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Research Papers

An Electrodiffusion Model for the Blood-Brain Barrier Permeability to Charged Molecules

[+] Author and Article Information
Guanglei Li

Department of Biomedical Engineering, The City College of the City University of New York, 160 Convent Avenue, New York, NY 10031

Bingmei M. Fu1

Department of Biomedical Engineering, The City College of the City University of New York, 160 Convent Avenue, New York, NY 10031fu@ccny.cuny.edu

1

Corresponding author.

J Biomech Eng 133(2), 021002 (Jan 24, 2011) (12 pages) doi:10.1115/1.4003309 History: Received September 14, 2009; Revised December 09, 2010; Posted December 21, 2010; Published January 24, 2011; Online January 24, 2011

The endothelial surface glycocalyx layer (SGL) and the basement membrane (BM) are two important components of the blood-brain barrier (BBB). They provide large resistance to solute transport across the BBB in addition to the tight junctions in the cleft between adjacent endothelial cells. Due to their glycosaminoglycan compositions, they carry negative charge under physiological conditions. To investigate the charge effect of the SGL and BM on the BBB permeability to charged solutes, we developed an electrodiffusion model for the transport of charged molecules across the BBB. In this model, constant charge densities were assumed in the SGL and in the BM. Both electrostatic and steric interaction and exclusion to charged molecules were considered within the SGL and the BM and at their interfaces with noncharged regions of the BBB. On the basis of permeability data for the positively charged ribonuclease (+4,radius=2.01nm) and negatively charged α-lactalbumin (10,radius=2.08nm) measured in intact rat mesenteric and pial microvessels, our model predicted that the charge density in both SGL and BM would be 30mEq/L, which is comparable to that in the SGL of mesenteric microvessels. Interestingly, our model also revealed that due to the largest concentration drop in the BM, there is a region with a higher concentration of negatively charged α-lactalbumin in the uncharged inter-endothelial cleft, although the concentration of α-lactalbumin is always lower than that of positively charged ribonuclease and that of a neutral solute in the charged SGL and BM.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic of the BBB (not in scale). (a) Three-dimensional view of the BBB. The capillary lumen is enclosed by endothelial cells with the surface glycocalyx and surrounded by the basement membrane. Pericytes attach to the abluminal membrane of the endothelium at irregular intervals, while the whole microvessel is ensheathed by astrocyte foot processes. Abbreviations: AP, astrocyte foot processes; BM, basement membrane; E, endothelium; L, lumen of microvessel; G, glycocalyx layer; N, nuclease of endothelial cell; P, pericyte; TJ, tight junction. (b) 2D cross-sectional view of the dash-lined region in (a). The curvature of the microvessel wall is neglected since the diameter of the microvessel is much larger than the width of the inter-endothelial cleft and the wall thickness. The paracellular transport pathway from the blood lumen to the brain tissue includes endothelial SGL, inter-endothelial cleft, BM, and the cleft between astrocyte foot processes. Due to symmetry, the dash-lined region is considered in our current model of the BBB and the periodic boundary condition is used at the side boundaries of the dash-lined region.

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Figure 3

Profiles of electrical potential (a) Ef in the SGL and (b) Eb in the BM for three different charge densities Cmf or Cmb=10 mEq/L (dotted line), 30 mEq/L (dashed line), 50 mEq/L (dash-dot-dash line), and 100 mEq/L (solid line).

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Figure 4

Dimensionless concentration distribution (a) Ci,f/CLumen in the SGL and (b) Ci,b/CLumen in the BM for a positively charged molecule, ribonuclease (charge number +4, dotted line), a neutral solute (solid line), and a negatively charged molecule, α-lactalbumin (charge number −10, dashed line). Here, Cmf=Cmb=30 mEq/L. Square and triangle in Fig. 4 indicate the concentrations of α-lactalbumin and ribonuclease, respectively, at the entrance to the BM.

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Figure 5

(a) Ratio of permeability of rat mesenteric microvessels to positively charged ribonuclease to that of negatively charged α-lactalbumin (Prib/Palpha) as a function of charge density in the SGL Cmf.The line is the model prediction, while the square symbol is in vivo experimental data (13). (b) Ratio of permeability of rat pial microvessels to positively charged ribonuclease to that of negatively charged α-lactalbumin (Prib/Palpha) as a function of charge density in the BM Cmb under two cases: (1) Cmf=30 mEq/L (dashed line) and (2) Cmf=Cm,b (solid line). The line is the model prediction, while the square symbol is in vivo experimental data (13). Lf=100 nm for all cases.

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Figure 7

Dimensionless permeabilities of (a) ribonuclease and (b) α-lactalbumin (P/PLb=12.5 nm,Cmf=Cmb=0) as a function of the BM thickness Lb under various charge density Cmf and Cmb(Cm=Cmb): 0 mEq/L (solid line), 25 mEq/L (dashed line), 50 mEq/L (short-dashed line), and 100 mEq/L (dash-dot-dash line).

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Figure 2

Enlarged view of dash-lined region of Fig. 1 (not in scale). At the luminal side, there is a SGL represented by a periodic array of cylindrical fibers (12,26-27,37-38). The radius of these fibers is rf, and the gap spacing between the fibers is Δ. The SGL has a thickness of Lf. Between two adjacent endothelial cells, there is an inter-endothelial cleft with a depth of L and a width of 2B. In the inter-endothelial cleft, there is an Ljun thick tight junction strand with a continuous slitlike opening of width 2Bs. The distance between the junction strand and luminal front of the cleft is L1. At the tissue side of the cleft, a BM separates the endothelium and the astrocyte foot processes ensheathing the brain blood vessels. The width of the basement membrane is 2Lb, and the length of the astrocyte foot processes is 2Wa. Between adjacent astrocyte foot processes, there is a cleft with length La and width 2Ba. These anatomical structures were drawn based on electron microscopy studies (17,22,28,30). The charge densities in the surface glycocalyx layer and basement membrane is Cmf and Cmb, respectively.

Grahic Jump Location
Figure 6

Dimensionless permeabilities of (a) ribonuclease and (b) α-lactalbumin (P/PLf=100 nm,Cmf=Cmb=0) as a function of the SGL thickness Lf under various charge density Cmf and Cmb(Cm=Cmb): 0 mEq/L (solid line), 25 mEq/L (dashed line), 50 mEq/L (short-dashed line), and 100 mEq/L (dash-dot-dash line).

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