The Statistical Theory of the Angiogenesis Equations

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Nonlinear Science Pub Date : 2024-01-22 DOI:10.1007/s00332-023-10006-2

Björn Birnir, Luis Bonilla, Manuel Carretero, Filippo Terragni

引用次数: 0

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.

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血管生成方程的统计理论

血管生成是一个多尺度的过程，通过这一过程，主血管发出次级血管芽，到达缺氧区域。血管生成可以是器官生长和发育的自然过程，也可以是癌症肿瘤诱发的病理过程。血管生成随机模型的均方场近似包括一个关于活跃血管尖端密度的偏微分方程（PDE）。在该方程中加入高斯和跳跃噪声项，就产生了一个随机偏微分方程，它定义了一个无穷维的莱维过程，是血管生成统计理论的基础。相关的函数方程已经求解，并获得了不变度量。该理论的结果与基础血管生成模型的直接数值模拟进行了比较。不变度量和矩是类似于 Korteweg-de Vries 孤子的函数，而 Korteweg-de Vries 孤子近似于活动血管尖端的确定性密度。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Nonlinear Science 数学-力学

CiteScore

5.00

自引率

3.30%

发文量

审稿时长

4.5 months

期刊介绍： The mission of the Journal of Nonlinear Science is to publish papers that augment the fundamental ways we describe, model, and predict nonlinear phenomena. Papers should make an original contribution to at least one technical area and should in addition illuminate issues beyond that area''s boundaries. Even excellent papers in a narrow field of interest are not appropriate for the journal. Papers can be oriented toward theory, experimentation, algorithms, numerical simulations, or applications as long as the work is creative and sound. Excessively theoretical work in which the application to natural phenomena is not apparent (at least through similar techniques) or in which the development of fundamental methodologies is not present is probably not appropriate. In turn, papers oriented toward experimentation, numerical simulations, or applications must not simply report results without an indication of what a theoretical explanation might be. All papers should be submitted in English and must meet common standards of usage and grammar. In addition, because ours is a multidisciplinary subject, at minimum the introduction to the paper should be readable to a broad range of scientists and not only to specialists in the subject area. The scientific importance of the paper and its conclusions should be made clear in the introduction-this means that not only should the problem you study be presented, but its historical background, its relevance to science and technology, the specific phenomena it can be used to describe or investigate, and the outstanding open issues related to it should be explained. Failure to achieve this could disqualify the paper.