Simulations of increased glomerular capillary wall strain in the 5/6‐nephrectomized rat

Abstract Objective Chronic glomerular hypertension is associated with glomerular injury and sclerosis; however, the mechanism by which increases in pressure damage glomerular podocytes remains unclear. We tested the hypothesis that increases in glomerular pressure may deleteriously affect podocyte structural integrity by increasing the strain of the glomerular capillary walls, and that glomerular capillary wall strain may play a significant role in the perpetuation of glomerular injury in disease states that are associated with glomerular hypertension. Methods We developed an anatomically accurate mathematical model of a compliant, filtering rat glomerulus to quantify the strain of the glomerular capillary walls in a remnant glomerulus of the 5/6‐nephrectomized rat model of chronic kidney disease. In terms of estimating the mechanical stresses and strains in the glomerular capillaries, this mathematical model is a substantial improvement over previous models which do not consider pressure‐induced alterations in glomerular capillary diameters in distributing plasma and erythrocytes throughout the network. Results Using previously reported data from experiments measuring the change of glomerular volume as a function of perfusion pressure, we estimated the Young's modulus of the glomerular capillary walls in both control and 5/6‐nephrectomized conditions. We found that in 5/6‐nephrectomized conditions, the Young's modulus increased to 8.6 MPa from 7.8 MPa in control conditions, but the compliance of the capillaries increased in 5/6‐nephrectomized conditions due to a 23.3% increase in the baseline glomerular capillary diameters. We found that glomerular capillary wall strain was increased approximately threefold in 5/6‐nephrectomized conditions over control, which may deleteriously affect both mesangial cells and podocytes. The magnitudes of strain in model simulations of 5/6‐nephrectomized conditions were consistent with magnitudes of strain that elicit podocyte hypertrophy and actin cytoskeleton reorganization in vitro. Conclusions Our findings indicate that glomerular capillary wall strain may deleteriously affect podocytes directly, as well as act in concert with other mechanical changes and environmental factors inherent to the in vivo setting to potentiate glomerular injury in severe renoprival conditions.


| INTRODUC TI ON
The renal glomerulus is a compliant structure that changes in volume depending on the pressure at which the blood enters the filtering capillaries. Multiple experimental strategies have been employed to estimate the compliance of the glomerulus in response to changes in perfusion pressure and investigate how this compliance is affected by disease. [1][2][3][4] While these studies provided a wealth of information in terms of characterizing the bulk response of the glomerular tuft to changes in perfusion pressure, the elasticity of the walls of individual glomerular capillaries remains to be rigorously quantified.
Furthermore, it is currently unknown how temporal changes in glomerular pressure translate to the magnitude of strain (stretch) of the individual glomerular capillary walls, and how these strains are distributed throughout the glomerular capillary network.
The magnitude of glomerular capillary wall strain is hypothesized to be of importance in numerous diseases in which glomerular capillary pressure is increased, such as diabetes, 5 hypertension 6 and in conditions involving the loss of functioning nephrons. 7 In vitro, cyclic mechanical strain deleteriously affects both podocytes [8][9][10][11][12][13][14] and mesangial cells, 12,15,16 and stretching acts synergistically with additional factors such as hyperglycemia 17  Estimating the magnitudes of mechanical strain throughout the glomerular capillary network necessitates quantification of the compliance or elasticity of the glomerular capillary walls such that changes in glomerular pressure may be translated to changes in individual glomerular capillary diameters. Rat glomerular compliance, in terms of the degree to which the glomerulus increases (decreases) in total volume as a result of increases (decreases) in perfusion pressure, has been estimated previously. 1,2 Past studies of glomerular compliance involved the removal of the renal corpuscle from the tissue sample, immersion of the corpuscle in an osmotic solution after removing Bowman's capsule, and perfusion of the same solution at varying pressures to measure changes in glomerular volume. 1 Experimentally, it is difficult to translate the results of these in vitro studies to in vivo conditions, in which Bowman's space pressure and a substantial colloid osmotic gradient hinder filtration across the capillary wall and the blood is, under physiological conditions, more viscous than the filtrate due to plasma protein concentration and hematocrit. Furthermore, this compliance relates to the change in total glomerular volume as a function of perfusion pressure and does not necessarily describe the compliance of individual glomerular capillaries.
Using a novel mathematical model of fluid flow and filtration in a compliant, anatomically accurate rat glomerular capillary network, 18 we estimated the elasticity and the strain of the individual glomerular capillaries in 5/6-nephrectomized (5/6-Nx) conditions based on results from studies that quantified glomerular volumetric compliance as a function of perfusion pressure. 1 We used our model to determine how these acute changes in the glomerular capillary diameters collectively affect glomerular volume, filtration and mechanical stress within the glomerular capillaries. We then compared our model predictions to the results of in vitro studies of podocyte strain to make inferences on the effect of 5/6-Nx-induced glomerular hypertension on podocyte structure and function.

| MATHEMATIC AL MODEL
We used a mathematical model of glomerular blood flow and filtra- Variables tracked on the length of the capillary included p ij (x), the axial pressure profile; Q ij (x), the blood flow; C ij (x), the plasma protein concentration; Π ij (x), the colloid osmotic pressure; and E ij , the erythrocyte volume assumed constant on the length of the capillary.
CSGFR ij , the capillary's filtration rate, was defined as: where R f ij is the resistance of the glomerular capillary wall to filtration, and p BS is Bowman's space pressure, which is assumed the same for all capillaries. To calculate the pressure profile on the length of the capillary, we solved the second-order linear differential equation which we derived in a previous work to model the change of pressure on the length of a permeable capillary 18 : The boundary conditions of equation 2 are p ij (-L ij /2) = p i and p ij (L ij /2) = p j , for p i and p j the pressure at nodes i and j, respectively; for R ij the resistance of the capillary. This equation was derived assuming that the filtration resistance R f ij is constant along the length of the capillary, such that the capillary may be sectioned into M sections of length L ij /M, each filtering with a filtration resistance of M f ij . By ensuring flux balance between each of these segments while taking into account fluid loss due to filtration at each segment, and taking M → ∞ we obtained equation 2, as described in our previous work. 18 Solving equation 2, we obtained the pressure profile on the length of the capillary: The relationship between p ij (x) and Q ij (x) was as follows:

Thus,
We modeled the filtration at each point on the length of the capillary as: for k the hydraulic conductivity of the capillary wall, defined as the permeability of the glomerular filtration barrier to water, and Π ij (x) the colloid osmotic pressure as a function of plasma protein concen- Assuming mass balance, Plasma proteins were assumed to not escape the glomerular capillary lumen 21 thus the colloid osmotic pressure only hinders filtration. To ensure consistency of the model equations 1 and 6, we required that: (3) TA B L E 1 Parameters used in the simulations of mechanical forces and filtration in the glomerulus in control and 5/6-Nx with associated references Abbreviations: C a , afferent plasma protein concentration; DBP, diastolic blood pressure in each case; k, glomerular capillary wall hydraulic conductivity; p BS , Bowman's Space pressure; R a , afferent resistance; R e , efferent resistance; SBP, systolic blood pressure in each case; t bm , glomerular basement membrane thickness; ϕ V , total relative glomerular volumetric compliance.
Additional parameters, including podocyte foot process height and width, are included in our previous work. 18 To maintain this equality, we note that R f ij was not fixed but could be iteratively updated so that this equality held, as shown in the model algorithm below. The permeability of the glomerular capillaries was determined by the hydraulic conductivity k, which was fixed and was assumed to be the same for all capillaries in the network (Table 1).

| Network model
We extrapolated equations 1-8 to the entire network of capillaries by imposing boundary conditions at the inlet of the afferent arteriole and outlet of the efferent arteriole, denoted by subscripts "a" and "e," respectively. Pressures were calculated at network nodes with boundary conditions p a and p e , set equal to blood pressure and peritubular capillary pressure, respectively. Afferent and efferent arterioles were assumed to have fixed resistances R a and R e that do not change, as the model is steady state. To calculate pressures at node i, we assumed conservation of blood flow at each network node. Thus for J the set of nodes j connected to node i, we solved the conservation of flow equation: For Q ij as described in equation 5. We evaluated Q ij at x = −L ij /2 in this equation because this position corresponded to the location at which blood is flowing into or out of node i as opposed to node j, at which x = +L ij /2. Equation 10 describes a linear system that we used to solve for all node pressures p i given pressure boundary conditions p a and p e . Using the node pressures as boundary conditions for each corresponding capillary, we obtained the filtration rate, pressure and flow profiles of each capillary using equations 1, 3 and 5, respectively.
We obtained the plasma protein concentration profiles on each capillary segment using the boundary condition C a to denote the plasma protein concentration at the inlet of the afferent arteriole, set equal to systemic plasma protein concentration (Table 1). We assumed conservation of plasma protein mass and perfect mixing at each network node, thus if we let K be the set of nodes k upstream of and connected to node i, and J be the set of nodes j downstream of and connected to node i, then for all nodes j in J we defined: Erythrocyte flow E ij was determined for each capillary segment using the boundary condition E a which denotes the flow of erythrocyte volume at the inlet of the afferent arteriole, calculated using systemic hematocrit. 22 A nonlinear function was used to distribute the erythrocyte flow at the network nodes. 19 This function was dependent upon daughter branch diameters and hematocrit in the feeding vessel. Capillary hematocrit (H t ) ij was defined as: Where the bar operator (-) indicates averaging Q ij (x) on the length of the vessel. Thus, given pressure boundary conditions p a and p e , plasma protein boundary condition C a and erythrocyte boundary condition E a , we calculated flow and filtration in each capillary segment in the glomerular capillary network.

| Apparent viscosity and mechanical stress equations
To determine the capillary resistance R ij used in the equations above, we took into account the non-Newtonian characteristics of blood, namely, its mutable viscosity. Assuming Poiseuille flow: where μ ij was the apparent viscosity of the blood as a function of hematocrit: In this formulation, pl ij had a linear relationship with the average plasma protein concentration on the length of the vessel 23 and λ was an empirically derived nonlinear function of the vessel diameter and hematocrit. 20 In addition to R f ij , μ ij was updated iteratively so that resistance of the capillaries was recalculated with each iteration. We estimated glomerular capillary shear stress, τ, by taking into account loss of flow due to filtration on the length of the vessel: Circumferential stress, also known as hoop stress, denoted by σ, was calculated using the Young-Laplace equation: where the bar operator again indicates averaging along the length of the capillary and t ij was the capillary wall thickness, taken to be a function of the endothelial cell layer thickness t e , the glomerular basement membrane thickness t bm , the minimum podocyte layer thickness t pod min , and the podocyte foot process width, w pod and height, h pod : The basement membrane thickness, t bm changed slightly in 5/6-Nx (Table 1). The remaining thicknesses and dimensions of the podocyte foot processes were the same as in our previous work. 18 A change in the circumferential and/or longitudinal stress from the baseline value was responsible for strain of the glomerular capillaries, as described below.

| Glomerular capillary compliance
To calculate strain of the glomerular capillary wall, we developed a constitutive relation assuming that the glomerular filtration barrier was an incompressible, neo-Hookean solid with the following strain energy density function 24 : for E the Young's modulus of the glomerular capillary wall, and ε r , ε θ , and ε x the radial, circumferential and longitudinal strains, respectively.
A neo-Hookean solid model was used based on the assumption that the nonlinear terms characteristic of soft tissue biomechanics were unnecessary when strains were assumed to be lower than 10%. 24 Nonzero ε r , ε θ , and ε x corresponded to pressure-induced changes in t ij , D ij , and L ij , respectively. From our strain energy density function, we defined the stresses σ r , σ θ , and σ x such that Using these relations, we estimated changes in t ij , D ij , and L ij due to the stresses imposed on the glomerular capillary walls by changes in pressure. To notate these changes, the superscript "0" is used to indicate the variable when the blood pressure is equal to MAP. Assuming incompressibility of the capillary wall, we determined the radial strain as a function of the circumferential and longitudinal strains: The circumferential and longitudinal stresses, σ θ and σ x , were estimated using equations of stress in a thin-walled cylinder. We assume a thin wall of the capillaries because the glomerular capillary diameters, measured in μm, are at least an order of magnitude larger than the wall thickness, measured in nm. For a given capillary segment ij, we let Using these equations we solved for t ij , D ij , and L ij as functions of t 0 ij , D 0 ij , and L 0 ij taking into account the change of pressure from p 0 ij to p ij : Thus t ij and L ij were functions of D ij , which was determined by solving for the real root of the cubic polynomial in equation 28.
Since E of the glomerular capillary walls was unknown, we

| Model algorithm
The algorithm employed in our model formulation was composed of two nested loops, as shown in Figure 1. The inner loop involved the iterative updating of the filtration resistance R f ij and the apparent blood viscosity μ ij until these variables converged, at which point the algorithm proceeded to the second loop which involved updating glomerular capillary diameters D ij , lengths L ij and thicknesses t ij taking into account glomerular capillary compliance. We briefly describe the main equations of the algorithm, with more details of the algorithm steps provided in our previous work. 18 The superscript "n" is used to indicate the iteration of the inner loop, and the superscript "m" is used to indicate the iteration of the outer loop. For the inner loop of the algorithm, equations 9 and 14 were used to calculate target values for R f ij and μ ij , respectively: These target values were then used to update R f ij and μ ij as follows: For α a smoothing parameter to control the magnitude by which R f ij and μ ij may change so as to avoid iterative oscillations in filtration and/or viscosity throughout the network. Convergence of the inner loop was satisfied when: In the outer loop of the algorithm, vessel diameters, lengths and wall thicknesses were updated according to equations 26-28:

| Estimating glomerular capillary wall E
The glomerular compliance data reported by Cortes et al. 1  We performed a sensitivity analysis to determine the relationship between E and ϕ V , and determined how changing the glomerular capillary wall E affects the compliance of the glomerular volume in response to alterations in PIP (Figure 2). We considered both control conditions and 5/6-Nx conditions in which hypertrophy increased glomerular capillary diameters by 23.3% at baseline pressure. 25 This increase in baseline diameter shifts the curve in Figure 2 to the right, which is why the glomerular capillaries are more compliant even with a higher E in 5/6-Nx over control (Table 2).

| 5/6-nephrectomy simulations
We quantified the magnitudes ( Figure 3) and location (Figure 4) of glomerular capillary wall strain throughout the glomerular capillary network in response to an acute shift in blood pressure from diastolic blood pressure (DBP) to systolic blood pressure (SBP).
Blood pressure was assumed to shift by SBP -DBP =23 mmHg and 55 mmHg in control and 5/6-Nx conditions, respectively. 26,27 MAP for control and 5/6-Nx conditions were 124 mmHg and 138 mmHg, respectively. 7 Additional parameters are available in Table 1. Other than the alteration of the baseline diameters as discussed above, we did not change the network topology or the number of glomerular capillaries to simulate 5/6-nephrectomized conditions, as to our knowledge there are no exhaustive tabulations of the glomerular capillary network structure in 5/6-nephrectomized conditions.
In the simulations whose results are depicted in Figure 3, we considered control and 5/6-Nx conditions (cases "Control, 23" and "5/6-Nx, 55") and examined the results in the case that the glomerulus undergoes structural changes associated with 5/6-Nx (changing E and baseline diameters, as in Table 1) but the difference between SBP and DBP remains at control levels (23 mmHg). We also considered the opposite case in which the glomerulus has control levels of elasticity and capillary diameters but the difference between SBP and DBP is increased to 5/6-Nx levels (55 mmHg). These cases are indicated by "5/6-Nx, 23" and "Control, 55," respectively. According to Figure 3, glomerular capillary wall strains (both circumferential and longitudinal) were greatly increased in 5/6-Nx, particularly in the vessels closest to the afferent arteriole (Figure 4), and this was primarily the result of a larger difference between SBP and DBP in the 5/6-Nx case. This is because when SBP-DBP =23 mmHg, the increased compliance of the glomerulus associated with 5/6-Nx conditions only meagerly increases strain over control conditions ( Figure 3). In general, longitudinal strain was roughly equal to half of the circumferential strain, thus we only show a comparison of the two groups based on their circumferential strain in Figure 4, having confirmed that the analogous figure with longitudinal strain is almost identical to Figure 4.
We compared control and 5/6-Nx groups with and without compliance to investigate the effect of glomerular capillary compliance on the mechanical stresses and hemodynamics in our simulations (Table 2). Error bars indicate the ranges of values that the variables take as blood pressure shifts from DBP to SBP in control and 5/6-Nx conditions. Compliance had a negligible effect on the mean and range of glomerular hemodynamic indices and mechanical stresses, thus results of noncompliant simulations were not included in Table 2.
To further probe the effects of pressure fluctuations on the localized function and mechanics of the glomerulus in control and 5/6-Nx conditions, we examined the changes in CSGFR ( Figure 5A) and shear stress ( Figure 5B). We found that in response to an increase in blood pressure, the capillaries that have the highest minimum CSGFR (denoted CSGFR − ) will experience a proportionally larger change in CSGFR. Since the blood pressure shift in 5/6-Nx was higher than that in control conditions, the acute changes in CSGFR in 5/6-Nx were larger. We investigated blood pressure shift-induced changes in shear stress throughout the glomerular capillary network as a function of length along the glomerulus ( Figure 5B). The relative change in blood flow, ΔQ, was highest closest to the afferent arteriole, whereas Δμ was highest closer to the efferent arteriole, both of which were results of increased filtration and thus concentration of plasma proteins and hematocrit. As expected, the combination of these opposite gradients resulted in a roughly uniform change in shear stress, τ along the length of the glomerulus. The change in shear stress in 5/6-Nx was larger due to the higher shift in blood pressure (Table 1) despite the increased diameters in the 5/6-Nx case.

| Sensitivity analysis
We conducted a sensitivity analysis to determine how glomerular capillary wall strain was influenced by parameters such as the wall  Abbreviations: E, Young's modulus of the glomerular capillary wall; P GC , glomerular capillary pressure; Q A , afferent plasma flow; SNGFR, single nephron glomerular filtration rate; V G , total glomerular capillary volume.

Control
Variables that change with changes in blood pressure are expressed as the mean value ±the range of values that that variable takes over the course of a change in blood pressure from DBP to SBP (SBP -DBP =23 mmHg for control and 55 mmHg for 5/6-Nx 26,27 ).
shaping the elasticity, however, because the wall thickness and E are uncoupled in our model, t 0 ij does not influence glomerular capillary wall elasticity on its own.

| DISCUSS ION
We developed a mathematical model of blood flow and filtration in a compliant glomerular capillary network to rigorously quantify the magnitudes of mechanical strain of the glomerular capillaries with and without loss of significant renal mass. Our model is novel in that it accounts for changes in glomerular capillary diameters in the distribution of erythrocytes and plasma throughout the capillary network, whereas previous anatomically accurate glomerular models have assumed fixed capillary diameters. 23,28 After parameterizing the model with data that related changes in perfusion pressure to glomerular volume, 1 we estimated the magnitude of the strain of the glomerular capillary walls in a remnant glomerulus of a 5/6-nephrectomized rat in response to physiological changes in blood pressure. Our simulations revealed that the median circumferential and longitudinal glomerular capillary wall strains were increased roughly threefold due to the increased glomerular capillary pressure and elasticity in 5/6-Nx, 7 and that these strains were highest in the vessels that branch off the afferent arteriole. The values of wall strain predicted by our model, particularly those pertaining to the vessels that branch off the afferent arteriole, are of a sufficient magnitude (approximately 4.9%) to affect podocyte structure and function in vitro. 14 Mechanical strain deleteriously affects podocytes through numerous mechanisms. 9,10 When subjected to 5% biaxial strain in vitro, podocytes hypertrophy 14 and reorganize their actin cytoskeleton F I G U R E 3 Capillary strain in control and 5/6-Nx conditions. Labels on the x-axis depict the condition in each simulation; "Control" and "5/6-Nx" refer to the choice of E and change in baseline capillary diameters (Table 1), while the number (23 or 55) indicates the difference between the SBP and DBP in the simulation, in mmHg. The actual, physiological difference between SBP and DBP in control and 5/6-Nx conditions is 23 and 55 mmHg, respectively, thus the first and fourth boxplots represent the model predictions in control and 5/6-Nx conditions, respectively. Both circumferential and longitudinal strains were considered, while radial strain was found to be insignificant (below 1%) and thus was not included. The range of values depicted with boxplots and data points correspond to the strain values for each capillary in the glomerulus based on a single simulation, thus no statistical comparison is included

5/6-Nx Control
to accommodate the enhanced mechanical load. 29,30 However, increased levels of strain such as 10% biaxial strain are necessary to cause podocyte detachment from the glomerular basement membrane in vitro. 8 Given this discrepancy, our simulations indicate that increases in strain may directly influence podocyte structure and function, and may also act in concert with other mechanical and biochemical changes in the glomerular environment to deleteriously affect podocyte stability.
Firstly, additional factors inherent to the in vivo setting, such as immune cell involvement, basement membrane stiffening and mesangial cell signaling, may play a role in the translation of glomerular capillary wall strain to podocyte apoptosis and foot process effacement. Since these factors were not present in the in vitro studies used to evaluate the podocyte reaction to strain, it is possible that 5% wall strain itself is sufficient to cause podocyte foot process effacement in vivo, when this magnitude of strain has not been found to be as deleterious to podocytes in vitro. Secondly, TGF-β1 has a similarly if not more deleterious effect on podocytes than even 20% biaxial strain in vitro, and cyclic strain increases podocyte expression of the TGF-β1 receptor in vitro. 8 Due to a pronounced increase in afferent blood flow, glomerular capillary wall shear stress increases substantially in 5/6-Nx (Table 2). In response to increased magnitudes of shear stress, endothelial cells increase secretion of TGF-β1. 31 As such, the combination of mechanical strain on podocytes and shear stress on endothelial cells may contribute to podocyte injury in 5/6-Nx.

F I G U R E 5
Localized changes in glomerular filtration and mechanics in control and 5/6-Nx. A, Changes in CSGFR in each capillary segment over the course of a change in blood pressure from DBP to SBP (SBP -DBP =23 mmHg for control and 55 mmHg for 5/6-Nx). Each data point corresponds to an individual capillary segment. Subscripts "+" and "−" indicate the value of CSGFR for blood pressure equal to SBP and DBP, respectively (Table 1) Finally, our results indicate that in 5/6-Nx, the average CSGFR more than doubles with each shift increase in blood pressure, and that the change in an individual capillary's CSGFR is proportional to its minimum CSGFR ( Figure 5A). The CSGFR is assumed to be proportional to the shear stress on the podocyte foot processes.
Shear stress on the podocyte foot processes has been implicated in hyperfiltration-mediated glomerular injury by destabilizing the podocyte actin cytoskeleton through the binding of prostaglandin

| PER S PEC TIVE
Our analyses indicate that there are substantial increases in strain of the glomerular capillary walls due to the increased pressure and remodeling of the remaining glomeruli that is associated with the significant loss of functional kidney mass. Our results indicate that this increased strain may deleteriously affect podocytes directly and may also act in concert with other factors inherent to the in vivo setting after significant loss of renal mass to disrupt podocyte structural integrity, thereby perpetuating glomerular injury and sclerosis.

CO N FLI C T O F I NTE R E S T
All authors declare no competing interests.

DATA AVA I L A B I L I T Y S TAT E M E N T
The data and mathematical models used to support the findings of this study are available upon request to the corresponding author.