A two-compartment model of VEGF distribution in the mouse

Abstract

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. In vivo preclinical experiments in the area of angiogenesis are typically performed in mouse models; this includes drug development targeting VEGF. Thus, to quantitatively interpret such experimental results, a computational model of VEGF distribution in the mouse can be beneficial. In this paper, we present an in silico model of VEGF distribution in mice, determine model parameters from existing experimental data, conduct sensitivity analysis, and test the validity of the model. The multiscale model is comprised of two compartments: blood and tissue. The model accounts for interactions between two major VEGF isoforms (VEGF(120) and VEGF(164)) 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.

Figure 1. Two compartment model.

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 q_v. VEGFR-1, VEGFR-2 and NRP-1 are localized on the luminal and abluminal surfaces of the endothelial cells. NRP-1 is also found on the myocyte cell surface. Inter-compartmental transport includes lymphatic drainage (k_L) and microvascular permeability (k­_p). Receptors and VEGF/receptor complexes on endothelial cells and myocytes can be internalized (k_int). VEGF is removed from the blood compartment via plasma clearance (c_v).

Figure 2. Molecular interactions.

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 k_on and k_off, respectively. k_c denotes the coupling of NRP-1 and VEGFR-1 and of NRP-1 to VEGFR-2. While only the internalization of NRP-1 is shown, all VEGF receptors and complexes can be internalized at a rate k_int. Similarly, while only the insertion of NRP-1 is shown, VEGFR-1 and VEGFR-2 also appear via insertion at a rate s. The blue bar is used to distinguish NRP-1 expressed on the myocytes from NRP-1 on the endothelial cells (orange bars).

Figure 3. Parameter optimization.

Model parameters were optimized computationally to fit simulation results (solid lines) to experimental data (open circles) for the concentration profiles of the unbound VEGF Trap and the VEGF/VEGF Trap complex. VEGF Trap was injected intravenously (into the blood compartment) at (A) 1 mg/kg, (B) 2.5 mg/kg, (C) 10 mg/kg, and (D) 25 mg/kg.

Figure 4. Concentration profiles following the injection of the anti-VEGF agent (VEGF Trap).

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.

Figure 5. Steady-state concentration of free VEGF.

(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 q_V164  =  0.063 molecules/cell/s (optimized value), plasma clearance rate c_V  =  0.23 min⁻¹, and permeability k_p  =  4.0×10⁻⁸ cm/s.

Figure 6. Normalized flows at steady-state.

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 Figure 1, internalization in the tissue compartment includes both that of VEGF/receptor complexes on the abluminal surfaces of endothelial cells as well as VEGF bound to NRP-1 on myocyte cell surfaces. There is a net flow of VEGF from the tissue into the blood compartment via microvascular permeability.

Figure 7. VEGF distributions and VEGFR occupancies at steady-state.

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.

Figure 8. Intercompartmental flows following the intravenous injection of the anti-VEGF agent (VEGF Trap).

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.

Figure 9. VEGF distributions upon injection of anti-VEGF (VEGF Trap).

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.

Figure 10. VEGF Receptor fractional occupancies upon injection of anti-VEGF (VEGF Trap).

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.

Figure 11. Effect of VEGF degradation on unbound VEGF levels.

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.