Biophysical Basis of Glomerular Filtration




Marcello Malpighi (1628–1694) discovered the renal corpuscle and proposed that each glomerular body embraces the ampullar extremity of a tubule to form a “glandular follicle”. Thereafter, progress toward understanding the structure and function of the nephron stalled for two centuries, until William Bowman finally established the proper anatomic relationship between the glomerular arterioles, capillary tuft, and uriniferous tubule in 1842. In that same year, Carl Ludwig, in his Habilitations thesis, addressed the driving force that separates the watery and crystalloid constituents of the plasma from its “proteid” constituents. He dismissed both the “nonexistent” vital force and chemical theories for converting blood to urine, and deduced from geometric considerations that local hydraulic forces drive filtration of blood plasma through porous glomerular capillary walls. Ludwig’s theory was not universally accepted at the time and other influential figures, such as Heidenhain, continued to advocate the secretory formation of urine. Ludwig also had the foresight to envision that the hyperproteinemia resulting from glomerular filtration causes concentration of the urine by endosmosis into the peritubular capillaries. Several decades later, vant Hoff and others began to describe osmosis in terms of pressure using thermodynamic principles, which inspired Ernest Henry Starling to contemplate a role for the osmotic pressure of the plasma colloids in glomerular filtration. Starling wondered whether the minimum blood pressure below which formation of urine ceases might equal the osmotic pressure of the plasma colloids that oppose filtration. In 1897, he tested this hypothesis using a colloid osmometer of his own design with which he estimated the osmotic pressure of the blood plasma protein to be 25–30 mmHg or about 0.4 mmHg-gram −1 /liter −1 . Then he observed that raising the ureteral pressure to within 30–45 mmHg of the arterial blood pressure would stop the flow of urine in a dog undergoing diuresis. Thus, the hydraulic pressure across the glomerular epithelium must exceed the plasma colloid osmotic pressure by some small amount in order for urine to form. On this basis, Ludwig’s filtration hypothesis was deemed credible. Further evidence for glomerular filtration was published in 1924 by Wearn and Richards, who directly visualized the passage of indigo carmine into Bowman’s space from the blood in the course of performing the first-ever micropuncture experiments. Wearn and Richards interpreted their own findings as “indirect evidence that the process in the glomerulus is physical”.


Introduction


Marcello Malpighi (1628–1694) discovered the renal corpuscle and proposed that each glomerular body embraces the ampullar extremity of a tubule to form a “glandular follicle”. Thereafter, progress toward understanding the structure and function of the nephron stalled for two centuries, until William Bowman finally established the proper anatomic relationship between the glomerular arterioles, capillary tuft, and uriniferous tubule in 1842. In that same year, Carl Ludwig, in his Habilitations thesis, addressed the driving force that separates the watery and crystalloid constituents of the plasma from its “proteid” constituents. He dismissed both the “nonexistent” vital force and chemical theories for converting blood to urine, and deduced from geometric considerations that local hydraulic forces drive filtration of blood plasma through porous glomerular capillary walls ( Figure 21.1 ). Ludwig’s theory was not universally accepted at the time and other influential figures, such as Heidenhain, continued to advocate the secretory formation of urine. Ludwig also had the foresight to envision that the hyperproteinemia resulting from glomerular filtration causes concentration of the urine by endosmosis into the peritubular capillaries. Several decades later, vant Hoff and others began to describe osmosis in terms of pressure using thermodynamic principles, which inspired Ernest Henry Starling to contemplate a role for the osmotic pressure of the plasma colloids in glomerular filtration. Starling wondered whether the minimum blood pressure below which formation of urine ceases might equal the osmotic pressure of the plasma colloids that oppose filtration. In 1897, he tested this hypothesis using a colloid osmometer of his own design with which he estimated the osmotic pressure of the blood plasma protein to be 25–30 mmHg or about 0.4 mmHg-gram −1 /liter −1 . Then he observed that raising the ureteral pressure to within 30–45 mmHg of the arterial blood pressure would stop the flow of urine in a dog undergoing diuresis. Thus, the hydraulic pressure across the glomerular epithelium must exceed the plasma colloid osmotic pressure by some small amount in order for urine to form. On this basis, Ludwig’s filtration hypothesis was deemed credible. Further evidence for glomerular filtration was published in 1924 by Wearn and Richards, who directly visualized the passage of indigo carmine into Bowman’s space from the blood in the course of performing the first-ever micropuncture experiments. Wearn and Richards interpreted their own findings as “indirect evidence that the process in the glomerulus is physical”.




Figure 21.1


Ludwig’s representation of renal microvasculature (a) and (b) pressure profiles along the glomerular capillary

(from ref. ).


Glomerular filtration eventually received theoretical consideration as a case of coupled transport, subject to the basic rules of non-equilibrium thermodynamics, which were articulated by Onsager in 1931 and adapted to describe the permeability of biological membranes by several investigators in the 1950s. Prior to the 1950s, the conventional description of transport through membranes simply combined Fick’s diffusion equation for solute flux with Darcy’s equation for water flux, such that the function of a membrane which is to “prescribe the road along which the system strives toward equilibrium”, was defined by two permeability coefficients, one for diffusion of solute and one for bulk flow of water. By the 1950s it had become clear that these “conventional” permeability equations for solute and volume flow could not fully describe the physical behavior of membranes, so attempts were made to supplement them. The most cited contribution in this area came from Kedem and Katchalsky, who pointed out that the prior approach was incomplete due to the fact that it included only two coefficients, whereas Onsager’s theory calls for exactly three coefficients to characterize permeability for a solute–solvent system. Qualitatively, the hydrodynamic resistance to free diffusion is due to friction between solute and solvent alone, and is determined by a single diffusion coefficient. But passage through a membrane involves two additional factors, namely, the friction between solute and membrane, and the friction between solvent and membrane. Hence, three processes are at play, and three coefficients are required to account for them all. Kedem and Katchalsky then proceeded with a formal argument, starting from the rate of entropy production and invoking Onsager’s theory for a solute–solvent system which is paraphrased as follows:


For present purposes, we are interested in the transmembrane flux of a two-component system consisting of a water (w) and a non-electrolyte solute (s). Each of these components is driven by a conjugate force equivalent to its difference in free energy across the membrane. The conjugate forces for water and non-electrolyte solute are:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-1-Frame class=MathJax style="POSITION: relative" data-mathml='Δμw=VwΔP+RTΔlnγwXw’>???=????+???ln????Δμw=VwΔP+RTΔlnγwXw
Δ μ w = V w Δ P + R T Δ ln γ w X w

<SPAN role=presentation tabIndex=0 id=MathJax-Element-2-Frame class=MathJax style="POSITION: relative" data-mathml='Δμs=VsΔP+RTΔlnγsXs’>???=????+???ln????Δμs=VsΔP+RTΔlnγsXs
Δ μ s = V s Δ P + R T Δ ln γ s X s
where V is a partial molar volume, Δ P is the pressure difference, X is the mole fraction, and γ is an activity function, empirically derived as a function of X . Since water flux affects X s and solute flux affects X w , the two conjugate forces and fluxes are coupled. For a small deviation from equilibrium this coupling can be taken into account by the following linear flux equations:
<SPAN role=presentation tabIndex=0 id=MathJax-Element-3-Frame class=MathJax style="POSITION: relative" data-mathml='Jw=L11Δμw+L12Δμs’>??=?11???+?12???Jw=L11Δμw+L12Δμs
J w = L 11 Δ μ w + L 12 Δ μ s

<SPAN role=presentation tabIndex=0 id=MathJax-Element-4-Frame class=MathJax style="POSITION: relative" data-mathml='Js=L21Δμw+L22Δμs’>??=?21???+?22???Js=L21Δμw+L22Δμs
J s = L 21 Δ μ w + L 22 Δ μ s
where Lxy are the so-called phenomenological constants. Onsager’s theory says that L 21 =L 12 . Therefore, if the three coefficients, L 11 , L 22 , and L 21 =L 12 are known, along with baseline values of J w and J s , then one can predict the changes in J w and J s that will arise from any alteration in Δ μ w or Δ μ s . However, the physical meanings of the phenomenological constants are difficult to appreciate, and a more familiar form of the Onsager equations was provided in 1958 by Kedem and Katchalsky to describe transport across biological membranes :
<SPAN role=presentation tabIndex=0 id=MathJax-Element-5-Frame class=MathJax style="POSITION: relative" data-mathml='Jv=LP⋅(ΔP−σsΔΠ)’>??=??(??????)Jv=LP⋅(ΔP−σsΔΠ)
J v = L P ⋅ ( Δ P − σ s Δ Π )

<SPAN role=presentation tabIndex=0 id=MathJax-Element-6-Frame class=MathJax style="POSITION: relative" data-mathml='Js=Ps⋅ΔCs+JV(1−σs)·Cs¯’>??=?????+??(1??)·??Js=Ps⋅ΔCs+JV(1−σs)·Cs¯
J s = P s ⋅ Δ C s + J V ( 1 − σ s ) · C s ¯


When applied to movement across a capillary wall, J v and J s denote respectively the flux of volume (substituting volume for water is allowable for dilute solutions) and solute; Δ P , Δ Π , and Δ C are differences in hydrostatic pressure, osmotic pressure, and concentration integrated across the membrane; σ s is the reflection coefficient of the membrane for s ; L p is the hydraulic permeability per unit area of membrane; and P s is the diffusive permeability of the membrane to s ; <SPAN role=presentation tabIndex=0 id=MathJax-Element-7-Frame class=MathJax style="POSITION: relative" data-mathml='Cs¯’>??Cs¯
C s ¯
is the mean concentration of s within the membrane. Π is a function of C . σ s assumes a value between zero and one. Three of these parameters, L p , P s , and σ s , are characteristics of the membrane, in keeping with the Onsager theory which requires exactly three coefficients to describe the coupled transport of the two entities, v and s . Equation (21.3a) is often referred to as the “Starling equation.” Equation (21.3b) expresses J s as the sum of diffusive and convective components. Equations (21.3a) and (21.3b) are coupled. J s explicitly depends on J v . J v depends on J s because J s affects Δ Π .


There are limitations to irreversible thermodynamics and to the simplified equations, beginning with the assumption of a linear relationship between fluxes and forces. For example, one can imagine how increasing Δ P might cause a capillary wall to stretch, thereby changing the geometry of its pores and altering L p . Also, <SPAN role=presentation tabIndex=0 id=MathJax-Element-8-Frame class=MathJax style="POSITION: relative" data-mathml='Cs¯’>??Cs¯
C s ¯
can take a variety of forms, depending on whether σ is taken to be active throughout the membrane or to be a membrane entrance phenomenon, and this will affect how J s is parsed into its diffusive and convective components. Finally, protein accumulates near the capillary wall during filtration, which could raise the local colloid osmotic pressure at the wall and cause L p to be underestimated when calculated based on Π for the bulk plasma. Nonetheless, these equations remain the basis for all current understanding of the physical factors that determine transport of water and solutes between the glomerular capillary plasma and the urinary space.


Depending on the context, different simplifying assumptions are made that streamline the description of capillary flux in the glomerulus. For example, when considering J v (i.e., glomerular filtration), the solutes are divided into two groups, large and small. Large solutes are the colloids, and it is assumed that P s for these is zero and σ s is unity. All other solutes are assumed to be small, and it is assumed that σ s (and Δ C ) for these is zero. Solutes with intermediate permeability are ignored. Therefore, Δ Π can be substituted by the colloid osmotic pressure of the glomerular plasma, and a full description of J v is provided by Eq. (21.3a) . This obviates the need to consider coupled transport. Although the contribution of filtered macromolecules to the transcapillary oncotic pressure may be negligible, there are times when it is critical to understand the sieving properties of the glomerulus for large molecules. In such cases, simplifying assumptions are made regarding the geometry of the filtration barrier and the shape of the solute molecules, so that the process can be conveniently described using hydrodynamic theory.




The Magnitude of Renal Blood Flow and Glomerular Filtration


In humans, the kidneys constitute 0.5% of the body weight, but receive 20% of the cardiac output. The low resistance to renal blood flow is owing to the large number of parallel conductances, with each human kidney containing about 1 million glomeruli. Approximately 8000 liters per day of blood plasma transits the extrarenal organs, of which about 20 liters is filtered into interstitial spaces and returned to the blood as lymph. In contrast, the kidneys form 180 liters per day of glomerular filtrate from 900 liters of blood plasma. The high rate of filtration by the kidney relative to other organs is due to a greater ultrafiltration coefficient, not to greater Starling force. The surface area available for filtration in the human kidney is in the order of 1.2 m 2 overall or 0.6 mm 2 per glomerulus. A meaningful number is difficult to assign to the capillary surface area in other major organs where the number of capillaries perfused at any given moment is highly variable. The hydraulic permeability L p of fenestrated glomerular capillaries has been estimated from 2.5–4.0 μl/min/mmHg/cm 2 in rats and humans, which is 50-fold higher than L p for non-fenestrated skeletal muscle.




Glomerular Hemodynamics by Inference


Having first identified inulin and PAH as a markers of GFR renal plasma flow (RPF), Homer Smith and colleagues used clearance of these markers to make logical judgments about the regulation of GFR. Smith observed a reciprocal relationship between RPF and filtration fraction in human subjects injected with pyrogens, and recognized that this is contrary to what should occur if the changes in RPF were mediated by a preglomerular resistance. He used equations to argue that the renal resistance changed in these experiments due to dilation and constriction of the efferent arteriole. His formulation required a strong inverse effect of efferent resistance on filtration fraction, which could be achieved by assuming that the net ultrafiltration pressure vanishes at some point along the capillary, as hydrostatic pressure declines and plasma oncotic pressure increases. Based on knowledge that this occurs in the mesenteric circulation, Smith was willing to assume that this also happens in the kidney, and coined the term “filtration pressure equilibrium” in reference to the phenomenon (see Figure 21.2 ). Smith later recanted his notion of filtration pressure equilibrium in the glomerular capillary, arguing on teleologic grounds that the hydrostatic null point should occur in the proximal portion of the efferent arteriole in order to promote maximal GFR and maximal reabsorption in the peritubular capillary. His revised thinking was likely influenced by the contemplations of Gomez.




Figure 21.2


Filtration equilibrium illustrated in the glomerulus

(from ref. ).




Glomerular Hemodynamics and Micropuncture


A full and direct assessment of the filtration forces and hydraulic permeability in a mammalian glomerulus was first published by Brenner et al. in 1971. Three developments made this possible. First, a mutant rat strain (Munich Wistar) was discovered with glomeruli on the kidney surface making them accessible for glomerular micropuncture. Second, a servo-null device was invented that enabled accurate and rapid pressure measurements in capillaries and tubules. Third, a microadaptation of the Lowry method was developed for measuring the protein concentration in a few nanoliters of plasma which could be obtained by micropuncture from a postglomerular arteriole.


Given values for the pressure in the glomerular capillary P GC and Bowman’s space P BS , pre- and postglomerular plasma protein concentrations c 0 and c 1 , single nephron 3 H-inulin clearance SNGFR , and a simple mathematical model for computing changes in the ultrafiltration pressure P UF along the glomerular capillary, it is possible to obtain values for the glomerular plasma flow Q 0 and ultrafiltration coefficient LpA . LpA is the product of the hydraulic permeability Lp (see Eq. (21.3a) ) and the filtration surface area A .


The mathematical model for computing the physical determinants of SNGFR from micropuncture data was developed by Deen, Robertson, and Brenner in 1972. This model treats the glomerular capillary as a circular cylinder of unit length and surface area, uniform permeability to water and small solutes, and zero permeability to protein (see Figure 21.3 ). As in Eq. (21.3a) , the filtration flux at any point along the capillary is equal to the product of the Starling force, Δ P −ΔΠ, and the hydraulic permeability, Lp . SNGFR is obtained by integrating the flux along the capillary length:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-9-Frame class=MathJax style="POSITION: relative" data-mathml='SNGFR=∫01Jv⋅dx=LpA∫01(ΔP−ΔΠ)dx=LpA⟨PUF⟩’>?????=10????=???10(????)??=??????SNGFR=∫01Jv⋅dx=LpA∫01(ΔP−ΔΠ)dx=LpA⟨PUF⟩
S N G F R = ∫ 0 1 J v ⋅ d x = L p A ∫ 0 1 ( Δ P − Δ Π ) d x = L p A ⟨ P U F ⟩
where Δ P = P GC P BS , ΔΠ+Π GC −Π BS and ⟨ P UF ⟩ is the mean ultrafiltration pressure. The term LpA represents the product of the hydraulic permeability Lp and filtration surface area A . For the non-dimensionalized capillary, A equals unity. For the real capillary, micropuncture data do not distinguish between changes in Lp and changes in A .


Figure 21.3


Glomerular capillary represented by a homogerous circular circular cylinder with unit length and surface area (Q: Plasma flow; Jv: Filtration water flux; X: Axial position along the capillary).


To perform the integration in Eq. (21. 4) it is necessary to know how the integrand varies along the capillary. In theory, both Δ P and Δ Π should change along the capillary, since P GC must decline due to axial flow resistance and Π GC must rise as water moves from the plasma into Bowman’s space. It has always been assumed that the decline in P GC along the capillary is small relative to the increase in Π GC . This assumption was eventually justified by a three-dimensional reconstruction of the rat glomerulus submitted to computational analysis. It is our custom to ignore the small axial pressure drop and represent Δ P as a constant, since including a 1–2 mmHg axial pressure drop in the model has a minimal effect on ⟨ P UF ⟩. However, to better illustrate certain principles in this chapter, we have incorporated a 1 mmHg decline in P GC from the beginning to the end of the glomerular capillary.


For the purposes of determining Δ Π it is assumed that all solutes in the system are either completely impermeant plasma proteins that exert their full osmotic potential ( σ =1, Ps =0) and reside solely in the plasma or small molecules that are freely filtered ( σ =0) and contribute nothing to Δ Π. Thus, Δ Π is reduced to the plasma oncotic pressure, Π GC . The oncotic pressure in a plasma sample is determined from the protein concentration c , according to an empiric relationship developed by Landis and Pappenheimer:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-10-Frame class=MathJax style="POSITION: relative" data-mathml='Π=α1c+α2c2′>?=?1?+?2?2Π=α1c+α2c2
Π = α 1 c + α 2 c 2


The values of α 1 and α 2 in Eq. (21.5) vary according to the ratio of albumin to globulin in the plasma. When Π is expressed in mmHg and c in grams per 100 ml, for rat plasma, α 1 and α 2 are 1.73 and 0.28, respectively. According to Eq. (21.5) , Π GC will increase from 18 to 35 mmHg along the length of a glomerular capillary if the systemic plasma contains 6 g/dl of protein and the nephron filtration fraction is 0.29. Such values are typical of the rat.


LpA is computed from SNGFR and ⟨ P UF ⟩, according to Eq. (21.4) . To obtain ⟨ P UF ⟩ it is necessary to know the profile for Π GC along the capillary. This profile is computed from the following mass balance considerations for protein and water. First are three conservation of mass equations:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-11-Frame class=MathJax style="POSITION: relative" data-mathml='Q0=SNGFR(c1c1−c0)’>?0=?????(?1?1?0)Q0=SNGFR(c1c1−c0)
Q 0 = S N G F R ( c 1 c 1 − c 0 )

<SPAN role=presentation tabIndex=0 id=MathJax-Element-12-Frame class=MathJax style="POSITION: relative" data-mathml='cQ=c0Q0′>??=?0?0cQ=c0Q0
c Q = c 0 Q 0

<SPAN role=presentation tabIndex=0 id=MathJax-Element-13-Frame class=MathJax style="POSITION: relative" data-mathml='Jv=−dQdx’>??=????Jv=−dQdx
J v = − d Q d x
where Q 0 is the nephron plasma flow and c 0 and c 1 are the pre- and post-capillary plasma protein concentrations. Differentiating Eq. (21.6b) and substituting Eqs. (21.6c), (21.5), and (21.3a) :
<SPAN role=presentation tabIndex=0 id=MathJax-Element-14-Frame class=MathJax style="POSITION: relative" data-mathml='dcdx=c2c0Q0Jv’>????=?2?0?0??dcdx=c2c0Q0Jv
d c d x = c 2 c 0 Q 0 J v

<SPAN role=presentation tabIndex=0 id=MathJax-Element-15-Frame class=MathJax style="POSITION: relative" data-mathml='=Lp⋅c2c0Q0(ΔP−(α1c+α2c2))’>=???2?0?0(??(?1?+?2?2))=Lp⋅c2c0Q0(ΔP−(α1c+α2c2))
= L p ⋅ c 2 c 0 Q 0 ( Δ P − ( α 1 c + α 2 c 2 ) )


A standard root-finding algorithm is used to obtain a value for LpA by numerical integration of Eq. (21.7) along the entire capillary to obtain an estimate for the plasma protein concentration at the end of the capillary ( c 1 *) and adjust the value of LpA until c 1 * is arbitrarily close to the measured value of c 1 .


<SPAN role=presentation tabIndex=0 id=MathJax-Element-16-Frame class=MathJax style="POSITION: relative" data-mathml='c1*=c0+LPAc0Q0∫01c2(ΔP−(α1c+α2c2))⋅dx’>?1=?0+????0?010?2(??(?1?+?2?2))??c1*=c0+LPAc0Q0∫01c2(ΔP−(α1c+α2c2))⋅dx
c 1 * = c 0 + L P A c 0 Q 0 ∫ 0 1 c 2 ( Δ P − ( α 1 c + α 2 c 2 ) ) ⋅ d x


In a typical experiment, SNGFR , Δ P , and c 1 are measured in several nephrons. Most often, these parameters are not obtained from the same nephrons. The mean values for an experiment are inserted into the model to calculate the determinants of SNGFR for an idealized nephron.


From the foregoing description, we see that SNGFR is fully determined by Δ P , Q 0 , c 0 , and LpA . Typical values for these parameters are shown in Table 21.1 for Munich Wistar rats from two different breeding colonies under different volume states. Conceptually, SNGFR can be made to increase by raising Δ P , Q 0 or LpA or by reducing c 0 . But the magnitude of the dependence on each of the four determinants depends on the values of the other three. Some of these interactions are shown in Figures 21.4–21.7 and discussed below.



Table 21.1

Representative Micropuncture Data in Munich Wistar Rats from the Blantz Lab in San Diego and Brenner Lab in Boston.










































































Laboratory State of Hydration SNGFR (nl/min) Δ P (mmHg) Π 0 (mmHg) Q 0 (nl/min) LpA (nl/s/mmHg) Filtration Pressure Equilibrium Reference
Blantz Hydropenia 30 30.5 18.3 86 0.08 * Yes
Euvolemia 31 37.2 19.7 121 0.06 No
Acute 2.5% plasma volume expansion 45 42.2 18.2 177 0.05 No
Brenner Hydropenia 21 35.3 19.4 65 0.08 * Yes
Euvolemia 32 33.4 19.4 114 0.08 * Yes
Acute 5% plasma volume expansion 50 41.2 22.9 201 0.08 No

* Minimum estimate due to filtration pressure equilibrium. LpA values that show differently from the original papers were originally calculated based on a linear estimate of the oncotic pressure profile and are recalculated here using the non-linear model.




Figure 21.4


Pressure (left) and total flux (right) along the length of the glomerular capillary.

In left panel solid curves represent plasma oncotic pressure which rises due to removal of water, while dashed line represents Δ P , which declines by 1 mmHg along the capillary due to flow resistance. SNGFR is the total amount filtered at position 1. Other inputs include systemic plasma protein concentration 5.8 mg/dl, and LpA 0.08 nl/s/mmHg. Filtration ceases where oncotic pressure rises to equal Δ P , which occurs when incoming plasma flow, Q 0 , is low.



Figure 21.5


Δ P (lines) and Π GC (curves) along the length of a glomerular capillary for three different values of Δ P .

Left panel: Q 0 =50 nl/min. Right panel: Q 0 =150 nl/min. Other inputs include systemic plasma protein concentration 5.8 g/dl, and LpA=0.06 nl/s/mmHg. Δ P has little effect on whether or not filtration pressure equilibrium is achieved. In contrast, increasing Q 0 from 50 to 150 nl/min eliminates filtration equilibrium regardless of Δ P .



Figure 21.6


SNGFR , filtration fraction or mean ultra-filtration pressure (P UF ) as a function of nephron plasma flow, Q 0 .

Curves in top panels were generated for three different values of LpA. Curves in the bottom panel were generated for three values of Δ P . Unless otherwise stated, LpA=0.06 nl/s/mmHg; Δ P =35 mmHg; Systemic plasma oncotic pressure=19.5 mmHg. Values over this range occur in the rat under varying levels of volume expansion.



Figure 21.7


Effects of efferent arteriolar resistance ( R E ) on SNGFR , nephron plasma flow ( Q 0 ), and glomerular capillary pressure ( P GC ).

In top right panel, P GC is made an independent variable by manipulating R E . Effects on Q 0 and P GC are nonlinear, because the filtration fraction rises along with R E such that a lesser fraction of the blood flow transits the efferent arteriole. The Q 0 and P GC curves (bottom) are insensitive to LpA. The apex of the SNGFR curve occurs within the physiological range of P GC (48–52 mmHg). Medical treatments for glomerular capillary hypertension that reduce R E until P GC is normal will not reduce SNGFR . However, for P GC <48 mmHg, reducing R E will cause SNGFR to decline.


Ultrafiltration Coefficient, LpA , and Filtration Pressure Equilibrium


If the ratio of LpA to Q 0 is great enough, then Π GC will rise to become arbitrarily near to Δ P at some point along the glomerular capillary, resulting in filtration pressure equilibrium. The remaining capillary surface downstream from the equilibration point will not contribute to the flux. It is possible to infer the presence of filtration equilibrium from micropuncture data, but it is not possible to know at what point along the length of the capillary equilibrium occurs. Therefore, it is not possible to compute actual values for ⟨ P UF ⟩ or LpA for nephrons in filtration equilibrium. When Eqs. (21.4)–(21.8) are applied to data from a nephron in filtration pressure equilibrium, the values generated for LpA and ⟨ P UF ⟩ are respective minimum and maximum estimates for actual LpA and ⟨ P UF ⟩. If a change in LpA occurs while a nephron remains in filtration pressure equilibrium, the equilibrium point will shift along the capillary, but SNGFR will not be affected. In order for SNGFR to be affected by a change in LpA , the nephron must not be in filtration equilibrium.


A debate over whether filtration pressure equilibrium occurs dates back to Homer Smith, who used conjecture and teleology to argue both sides of the issue at different points in his career ( vide supra ). Brenner and colleagues found filtration equilibrium in each of 12 consecutive published series, suggesting that filtration equilibrium is universal for hydropenic or euvolemic Munich Wistar rats. However, contrary data were generated by other micropuncture laboratories. At one point, this led to consternation. The issue was resolved after experiments done with rats exchanged between different laboratories led to the conclusion that filtration equilibrium prevails in some rat strains or breeding colonies but not in others, and that the difference is attributable to differences in LpA . This finding detracts somewhat from teleologic arguments for or against filtration pressure equilibrium.


Nephron Plasma Flow, Q 0


Q 0 does not appear in the Starling equation for water flux ( Eq. (21.3a) ) or in the flux integral that defines SNGFR ( Eq. (21.4) ). Nonetheless, Q 0 is an important determinant of SNGFR . In fact, increased renal plasma flow underlies many of the physiologic increases in GFR that occur in the normal course of life, such as during pregnancy or after protein feeding. SNGFR is the simple product of LpA and ⟨ P UF ⟩ ( Eq. (21.4) ). ⟨ P UF ⟩ becomes greater if the average plasma oncotic pressure along the capillary is less. Removing a given amount of water from the plasma will cause a lesser increase in the plasma oncotic pressure if that water is subtracted from a larger initial plasma volume. Hence, increasing Q 0 will cause oncotic pressure to rise more slowly along the capillary. Therefore, increasing Q 0 causes ⟨ P UF ⟩ to increase. The precise effect of Q 0 on the rate of rise in plasma protein concentration along the nephron is described mathematically in Eq. (21.7). SNGFR will be most sensitive to changes in Q 0 under conditions of filtration pressure equilibrium where the filtration fraction remains constant as Q 0 increases. In filtration disequilibrium, c 1 , ergo filtration fraction, will decline with increasing Q 0 to reduce the impact of Q 0 on SNGFR . Homer Smith recognized that renal plasma flow should affect GFR by this mechanism, and that his experiments ( vide supra ) failed to confirm a plasma flow dependence of GFR only because the particular tools that he employed to manipulate the renal blood flow were confounded by offsetting effects on P GC .


Systemic Plasma Protein Concentration, c 0


In the idealized glomerulus, an isolated change in c 0 will cause opposite changes in ⟨ P UF ⟩ and, therefore, SNGFR . However, it is difficult to demonstrate this experimentally because it is nearly impossible to manipulate oncotic pressure of the arterial plasma without affecting the neurohumoral milieu of the entire body, thereby altering other determinants of SNGFR . In fact, the circumstances associated with low oncotic pressure in real life (e.g., generalized capillary leak, sepsis, malnutrition or nephrosis) are generally associated with a low GFR. When c 0 is manipulated by whatever means, changes in other determinants occur to offset the impact on SNGFR . These changes are discussed below under “Interactions Among the Determinants of SNGFR.”


Hydrostatic Pressure, P GC , and Δ P


Whereas SNGFR is insensitive to LpA when Q 0 is low and insensitive to Q 0 when LpA is low, SNGFR will always be sensitive to an isolated change in Δ P unless Δ P is so low as to be exceeded by the incoming plasma oncotic pressure, in which case SNGFR will be zero. This is true because the proportional increase in ⟨ P UF ⟩ brought about by any increment in Δ P−Π 0 is relatively insensitive to the other determinants of SNGFR. This is illustrated in Figure 21.5 and in the lower half of Figure 21.6 .


The interposition of the efferent arteriole between the glomerulus and peritubular capillary provides a simple mechanism for regulating Δ P independently of Q 0 . Furthermore, this arrangement provides an opportunity to elicit reciprocal changes in P GC and pressure in the downstream peritubular capillary P PTC . Tying an increase in P GC to a decrease in P PTC has teleologic appeal, as this will facilitate homeostasis of the effective circulating blood volume while stabilizing GFR. If the efferent arteriole reacts to sustain P GC and reduce P PTC during a decline in renal perfusion pressure or effective circulating blood volume, then GFR will be relatively spared from declining while filtration fraction will increase, thus affecting both the hydraulic and oncotic components of the Starling force that drives reabsorption by the peritubular capillary.


Regulating the efferent arteriole in this way is largely the purview of the renin–angiotensin system, which figures prominently among the myriad neurohumoral mechanisms contained in models of blood pressure and salt homeostasis. Angiotensin II is antinatriuretic and constricts arterioles throughout the body but, on balance, its effect on the glomerulus is always to elevate Δ P . Thus, in spite of being a renal vasoconstrictor, angiotensin II protects GFR from total decline when the arterial blood pressure is low or when the preglomerular resistance is high.


While unduly low P GC must impair glomerular filtration, P GC and SNGFR are poorly correlated under normal circumstances, as are P GC and arterial blood pressure. This implies that the kidney generally protects P GC against the influence of arterial blood pressure and employs determinants other than Δ P to effect physiologically those changes in SNGFR that normally occur throughout life. Furthermore, it has recently been demonstrated that the preglomerular myogenic elements, long associated with static renal blood flow autoregulation, efficiently buffer the glomerular capillary against systolic pressure pulses delivered at the heart-rate frequency. Teleologic reasoning behind sheltering the glomerular capillary from high pressure is that high P GC augments wall stress in the glomerular capillary, which elicits a trophic response. If unchecked, this response will ultimately sclerose and destroy the glomerulus. Therefore, high P GC is always pathologic, and treating glomerular capillary hypertension has been a cornerstone of nephrology practice for more than two decades. Some examples of glomerular capillary hypertension include angiotensin II-mediated hypertension, experimental glomerulonephritis, and residual nephrons after subtotal nephrectomy. It has been asserted, and commonly accepted, that glomerular capillary hypertension also underlies glomerular hyperfiltration in early diabetes mellitus. However, there are more than 10 published micropuncture studies in which diabetic hyperfiltration occurred in the absence of glomerular capillary hypertension or in which glomerular capillary hypertension was treated with no mitigating effect on diabetic hyperfiltration. This does not detract from the salutary effect of therapy to reduce P GC , which applies to all glomerular diseases.


Interactions Among the Determinants of SNGFR


According to the standard model of Deen and Brenner, SNGFR is completely determined by a set of four parameters, which include P , Q 0 , LpA , and c 0 (or Π 0 ). To state that SNGFR can be calculated from these four determinants is a mathematical truism which requires no consideration of how the four determinants might correlate in actual physiology. In fact, the individual components of the glomerular microvasculature that influence determinants of SNGFR generally affect more than one of them at a time. For example, an isolated increase in resistance of the preglomerular arteriole will directly reduce both Δ P and Q 0 and will reliably reduce SNGFR . In contrast, an isolated increase in resistance of the efferent arteriole will directly augment Δ P and reduce Q 0 . Since these two effects exert opposing influences on SNGFR , increasing efferent arteriolar resistance might cause SNGFR to increase, decrease or remain the same, depending on other circumstances. For example if P GC is low enough to be at or below Π GC , then raising the efferent resistance can only cause SNGFR to increase whereas, if efferent resistance increases toward infinity, Q 0 must tend toward zero while P GC cannot exceed the arterial blood pressure and, therefore, SNGFR must decline. The point where the impact of increasing the efferent resistance switches from positive to negative is within the domain of values that occur in vivo . Much of the acute renal failure encountered in contemporary medical practice occurs when drugs that reduce the ratio of efferent:afferent resistance are taken by patients who operate to the left of that point (see Figure 21.7 ).


There are other correlations between determinants of SNGFR that are more difficult to explain. For example, since LpA is computed as the ratio of SNGFR to ⟨ P UF ⟩, random uncorrelated errors in Δ P and SNGFR will cause an inverse correlation to appear between Δ P and LpA . Also, an isolated reduction in LpA , if sufficient to reduce SNGFR , will remove a shunt pathway for fluid to bypass the efferent arteriole, thereby increasing Δ P . Furthermore, the physical orientation of the glomerular mesangium relative to the intraglomerular portion of the efferent arteriole allows activation of the same contractile elements to simultaneously reduce LpA and constrict the efferent arteriole. This appears to explain why angiotensin II, the prototype effector of glomerular hemodynamics, simultaneously increases Δ P and reduces LpA , and why lyzing mesangial cells with an antibody negates the effects of angiotensin II on both Δ P and LpA .


Another interesting interaction among two determinants of SNGFR involves Δ P and the systemic plasma oncotic pressure Π 0. As mentioned above, experiments targeted at confirming the role of Π 0 as a determinant of SNGFR are encumbered by the difficulty in manipulating the plasma protein concentration independent of the neurohumoral environment. To get around this, Brenner and Blantz drew upon a wide variety of infusion and exchange protocols to alter the systemic plasma protein concentration in multiple ways that were likely to yield contrary effects on the effective circulating volume and hematocrit. These differences were roughly intended to cancel each other out and reveal the underlying impact of Π 0 on SNGFR , Δ P , Q 0 , and LpA . Both groups of investigators confirmed that the four determinants of SNGFR are not independent of one another. In particular, they discovered that, regardless of the experimental means for invoking a change in Π 0 , a change in Π 0 causes a parallel change in Δ P , and reciprocal change in LpA . In contrast, neither SNGFR nor Q 0 are predictably tied to Π 0 . Most remarkably, Δ P is so strongly dependent on Π 0 that the afferent effective filtration pressure, Δ P−Π 0 , is independent of Π 0 . A biophysical or anatomic explanation for this interaction between Π 0 and Δ P has not been forthcoming. It seems, instead, that a physiological mechanism is involved in autoregulating the afferent effective filtration pressure (see Table 21.2 ).



Table 21.2

Multivariate Regression Applied to Combined Micropuncture Data from Brenner and Blantz to Test for Interactions between Δ P and the other Determinants of SNGFR *






















Dependent Variable Independent Terms in Multivariate Regression
P (mmHg) Π 0 (mmHg) LpA (nl/sec/mmHg) Q 0 (nl/min)
Regression coefficient 0.84±0.09 −72±10 ~0
P-Value associated with regression coefficient 2×10 −10 1×10 −10 0.750

* The original regression model included the protocol for manipulating the systemic plasma protein and the lab where the work was performed, neither of which influenced the result.



Brenner and Blantz also both observed an inverse correlation between Π 0 and LpA . It is possible that this is an artifact of concentration polarization which causes plasma proteins to accumulate near the capillary wall. Concentration polarization will lead to an overestimate of P UF , because the calculation of P UF will be based on a lower value of Π than is present at the plasma interface with the capillary wall. Using an overestimate of ⟨ P UF ⟩ in Eq. (21.4) will lead to an underestimate for LpA. If concentration polarization occurs in the glomerular capillary, the effect will be greatest when c 0 is least. Hence, the appearance could arise of an inverse dependence of LpA on c 0 , even though c 0 has no actual effect on the capillary wall. However, there is likely to be enough scrubbing by red blood cells to prevent concentration polarization. Furthermore, concentration polarization cannot explain the correlation between Π 0 and Δ P . Finally, we have already discussed a mechanism for the inverse relationship between Δ P and LpA . Therefore, the inverse correlation of between LpA and Π 0 might arise because LpA is affected, as an innocent bystander, by a mechanism that is postulated to autoregulate afferent P UF .




The Filtration Barrier and Filtration of Macromolecules


A striking feature of glomerular filtration is the ability of the capillary wall to discriminate among molecules of varying size. Solutes up to the size of inulin pass freely from the plasma to Bowman’s space, while passage becomes progressively difficult for substances that are larger such that all but the smallest plasma proteins are screened almost entirely. Hence, when describing the determinants of SNGFR, the concentration of macromolecules in the filtrate is low enough that these contribute negligibly to the Starling forces. However, the filtration of small amounts of macromolecules is important for other reasons. For example, the plasma transiting a normal pair of human kidneys in one day contains 50,000 grams of protein, while the appearance of one gram per day of protein in the urine is sufficient to establish the presence of glomerular disease. Discerning how this small amount of protein winds up in Bowman’s space is key to explaining how the filtration barrier operates normally, and to understanding the physical aspects of glomerular disease.


The filtration of a macromolecule is most often quantified in terms of its sieving coefficient, Θ , which is the ratio of solute concentration in the filtrate relative to filtrand. The earliest direct test for protein in mammalian glomerular filtrate was performed by Walker in 1941, who reported that micropuncture fluid obtained from Bowman’s space in rat, guinea pig or opossum contained “either no protein or, at most, very small amounts.” The assay in use at the time would have detected an overall sieving coefficient for protein of 0.4%, which made it insensitive by later standards. Several subsequent micropuncture studies during the 1970s yielded widely varying amounts of albumin in proximal tubular fluid, even within an experiment. This variability was reasonably ascribed to contamination of samples by extratubular proteins, since only 1% contamination of a tubular fluid sample with plasma from the peritubular capillary would markedly alter the apparent result. Performing micropuncture with a system of concentric pipettes to reduce contamination, Tojo succeeded in confirming a strong inverse relationship between TF/P inulin and albumin concentration in fluid collected from rat proximal tubules due to removal of filtered albumin along the proximal tubule. By linear extrapolation to TF/P inulin of unity, which represents Bowman’s space, the sieving coefficient for albumin was estimated at 0.062%. Meanwhile, the sieving coefficient of low molecular weight proteins was almost 99%, confirming the size-selective nature of protein sieving (see Figure 21.8 ). However, micropuncture remains an unwieldy technique for studying glomerular sieving of macromolecules, and most of what is known about glomerular sieving has been learned by other means ( vide infra ).




Figure 21.8


Albumin (a) and low molecular weight protein (LMWP) delivery (b) along the nephron.

Linear regression over the domain of TF/Pinulin from 1 to 2 was used to calculate the protein concentration in Bowman’s space

(from ref. ).


Pore Theory


The glomerular capillary wall clearly screens solutes according to size, and it is usually taught that the glomerulus also screens macromolecules according to charge. The physical basis of molecular sieving is often modeled using pore theory. In pore theory, the coupled flux equations ( Eq. (21.3a,b) ) are modified to fit an idealized model of the glomerular capillary wall. The usual model consists of a solid barrier perforated by cylindrical pores. In some models, the pores form a homogenous population, while other models employ a mixture of pores of different sizes. This paradigm for describing the flow of material through porous membranes was applied to diffusion and filtration of solutes by capillaries, and an early review of the subject was published by Pappenheimer in 1953. Mathematical descriptions were developed for pores shaped like circular cylinders or rectangular slits. At about the same time, theories were developed by others to explain the migration of large solutes through fibrous gels. Given what is now known about the physical structure of the glomerular capillary wall, pores, slits, and fibrous gels are all relevant to glomerular sieving. Hence, descriptions based solely on cylindrical pores have become more obviously phenomenological. Nonetheless, pore theory remains the most popular paradigm for describing nuances of glomerular sieving in humans.


Pore theory builds on the Kedem–Katchalsky flux equations ( Eq. (21.3a,b) ) by associating the three membrane parameters, Lp , P s , and σ s with an idealized physical structure. This structure incorporates the membrane geometry, the Stokes–Einstein radius of the solute molecules, temperature, viscosity, and the Boltzman constant. The solutes are represented by rigid spheres that interact with the solvent medium and with the pores, but not with each other. This makes the problem mathematically tractable. The filtration properties of a membrane with circular pores turn out to depend on two things: (1) the ratio of pore diameter to membrane thickness; and (2) the overall fraction of the membrane area covered by pores.


An explanation begins with Fick’s first law of diffusion for a solute s :


<SPAN role=presentation tabIndex=0 id=MathJax-Element-17-Frame class=MathJax style="POSITION: relative" data-mathml='Jsdiffusion=−DsdCsdx’>???????????=???????Jsdiffusion=−DsdCsdx
J s d i f f u s i o n = − D s d C s d x
where:
<SPAN role=presentation tabIndex=0 id=MathJax-Element-18-Frame class=MathJax style="POSITION: relative" data-mathml='Ds=RTfN’>??=????Ds=RTfN
D s = R T f N
is the diffusion coefficient, N is Avogadro’s number, and f is the frictional force that opposes diffusion of s .


According to Stokes’ law, a sphere of radius a falling at unit velocity in a medium of viscosity:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-19-Frame class=MathJax style="POSITION: relative" data-mathml='Jsdiffusion=D0ApδΔCs’>???????????=?0??????Jsdiffusion=D0ApδΔCs
J s d i f f u s i o n = D 0 A p δ Δ C s
η faces a frictional force given by:
<SPAN role=presentation tabIndex=0 id=MathJax-Element-20-Frame class=MathJax style="POSITION: relative" data-mathml='f=6πηa’>?=6???f=6πηa
f = 6 π η a


Einstein combined Fick’s law of diffusion and Stokes law to derive the diffusion coefficient for a spherical molecule in free solution, D 0 , in terms of its molecular radius a :


<SPAN role=presentation tabIndex=0 id=MathJax-Element-21-Frame class=MathJax style="POSITION: relative" data-mathml='D0=RT6πηaN’>?0=??6????D0=RT6πηaN
D 0 = R T 6 π η a N


Since actual molecules are not spherical, the Stokes–Einstein radius, a , of a molecule is a virtual quantity represented by a sphere of equivalent radius. Using the Stokes–Einstein radius of a marker solute to estimate the size of pores in a membrane will overestimate the actual pore radius if the marker solute is flexible and can squeeze through a smaller pore than a rigid sphere with the same Stokes-Einstein radius.


For the diffusion of small molecules through a membrane that contains large pores, Fick’s first law is rewritten:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-22-Frame class=MathJax style="POSITION: relative" data-mathml='Jsdiffusion=D0ApδΔCs’>???????????=?0??????Jsdiffusion=D0ApδΔCs
J s d i f f u s i o n = D 0 A p δ Δ C s
where A p is the fraction of the membrane surface covered by pores and δ is the membrane thickness. Therefore, the diffusion through a membrane of a small solute with known D 0 is a convenient method to determine the ratio of pore area to thickness for the membrane.


However, when the molecular radius of the solute molecule is on the same scale as the pore size, diffusion through the membrane is less than predictable from Eq. (21.13) . In other words, mobility of the solute is restricted. There are two factors that contribute to restricted mobility. First, there is steric hindrance to the solute entering or residing within the pore. Second, solute molecules within the pore experience greater friction than predicted by Stokes’ law for solute molecules in free solution. While it makes sense to us to represent the steric hindrance by a reduced effective pore area and the increased friction as a reduced effective diffusion coefficient, most of the literature combines both effects into an expression for the effective pore area. This is described below.


Diffusion through a Porous Membrane: Steric Hindrance and Altered Friction


Restricted passage of solutes of increasing molecular radius is a basic property of membrane structures made of impermeable matrix with pores or fibrous gels. The basis for molecular sieving in all cases is the exclusion of large solute molecules from a portion of the membrane that is otherwise available to be occupied by water and other small molecules. The formulae for describing steric hindrance are different for pores than for gels. Here we will describe the phenomenon for cylindrical pores.


The center of a spherical molecule cannot approach any closer than its own radius to the edge of any pore. Hence, the fraction of a cylindrical pore, area Vp and radius r , that is available to be occupied by a solute with molecular radius a , is:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-23-Frame class=MathJax style="POSITION: relative" data-mathml='VVp=π(r−a)2πr2=(1−ar)2′>???=?(??)2??2=(1??)2VVp=π(r−a)2πr2=(1−ar)2
V V p = π ( r − a ) 2 π r 2 = ( 1 − a r ) 2


Introducing solvent flow makes the steric hindrance more complex. This was first addressed by Ferry who added a term to account for laminar flow within the pore :


<SPAN role=presentation tabIndex=0 id=MathJax-Element-24-Frame class=MathJax style="POSITION: relative" data-mathml='VVp=(1−ar)2⋅(2−(1−ar)2)’>???=(1??)2(2(1??)2)VVp=(1−ar)2⋅(2−(1−ar)2)
V V p = ( 1 − a r ) 2 ⋅ ( 2 − ( 1 − a r ) 2 )


The frictional drag on a solute molecule moving through a pore is also different from that described by Stokes’ law for a solute in an unbounded free solution. The drag according to Stokes’ law for the unbounded condition and the drag encountered in a pore are given by Eq. (21.16a,b) for a solute moving with velocity u in a fluid with velocity v . k 1 and k 2 are component drag coefficients that weight the contributions of the particle and fluid velocities. k 1 and k 2 are functions of a/r .


Theoretical treatments have provided approximate solutions for k 1 and k 2 for particles in cylindrical tubes. Determining values for k 1 and k 2 is computationally intense. For this reason, k 1 and k 2 were originally provided for only a few values of a/r and a polynomial equation was fit to these points to allow interpolation. This approximation expression was inaccurate for a/r >0.6, but this was the only method available until better computers were built in the 1970s.


<SPAN role=presentation tabIndex=0 id=MathJax-Element-25-Frame class=MathJax style="POSITION: relative" data-mathml='funbounded=6πηa(u−v)’>??????????=6???(??)funbounded=6πηa(u−v)
f u n b o u n d e d = 6 π η a ( u − v )

<SPAN role=presentation tabIndex=0 id=MathJax-Element-26-Frame class=MathJax style="POSITION: relative" data-mathml='fpore=6πηa(u⋅k1−v⋅k2)’>?????=6???(??1??2)fpore=6πηa(u⋅k1−v⋅k2)
f p o r e = 6 π η a ( u ⋅ k 1 − v ⋅ k 2 )


Accounting for steric hindrance and dividing Eq. (21.16b) by Eq. (21.16a) to correct for the departure from Stokes’ law, the diffusive flux is rewritten from Eq. (13) to become:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-27-Frame class=MathJax style="POSITION: relative" data-mathml='Jsdiffusion=D0Apδ((1−ar)2(2−(1−ar)2))·(u−vuk1−vk2)ΔCs=D0AeffδΔCs’>???????????=?0???((1??)2(2(1??)2))·(????1??2)???=?0????????Jsdiffusion=D0Apδ((1−ar)2(2−(1−ar)2))·(u−vuk1−vk2)ΔCs=D0AeffδΔCs
J s d i f f u s i o n = D 0 A p δ ( ( 1 − a r ) 2 ( 2 − ( 1 − a r ) 2 ) ) · ( u − v u k 1 − v k 2 ) Δ C s = D 0 A e f f δ Δ C s
where A eff represents the “effective” pore area and depends on a/r as well as the solute and bulk flow velocities. The combined effects of steric hindrance and friction on A eff are illustrated in Figure 21.9 using published values for k 1 and k 2 . The approximation equation used by Landis and Pappenheimer in 1963 is also shown where:
<SPAN role=presentation tabIndex=0 id=MathJax-Element-28-Frame class=MathJax style="POSITION: relative" data-mathml='funboundedfpore=1−2.104⋅(ar)+2.09(ar)3−0.95(ar)5+⋯’>???????????????=12.104(??)+2.09(??)30.95(??)5+funboundedfpore=1−2.104⋅(ar)+2.09(ar)3−0.95(ar)5+⋯
f u n b o u n d e d f p o r e = 1 − 2.104 ⋅ ( a r ) + 2.09 ( a r ) 3 − 0.95 ( a r ) 5 + ⋯



Figure 21.9


Ratio of effective pore area A eff to physical pore area A P that applies to diffusion of solute with radius a through circular pore with radius r .

The model accounts for steric hindrance and for friction; u and v are respective solute and bulk flow velocities. Friction is calculated based on published coefficients. L–P refers to result generated by older method of Landis and Pappenheimer, which works well for a/r <0.6. Results shown in semi-log (left) and linear (right) formats.


From Figure 21.9 it is clear that restricted diffusion will cause a membrane to discriminate between two solute molecules of different radii, even when the radii of both are considerably less than the radius of the pore. Also, note that Eq. (21.17) reduces to Eq. (21.13) in the absence of bulk flow and as a/r approaches zero.


Pore Theory and Hydrodynamic Flow


It is allowable to describe bulk flow within a cylindrical pore using Poiseulle’s law as long as the pore radius is several-fold the radius of a water molecule. Accordingly:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-29-Frame class=MathJax style="POSITION: relative" data-mathml='q=−πr48ηdPdy’>?=??48?????q=−πr48ηdPdy
q = − π r 4 8 η d P d y
where q represents bulk flow within a single pore and dP/dy is the axial pressure gradient along the pore. For flow per unit area across a membrane, pressure is replaced by the Starling forces such that:
<SPAN role=presentation tabIndex=0 id=MathJax-Element-30-Frame class=MathJax style="POSITION: relative" data-mathml='Jv=nπr48ηδ(ΔP−ΔΠ)=Awr28ηδ(ΔP−ΔΠ)’>??=???48??(????)=???28??(????)Jv=nπr48ηδ(ΔP−ΔΠ)=Awr28ηδ(ΔP−ΔΠ)
J v = n π r 4 8 η δ ( Δ P − Δ Π ) = A w r 2 8 η δ ( Δ P − Δ Π )
where n is the number of pores per unit area and A w is the restricted pore area available to water. Comparing Eq. (20) to Eq. (3a), the hydraulic permeability L p for an isoporous membrane is given in terms of the pore radius r and the ratio of total pore area to membrane thickness A w /δ.
<SPAN role=presentation tabIndex=0 id=MathJax-Element-31-Frame class=MathJax style="POSITION: relative" data-mathml='Lp=Awr28ηδ’>??=???28??Lp=Awr28ηδ
L p = A w r 2 8 η δ


Combining Bulk Flow and Restricted Diffusion


The solute flux equation ( Eq. (21.3b) ) includes terms for diffusion and advection. Advective transport and restricted diffusion occur simultaneously in the glomerulus, and each contributes to the presence of large molecules in the filtrate. Furthermore, convection and diffusion are coupled, and this coupling must be unraveled for a full understanding of glomerular sieving. We shall present two approaches to this that are both based on pore theory and rely on the sieving coefficient, Θ , to draw inferences regarding the filtration barrier. The first approach is the early work of Pappenheimer, and the second approach is that of Chang and Deen who based their method on the prior work of Patlak. It is the latter approach to interpreting sieving data that is used by most authors who publish in the physiology or clinical literature nowadays.


There are some intuitive features of glomerular sieving, and some that are not so intuitive. First, it is intuitive that Θ cannot be a negative number, nor can it exceed unity. If Θ exceeds unity, a model other than pore theory is required to explain the flux. It is also intuitive that Θ will approach unity for small solutes, and that Θ will equal zero for any solute that is larger than the largest pore. A feature that is not so intuitive is how intermediate Θ can arise from an isoporous membrane, although we have shown intermediate A eff for an isoporous membrane in Figure 21.9 . Intermediate Θ also owes to the effect of filtration rate on molecular sieving, which appears as the second term in Eq. (21.3b) . If the passage of a solute is restricted relative to the passage of water, then the filtrate will become diluted during filtration. This will give rise to a concentration gradient for diffusion. Thus, the overall sieving coefficient is determined by competition between the filtration rate, which tends to dilute the filtrate, and the restricted diffusion, which fights to reduce the concentration difference that arises from molecular sieving.


It is intuitive that, if A eff is non-zero and J v is low enough, then solute will eventually equilibrate between the plasma and Bowman’s space ( Θ =1). It is also intuitive that, at high rates of J v , Θ will approach the ratio of restricted pore area for solute relative to water. Pappenheimer provided a quantitative expression for Θ that satisfies these two conditions. This derivation begins with a bulk-flow sieving step to create an initial filtrate to plasma concentration ratio:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-32-Frame class=MathJax style="POSITION: relative" data-mathml='CfitrateCplasma=(1−σs)=AsAw’>???????????????=(1??)=????CfitrateCplasma=(1−σs)=AsAw
C f i t r a t e C p l a s m a = ( 1 − σ s ) = A s A w

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Jun 6, 2019 | Posted by in NEPHROLOGY | Comments Off on Biophysical Basis of Glomerular Filtration
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