Dynamic Model of Tumor Growth (1)

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).