Mathematical Model

In the following, we decribe (in rather mathemathical terms) the actual model problem that we want to solve.

Continuous Problem

In principle, we want to describe processes that are continuously observable in time and space (e.g. continuum mechanics). Additionally, we are interested in systems of reaction-diffusion equations that compartmentalized. That is, where different compartments have different reaction-diffusion processes and the interaction between compartments happens through their compartment intesection (e.g. a living cell).

Compartments

We say that each compartment, denoted by $\Omega^k$, is an open, bounded, connected subset of Euclidean space $\mathbb{R}^d$, with Lipschitz boundary $\partial\Omega^k$ and space dimension $d=2$ or $d=3$. Moreover, we require that the compartments are the non-overlapping decomposition of an open, bounded, and connected domain $\Omega\subset\mathbb{R}^d$, such that,

$$\overline{\Omega}=\bigcup_{k=1}^{K}\overline{\Omega}^k\quad\text{and}\quad\Omega^k\cap\Omega^l=\emptyset\quad \text{with}\quad k\ne l,$$

where $K$ is the total number of compartments.

Species

We will distinguish species within the same compartment with a subscript, usually $i$, and species on different compartments with a superscript, usually $k$. For example, $u_i^k$ is the $i$-th species on the $k$-th compartment. Furthermore, we will denote bold letters to refer to all $N^k$ species in the $k$-th compartment, i.e, $\bm{u}^k:=[u^k_1,\dots,u_{N^k}^k]^T$.

Transport

We then characterize transport with a diffusive flux operator, $\mathcal{D}_i^k$, for the $i$-th species by

$$\mathcal{D}^k_i\left(\bm{u}^k\right):=-\mathsf{D}^k_{i\star} \nabla \bm{u}^k,$$

where $\mathsf{D}^k_{i\star}$ represents the $i$-th row of the self and cross-dispersion tensor $\mathsf{D}^k$, in other words, $\mathsf{D}^k_{i\star}\nabla\bm{u}^k:=\sum_{i=1}^{N^k}\mathsf{D}^k_{ij}\nabla u^k_j$.

Reaction Network

A (bio-)chemical reaction is the transformation of one species to another, i.e, $u_i^k\to u_j^k$. There are many natural processes that present a rich reaction network. Particularly, we will represent the deterministic rate of change of $u^k_i$ caused by chemical reactions with the reaction operator $\mathcal{R}_i^k\left(\bm{u}^k\right)$. Such an operator is required to be a Lipschitz function and be mass conservative.

Membrane

We call the boundaries of the compartments as membranes. In particular, we designate its geometry to be a $(d-1)$-manifold defined with respect to the boundary of the compartments, i.e.,

$$\Gamma^{kl}:= \left\{ \begin{matrix} \partial\Omega_k\cap\partial\Omega_l, \quad& \text{if }k\ne l\quad& \text{interior boundaries}\\ \partial\Omega_k\cap\partial\Omega, \quad& \text{if }k = l\quad& \text{exterior boundaries}. \end{matrix} \right.$$

where $\partial\Omega$ represents the boundaries of the domain $\Omega$.

Transmission Conditions

The flux rate at which compartment species are transformed and moved across the membrane depends on its concentration and the dispersion coefficients at which species can move on the surroundings of the membrane. In our model, we say that such a transport is equal to the outer flux $\mathcal{D}_i^k$ of the species $i$ leaving the compartment $k$ and is defined by a general transmission condition, $\mathcal{T}^{kl}_i$, that may take the form of typical Dirichlet, Neumann, and Robin boundary condition as well as complex chemical reaction networks, i.e.,

$$\mathcal{D}_i^{k}\left(\bm{u}^k\right)\cdot\mathbf{n}^k = \mathcal{T}_i^{kl}\left(\bm{u}^k,\bm{u}^l\right)\qquad \text{on }\Gamma^{kl},$$

where $\mathbf{n}^k$ is the outer normal vector on $\Omega^k$. As is natural, the transmission conditions must be mass conservative.

Strong Formulation

Joining all definitions from above, we obtain a boundary value problem (BVP) which it reads as follow:

Given an initial condition ${u_{(0)}}_i^k$ and a final time $T$, find $u_i^k$ such that

$$\begin{aligned} \partial_t u_i^k = -\nabla\cdot\mathcal{D}_i^k\left(\bm{u}^k\right) + \mathcal{R}_i^k\left(\bm{u}^k\right) &\qquad \text{in }\Omega^k\times(0,T),\\ \mathcal{D}_i^k\left(\bm{u}^k\right)\cdot \mathbf{n}^k = \mathcal{T}^{kl}_i(\bm{u}^k,\bm{u}^l) &\qquad \text{on }\Gamma^{kl}\times(0,T),\\ u_i^k = {u_{(0)}}_i^k &\qquad \text{in }\Omega^k, \end{aligned}$$

for every $k,l=1,\ldots,K$, and $i=1,\ldots,N^k$.