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Urinary Concentration and Dilution: Models

John L. Stephenson

10.1002/cphy.cp080230

Source: Supplement 25: Handbook of Physiology, Renal Physiology

Originally published: 1992

Published online: January 2011

Full Article on Wiley Online Library



Abstract

The sections in this article are:

1 Whole‐Body Water and Solute Balance
1.1 Osmolal Clearance
1.2 Effect on Plasma Osmolality of Urine Flow of a Given Osmolality
1.3 Hypertonic Urine Implies a Hypotonic Tubular Reabsorbate and Vice Versa
1.4 Free Water Clearance
1.5 Antidiuretic Hormone Controls Water Excretion
2 Functional Implications of Renal Architecture
2.1 Production of Hyperosmotic Urine Is Related to the Loop of Henle
2.2 Significance of the S‐Shaped Configuration of the Mammalian Nephron for the Concentration of Urine
2.3 New Functional Features Introduced by the S‐Shaped Configuration
3 A Short History of Models of the Renal Counterflow System
3.1 Active Water Transport
3.2 Concentration by Differential Permeabilities and Counterflow
3.3 Concentration by Multiplication of a “Single Effect” along the Loop of Henle
3.4 Active Salt Transport out of the AHL Is the Probable Single Effect
3.5 Experimental Predictions of the Kuhn Models
3.6 Problems with the Kuhn Models
3.7 Vasa Recta Models
3.8 Nephrovascular Models
4 Equations for the Renal Medulla
4.1 Differential Equations
4.2 Integrated Mass Balance Equations
4.3 Generalized Multiplication Rule
4.4 Nephrovascular Coupling
4.5 Transmural Fluxes
5 Analytical Solution of the Differential Equations
5.1 Solutions for Single Flow Tubes
5.2 Analytical Solutions of Multitube Counterflow Problems
5.3 Solutions for Central Core Model
5.4 Externally Driven Central Core Multiplier
5.5 Multiplication in Inner and Outer Medullae
5.6 Non‐Zero Diffusion Coefficient
5.7 Solution of Time‐Dependent Equations
6 Numerical Solution of the Differential Equations
6.1 Method of Lines
6.2 Shooting Method
6.3 Quasilinearization
6.4 Finite Difference Methods
7 Results of Computer Simulation
7.1 Origin of Inner Medullary Concentration Gradient
7.2 Role of Collecting Duct
7.3 Role of Vasa Recta
7.4 Transition from Diuresis to Antidiuresis
8 Free‐Energy Balance in Renal Counterflow Systems
9 Summary and Future Directions
Figure 1. Figure 1.

Response of whole kidney to varying rates of vasopressin infusion in rats. Note the dramatic decrease in urine flow rate at higher rates of vasopressin infusion with minimal change in osmolar clearance

Data from Atherton et al. 3. From Knepper and Stephenson 111
Figure 2. Figure 2.

Relation between relative medullary thickness and maximum urinary osmolality. Long loops of Henle are required to make highly concentrated urine

Data from Schmidt‐Nielsen and O'Dell 179. From Burg and Stephenson 20
Figure 3. Figure 3.

Prototype three‐section glomerular nephron. With its straight configuration, solute and water transport from different segments is uncoupled. In the proximal tubule (PT) with a large effective hydraulic permeability, Lp, absorption is nearly isotonic; in the diluting segment (DIL) a smaller Lp limits water absorption relative to solute absorption, and tubule fluid is diluted; in final segment, hydraulic permeability is under control of antidiuretic hormone (ADH). In the absence of ADH tubule fluid remains dilute, and in the presence of ADH it equilibrates osmotically with the surrounding interstitium. Thus final urine can vary from markedly hypotonic to isotonic relative to plasma. Black arrows indicate solute movement; white arrows, volume flow.

Reproduced, with permission, from Stephenson 201, Annual Review of Biophysics and Bioengineering, Vol. 7. © 1978 by Annual Reviews, Inc
Figure 4. Figure 4.

The S‐shaped mammalian nephron. The essential functional feature that allows concentration of urine is juxtaposition of the diluting segment (DIL; now ascending limb of Henle's loop) with the water‐permeable descending limb and antidiuretic hormone (ADH)‐sensitive collecting duct. This permits solute supplied by ascending limb to extract water from collecting duct and descending limb. Black arrows indicate solute movement; white arrows, volume flow; PT, proximal tubule.

Reproduced, with permission, from Stephenson 201, Annual Review of Biophysics and Bioengineering, Vol. 7. © 1978 by Annual Reviews, Inc
Figure 5. Figure 5.

Initial stage of basic scheme for concentrating by permselective membranes. A very large volume of phenol solution at concentration Co is separated by membrane P permeable to phenol but not to water (rubber in experiments of Kuhn and Ryffel) from a large volume of sucrose solution, also at concentration Co. The sucrose solution is separated by membrane W permeable to water but not to sucrose or to phenol (copper ferrocyanide) from smaller volumes of phenol and sucrose solutions, also at concentration Co. As indicated by arrows, phenol diffuses into sucrose solution, increasing its osmolality and causing water to be osmotically extracted from the smaller volume solutions.

Adapted from Kuhn and Ryffel 122
Figure 6. Figure 6.

Basic scheme for concentrating by permselective membranes at equilibrium. Phenol has been added to the original sucrose solution so that its concentration has approximately doubled. Water has been extracted from the smaller volumes of sucrose and phenol so that their volumes are halved and concentrations doubled.

Adapted from Kuhn and Ryffel 122
Figure 7. Figure 7.

Arrangement for parallel processing of counterflowing sucrose and phenol solutions to yield progressive concentration of the sucrose solution. In the first apparatus on the left, sucrose and phenol, both at concentration Co, interact through a phenol‐permeable membrane P to give a combined solution of concentration 2Co. This interacts with counterflowing sucrose solution in the first apparatus on the right through a water‐permeable membrane W to give a sucrose solution of concentration 2Co. The phenol is then removed by diffusion into a concurrent flow of any solution at concentration Co through a second membrane P. As the solution is processed from left to right, there is progressive concentration of the sucrose solution. In further stages to the right, phenol reservoirs are not shown in order to simplify the diagram. Kuhn and Ryffel suggested that this scheme might mimic the operation of the kidney.

Adapted from Kuhn and Ryffel 122
Figure 8. Figure 8.

Actual scheme used by Kuhn and Ryffel. See text for explanation.

Adapted from Kuhn and Ryffel 122
Figure 9. Figure 9.

Hargitay and Kuhn countercurrent multiplier without withdrawal. Solution at concentration C1 enters at the left and reverses flow through hairpin constriction at the right to return to the left. The two counterflowing solutions are separated by a waterpermeable and solute‐impermeable membrane. Mass balance requires that at any position x along the tube C1 = C2 and F1 = ‐ F2. This means that the pressure difference Δp, the single effect, is unapposed by any osmotic force and will drive water from the right‐flowing to the left‐flowing stream at a rate that depends on the permeability of the membrane, thus concentrating the right‐flowing and diluting the left‐flowing stream.

Adapted from Hargitay and Kuhn 65
Figure 10. Figure 10.

Hargitay and Kuhn multiplier with withdrawal. A fraction fu of the combined descending limb (DL) and collecting duct (CD) flow is withdrawn at the loop. Mass balance now requires that C1 = C3 ≫ C2. This creates an osmotic force opposing the single effect and reduces the concentration. The degree of the reduction depends on the ratio of the single effect to the withdrawal. AL, ascending limb.
Figure 11. Figure 11.

Comparison of classic representation of concentration build‐up in solute cycling multiplier (A) with calculated profiles (B). Multiplication by counterflow is apparent, but the concentration difference between counterflowing limbs varies with both time and position. “Pump” and “leak” were adjusted to give a limiting “single effect” of 200 mOsm/liter.

Reprinted with permission from Garner et al. 47, Bulletin of Mathematical Biology, Vol. 40, “Transient behaviour of the single loop solute cycling model of the renal medulla.” © 1978, Pergamon Press
Figure 12. Figure 12.

Relative osmolalities of tubular fluid in slices from kidneys of hydropenic rats. Values are given as percent of maximum. O.Z, outer zone; I.Z, inner zone (as modified for ref. 111 from ref. 260).

From Knepper and Stephenson 111
Figure 13. Figure 13.

Composition of renal medulla and urine during antidiuresis and water diuresis (adapted from Hai and Thomas 63 for 20.

From Knepper and Stephenson 111
Figure 14. Figure 14.

Comparison of osmolalities of fluids collected by micropuncture near the tip of the inner medulla in nine hamsters, one kangaroo rat, and one Psammomys.

From Gottschalk and Mylle 58
Figure 15. Figure 15.

Osmolality ratio (tubule fluid/plasma) of fluid collected by micropuncture from seven rats during antidiuresis. Early tubule fluid was dilute even though urine was concentrated. Different symbols refer to different rats.

From Gottschalk and Mylle 58
Figure 16. Figure 16.

Countercurrent exchanger consisting of ascending vas rectum (AVR) and descending vas rectum (DVR). Solute is added to surrounding interstitium at rate J per unit length. Solute entering AVR from interstitium is recycled diffusively to DVR, with resulting build‐up of concentration at the loop.
Figure 17. Figure 17.

Diagram of two‐loop counterflow system with parallel Henle's loop and vas rectum.

Reprinted by permission from Stephenson 190, Nature, Vol. 206, pp. 1215–1219. © 1965 Macmillan Magazines Ltd
Figure 18. Figure 18.

A model of a vasa recta bundle. Direction of flow is indicated by arrows.

From Marsh and Segel 138
Figure 19. Figure 19.

Schematic representation of the intricate counterflow system of the renal medulla. Note that in this figure AVR denotes arterial vas rectum (or descending vas rectum; VVR, venous vas rectum (or ascending vas rectum); DL, descending limb of Henle's loop; AL, ascending limb of Henle's loop; CD, collecting duct; CAP, capillary.

From Kriz and Lever 119
Figure 20. Figure 20.

a: Central core model of the renal medulla. Black arrows indicate salt movement; open arrows, water movement; hatched arrows, urea movement. With sufficiently large solute and water permeabilities of ascending vasa recta (AVR) and descending vasa recta (DVR), concentrations in AVR, DVR, and interstitium become nearly identical at a given medullary level. Solute removal is then given by the product of the concentration and the volume flow difference in AVR and DVR. Functionally, AVR, DVR, and interstitium are merged into a single tube closed at the papillary tip and open at the junction of medulla and cortex. CD, collecting duct; AHL, ascending limb of Henle's loop; DHL, descending limb of Henle's loop. b: shows various core components being hypothetically merged to give the final cross‐sectional configuration shown in c.

Reprinted from Stephenson 193, Kidney International, with permission
Figure 21. Figure 21.

Skeletonized central core model. Black arrows indicate solute movement; white arrows, volume flow. AHL, ascending limb; DHL, descending limb, CD, collecting duct.

From Stephenson 206
Figure 22. Figure 22.

Single osmotic effect in two modes of operation. Mode I, active transport (upper titles); mode II, mixing (bracketed); initial (a) and final (b) equilibrium states are shown. Open arrow indicates water movement; closed arrow, salt movement.

Reprinted from Stephenson 193, Kidney International, with permission
Figure 23. Figure 23.

Configuration for Peskin‐Layton single nephron model. By assumption, C1(γ) = C3(γ) = C(γ) = J2s/Jv.

From Layton 124
Figure 24. Figure 24.

Structure of model 1 of Wexler, Kalaba, and Marsh. Top: cross‐section showing connections between tubules, blood vessels, and interstitial space. Bottom: vertical aspect. DVR, descending vasa recta; AVR, ascending vasa recta; DHL, descending limb of Henle's loop; AHL, ascending limb of Henle's loop; DT, distal tubule; CD, collecting duct; I, interstitium. OM, outer medulla; IM, inner medulla.

From Wexler et al. 252
Figure 25. Figure 25.

Structure of model 3 of Wexler, Kalaba, and Marsh. Top: cross‐section showing connections between tubules, blood vessels, and interstitial space. Bottom: vertical aspect. DVR, descending vasa recta; AVR, ascending vasa recta; DHL, descending limb of Henle's loop; AHL, ascending limb of Henle's loop; DT, distal tubule; CD, collecting duct; I, interstitium; OM, outer medulla; IM, inner medulla.

From Wexler et al. 252
Figure 26. Figure 26.

Interstitial concentrations of sodium chloride and urea predicted by model 3 of Wexler, Kalaba, and Marsh.

From Wexler et al. 252
Figure 27. Figure 27.

Configuration of two‐nephron, central core, electrolyte model of renal medulla of Stephenson, Zhang, and Tewarson. DHL, descending limb of Henle's loop; AHL, ascending limb of Henle's loop; CD, collecting duct; DN, distal nephron.

From Stephenson et al. 213
Figure 28. Figure 28.

Core osmolalities for various modifications of rabbit and hamster data. Top: A, unmodified; B, urea permeability of descending limb reduced; C, urea permeability of ascending limb reduced; D, urea permeability of both descending and ascending limbs reduced. Bottom: E, unmodified; F, urea permeability of both descending and ascending limbs reduced; G, urea permeabilities as in F, descending limb electrolyte permeabilities also reduced.

From Stephenson et al. 213
Figure 29. Figure 29.

Effect of varying urea permeability of thin descending limb of Henle's loop on core osmolality at papilla and at junction of inner and outer medullae.

From Stephenson et al. 213
Figure 30. Figure 30.

Model of Chandhoke, Saidel, and Knepper. Renal cortex is divided into a medullary ray and a labyrinth. Renal arterial blood (afferents) flows to four groups of glomeruli in cortical labyrinth: those of superficial nephrons (G1), midcortical short nephrons (G2), midcortical long nephrons (G3), and juxtamedullary nephrons (G4). G1 and G2 give rise to all short nephrons (SN), whereas G3 and G4 yield all long nephrons (LN). Efferent blood from G1, G2, and G3 flows to capillary plexuses of both labyrinth and medullary ray. Capillary blood of medullary ray drains into labyrinth. Efferent blood from G4 that flows to descending vasa recta (DVR) is sole blood supply to medulla. Proximal convoluted tubules of short nephrons (PCTSN) give rise to their straight counterparts (PCTSN) found within the medullary ray. However, proximal straight tubules of long nephrons (PSTLN) enter medulla without traversing medullary ray. Transition of proximal straight tubules to descending limbs of Henle (DLSN and DLLN, respectively) occurs at outer stripe‐inner stripe border. All DLSN turn to form medullary thick ascending limbs of short nephrons (MALSN) at inner stripe‐inner zone border; descending limbs of long nephrons (DLLN) turn at various depths along inner zone to form thin ascending limbs of Henle (TNAL.) At inner‐outer zone border, thin ascending limbs become medullary thick ascending limbs of long nephrons (MALLN). In medullary ray, medullary thick ascending limbs of short nephrons become cortical thick ascending of short nephrons (CALSN), which lead to distal convoluted tubules of short nephrons (DTSN) found in labyrinth. Flow from MALLN is assumed to go directly into distal convoluted tubules of long nephrons (DTLN). DTLN fluid feeds into connecting tubules (CT) that rise as arcades within labyrinth and, together with fluid delivered from DTSN, empties into cortical collecting ducts (CCD) of medullary ray. In the medulla, collecting ducts become outer medullary ducts (OMCD) and then inner medullary collecting ducts (IMCD). IMCD are assumed to have a converging structure. Blood from descending vasa recta (DVR) enters medullary capillary plexus and then flows into ascending vasa recta (AVR). At cortico‐medullary junction, the blood in AVR constitutes the medullary venous return, and the blood from cortical labyrinth capillary plexus, the cortical venous return.

From Chandhoke et al. 26
Figure 31. Figure 31.

Simulated tubule fluid/plasma (TF/P) sodium chloride in fluid of the medullary collecting ducts for no active sodium chloride transport (SO), uniform active sodium chloride transport (SU), and nonuniform active sodium chloride transport (SN).

From Chandhoke et al. 26
Figure 32. Figure 32.

Simulated tubule fluid/plasma (TF/P) urea in fluid along the medullary collecting ducts for different cases of collecting duct active sodium chloride transport: none (SO), uniform (SU), and nonuniform (SN).

From Chandhoke et al. 26
Figure 33. Figure 33.

Simulated tubule fluid/plasma (TF/P) osmolarity in fluid along the medullary collecting ducts for different cases of collecting duct active sodium chloride transport: none (SO), uniform (SU), and non‐uniform (SN).

From Chandhoke et al. 26
Figure 34. Figure 34.

Concentration ratio of three‐tube model, consisting of descending thin limb coupled to ascending and descending vasa recta, as a function of exchange efficiency K, for both an approximate analytical solution (—) and a numerical solution (•) 211.
Figure 35. Figure 35.

Effect of vasa recta plasma flow rate on solute accumulation in inner medulla. Fraction of total solute load entering medulla via descending limb lost through solute washout arising from imperfect countercurrent exchange is given in parentheses.

From Moore and Marsh 147
Figure 36. Figure 36.

Time course of distal tubule and collecting duct water permeability changes used in model of Moore, Marsh, and Martin to cause transition from steady‐state diuresis to antidiuresis. Initial 30 min were used to allow system to achieve a steady state. DT, distal tubule; CD, collecting duct

From Moore et al. 149
Figure 37. Figure 37.

Predicted change in solute concentrations in descending limb (DLH) tubular fluid following antidiuretic hormone. Zero time corresponds to 30 min in Figure 36. Concentrations expressed as fractional increase in concentration difference. Left: predicted sodium chloride concentrations. Right: predicted urea concentrations.

From Moore et al. 149


Figure 1.

Response of whole kidney to varying rates of vasopressin infusion in rats. Note the dramatic decrease in urine flow rate at higher rates of vasopressin infusion with minimal change in osmolar clearance

Data from Atherton et al. 3. From Knepper and Stephenson 111


Figure 2.

Relation between relative medullary thickness and maximum urinary osmolality. Long loops of Henle are required to make highly concentrated urine

Data from Schmidt‐Nielsen and O'Dell 179. From Burg and Stephenson 20


Figure 3.

Prototype three‐section glomerular nephron. With its straight configuration, solute and water transport from different segments is uncoupled. In the proximal tubule (PT) with a large effective hydraulic permeability, Lp, absorption is nearly isotonic; in the diluting segment (DIL) a smaller Lp limits water absorption relative to solute absorption, and tubule fluid is diluted; in final segment, hydraulic permeability is under control of antidiuretic hormone (ADH). In the absence of ADH tubule fluid remains dilute, and in the presence of ADH it equilibrates osmotically with the surrounding interstitium. Thus final urine can vary from markedly hypotonic to isotonic relative to plasma. Black arrows indicate solute movement; white arrows, volume flow.

Reproduced, with permission, from Stephenson 201, Annual Review of Biophysics and Bioengineering, Vol. 7. © 1978 by Annual Reviews, Inc


Figure 4.

The S‐shaped mammalian nephron. The essential functional feature that allows concentration of urine is juxtaposition of the diluting segment (DIL; now ascending limb of Henle's loop) with the water‐permeable descending limb and antidiuretic hormone (ADH)‐sensitive collecting duct. This permits solute supplied by ascending limb to extract water from collecting duct and descending limb. Black arrows indicate solute movement; white arrows, volume flow; PT, proximal tubule.

Reproduced, with permission, from Stephenson 201, Annual Review of Biophysics and Bioengineering, Vol. 7. © 1978 by Annual Reviews, Inc


Figure 5.

Initial stage of basic scheme for concentrating by permselective membranes. A very large volume of phenol solution at concentration Co is separated by membrane P permeable to phenol but not to water (rubber in experiments of Kuhn and Ryffel) from a large volume of sucrose solution, also at concentration Co. The sucrose solution is separated by membrane W permeable to water but not to sucrose or to phenol (copper ferrocyanide) from smaller volumes of phenol and sucrose solutions, also at concentration Co. As indicated by arrows, phenol diffuses into sucrose solution, increasing its osmolality and causing water to be osmotically extracted from the smaller volume solutions.

Adapted from Kuhn and Ryffel 122


Figure 6.

Basic scheme for concentrating by permselective membranes at equilibrium. Phenol has been added to the original sucrose solution so that its concentration has approximately doubled. Water has been extracted from the smaller volumes of sucrose and phenol so that their volumes are halved and concentrations doubled.

Adapted from Kuhn and Ryffel 122


Figure 7.

Arrangement for parallel processing of counterflowing sucrose and phenol solutions to yield progressive concentration of the sucrose solution. In the first apparatus on the left, sucrose and phenol, both at concentration Co, interact through a phenol‐permeable membrane P to give a combined solution of concentration 2Co. This interacts with counterflowing sucrose solution in the first apparatus on the right through a water‐permeable membrane W to give a sucrose solution of concentration 2Co. The phenol is then removed by diffusion into a concurrent flow of any solution at concentration Co through a second membrane P. As the solution is processed from left to right, there is progressive concentration of the sucrose solution. In further stages to the right, phenol reservoirs are not shown in order to simplify the diagram. Kuhn and Ryffel suggested that this scheme might mimic the operation of the kidney.

Adapted from Kuhn and Ryffel 122


Figure 8.

Actual scheme used by Kuhn and Ryffel. See text for explanation.

Adapted from Kuhn and Ryffel 122


Figure 9.

Hargitay and Kuhn countercurrent multiplier without withdrawal. Solution at concentration C1 enters at the left and reverses flow through hairpin constriction at the right to return to the left. The two counterflowing solutions are separated by a waterpermeable and solute‐impermeable membrane. Mass balance requires that at any position x along the tube C1 = C2 and F1 = ‐ F2. This means that the pressure difference Δp, the single effect, is unapposed by any osmotic force and will drive water from the right‐flowing to the left‐flowing stream at a rate that depends on the permeability of the membrane, thus concentrating the right‐flowing and diluting the left‐flowing stream.

Adapted from Hargitay and Kuhn 65


Figure 10.

Hargitay and Kuhn multiplier with withdrawal. A fraction fu of the combined descending limb (DL) and collecting duct (CD) flow is withdrawn at the loop. Mass balance now requires that C1 = C3 ≫ C2. This creates an osmotic force opposing the single effect and reduces the concentration. The degree of the reduction depends on the ratio of the single effect to the withdrawal. AL, ascending limb.



Figure 11.

Comparison of classic representation of concentration build‐up in solute cycling multiplier (A) with calculated profiles (B). Multiplication by counterflow is apparent, but the concentration difference between counterflowing limbs varies with both time and position. “Pump” and “leak” were adjusted to give a limiting “single effect” of 200 mOsm/liter.

Reprinted with permission from Garner et al. 47, Bulletin of Mathematical Biology, Vol. 40, “Transient behaviour of the single loop solute cycling model of the renal medulla.” © 1978, Pergamon Press


Figure 12.

Relative osmolalities of tubular fluid in slices from kidneys of hydropenic rats. Values are given as percent of maximum. O.Z, outer zone; I.Z, inner zone (as modified for ref. 111 from ref. 260).

From Knepper and Stephenson 111


Figure 13.

Composition of renal medulla and urine during antidiuresis and water diuresis (adapted from Hai and Thomas 63 for 20.

From Knepper and Stephenson 111


Figure 14.

Comparison of osmolalities of fluids collected by micropuncture near the tip of the inner medulla in nine hamsters, one kangaroo rat, and one Psammomys.

From Gottschalk and Mylle 58


Figure 15.

Osmolality ratio (tubule fluid/plasma) of fluid collected by micropuncture from seven rats during antidiuresis. Early tubule fluid was dilute even though urine was concentrated. Different symbols refer to different rats.

From Gottschalk and Mylle 58


Figure 16.

Countercurrent exchanger consisting of ascending vas rectum (AVR) and descending vas rectum (DVR). Solute is added to surrounding interstitium at rate J per unit length. Solute entering AVR from interstitium is recycled diffusively to DVR, with resulting build‐up of concentration at the loop.



Figure 17.

Diagram of two‐loop counterflow system with parallel Henle's loop and vas rectum.

Reprinted by permission from Stephenson 190, Nature, Vol. 206, pp. 1215–1219. © 1965 Macmillan Magazines Ltd


Figure 18.

A model of a vasa recta bundle. Direction of flow is indicated by arrows.

From Marsh and Segel 138


Figure 19.

Schematic representation of the intricate counterflow system of the renal medulla. Note that in this figure AVR denotes arterial vas rectum (or descending vas rectum; VVR, venous vas rectum (or ascending vas rectum); DL, descending limb of Henle's loop; AL, ascending limb of Henle's loop; CD, collecting duct; CAP, capillary.

From Kriz and Lever 119


Figure 20.

a: Central core model of the renal medulla. Black arrows indicate salt movement; open arrows, water movement; hatched arrows, urea movement. With sufficiently large solute and water permeabilities of ascending vasa recta (AVR) and descending vasa recta (DVR), concentrations in AVR, DVR, and interstitium become nearly identical at a given medullary level. Solute removal is then given by the product of the concentration and the volume flow difference in AVR and DVR. Functionally, AVR, DVR, and interstitium are merged into a single tube closed at the papillary tip and open at the junction of medulla and cortex. CD, collecting duct; AHL, ascending limb of Henle's loop; DHL, descending limb of Henle's loop. b: shows various core components being hypothetically merged to give the final cross‐sectional configuration shown in c.

Reprinted from Stephenson 193, Kidney International, with permission


Figure 21.

Skeletonized central core model. Black arrows indicate solute movement; white arrows, volume flow. AHL, ascending limb; DHL, descending limb, CD, collecting duct.

From Stephenson 206


Figure 22.

Single osmotic effect in two modes of operation. Mode I, active transport (upper titles); mode II, mixing (bracketed); initial (a) and final (b) equilibrium states are shown. Open arrow indicates water movement; closed arrow, salt movement.

Reprinted from Stephenson 193, Kidney International, with permission


Figure 23.

Configuration for Peskin‐Layton single nephron model. By assumption, C1(γ) = C3(γ) = C(γ) = J2s/Jv.

From Layton 124


Figure 24.

Structure of model 1 of Wexler, Kalaba, and Marsh. Top: cross‐section showing connections between tubules, blood vessels, and interstitial space. Bottom: vertical aspect. DVR, descending vasa recta; AVR, ascending vasa recta; DHL, descending limb of Henle's loop; AHL, ascending limb of Henle's loop; DT, distal tubule; CD, collecting duct; I, interstitium. OM, outer medulla; IM, inner medulla.

From Wexler et al. 252


Figure 25.

Structure of model 3 of Wexler, Kalaba, and Marsh. Top: cross‐section showing connections between tubules, blood vessels, and interstitial space. Bottom: vertical aspect. DVR, descending vasa recta; AVR, ascending vasa recta; DHL, descending limb of Henle's loop; AHL, ascending limb of Henle's loop; DT, distal tubule; CD, collecting duct; I, interstitium; OM, outer medulla; IM, inner medulla.

From Wexler et al. 252


Figure 26.

Interstitial concentrations of sodium chloride and urea predicted by model 3 of Wexler, Kalaba, and Marsh.

From Wexler et al. 252


Figure 27.

Configuration of two‐nephron, central core, electrolyte model of renal medulla of Stephenson, Zhang, and Tewarson. DHL, descending limb of Henle's loop; AHL, ascending limb of Henle's loop; CD, collecting duct; DN, distal nephron.

From Stephenson et al. 213


Figure 28.

Core osmolalities for various modifications of rabbit and hamster data. Top: A, unmodified; B, urea permeability of descending limb reduced; C, urea permeability of ascending limb reduced; D, urea permeability of both descending and ascending limbs reduced. Bottom: E, unmodified; F, urea permeability of both descending and ascending limbs reduced; G, urea permeabilities as in F, descending limb electrolyte permeabilities also reduced.

From Stephenson et al. 213


Figure 29.

Effect of varying urea permeability of thin descending limb of Henle's loop on core osmolality at papilla and at junction of inner and outer medullae.

From Stephenson et al. 213


Figure 30.

Model of Chandhoke, Saidel, and Knepper. Renal cortex is divided into a medullary ray and a labyrinth. Renal arterial blood (afferents) flows to four groups of glomeruli in cortical labyrinth: those of superficial nephrons (G1), midcortical short nephrons (G2), midcortical long nephrons (G3), and juxtamedullary nephrons (G4). G1 and G2 give rise to all short nephrons (SN), whereas G3 and G4 yield all long nephrons (LN). Efferent blood from G1, G2, and G3 flows to capillary plexuses of both labyrinth and medullary ray. Capillary blood of medullary ray drains into labyrinth. Efferent blood from G4 that flows to descending vasa recta (DVR) is sole blood supply to medulla. Proximal convoluted tubules of short nephrons (PCTSN) give rise to their straight counterparts (PCTSN) found within the medullary ray. However, proximal straight tubules of long nephrons (PSTLN) enter medulla without traversing medullary ray. Transition of proximal straight tubules to descending limbs of Henle (DLSN and DLLN, respectively) occurs at outer stripe‐inner stripe border. All DLSN turn to form medullary thick ascending limbs of short nephrons (MALSN) at inner stripe‐inner zone border; descending limbs of long nephrons (DLLN) turn at various depths along inner zone to form thin ascending limbs of Henle (TNAL.) At inner‐outer zone border, thin ascending limbs become medullary thick ascending limbs of long nephrons (MALLN). In medullary ray, medullary thick ascending limbs of short nephrons become cortical thick ascending of short nephrons (CALSN), which lead to distal convoluted tubules of short nephrons (DTSN) found in labyrinth. Flow from MALLN is assumed to go directly into distal convoluted tubules of long nephrons (DTLN). DTLN fluid feeds into connecting tubules (CT) that rise as arcades within labyrinth and, together with fluid delivered from DTSN, empties into cortical collecting ducts (CCD) of medullary ray. In the medulla, collecting ducts become outer medullary ducts (OMCD) and then inner medullary collecting ducts (IMCD). IMCD are assumed to have a converging structure. Blood from descending vasa recta (DVR) enters medullary capillary plexus and then flows into ascending vasa recta (AVR). At cortico‐medullary junction, the blood in AVR constitutes the medullary venous return, and the blood from cortical labyrinth capillary plexus, the cortical venous return.

From Chandhoke et al. 26


Figure 31.

Simulated tubule fluid/plasma (TF/P) sodium chloride in fluid of the medullary collecting ducts for no active sodium chloride transport (SO), uniform active sodium chloride transport (SU), and nonuniform active sodium chloride transport (SN).

From Chandhoke et al. 26


Figure 32.

Simulated tubule fluid/plasma (TF/P) urea in fluid along the medullary collecting ducts for different cases of collecting duct active sodium chloride transport: none (SO), uniform (SU), and nonuniform (SN).

From Chandhoke et al. 26


Figure 33.

Simulated tubule fluid/plasma (TF/P) osmolarity in fluid along the medullary collecting ducts for different cases of collecting duct active sodium chloride transport: none (SO), uniform (SU), and non‐uniform (SN).

From Chandhoke et al. 26


Figure 34.

Concentration ratio of three‐tube model, consisting of descending thin limb coupled to ascending and descending vasa recta, as a function of exchange efficiency K, for both an approximate analytical solution (—) and a numerical solution (•) 211.



Figure 35.

Effect of vasa recta plasma flow rate on solute accumulation in inner medulla. Fraction of total solute load entering medulla via descending limb lost through solute washout arising from imperfect countercurrent exchange is given in parentheses.

From Moore and Marsh 147


Figure 36.

Time course of distal tubule and collecting duct water permeability changes used in model of Moore, Marsh, and Martin to cause transition from steady‐state diuresis to antidiuresis. Initial 30 min were used to allow system to achieve a steady state. DT, distal tubule; CD, collecting duct

From Moore et al. 149


Figure 37.

Predicted change in solute concentrations in descending limb (DLH) tubular fluid following antidiuretic hormone. Zero time corresponds to 30 min in Figure 36. Concentrations expressed as fractional increase in concentration difference. Left: predicted sodium chloride concentrations. Right: predicted urea concentrations.

From Moore et al. 149
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Article Title: Urinary Concentration and Dilution: Models
 

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John L. Stephenson. Urinary Concentration and Dilution: Models. Compr Physiol 2011, Supplement 25: Handbook of Physiology, Renal Physiology: 1349-1408. First published in print 1992. doi: 10.1002/cphy.cp080230