The basis of this mouse model is the human model developed by Stefanini and co-workers that was used to explore the VEGF distributions in humans in health and disease 
; the model is significantly expanded as explained below. The mouse is divided into the blood compartment and tissue compartment (). As an approximation, the tissue compartment is represented by skeletal muscle that comprises the majority of its mass (43% of total body mass 
). To build the tissue compartment, the skeletal muscle is represented as cylindrical fibers (myocytes) aligned in parallel with cylindrical microvessels dispersed between the muscle fibers. The space between the muscle fibers and microvessels is designated as the interstitial space, which is itself composed of the basement membranes of the parenchymal cells/myocytes (PBM) and endothelial cells (EBM), in addition to the extracellular matrix (ECM). VEGF is secreted by the myocytes into the interstitial space where it can bind to VEGF receptors VEGFR-1, VEGFR-2, and NRP-1 on the abluminal surface of the endothelial cells, as well as to glycosaminoglycan chains (GAG) in the basement membranes and extracellular matrix. The binding of VEGF to NRP-1 on the surface of the myocytes is also included. VEGF can be transported to the blood via the lymphatics at a rate kL
and can be exchanged between the blood and interstitium via microvascular permeability at a rate kp
. VEGF in the blood can bind to receptors on the luminal surface of the endothelial cells and can be removed via plasma clearance at a rate cV
. Receptors and VEGF/receptor complexes can be internalized at a rate kint.
summarizes the molecular interactions of VEGF120
with their receptors and with GAG chains in the PBM, EBM, and ECM.
In our model, we include an anti-VEGF agent that can bind to and form a complex with VEGF in both the blood and tissue. The unbound anti-VEGF agent and the complex are also subject to intercompartmental transport via permeability and lymphatic drainage, and can also be cleared from the blood. The molecular interactions between the two VEGF isoforms and the anti-VEGF agent are illustrated in .
We incorporate pore theory in modeling the interstitial space to reflect the available volume for VEGF to diffuse. The VEGF molecules are free to diffuse in the available interstitial fluid volume, denoted KAV
, which is defined as the available fluid volume (UAV
) divided by the total tissue volume (U
). Based on the geometry of the pores in the basement membranes and extracellular matrix, the partition coefficient and available fluid volume can be calculated as follows:
The overall model is parameterized for a 25-gram mouse and is not currently strain specific. Model parameters are summarized in , , , , and .
Whole-body mouse parameters mouse.
Geometric parameters of mouse gastrocnemius muscle.
Kinetic parameters of VEGF and anti-VEGF.
The formulation of the two-compartment mouse model follows the scheme we previously applied to the human model 
. That is, we first begin by determining parameters of the whole mouse, such as the mass and blood volume. The whole-mouse is then divided into the blood and tissue compartments. Then the blood and tissue compartments are further characterized by parameters detailing the plasma volume and tissue geometry, respectively.
The parameters describing the whole mouse are presented in . The volume of blood for a mouse is 6–8 mL per 100 g body weight 
, yielding 1.75 mL of blood for a 25-gram mouse. Plasma volume used in our model is 0.85 mL, based on 3.42 mL of plasma per 100 g body weight 
. Hence, the plasma volume accounts for 49% of the blood volume. We consider the volume of the tissue to be the difference between the total volume of the mouse and the volume of the blood. The density of whole blood is 1.002 g/cm3 
; thus the mass of blood is calculated to be 1.75 g. This yields 23.25 g as the mass of the tissue, and using 1.06 g/cm3
as the density of skeletal muscle 
, we calculate the volume of the tissue to be 21.93 cm3
To model the tissue compartment, we used many properties of the mouse gastrocnemius muscle since this muscle is extensively studied and characterized (). However, not all necessary parameter values for the mouse gastrocnemius are available and we had to use available parameters from other muscles. The interstitial space in the mouse pectoral muscle is 14.4 µL/100 mg wet weight 
and thus the fraction of tissue that is interstitial space is calculated to be 0.15, using 1.06 g/mL as the density of skeletal muscle 
. The capillary density of the mouse gastrocnemius has been measured to range from 300 to 1,700 capillaries/mm2 
. This wide range is due to different factors such as mouse strain, age and exercise level.
Because we first constrained the fractional volume of interstitial space, other tissue parameters needed to be adjusted accordingly to yield reasonable values for fractional volumes of muscle fibers and blood. Particularly, the capillary density, capillary/fiber ratio, and fiber cross-sectional area determine the remaining parameters required to characterize the mouse model. We used a capillary density of 650 capillaries/mm2
, a capillary/fiber ratio of 1.95, and a fiber cross-sectional area of 2,500 µm2
, which are consistent with experimental measurements 
. This then yields a fiber density of 333 fibers/mm2
and a fiber fractional volume of 0.83. This leaves a blood fractional volume of 0.014, which requires the luminal capillary diameter to be 5.25 µm. Capillary diameters in the mouse have been measured to range from 3.6 µm in the calf muscle to 5.9 µm in the skin 
. Assuming the capillary wall thickness to be 0.5 µm, the external capillary diameter is then 6.25 µm and the capillary volume is calculated to be 2% of the total tissue volume. The capillary cross-sectional area is 30.67 µm2
and using a capillary perimeter to cross-sectional area correction factor of 1.23 
, the capillary perimeter is calculated as 21.77 µm. A capillary surface area correction factor of 1.1 
yields the capillary surface area of 155.68 cm2
tissue. The abluminal and luminal surface areas of one endothelial cell are each taken to be 1,000 µm2 
The fiber volume fraction is corrected to account for the capillary wall thickness and is calculated to be 82.74%. Using a fiber perimeter correction factor of 1.21 
, the fiber perimeter is calculated to be 214.10 µm and the surface area is then 713.68 cm2
tissue. The gastrocnemius is a mixed muscle; however, it has been reported that the outer zone comprises most of the mouse gastrocnemius, and that it is comprised of 94% type IIA fibers 
. Hence in the model, we base the fiber geometry on type IIA fibers. In particular, the myonuclear domain size for mouse type IIA fibers is 21,400 µm3
. Using this estimate and a fiber cross-sectional area of 2,500 µm2
, the calculated length of one muscle fiber myonuclear domain is 8.56 µm. The surface area of a myonuclear domain is then calculated to be 1.83×10−5
. Although the calculated muscle fiber length is shorter than that calculated for other animals and muscle types, similar measurements have been observed experimentally. For example, in the rat gastrocnemius, type IIA muscle fibers were found to have approximately 50 nuclei/mm (myonuclear fiber length of 20 µm) and cross-sectional areas of 2500 µm2 
. In the mouse, the extensor digitorum longus (EDL) and soleus have approximately 40 and 60 nuclei/mm (myonuclear fiber lengths of 17 and 25 µm), respectively 
. However, the EDL and soleus muscles are primarily composed of type I and type IIB fibers, respectively 
, and have cross-sectional areas of less than 2,000 µm2 
The interstitial space is assumed to be composed of the extracellular matrix (ECM), parenchymal basement membrane (PBM) and endothelial basement membrane (EBM). Although VEGF is able to diffuse in the interstitial space, part of this volume is inaccessible to VEGF. The thicknesses of the basement membranes are 154 nm 
, yielding volume fractions of 0.0020 and 0.0091 cm3
tissue for the EBM and PBM, respectively. The remaining interstitial space volume of 0.14 cm3
tissue is taken to be the extracellular matrix. These three elements of the interstitial space are assumed to have a solid fraction composed primarily of collagen, which is unavailable to VEGF, and a fluid fraction that is accessible to VEGF. The fraction of body weight that is composed of collagen in the mouse is estimated to be 2.5% 
. Using a density of 1.41 g/cm3
for collagen 
, the total volume of collagen in the interstitial space in a 25-gram mouse is 0.443 cm3
. The ratio of basement membrane collagen to total body collagen is 0.0083 
. Using this ratio, the total collagen in the ECM, EBM, and PBM is then calculated to be 0.4396 cm3
, 0.0007 cm3
, and 0.0030 cm3
, respectively. Hence, the non-collagen fractions of the ECM, EBM, and PBM volumes are 86%, 98%, and 98%, respectively.
We further consider pores in the ECM, EBM, and PBM, which may be inaccessible to freely diffusible molecules in the interstitial space. The EBM pore size for rat brain capillaries has been measured to be 7 nm 
and the ECM pore size is 66 nm in humans 
. With these pore sizes, the partition coefficients of the EBM and PBM are 0.35, and the partition coefficient for the ECM is 0.90 
. The available space for the ECM and basement membranes for VEGF to diffuse is then calculated as the product of the volume, fluid fraction, and partition coefficient. The total available space in the interstitial space (KAV)
is then calculated to be 0.113 cm3
Receptor densities and ECM binding site densities are listed in . VEGF receptors VEGFR-1, VEGFR-2, and co-receptor NRP-1 are assumed to be evenly distributed along both the luminal and abluminal surfaces of the microvessels 
. Furthermore, NRP-1 is distributed on the surface of myocytes. Based on quantitative flow cytometry measurements in the mouse gastrocnemius in vivo
, the concentrations of VEGFR-1, VEGFR-2 are 1,050 and 700 dimerized receptors per endothelial cell, respectively (Imoukhuede and Popel, unpublished observations). The number of NRP-1 dimers per endothelial cell is approximately 35,000 based on in vitro
. Additionally, in vitro
characterization of human quadriceps muscles estimated 35,000 neuropilin-1 dimers per myonuclear domain (Imoukhuede and Popel, unpublished observations). To our knowledge, there are currently no quantitative measurements in vivo
of NRP-1 on endothelial and myocyte cell surfaces.
It is known that VEGF164
binds to the glycosaminoglycan (GAG) chains of the heparan sulfate proteoglycans in the extracellular matrix 
; however to our knowledge, there are currently no direct experimental measurements of binding affinities of VEGF to the ECM. Based on the binding affinities of basic fibroblast growth factor to GAG chains in the ECM, which are assumed to be similar to VEGF binding affinities, the ECM, PBM, and EBM binding site densities are taken to be 0.75, 13, and 13 µM respectively 
Transport parameters for VEGF, anti-VEGF, and the VEGF/anti-VEGF complex are listed in . In mice, the half-life of VEGF in the circulation has been reported to be approximately 3 min 
. The VEGF clearance rate in plasma is then 0.23 min-1
, which is equal to ln(2)/half-life. VEGF is assumed to be a 45-kDa globular molecule and its microvascular permeability is taken to be 4.0×10−8
cm/s in accordance with previous models based on the molecular weight 
. In previous models, the microvascular permeabilities for molecules were determined based on their molecular weights and Stokes-Einstein radii 
. In particular the permeability for bevacizumab, a VEGF antibody, was 3×10−8
cm/s based on a molecular weight of 150 kDa 
. VEGF Trap, which has a molecular weight of 115 kDa 
, is taken to have a comparable size compared to bevacizumab. Thus the permeability of VEGF Trap is also chosen to be 3×10−8
cm/s. The VEGF secretion rate, lymphatic drainage rate, and clearance rates of anti-VEGF and the VEGF/anti-VEGF complex were determined using experimental data and are discussed in the “Free parameters” subsection of the Results.
The kinetic parameters for the binding and unbinding of VEGF to VEGFR-1, VEGFR-2, NRP-1, and GAG chains are listed in and are identical to those used in previous models 
. Kinetic parameters for the binding and unbinding of VEGF to the anti-VEGF agent were fitted parameters and are discussed in the “Free parameters” subsection of the Results. Because VEGF Trap is the anti-VEGF agent used in the model presented here, VEGF Trap and anti-VEGF are used interchangeably throughout this text.