In 1958, Edelman and colleagues used isotopes to measure exchangeable sodium (Na^{+}_{e}), exchangeable potassium (K^{+}_{e}), and total body water (TBW) in a heterogenous group of chronically ill patients; the experiment described the relation between these variables and plasma sodium concentration ([Na^{+}])^{1}:

Edelman's experiment confirmed previous observations by other investigators that changes in serum sodium concentration parallel net changes of sodium and potassium balance in relation to water balance^{2} (Figure 1A). Four years earlier, Edelman's group obtained serial measurements of Na^{+}_{e}, K^{+}_{e}, and TBW in 12 patients before and after surgery, along with meticulous measurements of fluid and electrolyte balance in three; his 1958 study reanalyzed those data, combined them with measurements in three additional patients, and found that the change (Δ) in serum sodium concentration was linearly correlated with Δ (Na^{+}_{e}+K^{+}_{e})/TBW (Figure 1B).^{1}

The discussion section of the 1958 study by Edelman *et al.* algebraically analyzed previously published formulas used to guide the correction of hyponatremia and concluded,^{1} “Our data, therefore, provide an experimental basis for the formulas used in clinical manipulations and demonstrate the coordinated influence of the three major components of body composition on serum sodium concentration.”

Edelman *et al.* recognized that the intercept derived from the linear regression of their data might have physiological significance; they observed,^{1} “The zero intercept of the regression equation for plasma sodium concentration in plasma water versus (Na^{+}_{e}+K^{+}_{e})/TBW is a negative constant (−25.6 mEq/L), which probably is a measure of the quantity of osmotically inactive exchangeable sodium and potassium per unit of body water. Since body water in this group of patients averaged 36.1 L …, the estimated osmotically inactive Na+K is approximately 920 mEq. Osmotically inactive exchangeable bone sodium, which is about 750 mEq, probably accounts for almost all of this quantity.”

As reviewed by Wagner and coworkers^{3} in this issue of the Journal, the recent literature has seen a flurry of publications comparing potential benefits and flaws of several new eponymous equations that have been used in the management of hypo- and hypernatremia (*e.g.*, the “formulas” or “equations” of “Adrogue-Madias,” “Barsoum-Levine,” and “Nguyen-Kurtz”).^{3} Because these are mere algebraic manipulations of concepts established in the 1950s, we should ask whether labeling them with eponyms is warranted.

To make the Edelman data more approachable, Rose rounded the slope of the linear regression to one and omitted its intercept; to make it clinically applicable, he expressed it in serum sodium rather than the concentration of sodium in plasma water^{4}:

Adrogue and Madias adopted Rose's simplified version and rearranged it algebraically to offer a prediction of the Δ Serum [Na+]to be expected from infusion of 1 L of intravenous fluid; the change depends on the starting serum sodium level ([Na+]serum) and the sodium and potassium concentrations of the infusate ([Na++K+]inf)^{5}:

Noting that this algebraic formulation was only valid for a 1-L infusion, Chen and coworkers proposed an improved equation that could be applied to any volume of intravenous fluid (V_{inf})^{6}:

The improved equation of Chen, which, as of this writing, has not yet achieved eponymous recognition, can be mathematically reduced to the simplified Edelman relationship (with a slope of 1 and no intercept).

Because an accurate prediction of the change in serum sodium concentration (Δ [Na^{+}]_{s}) requires more inclusive measurements of cation and fluid balance, Barsoum and Levine offered a more complicated equation that included volumes of fluid intake (V_{inf}), urine volume (V_{u}), extrarenal output (V_{o}), and net fluid balance (ΔV) as well as cation concentrations of urine ([Na^{+}+K^{+}]_{u}), extrarenal fluid ([Na^{+}+K^{+}]_{o}), and the infusate ([Na^{+}+K^{+}]_{inf})^{7}:

If all output is ignored (so that ΔV becomes V_{inf}), the equation published by Barsoum and Levine reduces to Chen's improvement of the equation published by Adrogue and Madias.

Nguyen and Kurtz offered an even more complicated formula (which will not be shown here), which, in addition to considering all intake and output volumes and cation concentrations, also included the intercept and slope of Edelman's linear regression (1).^{8} It is mathematically identical to the Edelman relationship (with a slope of 1.11 and an intercept of −25.6), if one assumes that added cations increase the numerator of the equation and that net fluid balance alters TBW.

Inclusion of the intercept of Edelman's linear regression acknowledges the role that osmotically inactive sodium storage plays in determining the response of the serum sodium concentration to hypertonic saline, a topic that is extensively discussed in the review by Wagner.^{2} While rapid internal exchanges between osmotically active and inactive sodium pools are of great interest to physiologists, they remain unproven and are of little importance to clinicians attempting to manage patients with abnormal serum sodium concentrations.

Balance studies lasting several weeks have shown, rather convincingly, that when normal participants are fed a very high-sodium diet, retained sodium seems to disappear with a minimal increase in serum sodium concentration or extracellular volume. Does such a phenomenon occur within hours?

A report of 12 normal volunteers studied after 8 days on a low-sodium diet was said to show that “healthy individuals are able to osmotically inactivate significant amounts of sodium after hypertonic saline infusion.”^{9} However, evidence supporting this conclusion is far from convincing. Rather than the pre-infusion value, the serum sodium obtained 5 minutes after completion of the infusion of 3% saline was the reference point for subsequent changes; the fall in serum sodium between 5 and 240 minutes was the basis for the investigators' conclusion. At 5 minutes, the serum sodium had increased by 4 mEq/L in six participants and by 5 mEq/L in two, considerably more than the 3-mEq/L increase expected for an apparent volume of distribution equaling TBW. On the basis of measured urine losses of sodium and potassium, the investigators observed that the expected decrease in serum sodium between 5 and 240 minutes should have been 0.9±0.1 mEq/L, rather than the observed 1.8-mEq/L decrease.^{9} However, that discrepancy can be explained by the unexpectedly high serum sodium value at 5 minutes.

Why was the 5-minute sodium value too high? It seems unlikely that sodium was released from osmotically inactive sodium stores in the first 5 minutes after infusion of hypertonic saline only to be taken up again along with infused sodium in the subsequent 4 hours. For a better explanation, consider why sodium, an extracellular solute, seems to distribute in TBW. Immediately after infusion of hypertonic saline, an osmotic disequilibrium is created that is eventually, but not instantaneously, abolished by water movement out of cells.^{2} Owing to the large mass of skeletal muscles (approximately 24 kg in a 70-kg adult) relative to its blood supply at rest (approximately 1 L/min), osmotic equilibrium (and a volume of distribution equal to TBW) is delayed for several minutes.^{10} The study's findings cannot be taken as a proof of acute storage of sodium in an osmotically inactive pool.

In the 1950s, meticulous balance studies and serial measurements of exchangeable sodium and potassium and TBW—unlikely to be repeated in our lifetimes—showed that short-term predictive changes in serum sodium concentration on the basis of cation and fluid balance are reasonably accurate (Figure 1).^{1}^{,}^{2} These studies provide little cause for worry that acute storage or release of sodium from osmotically inactive sodium stores will affect our treatment of hypo- or hypernatremia in a clinically meaningful way.

A simple ratio of total body cations to TBW modified by changes to the numerator and denominator is conceptually easy, whereas algebraic transformations of the ratio are not. All predictive formulas reduce to the following simple concept:

Applying simple algebra to (2),

We have developed an online calculator (https://www.richardsternsmd.com/calculator) preserving the concept that serum sodium concentration equals total body (Na+K) (the numerator) divided by TBW (the denominator). TBW is estimated from the patient's sex, weight, age, and height. Net (Na^{+}+K^{+}) balance (changing the numerator) and fluid balance (changing the denominator) are calculated from volumes and (Na^{+}+K^{+}) concentrations of intravenous fluids and urine. The calculator compares the expected change in serum sodium with its actual change (which should be measured frequently). Differences between observed and expected values suggest a change in urine output and composition, which should be re-evaluated. The dose of hypertonic saline or 5% dextrose in water needed to attain a desired rate of correction can also be calculated without troubling the user with algebraic transformations of classic equations.

### Disclosures

A.M. Sterns reports the following: Employer: LIFTEH2 and Sterns Tech LLC. R.H.Sterns reports the following: Employer: Rochester General Hospital and University of Rochester School of Medicine and Dentistry; and Other Interests or Relationships: UpToDate.

### Funding

None.

## Acknowledgments

The content of this article reflects the personal experience and views of the author(s) and should not be considered medical advice or recommendation. The content does not reflect the views or opinions of the American Society of Nephrology (ASN) or *Kidney360*. Responsibility for the information and views expressed herein lies entirely with the author(s). The authors dedicate this editorial to the memory of Gwen K. Sterns, MD.

### Author Contributions

**Conceptualization:** Richard H. Sterns.

**Formal analysis:** Richard H. Sterns, Adam M. Sterns.

**Methodology:** Richard H. Sterns.

**Software:** Adam M. Sterns.

**Supervision:** Richard H. Sterns.

**Validation:** Adam M. Sterns.

**Writing – original draft:** Richard H. Sterns.

**Writing – review & editing:** Richard H. Sterns, Adam M. Sterns.

## References

**Keywords:**

clinical nephrology; electrolytes; hypernatremia; hypokalemia; hyponatremia; intravenous; osmolality; vasopressin; water-electrolyte balance