Consider a random lattice where the position of each vertex $i$ is governed by $$ \eta \frac{d \mathbf{x}_i}{dt}=\mathbf{F}_i(t), $$ where $\mathbf{F}_i(t)$ denotes the total force acting on vertex $i$ at time $t$. This is known in epithelial biology as the vertex model.
In this research paper, $\mathbf{F}_i$ is explicitly given as $$ \mathbf{F}_i=\sum_{\alpha} \left(-K (A_\alpha(t) -A^{(0)})\nabla_iA_\alpha(t)-\Gamma P_\alpha(t) \nabla_i P_\alpha(t) \right)-\sum_j\Lambda \nabla_il_{ij}(t), $$ where $K,A^{(0)},\Gamma$ and $\Lambda$ are constants, $A_\alpha$ and $P_\alpha$ are the area and perimeter of cell (face between edges) $\alpha$ and $l_{ij}$ is the length of the cell edge (or bond) shared by vertices $i$ and $j$. $\nabla_i$ denotes the gradient operator evaluated at $\mathbf{x}_i$.
My question is: given such $\mathbf{F}_i$, how can I solve the first ODE (via Euler's method, for example)? Shouldn't I need to have $A_\alpha, P_\alpha$ and $l_{ij}$ written explicitly in terms of $\mathbf{x}_i$? Can I even do this?
I'm missing something but I'm not quite sure what it is. Any ideas are appreciated!