## 2 Mechanics Modeling
### Simulation of Applications of Interest  
In silico modeling in BTE is becoming a predictive tool for bone regeneration simulation within the scaffold reducing the number of in vivo/in vitro experiments that need to be performed. Bone tissue regeneration in vivo using scaffolds inherently attends to two well-differentiated spatial and temporal scales. One is the tissue level or macroscopic scale, with the other one being  
Figure 1: Potential lattice-based scaffold microstructures analyzed for bone tissue engineering (BTE) applications. Reprinted with permission from [37]. named as pore level or micro/mesoscopic scale. Therefore, the mathematical and in silico modeling of tissue growth within scaffolds has been usually restricted to one of these scales. At the macroscopic scale, finite element analyses (FEA) has been used to analyze the solid mechanical behavior of the scaffold. Two different modeling approaches can be found in the literature at this scale: FEA models, that simulate the fluid flow with poroelastic materials (in which the flow is defined according to Darcy's law) and computational fluid dynamics (CFD) models, which solve the governing equations of fluid flow dynamics. On the one hand, FEA have tried to investigate the effect of scaffold architecture on the neotissue formed within the scaffold. The mathematical model of cell/tissue differentiation proposed by [PERSON] et al. [39] has been widely used in bone tissue engineering applications [40; 41; 42; 43; 44; 45; 46; 47]. In these studies, cell proliferation is modeled as a diffusion problem and macroscopic variables, such as the shear strain and fluid flow, are considered as stimuli for cell differentiation. Osteochondral defect repair using scaffolds has been simulated in [40]. [PERSON] et al. [42] predicted tissue differentiation with FEA by means of a lattice modeling approach in a bone chamber and compared the numerical results with in vivo bone formation. [PERSON] et al. [41] used a CAD-based model to analyze the effects of scaffold pore morphology on cell differentiation stimulus values. These models are either based on computer aided design (CAD) [42] or on micro-computed tomography (\(\upmu\)CT) [43; 44; 47]. However, CAD models are not able to capture the strain distributions seen in the \(\upmu\)CT [48]. On the other hand, CFD characterizes the hydrodynamic field imposed on cells within different types of scaffolds [48; 49; 50; 51; 52; 53]. They allow numerical prediction of the shear stresses induced at the internal walls in relation with the scaffold geometry and, thus, they qualitatively relate the flow of culture medium with the macroscopic 3D shear stresses [54; 55]. It is well known that flow-induced shear stress produces a significant stimulatory effect and plays a critical role in the physiological responses [56; 57; 58; 59]. More recent studies have predicted the new tissue formed and have related the wall shear stresses (WSS) with cellular processes such as proliferation, differentiation, or mineralization of the extracellular matrix by means of CFD analyses. In particular, [PERSON] et al. [60] investigated the influence of fluid-flow-induced shear stress on the proliferation, differentiation, and matrix deposition of human periorestal-derived cells in 3D Ti6 Al4V scaffolds. [PERSON] et al. [61] numerically described the growth of a neotissue in a perfusion bioreactor where the growth was coupled to oxygen concentration and shear stress. [PERSON] et al. [62] calculated the optimal fluid flow rate to be applied to the bioreactor for BTE experiments by means of CFD for different scaffold pore shapes, pore diameters, and porosities. They employed a combination of CFD with mechano-regulation theories to optimize the external flow rate for a perfusion bioreactor. Such flow rate would maximize the scaffold surface fraction, whose WSS was in the range required for mineralization. In the microscopic scale, [PERSON] et al. [63] have presented a mechanobiological model applied to a periodic unit cell of the scaffold microstructure.