Question

A 32-year-old woman with a history of type 1 diabetes mellitus is brought to the emergency department by her husband due to altered mental status and 3 days of severe nausea, recurrent vomiting, and generalized abdominal pain. She has been unable to retain any oral intake and missed her last three doses of insulin.

On physical examination, her temperature is 37.1°C (98.8°F), blood pressure is 98/60 mm Hg, pulse is 122/min, and respirations are 24/min and deep. Her mucous membranes are remarkably dry, and skin turgor is decreased. Her breath has a fruity odor.

Laboratory studies show:
Serum sodium: $140 \text{ mEq/L}$
Serum potassium: $4.1 \text{ mEq/L}$
Serum chloride: $92 \text{ mEq/L}$
Serum bicarbonate: $14 \text{ mEq/L}$
Serum glucose: $540 \text{ mg/dL}$
Serum BUN: $45 \text{ mg/dL}$
Serum creatinine: $1.9 \text{ mg/dL}$
Serum ketones: Positive

Arterial blood gas on room air:
pH: 7.28
$\text{PaCO}_2$: $29 \text{ mmHg}$
$\text{PaO}_2$: $98 \text{ mmHg}$

Assuming a normal reference serum anion gap of $12 \text{ mEq/L}$ and a normal reference serum bicarbonate of $24 \text{ mEq/L}$, what is the calculated delta ratio ($\Delta\text{AG} / \Delta\text{HCO}_3^-$) for this patient?