## 5 Mathematical Models
### Models at the Meso Scale
#### 5.2.2 Cellular Particle Dynamics  
In the previous subsection, MCMC methods were introduced. While it is incredibly helpful for various statistical physics and biology problems, there are some situations that might not even be amenable to stochastic approaches [49; 117]. CPD is based, as in the Subcellular Element Model (SEM) approach [118], on the idea to \"_construct a cell from a cloud of \(\mathrm{N}_{CellularElements}\) sub-cellular entities_\" called elements [49; 118]. The elements, belonging to a single cell, all together define the same cell and, through their relative positions and in response to the cell's internal and external forces, determine the cell's shape (see Figure 6). The SEM approach (that takes place in a Lagrangian framework) is based on a stochastic assumption using Langevin equations [119]. The stochastic variables are the position vectors \(\mathbf{y}_{\alpha_{i}}\) of each element, where the notation \(\alpha_{i}\) denotes the \(\alpha\)-th element of the \(i\)-th cell, and where \(\mathbf{y}_{\alpha_{i}}=\left(y_{\alpha_{i}}^{1},\ldots,y_{\alpha_{i}}^{N}\right) \in\Re^{N}\). In the simplest case, in which the inertial terms in the equation of motion are discarded and in which the movement of elements is due to stochastic forces and biomechanical interactions, the following equation is valid for the generic element \(\alpha_{i}\):  
\[\mu\frac{\partial\mathbf{y}_{\alpha_{i}}}{\partial t}=\mathbf{\eta}_{\alpha_{i}}- \underbrace{\
abla_{\mathbf{y}_{\alpha_{i}}}\left[\sum_{\beta_{i}\
eq\alpha_{i}}V _{intra}\left(\left|\mathbf{y}_{\alpha_{i}}-\mathbf{y}_{\beta_{i}}\right|\right) \right]}_{\text{Internal forces }\mathbf{F}_{\alpha_{i}}^{intra}}\underbrace{-\
abla_{\mathbf{y}_{ \alpha_{i}}}\left[\sum_{i\
eq i}\sum_{\beta_{i}}V_{inter}\left(\left|\mathbf{y}_{ \alpha_{i}}-\mathbf{y}_{\beta_{i}}\right|\right)\right]}_{\text{External forces }\mathbf{F}_{\alpha_{i}}^{inter}} \tag{79}\]  
where \(\mu\) is the friction coefficient, \(\mathbf{\eta}_{\alpha_{i}}=\left(\eta_{\alpha_{i}}^{1},\ldots,\eta_{\alpha_{i}}^{ N}\right)\in\Re^{N}\) is a stochastic force modeled as a Gaussian white noise with zero mean and variance \(\left\langle\eta_{\alpha_{i}}^{n}(t),\eta_{\beta_{i}}^{m}(t^{\prime})\right\rangle\) with  
\[\left\langle\eta_{\alpha_{i}}^{n}(t),\eta_{\beta_{i}}^{m}(t^{\prime})\right\rangle =2D\delta_{n,m}\delta_{i,j}\delta_{\alpha_{i},\beta_{i}}\delta_{t,t^{\prime}}, \tag{80}\]  
Figure 6: A schematic figure showing two cells constructed from elements, with intra-cellular potentials binding elements of a given cell to each other, and inter-cellular potentials binding adjacent elements from neighboring cells. where \(D\) is the self-diffusion coefficient. The Equation (80) indicates that stochastic forces are uncorrelated in all the possible variables: between cells, between elements within a cell, between spatial dimensions, and between two different instants. The vectors \(\mathbf{F}_{\alpha_{i}}^{Intra}=\left(F^{Intra}_{\alpha_{i}}^{1},\ldots,F^{Intra}_{ \alpha_{i}}^{N}\right)\in\Re^{N}\) and \(\mathbf{F}_{\alpha_{i}}^{Inter}=\left(F^{Inter}_{\alpha_{i}}^{1},\ldots,F^{Inter}_{ \alpha_{i}}^{N}\right)\in\Re^{N}\) represent the internal and external biomechanical forces, respectively, acting on the generic element \(\alpha_{i}\). The functions \(V_{intra}\Big{(}d_{\alpha_{i},\beta_{j}}\Big{)}\) and \(V_{inter}\Big{(}d_{\alpha_{i},\beta_{j}}\Big{)}\), which depend on the distance  
\[d_{\alpha_{i},\beta_{j}}=\left|\mathbf{y}_{\alpha_{i}}-\mathbf{y}_{\beta_{j}}\right|= \left(\sum_{n=1}^{N}\left(y_{\alpha_{i}}^{n}-y_{\beta_{j}}^{n}\right)^{2} \right)^{\frac{1}{2}}\]  
between two elements, represent the intra- and inter-cellular interaction potential between two elements in the same and different cells, respectively. In [49], Flenner et al.