This extension of our previous model 
is useful for exploring the effects of luminal vs. abluminal distribution of VEGF receptors on the endothelial surfaces. We have shown that such configurations can drastically affect the VEGF profile in the tissue and in the blood.
First, we have shown that the removal of clearance in the presence of lymphatics could reverse the free VEGF gradient between the tissue and the blood compartments. Such a situation might correspond to certain pathological conditions, but the simulation is also instructive as a characterization of the VEGF transport system. However, it is important to note that our current model does not explicitly include the convective component of transvascular permeability and such addition could attenuate the predicted gradient reversal. Secondly, at a fixed VEGF secretion rate, the free VEGF in the available interstitial fluid is much higher than that in the plasma. When the free VEGF concentration in the plasma is constant (~1 pM), VEGF extravasation and plasma VEGF clearance over time are constant over the range of receptors we studied. We have found that the amount of VEGF disappearing by internalization of luminal receptors to which it binds, the amount of VEGF extravasating and the amount of VEGF removal from lymphatic drainage are all proportional to the luminal receptor density but insensitive to the abluminal receptor density. We have established a mathematical relationship between the amount of VEGF secreted and VEGF disappearing by internalization of abluminal receptors. Thirdly, we can summarize the VEGF transport between the tissue and the blood as shown in . VEGF is secreted in the tissue. Depending on the receptor density on the abluminal and luminal endothelial surfaces, VEGF is mainly either sequestered by the matrix or binds to abluminal receptors. Upon binding, VEGF disappears by internalization of the abluminal receptors it has bound to. Only a small fraction (free ligands) enters the blood compartment (mainly by intravasation rather than lymphatic drainage). VEGF then disappears either by internalization of receptors located on the luminal endothelial surface to which they bind or, when the receptor densities are very low, by plasma clearance. This overall transport explains why, regardless of where the receptors are expressed on the endothelial cells (abluminal vs. luminal surfaces), the binding to the receptors occurs more in the tissue than in the plasma (since a higher concentration of free ligands is available in this compartment – due to secretion – as compared to the free VEGF in the blood). However, our simulations have revealed that for high abluminal and low luminal receptor densities, VEGF can bind “preferentially” to VEGFR1 on the abluminal surface and to VEGFR2 on the luminal surface of the endothelial cells. This result requires experimental exploration. In particular, this result shows that quantification of luminal vs. abluminal receptors can be crucial in understanding VEGF signaling in both physiological and pathological conditions. Finally, our simulations reveal that VEGF binds “preferentially” to VEGFR2 compared to VEGFR1. If VEGFR2 is shown to be pro-angiogenic and VEGFR1 is shown to be anti-angiogenic, then we can conclude that, overall, the signaling is mainly pro-angiogenic regardless of the receptor distribution on the endothelial cells.
Summary of VEGF transport in the body.
Since VEGF receptor distribution between the abluminal and luminal endothelial surfaces plays such an important role, it would be interesting to investigate if some pathologies could be explained by decreased receptor expression or internalization. For example, in our previous model, we had shown that an increase in VEGF vascular permeability or secretion could not solely explain the increase of free VEGF concentration in plasma seen in cancer patients 
. It could be interesting to see if deregulated receptor expression could explain the plasma VEGF increase in cancer (as compared to healthy subjects). The present model suggests, for example, that VEGF could intravasate in high proportion if the amount of VEGF disappearing by internalization of bound receptors decreases, i.e., if the internalization rate of the receptors or if the receptors expression decreases.
The present model also suggests that, since most of VEGF disappears via internalization of bound receptors (whether on the luminal or abluminal endothelial surface), the increase of internalization of receptors could potentially decrease VEGF signal transduction. This could be done either by increasing the internalization rate of the already-existing receptors or by bioengineering cells expressing VEGF receptors which would have the property of having a high binding affinity for VEGF as well as a higher internalization rates than endothelial cells. Decreasing the VEGF signal transduction of endothelial cells could have potential therapeutic applications.
For a complex system such as the VEGF receptor-ligand interactions and transport considered, it is necessary to add elements and further increase the degree of complexity step by step in order to understand the effect of each factor. We can outline further steps in refining the model. First, the model has looked at the effect of the receptors in the proportion 1
1 for VEGFR1
NRP1. It would also be of interest to see how unequal ratios of receptors can influence the distribution and concentration of VEGF, especially when experimental data on receptor distribution in vivo become available. Secondly, at the moment, the model considers two isoforms of VEGF: VEGF121
. Other isoforms could be added to the computational model when new quantitative information becomes available. The model could also include neuropilin-2 which could compete for VEGF. Thirdly, the introduction of soluble VEGFR1 (sFlt-1) would also be of interest, especially since recent results have shown that sFlt-1 can serve as an additional means for VEGF to be transported from the plasma into the tissue 
. In that study, we hypothesized that the anti-angiogenic potential of sVEGFR1 may stem from its dominant-negative heterodimerization with cell surface VEGFRs and predicted that the circulating (plasma) level of sVEGFR1 is significantly higher than its interstitial concentrations, which could imply that sVEGFR1 may have a greater modulatory influence on luminal VEGFRs than abluminal VEGFRs 
Platelets have been shown to be significant reservoirs of VEGF in the blood circulation. It would be interesting to include such elements into the model. Again, quantification of luminal receptors would be crucial, especially since platelets have been shown to sequester large amounts of VEGF and release VEGF from α-granules 
Similarly, the body tissue compartment was considered to have the properties of skeletal muscle. It could be important to distinguish between highly vascularized and relatively avascular organs, as well as elements with varying rates of lymphatic drainage. This would require experimental data on VEGF secretion and other tissue characteristics that at present are poorly known. Furthermore, luminal and abluminal receptors may not be equally accessible by VEGF possibly because of endothelial cell polarity: basement membrane on the abluminal side and glycocalyx on the luminal side.
A current assumption was the conservation of total (free and bound) density of receptors at each time step. In other words, we assumed that the internalization of receptors was equal to the receptor insertion per abluminal or luminal endothelial surface for each time point. Relaxing such assumptions and replacing them by the experimentally-based receptor dynamics would make the model more accurate.
In our model, we assumed that the vascular permeability was fixed. In reality, VEGF, also known as VPF (vascular permeability factor), plays an important role in regulating permeability 
. An addition to the model would be to determine a quantitative relationship between the vascular permeability and the concentration of VEGF and include that relationship in the model.
Our study has shown that quantification of luminal vs. abluminal receptors could be very useful to better understand VEGF signaling and the mechanisms underlying VEGF-dependent diseases as well as angiogenesis and will motivate experimental exploration.