As a new capillary grows from a blood vessel, a series of cellular processes occur. These vascularization events have been extensively studied experimentally, however the whole angiogenic sequence has yet to be characterized by any experiment or model, and numerous unknowns remain. What is known is that an endothelial cell from an existing vessel becomes activated. The activated cell starts to migrate into the extracellular matrix by degrading it; this unique, spindle-shaped cell is called the tip cell. Cells adjacent to the tip cell begin to proliferate, and follow the tip cell; they are referred to as stalk cells. These processes result in formation of a sprout [1
]. This capillary sprout moves towards a stimulus, in response to chemical cues, mechanical factors, and a degree of random motility. Finally, the sprout joins an adjacent capillary. Together these events define the process of sprouting angiogenesis.
Hypoxia is a main stimulus for angiogenesis during ischemia, exercise, inflammation, and cancer. In response to hypoxia, a transcription factor hypoxia-inducible factor 1 (HIF1) activates hundreds of genes in cells exposed to low oxygen [2
]. These genes include vascular endothelial growth factor, VEGF. Overall, VEGF protein stimulates chemotaxis and proliferation of endothelial cells. There are seven known isoforms of VEGF, each with a different effect on cell behavior, and ultimately, on vascular pattern formation [4
]; additionally, there are splice variants of the VEGF isoforms, VEGFxxx
]. Here, we first refer to HIF1-dependent expression of VEGF and represent the effect of the VEGF-A (VEGF165
) isoform on cells, unless otherwise specified.
Along with VEGF, another ligand, Delta, and its receptor play a key role in angiogenic tip cell formation and proliferation, and the integrity of a microvascular network. Recent studies have focused on the multiple effects of Notch-Delta signaling on vascular sprout formation. Delta-like ligand 4 (Dll4) is a transmembrane ligand for Notch receptors, and it is critical to vascular development. So important is Dll4, that like VEGF, haploinsufficiency of the Dll4 gene is embryonically lethal in many mouse strains, as a result of extensive vascular defects [7
]. Dll4 is primarily expressed in endothelial cells, and correlated to the local concentration of VEGF [10
], as well as VEGF receptor concentrations. A blockade of VEGF leads to a decrease of Dll4 [11
], while Notch-Delta signaling downregulates VEGFR2 [12
]. One study showed the presence of Dll4 reduced tip cell formation as a function of VEGF [13
], and another demonstrated Notch suppressed branching and proliferation at the sprout tip [7
]. A Dll4 deficiency causes an increase in sprout formation but vessels appear nonproductive, with less capability of carrying blood or reducing hypoxia in surrounding tissue [11
]. Overexpression of Dll4 diminishes the growth of new sprout tips. In the computational research presented here, we focus on the effects of VEGF protein concentrations and Dll4 haploinsufficiency on endothelial cells and how this cell level behavior contributes to differences in capillary network formation.
Mathematical representations of angiogenesis date to the 1970's, and their numbers continue to expand rapidly. Some of the first models were differential equations representing a generic growth factor as a chemotactic stimulus, produced and released by a tumor mass, and inducing growth of vessels into the tumor [14
]. Models have since included detailed equation-based network models of tumor-induced angiogenesis [17
], a model of capillary growth through a corneal pocket assay [18
], molecular level interactions of VEGF complexes coupled to vessel oxygenation [19
], a cell-level rule-based model of network growth in mesenteric tissue [20
], Potts models of angiogenic growth [21
], a model of tip cell selection as a function of notch-signaling [23
], network formation stemming from capillary movement through a matrix composed of aligned collagen fibers [24
], and VEGF-driven angiogenic growth applied to a vascular engineering construct environment [25
], among many others [26
]. Each model brings a new perspective on the biological phenomenon behind neovascularization, and together they give insight on multiple conditions affecting angiogenesis, and multiple conditions affected by angiogenesis. The goal of the current modeling effort is to provide a framework where many of these models could be employed or integrated, or at the very least their hypotheses tested in a new context, specifically related to biological observations and experimental data.
For this reason, we chose to design the model using the broad framework of three-dimensional agent-based modeling. Agent-based programming has roots in social science, game theory, economics, evolution and public health. More recently it has emerged as a tool useful for a range of biomedical applications, including angiogenesis [20
], membrane transport [31
], inflammatory response [33
], and tumor growth [36
Agents are objects that can interact with their environment, and modify their surroundings. They are analogous to digital organisms familiar to evolutionary biology in that they carry a computational genome, or a sequence of instructions (henceforth called rules). These rules determine the agents' response to logical functions. The rules of agents require listing the factors that influence cell behavior as events, with direct counterparts in biology. Unlike digital organisms, agents used in the model are not inherently self-replicating. Agents' rules may evolve, and they may copy their instructions when modified to represent growth.
In agent-based modeling, global functions (e.g., global ischemia) and sophisticated rules can govern agent behavior. Agent interactions with one another and their environment can also be asynchronous. The rule-based modeling we describe is a continual, iterative process, much like perfecting in vitro or in vivo experiments. As more knowledge is gained, the current assumptions may change, and a cycle of improvements is needed to keep pace with current biological information. Furthermore, we start with a very general model – its parameters will be changed to represent specific species, tissues and conditions.
We employed this agent-based approach to develop a three-dimensional, computational model that simulates cellular sprouting at the onset of angiogenesis. We use the model to determine and weigh the critical events in angiogenesis; and differentiate under what microenvironments, which factors dominate and result in a particular vessel and capillary network phenotype.
The model is based on experimental work found from extensive literature research, and methods in the model are closely governed by biological mechanisms. Currently, the model is applied to conditions that might occur in a three-dimensional in vitro setting. We represent physiological changes at the cell level; visually simulate in three dimensions assumptions behind cell activation, migration, elongation, proliferation and branching; and test cell level behavior in response to different stimuli, focusing in the current model on activation by a threshold change in VEGF and changes in ligand presence. A novel capillary network emerges out of this complex interaction of single cells.
Results of the model show the relationship between growth factor gradients, cell sprouting, cell migration and cell proliferation. Results also showed how variations in the mechanisms of cellular persistence alter vessel growth. We predicted the effects of migration separate from proliferation on tip cell and stalk cell movement, and on total vessel growth. Furthermore, the model represents novel findings of how Delta ligand changes influence capillary phenotype. Overall, the model represents a three-dimensional framework upon which to test and develop biologically realistic mechanisms underlying blood vessel growth.