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

Panadda Dechadilok

Department of Physics,
Faculty of Science,
Chulalongkorn University,
6th Floor, Mahamakut Building, Payathai Road,
Pathumwan, Bangkok 10330, Thailand
e-mail: panadda.D@chula.ac.th

Pisut Katavetin

Division of Nephrology,
Department of Medicine,
Faculty of Medicine,
Chulalongkorn University, Rama IV Road,
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

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.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Mattern, K. J. , 2008, “ Permeability Studies in Biomimetic Glycosaminoglycan-Hydrogel Membranes,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
Haraldsson, B. , Nystrom, J. , and Deen, W. M. , 2008, “ Properties of the Glomerular Barrier and Mechanisms of Proteinuria,” Physiol. Rev., 88(2), pp. 451–487. [CrossRef] [PubMed]
Deen, W. M. , Lazzara, M. J. , and Myers, B. D. , 2001, “ Structural Determinants of Glomerular Permeability,” Am. J. Physiol. Renal Physiol., 281(4), pp. 579–596.
Layton, A. T. , and Edwards, A. , 2014, Mathematical Modeling in Renal Physiology, Springer, Berlin. [CrossRef]
Maddox, D. A. , Deen, W. M. , and Brenner, B. M. , 1992, “ Glomerular Filtration,” Handbook of Physiology, Renal Physiology, E. E. Windhager, ed., Oxford University Press, New York, pp. 545–638. [CrossRef]
Deen, W. M. , Bridges, C. R. , Brenner, B. M. , and Myers, B. D. , 1985, “ Heteroporous Model of Glomerular Size Selectivity: Application to Normal and Nephrotic Humans,” Am. J. Physiol., 249(3), pp. F374–F389. [PubMed]
Rippe, B. , and Haraldsson, B. , 1994, “ Transport of Macromolecules Across Microvascular Walls: The Two-Pore Theory,” Physiol. Rev., 74(1), pp. 163–219. [PubMed]
Tencer, J. , Frick, I.-M. , Oquist, B. W. , Alm, P. , and Rippe, B. , 1998, “ Size-Selectivity of the Glomerular Barrier to High Molecular Weight Proteins: Upper Size Limitations of Shunt Pathways,” Kidney Int., 53(3), pp. 709–715. [CrossRef] [PubMed]
Öberg, C. M. , and Rippe, B. , 2014, “ A Distributed Two-Pore Model: Theoretical Implications and Practical Application to the Glomerular Sieving of Ficoll,” Am. J. Physiol. Renal Physiol., 306(8), pp. F844–F854. [CrossRef] [PubMed]
Curry, F. E. , and Michel, C. C. , 1980, “ A Fiber Matrix Model of Capillary Permeability,” Microvasc. Res., 20(1), pp. 96–99. [CrossRef] [PubMed]
Ohlson, M. , Sorensson, J. , Lindstrom, K. , Blom, A. M. , Fries, E. , and Haraldsson, B. , 2001, “ Effects of Filtration Rate on the Glomerular Barrier and Clearance of Four Differently Shaped Molecules,” Am. J. Physiol. Renal Physiol., 281(1), pp. F103–F113. [PubMed]
Öberg, C. M. , and Rippe, B. , 2013, “ Quantification of the Electrostatic Properties of the Glomerular Filtration Barrier Modeled as a Charged Fiber Matrix Separating Anionic From Neutral Ficoll,” Am. J. Physiol. Renal Physiol., 304(6), pp. F781–F787. [CrossRef] [PubMed]
Drumond, M. C. , and Deen, W. M. , 1994, “ Structural Determinants of Glomerular Hydraulic Permeability,” Am. J. Physiol., 266(1), pp. F1–F12. [PubMed]
Drumond, M. C. , and Deen, W. M. , 1995, “ Hindered Transport of Macromolecules Through a Single Row of Cylinders: Application to Glomerular Filtration,” ASME J. Biomech. Eng., 117(4), pp. 414–422. [CrossRef]
Edwards, A. , Daniels, B. S. , and Deen, W. M. , 1999, “ Ultrastructural Model for Size Selectivity in Glomerular Filtration,” Am. J. Physiol. Renal Physiol., 276(6), pp. F892–F902.
Rodewald, R. , and Karnovsky, M. J. , 1974, “ Porous Substructure of the Glomerular Slit Diaphragm in the Rat and Mouse,” J. Cell Biol., 60(2), pp. 423–433. [CrossRef] [PubMed]
Blouch, K. , Deen, W. M. , Fauvel, J.-P. , Bialek, J. , Derby, G. , and Myers, B. D. , 1997, “ Molecular Configuration and Glomerular Size Selectivity in Healthy and Nephrotic Humans,” Am. J. Physiol., 273(3), pp. F430–F437. [PubMed]
Gagliardini, E. , Conti, S. , Benigni, A. , Remuzzi, G. , and Remuzzi, A. , 2010, “ Imaging of the Porous Ultrastructure of the Glomerular Epithelial Filtration Slit,” J. Am. Soc. Nephrol., 21(12), pp. 2081–2089. [CrossRef] [PubMed]
Rice, W. L. , Hoek, A. N. V. , Paunescu, T. G. , Huynh, C. , Goetze, B. , Singh, B. , Scipioni, L. , Stern, L. A. , and Brown, D. , 2013, “ High Resolution Helium Ion Scanning Microscopy of the Rat Kidney,” PLoS One, 8(3), p. e57051. [CrossRef] [PubMed]
Deen, W. M. , Robertson, C. R. , and Brenner, B. M. , 1974, “ Concentration Polarization in an Ultrafiltration Capillary,” Biophys. J., 14(5), pp. 412–431. [CrossRef] [PubMed]
Brady, J. F. , 1994, “ Hindered Diffusion,” American Institute of Chemical Engineers Annual Meeting, San Francisco, CA, Nov. 13–18, p. 320.
Clague, D. S. , and Phillips, R. J. , 1996, “ Hindered Diffusion of Spherical Macromolecules Through Dilute Fibrous Media,” Phys. Fluids, 8(7), pp. 1720–1731. [CrossRef]
Phillips, R. J. , 2000, “ A Hydrodynamic Model for Hindered Diffusion of Proteins and Micelles in Hydrogels,” Biophys. J., 79(6), pp. 3350–3354. [CrossRef] [PubMed]
Oseen, C. W. , 1927, Neuere Methoden und Ergebnisse in der Hydrodynamik, Akademische Verlagsgesellschaft, Leipzig, Germany.
Dufresne, E. R. , Altman, D. , and Grier, D. G. , 2001, “ Brownian Dynamics of a Sphere Between Parallel Walls,” Europhys. Lett., 53(2), pp. 264–270. [CrossRef]
Johnson, E. M. , Berk, D. A. , Jain, R. K. , and Deen, W. M. , 1996, “ Hindered Diffusion in Agarose Gels: Test of Effective Medium Model,” Biophys. J., 70(2), pp. 1017–1026. [CrossRef] [PubMed]
Johansson, L. , and Löfroth, J. E. , 1993, “ Diffusion and Interaction in Gels and Solutions IV: Hard Sphere Brownian Dynamics Simulations,” J. Chem. Phys., 98(9), pp. 7471–7479. [CrossRef]
Tsai, D. S. , and Strieder, W. , 1986, “ Effective Conductivities of Random Fiber Beds,” Chem. Eng. Commun., 40(1–6), pp. 207–218. [CrossRef]
Kosar, T. F. , and Phillips, R. J. , 1995, “ Measurement of Protein Diffusion in Dextran Solutions by Holographic Interferometry,” AIChE J., 41(3), pp. 701–711. [CrossRef]
Kosto, K. B. , and Deen, W. M. , 2004, “ Diffusivities of Macromolecules in Composite Hydrogels,” AIChE J., 50(11), pp. 2648–2658. [CrossRef]
Bolton, G. R. , Deen, W. M. , and Daniels, B. S. , 1998, “ Assessment of the Charge Selectivity of Glomerular Basement Membrane Using Ficoll Sulfate,” Am. J. Physiol., 274(5), pp. F889–F896. [PubMed]
Edwards, A. , Deen, W. M. , and Daniels, B. S. , 1997, “ Hindered Transport of Macromolecules in Isolated Glomeruli I: Diffusion Across Intact and Cell-Free Capillaries,” Biophys. J., 72(1), pp. 204–213. [CrossRef] [PubMed]
Brinkman, H. C. , 1947, “ A Calculation of the Viscous Force Exerted by A Flowing Fluid on a Dense Swarm of Particles,” Appl. Sci. Res. A, 1, pp. 27–34. [CrossRef]
Sugihara-Seki, M. , 2004, “ Motion of a Sphere in a Cylindrical Tube Filled With a Brinkman Medium,” Fluid. Dyn. Res., 34(1), pp. 59–76. [CrossRef]
Edwards, A. , Daniels, B. S. , and Deen, W. M. , 1997, “ Hindered Transport of Macromolecules in Isolated Glomeruli—II: Convection and Pressure Effects in Basement Membrane,” Biophys. J., 72(1), pp. 214–222. [CrossRef] [PubMed]
Punyaratabandhu, N. , 2015, “ Simulation of Transport of Spherical Particles Through Hydrogel and Row of Parallel Fibers for Applications in Glomerular Filtration in Normal and Nephrotic Humans,” M.Sc. thesis, Chulalongkorn University, Bangkok, Thailand.
Boyd, R. F. , and Zydney, A. L. , 1997, “ Sieving Characteristics of Multilayer Ultrafiltration Membranes,” J. Membr. Sci., 131(1–3), pp. 155–165. [CrossRef]
Hora, K. , Ohno, S. , Oguchi, H. , Furukawa, T. , and Furuta, S. , 1990, “ Three-Dimensional Study of Glomerular Slit Diaphragm by Quick-Freezing and Deep-Etching Replica Method,” Eur. J. Cell Biol., 53(2), pp. 402–406. [PubMed]
Tryggvason, K. , and Wartiovaara, J. , 2005, “ How Does the Kidney Filter Plasma?,” Physiology, 20(2), pp. 96–101. [CrossRef] [PubMed]
Amsden, B. , 1998, “ Solute Diffusion in Hydrogels. An Examination of the Retardation Effect,” Polym. Gels Networks, 6(1), pp. 13–43. [CrossRef]
Drumond, M. C. , and Deen, W. M. , 1994, “ Stokes Flow Through a Row of Cylinders Between Parallel Walls: Model for the Glomerular Slit Diaphragm,” ASME J. Biomech. Eng., 116(2), pp. 184–189. [CrossRef]
Drumond, M. C. , Kristal, B. , Myers, B. D. , and Deen, W. M. , 1994, “ Structural Basis for Reduced Glomerular Filtration Capacity in Nephrotic Humans,” J. Clin. Invest., 94(3), pp. 1187–1195. [CrossRef] [PubMed]
Lazzara, M. J. , Blankschtein, D. , and Deen, W. M. , 2000, “ Effects of Multisolute Steric Interactions on Membrane Partition Coefficients,” J. Colloid Interface Sci., 226(1), pp. 112–122. [CrossRef] [PubMed]
Satchell, S. , 2013, “ The Role of the Glomerular Endothelium in Albumin Handling,” Nat. Rev. Nephrol., 9(12), pp. 717–725. [CrossRef] [PubMed]
Sugimoto, H. , Hamano, Y. , Charytan, D. , Cosgrove, D. , Kieran, M. , Sudhakar, A. , and Kalluri, R. , 2003, “ Neutralization of Circulating Vascular Endothelial Growth Factor (VEGF) by Anti-VEGF Antibodies and Soluble VEGF Receptor 1 (sFlt-1) Induces Proteinuria,” J. Biol. Chem., 278(15), pp. 12605–12608. [CrossRef] [PubMed]
Davis, B. , Cas, A. D. , Long, D. A. , White, K. E. , Hayward, A. , Ku, C.-H. , Woolf, A. S. , Bilous, R. , Viberti, G. , and Gnudi, L. , 2007, “ Podocyte-Specific Expression of Angiopoietin-2 Causes Proteinuria and Apoptosis of Glomerular Endothelia,” J. Am. Soc. Nephrol., 18(8), pp. 2320–2329. [CrossRef] [PubMed]
Wartiovaara, J. , Öfverstedt, L.-G. , Khoshnoodi, J. , Zhang, J. , Mäkelä, E. , Sandin, S. , Ruotsalainen, V. , Cheng, R. H. , Jalanko, H. , Skoglund, U. , and Tryggvason, K. , 2004, “ Nephrin Strands Contribute to a Porous Slit Diaphragm Scaffold as Revealed by Electron Tomography,” J. Clin. Invest., 114(10), pp. 1475–1483. [CrossRef] [PubMed]
Lazzara, M. J. , and Deen, W. M. , 2001, “ Effects of Plasma Proteins on Sieving of Tracer Macromolecules in Glomerular Basement Membrane,” Am. J. Physiol. Renal Physiol., 281(5), pp. F860–F868. [PubMed]
Neal, C. R. , Muston, P. R. , Njegovan, D. , Verrill, R. , Harper, S. J. , Deen, W. M. , and Bates, D. O. , 2007, “ Glomerular Filtration Into the Subpodocyte Space Is Highly Restricted Under Physiological Perfusion Conditions,” Am. J. Physiol. Renal Physiol., 293(6), pp. F1787–F1798. [CrossRef] [PubMed]
Pagtalunan, M. E. , Miller, P. L. , Jumping-Eagle, S. , Nelson, R. G. , Myers, B. D. , Rennke, H. G. , Coplon, N. S. , Sun, L. , and Meyer, T. W. , 1997, “ Podocyte Loss and Progressive Glomerular Injury in Type II Diabetes,” J. Clin. Invest., 99(2), pp. 342–348. [CrossRef] [PubMed]
Stillman, I. E. , and Karumanchi, S. A. , 2007, “ The Glomerular Injury of Preeclampsia,” J. Am. Soc. Nephrol., 18(8), pp. 2281–2284. [CrossRef] [PubMed]
van den Berg, J. G. , van den Bergh Weerman, M. A. , Assmann, K. J. M. , Weening, J. J. , and Florquin, S. , 2004, “ Podocyte Foot Process Effacement Is Not Correlated With the Level of Proteinuria in Human Glomerulopathies,” Kidney Int., 66(5), pp. 1901–1906. [CrossRef] [PubMed]
Zhang, A. , and Huang, S. , 2012, “ Progress in Pathogenesis of Proteinuria,” Int. J. Nephrol., 2012, p. 314251.
Nieuwdorp, M. , Mooij, H. L. , Kroon, J. , Atasever, B. , Spaan, J. A. E. , Ince, C. , Holleman, F. , Diamant, M. , Heine, R. J. , Hoekstra, J. B. L. , Kastelein, J. J. P. , Stroes, E. S. G. , and Vink, H. , 2006, “ Endothelial Glycocalyx Damage Coincides With Microalbuminuria in Type 1 Diabetes,” Diabetes, 55(4), pp. 1127–1132. [CrossRef] [PubMed]
Satoh, T. , Kato, H. , Kumagai, Y. , Yoneyama, M. , Sato, S. , Matsushita, K. , Tsujimura, T. , Fujita, T. , Akira, S. , and Takeuchi, O. , 2010, “ LGP2 Is a Positive Regulator of RIG-I– and MDA5-Mediated Antiviral Responses,” PNAS, 107(4), pp. 1512–1517. [CrossRef] [PubMed]
Jeansson, M. , and Haraldsson, B. , 2003, “ Glomerular Size and Charge Selectivity in the Mouse After Exposure to Glucosaminoglycan-Degrading Enzymes,” J. Am. Soc. Nephrol., 14(7), pp. 1756–1765. [CrossRef] [PubMed]
van den Born, J. , van den Heuvel, L. P. , Bakker, M. A. , Veerkamp, J. H. , Assmann, K. J. , and Berden, J. H. , 1992, “ A Monoclonal Antibody Against GBM Heparan-Sulfate Induces an Acute Selective Proteinuria in Rats,” Kidney Int., 41(1), pp. 115–123. [CrossRef] [PubMed]
Kanwar, Y. S. , Rosenzweig, L. J. , and Kerjasachki, D. I. , 1981, “ Glycosaminoglycans of the Glomerular Basement Membrane in Normal and Nephrotic States,” Renal Physiol., 4, pp. 121–130.
Wijinhoven, T. J. M. , Lensen, J. F. M. , Wismans, R. G. P. , Lamrani, M. , Momnens, L. A. H. , Wevers, R. A. , Rops, A. L. W. M. M. , van der Vlag, J. , Berden, J. H. M. , van den Heuvel, L. P. W. J. , and van Kuppevelt, T. H. , 2007, “ In Vivo Degradation of Heperan Sulfates in the Glomerular Basement Membrane Does Not Result in Proteinuria,” J. Am. Soc. Nephrol., 18(3), pp. 823–832. [CrossRef] [PubMed]
Rossi, M. , Morita, H. , Sormunen, R. , Airenne, S. , Kreivi, M. , Wang, L. , Fukai, N. , Olsen, B. R. , Tryggvason, K. , and Soininen, R. , 2003, “ Heparan Sulfate Chains of Perlecan are Indispensable in the Lens and Capsule But Not in Kidney,” EMBO J., 22(2), pp. 236–245. [CrossRef] [PubMed]
Sangani, A. S. , and Acrivos, A. , 1982, “ Slow Flow Past Periodic Arrays of Cylinders With Application to Heat Transfer,” Int. J. Multiphase Flow, 8(3), pp. 193–206. [CrossRef]


Grahic Jump Location
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.

Grahic Jump Location
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).

Grahic Jump Location
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).

Grahic Jump Location
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).

Grahic Jump Location
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].

Grahic Jump Location
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).



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In