Numerical Flow Simulations of the Shear Stress Forces Arising in Filtration Slits during Glomerular Filtration in Rat Kidney

Abstract

Key Points:

1. Computational fluid dynamics were applied to estimate the shear stress challenge to the filtration barrier during glomerular filtration in rats.

2. Shear forces were especially relevant in pathologic situations where they contribute to the loss of viable podocytes.

Background: The flow dynamic forces during glomerular filtration challenging the fixation of podocytes to the glomerular basement membrane (GBM) are insufficiently understood.

Methods: Numerical flow simulations were used to estimate these forces in the rat kidney. Simulations were run with a three-dimensional (3D) model of the slit diaphragm as a zipper structure according to Rodewald and Karnovsky. The GBM was modeled as a porous medium.

Results: Filtrate flow exerted a mean wall shear stress of 39 Pa with a maximum of 152 Pa on the plasma membrane of foot processes and up to 250 Pa on internal surfaces of the slit diaphragm. The slit diaphragm accounted for 25% of the hydrodynamic resistance of the glomerular filtration barrier. Based on the results of the 3D model, we developed a two-dimensional (2D) model that allowed us to perform extensive parameter variations. Reducing the filtration slit width from 40 to 30 nm almost doubled wall shear stress. Furthermore, increasing filtrate flow velocity by 50% increased wall shear stress by 47%. When increasing the viscous resistance of the slit diaphragm, the pressure drop across the slit diaphragm increased to intolerably high values. A lower viscous resistance of the slit diaphragm than that of the GBM accounted for a gradual pressure decline along the filtration barrier. The subpodocyte space tempered these challenges in circumscribed areas of filtration surface but had only a marginal impact on overall forces.

Conclusions: The filtration barrier experiences high levels of shear and pressure stress accounting for the detachment of injured but viable podocytes from the GBM—a hallmark in many glomerular diseases.

Graphical abstract

Figure 1

Glomerular filtration barrier and subunit. (A) SEM showing the surface of a glomerular tuft of a healthy rat glomerulus. The major part of the surface is covered by the layer of interdigitated foot processes, podocyte cell bodies cover only a limited portion. Bar: 3 µm. (B) TEM of the glomerular filtration barrier in rats. The barrier consists of three components (from bottom to top): (i) the porous layer of the capillary endothelium and (ii) the GBM, as seen after glutaraldehyde fixation seemingly consisting of three layers. Specimens after deep freezing uncover a homogenous structure of the GBM (not shown) and (iii) the layer of interdigitating podocyte foot processes bridged by the slit diaphragms. Bar: 0.1 μm. (C) Schematic showing the filtrate flow through the filtration barrier. Endothelial pores and filtration slits are asymmetrically arranged. (D) Three-dimensional (3D) reproduction of the structural organization of the slit diaphragm adapted to Rodewald and Karnovsky. Units are represented in μm. (E) Subunit consisting of two foot process halves and one slit diaphragm; periodic conditions are applied in X-direction so that slit diaphragm is in the middle of the subunit; the slit diaphragm and the endothelial pore are asymmetrically placed. Components are indicated. Filtrate flows in positive Y-direction. BS, Bowman's space; GBM, glomerular basement membrane; SD, slit diaphragm; SEM, scanning electron micrograph; SPS, subpodocyte space; TEM, transmission electron micrograph.

Figure 2

Meshes. (A–C) The distribution of control volumes for the 3D representation of the filtration barrier. (D–F) The distribution of control volumes for the 2D representation of the filtration barrier.

Figure 3

Wall shear stress to the slit diaphragm. (A) Slit diaphragm structure based on the zipper model of Rodewald and Karnovsky. The shear stress is effective at the walls of the rectangular pores to the foot processes, with peak values around 150 Pa. (B) Filtrate flow through structural subdivisions within the rectangular pores of the slit diaphragm. Within these pores, the velocity is increased to around 200 µm/s. (C) Wall shear stress to the walls of the rectangular pores of the slit diaphragm zipper structure. The wall shear stress reaches values around 250 Pa. WSS, wall shear stress.

Figure 4

Pressure profiles (net filtration pressure) along the filtration barrier comparing 3D and 2D simulations. The 3D profile (black line) correlates satisfactorily with the 2D profile (red line) with VR_SD=2.2×10¹⁸ m⁻². In comparison, the situation of VR_SD=VR_GBM=0.7×10¹⁸ m⁻² is represented as the green line, which resulted in a higher mean velocity in slit diaphragm as targeted and an incorrect pressure distribution. VR, viscous resistance.

Figure 5

Contour plots of the velocity of filtrate flow along the filtration barrier: basic conditions (case 3 in Table 3). (A) Integrated flow profile. The flow velocity increases steeply toward the slit diaphragm, where it reaches an average value of 31.5 μm/s. (B) Flow profile within the filtration slit in increased resolution. The more uniform profile in the porous regions changes into a parabolic-like profile above the slit diaphragm. (C) Isolated flow profile in X-direction. Significant flows take place in the X-direction on exiting the endothelium and entering the filtration slit. However, these flows are one order of magnitude smaller when compared with the Y-component.

Figure 6

Shear stress to foot processes under basic conditions (case 3 in Table 3). Color-coded velocity distribution together with the increase and decrease of the shear stress of filtrate flow to the lateral wall of a foot process before, at, and after the slit diaphragm (red, purple, and green, respectively).

Figure 7

Pressure profiles (mean net filtration pressure) along the filtration barrier in relation to various proportions of the VR_SD keeping the mean velocity in slit diaphragm constant (cases 3 and 10–13 in Table 3). (A) 1D pressure distribution along the filtration barrier. Note that increases of the viscous resistance of the slit diaphragm above basic conditions (red line) lead to steep pressure drops across the slit diaphragm (blue and black lines). (B) 2D contour plots of the pressure distribution along the filtration barrier. Case 3 in Table 3, basic conditions with VR_SD=2.2×10¹⁸ m⁻² and VR_GBM=0.7×10¹⁸ m⁻². (C) 2D contour plots of the pressure distribution along the filtration barrier. Case 13 in Table 3, four-fold increased viscous resistance of the slit diaphragm with VR_SD=8.8×10¹⁸ m⁻² and reduced VR_GBM=0.13×10¹⁸ m⁻². Note the significant difference between situations (B) and (C).