## Abstract

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

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.

Therefore: (1) where *g* is plasma osmolality/total body osmolality.

Since: (2) (3) where θ_{pw} is plasma osmolality, V_{pw} is plasma water volume, θ_{ISF} is ISF osmolality, V_{ISF} is ISF volume, θ_{ICF} is ICF osmolality, V_{ICF} is ICF volume, and TBW 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: (4)

We now define the components of plasma osmolality and total body osmolality: (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. (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: (1)

Therefore: (7)

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

Since: (8)

Rearranging: (9) where:

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} (10) (11) 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): (12) where [Na^{+}]_{pw} is expressed in millimoles per liter H_{2}O, Ø is osmotic coefficient of Na^{+} salts, and θ is osmolality;

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 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})/TBW − (Na_{osm inactive} + K_{osm inactive})/TBW 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 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.

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} (11)

Since the terms (Na_{e} + K_{e})/TBW − (Na_{osm inactive} + K_{osm inactive}/TBW) represent the osmotically active Na^{+} and K^{+} ions:

Since osmol_{ECF} = osmol_{pw} + osmol_{ISF}:

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 θ_{pw} and θ_{ISF} in the Gibbs-Donnan correction factor *g* (i.e., θ_{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 θ_{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})/TBW 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 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

Historically, the pathogenesis of the dysnatremias has focused on the effect of changes in the mass balance of Na_{e}, K_{e}, and TBW on the [Na^{+}]_{pw}. This pathophysiological approach is based on the empirical relationship between the [Na^{+}]_{pw} and Na_{e}, K_{e}, and TBW originally demonstrated by Edelman et al. (10) where [Na^{+}]_{pw} = 1.11(Na_{e} + K_{e})/TBW − 25.6. 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})/TBW 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:

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:

Therefore, *Eq. 12* is especially helpful in explaining quantitatively how physiological factors that do not alter the value of the (Na_{e} + K_{e})/TBW 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 θ_{pw} and θ_{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}.

### 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})/TBW 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})/TBW. During the non-Na^{+} and non-K^{+} osmoles-induced osmotic shift of water from the intracellular compartment to the extracellular space, the TBW remains constant because the change in intracellular volume is equal to the change in extracellular volume. The (osmol_{ECF} + osmol_{ICF})/TBW term in *Eq. 12* increases because non-Na^{+} and non-K^{+} osmoles increase the osmol_{ECF} term whereas the TBW 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}/V_{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: Multiplying both sides of the equation by 0.93 to convert [Na^{+}]_{pw} to [Na^{+}]_{p} (1, 7):

Because 0.93 × [Na^{+}]_{pw} = [Na^{+}]_{p} (1, 7):

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): where Δ[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): 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})/TBW 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 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:

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 how physiological factors that do not alter the value of the (Na_{e} + K_{e})/TBW 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-07789.

## Footnotes

^{a}The osmolal quantity of plasma osmotically active Na^{+}can also be determined by calculating the product of V_{pw}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) × V_{pw}. 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}.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.

- Copyright © 2006 the American Physiological Society