Dynamic Model of Tumor Growth (2)

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.