Model description:
This example deals with a model describing the control of insulin on glucose utilization in humans. The model is shown in Fig. 2. The experiment consists of an impulse input of glucose labeled with a tracer and of the measurement in plasma of glucose, labeled glucose and insulin concentrations. The measured insulin concentration acts as model input $u$, while the model output $y$ is the measured tracer glucose concentration. The control by insulin on the glucose system is exerted by insulin in a remote compartment $(x_3)$. The glucose system is described by two compartments which represent, respectively, glucose in rapidly $(x_1)$ and slowly equilibrating tissues $(x_2)$ which include the muscle tissues. Insulin control is exerted on glucose utilization in compartment 3 (insulin-dependent tissues) while glucose utilization in compartment 1 refers to insulin-independent tissues.
The system-experiment model is
$$\begin{align*} \dot{x}_1(t) &= - \left(k_p + \frac{F_{01}/V_1}{g(t)} + k_{21}\right)x_1(t) + k_{12}x_2(t)\\ \dot{x}_2(t) &= k_{21}x_1(t) - (k_{02} + x_3(t) + k_{12})x_2(t) \\ \dot{x}_3(t) &= -k_bx_3(t) + k_au(t)\\ y_1(t) &= x_1(t)/V \\ \end{align*}$$
The initial conditions are $x_1(0)=i_1,x_2(0)$, and $x_3(0)=0$. System parameters are presented in the table below.
| $x_1,x_2,x_3$ | glucose masses in compartments 1 and 2 and the concentration of insulin in a remote compartment 3, respectively; |
| $u$ | plasma insulin concentration; |
| $g$ | known plasma glucose concentration; |
| $y$ | plasma tracer glucose concentration; |
| $V_1$ | volume of the accessible compartment; |
| $F_{01}$ | constant component of insulin-independent glucose uptake with $k_p$ accounting for the proportional component; |
| $k_a,k_b,k_{21},k_{12},k_{02}$ | constant rate parameters |
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Publication details:
| Title | Global identifiability of nonlinear models of biological systems |
| Publication Type | Journal Article |
| Year of Publication | 2001 |
| Authors | Audoly, S., Bellu G., D'Angio L., Saccomani M.P., and Cobelli C. |
| Journal | IEEE Transactions on Biomedical Engineering |
| Volume | 48 |
| Pagination | 55-65 |
| Date Published | Jan |
| ISSN | 0018-9294 |
| Keywords | a priori global identifiability, algebra, algorithm, Algorithm design and analysis, Algorithms, Biological, Biological system modeling, biological system models, Biological systems, Biology computing, characteristic set, computer algebra techniques, differential algebra, Glucose, Humans, Insulin, Models, nonlinear dynamic models, Nonlinear dynamical systems, Nonlinear Dynamics, Nonlinear equations, nonlinear models, parameter estimation, Pharmacokinetics, physiological models, physiological systems, solution uniqueness, Testing, Time varying systems, time-varying parameters, zero initial conditions |
| Abstract | A prerequisite for well-posedness of parameter estimation of biological and physiological systems is a priori global identifiability, a property which concerns uniqueness of the solution for the unknown model parameters. 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. Here, the authors present a new algorithm for testing global identifiability of nonlinear dynamic models, based on differential algebra. The characteristic set associated to the dynamic equations is calculated in an efficient way and computer algebra techniques are used to solve the resulting set of nonlinear algebraic equations. The algorithm is capable of handling many features arising in biological system models, including zero initial conditions and time-varying parameters. Examples of usage of the algorithm for analyzing a priori global identifiability of nonlinear models of biological and physiological systems are presented. |
| DOI | 10.1109/10.900248 |