User:Odowdall/Multi-compartment model

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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 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 stochastic or deterministic.^[1]

Multi-compartment models are used in many fields including pharmacokinetics, epidemiology, biomedicine, systems theory, complexity theory, engineering, physics, information science and social science. 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.

Discrete Models

Discrete models are concerned with discrete variables, often a time interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \Delta t} . An example of a discrete stochastic multi-compartmental model is a discrete version of the Lotka–Volterra Model. Here consider two compartments prey and predators denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle x(t)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle y(t)} respectively. The compartments are coupled to each other by mass action terms in each equation. Over a discrete time-step Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \Delta t} , we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \begin{align} 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{align}}

Here

the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle x(t)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle y(t)} terms represent the number of that population at a given time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle t} ;
the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \alpha x(t)\Delta t} term represents the birth of prey;
the mass action term Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \beta x(t) y(t) \Delta t} is the number of prey dying due to predators;
the mass action term Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \delta x(t) y(t) \Delta t} represents the birth of predators as a function of prey eaten;
the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \gamma y(t) \Delta t} term is the death of predators;
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \alpha, \beta, \delta, } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \Delta t \rightarrow 0} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \begin{align} &\lim_{\Delta t \rightarrow 0} \frac{x(t + \Delta t)-x(t)}{\Delta t} \equiv \frac{d x}{dt} = \alpha x - \beta x y\\ &\lim_{\Delta t \rightarrow 0}\frac{y(t + \Delta t)-y(t)}{\Delta t}\equiv \frac{d y}{dt} = \delta x y - \gamma y \end{align} }

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.

References

^ Sanft, Rebecca (2020). Exploring mathematical modeling in biology through case studies and experimental activities. Anne Walter. London. ISBN 978-0-12-819596-3. OCLC 1148203409.{{cite book}}: CS1 maint: location missing publisher (link)

[1] Sanft, Rebecca (2020). Exploring mathematical modeling in biology through case studies and experimental activities. Anne Walter. London. ISBN 978-0-12-819596-3. OCLC 1148203409.{{cite book}}: CS1 maint: location missing publisher (link)

[1]