This is a full polynomial form of the model, given in Enzyme kinetics:

$$\begin{align*} \dot{x}_1 &= -bx_1 + ax_5\\ \dot{x}_2 &= \alpha x_1 - \beta x_2\\ \dot{x}_3 &= \gamma x_2 - \delta x_3\\ \dot{x}_4 &= \sigma x_4 x_6 ( \gamma x_2 - \delta x_3) \\ \dot{x}_5 &= -\sigma x_4 x_5^2 x_6 ( \gamma x_2 - \delta x_3)\\ \dot{x}_6 &= -x_6^2( \gamma x_2 - \delta x_3) \end{align*}$$

with $x_1(0) = 0.3617$, $x_2(0) = 0.9137$, $x_3(0)=1.3934$, $x_4(0) = x_3(0)^{\sigma}$, $x_5(0)=\dfrac{1}{A+x_3(0)^{\sigma}}$, $x_6(0)=\dfrac{1}{x_3(0)}.$

The model describes the oscillations in enzyme kinetics. The state variable $x_1$ represents an enzyme concentration whose rate of synthesis is regulated by feedback control via a metabolite $x_3$, and $x_2$ regulates the synthesis of $x_3$. It is characterised by a rational kinetics consisting of a Hill-like term, and it is given by:

$$\begin{align*} \dot{x}_1 &= -bx_1 + \dfrac{a}{A+X_3^\sigma},\\ \dot{x}_2 &= \alpha x_1 - \beta x_2,\\ \dot{x}_3 &= \gamma x_2 - \delta x_3 \end{align*}$$

with $x_1(0) = 0.3617$, $x_2(0) = 0.9137$, $x_3(0)=1.3934.$

Consider the model from Dynamic Model of Tumor Growth (1). The complete model formulation describes the phenomenology of tumor growth slowdown, as the tumor consumes its available support; stimulatory and inhibitory influences from the tumor cells; inhibition due to administered inhibitors; and the clearance of the administered inhibitor. In the simplified model, the latter effect is not described, only the serum level of the inhibitor to be maintained is represented, so a second-order system is to be analyzed:

$$\begin{align*} \dot{x}_{1} &=-\lambda x_{1}\ln\left(\dfrac{x_{1}}{x_{2}}\right) \\ \dot{x}_2 &= b_x1 - dx_1^{{2}\over{3}}x_2 - ex_2u \\ y&=x_1, \end{align*}$$

where $x_1$ is the tumor volume (mm$^3$), $x_2$ is the vasculature volume (mm$^3$), and $u$ is the serum level of the inhibitor (mg/kg). The last equation represents that tumor volume is the measured output of the system. The characteristics of the parameters for the Lewis lung carcinoma and the mice used in the experiment are: $\lambda = 0.192($day$^{-1})$, $b = 5.85 ($day$^{−1}),$ $d =0.00873 ($day$^{−1}$mm$^{−2}),$ while the parameter characteristic for the inhibitor (endostatin) is: $e = 0.66 ($day$^{−1} ($mg/kg$)^{−1}).$ Attached figure shows the nonlinear behavior of the simplified model.

In 1999, a research was carried out at the Harvard Medical University by Philip Hahnfeldt et al. to investigate experimentally and theoretically the effects of angiogenic inhibitors on tumor growth dynamics. They posed a quantitative theory for tumor growth under angiogenic stimulator/inhibitor control. In their experiments, mice were injected with Lewis lung carcinoma cells. The following equations comprise the entire model formulation:

$$\begin{align*} \dot{x}_1 &=-\lambda_1x_1\ln\left(\frac{x_1}{x_2}\right) \\ \dot{x}_2 &=bx_1-dx_1^{\frac{2}{3}}x_2-ex_2x_3 \\ \dot{x}_3 &=\int_0^tu(t^{\prime})\exp(-\lambda_{3}(t-t^{\prime})){\mathrm d}t^{\prime} \\ y &=x_{1}, \end{align*}$$

where $x_1$is the tumor volume (mm$^3$), $x_2$is the supporting vasculature volume (mm$^3$), $x_3$ is the inhibitor serum level (mg/kg), and $u$ is the inhibitor administration rate (mg/kg/day).

The model we discuss here has been proposed to study glucose metabolism in the brain from positron emission tomography (PET) [$^{18}$F]-Fluoro-Deoxy-Glucose($^{18}$F-FDG) data K. Schmidt, G. Mies, and L. Sokoloff, “Model of kinetic behavior of deoxyglucose in heterogeneous tissues in brain: A reinterpretation of the significance of parameters fitted to homogeneous tissue models,” J. Cereb. Blood Flow Metab., vol. 11, pp. 10–24, 1991. The model is shown in Fig. 4. It is a two compartment model with two time-varying parameters which account for brain tissue heterogeneity.

The system-experiment model is

$$\begin{align*} \dot{x}_2(t) &= k_{21}u(t) - (k_{12} + k_{32}(t))x_2(t) \\ \dot{x}_3(t) &= k_{32}(t)x_2(t)\\ y_1(t) &= x_2(t) + x_3(t)\\ \end{align*}$$

System parameters are presented in the table below.