Research Article Modeling and Simulation of VEGF Receptors Recruitment in Angiogenesis A. Salvadori , 1,2 V. Damioli, 1 C. Ravelli, 2,3 and S. Mitola 2,3 1 DIMI, Universit` a degli Studi di Brescia, Brescia 25123, Italy 2 Laboratory for Preventive and Personalized Medicine (MPP Lab), Universit` a degli Studi di Brescia, Brescia 25123, Italy 3 DMMT, Universit` a degli Studi di Brescia, Brescia 25123, Italy Correspondence should be addressed to A. Salvadori; alberto.salvadori@unibs.it 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. Tis 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. 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 diferent 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. Te 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. Te 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. Te interaction between VEGFR2 and extracellular ligands, produced by tumor cells, is essential to cancer growth. Specif- ically, ligand stimulation causes the relocation of VEGFR2 in the basal aspect in cells plated on ligand-enriched extra- cellular 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 stud- ied and mathematical models have been proposed. Some concerned the estimation of the reaction rates for membrane- bound reactants [1–3]; a few models with diferent level of complexity account for adhesion receptor-ligand (as inte- grins and fbronectin) kinetics, receptor-ligand densities, cell rheology, and cytoskeletal force generation [4]. Only a few investigations concerned specifcally VEGFR2 [5, 6]. Codesigned experiments and simulations for VEGFR2 have been recently developed, with biological fndings 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. Te mathematical model here proposed accounts for difusion 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]. Te efect of the cell deformation on the difusion- reaction process on the membrane is here strongly simplifed, surrogating the efects of the change in geometry on the chemodifusive equations with a fctitious source term of ligands, detailed in Section 2.2. Te model stems from continuity equations (for mass, energy, and entropy; see Section 2.1), standard chemical kinetics, summarized in Section 2.6, thermodynamic restric- tions, and constitutive specifcations, detailed in Section 2.4. Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 4705472, 10 pages https://doi.org/10.1155/2018/4705472