## Abstract

In evaluating the renal mechanisms responsible for the generation of the dysnatremias, an analysis of free water clearance (FWC) and electrolyte-free water clearance (EFWC) is often utilized to characterize the rate of urinary free water excretion in these disorders. Previous analyses of FWC and EFWC have failed to consider the relationship among plasma water Na^{+} concentration ([Na^{+}]_{pw}), total exchangeable Na^{+} (Na_{e}), total exchangeable K^{+} (K_{e}), and total body water (TBW); (Edelman IS, Leibman J, O'Meara MP, and Birkenfeld LW. *J Clin Invest* 37: 1236–1256, 1958). In their derivations, the classic FWC and EFWC formulas fail to consider the quantitative and physiological significance of the slope and *y*-intercept in this equation. Consequently, previous EFWC formulas incorrectly assume that urine is isonatric when [Na^{+} + K^{+}]_{urine} = [Na^{+}]_{p} or [Na^{+} + K^{+}]_{urine} = [Na^{+}]_{p} + [K^{+}]_{p} (where [Na^{+}]_{p} and [K^{+}]_{p} represent plasma Na^{+} and K^{+} concentrations, respectively). Moreover, previous formulas cannot be utilized in the setting of hyperglycemia. In this article, we have derived a new formula termed modified electrolyte-free water clearance (MEFWC) for determining the electrolyte-free water clearance, taking into consideration the empirical relationship between the [Na^{+}]_{pw} and Na_{e}, K_{e}, and TBW: MEFWC = V [1 − 1.03[Na^{+} + K^{+}]_{urine}/([Na^{+}]_{p} + 23.8)]. MEFWC, unlike previous formulas, is derived based on the requirement of the Edelman equation that urine is isonatric only when [Na^{+} + K^{+}]_{urine} = (Na_{e} + K_{e})/TBW = 0.97[Na^{+}]_{p} + 23.1. Furthermore, since we have shown that the *y*-intercept in the Edelman equation varies directly with the plasma glucose concentration, in patients with hyperglycemia, MEFWC = V [1 − 1.03[Na^{+} + K^{+}]_{urine}/{[Na^{+}]_{p} + 23.8 + (1.6/100)([glucose]_{p} − 120)}]. The MEFWC formula will be especially useful in assessing the renal contribution to the generation of the dysnatremias.

- sodium
- potassium
- free water clearance
- electrolyte-free water clearance

## CLASSIC FORMULAS UTILIZED TO CALCULATE THE URINARY FREE WATER CLEARANCE

in previous analyses of the mechanisms responsible for changes in the plasma Na^{+} concentration ([Na^{+}]_{p}), the concepts of free water clearance (FWC) and electrolyte-free water clearance (EFWC) were utilized to characterize and predict the effect of an abnormal rate of urinary free water excretion on the [Na^{+}]_{p} (4, 8, 17, 19, 21). FWC was originally defined quantitatively as V (1 − U_{osm}/P_{osm}), where V = urinary flow rate, U_{osm} = urinary osmolality, and P_{osm} = plasma osmolality (21). FWC is an analysis based on a comparison of urine to plasma osmolality to determine whether the kidney is excreting dilute urine and to quantify the rate of urinary free water excretion. In 1981, Goldberg (4) emphasized that although urea is a component of the measured plasma and urine osmolality, since it has a high permeability across cell membranes, urea does not alter the [Na^{+}]_{p} by modulating the distribution of water between body fluid compartments. Accordingly, Goldberg suggested that a new formula be used termed EFWC, where V(1 − [Na^{+} + K^{+}]_{urine} /[Na^{+}]_{p}) (4). To account for the effect of K^{+} on the [Na^{+}]_{p}, Shoker (19) and subsequently Mallie et al. (8) suggested that EFWC be calculated as V{1 − [Na^{+} + K^{+}]_{urine}/([Na^{+}]_{p} + [K^{+}]_{p})}. Furthermore, since glucose can alter the [Na^{+}]_{p} by inducing the shift of water between body fluid compartments (5), Shoker (19) revised the calculation of EFWC to include effective osmoles other than Na^{+} and K^{+} as follows: V{1 − (2 [Na^{+} + K^{+}]_{urine} + [other effective osmoles])/(2 ([Na^{+}]_{p} + [K^{+}]_{p}) + [other effective osmoles])} (19). These formulas are summarized in Table 1.

## EMPIRICAL AND THEORETICAL REASONS FOR ACCEPTING THE EDELMAN EQUATION AS THE BASIS FOR MODIFYING THE CLASSIC EFWC FORMULAS

It has been suggested that the EFWC analysis is superior to the calculation of FWC to document the role of the kidney in generating the dysnatremias, since the EFWC takes into consideration the fact that urea is an ineffective osmole (4, 8, 17, 19). However, neither the FWC nor EFWC formula considers the empirical relationship between the plasma water Na^{+} concentration ([Na^{+}]_{pw}) and Na_{e}, K_{e}, and total body water (TBW) originally demonstrated by Edelman et al. (3): [Na^{+}]_{pw} = 1.11(Na_{e} + K_{e})/TBW − 25.6 (*EQ. 1*), where Na_{e} and K_{e} are total exchangeable Na^{+} and K^{+}, respectively. Specifically, these previous analyses fail to consider the quantitative and physiological significance of the slope and *y*-intercept in the Edelman equation in their derivations.

Recently, we have shown quantitatively the necessity for the slope and *y*-intercept in the Edelman equation and their physiological and clinical significance (6, 9–14). Our analysis demonstrated that the empirically determined slope of 1.11 can be theoretically predicted by considering the combined effect of the osmotic coefficient of Na^{+} salts at physiological concentrations and Gibbs-Donnan equilibrium (12, 13). Our analysis indicated that ionic interactions between Na^{+} and its associated anions (as reflected by the osmotic coefficient of Na^{+} salts) have a modulating effect on the [Na^{+}]_{pw}. Moreover, our results demonstrated that Gibbs-Donnan equilibrium has an incremental effect on the [Na^{+}]_{pw}. Since the presence of negatively charged, impermeant proteins in the plasma space alters the distribution of Na^{+} and its associated anions between the plasma and interstitial fluid to preserve electroneutrality, the Gibbs-Donnan effect raises the [Na^{+}]_{pw} at any given quantity of (Na_{e} + K_{e})/TBW. Furthermore, we also demonstrated that there are several determinants of the *y*-intercept in the Edelman equation which independently alter the [Na^{+}]_{pw}: the osmotically inactive exchangeable Na^{+} and K^{+}, the plasma water [K^{+}], and the osmotically active non-Na^{+} and non-K^{+} osmoles (9, 11–13). The components of the *y*-intercept reflect the fact that not all exchangeable Na^{+} and K^{+} are osmotically active and that non-Na^{+} osmoles are also involved in the distribution of water between the body fluid compartments. Therefore, the components of the *y*-intercept reflect the role of osmotic equilibrium in the modulation of the [Na^{+}]_{pw}.

The role of osmotic equilibrium in the modulation of the [Na^{+}]_{p} is best exemplified by the effect of hyperglycemia on the [Na^{+}]_{p}. It is well known that there is an expected decrease of 1.6 meq/l in the [Na^{+}]_{p} for each 100 mg/dl increment in the plasma glucose concentration ([glucose]_{p}) resulting from the osmotic shift of water between the intracellular fluid compartment and the extracellular fluid compartment (5). Indeed, we have demonstrated that the *y*-intercept is not constant in hyperglycemia-induced hyponatremia and will vary directly with the [glucose]_{p} (6, 11–13). Our analysis also indicated that the following formula can be used to predict the effect of changes in Na_{e}, K_{e}, and TBW as well as the dilutional effect of hyperglycemia on the [Na^{+}]_{p} attributable to the osmotic shift of water where [Na^{+}]_{p} = 1.03(Na_{e} + K_{e})/TBW − 23.8 − (1.6/100)([glucose]_{p} − 120) (11–13). Thus any analysis of the pathophysiology of the dysnatremias in the setting of hyperglycemia must take into consideration the effect of [glucose]_{p} on the magnitude of the *y*-intercept.

## DEFINITION OF AN ISONATRIC SOLUTION DICTATED BY THE EDELMAN EQUATION

A solution is defined as isonatric when its addition or loss from the plasma will not result in an alteration in the [Na^{+}]_{p}. Quantitatively, it is commonly assumed that a solution is isonatric when its [Na^{+} + K^{+}] = [Na^{+}]_{p} (4, 17). Moreover, according to this definition, it was implicitly assumed that the [Na^{+}]_{p} = (Na_{e} + K_{e})/TBW. The latter equation is a simplified version of *EQ. 1*, where the slope and *y*-intercept of the Edelman equation were erroneously assigned values of one and zero, respectively. However, given the empirical and theoretical basis for the non-zero values of the slope and *y*-intercept in the Edelman equation (3, 11–13), the definition of an isonatric solution requires modification. Importantly, it can be demonstrated that the Edelman equation dictates that a solution is isonatric if its [Na^{+} + K^{+}] = (Na_{e} + K_{e})/TBW. The addition or loss from the plasma of a solution with this property does not result in an alteration in the [Na^{+}]_{p}. Mathematically, the fact that a solution is isonatric when its [Na^{+} + K^{+}] = (Na_{e} + K_{e})/TBW can be demonstrated as follows: (1)

Multiplying both sides of *EQ. 1* by 0.93 (1, 2) to convert [Na^{+}]_{pw} to [Na^{+}]_{p}

Since 0.93 × [Na^{+}]_{pw} = [Na^{+}]_{p} (1, 2) (2)

If [Na^{+}]_{p1} = [Na^{+}]_{p2}, then (3)

Since (4)

Assuming that there is no input and only urinary loss (5)

Substituting *EQ. 5* for (Na_{e2} + K_{e2} )/TBW_{2} in *EQ. 3*

Rearranging

Rearranging

Therefore

Solving for [Na^{+} + K^{+}]_{urine}

Thus a solution is isonatric when its [Na^{+} + K^{+}] = (Na_{e} + K_{e})/TBW. Since (Na_{e} + K_{e})/TBW = ([Na^{+}]_{p} + 23.8)/1.03 according to *EQ. 6*, urine is isonatric when its [Na^{+} + K^{+}] = ([Na^{+}]_{p} + 23.8)/1.03 = 0.97[Na^{+}]_{p} + 23.1 (Fig. 1). In contrast, previous EFWC formulas incorrectly assume that urine is isonatric when [Na^{+} + K^{+}]_{urine} = [Na^{+}]_{p} (4, 17) or [Na^{+} + K^{+}]_{urine} = [Na^{+}]_{p} + [K^{+}]_{p} (8, 19).

## DERIVATION OF A NEW FORMULA FOR CALCULATING EFWC: THE MODIFIED EFWC EQUATION

By failing to incorporate the complete Edelman equation in their derivations, previous formulas suffer from the limitation that the Edelman equation dictates that urine be considered isonatric (incapable of changing the [Na^{+}]_{p}) only when [Na^{+} + K^{+}]_{urine} = (Na_{e} + K_{e})/TBW. In contrast, current formulas implicitly assume that urine is isonatric when [Na^{+} + K^{+}]_{urine} = [Na^{+}]_{p} or [Na^{+} + K^{+}]_{urine} = [Na^{+}]_{p} + [K^{+}]_{p}. We now demonstrate the mathematical derivation of the modified EFWC (MEFWC) equation that is consistent with the implications of the Edelman equation.

(2)

Rearranging (6)

Urine can be viewed conceptually as having two components: one component containing a concentration of Na^{+} + K^{+} that is isonatric, and a second component that does not contain Na^{+} and K^{+} salts and is termed electrolyte-free water. The isonatric urine component by definition will not change the [Na^{+}]_{p} if excreted or absorbed, whereas the electrolyte-free water component will change the [Na^{+}]_{p} if excreted or absorbed. According to *EQ. 2*, the isonatric component must have a [Na^{+} + K^{+}] = (Na_{e} + K_{e})/TBW. Specifically, when [Na^{+} + K^{+}]_{urine} = (Na_{e} + K_{e})/TBW, the excretion of urine will not change [Na^{+}]_{p} from its current value.

These two urine components can be represented algebraically as (7)

where V = urine flow rate, *E* = [Na^{+} + K^{+}], IEC = isonatric electrolyte clearance, and MEFWC = modified electrolyte-free water clearance.

Furthermore (8)

Since according to *EQ. 6* (Na_{e} + K_{e})/TBW = ([Na^{+}]_{p} + 23.8)/1.03; *EQ. 8* can be rewritten as (9)

Rearranging *EQ. 7* (10)

Since according to *EQ. 9*

*EQ. 10* can be rewritten as (11)

Rearranging (12)

Taking into consideration the quantitative and physiological significance of the slope and *y*-intercept in *EQ. 1*, we have therefore derived a new formula for determining EFWC:

This new formula incorporates the known empirical relationship between the [Na^{+}]_{pw} and Na_{e}, K_{e}, and TBW in its derivation. By accounting for the non-zero values of the slope and *y*-intercept in the Edelman equation, this formula takes into consideration the effects of the osmotic coefficient of Na^{+} salts at physiological concentrations and Gibbs-Donnan and osmotic equilibrium on the [Na^{+}]_{pw}. Unlike previous formulas, MEFWC incorporates in its derivation the fact that plasma is 93% water (1, 2). In addition, MEFWC is mathematically derived based on the Edelman equation and therefore predicts correctly that urine is isonatric only when [Na^{+} + K^{+}]_{urine} = (Na_{e} + K_{e})/TBW. Moreover, MEFWC accounts for the ineffectiveness of urea in altering the distribution of water between the cells and the extracellular fluid by incorporating the electrolyte clearance (rather than osmolar clearance) in its derivation. Finally, in a euglycemic patient, MEFWC has three determinants which can vary: V, [Na^{+} + K^{+}]_{urine}, and [Na^{+}]_{p}. MEFWC increases linearly as V increases, and curvilinearly as [Na^{+}]_{p} increases. In contrast, MEFWC varies inversely with the [Na^{+} + K^{+}]_{urine} (Fig. 2).

## CLINICAL UTILITY OF THE MEFWC FORMULA

Based on the empirical relationship between the [Na^{+}]_{pw} and Na_{e}, K_{e}, and TBW empirically demonstrated by Edelman et al. (3) (*EQ. 1*), we now proceed to quantitatively compare the clinical validity of the MEFWC formula and previous free-water clearance formulas. According to *EQ. 1*, as long as the [Na^{+} + K^{+}]_{urine} = (Na_{e} + K_{e})/TBW = 0.97[Na^{+}]_{p} + 23.1, the [Na^{+}]_{p} remains unaltered. Since there is no change in the [Na^{+}]_{p}, the urinary EFWC must be equal to zero. Therefore, one can easily assess the clinical validity of the various free-water clearance formulas by calculating the urinary free-water clearance in a hypothetical patient with a [Na^{+} + K^{+}]_{urine} = (Na_{e} + K_{e})/TBW = 0.97[Na^{+}]_{p} + 23.1.

Using the various free-water clearance formulas (Table 1), we will now calculate the urinary free-water clearance in our patient: [Na^{+} + K^{+}]_{urine}= 130 mmol/l, urine flow rate = 1.5 l/day, urine osmolality = 540 mosmol/kgH_{2}O, [Na^{+}]_{p} = 110 mmol/l, [K^{+}]_{p} = 4.0 mmol/l, and plasma osmolality = 256 mosmol/kgH_{2}O:

As required by *EQ. 1*, when [Na^{+} + K^{+}]_{urine} = 0.97[Na^{+}]_{p} + 23.1 as in this patient example, the [Na^{+}]_{p} remains constant. Only the MEFWC formula that mathematically incorporates this equality in its derivation predicts the expected result that the urinary free water clearance is zero. In contrast, free water clearance as calculated by FWC (21), EFWC_{1} (4, 17), and EFWC_{2} formulas (8, 19) predict incorrectly a non-zero value for urinary free water clearance.

Similarly, according to *EQ. 6*, in a patient with a [Na^{+}]_{p} of 140 mmol/l and [K^{+}]_{p} of 4 mmol/l, if urinary [Na^{+} + K^{+}] is 159 mmol/l (i.e. [Na^{+} + K^{+}]_{urine} = 0.97[Na^{+}]_{p} + 23.1), the [Na^{+}]_{p} will remain constant and the urinary free water clearance must be zero. As shown in Table 2, the urinary free water clearance as calculated by the MEFWC formula is zero, whereas a non-zero value is incorrectly derived using the previous free-water clearance formulas. In contrast, if urinary [Na^{+} + K^{+}] is 140 mmol/l (i.e. [Na^{+} + K^{+}]_{urine} = [Na^{+}]_{p}), or if urinary [Na^{+} + K^{+}] is 144 mmol/l (i.e. [Na^{+} + K^{+}]_{urine} = [Na^{+}]_{p} + [K^{+}]_{p}), the loss of such a solution from the plasma must result in a change in the [Na^{+}]_{p}. Since there is an alteration in the [Na^{+}]_{p}, the urinary free water clearance cannot be zero as predicted by the MEFWC formula, whereas a zero value is inaccurately predicted by the previous free-water clearance formulas (Table 2). Therefore, if the [Na^{+}]_{p} is 140 mmol/l, a solution that is isonatric to the [Na^{+}]_{p} must have a [Na^{+} + K^{+}] equal to 159 mmol/l. In contrast, if a solution's [Na^{+} + K^{+}] is equal to the [Na^{+}]_{p} (140 mmol/l) or [Na^{+}]_{p} + [K^{+}]_{p} (144 mmol/l), its addition or loss from the plasma will lead to a change in the [Na^{+}]_{p}. Such a solution would be hyposmotic as there are other effective non-Na^{+} and non-K^{+} osmoles in the plasma (i.e., glucose, Ca^{+2}, Mg^{+2}). Thus an alteration in the [Na^{+}]_{p} will ensue due to the osmotic shift of water between body fluid compartments. Interestingly, it has been demonstrated that a NaCl solution with a [Na^{+}] of 160 mmol/l has an equivalent osmotic pressure to that of normal plasma with an osmolality of 298 mosmol/kgH_{2}O (7).

Finally, the inaccuracies of the EFWC_{1} and EFWC_{2} formulas are more exaggerated in clinical conditions characterized by a high urinary flow rate. As demonstrated in Fig. 3, in a patient with a [Na^{+}]_{p} = 110 mmol/l, [K^{+}]_{p} = 3 mmol/l, and [Na^{+} + K^{+}]_{urine} = 100 mmol/l, calculations of EFWC based on the EFWC_{1} and EFWC_{2} formulas result in greater errors at higher urinary flow rates compared with that calculated according to the MEFWC formula.

## FACTORS MODULATING THE SLOPE AND *Y*-INTERCEPT IN THE EDELMAN EQUATION

As the slope and *y*-intercept in the Edelman equation have several physiological determinants, alterations in these parameters could result in changes in the slope and *y*-intercept in *EQ. 1*. Since the slope of *EQ. 1* is determined by the combined effect of the osmotic coefficient of Na^{+} salts at physiological concentrations and Gibbs-Donnan equilibrium (12, 13), clinical conditions characterized by hemoconcentration or hemodilution would be expected to change the value of the slope in *EQ. 1* by altering Gibbs-Donnan equilibrium. Similarly, alterations in the magnitude of the parameters comprising the *y*-intercept could lead to a change in its value. For instance, the quantities of Na^{+} lost and water retained in the syndrome of antidiuretic hormone secretion (SIADH) are insufficient to account for the magnitude of the observed reduction in [Na^{+}]_{p} in severely hyponatremic patients (15, 18). This discrepancy has been attributed to loss or inactivation of an osmotically active solute. A change in the quantity of osmotically inactive Na_{e} and K_{e} would, therefore, lead to a change in the magnitude of the *y*-intercept. Moreover, changes in the quantity of osmotically active non-Na^{+} and non-K^{+} osmoles would also alter the magnitude of the *y*-intercept. Indeed, we have previously demonstrated that the *y*-intercept is not constant in hyperglycemia-induced dilutional hyponatremia resulting from the translocation of water and will vary directly with the [glucose]_{p} (6, 11–13).

Although the exact slope and *y*-intercept may not be known in any given individual, Edelman et al. (3) demonstrated that the slope of 1.11 and *y*-intercept of −25.6 in the empirically derived regression equation (*EQ. 1*) provide an excellent characterization of the relationship between the [Na^{+}]_{pw} and Na_{e}, K_{e}, and TBW in euglycemic clinical conditions. In the derivation of the MEFWC formula, the slope of 1.03 and *y*-intercept of −23.8 are utilized instead to account for the fact that plasma is 93% water. It is therefore important to realize that the slope of 1.03 and *y*-intercept of −23.8 are not applicable in clinical conditions characterized by an increase in the lipid and protein fraction of plasma as in hyperlipidemia and multiple myeloma (20). Finally, a modified *y*-intercept must be utilized in the setting of hyperglycemia-induced hyponatremia because the *y*-intercept will vary directly with the [glucose]_{p} (6, 11–13).

## MEFWC IN HYPERGLYCEMIC STATES

In the setting of hyperglycemia, *EQ. 12* must be modified to account for the dilutional effect of blood glucose on the [Na^{+}]_{p} (5). We have previously demonstrated that the *y*-intercept in the Edelman equation is not constant and will vary predictably with the [glucose]_{p} (6, 11–13). Moreover, we have previously shown (11–13) that the [Na^{+}]_{p} varies with the [glucose]_{p} according to (13)

Therefore, in the setting of hyperglycemia, the MEFWC formula must be generalized as follows: (14)

Hyperglycemia-induced hyponatremia results from changes in the mass balance of Na^{+}, K^{+}, and H_{2}O (osmotic diuresis) and from the dilutional effect of hyperglycemia induced by the translocation of water (6, 11–13). *Equation 14* takes into consideration the dilutional effect of hyperglycemia on the [Na^{+}]_{p} by accounting for the fact that there is an expected decrease of 1.6 meq/l in the [Na^{+}]_{p} for each 100 mg/dl increment in the [glucose]_{p} (5). In addition, *EQ. 14* accounts for the increase in urinary Na^{+}, K^{+}, and H_{2}O excretion resulting from the glucosuria-induced osmotic diuresis as reflected by the terms [Na^{+} + K^{+}]_{urine} and V in *EQ. 14*. Since glucosuria can only affect [Na^{+}]_{p} by altering urinary Na^{+}, K^{+}, and H_{2}O excretion, the incorporation of urinary glucose excretion in the EFWC formula as previously suggested (19) has no mathematical basis. Moreover, none of the previous free water clearance formulas considers the dilutional effect of hyperglycemia on the [Na^{+}]_{p} induced by the translocation of water, and therefore they are not applicable in the setting of hyperglycemia. Finally, according to *EQ. 14*, MEFWC is zero (i.e. urine is isonatric to the [Na^{+}]_{p}) when the [Na^{+} + K^{+}]_{urine} = (Na_{e} + K_{e})/TBW = {[Na^{+}]_{p} + 23.8 + (1.6/100)([glucose]_{p} − 120)}/1.03 = 0.97 [Na^{+}]_{p} + 23.1 + 0.0155 ([glucose]_{p} − 120) (Fig. 4). Since *EQ. 14* already accounts for the dilutional effect of hyperglycemia on the [Na^{+}]_{p}, the actual measured [Na^{+}]_{p} should be employed when *EQ. 14* is utilized in calculating the EFWC.

Using the patient data from Shoker's analysis (19), we now illustrate the utility of this formula in a patient with diabetic ketoacidosis with a urinary [Na^{+} + K^{+}] = 100 mmol/l, [glucose]_{urine} = 270 mg/dl (15 mmol/l), V = 2 l/day, urine osmolality = 600 mosmol/kgH_{2}O, [Na^{+}]_{p} = 120 mmol/l, [K^{+}]_{p} = 3 mmol/l, [glucose]_{p} = 720 mg/dl (40 mmol/l), and plasma osmolality = 315 mosmol/kgH_{2}O:

The urinary free-water clearance as calculated by the previous formulas is as follows:

In the setting of hyperglycemia, Shoker suggested that the renal clearance of glucose should also be incorporated in the calculation of EFWC as V{1 − (2 [Na^{+} + K^{+}]_{urine} + [glucose]_{urine})/(2([Na^{+}]_{p} + [K^{+}]_{p}) + [glucose]_{p})} (19). EFWC as calculated according to this formula is −0.38 l/day. This formula cannot be correct because it does not incorporate in its derivation the fact that there is an expected decrease of 1.6 meq/l in the [Na^{+}]_{p} for each 100 mg/dl increment in the [glucose]_{p} (5). Furthermore, as discussed, there is no theoretical basis for incorporating the urinary glucose excretion rate into the EFWC formula since glucosuria can only affect [Na^{+}]_{p} by altering urinary excretion of Na^{+}, K^{+}, and H_{2}O, which are already accounted for mathematically. In addition, it is well known that glucosuria-induced osmotic diuresis results in the loss of H_{2}O in excess of Na^{+} + K^{+} (16); hence, EFWC in the setting of osmotic diuresis cannot be a negative value as is incorrectly predicted by the formula derived by Shoker (19). Therefore, as shown in this clinical example, calculations of free-water clearance based on previous formulas will result in an inaccurate estimation of the rate of urinary free water excretion since these formulas cannot account for the dilutional effect of blood glucose on the [Na^{+}]_{p} and fail to incorporate the parameters of the Edelman equation in their derivations.

## SUMMARY

The classic FWC and EFWC formulas used to assess the rate of urinary free water clearance fail to incorporate in their derivations the empirical relationship between the [Na^{+}]_{pw} and Na_{e}, K_{e}, and TBW originally demonstrated by Edelman et al. (3). Because previous EFWC formulas do not consider the quantitative and physiological significance of the slope and *y*-intercept in the Edelman equation, they implicitly assume in their derivation that urine is isonatric to the [Na^{+}]_{p} when [Na^{+} + K^{+}]_{urine} is equal to the [Na^{+}]_{p} or [Na^{+}]_{p} + [K^{+}]_{p}. In this article, we present a new formula, MEFWC, for determining the EFWC, taking into consideration the relationship between the [Na^{+}]_{pw} and Na_{e}, K_{e}, and TBW empirically demonstrated by Edelman et al. (3). As required by the Edelman equation, we demonstrate that urine is isonatric to the [Na^{+}]_{p} if [Na^{+} + K^{+}]_{urine} is equal to 0.97[Na^{+}]_{p} + 23.1. Our new formula incorporates this fact in its derivation, and it also takes into consideration the quantitative and physiological significance of the slope and *y*-intercept in the Edelman equation. This new formula will be especially useful in the evaluation of the urinary diluting defect in hyponatremic disorders as well as the urinary concentrating defect that contributes to the development of hypernatremia in diabetes insipidus. Moreover, we have derived a generalized formula for calculating the MEFWC in the setting of hyperglycemia, which can be utilized to quantify the rate of urinary free water excretion in patients with diabetic ketoacidosis and hyperglycemic nonketotic coma.

## GRANTS

This work was supported by the Max Factor Family Foundation, the Richard and Hinda Rosenthal Foundation, and the Fredericka Taubitz fund to I. Kurtz.

## Footnotes

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

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- Copyright © 2005 the American Physiological Society