The reconstitution identity

Four steps, one of which almost every calculator in this category gets wrong in the same way.

Educational arithmetic only. Not medical advice and not a dosing protocol.

The whole thing in one line

For a vial of mass m dissolved to a finished volume V, measured into a barrel calibrated at k units per millilitre, the reading for an amount d is:

units = d × V / (m × k-1), or more usefully in the order it is worked: c = m / V, then v = d / c, then units = v × k.

With everything in consistent units, a 5 mg vial in 2 mL measured for 250 mcg on a U-100 barrel: c = 5 mg / 2 mL = 2.5 mg/mL = 2,500 mcg/mL; v = 250 / 2,500 = 0.1 mL; units = 0.1 × 100 = 10.

Step one is where the error lives: V is not the diluent

A lyophilised solid occupies space. Once dissolved, the finished solution is the diluent plus the volume the solid itself contributes:

V = diluent + displacement

Concentration is mass over the FINISHED volume. Computing mass over diluent gives a figure that is too high, because the denominator is too small. Nearly every calculator on the open web does exactly that and presents the result as a measurement.

The displacement for a few milligrams of peptide is small, and often genuinely negligible against the error of reading a 1 mL barrel. That is an argument for not worrying much about it. It is not an argument for pretending it is zero, and the difference between those two positions is the difference between a tool that tells a reader what it knows and one that quietly substitutes a convenient default.

So what can honestly be said when the displacement is unknown

Since V ≥ diluent, it follows that m / V ≤ m / diluent. The figure from mass over diluent is therefore the highest the concentration can be: an upper bound, true for every possible displacement, and the strongest true statement the inputs support.

That bound also points the useful way. A volume derived from an upper-bound concentration is the smallest volume that could hold the amount wanted, so a reader drawing it has not drawn more than intended. Getting the direction of an uncertainty right is more useful than hiding that there was one.

Measuring the finished volume removes the question entirely. Mark the barrel after mixing, enter that figure, and the concentration is a measurement rather than a bound. The main calculator has a field for it and switches the label when it is used.

Why this site holds exact fractions rather than decimals

The identity chains four user-entered quantities through a divide, a divide and a multiply. In ordinary floating-point arithmetic 0.1 + 0.2 does not equal 0.3, and that error compounds through a chain. It is small, but it lands in the third or fourth decimal place of a syringe reading, which is the digit a reader acts on.

So every quantity here is held as an exact fraction of whole numbers, and rounding happens once, when the answer is printed. 5 mg in 3 mL is held as exactly five thirds of a milligram per millilitre, not as 1.6666666666666667.

Related

• The calculator that applies this
• Concentration chart

  The first step, pre-computed for common vials.

• Graduations and readability

  What the last step can and cannot show.