**6. Mechanobiological models treating bone healing and drugs effect**

Mechanobiological mathematical models of bone remodeling are not only used to present diseases and mechanical effects on the process. Actually, the main target of all researchers, by developing these models, is to provide ideas and therapeutic solutions that could be applied in the real life after a clinical validation. Hence, in this section, some models considering bone healing and the effect of some bone diseases' treatments are discussed.

A mathematical model integrating the biological and mechanical aspects to describe the process of osteointegration has been described in [87]. The osteointegration is the direct structural and functional connection between a living bone and an artificial implant surface [88]. Knowing that many factors are involved into healing the interface between the non-biological material and bone, a mechanobiological modeling of the concerned surface is mandatory. In this study, authors have subdivided the bone healing process into four stages: (i) Blood clotting, (ii) Cell migration, (iii) Granulation tissue, and finally (iv) bone formation. The formulated mathematical model, describing the process, took into consideration all the stages of bone-implant healing. Yet, in this chapter, we have only shed the light on bone formation phase. Researchers have suggested a differential equation to schematize the osteogenesis stage, that depends on ontogenetic cells' migration and the osteogenic chemical variables. These two component values have been calculated based on previous equations describing the former phases. Concerning the mechanical stimulus, it has been presented through a displacement matrix.

In another study, authors have developed a mathematical bone remodeling model gathering the biological factors and the mechanical stimuli's effect on bone cells dynamics [89]. This research has been devoted to assess the bone mechanobiological response and the cellular activity variation over time depending on specific types of physical activities. Based on the previous works [88, 89], the bone cells concentrations have been calculated. The mechanical stimuli have been added to promote the proliferation of the preosteoblast and to regulate RANKL

production. As well as the majority of studies [80, 90, 91], the SED has been used to schematize the biomechanical stimuli's influence on cellular dynamics. The SED is calculated based on the stress and deformation tensors. In this study, these tensors were decomposed into deviatoric and hydrostatic parts, in order to get both hydrostatic and deviatoric SED quantities. Considering three stimuli in different remodeling periods, researchers have performed their finite element isotropic models. Seeking to know the formation performance, cortical and trabecular bone densities in the diaphysis and epiphysis regions of a proximal femur have been investigated. The comparison between the three mechanical stimuli (SED, deviatoric SED, and hydrostatic SED) revealed that: the formation of cortical and trabecular bone is higher for hydrostatic SED, followed by deviatoric SED, and finally by the whole SED stimuli. It has been remarked that the thickness of cortical bone was more significant applying Hydrostatic SED compared with the other stimulus type, while the trabecular density was higher under deviatoric SED stimulus. This study investigated then many mechanical stimuli and their particularity and effect on the development of bone density in many regions. Thus, it gave some insight about bone remodeling especially bone formation phase, which is also a primordial phase in bone healing process.

Concerning, the investigation of drugs dose effect on bone remodeling, the traditional mathematical models were not widely used in the literature. Instead, researchers opted for pharmacokinetic pharmacodynamic (PK/PD) models, which permit to predict the behavior of the treatment in different compartment of the body before reaching the blood. The results are then coupled with a traditional bone remodeling mathematical model. Various studies have been done in this respect, but which mostly treat the same type of drugs used for bone reparation such as denosumab. Studding denosumab effect on bone remodeling, this treatment has been generally included by modifying the function dedicated to calculate RANKL concentration in such a way the RANKL-binding denosumab is taken into consideration [82, 92]. In the work of [83], the mechanobiological feedback function has been incorporated to regulate preosteoclasts proliferation as previously done in [80] and the denosumab effect has been added by adjusting RANKL concentration. Other studies have been interested in PTH drug's effect, which is usually used for postmenopausal women. In the work of [86] for example, authors have created an equation representing PTH amount in the blood depending on the serum concentration injected. Based on the outcomes, they formulated another function that will control preosteoblast proliferation, lining cells differentiation, and active osteoclast apoptosis. Another recent study [85] has been based on [86], have investigated the same thing, which is PTH effect on bone density. This time, authors have included PTH into their bone remodeling mathematical model by modifying active osteoblasts apoptosis rate multiplying the normal parameter value by the function representing PTH drug concentration. In contrary to the work of [86], this model considered the mechanical feedback triggering preosteoblast proliferation. The term representing this mechanical feedback is the same used in [80].

#### **7. Conclusion**

To maintain the multiple functions of a skeletal, this last should be regularly renewed in such a way the old bone tissue is substituted with a new one. Applying cyclic mechanical loading on the bone helps to trigger the bone remodeling. Thus, people doing physical activities generally have stronger bones. On the other hand, the occurrence of some diseases because of age, hormonal deficiency or bad dietary habits leads to bone deterioration. The present chapter shed the light on the most

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

popular bone diseases and the different mechanobiological bone remodeling that investigate the evolution of these diseases and some drugs effect on the evolution of bone mass over time. Based on the forgoing reviewed articles, some conclusions could be extracted:

