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Department of Biomedical Engineering, Johns Hopkins University School of Medicine, Baltimore, Maryland
Submitted 20 July 2006 ; accepted in final form 4 October 2006
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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transcription factor. We estimate VEGF secretion and its concentrations and gradients in resting muscle and for different levels of exercise. The effects of low levels of inspired oxygen are also studied. We predict that the high spatial heterogeneity of muscle fiber VEGF secretion in hypoxic tissue leads to significant gradients of VEGF concentration and VEGF receptor activation. VEGF concentration gradients are predicted to be significant in both resting and exercising muscle (4% and 6–8% change in VEGF over 10 µm, respectively), sufficient for chemotactic guidance of 50-µm-long sprout tip cells. VEGF gradients also result in heterogeneity in VEGF receptor activation—a possible explanation for the stochasticity of sprout location. In the absence of interstitial flow, gradients are 10-fold steeper in the transverse direction (i.e., perpendicular to the muscle fibers) than in the longitudinal direction. This may explain observed perpendicular anastomoses in skeletal muscle. angiogenesis; cytokine; vascular endothelial; growth factor; mathematical model
VEGF isoforms are secreted by a variety of cells in the body, including myocytes, or skeletal muscle fibers. Secretion rates vary for each cell type (68). In rats, there are five main protein splice isoforms produced from the VEGF gene, denoted 120, 144, 164, 188, and 205 (corresponding to the number of amino acids). Isoforms VEGF120 and VEGF164 are the most frequently expressed (44) and each causes different intracellular signaling in angiogenesis (34, 49). VEGF164 is a 45-kDa homodimeric glycoprotein and contains an exon-7 encoded domain that confers binding to heparin and neuropilins. VEGF120 is a 36-kDa homodimeric glycoprotein and is missing the exon-7 encoded domain. Thus only VEGF164 binds the heparan sulfate proteoglycans (HSPG) present in high concentrations in the extracellular matrix (ECM) and basement membranes (BM) in the interstitial space. Neuropilins, coreceptors for VEGF164, are not included in this analysis; they have yet to be quantified in skeletal muscle. The secretion of VEGF by one subset of cells and VEGF internalization by a second subset of cells may result in a steep VEGF164 gradient in tissue (60). The response of endothelial cells to VEGF occurs as a result of intracellular signaling initiated by the binding of VEGF family members to their cell surface receptor tyrosine kinases: VEGFR1 and VEGFR2. These two VEGF receptors also serve to deplete VEGF from the interstitial space after binding by internalization of the receptor-ligand complex. The receptors and their interactions are discussed in depth in Refs. 45 and 57.
Use of VEGF for proangiogenic therapy for cardiac and limb ischemia has generated considerable interest, but direct administration of VEGF has not yet produced effective results compared with placebo trials. Initial trials of transplantation of proangiogenic cells show promising results but require further investigations to confirm their effectiveness and safety (15, 16, 59). Exercise-induced proangiogenic therapy is an alternative with promising initial results and a low potential for harmful side effects (6, 33, 66). However, to create an effective VEGF-driven angiogenic therapy for ischemia, a better understanding of the mechanisms of angiogenesis is still required.
Experimental evidence shows that increased microvascular shear stress can be sufficient for induction of VEGF upregulation (42, 67). However, hypoxia also stimulates VEGF upregulation as a result of varying levels of the transcription factor hypoxia inducible factor (HIF-1), an oxygen sensing molecule (61). The angiogenic response to VEGF depends not only on total VEGF concentration but also the spatial distribution and extracellular gradients of VEGF, which acts as a chemoattractant and directs capillary growth (21, 24). Regulation of microenvironmental levels of VEGF (as distinct from whole tissue levels of VEGF) is necessary for prevention of leaky, malformed vessels and hemangiomas (1, 47).
In the present study, we constructed a three-dimensional computational model to study the transport of VEGF during exercise in vivo using the well characterized rat EDL as a sample environment. We use an empirical relationship between VEGF secretion and oxygen concentration consistent with experimental data on HIF-1 protein levels vs. oxygen tension (29) and with experimental data on VEGF secretion vs. HIF-1 protein (61). We previously constructed models studying the kinetics of VEGF binding to receptors and the effects of the presence of neuropilin-1 or placental growth factor (PlGF; Refs. 39, 40), and a model of VEGF transport in two-dimensional slices of resting muscle (38). However, to our knowledge, this is the first model of VEGF transport in vivo under conditions of exercise that can predict VEGF distribution in three dimensions at resolutions currently impossible to achieve experimentally. Use of this model will aid in understanding mechanisms of exercise-induced VEGF upregulation and angiogenesis, including hypoxia-induced VEGF upregulation, formation of VEGF gradients in three dimensions, and activation of angiogenesis processes of endothelial cells by VEGF signaling. In addition, predictions of VEGF distributions in three dimensions will be a valuable tool for future studies of VEGF and angiogenesis. This model is also general enough to be applied to simulation of other tissue types, including myocardium and solid tumors.
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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Rat EDL muscle geometry has been well characterized in resting and exercise states and in its responses to hypoxia and electrical stimulation (26, 42, 61). We constructed a three-dimensional model of EDL tissue with dimensions 200x208x800 µm3; muscle fibers and blood vessels are separated by interstitial space (Fig. 1). At the edges of the model tissue volume, periodic boundary conditions for the molecular species under consideration are applied, (i.e., this is a piece of muscle surrounded by similar tissue). The dimensions of the area were chosen to avoid geometric discontinuities.
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Two venules and two arterioles (10 µm in diameter) were added to each 100x104x800 µm3 volume, for a total of eight arterioles and eight venules placed in a staggered fashion along the longitudinal axis of the tissue (i.e., parallel to the muscle fibers). For an example of arteriole and venule placement, refer to Ref. 28. Capillaries of 6 µm external diameter and 0.5 µm wall thickness (5 µm in lumen diameter) were added randomly into the interstitial space between the muscle fibers (Fig. 1B). The endothelial cells that line these vessels express VEGFR1 and VEGFR2 uniformly on their abluminal surface. A total of 66 capillaries was placed randomly in the tissue, each running approximately along the longitudinal axis of a muscle fiber. Each capillary is 
350 µm in length and connects to an arteriole at one end and a venule at the other. The placement of capillaries was constrained by a minimum capillary to capillary distance parameter of 5 µm (measured from the outer edge of each capillary) and a minimum capillary to muscle fiber distance of 1 µm (measured from the outer edge of a capillary to the outer edge of a muscle fiber). Approximately 33 capillaries cross any tissue cross section consistent with experimental observations of 800/mm2 capillary density in resting EDL (62). By taking tissue cross sections and measuring the distance from random points in the tissue to the center of the nearest capillary, we noted that our network closely matches experimentally measured vascular distribution [Ref. 31; mean distance to capillary: 15.5 µm (model network) vs. 15.9 µm (experimental measurement); 95th percentile distance to capillary, 29.8 µm (model) vs. 28.6 µm (experiment)]. Each arteriole, venule, or capillary vessel is constructed as a series of small cylindrical segments, each 10 µm in length, to allow for non-straight vessels. Tortuosity increases total capillary length by 1% and no anastomoses were added to the vasculature. The vessels accounted for 2.5% of the tissue volume and the interstitial space totals 17.8% of the tissue volume.
For the two-dimensional simulations, a cross section was extracted from the model at a position without arterioles and venules (see Fig. 4A).
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Each of the above parameters can be stated in different units; the units given above are consistent for the model equations presented. The parameters should be converted to other units for comparison with appropriate experimental data, e.g., molecules per cell or picomoles per cubic micron of tissue. These conversions can be achieved using the surface area-to-tissue volume ratios for myocytes (850 cm2/cm3) or blood vessels (150 cm2/cm3) in this tissue and the area of the cell (1,000 µm2/endothelial cell; myocyte surface area is discussed later). For example, VEGF binding to endothelial cells is typically reported here as 10–7 pmol/cm2; 1,000 10–7 pmol/cm2 is equivalent to 600 molecules/endothelial cell.
Transport Calculations and Reaction Kinetics
VEGF diffusion occurs within interstitial space (ECM and BM), and muscle fibers and capillaries are explicitly represented in the model as a separate phase and thus are boundaries for VEGF transport. BMs are thin and, 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 binding to receptors comes from adjacent EBM; VEGF dissociating from receptors is released into the EBM; VEGF produced by myocytes is secreted into adjacent MBM space, from which diffusion into the ECM can then occur.
The transport of VEGF within the ECM (diffusion and binding to ECM proteoglycans) is described by mass balance equations:
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and Jout have different values. VEGF transport between ECM and EBM is described by mass balance equations:
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On the endothelial cell surface, VEGF can be bound to receptors and both free and bound receptors can be internalized, following our previous study (40). Equations 7–12 apply at the surface of endothelial cells only, as muscle fibers are assumed not to express significant levels of VEGF receptors. The VEGF concentrations in Eqs. 7–12 are the free VEGF concentrations in the EBM.
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Thus Eqs. 1, 2, 13, and 14 govern the concentration of VEGF in the ECM space, whereas Equations 3–6 and 7–12 together form the boundary conditions at the myocyte cell surface (Eqs. 3–4) and at the endothelial cell surface (Eqs. 5–12).
Oxygen-Dependent VEGF Secretion
VEGF secretion from each nucleus in the skeletal myocyte is dependent on the average oxygen tension (PO2), although the exact relationship has not been experimentally measured (Fig. 2; Ref. 48). Cellular oxygen sensing occurs through HIF-1 protein, a transcription factor (55), and HIF-1 response to oxygen has been measured in vitro in HeLa (immortal cervical cancer line) cells for physiologically relevant O2 ranges (29). HIF-1 concentration and activity increased exponentially as intracellular PO2 decreases from 20 to 3 mmHg but then decreased as PO2 falls below 3 mmHg (29). Experimental evidence shows that VEGF mRNA is upregulated up to sixfold with increased concentration of HIF-1 during exercise (61). However, experimental measurements are not available for intracellular oxygen tension in the control animals, and measurements of VEGF mRNA at one- to twofold elevation of HIF-1 concentration are not sufficient to develop a definitive quantitative relationship between oxygen tension and VEGF mRNA levels. Therefore, a graded increase in VEGF mRNA is expected for decreasing muscle PO2, but the exact O2-HIF-VEGF secretion relationship has not been quantified.
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20 mmHg) secretes VEGF at basal (resting) levels (see Estimates of VEGF Secretion Rate) because intracellular oxygen sensing by HIF-1 does not change significantly above that level (11% difference between 20 and 30 mmHg; Ref. 29). When skeletal muscle is electrically stimulated or exercised, the muscle fibers become hypoxic and VEGF secretion is upregulated. The present model assumes that under severely hypoxic conditions (PO2 
1 mmHg), skeletal myocytes secrete VEGF at sixfold basal level, consistent with Ref. 61. Because in vivo VEGF secretion at intermediate degrees of hypoxia has not been experimentally measured, VEGF upregulation dependence on oxygen is modeled by the following equation (Fig. 3) :
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is a coefficient describing the shape of the O2-VEGF secretion curve. When = 1, VEGF secretion is linearly dependent on PO2; as increases, VEGF secretion increases modestly as PO2 drops below 20 mmHg and increases steeply as PO2 nears severe hypoxic levels (near 1 mmHg; Fig. 4). We assumed = 3 for our model and also explored the effect of setting = 1.
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2,500 µm2. For this alternative mode of VEGF secretion, each MD secretes VEGF uniformly and responds to the average PO2 within that local volume. No study has yet determined the spatial range of oxygen sensing within each myocyte around each nucleus and the range of transport of VEGF from each nucleus to the myocyte surface. Therefore, both secretion modes are explored here to determine whether significantly different VEGF gradients are produced [supplementary Fig. S2 (the online version of this article contains supplemental data)].
Oxygen Transport
Spatial oxygen distribution is predicted using our previously published model of oxygen transport in EDL (28), with the amendment that transport is periodic at boundaries in all three dimensions to remain consistent with VEGF transport. Exercise imposes higher oxygen consumption (up to 50-fold above resting levels) and higher blood flow (up to 25-fold above resting). This model used consumption parameters for low or moderate intensity exercise, 10–3 and 2·10–3 ml O2·ml tissue–1·s–1, respectively (4, 26, 28). These values correspond to six- and twelve-fold basal (resting) consumption. This study simulated oxygen transport for rest or exercise while breathing in a normal environment (21% O2) or in a low oxygen environment (hypoxic hypoxia). These conditions are interpreted in the model as normal or low blood oxygen saturation of arterioles feeding into capillaries (SO2A). Under resting conditions, SO2A has been measured to vary between 0.6 and 0.8 (27, 65). Our model uses SO2A values of 0.6 and 0.3 for simulations under normal and hypoxic hypoxia environments, respectively. The model of microvascular blood flow takes into account the Fahraeus-Lindqvist effect and non-uniform hematocrit distribution due to phase separation at vascular bifurcations (28). Using parameters for exercising skeletal muscle (28), the hydrodynamic pressure drop from an arteriole to a venule is 10 mmHg, blood feeding into capillaries have discharge hematocrits of 0.4, the blood flow model predicts the average blood velocity in capillaries of 1,100 µm/s (10 times the resting velocity; Ref. 42). These conditions underestimate the heterogeneities in flow and oxygen transport because the hydrodynamic pressure drop, entrance discharge hematocrit, and entrance SO2 are likely to vary (13); however, we will show that spatial gradients of VEGF and its secretion are significant even under these conditions.
Numerical Solution
Equations 1–14 were discretized using a fully implicit finite difference algorithm; first-order spatial and temporal derivatives were expressed with a forward difference scheme and second-order spatial derivatives were expressed with a central difference scheme. An orthogonal grid with uniform spacing of 1 µm in each spatial coordinate was used; decreasing the grid spacing to 0.5 µm did not significantly alter the results. Areas of BM are identified by intersections between a grid point in ECM and a grid point within a muscle fiber or capillary. The BM thickness is less than one-tenth of the grid size and thus its effect is included in the lumped boundary condition (Equations 3–6 and 7–14). Each BM point lies between two grid points and the BM in the model is discretized into a number volumes equal to the total number of grid intersections passing through BM space in the model. Discretized BM areas surrounding each capillary or muscle fiber are assumed to be equal in size. At the boundaries of the tissue, periodic boundary conditions were applied.
In this study we are interested in the steady-state solution, not the transients. At each simulation step, free VEGF concentration was first obtained using Equations 1–6, and then binding, insertion, and internalization of VEGF receptors at the cell surface and binding to HSPG were solved independently using Equations 7–14. The solution is iterated until VEGF concentrations converge to a steady state. A red-black successive over-relaxation scheme was used in which the inner iterations did not need to proceed to convergence because steady state is the only result of interest and the solutions at intermediate "time" steps are not of importance (17). The convergence criterion used was a maximum fractional change of 10–5 for VEGF120 and VEGF164 at each grid point per step. The model is coded in C++ and was run on 2.5-Ghz Itanium2 processor with 2 GB of RAM. Two-dimensional simulations require 10 min to reach a steady-state solution and three-dimensional simulations require 48 h.
VEGF Transport Parameters
The physiological parameters used in this model are summarized in Table 1. While experimentally measured parameters for VEGF transport in rat EDL are preferable, many parameters were estimated from experiments performed on rats and other species.
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Aqueous diffusivities of 142 and 133 µm2/s were calculated for VEGF120 and VEGF164, respectively, using a molecular weight-based relationship for globular proteins (3) and adjusted to 37°C using the Stokes-Einstein relation. To obtain in vivo values of diffusivity, we followed a method previously outlined (18) to calculate diffusive hindrance in the interstitial space. In skeletal muscle ECM, VEGF diffusion is hindered by collagen fibers (concentration: 75 mg/g ECM; fiber radius: 20 nm; fiber volume fraction: 0.14) and glycosaminoglycan (GAG) chains (5 mg/g ECM, 0.55 nm; volume fraction: 7.8 10–4; Ref. 35). Using these values, we predict in vivo diffusivities of 113 and 104 µm2/s for VEGF120 and VEGF164, respectively.
HSPG and VEGF Receptor Density and Kinetics
No in vivo measurements are available for the HSPG concentration in skeletal muscle, so values in ECM and BM were obtained from measurements in human myocardium (30, 53) summarized in Ref. 18. Effective VEGF-HSPG binding kinetics in vivo has not been measured, so we use those determined from in vitro studies for reactions of VEGF binding to HSPG (9).
VEGF receptor concentrations were based on in vivo measurements of total VEGFR2 protein concentration and capillary density in human vastus lateralis skeletal muscle (10, 54). Assuming that total protein concentration is 150 mg/g muscle in skeletal muscle (7), surface area is 1,000 µm2/endothelial cell, and human capillary diameter is 7 µm, the in vivo measurements correspond to an average VEGFR2 density of 20,000 receptors per cell. This is of the same order of magnitude as previously measured in vitro values of 50,000 receptors/cell (40). However, no study has yet determined the percentage of total VEGF receptors that are expressed: on the abluminal capillary surface, within the cell, and on the luminal cell surface. In this study, we assume as a baseline that 50% of total receptors (10,000 VEGFR2/cell) are expressed on the abluminal cell surface (i.e., in a location available for binding of interstitial VEGF).
In vivo measurements in human muscle find approximately ten times as much VEGFR1 expressed as VEGFR2 (10, 54); however, soluble VEGFR1 (sVEGFR1) is present in interstitial spaces of skeletal muscle (although we have not included it in this model) and no studies in skeletal muscle have yet reported the ratio of VEGFR1 to sVEGFR1. We assume that an amount of VEGFR1 is expressed on the abluminal extracellular surface equal to that of VEGFR2 (10,000 VEGFR1/cell). Neuropilin density has not yet been measured in skeletal muscle and is not included here. VEGF receptor kinetic rates that were obtained from in vitro binding studies (40) were assumed to be valid for the in vivo model.
Estimates of VEGF Secretion Rate
Currently, only in vitro (68) and explanted tissue (32, 36, 41, 58) measurements of VEGF protein secretion have been published. Measurements of total VEGF in rat include VEGF located both intracellularly and extracellularly, which is sensitive to HSPG and sVEGFR1 concentrations (42). To simulate in vivo conditions, secretion rates for VEGF120 and VEGF164 were obtained by finding values that result in the model matching experimental in vivo measurements of free (unbound) interstitial VEGF in human skeletal muscle (25). Under resting conditions, unbound VEGF concentrations range between 0.6 and 1.5 pM; in our study we target a concentration of 1 pM for muscle at rest. To meet this requirement, muscle fibers must express VEGF at a rate of 3.2 10–17 pmol·µm–2·s–1 (2.7 fmol·l tissue–1·s–1). Different receptor densities would require different secretion rates to achieve the same unbound VEGF concentration.
Protein production and secretion of skeletal myocytes can be compared with that of mononucleate cells, using the MD, as defined above. The VEGF secretion rate used in this model corresponds to 0.048 molecules·MD–1·s–1, comparable to explanted tissue measurements in various explanted tissues including rat skeletal muscle: 0.01 in rat tibialis anterior (58), 0.08 in rat adipocyte (41), and 0.10 in rat cavernous smooth muscle (36) (all in molecules·cell–1·s–1). Using relative mRNA abundances for splice variants of VEGF in mouse skeletal muscle (44) and assuming a linear relationship between VEGF protein secretion and mRNA levels (61), we infer that myocytes secrete VEGF164 at nearly 12 times the rate of VEGF120 (2.95 vs. 0.25 10–17 pmol·µm–2·s–1, or 0.044 vs. 0.004 molecules·MD–1·s–1).
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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The model described above was used to perform simulations of VEGF secretion, diffusion, binding, and internalization in skeletal muscle representative of rat extensor digitorum longus. First, two-dimensional simulations were performed for cross sections of the muscle perpendicular to the alignment of the muscle fibers. Then, to estimate VEGF distribution and concentration gradients in the longitudinal direction, three-dimensional simulations of VEGF distribution in the muscle were performed. For each case, uniform VEGF secretion from all fibers was simulated. Then, with the use of the predicted oxygen distribution in the tissue, fiber-specific VEGF secretion was calculated for several cases of increased oxygen consumption (e.g., exercise) and a comparison was made between normal and reduced levels of inspired oxygen.
Two-Dimensional Simulations of VEGF Distribution for Uniform VEGF Secretion
The oxygen distribution in the tissue at rest is close to uniform and is above 20 mmHg across the cross section (21.8–35.2 mmHg). We assume that this level of oxygen will result in a basal (or resting) level of VEGF secretion uniformly from all the fibers in the cross section (Fig. 3A).
The uniform secretion of VEGF from the fibers results in relative VEGF gradients in the cross section of up to 12% VEGF/10 µm, with mean gradients of 3% VEGF/10 µm (Fig. 3B). 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. The gradients are largely due to the heterogeneous placement of the capillaries. The concentration is highest (red) in regions of low capillarity and lowest (blue) in regions of high capillarity (Fig. 3B). Note that the gradients in the free VEGF concentration and in the total (free plus HSPG-bound) VEGF are the same in percentage terms because the low fractional occupancy of HSPG (<1%) creates a linear relationship between free and HSPG-bound VEGF concentration. The gradients in VEGF concentration are reflected in the VEGF receptor binding level on the surface of the capillaries, as estimated by the VEGF bound to VEGF receptors on the microvessels (Fig. 3A). The mean surface concentration of bound VEGF is 516 x 10–7pmol VEGF/cm2 with a standard deviation of 22 x 10–7 pmol VEGF/cm2; VEGF bound to VEGFR2 only is 139 x 10–7 pmol VEGF/cm2 with a standard deviation of 5.5 x 10–7 pmol VEGF/cm2.
Of the total VEGF in the tissue, 50% is bound to receptors (VEGFR1: 37%; VEGFR2: 13%), 49% is bound to HSPG (ECM: 36%; MBM: 9%; EBM: 4%), and 1% exists as freely diffusible VEGF. Of this unbound VEGF, 7% VEGF is VEGF120 and the other 93% is VEGF164. The concentration of total (bound and free) VEGF is 17.5-fold higher in the MBM and 16.6-fold higher in the EBM compared with the ECM due to the higher concentration of HSPG in BM. On the endothelial cells, 2.3% of VEGFR1 and 0.8% of VEGFR2 are occupied by VEGF (230 and 80 molecules bound per cell, respectively).
As a first approximation of the changes in exercise, VEGF secretion from the muscle fibers is increased uniformly to fivefold the resting secretion levels (Fig. 3C). Although this results in a significant increase in the average interstitial VEGF concentration, there is no change in the relative concentration gradients (Fig. 3D). The absolute gradients (e.g., pM VEGF/10 µm) are higher and can be obtained by multiplying the relative gradient by the average interstitial VEGF concentration. The cell surface VEGF binding level of the capillaries is higher on average but maintains the same distribution as the resting muscle [total VEGF binding, mean 2,580 ± 85 (SD) 10–7 pmol/cm2; VEGFR2 only, mean 697 ± 26.7 (SD) 10–7 pmol/cm2 or 418 ± 16 VEGF molecules/endothelial cell; Fig. 3C].
Two-Dimensional Simulations of VEGF Distribution for Oxygen-Dependent VEGF Secretion
The distribution of oxygen in the muscle depends on the blood flow and oxygen content of the capillary network, as well as the consumption rate in the muscle fibers (Fig. 4A). The cross-sectional slice of the tissue that is used in the two-dimensional simulations is marked. The average oxygen concentration across the cross-sectional area of each muscle fiber is used to calculate the secretion rate of VEGF from that fiber (Fig. 4B). For oxygen tensions above 20 mmHg, VEGF secretion rate is unchanged from the basal rate. There is a maximal sixfold upregulation of oxygen-dependent VEGF secretion rate when PO2 falls below 1 mmHg. This does not preclude higher VEGF secretion rates due to tumorigenesis or genetic manipulation. Between these two extremes, the VEGF secretion rate is a function of the oxygen tension across the fibers as described in Oxygen-Dependent VEGF Secretion.
For light exercise (oxygen consumption in the muscle 6-fold above resting), the oxygen distribution is similar to resting conditions (data not shown). Thus the secretion level of VEGF is similar to the resting muscle (Fig. 3, A and B). The mean VEGF gradient is 3.1% VEGF/10 µm and the VEGF-VEGFR2 binding is 143 ± 5.5 10–7 pmol/cm2.
The oxygen gradients across the tissue are steeper for light exercise when the simulation is performed for lower values (one-half normal) of arteriolar hemoglobin saturation, SO2A, corresponding to lower inspired oxygen (Fig. 5A), which results in lower oxygen content of the capillaries and is a condition used in some physiological exercise experiments (61). This condition is referred to as hypoxic hypoxia. This oxygen distribution results in non-uniform secretion of VEGF from the fibers in the muscle (Fig. 5B). The average VEGF secretion increase from the fibers is now 3.6-fold basal level, but the range is 2.1-fold to 4.8-fold. The gradients of interstitial VEGF (Fig. 5C) are larger than in resting muscle (mean 4.9% VEGF/10 µm). These increased gradients and average VEGF concentration (98 pM) correspond to a more heterogeneous distribution of VEGF receptor binding on the vessels (mean binding 1,600 ± 79 10–7 pmol/cm2; VEGFR2 only 401 ± 18.3 10–7 pmol/cm2; Fig. 5B).
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Combining the moderate exercise with the lower inspired oxygen content results in severely limited oxygen delivery to the muscle fibers (Fig. 5G). This results in upregulation of VEGF secretion by the muscle fibers (mean 5.7-fold, range 4.0- to 6.0-fold; Fig. 5H). Some fibers are surrounded by five adjacent capillaries so they remain oxygenated (7–8 mmHg) even under exercise conditions, and these fibers have smaller increases in VEGF secretion. The gradients of VEGF concentration are increased more (mean 6.4% VEGF/10 µm; Fig. 5I), and both the mean capillary receptor binding and its variability are higher (mean 2,973 ± 122 10–7 pmol/cm2; VEGRF2 only 788 ± 32.4 10–7 pmol/cm2; Fig. 5H). Results of simulations for linear oxygen-VEGF secretion relationship are presented in the supplement data.
Three-Dimensional Simulations of VEGF Distribution for Uniform VEGF Secretion
In resting muscle, oxygen concentration is high across the tissue and we assume VEGF secretion from the fibers to be uniform (2.7 fmol·liter tissue–1·s–1). Despite this uniform basal secretion level, three-dimensional simulation of VEGF distribution in muscle reveals gradients of VEGF concentration both in cross section and along the length of the fibers (Fig. 6A). These gradients are predicted to be present even without oxygen gradients in the tissue, such as may be established in exercise (Figs. 5, A, D, G, and 7, A and E). Two-dimensional cross sections of the interstitial VEGF concentration reveal similar patterns of VEGF distribution at different points along the length of the muscle (Fig. 6B). The VEGF gradients in the longitudinal (along the fibers) direction are less significant than in the cross-sectional planes, as demonstrated by the similarity in the histograms of VEGF concentration (Fig. 6C). The mean gradient (in three dimensions) of VEGF concentration is 3.7% VEGF/10 µm; the gradient in the longitudinal direction averages 0.4% VEGF/10 µm (Fig. 8). VEGF binding to cell surface receptors is nonuniform across the muscle (Figs. 6D and 9).
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In moderate exercise, oxygen distribution in the muscle fibers is less uniform than in resting muscle (Fig. 7A), and this is particularly true when the subject is additionally inspiring reduced-oxygen air (Fig. 7E).
The low-oxygen distribution in the muscle results in VEGF upregulation by the muscle fibers, and the difference between the normoxic (mean, 4.0-fold; range, 1.4–5.7-fold) and hypoxic (mean, 5.8-fold; range, 3.1–6.0-fold) conditions is stark (Fig. 7, B and F). The average interstitial VEGF concentration in the tissue is increased (Fig. 7, C and G), along with the gradients (normoxic, 5.6% VEGF/10 µm; hypoxic, 7.1% VEGF/10 µm; Fig. 8). The average VEGF binding to receptors on surfaces of microvessels is significantly increased under hypoxic breathing conditions (Fig. 7, D and G). The variability in capillary receptor binding is also increased significantly (moderate exercise; mean total binding 1650 ± 103 10–7 pmol/cm2 of which VEGF binding to VEGFR2 is 412 ± 25.4 10–7 pmol/cm2; moderate exercise + hypoxic hypoxia: mean total binding 3000 ± 151 10–7 pmol/cm2 of which VEGF binding to VEGFR2 is 780 ± 39.2 10–7 pmol/cm2; Fig. 9).
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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The activation of sprouting at the existing blood vessels (the initiation step of daughter vessel outgrowth) may require local VEGF concentration (as measured by binding to the receptor tyrosine kinases) to reach a threshold level. The presence of significant gradients of VEGF can result in a small number of vessels crossing the activation threshold (Fig. 9), even where the average VEGF concentration in the tissue is not sufficiently high for global angiogenic activation. These gradients may limit the number of angiogenic sprouts that begin and define their origins at specific locations in the blood vessel network. If all vessels were activated simultaneously, many new vessels would grow close to each other, and ultimately oversupply the tissue, an inefficient solution.
Our simulations show that longitudinal VEGF gradients (along the muscle fiber orientation) are significant but that transverse gradients (perpendicular to muscle fiber orientation) of VEGF are predicted to be 10 times steeper on average over 10 µm distances. Our simulations show that longitudinal gradients are driven predominantly by spatial heterogeneity of oxygen distribution and VEGF secretion, whereas transverse VEGF gradients are driven by both VEGF secretion heterogeneity and heterogeneity of capillary placement. Therefore, we predict that angiogenic activation of capillary endothelial cells is driven mainly by transverse gradients that are predicted to guide angiogenic sprouts to grow away from the parent vessel at nearly 90 degrees. Although most angiogenic sprouts range 70–90 degrees from the parent vessel, some sprouts can be 45 degrees or less (5, 23), so we predict that other factors must help guide capillaries toward a more acute sprouting angle. Factors such as interstitial fluid flow (not included in our model because their directionality cannot be precisely measured in vivo) can have a powerful effect on the magnitude and orientation of VEGF gradients (24). Furthermore, endothelial tip cells also respond to other cues such as collagen orientation for "contact guidance" (8) and to extracellular matrix density (22) during angiogenic sprouting.
The gradients of VEGF result in the highest likelihood of vessel angiogenic activation being in areas of sparse vascularization, i.e., areas that experience higher levels of hypoxia (Figs. 5, 7). This raises an interesting question: does neovascularization in hypoxic tissues begin by sprouting from vessels in the hypoxic area and proceed by forming connections with vessels in less hypoxic zones? And, if so, how does the growing sprout move down the VEGF gradient, rather than up as would be the case in classic chemotaxis, e.g., in tumor angiogenesis, where the tumor is initially devoid of vessels and the VEGF induces angiogenic sprouting from vessels in normal, normoxic adjacent tissue? Recent studies suggest that this may be explained by local microgradients resulting from the combination of proteases that release VEGF164 from the matrix and slow interstitial convective flows (20, 24).
The small arterioles and venules in the microvascular network analyzed here are assumed to express VEGF receptors at the same surface density as the capillaries. There is insufficient experimental detail on the spatial distribution of VEGF receptors in the microcirculation. In addition, neuropilin is not included in this analysis, as the density in skeletal muscle is not known. VEGF receptor expression is assumed not to change during the acute exercise modeled here.
All muscle fibers in our simulations are considered equal, but heterogeneity of muscle fiber activation exists for large volumes of muscle. Under exercising conditions, muscle fibers are activated in groups by motoneurons, and different fibers can be activated at different, possibly irregular, intervals (37). However, our model is concerned with steady-state distributions of VEGF for a small tissue volume consisting of 30 muscle fibers; our assumption that all muscle fibers in this volume are activated equally will not affect our analysis of VEGF gradients.
The form of the oxygen-HIF-VEGF secretion rate relationship has not been defined in vivo. We assume a nonlinear relationship, where the induction of VEGF upregulation is skewed toward conditions of severe hypoxia (Fig. 4B). For comparison, we performed the two-dimensional oxygen-dependent simulations for a linear VEGF response to hypoxia (supplemental Fig. S1). The linear VEGF-O2 response results in near-maximum VEGF secretion rates under non-hypoxic conditions (at 5 mmHg, secretion is 5-fold basal for the linear relationship and 3.2 for the hyperbolic relationship). Overall, VEGF secretion is increased and VEGF secretion is more homogeneous between fibers for the linear vs. hyperbolic relationship.
The three-dimensional simulations of oxygen-dependent VEGF secretion presented here assume that the secretion rate from the surface of each 2-µm-thick slice of myocyte is based on the average oxygen tension across that slice. We also performed simulations in which the oxygen tension is averaged over the average size of myonuclear domains (20 µm long). The VEGF concentrations and gradients are not significantly different (supplemental Fig. S2). VEGF gradients in all exercise cases for these two modes of secretion differ by <0.1% VEGF/10 µm for transverse gradients and <0.01% for longitudinal gradients.
Our study provides quantitative predictions of VEGF gradients in vivo at a resolution, below cell dimensions, that cannot be achieved by experimental techniques. Our results show that hypoxia can create significant VEGF gradients, which can signal angiogenic sprouting on specific capillaries. To test the predictions of the model, hypoxia could be imaged, for example, using EF5/Cy3 immunohistochemical staining. Originally developed to image hypoxia in tumors, this technique can be applied to muscle environments (64). In addition, imaging and staining for angiogenic activity can be achieved on the level of tip cells and filopodia (21). Costaining experiments for angiogenic responses for hypoxia, VEGF localization, and angiogenic activity would greatly enhance the understanding of the mechanisms of angiogenesis.
In summary, this study makes several significant findings. First, significant extracellular VEGF concentration gradients (mean 3.7% VEGF/10 µm at rest, 5.6% VEGF/10 µm for moderate exercise) are predicted to exist in skeletal muscle. The length of a typical retinal or hindbrain endothelial tip cell is 50 µm (21, 52), and thus the change in VEGF concentration across the cell may be up to five times higher. Second, VEGF concentration gradients in the tissue result in significant heterogeneity in the activation of VEGF endothelial cell surface receptors on blood vessels. VEGF gradients may thus be a significant contributor to the stochasticity in sprout initiation. Third, in skeletal muscle VEGF concentration, gradients are 10-fold steeper in the transverse direction (i.e., perpendicular to the muscle fibers) than in the longitudinal direction (parallel to fibers), excluding the effects of interstitial flow. This is a possible cause of the observed angles (45–90°) of anastomoses between parallel capillaries in skeletal muscle (5, 23), with sprouts being guided perpendicular to the muscle fibers (and existing capillaries).
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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O2 max. Am J Physiol Heart Circ Physiol 271: H721–H737, 1996.This article has been cited by other articles:
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  J. W. Ji, F. Mac Gabhann, and A. S. Popel Skeletal muscle VEGF gradients in peripheral arterial disease: simulations of rest and exercise Am J Physiol Heart Circ Physiol, December 1, 2007; 293(6): H3740 - H3749. [Abstract] [Full Text] [PDF]
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 B. C. Thorne, A. M. Bailey, and S. M. Peirce Combining experiments with multi-cell agent-based modeling to study biological tissue patterning Brief Bioinform, July 1, 2007; 8(4): 245 - 257. [Abstract] [Full Text] [PDF]
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), VEGF bound to VEGF receptors on endothelial cells of blood vessels (
), and total VEGF concentration (free and matrix bound) for uniform VEGF secretion from each fiber. The VEGF secretion rate is 3.2 10–17 pmol·µm–2·s–1, or 0.048 molecules·MD–1·s–1; see Oxygen-Dependent VEGF Secretion. C and D: uniform 5-fold upregulation of VEGF secretion from each fiber. Notation as in A and B. 1,000 10–7 pmol/cm2 = 600 VEGF molecules/endothelial cell.
