[Paper Review] A novel nonlocal partial differential equation model of endothelial progenitor cell cluster formation during the early stages of vasculogenesis

Chiara Villa, Alf Gerisch+1 more|arXiv (Cornell University)|May 24, 2021

Mathematical Biology Tumor Growth120 references8 citations

TL;DR

This paper proposes a novel nonlocal partial differential equation model to simulate endothelial progenitor cell (EPC) cluster formation during early vasculogenesis, integrating endogenous chemotaxis, matrix degradation, proliferation, and nonlocal cell adhesion. The model reveals that chemotaxis drives cluster topology while matrix degradation accelerates cluster formation speed, and nonlocal adhesion alone fails to stabilize clusters long-term, necessitating improved adhesion modeling.

ABSTRACT

Neovascularisation is essential for tissue development and regeneration, in addition to playing a key role in pathological settings such as ischemia and tumour development. Experimental findings in the past two decades have led to the identification of a new mechanism of neovascularisation, cluster-based vasculogenesis, during which endothelial progenitor cells (EPCs) mobilised from the bone marrow are capable of bridging distant vascular beds in a variety of hypoxic settings in vivo. This process is characterised by the formation of EPC clusters during its early stages and, while much progress has been made in identifying various mechanisms underlying cluster formation, we are still far from a comprehensive description of such spatio-temporal dynamics. In order to achieve this, we propose a novel mathematical model of the early stages of cluster-based vasculogenesis, comprising of a system of nonlocal partial differential equations including key mechanisms such as endogenous chemotaxis, matrix degradation, cell proliferation and cell-to-cell adhesion. We conduct a linear stability analysis on the system, solve the equations numerically, conduct a parametric analysis of the numerical solutions of the 1D problem to investigate the role of underlying dynamics on the speed of cluster formation and the size of clusters, and verify the key results of the parametric analysis with simulations of the 2D problem. Our results, which qualitatively compare with data from in vitro experiments, elucidate the complementary role played by endogenous chemotaxis and matrix degradation in the formation of clusters, and they indicate that previous approaches to the nonlocal modelling of cell-to-cell adhesion, while they capture the aggregating effect of cell-to-cell adhesion, are not sufficient to capture its stabilising effect on clusters, and new continuum cell-adhesion modelling strategies are required.

Motivation & Objective

• To develop a comprehensive mathematical model of early-stage cluster-based vasculogenesis, a recently identified mechanism of neovascularization.
• To investigate the spatio-temporal dynamics of endothelial progenitor cell (EPC) clustering under hypoxic conditions.
• To determine the relative contributions of endogenous chemotaxis, matrix degradation, proliferation, and nonlocal cell adhesion to cluster formation.
• To identify key parameters influencing cluster speed, width, and compactness through parametric analysis.
• To guide future experimental and modeling efforts by identifying limitations in current adhesion mechanisms and suggesting new strategies.

Proposed method

• Formulates a system of nonlocal partial differential equations (PDEs) to describe EPC density, matrix density, MMP concentration, and chemoattractant (VEGF) dynamics.
• Incorporates nonlocal terms to model cell-to-cell and cell-to-matrix adhesion, reflecting long-range interactions at the tissue scale.
• Integrates key biological processes: endogenous chemotaxis toward VEGF, MMP-mediated matrix degradation, EPC proliferation, and matrix remodelling.
• Performs linear stability analysis to assess the emergence of spatial patterns from homogeneous states.
• Solves the 1D system numerically and conducts parametric analysis on cluster width and compactness metrics.
• Validates key 1D findings in 2D simulations to confirm robustness and biological relevance.

Experimental results

Research questions

• RQ1 How do endogenous chemotaxis and matrix degradation jointly influence the speed and topology of EPC cluster formation?
• RQ2 To what extent does nonlocal cell adhesion contribute to cluster stability over time?
• RQ3 What are the quantitative effects of varying proliferation and degradation rates on cluster size and compactness?
• RQ4 How do the model's predictions compare with in vitro experimental data on EPC cluster formation?
• RQ5 What are the limitations of current nonlocal adhesion terms in maintaining long-term cluster integrity?

Key findings

• Endogenous chemotaxis is primarily responsible for determining the spatial topology of EPC clusters, guiding cell aggregation into distinct clusters.
• Matrix degradation, driven by MMPs, is the dominant factor accelerating the speed of cluster formation, with faster degradation leading to earlier cluster emergence.
• The nonlocal cell-to-cell adhesion term promotes initial aggregation but is insufficient to stabilize clusters over long time periods, indicating a need for improved adhesion modeling.
• Parametric analysis in 1D shows that increasing proliferation rates leads to wider and more compact clusters, while higher degradation rates reduce cluster width and increase formation speed.
• 2D simulations confirm that the 1D parametric results are robust, with clusters forming in spatially coherent patterns consistent with in vitro observations.
• The model's predictions qualitatively match experimental data, including cluster diameters of 100–400 μm and network lengths up to 500 μm after 72 hours.