Drug X: Dose, Vials and Cost
A 60 kg patient is prescribed drug X at 5 mg/kg per day, given in two divided doses. The formulary requires the total daily dose to be reduced by 25% because of the patient's renal function. Drug X is supplied in 5 mL vials containing 250 mg, and ward policy is that any vial opened must be discarded after 24 hours. A 10 mL vial containing 500 mg is also stocked, priced at $32.20; a 5 mL vial costs $18.40. The course is planned for 7 days, after which the patient's weight and renal function are to be reassessed. All doses are rounded to two decimal places only where stated.
1. Calculate the volume of each individual dose in mL, and the number of 5 mL vials required for the 7-day course.
Read the worked answer
Step 1 - daily dose before adjustment: 5 mg/kg × 60 kg = 300 mg/day.
Step 2 - apply the renal reduction: 300 × 0.75 = 225 mg/day.
Step 3 - per dose: 225 ÷ 2 = 112.5 mg per dose.
Step 4 - concentration: 250 mg ÷ 5 mL = 50 mg/mL.
Step 5 - dose volume: 112.5 ÷ 50 = 2.25 mL per dose.
Step 6 - course volume: 2.25 × 2 = 4.5 mL/day; 4.5 × 7 = 31.5 mL.
Step 7 - vials: 31.5 ÷ 5 = 6.3, so 7 vials on volume alone. Check against the 24-hour discard rule: each day needs 4.5 mL, which fits inside one 5 mL vial, so exactly one vial is opened per day - 7 vials. Both methods agree.
The common error here is applying the 25% reduction to the per-dose figure after dividing, or forgetting that the reduction applies to the total daily dose. Both give 225 mg/day, so they coincide; they would not if the reduction were applied to one dose only.
2. Using the 24-hour discard rule, compare the cost of the 7-day course using 5 mL vials versus 10 mL vials, and comment on which presentation is more economical per mg.
Read the worked answer
5 mL vials: daily requirement 4.5 mL ≤ 5 mL, so one vial per day → 7 × $18.40 = $128.80.
10 mL vials: daily requirement 4.5 mL ≤ 10 mL, and the remainder cannot be kept beyond 24 hours, so still one vial per day → 7 × $32.20 = $225.40.
Difference: $96.60 more expensive with the 10 mL vials.
Per-mg cost: 5 mL vial = $18.40 ÷ 250 mg = $0.0736/mg. 10 mL vial = $32.20 ÷ 500 mg = $0.0644/mg.
So the larger vial is cheaper per mg yet more expensive for this patient, because the discard rule destroys the unused 5.5 mL each day. The lesson: unit-price comparisons are only valid once the wastage constraint is applied. The 10 mL vial would only win if the daily volume exceeded 5 mL, or if vials could be shared between patients on the same day.
3. After 7 days the patient's weight is re-measured at 66 kg and renal function is unchanged. What is the new daily volume, and does the number of 5 mL vials needed per day change?
Read the worked answer
New daily dose: 5 × 66 = 330 mg/day; × 0.75 = 247.5 mg/day.
New daily volume: 247.5 ÷ 50 = 4.95 mL/day.
Per dose: 4.95 ÷ 2 = 2.475 mL per dose.
Vials per day: 4.95 mL still fits within a single 5 mL vial, so the number of vials opened per day is unchanged at 1 per day (7 vials per week), and the cost is unchanged at $128.80.
However, the margin is now only 0.05 mL. Any further weight gain pushes the daily volume above 5 mL, at which point a second 5 mL vial would have to be opened each day and the cost would double to $257.60. The correct answer therefore flags both the arithmetic and the fragility: the answer 'no change' is right for 66 kg but is on a knife-edge, and a clinician should look again if weight rises further.