## 4 Modeling of Physical Phenomena  
In this section, we review the modeling and simulation of the most relevant physical phenomena that take place in biomaterials in BTE applications, such as biodegradation. Most BTE products exploit the ability of biomaterials to dissolve over time, finally resulting in the removal of the scaffold implant. This class of biomaterials are synthetic polymers: Lactic polymer (trimethylene carbonate p,t-lactide (TMCDLLA)) [90], poly-e-caprolactone (PCL) [91; 92], polylactic acid (PLA) [93], and polyglycolic acid (PGA) [94; 95], to cite a few. Even though the degradation products are naturally evacuated from the human body, some secondary problems have been reported in the reaction of these residuals with the living tissues [96]. Modeling and simulation of biodegradation of polymeric biomaterials were mainly developed after the theory proposed by [PERSON] [97]. This modeling was the basis for the development of a number of applications in BTE using polymeric scaffolds and it considers water diffusion within the bulk polymeric biomaterial according to a Fickean law as follows,  
\[\dot{d}=\alpha\Delta d\text{ in }\Omega+\text{boundary and initial conditions}, \tag{3}\]  
where \(d\) is the water (aqueous) concentration, \(\alpha\) the diffusion coefficient, and \(\Delta\) the Laplacian operator. The dot on \(d\) denotes time derivative and \(\Omega\) the biomaterial domain. Water concentration, \(W\), within the polymer is then related to the change rate of the molecular weight of the biomaterial due to hydrolysis and assumed to depend on the local water content,  
\[W=-\beta d\text{ in }\Omega. \tag{4}\]  
where \(\beta\) is a constant property of the material. The model introduced above was applied to the study of the degradation of a unit cell of the scaffold microstructure [66] and coupled with a bone growth multiscale model [68; 69; 73]. Other researchers [98; 99] presented a similar but extended model to analyze degradation of biopolymers, which takes into consideration crystallization. Further studies [100; 101] introduced a mathematical continuum formulation based on the mixture theory to model degradation, transport of molecules across the extracellular matrix, and swelling in a hydrogel scaffold. The model was useful to investigate hydrolytic and enzymatic degradation. Moreover, a similar model for the simulation of the hydrolysis phenomena in polymeric biomaterials was presented in [102]. In this approach, the hydrolysis reaction was modeled by a fundamental stochastic process and an additional autocatalytic effect. Results were shown over different polymeric matrices providing a good agreement with experimental data in the literature. Inorganic bioactive materials, such as bioglasses, are widely used in BTE due to their ability to react with the body fluid and dissolve, finally resulting in the formation of a hydroxyapatite surface layer. This layer can form stable bonds with the adjacent living tissue, which is specially well suited to bone implants and BTE [11; 103; 104; 105; 106; 107]. Dissolution of bioactive glasses was macroscopically modeled using a physical-chemical model which turned into a reaction-diffusion continuum model [108]. Dissolution was represented using the Voxel-FEM approach. Additionally, a reaction-diffusion modeling approach was proposed for the simulation of degradation of calcium phosphate scaffolds [109]. Recently, a generic mathematical framework for the simulation and design of dissolution of biomaterials for tissue engineering and drug delivery applications has been introduced [110]. The model was experimentally validated by means of a straightforward ad hoc setup that considered the dissolution of bicarbonate pellets. Even though magnesium implants have been used since several decades ago, they have gained an increasing interest in the last years. The main motivations are the great advantages they offer versus traditional metallic, bioceramic, or polymeric implants.