Mathematical Problems in Engineering

Volume 2018, Article ID 4705472, 10 pages

https://doi.org/10.1155/2018/4705472

## Modeling and Simulation of VEGF Receptors Recruitment in Angiogenesis

^{1}DIMI, Università degli Studi di Brescia, Brescia 25123, Italy^{2}Laboratory for Preventive and Personalized Medicine (MPP Lab), Università degli Studi di Brescia, Brescia 25123, Italy^{3}DMMT, Università degli Studi di Brescia, Brescia 25123, Italy

Correspondence should be addressed to A. Salvadori; ti.sbinu@irodavlas.otrebla

Received 21 December 2017; Revised 3 May 2018; Accepted 6 May 2018; Published 26 August 2018

Academic Editor: Anna Vila

Copyright © 2018 A. Salvadori et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Angiogenesis, the process of new blood vessel formation from preexisting ones, plays a pivotal role in tumor growth. Vascular endothelial growth factor receptor-2 (VEGFR2) is the main proangiogenic tyrosine kinase receptor expressed by endothelial cells (ECs). VEGFR2 binds different ligands triggering vascular permeability and growth. VEGFR2-ligands accumulate in the extracellular matrix (ECM) and induce the polarization of ECs as well as the relocation of VEGFR2 in the basal cell membrane in contact with ECM. We propose here a multiphysical model to describe the dynamic of VEGFR2 on the plasma membrane. The governing equations for the relocation of VEGFR2 on the membrane stem from a rigorous thermodynamic setting, whereby strong simplifying assumptions are here taken and discussed. The multiphysics model is validated against experimental investigations.

#### 1. Introduction

Vascular Endothelial Growth Factor Receptor-2 (VEGFR2) is a proangiogenic receptor expressed on endothelial cells (ECs) and is the main mediator of the angiogenic response. The interaction between VEGFR2 and extracellular ligands, produced by tumor cells, is essential to cancer growth. Specifically, ligand stimulation causes the relocation of VEGFR2 in the basal aspect in cells plated on ligand-enriched extracellular matrix both in vitro and in vivo, and ultimately receptors-ligands interaction activates the ECs division and proliferation towards tumor cells. Upon release, growth factors associate with the extracellular matrix and act as ECs guidance during neo-vessel formation.

Receptor-ligand interactions have been extensively studied and mathematical models have been proposed. Some concerned the estimation of the reaction rates for membrane-bound reactants [1–3]; a few models with different level of complexity account for adhesion receptor-ligand (as integrins and fibronectin) kinetics, receptor-ligand densities, cell rheology, and cytoskeletal force generation [4]. Only a few investigations concerned specifically VEGFR2 [5, 6].

Codesigned experiments and simulations for VEGFR2 have been recently developed, with biological findings and the predictive ability of the model extensively discussed in [7]. Here, we profoundly describe the modeling of VEGFR2 recruitment in angiogenesis, detailing the thermodynamic description, the weak formulation, and the algorithms for the numerical solution.

The mathematical model here proposed accounts for diffusion of VEGFR2 along the cellular membrane and for ligands-receptors chemical reactions. It is framed in the mechanics and thermodynamics of continua, following a general description proposed in [8], and takes advantage of successful descriptions of physically similar systems [9, 10]. The effect of the cell deformation on the diffusion-reaction process on the membrane is here strongly simplified, surrogating the effects of the change in geometry on the chemodiffusive equations with a fictitious source term of ligands, detailed in Section 2.2.

The model stems from continuity equations (for mass, energy, and entropy; see Section 2.1), standard chemical kinetics, summarized in Section 2.6, thermodynamic restrictions, and constitutive specifications, detailed in Section 2.4. This sequence provides the governing equations in a strong form in Section 2.7, which is converted in a weak form in Section 2.8 prior to the numerical approximation via the Finite Element Method (FEM). The partial differential equations of the model have been implemented in a computer code, with the ultimate goal to predict conditions for angiogenesis. The FEM code, implemented in the deal.ii open source finite element library, has been validated against codesigned experiments partially discussed in the companion paper [7].

#### 2. Modeling VEGFR2 Diffusion Driven by Its Specific Ligand

##### 2.1. Mass Balance Equations

A general formulation for the chemo-transport-mechanics problem is here tailored to model the relocation of VEGFR-2 driven by its specific ligand on the lipid bilayer membrane (henceforth denoted with ). The interaction between receptors and ligands is described as a chemical reaction, which produces a receptor-ligand complex where and are the kinetic constants of the forward and backward reaction, respectively. The reaction rate , measured in , quantifies the net formation of (C) as the difference between the forward and backward reaction rates.

Complex internalization and its return back to the surface are not considered in this model. Therefore, the mass balance equations are defined on the membrane as follows:Symbols in (2a)–(2c) have the following meaning (concentrations are defined in space and time, i.e., . The same holds for , , and . Functional dependence is specified when necessary only, to favor readability): (with ) is the molarity (i.e., the number of moles per unit area) of a generic species ; is the mass flux in terms of molecules, i.e., the number of molecules of species measured per unit length per unit time, and is a tangent vector field on the membrane; is the rate in number of molecules per unit volume per unit time at which species is generated by sources, and is the time.

Ligands, whose degradation is negligible, are immobilized in the substrate as they are in vitro. The complex are assumed to be immobile as well, i.e.,Since receptors are free to move along the membrane, reaction (1) portrays a conversion of mobile to trapped receptors and vice versa.

Equations (2a)–(2c) are defined on the cell membrane. Accordingly, the divergence operator has to be defined on the same surface. Denoting with the cell membrane unit normal,

Mass balance equations (2a)–(2c) shall be accompanied by the balance of force in order to model the mechanical deformation of the cell, whose boundary, the membrane, is the geometrical support of (2a)–(2c). Modeling the evolution of the Laplace-Beltrami operator that presides formulation concurrently with the large deformation of the cell is a phenomenally ambitious task, which is in progress motivated by the promising outcomes here shown. In the present work, we surrogate the mechanics with some simplifying assumptions.

##### 2.2. Surrogated Mechanics

During the codesigned experimental test, the cell progressively spreads out on the substrate. Since the latter is enriched with immobilized ligands, the cell surface in contact with the support increases with time and results in a supply of available ligands for the chemical reaction (1) to occur. Mechanical models for cell spreading involve very sophisticated descriptions of active and passive behavior of cells [11–13], leading to simulations of impressive computational burden. In the present, seminal works do not account explicitly for the mechanical evolution of the cell, which keeps its original shape. Rather, we surrogate the effects of its change in geometry on the chemodiffusive equations (2a)–(2c) by introducing a source term of ligands whose expression is calibrated from experimental evidence [14]. The following expression for in (2b) is taken [7]:The path of reasoning beyond (5) is equivalent to consider the cell as rigid and the substrate much more deformable, so that the latter envelopes the spherical cell, as depicted in Figure 1.