Type of mathematical 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. Sometimes, the physical system that we try to model in equations is too complex, so it is much easier to discretize the problem and reduce the number of parameters. Each compartment is assumed to be a homogeneous entity within which the entities being modeled are equivalent. A multi-compartment model is classified as a lumped parameters model. Similar to more general mathematical models, multi-compartment models can treat variables as continuous, such as a differential equation, or as discrete, such as a Markov chain. Depending on the system being modeled, they can be treated as stochastic or deterministic.
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. Most commonly, the mathematics of multi-compartment models is simplified to provide only a single parameter—such as concentration—within a compartment.
In Systems Theory.
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.
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
formula_1
formula_2
Where "k" is the proportionality.
Software.
Simulation Analysis and Modeling 2 SAAM II is a software system designed specifically to aid in the development and testing of multi-compartment models. It has a user-friendly graphical user interface
wherein compartmental models are constructed by creating a visual representation of the model. From this model, the program automatically creates systems of ordinary differential equations. The program can both
simulate and fit models to data, returning optimal parameter estimates and associated statistics. It was developed by scientists working on metabolism and hormones kinetics (e.g., glucose, lipids, or insulin). It was then used for tracer studies and pharmacokinetics. Albeit a multi-compartment model can in principle be developed and run via other software, like MATLAB or C++ languages, the user interface offered by SAAM II allows the modeler (and non-modelers) to better control the system, especially when the complexity increases.
Discrete Compartmental Model.
Discrete models are concerned with discrete variables, often a time interval formula_3. An example of a discrete multi-compartmental model is a discrete version of the Lotka–Volterra Model. Here consider two compartments prey and predators denoted by formula_4 and formula_5 respectively. The compartments are coupled to each other by mass action terms in each equation. Over a discrete time-step formula_3, we get
formula_7
Here
These equations are easily solved iteratively.
Continuous Compartmental Model.
The discrete Lotka-Volterra example above can be turned into a continuous version by rearranging and taking the limit as formula_17.
formula_18
This yields a system of ordinary differential equations. Treating this model as differential equations allows the implementation of calculus methods to study the dynamics of the system more in-depth.
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:
 formula_19
Where 
formula_20 for formula_21 (as the total 'contents' of all compartments is constant in a closed system)
Or in matrix forms:
 formula_22
Where
formula_23 and formula_24 (as the total 'contents' of all compartments is constant in a closed system)
In the special case of a closed system (see below) i.e. where formula_25 then there is a general solution.
 formula_26
Where formula_27, formula_28, ... and formula_29 are the eigenvalues of formula_30; formula_31, formula_32, ... and formula_33 are the respective eigenvectors of formula_30; and formula_35, formula_36, ... and formula_37 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_38 or formula_39 and also that one eigenvalue is 0, say formula_27
So
 formula_41
Where
 formula_42 for formula_43
This solution can be rearranged:
 formula_44
This somewhat inelegant equation demonstrates that all solutions of an "n-cell" multi-compartment model with constant or no inputs are of the form:
 formula_45
Where formula_46 is a "nxn" matrix and formula_28, formula_48, ... and formula_29 are constants.
Where formula_50
Model topologies.
Generally speaking, as the number of compartments increases, it is challenging both to find the algebraic and numerical solutions to 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:
References.
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