How many vials from a BCH lot should I send for identity confirmation by mass spectrometry?

Setup, so nobody has to ask: BCH · identity confirmation by mass spectrometry.

I have worked this out and I would like someone to find the error, because I suspect there is one.

My working so far, for the record, is below, and I am fairly sure the error is in the unit conversion rather than the algebra.

How many significant figures are actually justified here?

A certificate that reports one test result on one vial extrapolates to claim that all two hundred vials in the lot are identical, which is an assumption worth questioning.

Under AQL sampling plans, testing two vials from a fifty-vial lot gives you an operating characteristic curve that tells you what risks you are accepting.

More usefully, the statistical foundation here is well-established, which is why sampling plans from decades ago are still valid.

Sampling plans for pharmaceutical manufacturing are defined in ISO 2859 and ANSI Z1.4, and they are based on statistical sampling theory.

The caveat is that sampling is a trade-off between cost and confidence, and neither test nor assumption is cost-free.

Assume segregation is possible, and design your sampling to catch it if it exists.

The honest statement is that unless you have tested multiple vials or have segregation data, you are making an assumption about lot homogeneity that may not hold.

If the entire lot failed qualification, a retest on a different vial is sometimes done, but reporting a retest result under the same lot number is misleading.

Concretely, the sample size determination requires choosing a confidence level and an acceptable error rate, and the smaller the error rate you want, the larger your sample must be.

The practical summary: a lot number without a sampling statement is a lot number without meaning.

If a lot has visibly segregated — some vials showing different appearance — then sampling the top and bottom of the shipment is worth doing.

A statement that "lot 20260412 complies with specifications" is meaningless without stating which vials from the lot were tested and how many there were.

Lyophilised peptide homogeneity studies show that vial-to-vial variation is usually small but occasionally large, depending on the distribution in the freeze-dryer.

If testing multiple vials, state how many you tested and why you chose those vials.

Thermal history during shipping is different for every vial, so a lot that experienced thermal abuse may have internal variation even if it was originally homogeneous.

Published segregation failures show that even modern automated processes sometimes produce lots with measurable vial-to-vial variation.

One qualification: testing more vials gives better confidence, but at some point the cost outweighs the benefit.

Assume segregation is possible, and design your sampling to catch it if it exists.