How many vial-days does a 2.5 mg vial give at 3.33 mg/mL on a weekly schedule?

The case in front of me: 2.5 mg · 3.33 mg/mL.

I want the working, not the result — I need to be able to redo it with different numbers.

I care about the precision as well as the value — I want to know how many figures are real.

Where is my error, and what is the correct working?

0.75 mL of solution, and the rest depends on your dose. A 2.5 mg vial reconstituted to 3.33 mg/mL occupies 2.5 ÷ 3.33 = 0.75 mL. At a 1 mg weekly dose that is 2 weeks; at 2.4 mg weekly it is 1 weeks — and both of those assume the vial contains its label claim, which is the assumption a content assay exists to test. Subtract one draw's dead space per dose: a few microlitres on a fixed-needle syringe, up to a hundred on a luer one.

On the detail: the single most useful thing to do is write the arithmetic on the vial label, because you will reconstruct it from memory at an inconvenient moment if you do not.

Rotation of injection site is a tolerability measure, not a pharmacokinetic one, but if you are going to do it you might as well do it right.

On filtration: a 0.22 µm syringe filter will remove particulates and organisms, and it will also adsorb a fraction of your peptide onto the membrane — with a low-binding PVDF or PES membrane the loss is typically a few per cent.

Published data on syringe dead space quantifies low-dead-space designs as retaining under 2 µL against 35 µL or more for conventional detachable-needle syringes.

One qualification: if your arithmetic and someone else's disagree by a factor of ten, one of you has made a unit error, and writing out the units at every step is the diagnostic.

If in doubt, use more diluent and accept the shorter usable window.

The distinction that resolves most of these questions is understanding what concentration actually means and why it is not the same as label claim.

Number of stopper piercings matters less than the gauge doing the piercing. A 30G or 31G needle through a butyl stopper leaves a track that reseals; a 21G or 18G drawing needle punches a core and can drop it into the solution.

Do not use the same needle to pierce the stopper and to administer. The tip is blunted by the stopper, and the hub now contains a dose you are about to lose to dead space anyway.

The insulin-unit standard U-100 means 100 units per millilitre, so one unit is 0.01 mL — this is the conversion that trips up more people here than any other single piece of arithmetic.

The limitation is that technique reduces risk, it does not remove it, and nothing you can do outside a controlled environment makes a non-sterile preparation sterile.

Write the arithmetic on the vial label. It costs nothing and removes the step where you reconstruct it from memory.