Am J Physiol Renal Physiol 288: F1–F7, 2005;
doi:10.1152/ajprenal.00259.2004.
TRANSLATIONAL PHYSIOLOGY
Derivation of a new formula for calculating urinary electrolyte-free water
clearance based on the Edelman equation
Minhtri K. Nguyen and Ira Kurtz
Division of Nephrology, David Geffen School of Medicine at UCLA, Los Angeles, California
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 ⫺ Uosm/Posm), where V ⫽ urinary flow rate,
Address for reprint requests and other correspondence: M. K. Nguyen, Div.
of Nephrology, David Geffen School of Medicine at UCLA, 10833 Le Conte
Ave., Rm. 7–155 Factor Bldg., Los Angeles, CA 90095 (E-mail:
[email protected]).
http://www.ajprenal.org
Uosm ⫽ urinary osmolality, and Posm ⫽ 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 Nae, Ke, and total body water
(TBW) originally demonstrated by Edelman et al. (3): [Na⫹]pw
⫽ 1.11(Nae ⫹ Ke)/TBW ⫺ 25.6 (Eq.1), where Nae and Ke 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 beThe 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.
0363-6127/05 $8.00 Copyright © 2005 the American Physiological Society
F1
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Nguyen, Minhtri K., and Ira Kurtz. Derivation of a new formula
for calculating urinary electrolyte-free water clearance based on the
Edelman equation. Am J Physiol Renal Physiol 288: F1–F7, 2005;
doi:10.1152/ajprenal.00259.2004.—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⫹ (Nae), total exchangeable K⫹ (Ke), 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 Nae, Ke, 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 ⫽ (Nae ⫹ Ke)/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.
F2
ELECTROLYTE-FREE WATER CLEARANCE
Table 1. Free water clearance formulas
FWC
V(1⫺Uosm/Posm)
EFWC1
冉
V 1 ⫺
EFWC2*
冊
冉
关Na⫹ ⫹ K⫹兴urine
关Na⫹兴p
V 1 ⫺
MEFWC†
关Na⫹ ⫹ K⫹兴urine
关Na⫹]p ⫹ 关K⫹兴p
冊
冉
V 1 ⫺
冊
1.03关Na⫹ ⫹ K⫹兴urine
关Na⫹兴p ⫹ 23.8
FWC, free water clearance; EFWC, electrolyte-free water clearance; MEFWC, modified EFWC; V, urinary flow rate; [Na⫹]p and [K⫹]p, plasma Na⫹ and K⫹
concentration, respectively; Uosm and Posm, urine and plasma osmolality, respectively. *To account for effective osmoles other than Na⫹ and K⫹, Shoker (19)
revised the calculation of EFWC2 as follows: V{1 ⫺ (2 [Na⫹ ⫹ K⫹]urine ⫹ [other effective osmoles])/(2([Na⫹]p ⫹ [K⫹]p) ⫹ [other effective osmoles])}. †In
the setting of hyperglycemia, the generalized MEFWC formula is utilized where MEFWC ⫽ V[1 ⫺ 1.03 [Na⫹ ⫹ K⫹]urine/{[Na⫹]p ⫹ 23.8 ⫹ (1.6/100)([glucose]p
⫺ 120)}]
DEFINITION OF AN ISONATRIC SOLUTION DICTATED BY
THE EDELMAN EQUATION
关Na ⫹兴pw ⫽ 1.11共Nae ⫹ Ke兲/TBW ⫺ 25.6
(1)
Multiplying both sides of Eq. 1 by 0.93 (1, 2) to convert
[Na⫹]pw to [Na⫹]p
0.93 ⫻ 关Na ⫹兴pw ⫽ 1.03
共Nae ⫹ Ke兲
⫺ 23.8
TBW
Since 0.93 ⫻ [Na⫹]pw ⫽ [Na⫹]p (1 , 2)
关Na ⫹兴p ⫽ 1.03
共Nae ⫹ Ke兲
⫺ 23.8
TBW
关Na ⫹兴p1 ⫽ 1.03
共Nae1 ⫹ Ke1兲
⫺ 23.8
TBW1
关Na ⫹兴p2 ⫽ 1.03
共Nae2 ⫹ Ke2兲
⫺ 23.8
TBW2
(2)
If [Na⫹]p1 ⫽ [Na⫹]p2, then
Na e2 ⫹ Ke2 Nae1 ⫹ Ke1
⫽
TBW2
TBW1
(3)
Since
Nae2 ⫹ Ke2 Nae1 ⫹ Ke1 ⫹ (Na⫹ ⫹ K⫹)input⫺output
⫽
TBW2
TBW1 ⫹ V共input⫺output兲
(4)
Assuming that there is no input and only urinary loss
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
AJP-Renal Physiol • VOL
[Na⫹]p ⫽ (Nae ⫹ Ke)/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⫹] ⫽ (Nae ⫹ Ke)/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⫹] ⫽ (Nae ⫹
Ke)/TBW can be demonstrated as follows:
Na e2 ⫹ Ke2 Nae1 ⫹ Ke1 ⫺ 关Na⫹ ⫹ K⫹兴urine ⫻ Vurine
⫽
TBW2
TBW1 ⫺ Vurine
(5)
Substituting Eq. 5 for (Nae2 ⫹ Ke2 )/TBW2 in Eq. 3
288 • JANUARY 2005 •
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tween 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 GibbsDonnan effect raises the [Na⫹]pw at any given quantity of
(Nae ⫹ Ke)/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 yintercept 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 yintercept 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 yintercept 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 Nae, Ke, 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(Nae ⫹ Ke)/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.
F3
ELECTROLYTE-FREE WATER CLEARANCE
[Na⫹ ⫹ K⫹]urine ⫽ (Nae ⫹ Ke)/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.
Na e1 ⫹ Ke1 ⫺ 关Na⫹ ⫹ K⫹兴urine ⫻ Vurine Nae1 ⫹ Ke1
⫽
TBW1 ⫺ Vurine
TBW1
Rearranging
共Na e1 ⫹ Ke1 ⫺ 关Na⫹ ⫹ K⫹兴urine ⫻ Vurine兲 ⫻ TBW1
⫽ 共Nae1 ⫹ Ke1兲共TBW1 ⫺ Vurine兲
关Na ⫹兴p ⫽ 1.03
Rearranging
共Na e1 ⫹ Ke1兲 ⫻ TBW1 ⫺ 共关Na⫹ ⫹ K⫹兴urine ⫻ Vurine兲 ⫻ TBW1
⫽ 共Nae1 ⫹ Ke1兲 ⫻ TBW1 ⫺ 共Nae1 ⫹ Ke1兲 ⫻ Vurine
共Na e ⫹ Ke兲/TBW ⫽ 共关Na⫹兴p ⫹ 23.8兲/1.03
关Na ⫹ ⫹ K⫹兴urine ⫽
共Nae1 ⫹ Ke1兲
TBW1
Thus a solution is isonatric when its [Na⫹ ⫹ K⫹] ⫽ (Nae ⫹
Ke)/TBW. Since (Nae ⫹ Ke)/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).
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⫹] ⫽ (Nae ⫹ Ke)/TBW. Specifically, when [Na⫹ ⫹
K⫹]urine ⫽ (Nae ⫹ Ke)/TBW, the excretion of urine will not
change [Na⫹]p from its current value.
These two urine components can be represented algebraically as
V ⫽ IEC ⫹ MEFWC
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
(6)
(7)
where V ⫽ urine flow rate, E ⫽ [Na⫹ ⫹ K⫹], IEC ⫽ isonatric
electrolyte clearance, and MEFWC ⫽ modified electrolyte-free
water clearance.
Furthermore
IEC ⫽
关Na⫹ ⫹ K⫹兴urine ⫻ V
共Nae ⫹ Ke兲/TBW
(8)
Since according to Eq. 6 (Nae ⫹ Ke)/TBW ⫽ ([Na⫹]p ⫹
23.8)/1.03; Eq. 8 can be rewritten as
IEC ⫽
1.03关Na⫹ ⫹ K⫹兴urine ⫻ V
关Na⫹兴p ⫹ 23.8
(9)
Rearranging Eq. 7
MEFWC ⫽ V ⫺ IEC
(10)
Since according to Eq. 9
IEC ⫽
1.03关Na⫹ ⫹ K⫹兴urine ⫻ V
,
关Na⫹兴p ⫹ 23.8
Eq. 10 can be rewritten as
MEFWC ⫽ V ⫺
Fig. 1. Definition of isonatric urine. In comparing [Na⫹]p (mmol/l) with
[Na⫹ ⫹ K⫹]urine (mmol/l), according to previous electrolyte-free water
clearance formulas, urine is isonatric to the [Na⫹]p when [Na⫹ ⫹ K⫹]urine ⫽
[Na⫹]p (dotted line) or [Na⫹ ⫹ K⫹]urine ⫽ [Na⫹]p ⫹ [K⫹]p (dashed line).
According to the MEFWC formula, urine is isonatric to the [Na⫹]p when
[Na⫹ ⫹ K⫹]urine ⫽ 0.97[Na⫹]p ⫹ 23.1 (solid line).
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Rearranging
288 • JANUARY 2005 •
冉
1.03关Na⫹ ⫹ K⫹兴urine ⫻ V
关Na⫹兴p ⫹ 23.8
(11)
冊
(12)
MEFWC ⫽ V 1 ⫺
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1.03关Na⫹ ⫹ K⫹兴urine
关Na⫹兴p ⫹ 23.8
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Solving for [Na⫹ ⫹ K⫹]urine
(2)
Rearranging
Therefore
共关Na ⫹ ⫹ K⫹兴urine ⫻ Vurine兲 ⫻ TBW1 ⫽ 共Nae1 ⫹ Ke1兲 ⫻ Vurine
共Nae ⫹ Ke兲
⫺ 23.8
TBW
F4
ELECTROLYTE-FREE WATER CLEARANCE
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:
冉
MEFWC ⫽ V 1 ⫺
冊
1.03关Na⫹ ⫹ K⫹兴urine
关Na⫹兴p ⫹ 23.8
FWC ⫽ V共1 ⫺ Uosm/Posm兲 ⫽ 1.5共1 ⫺ 540/256兲 ⫽ ⫺ 1.7 l/day
EFWC 1 ⫽ V关1 ⫺ 共UNa ⫹ UK兲/关Na⫹兴p兴
⫽ 1.5共1 ⫺ 130/110兲 ⫽ ⫺ 0.27 l/day
EFWC 2 ⫽ V关1 ⫺ 共UNa ⫹ UK兲/共关Na⫹兴p ⫹ 关K⫹兴p兲兴
⫽ 1.5共1 ⫺ 130/114兲 ⫽ ⫺ 0.21 l/day
冉
冉
MEFWC ⫽ V 1 ⫺
⫽ 1.5 1 ⫺
CLINICAL UTILITY OF THE MEFWC FORMULA
Based on the empirical relationship between the [Na⫹]pw
and Nae, Ke, 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-
冊
1.03 ⫻ 130
⫽ 0 l/day
110 ⫹ 23.8
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
Fig. 2. Determinants of modified electrolyte-free water clearance (MEFWC). MEFWC has 3 determinants: urinary flow rate
(V; l/day), [Na⫹ ⫹ K⫹]urine (mmol/l), and plasma Na⫹ concentration ([Na⫹]p; mmol/l). To determine the effect of changes
in each of these 3 variables on the MEFWC (l/day), each
variable is altered while the other two variables are maintained
constant. As shown in A, MEFWC increases proportionally
with changes in V (where [Na⫹ ⫹ K⫹]urine ⫽ 40 mmol/l and
[Na⫹]p ⫽ 130 mmol/l). B: MEFWC varies inversely with
changes in [Na⫹ ⫹ K⫹]urine (where V ⫽ 2 l/day and [Na⫹]p ⫽
130 mmol/l). C: as [Na⫹]p increases, MEFWC increases in a
curvilinear fashion (where V ⫽ 2 l/day and [Na⫹ ⫹ K⫹]urine ⫽
40 mmol/l).
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冊
1.03关Na⫹ ⫹ K⫹兴urine
关Na⫹兴p ⫹ 23.8
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This new formula incorporates the known empirical relationship between the [Na⫹]pw and Nae, Ke, 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 ⫽
(Nae ⫹ Ke)/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).
water clearance formulas. According to Eq.1, as long as the
[Na⫹ ⫹ K⫹]urine ⫽ (Nae ⫹ Ke)/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 freewater clearance in a hypothetical patient with a [Na⫹ ⫹
K⫹]urine ⫽ (Nae ⫹ Ke)/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/kgH2O, [Na⫹]p ⫽ 110
mmol/l, [K⫹]p ⫽ 4.0 mmol/l, and plasma osmolality ⫽ 256
mosmol/kgH2O:
ELECTROLYTE-FREE WATER CLEARANCE
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), cliniTable 2. Isonatric urine and EFWC
[Na⫹]p,
mmol/l
[K⫹]p,
mmol/l
[Na⫹ ⫹ K⫹]urine,
mmol/l
Vurine,
l/day
EFWC1,
l/day
EFWC2,
l/day
MEFWC,
l/day
140
140
140
4.0
4.0
4.0
140
144
159
4
4
4
0
⫺0.114
⫺0.543
0.111
0
⫺0.417
0.479
0.378
0
The value of [Na⫹ ⫹ K⫹]urine that is isonatric to the [Na⫹]p varies with the
formula utilized to calculate EFWC. Only the MEFWC formula predicts that
urine is isonatric to the [Na⫹]p of 140 mmol/l when [Na⫹ ⫹ K⫹]urine ⫽ 159
mmol/l as dictated by the Edelman equation.
AJP-Renal Physiol • VOL
Fig. 3. Comparison of EFWC (l/day) as calculated according to EFWC1 (solid
line), EFWC2 (dotted line), and MEFWC (dashed line). In this example,
[Na⫹]p ⫽ 110 mmol/l, [K⫹]p ⫽ 3 mmol/l, and [Na⫹ ⫹ K⫹]urine ⫽ 100 mmol/l.
EFWC is calculated according to the various EFWC formulas at an increasing
V (l/day). The magnitude of the difference between EFWC1 and EFWC2
compared with MEFWC varies proportionately with V.
cal 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
Nae and Ke 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 Nae,
Ke, 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
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the urinary free water clearance is zero. In contrast, free water
clearance as calculated by FWC (21), EFWC1 (4, 17), and
EFWC2 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/kgH2O (7).
Finally, the inaccuracies of the EFWC1 and EFWC2 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
EFWC1 and EFWC2 formulas result in greater errors at higher
urinary flow rates compared with that calculated according to
the MEFWC formula.
F5
F6
ELECTROLYTE-FREE WATER CLEARANCE
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
关Na ⫹兴p ⫽ 1.03共Nae ⫹ Ke兲/TBW ⫺ 23.8
⫺ 共1.6/100兲共关glucose兴p ⫺ 120兲
(13)
Therefore, in the setting of hyperglycemia, the MEFWC formula must be generalized as follows:
MEFWC
冉
⫽V 1⫺
1.03关Na⫹ ⫹ K⫹兴urine
关Na 兴p ⫹ 23.8 ⫹ 共1.6/100兲共关glucose兴p ⫺ 120兲
⫹
冊
Hyperglycemia-induced hyponatremia results from changes in
the mass balance of Na⫹, K⫹, and H2O (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 H2O 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 H2O
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
冉
MEFWC ⫽ 2 1 ⫺
冊
1.03 ⫻ 100
120 ⫹ 23.8 ⫹ 共1.6/100兲共720 ⫺ 120兲
⫽ 0.66 l/day
The urinary free-water clearance as calculated by the previous
formulas is as follows:
FWC ⫽ V共1 ⫺ Uosm/Posm兲 ⫽ 2共1 ⫺ 600/315兲 ⫽ ⫺ 1.8l/day
EFWC 1 ⫽ V关1 ⫺ 共UNa ⫹ UK兲/关Na⫹兴p兴
⫽ 2共1 ⫺ 100/120兲 ⫽ 0.33 l/day
EFWC 2 ⫽ V关1 ⫺ 共UNa ⫹ UK兲/共关Na⫹兴p ⫹ 关K⫹兴p兲兴
⫽ 2共1 ⫺ 100/123兲 ⫽ 0.37 l/day
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 H2O, which are
already accounted for mathematically. In addition, it is well
known that glucosuria-induced osmotic diuresis results in the
loss of H2O 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
Fig. 4. Definition of isonatric urine in the setting of hyperglycemia. During
hyperglycemia, urine is isonatric to the [Na⫹]p (mmol/l) when [Na⫹ ⫹ K⫹]urine
(mmol/l) ⫽ 0.97[Na⫹]p ⫹ 23.1 ⫹ 0.0155 ([glucose]p ⫺ 120), where [glucose]p
is plasma glucose concentration. At any given [Na⫹]p, increases in [glucose]p
result in a corresponding increment in the isonatric [Na⫹ ⫹ K⫹]urine. Solid
line, [glucose]p ⫽ 120 mg/dl; dotted line: [glucose]p ⫽ 500 mg/dl; dashed line:
[glucose]p ⫽ 1,000 mg/dl.
AJP-Renal Physiol • VOL
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
Nae, Ke, and TBW originally demonstrated by Edelman et al.
(3). Because previous EFWC formulas do not consider the
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(14)
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 ⫽ (Nae ⫹ Ke)/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/kgH2O, [Na⫹]p ⫽ 120 mmol/l,
[K⫹]p ⫽ 3 mmol/l, [glucose]p ⫽ 720 mg/dl (40 mmol/l), and
plasma osmolality ⫽ 315 mosmol/kgH2O:
ELECTROLYTE-FREE WATER CLEARANCE
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.
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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 Nae, Ke, 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.
F7