Peptide basics

How peptide reconstitution maths works

This page covers the arithmetic and nothing else. It does not describe a procedure, a volume of liquid to add, an amount to prepare, or a schedule. Each of those is decided elsewhere, by a label or by a professional who can inspect the exact materials. What follows is only the shape of the calculation those decisions get fed into.

Concentration is a division

Write A for the amount of drug stated on the vial label, and V for the volume of liquid, in millilitres, specified by whichever instruction the reader is following. The concentration C is A divided by V. That is the whole operation. Everything else is either a unit conversion feeding into that division or a second division coming out of it.

The second division is the draw. Write T for the target amount, again a value supplied by an authoritative instruction rather than produced here. The volume D that contains T is T divided by C. Two divisions, in a fixed order, with nothing in between.

The units travel with the numbers and are part of the answer. If A carries an amount unit and V carries millilitres, then C carries amount-per-millilitre, and dividing an amount by that concentration returns millilitres. The SI defines these units and their relationships precisely, which is what makes the cancellation reliable.¹

Notice what is absent. Nothing in A = C × V tells you what V should be, or what T should be. The equation relates numbers you already have; it cannot manufacture one that is missing.

Milligrams, micrograms, and the metric prefix

The prefixes are defined multipliers, not conventions that vary by context. Milli- means a factor of one thousandth and micro- a factor of one millionth, so one milligram equals one thousand micrograms. Going from mg to mcg multiplies by 1000; going the other way divides by 1000.¹

Convert before you divide

The order matters. A and T frequently arrive in different units — one printed on a vial in milligrams, the other written in an instruction in micrograms. Both must be expressed in the same unit before either division is performed. Converting afterwards, on the result, is the standard way this calculation goes wrong by a factor of a thousand, and a factor of a thousand does not look obviously wrong on a page.

As pure arithmetic with arbitrary placeholder digits, attached to no product: an A of 4 mg is 4000 mcg; divided by a V of 2 mL that gives a C of 2000 mcg/mL, and a T of 500 mcg divided by that C gives a D of 0.25 mL. The digits mean nothing outside this paragraph. What carries over is the sequence — convert, divide, divide.

The abbreviations are the hazard

The Institute for Safe Medication Practices maintains a list of abbreviations, symbols, and dose designations associated with reported medication errors. The Greek symbol for micro appears on it, with mcg given as the safer written form, because a handwritten or poorly rendered µ has been misread as m. The standalone letter U for unit appears as well, having been misread as a digit. These are documented failure modes, recorded because they produced harm.²

The practical consequence for a home calculation is narrow: transcribe the unit in full, in the form the source printed it, and treat any character you had to guess at as a value you do not have.²

A label states an amount, not a volume

A millilitre is a unit of volume — how much space a liquid occupies. It says nothing about how much drug that liquid contains. This distinction collapses easily in reading, and when it collapses the arithmetic that follows is structurally wrong rather than slightly off.¹

A vial label states an amount of drug, or a concentration expressed as an amount per unit volume. Approved prescribing information for injectable products follows this convention: strength is given as mass, or as mass per millilitre. It is not a volume to withdraw, and it is not an instruction about how much liquid to add. Those are separate statements, in separate places, or absent entirely.³

So A is an amount and D is a volume — different kinds of quantity, connected only by C. If you find yourself using a figure from a label directly as a volume, the units have stopped cancelling and the calculation has already failed.

The U-100 scale is a concentration convention

A U-100 syringe is graduated in marks labelled "units", which is misleading if read generally. On that scale, one mark is defined as one hundredth of a millilitre — 0.01 mL — because the scale was built around insulin preparations supplied at 100 units per millilitre. Labels for such products state that concentration explicitly.⁴

That is the entire content of the word "unit" here: a graduation on a barrel, defined by a concentration convention for a different drug class. It is not a universal measure of drug quantity, and a number of units converts to a volume only through the scale it was read on.⁴

Taking a number read on a U-100 barrel and applying it to a syringe with a different scale is a category error, not an approximation. The same mark position means a different volume on a different barrel — and U itself is a documented error-prone abbreviation, so the notation gives no protection against the mistake.²⁴

Rounding, precision, and what a device can represent

Rounding is a display decision and it belongs at the end. An intermediate value — C, most often — should be carried at full precision into the next step. Rounding C and then dividing by the rounded figure propagates the error into D, and the propagated error is larger than the rounding that caused it.

Calculation precision and display precision are two separate settings that look like one. Keep the unrounded result available beside whatever is shown, so the shown figure can be checked against the value it came from rather than replacing it.

• Carry intermediate values unrounded; round once, at the point of display.
• State the precision being displayed rather than letting a truncated figure imply exactness.
• Keep the unrounded millilitre value beside any graduation or mark, so the two can be compared.
• Preserve the units when copying a result anywhere else; a bare number is not a result.

There is a further constraint arithmetic will not surface. A computed volume can fall between two printed graduations. A position between marks is not a value the device can represent, and estimating it by eye converts a calculated number into a guess while leaving it looking calculated.

The round-trip check

There is one check worth doing every time, and it costs a single multiplication. Having computed D as T divided by C, recompute the amount by multiplying D by C. The result should be T — the same T you started with, in the same unit.

It works because the two operations are inverses, so a discrepancy has to come from outside the arithmetic: a unit that changed between steps, a value transcribed differently the second time, an intermediate figure rounded too early. Those errors are invisible in a single forward calculation, because a wrong answer computed correctly looks exactly like a right one.

Be clear about what the round trip certifies. It confirms internal consistency among the numbers as entered. If A was misread off the label, or T came from a source that does not apply, the round trip closes perfectly on the wrong values. Consistency is a floor, not a verification.

What the arithmetic cannot establish

The calculation operates on numbers, so its reach ends where numbers end. Everything that determines whether a preparation is fit to use is a physical or clinical property, and none of it is visible to a division.

• Identity — whether the contents are the substance the label names.
• Purity — what else is present, and in what quantity.
• Sterility — whether the preparation is free of microbial contamination.
• Stability — whether the substance remains intact after reconstitution, and for how long.
• Compatibility — whether the substance and the diluent behave acceptably together.
• Appropriateness — whether any of this suits a particular person, which is a clinical judgement, not a computation.⁵

Sterile compounding is a regulated discipline for exactly this reason. Published standards for sterile preparations address environment, personnel training, technique, testing, and beyond-use dating — an apparatus built because correct arithmetic establishes none of those things and never could.⁵

The arithmetic here is genuinely simple, and that is the part worth being careful about. Simplicity makes it feel finished. The division is finished; whether the numbers going into it were the right numbers is a separate question, answered by labels, pharmacists, and clinicians rather than by a calculation.