# Nonlinear Models of Biological Systems (3)

## Model description:

This example deals with a model described in E. Walter, Identifiability of State Space Models. Berlin, Germany:Springer-Verlag, 1982. The model is shown in Fig. 3. All fluxes are assumed linear except some leaving compartment 3. In particular, parameters $k_{13}$, $k_{23}$, and $k_{43}$ are assumed to be parametrically linearly controlled by the arrival compartment, i.e. $k_{13}(x_1) = k_{13}x_1$, $k_{23}(x_2) = k_{23}x_2$ and $k_{43}(x_4) = k_{43}x_4$.

The system-experiment model is

\begin{align*} \dot{x}_1(t) &= - (k_{01} + k_{21})x_1(t) + k_{12}x_2(t) + k_{13}x_1(t)x_3(t)\\ \dot{x}_2(t) &= k_{21}x_1(t) - (k_{02} + k_{12} + k_{32} +k_{42})x_2(t) - k_{23}x_3(t)x_2(t) + k_{24}x_4(t)\\ \dot{x}_3(t) &= k_{32}x_2(t) - [k_{03} + k_{13}x_1(t) + k_{23}x_2(t) - k_{43}x_4(t)]x_3(t) + k_{34}x_4(t) + u(t)\\ \dot{x}_4(t) &= k_{43}x_2(t) + k_{43}x_3(t)x_4(t) - (k_{24} + k_{34})x_4(t)\\ y_1(t) &= x_1(t) \\ y_2(t) &= x_2(t) \end{align*}

4

## Publication details:

 Title Global identifiability of nonlinear models of biological systems Publication Type Journal Article Authors Audoly, S., Bellu G., D'Angio L., Saccomani M.P., and Cobelli C.

# Nonlinear Models of Biological Systems (2)

## 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 Authors Audoly, S., Bellu G., D'Angio L., Saccomani M.P., and Cobelli C.

# Nonlinear Models of Biological Systems (1)

## Model description:

The two compartment model describes the kinetics of a drug in the human body. The drug is injected into the blood (compartment 1) where it exchanges linearly with the tissues (compartment 2); the drug is irreversibly removed with a nonlinear saturative characteristic from compartment 1 and with a linear one from compartment 2. The I/O experiment takes place in compartment 1.

The system-experiment model is

\begin{align*} \dot{x}_1(t) &= - \left(k_{21} + \frac{V_M}{K_m + x_1}\right)x_1(t) + k_{12}x_2(t) + b_1u(t) \\ \dot{x}_2(t) &= k_{21}x_1(t) - (k_{02} + k_{12})x_2(t) \\ y(t) &= c_1x_1(t) \\ \end{align*}

The initial conditions are $x_1(0) = 0$ and $x_2(0) = 0$. System parameters are presented in the table below.

 $x_1$, $x_2$ drug masses in compartment 1 and 2, respectively; $u$ drug input; $y$ measured drug outtup; $k_{12}$, $k_{21}$ and $k_{02}$ constant rate parameters; $V_M$ and $K_m$ classical Michaelis-Mentel parameters; $b_1$ and $c_1$ input and output parameters, respectively

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## Publication details:

 Title Global identifiability of nonlinear models of biological systems Publication Type Journal Article Authors Audoly, S., Bellu G., D'Angio L., Saccomani M.P., and Cobelli C.

# A nonlinear MIMO system

## Model description:

The system has two inputs and two outputs and is described by the following set of equations:

\begin{align*} y_1(k)&=0.21y_1(k-1)-0.12y_2(k-2)+0.3y_1(k-1)u_2(k-1) \\ &-1.6u_2(k-1)+1.2u_1(k-1)\\ y_2(k)&=0.25y_2(k-1)-0.1y_1(k-2)-0.2y_2(k-1)u_1(k-1) \\ &-2.6u_1(k-1)-1.2u_2(k-1). \end{align*}

## Publication details:

 Title U-model Based Adaptive Tracking Scheme for Unknown MIMO Bilinear Systems Publication Type Conference Paper Authors Azhar, A.S.S., Al-Sunni F.M., and Shafiq M.