Conceived and designed the experiments: PY SDF MOS ASP. Performed the experiments: PY. Analyzed the data: PY SDF MOS ASP. Wrote the paper: PY SDF MOS ASP.
The authors have declared that no competing interests exist.
Vascular endothelial growth factor (VEGF) is a key regulator of angiogenesis – the growth of new microvessels from existing microvasculature. Angiogenesis is a complex process involving numerous molecular species, and to better understand it, a systems biology approach is necessary.
The multiscale model is comprised of two compartments: blood and tissue. The model accounts for interactions between two major VEGF isoforms (VEGF120 and VEGF164) and their endothelial cell receptors VEGFR-1, VEGFR-2, and co-receptor neuropilin-1. Neuropilin-1 is also expressed on the surface of parenchymal cells. The model includes transcapillary macromolecular permeability, lymphatic transport, and macromolecular plasma clearance. Simulations predict that the concentration of unbound VEGF in the tissue is approximately 50-fold greater than in the blood. These concentrations are highly dependent on the VEGF secretion rate. Parameter estimation was performed to fit the simulation results to available experimental data, and permitted the estimation of VEGF secretion rate in healthy tissue, which is difficult to measure experimentally. The model can provide quantitative interpretation of preclinical animal data and may be used in conjunction with experimental studies in the development of pro- and anti-angiogenic agents. The model approximates the normal tissue as skeletal muscle and includes endothelial cells to represent the vasculature. As the VEGF system becomes better characterized in other tissues and cell types, the model can be expanded to include additional compartments and vascular elements.
Vascular endothelial growth factor (VEGF) belongs to a family of cytokines that play an important role in angiogenesis – the formation of new capillaries from pre-existing vessels. The VEGF family in mammals is composed of VEGF-A, VEGF-B, VEGF-C, VEGF-D and placental growth factor (PlGF). The most well-studied member is VEGF-A (generally referred to as VEGF) that consists of several splice isoforms including VEGF121, VEGF145, VEGF165, VEGF189 and VEGF206 in humans, where the subscripted number indicates the number of amino acids
The tyrosine-kinase receptors of VEGF include VEGFR-1 (Flt-1), VEGFR-2 (Flk-1 or KDR in humans), and VEGFR-3 (Flt-4). VEGFR-1 and VEGFR-2 are the primary receptors for VEGF-A and play a major role in angiogenesis, while VEGFR-3 binds VEGF-C and VEGF-D and plays a major role in lymphangiogenesis. VEGFR-1 and VEGFR-2 are predominantly expressed on endothelial cells; however, these receptors have also been shown to be present on bone marrow-derived cells
Computational models of VEGF-mediated angiogenesis have been developed to study various aspects of the angiogenic process
Under physiological conditions, VEGF level in the mouse blood is low (<1.5 pM)
Due to the large number of experimental studies performed in mice, including the work of Rudge
The basis of this mouse model is the human model developed by Stefanini and co-workers that was used to explore the VEGF distributions in humans in health and disease
The model is divided into the tissue and blood compartments. VEGF120 and VEGF164 are secreted by the parenchymal cells (myocytes) into the available interstitial space at rate
The binding interactions of VEGF120 and VEGF164 are different. VEGF120 binds to VEGFR-1 and VEGFR-2 but not to NRP-1. VEGF164 binds to VEGFR-1, VEGFR-2, NRP-1, and glycosaminoglycan (GAG) chains in the extracellular matrix. In simulations where the anti-VEGF agent (VEGF Trap) is added, both isoforms bind to the anti-VEGF agent to form a complex. Binding and unbinding of VEGF to receptors are denoted as
In our model, we include an anti-VEGF agent that can bind to and form a complex with VEGF in both the blood and tissue. The unbound anti-VEGF agent and the complex are also subject to intercompartmental transport via permeability and lymphatic drainage, and can also be cleared from the blood. The molecular interactions between the two VEGF isoforms and the anti-VEGF agent are illustrated in
We incorporate pore theory in modeling the interstitial space to reflect the available volume for VEGF to diffuse. The VEGF molecules are free to diffuse in the available interstitial fluid volume, denoted
The model is fully described with forty coupled ordinary differential equations (ODEs) including 24 molecular species in the tissue and 16 molecular species in the blood, representing a total of 94 chemical reactions. The complete set of equations and chemical reactions are presented in
In these equations,
The forty differential equations were implemented in MATLAB® (v7.10.0.499 R2010a, Mathworks®) using the SimBiology® toolbox and the simulations were run on a laptop PC. All simulations were performed using the sundials solver routine with an absolute tolerance of 10−20 and a relative tolerance of 10−5.
The overall model is parameterized for a 25-gram mouse and is not currently strain specific. Model parameters are summarized in
The formulation of the two-compartment mouse model follows the scheme we previously applied to the human model
The parameters describing the whole mouse are presented in
Value | Unit | Reference | |
|
25 | g |
|
|
1.75 | mL |
|
|
0.85 | mL |
|
|
0.49 | - | Calculated (see manuscript) |
|
1.75 | g | Calculated (see manuscript) |
|
23.25 | g | Calculated (see manuscript) |
|
21.93 | cm3 | Calculated (see manuscript) |
To model the tissue compartment, we used many properties of the mouse gastrocnemius muscle since this muscle is extensively studied and characterized (
Value | Unit | Reference | |
|
650 | #/mm2 | |
|
1.95 | - | |
|
2,500 | µm2 | |
|
33,333 | #/cm2 | |
|
5.25 | µm | |
|
0.5 | µm |
|
|
6.25 | µm | Calculated (see manuscript) |
|
1.99% | cm3/cm3 tissue | Calculated (see manuscript) |
|
21.77 | µm | Calculated (see manuscript) |
|
155.68 | cm2/cm3 tissue | Calculated (see manuscript) |
|
82.74% | cm3/cm3 tissue | Calculated (see manuscript) |
|
214.10 | µm | Calculated (see manuscript) |
|
8.56 | µm | Calculated (see manuscript) |
|
1.83×10−5 | cm2 | Calculated (see manuscript) |
|
154 | nm |
|
|
154 | nm |
|
|
7 | nm |
|
|
66 | nm |
|
|
0.002014 | cm3/cm3 tissue | Calculated (see manuscript) |
of which available to VEGF | 0.001983 | cm3/cm3 tissue | Calculated (see manuscript) |
|
0.009123 | cm3/cm3 tissue | Calculated (see manuscript) |
of which available to VEGF | 0.008986 | cm3/cm3 tissue | Calculated (see manuscript) |
|
0.141463 | cm3/cm3 tissue | Calculated (see manuscript) |
of which available to VEGF | 0.121419 | cm3/cm3 tissue | Calculated (see manuscript) |
|
0.113117 | cm3/cm3 tissue | Calculated (see manuscript) |
Because we first constrained the fractional volume of interstitial space, other tissue parameters needed to be adjusted accordingly to yield reasonable values for fractional volumes of muscle fibers and blood. Particularly, the capillary density, capillary/fiber ratio, and fiber cross-sectional area determine the remaining parameters required to characterize the mouse model. We used a capillary density of 650 capillaries/mm2, a capillary/fiber ratio of 1.95, and a fiber cross-sectional area of 2,500 µm2, which are consistent with experimental measurements
The fiber volume fraction is corrected to account for the capillary wall thickness and is calculated to be 82.74%. Using a fiber perimeter correction factor of 1.21
The interstitial space is assumed to be composed of the extracellular matrix (ECM), parenchymal basement membrane (PBM) and endothelial basement membrane (EBM). Although VEGF is able to diffuse in the interstitial space, part of this volume is inaccessible to VEGF. The thicknesses of the basement membranes are 154 nm
We further consider pores in the ECM, EBM, and PBM, which may be inaccessible to freely diffusible molecules in the interstitial space. The EBM pore size for rat brain capillaries has been measured to be 7 nm
Receptor densities and ECM binding site densities are listed in
Model parameters | Tissue parameters | ||||
Value | Unit | Value (Tissue) | Value (Blood) | Unit | |
|
1,050 | dimers/EC | pmol/cm3 tissue | ||
Luminal EC | 525 | dimers/EC | - | 1.70×10−1 | pmol/cm3 tissue |
Abluminal EC | 525 | dimers/EC | 1.36×10−2 | - | pmol/cm3 tissue |
|
700 | dimers/EC | pmol/cm3 tissue | ||
Luminal EC | 350 | dimers/EC | - | 1.13×10−1 | pmol/cm3 tissue |
Abluminal EC | 350 | dimers/EC | 9.07×10−3 | - | pmol/cm3 tissue |
|
35,000 | dimers/EC | pmol/cm3 tissue | ||
Luminal EC | 17,500 | dimers/EC | - | 5.67 | pmol/cm3 tissue |
Abluminal EC | 17,500 | dimers/EC | 4.53×10−1 | - | pmol/cm3 tissue |
Myocytes | 35,000 | dimers/myocyte | 2.26 | - | pmol/cm3 tissue |
|
0.75 | µM | 82.50 | - | pmol/cm3 tissue |
|
13 | µM | 9.02 | - | pmol/cm3 tissue |
|
13 | µM | 40.95 | - | pmol/cm3 tissue |
EC = endothelial cell.
Conversions:
Abluminal EC receptors: 2.59×10−5 (pmol/cm3 tissue)/(dimers/EC).
Luminal EC receptors: 3.24×10−4 (pmol/cm3 tissue)/(dimers/EC).
Myocyte receptors: 6.46×10−5 (pmol/cm3 tissue)/(dimers/myocyte).
ECM: 1.10×108 (pmol/cm3 tissue)/M.
EBM: 6.94×105 (pmol/cm3 tissue)/M.
PBM: 3.15×106 (pmol/cm3 tissue)/M.
It is known that VEGF164 binds to the glycosaminoglycan (GAG) chains of the heparan sulfate proteoglycans in the extracellular matrix
Transport parameters for VEGF, anti-VEGF, and the VEGF/anti-VEGF complex are listed in
Value | Unit | Reference | |
|
0.0680 | molecules/cell/s | see manuscript |
VEGF164 secretion rate | 0.0626 | molecules/cell/s | see manuscript |
VEGF120 secretion rate | 0.0054 | molecules/cell/s | see manuscript |
|
7.00×10−6 | cm3/s | |
|
|||
VEGF | 4.00×10−8 | cm/s |
|
anti-VEGF & VEGF/anti-VEGF complex | 3.00×10−8 | cm/s |
|
|
|||
VEGF | 0.23 | min−1 |
|
anti-VEGF |
8.86×10−4 | min−1 |
|
VEGF/anti-VEGF complex |
2.79×10−4 | min−1 |
|
*Optimized parameter values.
The kinetic parameters for the binding and unbinding of VEGF to VEGFR-1, VEGFR-2, NRP-1, and GAG chains are listed in
Measured parameters | Tissue parameters | ||||
|
Value | Unit | Value (Tissue) | Value (Blood) | Unit |
kon | 3.00×107 | M−1s−1 | 2.65×10−1 | 6.15×10−2 | (pmol/cm3 tissue)−1s−1 |
koff | 1.00×10−3 | s−1 | |||
Kd | 33 | pM | 3.73×10−3 | 1.61×10−2 | pmol/cm3 tissue |
|
|||||
kon | 1.00×107 | M−1s−1 | 8.84×10−2 | 2.05×10−2 | (pmol/cm3 tissue)−1s−1 |
koff | 1.00×10−3 | s−1 | |||
Kd | 1.00×102 | pM | 1.13×10−2 | 4.88×10−2 | pmol/cm3 tissue |
|
|||||
kon | 3.13×106 | M−1s−1 | 2.77×10−2 | 6.41×10−3 | (pmol/cm3 tissue)−1s−1 |
koff | 1.00×10−3 | s−1 | |||
Kd | 3.12×102 | pM | 3.52×10−2 | 1.52×10−1 | pmol/cm3 tissue |
|
|||||
kon | 4.20×105 | M−1s−1 | 3.72×10−3 | - | (pmol/cm3 tissue)−1s−1 |
koff | 1.00×10−2 | s−1 | |||
Kd | 2.40×101 | nM | 2.71 | - | pmol/cm3 tissue |
|
|||||
kc V164R2,N1 | 3.10×1013 | (mol/cm2)−1s−1 | 1.99×10−1 | 1.59×10−2 | (pmol/cm3 tissue)−1s−1 |
koff V164R2,N1 | 1.00×10−3 | s−1 | |||
kc V164N1,R2 | 1.00×1014 | (mol/cm2)−1s−1 | 6.41×10−1 | 5.13×10−2 | (pmol/cm3 tissue)−1s−1 |
koff V164N1,R2 | 1.00×10−3 | s−1 | |||
|
|||||
kc R1,N1 | 1.00×1014 | (mol/cm2)−1s−1 | 6.41×10−1 | 5.13×10−2 | (pmol/cm3 tissue)−1s−1 |
kdissoc R1,N1 | 1.00×10−2 | s−1 | |||
|
|||||
kint, R | 2.80×10−4 | s−1 | |||
kint, C | 2.80×10−4 | s−1 | |||
|
|||||
kon | 1.13×108 | M−1s−1 | 1.00 | 2.32×10−1 | (pmol/cm3 tissue)−1s−1 |
koff | 4.23×10−5 | s−1 | |||
Kd | 0.37 | pM | 4.18×10−5 | 1.81×10−4 | pmol/cm3 tissue |
*Optimized parameter values.
Conversions:
Tissue: 1.13×108 (pmol/cm3 tissue)/M and 1.56×1014 (pmol/cm3 tissue)/(mol/cm2 EC).
Blood: 4.88×108 (pmol/cm3 tissue)/M and 1.95×1015 (pmol/cm3 tissue)/(mol/cm2 EC).
The experiments of Rudge
Model parameters were optimized computationally to fit simulation results (solid lines) to experimental data (open circles)
In order to fit the simulation results to the experimental concentration profiles, the values of a subset of model parameters were chosen to be optimized, specifically the following five parameters: VEGF secretion rate, lymphatic drainage rate, clearance rate of VEGF Trap, clearance rate of the VEGF/VEGF Trap complex, and the dissociation constant of VEGF and VEGF Trap. These five parameters are denoted as the free parameters. We chose to optimize these free parameters based on the uncertainty of the parameter values in the literature and our sensitivity analysis. The sensitivity analysis systematically investigated the impact of individual parameters on the model output (namely, the concentration profiles of VEGF, VEGF Trap, and the complex; data not shown).
The multiple VEGF isoforms have different expression levels depending on the species and tissue type
The lymphatic drainage rate has been measured to be 0.2–0.3 mL/hr in suckling rats weighing 35–45 g
The half-life of VEGF Trap in mouse serum was reported as 72 h
Kinetic parameters for the binding and unbinding of VEGF to the anti-VEGF agent are based on the pharmacokinetic parameters for VEGF Trap: The unbinding rate (
The estimation of the free parameters was treated as a non-linear optimization problem. We used an automated optimization approach that explored the free parameter space and returned the combination of free parameter values that allowed the simulation results to best fit the experimental data from Rudge
where
Because the optimization function only minimizes the objective function locally, twenty optimization trials were performed such that for each trial, the initial value of each of the free parameters was randomly generated within
Lower bound | Upper bound | Min | Max | Optimal |
Unit | |
VEGF164 secretion rate | 0.01 | 0.20 | 0.0544 | 0.0647 | 0.0626 | molecules/cell/s |
Lymphatic drainage | 7.00×10−6 | 7.00×10−4 | 7.00×10−6 | 2.73×10−5 | 7.00×10−6 | cm3/s |
Clearance of VEGF Trap | 1.60×10−5 | 1.60×10−3 | 8.45×10−4 | 9.38×10−4 | 8.86×10−4 | min−1 |
Clearance of VEGF/VEGF Trap complex | 1.60×10−5 | 1.60×10−3 | 2.16×10−4 | 3.20×10−4 | 2.79×10−4 | min−1 |
Kd of VEGF and VEGF Trap | 0.25 | 5.00 | 0.32 | 0.48 | 0.37 | pM |
*Of the 20 trials, the optimal trial was the one that yielded the smallest weighted sum of squared residuals.
The optimized model was first simulated to achieve steady-state concentrations. The steady-state concentrations of VEGF were 0.098 pM and 5.27 pM in the blood and tissue, respectively. This plasma concentration is consistent with experimental measurements in mice. Multiple studies have shown that the VEGF plasma concentration is low in mice. It ranges from 13.7 pg/mL
After the steady-state was achieved, we simulated an injection of 25 mg/kg VEGF Trap into the blood compartment, which lasted for one minute. The concentration of unbound VEGF in the blood and tissue compartments decreases after the injection as VEGF Trap and VEGF bind to form a complex (
A 25 mg/kg injection of anti-VEGF in the blood at time 0 was simulated using the optimized parameters. The profiles of (A) unbound VEGF, (B) unbound anti-VEGF, and (C) VEGF/anti-VEGF complex are shown in the blood (red) and tissue (blue) compartments. In (A), VEGF level in the tissue drops significantly after injection. Blood VEGF concentration increases to levels greater than steady-state reaching a maximum at 2.7 weeks post-injection, and returns to original steady-state levels by approximately 10 weeks after injection. Note that the axes of the panels are on different scales.
It is interesting to note that while the increase in blood VEGF levels occurs over the course of a few weeks in this mouse model, the human model predicts this similar increase over the course of only a few days. The difference in the time scale of the VEGF increase can be attributed to the fact that in the human model, the kinetic parameters of the anti-VEGF agent were based on those of bevacizumab (Avastin), which are different from those of VEGF Trap
While the parameter optimization resulted in the optimized model, single parameter values can be varied individually to explore the robustness of the model output with respect to these individual parameters. To investigate the sensitivity of VEGF levels to systemic model parameters, VEGF secretion rate, VEGF clearance rate, and microvascular permeability to VEGF were varied individually. VEGF164 secretion rate was varied from 0.01 to 0.20 molecules/cell/s (
(A) The steady-state concentration of VEGF in the tissue compartment but not in the blood is highly dependent on the VEGF secretion rate. (B) The VEGF concentration in the blood is more sensitive to the VEGF plasma clearance rate than the VEGF concentration in the tissue. (C) As microvascular permeability of VEGF increases, VEGF concentrations in the tissue and blood compartments equilibrate to 2.25 pM. For all simulations, unless the parameter is varied: VEGF164 secretion rate
In addition to systemic model parameters, geometric parameters were varied to examine their effect on steady-state VEGF levels. In particular, the length of the myonuclear domain was changed from 8.56 µm to 20 µm, as the latter value is closer to the upper limit reported in the literature
The model predicted the flows of VEGF in the mouse at steady state (no injection of VEGF Trap) normalized to that of the secretion. The majority (99.2%) of VEGF produced by secretion was removed from the tissue compartment via internalization of the VEGF/receptor complexes (
Flows of VEGF are normalized to that of the secretion flow. Most of the VEGF/receptor complexes are removed from the tissue through internalization. As with
The model provided an estimate of the steady-state fraction of total VEGF bound to receptors and to GAG chains located in the extracellular matrix and basement membranes. In the blood, 39.5% and 36.4% of total VEGF was in the form of the VEGFR-2/VEGF164/NRP-1 and VEGF120/VEGFR-1/NRP-1 ternary complexes, respectively (
The distributions of total VEGF and of the two individual isoforms are shown for the (A) blood and (B) tissue compartments. In the blood compartment, the majority of VEGF is found in the form of the VEGFR-2/VEGF164/NRP-1 and VEGF120/VEGFR-1/NRP-1 ternary complexes. In the tissue compartment, most of the VEGF is in the form of the VEGF164 isoform bound to NRP-1 on the myocytes. The occupancies of total VEGF receptors and of the individual receptors are shown for the (C) blood and (D) tissue compartments. Unbound NRP-1 on the luminal endothelial cell surface makes up the majority of total receptors and complexes in the blood compartment. Similarly, unbound NRP-1 on the abluminal endothelial cell and myocyte cell surfaces makes up the majority of total receptors and complexes in the tissue compartment.
Fractional occupancies of VEGF receptors at steady state were calculated. In the blood, unbound NRP-1 constituted 95.1% of the receptors on the luminal surface of endothelial cells (
Using the optimized model, we explored the effects of injecting VEGF Trap into the blood as a function of time, after an initial steady state had been achieved. To visualize the relative amounts of VEGF, VEGF Trap and VEGF/VEGF Trap complex moving between two compartments, the net flows of these molecular species after a 25 mg/kg injection of VEGF Trap into the blood were calculated. The net flow for each molecular species is the summation of the flows from intravasation, extravasation, and lymphatic drainage. The sign of the net flow indicates the net direction of flow of the molecule. Flow of a molecule from the tissue to the blood via intravasation and lymphatic drainage has a positive sign, while flow from the blood to the tissue via extravasation is considered to be negative. The flow of unbound VEGF remained positive during the course of the simulation and is smaller relative to the flow of the VEGF/VEGF Trap complex (
Instantaneous net flow rates of (A) unbound VEGF, (B) anti-VEGF, and (C) VEGF/anti-VEGF complex are calculated upon a 25 mg/kg injection of anti-VEGF into the blood. A positive net flow indicates movement from the tissue into the blood via intravasation and lymphatics, and a negative net flow indicates movement from the blood into the tissue via extravasation. Note that the y-axes of the panels are on different scales.
The percentages of total VEGF in the two compartments were calculated after a 25 mg/kg intravenous injection of VEGF Trap. In the blood, most of VEGF became sequestered by VEGF Trap shortly after the injection (
The distributions of VEGF in the (A) blood and (B) tissue compartments are shown subject to a 25 mg/kg injection of the anti-VEGF agent into the blood compartment at 0 weeks. Before the injection, 95% of VEGF in the blood compartment is receptor bound. In the tissue compartment, 38% of VEGF is sequestered in the extracellular matrix and endothelial and parenchymal basement membranes. 60% of VEGF is receptor-bound. Shortly following the injection of the anti-VEGF agent, essentially all of the VEGF in both compartments becomes sequestered by the anti-VEGF agent. By 14 weeks, VEGF distributions return to original steady-state levels.
The receptor occupancies of the VEGF receptors were calculated following the injection of 25 mg/kg VEGF Trap into the blood compartment. The percent of ligated VEGFR-1 decreased in both tissues (
The fractional occupancies of (A) VEGFR-1, (B) VEGFR-2, and (C) NRP-1 are shown for the blood (red) and tissue (blue) compartments following a 25 mg/kg injection of the anti-VEGF agent into the blood compartment at 0 weeks. For all three receptors in the blood compartment, the percent of receptors ligated with VEGF decreases to essentially zero quickly after the injection of the anti-VEGF; however, the percent of ligated receptors then increases to values above pre-injection levels before returning to pre-injection levels. In the tissue compartment, this effect is not seen as the percent of ligated receptors decreases quickly after injection and then returns to pre-injection levels. Note that the y-axes of the panels are on different scales.
The VEGF secretion rate from the muscle fibers was calculated by parameter optimization. From the twenty optimization trials that were performed, the minimum and maximum optimized VEGF164 secretion rates were 0.0544 and 0.0647 molecules/cell/s, respectively. In the trial that yielded the smallest WSSR, the VEGF164 secretion rate was 0.0626 molecules/cell/s, which was the value used in the optimized model. Since the expression ratio of VEGF164:VEGF120 is taken to be 92%:8% in accordance to Ee
Up to this point, we have not included any specific degradation mechanisms of VEGF in the model. However,
Steady-state concentrations of unbound VEGF were simulated using the optimal free parameter values obtained with the inclusion of VEGF degradation. The steady-state concentrations of VEGF in the normal tissue and blood compartments were 4.61 pM and 0.082 pM, respectively. These constitute a 12% and 16% decrease in the steady-state concentrations of VEGF in the normal tissue and blood compartments, respectively, from the optimized model obtained before the addition of VEGF degradation. Additionally, the steady-state VEGF concentrations were determined as the VEGF degradation rate was varied (
A VEGF degradation rate constant of 1.16×10−2 min−1 (corresponding to half-life of 60 minutes) in the normal tissue was added before re-performing the estimation of the free parameters. Using the new set of optimized parameter values, the steady-state concentrations of unbound VEGF were calculated as the degradation rate was varied. The concentration of unbound VEGF in the normal tissue decreases as the degradation rate increases.
We have extended the previously developed compartmental model of VEGF distribution in humans to investigate the VEGF distribution in the mouse. As with the previous version of the human model
The effects of VEGF Trap on the distribution of VEGF in mice have been measured experimentally by Rudge
Of these free parameters, the endogenous VEGF secretion rate by the myocytes and its predicted value was of particular interest, as direct experimental measurement of this parameter is non-trivial, and it is predicted to have a drastic effect on the concentration of unbound VEGF in the tissue. VEGF levels are typically measured from blood samples experimentally; however, different transport routes aside from VEGF secretion such as lymphatic drainage, microvascular permeability, internalization, and plasma clearance all contribute to the VEGF concentration measured in the blood. By allowing the VEGF secretion rate to be one of the free parameters, we were able to find the best fitted value for the secretion rate of VEGF by the myocytes, which is not significantly changed by the inclusion of VEGF degradation in the normal tissue.
Several assumptions used in the model contribute to model limitations. The different mouse tissues were represented by a spatially averaged compartment denoted as the tissue compartment. Skeletal muscle is the primary tissue for which the VEGF system is well characterized. Additionally, skeletal muscle constitutes a large percentage of the mouse body weight. Therefore, the majority of the properties of the tissue were based on available literature characterization of the mouse gastrocnemius muscle. To more accurately describe the mouse, specialized compartments may be added to better detail the distribution and flows of VEGF throughout the animal body. For example, it may be beneficial to distinguish a compartment representing the liver or the kidneys, as these organs play a role in the clearance of VEGF and may affect the estimated model parameters. Although vascular elements, such as pericytes, are also important in angiogenesis, for simplicity we have represented the vasculature using only endothelial cells. Including additional compartments and vascular structures would require experimental data for VEGF secretion and the density of VEGF receptors and co-receptors. We can easily expand the model as these data become available. Similarly, we could extend the model to include additional receptors and co-receptors, such as soluble VEGFR-1 or neuropilin-2, which also influence angiogenesis
VEGF120 and VEGF164 are the two VEGF-A isoforms that are predominantly expressed and hence are the two isoforms included in our model. A human VEGF isoform, VEGF189 has been shown to correlate with xenotransplantability of colon cancer tumor in mice, where increased expression of this isoform was correlated with successful transplantations
Degranulation of platelets have been shown to be a source of growth factors, such as VEGF, in the blood
The
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The authors thank Dr. Gang Liu and the other members of the Popel Lab for their helpful discussions. The authors thank Dr. Princess Imoukhuede for providing experimental values of receptor densities.