Simulating sprouting angiogenesis

Abstract

Angiogenesis1 is the biological mechanism by which new blood vessels sprout from existing ones. It differs from vasculogenesis, which is the de novo growth of the primary vascular network from initially dispersed endothelial cells (ECs). Vasculogenesis is predominant in embryonic tissue whilst new vasculature in the adult body arises mostly from angiogenesis. ECs, lining the inside of blood vessels, react to different angiogenic stimuli and inhibitors. Among the stimuli is the vascular endothelial growth factor (VEGF) which is up-regulated in tissue where the vascular structure is damaged or insufficiently developed to meet oxygen demand. The identification of the processes involved in angiogenesis is quite recent and has stirred increased interest in therapeutic and clinical applications according to Carmeliet et al. [1]. One can think of tissue repair in wound beds, inhibition of growth of tumorous tissue or vascular reform during the female reproductive cycle. Rossiter et al. [2] showed that VEGF induced angiogenesis is crucial for wound healing in an experiment where wounds were inflicted upon normal and VEGF-deficient mice. New vasculature ensures supply of oxygen and lymphocytes and disposal of carbon dioxide and lactates, accelerating wound healing and tissue reconstruction. The increased creation of new vasculature around tumorous tissue is believed to follow the same process and inhibiting angiogenesis is therefore an important topic in clinical studies on cancer treatment. Biochemical laboratory experiments can be hard, time consuming, expensive or unethical. Computational models can be used to provide an easy, quick and cheap way to get insights that would otherwise require laboratory experiments. The understanding of biological processes needs quantification and in this sense mathematical formulation of the relations involved becomes useful. Their mathematical interpretation and experimental verification is an iterative process resulting in better understanding of the process itself. Computer simulation will never make laboratory experiments obsolete, but it can provide guidance in targeting viable hypotheses before conducting in vitro or in vivo experiments. Mathematical modeling of biological cellular processes dates back to the simulation by Glazier and Graner in 1992. They describe natural sorting behavior of different cell types [3] and different re-arrangement patterns driven by the differential adhesion hypothesis [4]. This hypothesis states that cells of different types have specific potential energies upon adhesion, driving sorting behavior. In these simulations, the cellular Potts model2 (CPM) is used. A CPM for vasculogenesis based on this work was made byMerks et al. [5, 6] in which a layer of partial differential equations (PDEs) models the chemoattractants. Later, Merks added Vascular Endothelial cadherin (VE-cadherin) caused contact-inhibited chemotaxis to simulate angiogenic-like sprout formation [7]. From an initial clump of ECs in the model sprouting behavior appears. Merks postulates that both vasculogenesis and angiogenesis must be driven by the same principles. To produce these results, a generic library called the Tissue Simulation Toolkit (TST) was written in C++ starting from 2004 modeling the CPM described by Glazier et al. [4] in a generic way. Merks [7] extensively describes the advantages of a cell based approach over a continuum approach that is widely used in mathematical biology. Although his CPM is a nice method that increases insight in the angiogenic process, it is computationally heavy, limiting the scalability of the tractable problem domain. Vermolen and Gefen [8] described tissue behavior using a semi-stochastic cell-based formalism to model the migration of cells in colonies in the context of wound healing, tumor growth, bone ingrowth and contraction formation. Movement of cells is assumed to be the result of a strain energy density working as a mechanical stimulus. Like the CPM, the model tracks displacement and viability of individual cells. The aim of this study is to adapt this semi-stochastic cell-based formalism to describe angiogenesis, hence connecting this modeling approach to the subject ofMerks’ work. The need for such a model is clearly stated in the discussion of Vermolen’s work [9]. Thanks to the computational less heavy character in comparison with the CPM, we hope to be able to simulate larger areas to get a better glance at large scale behavior whilst still being able to benefit from the cell-based character of the model. We also improve the biochemical model for the degrading of the substrate by the cells and formulate all relevant parameters based on local properties. The challenge is to translate the advantages of Merks’ CPM, like cell shape specific behavior, tracking of elongation patterns and cell-cell contact behavior, to this new formalism without compromising the computational simplicity. To verify our simulation results with biochemical experiments, this study is performed in collaboration with the Dermatology Department of the VU Medical Center. This department does in vitro laboratory research on many processes that occur in the skin, for example the role of endothelial cells during skin wound healing. The first aim of this research is tomimic their in vitro angiogenesis sprouting assay using our computational model, simulating the response to different chemical stimuli like VEGF. Formulating a way to visually and numerically compare the laboratory work to the simulated results is key to making the model applicable in practice.