Compartmental Model
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A multi-compartment model is a type of
mathematical model A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
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 The lumped-element model (also called lumped-parameter model, or lumped-component model) simplifies the description of the behaviour of spatially distributed physical systems, such as electrical circuits, into a Topology (electrical circuits), t ...
model. Similar to more general
mathematical model A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
s, multi-compartment models can treat variables as continuous, such as a
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
, or as discrete, such as a
Markov chain A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
. Depending on the system being modeled, they can be treated as stochastic or deterministic. Multi-compartment models are used in many fields including
pharmacokinetics Pharmacokinetics (from Ancient Greek ''pharmakon'' "drug" and ''kinetikos'' "moving, putting in motion"; see chemical kinetics), sometimes abbreviated as PK, is a branch of pharmacology dedicated to determining the fate of substances administered ...
,
epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and determinants of health and disease conditions in a defined population. It is a cornerstone of public health, and shapes policy decisions and evidenc ...
,
biomedicine Biomedicine (also referred to as Western medicine, mainstream medicine or conventional medicine)
,
systems theory Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has causal boundaries, is influenced by its context, defined by its structu ...
, 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 Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
of
solute In chemistry, a solution is a special type of homogeneous mixture composed of two or more substances. In such a mixture, a solute is a substance dissolved in another substance, known as a solvent. If the attractive forces between the solvent ...
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 :\frac=u(t)-kq :C=\frac Where ''k'' is the proportionality.


Software

Simulation Analysis and Modeling 2
SAAM II SAAM II, short for "Simulation Analysis and Modeling" version 2.0, is a renowned computer program designed for scientific research in the field of bioscience. It is a descriptive and exploratory tool in drug development, tracers, metabolic diso ...
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 MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...
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 \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) and y(t) respectively. The compartments are coupled to each other by mass action terms in each equation. Over a discrete time-step \Delta t, we get \begin 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 Here * the x(t) and y(t) terms represent the number of that population at a given time t; * the \alpha x(t)\Delta t term represents the birth of prey; * the mass action term \beta x(t) y(t) \Delta t is the number of prey dying due to predators; * the mass action term \delta x(t) y(t) \Delta t represents the birth of predators as a function of prey eaten; * the \gamma y(t) \Delta t term is the death of predators; * \alpha, \beta, \delta, and \gamma are real valued parameters determining the weights of each transitioning term. 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 \Delta t \rightarrow 0. \begin &\lim_ \frac \equiv \frac = \alpha x - \beta x y\\ &\lim_\frac\equiv \frac = \delta x y - \gamma y \end 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: : \begin \dot_1=q_1 k_+q_2 k_+\cdots+q_n k_+u_1(t) \\ \dot_2=q_1 k_+q_2 k_+\cdots+q_n k_+u_2(t) \\ \vdots\\ \dot_n=q_1 k_+q_2 k_+\cdots+q_n k_+u_n(t) \end Where :0=\sum^n_ for j=1,2,\dots,n (as the total 'contents' of all compartments is constant in a closed system) Or in matrix forms: : \mathbf=\mathbf+\mathbf Where :\mathbf=\begin k_& k_ &\cdots &k_\\ k_& k_ & \cdots&k_\\ \vdots&\vdots&\ddots&\vdots \\ k_& k_ &\cdots &k_\\ \end \mathbf=\begin q_1 \\ q_2 \\ \vdots \\ q_n \end \mathbf=\begin u_1(t) \\ u_2(t) \\ \vdots \\ u_n(t) \end and \begin 1 & 1 &\cdots & 1\\ \end\mathbf=\begin 0 & 0 &\cdots & 0\\ \end (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 \mathbf=0 then there is a general solution. : \mathbf = c_1 e^ \mathbf + c_2 e^ \mathbf + \cdots + c_n e^ \mathbf Where \lambda_1, \lambda_2, ... and \lambda_n are the
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of \mathbf; \mathbf, \mathbf, ... and \mathbf are the respective
eigenvectors In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of \mathbf; and c_1, c_2, .... and c_n 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 In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
and
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
then either \lambda=0 or \begin 1 & 1 &\cdots & 1\\ \end\mathbf=0 and also that one
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
is 0, say \lambda_1 So : \mathbf = c_1 \mathbf + c_2 e^ \mathbf + \cdots + c_n e^ \mathbf Where : \begin 1 & 1 &\cdots & 1\\ \end\mathbf=0 for \mathbf=2, 3, \dots n This solution can be rearranged: : \mathbf = \Bigg \mathbf\begin c_1 & 0 & \cdots & 0 \\ \end + \mathbf\begin 0 & c_2 & \cdots & 0 \\ \end + \dots + \mathbf\begin 0 & 0 & \cdots & c_n \\ \end \Bigg\begin 1 \\ e^ \\ \vdots \\ e^ \\ \end This somewhat inelegant equation demonstrates that all solutions of an ''n-cell'' multi-compartment model with constant or no inputs are of the form: : \mathbf = \mathbf \begin 1 \\ e^ \\ \vdots \\ e^ \\ \end Where \mathbf is a ''nxn'' matrix and \lambda_2, \lambda_3, ... and \lambda_n are constants. Where \begin 1 & 1 &\cdots & 1\\ \end\mathbf=\begin a & 0 & \cdots & 0 \\ \end


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: #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''1''n'' ≠ 0 or/and ''k''''n''1 ≠ 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 A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
*
Biomedical engineering Biomedical engineering (BME) or medical engineering is the application of engineering principles and design concepts to medicine and biology for healthcare purposes (e.g., diagnostic or therapeutic). BME is also traditionally logical sciences ...
*
Biological neuron models Biological neuron models, also known as a spiking neuron models, are mathematical descriptions of the properties of certain cells in the nervous system that generate sharp electrical potentials across their cell membrane, roughly one millisecon ...
*
Compartmental models in epidemiology Compartmental models are a very general modelling technique. They are often applied to the mathematical modelling of infectious diseases. The population is assigned to compartments with labels – for example, S, I, or R, (Susceptible, Infectious, ...
*
Physiologically-based pharmacokinetic modelling Physiologically based pharmacokinetic (PBPK) modeling is a mathematical modeling technique for predicting the absorption, distribution, metabolism and excretion (ADME) of synthetic or natural chemical substances in humans and other animal species. ...


References

*Godfrey, K., ''Compartmental Models and Their Application'', Academic Press, 1983 (). *Anderson, D. H., ''Compartmental Modeling and Tracer Kinetics'', Springer-Verlag Lecture Notes in Biomathematics #50, 1983 (). *Jacquez, J. A, ''Compartmental Analysis in Biology and Medicine'', 2nd ed., The University of Michigan Press, 1985. *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, pp. 239–262, 1996 ({{ISBN, 0-8031-2030-3). Mathematical modeling Systems theory