 Multicompartment model

A multicompartment 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 homogenous 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 multicompartment model is a lumped parameters model.
Multicompartment 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 multicompartment 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.
It was introduced in the field of Operations research by Jay Forrester as a methodology that became known as System Dynamics.
Contents
Assumptions
Multicompartment modeling requires the adoption of several assumptions, such that systems in physical existence can be modeled mathematically:
 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.
Most commonly, the mathematics of multicompartment models is simplified to provide only a single parametersuch as concentrationwithin a compartment.
Singlecompartment model
Possibly the simplest application of multicompartment model is in the singlecell 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.
Multicompartment model
As the number of compartments increases, the model can be very complex and the solutions usually beyond ordinary calculation. Below shows a threecell model with interlinks among each other.
The formula for ncell multicompartment models become:where
 for
Or in matrix forms:
where
 and
 and
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:
 Closed model: No sinks or source, lit. all k_{oi} = 0 and u_{i} = 0;
 Open model: There are sinks or/and sources among cells.
 Catenary model: All compartments are arranged in a chain, with each pool connecting only to its neighbors. This model has two or more cells.
 Cyclic model: It's a special case of the catenary model, with three or more cells, in which the first and last cell are connected, i.e. k_{1n} ≠ 0 or/and k_{n1} ≠ 0.
 Mammillary model: Consists of a central compartment with peripheral compartments connecting to it. There are no interconnections among other compartments.
 Reducible model: It's a set of unconnected models. It bears great resemblance to the computer concept of forest as against trees.
See also
 Mathematical model
 Biomedical engineering
 Biological neuron models
 Compartmental models in epidemiology
 Physiologicallybased pharmacokinetic modelling
References
 Godfrey, K., Compartmental Models and Their Application, Academic (1983), ISBN 0122869702
 Anderson, D. H., Compartmental Modeling and Tracer Kinetics, SpringerVerlag Lecture Notes in Biomathematics #50 (1983), ISBN 0387123032
 Evans, W. C., Linear Systems, Compartmental Modeling, and Estimability Issues in IAQ Studies, in Tichenor, B., Characterizing Sources of Indoor Air Pollution and Related Sink Effects, ASTM STP 1287 (1996), pp. 239262 ISBN 0803120303
Categories: Mathematical modeling
 Systems theory
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