A 39-year-old previously healthy woman is brought to the emergency department after her ro… — USMLE Step 3

A 39-year-old previously healthy woman is brought to the emergency department after her roommates found her lethargic and confused. Empty bottles of an unknown liquid were found in her room. On arrival, she is difficult to rouse and responds poorly to stimuli. Vital signs show a temperature of $36.8^\circ\text{C}$ ( $98.2^\circ\text{F}$ ), blood pressure of $115/75\text{ mm Hg}$ , pulse of $110/\text{min}$ , and respiratory rate of $28/\text{min}$ with deep, rapid breaths. Laboratory studies report the following:

Parameter	Value	Reference Range
Sodium	$138\text{ mEq/L}$	$136-145\text{ mEq/L}$
Potassium	$4.8\text{ mEq/L}$	$3.5-5.0\text{ mEq/L}$
Chloride	$102\text{ mEq/L}$	$98-106\text{ mEq/L}$
Bicarbonate	$8\text{ mEq/L}$	$22-28\text{ mEq/L}$
Blood urea nitrogen (BUN)	$28\text{ mg/dL}$	$7-18\text{ mg/dL}$
Creatinine	$1.5\text{ mg/dL}$	$0.6-1.2\text{ mg/dL}$
Glucose	$108\text{ mg/dL}$	$70-110\text{ mg/dL}$

Additional laboratory analysis:
* Measured serum osmolality: $340\text{ mOsm/kg } H_2O$
* Arterial blood gas (room air) pH: $7.15$
* Arterial PaCO2: $20\text{ mm Hg}$

Based on these laboratory results, what is this patient's osmolar gap? (Express the result in $\text{mOsm/kg } H_2O$ .)

Answer: 48 mOsm/kg

Answer

The correct osmolar gap is

48\text{ mOsm/kg}

. To find this, first determine the calculated serum osmolality using the standard formula:

2 \times [\text{Na}^+] + \frac{\text{BUN}}{2.8} + \frac{\text{Glucose}}{18}

. Substituting the given values yields

2 \times 138 + \frac{28}{2.8} + \frac{108}{18} = 276 + 10 + 6 = 292\text{ mOsm/kg}

. The osmolar gap is then found by subtracting this calculated value from the measured serum osmolality provided by the laboratory:

340 - 292 = 48\text{ mOsm/kg}

. A gap greater than

10-15\text{ mOsm/kg}

, combined with an elevated anion gap metabolic acidosis, strongly suggests the presence of unmeasured low-molecular-weight substances, such as methanol or ethylene glycol (antifreeze).

Step-by-Step Solution

Identify the formula for calculated serum osmolality.

Calculated Osmolality =

2 \times [\text{Na}^+] + \frac{\text{BUN}}{2.8} + \frac{\text{Glucose}}{18}

This formula accounts for the major osmotically active solutes in the serum (sodium, urea, and glucose) to estimate the baseline osmolality.

Substitute the patient's sodium, BUN, and glucose values into the formula.

2 \times 138 + \frac{28}{2.8} + \frac{108}{18} = 276 + 10 + 6 = 292\text{ mOsm/kg}

Converting BUN and glucose from mg/dL to mOsm/L requires the specific division constants (2.8 for BUN, 18 for glucose) to determine their exact osmotic contribution.

Calculate the osmolar gap.

Measured osmolality (

340

) - Calculated osmolality (

292

) =

48\text{ mOsm/kg}

The osmolar gap measures the difference between the directly measured osmolality and the calculated baseline, revealing the presence of unmeasured osmotically active substances like toxic alcohols.

Key Concept

Calculation and interpretation of the osmolar gap in the evaluation of a patient with suspected toxic alcohol ingestion.

Estimated Time:1m 30s

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