# Dynamic Model of Tumor Growth (1)

## Model description:

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).

## Publication details:

 Title Model-based Angiogenic Inhibition of Tumor Growth using Feedback Linearization Publication Type Conference Paper Year of Publication 2013 Authors Szeles, A., Drexler D.A., Sapi J., Harmati I., and Kovacs L. Conference Name IEEE 52nd Annual Conference on Decision and Control (CDC), 2013 Date Published 12/2013 Publisher IEEE Conference Location Firenze ISBN Number 978-1-4673-5714-2 Accession Number 14158507 Keywords cancer, feedback, linearisation techniques, medical control systems, nonlinear control systems, patient treatment, time-varying systems, tumours Abstract In the last decades beside conventional cancer treatment methods, molecular targeted therapies show prosperous results. These therapies have limited side-effects, and in comparison to chemotherapy, tumorous cells show lower tendency of becoming resistant to the applied antiangiogenic drugs. In clinical research, antiangiogenic therapy is one of the most promising cancer treatment methods. Using a simplified model of the reference dynamical model for tumor growth under angiogenic inhibition from the literature, exact linearization is performed in the paper to handle the nonlinear behavior of the model. Two different control methods are applied on the linearized model: flat control and switching control. Simulations are performed on the nonlinear model to show the characteristics of the therapies carried out using the presented control methods. DOI 10.1109/CDC.2013.6760184