone matrix and induces MNOC differentiation from precursor cells, estrogen has complex effects with final outcome in decreasing bone resorption by MNOC, calcitonin decreases levels of blood calcium by inhibiting MNOC function, and osteocalcin inhibits mineralisation (Sikavitsas et al, 2001). The discovery of the RANKL-RANK-OPG pathway enabled a more detailed study of the control mechanisms of bone remodelling. Robling et al. states that all PTH, PGE (prostaglandin), IL (interleukin), and vitamin D are "translated" by corresponding cells (osteoblasts) into RANKL levels (Robling et al, 2006). Further, nitric oxide NO is known to be a strong inhibitor of bone resorption and recently it has been known that it works in part by suppressing the expression of RANKL and, moreover, by promoting the expression of OPG (Robling et al, 2006). Both these effects eventually lead to a decrease of numbers of active osteoclasts MNOC, which in turn causes decrease of bone resorption. Kong and Penninger mention that the OPG expression is induced by estrogen (Kong and Penninger, 2000). Boyle et al. add that OPG production by osteoblasts is based on anabolic stimulation from TGF-β or estrogen (Boyle et al, 2003). Martin also deals with the question how hormones and cytokines influence contact-dependent regulation of MNOC by osteoblasts. He summaries results from the carried out experiments (mainly in vitro) that PTH, IL-11, and vitamin D (1.25(OH)2D3 more precisely) promotes RANKL formation, which in turn increases osteoclastogenesis (Martin, 2004).

RANKL-RANK-OPG pathway mediates many of these above mention biochemical factors. Moreover, RANKL levels also reflect microcrack density. Hence, it is essential to incorporate

$$\begin{array}{ll} RANKL + RANK & \stackrel{\mathbf{k}\_{\pm 1}}{\leftrightarrow} RR, \\ RANKL + OPG & \stackrel{\mathbf{k}\_{\pm 2}}{\leftrightarrow} RO\_{\text{Inactive}}. \end{array} \tag{1}$$

$$\begin{split} \frac{d\mathbf{n}\_{RANKL}}{d\tau} &= -n\_{RANKL} (\mathcal{J}\_{\text{RK}}^{\text{ROO}} + n\_{RANKL} - n\_{OPG}) + \\ &+ \delta\_{-1}^{\text{ROO}} (\mathcal{J}\_{\text{RR}}^{\text{ROO}} - n\_{RANKL} + n\_{OPG}) - \\ &- \delta\_{+2}^{\text{ROO}} n\_{RANKL} n\_{OPG} + \delta\_{-2}^{\text{ROO}} (\mathcal{J}\_{\text{RO}}^{\text{ROO}} - n\_{OPG}), \\ \frac{d\mathbf{n}\_{OPG}}{d\tau} &= -\delta\_{+2}^{\text{ROO}} n\_{RANKL} n\_{OPG} + \delta\_{-2}^{\text{ROO}} (\mathcal{J}\_{\text{RO}}^{\text{ROO}} - n\_{OPG}), \end{split} \tag{2}$$

$$\begin{array}{lcl} \mathcal{S}\_{-1}^{\text{ROO}} &=& \frac{k\_{-1}}{k\_{+1}[\text{RANKL}\_{\text{stand}}]},\\ \mathcal{S}\_{-2}^{\text{ROO}} &=& \frac{k\_{-2}}{k\_{+1}[\text{RANKL}\_{\text{stand}}]},\\ \mathcal{S}\_{+2}^{\text{ROO}} &=& \frac{k\_{+2}}{k\_{+1}},\\ \mathcal{S}\_{\text{ROO}}^{\text{ROO}} &=& \frac{\mathcal{C}\_{\text{RO}}}{[\text{RANKL}\_{\text{stand}}]} = \frac{[\text{RO}\_{0}] + [\text{OPG}\_{0}]}{[\text{RANKL}\_{\text{stand}}]},\\ \mathcal{P}\_{\text{RR}}^{\text{ROO}} &=& \frac{\mathcal{C}\_{\text{RR}}}{[\text{RANKL}\_{\text{stand}}]} = \frac{[\text{RR}\_{0}] + [\text{RANKL}\_{0}] - [\text{OPG}\_{0}]}{[\text{RANKL}\_{\text{stand}}]}. \end{array} \tag{3}$$

$$
\dot{\mathfrak{x}} = -A\mathfrak{x}^2 - B\mathfrak{x} + \mathcal{C}, \quad A > 0, \mathcal{C} > 0,\tag{4}
$$

$$\begin{split} \chi(\mathbf{r}) &= \left[ \frac{2A}{\sqrt{B^2 + 4AC}} \left( 1 + \frac{1 + \frac{2A}{\sqrt{B^2 + 4AC}} \left( \mathbf{x}\_0 + \frac{B}{2A} \right)}{1 - \frac{2A}{\sqrt{B^2 + 4AC}} \left( \mathbf{x}\_0 + \frac{B}{2A} \right)} e^{\sqrt{B^2 + 4AC} \mathbf{r}} \right) \right]^{-1} . \\ &\cdot \left[ \left( 1 - \frac{B}{\sqrt{B^2 + 4AC}} \right) \frac{1 + \frac{2A}{\sqrt{B^2 + 4AC}} \left( \mathbf{x}\_0 + \frac{B}{2A} \right)}{1 - \frac{2A}{\sqrt{B^2 + 4AC}} \left( \mathbf{x}\_0 + \frac{B}{2A} \right)} e^{\sqrt{B^2 + 4AC} \mathbf{r}} - \frac{B}{\sqrt{B^2 + 4AC}} - 1 \right]. \end{split} (5)$$

$$
\delta\_{-1}^{\text{RRO}}, \mathfrak{r}\_{7\text{days}}^{\text{RRO}}, \mathfrak{n}\_{\text{RK}\_{\text{0}}}, \mathfrak{n}\_{\text{RR}\_{\text{0}}\ell}
$$

$$\begin{aligned} \tau\_{7\text{days}}^{\text{RRO}} &= t k\_{+1} \mathcal{C}\_{\text{RR}}|\_{t=7\text{days}} = 6 \, 10^5 \, 10^7 \, 10^{-12} \approx 10^0 \\ n\_{\mathcal{R}R\_0} &\approx 10^0, \\ n\_{\mathcal{R}K\_0} &\approx 10^0, \\ \mathcal{S}\_{-1^{\text{RRO}}} &= \frac{k\_{-1}}{k\_{+1} [\text{RANKL}\_{\text{stand}}]} \approx k\_{-1} \mathbf{1} \, \mathbf{0}^5, \end{aligned}$$

$$
\delta\_{-1}^{\rm RRO} = 4.9210^{-6}, \tau\_{7\rm days}^{\rm RRO} = 4.64, n\_{\rm RK\_0} = 1.037, n\_{\rm RR\_0} = 0.0947. \tag{6}
$$

$$2^{\text{nd}} \text{reaction in (1)} \rightarrow [\text{OPG}] (t).$$

$$1^{\text{st}} \text{reaction of (1)} \rightarrow \text{[RR]} [\tau\_{\text{7daws}}]$$

$$
\delta\_{-2}^{\mathrm{RRO}}, \delta\_{+2}^{\mathrm{RRO}}, \tau\_{\mathrm{OPG}}^{\mathrm{RRO}}, n\_{RO\_0},
$$

$$
\delta\_{-2}^{\rm RRO} = 5.86 \, 10^{-19}, \delta\_{+2}^{\rm RRO} = 12.96, \tau\_{\rm OPG}^{\rm RRO} = 11.36, n\_{\rm RO\_0} = 6.135. \tag{7}
$$

$$\begin{aligned} \delta\_{\Sigma\_{1}^{\text{RO}}}^{\text{RO}} &= \frac{k\_{-1}}{k\_{\text{-1}}[\text{RANKL}\_{\text{stand}}]} = 4.9210^{-6}, \\ \delta\_{\Sigma\_{2}^{\text{RO}}}^{\text{RO}} &= \frac{k\_{-2}}{k\_{\text{-1}}[\text{RANKL}\_{\text{stand}}]} = 5.8610^{-19}, \\ \delta\_{\Sigma\_{2}^{\text{RO}}}^{\text{RO}} &= \frac{k\_{\text{-2}}}{k\_{\text{+1}}} = 12.96, \\ \delta\_{\text{RO}}^{\text{RO}} &= \frac{\mathcal{C}\_{\text{RO}}}{[\text{RANKL}\_{\text{stand}}]} = \frac{[\text{RO}\_{0}] + [\text{OPG}\_{0}]}{[\text{RANKL}\_{\text{stand}}]} = 6.135 + n\_{\text{OPG}\_{0}}, \\ \delta\_{\text{RR}}^{\text{RO}} &= \frac{\mathcal{C}\_{\text{RR}}}{[\text{RANKL}\_{\text{stand}}]} = \frac{[\text{RR}\_{0}] + [\text{RANKL}\_{0}] - [\text{OPG}\_{0}]}{[\text{RANKL}\_{\text{stand}}]} = 0.0947 + n\_{\text{RANKL}\_{0}} - [\text{OPG}\_{0}], \\ \delta\_{\text{RR}}^{\text{RO}} &= \frac{\mathcal{C}\_{\text{RO}}}{[\text{RANKL}\_{\text{stand}}]} = \frac{[\text{RR}\_{0}] - [\text{RANKL}\_{0}] + [\text{OPG}\_{0}]}{[\text{RANKL}\_{\text{stand}}]} = 1.037 - n\_{\text{RANKL}\_{0}} + n\_{\text{OPG}\_{0}}, \\ \delta\_{\text{R}}^{\text{RO}} &= \frac{\eta\_{\text{R$$


$$
\beta\_1 = 1.41/0.79n\_{RR} - 0.81,\tag{9}
$$

$$\text{PTH} + \text{RANKL}\_{\text{roducers}} + \text{Substratum} \overset{\text{h} \approx 1}{\longleftrightarrow} \text{RANKL} + \text{RANKL}\_{\text{roducers}} \tag{10}$$

$$\frac{\text{d[PTH]}}{\text{d}\tau} = -[\text{PTH}](\mathcal{J}\_{\text{Substr}}^{\text{PTH}} + [\text{PTH}]) + \delta\_{-1}^{\text{PTH}}(\mathcal{J}\_{\text{RANKL}}^{\text{PTH}} - [\text{PTH}]),\tag{11}$$

$$\begin{array}{l} \mathcal{S}\_{-1}^{\text{PTH}} &= \frac{k\_{-1}}{k\_{+1} \text{[RANKL\_{stand}]}},\\ \mathcal{B}\_{\text{RANKL}}^{\text{PTH}} &= \frac{\mathcal{C}\_{\text{RANKL}}}{\text{[RANKL\_{stand}]}} = \frac{\text{[RANKL\_0]} + \text{[PTH}\_0]}{\text{[RANKL\_{stand}]}},\\ \mathcal{B}\_{\text{Substr}}^{\text{PTH}} &= \frac{\mathcal{C}\_{\text{Substr}}}{\text{[RANKL\_{stand}]}} = \frac{\text{[Substr}\_0] - \text{[PTH}\_0]}{\text{[RANKL\_{stand}]}}. \end{array} \tag{12}$$

$$[\text{RANKL}] = \beta\_{\text{RANKL}}^{\text{PTH}} - [\text{PTH}].$$

$$
\delta\_{-1}^{\rm PTH} = 0.145, \tau\_{10 \text{days}}^{\rm PTH} = 26.17, \eta\_{\text{Substr}\_{\alpha}}^{\rm PTH} = 0.018. \tag{13}
$$


$$\begin{array}{ll} NO + RANKL & \stackrel{\kappa\_{\pm 1}}{\longleftrightarrow} Remaining\\_product, \\ NO + OPG\_{\text{products}} + Substantum & \stackrel{\text{k\_{\pm 2}}}{\longleftrightarrow} OPG + OPG\_{\text{products}}. \end{array} \tag{14}$$

$$\begin{aligned} \text{RANKL:} \, \delta\_{-1}^{\text{NO,K}} &= 2.06 \, 10^{-12}, \tau\_{24\text{h}}^{\text{NO,K}} = 1.036, n\_{\text{Remaining\\_product}\_0} = 0.930, \\ \text{OPG:} \, \delta\_{-1}^{\text{NO,O}} &= 528.2, \tau\_{24\text{h}}^{\text{NO,O}} = 1.6 \, 10^{-3}, n\_{\text{Substr}\_0} = 8.50. \end{aligned} \tag{15}$$

$$\begin{split} [\text{OPG}\_0]^{\text{RRO}} &= \frac{n\_{OPG\_{\text{stand}}}}{\\ \cdot \text{ } fitNOOPG \{ [\text{NO}\_{\text{ln\\_vitro, standard}}] \}} \cdot \\ \cdot \text{ } fitNOOPG \{ (1 + \ln([\text{NO}\_{\text{ln\\_vvive}}]/[\text{NO}\_{\text{ln\\_vvive}, \text{stand}}]) \} \text{[} \text{NO}\_{\text{ln\\_vvive}, \text{stand}} \text{]} \} \end{split}$$

$$\begin{aligned} \text{RANKL: } [\text{NO}\_{\text{ln\\_vtrro\\_stand}}]^{\text{NO}, \text{R}} &= 0.036, \\ \text{OPG: } [\text{NO}\_{\text{ln\\_vtrro\\_stand}}]^{\text{NO}, \text{O}} &= \text{1.01.} \end{aligned} \tag{16}$$


Feasible Simulation of Diseases Related to Bone Remodelling and of Their Treatment 203

observed that, together with estradiol, there is a decline in nitric oxide levels - see section 2.3. An example of a woman who is physically active (correct mechanical stimuli on regular daily basis, i.e. approximately 20 000 steps per day) but in a consequence of menopause has decreased serum levels of estradiol and nitric oxide (as mentioned above - estradiol: 2.5 ��

model predicts a decrease of 8% in bone tissue density, which does not seem to be osteoporosis yet. This may be because menopause is accompanied by more effects than these two mentioned and also most probably because they are less physically active (may be caused by pain). If we combine the 8% decrease (Fig. 6) caused by menopause alone with another 9% decline (Fig. 4) caused by improper loading, we get a significant drop by almost 20% in the overall bone density of the femur, which can be considered as osteoporotic state. One possible treatment of bone loss connected with menopause is treated with hormone therapy (HRT). Simulation of such a treatment that increased estradiol serum levels to <sup>20</sup> ��

in Fig. 6. Again, the importance of mechanical stimulation shown when increased physical activity (running 30 minutes every other day) increases bone density in similar fashion as HRT treatment (the same figure). And best results are reached when both effects are

A natural goal of the modelling of a process in the human body is to help in understanding its mechanisms and ideally to help in the treatment of diseases related to this phenomenon. For this reason, more detailed influences of various biochemical factors were added. Nowadays, the RANKL/RANK/OPG chain is deemed to be one of the most important biochemical controls of the bone remodelling process. The direct cellular contact of osteoclast precursor with stromal cells is needed for osteoclastogenesis. This contact is mediated by the receptor on osteoclasts and their precursor, RANK, and ligand RANKL on osteoblasts. Osteoprotegerin binds with higher affinity to RANK which inhibits the receptor-ligand interaction and as a result, it reduces osteoclastogenesis. Thus, the raise in OPG concentration results in a smaller number of resorbing osteoclasts, which leads to higher bone tissue density. The results discussed in the presented work have exactly the same behaviour. Similarly, the effects of RANKL, RANK, and estradiol were added to the mentioned model. Consequently, simulation capabilities were demonstrated on examples of diseases and their treatment. These results

However, the impression that the presented model is able to simulate the bone remodelling process in the whole complexity is not correct. It has limitations, as mentioned below, in the spatial precision of the results (i.e. actual structure of bone tissue) and also some control mechanisms cannot be included (e.g. TGF-β effects, as was mentioned at the end of section 2.4). But still, the model can be at least considered as a summary of known important factors, comprising the fundamental aspects of the currently known knowledge of the bone remodelling phenomenon, with some predictive capabilities and encouraging predictive simulations. Since the presented model is a concentration model, it cannot be used arbitrarily. The limitation is, of course, in the spatial precision of results. The minimal volume unit (finite element) should be sufficiently large to contain enough of all the

combined and even the original bone tissue density can be restored - Fig. 6.

were partially validated by clinical studies found in literature.

�� per day of nitroglycerin treatment which is actually less expensive) is given

��), see (Klika et al, 2010). Further it was

�� per day of nitroglycerin) is depicted in Fig. 6. The presented

��,

��

in some not (serum level remains above <sup>20</sup> ��

NO level correspond to 0.02 ��

(or by 0.107 ��

**4. Conclusion** 

### **3. Examples of predictions of bone remodelling based on the presented model**

We may now simulate the response of bone remodelling to changing environment, both mechanical and biochemical. Similarly, as was described in (Maršík et al, 2010), density distribution patterns may be obtained using Finite Elements Method (FEM). The results from the previous section will be used.

To demonstrate the usage of the presented model, a prediction of density distribution in human femur was carried out (the FE model of femur consists of 25636 3D 10 node tetrahedron elements). As an initial state, a homogeneous distribution of apparent density throughout the whole bone was used (1.8 g/cm3). Since each iteration is calculated by solving differential equations representing the whole model of bone remodelling, it is actually a time evolution of bone remodelling in bone (Ansys v11.0 program was used to calculate the mechanical stimuli in each element). Real geometry was gained from a CT-scan and external forces were applied in the usual directions and magnitude - including muscle forces acting on the femur (Heller et al., 2005). This technique was used for all computed results included in this chapter. One may observe that cortical bone (the denser part of bone) and cancellous bone are formed (and maintained) as a consequence of mechanical stimuli distribution in human bone.

It was needed to use a relation between predicted bone density and mechanical properties (elastic moduli). There is a great disparity in the proposed elastic-density relationships (Helgason et al, 2008; Rice et al, 1988; Rho et al, 1993; Hodgskinson and Currey, 1992). We used a most common relationship for the human femur because the relation seems to be site-specific (Helgason et al, 2008):�� � ��, which defines through bone density the constitutive relation being considered. If a different power law is used, the pattern of density distribution still remains the same.

As was mentioned in the introduction, the most important mechanical stimulus for maintaining bone tissue is the most common daily activity - walking. Nowadays, many people have sedentary jobs and that is why they spent less hours by walking than would be appropriate. An example of a person whose walking activity is around 55% of a healthy standard (1.5 hours of walking compared to 2.75h) is shown - inappropriate loading leads to decrease in bone density, see Fig. 4. Naturally, a possible treatment would be in spending more time walking (1.25h more) or similar effect can be reached by running, which is a higher osteogenic stimuli - 30 minutes of running every other day will compensate for the disuse - see Fig. 4.

Hyperparathyroidism was chosen as an example of biochemical control of bone remodelling. As was discussed in section 2.2, during this disease, the PTH serum levels are <sup>97</sup> �� ��, whereas a standard value is <sup>34</sup> �� ��. This should lead to a significant decrease in bone density. An example of a person who is physically active (correct mechanical stimuli on regular daily basis, i.e. approximately 20 000 steps per day) but suffers from hyperparathyroidism (serum level of PTH is <sup>97</sup> �� ��) is depicted on Fig. 5. We may observe an expected decrease in bone density. Because mechanical loading has an osteogenic effect even in a disease state, we may try to increase bone density (possibly insufficiently) only with an increase in mechanical stimulation, 30 minute-running per every other day was chosen. The predicted improvement is depicted in Fig.5.

During menopause, a decline in estradiol levels occur. In some women, the decrease is very dramatic (a drop bellow <sup>5</sup> �� �� is observed, whereas a standard serum level is 40-<sup>60</sup> �� ��) while combined and even the original bone tissue density can be restored - Fig. 6.

in some not (serum level remains above <sup>20</sup> �� ��), see (Klika et al, 2010). Further it was observed that, together with estradiol, there is a decline in nitric oxide levels - see section 2.3. An example of a woman who is physically active (correct mechanical stimuli on regular daily basis, i.e. approximately 20 000 steps per day) but in a consequence of menopause has decreased serum levels of estradiol and nitric oxide (as mentioned above - estradiol: 2.5 �� ��, NO level correspond to 0.02 �� �� per day of nitroglycerin) is depicted in Fig. 6. The presented model predicts a decrease of 8% in bone tissue density, which does not seem to be osteoporosis yet. This may be because menopause is accompanied by more effects than these two mentioned and also most probably because they are less physically active (may be caused by pain). If we combine the 8% decrease (Fig. 6) caused by menopause alone with another 9% decline (Fig. 4) caused by improper loading, we get a significant drop by almost 20% in the overall bone density of the femur, which can be considered as osteoporotic state. One possible treatment of bone loss connected with menopause is treated with hormone therapy (HRT). Simulation of such a treatment that increased estradiol serum levels to <sup>20</sup> �� �� (or by 0.107 �� �� per day of nitroglycerin treatment which is actually less expensive) is given in Fig. 6. Again, the importance of mechanical stimulation shown when increased physical activity (running 30 minutes every other day) increases bone density in similar fashion as HRT treatment (the same figure). And best results are reached when both effects are

### **4. Conclusion**

202 Theoretical Biomechanics

We may now simulate the response of bone remodelling to changing environment, both mechanical and biochemical. Similarly, as was described in (Maršík et al, 2010), density distribution patterns may be obtained using Finite Elements Method (FEM). The results

To demonstrate the usage of the presented model, a prediction of density distribution in human femur was carried out (the FE model of femur consists of 25636 3D 10 node tetrahedron elements). As an initial state, a homogeneous distribution of apparent density throughout the whole bone was used (1.8 g/cm3). Since each iteration is calculated by solving differential equations representing the whole model of bone remodelling, it is actually a time evolution of bone remodelling in bone (Ansys v11.0 program was used to calculate the mechanical stimuli in each element). Real geometry was gained from a CT-scan and external forces were applied in the usual directions and magnitude - including muscle forces acting on the femur (Heller et al., 2005). This technique was used for all computed results included in this chapter. One may observe that cortical bone (the denser part of bone) and cancellous bone are formed (and maintained) as a consequence of mechanical stimuli

It was needed to use a relation between predicted bone density and mechanical properties (elastic moduli). There is a great disparity in the proposed elastic-density relationships (Helgason et al, 2008; Rice et al, 1988; Rho et al, 1993; Hodgskinson and Currey, 1992). We used a most common relationship for the human femur because the relation seems to be site-specific (Helgason et al, 2008):�� � ��, which defines through bone density the constitutive relation being considered. If a different power law is used, the pattern of

As was mentioned in the introduction, the most important mechanical stimulus for maintaining bone tissue is the most common daily activity - walking. Nowadays, many people have sedentary jobs and that is why they spent less hours by walking than would be appropriate. An example of a person whose walking activity is around 55% of a healthy standard (1.5 hours of walking compared to 2.75h) is shown - inappropriate loading leads to decrease in bone density, see Fig. 4. Naturally, a possible treatment would be in spending more time walking (1.25h more) or similar effect can be reached by running, which is a higher osteogenic stimuli - 30 minutes of running every other day will compensate for the

Hyperparathyroidism was chosen as an example of biochemical control of bone remodelling. As was discussed in section 2.2, during this disease, the PTH serum levels are

density. An example of a person who is physically active (correct mechanical stimuli on regular daily basis, i.e. approximately 20 000 steps per day) but suffers from

expected decrease in bone density. Because mechanical loading has an osteogenic effect even in a disease state, we may try to increase bone density (possibly insufficiently) only with an increase in mechanical stimulation, 30 minute-running per every other day was

During menopause, a decline in estradiol levels occur. In some women, the decrease is very

�� is observed, whereas a standard serum level is 40-<sup>60</sup> ��

��. This should lead to a significant decrease in bone

��) is depicted on Fig. 5. We may observe an

��) while

**3. Examples of predictions of bone remodelling based on the presented** 

**model** 

from the previous section will be used.

distribution in human bone.

disuse - see Fig. 4.

dramatic (a drop bellow <sup>5</sup> ��

<sup>97</sup> ��

density distribution still remains the same.

��, whereas a standard value is <sup>34</sup> ��

hyperparathyroidism (serum level of PTH is <sup>97</sup> ��

chosen. The predicted improvement is depicted in Fig.5.

A natural goal of the modelling of a process in the human body is to help in understanding its mechanisms and ideally to help in the treatment of diseases related to this phenomenon. For this reason, more detailed influences of various biochemical factors were added. Nowadays, the RANKL/RANK/OPG chain is deemed to be one of the most important biochemical controls of the bone remodelling process. The direct cellular contact of osteoclast precursor with stromal cells is needed for osteoclastogenesis. This contact is mediated by the receptor on osteoclasts and their precursor, RANK, and ligand RANKL on osteoblasts. Osteoprotegerin binds with higher affinity to RANK which inhibits the receptor-ligand interaction and as a result, it reduces osteoclastogenesis. Thus, the raise in OPG concentration results in a smaller number of resorbing osteoclasts, which leads to higher bone tissue density. The results discussed in the presented work have exactly the same behaviour. Similarly, the effects of RANKL, RANK, and estradiol were added to the mentioned model. Consequently, simulation capabilities were demonstrated on examples of diseases and their treatment. These results were partially validated by clinical studies found in literature.

However, the impression that the presented model is able to simulate the bone remodelling process in the whole complexity is not correct. It has limitations, as mentioned below, in the spatial precision of the results (i.e. actual structure of bone tissue) and also some control mechanisms cannot be included (e.g. TGF-β effects, as was mentioned at the end of section 2.4). But still, the model can be at least considered as a summary of known important factors, comprising the fundamental aspects of the currently known knowledge of the bone remodelling phenomenon, with some predictive capabilities and encouraging predictive simulations. Since the presented model is a concentration model, it cannot be used arbitrarily. The limitation is, of course, in the spatial precision of results. The minimal volume unit (finite element) should be sufficiently large to contain enough of all the

Feasible Simulation of Diseases Related to Bone Remodelling and of Their Treatment 205

Fig. 5. Prediction of the PTH effect on bone quality - simulation of hyperparathyroidism

other day), and its simulation. Notice the change of bone mass (BM) of the whole femur.

��), treatment proposal - running (30 minutes every

(<sup>97</sup> ��

�� where normal PTH levels are <sup>34</sup> ��

Fig. 4. Prediction of disuse effect on bone quality - simulation of insufficient loading (half of the recommended daily stimulation), treatment proposal - running (30 minutes every other day), and its simulation. Notice the change of bone mass (BM) of the whole femur.

Fig. 4. Prediction of disuse effect on bone quality - simulation of insufficient loading (half of the recommended daily stimulation), treatment proposal - running (30 minutes every other

day), and its simulation. Notice the change of bone mass (BM) of the whole femur.

Fig. 5. Prediction of the PTH effect on bone quality - simulation of hyperparathyroidism (<sup>97</sup> �� �� where normal PTH levels are <sup>34</sup> �� ��), treatment proposal - running (30 minutes every other day), and its simulation. Notice the change of bone mass (BM) of the whole femur.

Feasible Simulation of Diseases Related to Bone Remodelling and of Their Treatment 207

substances entering the reaction schemes, namely osteoclasts and osteoblasts. It surely cannot be used on the length scales of BMU where it is no longer guaranteed that any osteoclast is present. There are approximately 107 BMU in a human skeleton present at any moment (Klika and Maršík, 2010) and, because bones have a total volume of 1.75l, there is 1 BMU per 0.175 mm3 on average at any moment. In other words, the presented model cannot

Ongoing applications of the model include simulations of the 3D geometries of the femur and vertebrae (FE models) under various conditions (both biochemical and mechanical). The preliminary results are encouraging and show the correct density distribution. Currently, we are working on bone modelling (change of shape of bone) model that would add the possibility to adapt bone to its mechanical environment as it is observed in vivo. Further, we would like to have a more detailed description of the inner structure of bone as an outcome of the model. Most probably, a homogenisation technique will be used for addressing this goal. Most importantly, a validation of the model's predictions (or finding limits of its application) should start in near future in cooperation with Ambulant Centre for Defects of

This research has been supported by the Czech Science Foundation project no. 106/08/0557, by Research Plan No. AV0Z20760514 of the Institute of Thermomechanics AS CR, and by Research Plan MSM 6840770010 "Applied Mathematics in Technical and Physical Sciences"

Armour, K. E., van't Hof, R. J., Grabowski, et al. Evidence for a pathogenic role of nitric

Beaupre GS, Orr TE, and Carter DR. An approach for time-dependent bone modeling and

Boyle WJ, Simonet WS, and Lacey DL. Osteoclast differentiation and activation. Nature,

Carter DR. Mechanical loading history and skeletal biology. Journal of Biomechanics,

Carter GD, Carter R, Jones J et al. How accurate are assays for 25-hydroxyvitamin D data

Cranney A., Horsley T., O'Donnell S. et al. Effectiveness and Safety of Vitamin D in Relation

deGroot R. S., Mazur P. Non-equilibrium Thermodynamics. North-Holland, Amsterdam,

oxide in inflammation-induced osteoporosis.J Bone Miner Res,14(12): 2137-42, 1997.

remodeling-application: a preliminary remodeling simulation. J Orthopaed Res,

from the international vitamin D external quality assessment scheme, Clin Chem 50

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mm3 and we recommend that it is not used on

be used for length scales smaller than √0.175 �

mm3 ≈ 0.8 mm.

of the Ministry of Education, Youth and Sports of the Czech Republic.

length scales below √0.5 �

Locomotor Aparatus in Prague.

8:662–670, 1990.

423(3):337–342, 2003.

20:1095–1109, 1987.

(11), 2195-2197, 2004

Research and Quality.

1962.

**5. Acknowledgment** 

**6. References** 

Fig. 6. Prediction of the menopause effect on bone quality (estradiol levels decreased to 2.5 �� �� and NO to half of its normal level), treatment proposal, and its simulation - hormonal treatment (HRT), running (30 minutes every other day). Notice the change of bone mass (BM) of the whole femur.

substances entering the reaction schemes, namely osteoclasts and osteoblasts. It surely cannot be used on the length scales of BMU where it is no longer guaranteed that any osteoclast is present. There are approximately 107 BMU in a human skeleton present at any moment (Klika and Maršík, 2010) and, because bones have a total volume of 1.75l, there is 1 BMU per 0.175 mm3 on average at any moment. In other words, the presented model cannot be used for length scales smaller than √0.175 � mm3 and we recommend that it is not used on length scales below √0.5 � mm3 ≈ 0.8 mm.

Ongoing applications of the model include simulations of the 3D geometries of the femur and vertebrae (FE models) under various conditions (both biochemical and mechanical). The preliminary results are encouraging and show the correct density distribution. Currently, we are working on bone modelling (change of shape of bone) model that would add the possibility to adapt bone to its mechanical environment as it is observed in vivo. Further, we would like to have a more detailed description of the inner structure of bone as an outcome of the model. Most probably, a homogenisation technique will be used for addressing this goal. Most importantly, a validation of the model's predictions (or finding limits of its application) should start in near future in cooperation with Ambulant Centre for Defects of Locomotor Aparatus in Prague.

### **5. Acknowledgment**

206 Theoretical Biomechanics

Fig. 6. Prediction of the menopause effect on bone quality (estradiol levels decreased to

�� and NO to half of its normal level), treatment proposal, and its simulation - hormonal treatment (HRT), running (30 minutes every other day). Notice the change of bone mass

2.5 ��

(BM) of the whole femur.

This research has been supported by the Czech Science Foundation project no. 106/08/0557, by Research Plan No. AV0Z20760514 of the Institute of Thermomechanics AS CR, and by Research Plan MSM 6840770010 "Applied Mathematics in Technical and Physical Sciences" of the Ministry of Education, Youth and Sports of the Czech Republic.

### **6. References**


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**10** 

*Spain* 

**Towards a Realistic and Self-Contained** 

Most of human mechanical interactions with the surrounding world are performed by the hands. They allow us to perform very different tasks; from exerting high forces (e.g. using a hammer) to executing very precise movements (e.g. cutting with a surgical tool). This versatility is possible because of a very complex constitution: a great number of bones connected through different joints, a complicated musculature and a dense nervous system. This complexity is already evident from the kinematics point of view, with more than 20

Mathematical representations are used in order to perform qualitative or quantitative analyses on this complex reality. These representations are known as biomechanical models of the hand. In biomechanics, their use allows studying problems that cannot be analysed directly on humans or that have an experimental cost that is too high; e.g., the study of new alternatives for restoring hand pathologies. Biomechanical models are a description of the hand as a mechanical device: the different elements of the hand are defined in terms of rigid bodies, joints and actuators, and the mechanical laws are applied. As they are simplified mathematical

models of the reality, their use and validity depends on the simplifications considered.

The first biomechanical models of the hand were developed to explain and clarify the functionality of different anatomical elements. In this regard, we can find many works that studied the function of the intrinsic muscles (Leijnse & Kalker, 1995; Spoor, 1983; Spoor & Landsmeer, 1976; Storace & Wolf, 1979, 1982; Thomas et al., 1968) and many others that tried to give an insight into the movement coordination of the interphalangeal joints (Buchner et al., 1988; Lee & Rim, 1990). Models for studying the causes and effects of different pathologies of the hand also appeared early on, such as the swan neck and boutonnière deformities or the rupture of the triangular ligament or the volar displacement of the extensor tendon (Smith et al., 1964; Storace & Wolf, 1979, 1982). All these models were, though, very limited, twodimensional models allowing only the study of flexion-extension movements, they modelled only one finger, and they included important simplifications. By the year 2000, few threedimensional models had been developed (Biryukova & Yourovskaya, 1994; Casolo & Lorenzi,

Antonio Pérez-González, Marta C. Mora, Beatriz E. León, Margarita Vergara, José L. Iserte,Pablo J.

degrees of freedom (DOF) controlled by muscles, tendons and ligaments.

**1. Introduction** 

 \*

Rodríguez-Cervantes and Antonio Morales

*Universitat Jaume I, Spain* 

**Biomechanical Model of the Hand** 

Joaquín L. Sancho-Bru et al.\*

*Universitat Jaume I* 

