Free water deficit calculator

The calculator above estimates the free water deficit in hypernatremia from serum sodium, weight, sex and age. This article explains where the formula comes from, which total body water (TBW) fractions it uses, how to check the arithmetic, and why the result is only a starting estimate.

Correction-rate decisions belong to clinicians. The calculator and this page give no infusion rate, volume per hour, fluid choice or correction schedule. Hypernatremia treatment is planned by the treating team, using serial sodium measurements and the patient’s clinical state.

Units. Sodium in mmol/L (= mEq/L). Weight in kg. Result in liters (L), displayed to 1 decimal place.

The formula

Free water deficit (L) = TBW × (Na ÷ 140 − 1), where TBW (L) = fraction × weight (kg) [2, 3]

Where it comes from. The formula assumes that hypernatremia is pure water loss and that total body sodium stays the same. Sodium concentration × body water is then constant, so the normal body water is TBW × Na ÷ 140. The deficit is the difference: TBW × (Na ÷ 140 − 1).

Some sources, including Adrogué and Madias, write the conventional formula as TBW × (1 − 140 ÷ Na), which gives a smaller value (5.25 L instead of 6.0 L in example 1). They did not recommend the conventional formula for planning treatment [1].

• Reference sodium: 140 mmol/L, the constant in the formula. It is not a treatment target. Merck writes the same idea with a “desired Na”: TBW × (Na − desired) ÷ desired [3]. With desired = 140, the two forms are identical.
• When it does not apply: if sodium is below 140 mmol/L, the calculator shows “not applicable” and never a negative volume. At exactly 140 it shows 0.0 L, “no deficit”.
• Precision: full precision internally, display rounded to 1 decimal place (half away from zero).

TBW fractions used

These fractions are from Adrogué and Madias [1], as also used by the Merck calculator [3]. The NEJM review does not give a number for “elderly”, so the calculator uses ≥ 65 years, Merck’s cut-off [3]. In water-depleted hypernatremia, some authors use lower fractions (0.5 for men, 0.4 for women) [2]. That gives a smaller deficit. Yun et al. also describe a simplification of 60% for men and 50% for women with 5% deducted for elderly patients [2], which would make elderly men 0.55 rather than 0.5. This is one reason calculators differ.

Worked example 1: adult man (arithmetic only)

Man, 40 years, 70 kg, Na 160 mmol/L.

1. TBW: 0.6 × 70 = 42 L
2. Ratio: 160 ÷ 140 − 1 = 0.142857
3. Deficit: 42 × 0.142857 = 6.0 L

Worked example 2: the age cut-off matters (arithmetic only)

Woman, 70 years, 70 kg, Na 160 mmol/L.

1. TBW: 0.45 × 70 = 31.5 L
2. Deficit: 31.5 × 0.142857 = 4.5 L

Man, 65 years, 80 kg, Na 155 mmol/L. At 65 he counts as elderly (≥ 65), so the fraction is 0.5.

1. TBW: 0.5 × 80 = 40 L
2. Ratio: 155 ÷ 140 − 1 = 0.107143
3. Deficit: 40 × 0.107143 = 4.286 → 4.3 L

With 0.6 instead, the deficit would be 48 × 0.107143 = 5.1 L. That is nearly a liter different for the same patient, from one assumption alone.

Worked example 3: a published case (arithmetic only)

Woman, 40 years, 60 kg, Na 166 mmol/L.

1. TBW: 0.5 × 60 = 30 L
2. Ratio: 166 ÷ 140 − 1 = 0.185714
3. Deficit: 30 × 0.185714 = 5.571 → 5.6 L

This matches the worked example in Yun et al. [2].

These deficits are formula outputs, used here to show the arithmetic. They are not a volume to give, and they say nothing about how fast or over how long. Those decisions belong to the treating clinician.

Worked example 4: sodium below 140 (arithmetic only)

Man, 70 kg, Na 135 mmol/L. The arithmetic would give 42 × (135 ÷ 140 − 1) = −1.5 L. A negative “deficit” has no meaning here, so the calculator shows “not applicable”.

The Adrogué–Madias alternative

Adrogué and Madias did not recommend relying on the conventional formula: “We do not recommend using the conventional formula … it underestimates the deficit in patients with hypotonic fluid loss” [1]. They preferred estimating how much one liter of a given infusate would change the serum sodium [1]. The formula is [1, 2]:

ΔNa per L = (infusate Na + infusate K − serum Na) ÷ (TBW + 1)

Worked example of the Adrogué–Madias formula (arithmetic only)

Man, 70 kg (TBW 42 L), serum Na 160 mmol/L.

• Input: infusate Na 0, K 0: (0 + 0 − 160) ÷ (42 + 1) = −160 ÷ 43 = −3.72 → −3.7 mmol/L per L
• Input: infusate Na 77, K 0: (77 + 0 − 160) ÷ 43 = −83 ÷ 43 = −1.93 → −1.9 mmol/L per L

These numbers show the arithmetic only. They are not a choice of fluid, a volume or a rate. Infusate choice, volume and rate are decided by the treating clinician. The formula’s predictions can be inaccurate, for example in severe volume depletion or reduced kidney function, and sodium is remeasured during treatment [2].

Infusate tonicity is covered in isotonic, hypotonic, hypertonic and IV fluid osmolarity.

Limitations

• Current deficit only. The formula ignores ongoing losses: insensible, renal and gastrointestinal. Yun et al. note that these must be added when planning water replacement [2].
• Pure water loss assumed. With hypotonic fluid losses (sodium and water together) the conventional formula underestimates the deficit [1]. It cannot account for the sodium and potassium in an infusate. The Adrogué–Madias formula can.
• TBW is a population estimate. Body composition, obesity, edema and age change it. Merck applies its factor to normal (pre-illness) body weight [3], while most users enter current weight.
• Hyperglycemia. High glucose lowers the measured sodium and can hide a water deficit. Hypernatremia work-ups recommend confirming the glucose-corrected sodium [2]. The calculator does not do that for you. See corrected sodium for hyperglycemia.
• Not for infants or neonates. Children are given a single fraction (0.6). Neonatal water balance is outside this tool’s scope.
• Serial values matter more than one estimate. Sodium is remeasured during treatment, and decisions follow those values. The calculator does not.

FAQs

Why does the calculator say “not applicable” when sodium is low? The formula estimates a water deficit. Below 140 mmol/L there is none to estimate, and a negative volume would be misleading.

Why do the TBW fractions differ by sex and age? The published fractions are lower for women and for older adults [1, 3]. The calculator states its ≥ 65-year cut-off because the original review does not define “elderly”.

Does the result tell me how much fluid to give, or how fast? No. It is an estimate of the current deficit only. Volumes, rates and fluid choice are clinical decisions, and they must also cover ongoing losses [2].

What is the difference between this and the Adrogué–Madias formula? The conventional formula estimates a total deficit in liters. Adrogué–Madias predicts how one liter of a specific infusate changes serum sodium, and its authors prefer it [1].

Should I use measured or glucose-corrected sodium? In marked hyperglycemia, the measured sodium understates the water deficit. Work-ups recommend checking the glucose-corrected value [2]. Discuss with the treating team which value to use.

Educational content, not medical advice. The free water deficit is an estimate for learning. It is not a fluid order. Correction-rate decisions, fluid choice and monitoring belong to the treating clinician. Verify against your institution’s protocols and your laboratory’s reference ranges. See our disclaimer and how we check content.

Related: Corrected sodium for hyperglycemia · Serum osmolality calculator · Normal osmolality range · Isotonic, hypotonic, hypertonic · IV fluid osmolarity · Osmolality vs osmolarity

References

1. Adrogué HJ, Madias NE. Hypernatremia. N Engl J Med. 2000;342(20):1493–1499. doi:10.1056/NEJM200005183422006. https://pubmed.ncbi.nlm.nih.gov/10816188/
2. Yun G, Baek SH, Kim S. Evaluation and management of hypernatremia in adults: clinical perspectives. Korean J Intern Med. 2023;38(3):290–302. doi:10.3904/kjim.2022.346. https://pmc.ncbi.nlm.nih.gov/articles/PMC10175862/
3. Merck Manual Professional Edition. Water Deficit in Hypernatremia (medical calculator). https://www.merckmanuals.com/medical-calculators/WaterDeficit.htm