**6. Finite ellement modeling of bone mechanobiological behavior**

This section presents the hybrid multiscale modeling ap-proach (Finite Element EF/Neural Network NN) of bone ultrastructure (**Figure 7**).

This approach, which is summarized in **Figure 8**, is composed of four steps: (i) development and simulation of geometric FE models for each level scale separately (microfi- bril, fibril and fiber), (ii) use of the results obtained from FE simulation in each scale level for the neural network program training phase, (iii) generalization of the results in neural network prediction phase, (iv) transition between the different scales using the same NN program.

The forth step is constituted from three NN blocks as Sembled in series (NN block for each scale level) so each NN (i + 1) Block uses as inputs the Nni block outputs (being i = 1,2). Finally, NN3 outputs allow obtaining MCFR elastic properties.

#### **Figure 7.**

*Schematic representation of bone remodeling based on BMU activity coupled to mechanical stimulus: at the remodeling cycle (n), the applied load generates mechanical stress, strain, and fatigue damage states at every FE of the mesh.*

*Multi-Scale Modeling of Mechanobiological Behavior of Bone DOI: http://dx.doi.org/10.5772/intechopen.95035*

**Figure 8.**

*Hybrid (FE/NN) multiscale modeling of bone ultrastructure [7, 12].*

#### **6.1 Results and discussions**

In this study, the main results are focused on the new multi-scale hybrid modeling developed during the numerical simulation. By showing the synthetic results obtained with the NN method according to three levels of scale MCM, MCF and MCFR (**Figure 9**). By showing the variation of the generalized Young's modulus MCM as a function of the Young's modulus of the two essential bone constituents according to this scale (mineral and collagen) (**Figure 9a**). In the light of various investigations, these results found under the NN1 analysis are integrated in **Figure 9a** to construct the generalized Young's modulus MCF, obviously by integrating the elastic properties of the mineral. By analogy, the results of NN2 are likewise used to evaluate the Young MCFR's modulus (**Figure 9c**). It must be taken into account that the quantity of the mineral NN degrades in the matrix depending on the level of the scale. Our studies are flexible to have flexibility on the variation of the Young's modulus of the mineral under the effect of a significant difference between the subject amorphous structure [13]. **Figure 9a** shows that the Young's modulus of mineral has a very significant effect compared to the effect of TC modulus of Young's molecules for the MCM level scale. On the other hand, the synthesis in **Figure 9b** shows us that the elastic properties of the mineral are also slightly influential, compare to the Young's modulus of MCMs for the different scale levels of MCF. These results found, are comparable with other studies in literature [12, 14]. On the other hand, the curve of **Figure 9c** shows that the Young's modulus MCFs has a remarkable effect on the equivalent Young's modulus of the MCFR compared to the Young's modulus of the mineral.

Generally, rehological properties of the scale levels of the bone ultrastructure are related to several geometric and mechanical parameters such as the elementary bone compounds of Young's modulus (mineral, collagen), the nature of collagen (dry, wet), the size of the mineral crystal and the number of crossed links. Therefore, the estimated elastic constants have been shown in **Table 2** to compare our results with the experimental and numerical results of other studies in the literature. The literature study dealing with the characterization of the mechanical properties of MCFR are very little and even are not sufficiently developed, for this reason, the comparison of the results is limited to the MCM and MCF scale levels (see **Figures 10** and **11**, respectively).

**Figure 10**, configures a good correlation between predicted NN (our numerical study) and the experimental study for the small strain margin based on X-ray

#### **Figure 9.**

*Evolution of elastic moduli (GPa) of MCM and MCF as function of the mineral Young's modulus and passage between the MCM and MCF [7].*

diffraction, atomic force microscopy (AFM) and the calculation of molecular dynamics. (MD).

Also, **Figure 11** records a good agreement between our study predicted NN for the MCF and the numerical and experimental results: analysis by DRX, calculation of molecular dynamics (MD) and calculation by the finite element method. However, there is a slight uncertainty which can be introduced by different sources: the different methods used, the size and nature (hydrated or dehydrated) of the MCM and MCF tested and the hypotheses considered by each. As an indication during this work, consider the mineral as a homogeneous matrix without taking into account the presence of water (PCNs are negligible). These assumptions may explain the differences mentioned above. However, due to the living nature of the materials studied (bone), the Young's modulus of MCM can be on average about 1 ± 0.2 GPa and 40 ± 2 GPa for the Young's modulus of MCF.

Moreover, the comparative study of NN predicted the average Young's modulus of MCF and also comparable to the study carried out in the literature.

The elastic properties are closely linked to several material and structural parameters at the level of our study scale. This translates into a great sensitivity in terms of mechanical properties and service life. Indeed, if the boundary conditions are specified, we can assign to each scale level a unique value of the elastic properties of the bone. As a result, carrying out experimental tests or numerical


#### *Multi-Scale Modeling of Mechanobiological Behavior of Bone DOI: http://dx.doi.org/10.5772/intechopen.95035*


**Table 2.** *Material properties for bone used for the remodeling simulation [15–17].*

#### *Biomechanics and Functional Tissue Engineering*

*Multi-Scale Modeling of Mechanobiological Behavior of Bone DOI: http://dx.doi.org/10.5772/intechopen.95035*

**Figure 10.**

*Comparison between NN predicted average Young's modulus of MCM and literature results [7].*

#### **Figure 11.**

*Comparison between NN predicted average Young's modulus of MCF and literature results [7].*

simulations on a case-by-case basis can waste time and become more expensive, hence the growing interest in the use of intelligent digital methods, such as the artificial neural networks method. This method offers a good balance between cost/ quality/performance. In this study, the combinations of artificial neural network method and finite element analysis were implemented and used to determine the elastic mechanical properties at different scale levels of the bone tissue nanostructure. Second step, an approach multi-scale using neural networks has been developed. This approach uses the results of finite element analysis for the learning phase. It makes it possible to generalize the results obtained by finite elements and to make the transition between the different scale levels. The results were compared and validated by other studies in the literature and good agreement was observed. This hybrid multi-scale approach makes it possible to quickly determine (a few seconds) the equivalent mechanical properties according to the parameters entered. Here, the method was only used to determine elastic properties but can be approved to identify equivalent mechanical properties related to fracture behavior.

#### **6.2 Relationship between the mechanics and the activities of bone cells in the process of bone remodeling**

On the other hand, the complement of this work aims to develop an FE model to show the methodology of bone remodeling, by considering the activities of osteoclasts and osteoblasts (**Figure 12**). The mechanical properties of bone are demonstrated by carefully considering the accumulation and mineralization of

**Figure 12.**

*Predicted bone adaptation sequences in the form of apparent bone density variation in gram per cubic centimeter [18].*

failures under the effect of fatigue of the bone material. The strain-damage coupled stimulation phase is shown, which monitors the level of autocrine and paracrine factors. Cell phones and their behavior are based on the dynamic equation of, who describes autocrine and paracrine interactions between osteoblasts and osteoclasts and calculates cell population dynamics and bone mass changes at a discrete site of bone remodeling (**Figure 13**). The FE model developed was implemented in the FE Abaqus code. An example of a human proximal femur is studied using the developed model. The model was able to predict the final adaptation of the human proximal femur similar to models seen in a human proximal femur. The results obtained reveal a complex spatio-temporal bone adaptation [18]. The proposed FEM model provides insight into how bone cells adapt their architecture to the mechanical and biological environment.

The loads on the hip joint sacrificed in the current work constitute the majority of load located on the mediolateral plane of a femur and its margin is clearly greater than the other loads. Therefore, the 2D femur was a reasonable representation of the 3D remodeling behavior. Furthermore, the simulations employed the fixed model parameters given in **Table 2**. The recorded values may be subject to change due to several factors (disease, age, drugs, sex, bone sites, etc.). Finally, the analysis considers only one factor for a single parameter value of the model has been modified of all data, unlike the other parameters, which were fixed. This hypothesis showed in particular that the levels of bone cells play an important role in the process of bone adaptation, which can be modulated by specific bone drugs. Nevertheless, for a future general analysis of AS, it is necessary to consider the full variation of the factor parameters simultaneously for different geometries of femurs.

#### **6.3 Examples of multiscale and multiphysics numerical modeling of biological tissues**

The theoretical-numerical simulation and predictive modeling of the behavior and growth of biological tissues is a strange and new technique. As a result, different and multiple knowledge.

tools necessary, it is an overlap between experimentation, digital and also the theoretical, which are not yet well studied or even understood. George D. et al. [19] presented some specific multiscale multiphysics techniques and analyzes for biological tissues applied to the predictive behavior of cortical veins as a function of microstrural properties, taking into account bone remodeling and growth as a

*Multi-Scale Modeling of Mechanobiological Behavior of Bone DOI: http://dx.doi.org/10.5772/intechopen.95035*

**Figure 13.** *Sequences of predicted density distributions (gray level) for three different remodeling load levels [18].*

**Figure 14.**

*Macroscopic evolution from applied external mechanical boundary conditions [19].*

function of local mechanobiology (**Figure 14**). The hypotheses and the approaches used are well mastered to discover and understand the mechanical-biological phenomena, as well as a clear vision for different life periods of biological changes and their use for industrial applications.

Comparable studies, taking into account local mechanics to biology, have been developed by Ruimermann [20] to develop specific techniques of bone remodeling of patients for 3D micro-architectures but no study has shown a real link with medical applications. Real. The model of this study will allow coupling between these different scales to be able to obtain a macroscopic mechanobiological model predictive of bone changes as a function of local biological constituents.

**Figure 15** shows a complementary analysis, which was carried out based on different boundary conditions. Through a triangular compressive load applied to

#### **Figure 15.**

*Mesoscopic (trabecular size) evolution for the triangular load condition [20]. (a) Trabecular bone density obtained with current 3D model and cell activation. (b) Schematic of trabecular bone density obtained froma 2D phenomenological model with same loading conditions from Weinans [21].*

#### **Figure 16.**

*Comparison of bone density distribution between current mechanobiological model and phenomenological model after applying triangular compressive mechanical load.*

#### *Multi-Scale Modeling of Mechanobiological Behavior of Bone DOI: http://dx.doi.org/10.5772/intechopen.95035*

the entire basic bone microstructure. As a result, the trabeculae are mostly directed along the main load directions with an increasingly smaller size and less intense loads. The localization of cell activation is linked to the calculated mechanobiological stimulus and its quantification is done in the same way.

The results obtained are compared with data extracted from a phenomenological model developed in the literature [21]. Here, comparable results are observed at the level of bone density distribution (**Figure 16**). As a result, the model developed has a good estimate of biological activations and a consistency more adaptable to the patient. The microstructural properties of the bone obtained is strongly related to mechanical and quantifiable biological parameters. Mechanical stability is obtained by provocation and localization of biological cells, and can be physically close to real cases.
