BIRS Workshop Lecture Videos
Tumor induced angiogenesis Bonilla, Luis
Angiogenesis is a multiscale process by which blood vessels grow from existing ones and carry oxygen to distant organs. Angiogenesis is essential for normal organ growth and wounded tissue repair but it may also be induced by tumors to amplify their own growth. Mathematical and computational models contribute to understand- ing angiogenesis and developing anti-angiogenic drugs, but most work only involves numerical simulations and analysis has lagged. A recent stochastic model of tumor induced angiogenesis including branching, elongation, and anastomosis (fusion) of blood vessels captures some of its intrinsic multiscale structures, yet allows one to extract a deterministic integropartial differential description of the vessel tip density . Vessel tips proliferate due to branching, elongate following Langevin dynamics and, when they meet other vessels, join them by anastomosis and stop moving. Stalk endothelial cells follow the tip cells, so that the trajec- tories thereof constitute the advancing blood vessel. Anastomosis keeps the number of vessel tips relatively small, so that we cannot use the law of large numbers to derive equations for their density. Nevertheless, we show that ensemble averages over many replicas of the stochastic process correspond to the solution of the deterministic equations with appropriate boundary conditions . Most of the time, the density of tips sprouting from a primary blood vessel advances chemotactically towards the tumor driven by a soliton similar to the famous Korteweg-de Vries soliton. There are two collective coordinates whose slow dynamics changes the shape and velocity of the soliton. Analyzing the equations for the collective coordinates paves the way for controlling angiogenesis through the soliton, the engine that drives this process . References  L.L. Bonilla, V. Capasso, M. Alvaro, and M. Carretero, Hybrid modeling of tumor-induced angiogenesis, Phys. Rev. E 90, 062716 (2014).  F.Terragni,M.Carretero,V.CapassoandL.L.Bonilla,StochasticModelofTumor-inducedAngiogenesis:EnsembleAveragesand Deterministic Equations, Phys. Rev. E 93, 022413 (2016).  L.L.Bonilla,M.Carretero,F.Terragni,andB.Birnir,Solitondrivenangiogenesis,submittedforpublication,2016.
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