Quantitative interrelationship between Gibbs-Donnan equilibrium, osmolality of body fluid compartments, and plasma water sodium concentration

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1 J Appl Physiol 100: , First published December 15, 2005; doi: /japplphysiol TRANSLATIONAL PHYSIOLOGY Quantitative interrelationship between Gibbs-Donnan equilibrium, osmolality of body fluid compartments, and plasma water sodium concentration Minhtri K. Nguyen and Ira Kurtz Division of Nephrology, David Geffen School of Medicine at UCLA, Los Angeles, California Submitted 4 October 2005; accepted in final form 12 December 2005 Nguyen, Minhtri K., and Ira Kurtz. Quantitative interrelationship between Gibbs-Donnan equilibrium, osmolality of body fluid compartments, and plasma water sodium concentration. J Appl Physiol 100: , First published December 15, 2005; doi: /japplphysiol The presence of negatively charged, impermeant proteins in the plasma space alters the distribution of diffusible ions in the plasma and interstitial fluid (ISF) compartments to preserve electroneutrality. We have derived a new mathematical model to define the quantitative interrelationship between the Gibbs-Donnan equilibrium, the osmolality of body fluid compartments, and the plasma water Na concentration ([Na ] pw ) and validated the model using empirical data from the literature. The new model can account for the alterations in all ionic concentrations (Na and non-na ions) between the plasma and ISF due to Gibbs- Donnan equilibrium. In addition to the effect of Gibbs-Donnan equilibrium on Na distribution between plasma and ISF, our model predicts that the altered distribution of osmotically active non-na ions will also have a modulating effect on the [Na ] pw by affecting the distribution of H 2 O between the plasma and ISF. The new physiological insights provided by this model can for the first time provide a basis for understanding quantitatively how changes in the plasma protein concentration modulate the [Na ] pw. Moreover, this model defines all known physiological factors that may modulate the [Na ] pw and is especially helpful in conceptually understanding the pathophysiological basis of the dysnatremias. plasma water sodium concentration; hyponatremia Address for reprint requests and other correspondence: M. K. Nguyen, Division of Nephrology, David Geffen School of Medicine at UCLA, Le Conte Ave., Rm Factor Bldg., Los Angeles, CA ( mtnguyen@mednet.ucla.edu). IT IS WELL RECOGNIZED THAT the plasma water Na and Cl concentrations and interstitial fluid (ISF) Na and Cl concentrations are different despite the high permeability of Na and Cl ions across the capillary membrane, which separates these two fluid compartments (23). This difference in ionic concentrations between the plasma and the ISF is attributed to the much higher concentration of proteins in the plasma compared with the ISF. Proteins are large-molecular-weight substances and therefore do not cross the capillary membrane easily. The low protein permeability across capillary membranes is responsible for causing ionic concentration differences between the plasma and ISF and is known as the Gibbs-Donnan effect or Gibbs-Donnan equilibrium (23). Negatively charged, nonpermeant proteins present predominantly in the plasma space will attract positively charged ions and repel negatively charged ions (23). The passive distribution of cations and anions is altered to preserve electroneutrality in the plasma and ISF. As a result, the diffusible cation concentration is higher in the compartment containing nondiffusible, anionic proteins, whereas diffusible anion concentration is lower in the protein-containing compartment. Gibbs- Donnan equilibrium is established when the altered distribution of cations and anions results in electrochemical equilibrium. It is also well recognized that another consequence of the Gibbs- Donnan effect is that there are more osmotically active particles in the plasma space than in the ISF at equilibrium (13, 23). Consequently, the plasma osmolality is slightly greater than the osmolality of the ISF and intracellular fluid (ICF). Indeed, the plasma osmolality is typically 1 mosmol/l H 2 O greater than that of the ISF and ICF (13). In addition to the modulating effect of Gibbs-Donnan equilibrium on the [Na ] pw and plasma osmolality, alterations in the osmolality of the ISF and ICF will also lead to changes in the [Na ] pw and plasma osmolality due to intercompartmental H 2 O shift since the body fluid compartments are in osmotic equilibrium. Presently, there are no formulas in the literature that have determined the mathematical relationships between Gibbs-Donnan equilibrium, osmolality of body fluids (plasma, ISF, and ICF), and [Na ] pw. In this article, based on the principles of Gibbs- Donnan and osmotic equilibrium, we derive for the first time a new equation that quantitatively predicts the effect of changes in negatively charged plasma proteins on the osmolality of all body fluid compartments and the [Na ] pw. MATHEMATICAL DERIVATION Quantification of the Effect of Gibbs-Donnan Equilibrium on the [Na ] pw by Modulating the Osmolality of the Body Fluid Compartments It is well known that the plasma osmolality is slightly greater than the osmolality of the ISF and ICF owing to the Gibbs- Donnan equilibrium (13, 23). The plasma osmolality is typically 1 mosmol/l H 2 O greater than that of the ISF and intracellular compartment (13). Therefore: plasma osmolality total body osmolality To equate plasma osmolality to the total body osmolality, one has to introduce a correction factor for the incremental effect of Gibbs-Donnan equilibrium on the plasma osmolality. This correction factor, which will be termed g, is equal to the ratio of plasma osmolality/total body osmolality and is therefore unitless. The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact /06 $8.00 Copyright 2006 the American Physiological Society 1293

2 1294 GIBBS-DONNAN EQUILIBRIUM, OSMOLALITY, AND SODIUM CONCENTRATION Therefore: Plasma osmolality g total body osmolality (1) where g is plasma osmolality/total body osmolality. Since: Total number of osmoles Total body osmolality Total body water Total body osmolality pw ISF V ISF ICF V ICF (3) where pw is plasma osmolality, is plasma water volume, ISF is ISF osmolality, V ISF is ISF volume, ICF is ICF osmolality, V ICF is ICF volume, and is total body water. The unit of osmolality is expressed throughout the derivation in milliosmoles per liter H 2 O rather than milliosmoles per kilogram H 2 O assuming the density of water is 1 kg/l. Therefore: pw g (4) pw ISF V ISF ICF V ICF We now define the components of plasma osmolality and total body osmolality: Plasma osmolality Na pw K pw osmol pw (5) It is well known that not all exchangeable Na (Na e ) and exchangeable K (K e ) are osmotically active because there is abundant evidence for the existence of osmotically inactive Na and K storage in bone and skin (3, 5, 8 10, 14, 25, 27 29). Hence, only osmotically active exchangeable Na and K contribute to the distribution of water between the extracellular and intracellular spaces. Total body osmolality Na osm active K osm active osmol ECF osmol ICF (6) where Na pw is plasma water Na,K pw is plasma water K, osmol pw is osmotically active plasma water non-na non-k osmoles, Na osm active is osmotically active Na,K osm active is osmotically active K, osmol ECF is osmotically active extracellular non-na and non-k osmoles, and osmol ICF is osmotically active intracellular non-na and non-k osmoles. Since: Plasma osmolality g total body osmolality (1) Therefore: Na pw K pw osmol pw g (7) Na osm active K osm active osmol ECF osmol ICF Let [Na ] pw plasma water Na concentration and [K ] pw plasma water K concentration. Since: Na pw / Na pw and K pw / K pw (2) Na pw K pw osmol pw g Na osm active K osm active Rearranging: Na pw g Na osm active K osm active where: g osmol ECF osmol ICF osmol ECF osmol ICF pw K pw osmol pw pw ISF V ISF ICF V ICF Since total exchangeable Na (Na e ) and exchangeable K (K e ) consists of osmotically active and osmotically inactive exchangeable Na and K : Let Na e total exchangeable Na ;K e total exchangeable K ;Na osm inactive osmotically inactive Na ; and K osm inactive osmotically inactive K. Hence: Na e Na osm active Na osm inactive and K e K osm active K osm inactive Na pw g Na e K e Na osm inactive K osm inactive (10) osmol ECF osmol ICF K pw osmol pw Na pw g Na e K e Na osm inactive K osm inactive (11) osmol ECF osmol ICF K pw osmol pw where [Na ] pw is expressed in milliosmoles per liter H 2 O. To convert [Na ] pw in Eq. 11 from milliosmoles per liter H 2 O to millimoles per liter H 2 O, one needs to divide both sides of Eq. 11 by the osmotic coefficient Ø (13, 16): Na * pw g/ø Na e K e Na osm inactive K osm inactive (12) osmol ECF osmol ICF 1/ØK pw osmol pw where [Na ]* pw is expressed in millimoles per liter H 2 O,Øis osmotic coefficient of Na salts, and is osmolality; pw g pw ISF V ISF ICF V ICF This equation quantitatively predicts the effect of changes in negatively charged plasma proteins on the osmolality of all body fluid compartments and the [Na ] pw. Clinical Validity of Equation 11 Using the data from Table 1, one can demonstrate the clinical validity of Eq. 11. Because the solutes in Table 1 are (8) (9)

3 GIBBS-DONNAN EQUILIBRIUM, OSMOLALITY, AND SODIUM CONCENTRATION Table 1. Osmolar substances in the extracellular and intracellular compartments Plasma, mosmol/l H2O Interstitial, mosmol/l H2O Intracellular, mosmol/l H2O Na K Ca Mg Cl HCO HPO 2 4,H 2PO SO Phosphocreatine 45 Carnosine 14 Amino acids Creatine Lactate Adenosine triphosphate 5 Hexose monophosphate 3.7 Glucose Protein Urea Others Total mosmol/l Corrected osmolar activity, mosmol/l* Total osmotic pressure at 37 C, mmhg 5,443 5,423 5,423 Data are adapted from Ref. 13. *Corrected osmolar activity accounts for the reduced osmotic activity of ionic particles in solution. expressed in milliosmoles per liter H 2 O, Eq. 11 will be utilized in this example. It is also important to realize that the measured Na and K ions in the plasma, ISF, and ICF in Table 1 include only the osmotically active Na and K ions, which is reflected by the terms (Na e K e )/ (Na osm inactive K osm inactive )/ in Eq. 11. Table 1 includes only the osmotically active Na and K ions because the distribution of H 2 O depends solely on osmotically active solute particles, as reflected by the fact that the osmolality of the plasma, ISF, and ICF are essentially equal (with the exception that the plasma osmolality is slightly greater than the ISF osmolality owing to the Gibbs-Donnan equilibrium). In other words, if the measured Na and K ions in the plasma, ISF, and ICF in Table 1 were to include both osmotically active and inactive Na and K ions, the calculated osmolality of the plasma, ISF, and ICF cannot be equal to one another. Moreover, the determination of the [Na ] and [K ] in the plasma, ISF, and ICF by conventional laboratory techniques measures only the osmotically active Na and K ions. Exchangeable but osmotically inactive bound Na and K ions (e.g., exchangeable bound, osmotically inactive Na in bone) are not measured by these techniques. The measurement of total exchangeable Na and total exchangeable K would require isotope dilution methodology. In this example, we will calculate the [Na ] pw in an average 70-kg man in whom the is 42 liters, of which 25 liters are in the ICF, 14 liters are in the ISF, and 3 liters are in the plasma space. Since: Na e Na osm active Na osm inactive and K e K osm active K osm inactive Therefore: Na e K e (Na osm inactive K osm inactive ) Na osm active K osm active Na osm active Na pw a Na ISF V ISF Na ICF V ICF 2,722 mosmol K osm active K pw K ISF V ISF K ICF V ICF 3,568.6 mosmol Na osm active K osm active 2,722 mosmol 3,568.6 mosmol 6,290.6 mosmol osmol b pw pw Na K pw mosmol/l mosmol/l 3 liters mosmol osmol ISF ISF Na K ISF V ISF mosmol/l 143 mosmol/l 14 liters 2,209.2 mosmol osmol ICF ICF Na K ICF V ICF mosmol/l 154 mosmol/l 25 liters 3,680 mosmol osmol pw pw Na K pw mosmol/l mosmol/l155.6 mosmol/l pw g pw ISF V ISF ICF V ICF / 301.8/301.1 Na pw g Na e K e Na osm inactive K osm inactive (11) osmol ECF osmol ICF K pw osmol pw Since the terms (Na e K e )/ (Na osm inactive K osm inactive /) represent the osmotically active Na and K ions: Na pw g Na osm active K osm active 1295 osmol ECF osmol ICF K pw osmol pw Since osmol ECF osmol pw osmol ISF : Na pw g Na osm active K osm active K pw osmol pw osmol pw osmol ISF osmol ICF Na pw 301.8/301.16, , ,680/ Na pw 142 mosm/l a The osmolal quantity of plasma osmotically active Na can also be determined by calculating the product of and the difference between the plasma osmolality and the sum of the osmolality of all the non-na osmoles, i.e. (plasma osmolality non-na osmolality). This calculation does not require the value of the [Na ] pw. b The plasma non-na and non-k osmoles can also be calculated by simply adding the quantity of all the individual plasma non-na and non-k osmoles, a calculation that does not require the value of the [Na ] pw.

4 1296 GIBBS-DONNAN EQUILIBRIUM, OSMOLALITY, AND SODIUM CONCENTRATION As demonstrated in this example, Eq. 11 precisely predicts that the [Na ] pw is 142 mosmol/l H 2 O. Because Gibbs-Donnan equilibrium alters the distribution of Na and non-na ions between the plasma and ISF, Eqs. 11 and 12 account for the alterations in all ionic concentrations (Na and non-na ions) between the plasma and ISF (as reflected by the changes in the osmolality of plasma and ISF). Equations 11 and 12 therefore take into consideration the fact that the altered distribution of osmotically active non-na ions due to the Gibbs-Donnan equilibrium will also have a modulating effect on the [Na ] pw by affecting the distribution of H 2 O between the plasma and ISF. Moreover, in contrast to our previous analysis (18), Eqs. 11 and 12 account for the fact that alterations in the plasma protein concentration will result in changes in the distribution of Na and non-na ions between the plasma and ISF due to Gibbs-Donnan equilibrium. The alterations in the concentrations of Na and non-na ions between the plasma and ISF due to changes in the plasma protein concentration will be reflected by the changes in the osmolality of the plasma and ISF, which are accounted for by the terms pw and ISF in the Gibbs-Donnan correction factor g (i.e., pw [Na ] pw [K ] pw [Ca 2 ] pw [Mg 2 ] pw [Cl ] pw [HCO 3 ] pw [H 2 PO 4 ] pw [SO 4 2 ] pw [protein] pw etc.; and ISF [Na ] ISF [K ] ISF [Ca 2 ] ISF [Mg 2 ] ISF [Cl ] ISF [HCO 3 ] ISF [H 2 PO 4 ] ISF [SO 4 2 ] ISF etc.). DISCUSSION Modulation of [Na ] pw by Alterations in the Osmolality of Body Fluids In this article, we examine quantitatively the interrelationship between the Gibbs-Donnan equilibrium, osmolality of the body fluids (plasma, ISF, and ICF), and [Na ] pw. By altering the distribution of diffusible ions between the plasma and ISF, Gibbs-Donnan equilibrium has a modulating effect on the osmolality of the plasma and ISF. Because the body fluid compartments are in osmotic equilibrium (13), changes in the osmolality in any one compartment will alter the distribution of H 2 O in the other two compartments. Consequently, the Gibbs- Donnan equilibrium will also have a modulating effect on the osmolality of the ICF as well. Because the [Na ] pw is determined by the quantity of Na ions in the plasma space and the plasma water volume, changes in the osmolality in any body fluid compartment will have a modulating effect on the [Na ] pw by altering the distribution of H 2 O in the plasma space. It is therefore not surprising that a determinant of the [Na ] pw is the osmolality of the body fluid compartments. Indeed, as depicted in Eq. 12, the osmolality of the plasma, ISF, and ICF compartments has a modulating effect on the [Na ] pw as reflected by the term g. Moreover, although the plasma, ISF, and ICF osmolality can have a direct effect on the [Na ] pw completely independent of g, the osmolality of the plasma, ISF, and ICF also has a separate quantitative effect on the [Na ] pw mediated through the Gibbs- Donnan equilibrium. Indeed, the plasma, ISF, and ICF osmolality can affect the [Na ] pw completely independent of g because the body fluid compartments are in osmotic equilibrium. However, if osmotic equilibrium is the only mechanism modulating the [Na ] pw, then the [Na ] pw will be equal to the [Na ] ISF and the plasma and ISF osmolality will be exactly equal. In reality, the [Na ] pw is greater than the [Na ] ISF and the plasma osmolality is slightly greater than the ISF osmolality owing to the effect of Gibbs-Donnan equilibrium in modulating the distribution of Na and non-na ions between the plasma and ISF. This fact is accounted for by the Gibbs- Donnan correction factor g in Eq. 12. Osmotic Equilibrium of Body Fluid Compartments: Osmotically Active and Inactive Exchangeable Na and K Because osmotic equilibration depends only on osmotically active solutes, the presence of osmotically inactive Na e and K e has to be accounted for in Eq. 12. Indeed, there is convincing evidence for the existence of an osmotically inactive Na and K reservoir (3, 5, 8 10, 14, 25, 27 29). Recently, Titze et al. (28) reported Na accumulation in an osmotically inactive form in human subjects in a terrestrial space station simulation study and suggested the existence of an osmotically inactive Na reservoir that exchanges Na with the extracellular space. Titze et al. (27) also showed that salt-sensitive Dahl rats (which developed hypertension if fed a high-sodium diet) were characterized by a reduced osmotically inactive sodium storage capacity compared with Sprague-Dawley rats, thereby resulting in fluid accumulation and high blood pressure. In addition, Heer et al. (14) demonstrated positive Na balance in healthy subjects on a metabolic ward without increases in body weight, expansion of the extracellular space, or plasma sodium concentration. These authors, therefore, suggested that there is osmotic inactivation of exchangeable Na. Moreover, there is also evidence that a portion of intracellular K is bound and is therefore osmotically inactive (5). Together, these findings provide convincing evidence for the existence of an osmotically inactive Na and K reservoir. Because osmotically inactive Na e and K e cannot contribute to the distribution of water between the extracellular and intracellular spaces, osmotically inactive exchangeable Na and K cannot contribute to the modulation of the [Na ] pw and therefore are accounted for quantitatively by the (Na osm inactive K osm inactive )/ term in Eq. 12. It is also evident that any osmotically active non-na osmoles are involved in the distribution of H 2 O between the body fluid compartments. The terms K e,[k ] pw, osmol ECF, osmol ICF, and osmol pw account for the fact that non-na osmoles also alter the distribution of H 2 O between the body fluid compartments. Moreover, because the osmotic activity of most ionic particles is slightly less than one owing to electrical interactions between the ions (16), the osmotic coefficient Ø in Eq. 12 accounts for the effectiveness of Na ions as independent osmotically active particles under physiological conditions. Role of Gibbs-Donnan Equilibrium in Modulating the Osmolality of the Body Fluid Compartments It is well recognized that Gibbs-Donnan equilibrium modulates the plasma osmolality by altering the distribution of Na and Cl ions in the plasma and ISF to preserve electroneutrality (13, 23). Because of the presence of negatively charged, impermeant proteins in the plasma space, diffusible cation concentration is higher and diffusible anion concentration is lower in the plasma compartment. As a result of this Gibbs- Donnan effect, the total osmolar concentration is slightly

5 greater in the plasma compartment; this extra osmotic force from diffusible ions is added to the osmotic forces exerted by the anionic proteins. The end result of the Gibbs-Donnan effect is that more water moves into the plasma compartment than would be predicted on the basis of the protein concentration alone. A steady state is achieved in which the plasma osmolality is maintained at a slightly greater osmolality than the ISF because the capillary hydrostatic pressure opposes the osmotic movement of water into the plasma space. Interestingly, cells contain a significant concentration of large-molecular-weight, impermeant anionic proteins that also lead to the Gibbs-Donnan equilibrium across the cell membrane. As a result of the Gibbs-Donnan equilibrium, electroneutrality would be preserved but there will be more particles intracellularly. The higher intracellular osmolality would lead to osmotic water movement down its concentration gradient, and the cell would tend to swell. This osmotic water movement would upset the Gibbs-Donnan effect, and the ions would diffuse to reestablish the Gibbs-Donnan equilibrium. There would again be an osmotic gradient across the cell membrane, resulting in further water movement down its concentration gradient. Consequently, this is an unstable situation that, if unopposed, would lead to significant cell swelling. In preventing cell swelling, cells pump Na ions out of the cell actively to compensate for the Gibbs-Donnan effect due to impermeant intracellular proteins. The Na -K ATPase renders the cell membrane effectively impermeant to Na ions, thereby creating a double-donnan effect (21). Therefore, in maintaining cell volume, the Gibbs-Donnan effect due to ISF Na balances the Gibbs-Donnan effect due to impermeant intracellular proteins (21). As a result of this double-donnan effect, there is no net osmotic pressure across the cell membrane, and the system is stable (21). The net result of the Gibbs-Donnan effect is that the plasma osmolality is typically 1 mosmol/l H 2 O greater than that of the ISF and intracellular compartment, which have the same osmolality (13). Therefore, a correction factor for the Gibbs- Donnan effect termed g is incorporated in Eq. 12 to account for this small osmotic inequality. Not surprisingly, the determinants of g are the osmolality of the plasma, ISF, and ICF. Analysis of the Pathogenesis of the Dysnatremias GIBBS-DONNAN EQUILIBRIUM, OSMOLALITY, AND SODIUM CONCENTRATION Historically, the pathogenesis of the dysnatremias has focused on the effect of changes in the mass balance of Na e,k e, and on the [Na ] pw. This pathophysiological approach is based on the empirical relationship between the [Na ] pw and Na e,k e, and originally demonstrated by Edelman et al. (10) where [Na ] pw 1.11(Na e K e )/ However, this approach is inadequate in that it fails to explain quantitatively how several known physiological factors that do not alter the value of the (Na e K e )/ term are capable of modulating the [Na ] pw. These physiological parameters that modulate the [Na ] pw are components of the slope and y- intercept of the Edelman equation. It is important to appreciate that Eq. 12 is in essence a mathematical analysis of all the physiological parameters modulating the [Na ] pw as implicated in the Edelman equation. Equation 12 can be rearranged as follows: Na * pw g/ø Na e K e g/ø Na osm inactive K osm inactive osmol ECF osmol ICF 1/ØK pw osmol pw Therefore, the slope of 1.11 in Edelman s equation is represented by the term g/ø in Eq. 12, whereas the y-intercept of 25.6 is represented by the terms: g/ø Na osm inactive K osm inactive (osmol ECF osmol ICF ) 1/ØK pw osmol pw Therefore, Eq. 12 is especially helpful in explaining quantitatively how physiological factors that do not alter the value of the (Na e K e )/ term are capable of modulating the [Na ] pw by altering the value of the slope and y-intercept in the Edelman equation. Association of Hypoalbuminemia and Hyponatremia Alterations in the distribution of Na and non-na ions between the plasma and ISF resulting from changes in the plasma protein concentration will be reflected by the changes in the plasma and ISF osmolality. Equation 12 accounts for the effect of changes in the plasma protein concentration on the Gibbs-Donnan equilibrium since the Gibbs-Donnan correction factor g includes the terms pw and ISF. Therefore, clinical conditions characterized by hypoalbuminemia would be expected to change the [Na ] pw by altering the slope of the Edelman equation, which will be reflected by changes in the term g in Eq. 12. Indeed, Ferreira da Cunha et al. (11) demonstrated a significant association between hypoalbuminemia and hyponatremia and that the plasma sodium concentration ([Na ] p ) varies proportionally with incremental changes in the albumin concentration. Similarly, Dandona et al. (6) also reported a strong association between low [Na ] p and albumin concentration. These authors also reported several case reports of hypoalbuminemic hyponatremia in patients in whom correction of the hypoalbuminemia with albumin infusion resulted in significant improvement in their hyponatremia. The authors attributed the improvement in the hyponatremia with albumin infusions to the nonosmotic suppression of antidiuretic hormone secretion. However, the marked improvement in the hyponatremia in these patients cannot be due to a significant increase in the urinary free water excretion because the urinary osmolality remained inappropriately elevated after albumin infusion in these hyponatremic patients (urine osmolality ranged from 350 to 430 mosmol/l H 2 O after albumin infusion). Therefore, the improvement in the hyponatremia with the correction of the hypoalbuminemia in these patients likely reflected the alterations in the distribution of Na ions between the plasma and ISF resulting from the Gibbs-Donnan effect. Specifically, an increase in the negatively charged albumin concentration will attract positively charged Na ions into the plasma space, thereby leading to an increase in the [Na ] p. 1297

6 1298 GIBBS-DONNAN EQUILIBRIUM, OSMOLALITY, AND SODIUM CONCENTRATION Non-Na and Non-K Osmoles-Induced Hyponatremia Non-Na and non-k osmoles are capable of modulating the [Na ] pw by altering the value of the y-intercept in the Edelman equation. For example, mannitol administration in patients with impaired renal function has been shown to result in the development of hyponatremia (2, 4). Mannitol is confined to the extracellular fluid after intravenous administration. The resultant increased osmolality of the extracellular fluid will in turn promote the osmotic shift of water from the intracellular compartment to the extracellular compartment. Therefore, hyponatremia may result when mannitol is given to patients with impaired renal function because the mannitol is poorly cleared by the kidney. Similarly, contrast-induced translocational hyponatremia has been described in patients with advanced kidney disease undergoing cardiac catheterization (26). In addition, maltose and sucrose used in commercial intravenous immunoglobulin preparations as well as hyperglycemia can promote the translocation of water from the intracellular compartment to the extracellular compartment, thereby resulting in dilutional hyponatremia (15, 20, 22). Because the osmotic shift of water from the ICF to the ECF does not alter the value of the (Na e K e )/ term in the Edelman equation, non-na and non-k osmoles therefore induce changes in the [Na ] pw by altering the magnitude of the y-intercept in the Edelman equation. Specifically, non-na and non-k osmoles will lead to an increase in the ratio (osmol ECF osmol ICF )/. During the non-na and non-k osmolesinduced osmotic shift of water from the intracellular compartment to the extracellular space, the remains constant because the change in intracellular volume is equal to the change in extracellular volume. The (osmol ECF osmol ICF )/ term in Eq. 12 increases because non-na and non-k osmoles increase the osmol ECF term whereas the remains unchanged. Secondly, the [K ] pw will also be affected by the non-na and non-k osmoles-induced osmotic shift of water and subsequent cellular K efflux induced by the increase in intracellular K concentration and solvent drag induced by hyperosmolality. Finally, non-na and non-k osmoles will increase the term osmol pw /, thereby lowering the [Na ] pw by promoting the osmotic movement of water into the plasma space. Non-Na and non-k osmoles therefore induce changes in the [Na ] pw by altering the magnitude of the y-intercept in the Edelman equation. In maltose-induced hyponatremia, it has been shown that there is a decrease in the [Na ] p of 0.59 mm/l for each millimolar increase in the plasma maltose concentration (22). As a result, to predict the dilutional effect of maltose on the [Na ] p attributable to the osmotic shift of water, the correction factor of 0.59 can be incorporated into the Edelman equation: Na pw 1.11 Na e K e 25.6 Multiplying both sides of the equation by 0.93 to convert [Na ] pw to [Na ] p (1, 7): 0.93 Na pw 1.03 Na e K e 23.8 Because 0.93 [Na ] pw [Na ] p (1, 7): Na p 1.03 Na e K e 23.8 Because there is an expected decrease of 0.59 mm/l in the [Na ] p for each millimolar increment in the plasma maltose concentration (22): Na p 1.03 Na e K e maltose p where [maltose] p is change in plasma maltose concentration (mm/l). Similarly, the y-intercept is not constant in hyperglycemia-induced dilutional hyponatremia resulting from the translocation of water and will vary directly with the plasma glucose concentration. Previously, it has been shown that there is an expected decrease of 1.6 mm/l in the [Na ] p for each 100 mg/dl increment in the plasma glucose concentration due to the translocation of water from the ICF to the ECF (15). Hence, to predict the dilutional effect of hyperglycemia on the [Na ] p attributable to the osmotic shift of water, the [Na ] p can be predicted from the following equation (17): Na p 1.03Na e K e / /100glucose p 120 where [glucose] p is plasma glucose concentration (mg/dl). Determinants of Osmotic Activity The osmotic activity of a solute depends on its ability to move randomly in solution. Therefore, any factors that reduce the random movement of a solute will reduce its osmotic activity. It is well known that not all Na e and K e are osmotically active (3, 5, 8 10, 14, 25, 27 29). Indeed, there is abundant evidence that a portion of Na e is bound in bone and skin and is therefore rendered osmotically inactive (3, 8 10, 14, 25, 27 29). Likewise, a portion of cellular K is reduced in its mobility and in its osmotic activity owing to its association with anionic groups such as carboxyl groups on proteins or to phosphate groups in creatine phosphate, ATP, proteins, and nucleic acids (5). Because osmotically inactive Na e and K e cannot contribute to the distribution of water between the extracellular and intracellular compartments, osmotically inactive Na e and K e cannot contribute to the modulation of the [Na ] pw and therefore are accounted for quantitatively by the (Na osm inactive K osm inactive )/ term in Eq. 12. It is also well known that the osmotic activity of most ions is slightly less than one owing to the electrostatic interactions between ions (16). Because ionic interactions in plasma reduce the random movement of Na ions, each mole of Na ions does not exert exactly one osmole of osmotic activity (16). The osmotic coefficient Ø in Eq. 12 therefore accounts for the effectiveness of Na salts as independent osmotically active particles under physiological conditions and is reflected in the slope of the Edelman equation. Finally, the osmotic activity of a solute will depend on its impermeability to cross cell membranes. For instance, urea is considered to be an ineffective osmole

7 because its concentration is equal throughout the body fluid compartments owing to its high permeability across cell membranes (12). This fact can be quantitatively demonstrated by calculating the [Na ] pw assuming that the plasma urea concentration increases from a value of 4 to 14 mosmol/l H 2 O. By using Table 1, the [Na ] pw is calculated as follows: Na osm active 2,722 mosmol K osm active 3,568.6 mosmol Na osm active K osm active 2,722 mosmol 3,568.6 mosmol 6,290.6 mosmol osmol pw mosmol osmol ISF 2,349.2 mosmol osmol ICF 3,930 mosmol osmol pw mosm/l GIBBS-DONNAN EQUILIBRIUM, OSMOLALITY, AND SODIUM CONCENTRATION g 311.8/311.1 Na pw g Na osm active K osm active osmol pw osmol ISF osmol ICF K pw osmol pw Na pw 311.8/311.16, , ,930/ Na pw 142 mosmol/l Changes in the plasma urea concentration therefore do not alter the [Na ] pw as demonstrated quantitatively by the application of Eq. 11. Summary Changes in the osmolality of the body fluid compartments have long been known to result in intercompartmental H 2 O shifts, thereby leading to alterations in the [Na ] pw. In this article, we derive an equation that depicts the quantitative interrelationship between the Gibbs-Donnan equilibrium, osmolality of body fluids (plasma, ISF, and ICF), and [Na ] pw. This equation also defines all the known factors that determine the magnitude of the [Na ] pw. Moreover, Eq. 12 accounts for the alterations in all ionic concentrations (Na and non-na ions) between the plasma and ISF. It is important to take into consideration the fact that the altered distribution of osmotically active non-na ions due to the Gibbs-Donnan equilibrium will also have a modulating effect on the [Na ] pw by affecting the distribution of H 2 O between the plasma and ISF. Furthermore, Eq. 12 can account for alterations in the distribution of Na and non-na ions between the plasma and ISF resulting from changes in the Gibbs-Donnan equilibrium induced by changes in the plasma protein concentration. Importantly, this new equation can be an indispensable tool in the analysis of the pathophysiology of the dysnatremias. In particular, Eq. 12 is especially helpful in explaining quantitatively 1299 how physiological factors that do not alter the value of the (Na e K e )/ term are capable of modulating the [Na ] pw by altering the value of the slope and y-intercept in the Edelman equation. GRANTS This work was supported by grants to I. Kurtz from the Max Factor Family Foundation, Richard and Hinda Rosenthal Foundation, Fredricka Taubitz fund, and National Institute of Diabetes and Digestive and Kidney Diseases Grants DK-63125, DK-58563, and DK REFERENCES 1. Albrink M, Hold PM, Man EB, and Peters JP. The displacement of serum water by the lipids of hyperlipemic serum. A new method for rapid determination of serum water. J Clin Invest 34: , Aviram A, Pfau A, Czaczkes JW, and Ullmann TD. Hyperosmolality with hyponatremia caused by inappropriate administration of mannitol. Am J Med 42: , Bartter FC and Schwartz WB. The syndrome of inappropriate secretion of antidiuretic hormone. Am J Med 42: , Berry AJ and Peterson ML. Hyponatremia after mannitol administration in the presence of renal failure. Anesth Analg 60: , Cameron IL, Hardman WE, Hunter KE, Haskin C, Smith NKR, and Fullerton GD. Evidence that a major portion of cellular potassium is bound. Scan Micros 4: , Dandona P, Fonseca V, and Baron DN. Hypoalbuminemic hyponatremia: a new syndrome? Br Med J 291: , Davis FE, Kenyon K, and Kirk J. A rapid titrimetric method for determining the water content of human blood. Science 118: , Edelman IS, James AH, Baden H, and Moore FD. Electrolyte composition of bone and the penetration of radiosodium and deuterium oxide into dog and human bone. J Clin Invest 33: , Edelman IS, James AH, Brooks L, and Moore FD. Body sodium and potassium. IV. The normal total exchangeable sodium; its measurement and magnitude. Metabolism 3: , Edelman IS, Leibman J, O Meara MP, and Birkenfeld LW. Interrelations between serum sodium concentration, serum osmolality and total exchangeable sodium, total exchangeable potassium and total body water. J Clin Invest 37: , Ferreira da Cunha D, Monteiro JP, Modesto dos Santos V, Oliveira FA, and Carvalho da Cunha SF. Hyponatremia in acute-phase response syndrome patients in general surgical wards. Am J Nephrol 20: 37 41, Goldberg M. Hyponatremia. Med Clin North Am 65: , Guyton AC and Hall JE. The body fluid compartments: extracellular and intracellular fluids; interstitial fluid and edema. In: Textbook of Medical Physiology, edited by Guyton AC and Hall JE. Philadelphia, PA: Saunders, 2000, p Heer M, Baisch F, Kropp J, Gerzer R, and Drummer C. High dietary sodium chloride consumption may not induce body fluid retention in humans. Am J Physiol Renal Physiol 278: F585 F595, Katz M. Hyperglycemia-induced hyponatremia. Calculation of expected sodium depression. N Engl J Med 289: , Kutchai HC. Cellular membranes and transmembrane transport of solutes and water. In: Physiology, edited by Berne RM, Levy MN, Koeppen BM, and Stanton BA. St. Louis, MO: Mosby, 2004, p Nguyen MK and Kurtz I. Are the total exchangeable sodium, total exchangeable potassium and total body water the only determinants of the plasma water sodium concentration? Nephrol Dial Transplant 18: , Nguyen MK and Kurtz I. Determinants of the plasma water sodium concentration as reflected in the Edelman equation: role of osmotic and Gibbs-Donnan equilibrium. Am J Physiol Renal Physiol 286: F828 F837, Nguyen MK and Kurtz I. New insights into the pathophysiology of the dysnatremias: a quantitative analysis. Am J Physiol Renal Physiol 287: F172 F180, Nguyen MK and Kurtz I. True hyponatremia secondary to intravenous immunoglobulin. J Am Soc Nephrol 16: 532, 2005.

8 1300 GIBBS-DONNAN EQUILIBRIUM, OSMOLALITY, AND SODIUM CONCENTRATION 21. Sperelakis N. Gibbs-Donnan equilibrium potentials. In: Cell Physiology Sourcebook: A Molecular Approach, edited by Sperelakis N. San Diego, CA: Academic, 2001, p Palevsky PM, Rendulic D, and Diven WF. Maltose-induced hyponatremia. Ann Intern Med 118: , Pitts RF. Volume and composition of the body fluids. In: Physiology of the Kidney and Body Fluids, edited by Pitts RF. Chicago, IL: Year Book Medical, 1974, p Rose BD. Introduction to disorders of osmolality. In: Clinical Physiology of Acid-Base and Electrolyte Disorders, edited by Rose BD. New York: McGraw-Hill, 1994, p Schwartz WB, Bennett W, Curelop S, and Bartter FC. A syndrome of renal sodium loss and hyponatremia probably resulting from inappropriate secretion of antidiuretic hormone. Am J Med 23: , Sirken G, Raja R, Garces J, and Bloom E. Contrast-induced translocational hyponatremia and hyperkalemia in advanced kidney disease. Am J Kidney Dis 43: E31 E35, Titze J, Krause H, Hecht H, Dietsch JR, Lang R, Kirsch KA, and Hilgers KF. Reduced osmotically inactive Na storage capacity and hypertension in the Dahl model. Am J Physiol Renal Physiol 283: F134 F141, Titze J, Maillet A, Lang R, Gunga HC, Johannes B, Gauquelin-Koch G, Kihm E, Larina I, Gharib C, and Kirsch KA. Long-term sodium balance in humans in a terrestrial space station simulation study. Am J Kidney Dis 40: , Titze J, Shakibaei M, Schafflhuber M, Schulze-Tanzil G, Porst M, Schwind KH, Dietsch P, and Hilgers KF. Glycosaminoglycan polymerization may enable osmotically inactive Na storage in the skin. Am J Physiol Heart Circ Physiol 287: H203 H208, 2004.