Stochastic modeling and analysis of tumor-induced blood vessel formation*

Imbalance of angiogenesis, the process of blood vessel formation and growth, is behind many diseases, including cancer. Stochastic models of tumor-induced angiogenesis have been proposed since 1990 but most work has been heavily computational. Recently, we have analyzed angiogenesis models that represent cells at the tips of new blood vessels as active particles whose trajectories are the blood vessels. Vessel tips are subject to chemotaxis and haptotaxis forces in Langevin equations and branch stochastically producing new tips. When one active tip meets a preexisting vessel (trajectory of another tip) joins it and ceases to be active, a process called anastomosis. The same occurs when it arrives at the tumor [1,2,3]. Thus anastomosis is a killing point process that depends on the past history of the given realization. For the ensemble averaged density of active tips, we have derived a Fokker-Planck equation that contains source terms with memory characterizing anastomosis [1,2]. For simple geometries, the density of active tips evolves to a soliton-like wave whose shape and velocity follow simple differential equations. Numerical simulations of the stochastic process confirm our findings, which are a step toward controlling angiogenesis [3,4,5].
[1] L.L. Bonilla, et al, Phys. Rev. E 90, 062716 (2014).
[2] F. Terragni, et al, Phys. Rev. E 93, 022413 (2016).
[3] L.L. Bonilla, M. Carretero, and F. Terragni, Entropy 19, 209 (2017).
[4] L.L. Bonilla, et al, Sci. Rep. 6, 31296 (2016).
[5] L.L. Bonilla, M. Carretero, and F. Terragni, Phys. Rev. E 94, 062415 (2016).