Free water deficit in hypernatraemia

Clinical background

Hypernatraemia arises when water loss exceeds sodium loss, so that the serum sodium rises above 145 mEq/L. The decision the tool serves is how large the accumulated water deficit is, and hence how much free water must be given to restore the sodium concentration to normal. Without an estimate of the deficit, one risks either giving too little, leaving the hypernatraemia uncorrected, or giving too much too quickly, with a risk of cerebral oedema in chronic hypernatraemia.

The calculation is particularly relevant in intensive care units and emergency departments, where patients may have large insensible losses, an impaired thirst mechanism or diabetes insipidus. The clinical difficulty lies not in producing a number, but in interpreting it in the light of ongoing losses and the rate of correction that is safe for the individual patient.

Calculating the free water deficit

The formula rests on the principle that the serum sodium concentration reflects the ratio of total body sodium to total body water. If water has been lost without a corresponding loss of sodium, the missing water can be calculated from the raised sodium concentration and the patient's estimated total body water.

$$\text{Free water deficit (L)} = f \times \text{weight (kg)} \times \left(\frac{[\text{Na}^+]_{\text{serum}}}{140} - 1\right)$$

where $f$ is the total body water fraction and $[\text{Na}^+]_{\text{serum}}$ is the measured serum sodium in mEq/L.

The variables in the formula:

• Total body water fraction ($f$): an estimate of the body's water fraction. For an adult man $f = 0{.}6$, for an adult woman or an older man $f = 0{.}5$, for an older woman $f = 0{.}45$, and for children $f = 0{.}6$. These values rest on classical dilution studies of total body water, in which Watson et al. derived anthropometric prediction equations based on 458 adult men and 265 adult women [5]. The percentages are simplified averages used clinically; the actual body water fraction varies between individuals with body composition, particularly with the degree of adiposity.
• Body weight: the patient's current weight in kg.
• Measured serum sodium: the actual serum sodium value in mEq/L. The normal value of 140 in the denominator is the sodium concentration to which the water deficit is referenced.

The formula was derived and popularised by Adrogué and Madias in a review article in the New England Journal of Medicine in 2000, which summarised the quantitative principles for treating the dysnatraemias [1]. It is not an empirically derived risk model from a defined cohort with a specific outcome, but a theoretical mass balance equation based on physiological principles of water and sodium balance.

Interpretation in practice

The result gives the volume of water, in litres, theoretically required to lower the serum sodium to 140 mEq/L. It is a starting point for planning fluid replacement, not a prescription to be infused all at once.

Once the replacement volume has been calculated, the choice of fluid must be made. Free water is given orally or by tube if the patient can tolerate it. Intravenously, 5% glucose or, more commonly in practice, hypotonic saline (0.45%) is used. The choice affects how the serum sodium changes, and the Adrogué–Madias formula for the change in serum sodium per litre of infusate can be used as a complement to estimate the effect of the chosen fluid [1].

The serum sodium should be measured every two to four hours during correction, and the infusion rate adjusted according to the measured change rather than according to the initial calculation.

Validation and performance

The theoretical basis on which the formula rests has never been validated in a prospective study systematically comparing the predicted water deficit with the measured serum sodium response. The most relevant external evaluation was performed by Lindner et al., who compared four formulae for predicting the change in serum sodium (Adrogué–Madias, Barsoum–Levine, Kurtz–Nguyen and their own electrolyte-free water clearance-based formula) in 66 patients in an intensive care unit with a total of 681 patient days, 194 of them hypernatraemic [2]. All the formulae correlated significantly with the measured changes, but individual variation was wide. The mean difference between predicted and measured serum sodium was 3.4 to 4.5 (±4.4 to 4.7) mEq/L for the published formulae. During active hypernatraemia the deviations were larger still: 5.0 to 6.7 (±3.9 to 4.3) mEq/L. The authors concluded that the available formulae do not accurately predict changes in serum sodium in the individual ICU patient, and that infusion therapy should be guided by serial measurements rather than by formula calculations alone [2].

A methodological limitation highlighted by Chen et al. is that the Adrogué–Madias formula for the change in serum sodium per litre of infusate is exact only for 1 L; proportional scaling to other volumes is not linear, and the deviation grows with volume [4]. This concerns the related Adrogué–Madias infusate formula, but the principle that these formulae rest on the assumption of an unchanged total body water during treatment is common to both.

As regards the safety of correction, the traditional recommendation not to exceed 0.5 mEq/L per hour has been questioned by a retrospective study by Chauhan et al., which analysed 122 patients with severe hypernatraemia (>155 mmol/L) on admission and 327 with hospital-acquired hypernatraemia in the ICU [3]. Rapid correction (>0.5 mmol/L per hour) was not associated with increased mortality, seizures, reduced consciousness or cerebral oedema compared with slower correction. Not a single case of cerebral oedema could be attributed to rapid correction on manual review of the records. The study is retrospective and observational, and the results concern intensive care patients with, in many cases, acute hypernatraemia. It should not be read as meaning that the rate of correction does not matter, but it does show that the evidential support for the 0.5 mEq/L per hour threshold is weak [3].

Limitations

The free water deficit formula is valid only when the hypernatraemia is caused by pure water loss. With hypotonic fluid loss — that is, when both water and sodium have been lost but proportionally more water — the formula does not give complete information about how different infusates with varying sodium and potassium concentrations will affect the serum sodium [6]. This is the commonest clinical situation, particularly with gastrointestinal losses or diuretic use.

Other important limitations:

• The assumption of unchanged total body water: the formula calculates the static deficit at a given moment and does not take account of ongoing insensible losses, urine output or drain losses during treatment. These must be added separately to the daily fluid plan.
• The body water fraction is a rough estimate: the fractions 0.6, 0.5 and 0.45 are population averages. In patients with marked adiposity the actual water fraction is lower, and the formula then overestimates the water deficit. The converse applies to patients with a low body weight and a large muscle mass.
• Not applicable in hyperglycaemia: a raised blood glucose shifts water into the intravascular compartment and lowers the serum sodium (translocational hyponatraemia), which does not reflect a genuine sodium or water deficit. Correct the serum sodium for glucose before using the formula.
• Not validated for children of low body weight: the fraction of 0.6 for children is an approximation; infants have a higher water fraction (up to 0.7) that gradually falls over the first years of life.

References

1. Adrogué HJ, Madias NE. Hypernatremia. N Engl J Med. 2000;342(20):1493-9. PMID: 10816188
2. Lindner G, Schwarz C, Kneidinger N, et al. Can we really predict the change in serum sodium levels? An analysis of currently proposed formulae in hypernatraemic patients. Nephrol Dial Transplant. 2008;23(11):3501-8. PMID: 18723567
3. Chauhan K, Pattharanitima P, Patel N, et al. Rate of Correction of Hypernatremia and Health Outcomes in Critically Ill Patients. Clin J Am Soc Nephrol. 2019;14(5):656-663. PMID: 30948456
4. Chen S, Shieh M, Chiaramonte R, et al. Improving on the Adrogué-Madias Formula. Kidney360. 2021;2(2):365-370. PMID: 35373033
5. Watson PE, Watson ID, Batt RD. Total body water volumes for adult males and females estimated from simple anthropometric measurements. Am J Clin Nutr. 1980;33(1):27-39. PMID: 6986753
6. Nguyen MK, Kurtz I. A new quantitative approach to the treatment of the dysnatremias. Clin Exp Nephrol. 2003;7(2):125-37. PMID: 14586731