**2.3 Bone remodeling cycle**

Bone remodeling process (**Figure 2**) as mentioned before is a succession of events governed by many biochemical factors. In this section, the most important interactions, actions and reactions are described in summary to show the bone cells dynamics and their effect on the bone mass changes.

#### *2.3.1 Activation*


*Mechanobiological Behavior of a Pathological Bone DOI: http://dx.doi.org/10.5772/intechopen.97029*

**Figure 2.**

*Schematic representation of bone cells main biochemical interactions during bone remodeling process.*

	- 1.Differentiation, proliferation, and activity of osteoclasts. The osteoclastogenesis is mediated by RANK-RANKL binding.
	- 2.Active osteoclasts secrete hydrogen ions and acid phosphatases
	- 3.Mineral phase of the bone matrix is dissolved
	- 4.Active osteoclasts secrete enzymes
	- 5.Organic phase of the bone is resorbed, and embedded biochemical factors are released (e.g. TGFβ and BMP)
	- 6.Osteoblastogenesis is stimulated and osteoprotegerin (OPG) amount increases inducing RANKL decrease
	- 7.Osteoclasts undergo apoptosis
	- 1.Macrophages clean the bone lacunae from bone matrix debris
	- 2.Active osteoblasts adhere to bone lacunae
	- 1.Active osteoblasts synthetize the collagen to produce the osteoid that gets gradually mineralized.
	- 1.Signals exciting bone cells decreases in the area
	- 2.BMU recruitment gets smaller

Any interruption of the biological pathways, during remodeling process, can lead to enormous consequences on the bone cells functioning, which in its turn leads to very dangerous diseases.

#### **2.4 Bone mechanobiology**

The mechanobiology particularity of bone tissue is the base source of its renewing ability and its capacity to be adapted to external charges. Based on many experimentations, researchers have found noticeable changes in bone mass, while variating the external applied charges on the skeleton [34–39]. Thanks to the bone microarchitecture and its tissue composition, the mechanical loads are transmitted at the cells' level as a combination of fluid shear stress and extracellular strain matrix [40]. Then, thanks to the osteocytes' mechanosensing mechanisms like ion channels, integrins, gap junction, and actin cytoskeleton, these cells become able to release biochemical components (e.g. proteins and cytokines) that regulates osteoblasts and osteoclasts' activity (e.g. increasing their proliferation/differentiation and inhibiting their apoptosis). For the sake of illustration, we mention the most known biochemical factors involved: PGE2 [41], NO [42], and SCLR [43]. In addition to osteocytes, osteoblasts have also been classified as a mechanosensory cell as their respond to the mechanical signal mediated by the fluid flow by upregulating the cyclooxygenase-2 (Cox-2) and c-fos and producing intracellular calcium (Ca2+).

### **3. Bone deterioration causes**

#### **3.1 Normal bone diseases**

#### *3.1.1 Osteoporosis*

Osteoporotic bone is a fragile bone characterized by a low mass (**Figure 3**) and deteriorated microstructure. These features are making this bone highly susceptible to fractures. Osteoporosis is a biochemical problem resulted from both osteoblast and osteoclast behaviors' dysregulation that leads to excessive bone resorption. It affects a large slice of the world's population, especially women, and it causes physical debilitation and frequent fracture incidence in patients. Osteoporotic fractures are generally occurring in the spine, hips, femur, and forearm; and they can be detected in case of a drop in the bone mineral density (BMD) value in the fractured area [44]. Based on the disease type, scientists have subdivided osteoporosis into two categories: (i) the primary osteoporosis, which is related to age and hormonal dysregulation, or unknown causes. This latter case is called idiopathic primary osteoporosis [45]. Then, (ii) the secondary osteoporosis, which is related to other diseases appearance (e.g. cancer, hematologic, and gastrointestinal diseases), or to some treatments use (e.g. Cancer chemotherapeutic drugs, glucocorticoids, and anticonvulsants [45]).

*Mechanobiological Behavior of a Pathological Bone DOI: http://dx.doi.org/10.5772/intechopen.97029*

#### **Figure 3.**

*Difference between normal proximal femur and femur affected by osteoporosis, Paget's disease, and cancer, based on medical images.*

#### *3.1.2 Paget's disease of bone*

Paget's disease of bone is a chronic bone disease, that affect either a single or multiples parts of the skeletal. The concerned areas are characterized by increased bone resorption accompanied with increased but disorganized bone formation [46] (**Figure 3**). These problems in the bone remodeling process causes deformed and weak bones. Generally, this disease affect men and the elderly more than women and young people [47]. The real etiology of this disease is still unknown, however, 40% of patient affected by Paget's disease have been detected to have a family history of SQSTM1 gene mutation with a protein regulating osteoclasts called p62 [48]. Advanced Paget's disease of bone could cause several complication to the patient, among which, bone pain, bone fractures, and hypercalcemia [49].

#### *3.1.3 Osteogenesis imperfecta*

Also known as brittle bone disease, osteogenesis imperfecta is a group of rare bone diseases characterized by heterogeneous disturbance of the cognitive tissue. All these diseases are associated with bone mass diminution, increased bone fragility, bone disfigurement, and bone formation insufficiency [50]. Osteogenesis imperfecta etiology-associated differ from a disease to another as it depends mainly on the onsets and intensity of each one. Genetic, phenotypic and functional classification have been adopted to find out the new causative mutation of osteogenesis imperfecta onset [51]. The most commonly known osteogenesis imperfecta diseases are: (i) X-linked hyposphantaemia (XLH), which is characterized by a mutation of phosphate regulating endopeptidase (PHEX) leading to dysregulate 1,25 OH vitamin D levels; (ii) Hypoparathyroidism, which is characterized by PTH deficit that regulate calcium homeostasis; (iii) Hypophosphatasia, which results from mutation of a responsible gene of encoding ALP called ALPL. The dysregulation of the enzyme disrupts its function to prompt adequate mineralization at an appropriate time in bone tissue; and (vi) Osteopetrosis caused by gene mutation code for RANK and RANKL, leading to an impairment of osteoclasts and thus increased bone mass with a deterioration of bone quality resulting in atraumatic fractures.

#### **3.2 Metastatic bone diseases**

Bone degradation is related to several different factors among which, cancer is the most discussed issue. Regardless of the cancer type, cancer patients have a

higher risk of bone loss than the rest of the population [52]. In fact, bone microenvironment enables tumor cells to home, proliferate, and colonize [5]. Thus, generally, cancer cells migrate form their inherent location to bone microenvironment, were osteoblasts and osteoclasts secrete important biochemical factors for their survival [53]. Each cancer type secretes special biochemical factors affecting bone environment and stimulating their growth. The spread of tumor causes the so-called bone metastasis disease either if it is a solid tumor (e.g. prostate cancer, breast cancer) or a liquid tumor (e.g. multiple myeloma). Bone metastasis incidence is mostly occurring in multiple myeloma patients (95–70%), breast cancer patients (75–65%), and prostate cancer patients (75–65%) [54].

#### *3.2.1 Multiple myeloma*

Multiple myeloma is a blood cancer disease characterized by an abnormal growth of B-lymphocytes, while differentiating into plasma cells. This cancer is characterized by renal impairment (creatinine >2 mg/dL), hypercalcemia (calcium >11 mg/dL) anemia (hemoglobin <10 mg/dL), the infiltration of clonal plasma, and end organ damage such as lytic lesions in the bone [55, 56]. As mentioned before, this disease is tightly related to bone degradation. Located in the bone marrow, multiple myeloma cells are able to interact with bone remodeling cells by means of adhesion molecules to proliferate and survive [57]. Indeed, multiple myeloma adhesion incite osteoblastic cells to release interleukin- 6 (IL-6), which has an essential role in tumor growth and survival [5, 58]. Studies found that multiple myeloma growth affect both bone formation and bone resorption. On one hand, they stimulate the differentiation and activity of osteoclasts by secreting RANKL, the interleukins IL-1, IL-3, IL-6, and the parathyroid hormone-related protein (PTHrP) that in turn stimulate RANKL expression by osteoblasts. Additionally, they express syndecan-1, which is a type 1 transmembrane proteoglycan, that binds OPG [59]. This multiple effects on RANK/RANKL/OPG pathway enhances osteoclastogenesis and consequently bone resorption. On the other hand, multiple myeloma cells inhibit the osteoblastogenesis by expressing some biochemical factors such like Wnt and dickkopf-related protein 1 (DKK-1) [60].

#### *3.2.2 Breast cancer*

Breast cancer is a common disease among postmenopausal women. It is characterized by an abnormal growth of breast epithelial cells. Multiple factors are related to this disease incidence including age, as advancing age increases the risk of getting breast cancer, gender, as women are the most concerned, personal or family history of breast cancer, and exogenous hormone use, as some treatments increasing sex hormones notably estrogen and progesterone [61]. Brest cancer develops because of DNA damage that could be the result of sex hormones exposure [62]. Epithelial tumor cells tend as well as the other cancer types, to migrate into the bone microenvironment and interact with bone cells. Actually, breast cancer tumor cells stimulate RANKL expression and inhibit the OPG one by expressing a multitude of biochemical factors, among which we note the interukins IL-1, IL-6, IL-8, IL-11, M-CSF, BMP, DKK-1, PGE2, PTHrP [63]. As a consequence, tumor prevalence in the bone area induces bone loss.

#### *3.2.3 Prostate cancer*

Prostate cancer is a common disease among elderly men, which usually leads to death as it is the case in the USA [64]. It is characterized by an abnormal growth of *Mechanobiological Behavior of a Pathological Bone DOI: http://dx.doi.org/10.5772/intechopen.97029*

the prostate basal and luminal epithelial cells [65]. The real etiology of prostate cancer is still unknown, however, there is some factors related to its incidence. For instance, age, as advancing age increases the risk of getting prostate cancer, family history of prostate cancer, and using dietary supplement rich of vitamin E [66]. Similarly to multiple myeloma and breast cancer, prostate tumor cells also adhere to bone microenvironment and interact with bone cells. It has been reported that prostate tumor cells secrete PTHrP, Wnt and DKK-1 that regulates osteoblast and osteoclast behavior once settled in the bone [67, 68]. Thereafter, when prostate cancer cells adapt to the new environment, they start secreting the prostate- specific antigen (PSA), that inhibit PTHrP. The high amount of Wnt stimulate osteoblasts differentiation, and thus RANKL amount increase. When RANKL increases, bone resorptions is enhanced as result, and the biochemical factors incorporated in the old bone are released stimulating tumor cells proliferation [7].

#### **4. Mathematical models of a pathological bone**

Several studies have been interested in bone diseases' effect on bone remodeling process [5]. A large number of the mathematical models, developed in this regard have been concentrated on the biological aspect related to bone remodeling, so that they can provide a better understanding of the disease biological effect on the process.

#### **4.1 Normal bone diseases**

Osteoporosis, as it is the most widespread bone disease and the most known bone problem, it has captured the attention of many authors, especially those developing bone remodeling mathematical models. Through their mathematical models, the authors try to schematize as best as possible the osteoporosis disease in such a way the results could fit the experimental data.

In their study [69], Lemaire and co-authors have investigated the effect of osteoporosis on the normal biochemical interactions during remodeling process. They opted for the implementation of three osteoporosis causes: (i) estrogen deficiency, (ii) vitamin D hormone calcitriol (1*:*25ð Þ *OH* <sup>2</sup>*D*3) deficiency, and (iii) glucocorticoid excess. These osteoporosis' causes have been implemented by changing some parameters' value in the principal bone remodeling mathematical model. According to authors, estrogen deficiency could be represented by decreasing OPG's production rate parameter until osteoclast to osteoblast concentration ratio reaches 5 in the steady state. Then, concerning vitamin D deficiency, it could be represented by increasing the PTH production rate, while glucocorticoid excess could be schematized by decreasing osteoblast progenitors' differentiation rate. The choices made by the authors are inspired from the literature: (i) authors have choose to decrease PTH production, as originally estrogen expression is stimulating OPG production [70], (ii) PTH effect on bone cells changes with vitamin D3 deficiency, thus PTH production rate alteration has been based on experimental observations [71], and (iii) Glucocorticoid induces a decrease in the core binding factor A1 (Cbfa) that is important for osteoblast differentiation, thus, the rate of preosteoblast differentiation has been reduced to mimic the glucocorticoid excess effect osteoblastogenesis [72].

Targeting the same objective, Pivonka et al. [73] have studied the RANK-RANKL-OPG signaling pathway to model osteoporosis disease in postmenopausal women. In their research study, the catabolic bone disease has been represented

firstly by decreasing the OPG production value, then by testing a combination of changed parameter related to the components RANK-RANKL-OPG. The combination used in the model is based on [74]'s work, where RANK responsiveness and RANKL production are up-regulated, while OPG production is down-regulated. These changes enhanced osteoclasts' response comparably to the osteoblasts' one. However, the intensity of bone resorption resulted is different from changing one parameter to three parameters. Indeed, it has been observed that a change in one single parameter leads to less sever bone resorption. Overall, the obtained numerical results were qualitatively consistent with the experimental observation related to RANK/RANKL/OPG system changes in the case of postmenopausal osteoporosis (PMO) bone disorder.

Apart from osteoporosis, Paget's disease has also interested some researchers, but another time the model developed considering this bone problem have only treated the biological side of the remodeling process. The work of Komarova et al. [75] have illustrated the bone cells dynamics in the presence of Paget's disease. To proceed, authors have altered the parameters representing the normalized activity of resorption and formation parameters in addition to the autocrine parameters. Their model has demonstrated more sensitivity to changes made on the autocrine parameters' values. These changes lead to increase bone resorption succeeded by increased bone formation and represented nearly the same effect of Paget's disease of bone. Nevertheless, the real biological process or the particular effects inducing this type of bone diseases have not been deemed.

#### **4.2 Metastatic bone diseases**

Being one of bone metastasis causes, cancer seems to be an important subject to treat for the clinical interest. As shown before, different types of cancer are causing bone loss, that could be sometimes escorted by bone formation. These bone changes are due to the cancer cells growing inside the bone, that represents a rich area assuring their expansion.

The paper of [76] have treated the problem of cancer's effect on bone health independently of the cancer type. Based on a mathematical model composed of differential equations, authors have calculated bone cells number's variation, which depend either from a normal biological process or a disrupted process controlled by tumor cells growth. According to the tumor growth, the type of bone disorder (i.e. osteolysis or osteosclerosis) is deducted. Therefore, the types of biochemical released factors controlling the process are identified (i.e. TGFβ and PTHrP for osteolysis case/TGFβ and IGF for osteosclerosis case).

In more detailed studies, authors have opted for specific cancer's type effect on the remodeling process. For instance, a mathematical model have been described by Farhat et al. [7] to quantifying bone remodeling changes within the existence of prostate tumor cells in the bone microenvironment. Based on a large literature study, the authors managed to detect the main biological factor controlling the interactions between bone cells and the tumor's one. In summary, authors have made the following assumptions:


*Mechanobiological Behavior of a Pathological Bone DOI: http://dx.doi.org/10.5772/intechopen.97029*


Through this article, prostate cancer growth's impact on the bone remodeling process has been properly established as each influencing factor has been simply explained. According to the results, prostate cancer cells existence induces an increase of RANKL-OPG ratio and bone production. Besides, they found that there are two osteogenic states over the course of disease.

Multiple myeloma's influence on bone remodeling has captured more attention than the other cancers. The research article [6] has provided a model schematizing multiple myeloma cells' growth impact on bone cells dynamics. The model has not clearly shown the biochemical effect of these malignant cells on the normal process. Although, it has shown their effect on the autocrine and paracrine parameters described previously in [75]. Authors have suggested a tumor density differential equation to schematize the metastasis evolution, and they have implemented it mathematically into the paracrine and autocrine variables to disrupt the normal oscillation of bone cells during remodeling cycles. This study findings have completely represented multiple myeloma effect on bone mass evolution. However, the biological interaction occurring between tumor cells and bone cells have not been explicitly shown. Thus, the effect of biochemical factors is not clearly presented.

One year later, a study considered precisely the main biological interactions occurring during a remodeling process affected by multiple myeloma's actions. In the model suggested by Wang et al. [77], multiple myeloma cells behavior have been modeled by differential equations, which permit to calculate the concentration of the principal biochemical factors affecting the bone cells. These factors are the interleukin 6 IL-6, which is secreted by uncommitted osteoblasts promoting multiple myeloma tumor cells' proliferation, and the very-late antigen 4 (VLA-4), which mediate Multiple myeloma-Bone marrow-derived mesenchymal stem cells adhesion and promote IL-6 expression. This antigen effect appears after VLA-4 binding vascular cell adhesion molecule 1 (VCAM-1) expressed by the uncommitted osteoblasts. A further investigation of the problem has been done by [78], where, additionally to the previous biochemical factors incorporated in the model of [77], they have added the effect of small leucine-rich proteoglycan (SLRPs). This regulator is secreted by active osteoblasts. Its main function, in the case on multiple myeloma tumor cells existence, is to inhibit their proliferation. Through this work, the biochemical mechanism controlling the bone remodeling is clearly described and properly established by means of the proposed system of differential equations.

#### **5. Mechanobiological mathematical models of a pathological bone**

In contrary to strictly biological mathematical models, bone diseases have been also incorporated into mechanobiological mathematical models, where the mechanical aspect is taken into consideration. Nevertheless, very few bone diseases have been discussed through these models. In this chapter, we are going to present the mechanobiological models treating osteoporosis problem. All the presented models are based on biological bone remodeling models that represent bone cell

population through differential Equations [79]. These equations permit to determine bone cells' concentration variation over time. The behavior of these cells is controlled by the most influencing biochemical factors that exist in bone area within remodeling process.

In the work [80], authors were interested in explaining the experimentally changes of bone mass in postmenopausal women suffering from osteoporosis, without neglecting the mechanical influence on the process. Indeed, bone cells concentration variation over time has been controlled by biochemical factors and also by the mechanical strains in the extravascular cortical bone matrix. An activation function *Πmech act*,*OBa* (Eq. (1)), representing strains in the extravascular area, was used to promote preosteoblast proliferation and RANKL production. The mechanical stimulus represented by strain energy density (SED) has a minimum and maximum values that correspond respectively to the threshold SED values *Ψbm*, *min* and *Ψbm*, *max* . Indeed, according to the literature, RANKL/OPG ratio is affected by hydraulic pressure at the microscopic level [81]. Therefore, RANKL production function has been formulated as a function of the SED sensed at the bone matrix's level. On the other hand, the effect of postmenopausal osteoporosis has been implemented by considering the effect of mechanical feedback on the progress of PMO. First, authors have fixed a production rate of PTH that reflect the PMO and then they variated the maximum proliferation rate of preosteoblasts depending on the mechanical stimulus' level. Based on this, various PMO-scenarios have been established by changing the value of the strength of anabolic strength parameter . This parameter represents the slope of the activation function. Hence, it determines the degree of which the SED should be increased to reach the maximum value of the activation function that allows a maximum proliferation rate to the preostoblasts.

$$\Pi\_{\text{act,OBa}}^{\text{mech}} = \begin{cases} 1/2 & \Psi\_{bm} < \Psi\_{bm, min} \\ 1/2 \left( 1 + \lambda \left( \frac{\Psi\_{bm}}{\Psi\_{bm, min}} - 1 \right) \right) & \Psi\_{bm, min} < \Psi\_{bm} < \Psi\_{bm, max} \\ 1 & \Psi\_{bm, max} \le \Psi\_{bm} \end{cases} \tag{1}$$

Based on the results, bone porosity, calculated based on the bone remodeling mathematical model, was able to show the severity of PMO evolution over time. According to the authors, the negligence of the mechanical feedback in the presence of PMO disease, lead to unbounded bone resorption, which doesn't reflect the reality. Thus, every study dealing with biological aspects of bone remodeling should take into consideration the mechanical aspect as well. With the same perspective, Pivonka et al. [82], have addressed the geometrical feedback effect upon bone volume evolution throughout the remodeling process and its capacity of accelerating osteoporotic porosity evolution. In this study, the osteoporosis origin has not been mentioned. Yet, its effect has been incorporated by increasing the PTH concentration by applying a continuous PTH administration rate of 500 pM/day, as previously done in [69], which induces an increase of RANKL/OPG's ratio and perturbs consequently its homeostatic steady-state.

Several other studies have been interested in osteoporosis effect on bone remodeling. For instance, [83, 84] are studies where the mechanobiological model developed by [80] has been modified to include PMO disease effect on bone remodeling process. RANKL production *PRANKl* has been modified by adding the RANKL produced because of the PMO *PPMO RANKL* (Eq. (2)). This term leads to increase RANKL concentration and enhance preosteoclasts differentiation. The excess of RANKL production is defined by (Eq. (3)), where *PPMO*,*ini RANKL* is PMO-initiating excess production rate of RANKL and *φRANKL PMO* is reduced factor of *<sup>P</sup>PMO*,*ini RANKL* which is

formulated as presented in (Eq. (4)). In (Eq. (4)), *ξ* represents the characteristic time of the RANKL production decrease, *t RANKL PMO* determine the shape of Lorentztype function (Eq. (4)), and *tPMO*,*ini* the time corresponding to PMO onset.

$$P\_{RANKL} = P\_{RANKL}^{mech} + P\_{RANKL}^{PMO} \tag{2}$$

$$P\_{\text{RANKL}}^{\text{PMO}} = P\_{\text{RANKL}}^{\text{PMO}, \text{ini}} \rho\_{\text{PMO}}^{\text{RANKL}} \tag{3}$$

$$\rho\_{\text{PMO}}^{\text{RANKL}} = \frac{\xi^2}{\xi^2 + \left(\frac{t - t\_{\text{PMO},ini}}{t\_{\text{PMO}}^{\text{RANKL}}}\right)2} \tag{4}$$

Furthermore, PMO effect has been mediated by decreasing the anabolic strength parameter, as previously done in [80].

Recently, PMO problem has been further investigated in the mechanobiological model described in [85]. PMO effect has been incorporated while taking into consideration the mechanical loading applied on the bone. Indeed, the model mechanical aspect has been similar to the previous discussed work of [80] and PMO effect was incorporated in a similar way to the work [86], where authors have imposed an increase in RANKL concentration by including a dosage term that increases RANKL production. This term is derived from experimental data of ovariectomized rats.
