The study of VEGF binding to receptors on cells in vitro, and the validation of the VEGF kinetic interaction network between multiple ligands and multiple receptors, leads us to ask the question: how does this network behave in vivo? In sections 4 and 5 we will discuss the transport of VEGF between tissues and around the body, but here we will focus first on the behavior of VEGF in a local volume of tissue. This multicellular milieu requires significant additions to our model in order to accurately simulate the local transport of VEGF, including diffusion of VEGF ligands over significant distances, extracellular matrix sequestration and variable production rates of VEGF throughout the tissue. We place these all in an anatomically-based 2D or 3D multicellular tissue geometry.
The models can predict the creation of interstitial VEGF gradients due to the non-uniform nature of the tissue anatomy. This is of particular interest because VEGF is believed to be a chemotactic guiding agent for blood vessels, but also because local variability in VEGF concentration can lead to local variation in VEGF receptor ligation and signaling, allowing for focal activation of endothelial cells. The model framework can be adapted to most tissues; here we present a case with parameters specifically selected to represent a skeletal muscle experiencing ischemia (specifically, rat extensor digitorum longus, or EDL, for a rodent model of hindlimb artery ligation), and we describe how to computationally test several therapeutic interventions including gene therapy and exercise.
3.1 Mathematical Framework for Tissue Architecture, Blood Flow & Tissue Oxygenation
2D and 3D tissue geometry based on microanatomy
A cross-section (for 2D) or a volume of tissue (for 3D) is reconstructed from histological and other microanatomical information (). The major relevant features of the tissue are the blood vessels, the parenchymal cells (from here on, we will assume these are skeletal myocytes, i.e. long multinucleated cells) and the interstitial space between them. From a computational modeling point of view, the tissue comprised of volumes and surfaces, defined as those portions of the tissue where molecules can move in all directions (volumes) and those portions where the movement of molecules is restricted to a plane (e.g. receptors inserted in cell membrane can move only laterally). There are three major volumes of the tissue for our purposes: the vascular space (i.e. inside the blood vessels, determined by the density of blood vessels and their diameters); the intracellular space (whether inside of parenchymal cells or endothelial cells); and the interstitial space between cells, which is itself divided into three volumes (each of which is not contiguous), based on the density of the fibrous matrix present: the extracellular matrix, and basement membrane regions surrounding the endothelial cells and the myocytes.
There are two major surfaces; again, these are not contiguous. First, the combined cell surfaces of the skeletal myocytes, which are assumed to be cylindrical (diameter 37.5 µm, consistent with rat histology), and arranged in a regular hexagonal grid formation, accounting for almost 80% of total tissue cross-sectional area. VEGF is secreted from the myocytes’ surfaces. Second, the surface of the endothelial cells that make up the blood vessels, specifically, the abluminal surface (the luminal surface faces the blood stream and we neglect it for now). Again, the blood vessels are assumed to be cylindrical, and although most (but not all) are parallel to the muscle fibers, they do not occupy every possible position between fibers, but instead have a stochastic, nonuniform arrangement (based on experimentally measured capillary-to-fiber ratios, capillary-to-fiber distances and histology) occupying 2.5% of total tissue volume (leaving ~18% as interstitial space). On this endothelial surface, VEGF receptors are expressed. Thus, VEGF must diffuse from the myocyte surface where it is secreted, through basement membranes and extracellular matrix, to the endothelial surface where it ligates its cognate receptors.
To model tissues at the meso-scale, we use the above microanatomical information as an input to a set of integrated models of blood flow, oxygen transport, and VEGF transport ().
Volumes: Blood Flow
Blood flow and hematocrit value calculations are based on Pries et al.
’s two-phase continuum model (Pries and Secomb, 2005
), and reduces to a system of nonlinear algebraic equations (two per vessel) that are solved iteratively (Ji et al., 2006
). The Fahraeus-Lindqvist effect and non-uniform hematocrit distribution at vascular bifurcations are included in the blood flow model. Higher blood flow rates are used for exercising conditions, to represent the increased perfusion (and enhanced oxygen delivery) to exercising muscles. In addition, exercise-trained rats have higher average capillary blood velocity.
Volumes: Diffusion and consumption of Oxygen
Oxygen transport in the tissue in detailed in (Ji et al., 2006
) and (Goldman and Popel, 2000
). Oxygen arrives in the tissue via the blood vessel network, and the partial pressure of oxygen in the vessels, Pb
, is described by
is mean blood velocity; αb
is oxygen solubility in blood; HD
is the discharge hematocrit;
are the oxygen binding capacity and oxygen saturation of the red blood cell; ξ is the longitudinal position in the vessel; R is vessel radius; Jwall
is the oxygen flux across the vessel walls (i.e. into the tissue). Oxygen diffuses across the endothelial cells, and freely throughout the tissue (both interstitial and intracellular). Within the cells, it may also be consumed by binding to Myoglobin (Mb). The local partial pressure of oxygen in the tissue, P, is described by
are the diffusivities of oxygen and myoglobin; α is the oxygen solubility in tissue; CMbbind
are the oxygen binding capacity and oxygen saturation of myoglobin; and M(P) represents Michaelis-Menten kinetic consumption of oxygen.
Volumes: Diffusion of VEGF and sequestration by ECM
The VEGF ligands, VEGF120
(rodent form of VEGF121
) and VEGF164
(rodent form of VEGF165
), can both diffuse through the interstitium following secretion, however the longer isoform also binds to glycoproteins in the extracellular matrix, becoming reversibly sequestered. The equations are thus identical to those of section 2, with the addition of binding and unbinding terms:
where Ci and Cj are concentrations of two interstitial molecules, i and j. In the rat EDL model, Ci
] or [V164
=[GAG]. Concentration of proteins in the thin endothelial or myocyte basement membranes is given by an equation of the form:
is the secretion rate from the cell (typically from myocytes); Rm
are the concentrations of receptor m and of the i-m complex on the cell surface (typically on endothelial cells); Jout
is the Fickian diffusive flux from BM to ECM of VEGF; and dBM
is the basement membrane thickness.
Surfaces: Receptor-ligand interactions
The ligand-receptor interactions that take place are precisely those that were outlined in section 2, and that will be used in Sections 4 and 5: VEGF120
bind to VEGFR1 and VEGFR2, while only the longer isoform binds Neuropilin-1 and extracellular matrix. The general form of the receptor and receptor complex equations is therefore:
are the membrane insertion rate and internalization rate of receptor m; kcouple,m,n
are the kinetic rate of binding and unbinding of two surface receptors m and n to each other. Note in particular that the concentration of the ligand (Ci
) in each case is the concentration in the basement membrane region closest to the receptor. Thus, the receptor occupancy varies from cell to cell across the capillary network. Examples of specific individual equations can be found in section 2.
Surfaces: VEGF production/secretion rates What is not included in these models?
Intracellular VEGF is not included in these simulations; that includes both post-internalization VEGF and pre-secretion VEGF. In addition, we neglect the intravasation of VEGF into the bloodstream, either by endothelial cell secretion or through paracellular routes, e.g. permeability. Lymphatic transport of VEGF is also neglected. These additional transport routes could be accommodated in the above model structure with the addition of new surfaces or terms. Although endothelial VEGF production and parenchymal VEGFR expression have been observed in recent years (Lee et al., 2007
; Bogaert et al., 2009
), these are not included as part of these simulations; there is no technical obstacle to doing so.
Relationship to single-compartment models
It is important to note that the spatial averages of VEGF concentrations at the endothelial cell surface and of VEGFR activation in the meso-scale models match well with the values in single-compartment models (section 4) that do not include diffusion or VEGF gradients. Thus, it may be possible to calculate the average receptor activation using less computationally intensive compartment models, and use the meso-scale models to estimate the spatial gradients.
3.2 Case Study: Pro-angiogenic VEGF gene therapy for muscle ischemia
In order to improve the perfusion and healing of ischemic muscle tissue with impaired angiogenic response, several therapies have been suggested, typically involving the delivery of VEGF (one or more isoforms) to the muscle.
The first of these, gene therapy, increases the VEGF secretion by adding additional VEGF-encoding genes to the cells that are transfected. By transfecting multiple copies, or by judicious choice of VEGF promoters and enhancers in the new construct, significant increases in VEGF secretion can be obtained. We have modeled both uniform upregulation of VEGF (increasing VEGF secretion at every myocyte surface point in the model) and stochastic upregulation, in which each cell has a randomly increased VEGF production within a certain range (using the myonuclear density, we know the size of the myocyte surface that is under the control of each nucleus; thus, we can assign a random number to each region, that stays constant through the simulation) (Mac Gabhann et al., 2007a
). These increases in VEGF production result in increased VEGFR2 activation, however the VEGF gradients are not significantly increased (); in this case blood vessels might be induced to sprout, but have no directional cues. Further simulations restricting the VEGF transfection to a specific region of the muscle demonstrates increased VEGFR2 activation coupled with very high VEGF gradients towards the transfected tissue, but only in a narrow region between transfected and nontransfected tissue (Mac Gabhann et al., 2007a
). This suggests that VEGF gene delivery needs to be effectively localized with a high degree of spatial accuracy to allow the gradients of VEGF to bring the new vessels to the affected volume.
3.3 Case Study: Pro-angiogenic VEGF cell-based therapy for muscle ischemia
Another route to bringing more VEGF to the tissue, and one which may allow for more spatial specificity, is the delivery of VEGF-overexpressing cells, e.g. myoblasts that will effectively integrated into the existing muscle and produce excess VEGF locally. To simulate this, we select specific myocytes in the model to overexpress VEGF, and distribute these distantly or close together (Mac Gabhann et al., 2006
; Mac Gabhann et al., 2007a
). That is, since the secretion rate of VEGF can have a different value for every spatial location on the myocytes surface in our model, we can upregulate VEGF in a specific subset of these cells.
For this therapy, we observe in the simulations both increased VEGFR2 binding and increased VEGF gradients (), but only within approximately one to two myocyte diameters from the new VEGF overexpressing cells (Mac Gabhann et al., 2006
; Mac Gabhann et al., 2007a
). In addition, cells close together synergize while distant ones do not. In this way, we can see that a small number of cells, or cells distributed too broadly, would have a low probability of attracting perfusion from a neighboring region; however, a large mass of cells, at the right location, could serve as a local chemoattractant.
The results described in sections 3.2 and 3.3, for therapies reliant on VEGF upregulation alone, mirror the outcome of the several clinical trials of VEGF isoforms in humans for coronary artery disease (CAD) or peripheral artery disease (PAD); these trials have not had the success that was expected of them. Instead, the standard of care for PAD continues to be exercise, and it is this therapy that we consider next.
3.4 Case Study: Pro-angiogenic exercise therapy for muscle ischemia
Exercise training in rats has been shown to not only restore the ability of hypoxic, ischemic tissue to upregulate VEGF following injury, but also increases the expression levels of the VEGF receptors (Lloyd et al., 2003
). Thus, we used our model to simulate the exercise-dependent upregulation of both the ligands and the receptors, using experimentally measured increases (Ji et al., 2007
; Mac Gabhann et al., 2007a
; Mac Gabhann et al., 2007b
In this case, we increase the secretion rate of VEGF isoforms from each point on the myocyte surface, during exercise; in addition, we increase the insertion rate of the VEGF receptors at every point on the endothelial cell surface at all times (as a result of exercise training). The results of these simulations are quite different from those before: first, during exercise, both the VEGFR2 activation and the VEGF gradients are increased, not just locally but across the upregulated tissue (Ji et al., 2007
; Mac Gabhann et al., 2007a
); second, during rest periods, while VEGF upregulation ceases and the occupancy of VEGFR2 returns to lower levels, the high VEGF gradients are maintained (). This suggests that the activation step for attracting new blood vessels may be during a smaller window of time, while the guidance of the new vessel to its destination can take place continuously.
This observation that our current best strategy for PAD, exercise, increases both ligand expression and receptor activation, leaves us with the possibility of developing combined ligand-receptor therapy (especially for those who cannot exercise).