Integrating multiscale angiogenesis models presents a way to maintain and capitalize on biomedical knowledge in a quantitative form. It is a complex endeavor. Models were built to answer different biological questions and generate hypotheses that may have been very specific. As such, the models often use different algorithms, employ diverse languages, and focus on specific angiogenic factors, cell types or molecular species. They can be on different time and spatial scales.
Requirements then for integration are many. The most general form of integration must allow data exchange between models, feedback, asynchronous and synchronous running of models, and variable temporal and spatial scales. Furthermore, the integration must be designed so that the desired integrated or global biological phenomenon is best represented, while still maintaining the integrity of individual models.
With these criteria in mind, we began the development of a multiscale integrated model, with a goal of building a prototype from five existing models (called modules to distinguish them from the integrated model): a tissue geometry module, a blood flow module, an O2 transport module, a VEGF module, and a cell module ().
Figure 1 Schematic for the integration of five modules. Blood flow and the O2 transport module are at the tissue level, while the VEGF module and the cell module are focused on the cellular level. Time scales range from seconds (O2 and blood flow modules) to hours (more ...)
To introduce the design of the integration in the context of the biology, as well as the capabilities of individual modules, we briefly describe each module, and follow-up by defining the controller.
2.1 Geometry, Blood Flow & Oxygen Transport Modules
A blood flow model was previously built and validated for a given geometry of microvascular network in 3D tissue, specifically applied to skeletal muscle (for a review, see [12
]). The model predicts the amount of blood flow and hematocrit through capillaries in the network, by solving a set of nonlinear algebraic equations for pressure at the network nodes (bifurcations) and blood flow rate and hematocrit in the vascular segments (). From blood flow, the relative oxygen distribution through the network is then predicted by convection-diffusion-reaction partial differential equations governing oxygen transport.
Figure 2 Velocity (A) and hematocrit (B) distributions for a representative 100 μm by 100 μm by 800 μm capillary network in rat skeletal muscle. The figure is adapted from , where the equation-based models representing blood flow and (more ...)
2.2 VEGF Module
The 3D model of the rat skeletal muscle introduced above was used in development of a model predicting VEGF distributions and VEGF receptor (VEGFR) occupancy in 3D tissue during exercise [14
]. In the model, VEGF secretion is a function of myocyte oxygen pressure and an experimentally observed oxygen-HIF-VEGF relationship. Within the muscle, the capillaries are nonuniformly spaced, and this leads to nonuniform oxygen distribution and VEGFR expression within the tissue (). VEGFR activation and VEGF binding is also predicted in the 3D model from a series of developed molecular-based chemical kinetic models [15
(right). Distribution of VEGF receptor activation along microvessels in skeletal muscle, during moderate exercise (top panel) and moderate exercise with low inspired oxygen (bottom panel).
2.3 Cell Module
Interaction of VEGF, MMPs, and the extracellular matrix at the molecular level trigger events at the cell level. These events include tip cell activation, endothelial cell migration, chemotaxis and haptotaxis and cell proliferation. We developed a computational model to mimic cellular sprouting at the onset of angiogenesis using agent-based methodology, governed by rules ().
Figure 4 Schematic of the 3D cell module. Capillaries are represented by endothelial cells. An example of a growing network with four capillaries is shown in the gray inset. Cells are divided into segments. Each segment is represented by two nodes. Currently, (more ...)
Examples of logical rules that guide cell activity in the model include migration rates as a function of local VEGF levels; or Boolean behavior for whether cellular elongation is allowed or not. Two examples of rules, in a restricted 2D case, are shown in . The module rules based on experimental work compiled from extensive literature research, and applied to conditions that might occur in a 2D or 3D in vitro setting. Individual cell behavior (activation, elongation, migration, and proliferation) combine to produce a novel capillary network, emerging out of combinatorially complex interactions of single cells. A particular benefit of the rule or event-based modeling is the ability to easily produce in silico knockouts on multiple biological levels. Here we describe an example, an in silico molecular level knockout of the Dll4 ligand.
Figure 5 Illustrations of cell movement represented by rules in the cell module. The tip cell is represented by a red node and segment; the node shared between the tip and stalk cells is yellow; and the blue node and segment is the adjacent stalk cell segment. (more ...) 2.3.1. Dll4 Knockouts
Delta-like ligand 4 (Dll4) is a transmembrane ligand for Notch receptors. It is a critical ligand for vascular development; haploinsufficiency of the Dll4 gene is embryonically lethal in many mouse strains [16
]. Dll4 is primarily expressed in endothelial cells, and it is correlated to the local concentration of VEGF [17
], as well as to VEGF receptor concentrations; a blockade of VEGF leads to a decrease of Dll4 [18
], while Notch-Delta signaling downregulates VEGFR2 [19
]. 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 [18
]. Overexpression of Dll4 diminishes the growth of new sprout tips.
The cell module predicts the effects of VEGF protein concentrations and Dll4 haploinsufficiency on endothelial cells and capillary growth. In the module, cell-level behavior alone contributes to differences in capillary network formation, and the vascular networks resulting from the model appear similar in vascular density, branching and tortuosity to networks found in Dll4 experiments ().
Snapshot of the model output directly after angiogenic stimulation (A) and for simulated hypoxia in Dll4+/+ conditions, after 160 hrs (B) and in Dll4−/+ conditions, after 160 hrs (C).
2.4 The Controller: Running Models, Data Handling& Feedback
The controller provides a platform to seamlessly integrate the modules mentioned above and simulate angiogenesis at multiple temporal and spatial scales. These modules can be, and currently are, rewritten as different Application Programming Interface (API) libraries and provide language binding for a variety of languages (such as C, C++, FORTRAN, JAVA, MATLAB). By this means, libraries can be called from different languages and shared by the community as well.
Integration revolves around the interaction of the controller and the modules, and between the controller and parameter databases or parameter text files. Parameters include biochemical parameters (such as kinetic rate constants) and physical parameters (such as vascular geometry). The parameters, currently inputted via text files, are being compiled into a database format. In this way, parameter sensitivity analysis can be performed to generate experimental hypotheses. Modules are compiled into the library files. Parameters dependent on a biological microenvironment are maintained outside of the library files, so that the modules are general, and function independent of their application to a specific tissue or cell. The controller reads the parameters from the database or text files, and invokes the modules. Each module takes input from the controller and outputs its results to the controller, which then passes the relevant data on to subsequent modules. The forward and feedback interactions between different modules are regulated through the controller.