The Statistical Theory of the Angiogenesis Equations Articles uri icon

publication date

  • January 2024

issue

  • 29

volume

  • 34

International Standard Serial Number (ISSN)

  • 0938-8974

Electronic International Standard Serial Number (EISSN)

  • 1432-1467

abstract

  • Angiogenesis is a multiscale process by which a primary blood vessel issues secondary vessel sprouts that reach regions lacking oxygen. Angiogenesis can be a natural process of organ growth and development or a pathological one induced by a cancerous tumor. A mean-field approximation for a stochastic model of angiogenesis consists of a partial differential equation (PDE) for the density of active vessel tips. Addition of Gaussian and jump noise terms to this equation produces a stochastic PDE that defines an infinite-dimensional Lévy process and is the basis of a statistical theory of angiogenesis. The associated functional equation has been solved and the invariant measure obtained. The results of this theory are compared to direct numerical simulations of the underlying angiogenesis model. The invariant measure and the moments are functions of a Korteweg–de Vries-like soliton, which approximates the deterministic density of active vessel tips.

subjects

  • Mathematics

keywords

  • angiogenesis; stochastic pdes; statistical theory of nonlinear pdes; moments and structure functions; mathematical biology