Multi-compartment Model - Multi-compartment Model

Multi-compartment Model

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:

$\begin{align} \dot{q}_1=q_1 k_{11}+q_2 k_{12}+\cdots+q_n k_{1n}+u_1(t) \\ \dot{q}_2=q_1 k_{21}+q_2 k_{22}+\cdots+q_n k_{2n}+u_2(t) \\ \vdots\\ \dot{q}_n=q_1 k_{n1}+q_2 k_{n2}+\cdots+q_n k_{nn}+u_n(t) \end{align}$

where

for

Or in matrix forms:

$\mathbf{\dot{q}}=\mathbf{Kq}+\mathbf{u}$

where $\mathbf{K}=\begin{bmatrix} k_{11}& k_{12} &\cdots &k_{1n}\\ k_{21}& k_{22} & \cdots&k_{2n}\\ \vdots&\vdots&\ddots&\vdots \\ k_{n1}& k_{n2} &\cdots &k_{nn}\\ \end{bmatrix} \mathbf{q}=\begin{bmatrix} q_1 \\ q_2 \\ \vdots \\ q_n \end{bmatrix}$ and $\mathbf{u}=\begin{bmatrix} u_1(t) \\ u_2(t) \\ \vdots \\ u_n(t) \end{bmatrix}$

In the special case where the elements of are constants (or zero) then there is a general solution.

where, ... and are the eigenvalues of ;, ... and are the respective eigenvectors of ;, .... and are constants and is the inverse matrix of .

This solution can be rearranged:

$\mathbf{q} = \Bigg[ \mathbf{v_1}\begin{bmatrix} c_1 & 0 & \cdots & 0 \\ \end{bmatrix} + \mathbf{v_2}\begin{bmatrix} 0 & c_2 & \cdots & 0 \\ \end{bmatrix} + \dots + \mathbf{v_n}\begin{bmatrix} 0 & 0 & \cdots & c_n \\ \end{bmatrix} \Bigg] \begin{bmatrix} e^{\lambda_1t} \\ e^{\lambda_2t} \\ \vdots \\ e^{\lambda_nt} \\ \end{bmatrix} - \mathbf{K}^{-1}\mathbf{u}$

This somewhat inelegant equation demonstrates that all solutions of an n-cell multi-compartment model with constant or no inputs are of the form:

$\mathbf{q} = \mathbf{A} \begin{bmatrix} e^{\lambda_1t} \\ e^{\lambda_2t} \\ \vdots \\ e^{\lambda_nt} \\ \end{bmatrix} + \mathbf{b}$

Where is a nxn matrix, is a nx1 vector and, ... and are constants.

Read more about this topic: Multi-compartment Model

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