## Abstract

Edelman et al. have empirically shown that plasma water sodium concentration ([Na^{+}]_{pw}) is equal to 1.11(Na_{e} + K_{e})/TBW - 25.6 (Edelman IS, Leibman J, O'Meara MP, Birkenfeld LW. *J Clin Invest* 37: 1236–1256, 1958). However, the physiological significance of the slope and *y*-intercept in this equation has not been previously considered. Our analysis demonstrates that there are several clinically relevant parameters determining the magnitude of the *y*-intercept that independently alter [Na^{+}]_{pw}: *1*) osmotically inactive exchangeable Na^{+} and K^{+}; *2*) plasma water K^{+} concentration; and *3*) osmotically active non-Na^{+} and non-K^{+} osmoles. In addition, we demonstrate quantitatively the physiological significance of the slope in the Edelman equation and its role in modulating [Na^{+}]_{pw}. The slope of 1.11 in this equation which Edelman et al. determined empirically can be theoretically predicted by considering the combined effect of the osmotic coefficient of Na^{+} salts at physiological concentrations and Gibbs-Donnan equilibrium. In addition, our results demonstrate that the slope has an independent quantitative impact on the magnitude of the *y*-intercept in the Edelman equation. From a physiological standpoint, the components of both the slope and the *y*-intercept need to be addressed when considering the factors that modulate [Na^{+}]_{pw}.

- osmotic equilibrium
- dysnatremia
- osmotic coefficient
- potassium

total exchangeable sodium (Na_{e}), total exchangeable potassium (K_{e}), and total body water (TBW) are the major determinants of plasma water sodium concentration ([Na^{+}]_{pw}) (6). The relationship between [Na^{+}]_{pw} and Na_{e}, K_{e}, and TBW was originally demonstrated empirically by Edelman et al.: [Na^{+}]_{pw} = 1.11(Na_{e} + K_{e})/TBW - 25.6. Changes in the mass balance of Na^{+}, K^{+} and H_{2}O, therefore, have an important impact on [Na^{+}]_{pw}. However, previous analyses of the pathogenesis and treatment of the dysnatremias have failed to consider the quantitative and biological significance of the slope (1.11) and *y*-intercept (-25.6) in this equation (10, 18). Recently, we have demonstrated the necessity for a *y*-intercept in the Edelman equation (15, 16). In this article, we characterize the role of osmotic equilibrium in modulating [Na^{+}]_{pw} by determining the physiological relevance of the components of the *y*-intercept in the Edelman equation. Moreover, we demonstrate quantitatively for the first time the role of the osmotic coefficient of Na^{+} salts and Gibbs-Donnan equilibrium in modulating [Na^{+}]_{pw} as reflected in the Edelman equation. Our results indicate that the empirically derived slope of 1.11 in this equation is in excellent agreement with our calculated theoretical value and is mathematically determined by the combined effect of Gibbs-Donnan equilibrium and the osmotic coefficient of Na^{+} salts at physiological concentrations.

## PHYSIOLOGICAL DETERMINANTS AND SIGNIFICANCE OF THE *Y*-INTERCEPT IN THE EDELMAN EQUATION

Na_{e}, K_{e}, and TBW are the primary determinants of [Na^{+}]_{pw} (6). Therefore, alterations in the mass balance of Na^{+}, K^{+} and H_{2}O lead to changes in [Na^{+}]_{pw}. The Edelman equation delineates the relationship between [Na^{+}]_{pw} and Na_{e}, K_{e}, and TBW. However, in this equation, [Na^{+}]_{pw} is not equal to the ratio (Na_{e} + K_{e})/TBW as is commonly assumed. Rather, [Na^{+}]_{pw} is equal to 1.11(Na_{e} + K_{e})/TBW - 25.6 (6). To determine the physiological parameters of the *y*-intercept (-25.6) that can independently alter [Na^{+}]_{pw} (15, 16), we start with the assumption that body fluid compartments are in osmotic equilibrium

Therefore where Na_{osm active} = osmotically active Na^{+}; K_{osm active} = osmotically active K^{+}; osmol_{ECF} = osmotically active, extracellular non-Na^{+} and non-K^{+} osmoles; osmol_{ICF} = osmotically active, intracellular non-Na^{+} and non-K^{+} osmoles; Na_{pw} = plasma water Na^{+}; K_{pw} = plasma water K^{+}; osmol_{pw} = osmotically active, plasma water non-Na^{+} and non-K^{+} osmoles; and V_{pw} = plasma water volume.

Let [Na^{+}]_{pw} = plasma water Na^{+} concentration and [K^{+}]_{pw} = plasma water K^{+} concentration.

Since Na_{pw}/V_{pw} = [Na^{+}]_{pw} and K_{pw}/V_{pw} = [K^{+}]_{pw} Rearranging

It has been shown that exchangeable Na^{+} in bone is bound and is, therefore, osmotically inactive (4–6). More recently, Cameron et al. (3) provided evidence that a major portion of intracellular K^{+} is bound as well and is also osmotically inactive. Hence, not all exchangeable Na^{+} and exchangeable K^{+} are osmotically active.

Therefore, let Na_{e} = total exchangeable Na^{+}; K_{e} = total exchangeable K^{+}; Na_{osm inactive} = osmotically inactive Na^{+}; and K_{osm inactive} = osmotically inactive K^{+}.

Since Na_{e} = Na_{osm active} + Na_{osm inactive} and K_{e} = K_{osm active} + K_{osm inactive} 2

Hence, we have demonstrated mathematically that the *y*-intercept in the Edelman equation is composed of

We now address the role of each of these parameters in modulating [Na^{+}]_{pw}.

### Na_{e} + K_{e}/TBW and [K^{+}]_{pw}

Although it is evident why changes in Na_{e} and TBW can result in changes in [Na^{+}]_{pw}, the mechanism for the effect of K^{+} on [Na^{+}]_{pw} is less apparent from *Eq. 2* because K^{+} appears in two terms: Na_{e} + K_{e}/TBW and [K^{+}]_{pw}. Based on our analysis, both K_{e} and [K^{+}]_{pw} must independently affect [Na^{+}]_{pw}. Indeed, the effect of K^{+} on [Na^{+}]_{pw} differs, depending on whether K^{+} is restricted to the interstitial space (interstitial fluid; ISF), intracellular compartment (intracellular fluid; ICF), or the plasma space (Fig. 1*A*). Because K^{+} is as osmotically active as Na^{+}, interstitial and intracellular K^{+} will tend to increase [Na^{+}]_{pw} by promoting the movement of water from the plasma space into the interstitial space and intracellular compartment, respectively. In contrast, the presence of K^{+} in the plasma space will have a dilutional effect on [Na^{+}]_{pw} by obligating the retention of H_{2}O in the plasma space. Therefore, plasma K^{+} will have a dilutional effect on [Na^{+}]_{pw} by acting as an osmotically active, plasma non-Na^{+} osmole. This dilutional effect of plasma K^{+} on [Na^{+}]_{pw} is, therefore, analogous to that of blood glucose (which is also an osmotically active, plasma non-Na^{+} osmole). Although K_{e} includes intracellular, interstitial, and plasma K^{+}, K_{e} per se will have a net incremental effect on [Na^{+}]_{pw} because the majority of K^{+} ions are in the intracellular and interstitial compartments and therefore the effect of K^{+} in these latter compartments will predominate. For instance, in an average 70-kg man, TBW 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 (18). Therefore, the quantities of exchangeable K^{+} in the ICF, ISF, and plasma space are 3,750 mmol (150 mmol/l × 25 liters), 62 mmol (4.4 mmol/l × 14 liters), and 14 mmol (4.6 mmol/l × 3 liters), respectively. Quantitatively, K_{e} will have a net incremental effect on the [Na^{+}]_{pw} because the effect of K^{+} in the intracellular and interstitial compartments will predominate over that of the plasma compartment.

### (Na_{osm} inactive + K_{osm inactive})/TBW

This parameter reflects the quantity of osmotically inactive exchangeable Na^{+} and K^{+} per unit of TBW. It has been shown that exchangeable Na^{+} in bone and a major portion of intracellular K^{+} are bound and are, therefore, osmotically inactive (3–6). However, the significance of osmotically inactive Na_{e} and K_{e} in the determination of [Na^{+}]_{pw} has not been well appreciated. In the syndrome of inappropriate secretion of antidiuretic hormone, the quantities of Na^{+} lost and H_{2}O retained are insufficient to account for the magnitude of the observed reduction in [Na^{+}]_{pw} in severely hyponatremic patients (2, 19). This discrepancy has been attributed to loss or inactivation of osmotically active solute. More recently, Heer et al. (11) 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^{+}. Similarly, Titze et al. (21) 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. (20) also showed that salt-sensitive Dahl rats (which developed hypertension if fed a high-sodium diet) were characterized by a reduced osmotically inactive Na^{+} storage capacity compared with Sprague-Dawley rats, thereby resulting in fluid accumulation and high blood pressure. Together, these findings provide convincing evidence for the existence of an osmotically inactive Na^{+} and K^{+} reservoir. These osmotically inactive Na_{e} and K_{e} are “ineffective osmoles,” and they do not contribute to the distribution of water between the extracellular and intracellular spaces. Because Na_{e} and K_{e} in the Edelman equation include osmotically active as well as osmotically inactive components, failure to consider osmotically inactive Na_{e} and K_{e} will result in an overestimation of [Na^{+}]_{pw}.

### (Osmol_{ECF} + osmol_{ICF})/TBW

The *y*-intercept is also a reflection of the effect of extracellular and intracellular osmotically active, non-Na^{+} and non-K^{+} osmoles on [Na^{+}]_{pw}. Hence, the presence of osmotically active, non-Na^{+} and non-K^{+} osmoles in the intracellular compartment (osmol_{ICF}) will tend to increase [Na^{+}]_{pw} by promoting the osmotic shift of water from the plasma space into the intracellular compartment (Fig. 1*B*). However, it may seem counterintuitive at first glance that the presence of osmotically active, non-Na^{+} and non-K^{+} osmoles in the extracellular compartment (osmol_{ECF}) will result in an increment in [Na^{+}]_{pw}. On the other hand, it is important to realize that the osmotically active, non-Na^{+} and non-K^{+} osmoles in the extracellular compartment (osmol_{ECF}) include osmoles in both the plasma space and the ISF. The presence of osmotically active, non-Na^{+} and non-K^{+} osmoles in the ISF will, therefore, increase [Na^{+}]_{pw} by inducing the osmotic shift of water from the plasma space into the interstitial compartment. In contrast, the presence of osmotically active, non-Na^{+} and non-K^{+} osmoles in the plasma space will promote water shift from the ISF and intracellular compartment to the plasma space, thereby lowering the [Na^{+}]_{pw}. Because only one-fifth of the extracellular fluid (ECF) is confined to the plasma space (18), it is not surprising that quantitatively the presence of osmotically active, non-Na^{+} and non-K^{+} osmoles in the extracellular compartment (osmol_{ECF}) will have a net effect of increasing [Na^{+}]_{pw} (Fig. 1*C*).

Furthermore, the (osmol_{ECF} + osmol_{ICF})/TBW term accounts for the contribution of osmotically active, non-Na^{+} and non-K^{+} osmoles to total body osmolality. Because TBW is passively distributed in proportion to osmotic activity, total body osmolality will determine the distribution of water allotted to each millimole of osmotically active particle in the body. Specifically, for a given TBW, as the total number of osmotically active particles present in the body increases, the distribution of water allotted to each millimole of osmotically active particle will correspondingly be less. Therefore, for a given quantity of osmotically active particles in a body fluid compartment, the distribution of water within that compartment will be determined by total body osmolality.

### Osmol_{pw}/V_{pw}

The osmotically active, plasma water non-Na^{+} and non-K^{+} osmoles contribute to the movement of water between the plasma space, ISF, and intracellular compartment. Specifically, the presence of osmotically active, plasma water non-Na^{+} and non-K^{+} osmoles (for example, glucose, Ca^{2+}, Mg^{2+}, Cl^{-}, ) will have a dilutional effect on [Na^{+}]_{pw} by obligating the retention of water in the plasma space (Fig. 1*C*).

## PHYSIOLOGICAL DETERMINANTS AND SIGNIFICANCE OF THE SLOPE IN THE EDELMAN EQUATION

The above analysis, therefore, demonstrates that there are several determinants of the *y*-intercept that independently alter [Na^{+}]_{pw}: osmotically inactive exchangeable Na^{+} and K^{+}, plasma water K^{+} concentration ([K^{+}]_{pw}), and osmotically active non-Na^{+} and non-K^{+} osmoles. We now address the significance of the slope of 1.11 in the Edelman equation.

### Gibbs-Donnan Equilibrium

It is well known that [Na^{+}]_{pw} and plasma water chloride concentration ([Cl^{-}]_{pw}) and ISF Na^{+} ([Na^{+}]_{ISF}) and Cl^{-} concentrations ([Cl^{-}]_{ISF}) are different despite the high permeability of Na^{+} and Cl^{-} ions across the capillary membrane, which separates these two fluid compartments (17, 18). This difference in ionic concentrations between plasma and ISF is due to the much higher concentration of proteins in the plasma compared with ISF. Proteins are large molecular weight substances and therefore do not pass the capillary membrane easily. This lack of 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 (17, 18).

Because proteins are negatively charged, nonpermeant molecules present predominantly in the plasma space, their presence will attract positively charged Na^{+} ions and repel negatively charged Cl^{-} ions (Fig. 2). The distribution of Na^{+} and Cl^{-} ions is altered to preserve electroneutrality in the plasma and ISF. As a result, [Na^{+}]_{pw} is typically greater than [Na^{+}]_{ISF}, whereas [Cl^{-}]_{pw} is lower than [Cl^{-}]_{ISF} (Fig. 3) (17, 18). According to Gibbs-Donnan equilibrium, the diffusible Na^{+} and Cl^{-} ions will distribute in such a manner that at equilibrium, the product of the concentrations of Na^{+} and Cl^{-} ions is the same on both sides of the membrane (17, 18) 3

As a result of this Gibbs-Donnan effect, there are more osmotically active particles in the plasma space than that in the ISF at equilibrium (9, 17, 18). Consequently, plasma osmolality is slightly greater than the osmolality of the ISF and intracellular compartment. Indeed, plasma osmolality is typically 1 mosmol/l greater than that of the ISF and intracellular compartment (9). Because H_{2}O is freely permeable across the capillary membrane, the fact that plasma osmolality is maintained at a slightly greater osmolality than that of ISF indicates that the capillary hydrostatic pressure must play a role in opposing the osmotic movement of water into the plasma space.

### Derivation of Determinants of the Slope in the Edelman Equation

To derive the determinants of the slope in the Edelman equation, one needs to consider the effect of Gibbs-Donnan equilibrium on [Na^{+}]_{pw}. In the presence of negatively charged, impermeant plasma proteins, the diffusible Na^{+} and Cl^{-} ions will distribute in such a manner that at equilibrium, the product of the concentrations of Na^{+} and Cl^{-} ions is the same on both sides of the membrane (17, 18) 3 where [Na^{+}]_{pw} and [Na^{+}]_{ISF} = plasma water and ISF Na^{+} concentration in the presence of the Gibbs-Donnan effect, respectively, and [Cl^{-}]_{pw} and [Cl^{-}]_{ISF} = plasma water and ISF chloride concentration in the presence of the Gibbs-Donnan effect, respectively. 4

It is known that the Gibbs-Donnan ratio for the distribution of univalent cations between the plasma and interstitial fluid is 100:95 (17). Therefore, [Na^{+}]_{ISF} should be 0.95 times that of [Na^{+}]_{pw} 4A

Now, we consider in the absence of the Gibbs-Donnan effect. In the absence of the Gibbs-Donnan effect, must be equal to , and must be equal to at equilibrium where and plasma water and ISF [Na] in the absence of the Gibbs-Donnan effect, respectively, and and plasma water and ISF [Cl] in the absence of the Gibbs-Donnan effect, respectively.

Because the total number of Na^{+} ions and total volume of fluid in the plasma and ISF are the same in the absence or presence of the Gibbs-Donnan effect where plasma water and ISF volume in the absence of the Gibbs-Donnan effect, respectively, and V_{pw} and V_{ISF} = plasma water and ISF volume in the presence of the Gibbs-Donnan effect, respectively.

Because at equilibrium and substituting

Therefore, in the absence of the Gibbs-Donnan effect, the and at equilibrium would be as follows 5

Because [Na^{+}]_{ISF} = 0.95 × [Na^{+}]_{pw} (*Eq. 4A*) and substituting *Eq. 4A* into *Eq. 5*

Let Therefore 6

In deriving the physiological parameters comprising the *y*-intercept in the Edelman equation, we have in effect determined the by assuming that the osmolality in all fluid compartments is exactly equal. This assumption is only true if negatively charged, nonpermeant proteins are not present in the plasma. Actually, plasma osmolality is typically 1 mosmol/l greater than that of the ISF and ICF due to the Gibbs-Donnan effect (9). Therefore, *Eq. 2* depicts the in the absence of the Gibbs-Donnan effect 2 where is expressed in mOsm/L.

Because *Eq. 2* is derived based on the assumption that the body fluid compartments are in osmotic equilibrium, is expressed in milliosmoles per liter in this equation. In the Edelman equation, the [Na^{+}]_{pw} is expressed in millimoles per liter. Although both moles and osmoles refer to the number of particles, the two values differ in that moles simply refers to the number of particles, whereas osmoles take into account how effective the molecule is as an independent osmotically active particle. Importantly, each mole of ionic particles does not exert exactly one osmole of osmotic activity due to the electrical interactions between the ions (i.e., the osmotic activity of most ionic particles is slightly <1). Because there are intermolecular forces between particles, the osmotic coefficient Ø is used to account for the effectiveness of ions as independent osmotically active particles at physiological concentrations (13). Specifically, NaCl has an osmotic coefficient of 0.93, whereas NaHCO_{3} has an osmotic coefficient of 0.96 (13). Because Na^{+} is present in the plasma predominantly as NaCl and NaHCO_{3}, and the normal plasma [Cl^{-}] and [] are 104 and 24 mmol/l, respectively, the average osmotic coefficient of Na^{+} salts is estimated to be 0.94 (104/128 × 0.93 + 24/128 × 0.96). Therefore, to convert in *Eq. 2* from milliosmoles per liter to millimoles per liter, one needs to multiply both sides of *Eq. 2* by 1.064 (1/0.94) 7 where is expressed in millimoles per liter.

Because (*Eq. 6*) 8 where

Hence, (1.064)*G* or *G*/Ø (where Ø represents the average osmotic coefficient of Na^{+} salts, and the term *G* represents the Gibbs-Donnan effect) in *Eq. 8* corresponds to the slope of 1.11 in the Edelman equation, whereas the *y*-intercept of -25.6 in the Edelman equation as discussed is represented by the terms

The slope of 1.11 in the Edelman equation, therefore, is a reflection of Ø and *G* and their effect on [Na^{+}]_{pw}. Importantly, *G* is also a function of the volumes of plasma and ISF (V_{pw} and V_{ISF}). Because Na^{+} ions can freely diffuse across capillary membranes, their distribution between the plasma and ISF at equilibrium will depend on the respective volumes of plasma and ISF. In other words, there will be more Na^{+} ions distributed in ISF at equilibrium than that in plasma by virtue of the fact that ISF volume is much greater than that of plasma. Furthermore, it is evident from *Eq. 8* that the physiological parameters in the *y*-intercept in the Edelman equation are multiplied by *G*, and therefore *G* has an additional independent effect on [Na^{+}]_{pw} because it is also a determinant of the magnitude of the *y*-intercept. This is not surprising because the components of the *y*-intercept must be similarly affected by the Gibbs-Donnan effect. Specifically, the distribution of non-Na^{+} cations and anions between the plasma and ISF (as represented by the terms osmol_{ECF}, [K^{+}]_{pw}, and osmol_{pw}) will also be altered by the presence of negatively charged, impermeant plasma proteins.

Regarding the magnitude of *G*, because the denominator of this ratio (V_{pw} + 0.95 × V_{ISF}) is less than the numerator (V_{pw} + V_{ISF}), then *G* must be >1. Because roughly one-fifth of ECF is confined to the plasma space (18), *G* can be calculated as follows

Therefore, *G* accounts for the altered distribution of Na^{+} ions between the plasma and ISF due to the presence of negatively charged, nonpermeant proteins in the plasma space. Indeed, *G* takes into consideration the fact that [Na^{+}]_{pw} must be greater than because the presence of negatively charged, impermeant plasma proteins will attract more positively charged Na^{+} ions within the plasma space.

Furthermore, according to *Eq. 8*, the slope of the Edelman equation is represented by the term (1.064)*G.* Because the value of *G* is ∼1.04, the predicted mathematically derived slope in *Eq. 8* must be equal to 1.11, which is identical to the empirically determined slope in the Edelman equation. Therefore, the slope of 1.11 in the Edelman equation reflects not only the Gibbs-Donnan effect on [Na^{+}]_{pw} but is also determined by the osmotic coefficient of Na^{+} salts.

## PATHOPHYSIOLOGICAL IMPLICATIONS

### Glucose and [Na^{+}]_{pw}

Changes in the plasma glucose concentration alter [Na^{+}]_{pw} because of a corresponding change in the magnitude of several components of the *y*-intercept in the Edelman equation. Previously, it has been shown that there is an expected decrease of 1.6 meq/l in the plasma sodium concentration for each 100 mg/dl increment in the plasma glucose concentration assuming no change in the mass balance of Na^{+}, K^{+}, and H_{2}O (12). It is commonly assumed that this correction factor of 1.6 is solely attributable to the effect of hyperglycemia on plasma [Na^{+}] ([Na^{+}]_{p}). While this concept is not incorrect, it is, however, incomplete. In fact, the correction factor of 1.6 results from simultaneous changes in three of the four parameters determining the *y*-intercept. First, hyperglycemia results in an increase in the ratio (osmol_{ECF} + osmol_{ICF})/TBW, which accounts for the hyperglycemia-induced increase in total body osmolality. This ratio increases because hyperglycemia increases the osmol_{ECF} term, whereas TBW remains unchanged. In the hyperglycemia-induced osmotic shift of water from the intracellular compartment to the extracellular space, TBW remains constant because the change in intracellular volume (ΔV_{ICF}) is equal to the change in extracellular volume (ΔV_{ECF}). Second, the hyperglycemia-induced osmotic shift of water results in a decrease in [K^{+}]_{pw}. The final [K^{+}]_{pw} will also be affected by the magnitude of subsequent cellular K^{+} efflux induced by the decrease in [K^{+}]_{pw} and hyperosmolality. Finally, the plasma water concentration of osmotically active non-Na^{+} and non-K^{+} osmoles represented by the term osmol_{pw}/V_{pw} increases in hyperglycemia, thereby lowering [Na^{+}]_{pw} by dilution by promoting the osmotic movement of water into the plasma space.

In the setting of hyperglycemia, changes in [Na^{+}]_{pw} result not only from the dilutional effect of hyperglycemia induced by the translocation of water but also to changes in the mass balance of Na^{+}, K^{+}, and TBW. To predict the effect of changes in Na_{e}, K_{e}, and TBW as well as the dilutional effect of hyperglycemia on [Na^{+}]_{p} attributable to the osmotic shift of water, the correction factor of 1.6 can be incorporated into the Edelman equation

Multiplying both sides of the equation by 0.93 to convert [Na^{+}]_{pw} to [Na^{+}]_{p} (1)

Because 0.93 × [Na^{+}]_{pw} = [Na^{+}]_{p} (1)

In the study by Edelman et al. (6), the average plasma glucose concentration was 120 mg/dl. Because there is an expected decrease of 1.6 meq/l in [Na^{+}]_{p} for each 100 mg/dl increment in the plasma glucose concentration (12)

Because hyperglycemia-induced hyponatremia can result from changes in the mass balance of Na^{+}, K^{+}, and TBW as well as to the dilutional effect of hyperglycemia induced by the translocation of water, [Na^{+}]_{p} can be predicted from the following equation: [Na^{+}]_{p} = 1.03(Na_{e} + K_{e})/TBW - 23.8 - (1.6/100)([glucose] - 120). Therefore, the *y*-intercept is not constant and will vary directly with the plasma glucose concentration.

### K^{+} and [Na^{+}]_{pw}

Previous studies have invoked a role for direct cellular Na^{+} uptake (to replace intracellular K^{+}) as a mechanism for the modulation of [Na^{+}]_{pw} during hypokalemia resulting from negative K^{+} balance (7, 8, 14). Indeed, the concept that a deficit of intracellular K^{+} may result in the entry of Na^{+} into cells to restore electroneutrality is supported by the finding of an increase in nonextracellular Na^{+} and an increased ratio of nonextracellular Na^{+} to exchangeable Na^{+} in patients with diuretic-induced hyponatremia (7). However, the cellular uptake of extracellular Na^{+} per se cannot be responsible for the depression of [Na^{+}]_{pw} in patients with hypokalemia due to negative mass balance of K^{+}. The cellular uptake of extracellular Na^{+} without concomitant K^{+} exit will not result in a fall in [Na^{+}]_{pw} because osmotic equilibrium must be restored by a shift of extracellular water into the cells, thereby raising [Na^{+}]_{pw} to normal. This is demonstrated quantitatively by the fact that the cellular uptake of extracellular Na^{+} without concomitant cellular K^{+} efflux will not change the ratio (Na_{e} + K_{e})/TBW or any other terms in *Eq. 8*. In fact, from the standpoint of [Na^{+}]_{pw}, it is irrelevant whether Na^{+} is restricted to the intracellular or extracellular compartments. Intracellular Na^{+}, which is as osmotically active as intracellular K^{+}, acts to raise [Na^{+}]_{pw} by promoting the shift of water from ECF into ICF. Similarly, interstitial Na^{+} will also cause a shift of water from the plasma space into the interstitial space, thereby increasing [Na^{+}]_{pw}.

Given that cellular Na^{+} influx per se will not alter the [Na^{+}]_{pw}, does *Eq. 8* predict that the cellular loss of K^{+} accompanied by an equimolar Na^{+} influx will result in a decrement in the [Na^{+}]_{pw}? Because in this scenario intracellular K^{+} is replaced by extracellular Na^{+}, there will be no change in the ratio (Na_{e} + K_{e})/TBW. However, the equimolar exchange of cellular K^{+} for Na^{+} will lead to an increase in [K^{+}]_{pw} and therefore, according to *Eq. 8*, [Na^{+}]_{pw} must decrease. Because K^{+} is as osmotically active as Na^{+}, the increase in plasma K^{+} obligates the retention of water in the plasma space, thereby diluting [Na^{+}]_{pw}. Specifically, the reason for the decrease in [Na^{+}]_{pw} is not that Na^{+} enters the ICF per se but rather that K^{+} efflux into the ECF prevents water from entering the ICF with Na^{+}, resulting in a dilution of [Na^{+}]_{pw}. Finally, regardless of the mechanisms of cellular K^{+} efflux, it is important to appreciate that the flux of K^{+} into the plasma space lessens the magnitude of the hypokalemia. However, because cellular K^{+} efflux ameliorates the hypokalemia, as [K^{+}]_{pw} increases, the magnitude of water flux out of the plasma space will diminish resulting in a decrease in [Na^{+}]_{pw}.

### Effect of Gibbs-Donnan Equilibrium and Osmotic Coefficient of Na^{+} Salts on [Na^{+}]_{pw}

The effect of Gibbs-Donnan equilibrium and osmotic coefficient of Na^{+} salts on the [Na^{+}]_{pw} can be illustrated graphically in Fig. 4. By assuming that [Na^{+}]_{pw} is exactly equal to the ratio (Na_{e} + K_{e})/TBW (10, 18), one cannot account for *1*) the net depressive effect of osmotically inactive exchangeable Na^{+} and K^{+}, [K^{+}]_{pw} and osmotically active, non-Na^{+} and non-K^{+} osmoles on [Na^{+}]_{pw}; *2*) the effect of electrical interactions between Na^{+} and its associated anions under physiological conditions (as reflected by the osmotic coefficient of Na^{+} salts); and *3*) the incremental effect of the Gibbs-Donnan effect (*G*) on the [Na^{+}]_{pw}. The effect of Gibbs-Donnan equilibrium on [Na^{+}]_{pw} can be demonstrated graphically by comparing the [Na^{+}]_{pw} in the presence of the Gibbs-Donnan effect to [Na^{+}]_{pw} in the absence of the Gibbs-Donnan effect. [Na^{+}]_{pw} in the absence of Gibbs-Donnan equilibrium can be predicted by dividing the Edelman equation by the factor *G* of 1.04 {i.e., [Na^{+}]_{pw} = 1.067(Na_{e} + K_{e})/TBW - 24.6}. As illustrated in Fig. 4, Gibbs-Donnan equilibrium has an incremental effect on the [Na^{+}]_{pw}. Similarly, [Na^{+}]_{pw} in the absence of the Gibbs-Donnan effect and electrical interactions between Na^{+} and its associated anions can be predicted by dividing the equation [Na^{+}]_{pw} = 1.067(Na_{e} + K_{e})/TBW - 24.6 by the factor 1/Ø of 1.064 {i.e., [Na^{+}]_{pw} = 1.00(Na_{e} + K_{e})/TBW - 23.1}. The effect of intermolecular forces between Na^{+} and its associated anions (as reflected by the osmotic coefficient) on the [Na^{+}]_{pw} can be illustrated by comparing these last two equations. Finally, Fig. 4 also demonstrates that at any given (Na_{e} + K_{e})/TBW, the components of the *y*-intercept, which include the osmotically inactive exchangeable Na^{+} and K^{+}, [K^{+}]_{pw}, and osmotically active, non-Na^{+} and non-K^{+} osmoles, result in a net depressive effect on [Na^{+}]_{pw}.

### Effect of Changes in Na_{e}, K_{e}, and TBW on [Na^{+}]_{p}

Changes in the mass balance of Na^{+}, K^{+}, and H_{2}O can lead to alterations in [Na^{+}]_{p} (6). To determine the effect of a given mass balance of Na^{+}, K^{+}, and H_{2}O on [Na^{+}]_{p}, one needs to estimate the initial Na_{e} + K_{e}. Taking into account the quantitative significance of the slope and *y*-intercept in the Edelman equation, the initial Na_{e} + K_{e} can be determined as follows

Multiplying both sides of the equation by 0.93 to convert [Na^{+}]_{pw} to [Na^{+}]_{p} (1)

Because 0.93 × [Na^{+}]_{pw} = [Na^{+}]_{p} (1)

Therefore

Hence, the effect of a given mass balance of Na^{+}, K^{+}, and H_{2}O on [Na^{+}]_{p} can be predicted by the following equation where

## APPARENT IDENTITY OF [Na^{+}]_{P} AND [Na^{+}]_{ISF}

According to Gibbs-Donnan equilibrium, [Na^{+}]_{pw} must be greater than [Na^{+}]_{ISF}. However, it is well known that [Na^{+}]_{p} is about the same as [Na^{+}]_{ISF} (17). The reason [Na^{+}]_{p} approximates [Na^{+}]_{ISF} has been attributed to the nonwater phase of plasma. Only 93% of plasma is normally composed of water, whereas fat and proteins account for the remaining 7% (1). Therefore, [Na^{+}]_{p} = 0.93 × [Na^{+}]_{pw}. As discussed previously, [Na^{+}]_{ISF} = 0.95 × [Na^{+}]_{pw} based on the Gibbs-Donnan ratio for the distribution of univalent cations between the plasma and ISF (17). Hence, the apparent identity of [Na^{+}]_{p} and [Na^{+}]_{ISF} is due to the fortuitous coincidence that the effect of the nonwater phase of plasma, 0.93, and the Gibbs-Donnan factor of 0.95 for Na^{+} are almost identical.

## EFFECT OF GIBBS-DONNAN EQUILIBRIUM ON PLASMA ONCOTIC PRESSURE

It has been empirically shown that the plasma oncotic pressure generated by plasma proteins is greater than that predicted on the basis of the protein concentration (17, 18). This difference is thought to be due to Gibbs-Donnan equilibrium, because more particles are present in the protein-containing compartment even if one disregards the nondiffusible protein anions. According to Gibbs-Donnan equilibrium (*Eq. 3*), if Na^{+} and Cl^{-} are the only diffusible ions in plasma water and ISF, it is predicted that the total number of millimoles of Na^{+} and Cl^{-} per liter in the plasma water will exceed that in the interstitial fluid by 0.4 mmol/l (18). Although this difference appears to be small, it has a significant hydrostatic effect on capillary oncotic pressure. Because normal plasma protein concentration is typically 0.9 mmol/l and 1 mmol/kg generates an osmotic pressure of 19.3 mmHg, the Gibbs-Donnan effect increases the capillary oncotic pressure from 17.4 mmHg (0.9 × 19.3) to 25 mmHg (1.3 × 19.3) (18). Therefore, due to Gibbs-Donnan equilibrium, colloid osmotic pressure has its origin not only in the effect of protein particles per se on the activity of water but also in the effect of the excess of diffusible ions (17, 18) (Fig. 2).

In summary, although Edelman et al. (6) have empirically shown that [Na^{+}]_{pw} is equal to 1.11(Na_{e} + K_{e})/TBW - 25.6, neither the physiological significance nor the theoretical basis of the slope and *y*-intercept in this equation was previously appreciated. Our analysis demonstrates that there are several physiologically relevant parameters determining the magnitude of the *y*-intercept that independently alter [Na^{+}]_{pw}: *1*) osmotically inactive Na_{e} and K_{e}; *2*) [K^{+}]_{pw}; and *3*) osmotically active, non-Na^{+} and non-K^{+} osmoles. In addition, we demonstrate quantitatively the physiological significance of the slope in the Edelman equation and its role in modulating [Na^{+}]_{pw}. The slope is shown mathematically to be determined by Ø and *G*. Furthermore, our results demonstrate that the slope has an independent quantitative impact on the magnitude of the *y*-intercept in the Edelman equation as well.

## APPENDIX

### Derivation of the Components of the Edelman Equation

Total body water osmolality = plasma water osmolality Therefore where Na_{osm active} = osmotically active Na^{+}; K_{osm active} = osmotically active K^{+}; osmol_{ECF} = osmotically active, extracellular non-Na^{+} and non-K^{+} osmoles; osmol_{ICF} = osmotically active, intracellular non-Na^{+} and non-K^{+} osmoles; Na_{pw} = plasma water Na^{+}; K_{pw} = plasma water K^{+}; osmol_{pw} = osmotically active, plasma water non-Na^{+} non-K^{+} osmoles; and V_{pw} = plasma water volume.

Let [Na^{+}]_{pw} = plasma water Na^{+} concentration and [K^{+}]_{pw} = plasma water K^{+} concentration.

Since Na_{pw}/V_{pw} = [Na^{+}]_{pw} and K_{pw}/V_{pw} = [K^{+}]_{pw}

Rearranging

Let Na_{e} = total exchangeable Na^{+}; K_{e} = total exchangeable K^{+}; Na_{osm inactive} = osmotically inactive Na^{+}; and K_{osm inactive} = osmotically inactive K^{+}.

Since Na_{e} = Na_{osm active} + Na_{osm inactive} and K_{e} = K_{osm active} + K_{osm inactive}

According to Gibbs-Donnan equilibrium 3 where [Na^{+}]_{pw} and [Na^{+}]_{ISF} = plasma water and ISF Na^{+} concentration in the presence of Gibbs-Donnan effect respectively; [Cl^{-}]_{pw} and [Cl^{-}]_{ISF} = plasma water and ISF Cl^{-} concentration in the presence of the Gibbs-Donnan effect, respectively.

Rearranging 4

It is known that the Gibbs-Donnan ratio for the distribution of univalent cations between the plasma and interstitial fluid is 100:95. Therefore, [Na^{+}]_{ISF} should be 0.95 times that of [Na^{+}]_{pw} 4A

In the absence of the Gibbs-Donnan effect (at equilibrium) where and = plasma water and ISF Na concentration in the absence of the Gibbs-Donnan effect, respectively; and plasma water and ISF Cl concentration in the absence of the Gibbs-Donnan effect, respectively.

Since the total number of Na^{+} ions and total volume of fluid in the plasma and ISF are the same in the absence or presence of the Gibbs-Donnan effect where and plasma water and ISF volume in the absence of the Gibbs-Donnan effect, respectively; V_{pw} and V_{ISF} = plasma water and ISF volume in the presence of the Gibbs-Donnan effect, respectively.

Since at equilibrium and substituting for

Rearranging

Therefore, in the absence of the Gibbs-Donnan effect, the and at equilibrium would be as follows 5

Since [Na^{+}]_{ISF} = 0.95 × [Na^{+}]_{pw} (*Eq. 4A*) and substituting *Eq. 4A* into *Eq. 5* Rearranging Rearranging

Let Therefore

In deriving the physiological parameters comprising the *y*-intercept in the Edelman equation, we have in effect determined the by assuming that the osmolality in all fluid compartments is exactly equal. Actually, the plasma osmolality is typically 1 mosm/l greater than that of the ISF and intracellular compartment due to the Gibbs-Donnan effect. Therefore, *Eq. 2* depicts in the absence of the Gibbs-Donnan effect where is expressed in milliosmoles per liter.

Because there are intermolecular forces between particles, NaCl has an osmotic coefficient (Ø) of 0.93, whereas NaHCO_{3} has an Ø of 0.96. Because Na^{+} is present in the plasma predominantly as NaCl and NaHCO_{3}, and the normal plasma [Cl^{-}] and [] are 104 and 24 mmol/l, respectively, the average Ø of Na^{+} salts is estimated to be 0.94 (104/128 × 0.93 + 24/128 × 0.96). Therefore, to convert in *Eq. 2* from milliosmoles per liter to millimoles per liter, one needs to multiply both sides by 1.064 (1/0.94) 7 where is expressed in millimoles per liter. Since (6) Where

Hence, (1.064)*G* or *G*/Ø (where Ø represents the average osmotic coefficient of Na^{+} salts) in *Eq. 8* corresponds to the slope of 1.11 in the Edelman equation, whereas the *y*-intercept of -25.6 in the Edelman equation is represented by the terms

## Acknowledgments

This work was supported by the Max Factor Family Foundation, the Richard and Hinda Rosenthal Foundation, the Fredericka Taubitz Fund, and the National Kidney Foundation of Southern California (J891002).

## Footnotes

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- Copyright © 2004 the American Physiological Society