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Conference Papers Year : 2014

A mathematical model of systemic inhibition of angiogenesis in metastatic development

Abstract

We present a mathematical model describing the time development of a population of tumors subject to mutual angiogenic inhibitory signaling. Based on biophysical derivations, it describes organism-scale population dynamics under the influence of three processes: birth (dissemination of secondary tumors), growth and inhibition (through angiogenesis). The resulting model is a nonlinear partial differential transport equation with nonlocal boundary condition. The nonlinearity stands in the velocity through a nonlocal quantity of the model (the total metastatic volume). The asymptotic behavior of the model is numerically investigated and reveals interesting dynamics ranging from convergence to a steady state to bounded non-periodic or periodic behaviors, possibly with complex repeated patterns. Numerical simulations are performed with the intent to theoretically study the relative impact of potentiation or impairment of each process of the birth/growth/inhibition balance. Biological insights on possible implications for the phenomenon of ''cancer without disease" are also discussed.
Nous pr\'{e}sentons un mod\'{e}le math\'{e}matique d\'{e}crivant le d\'{e}veloppement temporel d'une population de tumeurs en interactions mutuelles \textit{via} des signaux d'inhibition de l'angiog\'{e}n\'{e}se. Bas\'{e} sur une d\'{e}rivation biophysique, il d\'{e}crit la dynamique, \'{a} l'\'{e}chelle de l'organisme, qui r\'{e}sulte de l'influence relative de trois processus: naissance (diss\'{e}mination de tumeurs secondaires), croissance et inhibition (de l'angiog\'{e}n\'{e}se). Le mod\'{e}le r\'{e}sultant est une \'{e}quation aux d\'{e}riv\'{e}es partielles de transport non-lin\'{e}aire avec condition aux limites non-locale. La non-lin\'{e}arit\'{e} r\'{e}side dans une quantit\'{e} non-locale (le volume m\'{e}tastatique global) pr\'{e}sente dans la vitesse de croissance. Le comportement asymptotique du mod\'{e}le est \'{e}tudi\'{e} num\'{e}riquement et r\'{e}v\'{e}le des dynamiques int\'{e}ressantes, allant de la convergence vers un point d'\'{e}quilibre \'{a} des solutions p\'{e}riodiques exhibant des comportements non-triviaux. Des solutions born\'{e}es et non-p\'{e}riodiques sont aussi observ\'{e}es. Nous pr\'{e}sentons une exploration num\'{e}rique de l'espace des param\'{e}tres dans le but d'\'{e}tudier th\'{e}oriquement l'effet de la balance naissance/croissance/inhibition. De potentielles implications biologiques concernant le ph\'{e}nom\'{e}ne de ''cancer sans maladie" sont aussi discut\'{e}es.
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Dates and versions

hal-00867992 , version 1 (30-09-2013)

Identifiers

Cite

Sébastien Benzekry, Alberto Gandolfi, Philip Hahnfeldt. A mathematical model of systemic inhibition of angiogenesis in metastatic development. Congres SMAI 2013, May 2013, Seignosse, France. pp.75-87, ⟨10.1051.201445008⟩. ⟨hal-00867992⟩
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