A 54-year-old woman is admitted to the intensive care unit for a severe exacerbation of ch… — USMLE Step 3

A 54-year-old woman is admitted to the intensive care unit for a severe exacerbation of chronic obstructive pulmonary disease. She is started on a continuous intravenous infusion of theophylline. After 48 hours, her plasma theophylline concentration reaches a steady state of $8 \text{ mg/L}$ . To aggressively optimize bronchodilation, the intensivist triples the infusion rate. Simultaneous with this rate increase, the patient is diagnosed with a suspected multidrug-resistant infection, and rifampin, a potent cytochrome P450 inducer, is added to her regimen. Pharmacokinetic analysis indicates that rifampin therapy will precisely double the hepatic clearance of theophylline without altering its volume of distribution.

If the new, tripled theophylline infusion rate is maintained, which of the following accurately describes the patient's new theophylline steady-state concentration and the time required to reach it (compared to the original time to reach steady state)?

A
The steady-state concentration will decrease to $4 \text{ mg/L}$ , and the time required to reach it will be longer.
B
The steady-state concentration will increase to $12 \text{ mg/L}$ , and the time required to reach it will be unchanged.
The steady-state concentration will increase to $12 \text{ mg/L}$ , and the time required to reach it will be shorter.Answer
D
The steady-state concentration will increase to $24 \text{ mg/L}$ , and the time required to reach it will be shorter.
E
The steady-state concentration will increase to $24 \text{ mg/L}$ , and the time required to reach it will be unchanged.

Answer

The steady-state concentration will increase to

12 \text{ mg/L}

, and the time required to reach it will be shorter.

Steady-state concentration (

C_{ss}

) during a continuous intravenous infusion is calculated using the formula

C_{ss} = \frac{R_0}{Cl}

, where

R_0

is the infusion rate and

Cl

is the clearance. Initially, the patient is at a steady state of

8 \text{ mg/L}

. When the physician triples the infusion rate (making it

3R_0

) and rifampin concurrently doubles the clearance (making it

2Cl

), the new steady state becomes

\frac{3R_0}{2Cl}

, which equals

1.5 \times \frac{R_0}{Cl}

. Therefore, the new steady-state concentration will be

1.5 \times 8 \text{ mg/L} = 12 \text{ mg/L}

. Furthermore, the time required to reach a new steady state is independent of the infusion rate and depends entirely on the drug's half-life (

t_{1/2}

). Half-life is calculated as

t_{1/2} = \frac{0.693 \times V_d}{Cl}

. Because the volume of distribution (

V_d

) is unchanged and clearance is doubled, the half-life is reduced by half. Consequently, the time to reach the new steady state will be proportionally shorter.

Step-by-Step Solution

Calculate the effect of the interventions on the steady-state concentration.

The new steady-state concentration becomes

12 \text{ mg/L}

Steady-state concentration (

C_{ss}

) is proportional to the infusion rate (

R_0

) and inversely proportional to clearance (

Cl

). With a tripled

R_0

and doubled

Cl

, the new concentration is

1.5

times the original (

1.5 \times 8 \text{ mg/L} = 12 \text{ mg/L}

Determine the effect of the interventions on the drug's half-life.

The half-life (

t_{1/2}

) is halved.

Half-life is inversely proportional to clearance (

t_{1/2} = \frac{0.693 \times V_d}{Cl}

). Doubling the clearance while volume of distribution (

V_d

) is unchanged cuts the half-life in half.

Evaluate how the change in half-life affects the time to reach steady state.

The time to reach steady state is shorter.

The time to reach steady state depends solely on the half-life (typically taking 4 to 5 half-lives). A shorter half-life means steady state is achieved faster.

Key Concept

During a continuous infusion, steady-state concentration is determined by the ratio of infusion rate to clearance, whereas the time to reach steady state is determined solely by the drug's half-life (which is inversely related to clearance).

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