## Abstract

In the present study, an extended two-pore theory is presented where the porous pathways are continuously distributed according to small- and large-pore mean radii and SDs. Experimental glomerular sieving data for Ficoll were analyzed using the model. In addition, several theoretical findings are presented along with analytic solutions to many of the equations used in distributed pore modeling. The results of the data analysis revealed a small-pore population in the glomerular capillary wall with a mean radius of 36.6 Å having a wide arithmetic SD of ∼5 Å and a large-pore radius of 98.6 Å with an even wider SD of ∼44 Å. The small-pore radius obtained in the analysis was close to that of human serum albumin (35.5 Å). By reanalyzing the data and setting the distribution spread of the model constant, we discovered that a narrow distribution is compensated by an increased mean pore radius and a decreased pore area-to-diffusion length ratio. The wide distribution of pore sizes obtained in the present analysis, even when considering electrostatic hindrance due to the negatively charged barrier, is inconsistent with the high selectivity to proteins typically characterizing the glomerular filtration barrier. We therefore hypothesize that a large portion of the variance in the distribution of pore sizes obtained is due to the molecular “flexibility” of Ficoll, implying that the true variance of the pore system is lower than that obtained using flexible probes. This would also, in part, explain the commonly noted discrepancy between the pore area-to-diffusion length ratio and the filtration coefficient.

- two-pore model
- log-normal distributed model
- capillary permeability
- Ficoll
- standard deviation

the two-pore model of capillary permeability has been successfully applied to describe the transport of water and plasma solutes in a large number of different organs (8, 19, 26, 32, 33, 35, 36) and, recently, dialyzer membranes as well (7). The classic two-pore model assumes that the transcapillary transport of plasma solutes and water occurs over two distinct porous pathways, small pores and large pores, each having a fixed (discrete) pore radius. In contrast, distributed pore models assume that the pore radii are continuously distributed around a mean radius with a distribution spread [standard deviation (SD)] (1, 7, 9, 11, 20, 21). In most distributed models, only the small-pore system has been considered, whereas the large-pore pathway has been modeled as an unselective shunt pathway, theoretically allowing free passage of all solutes regardless of molecular size [to our knowledge the only exception is (1), where both the small-pore and large-pore systems are distributed]. However, when analyzing the glomerular transport of large globular proteins using the discrete two-pore model, Tencer et al. (36) found that there seems to be an upper size limit for molecular radii on the order of ∼110–115 Å. This implies that very large plasma proteins, such as IgM (molecular radius: ∼120 Å), will not pass the glomerular filtration barrier (GFB) at all under normal circumstances (36). Similarly, while studying the glomerular transport of globular proteins, Lund and colleagues (24) found a good fit for a (discrete) large-pore radius (*r*_{L}) of 110 Å.

Distributed pore models usually assume that the underlying pore size distribution can be characterized by a log-normal distribution, which has the advantage that it is defined only for positive pore radii, in contrast to the standard (Gaussian) normal distribution, which, if applied, would (per definition) also include negative pore radii. Often, it has been noted that biological mechanisms induce log-normal distributions, as when, for instance, the causative effects are multiplicative rather than additive (22). Indeed, the seminal work by Deen and colleagues (11) indicated that the log-normal distribution provides a better fit to the experimental data than other continuous distributions (i.e., normal distribution or gamma-distribution). In addition, the classic (discrete) two-pore model (regarding the goodness of fit) is superior to either a log-normal distributed model or an isoporous (single discrete pore radius) model in describing glomerular transport data, especially in nephrotic patients (11). Despite a superior goodness of fit (low χ^{2}-value) to the experimental data (7, 11), the discrete two-pore model typically shows a poor visual fit in the region between ∼50 and 65 Å (cf. Ref. 34), which has been attributed to the molecular flexibility of the Ficoll molecule in this region. Due to the poor fit, data in this “knee” region are sometimes excluded from regression analysis (34).

Most models for diffusion and filtration of solute molecules over a porous (or fibrous) barrier assume that the solute molecules behave like rigid spheres. Thus, compared with a rigid sphere, flexible polysaccharide molecules, such as dextran and, to a lesser degree, Ficoll, are hyperpermeable across the GFB, whereas less flexible, globular proteins, such as albumin, behave more similar to a rigid sphere (13, 14, 38). In terms of the discrete pore model, this means that the small-pore radius (*r*_{S}) will be overestimated when analyzing the sieving of polysaccharide probe molecules (7), while, to the best of our knowledge, a similar overestimation of the mean pore radius (*u*) in the distributed model has not been observed. Hence, it has been noted that the mean small-pore radius for Ficoll in distributed models is ∼35–39 Å, which is very close to that obtained using the discrete two-pore model to describe protein sieving data (∼37 Å) (24). At the same time, the width of the small-pore distribution [usually specified in terms of the geometric SD (commonly denoted “*s*”)] has been quite large, 1.15–1.19, corresponding to an arithmetic SD (σ_{S}) of ∼5–7 Å. The reason behind the observed hyperpermeability of the Ficoll molecule is not known. Conceivably, Ficoll may have several intermediate shapes and/or an increased molecular deformability, which may give the molecules an ability to change shape under pressure (13). Interestingly, a recent study by Georgalis et al. (16) indicated the coexistence of two closely spaced diffusive modes in Ficoll 70 solutions.

A widely used parameter in pore theory is *A*_{0}/Δ*x* (sometimes denoted “*fS*/*l*”), which represents the total pore area available for diffusive transport (*A*_{0}) divided by the diffusion length (Δ*x*). From this parameter, the total volume flow can be approximated (e.g., by using Poiseuille's law). Often, it has been noted that the volume flow approximated from the diffusive solute transport (*A*_{0}/Δ*x*) differs from that actually measured experimentally [e.g., the glomerular filtration rate (GFR) for glomerular capillaries]. This discrepancy is particularly evident in nonglomerular capillaries, as previously reviewed (31, 33), and in distributed pore models (20, 21, 34). In an early pioneering experiment, Lambert et al. (21) studied the glomerular sieving of radiolabeled polyvinylpyrrolidone and calculated values for *A*_{0}/Δ*x* that were up to one order of magnitude larger for the log-normal distributed model than that of the discrete model. In addition, the mean pore radius was lower for the distributed model (21). Jeansson and Haraldsson (20) studied glomerular size selectivity in the mouse using Ficoll and found a similar discrepancy between the two-pore model and the log-normal distributed + shunt model. In line with these results, our own experiments have yielded comparable results for the log-normal distributed + shunt model (5, 7, 34). The inflation of the *A*_{0}/Δ*x* parameter leads to an inconsistency among *A*_{0}/Δ*x* [as calculated from the filtration coefficient (*L*_{p}*S*)] and the value for *A*_{0}/Δ*x* as determined by the model regression on sieving data. Indeed, similar inconsistencies have been a classic controversy in the field of capillary physiology over several decades [cf. “Pappenheimer's pore puzzle” (29)].

It is reasonable to assume that there should be at least some variation in the pore radii and also an upper size limitation in the large-pore system. Thus, the present study aimed to combine the two “classic” models for glomerular transport (two-pore and log-normal + shunt) by the application and theoretical analysis of a distributed two-pore model. The model was used to analyze sieving data from Axelsson et al. (6) to determine the mean pore radius and the distribution spread of both the small-pore and large-pore systems in the GFB. The discrete two-pore model typically yields lower values for *L*_{p}*S* and *A*_{0}/Δ*x* and usually a mean small-pore radius that is ∼7–8 Å higher than that of the distributed + shunt model. Analysis of data with the discrete two-pore model is equivalent to using the distributed two-pore model with the spread parameter set to unity. To discern the effects of using a constant spread, the data were analyzed again setting the spread of the model constant. This revealed if there were any other differences between distributed and discrete (or fixed spread) pore modeling apart from the previously noted differences in *A*_{0}/Δ*x* and small-pore radius.

## METHODS

Experimental sieving data from Axelsson et al. (controls) (6) for the rat glomerulus were analyzed using a novel extended two-pore model where each pore population is distributed around a central tendency, *u*_{S} and *u*_{L}, respectively, each with a small-pore and large-pore SD (*s*_{S} and *s*_{L}, respectively). In summary, the mathematical construction is identical to that of the discrete two-pore model with the exception that the hindrance factors are calculated for a log-normal distribution of pore sizes instead of single discrete pore sizes.

#### Theoretical background.

The steady-state solute transport (*J*_{s}, in mol/min) across a semipermeable barrier can be described by the convection-diffusion equation, as follows: (1)where *D* is the diffusion coefficient (in cm^{2}/min), *A* is the effective area (in cm^{2}), *J*_{v} is the volume flux (GFR), σ is the reflection coefficient, and c(*x*) is the concentration (in mol/ml) as a function of the distance *x* (in cm) from the plasma side of the barrier. Thus, *J*_{s} is the sum of the diffusive transport [according to Fick's first law, *J*_{diffusion} = −*DA* × dc/d*x* (in mol/min)] and the convective transport *J*_{convection} (in mol/min) [the effective volume flux *J*_{v} × (1 − σ) (in ml/min) times the concentration (in mol/ml)]. All coefficients are assumed to be constant. *Equation 1* is a separable first-order ordinary differential equation that can be integrated over the barrier [ad modum Patlak et al. (30); see the appendix), yielding (2)where C_{p} is the plasma concentration (in mol/ml) and C_{i} is the downstream (filtrate) concentration (in mol/ml). The Péclet number (*Pe*), the “convection-to-diffusion ratio” for a particular barrier-flux-solute combination, is defined as follows: (3)where the permeability-surface area coefficient (*PS*; in ml/min) is given by the following equation: (4)The latter equality is particularly useful since hydrodynamic estimates give an expression for *A*/*A*_{0} rather than *A*/Δ*x* (33). The diffusion restriction coefficient (*A*/*A*_{0}) represents the restriction in the diffusive transport due to the hindrance of the barrier. As an example, if *A*/*A*_{0} = 1/3, then the clearance of a solute from *x* = 0 to *x* = Δ*x* is only one-third of that which would occur if there was no diffusive transport restriction. Assuming an ideal spherical uncharged solute, *D* can be estimated using the following Stokes-Einstein equation: (5)where *k* is the Boltzmann constant, T is the temperature (in K), η the viscosity of the permeate (the permeate is assumed to have a viscosity close to that of water, i.e., ∼0.7 mPa·s). and *a*_{e} is the Stokes-Einstein radius of the solute (in Å). In the present article, the term “radius” is used synonymously with Stokes-Einstein radius unless otherwise specified. During ultrafiltration, *J*_{s} = *J*_{v} × C_{i}, so the sieving coefficient (θ; equal to C_{i}/C_{p}) can be derived from *Eq. 2*, leading to the following practical expression: (6)The solute clearance (in ml/min) is simply *J*_{v} × θ. Thus, it is possible to relate the ultrafiltrate concentration to the plasma concentration using the following equation: (7)It is important to note that the solution above (*Eq. 2*) is only valid if *J*_{v} ≠ 0. During zero flux conditions, Fick's first law can be applied directly, i.e., *J*_{s} = −*PS*(C_{i} − C_{p}).

#### Log-normal distribution.

The log-normal distribution has been widely used in the characterization of the pore size distribution of both synthetic and biological barriers. There has, however, been considerable differences regarding the proper functional form of the log-normal probability density function as well as concerning the interpretation of the distribution parameters (40). Arguably, the most widely adopted distribution parameters are *u* and *s*. The functional form for the probability density function for the log-normal distribution is then given as follows: (8)The corresponding arithmetic mean (μ) and standard deviation (σ) (in Å) can be calculated by μ = *u* × *s*^{1/2 ln s} and σ = *u* × (*s*^{2 ln s} − *s*^{ln s})^{1/2}, respectively (40). In this article, both arithmetic (μ and σ) and geometric (*u* and *s*) distribution parameters are presented in the data analysis. The arithmetic mean and SD must not be confused with the corresponding moments for the standard normal distribution. The log-normal distribution is an asymmetric distribution with a positive skew.

#### Solute flux.

The net θ value for the filtration barrier was calculated using the following equation: (9)where *f*_{L} = 1 − *f*_{S} is the fraction of fluid flow passing the barrier via the large-pore pathway (i.e., *f*_{L} = *J*_{vL}/GFR), σ_{S} and σ_{L} are the reflection coefficients for the small-pore and large-pore systems, respectively, and *Pe*_{S} and *Pe*_{L} are the Péclet numbers for the small-pore and large-pore system, respectively. σ_{S} and σ_{L} values were calculated using the following equations: (10)and (11)where σ_{h,L} and σ_{h,S} are the homoporous small-pore and large-pore reflection coefficients, respectively. *Pe*_{S} and *Pe*_{L} were calculated using the following equations: (12)and (13)*PS*_{S} and *PS*_{L} are calculated using the following equations: (14)and (15)where *A*_{0,S}/*A*_{0} = 1 − *A*_{0,L}/*A*_{0} is the fractional cross-sectional pore area of the small-pore system and (*A*/*A*_{0})_{h,S} and (*A*/*A*_{0})_{h,L} are the (homoporous) diffusive transport restriction coefficients (cf. *Eq. 21*) for the small-pore and large-pore systems, respectively.

#### Volume flux.

If no osmotic gradient exists over an heteroselective barrier (i.e., Δπ = 0), then the volume flux via the *i*th pathway is directly proportional to the hydraulic conductance (α_{i}*K*_{f}) of the pathway, as follows: (16)where α_{i} = *K*_{f,i}/*K*_{f} is the fractional hydraulic conductance of the *i*th pathway. If there is an osmotic gradient, then the Starling equilibrium can be applied directly to the pathway, as follows: (17)where σ_{i,net} is the net osmotic reflection coefficient (i.e., effective σ_{o} for all solutes for the *i*th pathway). Hence, if an osmotic gradient exists over a heteroselective barrier, then the volume flow through each pathway will be different from that predicted solely from the hydraulic conductance of that pathway. This difference, known as the isogravimetric flux (32), is thus: (18)where σ_{o,net} is the ensemble osmotic reflection coefficient for all solutes and all pathways. The volume flux over the *i*th pathway in a heteroselective filtration barrier is therefore: (19)In the calculations, the net reflection coefficients have been approximated to those of albumin and a net Δπ of 28 mmHg has been assumed. If the net reflection coefficient of a pathway is smaller than the net osmotic reflection coefficient (for the barrier), then *J*_{vi,iso} will be negative for that pathway and the flux (*J*_{vi}) will be smaller than that predicted solely from the hydraulic conductance of that pathway. For some pathways, *J*_{vi} may be negative, and a recirculation of volume occurs. For a barrier that is nearly homoporous, such as the GFB, *J*_{vi,iso} will typically be very small (28).

#### Hydrodynamic hindrance factors.

The hydrodynamic hindrance factors recommended by Dechadilok and Deen for porous membranes (10) were used and are repeated here for convenience. These expressions have the advantage that they are properly averaged over the entire pore section rather than being “centerline approximations.” The homoporous reflection coefficient was estimated from the following equation: (20)according to Ennis et al. (cf. *Eq. 41* in Ref. 12). The diffusive transport restriction coefficient was calculated from (21)for λ ≤ 0.95 (*Eq. 16* in *Ref. 10*). For more closely fitting solutes (λ > 0.95), we used the following equation: (22)
according to the estimate by Mavrovouniotis and Brenner (cf. *Eq. 71* in Ref. 25). The hydrodynamic hindrance factors are bounded functions (per definition) in the sense that *A*/*A*_{0} tends to zero and σ_{h} tends to unity when λ → 1 and, in addition, that *A*/*A*_{0} tends to unity and σ_{h} tends to zero when λ → 0. Since *A*/*A*_{0} vanishes for *r* < *a*_{e}, the integrals in the numerators of *Eqs. 14* and *15* need only be evaluated from *a*_{e} to ∞.

#### Analytic solutions.

For the improper integrals in the denominators of Eqs. *10*, *11*, *14*, and *15*, there exists the following analytic solution: (23)for any real number (*n*) that can be derived using integration by parts (see the appendix). The total cross-sectional area of a log-normal distributed porous barrier with mean radius *u* and spread *s* can then be calculated exactly as follows: (24)where *N* is the total number of pores per unit weight kidney (or unit area for a membrane). Analogously, using Poiseuille's law, *K*_{f} is given by the following equation: (25)
If *A*_{0}/Δ*x* is known, *N* cancels so that (26)With minor modifications of the proof of *Eq. 23* (see the appendix), one can derive the following equation: (27)where erf is the error function. This thus makes it possible to solve both *Eqs. 14* and *15* completely for λ ≤ 0.95, as follows:
(28)
Using *Eq. 28*, instead of approximating *Eqs. 14* and *15* numerically, reduced the computation time by ∼50%. If the distribution spread is set to unity, then the above equations (*Eqs. 23–28*) reduce to those of the discrete two-pore model. Thus far, it seems as if the distributed two-pore model is identical to the discrete two-pore model when *s* tends to unity. All that remains is to show that *Eqs. 10* and *11* are equal to σ_{h,S} and σ_{h,L}, respectively, when *s*_{S} = *s*_{L} = 1 and, in addition, that *Eqs. 14* and *15* reduce to the homoporous case also for λ > 0.95 (with the spread set to unity). This can be proven (see the appendix) using the dominated convergence theorem (39) using the fact that both σ_{h} and *A*/*A*_{0} are bounded functions. The improper integrals in the numerators of *Eqs. 10* and *11* (and *Eqs. 14* and *15* for λ > 0.95) were evaluated numerically using a 21-point Gauss-Kronrod quadrature. Numeric calculations were performed using the software package GNU Science Library (15).

#### Nonlinear regression.

The theoretical sieving coefficients for each model were fitted to the experimental data (327 data points) using the nonlinear least-squares algorithm according to Levenberg and Marquardt to calculate the optimal values of *u*_{S}, *s*_{S}, *u*_{L}, *s*_{L}, *A*_{0}/Δ*x*, and α_{L} that minimize the weighted sum of squares (objective function), as follows: (29)using the MINPACK library with standard settings (27). This objective function provided a better goodness of fit (lower χ^{2}) than using the ordinary sum of squares on log-transformed data. Because of the limited data for higher solute radii, the large-pore fit was dependent on the initial value (mainly of the large-pore spread *s*_{L}). Therefore, a large number of initial values were tried with *s*_{L} ranging from 1.10 to 1.50. From these results, the best fit (lowest χ^{2}-score) was selected (cf. also Ref. 11, where a similar method is used).

#### Statistical analysis.

Parameter values are presented as means ± SE. A Pearson χ^{2}-test was used for testing the goodness of fit for the data fitted to the discrete and distributed two-pore models. Statistical differences between the different models (discrete; narrow spread: *s*_{S} = 1.05 and *s*_{L} = 1.15; wide spread: *s*_{S} = 1.10 and *s*_{L} = 1.30) were tested using a nonparametric Friedman test followed by post hoc testing using a Wilcoxon-Nemenyi-McDonald-Thompson test. Holm-Bonferroni corrections for multiple comparisons were made. Significance levels were set at *P* < 0.05, *P* < 0.01, and *P* < 0.001. All statistical calculations were made using the computer software R (version 3.0.0) for Linux.

## RESULTS

#### Experimental data analysis.

The optimal parameters for the distributed two-pore model analysis of the Ficoll data are shown in Table 1. The corresponding arithmetic parameters are μ_{S} = 36.9 Å and σ_{S} = 4.7 Å for the small-pore system and μ_{L} = 106.6 Å and σ_{L} = 44.0 Å for the large-pore system. *L*_{p}*S* was calculated from *A*_{0}/Δ*x* (per g kidney) in the distributed model using *Eq. 26*. A value for *L*_{p}*S* of 0.44 ml·min^{−1}·mmHg^{−1}·g^{−1} corresponds to a mean pressure gradient of only ∼1.5 mmHg, which is very low compared with measured values (37). Figure 1 shows θ versus *a*_{e} for the experimental data (dashed line) and the best fit for the regression of the distributed two-pore model (solid line). The dotted line represents a simulated scenario where *1*) *A*_{0}/Δ*x* has been decreased to ∼3 × 10^{5} cm to match the θ value of myoglobin (the value of *s*_{S} has very little effect in this range of solute radii) and *2*) the spread of the small pore has been decreased to ∼1.017 to match the sieving data for four different proteins [human myeloma dimeric κ-chain (κ-dimer, 28.4 Å), nHRP (30.4 Å), HSA (35.5 Å), and neutralized HSA (35.0 Å)] and all other parameters were set according to the best model fit for the Ficoll sieving data. The fractional clearances of the proteins are according to Lund et al. (24). Again, using *Eq. 26* to calculate *L*_{p}*S* from the “protein curve” yields a filtration coefficient of ∼0.06 ml·min^{−1}·mmHg^{−1}·g^{−1}, corresponding to a mean pressure gradient of ∼10 mmHg. In this simulated scenario, the corresponding arithmetic parameters are μ_{S} = 36.6 Å and σ_{S} = 0.6 Å for the small-pore pathway. The probability density functions (*Eq. 8*) for the small-pore (dashed line) and large-pore (dotted line) size distributions are shown in Fig. 2.

The data were reanalyzed using distributed two-pore models where *s*_{S} and *s*_{L} were set to constant values. Figure 3 shows θ versus *a*_{e} for this analysis. Along with the data (dotted line), three different scenarios are shown where the spread of the distributions have been held constant during the regression: *s*_{S} = 1.00 and s_{L} = 1.00 (dashed line), *s*_{S} = 1.05 and *s*_{L} = 1.15 (solid line), and *s*_{S} = 1.10 and *s*_{L} = 1.30 (dashed-dotted line). The knee region is expanded as the spread is decreased, leading to an increasingly poorer fit in this region, as has been previously noted using the classic two-pore-model (34). Interestingly, for the “wide” scenario (*s*_{S} = 1.10 and *s*_{L} = 1.30), a very low χ^{2}-value (0.086) was obtained. The optimized model parameters for the three different scenarios are shown in Table 2. As expected, as the spread of the model increases, both the small- and large-pore radius get smaller and *A*_{0}/Δ*x* gets larger for the widest scenario. No other significant differences among the three “fixed-spread” models were found. Despite the difference between *A*_{0}/Δ*x* in the groups, the fractional large-pore area remains relatively constant. *A*_{0,L}/*A*_{0} is larger for the “constant-spread” models than that obtained in the full analysis above (∼0.4 × 10^{−6}).

#### Theoretical analysis.

Figure 4 shows the effect of altering the breadth of the distribution by plotting several different scenarios from the discrete case (*s*_{S} = 1.00 and *s*_{L} = 1.00, dotted line) to increasingly wider distributions (*s*_{S} = 1.05 and *s*_{L} = 1.15, dashed-dotted line; *s*_{S} = 1.10 and *s*_{L} = 1.30, dashed line; and *s*_{S} = 1.15 and *s*_{L} = 1.45, solid line). For the small-pore system, a more narrow distribution leads to a steeper cutoff. As expected, with a wide distribution in the large-pore system, the selectivity approaches that of the shunted models (a horizontal line). It can also be seen that the transport of smaller solutes below 25 Å (such as VEGF and many other small peptide hormones) is largely unaffected by the width of the distribution of pore sizes.

To quantify the theoretical increase in *A*_{0}/Δ*x* due to an increased distribution spread, *A*_{0}/Δ*x* was plotted as a function of the distribution spread using *Eq. 24* (Fig. 5*A*). Similarly, Fig. 5*B* shows the GFR per gram kidney versus the distribution spread plotted using *Eq. 25* and assuming a net filtration pressure of 10 mmHg. *N* was set to 10^{18} pores/g, and *u* was set at 36.6 Å. When these theoretical predictions of the increase of *A*_{0}/Δ*x* and/or *K*_{f}, due to the distribution spread only are compared, it can be seen that the theoretical increase is much smaller than that obtained from the analysis of the experimental data (Table 2).

## DISCUSSION

We have presented here an extended two-pore theory where the porous pathways are continuously distributed according to *u*_{S} and *u*_{L} and their corresponding *s*_{S} and *s*_{L}, thus considering effects caused by an apparent (or actual) distribution in pore sizes. The results of the data analysis revealed a small-pore population with a wide distribution in pore sizes having an arithmetic SD of ∼5 Å. Such a wide pore size distribution, even when considering the electrostatic hindrance due to a negative pore charge (28), would not be consistent with the high selectivity to proteins normally characterizing the GFB (24, 34). To account for this contradiction, we hypothesize the following:

*1*. A large part of the variance in the distribution of pore sizes in the present analysis is due to the molecular flexibility of the Ficoll molecule, implying that the true variance of the pore system is lower than that obtained when using a flexible probe molecule.

*2*. The mean pore radius is near that of the true effective radius of the GFB, implying that the highly selective filtration barrier favors the filtration of Ficoll molecules having a “mean radius” close to the actual mean pore radius.

*3*. The inflation of *A*_{0}/Δ*x* (due to the wide distribution) can be explained, almost entirely, by the flexibility of the molecule (see below). Thus, the surface increase in distributed models compared with discrete models is due to the flexibility of the molecule, not the wide distribution of the pore population.

We have shown that the classic (discrete) two-pore model represents a special case of the distributed two-pore model where both *s*_{S} and *s*_{L} are set equal to unity. With the use of constant values for the spread parameters (*s*_{S} and *s*_{L}), our results revealed that a smaller distribution spread leads to a larger mean pore radius and a lower *A*_{0}/Δ*x* parameter. This result may be expected since *1*) the discrete small-pore radius is usually ∼43–47 Å (2, 4, 5, 7, 28), whereas a common value for the mean pore radius of the log-normal + shunt model is only ∼35–39 Å (5, 7, 34), and *2*) *A*_{0}/Δ*x* is typically two to three times higher in the distributed models than what is commonly found using the discrete two-pore model (5, 7, 34). Thus, in line with previous results, the data analysis in the present article yielded a high value for *A*_{0}/Δ*x* (∼21 × 10^{5} cm/g) for the distributed model, which leads to an unreasonably high value for *L*_{p}*S* (calculated from *A*_{0}/Δ*x*). A more reasonable value (∼3 × 10^{5} cm/g) was obtained using a simulated scenario where the small-pore spread and *A*_{0}/Δ*x* were lowered to match the θ values of five proteins. We also derived a practical equation (see *Eq. 24*) showing that *A*_{0}/Δ*x* is, as expected, directly dependent on the pore size distribution (both spread and mean radius) so that an inflated distribution spread leads to inflation in *A*_{0}/Δ*x* as well. However, this theoretical increase in *A*_{0}/Δ*x* is much smaller than what could be expected from the measured differences between the discrete and distributed models. What is the reason behind the (often noted) inflated *A*_{0}/Δ*x* values in the distributed models? Interestingly, the larger pore radius for the discrete two-pore model (compared with the distributed model) leads to less restriction to the diffusive transport (i.e., larger *A*/*A*_{0}) in the small-pore system. As an example, for a 30-Å solute, the small-pore *A*/*A*_{0} parameter is ∼2.4 × 10^{−3} for the distributed two-pore model (using *Eq. 28* with *u* = 36.6 Å and *s* = 1.13) compared with ∼7.4 × 10^{−3} for the discrete two-pore model (using *Eq. 28* with *u* = 44.7 Å and *s* = 1.00 or *Eq. 21*). This gives a factor of ∼3 between the *A*/*A*_{0} parameters of the distributed and discrete models, which is sufficient to explain (see *Eq. 4*) the discrepancy between the distributed and discrete two-pore models in this article. In conclusion, if the distribution in pore sizes is caused by the flexibility of the solute, then the increased area is apparent and does not reflect the real pore area of the membrane. Conversely, if there is an actual distribution in pore sizes, the ultrafiltration coefficient of the barrier, as calculated theoretically (from *A*_{0}/Δ*x* using *Eq. 26*), should more closely match the measured ultrafiltration coefficient.

The selective mechanisms of the GFB are based on the separation of molecular species depending on size, charge, and conformation. Recently, we quantified the electrostatic properties of the GFB in terms of the surface charge density of the barrier (in C/m^{2}) and found it to be similar to that of known surface charge densities for many proteins in the body (∼5–20 mC/m^{2}) (28). This is a much lower value than that suggested by Haraldsson et al. (200 mC/m^{2}) (17). Indeed, the latter charge density would be required if most of the difference in the glomerular permeability between Ficoll and globular proteins were due to charge effects. If the wide distribution measured in the present investigation represents the actual pore size distribution in a weakly/moderately charged GFB (i.e., ∼5–20 mC/m^{2}), then one would have to add ∼15–19 Å to the effective radius of the albumin molecule to achieve the same fractional clearance as that actually measured. In contrast, if the distribution of pore sizes in the GFB is narrow, as is proposed in the present study, then a “threshold effect” is possible (due to the similarity of the mean small-pore radius, 36.6 Å, and molecular radius of albumin, 35.5 Å), so that even a moderate surface charge density on the barrier (similar to that on many plasma proteins) is sufficient exclude albumin from passing through the small-pore system. According to the hypothesis in this study, conformation plays a crucial part in how molecular species are transported across the GFB. As an example, bikunin, a 36-Å (radius) elongated protein, had ∼80 times higher fractional clearance (θ) than albumin despite similar size and charge (23).

The GFB is a dynamic barrier in which the permeability can change dramatically even in a short period of time (2–5). The physiological and pathophysiological mechanisms behind the changes in permeability seem to be mediated primarily by the large-pore system. Despite the apparent role of the large-pore pathway in the regulation of the permeability of the GFB, very little is known about the underlying mechanisms or the equivalent biological structure. The permeability of the large-pore system is typically increased when the glomeruli are injured by disease [*f*_{L} (equal to *J*_{vL}/GFR) can increase several orders of magnitude within minutes]. In contrast to peripheral capillaries, the permeability of the large-pore system in glomerular capillaries is normally very low. This means that, in healthy glomerular capillaries, only very small amounts of large probe molecules are filtered into the urinary space, making direct measurement of the selectivity of the large-pore system difficult. Indeed, the value for the large-pore parameters in the present analysis should be interpreted carefully due to the limited range of the data in the large-pore region. In addition, the large SD (∼44 Å) may involve other factors than those suggested for the small-pore pathway above. For example, the data itself show their greatest variation in the large-pore portion of the sieving curve, which should contribute to the observed variance in the pore size distribution. It has been suggested that Ficoll_{400} is more similar to a random coil than a hard sphere (13). In light of this, Ficoll_{400} may be an inappropriate probe for measuring the size-selective properties of the large-pore system. On the other hand, if the hypothesis in this article is correct, as shown in Fig. 4 (and from the fact that large Ficoll_{400} molecules have a similar θ value as that of HSA in Fanconi syndrome (37); see Fig. 1), *f*_{L} can be predicted with good accuracy using Ficoll_{400}.

What are the physiological and pathophysiological roles of the small-pore system? We (5) have recently reported measurements of glomerular permeability during systemic angiotensin II infusion in rats. The analysis (using the log-normal distributed + shunt model) showed that the width of the small-pore distribution was markedly increased (corresponding to an increase in σ_{S} from 4.9 to ∼7–8 Å) when high doses of angiotensin II were administered. This increased heteroporosity of the small-pore system can also be seen in puromycin aminonucleoside nephrosis and is usually accompanied by a marked increase in the large-pore permeability (*f*_{L}) (18). Indeed, the widening of the small-pore distribution may be a general phenomenon, occurring after large increases in glomerular permeability. In addition, widening of the small-pore distribution (e.g., as shown in Figs. 3 and 4) may be a major pathophysiological mechanism in selective proteinuria.

The distributed pore model presented in the present article assumes a very simple structure of the glomerular capillary wall with two different size-selective modalities: small- and large-pore populations. If the experimental data were produced using this “equivalent” barrier (i.e., instead of the rat glomerulus), they would have the properties shown in Table 1. Given what is known about the actual structure of the glomerular capillary wall, the use of pores is obviously phenomenological. Nonetheless, pore theory is arguably the most popular paradigm for describing glomerular sieving and remains one of the simplest ways to model the transport of both solvent (as laminar flow) and solutes (using hydrodynamic restriction coefficients). In addition, since the model in the present article is based on established models and concepts, it is possible to directly compare new results with previous findings. Despite the differences in physical structure between the glomerular capillary wall and the distributed two-pore model, both the goodness-of-fit analysis and visual fit suggest that, in a functional sense, the barriers are remarkably similar.

The actual distribution of the size-selective structures in the GFB is not known. However, as shown in Fig. 1, over the course of just a few angstroms, the θ value (for proteins) was reduced ∼200-fold from 0.13 (human myeloma κ-dimer, 28.4 Å) to 6 × 10^{−4} (HSA, 35.5 Å). Arguably, such an impressive cutoff is not consistent with a heteroselective small-pore pathway unless the size distribution of the selective elements (pores, fibers, slits, etc.) is very narrow. Thus, in the present investigation, a narrow distribution in the small-pore system (*s*_{S} ≈ 1.017 and *u*_{S} ≈ 36.6) matched the sieving data of five globular proteins, giving an estimated 95.5% confidence interval of 35.4–37.9 Å for the “real” pore radius, which has been estimated by Lund et al. (24) to be 37.4 Å. In comparison, the hydrodynamic (Stokes-Einstein) radius of HSA is ∼35.5 Å. We (28) have previously reported that the electrostatic repulsion between the negative electric charge on the GFB and the anionic sites on the albumin molecule may add only a few angstroms to the apparent size of the albumin molecule. Indeed, if the estimate of the actual pore radius in this article is accurate, this means that albumin is effectively excluded from the small-pore pathway in healthy glomerular capillaries. In an ideal model for glomerular permeability, the GFR, as predicted by the solute flow, should match the measured GFR [thus resolving Pappenheimer's pore puzzle (29)]. In the present study, the measured GFR (0.65 ml·min^{−1}·g^{−1}) did not match theoretical GFRs (calculated from GFR = *L*_{p}*S* × ΔP_{net}) unless a very low ΔP_{net} is assumed. Lowering *s*_{S} and *A*_{0}/Δ*x*, as in the protein sieving scenario, leads to a better match between the GFR as calculated from the solute flow and the measured GFR.

In summary, we have shown that the permeability of the GFB can be described by a distributed two-pore model, assuming that the size-selective structures of the glomerular capillary wall are log-normally distributed small-pore and large-pore populations. In the case of Ficoll, there seems to be *a distribution effect* related to the flexible structure of the molecule, since the wide distribution obtained is inconsistent with the high selectivity characterizing healthy glomerular capillaries. Furthermore, both *A*_{0}/Δ*x* and *K*_{f} are, as could be expected, directly dependent the pore size distribution (both spread and mean radius). Practical equations (*Eqs. 24–26*) for both of these entities have been proved analytically, eliminating the need for numeric approximations. These equations are by no means limited to a capillary wall but are actually a generalization of the Hagen-Poiseuille equation for any porous membrane with a log-normal distribution of pore sizes. We have also demonstrated that a smaller distribution spread leads to *1*) an increased mean pore radius and *2*) a decreased *A*_{0}/Δ*x*. The latter effect is mainly due to the increased diffusive hindrance of the small-pore pathway in the distributed two-pore model leading to an inflation of *A*_{0}/Δ*x*. Finally, we have shown that the classic (discrete) two-pore model is actually a special case of the distributed two-pore model where *s*_{S} = *s*_{L} = 1.00.

## GRANTS

This work was supported by Swedish Research Council Grant 08285, the Heart and Lung Foundation, and the Medical Faculty at Lund University (ALF Grant).

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## AUTHOR CONTRIBUTIONS

Author contributions: C.M.Ö. and B.R. conception and design of research; C.M.Ö. analyzed data; C.M.Ö. and B.R. interpreted results of experiments; C.M.Ö. prepared figures; C.M.Ö. drafted manuscript; C.M.Ö. and B.R. edited and revised manuscript; C.M.Ö. and B.R. approved final version of manuscript.

## ACKNOWLEDGMENTS

The authors gratefully thank Josefin Axelsson for generously providing the Ficoll sieving data and Kerstin Wihlborg for expert secretarial assistance. The authors also thank Fredrik Nilsson (Unit of Medical Statistics and Epidemiology, Region Skåne) for valuable advice.

## Appendix A

##### The Patlak equation.

Define a function *h*(c) = [*J*_{v}(1 − σ)c − *J*_{s}]/*DA*. *Equation 1* can then be rewritten as follows: (30)which can be integrated over the barrier, from the plasma concentration (C_{p}) to the interstitial concentration (C_{i}), as follows: (31)where Δ*x* is the membrane thickness [i.e., c(Δ*x*) = C_{i}]. The result of this integration is the following equation: (32)which can be rewritten as follows:

##### Analytic solution.

To find the primitive function of the integrand, we start by using the integration of parts: (33)
The change of variables *R* = ln(*r*) gives the following equation: (34)where erf is the error function. Inserting erf into *Eq. 33* gives the following equation: (35)so that (36)which can be rearranged to the following equation: (37)

##### When s tends to unity.

Suppose *f*(*r*) is an integrable real-valued bounded function. We wish to prove that (38)
The change of variables *X* = *k*[ln(*r*) − ln(*u*)], where *k* = 1/ln(*s*), gives the following equation: (39)
Since *f*(*r*) is bounded, there is a real number (*m*) such that (40)for all *k* and all *X*. Under these conditions, the dominated convergence theorem (39) is applicable so that the limit may be taken under the integral sign, as follows:
(41)

## Glossary

- α
_{i} - Fractional hydraulic conductance for the
*i*th pore population - α
_{L} - Fractional hydraulic conductance for the large-pore population (i.e.,
*K*_{fL}/*K*_{f}) - α
_{S} - Fractional hydraulic conductance for the small-pore population (i.e.,
*K*_{fS}/*K*_{f}) - η
- Viscosity of the solvent (in Pa·s) (e.g., η
_{water}≈ 0.7 mPa·s) - θ
- Sieving coefficient
- θ
_{j,data} - Experimental sieving coefficient of the data
- θ
_{j,model} - Theoretical sieving coefficient of the model
- λ
- Solute-to-pore radius ratio (e.g.,
*a*_{e}/*r*_{s}) - μ
_{L} - Arithmetic mean pore radius for the large-pore population
- μ
_{S} - Arithmetic mean pore radius for the small-pore population
- Δπ
- Osmotic pressure gradient (in mmHg)
- σ
- Reflection coefficient
- σ
_{f} - Solvent-drag reflection coefficient
- σ
_{h,L} - Homoporous large-pore reflection coefficient
- σ
_{h,S} - Homoporous small-pore reflection coefficient
- σ
_{i,net} - Net osmotic reflection coefficient
- σ
_{L} - Arithmetic SD for the large-pore population
- σ
_{o} - Osmotic reflection coefficient
- σ
_{o,net} - Ensemble osmotic reflection coefficient for all solutes and all pathways
- σ
_{S} - Arithmetic SD for the small-pore population
*a*_{e}- Molecular (Stokes-Einstein) radius (in Å)
*A*- Effective surface area available for restricted diffusion (i.e.,
*A*_{0}≥*A*) *A*_{0}- Total cross-sectional pore area
*A*/*A*_{0}- Diffusive transport restriction coefficient (effective-to-total area ratio)
*B*^{n}- Analytical solution
- c(
*x*) - Concentration profile along the length of the pore (in mol/ml)
- C
_{i} - Downstream (filtrate) concentration (in mol/ml)
- C
_{p} - Plasma concentration (in mol/ml)
*D*- Free diffusion coefficient (in cm
^{2}/min) - erf
- Error function
*f*_{L}- Fractional volume flux across the large-pore population (i.e.,
*J*_{vL}/*J*_{v}) *f*_{S}- Fractional volume flux across the small-pore population (i.e.,
*J*_{vS}/*J*_{v}) *g*(r)- Log-normal probability density function
*G*^{n}- Analytical solution (defined in
*Eq. 23*) - GFR
- Glomerular filtration rate
- GFB
- Glomerular filtration barrier
*H*- Alternate notation of
*A/A*_{0} - HSA
- Human serum albumin
*J*_{convection}- Flux through convective transport
*J*_{diffusion}- Flux through diffusive transport
*J*_{s}- Total solute flux across the entire barrier (in mol/min)
*J*_{v}- Total volume flux across the entire barrier (in ml/min)
*J*_{vi}- Volume flux across the
*i*th pore population in a heteroselective barrier (in ml/min) *J*_{vL}- Volume flux across the large-pore population (in ml/min)
*J*_{vL,iso}- Isogravimetric flux for large-pore population (in μl/min) (
*J*_{vS,iso}= −*J*_{vL,iso}) *J*_{vS}- Volume flux across the small-pore population (in ml/min)
*k*- Boltzmann constant
*K*_{f}- See
*L*_{p}S *L*_{p}- Total hydraulic conductivity (in ml·min
^{−1}·mmHg^{−1}·cm^{−2}) *L*_{p}S- Filtration coefficient; total hydraulic conductance (in ml·min
^{−1}·mmHg^{−1}) *n*- Any real number
*N*- Total number of pores per unit weight kidney (in g) (or unit area for a membrane)
- nHRP
- Neutral horseradish peroxidase
*P*- Permeability coefficient;
*D*/Δ*x*(in cm/min) *Pe*- Péclet number
*Pe*_{L}- Péclet numbers for the large-pore system
*Pe*_{S}- Péclet numbers for the small-pore system
*PS*- Permeability-surface (diffusion capacity) coefficient (in ml/min)
*PS*_{L}- Permeability-surface (diffusion capacity) coefficient for the large-pore system
*PS*_{S}- Permeability-surface (diffusion capacity) coefficient for the small-pore system
- ΔP
- Hydraulic pressure gradient (in mmHg)
- ΔP
_{net} - Net pressure gradient (ΔP − σΔπ)
*r*- Log-normally distributed radius
*R*- Natural logarithm of
*r* *s*- Geometric pore SD
*s*_{L}- Geometric large-pore SD
*s*_{S}- Geometric small-pore SD
*S*- Barrier surface area (in cm
^{2}) per unit weight kidney (or unit area for a membrane) - SD
- Standard deviation
- T
- Temperature (in K) (body temperature = 310 K)
*u*- Geometric mean pore radius
*u*_{L}- Geometric mean large-pore radius (in Å)
*u*_{S}- Geometric mean small-pore radius (in Å)
*W*- Alternative notation of 1-σ
*x*- Distance (in cm)
- Δ
*x* - Total barrier thickness (in cm)

- Copyright © 2014 the American Physiological Society