Stochastic model of tumor-induced angiogenesis: Ensemble averages and deterministic equations.

  title={Stochastic model of tumor-induced angiogenesis: Ensemble averages and deterministic equations.},
  author={Filippo Terragni and Manuel Carretero and Vincenzo Capasso and Luis L. Bonilla},
  journal={Physical review. E},
  volume={93 2},
A recent conceptual model of tumor-driven angiogenesis including branching, elongation, and anastomosis of blood vessels captures some of the intrinsic multiscale structures of this complex system, yet allowing one to extract a deterministic integro-partial-differential description of the vessel tip density [Phys. Rev. E 90, 062716 (2014)]. Here we solve the stochastic model, show that ensemble averages over many realizations correspond to the deterministic equations, and fit the anastomosis… 

Figures and Tables from this paper

On the mathematical modelling of tumor-induced angiogenesis.

The outcomes of relevant numerical simulations show that the proposed model, in presence of anastomosis, is not self-averaging, so that the ``propagation of chaos" assumption cannot be applied to obtain a deterministic mean field approximation.

Stochastic Models of Blood Vessel Growth

A hybrid mesoscale tip cell model involves stochastic branching, fusion and extension of active vessel tip cells with reaction-diffusion growth factor fields and adopts the form of an advancing soliton that can be characterized by ordinary differential equations for its position, velocity and a size parameter.

The statistical theory of the angiogenesis equations

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

Ensemble Averages, Soliton Dynamics and Influence of Haptotaxis in a Model of Tumor-Induced Angiogenesis

This work shows the delaying effect of haptotaxis on the advance of the angiogenic vessel network by direct numerical simulations of the stochastic process and by a study of the soliton motion.

Integrodifference master equation describing actively growing blood vessels in angiogenesis

This system models tumor-induced angiogenesis, the process of formation of blood vessels induced by a growth factor released by a tumor, by tracking the density of active tips, calculated as an ensemble average over many realizations of the stochastic process.

Solitonlike attractor for blood vessel tip density in angiogenesis.

The location of the maximum vessel tip density for different replicas follows closely the soliton peak position calculated either by ensemble averages or by solving an alternative deterministic description of the density.

A convergent numerical scheme for integrodifferential kinetic models of angiogenesis

Notch signaling and taxis mechanims regulate early stage angiogenesis: A mathematical and computational model

A mathematical model of early stage angiogenesis is presented that permits to explore the relative importance of mechanical, chemical and cellular cues in vessel growth and unravels the regulating role of Jagged, Notch and Delta dynamics in vascular cells.



Hybrid modeling of tumor-induced angiogenesis.

This work sets up a conceptual stochastic model including branching, elongation, and anastomosis of vessels and derives a mean field approximation for their densities, which leads to a deterministic integropartial differential system that describes the formation of the Stochastic vessel network.

Stochastic modelling of tumour-induced angiogenesis

A novel mathematical approach is proposed for reducing complexity by (locally) averaging the stochastic cell, or vessel densities in the evolution equations of the underlying fields, at the mesoscale, while keeping Stochasticity at lower scales, possibly at the level of individual cells or vessels.

A cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis.

This model provides a quantitative framework to test hypotheses on the biochemical and biomechanical mechanisms that control tumor-induced angiogenesis and shows inhomogeneities in the extravascular tissue lead to sprout branching and anastomosis, phenomena that emerge without any prescribed rules.

A hybrid model for three-dimensional simulations of sprouting angiogenesis.

This work presents the first three-dimensional model of sprouting angiogenesis that considers explicitly the effect of the extracellular matrix and of the soluble as well as matrix-bound growth factors on capillary growth.

Multiscale Angiogenesis Modeling Using Mixed Finite Element Methods

A deterministic two-scale tissue-cellular approach for modeling growth factor-induced angiogenesis and using mixed finite element methods and a point-to-point tracking method to simulate sprout branching and anastomosis is presented.

Lattice and non-lattice models of tumour angiogenesis

Computational models of sprouting angiogenesis and cell migration: towards multiscale mechanochemical models of angiogenesis

A need is identified for the inclusion of cell mechanical principles in models of angiogenesis for the description of cell migration, cell-matrix and cell-cell interaction, as the generation of cellular forces is key to cell migration.

A multiscale hybrid approach for vasculogenesis and related potential blocking therapies.