# Three Compartment Model Describing the Dynamics of a Drug in a Tissue

## Model description:

Consider the system

\begin{align*} \dot{x}_1 &= p_{13}x_3 + p_{12}x_2 - p_{21}x_1+u \\ \dot{x}_2 &= -p_{12}x_2 + p_{21}x_1 \\ \dot{x}_3 &= -p_{13}x_3 \\ y &= x_2, \end{align*}

where $x=[x_1, x_2, x_3]$ is the state vector, e.g. $x_1,x_2,x_3$ are drug masses in compartment 1, 2 and 3, respectively; initial conditions are $x_1(0)=0$, $x_2(0)=0$, $x_3(0)=0$; $u$ is the drug input; $y$ is the measured drug output; $p=[p_{12},p_{21},p_{13}]$ is the rate parameter vector (assumed constant).

3

## Publication details:

 Title A new differential algebra algorithm to test identifiability of nonlinear systems with given initial conditions Publication Type Conference Paper Year of Publication 2001 Authors Saccomani, M.P., Audoly S., Bellu G., and D'Angio L. Conference Name Proceedings of the 40th IEEE Conference on Decision and Control, 2001. Date Published 12/2001 Publisher IEEE Conference Location Orlando, FL ISBN Number 0-7803-7061-9 Accession Number 7212178 Keywords differential equations, identification, nonlinear systems, polynomials Abstract A priori global identifiability is a fundamental prerequisite for model identification. It concerns uniqueness of the parametric structure of a dynamic model describing given input and output functions measured during an experiment. Assessing a priori global identifiability is particularly difficult for nonlinear dynamic models. Various approaches have been proposed in the literature, but no solution exists in the general case. The introduction of concepts of differential algebra and in particular the concept of characteristic set of a differential ideal introduced by Ritt (1950) have proven very useful tools in identifiability analysis. Yet the construction of an efficient algorithm still remains a difficult task. An improvement on existing algorithms has been published by some of the present the authors (Saccomani et al., 2000). Unfortunately this algorithm, like all other algorithms based on differential algebra, may run into difficulties for systems which are started at certain specific initial conditions. We propose a new version of the algorithm which gives the correct answer even if the system is started at special states from which the accessibility property is not guaranteed DOI 10.1109/.2001.980295