A 45-year-old man is brought to the emergency department after being rescued from a remote… — USMLE Step 3

A 45-year-old man is brought to the emergency department after being rescued from a remote hiking trail. He was lost for two days without access to food or water. On presentation, he is lethargic and confused. Vital signs show a temperature of $37.8^\circ\text{C}$ ( $100.0^\circ\text{F}$ ), pulse of $118/\text{min}$ , blood pressure of $92/58\text{ mmHg}$ , and respiratory rate of $16/\text{min}$ . Physical examination reveals dry mucous membranes, decreased skin turgor, and delayed capillary refill. His estimated premorbid weight is $70\text{ kg}$ .

Initial laboratory evaluation reveals the following:

Test	Result	Reference Range
Serum Sodium	$154\text{ mEq/L}$	$136-145\text{ mEq/L}$
Serum Potassium	$4.1\text{ mEq/L}$	$3.5-5.0\text{ mEq/L}$
Serum Chloride	$118\text{ mEq/L}$	$95-105\text{ mEq/L}$
Serum Bicarbonate	$25\text{ mEq/L}$	$22-28\text{ mEq/L}$
Blood Urea Nitrogen	$50\text{ mg/dL}$	$7-18\text{ mg/dL}$
Serum Creatinine	$1.9\text{ mg/dL}$	$0.6-1.2\text{ mg/dL}$
Serum Glucose	$98\text{ mg/dL}$	$70-110\text{ mg/dL}$

Based on the patient's presentation and laboratory results, what is his estimated free water deficit in liters? (Assume a target normal serum sodium of $140\text{ mEq/L}$ .)

Answer: 4.2 Liters

Answer

4.2

The patient is experiencing severe hypovolemic hypernatremia due to unreplaced sensible and insensible free water losses in the setting of environmental exposure. To properly direct intravenous fluid resuscitation (typically with

5\%

dextrose in water or enteral free water), the free water deficit must be calculated. The standard formula is:

\text{Free Water Deficit} = \text{TBW} \times \left( \frac{\text{Measured } Na^+}{\text{Target } Na^+} - 1 \right)

. First, calculate Total Body Water (TBW), which is approximately

60\%

(

0.6

) of lean body weight in a young or middle-aged adult male.

\text{TBW} = 0.6 \times 70\text{ kg} = 42\text{ L}

. Next, insert the measured sodium (

154\text{ mEq/L}

) and the target normal sodium (

140\text{ mEq/L}

) into the formula:

42 \times \left(\frac{154}{140} - 1\right) = 42 \times (1.1 - 1) = 42 \times 0.1 = 4.2\text{ Liters}

Step-by-Step Solution

Determine the patient's estimated Total Body Water (TBW).

For an adult male,

\text{TBW} = 0.6 \times \text{body weight (kg)}

. Therefore,

0.6 \times 70\text{ kg} = 42\text{ L}

Free water deficit calculations rely on the patient's total body water, which varies by sex and age (

0.6

for adult men,

0.5

for adult women and elderly men,

0.45

for elderly women).

Calculate the ratio of measured serum sodium to the target normal serum sodium.

\frac{154\text{ mEq/L}}{140\text{ mEq/L}} = 1.1

This ratio determines the proportional deficit of free water relative to the body's solute concentration.

Apply the free water deficit formula.

\text{Deficit} = \text{TBW} \times (1.1 - 1) = 42 \times 0.1 = 4.2\text{ Liters}

Multiplying the TBW by the excess sodium fraction yields the absolute volume of free water that has been lost.

Key Concept

Calculating free water deficit in hypernatremia

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