Simulation of transport of spherical particles through hydrogel and row of parallel fibers for applications in glomerular filtration in normal and nephrotic humans

Abstract

The renal main function is to remove excess fluid and solutes as well as metabolic waste to keep the normal blood volume and composition. It is believed that the first step of renal urine formation in kidney is blood ultrafiltration through glomerular capillary wall consisting of three cellular layers: the endothelium, the glomerular basement membrane (GBM) and the epithelium. The objective of the work presented in this thesis is to investigate how particle sizes and particle interaction with the nanostructure of the three layers affect glomerular size-selectivity and fluid permeability through the glomerular barrier using a mathematical simulation. The ultrastructural model, developed by Edwards and Deen (1999) is employed but with the effect of the endothelial cell layer on solute restriction included in addition to the contributions of the epithelial cell layer and the GBM. In this work, the GBM was modeled as a hydrogel consisting of type IV collagen and glycosaminoglycan (GAG). The contribution of two fibers to hindered diffusion of solutes in GBM was investigated using existing theory regarding diffusion in fibrous media. It is found that only the GAG-solute interaction had an effect on drag and solute diffusivity due to its much higher number in the GBM. The reduced diffusivity in the GAG-filled endothelium fenestrae was calculated the same way. Convective hindrance factor in the GBM was determined by employing the theory of Brinkman medium using its Darcy permeability. The slit diaphragm in the epithelial cell layer was modeled as row of parallel fibers with non-uniform spacing between adjacent fibers following the log-normal distribution. Mean values of half of fiber spacing were set at 12.10 (from scanning electron microscopy observation), 22 nm (from helium ion microscopy) and 2 nm (from transmitting electron microscopy observation and electron tomography). Endothelial cell layer was modeled as a layer having fenestrae that were much larger than solute sizes and filled with GAG-riched glycocalyx. The assumption for this layer was that the drag on solute due to solute-fenestrae wall interaction can be negligible, and only GAG-solute interaction in fenestrae caused diffusivity reduction and change in convection rate of solutes. Diffusive and convective hindrance factor in endothelium were completed in the same way as those in the GBM, but its Darcy permeability was calculated using the Amsden expression by assuming that it was gel with random array of cylinders. The obtained total sieving coefficient calculated under the assumption above agreed with experimental data from in vivo urinalysis, and the calculated hydraulic permeability falls within the range estimated from human glomerular filtration rate. The calculated results showed that the endothelial cell layer and the GBM significantly contributed to solute and fluid restriction of glomerular barrier. The absence of GAGs that filled the endothelial fenestrae can cause more than an order of magnitude increase in total sieving coefficient, whereas the contribution of the epithelial slit to glomerular size-selectivity is likely to be smaller than previously believed.