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A model for interpreting the tracer labeling of interendothelial clefts

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Abstract

We extended the model describing the low molecular weight electron dense tracer wake in the interendothelial cleft and surrounding tissue to describe the time-dependent transport of intermediate size solutes of 1.0–3.5 nm radius by convection and diffusion in an interendothelial cleft containing a fiber matrix. This model provides a quantitative basis on which to reinterpret electron microscopic studies of the distribution of tracers such as horseradish peroxidase (HRP; molecular weight=40,000; Stokes radius =3.0 nm) along the interendothelial cell cleft from the lumen to the tissue for example, we show that, in contrast to our results with low molecular weight tracers, the wake of large molecular weight tracers on the abluminal side of the junctional strand is not likely to be detected, because the concentration of the tracer is predicted to be very low in most experiments. Thus the lack of a tracer such as HRP on the abluminal side of the junctional strand and in the tissue is not as strong evidence against the presence of a cleft pathway as suggested previously. The model does provide the basis for the design of experiments to locate both the principal molecular sieve and breaks in the junctional strand from the standing gradient on the luminal side of the junctional strand. An important experimental variable is the pressure in the vessel lumen which can be varied between 0 and 30 cm H2O to change the contributions of diffusive and convective transport to transcapillary exchange through the interendothelial cleft. This approach will also allow the testing of models for transcapillary pathways for large molecules by measuring the distribution of fluorescent tracers across the microvessel wall and in the tissue surrounding the microvessel using confocal microscopy.

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Abbreviations

a :

fiber radius

B :

half height of the cleft

b :

half height of the basic large orifice

b s :

half height of the continuous narrow slit in the junction strand

C (i) :

concentration in cleft regioni, i=1, 3

C t :

concentration in the tissue space

C L :

concentration in the lumen

C a :

tissue concentration at the vessel wall

D :

half distance between the centers of adjacent openings

D (i) :

effective diffusivity in regioni of the cleft,i=1, 3

D free :

free solute diffusivity

D iw :

diffusivity in the restricted region without fiber matrix

D i,eff :

effective diffusivity in the region with fiber matrix

D t :

diffusivity in the tissue

d :

half width of the basic large orifice

Ei:

exponential integral

H :

height of the tissue space

K :

complete elliptic integral of the first kind

L :

total cleft depth

L B :

characteristic length of the intermediate region B

L i :

depth of the cleft regioni, i=1, 2, 3

L f :

depth of the entrance fiber layer

N c :

number of clefts on half surface of the vessel

P (i) :

equivalent permeability in cleft regioni, i=1, 3

P i,eff :

effective permeability in the fiber filled region of the cleft

P iw :

permeability in the cleft region without fiber

p (i) :

pressure in cleft regioni, i=1, 3

p L :

pressure in the lumen

p A :

pressure in the tissue

Pe(i) :

Peclet number in cleft regioni, i=1, 3

Pe(t) :

Peclet number in the tissue

Q :

total flux per unit tissue thickness

Q c :

averaged flux from a single cleft

q :

source strength per unit cleft length

r :

coordinate in the intermediate region

r m :

solute radius

S f :

solid fraction of the fiber matrix

T :

characteristic time

t :

time

u 0,v 0 :

velocity components inx, y directions atz=0 plane

V 0 :

velocity atz=0 plane

V a :

averaged velocity in the junction orifice

x, y, z :

coordinate system

x t :

coordinate in the tissue space

α:

cleft exit coefficient

β:

D (3)/D (1)

γ:

cosh (πd/L)

λn :

eigenvalue=nπ/D, n=1,2,3...

μ:

viscosity

μ(i) :

equivalent viscosity in cleft regioni, i=1, 3

μeff :

effective viscosity in the fiber region

σ(i) :

equivalent reflection coefficient in regioni, i=1, 3

σ i,eff :

effective reflection coefficient in the fiber filled cleft region

σ iw :

effective reflection coefficient in the cleft region without fiber

Φ(i) :

velocity potential in cleft regioni, i=1, 3

Φ L :

velocity potential atx=L

Φ L 1 :

velocity potential atx=L 1

χ(i) :

equivalent retardation coefficient in cleft regioni, i=1, 3

χt:

effective retardation coefficient in the tissue

ψ(i) :

stream function in cleft regioni, i=1, 3

ψ D :

stream function aty=D

ω k :

kπ/ψ D ,k=1, 2, 3, ...

Δ:

gap distance between the fibers

≈:

variable with dimension

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Authors and Affiliations

  1. Department of Human Physiology, School of Medicine, University of California at Davis, Davis, CA

    Fitz-Roy E. Curry & Roger H. Adamson

  2. Department of Mechanical Engineering, The City College of the City University of New York, 10031, New York, NY, U.S.A.

    Bingmei Fu & Sheldon Weinbaum

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  1. Bingmei Fu
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  2. Fitz-Roy E. Curry
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  3. Roger H. Adamson
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  4. Sheldon Weinbaum
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Fu, B., Curry, FR.E., Adamson, R.H. et al. A model for interpreting the tracer labeling of interendothelial clefts. Ann Biomed Eng 25, 375–397 (1997). https://doi.org/10.1007/BF02648050

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  • DOI: https://doi.org/10.1007/BF02648050

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