# Tuning the Cell and Biological Tissue Environment through Magneto-Active Materials
## 3 Theoretical and Computational Frameworks for Mechanically Mediated Biological Processes
### Modelling Mechanical Effects on Bone Development and Healing  
The interconnected multiscale nature of bone reconstruction and growth makes their modelling very challenging. Regarding these links through the scales, bone, cartilage and other fibrous tissues provide a scaffold/environment that supports cellular scale behaviour, which in turn is responsible for tissue synthetisation [72]. Furthermore, the cellular function is governed by signalling molecules and gene activation at the intracellular level, whose development is highly influenced by the extracellular environment [73]. To overcome the experimental limitations and difficulties to deeply understand such mechanisms, theoretical and computational modelling highlight as tools for testing hypotheses and further study these processes through the different scales. In this regard, [PERSON] [36] neatly classified the historical stages of modelling techniques into mathematical, biological and mechano-biological. Mathematical models are primarily concerned with the influence of mechanical loads on bone development. These models have suggested a localised mechanical behaviour within the healing area and have provided direct relations with the resulting tissue [26]. Despite their evolution, mathematical models are thought to lead primarily to a qualitative understanding of the behaviour of the tissue. These models experienced a significant advance when, aiming at predicting cell behaviour, different biological characteristicsand behaviours were incorporated. To this end, they were extended to account for time varying processes representing the flux of conditions in the healing stage, independent cellular activity, growth factor production and angiogenesis [39]. These improved formulations have allowed for exploring the role of the cell, extracellular matrix and growth factors concentrations on haptotactic and haptokinetic migration speed, cell proliferation rate, and cell differentiation [74,75]. [PERSON] et al. [76] examined osteoblast activity and its effect on reconstruction until apoptosis, while [PERSON] et al [77, 78] added cellular and intracellular activity, further raising the possibility of incorporating cell movement and directional angiogenesis. Moreover, mechanobiological models seek to amend the lack of mechanical phenomena in the previous analyses. These approaches have helped with exploring the effect of tissue patterning, defined by the arrangement, location, migration and subsequent changes of cells and extracellular matrix. The accuracy of mechanobiological methods is hampered by the increase in complexity of the systems, often dealing with interconnected PDE, finite element (FE) problems, lattice-based models and the modelling of biological processes [79, 80, 81]. For the incorporation of mechanical stimuli, different variables have been proposed showing a good correlation with experimental data: intermittent hydrostatic stress, octahedral shear strain, deviatoric and dilatational strain [82], substituting fluid velocity for hydrostatic pressure [79, 83] or a combination of these [84, 85]. Other approaches based on experimental studies include differences in linearity during healing based on phenotype [86]. More complex approaches include sequentially two or more of the phases in bone healing: inflammation phase, soft callus, hard callus and remodelling, delays between cell stimulation and tissue formation, effect of callus reabsorption in geometry [36]. More recently, based on the pioneering work by [PERSON] and [PERSON] [87], other approaches explore the mechanics of bone growth and regeneration under fixed forces and additional biological constraints, accounting for bone growth dynamics, depending on the availability of nutrients and hormones. To this end, the authors developed a finite element (FE) formulation for finite deformations coupling nonlinear mechanics with a mass proliferation equation, whose dynamics are governed by mechanical stimuli and internal variables associated with the nutrients and hormones.