BACKGROUND
angiogenesis is the process by which new blood vessels grow from pre-existing vascular networks.
this is a dynamic physiological mechanism  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  <cit> .
vascular endothelial growth factor  is a potent cytokine involved in angiogenesis  <cit> .
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  and vegf-b are known to be involved in angiogenesis.
vegf-a and three members  have been shown to be implicated in both mechanisms.
the vegf-a family contains the different splice isoforms vegf <dig>  vegf121b, vegf <dig>  vegf145b, vegf <dig>  vegf <dig>  vegf <dig>  vegf165b, vegf <dig>  vegf <dig>  vegf <dig>  vegf <dig>  <cit> .
the major splice variants are vegf <dig>  vegf <dig>  vegf <dig> and vegf <dig>  because the role of vegf <dig> and vegf <dig> in vivo is not well understood and their expression is low compared to vegf <dig> and vegf <dig>  <cit> , we consider the two main isoforms vegf <dig> and vegf <dig> in our model.
these two ligands bind two receptor tyrosine kinases vegfr <dig>  and vegfr <dig> .
the vegf <dig> isoform binds to a non-signaling co-receptor called neuropilin- <dig>  and to heparan sulfate proteoglycans which are constituents of the extracellular matrix  and the cellular basement membranes.
nrp <dig> also independently couples with vegfr <dig>  to which vegf <dig> can bind to form a complex of all three.
another complex can be formed when vegf <dig> bound to nrp <dig> binds in turn to vegfr <dig> .
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-vegfr <dig> signaling  <cit> .
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 vegfr <dig> when vegfr <dig> populations are larger  <cit> .
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 nrp <dig> enhances the binding of vegf <dig> to vegfr <dig>  <cit> .
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  <cit> .
anatomically detailed tissue models have predicted heterogeneity in the activation of the endothelial vegf receptors and how this affects vegf gradients  <cit> .
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  <cit> .
three-dimensional vegf distribution was also predicted in ischemic muscle in a model of peripheral arterial disease  <cit> .
a meta-analysis of vegf levels in healthy subjects and cancer patients revealed that plasma vegf levels are several-fold higher in cancer  <cit> .
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  <cit>  and in peripheral arterial disease  <cit> .
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  <cit> .
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 , 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 , parenchymal basement membranes  and endothelial basement membranes .
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.
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  <cit>  and skeletal muscle  <cit> .
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 .
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   <cit>  and the rest of the body .
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  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 , the basement membrane surrounding the parenchymal cells  and the basement membrane surrounding the endothelial cells .
the two isoforms of the vegf-a family considered in the model are the freely diffusible heparin-binding vegf <dig>  and non-heparin-binding vegf <dig> .
because vegf <dig> binds to the heparan sulfate proteoglycans  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  vegf in the interstitial space is uniform within each compartment and spatial variability due to gradient formation is neglected  <cit> .
the interactions between vegf <dig> and vegf <dig> and their receptors  are illustrated in figure 1c and will be further described in the equation section.
to summarize, vegf <dig> and vegf <dig> bind to both vegfr <dig> and vegfr <dig> but only vegf <dig> is believed to bind to the non-enzymatic co-receptor nrp <dig>  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  <cit> , 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  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  <cit>  by estimating that the damkohler number  is significantly less than one.
free vegf is transported by transcapillary permeability between the blood and tissue compartments and cleared from the blood .
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  <cit> .
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  while the concentrations of the surface species are expressed per unit surface area of the corresponding cell .
interstitial space
we use the following notation:  is the concentration of vegf binding sites of the ecm, pbm or ebm ;  represents the concentration of any unbound vegf isoform  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
v165+mecm⇌koff,v <dig> mecmkon,v <dig> mecmv165mecm
v165+mebm⇌koff,v <dig> mebmkon,v <dig> mebmv165mebm
v165+mpbm⇌koff,v <dig> mpbmkon,v <dig> mpbmv165mpbm
and are governed by the equations
ddt=−kon,v <dig> mebm+koff,v165mebm
ddt=−kon,v <dig> mecm+koff,v165mecm
ddt=−kon,v <dig> mpbm+koff,v165mpbm
ddt=kon,v <dig> mebm−koff,v165mebm
ddt=kon,v <dig> mecm−koff,v165mecm
ddt=kon,v <dig> mpbm−koff,v165mpbm
cell surface
,  and  represent the densities of the unoccupied receptor tyrosine kinases , the unoccupied co-receptor , and their coupled complexes , respectively;  and  are the concentrations of vegf bound to the vegf receptors and nrp <dig>  respectively;   is the concentration of vegf-bound in the form of the ternary complex .
sr represents the rate at which the receptors are inserted into the cell membrane and kint the internalization rate of the receptors.
the vegf <dig> isoform interacts with the receptors through the following reactions
v165+r1⇌koff,v165r1kon,v <dig> r1v165r <dig>
v165+r2⇌koff,v165r2kon,v <dig> r2v165r <dig>
v165+n1⇌koff,v165n1kon,v <dig> n1v165n <dig>
v165n1+r2⇌koff,v165n <dig> r2kc,v165n <dig> r2r2v165n <dig>
v165r2+n1⇌koff,v165r <dig> n1kc,v165r <dig> n1r2v165n <dig>
while the vegf <dig> isoform binds to the receptors according to these reactions
v121+r1⇌koff,v121r1kon,v <dig> r1v121r <dig>
v121+r2⇌koff,v121r2kon,v <dig> r2v121r <dig>
r1+n1⇌kdissoc,r1n1kc,r <dig> n1r1n <dig>
v121+r1n1⇌koff,v121r1kon,v <dig> r1v121r1n <dig>
v121r1+n1⇌kdissoc,r1n1kc,r <dig> n1v121r1n <dig>
the  <dig> preceding chemical reactions are governed by the following equations:
ddt=sr1−kint,r1−kon,v <dig> r1+koff,v165r1−kon,v <dig> r1+koff,v121r1−kc,r <dig> n1+kdissoc,r1n <dig>
ddt=sr2−kint,r2−kon,v <dig> r2+koff,v121r2−kon,v <dig> r2+koff,v165r2−kc,v165n <dig> r2+koff,v165n <dig> r <dig>
ddt=sn1−kint,n1−kc,v121r <dig> n1+kdissoc,r1n1−kc,r <dig> n1+kdissoc,r1n1−kon,v <dig> n1−koff,v165n1−kc,v165r <dig> n1+koff,v165r <dig> n <dig>
ddt=−kint,v121r1+kon,v <dig> r1−koff,v121r1−kc,r <dig> n1+kdissoc,r1n <dig>
ddt=−kint,v121r2+kon,v <dig> r2−koff,v121r <dig>
ddt=−kint,v165r1+kon,v <dig> r1−koff,v165r <dig>
ddt=−kint,v165r2+kon,v <dig> r2−koff,v165r2−kc,v165r <dig> n1+koff,v165r <dig> n <dig>
ddt=−kint,v165n1+kon,v <dig> n1−koff,v165n1−kc,v165n <dig> r2+koff,v165n <dig> r <dig>
ddt=−kint,r1n1+kc,r <dig> n1−kdissoc,r1n1−kon,v <dig> r1+koff,v121r <dig>
ddt=−kint,v165r2n1+kc,v165r <dig> n1−koff,v165r <dig> n1+kc,v165n <dig> r2−koff,v165n <dig> r <dig>
ddt=−kint,v121r1n1+kc,v121r <dig> n1−kdissoc,v121n1+kon,v121r1n1−koff,v121r1n <dig>
ligands in the tissue compartments
the ligands vegf <dig> and vegf <dig> 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:
ddt=qv121−kon,v <dig> r1+koff,v121r1−kon,v <dig> r1n1+koff,v121r1n1−kon,v <dig> r2+koff,v121r <dig>
ddt=qv165−kon,v <dig> mebm+koff,v <dig> mebmn−kon,v <dig> mecm+koff,v165mecm−kon,v <dig> mpbm+koff,v165mpbm−kon,v <dig> r1+koff,v165r1−kon,v <dig> r2+koff,v165r2−kon,v <dig> n1+koff,v165n <dig>
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 , 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  <cit> .
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:
εis=interstitial spacetotal tissue volume
f=interstitial fluidinterstitial space
Φ=available fluid volumeinterstitial fluid
εif=interstitial fluidtotal tissue volume=f×εis
kav=available fluid volumetotal tissue volume=Φ×εif
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  × 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 av, is related to the free vegf concentration in the total tissue volume  by av=kav.
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  tissue, d for diseased tissue, b for blood and p for blood plasma .
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 .
the term kpvnb reads as "rate of permeability  of vegf  from the normal tissue compartment  to the blood ".
the free vegf molecules extravasate at a rate kpvbn .
the interface between the normal tissue and the blood is the total surface of the microvessels denoted as snb .
uav,ndav,ndt=−kpvnbsnbav,n+kpvbnsnbp
note that, av × uav =  × u in the tissue compartments and p × up = b × ub in the blood compartment.
equation  can be rewritten in terms of moles per unit volume of total tissue
dndt=−kpvnbsnbunnkav,n+kpvbnsnbunubupb
vegf binding to the receptors, the extracellular matrix and the basement membranes  and ) are added leading to the full equations governing each free isoform in the normal tissue:
dndt=qv121n−kon,v <dig> r1nnn+koff,v121r1nn−kon,v <dig> r1n1nnn+koff,v121r1n1nn−kon,v <dig> r2nnn+koff,v121r2nn−kpvnbsnbunnkav,n+kpvbnsnbunubupb
dndt=qv165n−kon,v <dig> mebmnnn+koff,v <dig> mebmnn−kon,v <dig> mecmnnn+koff,v165mecmnn−kon,v <dig> mpbmnnn+koff,v165mpbmnn−kon,v <dig> r1nnn+koff,v165r1nn−kon,v <dig> r2nnn+koff,v165r2nn−kon,v <dig> n1nnn+koff,v165n1nn−kpvnbsnbunnkav,n+kpvbnsnbunubupb
similarly, in the diseased tissue compartment, vegf <dig> and vegf <dig> 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 and extravasate at a rate kpvbd.
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:
dddt=qv121d−kon,v <dig> r1ddd+koff,v121r1dd−kon,v <dig> r1n1ddd+koff,v121r1n1dd−kon,v <dig> r2ddd+koff,v121r2dd−kpvdbsdbuddkav,d+kpvbdsdbudubupb
dddt=qv165d−kon,v <dig> mebmddd+koff,v <dig> mebmdd−kon,v <dig> mecmddd+koff,v165mecmdd−kon,v <dig> mpbmddd+koff,v165mpbmdd−kon,v <dig> r1ddd+koff,v165r1dd−kon,v <dig> r2ddd+koff,v165r2dd−kon,v <dig> n1ddd+koff,v165n1dd−kpvdbsdbuddkav,d+kpvbdsdbudubupb
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 vegfr <dig>  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 .
the model can be readily extended to include these factors.
finally, in the blood compartment, vegf is cleared at a rate cv  per unit volume of total blood via several organs  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:
updpdt=−kpvbnsnbp+kpvnbsnbnkav,n−kpvbdsdbp+kpvdbsdbdkav,d
this can be rewritten per unit volume of whole blood by:
dbdt=−kpvbnsnbupb+kpvnbsnbubnkav,n−kpvbdsdbupb+kpvdbsdbubdkav,d
therefore, the vegf isoforms are governed by the two following equations.
dbdt=−cv121b−kpvbnsnbupb+kpvnbsnbubnkav,n−kpvbdsdbupb+kpvdbsdbubdkav,d
dbdt=−cv165b−kpvbnsnbupb+kpvnbsnbubnkav,n−kpvbdsdbupb+kpvdbsdbubdkav,d
whole-body parameters
each tissue is uniquely characterized by its biochemical, biophysical and geometrical properties as input parameters into the model.
similar to the consideration of total tissue volume compared to available interstitial fluid volume, we distinguish between the total tissue volume of the ecm, pbm and ebm, noted uecm, upbm, uebm respectively, and the available interstitial fluid volumes of the ecm, pbm and ebm, noted uav, ecm, uav, pbm, uav, ebm respectively.
in terms of these variables various tissue characteristics can be calculated for the different tissue compartments, e.g., the total amount of vegf <dig> bound to the binding sites of the ecm is equal to uav, ecm × av, where av is the concentration of vegf <dig> bound to the ecm in moles per liter of available ecm fluid volume.
similarly, the total amount of vegf <dig> sequestered by the ebm is equal to uav, ebm × av, where av is the concentration of vegf <dig> bound to the endothelial basement membrane in moles per liter of available ebm fluid volume.
the conversion is given by
uav, ebm av =  u
uav, ecm av =  u
in addition, the total abluminal surface area of endothelial cells is sec.
the total amount of unligated vegfr <dig> is * × sec, where * is the number of receptors vegfr <dig> per unit of endothelial surface.
* sec =  u
note that the kinetic rates can also be transformed, e.g.,
kav, onu = k onuav
in the blood phase, we express all concentrations with respect to its total volume.
if we designate red blood cell fractional volume or hematocrit as ht and neglect the small volume of white blood cells and platelets, then the concentration of species in blood can be expressed in terms of plasma concentrations as follows
p = b ×
where p represents the concentration of free vegf <dig> in the plasma.
numerical implementation
the model, represented by  <dig> ordinary differential equations  and initial conditions, was implemented using visual fortran  <dig> software on a pc.
the equations were non-dimensionalized prior to numerical solution using appropriate characteristic values and once the solutions were obtained were transformed to the original dimensional variables.
transient solutions were calculated using an adaptive step-size runge-kutta 5th-order accuracy integrative scheme.
a relative error tolerance of 10- <dig> was used.
the steady state was defined when the concentrations changed by less than 1%.
model parameters
the parameters are summarized in tables  <dig>   <dig>   <dig>   <dig>   <dig>   <dig>
the derivation of these parameters is found in  <cit> .
in this table,  <dig>   <dig> /m and  <dig>   <dig> /.
here, m = moles/liter available interstitial fluid volume.
the conversion of receptor densities to tissue concentrations is based on the relationship mentioned in table  <dig>  and the surface area of an endothelial cell,  <dig> μm <dig>  vegf concentration is normalized based on the entire interstitial space, since it diffuses throughout:  <dig>   <dig> /m .
vegf binding sites in the ecm and bms are based on those volumes:  <dig>   <dig> /m,  <dig>   <dig> /m,  <dig>   <dig> /m .
for example, m = moles/liter endothelial basement membrane.
conversions from pg/mg protein are based on  <dig> mg protein/g of tissue and  <dig> kda vegf,  <dig> kda vegfr <dig>   <dig> kda vegfr <dig>  <cit> .
the derivation of these parameters can be found in  <cit> .
in this table,  <dig>   <dig> /m and  <dig>   <dig> /.
here, m = moles/liter interstitial space.
the conversion of receptor densities to tissue concentrations is based on the relationship mentioned in table  <dig>  and the surface area of an endothelial cell,  <dig> μm <dig>  vegf concentration is normalized based on the entire interstitial space, since it diffuses throughout:  <dig>   <dig> /m .
vegf binding sites in the ecm and bms are based on those volumes:  <dig>   <dig> /m,  <dig>   <dig> /m,  <dig>   <dig> /m  <cit> .
blood
we consider a human subject of  <dig> kg.
to calculate the typical plasma and total blood volumes in a 70-kg human being, we used a study of ninety healthy subjects by gibson and evans  <cit> .
the volumes were plotted against the weight of the volunteer.
linear regression was performed for males and females and both volumes were determined for  <dig> kg.
we therefore consider  <dig>  liters of total blood including  <dig>  liters of blood plasma, which constitute  <dig> % of total blood volume.
normal tissue
the volume of the normal tissue is the volume corresponding to a 70-kg subject with a vastus lateralis skeletal muscle density of  <dig>  g/cm <dig>  less  <dig> cm <dig> of whole blood .
the parameters characterizing the healthy tissue  are summarized in tables  <dig>   <dig>   <dig> and the properties and characteristics of the skeletal muscle have been described in  <cit> .
however, a few adjustments were made.
the interstitial space is composed of  <dig> % of collagen   <cit> .
this content is not accessible to the vegf molecules and thus does not account for the available interstitial fluid volume.
interstitial fluid of muscle has been measured at 7% of tissue volume  <cit> , thus the total interstitial space is  <dig> % .
the pore sizes of the basement membranes and the extracellular matrix in the skeletal muscle are estimated at  <dig> nm  <cit>  and  <dig> nm  <cit>  respectively.
for a molecular weight of  <dig> kda , this corresponds to a partition coefficient Φ of  <dig>  and a kav of  <dig>  for the pbm and ebm  <cit>  and a partition coefficient of Φ of  <dig>  and a kav of  <dig>  for the ecm  <cit> .
the available volumes for vegf transport in the extracellular matrix, parenchymal basement membrane and endothelial basement membrane are therefore  <dig> ,  <dig> , and  <dig>  cm3/cm <dig> tissue, respectively .
for the vegf receptors, we assume that, at any instant, the insertion of receptors equals the internalization.
as a result, the number of total  receptors is conserved.
this condition also applies to the diseased tissue.
regulation of vegf receptors represents an important poorly explored area; more complex receptor dynamics can be considered in the model as experimental information becomes available.
diseased tissue
as an example of diseased tissue, we consider a  <dig> cm-diameter breast tumor.
for this tissue, we use parameters reported in a previous one-compartment model  <cit> .
assuming the tumor to be a sphere, the volume of the diseased compartment is  <dig> cm <dig>  the parameters are summarized in tables  <dig>   <dig>   <dig> and the properties and characteristics of the tumor tissue have been described in  <cit> .
the 5% collagen content increases the interstitial space from  <dig> %  <cit>  to  <dig> %.
in breast tumor, the pore size of the endothelial basement membrane has been measured to be several hundred nanometers  which corresponds to a partition coefficient of  <dig>   <cit> .
it was shown that the basement membranes and the ecm are similar and difficult to distinguish in the context of mammary tumor  <cit> .
therefore, a partition coefficient of  <dig>  was also adopted for the ecm  and the pbm.
this corresponds to a kav of  <dig>  for the three regions  <cit> .
the available interstitial fluid volumes for vegf transport in the extracellular matrix, parenchymal and endothelial basement membranes are therefore  <dig> ,  <dig> , and  <dig>  cm3/cm <dig> tissue, respectively.
permeability between the normal tissue and the blood
to determine the permeability between the normal tissue and the capillaries, we first determine the stokes-einstein radius for a vegf molecule.
the molecular weight of vegf <dig> isoform is approximately  <dig> kda.
the stokes-einstein radius ae  is calculated by the formula for globular molecules given in venturoli and rippe  <cit> : ae =  <dig>  ×  <dig>  .
for  <dig> kda, we thus find a stokes-einstein radius of  <dig>  Å. with this effective molecular radius, it is then possible to determine the permeability-surface area product  <cit>  as  <dig>  × 10- <dig> ml/s ×  <dig> g. with a surface area of  <dig> cm2/g  <cit> , we deduce that the permeability between the normal tissue and the blood is  <dig>  × 10- <dig> cm/s.
a similar value was found using the permeability-surface area product using schmittmann and rohr's study  <cit> .
because there is paucity of experimental data on vegf-dependence of macromolecular permeability  <cit> , we consider a permeability range from  <dig> × 10- <dig> to  <dig> × 10- <dig> cm/s for sensitivity analyses; this range includes the 2– <dig> fold vegf-dependent increase in permeability reported in  <cit> .
when the permeability between the normal tissue and the blood is fixed, the value  <dig> × 10- <dig> cm/s is selected.
the vegf transport is assumed to be passive.
permeability between the diseased tissue and the blood
for the permeability between the tumor and the blood, we note that ovalbumin and the vegf homodimer have a similar molecular weight .
in human tumor xenografts, the microvascular permeability for ovalbumin, a globular molecule, was measured to be  <dig>  × 10- <dig> cm/s  <cit> .
studies of tumor microvascular permeability for macromolecules in vivo also provide useful insights  <cit> .
however, most of these experiments use dextrans that are linear molecules contrary to vegf.
this difference is crucial for permeability purposes.
to resolve this issue, the relationship between the stokes-einstein radius and the permeability is usually preferred.
for a stokes-einstein radius of about  <dig>  Å , the permeability is around  <dig> × 10- <dig> cm/s  <cit> .
therefore, a range from  <dig> × 10- <dig> to  <dig> × 10- <dig> cm/s is chosen for the vascular permeability to vegf in tumor when the sensitivity analysis is performed since little data on vegf-dependence to macromolecule permeability is available  <cit> .
when the tumor/blood permeability is fixed at a specific value,  <dig> × 10- <dig> cm/s is assumed.
summary of experimental measurements of vegf concentration
a meta-analysis was performed by kut et al.
<cit> .
in breast cancer, plasma vegf levels were  <dig> to  <dig> times higher that those in healthy subjects .
in prostate cancer, plasma vegf levels were  <dig> to  <dig> times higher and in colorectal cancer about two times higher that those in healthy controls.
for a vegf molecular weight of  <dig> kda, the plasma levels in breast cancer patients correspond to  <dig>  –  <dig>  pm while they are in the range of  <dig>  –  <dig>  pm for healthy subjects.
in our study, we therefore assume a vegf plasma level of around  <dig> pm in healthy subjects and several-fold higher in breast cancer patients.
the vegf secretion rates are then calculated to match the observed plasma level.
RESULTS
in all simulations, unless specified otherwise, the vascular permeability of healthy tissue is  <dig> × 10- <dig> cm/s, the vegf plasma clearance  <dig>  min- <dig>  <cit> , the vegf <dig> secretion rate  <dig>  molecule/cell/s in the normal tissue, the vegf isoform expression ratio vegf165:vegf <dig> 92%:8%  <cit>  and the density of vegfr <dig>  vegfr <dig> and nrp <dig> is  <dig>  molecules/endothelial cell.
these parameter values are based on currently available experimental data and they can be altered as additional data become available.
in many cases, we systematically explore the sensitivity of the results to the parameter variation.
vegf represents the total vegf, i.e., vegf <dig> and vegf <dig>  this means that the free vegf concentration corresponds to the sum of the free vegf <dig> and the free vegf <dig> concentrations.
similarly, the vegf secretion rate represents the sum of the secretion rates of the two vegf isoforms.
healthy subject
without vegf clearance, steady-state blood and tissue concentrations are the same
a range from  <dig>  to  <dig>  molecule/cell/s was tested for vegf <dig> secretion rate.
in the absence of plasma clearance, the free vegf concentration in both compartments is close to directly proportional  to the vegf secretion rate in the normal tissue for the range we tested .
in the absence of plasma clearance, the steady-state total vegf concentration in the blood plasma equals that in the available interstitial fluid of the healthy tissue .
this is in agreement with v equation  which, at steady state, becomes p=nkav,n=av,n.
if  <dig> pm  of free vegf concentration is present, at steady state, in the normal tissue available interstitial fluid, a vegf concentration in the blood will also be  <dig> pm .
in the absence of plasma clearance, the steady-state free vegf concentrations in both compartments are independent of the permeability value as long as this value is non-zero .
this is because the permeability affects the dynamics of the system by regulating how fast the compartments reach the steady state but not the steady-state free vegf concentrations themselves.
looking at equations  and , in the absence of the tumor compartment and of plasma clearance, the steady-state free vegf concentration does not depend on the permeability between the healthy tissue and the blood since kpvnb=kpvbn is assumed.
the equation reads b=upubnkav,n.
in other words, the relationship p=nkav,n=av,n holds independently of the permeability value as long as kpvnb=kpvbn holds true.
plasma clearance reduces blood vegf concentration
the vegf half-life in plasma has been shown to be  <dig>  ±  <dig>  min  <cit> .
to explore the effect of the half-life, the plasma clearance rate cv was chosen between  <dig>  and  <dig>  min- <dig>
with the introduction of a clearance rate cv of  <dig>  min- <dig> , the steady-state concentration of free vegf in plasma becomes  <dig>  times lower than in available interstitial fluid at a vascular permeability of  <dig> × 10- <dig> cm/s and  <dig>  times lower at a permeability of  <dig> × 10- <dig> cm/s .
the dependence on vegf secretion rate is approximately linear for the range studied.
unlike free vegf concentration in normal tissue, free vegf concentration in the blood is strongly dependent on the vascular permeability.
this is because, at steady state, the permeability coefficients do not cancel out in the equations  and .
b=nkav,n
next, the clearance rate is fixed at  <dig>  min- <dig>  the vegf <dig> secretion rate in the healthy tissue is chosen to be  <dig>  molecule/cell/s so that, at steady state, there is  <dig> pm of free vegf in the blood for a vascular permeability of  <dig> × 10- <dig> cm/s.
we investigate the dependence on permeability in a range  <dig> × 10- <dig> to  <dig> × 10- <dig> cm/s.
at steady state, the free vegf concentration in the available interstitial fluid in the normal tissue is around  <dig>  pm independent of the permeability .
at very low permeability , very few vegf molecules secreted in the normal tissue enter the blood compartment.
because the clearance is directly proportional to the vegf concentration in the blood plasma  and ), a few molecules of vegf are cleared from the blood.
therefore, the internalization is very high and accounts for most of the loss of vegf in the normal tissue.
at very high permeability , a high concentration of free vegf enters the blood compartment and is cleared.
the internalization is smaller and the plasma clearance is responsible for most of the loss of vegf.
in the physiological range , the free vegf concentration in the blood plasma is roughly proportional to the permeability between the two compartments.
in the healthy tissue, however, the change in the net flow  from the healthy tissue to the blood due to permeability changes is balanced by slight changes in the binding to and the internalization of vegf by the receptors; thus, the healthy tissue vegf concentration vary slightly with the vascular permeability.
next, we explore the sensitivity of the results to plasma clearance in a range between  <dig>  and  <dig>  min- <dig>  the internalization of vegf by the receptors in healthy tissue regulates vegf concentration in the compartment .
in the blood, however, the vegf molecules can accumulate if the clearance rate is decreased .
transient effects of acute exercise
given the clearance rate , vascular permeability  and vegf165:vegf <dig> expression ratio , the model predicts that a vegf <dig> secretion rate of  <dig>  molecule/cell/s is necessary to achieve a free vegf plasma level of  <dig> pm, as reported under physiological conditions  <cit> .
this leads to a free vegf concentration in the normal tissue of approximately  <dig>  pm.
to study the vegf transient effects, we simulate a physical exercise experiment.
jensen et al.
have shown that a 3-hour two-legged knee extension upregulates vegf mrna by about  <dig>  fold for at least  <dig> hours  <cit> .
vegf mrna levels return to baseline between  <dig> and  <dig> hours  <cit> .
we assume a direct correlation between mrna and vegf protein level.
figure 3a shows the transient effect on free vegf concentration in available interstitial fluid and in blood plasma.
the free vegf concentrations return to baseline within  <dig> hours of cessation of the secretion upregulation .
it is interesting to note that the fold increase in blood and tissue concentrations are the same .
however, there is a time lag between the blood and normal tissue; in transition periods, vegf levels measured in plasma and tissue samples could be significantly different.
higher permeability would decrease this time lag .
this provides important insights on monitoring vegf protein levels in the case of exercise training.
this simulation demonstrates that vegf tissue concentration can increase several-fold during prolonged exercise, thus providing a stimulus for exercise-induced angiogenesis.
cancer patient
in this simulation, a 4-cm diameter tumor located in the breast is represented by the diseased tissue compartment.
the characteristics of this new compartment are presented in tables  <dig>   <dig>   <dig>
the compilation of the vegf levels in healthy subjects and cancer patients  <cit>  has revealed that cancer patients show, on average, a several-fold increase in their free vegf blood plasma levels.
this part of the study investigates the possible origins of this increase.
blood and normal tissue vegf levels are not significantly sensitive to tumor vegf secretion
we performed a sensitivity analysis on the tumor vegf secretion rate.
for the selected parameters we found that regardless of the vascular permeability in the tumor, free vegf concentration in available interstitial fluid in the normal tissue is insensitive to the vegf secretion rate in the tumor and to the vascular permeability of the tumor .
this qualitative behavior is independent of the density of the nrp <dig> in the tumor.
at a vascular permeability in the tumor of  <dig> × 10- <dig> cm/s , the free vegf level in the available tumor interstitial fluid is approximately proportional to the secretion rate of vegf in the tumor while the vegf concentration in the blood plasma is rather insensitive to the change in the tumor vegf secretion.
when increasing the permeability by two orders of magnitude , more vegf molecules secreted from the tumor enter the blood.
this results in an increase of vegf level in the blood plasma and a decrease of vegf level in the tumor for a given vegf secretion rate in tumor.
however, even at a high vegf secretion rate in the tumor, the plasma vegf concentration increases by less that 50%.
thus, an increase in the vegf secretion rate alone cannot explain the several-fold increase reported for cancer patients  <cit> , at least for the selected parameters of the model.
for a given secretion rate in tumor, our calculations show that for the plasma vegf level to double, the tumor size would have to increase approximately to 25-cm diameter.
we now set the tumor vegf <dig> secretion rate at  <dig>  or  <dig>  molecule/cell/s for  <dig>  and  <dig>  nrp <dig> in the tumor respectively, so that, at a vascular permeability in tumor of  <dig> × 10- <dig> cm/s , the steady-state free vegf concentration in the available interstitial fluid in the tumor corresponds to that in the available interstitial fluid in the normal tissue .
increasing vascular permeability of tumor decreases tumor vegf levels
we then perform a sensitivity analysis on the vascular permeability in the tumor .
we find that the free vegf concentration in the normal tissue remains constant independently of the tumor vascular permeability .
the two main reasons are: the normal tissue volume overwhelms the effects of smaller volumes  and the vascular permeability in the healthy tissue is small.
however, the vascular permeability in tumor has a high impact on the free vegf concentration in the tumor as shown in figure 4b.
for permeability higher than  <dig> × 10- <dig> cm/s, the free vegf level drops drastically in the tumor.
this behavior is the result of the amount of vegf molecules being cleared from the blood as they are transported from the tumor to the blood compartment.
the plasma vegf level is not highly affected by the increase of vascular permeability in the tumor because the clearance is directly proportional to the concentration and because the volume of the healthy tissue is bigger than that of the tumor.
in any case, the plasma vegf level does not exhibit a several-fold increase compared to the absence of tumor, even at high permeability .
this means that the vascular permeability in tumor increase alone cannot explain the several-fold increase of plasma vegf level in cancer patients  <cit> .
over a range of clearance rates from  <dig>  and  <dig>  min- <dig>  <cit> , there is little variation in the free vegf concentration  and this also cannot explain the several-fold increase in plasma vegf reported in cancer patients  <cit> .
increasing the nrp <dig> density in the tumor by an order of magnitude only decreases the tumor vegf at high vascular permeability in tumor.
increasing vascular permeability of healthy tissue increases blood and tumor vegf levels
for high vascular permeability of healthy tissue , the free vegf concentration in plasma would be increased approximately four- to five-fold compared to baseline .
therefore, the vascular permeability of healthy tissue may increase at sites distant from the tumor , leading in turn to a total increase of vegf concentration in the plasma or the additive effects of increased vegf secretion, vascular permeability in tumor, and tumor mass may explain the several-fold increase in plasma vegf in cancer patients as reported in the literature  <cit> .
distribution of free vs. receptor- and hspg-bound vegf in healthy and diseased tissues
in the above sections, we presented results of computer simulations for free vegf in the tissue and blood compartments.
to understand the total balance of vegf in the body, it is also important to assess the amounts of vegf bound to the receptors on the endothelial cells and to the hspg sites in the ecm and basement membranes.
the vegf distribution is shown in figure 5a.
for the parameters specified in the legend, 93% of total vegf in the healthy tissue is vegf <dig>  the model revealed that up to half of the vegf distribution in the healthy tissue and the tumor with  <dig>  nrp <dig> per endothelial cell, is in the form of a complex where vegf <dig> is bound to vegfr <dig> and nrp <dig> simultaneously.
in the tumor,  <dig> to 68%  of the vegf population, is vegf <dig> bound to the ecm while it represented only a quarter in the healthy tissue.
finally, the vast majority of free vegf in the blood is vegf <dig> , regardless of the density of nrp <dig> in the tumor.
the majority of cell surface receptors in healthy tissue and tumor receptors are not ligated
the receptor occupancy in the tumor with density of  <dig>  nrp <dig> per endothelial cell is qualitatively similar to that in the healthy tissue .
in these tissues, most vegfr <dig> and most nrp <dig> are present as the vegfr1-nrp <dig> complex.
most vegfr <dig> is unbound.
the majority of ligated vegfr <dig> is in the vegf165-vegfr2-nrp <dig> complex.
a ten-fold increase in nrp <dig> density in the tumor causes uncomplexed nrp <dig> to dominate.
concentration of free receptors and matrix components
relative probabilities of vegf binding to receptors and matrix components
the relative probabilities are expressed as concentration divided by kd.
the propensity to bind to vegfr <dig> and vegfr <dig> is much higher in healthy tissue than in tumor .
in particular, the higher the nrp <dig> density in tumor, the less the binding to vegfr <dig> and vegfr <dig>  however, the propensity to bind to nrp <dig> is much higher in tumor for a density of  <dig>  nrp <dig> per endothelial cell.
the probability to bind to the ecm is much higher than that of binding to the pbm or ebm because the volume of ecm is bigger.
flows of vegf in the body at steady state
discussion
the compartment model described here is a useful tool to simulate physiological and pathological situations involving vegf.
it provides informative quantitative biological details such as vegf distributions in tissue and in blood that are currently not accessible by direct experiments, as well as the sensitivity of vegf distribution to specific biological parameters.
even though, at this stage, the model considers two isoforms of the vegf-a family, the model can be readily extended to introduce additional factors, such as the isoform vegf <dig> that could also play an important role in angiogenesis.
for example, it was shown that the vegf <dig> represents the highest increase in protein levels amongst the vegf-a isoforms in the course of exercise training   <cit> .
receptors and co-receptors could also be added: soluble vegfr <dig>  or neuropilin- <dig> , for example, could also play significant roles in angiogenesis.
molecules that do not play a role in angiogenesis but compete for the receptor binding and therefore alter the vegf distribution could also be taken into account.
for instance, the virus-encoded vegf-e proteins bind with high affinity to vegfr <dig> but do not bind to vegfr <dig>  and vegf-b and plgf compete with vegf-a for binding to vegfr <dig>  <cit> .
the model takes into account vegf receptors on the abluminal surface of the endothelial cells.
there is some evidence that these receptors may be present on their luminal surfaces as well  <cit> ; however, to our knowledge, no quantification is available.
to take these receptors into account would require the introduction of receptors in the blood compartment and could potentially change the dynamics of the system.
since the vegf secretion rates were fine-tuned based on the steady-state concentration of vegf in the plasma found in the literature  <cit> , the addition of the vegf receptors on the luminal surface of the endothelial cells may therefore require re-adjustment of the secretion rates values.
for simplicity, healthy tissue is represented as approximately  <dig> kg of tissue whose parameters are representative of human vastus lateralis skeletal muscle, since this is one of the few tissues characterized.
it could be of interest to add the kidneys or liver and explicitly illustrate clearance from these organs.
more generally, clearance from healthy tissue could be added.
the addition of bones and poorly-vascularized organs could also play an important role since secretion of vegf apparently does not occur in these tissues.
the bone marrow could also be an important component to add to the model as it is the site where pro- and anti-angiogenic factors are segregated into separate platelet α-granules to be transported within the bloodstream and possibly released at the site of the tumor  <cit>  thus protecting vegf from binding to receptors on the luminal side of the endothelial cells or binding to anti-vegf agents present in the bloodstream or extravasating.
this crucial point leads to the introduction of the platelets as well.
not only are platelets a vector for vegf transport in the bloodstream, they have also been shown to be a location where the binding with vegf monoclonal antibodies, used in anti-angiogenic therapies, takes place  <cit> .
due to lack of experimental data in vivo, we consider kint to be the same in tumor and normal tissues.
experimental evaluation of the internalization rates of the receptors in vivo in healthy tissue and tumor would improve the accuracy of the model.
in our current model, vascular permeability is independent of vegf concentration.
however, vegf increases permeability in pathological angiogenesis where the blood vessels become leaky.
therefore, the model should define the permeability as a function of the vegf concentration.
another factor that could be significant is the transport of vegf via the lymphatics.
CONCLUSIONS
a compartmental model was formulated to represent both vegf transport throughout the entire human body and the distribution of free and bound vegf at the molecular level in tissues.
blood and tissue are interconnected by vascular permeability for vegf transport.
in the healthy subject, in the absence of clearance, free vegf in the plasma follows that in the available interstitial fluid volume in the healthy tissue.
when clearance is introduced, free vegf levels in tissue and blood are still approximately proportional to the vegf secretion rate.
however, free vegf concentration is lower in plasma than in tissue.
we also demonstrated that the internalization of the receptors decreases as the permeability increases, maintaining the free vegf level constant in the normal tissue while the concentration in the blood increases as well.
the model was used to determine variations of vegf levels during an exercise training experiment.
we simulated a 3-hour two-legged knee extension, studied by jensen et al.
<cit> .
after a 6-hr upregulation of vegf secretion, the vegf concentration returns to baseline after another  <dig> hours, assuming proportionality between mrna and vegf protein level.
the model predicted a time lag in vegf levels between tissue and blood during the transition periods.
this has implications for measurements as blood samples could exhibit higher vegf levels than tissue samples.
in pathological cases, a third compartment representing diseased tissue was added.
in our simulations, this diseased tissue was chosen to be a 4-cm diameter tumor located in the breast.
we investigated the possible causes of the several-fold increase in plasma vegf in cancer patients reported in the literature.
free vegf concentration in healthy tissue remained constant during all the performed simulations because the volume of that tissue was much larger than that of the tumor and the vascular permeability of healthy tissue is low.
the tumor compartment was generally the most sensitive to the change of vegf secretion rates in the tumor and the vascular permeability in the tumor.
these variations did not affect the plasma vegf levels significantly.
however, increasing vascular permeability in healthy tissue increased plasma vegf levels, suggesting that, apart from additive effects , a change in vascular permeability in the healthy tissue could explain the plasma vegf increase in cancer patients.
the model revealed that about half of the vegf distribution is in the form of a ternary complex where vegf <dig> is bound to vegfr <dig> and nrp <dig>  in the tumor, most of the other half of the vegf population was vegf <dig> bound to the ecm while it represented only a quarter in the normal tissue.
this led to a low amount of vegf <dig> bound to vegfr <dig> and vegfr <dig> in the tumor whereas it represents about 20% in the normal tissue.
most of vegfr <dig> is in its free state while most vegfr <dig> and nrp <dig> are present as the vegfr1-nrp <dig> complex.
the available binding site concentrations of the matrix components are independent of the nrp <dig> density in the tumor.
finally, the model showed that the higher the nrp <dig> density in tumor, the less the binding to vegfr <dig> and vegfr <dig>
this model has provided new insights on molecular distribution and biological details that cannot be easily assessed experimentally.
the adaptability of the compartmental model allows the simulation of human or animal subjects and vegf-dependent diseases as long as the biological properties of the studied tissue are available.
this model can be extended by including new molecular species, taking into account platelets and leukocytes as vegf carriers, or biophysical processes that could intervene in vegf transport or vegf binding.
in particular, the model presented here could serve as a basis for devising pro- and anti-angiogenic therapies and testing their potential effects on the vegf distribution in the human body.
glossary
the units are given for the healthy tissue and tumor compartments unless specified.
concentration of species
qv vegf secretion rate
kpvij permeation rate of vegf from compartment i to compartment j
sr insertion rate of the receptors r
kon association rate constant -1/s)
koff dissociation rate constant
kc rate constant of cell surface receptor coupling -1/s)
kdissoc dissociation rate constant of coupled receptors
kint internalization rate constant for cell surface receptors
cv clearance rate of vegf from the plasma
sij surface between compartment i and compartment j
ui volume of compartment i
authors' contributions
mos carried out the calculations, performed the simulations and wrote the first version of the manuscript.
fthw, fmg and asp participated in the design of the study and the model formulation, analysis of the results and writing and editing the manuscript.
all authors read and approved the final manuscript.

 angiogenesis is the process by which new blood vessels grow from pre-existing vascular networks. this is a dynamic physiological mechanism 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. vascular endothelial growth factor is a potent cytokine involved in angiogenesis. 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 and vegf-b are known to be involved in angiogenesis. vegf-a and three members have been shown to be implicated in both mechanisms. the vegf-a family contains the different splice isoforms vegf vegf121b, vegf vegf145b, vegf vegf vegf vegf165b, vegf vegf vegf vegf. the major splice variants are vegf vegf vegf and vegf because the role of vegf and vegf in vivo is not well understood and their expression is low compared to vegf and vegf, we consider the two main isoforms vegf and vegf in our model. these two ligands bind two receptor tyrosine kinases vegfr and vegfr. the vegf isoform binds to a non-signaling co-receptor called neuropilin- and to heparan sulfate proteoglycans which are constituents of the extracellular matrix and the cellular basement membranes. nrp also independently couples with vegfr to which vegf can bind to form a complex of all three. another complex can be formed when vegf bound to nrp binds in turn to vegfr. 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-vegfr signaling. 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 vegfr when vegfr populations are larger. 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 nrp enhances the binding of vegf to vegfr. 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. anatomically detailed tissue models have predicted heterogeneity in the activation of the endothelial vegf receptors and how this affects vegf gradients. 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. three-dimensional vegf distribution was also predicted in ischemic muscle in a model of peripheral arterial disease. a meta-analysis of vegf levels in healthy subjects and cancer patients revealed that plasma vegf levels are several-fold higher in cancer. 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 and in peripheral arterial disease. 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. 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, 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, parenchymal basement membranes and endothelial basement membranes. 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. 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 and skeletal muscle. 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. 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 and the rest of the body. 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 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, the basement membrane surrounding the parenchymal cells and the basement membrane surrounding the endothelial cells. the two isoforms of the vegf-a family considered in the model are the freely diffusible heparin-binding vegf and non-heparin-binding vegf. because vegf binds to the heparan sulfate proteoglycans 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 vegf in the interstitial space is uniform within each compartment and spatial variability due to gradient formation is neglected. the interactions between vegf and vegf and their receptors are illustrated in figure 1c and will be further described in the equation section. to summarize, vegf and vegf bind to both vegfr and vegfr but only vegf is believed to bind to the non-enzymatic co-receptor nrp 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, 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 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 by estimating that the damkohler number is significantly less than one. free vegf is transported by transcapillary permeability between the blood and tissue compartments and cleared from the blood. 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. 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 while the concentrations of the surface species are expressed per unit surface area of the corresponding cell. interstitial space we use the following notation: is the concentration of vegf binding sites of the ecm, pbm or ebm; represents the concentration of any unbound vegf isoform 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 v165+mecm⇌koff,v mecmkon,v mecmv165mecm v165+mebm⇌koff,v mebmkon,v mebmv165mebm v165+mpbm⇌koff,v mpbmkon,v mpbmv165mpbm and are governed by the equations ddt=−kon,v mebm+koff,v165mebm ddt=−kon,v mecm+koff,v165mecm ddt=−kon,v mpbm+koff,v165mpbm ddt=kon,v mebm−koff,v165mebm ddt=kon,v mecm−koff,v165mecm ddt=kon,v mpbm−koff,v165mpbm cell surface, and represent the densities of the unoccupied receptor tyrosine kinases, the unoccupied co-receptor, and their coupled complexes, respectively; and are the concentrations of vegf bound to the vegf receptors and nrp respectively; is the concentration of vegf-bound in the form of the ternary complex. sr represents the rate at which the receptors are inserted into the cell membrane and kint the internalization rate of the receptors. the vegf isoform interacts with the receptors through the following reactions v165+r1⇌koff,v165r1kon,v r1v165r v165+r2⇌koff,v165r2kon,v r2v165r v165+n1⇌koff,v165n1kon,v n1v165n v165n1+r2⇌koff,v165n r2kc,v165n r2r2v165n v165r2+n1⇌koff,v165r n1kc,v165r n1r2v165n while the vegf isoform binds to the receptors according to these reactions v121+r1⇌koff,v121r1kon,v r1v121r v121+r2⇌koff,v121r2kon,v r2v121r r1+n1⇌kdissoc,r1n1kc,r n1r1n v121+r1n1⇌koff,v121r1kon,v r1v121r1n v121r1+n1⇌kdissoc,r1n1kc,r n1v121r1n the preceding chemical reactions are governed by the following equations: ddt=sr1−kint,r1−kon,v r1+koff,v165r1−kon,v r1+koff,v121r1−kc,r n1+kdissoc,r1n ddt=sr2−kint,r2−kon,v r2+koff,v121r2−kon,v r2+koff,v165r2−kc,v165n r2+koff,v165n r ddt=sn1−kint,n1−kc,v121r n1+kdissoc,r1n1−kc,r n1+kdissoc,r1n1−kon,v n1−koff,v165n1−kc,v165r n1+koff,v165r n ddt=−kint,v121r1+kon,v r1−koff,v121r1−kc,r n1+kdissoc,r1n ddt=−kint,v121r2+kon,v r2−koff,v121r ddt=−kint,v165r1+kon,v r1−koff,v165r ddt=−kint,v165r2+kon,v r2−koff,v165r2−kc,v165r n1+koff,v165r n ddt=−kint,v165n1+kon,v n1−koff,v165n1−kc,v165n r2+koff,v165n r ddt=−kint,r1n1+kc,r n1−kdissoc,r1n1−kon,v r1+koff,v121r ddt=−kint,v165r2n1+kc,v165r n1−koff,v165r n1+kc,v165n r2−koff,v165n r ddt=−kint,v121r1n1+kc,v121r n1−kdissoc,v121n1+kon,v121r1n1−koff,v121r1n ligands in the tissue compartments the ligands vegf and vegf 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: ddt=qv121−kon,v r1+koff,v121r1−kon,v r1n1+koff,v121r1n1−kon,v r2+koff,v121r ddt=qv165−kon,v mebm+koff,v mebmn−kon,v mecm+koff,v165mecm−kon,v mpbm+koff,v165mpbm−kon,v r1+koff,v165r1−kon,v r2+koff,v165r2−kon,v n1+koff,v165n 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, 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. 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: εis=interstitial spacetotal tissue volume f=interstitial fluidinterstitial space Φ=available fluid volumeinterstitial fluid εif=interstitial fluidtotal tissue volume=f×εis kav=available fluid volumetotal tissue volume=Φ×εif 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 × 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 av, is related to the free vegf concentration in the total tissue volume by av=kav. 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 tissue, d for diseased tissue, b for blood and p for blood plasma. 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. the term kpvnb reads as "rate of permeability of vegf from the normal tissue compartment to the blood ". the free vegf molecules extravasate at a rate kpvbn. the interface between the normal tissue and the blood is the total surface of the microvessels denoted as snb. uav,ndav,ndt=−kpvnbsnbav,n+kpvbnsnbp note that, av × uav = × u in the tissue compartments and p × up = b × ub in the blood compartment. equation can be rewritten in terms of moles per unit volume of total tissue dndt=−kpvnbsnbunnkav,n+kpvbnsnbunubupb vegf binding to the receptors, the extracellular matrix and the basement membranes and ) are added leading to the full equations governing each free isoform in the normal tissue: dndt=qv121n−kon,v r1nnn+koff,v121r1nn−kon,v r1n1nnn+koff,v121r1n1nn−kon,v r2nnn+koff,v121r2nn−kpvnbsnbunnkav,n+kpvbnsnbunubupb dndt=qv165n−kon,v mebmnnn+koff,v mebmnn−kon,v mecmnnn+koff,v165mecmnn−kon,v mpbmnnn+koff,v165mpbmnn−kon,v r1nnn+koff,v165r1nn−kon,v r2nnn+koff,v165r2nn−kon,v n1nnn+koff,v165n1nn−kpvnbsnbunnkav,n+kpvbnsnbunubupb similarly, in the diseased tissue compartment, vegf and vegf 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 and extravasate at a rate kpvbd. 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: dddt=qv121d−kon,v r1ddd+koff,v121r1dd−kon,v r1n1ddd+koff,v121r1n1dd−kon,v r2ddd+koff,v121r2dd−kpvdbsdbuddkav,d+kpvbdsdbudubupb dddt=qv165d−kon,v mebmddd+koff,v mebmdd−kon,v mecmddd+koff,v165mecmdd−kon,v mpbmddd+koff,v165mpbmdd−kon,v r1ddd+koff,v165r1dd−kon,v r2ddd+koff,v165r2dd−kon,v n1ddd+koff,v165n1dd−kpvdbsdbuddkav,d+kpvbdsdbudubupb 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 vegfr 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. the model can be readily extended to include these factors. finally, in the blood compartment, vegf is cleared at a rate cv per unit volume of total blood via several organs 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: updpdt=−kpvbnsnbp+kpvnbsnbnkav,n−kpvbdsdbp+kpvdbsdbdkav,d this can be rewritten per unit volume of whole blood by: dbdt=−kpvbnsnbupb+kpvnbsnbubnkav,n−kpvbdsdbupb+kpvdbsdbubdkav,d therefore, the vegf isoforms are governed by the two following equations. dbdt=−cv121b−kpvbnsnbupb+kpvnbsnbubnkav,n−kpvbdsdbupb+kpvdbsdbubdkav,d dbdt=−cv165b−kpvbnsnbupb+kpvnbsnbubnkav,n−kpvbdsdbupb+kpvdbsdbubdkav,d whole-body parameters each tissue is uniquely characterized by its biochemical, biophysical and geometrical properties as input parameters into the model. similar to the consideration of total tissue volume compared to available interstitial fluid volume, we distinguish between the total tissue volume of the ecm, pbm and ebm, noted uecm, upbm, uebm respectively, and the available interstitial fluid volumes of the ecm, pbm and ebm, noted uav, ecm, uav, pbm, uav, ebm respectively. in terms of these variables various tissue characteristics can be calculated for the different tissue compartments, e.g., the total amount of vegf bound to the binding sites of the ecm is equal to uav, ecm × av, where av is the concentration of vegf bound to the ecm in moles per liter of available ecm fluid volume. similarly, the total amount of vegf sequestered by the ebm is equal to uav, ebm × av, where av is the concentration of vegf bound to the endothelial basement membrane in moles per liter of available ebm fluid volume. the conversion is given by uav, ebm av = u uav, ecm av = u in addition, the total abluminal surface area of endothelial cells is sec. the total amount of unligated vegfr is * × sec, where * is the number of receptors vegfr per unit of endothelial surface. * sec = u note that the kinetic rates can also be transformed, e.g., kav, onu = k onuav in the blood phase, we express all concentrations with respect to its total volume. if we designate red blood cell fractional volume or hematocrit as ht and neglect the small volume of white blood cells and platelets, then the concentration of species in blood can be expressed in terms of plasma concentrations as follows p = b × where p represents the concentration of free vegf in the plasma. numerical implementation the model, represented by ordinary differential equations and initial conditions, was implemented using visual fortran software on a pc. the equations were non-dimensionalized prior to numerical solution using appropriate characteristic values and once the solutions were obtained were transformed to the original dimensional variables. transient solutions were calculated using an adaptive step-size runge-kutta 5th-order accuracy integrative scheme. a relative error tolerance of 10- was used. the steady state was defined when the concentrations changed by less than 1%. model parameters the parameters are summarized in tables the derivation of these parameters is found in. in this table, /m and /. here, m = moles/liter available interstitial fluid volume. the conversion of receptor densities to tissue concentrations is based on the relationship mentioned in table and the surface area of an endothelial cell, μm vegf concentration is normalized based on the entire interstitial space, since it diffuses throughout: /m. vegf binding sites in the ecm and bms are based on those volumes: /m, /m, /m. for example, m = moles/liter endothelial basement membrane. conversions from pg/mg protein are based on mg protein/g of tissue and kda vegf, kda vegfr kda vegfr. the derivation of these parameters can be found in. in this table, /m and /. here, m = moles/liter interstitial space. the conversion of receptor densities to tissue concentrations is based on the relationship mentioned in table and the surface area of an endothelial cell, μm vegf concentration is normalized based on the entire interstitial space, since it diffuses throughout: /m. vegf binding sites in the ecm and bms are based on those volumes: /m, /m, /m. blood we consider a human subject of kg. to calculate the typical plasma and total blood volumes in a 70-kg human being, we used a study of ninety healthy subjects by gibson and evans. the volumes were plotted against the weight of the volunteer. linear regression was performed for males and females and both volumes were determined for kg. we therefore consider liters of total blood including liters of blood plasma, which constitute % of total blood volume. normal tissue the volume of the normal tissue is the volume corresponding to a 70-kg subject with a vastus lateralis skeletal muscle density of g/cm less cm of whole blood. the parameters characterizing the healthy tissue are summarized in tables and the properties and characteristics of the skeletal muscle have been described in. however, a few adjustments were made. the interstitial space is composed of % of collagen. this content is not accessible to the vegf molecules and thus does not account for the available interstitial fluid volume. interstitial fluid of muscle has been measured at 7% of tissue volume, thus the total interstitial space is %. the pore sizes of the basement membranes and the extracellular matrix in the skeletal muscle are estimated at nm and nm respectively. for a molecular weight of kda, this corresponds to a partition coefficient Φ of and a kav of for the pbm and ebm and a partition coefficient of Φ of and a kav of for the ecm. the available volumes for vegf transport in the extracellular matrix, parenchymal basement membrane and endothelial basement membrane are therefore,, and cm3/cm tissue, respectively. for the vegf receptors, we assume that, at any instant, the insertion of receptors equals the internalization. as a result, the number of total receptors is conserved. this condition also applies to the diseased tissue. regulation of vegf receptors represents an important poorly explored area; more complex receptor dynamics can be considered in the model as experimental information becomes available. diseased tissue as an example of diseased tissue, we consider a cm-diameter breast tumor. for this tissue, we use parameters reported in a previous one-compartment model. assuming the tumor to be a sphere, the volume of the diseased compartment is cm the parameters are summarized in tables and the properties and characteristics of the tumor tissue have been described in. the 5% collagen content increases the interstitial space from % to %. in breast tumor, the pore size of the endothelial basement membrane has been measured to be several hundred nanometers which corresponds to a partition coefficient of. it was shown that the basement membranes and the ecm are similar and difficult to distinguish in the context of mammary tumor. therefore, a partition coefficient of was also adopted for the ecm and the pbm. this corresponds to a kav of for the three regions. the available interstitial fluid volumes for vegf transport in the extracellular matrix, parenchymal and endothelial basement membranes are therefore,, and cm3/cm tissue, respectively. permeability between the normal tissue and the blood to determine the permeability between the normal tissue and the capillaries, we first determine the stokes-einstein radius for a vegf molecule. the molecular weight of vegf isoform is approximately kda. the stokes-einstein radius ae is calculated by the formula for globular molecules given in venturoli and rippe: ae = ×. for kda, we thus find a stokes-einstein radius of Å. with this effective molecular radius, it is then possible to determine the permeability-surface area product as × 10- ml/s × g. with a surface area of cm2/g, we deduce that the permeability between the normal tissue and the blood is × 10- cm/s. a similar value was found using the permeability-surface area product using schmittmann and rohr's study. because there is paucity of experimental data on vegf-dependence of macromolecular permeability, we consider a permeability range from × 10- to × 10- cm/s for sensitivity analyses; this range includes the 2– fold vegf-dependent increase in permeability reported in. when the permeability between the normal tissue and the blood is fixed, the value × 10- cm/s is selected. the vegf transport is assumed to be passive. permeability between the diseased tissue and the blood for the permeability between the tumor and the blood, we note that ovalbumin and the vegf homodimer have a similar molecular weight. in human tumor xenografts, the microvascular permeability for ovalbumin, a globular molecule, was measured to be × 10- cm/s. studies of tumor microvascular permeability for macromolecules in vivo also provide useful insights. however, most of these experiments use dextrans that are linear molecules contrary to vegf. this difference is crucial for permeability purposes. to resolve this issue, the relationship between the stokes-einstein radius and the permeability is usually preferred. for a stokes-einstein radius of about Å, the permeability is around × 10- cm/s. therefore, a range from × 10- to × 10- cm/s is chosen for the vascular permeability to vegf in tumor when the sensitivity analysis is performed since little data on vegf-dependence to macromolecule permeability is available. when the tumor/blood permeability is fixed at a specific value, × 10- cm/s is assumed. summary of experimental measurements of vegf concentration a meta-analysis was performed by kut et al.. in breast cancer, plasma vegf levels were to times higher that those in healthy subjects. in prostate cancer, plasma vegf levels were to times higher and in colorectal cancer about two times higher that those in healthy controls. for a vegf molecular weight of kda, the plasma levels in breast cancer patients correspond to – pm while they are in the range of – pm for healthy subjects. in our study, we therefore assume a vegf plasma level of around pm in healthy subjects and several-fold higher in breast cancer patients. the vegf secretion rates are then calculated to match the observed plasma level.  in all simulations, unless specified otherwise, the vascular permeability of healthy tissue is × 10- cm/s, the vegf plasma clearance min-, the vegf secretion rate molecule/cell/s in the normal tissue, the vegf isoform expression ratio vegf165:vegf 92%:8% and the density of vegfr vegfr and nrp is molecules/endothelial cell. these parameter values are based on currently available experimental data and they can be altered as additional data become available. in many cases, we systematically explore the sensitivity of the results to the parameter variation. vegf represents the total vegf, i.e., vegf and vegf this means that the free vegf concentration corresponds to the sum of the free vegf and the free vegf concentrations. similarly, the vegf secretion rate represents the sum of the secretion rates of the two vegf isoforms. healthy subject without vegf clearance, steady-state blood and tissue concentrations are the same a range from to molecule/cell/s was tested for vegf secretion rate. in the absence of plasma clearance, the free vegf concentration in both compartments is close to directly proportional to the vegf secretion rate in the normal tissue for the range we tested. in the absence of plasma clearance, the steady-state total vegf concentration in the blood plasma equals that in the available interstitial fluid of the healthy tissue. this is in agreement with v equation which, at steady state, becomes p=nkav,n=av,n. if pm of free vegf concentration is present, at steady state, in the normal tissue available interstitial fluid, a vegf concentration in the blood will also be pm. in the absence of plasma clearance, the steady-state free vegf concentrations in both compartments are independent of the permeability value as long as this value is non-zero. this is because the permeability affects the dynamics of the system by regulating how fast the compartments reach the steady state but not the steady-state free vegf concentrations themselves. looking at equations and, in the absence of the tumor compartment and of plasma clearance, the steady-state free vegf concentration does not depend on the permeability between the healthy tissue and the blood since kpvnb=kpvbn is assumed. the equation reads b=upubnkav,n. in other words, the relationship p=nkav,n=av,n holds independently of the permeability value as long as kpvnb=kpvbn holds true. plasma clearance reduces blood vegf concentration the vegf half-life in plasma has been shown to be ± min. to explore the effect of the half-life, the plasma clearance rate cv was chosen between and min- with the introduction of a clearance rate cv of min-, the steady-state concentration of free vegf in plasma becomes times lower than in available interstitial fluid at a vascular permeability of × 10- cm/s and times lower at a permeability of × 10- cm/s. the dependence on vegf secretion rate is approximately linear for the range studied. unlike free vegf concentration in normal tissue, free vegf concentration in the blood is strongly dependent on the vascular permeability. this is because, at steady state, the permeability coefficients do not cancel out in the equations and. b=nkav,n next, the clearance rate is fixed at min- the vegf secretion rate in the healthy tissue is chosen to be molecule/cell/s so that, at steady state, there is pm of free vegf in the blood for a vascular permeability of × 10- cm/s. we investigate the dependence on permeability in a range × 10- to × 10- cm/s. at steady state, the free vegf concentration in the available interstitial fluid in the normal tissue is around pm independent of the permeability. at very low permeability, very few vegf molecules secreted in the normal tissue enter the blood compartment. because the clearance is directly proportional to the vegf concentration in the blood plasma and ), a few molecules of vegf are cleared from the blood. therefore, the internalization is very high and accounts for most of the loss of vegf in the normal tissue. at very high permeability, a high concentration of free vegf enters the blood compartment and is cleared. the internalization is smaller and the plasma clearance is responsible for most of the loss of vegf. in the physiological range, the free vegf concentration in the blood plasma is roughly proportional to the permeability between the two compartments. in the healthy tissue, however, the change in the net flow from the healthy tissue to the blood due to permeability changes is balanced by slight changes in the binding to and the internalization of vegf by the receptors; thus, the healthy tissue vegf concentration vary slightly with the vascular permeability. next, we explore the sensitivity of the results to plasma clearance in a range between and min- the internalization of vegf by the receptors in healthy tissue regulates vegf concentration in the compartment. in the blood, however, the vegf molecules can accumulate if the clearance rate is decreased. transient effects of acute exercise given the clearance rate, vascular permeability and vegf165:vegf expression ratio, the model predicts that a vegf secretion rate of molecule/cell/s is necessary to achieve a free vegf plasma level of pm, as reported under physiological conditions. this leads to a free vegf concentration in the normal tissue of approximately pm. to study the vegf transient effects, we simulate a physical exercise experiment. jensen et al. have shown that a 3-hour two-legged knee extension upregulates vegf mrna by about fold for at least hours. vegf mrna levels return to baseline between and hours. we assume a direct correlation between mrna and vegf protein level. figure 3a shows the transient effect on free vegf concentration in available interstitial fluid and in blood plasma. the free vegf concentrations return to baseline within hours of cessation of the secretion upregulation. it is interesting to note that the fold increase in blood and tissue concentrations are the same. however, there is a time lag between the blood and normal tissue; in transition periods, vegf levels measured in plasma and tissue samples could be significantly different. higher permeability would decrease this time lag. this provides important insights on monitoring vegf protein levels in the case of exercise training. this simulation demonstrates that vegf tissue concentration can increase several-fold during prolonged exercise, thus providing a stimulus for exercise-induced angiogenesis. cancer patient in this simulation, a 4-cm diameter tumor located in the breast is represented by the diseased tissue compartment. the characteristics of this new compartment are presented in tables the compilation of the vegf levels in healthy subjects and cancer patients has revealed that cancer patients show, on average, a several-fold increase in their free vegf blood plasma levels. this part of the study investigates the possible origins of this increase. blood and normal tissue vegf levels are not significantly sensitive to tumor vegf secretion we performed a sensitivity analysis on the tumor vegf secretion rate. for the selected parameters we found that regardless of the vascular permeability in the tumor, free vegf concentration in available interstitial fluid in the normal tissue is insensitive to the vegf secretion rate in the tumor and to the vascular permeability of the tumor. this qualitative behavior is independent of the density of the nrp in the tumor. at a vascular permeability in the tumor of × 10- cm/s, the free vegf level in the available tumor interstitial fluid is approximately proportional to the secretion rate of vegf in the tumor while the vegf concentration in the blood plasma is rather insensitive to the change in the tumor vegf secretion. when increasing the permeability by two orders of magnitude, more vegf molecules secreted from the tumor enter the blood. this results in an increase of vegf level in the blood plasma and a decrease of vegf level in the tumor for a given vegf secretion rate in tumor. however, even at a high vegf secretion rate in the tumor, the plasma vegf concentration increases by less that 50%. thus, an increase in the vegf secretion rate alone cannot explain the several-fold increase reported for cancer patients, at least for the selected parameters of the model. for a given secretion rate in tumor, our calculations show that for the plasma vegf level to double, the tumor size would have to increase approximately to 25-cm diameter. we now set the tumor vegf secretion rate at or molecule/cell/s for and nrp in the tumor respectively, so that, at a vascular permeability in tumor of × 10- cm/s, the steady-state free vegf concentration in the available interstitial fluid in the tumor corresponds to that in the available interstitial fluid in the normal tissue. increasing vascular permeability of tumor decreases tumor vegf levels we then perform a sensitivity analysis on the vascular permeability in the tumor. we find that the free vegf concentration in the normal tissue remains constant independently of the tumor vascular permeability. the two main reasons are: the normal tissue volume overwhelms the effects of smaller volumes and the vascular permeability in the healthy tissue is small. however, the vascular permeability in tumor has a high impact on the free vegf concentration in the tumor as shown in figure 4b. for permeability higher than × 10- cm/s, the free vegf level drops drastically in the tumor. this behavior is the result of the amount of vegf molecules being cleared from the blood as they are transported from the tumor to the blood compartment. the plasma vegf level is not highly affected by the increase of vascular permeability in the tumor because the clearance is directly proportional to the concentration and because the volume of the healthy tissue is bigger than that of the tumor. in any case, the plasma vegf level does not exhibit a several-fold increase compared to the absence of tumor, even at high permeability. this means that the vascular permeability in tumor increase alone cannot explain the several-fold increase of plasma vegf level in cancer patients. over a range of clearance rates from and min-, there is little variation in the free vegf concentration and this also cannot explain the several-fold increase in plasma vegf reported in cancer patients. increasing the nrp density in the tumor by an order of magnitude only decreases the tumor vegf at high vascular permeability in tumor. increasing vascular permeability of healthy tissue increases blood and tumor vegf levels for high vascular permeability of healthy tissue, the free vegf concentration in plasma would be increased approximately four- to five-fold compared to baseline. therefore, the vascular permeability of healthy tissue may increase at sites distant from the tumor, leading in turn to a total increase of vegf concentration in the plasma or the additive effects of increased vegf secretion, vascular permeability in tumor, and tumor mass may explain the several-fold increase in plasma vegf in cancer patients as reported in the literature. distribution of free vs. receptor- and hspg-bound vegf in healthy and diseased tissues in the above sections, we presented results of computer simulations for free vegf in the tissue and blood compartments. to understand the total balance of vegf in the body, it is also important to assess the amounts of vegf bound to the receptors on the endothelial cells and to the hspg sites in the ecm and basement membranes. the vegf distribution is shown in figure 5a. for the parameters specified in the legend, 93% of total vegf in the healthy tissue is vegf the model revealed that up to half of the vegf distribution in the healthy tissue and the tumor with nrp per endothelial cell, is in the form of a complex where vegf is bound to vegfr and nrp simultaneously. in the tumor, to 68% of the vegf population, is vegf bound to the ecm while it represented only a quarter in the healthy tissue. finally, the vast majority of free vegf in the blood is vegf, regardless of the density of nrp in the tumor. the majority of cell surface receptors in healthy tissue and tumor receptors are not ligated the receptor occupancy in the tumor with density of nrp per endothelial cell is qualitatively similar to that in the healthy tissue. in these tissues, most vegfr and most nrp are present as the vegfr1-nrp complex. most vegfr is unbound. the majority of ligated vegfr is in the vegf165-vegfr2-nrp complex. a ten-fold increase in nrp density in the tumor causes uncomplexed nrp to dominate. concentration of free receptors and matrix components relative probabilities of vegf binding to receptors and matrix components the relative probabilities are expressed as concentration divided by kd. the propensity to bind to vegfr and vegfr is much higher in healthy tissue than in tumor. in particular, the higher the nrp density in tumor, the less the binding to vegfr and vegfr however, the propensity to bind to nrp is much higher in tumor for a density of nrp per endothelial cell. the probability to bind to the ecm is much higher than that of binding to the pbm or ebm because the volume of ecm is bigger. flows of vegf in the body at steady state discussion the compartment model described here is a useful tool to simulate physiological and pathological situations involving vegf. it provides informative quantitative biological details such as vegf distributions in tissue and in blood that are currently not accessible by direct experiments, as well as the sensitivity of vegf distribution to specific biological parameters. even though, at this stage, the model considers two isoforms of the vegf-a family, the model can be readily extended to introduce additional factors, such as the isoform vegf that could also play an important role in angiogenesis. for example, it was shown that the vegf represents the highest increase in protein levels amongst the vegf-a isoforms in the course of exercise training. receptors and co-receptors could also be added: soluble vegfr or neuropilin-, for example, could also play significant roles in angiogenesis. molecules that do not play a role in angiogenesis but compete for the receptor binding and therefore alter the vegf distribution could also be taken into account. for instance, the virus-encoded vegf-e proteins bind with high affinity to vegfr but do not bind to vegfr and vegf-b and plgf compete with vegf-a for binding to vegfr. the model takes into account vegf receptors on the abluminal surface of the endothelial cells. there is some evidence that these receptors may be present on their luminal surfaces as well; however, to our knowledge, no quantification is available. to take these receptors into account would require the introduction of receptors in the blood compartment and could potentially change the dynamics of the system. since the vegf secretion rates were fine-tuned based on the steady-state concentration of vegf in the plasma found in the literature, the addition of the vegf receptors on the luminal surface of the endothelial cells may therefore require re-adjustment of the secretion rates values. for simplicity, healthy tissue is represented as approximately kg of tissue whose parameters are representative of human vastus lateralis skeletal muscle, since this is one of the few tissues characterized. it could be of interest to add the kidneys or liver and explicitly illustrate clearance from these organs. more generally, clearance from healthy tissue could be added. the addition of bones and poorly-vascularized organs could also play an important role since secretion of vegf apparently does not occur in these tissues. the bone marrow could also be an important component to add to the model as it is the site where pro- and anti-angiogenic factors are segregated into separate platelet α-granules to be transported within the bloodstream and possibly released at the site of the tumor thus protecting vegf from binding to receptors on the luminal side of the endothelial cells or binding to anti-vegf agents present in the bloodstream or extravasating. this crucial point leads to the introduction of the platelets as well. not only are platelets a vector for vegf transport in the bloodstream, they have also been shown to be a location where the binding with vegf monoclonal antibodies, used in anti-angiogenic therapies, takes place. due to lack of experimental data in vivo, we consider kint to be the same in tumor and normal tissues. experimental evaluation of the internalization rates of the receptors in vivo in healthy tissue and tumor would improve the accuracy of the model. in our current model, vascular permeability is independent of vegf concentration. however, vegf increases permeability in pathological angiogenesis where the blood vessels become leaky. therefore, the model should define the permeability as a function of the vegf concentration. another factor that could be significant is the transport of vegf via the lymphatics.  a compartmental model was formulated to represent both vegf transport throughout the entire human body and the distribution of free and bound vegf at the molecular level in tissues. blood and tissue are interconnected by vascular permeability for vegf transport. in the healthy subject, in the absence of clearance, free vegf in the plasma follows that in the available interstitial fluid volume in the healthy tissue. when clearance is introduced, free vegf levels in tissue and blood are still approximately proportional to the vegf secretion rate. however, free vegf concentration is lower in plasma than in tissue. we also demonstrated that the internalization of the receptors decreases as the permeability increases, maintaining the free vegf level constant in the normal tissue while the concentration in the blood increases as well. the model was used to determine variations of vegf levels during an exercise training experiment. we simulated a 3-hour two-legged knee extension, studied by jensen et al.. after a 6-hr upregulation of vegf secretion, the vegf concentration returns to baseline after another hours, assuming proportionality between mrna and vegf protein level. the model predicted a time lag in vegf levels between tissue and blood during the transition periods. this has implications for measurements as blood samples could exhibit higher vegf levels than tissue samples. in pathological cases, a third compartment representing diseased tissue was added. in our simulations, this diseased tissue was chosen to be a 4-cm diameter tumor located in the breast. we investigated the possible causes of the several-fold increase in plasma vegf in cancer patients reported in the literature. free vegf concentration in healthy tissue remained constant during all the performed simulations because the volume of that tissue was much larger than that of the tumor and the vascular permeability of healthy tissue is low. the tumor compartment was generally the most sensitive to the change of vegf secretion rates in the tumor and the vascular permeability in the tumor. these variations did not affect the plasma vegf levels significantly. however, increasing vascular permeability in healthy tissue increased plasma vegf levels, suggesting that, apart from additive effects, a change in vascular permeability in the healthy tissue could explain the plasma vegf increase in cancer patients. the model revealed that about half of the vegf distribution is in the form of a ternary complex where vegf is bound to vegfr and nrp in the tumor, most of the other half of the vegf population was vegf bound to the ecm while it represented only a quarter in the normal tissue. this led to a low amount of vegf bound to vegfr and vegfr in the tumor whereas it represents about 20% in the normal tissue. most of vegfr is in its free state while most vegfr and nrp are present as the vegfr1-nrp complex. the available binding site concentrations of the matrix components are independent of the nrp density in the tumor. finally, the model showed that the higher the nrp density in tumor, the less the binding to vegfr and vegfr this model has provided new insights on molecular distribution and biological details that cannot be easily assessed experimentally. the adaptability of the compartmental model allows the simulation of human or animal subjects and vegf-dependent diseases as long as the biological properties of the studied tissue are available. this model can be extended by including new molecular species, taking into account platelets and leukocytes as vegf carriers, or biophysical processes that could intervene in vegf transport or vegf binding. in particular, the model presented here could serve as a basis for devising pro- and anti-angiogenic therapies and testing their potential effects on the vegf distribution in the human body. glossary the units are given for the healthy tissue and tumor compartments unless specified. concentration of species qv vegf secretion rate kpvij permeation rate of vegf from compartment i to compartment j sr insertion rate of the receptors r kon association rate constant -1/s) koff dissociation rate constant kc rate constant of cell surface receptor coupling -1/s) kdissoc dissociation rate constant of coupled receptors kint internalization rate constant for cell surface receptors cv clearance rate of vegf from the plasma sij surface between compartment i and compartment j ui volume of compartment i authors' contributions mos carried out the calculations, performed the simulations and wrote the first version of the manuscript. fthw, fmg and asp participated in the design of the study and the model formulation, analysis of the results and writing and editing the manuscript. all authors read and approved the final manuscript. 
we analyze the sensitivity of this distribution to the secretion rate, clearance rate and vascular permeability of vegf. vegf concentration in blood but not in tissue is dependent on the vascular permeability of healthy tissue. in a pathological context, 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. we find that, in a physiological context, vegf concentration varies approximately linearly with the vegf secretion rate. finally, the vegf distribution profile in healthy tissue reveals that about half of the vegf is complexed with the receptor tyrosine kinase vegfr and the co-receptor neuropilin- in diseased tissues, this binding can be reduced to 15% while vegf bound to the extracellular matrix and basement membranes increases. the results are of importance for physiological conditions and pathological conditions. angiogenesis is a process by which new capillaries are formed from pre-existing blood vessels in physiological or pathological contexts. 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 and in the blood. 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.
 angiogenesis is a process by which new capillaries are formed from pre-existing blood vessels in physiological or pathological contexts. this neovascular mechanism is mediated by the vascular endothelial growth factor 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 and in the blood. we analyze the sensitivity of this distribution to the secretion rate, clearance rate and vascular permeability of vegf.  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, 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 vegfr and the co-receptor neuropilin- in diseased tissues, this binding can be reduced to 15% while vegf bound to the extracellular matrix and basement membranes increases.  the results are of importance for physiological conditions and pathological conditions. 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. 