## Abstract

Because changes in the plasma water sodium concentration ([Na^{+}]_{pw}) are clinically due to changes in the mass balance of Na^{+}, K^{+}, and H_{2}O, the analysis and treatment of the dysnatremias are dependent on the validity of the Edelman equation in defining the quantitative interrelationship between the [Na^{+}]_{pw} and the total exchangeable sodium (Na_{e}), total exchangeable potassium (K_{e}), and total body water (TBW) (Edelman IS, Leibman J, O'Meara MP, Birkenfeld LW. *J Clin Invest* 37: 1236–1256, 1958): [Na^{+}]_{pw} = 1.11(Na_{e} + K_{e})/TBW − 25.6. The interrelationship between [Na^{+}]_{pw} and Na_{e}, K_{e}, and TBW in the Edelman equation is empirically determined by accounting for measurement errors in all of these variables. In contrast, linear regression analysis of the same data set using [Na^{+}]_{pw} as the dependent variable yields the following equation: [Na^{+}]_{pw} = 0.93(Na_{e} + K_{e})/TBW + 1.37. Moreover, based on the study by Boling et al. (Boling EA, Lipkind JB. 18: 943–949, 1963), the [Na^{+}]_{pw} is related to the Na_{e}, K_{e}, and TBW by the following linear regression equation: [Na^{+}]_{pw} = 0.487(Na_{e} + K_{e})/TBW + 71.54. The disparities between the slope and *y*-intercept of these three equations are unknown. In this mathematical analysis, we demonstrate that the disparities between the slope and *y*-intercept in these three equations can be explained by how the osmotically inactive Na^{+} and K^{+} storage pool is quantitatively accounted for. Our analysis also indicates that the osmotically inactive Na^{+} and K^{+} storage pool is dynamically regulated and that changes in the [Na^{+}]_{pw} can be predicted based on changes in the Na_{e}, K_{e}, and TBW despite dynamic changes in the osmotically inactive Na^{+} and K^{+} storage pool.

- exchangeable sodium

in 1958, edelman et al. reported the quantitative interrelationship between the [Na^{+}]_{pw}, and Na_{e}, K_{e}, and TBW in the following equation (5):
(1)

This equation has been named the “Edelman equation” after its original discoverer. The slope and *y*-intercept in the Edelman equation have recently been shown to have quantitative and physiological significance (7, 8, 13). This analysis demonstrates that there are several physiologically relevant parameters determining the magnitude of the slope and *y*-intercept that independently alter the [Na^{+}]_{pw}:
(2)

where *G* = (V_{pw} + V_{ISF})/(V_{pw} + R × V_{ISF}), Ø is the average osmotic coefficient of Na^{+} salts, R is the Gibbs-Donnan ratio for the distribution of univalent cations between the plasma and interstitial fluid, V_{pw} is plasma water volume, V_{ISF} is interstitial fluid volume, Na_{osm inactive} is osmotically inactive Na^{+}, K_{osm inactive} is osmotically inactive K^{+}, osmol_{ECF} is osmotically active extracellular non-Na^{+} and non-K^{+} osmoles, osmol_{ICF} is osmotically active intracellular non-Na^{+} and non-K^{+} osmoles, [Na^{+}]_{pw} is plasma water Na^{+} concentration, [K^{+}]_{pw} is plasma water K^{+} concentration, and osmol_{pw} is osmotically active plasma water non-Na^{+} non-K^{+} osmoles.

*Equation 2* has the form of a linear equation: *y* = *mx* + *b* where: *y* = [Na^{+}]_{pw}, *x* = (Na_{e} + K_{e})/TBW, *m* = *G*/Ø, *b* = −*G*/Ø [(Na_{osm inactive} + K_{osm inactive})/TBW − (osmol_{ICF} + osmol_{ECF})/TBW + [K^{+}]_{pw} + (osmol_{pw}/V_{pw})].

*Equation 2* defines all of the physiological parameters that determine the magnitude of the [Na^{+}]_{pw} (7, 8, 13). Simplistically, the [Na^{+}]_{pw} is a function of the quantity of Na^{+} and volume of water in the plasma space:
(3)

Therefore, any physiological factor that alters the numerator and/or denominator of this ratio will modulate the [Na^{+}]_{pw} and is a determinant of the [Na^{+}]_{pw}. Indeed, the parameters of the slope and *y*-intercept in the Edelman equation have recently been shown to modulate both the numerator and denominator of the ratio in *Eq. 3* (7, 8, 13). The Edelman equation accounts for the fact that all osmotically active solutes in the body fluid compartments modulate the [Na^{+}]_{pw}. Because the body fluid compartments are in osmotic equilibrium with each other, all osmotically active solutes determine the distribution of water in the plasma space and modulate the denominator of *Eq. 3*. Moreover, because Gibbs-Donnan equilibrium alters the distribution of Na^{+} between the plasma and interstitial fluid, it will modulate the numerator of *Eq. 3*. Last, the Edelman equation accounts for the osmotic coefficient of Na^{+} salts in its slope, which reflects the osmotic activity of Na^{+} salts. Indeed, the osmotic activity of Na^{+} salts will determine the distribution of water within the plasma space and therefore modulate the denominator of *Eq. 3*.

It is also well appreciated that not all Na_{e} and K_{e} are osmotically active (2–5, 18, 19). There is convincing evidence that a portion of Na_{e} is bound in bone and skin tissues and is therefore rendered osmotically inactive (3–5, 18, 19). Likewise, a portion of cellular K^{+} is reduced in its mobility and in its osmotic activity due to its association with anionic groups such as carboxyl groups on proteins or to phosphate groups in creatine phosphate, ATP, proteins, and nucleic acids (2). Because osmotically inactive Na_{e} and K_{e} do not contribute to the distribution of water between the plasma and nonplasma compartments, osmotically inactive Na_{e} and K_{e} cannot contribute to the modulation of the [Na^{+}]_{pw}. Consequently, the terms (Na_{e} + K_{e})/TBW − (Na_{osm inactive} + K_{osm inactive})/TBW in *Eq. 2* are used to demonstrate that not all Na_{e} and K_{e} in the Edelman equation contribute to the modulation of the [Na^{+}]_{pw}.

In 1963, Boling et al. conducted a similar study in which the [Na^{+}]_{pw}, Na_{e}, K_{e}, and TBW were measured (1). Based on the data in this study (Table 2), the [Na^{+}]_{pw} is related to the Na_{e}, K_{e}, and TBW by the following linear regression equation: [Na^{+}]_{pw} = 0.487(Na_{e} + K_{e})/TBW + 71.54 (*Eq. 4*). Surprisingly, there are marked disparities in the quantitative interrelationship between the [Na^{+}]_{pw} and Na_{e}, K_{e}, and TBW in the Edelman equation and *Eq. 4*. In this mathematical analysis, we demonstrate that the disparities in the slope and *y*-intercept of the Edelman equation and *Eq. 4* can be explained by how the osmotically inactive Na^{+} and K^{+} storage pool is quantitatively accounted for.

#### Osmotically inactive sodium and potassium storage as reflected in the Edelman and Boling data.

In this mathematical analysis, the slope and *y*-intercept refer to “*m*” and “*b*,” respectively, in the linear equation *y* = *mx* + *b* where *y* = [Na^{+}]_{pw} and *x* = (Na_{e} + K_{e})/TBW. The disparities between the slope and *y*-intercept in the Edelman equation and *Eq. 4* can be explained by the hypothesis that the osmotically inactive Na^{+} and K^{+} storage pool is quantitatively reflected in the magnitude of the *y*-intercept in the Edelman equation and magnitude of the slope in *Eq. 4*. Because osmotically inactive Na_{e} and K_{e} do not contribute to the distribution of water between the plasma and nonplasma compartments, only the osmotically active Na_{e} and K_{e} contribute to the modulation of the [Na^{+}]_{pw}. The osmotically active Na_{e} and K_{e} can be accounted quantitatively by one of the two mathematical expressions:

or

where F_{osm active} represents the fraction of Na_{e} + K_{e} that is osmotically active.

The term (Na_{osm inactive} + K_{osm inactive})/TBW is a component of the *y*-intercept in *Eq. 2*:
(2)

Alternatively, *Eq. 2* can be re-expressed to account for the osmotically active Na_{e} and K_{e} as a component of the slope instead:
(5)

We hypothesize that the slope and *y*-intercept in the Edelman equation (*Eq. 1*) are represented by the following mathematical expressions in *Eq. 2*, respectively:

We hypothesize that the slope and *y*-intercept in *Eq. 4* are represented by the following mathematical expressions in *Eq. 5*, respectively:

One can test the hypothesis that *Eq. 4* accounts for the osmotically active Na_{e} and K_{e} as a component of the slope. Using the data in the study by Edelman et al. (5), if one assumes that the slope is 0.487, one can calculate the *y*-intercept of each data point as follows:

Assuming that the slope is 0.487, the average theoretical *y*-intercept ± SE (*y*^{5}) of all the data points in the Edelman study (Table 1) is 67.625 ± 0.74, which approximates the *y*-intercept of 71.536 in the Boling study. In other words, if the osmotically active Na_{e} and K_{e} were to be accounted for as a component of the slope, then one would expect the average theoretical *y*-intercept (*y*^{5}) without the term (Na_{osm inactive} + K_{osm inactive})/TBW based on the Edelman data to approximate the *y*-intercept in *Eq. 4*, which is indeed the case as shown above (67.625 vs. 71.536).

Similarly, one can test the hypothesis that the osmotically inactive Na_{e} and K_{e} is accounted for as a component of the *y*-intercept in the Edelman equation. Using the data in the study by Boling and Lipkind (1), if one assumes that the slope is 1.11, one can calculate the *y*-intercept of each data point as follows:

Assuming that the slope is 1.11, the average theoretical *y*-intercept ± SE (*y*^{2}) of all the data points in the Boling study (Table 2) is −24.2 ± 0.65, which approximates the *y*-intercept of −25.6 in the Edelman equation. In other words, if the osmotically inactive Na_{e} and K_{e} were to be accounted for as a component of the *y*-intercept, then one would expect the average theoretical *y*-intercept (*y*^{2}) with the additional term (Na_{osm inactive} + K_{osm inactive})/TBW based on the Boling data to approximate the *y*-intercept in the Edelman equation, which is indeed the case as shown above (−24.2 vs. −25.6).

Interestingly, based on the data in the study of Edelman et al. (5), linear regression analysis using [Na^{+}]_{pw} as the dependent variable yields the following equation: [Na^{+}]_{pw} = 0.93(Na_{e} + K_{e})/TBW + 1.37 (*Eq. 6*). Nguyen et al. reported that the reason for the discrepancy between *Eqs. 1* (Edelman equation) and *6* is due to the fact that Edelman et al. accounted for measurement errors in the *x* and *y* variables in obtaining the relation between *y* = [Na^{+}]_{pw} and *x* = (Na_{e} + K_{e})/TBW described in *Eq. 1* (15). Edelman et al. recognized that measurement errors in *x* (due to measurement errors in Na_{e}, K_{e}, and TBW) would attenuate estimated correlation coefficients, so that *Eq. 6* would underestimate the magnitude of the slope between true values of *y* and *x*. Nguyen et al. reported that Edelman et al. were specific in describing their method of bias correction for correlation coefficients and that taking into account the analogous bias correction for *Eq. 6* yields results comparable to *Eq. 1* (15). Nguyen et al. have also independently verified similar regression relations using more modern statistical methods that take into account errors in *x* and *y* variables. For example, Nguyen et al. demonstrated that Passing-Bablok regression analysis yields a slope of 1.13 and a *y*-intercept of −28.1, which is in excellent agreement with the bias-corrected slope and *y*-intercept in *Eq. 1* (15).

As an alternative explanation to the disparities in the slope and *y*-intercept in *Eqs. 1* and *6*, we hypothesize that the osmotically inactive Na^{+} and K^{+} storage pool is quantitatively reflected in both the slope and *y*-intercept in *Eq. 6*:
(7)

where is the fraction of total Na_{e} and K_{e} that consists of osmotically active Na_{e} and K_{e} and partial fraction of Na_{e} and K_{e} that is osmotically inactive.

We hypothesize that the slope and *y*-intercept in *Eq. 6* are represented by the following mathematical expressions in *Eq. 7*, respectively:

Therefore, the osmotically inactive Na^{+} and K^{+} storage pool is accounted for partly by the slope and partly by the *y*-intercept in *Eq. 6*. This is in contrast to the Edelman equation and *Eq. 4* in which the osmotically inactive Na^{+} and K^{+} storage pool is accounted solely by the *y*-intercept and slope, respectively.

We will now demonstrate the validity of the above hypothesis regarding the slope and *y*-intercept of *Eq. 6*.

Using the data from the study by Edelman et al. (5), the total osmotically active Na^{+} and K^{+} pool can be calculated as follows:

where F_{osm active} is the fraction of Na_{e} and K_{e} that is osmotically active.

Alternatively, the total osmotically active Na^{+} and K^{+} pool can be calculated as follows:

where is the fraction of total Na_{e} and K_{e} that consists of osmotically active Na_{e} and K_{e} and partial fraction of Na_{e} and K_{e} that is osmotically inactive

Therefore: (8)

Rearranging *Eq. 8*,
(9)

Based on the Edelman equation:

Based on *Eq. 4*:

Therefore, using the data from the study of Edelman et al. (5), the total osmotically active Na^{+} and K^{+} pool is calculated for each data point as follows:

Assuming that the slope in *Eq. 6* is represented by the following mathematical expression in *Eq. 7*:

Therefore, with the use of data from the study of Edelman et al. (5), the fraction of total Na_{e} and K_{e} that consists of osmotically active Na_{e} and K_{e} and partial fraction of Na_{e} and K_{e} that is osmotically inactive can be calculated for each data point as follows:

Therefore, based on *Eq. 9*:
(9)

With the use of data from the study of Edelman et al. (5), the partial amount of Na_{e} and K_{e} that is osmotically inactive, (Na_{osm inactive} + K_{osm inactive})*/TBW, is calculated for each data point based on *Eq. 9*. This value can then be used to calculate the theoretical *y*-intercept in *Eq. 7*.

Based on our hypothesis, the theoretical *y*-intercept of *Eq. 7* is:

With the use of data from the study of Edelman et al. (5), the theoretical *y*-intercept without the term for the osmotically inactive Na^{+} and K^{+} storage pool can be calculated for each data point as follows:

The term [Na^{+}]_{pw} − 0.487(Na_{e} + K_{e})/TBW represents the magnitude of the theoretical *y*-intercept without the term for the osmotically inactive Na^{+} and K^{+} storage pool, since the slope of 0.487 already accounts for the total osmotically inactive Na^{+} and K^{+} storage pool in its slope. This value, [Na^{+}]_{pw} − 0.487 (Na_{e} + K_{e})/TBW, is then added to the quantity −1.11 (Na_{osm inactive} + K_{osm inactive})*/TBW as calculated above to determine the theoretical *y*-intercept of *Eq. 7* for each data point. The average theoretical *y*-intercept ± SE (*y*^{7}) of *Eq. 7* of all of the data points in the study of Edelman et al. is 1.37 ± 0.60, which is similar to the actual *y*-intercept of 1.37 of *Eq. 6* using linear regression analysis (Table 1). Therefore, this finding strongly supports our hypothesis that the osmotically inactive Na^{+} and K^{+} storage pool is quantitatively reflected in both the slope and *y*-intercept in *Eq. 6*.

Last, our analysis indicates that a significant fraction of the total exchangeable Na^{+} and K^{+} pool is osmotically inactive. This can be demonstrated as follows.

The slope in the Boling equation is represented by the following mathematical expression in *Eq. 5*:

The slope in the Edelman equation is represented by the following mathematical expression in *Eq. 2*:

Therefore:

Hence, the fraction of osmotically active Na^{+} and K^{+} pool is 0.44 of the total exchangeable Na^{+} and K^{+} pool. Conversely, the fraction of osmotically inactive Na^{+} and K^{+} storage pool is 0.56 of the total exchangeable Na^{+} and K^{+} pool.

#### Dynamic regulation of osmotically inactive sodium storage.

In clinical conditions characterized by Na^{+} retention, Na^{+} can be accumulated without commensurate water retention, thereby resulting in less volume expansion. The experimental studies of Titze et al. suggested that the osmotically inactive Na^{+} storage pool is dynamically regulated during Na^{+} retention (18, 19). Titze et al. demonstrated that skin Na^{+} retention resulted in an increased skin (Na^{+} + K^{+})/H_{2}O ratio compared with serum [Na^{+}] in saline-treated rats compared with water-treated rats in both control and DOCA rats, thereby suggesting osmotically inactive Na^{+} storage in the skin tissue (18). These investigators demonstrated that Na^{+} can be accumulated without commensurate water retention in the interstitium of the skin and that this skin Na^{+} storage is paralleled by increased polymerization and sulfation of glycosaminoglycans (18, 19).

However, there is current controversy as to whether the osmotically inactive Na^{+} storage pool is fixed or variable in clinical conditions characterized by changes in Na^{+} and H_{2}O balance (9, 16, 17). In contrast to the experimental studies of Titze et al. demonstrating that the osmotically inactive Na^{+} storage pool is dynamically regulated during Na^{+} retention at the skin tissue level, Seeliger et al. performed Na^{+}, K^{+}, and H_{2}O balance studies of 4 days duration in dogs and demonstrated that changes in exchangeable Na^{+} were often accompanied by changes in exchangeable K^{+} and that Na^{+} storage was osmotically active during Na^{+} retention at the total body level (17). Indeed, these investigators demonstrated that the changes in total body Na^{+} and K^{+} were proportional to the changes to total body water (TBW). Therefore, by considering the mass balance of Na^{+}, K^{+}, and H_{2}O, these researchers argued that Na^{+} accumulation occurs in an osmotically active form during Na^{+} retention. Similarly, Overgaard-Steensen et al. recently demonstrated that the plasma [Na^{+}] can be predicted based on the mass balance of Na^{+}, K^{+}, and H_{2}O in acute hyponatremia in a porcine model (16). Because changes in the plasma [Na^{+}] can be predicted based on changes in the (Na_{e} + K_{e})/TBW ratio, the (Na_{osm inactive} + K_{osm inactive})/TBW ratio must have remained relatively constant. Consequently, these investigators concluded that there is no substantial osmotic activation or inactivation resulting from changes in the mass balance of Na^{+}, K^{+}, and H_{2}O at the total body level.

Together, the previous experimental findings demonstrate that the plasma [Na^{+}] can be predicted based on the mass balance of Na^{+}, K^{+}, and H_{2}O at the total body level despite the paradoxical finding that there are changes in the osmotically inactive Na^{+} storage pool at the skin tissue level. These seemingly paradoxical findings can be reconciled by closely examining the relation between [Na^{+}]_{pw}, (Na_{e} + K_{e})/TBW, and (Na_{osm inactive} + K_{osm inactive})/TBW in *Eq. 2*:
(2)

The plasma [Na^{+}] is a function of the total amount of osmotically active Na^{+} and K^{+} distributed within the TBW. According to *Eq. 2*, in order for the plasma [Na^{+}] to be predicted based on the mass balance of Na^{+}, K^{+}, and H_{2}O, the ratio (Na_{osm inactive} + K_{osm inactive})/TBW must remain relatively constant. Indeed, Overgaard-Steensen et al. demonstrated that the plasma [Na^{+}] can be predicted based on the mass balance of Na^{+}, K^{+}, and H_{2}O by assuming a constant intercept in the Edelman equation in the porcine model (16). By assuming a constant intercept in predicting changes in the plasma [Na^{+}] based on the mass balance of Na^{+}, K^{+}, and H_{2}O, this implies that the quantity of osmotically inactive Na^{+} and K^{+} storage (numerator) changes proportionally to changes in TBW (denominator) in order for the ratio (Na_{osm inactive} + K_{osm inactive})/TBW in the *y*-intercept to remain relatively constant. Furthermore, in the study of Titze et al., these investigators demonstrated that Na^{+} retention resulted in an increased skin (Na^{+} + K^{+})/H_{2}O ratio in saline-treated rats compared with water-treated rats in both control and DOCA rats without concomitant changes in the plasma [Na^{+}] and total body (Na^{+} + K^{+})/H_{2}O ratio (18). Given that the plasma [Na^{+}] and total body (Na^{+} + K^{+})/H_{2}O ratio remained constant with Na^{+} retention, the ratio (Na_{osm inactive} + K_{osm inactive})/TBW must have remained relatively constant according to *Eq. 2*. Because changes in Na^{+} and K^{+} balance (osmotically active and inactive Na^{+} and K^{+}) are often accompanied by corresponding changes in TBW (because of changes in osmotically active Na^{+} and K^{+}) and that the ratio (Na_{osm inactive} + K_{osm inactive})/TBW must remain relatively constant, the quantity of osmotically inactive Na^{+} and K^{+} storage must vary proportionally to changes in TBW. In other words, because changes in the quantity of osmotically inactive Na^{+} and K^{+} are proportional to the associated changes in TBW induced by changes in the quantity of osmotically active Na^{+} and K^{+}, the ratio (Na_{osm inactive} + K_{osm inactive})/TBW will remain relatively constant.

It is also well known that a portion of Na_{e} is bound in bone and is therefore rendered osmotically inactive (3–5). Titze et al. demonstrated that there was no significant difference in the bone (Na^{+} + K^{+})/TBW ratio in salt- and DOCA-treated rats (18). As a result, the ratio (Na_{osm inactive} + K_{osm inactive})/TBW in *Eq. 2* must remain relatively constant with osmotically inactive bone Na^{+} and K^{+} storage associated with salt and DOCA treatment as well.

In summary, our mathematical analysis demonstrates that the osmotically inactive Na^{+} storage pool is dynamically regulated at both the skin tissue level and total body level. Thus, in clinical conditions characterized by Na^{+} retention, current evidence suggests that Na^{+} can be accumulated without commensurate water retention in the interstitium of the skin, thereby resulting in less volume expansion.

#### Clinical implications.

The Edelman equation has clinical importance, since changes in the [Na^{+}]_{pw} are due to alterations in the Na_{e}, K_{e}, and TBW. Because changes in the [Na^{+}]_{pw} are clinically due to changes in the mass balance of Na^{+}, K^{+}, and H_{2}O, the analysis and treatment of the dysnatremias are dependent on the validity of the Edelman equation in defining the quantitative interrelationship between the [Na^{+}]_{pw} and Na_{e}, K_{e}, and TBW. Indeed, formulas used in the analysis and treatment of the dysnatremias have been derived based on the Edelman equation (6, 10–12, 14). Therefore, due to the marked disparities between the slope and *y*-intercept of the Edelman equation, *Eqs. 4* and *6*, the validity of the Edelman equation may be questioned. In this regard, our mathematical analysis is important in validating the quantitative interrelationship between the [Na^{+}]_{pw} and Na_{e}, K_{e}, and TBW in the Edelman equation.

Moreover, a fundamental question is whether changes in the [Na^{+}]_{pw} can be predicted based on changes in the mass balance of Na^{+}, K^{+}, and H_{2}O if there are dynamic changes in the osmotically inactive Na^{+} and K^{+} storage pool. If there are dynamic changes in the osmotically inactive Na^{+} and K^{+} storage pool, one would expect that changes in the [Na^{+}]_{pw} cannot be predicted based on alterations in Na_{e}, K_{e}, and TBW due to concomitant changes in the *y*-intercept resulting from changes in the (Na_{osm inactive} + K_{osm inactive})/TBW term. However, Overgaard-Steensen et al. demonstrated that the plasma [Na^{+}] can be predicted based on the mass balance of Na^{+}, K^{+}, and H_{2}O by assuming a constant intercept in the Edelman equation in the porcine model (16). By assuming a constant intercept in predicting changes in the plasma [Na^{+}], the ratio (Na_{osm inactive} + K_{osm inactive})/TBW in the *y*-intercept must remain relatively constant. Given that the ratio (Na_{osm inactive} + K_{osm inactive})/TBW in the *y*-intercept remains relatively constant, the quantity of osmotically inactive Na^{+} and K^{+} storage in the numerator of this ratio must have changed in proportion to changes in TBW in the denominator of this ratio. This finding has important clinical implications. In the study by Overgaard-Steensen et al. (16), acute hyponatremia was induced with desmopressin acetate and infusion of 2.5% glucose solution in anesthesized pigs. Because acute decreases in the plasma [Na^{+}] were induced by water retention, this implies that acute changes in the quantity of osmotically inactive Na^{+} and K^{+} occur with acute changes in the plasma [Na^{+}] and with water retention in the syndrome of inappropriate antidiuretic hormone secretion (SIADH). It is not surprising that the osmotically inactive Na^{+} and K^{+} storage pool is dynamically regulated in response to changes in TBW. In SIADH, it is well known that the increase in TBW expands the extracellular fluid volume, which will in turn trigger increased urinary Na^{+} excretion to return the extracellular fluid volume toward normal. Likewise, the increment in the quantity of osmotically inactive Na^{+} and K^{+} with water retention is likely a compensatory mechanism in response to the expansion in extracellular fluid volume accompanying the increase in TBW.

Similarly, the quantity of osmotically inactive Na^{+} and K^{+} storage varies proportionally to changes in TBW accompanying Na^{+} retention. In the study of Titze et al., Na^{+} retention resulted in an increased skin (Na^{+} + K^{+})/H_{2}O ratio in saline-treated rats compared with water-treated rats in both control and DOCA rats without concomitant changes in the plasma [Na^{+}] and total body (Na^{+} + K^{+})/H_{2}O ratio (18). Given that the plasma [Na^{+}] and total body (Na^{+} + K^{+})/H_{2}O ratio remained constant with Na^{+} retention, the ratio (Na_{osm inactive} + K_{osm inactive})/TBW must have remained relatively constant according to *Eq. 2*. Because the ratio (Na_{osm inactive} + K_{osm inactive})/TBW remains relatively constant, the quantity of osmotically inactive Na^{+} and K^{+} in the numerator of this ratio must increase proportionally to the increase in TBW (denominator) induced by changes in the quantity of osmotically active Na^{+} and K^{+}.

This can be further illustrated by the following hypothetical example. Let's assume that there are 340 mmol of total Na^{+} and K^{+} (300 mmol osmotically active Na^{+} + K^{+} and 40 mmol osmotically inactive Na^{+} + K^{+}). If the osmolality of this hypothetical solution is 300 mosmol/kgH_{2}O, the 300 mmol osmotically active Na^{+} + K^{+} and their associated anions (total osmol 600 mosmol) will retain 2 liters of H_{2}O:

Now, let's assume that the total Na^{+} and K^{+} increase by 340 mmol (increment of 300 mmol osmotically active Na^{+} + K^{+} and increment of 40 mmol osmotically inactive Na^{+} + K^{+}). Assuming that the osmolality of the solution remains the same at 300 mosmol/kgH_{2}O, 600 mmol osmotically active Na^{+} + K^{+} and their associated anions (total osmol 1,200 mosmol) will retain 4 liters of H_{2}O:

Therefore, in this hypothetical example, the quantity of osmotically inactive Na^{+} and K^{+} increases from 40 to 80 mmol since the quantity of osmotically inactive Na^{+} and K^{+} must increase in proportion to the increase in the H_{2}O content induced by changes in the osmotically active Na^{+} and K^{+} in order for the (Na_{osm inactive} + K_{osm inactive})/H_{2}O ratio to remain constant.

In summary, our mathematical analysis therefore demonstrates that changes in the [Na^{+}]_{pw} can be predicted based on changes in Na_{e}, K_{e}, and TBW despite dynamic changes in the osmotically inactive Na^{+} and K^{+} storage pool. This is so because the ratio (Na_{osm inactive} + K_{osm inactive})/TBW in the *y*-intercept of *Eq. 2* remains relatively constant despite changes in the quantity of osmotically inactive Na^{+} and K^{+} storage. Consequently, inaccuracies in the dysnatremia formulas used in the analysis and treatment of the dysnatremias are likely due to errors in the clinical estimation of TBW and inaccuracies in accounting for all sources of input of Na^{+}, K^{+}, and H_{2}O (oral intake, total parenteral nutrition, tube feeding, intravenous fluids, and water of oxidation) and ongoing body fluid losses of Na^{+}, K^{+}, and H_{2}O (cutaneous, gastrointestinal, and urinary losses). In addition, a limitation of all dysnatremia formulas is that ongoing body fluid losses of Na^{+}, K^{+}, and H_{2}O are clinically accounted for based on prior body fluid losses of Na^{+}, K^{+}, and H_{2}O, which may not equate to actual ongoing body fluid losses of Na^{+}, K^{+}, and H_{2}O.

In conclusion, by accounting for measurement errors in the *x* and *y* variables in obtaining the relation between *y* = [Na^{+}]_{pw} and *x* = (Na_{e} + K_{e})/TBW, the quantitative interrelationship between the [Na^{+}]_{pw} and Na_{e}, K_{e}, and TBW as defined by the Edelman equation has clinical significance, since changes in the [Na^{+}]_{pw} are clinically due to changes in the mass balance of Na^{+}, K^{+}, and H_{2}O. However, linear regression analysis of the Edelman data using [Na^{+}]_{pw} as the dependent variable yields the following equation: [Na^{+}]_{pw} = 0.93(Na_{e} + K_{e})/TBW + 1.37 (*Eq. 6*). Moreover, based on the study by Boling et al., the [Na^{+}]_{pw} is related to the Na_{e}, K_{e}, and TBW by the following linear regression equation: [Na^{+}]_{pw} = 0.487(Na_{e} + K_{e})/TBW + 71.54 (*Eq. 4*). In this mathematical analysis, the disparities between the slope and *y*-intercept of these three equations can be explained by the hypothesis that the osmotically inactive Na^{+} and K^{+} storage pool is quantitatively reflected in the magnitude of the *y*-intercept in the Edelman equation (*Eq. 1*), magnitude of slope in *Eq. 4*, and magnitude of both slope and *y*-intercept in *Eq. 6*. This mathematical analysis also indicates that the osmotically inactive Na^{+} and K^{+} storage pool is dynamically regulated at both the skin tissue level and total body level and that the fraction of osmotically inactive Na^{+} and K^{+} storage pool is 0.56 of the total exchangeable Na^{+} and K^{+} pool. Last, our analysis demonstrates that changes in the [Na^{+}]_{pw} can be predicted based on changes in the Na_{e}, K_{e}, and TBW despite dynamic changes in the osmotically inactive Na^{+} and K^{+} storage pool.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

## AUTHOR CONTRIBUTIONS

M.K.N. conception and design of research; M.K.N., D-S.N., and M-K.N. analyzed data; M.K.N. interpreted results of experiments; M.K.N. drafted manuscript; M.K.N. edited and revised manuscript; M.K.N. approved final version of manuscript; D-S.N. and M-K.N. prepared figures.

- Copyright © 2016 the American Physiological Society