# American Institute of Mathematical Sciences

August  2018, 15(4): 827-839. doi: 10.3934/mbe.2018037

## Closed-loop control of tumor growth by means of anti-angiogenic administration

 1 Università Campus Bio-medico di Roma, Roma, Via Álvaro del Portillo 21, 00128, Italy 2 Istituto di Analisi dei Sistemi ed Informatica A. Ruberti, Consiglio Nazionale delle Ricerche, Roma, Via dei Taurini 19, 00185, Italy 3 Dipartimento di Ingegneria e scienze dell'informazione e matematica, Università dell'Aquila, L'Aquila, Via Vetoio, 67100, Italy 4 Istituto di Analisi dei Sistemi ed Informatica A. Ruberti, Consiglio Nazionale delle Ricerche, Roma, Via dei Taurini 19, 00185, Italy

* Corresponding author: Valerio Cusimano

The authors contributed equally to the research and are listed in alphabetical order

Received  February 21, 2017 Accepted  December 20, 2017 Published  March 2018

A tumor growth model accounting for angiogenic stimulation and inhibition is here considered, and a closed-loop control law is presented with the aim of tumor volume reduction by means of anti-angiogenic administration. To this end the output-feedback linearization theory is exploited, with the feedback designed on the basis of a state observer for nonlinear systems. Measurements are supposed to be acquired at discrete sampling times, and a novel theoretical development in the area of time-delay systems is applied in order to derive a continuous-time observer in spite of the presence of sampled measurements. The overall control scheme allows to set independently the control and the observer parameters thanks to the structural properties of the tumor growth model. Simulations are carried out in order to mimic a real experimental framework on mice. These results seem extremely promising: they provide very good performances according to the measurements sampling interval suggested by the experimental literature, and show a noticeable level of robustness against the observer initial estimate, as well as against the uncertainties affecting the model parameters.

Citation: Filippo Cacace, Valerio Cusimano, Alfredo Germani, Pasquale Palumbo, Federico Papa. Closed-loop control of tumor growth by means of anti-angiogenic administration. Mathematical Biosciences & Engineering, 2018, 15 (4) : 827-839. doi: 10.3934/mbe.2018037
##### References:

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##### References:
Graphical comparison of the real and estimate state under the action of the closed loop control law
Percentage of successes for the three criteria, in a population of 1000 mice treated with endostatin
Model parameters
 $\lambda$ day$^{-1}$ $b$ day$^{-1}$ $d$ day$^{-1}$ $c$ day$^{-1}$ $\eta$ day$^{-1}$ 0.192 5.85 0.00873 0.66 1.7
 $\lambda$ day$^{-1}$ $b$ day$^{-1}$ $d$ day$^{-1}$ $c$ day$^{-1}$ $\eta$ day$^{-1}$ 0.192 5.85 0.00873 0.66 1.7
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