Mean kinetic temperature is not the average temperature

The most misused statistic in cold-chain discussion is the arithmetic mean of a temperature trace. Chemical degradation rates increase roughly exponentially with temperature, which means that time spent at the high end of a profile contributes disproportionately to the total change. An arithmetic average adds every reading with equal weight and therefore systematically understates the damage done by a short excursion. Mean kinetic temperature is the correction: a single temperature which, held constant over the same period, would produce the same total degradation as the varying profile actually did. It is never lower than the arithmetic mean and is often several degrees higher.

The storage conditions a real study uses

The harmonised guideline defines the conditions under which stability data must be generated for registration, and they are worth knowing because they are the vocabulary any serious stability claim will use. For a product intended for storage at room temperature, long-term testing runs at twenty-five degrees and sixty per cent relative humidity, or thirty degrees and sixty-five per cent in hotter climatic zones, for at least twelve months. Accelerated testing runs at forty degrees and seventy-five per cent humidity for six months.

For a product intended for refrigerated storage, long-term testing runs at five degrees plus or minus three, and the accelerated condition becomes twenty-five degrees at sixty per cent humidity. Significant change at the accelerated condition triggers testing at an intermediate condition. A product intended for frozen storage is tested long-term at minus twenty, and because accelerated testing is not meaningful there, the guidance instead requires a single-batch study of the effect of a short excursion above the intended condition.

That last provision is the interesting one for this trade, because a frozen-storage product with no excursion data has no basis for any statement about what a warm afternoon in transit did to it. Biotechnological products have their own parallel guidance, which additionally requires that the analytical methods be capable of detecting the degradation products characteristic of the molecule class.¹

Why accelerated data extrapolates badly for peptides

The temptation with any stability programme is to run the accelerated condition, fit an Arrhenius relationship to the rate constants, and extrapolate to the intended storage temperature. For a single reaction with a temperature-independent mechanism that is sound. For peptides it frequently is not, and the reason is that different pathways have different activation energies.

Suppose a peptide degrades at five degrees principally by deamidation and at forty degrees principally by hydrolysis, with the second having a higher activation energy. Measuring total degradation at forty degrees measures mostly hydrolysis; extrapolating that rate down to five degrees predicts almost nothing about the deamidation that will actually dominate. Aggregation is worse still, because it is frequently nucleated by interfaces and mechanical stress rather than by thermal energy alone, and does not obey a simple temperature relationship at all.

The practical rule the Journal applies when reading a stability claim is to ask what condition the data was generated at and whether the degradation products were identified as well as quantified. Accelerated data that shows which products form is genuinely useful as a warning of what to watch for. Accelerated data reduced to a single percentage and extrapolated to a shelf life is a projection dressed as a measurement, and for this class of molecule it is a poor projection.

Mean kinetic temperature, worked

Mean kinetic temperature is defined through the Arrhenius relationship: it is the temperature which, held constant, would produce the same degradation as the actual varying profile. Computing it requires converting each temperature to a rate through an exponential, averaging the rates, and converting back. The activation energy conventionally assumed for the purpose is around eighty-three kilojoules per mole, which is a compromise value rather than a property of any particular molecule.

The consequence is easiest to see in an example. A profile spending ninety hours at four degrees and six hours at thirty-eight degrees has an arithmetic mean of about six degrees. Its mean kinetic temperature, computed with the conventional activation energy, comes out several degrees higher, because the six hot hours contribute far more to the exponentially weighted average than their share of the elapsed time. Reporting the arithmetic mean of such a profile is not merely imprecise; it discards exactly the information the calculation exists to capture.

This is why the compendial definition of controlled room temperature is expressed both as a range and as a mean kinetic temperature not exceeding twenty-five degrees, and why any shipment assessment that quotes only an average should be read with suspicion. The original derivation of the working formula is now more than fifty years old and remains the basis of the calculation in current use.²³

What a logger records, and what it cannot

A single-use electronic logger is a thermistor, a clock, a small memory and a battery, and its output is a series of readings of the air adjacent to its housing at a configured interval. Four configuration choices determine what it can detect. The sampling interval sets the shortest excursion it can see: a device sampling every fifteen minutes will record a ten-minute spike only by luck. The memory depth sets the maximum journey length before it stops or overwrites. The start delay determines whether the packing operation appears in the trace. And the calibration interval determines whether the numbers mean anything at all.

Placement matters more than any of them. A logger in the outer carton records the transport environment; a logger inside the insulated payload against the vials records something much closer to what the product experienced; and the two can differ by twenty degrees during the same afternoon. A trace without a stated placement is close to uninterpretable, and placement is almost never stated.

Our own nine shipments used devices calibrated within the preceding twelve months, sampling at five-minute intervals, placed inside the insulated payload adjacent to the vials, started at the point of packing where we controlled it and at the point of receipt of the outbound label where we did not. We state all of that because a trace is a measurement and a measurement without its method is decoration.⁴

Nine parcels, instrumented

We ordered nine parcels of lyophilised material at catalogue prices, as ordinary customers, from suppliers we did not identify ourselves to, and arranged for a calibrated logger to travel inside the insulated payload of each. Eight arrived and produced complete traces. The ninth was held at a border for eleven days, exhausted its memory on day five, and arrived with a partial record and a cake that had visibly shrunk.

The headline results are in the table below. Transit times ran from two to nine days against quoted estimates of three to five. Five of the eight complete traces exceeded twenty-five degrees at some point; three exceeded thirty; the maximum recorded was thirty-eight degrees for three hours and forty minutes on a road leg. Mean kinetic temperature exceeded the arithmetic mean in every case, by between one and six degrees, with the largest gap on the trace with the shortest and hottest excursion — which is precisely the pattern the calculation is designed to expose.

Two of the eight arrived with their coolant packs still partly frozen, both on two-day domestic journeys. On the six longer journeys the coolant had fully melted and equilibrated, in most cases before the halfway point of the trip. We regard that as the central finding: for anything beyond about forty-eight hours, the gel pack is a delay rather than a control, and the shipment is relying on the material tolerating ambient temperature.

The chemistry, at least, is not in dispute. Deamidation, oxidation, aggregation and hydrolysis are as well characterised as anything in pharmaceutical science, they are predictable from a sequence, and they proceed at rates set by water, temperature and time. What this trade lacks is not the science but the four or five numbers that would connect it to the vial on the shelf.