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. Often times, this is done when the physical system is too complex to model with a full equation, 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.
Instant homogeneous distribution of materials or energies within a "compartment." The exchange rate of materials or energies among the compartments is related to the densities of these compartments. Usually, it is desirable that the materials do not undergo chemical reactions while transmitting among the compartments. When concentration of the cell is of interest, typically the volume is assumed to be constant over time, though this may not be totally true in reality.
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
d q d t = u ( t ) − k q {\displaystyle {\frac {\mathrm {d} q}{\mathrm {d} t}}=u(t)-kq}
C = q V {\displaystyle C={\frac {q}{V}}}
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 Δ t {\displaystyle \Delta t} . 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 x ( t ) {\displaystyle x(t)} and y ( t ) {\displaystyle y(t)} respectively. The compartments are coupled to each other by mass action terms in each equation. Over a discrete time-step Δ t {\displaystyle \Delta t} , we get
x ( t + Δ t ) = x ( t ) + α x ( t ) Δ t − β x ( t ) y ( t ) Δ t y ( t + Δ t ) = y ( t ) + δ x ( t ) y ( t ) Δ t − γ y ( t ) Δ t . {\displaystyle {\begin{aligned}x(t+\Delta t)&=x(t)+\alpha x(t)\Delta t-\beta x(t)y(t)\Delta t\\y(t+\Delta t)&=y(t)+\delta x(t)y(t)\Delta t-\gamma y(t)\Delta t.\end{aligned}}}
Here
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