## 5 Mathematical Models
### Models at the Macro-Scale
#### 5.1.3 Transport and Response of Biological and Chemical Species  
This subsection will describe models that are used to study how the tissue reacts to the chemical and mechanical environment in the culture medium. Although there are not many studies on this topic, this aspect is crucial for models that seek to create better culture protocols [64]. Creating a strict framework of conservation laws for mixes of interacting continua has been the subject of countless investigations [108]. According to this theory, the characteristics of composite biological tissues are assessed in relation to the volume percentage of the constituent parts (i.e., extracellular matrix, culture medium, and cells). A system of conservation equations that control the transfer of mass and momentum between the component phases is the foundation of the multi-phase models. For a system comprising incompressible phases, if inertial effects and body forces are negligible, then the equations governing the \(i\)-th phase may be expressed as  
\[\frac{\partial}{\partial t}(\rho_{i}\theta_{i})+\rho_{i}\text{div}( \theta_{i}\mathbf{v})-D_{i}\
abla^{2}\theta_{i} = S_{i}, \tag{62}\]\[\mathrm{div}(\theta_{i}\,\mathbf{T}_{i})+\sum_{j\
eq i}\mathbf{F}_{ij} = \mathbf{0} \tag{63}\]  
where \(\rho_{i}\), \(\theta_{i}\), \(\mathbf{v}\), \(D_{i}\), and \(\mathbf{T}_{i}\) are, respectively, the density, the volume fraction (a value in \([0\;1]\) interval), the velocity, the diffusive coefficient, and the stress tensor characterizing the \(i\)-th phase, where \(S_{i}\) is the net rate of mass transfer into the \(i\)-th phase and \(\mathbf{F}_{ij}\) denotes the force acting on phase \(i\) as a result of interactions with phase \(j\). The sum of all phases adds up to \(1\) (\(\sum_{i}\theta_{i}=1\)) following the conservation of mass. The biological component can also be modeled as a single-phase material, for instance, cells only, where Equation (62) can be used to estimate \(\theta_{c}\) as the cell density; in such a case, the conservation of mass law above does not apply; for example, Equation (63) does not apply. Depending on the type of biological phase, diffusive and convective transport are taken into account in Equation (62). For example, cells are typically thought of as a single phase that diffuses slowly and does not undergo convection. The force balance between the various phases is captured by the equation for the conservation of momentum (see Equation (63)). Interactions between cell populations, between the cells and the scaffold, and between the cells and the media are included in Equation (63). More complex processes, including tissue evolution through cell proliferation or ECM deposition (in biology, the extracellular matrix (ECM), also called the intercellular matrix (ICM), is a network consisting of extracellular macromolecules and minerals that provide structural and biochemical support to surrounding cells [109]), can be considered by using the term \(S_{i}\). The introduction of functions controlling cell proliferation or ECM formation is possible. Proliferation and death terms are typically included in cell growth functions, and they depend on the available cell volume fraction \(\theta_{c}\) and the concentration of nutrients like oxygen. Indeed, it is widely known that the migration, differentiation, and proliferation of cells are influenced by growth factors, waste products like lactate, and nutrients like oxygen (called _\"chemical species\"_). Mass conservation is used to describe the distribution of any solute, assuming Fickian diffusion and setting the initial and boundary conditions. \[\frac{\partial C_{i}}{\partial t}+\mathrm{div}(C_{i}\mathbf{v})-D_{i}\
abla^{2}C _{i} = M_{i}, \tag{64}\]  
where \(\mathbf{v}\) is the media flow velocity, \(D_{i}\) is the appropriate coefficient of diffusion, \(M_{i}\) is a term indicating the metabolic reaction, and \(C_{i}\) is the concentration of the \(i\)-th species (in, for example, oxygen).