Multi-vial stack calculator

A stack is a list of independent reconstitutions, not one larger reconstitution. Each row carries its own mass, its own diluent volume and its own target amount, and each is solved with the same three divisions using only its own numbers. No row can influence any other, which is a structural guarantee worth relying on when a table gets long: an error in the fourth row cannot corrupt the second, and a mistyped mass in one line stays contained in that line. That containment is a property of the arithmetic rather than of the table, and it holds however many rows there are and whatever units each row happens to use.

The corollary is that the only cross-row failure mode is human. Someone reads the wrong line for the vial in their hand, or carries a concentration from one row into the next instead of recalculating it, and the calculator cannot detect either, because from its point of view the table was correct when it was produced. That is why the rows are worth comparing rather than trusting. If two rows have mass and volume in the same proportion their concentrations are identical, and where they do not, the difference is exactly the ratio of the two ratios. The BAC-water calculator is the tool for choosing volumes that make rows comparable in the first place.

Some columns in a stack add up and some do not, and the distinction is easy to lose. Total mass adds, because it is a sum of masses. Concentrations do not add: they are mass per volume, and a sum of concentrations is not the concentration of anything. Draw volumes do not add either, since each is the volume of one draw rather than a volume in a shared container. The single cross-row figure this page reports is the total mass, and it is reported as a sum deliberately. Anything else on this page is per row and stays per row.

The tempting shortcut is to take the mean concentration across the rows and treat it as though it described the stack. It is correct only in a special case, and the case is easy to recognise: when every row was reconstituted to the same volume. Then the mass-weighted mean equals the plain mean, because the weights are equal, and the plain mean happens to be right. Change the volume in one row and the plain mean is wrong while still looking entirely reasonable, because a mean of concentrations is a quantity that never had physical meaning in the first place. The concentration after a volume change calculator does the weighted version properly.

Cost per vial is what was paid for that vial. It does not move when the mass inside it moves, when the diluent volume changes, or when the number of vials in the stack changes. Cost per unit of substance is the price divided by the mass, so it falls as the mass rises, exactly and proportionally, with the price held constant. Those two figures are the same information arranged differently, and the whole of the comparison problem lives in the gap between them: a larger vial is not a cheaper vial, it is a cheaper amount of substance, and only one of those usually appears on an invoice.

A stack is therefore a way of changing the amount of substance obtained per purchase while leaving the cost of any single purchase exactly where it was. It is not a way of reducing what a vial costs, and a stack whose rows were chosen for their price rather than their arithmetic will cost the same per vial as a lone vial and may well cost more per milligram. The cost per milligram calculator computes the unit price on its own, the guide to cost per dose explains why the operational figure differs from it, and the batch cost calculator is where the consumables belong. This page takes no price as an input and this paragraph quotes none, because a reference page that asserted a market rate would be asserting something it cannot check.

If the mass and the diluent volume of two rows stand in the same proportion, their concentrations are equal. With the same target amount their draw volumes are equal, and their barrel readings are equal too. A stack of proportional rows is therefore one row with a multiplier attached, and reading it as a table of independent results is wasted effort. That is useful as a check in its own right: a row that surprises you against the others is a row whose ratio of mass to volume differs, and the size of that difference tells you by how much.

The same reasoning applies inside a single row. The unit reading is the draw volume multiplied by the markings per millilitre of the syringe, so if two rows share a syringe and a target amount, identical concentrations mean identical readings and any difference in the reading is a difference in the concentration. None of this depends on how the stack was assembled or on anything the rows are for; it is the same division repeated. Where a row involves figures small enough for rounding to matter, the guide to rounding and small volumes shows how far that goes, and the inventory days calculator turns a total mass into a calendar figure once the arithmetic is settled.