Microvascular Permeability and Tumor Metastasis - Bioanalysis Advanced Materials Methods and Devices - page 54

described in terms of the additional drag on a spherical molecule moving within the
pore relative to movement in free solution, and the selectivity of the membrane in
terms of steric exclusion at the pore entrance [ 57 ].
In the fiber matrix theory, the principal hypothesis to describe the molecular filter of
the transvascular pathway is based on the presence of a glycocalyx layer at the
endothelial cell surface. Using the stochastic theory of Ogston et al. [ 61 ], Curry and
Michel [ 19 ] described the solute partition coefficient and the restricted solute diffusion
coefficient in terms of the fraction of thematrix volume occupied by fibers and the fiber
radius. The partition coefficient is defined as the space available to a solute of radius a
relative to the space available to water with a ¼ 0. In fact, their expressions for the

effective diffusivity of a solute only consider the steric exclusion of solutes by the fiber
array; they do not include the hydrodynamic interactions between the fibers and the
diffusing solute, which are important when the solute size is comparable to the gap
spacing between fibers. Using two alternative approaches, Philips et al. [ 63 ] calculated
the effects of hydrodynamic interactions on the hindered transport of solid spherical
macromolecules in ordered or disordered fibrous media. One approach was a rigorous
“Stokesian-dynamics” method or generalized Taylor dispersion theory [ 10 , 11 ], via
which local hydrodynamic coefficients can be calculated at any position in a fibrous
bed. But detailed information about the fiber configuration needs to be given. The other
approach was an effective medium theory based on Brinkman's equation. Comparing
predicted results with the experimental data for transport of several proteins in
hyaluronic acid solutions, they found that the use of Brinkman's equation was in
good agreement with more rigorous methods for a homogeneous fiber matrix, as
summarized in [ 57 ].
A simplified model of the endothelial surface glycocalyx (ESG) has been used by
Squire et al. [ 72 ], Sugihara-Seki [ 75 ], and Sugihara-Seki et al. [ 76 ], in which the core
proteins in the ESG were assumed to have a circular cylindrical shape and to be
aligned in parallel to form a hexagonal arrangement based on recent detailed struc-
tural analyses of the ESG. They analyzed the motion of solute and solvent to estimate
the filtration reflection coefficient as well as the diffusive permeability of the ESG.
Later, Zhang et al. [ 90 ] studied osmotic flow through the ESG using a method
developed by Anderson and Malone [ 4 ] for osmotic flow in porous membranes.
Instead of a rigorous treatment of the hexagonal geometry of the cylinders, they
adopted an approximation in which the geometry is replaced by an equivalent fluid
annulus around each cylinder and estimated the osmotic reflection coefficient of the
ESG. Further, Akinaga et al. [ 3 ] examined the charge effect on the osmotic flow for
membranes with circular cylindrical pores by extending the formulation of osmotic
flow developed by Anderson and Malone [ 4 ].

4.3.2 3D Models

1D models, used until 1984, were based on random section electron microscopy.
Bundgaard [ 13 ] was the first to attempt to reconstruct the 3D junction strand