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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Multi-compartment model ： ウィキペディア英語版 Multi-compartment 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. Each compartment is assumed to be a homogeneous 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 multi-compartment model is a lumped parameters model. 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. 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. Most commonly, the mathematics of multi-compartment models is simplified to provide only a single parameter—such as concentration—within a 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 :$\backslash frac=u(t)-kq$ :$C=\backslash frac$ where ''k'' is the proportionality. As the number of compartments increases, the model can be very complex and the solutions usually beyond ordinary calculation. Below shows a three-cell model with interlinks among each other. The formulae for n-cell multi-compartment models become: : $$ \begin \dot_1=q_1 k_+q_2 k_+\cdots+q_n k_+u_1(t) \\ \dot_2=q_1 k_+q_2 k_+\cdots+q_n k_+u_2(t) \\ \vdots\\ \dot_n=q_1 k_+q_2 k_+\cdots+q_n k_+u_n(t) \end Where :$0=\backslash sum^n\_\}=\backslash mathbf+\backslash mathbf$ Where :$\backslash mathbf=\backslash begin$ k_& k_ &\cdots &k_\\ k_& k_ & \cdots&k_\\ \vdots&\vdots&\ddots&\vdots \\ k_& k_ &\cdots &k_\\ \end \mathbf=\begin q_1 \\ q_2 \\ \vdots \\ q_n \end \mathbf=\begin u_1(t) \\ u_2(t) \\ \vdots \\ u_n(t) \end and $$ \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 $\backslash mathbf=0$ then there is a general solution. : $\backslash mathbf\; =\; c\_1\; e^\; \backslash mathbf\; +\; c\_2\; e^\; \backslash mathbf\; +\; \backslash cdots\; +\; c\_n\; e^\; \backslash mathbf$ Where $\backslash lambda\_1$, $\backslash lambda\_2$, ... and $\backslash lambda\_n$ are the eigenvalues of $\backslash mathbf$; $\backslash mathbf$, $\backslash mathbf$, ... and $\backslash mathbf$ are the respective eigenvectors of $\backslash mathbf$; 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 and eigenvector then either $\backslash lambda=0$ or $$ \begin 1 & 1 &\cdots & 1\\ \end\mathbf=0 and also that one eigenvalue is 0, say $\backslash lambda\_1$ So : $\backslash mathbf\; =\; c\_1\; \backslash mathbf\; +\; c\_2\; e^\; \backslash mathbf\; +\; \backslash cdots\; +\; c\_n\; e^\; \backslash mathbf$ Where : $$ \begin 1 & 1 &\cdots & 1\\ \end\mathbf=0 for $\backslash mathbf=2,\; 3,\; \backslash dots\; n$ 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: : $\backslash mathbf\; =\; \backslash mathbf$ \begin 1 \\ e^ \\ \vdots \\ e^ \\ \end Where $\backslash mathbf$ is a ''nxn'' matrix and $\backslash lambda\_2$, $\backslash lambda\_3$, ... and $\backslash lambda\_n$ are constants. Where $\backslash begin$ 1 & 1 &\cdots & 1\\ \end\mathbf=\begin a & 0 & \cdots & 0 \\ \end 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Multi-compartment model」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.