The Rate That Falls the Longer You Look

I.The First Known Thing — the rate that depends on the span

stratigraphy · sediment accumulation rates

Sediment settles and a record forms, and the obvious thing to ask of the record is how fast it formed. In 1981 Peter Sadler compiled nearly twenty-five thousand published rates of sediment accumulation, measured over spans running from hours to hundreds of millions of years, and plotted the rate against the span. The result was not a scatter around a typical value. It was a line on log-log paper, falling steadily across eleven orders of magnitude: the longer the interval over which a rate had been measured, the lower the rate came out, with no sign of a floor. Rates measured over a season and over a geological period were not estimates of one quantity taken with different precision. They were different quantities.

the explanation is the gaps

The explanation the field settled on is that deposition is intermittent. A bed is laid down in a flood or a storm and then nothing happens, sometimes for a very long time, until the next event; the stratigraphic record is mostly hiatus. A rate measured over a short span catches an interval in which something was happening and averages over few gaps. A rate measured over a long span necessarily incorporates the long gaps as well, and since long gaps are rarer than short ones but far longer, the longer the window, the larger the share of it that is silence. What the compiled data show is that this dilution never finishes. Schumer and Jerolmack made the mathematics explicit in 2009: if the quiet intervals between depositional events are drawn from a heavy-tailed distribution — one whose mean does not converge, so that there is always a longer gap not yet sampled — the observed rate falls as a power of the observation span, with a log-log slope of γ minus one, where γ is the tail index of the gap distribution. They fitted γ from the records themselves: about 0.17 for alluvial channels, 0.48 for floodplains, between 0.39 and 0.61 for carbonate platforms, 0.72 for the denudation of the Eastern Alps.

what the discipline did with it

Two things are worth holding on to. The first is that the field stopped treating the span dependence as a defect in its instruments. There is no true sedimentation rate that better sampling would reveal; there is a scaling law, and the exponent of the scaling law is a property of the environment — a channel is not a platform — recoverable from rates that were never intended to be compared. The second is that the exponent carries information a single rate cannot. It tells you how the gaps are distributed, which is to say how incomplete the record is at every scale, and it does so from the very thing that had looked like inconsistency in the literature. The Sadler effect has since been found wherever an intermittent process is averaged over a window: river incision, hillslope erosion, the fire record in lake sediments. Its lesson is general and has been drawn only in the earth sciences: when a rate keeps falling as you widen the window, measure the falling.

II.The Second Known Thing — the half-life that grows with the assay

pharmacokinetics · ADME and the terminal phase

Walk to a hall with no rock in it. A drug is given, its concentration in plasma is measured at intervals, and the last stretch of the falling curve is fitted to a straight line on semi-log paper. The slope of that line is the terminal elimination rate, and its reciprocal, dressed up, is the terminal half-life — the number that ends up in the label, sets the dosing interval, and predicts how long the drug takes to accumulate to steady state and to wash out again. It is, by the field's own account, the parameter most sensitive to the analytical method that produced it.

the phenomenon the field already knows

The reason is that the falling curve is not one straight line but a succession of them, each shallower than the last, and how many of them you see depends on how long you sample and how low your assay can read. Drug leaves the plasma quickly into tissue and returns from it slowly; the deepest reservoirs — kidney cortex for the aminoglycosides, fat and muscle for amiodarone, bone for the bisphosphonates, melanin, lysosomes — give the drug back over days or weeks at concentrations an ordinary assay cannot see. When the assay improves or the sampling is extended, a new and shallower phase appears, and the reported half-life lengthens. The review literature records the cases. Dexamethasone in the horse had a half-life of fifty-three minutes with a limit of quantification of a few nanograms per millilitre, and one of a hundred and fifty-eight minutes when the limit fell to two hundred picograms, with the clearance unchanged. Gentamicin had a half-life of about two hours until more sensitive assays found a terminal phase of a day or more, reflecting the renal accumulation that produces its toxicity. A ten-fold drop in the quantification limit can lengthen a terminal half-life ten-fold.

what the discipline does with it

It treats this as an artefact to be controlled. The textbook rule is to sample for at least three expected half-lives so that the slope is estimated accurately; the literature warns that a half-life reported from a short study is misleading; the remedy for a half-life that grew is to note which phase is pharmacologically relevant and report an effective half-life alongside the terminal one. The assumption underneath every one of these practices is that the curve eventually settles — that there is a last, slowest compartment, and that once the assay reaches it the half-life stops growing and the true terminal value has been found. The field has never compiled the reported half-lives of one drug against the span over which each was measured to ask whether the growth stops. Within the hall there is no reason to; a half-life is a property of a drug, and a value that depends on the study is a study that was done badly.

III.The Bridge

no pharmacokinetics paper cites a stratigraphic section

The halls share nothing. Stratigraphers do not read clinical pharmacology, the authors of half-life guidance do not read the sedimentary record, and no review spans the two. But both are estimating a rate from a process that is quiet for most of its duration, and both have found the same thing: the rate they report depends on the span over which they observed it, and the dependence does not go away.

X: a rate measured over a span averages in the dormancy the span contains

Call it hiatus incorporation. Any rate estimated over a window is an average over the window, and where the process alternates between activity and dormancy, the average includes every dormant interval the window happens to contain. Widen the window and it contains longer ones. If the dormant intervals have a characteristic length, the average settles once the window is a few times that length, and beyond it a longer window merely improves the precision — which is exactly what the three-half-lives rule assumes. But if the dormant intervals are heavy-tailed, so that intervals longer than any yet seen remain to be found at every scale, the average never settles; the rate falls as a power of the span, and the exponent of that fall is fixed by the tail of the dormancy distribution and by nothing about the instrument. From which follows the half that pharmacokinetics has not ported, because only stratigraphy has had to live with rates spanning eleven orders of magnitude: the span dependence of a rate is a measurement, and its exponent is a property of the process that no single rate, however carefully taken, contains. A molecule that returns from tissue with a broad distribution of residence times — bound, sequestered, redistributed, released — is a depositional system read backwards: the plasma is the section, the returns from tissue are the events, and the intervals in which the molecule sits in bone or in a lysosome are the hiatuses. Each new assay that lengthens a half-life is a longer core.

IV.The Question That Becomes Possible

neither discipline would ask this alone

Pharmacokinetics has the data and stratigraphy has the plot. For a drug that has been studied many times, the literature already holds reported terminal half-lives together with the last quantifiable sampling time, or the assay limit, of each study that produced them. These have never been plotted against one another because the field reads their disagreement as error. Sadler's move is to plot the disagreement and see whether it has a slope.

the scaling hypothesis

Does a drug's apparent terminal half-life scale with the span of observation as a power law whose exponent is stable across studies, assays and laboratories; and where it does, does that exponent predict accumulation on repeated dosing, and the time course of washout, better than any single reported half-life?

why it would matter

Because the two possibilities have different consequences and the field's practice presumes one of them. If the half-life saturates — if the curve of reported values against span flattens once sampling exceeds a few half-lives — then the compartmental picture is right, there is a last compartment, and the three-half-lives rule is sound. If instead the curve keeps rising as a straight line on log-log paper, then there is no last compartment to reach; the terminal half-life is a property of the study design in the same way a sedimentation rate is a property of its span, and every label value is a rate at one window. What replaces it is a pair: a scaling exponent, characterising how deep the reservoirs go, and a reference span at which it was read. The exponent would sort drugs by something no current parameter measures — not how large the tissue reservoir is, which the volume of distribution already gives, but how broadly its residence times are spread. It would also say, before a chronic dosing study is run, which compounds will still be accumulating at a year, and which will be found in tissue long after the plasma reads zero.

V.How It Could Be Proven — or Broken

a candidate must be breakable, or it is merely poetry

An intercognix candidate earns nothing until it names its own test. This one names two, and the first is a precondition for the second.

the honest status

Note the narrowness. The claim concerns small-molecule drugs with plasma concentration data from studies of differing duration and sensitivity; it says nothing about biologics, whose elimination is a different process, and nothing about drugs cleared cleanly from a single compartment, where the claim is that the plot flattens. Nor does it claim that the existing half-life is wrong for what it is used for — a dosing interval set from the phase that carries the effect is a reasonable thing — only that the number the field reports as a constant is a rate at one span, and that the spans have never been compared. And the scaling law itself is an assertion borrowed from rock: heavy-tailed hiatuses are well established in the sedimentary record and merely plausible in tissue, and the plot may turn out to be perfectly flat. Until it is drawn this remains exactly what its stamp says: a candidate. A structural echo between two halls of the library, waiting for someone with the tools of both to walk the corridor.