Abstract
A mathematical model for steady flow through a discontinuity in the tight junction of an endothelial intercellular cleft is presented. Subject to plausible assumptions the problem of calculating the flow in the cleft, in either the presence or the absence of a fibre matrix, reduces to the solution of Laplace's equation in a two-dimensional domain. For an idealized geometry representing a discontinuity between two semi-infinite tight junction regions, a general analytic solution is found by means of conformal mappings. The model geometry, unlike those assumed in previous studies, allows the tight junction regions to be out of alignment with each other, and even to overlap, modelling flow through a tortuous, rather than a direct, pathway. Useful asymptotic approximations for the flow rate are derived when the discontinuity is either very small or very large. For small discontinuities, the predicted flow rate is much greater than a naïve estimate based on uniform parallel flow through the discontinuity. For the special case where the tight junction regions are aligned with each other, comparison of our results with those of an approximate treatment due to Tsayet al. [Chem. Engng Commun. 82, 67–102 (1989)] shows generally very close agreement.
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Abbreviations
- d :
-
distance of tight junction region from luminal boundary
- K :
-
hydraulic conductance per unit length, defined by (4)
- K :
-
complete elliptic integral of the first kind, as defined by Gradshteyn and Ryzhik (1980)
- L p :
-
hydraulic permeability of the endothelium, given by (9)
- ℒ :
-
total length of cleft per unit area of endothelium
- l :
-
half the separation of the ends of the tight junction regions in thex-direction
- P :
-
hydrostatic pressure
- ΔP :
-
luminal-interstitial pressure difference
- Q :
-
dimensionless total flow rate through discontinuity, defined by (8)
- R :
-
distance between neighbouring discontinuities
- r :
-
length-scale representative of the discontinuity
- U :
-
total flow rate through discontinuity
- u :
-
z-averaged flow velocity
- v :
-
representative value of flow velocity in the discontinuity
- w :
-
width of the intercellular cleft
- x, y, z :
-
Cartesian coordinates (see Fig. 3a)
- Δy :
-
luminal-interstitial distance
- α, β:
-
constants defined by (14) and (15)
- γ 0 :
-
constant defined by (23)
- δ:
-
thickness of the tight junction region
- ɛ:
-
square of the distance between the ends of the tight junction regions
- η:
-
viscosity of fluid
- θ:
-
angle made by line joining the ends of the tight junction regions
- κ:
-
Darcy constant for flow through a fibre matrix
- λ:
-
dimensionless half-size of discontinuity inx-direction
- μ:
-
dimensionless distance of tight junction region from luminal boundary
- ν:
-
number of discontinuities per unit length of cleft
- ρ:
-
density of fluid
- max:
-
μ max denotes the larger ofμ 1 and 1-μ 1
- T :
-
Q T denotes the flow rate in the model of Tsayet al. (1989)
- 1, 2:
-
d 1,2 andμ 1,2 define the positions of the two ends of the tight junction regions
- ±:
-
β ± are the two roots of (15)
Literature
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Phillips, C.G., Parker, K.H. & Wang, W. A model for flow through discontinuities in the tight junction of the endothelial intercellular cleft. Bltn Mathcal Biology 56, 723–741 (1994). https://doi.org/10.1007/BF02460718
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DOI: https://doi.org/10.1007/BF02460718