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Mathematical Medicine and Biology Advance Access originally published online on November 25, 2008
Mathematical Medicine and Biology 2009 26(1):63-95; doi:10.1093/imammb/dqn024
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy

Alberto D'Onofrio{dagger}

Epidemiology and Biostatistics Division, European Institute of Oncology, Via Ripamonti 435, 20141 Milan, Italy

Alberto Gandolfi{ddagger}

Istituto di Analisi dei Sistemi ed Informatica "A. Ruberti" - CNR, Viale Manzoni 30, 00185 Rome, Italy

{dagger} Corresponding author. Email: alberto.donofrio{at}ieo.it

{ddagger} Email: gandolfi{at}iasi.cnr.it

Received on November 19, 2007. Revised on April 23, 2008. Accepted on October 19, 2008.

In this paper we propose a class of models that describe the mutual interaction between tumour growth and the development of tumour vasculature and that generalize existing models. The study is mainly focused on the effect of a therapy that induces tumour vessel loss (anti-angiogenic therapy), with the aim of finding conditions that asymptotically guarantee the eradication of the disease under constant infusion or periodic administration of the drug. Furthermore, if tumour and/or vessel dynamics exhibit time delays, we derive conditions for the existence of Hopf bifurcations. The destabilizing effect of delays on achieving the tumour eradication is also investigated. Finally, global conditions for stability and eradication in the presence of delays are given for some particular cases.

Keywords: angiogenesis; anti-angiogenesis; tumour eradication; stability; delay differential equations


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