1752-0509-2-77 1752-0509 Research article <p>A compartment model of VEGF distribution in blood, healthy and diseased tissues</p> Stefanini O Marianne stefanini@jhmi.edu Wu TH Florence florence.wu@jhu.edu Mac Gabhann Feilim feilim@jhu.edu Popel S Aleksander apopel@jhu.edu

Department of Biomedical Engineering, School of Medicine, Johns Hopkins University, Baltimore, Maryland 21205, USA

Department of Biomedical Engineering, University of Virginia, Charlottesville, VA 22908, USA

Robert M. Berne Cardiovascular Research Center, University of Virginia, Charlottesville, VA 22908, USA

 BMC Systems Biology 1752-0509 2008 2 1 77 http://www.biomedcentral.com/1752-0509/2/77 18713470 10.1186/1752-0509-2-77 19 2 2008 19 8 2008 19 8 2008 2008 Stefanini et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Background

Angiogenesis is a process by which new capillaries are formed from pre-existing blood vessels in physiological (e.g., exercise, wound healing) or pathological (e.g., ischemic limb as in peripheral arterial disease, cancer) contexts. This neovascular mechanism is mediated by the vascular endothelial growth factor (VEGF) family of cytokines. Although VEGF is often targeted in anti-angiogenic therapies, there is little knowledge about how its concentration may vary between tissues and the vascular system. A compartment model is constructed to study the VEGF distribution in the tissue (including matrix-bound, cell surface receptor-bound and free VEGF isoforms) and in the blood. We analyze the sensitivity of this distribution to the secretion rate, clearance rate and vascular permeability of VEGF.

Results

We find that, in a physiological context, VEGF concentration varies approximately linearly with the VEGF secretion rate. VEGF concentration in blood but not in tissue is dependent on the vascular permeability of healthy tissue. Model simulations suggest that relative VEGF increases are similar in blood and tissue during exercise and return to baseline within several hours. In a pathological context (tumor), we find that blood VEGF concentration is relatively insensitive to increased vascular permeability in tumors, to the secretion rate of VEGF by tumors and to the clearance. However, it is sensitive to the vascular permeability in the healthy tissue. Finally, the VEGF distribution profile in healthy tissue reveals that about half of the VEGF is complexed with the receptor tyrosine kinase VEGFR2 and the co-receptor Neuropilin-1. In diseased tissues, this binding can be reduced to 15% while VEGF bound to the extracellular matrix and basement membranes increases.

Conclusion

The results are of importance for physiological conditions (e.g., exercise) and pathological conditions (e.g., peripheral arterial disease, coronary artery disease, cancer). This mathematical model can serve as a tool for understanding the VEGF distribution in physiological and pathological contexts as well as a foundation to investigate pro- or anti-angiogenic strategies.

Background

Angiogenesis is the process by which new blood vessels grow from pre-existing vascular networks. This is a dynamic physiological mechanism (e.g., during wound healing, exercise training) but has been shown to be involved in pathological conditions such as age-related macular degeneration, rheumatoid arthritis, diabetic retinopathy, peripheral arterial disease, and cancer 1. Vascular endothelial growth factor (VEGF) is a potent cytokine involved in angiogenesis 2. This growth factor plays a role in transcapillary permeability and stimulates cell differentiation, proliferation, migration and survival. The human VEGF gene family is composed of five members. VEGF-A (also commonly referred as VEGF) and VEGF-B are known to be involved in angiogenesis. VEGF-A and three members (VEGF-C, VEGF-D and placental growth factor PlGF) have been shown to be implicated in both mechanisms. The VEGF-A family contains the different splice isoforms VEGF121, VEGF121b, VEGF145, VEGF145b, VEGF148, VEGF162, VEGF165, VEGF165b, VEGF183, VEGF189, VEGF206, VEGF232 3. The major splice variants are VEGF121, VEGF165, VEGF189 and VEGF206. Because the role of VEGF189 and VEGF206 in vivo is not well understood and their expression is low compared to VEGF121 and VEGF165 3, we consider the two main isoforms VEGF121 and VEGF165 in our model. These two ligands bind two receptor tyrosine kinases VEGFR1 (Flt-1, fms-like tyrosine kinase-1) and VEGFR2 (Flk-1/KDR, fetal liver kinase-1/Kinase Domain-containing Receptor). The VEGF165 isoform binds to a non-signaling co-receptor called Neuropilin-1 (NRP1) and to heparan sulfate proteoglycans which are constituents of the extracellular matrix (ECM) and the cellular basement membranes. NRP1 also independently couples with VEGFR1, to which VEGF121 can bind to form a complex of all three. Another complex can be formed when VEGF165 bound to NRP1 binds in turn to VEGFR2 (or vice versa).

VEGF interactions with these receptors in the context of angiogenesis have been extensively studied using computational models. For example, it was predicted that blockade of NRP1-VEGFR coupling would be a good strategy to decrease VEGF-VEGFR2 signaling 4. Other simulations showed that VEGFR heterodimers comprise 10–50% of the active, signaling VEGF receptor complexes, and that heterodimers form at the expense of homodimers of VEGFR1 when VEGFR2 populations are larger 5. However, in the absence of in vivo data, we do not include heterodimers in the model. Insights into the distribution of VEGF in tissues in vivo have been made; for example, it was predicted that the majority of the VEGF in muscle is bound to the cell surface receptors or to the extracellular matrix and that NRP1 enhances the binding of VEGF165 to VEGFR2 6. The models have also been used to design and test therapeutic approaches; for example, in peripheral arterial disease, exercise training results in increased VEGF secretion in hypoxic tissue and augmented VEGF receptor expression, and multi-scale computational models revealed that this leads to an increase in both VEGF concentration and VEGF gradients, a potentially effective therapeutic approach 7. Anatomically detailed tissue models have predicted heterogeneity in the activation of the endothelial VEGF receptors and how this affects VEGF gradients 8. Similarly, in hypoxic tissues, the high spatial heterogeneity of muscle fiber VEGF secretion leads to significant gradients of VEGF concentration and VEGF receptor activation in both resting and exercising muscle 9. Three-dimensional VEGF distribution was also predicted in ischemic muscle in a model of peripheral arterial disease 10.

A meta-analysis of VEGF levels in healthy subjects and cancer patients revealed that plasma VEGF levels are several-fold higher in cancer 11. This conclusion should be taken cautiously since VEGF levels vary with the tumor type, size and location; also, the ranges of VEGF in control subjects and cancer patients in some studies overlap. Plasma VEGF is also elevated during exercise 1213 and in peripheral arterial disease 14. To investigate the distribution of VEGF in human subjects under physiological and pathological conditions, we formulate a biophysically-accurate compartment model to describe the entire human body. Blood is represented by one compartment that communicates with two others representing healthy and diseased tissues. A sensitivity analysis is performed to investigate the role of parameters including secretion rate, clearance rate and vascular permeability of VEGF. The formulation is general and the model can be applied to both healthy human subjects and to subjects with a diseased tissue, e.g., ischemic limb or tumor. As an illustration, we consider a compartment representing a tumor to investigate the possible causes of the several-fold increase observed in plasma VEGF levels in cancer patients 11. The formulated model provides a foundation for studying various diseases where information about VEGF distribution in the body is important. It will also serve as a basis for simulating pro- and anti-angiogenic VEGF-related therapeutic procedures.

Methods

Geometry

As a first approximation, a tissue can be represented as a collection of capillaries (and small arterioles and venules), surrounded by parenchymal cells. For example, skeletal muscle is constituted of long fibers whose cross sections are approximately constant. A schematic of this configuration is shown in Figure 1A. Note that the stromal cells are not considered explicitly in the model, but rather lumped with parenchymal cells. Between the parenchymal cells and the capillaries lies the interstitial space composed of the extracellular matrix (ECM), parenchymal basement membranes (PBM) and endothelial basement membranes (EBM). In this study, these anatomical structures will be represented in a spatially-averaged manner: each structure will be represented by a distinct volume with specific VEGF binding properties, but VEGF gradients within the volume will not be considered.

<p>Figure 1</p>

Compartment model of VEGF transport in blood and tissues

Compartment model of VEGF transport in blood and tissues. A, Schematic of a tissue cross section. VEGF165 can bind to glycosaminoglycan chains (GAG) and be sequestered in the extracellular matrix whereas VEGF121 cannot. The isoforms have different cell surface receptor binding profiles. B, Compartment model set-up. Three compartments are used in our simulations: blood, healthy and diseased tissues. The diseased tissue compartment is not used in all simulations. VEGF is secreted by parenchymal cells in the healthy and diseased compartments (q). VEGF transport between the blood and the tissue compartments is via transcapillary permeability (kp). VEGF receptors are expressed on the abluminal side of the endothelial cells and VEGF binding to these receptors can lead to internalization (kint). Plasma clearance for VEGF is present in the blood compartment (cV). C, Interactions between VEGF, cell surface receptors, extracellular matrix and basement membranes. VEGF121 binds to VEGFR2 but does not bind to NRP1. VEGF165 interacts with VEGFR2 or NRP1. Once bound, it can form a ternary complex VEGFR2- VEGF165- NRP1. VEGF165 can be sequestered by the ECM, PBM or EBM by binding to GAG chains. VEGF121 binds to VEGFR1. This receptor couples with NRP1 to form VEGFR1-NRP1 complex or the ternary complex VEGF121-VEGFR1-NRP1 if previously occupied by VEGF121. VEGF165 also binds to VEGFR1 but not to the VEGFR1-NRP1 complex. The receptors (VEGFR1, VEGFR2 and NRP1) are inserted or internalized at the cell surface.

Computational model

We constructed a compartment model of VEGF transport and interactions with cell receptors representing the whole human body that is an extension of previous one-compartment models for breast tumor tissue 4 and skeletal muscle 6. This design can also be used to describe any animal if the parameters to characterize the tissues and blood are available. In physiological cases, we consider two compartments: tissue and blood. VEGF transport between the compartments is mediated by transcapillary permeability. In pathological cases, a third compartment represents the diseased tissue, e.g., a tumor in cancer or ischemic tissue in peripheral arterial disease (Figure 1B). Again vascular permeability allows free VEGF to move between the blood and the diseased tissue. Because the interacting surface between the healthy and the diseased tissues is, in most cases, much smaller than the trans-endothelial exchange surface between each tissue and blood, we do not include VEGF transport between the normal and diseased tissue directly. This additional parameter could be included in further studies.

Leaving aside the vascular system, the tissues of the body can be divided into two main groups: skeletal muscle (35%) 15 and the rest of the body (e.g., brain, heart, liver, kidney, bones, fat). As a first approximation, we assume that the rest of the body has the same characteristics as skeletal muscle. This assumption can be relaxed by formulating progressively more detailed compartment models containing different organs and tissues; however, the parameters describing VEGF-binding properties of different tissues are currently unavailable. Therefore, the healthy tissue compartment in the present model has the mass of the whole body (minus the blood) and the density of the skeletal muscle.

The interstitial space between the parenchymal cells and the vascular endothelial cells can be divided into three regions: the extracellular matrix (ECM), the basement membrane surrounding the parenchymal cells (PBM) and the basement membrane surrounding the endothelial cells (EBM). The two isoforms of the VEGF-A family considered in the model are the freely diffusible heparin-binding VEGF165 (VEGF164 in rodents) and non-heparin-binding VEGF121 (VEGF120 in rodents). Because VEGF165 binds to the heparan sulfate proteoglycans (HSPGs) that constitute a major component of the extracellular matrix and basement membranes, this isoform can be sequestered by the ECM and by cellular basement membranes. We assume that the distribution of unbound (free) VEGF in the interstitial space is uniform within each compartment and spatial variability due to gradient formation is neglected 4616.

The interactions between VEGF121 and VEGF165 and their receptors (VEGFR1, VEGFR2 and NRP1) are illustrated in Figure 1C and will be further described in the equation section. To summarize, VEGF121 and VEGF165 bind to both VEGFR1 and VEGFR2 but only VEGF165 is believed to bind to the non-enzymatic co-receptor NRP1. The receptors are inserted or internalized as illustrated in Figure 1C. We assume that the receptors are only present on the abluminal side of the blood vessels and therefore are only exposed to interstitial VEGF levels of the tissue compartments, though this assumption can be easily relaxed in the future. Although there is some qualitative evidence that receptors are present on the luminal side 17, to our knowledge, there is currently no quantitative data on the levels of receptor expression on the luminal vs. abluminal side of the endothelium.

VEGF molecules are secreted by the parenchymal cells present in the tissue. Depletion of VEGF molecules occurs if the cell surface receptor (VEGFR1, VEGFR2 or NRP1) that VEGF is bound to is internalized into the cellular membrane. The free molecules diffuse within the available interstitial fluid although here this diffusion is assumed to be fast compared to the biochemical reactions, the compartment well-mixed and the distribution of molecules uniform; the justification for this assumption was presented in 7 by estimating that the Damkohler number (ratio of diffusion time to reaction time) is significantly less than one. Free VEGF is transported by transcapillary permeability between the blood and tissue compartments and cleared from the blood (e.g., by the kidneys and liver).

Equations

The changes in species concentration over time are governed by mass balance relationships and represented by a system of coupled nonlinear ordinary differential equations; the details of the derivation are given in our previous publications 4616. Although the kinetics and properties may be different, the healthy and diseased tissues are governed by the same equations and constituted by the same molecular species. Therefore, unless specified otherwise, the following equations govern both healthy and diseased tissues. The biological and physical properties of the tissue, VEGF secretion rates and vascular permeability may vary between the normal and diseased tissues; the values of the parameters will be specified in a later section.

Each tissue compartment can be divided into subcompartments where distinct reactions take place: cell surfaces and interstitial space, the latter further subdivided into available interstitial fluid, ECM, PBM and EBM. Interstitial species include free VEGF, VEGF binding sites in the matrix and the complexes they form. Surface species consist of free receptors or VEGF-ligated receptors on the vascular endothelial abluminal surface. Unless specified otherwise, the concentrations of the interstitial and blood species are expressed per unit volume of the corresponding subcompartment (e.g., available fluid volume of the interstitial space, ECM, PBM or EBM) while the concentrations of the surface species are expressed per unit surface area of the corresponding cell (though it is possible to interconvert the units).

Interstitial space

We use the following notation: [M] is the concentration of VEGF binding sites of the ECM, PBM or EBM (the location is denoted by the corresponding subscript, e.g., [MECM]); [V] represents the concentration of any unbound VEGF isoform (unless specified by a subscript, e.g., [V121]) in the available interstitial fluid; kon and koff the kinetic rates for binding and unbinding respectively. In the interstitium of the normal and diseased tissues, the following reactions take place

V 165 + M E C M ⇌ k o f f , V 165 , M E C M k o n , V 165 , M E C M V 165 M E C M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6074@

V 165 + M E B M ⇌ k o f f , V 165 , M E B M k o n , V 165 , M E B M V 165 M E B M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@606C@

V 165 + M P B M ⇌ k o f f , V 165 , M P B M k o n , V 165 , M P B M V 165 M P B M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqGHRaWkcqWGnbqtdaWgaaWcbaGaemiuaaLaemOqaiKaemyta0eabeaakmaao4aaleaacqWGRbWAdaWgaaadbaGaem4Ba8MaemOBa4MaeiilaWIaemOvayLaeGymaeJaeGOnayJaeGynauJaeiilaWIaemyta0KaemiuaaLaemOqaiKaemyta0eabeaaaSqaaiabdUgaRnaaBaaameaacqWGVbWBcqWGMbGzcqWGMbGzcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqGGSaalcqWGnbqtcqWGqbaucqWGcbGqcqWGnbqtaeqaaaGccaGLahIaayzVHaGaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqWGnbqtdaWgaaWcbaGaemiuaaLaemOqaiKaemyta0eabeaaaaa@60C4@

and are governed by the equations

d [ M E B M ] d t = − k o n , V 165 , M E B M [ V 165 ] [ M E B M ] + k o f f , V 165 M E B M [ V 165 M E B M ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@71AD@

d [ M E C M ] d t = − k o n , V 165 , M E C M [ V 165 ] [ M E C M ] + k o f f , V 165 M E C M [ V 165 M E C M ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@71B7@

d [ M P B M ] d t = − k o n , V 165 , M P B M [ V 165 ] [ M P B M ] + k o f f , V 165 M P B M [ V 165 M P B M ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@721B@

d [ V 165 M E B M ] d t = k o n , V 165 , M E B M [ V 165 ] [ M E B M ] − k o f f , V 165 M E B M [ V 165 M E B M ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7503@

d [ V 165 M E C M ] d t = k o n , V 165 , M E C M [ V 165 ] [ M E C M ] − k o f f , V 165 M E C M [ V 165 M E C M ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@750D@

d [ V 165 M P B M ] d t = k o n , V 165 , M P B M [ V 165 ] [ M P B M ] − k o f f , V 165 M P B M [ V 165 M P B M ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7571@

Cell surface

[R], [N] and [RN] represent the densities of the unoccupied receptor tyrosine kinases (VEGFR1, VEGFR2), the unoccupied co-receptor (NRP1), and their coupled complexes (VEGFR1 coupled with NRP1), respectively; [VR] and [VN] are the concentrations of VEGF bound to the VEGF receptors and NRP1, respectively; [RVN] (or [VRN]) is the concentration of VEGF-bound in the form of the ternary complex (VEGFR2-VEGF165-NRP1 or VEGF121-VEGFR1-NRP1). sR represents the rate at which the receptors are inserted into the cell membrane and kint the internalization rate of the receptors.

The VEGF165 isoform interacts with the receptors through the following reactions

V 165 + R 1 ⇌ k o f f , V 165 R 1 k o n , V 165 , R 1 V 165 R 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqGHRaWkcqWGsbGudaWgaaWcbaGaeGymaedabeaakmaao4aaleaacqWGRbWAdaWgaaadbaGaem4Ba8MaemOBa4MaeiilaWIaemOvayLaeGymaeJaeGOnayJaeGynauJaeiilaWIaemOuaiLaeGymaedabeaaaSqaaiabdUgaRnaaBaaameaacqWGVbWBcqWGMbGzcqWGMbGzcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqWGsbGucqaIXaqmaeqaaaGccaGLahIaayzVHaGaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqWGsbGudaWgaaWcbaGaeGymaedabeaaaaa@5668@

V 165 + R 2 ⇌ k o f f , V 165 R 2 k o n , V 165 , R 2 V 165 R 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqGHRaWkcqWGsbGudaWgaaWcbaGaeGOmaidabeaakmaao4aaleaacqWGRbWAdaWgaaadbaGaem4Ba8MaemOBa4MaeiilaWIaemOvayLaeGymaeJaeGOnayJaeGynauJaeiilaWIaemOuaiLaeGOmaidabeaaaSqaaiabdUgaRnaaBaaameaacqWGVbWBcqWGMbGzcqWGMbGzcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqWGsbGucqaIYaGmaeqaaaGccaGLahIaayzVHaGaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqWGsbGudaWgaaWcbaGaeGOmaidabeaaaaa@5670@

V 165 + N 1 ⇌ k o f f , V 165 N 1 k o n , V 165 , N 1 V 165 N 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqGHRaWkcqWGobGtdaWgaaWcbaGaeGymaedabeaakmaao4aaleaacqWGRbWAdaWgaaadbaGaem4Ba8MaemOBa4MaeiilaWIaemOvayLaeGymaeJaeGOnayJaeGynauJaeiilaWIaemOta4KaeGymaedabeaaaSqaaiabdUgaRnaaBaaameaacqWGVbWBcqWGMbGzcqWGMbGzcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqWGobGtcqaIXaqmaeqaaaGccaGLahIaayzVHaGaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqWGobGtdaWgaaWcbaGaeGymaedabeaaaaa@5648@

V 165 N 1 + R 2 ⇌ k o f f , V 165 N 1 , R 2 k c , V 165 N 1 , R 2 R 2 V 165 N 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5E93@

V 165 R 2 + N 1 ⇌ k o f f , V 165 R 2 , N 1 k c , V 165 R 2 , N 1 R 2 V 165 N 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5E93@

while the VEGF121 isoform binds to the receptors according to these reactions

V 121 + R 1 ⇌ k o f f , V 121 R 1 k o n , V 121 , R 1 V 121 R 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOvay1aaSbaaSqaaiabigdaXiabikdaYiabigdaXaqabaGccqGHRaWkcqWGsbGudaWgaaWcbaGaeGymaedabeaakmaao4aaleaacqWGRbWAdaWgaaadbaGaem4Ba8MaemOBa4MaeiilaWIaemOvayLaeGymaeJaeGOmaiJaeGymaeJaeiilaWIaemOuaiLaeGymaedabeaaaSqaaiabdUgaRnaaBaaameaacqWGVbWBcqWGMbGzcqWGMbGzcqGGSaalcqWGwbGvcqaIXaqmcqaIYaGmcqaIXaqmcqWGsbGucqaIXaqmaeqaaaGccaGLahIaayzVHaGaemOvay1aaSbaaSqaaiabigdaXiabikdaYiabigdaXaqabaGccqWGsbGudaWgaaWcbaGaeGymaedabeaaaaa@5628@

V 121 + R 2 ⇌ k o f f , V 121 R 2 k o n , V 121 , R 2 V 121 R 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOvay1aaSbaaSqaaiabigdaXiabikdaYiabigdaXaqabaGccqGHRaWkcqWGsbGudaWgaaWcbaGaeGOmaidabeaakmaao4aaleaacqWGRbWAdaWgaaadbaGaem4Ba8MaemOBa4MaeiilaWIaemOvayLaeGymaeJaeGOmaiJaeGymaeJaeiilaWIaemOuaiLaeGOmaidabeaaaSqaaiabdUgaRnaaBaaameaacqWGVbWBcqWGMbGzcqWGMbGzcqGGSaalcqWGwbGvcqaIXaqmcqaIYaGmcqaIXaqmcqWGsbGucqaIYaGmaeqaaaGccaGLahIaayzVHaGaemOvay1aaSbaaSqaaiabigdaXiabikdaYiabigdaXaqabaGccqWGsbGudaWgaaWcbaGaeGOmaidabeaaaaa@5630@

R 1 + N 1 ⇌ k d i s s o c , R 1 N 1 k c , R 1 , N 1 R 1 N 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOuai1aaSbaaSqaaiabigdaXaqabaGccqGHRaWkcqWGobGtdaWgaaWcbaGaeGymaedabeaakmaao4aaleaacqWGRbWAdaWgaaadbaGaem4yamMaeiilaWIaemOuaiLaeGymaeJaeiilaWIaemOta4KaeGymaedabeaaaSqaaiabdUgaRnaaBaaameaacqWGKbazcqWGPbqAcqWGZbWCcqWGZbWCcqWGVbWBcqWGJbWycqGGSaalcqWGsbGucqaIXaqmcqWGobGtcqaIXaqmaeqaaaGccaGLahIaayzVHaGaemOuai1aaSbaaSqaaiabigdaXaqabaGccqWGobGtdaWgaaWcbaGaeGymaedabeaaaaa@5112@

V 121 + R 1 N 1 ⇌ k o f f , V 121 R 1 k o n , V 121 , R 1 V 121 R 1 N 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5ABE@

V 121 R 1 + N 1 ⇌ k d i s s o c , R 1 N 1 k c , R 1 , N 1 V 121 R 1 N 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@598C@

The 10 preceding chemical reactions are governed by the following equations:

d [ R 1 ] d t = s R 1 − k int , R 1 [ R 1 ] − k o n , V 165 , R 1 [ V 165 ] [ R 1 ] + k o f f , V 165 R 1 [ V 165 R 1 ] − k o n , V 121 , R 1 [ V 121 ] [ R 1 ] + k o f f , V 121 R 1 [ V 121 R 1 ] − k c , R 1 , N 1 [ N 1 ] [ R 1 ] + k d i s s o c , R 1 N 1 [ R 1 N 1 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@D23B@

d [ R 2 ] d t = s R 2 − k int , R 2 [ R 2 ] − k o n , V 121 , R 2 [ V 121 ] [ R 2 ] + k o f f , V 121 R 2 [ V 121 R 2 ] − k o n , V 165 , R 2 [ V 165 ] [ R 2 ] + k o f f , V 165 R 2 [ V 165 R 2 ] − k c , V 165 N 1 , R 2 [ V 165 N 1 ] [ R 2 ] + k o f f , V 165 N 1 , R 2 [ R 2 V 165 N 1 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@DFD4@

d [ N 1 ] d t = s N 1 − k int , N 1 [ N 1 ] − k c , V 121 R 1 , N 1 [ V 121 R 1 ] [ N 1 ] + k d i s s o c , R 1 N 1 [ V 121 R 1 N 1 ] − k c , R 1 , N 1 [ N 1 ] [ R 1 ] + k d i s s o c , R 1 N 1 [ R 1 N 1 ] − k o n , V 165 , N 1 [ V 165 ] [ N 1 ] − k o f f , V 165 N 1 [ V 165 N 1 ] − k c , V 165 R 2 , N 1 [ V 165 R 2 ] [ N 1 ] + k o f f , V 165 R 2 , N 1 [ R 2 V 165 N 1 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@113A@

d [ V 121 R 1 ] d t = − k int , V 121 R 1 [ V 121 R 1 ] + k o n , V 121 , R 1 [ V 121 ] [ R 1 ] − k o f f , V 121 R 1 [ V 121 R 1 ] − k c , R 1 , N 1 [ V 121 R 1 ] [ N 1 ] + k d i s s o c , R 1 N 1 [ V 121 R 1 N 1 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@B3BF@

d [ V 121 R 2 ] d t = − k int , V 121 R 2 [ V 121 R 2 ] + k o n , V 121 , R 2 [ V 121 ] [ R 2 ] − k o f f , V 121 R 2 [ V 121 R 2 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@80FC@

d [ V 165 R 1 ] d t = − k int , V 165 R 1 [ V 165 R 1 ] + k o n , V 165 , R 1 [ V 165 ] [ R 1 ] − k o f f , V 165 R 1 [ V 165 R 1 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@815E@

d [ V 165 R 2 ] d t = − k int , V 165 R 2 [ V 165 R 2 ] + k o n , V 165 , R 2 [ V 165 ] [ R 2 ] − k o f f , V 165 R 2 [ V 165 R 2 ] − k c , V 165 R 2 , N 1 [ V 165 R 2 ] [ N 1 ] + k o f f , V 165 R 2 , N 1 [ R 2 V 165 N 1 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@B942@

d [ V 165 N 1 ] d t = − k int , V 165 N 1 [ V 165 N 1 ] + k o n , V 165 , N 1 [ V 165 ] [ N 1 ] − k o f f , V 165 N 1 [ V 165 N 1 ] − k c , V 165 N 1 , R 2 [ V 165 N 1 ] [ R 2 ] + k o f f , V 165 N 1 , R 2 [ R 2 V 165 N 1 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@B8FE@

d [ R 1 N 1 ] d t = − k int , R 1 N 1 [ R 1 N 1 ] + k c , R 1 , N 1 [ N 1 ] [ R 1 ] − k d i s s o c , R 1 N 1 [ R 1 N 1 ] − k o n , V 121 , R 1 [ V 121 ] [ R 1 N 1 ] + k o f f , V 121 R 1 [ V 121 R 1 N 1 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@AA05@

d [ R 2 V 165 N 1 ] d t = − k int , V 165 R 2 N 1 [ R 2 V 165 N 1 ] + k c , V 165 R 2 , N 1 [ V 165 R 2 ] [ N 1 ] − k o f f , V 165 R 2 , N 1 [ R 2 V 165 N 1 ] + k c , V 165 N 1 , R 2 [ V 165 N 1 ] [ R 2 ] − k o f f , V 165 N 1 , R 2 [ R 2 V 165 N 1 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@C7FF@

d [ V 121 R 1 N 1 ] d t = − k int , V 121 R 1 N 1 [ V 121 R 1 N 1 ] + k c , V 121 R 1 , N 1 [ V 121 R 1 ] [ N 1 ] − k d i s s o c , V 121 N 1 [ V 121 R 1 N 1 ] + k o n , V 121 R 1 N 1 [ V 121 ] [ R 1 N 1 ] − k o f f , V 121 R 1 N 1 [ V 121 R 1 N 1 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@C828@

Ligands in the tissue compartments

The ligands VEGF121 and VEGF165 are secreted by the cells at a rate qV per unit volume of total tissue. The secretion rate is assumed to be constant. The concentrations of the ligands in tissue follow these equations:

d [ V 121 ] d t = q V 121 − k o n , V 121 , R 1 [ V 121 ] [ R 1 ] + k o f f , V 121 R 1 [ V 121 R 1 ] − k o n , V 121 , R 1 N 1 [ V 121 ] [ R 1 N 1 ] + k o f f , V 121 R 1 N 1 [ V 121 R 1 N 1 ] − k o n , V 121 , R 2 [ V 121 ] [ R 2 ] + k o f f , V 121 R 2 [ V 121 R 2 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@D525@

d [ V 165 ] d t = q V 165 − k o n , V 165 , M E B M [ V 165 ] [ M E B M ] + k o f f , V 165 , M E B M N [ V 165 M E B M ] − k o n , V 165 , M E C M [ V 165 ] [ M E C M ] + k o f f , V 165 M E C M [ V 165 M E C M ] − k o n , V 165 , M P B M [ V 165 ] [ M P B M ] + k o f f , V 165 M P B M [ V 165 M P B M ] − k o n , V 165 , R 1 [ V 165 ] [ R 1 ] + k o f f , V 165 R 1 [ V 165 R 1 ] − k o n , V 165 , R 2 [ V 165 ] [ R 2 ] + k o f f , V 165 R 2 [ V 165 R 2 ] − k o n , V 165 , N 1 [ V 165 ] [ N 1 ] + k o f f , V 165 N 1 [ V 165 N 1 ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeWabuqaaaaabaqcfa4aaSaaaeaacqWGKbazcqGGBbWwcqWGwbGvdaWgaaqaaiabigdaXiabiAda2iabiwda1aqabaGaeiyxa0fabaGaemizaqMaemiDaqhaaOGaeyypa0JaemyCae3aaSbaaSqaaiabdAfawjabigdaXiabiAda2iabiwda1aqabaGccqGHsislcqWGRbWAdaWgaaWcbaGaem4Ba8MaemOBa4MaeiilaWIaemOvayLaeGymaeJaeGOnayJaeGynauJaeiilaWIaemyta0KaemyrauKaemOqaiKaemyta0eabeaakiabcUfaBjabdAfawnaaBaaaleaacqaIXaqmcqaI2aGncqaI1aqnaeqaaOGaeiyxa0Laei4waSLaemyta00aaSbaaSqaaiabdweafjabdkeacjabd2eanbqabaGccqGGDbqxcqGHRaWkcqWGRbWAdaqhaaWcbaGaem4Ba8MaemOzayMaemOzayMaeiilaWIaemOvayLaeGymaeJaeGOnayJaeGynauJaeiilaWIaemyta0KaemyrauKaemOqaiKaemyta0eabaGaemOta4eaaOGaei4waSLaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqWGnbqtdaWgaaWcbaGaemyrauKaemOqaiKaemyta0eabeaakiabc2faDbqaaiabgkHiTiabdUgaRnaaBaaaleaacqWGVbWBcqWGUbGBcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqGGSaalcqWGnbqtcqWGfbqrcqWGdbWqcqWGnbqtaeqaaOGaei4waSLaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqGGDbqxcqGGBbWwcqWGnbqtdaWgaaWcbaGaemyrauKaem4qamKaemyta0eabeaakiabc2faDjabgUcaRiabdUgaRnaaBaaaleaacqWGVbWBcqWGMbGzcqWGMbGzcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqWGnbqtcqWGfbqrcqWGdbWqcqWGnbqtaeqaaOGaei4waSLaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqWGnbqtdaWgaaWcbaGaemyrauKaem4qamKaemyta0eabeaakiabc2faDbqaaiabgkHiTiabdUgaRnaaBaaaleaacqWGVbWBcqWGUbGBcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqGGSaalcqWGnbqtcqWGqbaucqWGcbGqcqWGnbqtaeqaaOGaei4waSLaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqGGDbqxcqGGBbWwcqWGnbqtdaWgaaWcbaGaemiuaaLaemOqaiKaemyta0eabeaakiabc2faDjabgUcaRiabdUgaRnaaBaaaleaacqWGVbWBcqWGMbGzcqWGMbGzcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqWGnbqtcqWGqbaucqWGcbGqcqWGnbqtaeqaaOGaei4waSLaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqWGnbqtdaWgaaWcbaGaemiuaaLaemOqaiKaemyta0eabeaakiabc2faDbqaaiabgkHiTiabdUgaRnaaBaaaleaacqWGVbWBcqWGUbGBcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqGGSaalcqWGsbGucqaIXaqmaeqaaOGaei4waSLaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqGGDbqxcqGGBbWwcqWGsbGudaWgaaWcbaGaeGymaedabeaakiabc2faDjabgUcaRiabdUgaRnaaBaaaleaacqWGVbWBcqWGMbGzcqWGMbGzcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqWGsbGucqaIXaqmaeqaaOGaei4waSLaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqWGsbGudaWgaaWcbaGaeGymaedabeaakiabc2faDjabgkHiTiabdUgaRnaaBaaaleaacqWGVbWBcqWGUbGBcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqGGSaalcqWGsbGucqaIYaGmaeqaaOGaei4waSLaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqGGDbqxcqGGBbWwcqWGsbGudaWgaaWcbaGaeGOmaidabeaakiabc2faDbqaaiabgUcaRiabdUgaRnaaBaaaleaacqWGVbWBcqWGMbGzcqWGMbGzcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqWGsbGucqaIYaGmaeqaaOGaei4waSLaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqWGsbGudaWgaaWcbaGaeGOmaidabeaakiabc2faDjabgkHiTiabdUgaRnaaBaaaleaacqWGVbWBcqWGUbGBcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqGGSaalcqWGobGtcqaIXaqmaeqaaOGaei4waSLaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqGGDbqxcqGGBbWwcqWGobGtdaWgaaWcbaGaeGymaedabeaakiabc2faDjabgUcaRiabdUgaRnaaBaaaleaacqWGVbWBcqWGMbGzcqWGMbGzcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqWGobGtcqaIXaqmaeqaaOGaei4waSLaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqWGobGtdaWgaaWcbaGaeGymaedabeaakiabc2faDbaaaaa@79D7@

Note that, here, the vascular permeability has yet to appear in the equations.

Transport between the compartments

When two tissue volumes containing different amount of extracellular matrix are separated by a semipermeable membrane (endothelium), in thermodynamic equilibrium the volume concentrations of ligand would not be the same. To correctly describe this phenomenon, a volumetric correction has to be made in order to take into account that VEGF can move only in a restricted region in the interstitial space. We follow the formalism introduced by Truskey et al. to describe macromolecular drug distribution in tissue 18. The extracellular matrix is a porous medium composed of proteins and polysaccharides and can deform under mechanical stress. This deformation can affect the volume accessible to the solutes. In addition, some pores are inaccessible to the freely diffusible molecules, because of their sizes or because of the tissue configuration. Finally, there is a steric exclusion of solutes near the surface of the solid phase of the extracellular matrix and the basement membranes. To reflect these properties in the model, the following parameters are introduced: Φ the partition coefficient, εIF the porosity, εIS the interstitial fraction, f the fluid fraction of the interstitial space and KAV the available volume fraction. These parameters are defined by and related to one another through the following equations:

ε I S = interstitial space total tissue volume MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqyTdu2aaSbaaSqaaiabdMeajjabdofatbqabaGccqGH9aqpjuaGdaWcaaqaaiabbMgaPjabb6gaUjabbsha0jabbwgaLjabbkhaYjabbohaZjabbsha0jabbMgaPjabbsha0jabbMgaPjabbggaHjabbYgaSjabbccaGiabbohaZjabbchaWjabbggaHjabbogaJjabbwgaLbqaaiabbsha0jabb+gaVjabbsha0jabbggaHjabbYgaSjabbccaGiabbsha0jabbMgaPjabbohaZjabbohaZjabbwha1jabbwgaLjabbccaGiabbAha2jabb+gaVjabbYgaSjabbwha1jabb2gaTjabbwgaLbaaaaa@6333@

f = interstitial fluid interstitial space MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOzayMaeyypa0tcfa4aaSaaaeaacqqGPbqAcqqGUbGBcqqG0baDcqqGLbqzcqqGYbGCcqqGZbWCcqqG0baDcqqGPbqAcqqG0baDcqqGPbqAcqqGHbqycqqGSbaBcqqGGaaicqqGMbGzcqqGSbaBcqqG1bqDcqqGPbqAcqqGKbazaeaacqqGPbqAcqqGUbGBcqqG0baDcqqGLbqzcqqGYbGCcqqGZbWCcqqG0baDcqqGPbqAcqqG0baDcqqGPbqAcqqGHbqycqqGSbaBcqqGGaaicqqGZbWCcqqGWbaCcqqGHbqycqqGJbWycqqGLbqzaaaaaa@5F48@

Φ = available fluid volume interstitial fluid MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuOPdyKaeyypa0tcfa4aaSaaaeaacqqGHbqycqqG2bGDcqqGHbqycqqGPbqAcqqGSbaBcqqGHbqycqqGIbGycqqGSbaBcqqGLbqzcqqGGaaicqqGMbGzcqqGSbaBcqqG1bqDcqqGPbqAcqqGKbazcqqGGaaicqqG2bGDcqqGVbWBcqqGSbaBcqqG1bqDcqqGTbqBcqqGLbqzaeaacqqGPbqAcqqGUbGBcqqG0baDcqqGLbqzcqqGYbGCcqqGZbWCcqqG0baDcqqGPbqAcqqG0baDcqqGPbqAcqqGHbqycqqGSbaBcqqGGaaicqqGMbGzcqqGSbaBcqqG1bqDcqqGPbqAcqqGKbazaaaaaa@6413@

ε I F = interstitial fluid total tissue volume = f × ε I S MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqyTdu2aaSbaaSqaaiabdMeajjabdAeagbqabaGccqGH9aqpjuaGdaWcaaqaaiabbMgaPjabb6gaUjabbsha0jabbwgaLjabbkhaYjabbohaZjabbsha0jabbMgaPjabbsha0jabbMgaPjabbggaHjabbYgaSjabbccaGiabbAgaMjabbYgaSjabbwha1jabbMgaPjabbsgaKbqaaiabbsha0jabb+gaVjabbsha0jabbggaHjabbYgaSjabbccaGiabbsha0jabbMgaPjabbohaZjabbohaZjabbwha1jabbwgaLjabbccaGiabbAha2jabb+gaVjabbYgaSjabbwha1jabb2gaTjabbwgaLbaakiabg2da9iabdAgaMjabgEna0kabew7aLnaaBaaaleaacqWGjbqscqWGtbWuaeqaaaaa@6BC2@

K A V = available fluid volume total tissue volume = Φ × ε I F MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6FEB@

The free VEGF molecules that diffuse in the interstitial space are effectively only present in the available interstitial fluid volume. In other words, the interstitial space is a part of the total tissue volume U and has a volume of εIS × U . It is composed of a solid phase of volume (εIS - εIF) × U and a fluid phase of volume εIF × U in which VEGF can circulate freely. However, given the pore size of the ECM and basement membranes and the molecular radius of VEGF, some regions are excluded because the growth factor molecules cannot access them. In conclusion, the VEGF molecules can diffuse in the available interstitial fluid volume UAV = KAV × U . Similarly, the free VEGF concentration in the available interstitial fluid volume, noted [V]AV, is related to the free VEGF concentration in the total tissue volume [V] by [V]AV=[V]KAV MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaei4waSLaemOvayLaeiyxa01aaSbaaSqaaiabdgeabjabdAfawbqabaGccqGH9aqpjuaGdaWcaaqaaiabcUfaBjabdAfawjabc2faDbqaaiabdUealnaaBaaabaGaemyqaeKaemOvayfabeaaaaaaaa@3AD9@.

Until now, all the equations have been expressed per unit volume of total tissue. However, because of volume exclusion, we now introduce explicit available interstitial volumes. We assign the subscripts N for healthy (or normal) tissue, D for diseased tissue, B for blood and p for blood plasma (available fluid volume to VEGF in the blood). Note that these subscripts are written as superscripts in the kinetic parameters for clarity reasons. The transport of free VEGF molecules from normal tissue to the blood occurs at a rate kpVNB MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4AaS2aa0baaSqaaiabdchaWjabdAfawbqaaiabd6eaojabdkeacbaaaaa@3231@ (units: cm/s). The term kpVNB MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4AaS2aa0baaSqaaiabdchaWjabdAfawbqaaiabd6eaojabdkeacbaaaaa@3231@ reads as "rate of permeability (kp) of VEGF (V) from the normal tissue compartment (N) to the blood (B)". The free VEGF molecules extravasate at a rate kpVBN MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4AaS2aa0baaSqaaiabdchaWjabdAfawbqaaiabdkeacjabd6eaobaaaaa@3231@ (units: cm/s). The interface between the normal tissue and the blood is the total surface of the microvessels denoted as SNB (units: cm2).

U A V , N d [ V ] A V , N d t = − k p V N B S N B [ V ] A V , N + k p V B N S N B [ V ] p MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@62E6@

Note that, [V]AV × UAV = [V] × U in the tissue compartments and [V]p × Up = [V]B × UB in the blood compartment. Equation (20) can be rewritten in terms of moles per unit volume of total tissue

d [ V ] N d t = − k p V N B S N B U N [ V ] N K A V , N + k p V B N S N B U N U B U p [ V ] B MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6758@

VEGF binding to the receptors, the extracellular matrix and the basement membranes (Eq. (18) and (19)) are added leading to the full equations governing each free isoform in the normal tissue:

d [ V 121 ] N d t = q V 121 N − k o n , V 121 , R 1 N [ V 121 ] N [ R 1 ] N + k o f f , V 121 R 1 N [ V 121 R 1 ] N − k o n , V 121 , R 1 N 1 N [ V 121 ] N [ R 1 N 1 ] N + k o f f , V 121 R 1 N 1 N [ V 121 R 1 N 1 ] N − k o n , V 121 , R 2 N [ V 121 ] N [ R 2 ] N + k o f f , V 121 R 2 N [ V 121 R 2 ] N − k p V N B S N B U N [ V 121 ] N K A V , N + k p V B N S N B U N U B U p [ V 121 ] B MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@1A78@

d [ V 165 ] N d t = q V 165 N − k o n , V 165 , M E B M N [ V 165 ] N [ M E B M ] N + k o f f , V 165 , M E B M N [ V 165 M E B M ] N − k o n , V 165 , M E C M N [ V 165 ] N [ M E C M ] N + k o f f , V 165 M E C M N [ V 165 M E C M ] N − k o n , V 165 , M P B M N [ V 165 ] N [ M P B M ] N + k o f f , V 165 M P B M N [ V 165 M P B M ] N − k o n , V 165 , R 1 N [ V 165 ] N [ R 1 ] N + k o f f , V 165 R 1 N [ V 165 R 1 ] N − k o n , V 165 , R 2 N [ V 165 ] N [ R 2 ] N + k o f f , V 165 R 2 N [ V 165 R 2 ] N − k o n , V 165 , N 1 N [ V 165 ] N [ N 1 ] N + k o f f , V 165 N 1 N [ V 165 N 1 ] N − k p V N B S N B U N [ V 165 ] N K A V , N + k p V B N S N B U N U B U p [ V 165 ] B MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeWabyqaaaaabaqcfa4aaSaaaeaacqWGKbazdaWadaqaaiabdAfawnaaBaaabaGaeGymaeJaeGOnayJaeGynaudabeaaaiaawUfacaGLDbaadaWgaaqaaiabd6eaobqabaaabaGaemizaqMaemiDaqhaaOGaeyypa0JaemyCae3aa0baaSqaaiabdAfawjabigdaXiabiAda2iabiwda1aqaaiabd6eaobaakiabgkHiTiabdUgaRnaaDaaaleaacqWGVbWBcqWGUbGBcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqGGSaalcqWGnbqtcqWGfbqrcqWGcbGqcqWGnbqtaeaacqWGobGtaaGcdaWadaqaaiabdAfawnaaBaaaleaacqaIXaqmcqaI2aGncqaI1aqnaeqaaaGccaGLBbGaayzxaaWaaSbaaSqaaiabd6eaobqabaGcdaWadaqaaiabd2eannaaBaaaleaacqWGfbqrcqWGcbGqcqWGnbqtaeqaaaGccaGLBbGaayzxaaWaaSbaaSqaaiabd6eaobqabaGccqGHRaWkcqWGRbWAdaqhaaWcbaGaem4Ba8MaemOzayMaemOzayMaeiilaWIaemOvayLaeGymaeJaeGOnayJaeGynauJaeiilaWIaemyta0KaemyrauKaemOqaiKaemyta0eabaGaemOta4eaaOWaamWaaeaacqWGwbGvdaWgaaWcbaGaeGymaeJaeGOnayJaeGynaudabeaakiabd2eannaaBaaaleaacqWGfbqrcqWGcbGqcqWGnbqtaeqaaaGccaGLBbGaayzxaaWaaSbaaSqaaiabd6eaobqabaaakeaacqGHsislcqWGRbWAdaqhaaWcbaGaem4Ba8MaemOBa4MaeiilaWIaemOvayLaeGymaeJaeGOnayJaeGynauJaeiilaWIaemyta0KaemyrauKaem4qamKaemyta0eabaGaemOta4eaaOWaamWaaeaacqWGwbGvdaWgaaWcbaGaeGymaeJaeGOnayJaeGynaudabeaaaOGaay5waiaaw2faamaaBaaaleaacqWGobGtaeqaaOWaamWaaeaacqWGnbqtdaWgaaWcbaGaemyrauKaem4qamKaemyta0eabeaaaOGaay5waiaaw2faamaaBaaaleaacqWGobGtaeqaaOGaey4kaSIaem4AaS2aa0baaSqaaiabd+gaVjabdAgaMjabdAgaMjabcYcaSiabdAfawjabigdaXiabiAda2iabiwda1iabd2eanjabdweafjabdoeadjabd2eanbqaaiabd6eaobaakmaadmaabaGaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqWGnbqtdaWgaaWcbaGaemyrauKaem4qamKaemyta0eabeaaaOGaay5waiaaw2faamaaBaaaleaacqWGobGtaeqaaaGcbaGaeyOeI0Iaem4AaS2aa0baaSqaaiabd+gaVjabd6gaUjabcYcaSiabdAfawjabigdaXiabiAda2iabiwda1iabcYcaSiabd2eanjabdcfaqjabdkeacjabd2eanbqaaiabd6eaobaakmaadmaabaGaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaaakiaawUfacaGLDbaadaWgaaWcbaGaemOta4eabeaakmaadmaabaGaemyta00aaSbaaSqaaiabdcfaqjabdkeacjabd2eanbqabaaakiaawUfacaGLDbaadaWgaaWcbaGaemOta4eabeaakiabgUcaRiabdUgaRnaaDaaaleaacqWGVbWBcqWGMbGzcqWGMbGzcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqWGnbqtcqWGqbaucqWGcbGqcqWGnbqtaeaacqWGobGtaaGcdaWadaqaaiabdAfawnaaBaaaleaacqaIXaqmcqaI2aGncqaI1aqnaeqaaOGaemyta00aaSbaaSqaaiabdcfaqjabdkeacjabd2eanbqabaaakiaawUfacaGLDbaadaWgaaWcbaGaemOta4eabeaaaOqaaiabgkHiTiabdUgaRnaaDaaaleaacqWGVbWBcqWGUbGBcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqGGSaalcqWGsbGucqaIXaqmaeaacqWGobGtaaGcdaWadaqaaiabdAfawnaaBaaaleaacqaIXaqmcqaI2aGncqaI1aqnaeqaaaGccaGLBbGaayzxaaWaaSbaaSqaaiabd6eaobqabaGcdaWadaqaaiabdkfasnaaBaaaleaacqaIXaqmaeqaaaGccaGLBbGaayzxaaWaaSbaaSqaaiabd6eaobqabaGccqGHRaWkcqWGRbWAdaqhaaWcbaGaem4Ba8MaemOzayMaemOzayMaeiilaWIaemOvayLaeGymaeJaeGOnayJaeGynauJaemOuaiLaeGymaedabaGaemOta4eaaOWaamWaaeaacqWGwbGvdaWgaaWcbaGaeGymaeJaeGOnayJaeGynaudabeaakiabdkfasnaaBaaaleaacqaIXaqmaeqaaaGccaGLBbGaayzxaaWaaSbaaSqaaiabd6eaobqabaGccqGHsislcqWGRbWAdaqhaaWcbaGaem4Ba8MaemOBa4MaeiilaWIaemOvayLaeGymaeJaeGOnayJaeGynauJaeiilaWIaemOuaiLaeGOmaidabaGaemOta4eaaOWaamWaaeaacqWGwbGvdaWgaaWcbaGaeGymaeJaeGOnayJaeGynaudabeaaaOGaay5waiaaw2faamaaBaaaleaacqWGobGtaeqaaOWaamWaaeaacqWGsbGudaWgaaWcbaGaeGOmaidabeaaaOGaay5waiaaw2faamaaBaaaleaacqWGobGtaeqaaaGcbaGaey4kaSIaem4AaS2aa0baaSqaaiabd+gaVjabdAgaMjabdAgaMjabcYcaSiabdAfawjabigdaXiabiAda2iabiwda1iabdkfasjabikdaYaqaaiabd6eaobaakmaadmaabaGaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqWGsbGudaWgaaWcbaGaeGOmaidabeaaaOGaay5waiaaw2faamaaBaaaleaacqWGobGtaeqaaOGaeyOeI0Iaem4AaS2aa0baaSqaaiabd+gaVjabd6gaUjabcYcaSiabdAfawjabigdaXiabiAda2iabiwda1iabcYcaSiabd6eaojabigdaXaqaaiabd6eaobaakmaadmaabaGaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaaakiaawUfacaGLDbaadaWgaaWcbaGaemOta4eabeaakmaadmaabaGaemOta40aaSbaaSqaaiabigdaXaqabaaakiaawUfacaGLDbaadaWgaaWcbaGaemOta4eabeaakiabgUcaRiabdUgaRnaaDaaaleaacqWGVbWBcqWGMbGzcqWGMbGzcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqWGobGtcqaIXaqmaeaacqWGobGtaaGcdaWadaqaaiabdAfawnaaBaaaleaacqaIXaqmcqaI2aGncqaI1aqnaeqaaOGaemOta40aaSbaaSqaaiabigdaXaqabaaakiaawUfacaGLDbaadaWgaaWcbaGaemOta4eabeaaaOqaaiabgkHiTiabdUgaRnaaDaaaleaacqWGWbaCcqWGwbGvaeaacqWGobGtcqWGcbGqaaqcfa4aaSaaaeaacqWGtbWudaWgaaqaaiabd6eaojabdkeacbqabaaabaGaemyvau1aaSbaaeaacqWGobGtaeqaaaaadaWcaaqaamaadmaabaGaemOvay1aaSbaaeaacqaIXaqmcqaI2aGncqaI1aqnaeqaaaGaay5waiaaw2faamaaBaaabaGaemOta4eabeaaaeaacqWGlbWsdaWgaaqaaiabdgeabjabdAfawjabcYcaSiabd6eaobqabaaaaOGaey4kaSIaem4AaS2aa0baaSqaaiabdchaWjabdAfawbqaaiabdkeacjabd6eaobaajuaGdaWcaaqaaiabdofatnaaBaaabaGaemOta4KaemOqaieabeaaaeaacqWGvbqvdaWgaaqaaiabd6eaobqabaaaamaalaaabaGaemyvau1aaSbaaeaacqWGcbGqaeqaaaqaaiabdwfavnaaBaaabaGaemiCaahabeaaaaGcdaWadaqaaiabdAfawnaaBaaaleaacqaIXaqmcqaI2aGncqaI1aqnaeqaaaGccaGLBbGaayzxaaWaaSbaaSqaaiabdkeacbqabaaaaaaa@CC3B@

Similarly, in the diseased tissue compartment, VEGF121 and VEGF165 are secreted by cells at a rate qV . The secretion rate is assumed constant. Free VEGF ligands can enter the blood at a rate kpVDB MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4AaS2aa0baaSqaaiabdchaWjabdAfawbqaaiabdseaejabdkeacbaaaaa@321D@ and extravasate at a rate kpVBD MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4AaS2aa0baaSqaaiabdchaWjabdAfawbqaaiabdkeacjabdseaebaaaaa@321D@. The interface between the diseased tissue and the blood is the total surface of the microvessels denoted as SDB. The equations governing each VEGF isoform read:

d [ V 121 ] D d t = q V 121 D − k o n , V 121 , R 1 D [ V 121 ] D [ R 1 ] D + k o f f , V 121 R 1 D [ V 121 R 1 ] D − k o n , V 121 , R 1 N 1 D [ V 121 ] D [ R 1 N 1 ] D + k o f f , V 121 R 1 N 1 D [ V 121 R 1 N 1 ] D − k o n , V 121 , R 2 D [ V 121 ] D [ R 2 ] D + k o f f , V 121 R 2 D [ V 121 R 2 ] D − k p V D B S D B U D [ V 121 ] D K A V , D + k p V B D S D B U D U B U p [ V 121 ] B MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@1884@

d [ V 165 ] D d t = q V 165 D − k o n , V 165 , M E B M D [ V 165 ] D [ M E B M ] D + k o f f , V 165 , M E B M D [ V 165 M E B M ] D − k o n , V 165 , M E C M D [ V 165 ] D [ M E C M ] D + k o f f , V 165 M E C M D [ V 165 M E C M ] D − k o n , V 165 , M P B M D [ V 165 ] D [ M P B M ] D + k o f f , V 165 M P B M D [ V 165 M P B M ] D − k o n , V 165 , R 1 D [ V 165 ] D [ R 1 ] D + k o f f , V 165 R 1 D [ V 165 R 1 ] D − k o n , V 165 , R 2 D [ V 165 ] D [ R 2 ] D + k o f f , V 165 R 2 D [ V 165 R 2 ] D − k o n , V 165 , N 1 D [ V 165 ] D [ N 1 ] D + k o f f , V 165 N 1 D [ V 165 N 1 ] D − k p V D B S D B U D [ V 165 ] D K A V , D + k p V B D S D B U D U B U p [ V 165 ] B MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeWabyqaaaaabaqcfa4aaSaaaeaacqWGKbazdaWadaqaaiabdAfawnaaBaaabaGaeGymaeJaeGOnayJaeGynaudabeaaaiaawUfacaGLDbaadaWgaaqaaiabdseaebqabaaabaGaemizaqMaemiDaqhaaOGaeyypa0JaemyCae3aa0baaSqaaiabdAfawjabigdaXiabiAda2iabiwda1aqaaiabdseaebaakiabgkHiTiabdUgaRnaaDaaaleaacqWGVbWBcqWGUbGBcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqGGSaalcqWGnbqtcqWGfbqrcqWGcbGqcqWGnbqtaeaacqWGebaraaGcdaWadaqaaiabdAfawnaaBaaaleaacqaIXaqmcqaI2aGncqaI1aqnaeqaaaGccaGLBbGaayzxaaWaaSbaaSqaaiabdseaebqabaGcdaWadaqaaiabd2eannaaBaaaleaacqWGfbqrcqWGcbGqcqWGnbqtaeqaaaGccaGLBbGaayzxaaWaaSbaaSqaaiabdseaebqabaGccqGHRaWkcqWGRbWAdaqhaaWcbaGaem4Ba8MaemOzayMaemOzayMaeiilaWIaemOvayLaeGymaeJaeGOnayJaeGynauJaeiilaWIaemyta0KaemyrauKaemOqaiKaemyta0eabaGaemiraqeaaOWaamWaaeaacqWGwbGvdaWgaaWcbaGaeGymaeJaeGOnayJaeGynaudabeaakiabd2eannaaBaaaleaacqWGfbqrcqWGcbGqcqWGnbqtaeqaaaGccaGLBbGaayzxaaWaaSbaaSqaaiabdseaebqabaaakeaacqGHsislcqWGRbWAdaqhaaWcbaGaem4Ba8MaemOBa4MaeiilaWIaemOvayLaeGymaeJaeGOnayJaeGynauJaeiilaWIaemyta0KaemyrauKaem4qamKaemyta0eabaGaemiraqeaaOWaamWaaeaacqWGwbGvdaWgaaWcbaGaeGymaeJaeGOnayJaeGynaudabeaaaOGaay5waiaaw2faamaaBaaaleaacqWGebaraeqaaOWaamWaaeaacqWGnbqtdaWgaaWcbaGaemyrauKaem4qamKaemyta0eabeaaaOGaay5waiaaw2faamaaBaaaleaacqWGebaraeqaaOGaey4kaSIaem4AaS2aa0baaSqaaiabd+gaVjabdAgaMjabdAgaMjabcYcaSiabdAfawjabigdaXiabiAda2iabiwda1iabd2eanjabdweafjabdoeadjabd2eanbqaaiabdseaebaakmaadmaabaGaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqWGnbqtdaWgaaWcbaGaemyrauKaem4qamKaemyta0eabeaaaOGaay5waiaaw2faamaaBaaaleaacqWGebaraeqaaaGcbaGaeyOeI0Iaem4AaS2aa0baaSqaaiabd+gaVjabd6gaUjabcYcaSiabdAfawjabigdaXiabiAda2iabiwda1iabcYcaSiabd2eanjabdcfaqjabdkeacjabd2eanbqaaiabdseaebaakmaadmaabaGaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaaakiaawUfacaGLDbaadaWgaaWcbaGaemiraqeabeaakmaadmaabaGaemyta00aaSbaaSqaaiabdcfaqjabdkeacjabd2eanbqabaaakiaawUfacaGLDbaadaWgaaWcbaGaemiraqeabeaakiabgUcaRiabdUgaRnaaDaaaleaacqWGVbWBcqWGMbGzcqWGMbGzcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqWGnbqtcqWGqbaucqWGcbGqcqWGnbqtaeaacqWGebaraaGcdaWadaqaaiabdAfawnaaBaaaleaacqaIXaqmcqaI2aGncqaI1aqnaeqaaOGaemyta00aaSbaaSqaaiabdcfaqjabdkeacjabd2eanbqabaaakiaawUfacaGLDbaadaWgaaWcbaGaemiraqeabeaaaOqaaiabgkHiTiabdUgaRnaaDaaaleaacqWGVbWBcqWGUbGBcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqGGSaalcqWGsbGucqaIXaqmaeaacqWGebaraaGcdaWadaqaaiabdAfawnaaBaaaleaacqaIXaqmcqaI2aGncqaI1aqnaeqaaaGccaGLBbGaayzxaaWaaSbaaSqaaiabdseaebqabaGcdaWadaqaaiabdkfasnaaBaaaleaacqaIXaqmaeqaaaGccaGLBbGaayzxaaWaaSbaaSqaaiabdseaebqabaGccqGHRaWkcqWGRbWAdaqhaaWcbaGaem4Ba8MaemOzayMaemOzayMaeiilaWIaemOvayLaeGymaeJaeGOnayJaeGynauJaemOuaiLaeGymaedabaGaemiraqeaaOWaamWaaeaacqWGwbGvdaWgaaWcbaGaeGymaeJaeGOnayJaeGynaudabeaakiabdkfasnaaBaaaleaacqaIXaqmaeqaaaGccaGLBbGaayzxaaWaaSbaaSqaaiabdseaebqabaGccqGHsislcqWGRbWAdaqhaaWcbaGaem4Ba8MaemOBa4MaeiilaWIaemOvayLaeGymaeJaeGOnayJaeGynauJaeiilaWIaemOuaiLaeGOmaidabaGaemiraqeaaOWaamWaaeaacqWGwbGvdaWgaaWcbaGaeGymaeJaeGOnayJaeGynaudabeaaaOGaay5waiaaw2faamaaBaaaleaacqWGebaraeqaaOWaamWaaeaacqWGsbGudaWgaaWcbaGaeGOmaidabeaaaOGaay5waiaaw2faamaaBaaaleaacqWGebaraeqaaaGcbaGaey4kaSIaem4AaS2aa0baaSqaaiabd+gaVjabdAgaMjabdAgaMjabcYcaSiabdAfawjabigdaXiabiAda2iabiwda1iabdkfasjabikdaYaqaaiabdseaebaakmaadmaabaGaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaGccqWGsbGudaWgaaWcbaGaeGOmaidabeaaaOGaay5waiaaw2faamaaBaaaleaacqWGebaraeqaaOGaeyOeI0Iaem4AaS2aa0baaSqaaiabd+gaVjabd6gaUjabcYcaSiabdAfawjabigdaXiabiAda2iabiwda1iabcYcaSiabd6eaojabigdaXaqaaiabdseaebaakmaadmaabaGaemOvay1aaSbaaSqaaiabigdaXiabiAda2iabiwda1aqabaaakiaawUfacaGLDbaadaWgaaWcbaGaemiraqeabeaakmaadmaabaGaemOta40aaSbaaSqaaiabigdaXaqabaaakiaawUfacaGLDbaadaWgaaWcbaGaemiraqeabeaakiabgUcaRiabdUgaRnaaDaaaleaacqWGVbWBcqWGMbGzcqWGMbGzcqGGSaalcqWGwbGvcqaIXaqmcqaI2aGncqaI1aqncqWGobGtcqaIXaqmaeaacqWGebaraaGcdaWadaqaaiabdAfawnaaBaaaleaacqaIXaqmcqaI2aGncqaI1aqnaeqaaOGaemOta40aaSbaaSqaaiabigdaXaqabaaakiaawUfacaGLDbaadaWgaaWcbaGaemiraqeabeaaaOqaaiabgkHiTiabdUgaRnaaDaaaleaacqWGWbaCcqWGwbGvaeaacqWGebarcqWGcbGqaaqcfa4aaSaaaeaacqWGtbWudaWgaaqaaiabdseaejabdkeacbqabaaabaGaemyvau1aaSbaaeaacqWGebaraeqaaaaadaWcaaqaamaadmaabaGaemOvay1aaSbaaeaacqaIXaqmcqaI2aGncqaI1aqnaeqaaaGaay5waiaaw2faamaaBaaabaGaemiraqeabeaaaeaacqWGlbWsdaWgaaqaaiabdgeabjabdAfawjabcYcaSiabdseaebqabaaaaOGaey4kaSIaem4AaS2aa0baaSqaaiabdchaWjabdAfawbqaaiabdkeacjabdseaebaajuaGdaWcaaqaaiabdofatnaaBaaabaGaemiraqKaemOqaieabeaaaeaacqWGvbqvdaWgaaqaaiabdseaebqabaaaamaalaaabaGaemyvau1aaSbaaeaacqWGcbGqaeqaaaqaaiabdwfavnaaBaaabaGaemiCaahabeaaaaGcdaWadaqaaiabdAfawnaaBaaaleaacqaIXaqmcqaI2aGncqaI1aqnaeqaaaGccaGLBbGaayzxaaWaaSbaaSqaaiabdkeacbqabaaaaaaa@C91B@

We assume that no receptors are present on the luminal side of the endothelial cells. Therefore, only free ligands exist in the blood compartment. Nevertheless, receptors in this compartment could be added to the model as experimental knowledge emerges and the density of receptors on the luminal side of the endothelial cells is quantified. The presence of platelets and leukocytes as potential VEGF carriers, is neglected in the current model as well as soluble VEGFR1 (sFlt) as a VEGF-neutralizing molecule. Given the complexity of the model, it is preferable to introduce additional factors one at a time and investigate their effects (a computational equivalent of knock-in or knock-out procedures). The model can be readily extended to include these factors.

Finally, in the blood compartment, VEGF is cleared at a rate cV (units: s-1) per unit volume of total blood via several organs (e.g., kidneys) that are not explicitly represented here. This process is described as a first-order reaction. Similarly to the equations for free VEGF in normal and diseased tissues, the governing equation for each free isoform in the blood is:

U p d [ V ] p d t = − k p V B N S N B [ V ] p + k p V N B S N B [ V ] N K A V , N − k p V B D S D B [ V ] p + k p V D B S D B [ V ] D K A V , D MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@8319@

This can be rewritten per unit volume of whole blood by:

d [ V ] B d t = − k p V B N S N B U p [ V ] B + k p V N B S N B U B [ V ] N K A V , N − k p V B D S D B U p [ V ] B + k p V D B S D B U B [ V ] D K A V , D MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@8A81@

Therefore, the VEGF isoforms are governed by the two following equations.

d [ V 121 ] B d t = − c V 121 [ V 121 ] B − k p V B N S N B U p [ V 121 ] B + k p V N B S N B U B [ V 121 ] N K A V , N − k p V B D S D B U p [ V 121 ] B + k p V D B S D B U B [ V 121