Multi-compartment model

A multi-compartment model is a type of mathematical model used for describing the way materials or energies are transmitted among the "compartments" of a system. Each compartment is assumed to be a homogeneous entity within which the entities being modelled are equivalent. For instance, in a pharmacokinetic model, the compartments may represent different sections of a body within which the concentration of a drug is assumed to be uniformly equal.
Hence a multi-compartment model is a lumped parameters model.
Multi-compartment models are used in many fields including pharmacokinetics, epidemiology, biomedicine, systems theory, complexity theory, engineering, physics, information science and social science. The circuits systems can be viewed as a multi-compartment model as well.
In systems theory, it involves the description of a network whose components are compartments that represent a population of elements that are equivalent with respect to the manner in which they process input signals to the compartment.
Most commonly, the mathematics of multi-compartment models is simplified to provide only a single parameter—such as concentration—within a compartment.
Single-compartment model.
Possibly the simplest application of multi-compartment model is in the single-cell concentration monitoring (see the figure above). If the volume of a cell is "V", the mass of solute is "q", the input is "u"("t") and the secretion of the solution is proportional to the density of it within the cell, then the concentration of the solution "C" within the cell over time is given by
where "k" is the proportionality.
Multi-Compartment Model.
As the number of compartments increases, the model can be very complex and the solutions usually beyond ordinary calculation.
The formulae for n-cell multi-compartment models become:
Where 
Or in matrix forms:
Where
In the special case of a closed system (see below) i.e. where formula_9 then there is a general solution.
Where formula_11, formula_12, ... and formula_13 are the eigenvalues of formula_14; formula_15, formula_16, ... and formula_17 are the respective eigenvectors of formula_14; and formula_19, formula_20, ... and formula_21 are constants.
However it can be shown that given the above requirement to ensure the 'contents' of a closed system are constant, then for every pair of eigenvalue and eigenvector then either formula_22 or formula_23 and also that one eigenvalue is 0, say formula_11
So
Where
This solution can be rearranged:
This somewhat inelegant equation demonstrates that all solutions of an "n-cell" multi-compartment model with constant or no inputs are of the form:
Where formula_30 is a "nxn" matrix and formula_12, formula_32, ... and formula_13 are constants.
Where formula_34
Model topologies.
Generally speaking, as the number of compartments increase, it is challenging both to find the algebraic and numerical solutions of the model. However, there are special cases of models, which rarely exist in nature, when the topologies exhibit certain regularities that the solutions become easier to find. The model can be classified according to the interconnection of cells and input/output characteristics: