How many vials from an FGP lot should I send for identity confirmation by mass spectrometry?

Setup, so nobody has to ask: FGP · 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.

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

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.

If the lot was manufactured in multiple batches, testing vials from each batch separately establishes whether batch-to-batch variation is acceptable.

Stratified sampling — testing one vial from the top, one from the middle, and one from the bottom of a shipment — is cheap insurance against segregation.

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

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

edited 8 Sept 2024 by Dr_Colm_Fitzhenry — clarified the distinction between purity and content

In practice, start from the question: how many vials from this lot do I need to test to claim that the lot meets specification, and the answer depends on both the lot size and the acceptable risk.

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

It helps to be literal here: for a quantitative result like content, the acceptable range determines how many vials you need to test to establish the lot complies.

The limitation is that you cannot know for certain without testing every vial, and you almost never can afford to do that.

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

Mechanically, the practical consequence is that spot-testing one vial from a new supplier is better than assuming they are all the same.

If you have reason to suspect inhomogeneity — different appearance in different vials, or a long or warm shipment — testing more vials is the diagnostic move.

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

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.

To be exact about it, the single most misleading statement on a research-grade certificate is a lot number with no statement of how many vials from that lot were tested.

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

Published data on lot homogeneity from manufacturers who sample multiple vials consistently find variation below the published specifications, suggesting the sampling plans work.

I would treat a "complies with" statement without sampling details as a claim rather than as evidence.

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