FMG, JWJ, and ASP conceived and designed the experiments. FMG and JWJ performed the experiments. FMG, JWJ, and ASP analyzed the data. FMG, JWJ, and ASP contributed reagents/materials/analysis tools. FMG, JWJ, and ASP wrote the paper.
The authors have declared that no competing interests exist.
Members of the vascular endothelial growth factor (VEGF) family of proteins are critical regulators of angiogenesis. VEGF concentration gradients are important for activation and chemotactic guidance of capillary sprouting, but measurement of these gradients in vivo is not currently possible. We have constructed a biophysically and molecularly detailed computational model to study microenvironmental transport of two isoforms of VEGF in rat extensor digitorum longus skeletal muscle under in vivo conditions. Using parameters based on experimental measurements, the model includes: VEGF secretion from muscle fibers; binding to the extracellular matrix; binding to and activation of endothelial cell surface VEGF receptors; and internalization. For 2-D cross sections of tissue, we analyzed predicted VEGF distributions, gradients, and receptor binding. Significant VEGF gradients (up to 12% change in VEGF concentration over 10
It is not currently possible to experimentally quantify the gradients of protein concentration in the extracellular space in vivo. However, the concentration gradients of vascular endothelial growth factor (VEGF) are essential for both initiation and directed guidance of new blood vessels. The authors develop a computational model of VEGF transport in tissue in vivo (skeletal muscle, though the method is applicable to other tissues and other proteins) with realistic geometry and including biophysical interactions of VEGF, its receptors, and the extracellular matrix. Using this model, the authors predict for the first time the distribution of VEGF concentration and VEGF receptor activation throughout the tissue. VEGF concentration gradients are significant, up to 12% change in VEGF concentration over 10 μm in resting muscle. Transplanting VEGF-overexpressing myocytes (for therapeutic induction of blood vessel growth) increases the gradients significantly. Endothelial cells in sprouting vessels are approximately 50 μm long, and therefore the predicted gradients across the cell are high and sufficient for chemotactic guidance of the new vessels. The VEGF concentration gradients also result in significant heterogeneity in the activation of VEGF receptors on blood vessels throughout the tissue, a possible reason for the sporadic nature of sprout initiation.
Vascular endothelial growth factor (VEGF) is a key promoter of angiogenesis in vivo and it increases proliferation and migration of endothelial cells cultured in vitro [
VEGF is involved in both physiological (e.g., during exercise and wound healing) and pathological angiogenesis (e.g., in ~60% of human tumors and in age-related macular degeneration). VEGF upregulation is necessary for physiological angiogenesis under conditions of hypoxia via oxygen-sensing mechanisms in the HIF-1 pathway [
To create an effective VEGF-driven pro-angiogenic therapy, better understanding is needed of both physiological and pathological VEGF-induced angiogenesis. Increasing VEGF concentration leads to angiogenesis, but concentrations of VEGF beyond critical levels may result in formation of abnormal vessels (leaky, tumor-like, and with larger lumens) and hemangiomas [
In the present study, we constructed a computational model to study extracellular diffusion of VEGF in vivo and effects of VEGF upregulation on gradients and receptor binding using the well-characterized rat EDL tissue as a sample environment. We have previously constructed models studying the kinetics of VEGF binding to receptors on cells in vitro and the effect of the presence of NRP-1 or placental growth factor [
Rat EDL muscle geometry has been well-characterized physiologically and in its responses to hypoxia, hemodynamic shear stress, and electrical stimulation [
(A) Cross-sectional view of EDL tissue: red-filled circles represent muscle fibers, black unfilled circles represent capillaries located in the interstitium of the tissue. The fibers are assumed to be regularly spaced and hexagonally packed.
(B) Interstitial space near a capillary: muscle fibers are surrounded by a thin MBM; capillaries formed by endothelial cells are surrounded by an EBM. The ECM lies in between the EBM and MBM, and VEGF diffuses throughout the interstitial space.
(C) Diffusion and binding: two VEGF isoforms are secreted from the skeletal muscle myocyte into the MBM: VEGF120 and VEGF164 diffuse through the MBM, EBM, and ECM, but only VEGF164 is able to bind with HSPG in each layer. Near the endothelial cell surface (located in the EBM), VEGF can interact with VEGFR1 and VEGFR2, and both can be internalized whether bound to VEGF or unbound.
Capillaries of 6 μm in external diameter and 0.5 μm wall thickness (5 μm in lumen diameter) were added randomly into the interstitial space between the muscle fibers (
Each muscle fiber in the model is surrounded by a uniform thin myocyte basement membrane (MBM) layer (
Different units can be used to define each of the above parameters; the units given in the glossary (
Glossary
In BM spaces, the membrane thickness is no more than 85 nm, and gradients across this distance are expected to be negligible, therefore free VEGF concentration is assumed to be uniform perpendicular to the capillary surface in the BM. Thus, at endothelial cell surfaces, the free VEGF concentrations are equal to those of its adjacent EBM spaces. VEGF that binds to receptors comes from the adjacent EBM, and VEGF that dissociates from receptors is released into the EBM; VEGF expressed by myocytes is secreted into an adjacent MBM space from which diffusion can then occur.
Transport within the ECM is described by mass balance equations:
The mass balance equations describing VEGF transport between ECM and EBM are:
On the endothelial cell surface, the binding kinetics between VEGF and receptors follows our previous study [
Thus,
The physiological parameters used in this model are summarized in
Parameters for VEGF Transport and Binding
VEGF diffusivity was calculated using a molecular-weight–based relationship for globular proteins [
HSPG concentration in ECM and BM spaces were obtained from values measured in human myocardium [
VEGF receptor concentrations and kinetic rates were chosen based on in vivo measurements of total VEGFR2 protein concentration and capillary density in human vastus lateralis skeletal muscle [
In vivo measurements in human show that approximately ten times as much VEGFR1 is expressed as VEGFR2 [
Currently, only in vitro measurements of VEGF protein secretion have been published [
Skeletal myocytes are multinucleate cells, hence they cannot be directly compared with the mononucleate cells used to measure VEGF secretion in vitro experimentally. The myonuclear domain (MD) is the volume and associated cell membrane surface area corresponding to (and under the control of) each nucleus of the cell. To make a valid comparison, secretion from this surface area is assumed to be under the control of a single nucleus. By counting the number of nuclei per unit length of the fiber, and the cross-sectional area, the fiber surface area of each MD was shown to decrease with the cross-sectional area of the fiber [
Using relative mRNA abundances for splice variants of VEGF in mouse skeletal muscle [
Under resting physiological conditions, each skeletal muscle fiber is assumed to secrete VEGF at the same rate because all muscle fibers are well-oxygenated. The uniform secretion simulation results are summarized in
(A) VEGF concentration variations in skeletal muscle. The surface represents the total VEGF concentration (free plus HSPG-bound) across the interstitial space.
(B) Graphical representation of VEGF binding: large gray circles represent muscle fibers. Small circles represent capillaries and are color-coded to show the amount of VEGF bound to the surface of the capillary.
(C) Histogram of average VEGF binding to capillaries. Each capillary has a different amount of bound VEGF due to the spatial variations of VEGF. Some capillaries may be activated while others are not.
(D) Histogram of VEGF gradients. The percentage of tissue that experiences VEGF gradient of a certain magnitude. Gradient is defined as the change in VEGF concentration over 10 μm divided by the mean VEGF concentration in the tissue. Capillary distribution in subsequent figures (except
(A) Uniform capillary distribution results in a decrease in the average VEGF gradients in the tissue.
(B,C) VEGF gradients at steady state are invariant (in percentage terms) to changes in the extracellular matrix composition. Increased density or affinity of VEGF binding sites results in increased bound VEGF content (and thus increased absolute values of the gradient), but no change in the relative gradient.
Under conditions of uniform secretion, relative VEGF gradients average 3.0% VEGF/10 μm and reach a maximum of 12.2% VEGF/10 μm in the ECM (relative gradients are measured as change in VEGF concentration across 10 μm, divided by mean VEGF tissue concentration); the scale of 10 μm is chosen as relevant to endothelial cell sensing of chemotactic gradients during sprout formation. VEGF concentrations and an analysis of gradients are visualized in
When HSPG binding affinity was increased to 10-fold basal value (by increasing
To study the effect of changing VEGF receptor concentrations on VEGF gradient formation, VEGF secretion levels were changed to maintain a mean unbound VEGF concentration of 1 pM. Decreasing VEGFR1 and VEGFR2 insertion rates to 20% each effectively decreases cell surface receptor concentrations to approximately 2,000 receptors/cell each (
Increasing VEGFR2 density (A) and increasing VEGFR1 density (B) increase the gradients of VEGF concentration in tissue. Surfaces represent concentration of total VEGF (pM, free plus HSPG-bound) in the ECM across the cross section of tissue. Histograms of VEGF gradients and the percentage of tissue involved. Gradient defined as in
Altering the ratio of VEGFR1 to VEGFR2 cell surface insertion rates follows the same trend (
When VEGF receptor affinity is increased 10-fold, VEGF gradients increase to an average of 9.8% VEGF/10 μm and a maximum of 60.1% VEGF/10 μm (
(A) Increased binding affinity (increased on-rate) has a similar outcome to increases in the receptor density: the VEGF gradients are magnified.
(B) Increased internalization rate of VEGF-receptor complexes results in an increase in the VEGF gradients.
VEGF expression is upregulated 4-fold in exercise [
Under uniform secretion, the bound VEGF concentration on capillaries fits an approximately normal distribution with a mean and standard deviation of 511 and 26.1 10−7 pmol/cm2, respectively (
To predict the impact of cell-based therapy in delivering VEGF to muscle, we examined the effects of overexpression in one fiber versus distributed overexpression. Specific muscle fibers were chosen to chronically overexpress VEGF such that the number and spatial positions of overexpressing fibers varied while total secretion in the tissue was kept constant. Six conditions were studied: single fiber at 40-fold overexpression, two adjacent fibers at 20-fold, two distant fibers at 20-fold, three adjacent fibers at 13.3-fold, three distant fibers at 13.3-fold, and uniform secretion at 1.3-fold above basal level (while remaining fibers secrete at basal level), and the results are summarized in
The total VEGF expression in each of the tissues (A–F) is the same; the arrangement delivery of VEGF from cells incorporated into each tissue is different. Stars mark the fiber(s) that overexpress VEGF. VEGF gradients and capillary activation graphs are as for
(A) 40-Fold overexpression of VEGF in one fiber.
(B) Uniform 1.33-fold overexpression of VEGF by all fibers.
(C,D) 20-Fold overexpression of VEGF in two fibers, close together (C) or distant (D).
(E,F) 13.3-Fold overexpression of VEGF in three fibers, close together (E) or distant (F).
Total VEGF increases equally in all cases but much larger gradients are generated by local overexpression. Under single-fiber 40-fold overexpression (
The differences in VEGF gradients for the overexpression models directly affect cell surface receptor VEGF binding. For single-fiber overexpression (
Each vessel in each tissue experiences a different level of VEGF binding. Total VEGF expression level is the same in each tissue, and the mean VEGF binding to capillaries is the same, but the concentrating of VEGF overexpression into a small number of adjacent fibers results in increased variability of binding. The tissues are arranged in decreasing order of standard deviation of VEGF-capillary binding, as a metric of the variability in capillary activation within the tissue. The more concentrated the VEGF overexpression, the higher the variability. Each tissue is labeled with the number of fibers overexpressing VEGF, the level of overexpression in each fiber, and the panel in
Overexpression from a single fiber produces a distinct threshold of VEGF binding between capillaries adjacent to overexpressing fibers and non-adjacent capillaries (
To determine the effect of increasing the total VEGF dose delivered to the tissue, we change the density of VEGF-expressing fibers. One, two, or three muscle fibers were each chosen to overexpress VEGF at 40-fold over basal level to study the effect of overexpression density on VEGF distribution (
The delivery of two or three fibers overexpressing VEGF 40-fold is compared with an equivalent uniform overexpression.
(A,B) VEGF concentration and gradients for three 40-fold overexpressing fibers close together (A) or distant (B).
(C) Uniform VEGF overexpression of 4-fold.
(D,E) VEGF concentration and gradients for two 40-fold overexpressing fibers close together (D) or distant (E).
(F) Uniform VEGF overexpression of 2.67-fold.
The distribution of VEGF binding on vessels for one, two, and three 40-fold overexpressing fibers, and the equivalent uniform overexpression. Each tissue is labeled with the number of fibers overexpressing VEGF, the level of overexpression in each fiber, and the panel in
This study establishes a conceptual framework for the study and prediction of VEGF gradients in tissue, which cannot be measured with present experimental techniques. The effects of VEGF on angiogenesis are widely studied, and experiments have shown that VEGF gradients are necessary for endothelial tip cell migration and guidance during sprouting angiogenesis to prevent malformation of microvasculature [
The gradients of VEGF concentration result in significant heterogeneity in the binding of VEGF to VEGF receptors on the capillaries; this may be a reason for the stochasticity of sprout formation—some capillaries become angiogenically active, while others do not. In addition, non-uniform overexpression of VEGF leads to increased gradients and to a different pattern of capillary activation by VEGF compared with uniform overexpression.
Our model is constructed from published rat EDL skeletal muscle data, but parameters such as binding affinity of receptors have never been measured in vivo and others such as free VEGF concentration have never been measured specifically for rat EDL tissue. Receptor expression on endothelial cells has not been measured for rat EDL so our model uses measurements for human skeletal muscle, and we assumed that the same number of receptors per endothelial cell is expressed in human and rat. Our sensitivity analysis of the effect of VEGF receptor concentration on the formation of VEGF gradients predicts that gradients will be smaller if receptor expression is lower in rat than human. Also, we have previously shown that the presence of NRP-1 has a strong effect on VEGF binding kinetics [
Our analyses of both VEGF receptor quantity and receptor binding kinetics can also be interpreted as alternative model parameters for different tissues types or demonstrate the response of VEGF gradients to a change in muscle state (e.g., increase in VEGFR2 in response to exercise [
Both total and unbound VEGF concentrations have been measured in vivo [
Currently, there are no measurements of VEGF receptor binding or VEGF distribution in vivo. Our model predicts that approximately half of total VEGF is bound to the cell surface receptors and that less than 3% of VEGF receptors (both VEGFR1 and VEGFR2) are occupied. VEGF in the interstitium is essential to the guidance of capillary sprouts during angiogenesis, and only 2.5% of interstitial (non- cell surface–bound) VEGF is freely diffusible. It is unknown whether HSPG-bound VEGF plays a direct role in guiding capillary sprouts, but studies show that heparin-binding isoforms of VEGF are essential to proper vessel formation [
The low fractional occupancy of VEGF receptors on endothelial cells indicates that in resting skeletal muscle, approximately 310 molecules of VEGF are bound per cell on average. Endothelial cells may use a system of VEGF signaling amplification to produce angiogenic events. Despite these small quantities of bound VEGF, we have previously shown through Monte Carlo simulations that a continuum model gives valid results for VEGF interactions [
Our sensitivity analysis shows that VEGF gradients are highly sensitive to parameters for VEGF receptor kinetics and concentration values. In our model, HSPG concentration and binding kinetics do not affect relative gradients, and increasing HSPG concentration causes more VEGF to be bound without affecting free VEGF content. Experiments show that HSPG has a significant effect on the mitogenic activity of VEGF [
Our model shows that even under resting physiological conditions, VEGF gradients of up to 12% VEGF/10 μm can exist throughout skeletal muscle. The gradients exist due to the spatial heterogeneity of capillaries within the tissue. When capillaries are uniformly spatially distributed, gradients do not exceed 4% VEGF/10 μm at any location. The model's random capillary placement is typical of capillary distribution in skeletal muscle [
Our model predicts the angiogenic effect of VEGF using the density of bound VEGF receptors on capillaries as an indication of the probability of capillary proliferation or activation which leads to angiogenesis. Because VEGF receptor binding has low heterogeneity (i.e., low standard deviation) under uniform expression (
Local overexpression simulations (where only one to three fibers overexpress VEGF) follow experimental observations of transgenic myocyte transplants [
Our model provides a framework for future models to study VEGF gradients, and can be applied to hosts of other physiological and drug-induced conditions (including exercise, ischemia, and VEGF pellet administration), as well as to other tissues (including tumor). The 2-D model can be extended to a 3-D model in order to analyze longitudinal VEGF gradients at the cost of computational complexity. This study also provides a valuable tool to biologists by raising questions concerning both the importance of intracellular VEGF signaling and biophysical changes to the tissue environment when stress is applied to tissue. Many questions still remain unanswered such as whether large quantities of VEGF receptors are expressed on myocytes or on the intraluminal surface of capillaries. These key questions may affect the results we present in this study and provide valuable insight into the complexity of VEGF signaling and interactions with other pro- and anti-angiogenic factors.
In summary, the key findings of this study are: significant VEGF gradients are predicted to exist in tissue in vivo, sensible at the single-cell level; VEGF concentration gradients lead to blood vessel VEGF receptor activation heterogeneity; VEGF gradients increase following VEGF overexpression; and the arrangement of VEGF-overexpressing cells affects VEGF gradients and VEGF receptor activation.
In this study we are interested in the steady state solution. At each simulation step, free VEGF concentration was first obtained using
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The authors thank Dr. Olga Hudlicka and Dr. Margaret D. Brown for critical discussions of angiogenesis in skeletal muscle, Dr. Patricia A. D'Amore for discussions of VEGF isoforms, and Dr. Brian H. Annex and Dr. Christopher D. Kontos for discussions of pro-angiogenic therapies.
basement membrane
endothelial basement membrane
extracellular matrix
extensor digitorum longus
fibroblast growth factor
heparan sulfate proteoglycans
myocyte basement membrane
myonuclear domain
vascular endothelial growth factor