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Research Papers

# Transport of Spherical Particles Through Fibrous Media and a Row of Parallel Cylinders: Applications to Glomerular Filtration

[+] Author and Article Information
Numpong Punyaratabandhu

Department of Physics,
Faculty of Science,
Chulalongkorn University,
6th Floor, Mahamakut Building, Payathai Road,
Pathumwan, Bangkok 10330, Thailand
e-mail: a.pong.bandhu@gmail.com

Pimkhwan Kongoup

Department of Physics,
Faculty of Science,
Chulalongkorn University,
6th Floor, Mahamakut Building, Payathai Road,
Pathumwan, Bangkok 10330, Thailand
e-mail: khwankongoup@gmail.com

Department of Physics,
Faculty of Science,
Chulalongkorn University,
6th Floor, Mahamakut Building, Payathai Road,
Pathumwan, Bangkok 10330, Thailand

Pisut Katavetin

Division of Nephrology,
Department of Medicine,
Faculty of Medicine,
Pathumwan, Bangkok 10330, Thailand
e-mail: pkatavetin@yahoo.com

Wannapong Triampo

Biophysics Group,
Department of Physics,
Faculty of Science,
Mahidol University,
Rama 6,
Bangkok 10400, Thailand
e-mail: wtriampo@gmail.com

1Corresponding author.

Manuscript received February 13, 2017; final manuscript received July 4, 2017; published online September 28, 2017. Assoc. Editor: Jeffrey Ruberti.

J Biomech Eng 139(12), 121005 (Sep 28, 2017) (13 pages) Paper No: BIO-17-1062; doi: 10.1115/1.4037550 History: Received February 13, 2017; Revised July 04, 2017

## Abstract

Viewed in renal physiology as a refined filtration device, the glomerulus filters large volumes of blood plasma while keeping proteins within blood circulation. Effects of macromolecule size and macromolecule hydrodynamic interaction with the nanostructure of the cellular layers of the glomerular capillary wall on the glomerular size selectivity are investigated through a mathematical simulation based on an ultrastructural model. The epithelial slit, a planar arrangement of fibers connecting the epithelial podocytes, is represented as a row of parallel cylinders with nonuniform spacing between adjacent fibers. The mean and standard deviation of gap half-width between its fibers are based on values recently reported from electron microscopy. The glomerular basement membrane (GBM) is represented as a fibrous medium containing fibers of two different sizes: the size of type IV collagens and that of glycosaminoglycans (GAGs). The endothelial cell layer is modeled as a layer full of fenestrae that are much larger than solute size and filled with GAGs. The calculated total sieving coefficient agrees well with the sieving coefficients of ficolls obtained from in vivo urinalysis in humans, whereas the computed glomerular hydraulic permeability also falls within the range estimated from human glomerular filtration rate (GFR). Our result indicates that the endothelial cell layer and GBM significantly contribute to solute and fluid restriction of the glomerular barrier, whereas, based on the structure of the epithelial slit obtained from electron microscopy, the contribution of the epithelial slit could be smaller than previously believed.

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## Figures

Fig. 1

Schematic drawing of an idealized structural unit of the glomerular barrier corresponding to one slit diaphragm. Based on the ultrastructural model introduced by Drumond and Deen [13,14], the glomerular capillary wall is viewed as an assembly of many repeating structural units along the length of the capillary. Filtrated fluid and macromolecules are transported from the capillary lumen through the endothelial fenestrae, across the GBM, through the slit diaphragm into the primary urine in Bowman's Space. W is the width of the structural unit, whereas w is the distance between two adjacent epithelial podocytes. LGBM is the GBM thickness, and δ is the distance between the GBM downstream surface and the slit diaphragm connecting the podocytes. Figure is not drawn to scale.

Fig. 2

Diffusive permeability of an isolated GBM (ΦGBM-FluidKd) as a function of radii of uncharged test solutes (rs). Diffusive permeability calculated by setting ϕcollagen = ϕGAG = 0.05 (1:1 mixture of large and small fibers) in the calculation of ΦGBM-Fluid and Kd (computed from solute–GAG and solute–collagen interactions using the approximation analogous to that proposed by Oseen (1927)) are compared to ΦGBM-FluidKd calculated by taking into account of the presence of collagens (large fibers) alone (ϕcollagen = 0.05 and ϕGAG = 0), and ΦGBM-FluidKd calculated by including only the presence of the small fibers the size of GAGs (ϕcollagen = 0 and ϕGAG = 0.05). Also presented is ΦGBM-FluidKd from an empirical expression of Deen et al. [3] (filled circles).

Fig. 3

Sieving coefficient across an epithelial slit (θep) as a function of radii of uncharged test solutes (rs). Presented results include θep obtained by assuming that 〈u〉 = 12.10 nm and 〈u2〉 = 1.39 nm based on images from an SEM observation of Gagliardini et al. [18] (dashed line), and θep calculated under an assumption that 〈u〉 = 22 nm and 〈u2〉 = 4 nm based on the observation using helium ion microscopy by Rice et al. [19] (solid line).

Fig. 4

Ratio between the test solute concentration at the upstream end of the filtering epithelial slit adjacent to the GBM surface and the test solute concentration in the capillary lumen (CGBM-slit/CP), obtained as the product of the sieving coefficient across GBM (θGBM) and the endothelial fenestrae (θen), as a function of volume fraction of GAGs in the endothelial fenestrae (ϕGAG,en). Results are plotted for rs = 2.6 nm and 3.6 nm. Dashed lines are CGBM-slit/CP obtained by assuming that 〈u〉 = 12.10 nm and 〈u2〉 = 1.39 nm [18], whereas solid lines are CGBM-slit/CP calculated by assuming that 〈u〉 = 22 nm and 〈u2〉 = 4 nm [19].

Fig. 5

Total sieving coefficient across the glomerular capillary wall (θ) as a function of radii of uncharged test solutes (rs). Presented results include (a) the sieving coefficient computed by assuming that 〈u〉 = 12.10 nm and 〈u2〉 = 1.39 nm [18], and (b) 〈u〉 = 22 nm and 〈u2〉 = 4 nm [19]. Results are presented for various values of GAG volume fraction in the endothelial fenestrae: ϕGAG,en = 0.06 (solid lines), ϕGAG,en = 0.07 (dot dashed lines), ϕGAG,en = 0.08 (dashed lines) and ϕGAG,en = 0.09 (dashed lines). Also presented are the sieving coefficients of ficoll solutes obtained from in vivo urinalysis in humans by Blouch et al. [17] (filled circles).

Fig. 6

Total sieving coefficient across the glomerular capillary wall (θ) calculated with the three cellular layers intact (solid line) as a function of radii of uncharged test solutes (rs) is compared to θ calculated by assuming that there is no slit diaphragm (dashed line) and θ obtained by assuming that ϕGAG,en = 0 (dot-dashed line). Also presented are the sieving coefficients of ficoll solutes obtained from in vivo urinalysis in humans by Blouch et al. [17] (filled circles).

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