The Ore the Bottleneck Decides

I.The First Known Thing — the grade at which rock becomes ore

mining engineering · the economic definition of ore

A deposit is not a body of ore. It is a body of rock in which a metal is distributed at every concentration from nothing to rich, and the mine draws a line through that distribution: rock above the line is ore and goes to the mill, rock below it is waste and goes to the dump. The line is the cut-off grade, and the first thing to know about it is that it is not a geological quantity. Kenneth Lane, working for the Rio Tinto group in the late 1960s and publishing the theory in full in 1988 under the title The Economic Definition of Ore, showed that the cut-off is a decision variable, that it has an optimum, and that the optimum is set by the shape of the operation rather than by the shape of the deposit. The deposit contributes only its grade–tonnage curve: how much rock there is at each grade. Everything else that fixes the line belongs to the mine.

three stages, three capacities, three cut-offs

Lane's mine has three stages in series, each with a fixed capacity. The mine itself can extract so many tonnes of rock a year. The treatment plant can process so many tonnes of ore. The market, or the refinery, can absorb so much metal. Raising the cut-off sends less rock to the mill and more to the dump, so the mill handles richer feed and less of it, and the mine must dig faster to keep the mill full; lowering it does the reverse. Each stage therefore has its own preferred line — the grade at which the last tonne admitted just pays for the capacity it consumes at that stage — and Lane calls these the three limiting cut-offs. They are different numbers. When the plant is the bottleneck, the operation should feed it only rich rock and the cut-off is high; when the pit is the bottleneck, every tonne dug is dear and the cut-off is low, because rock already lifted should not be thrown away; when the market is the bottleneck the line sits somewhere else again. The rock has not changed. What has changed is which stage is full.

the balancing cut-offs, and why the optimum sits at a kink

Between any two of the limiting cut-offs lies a third kind, which Lane calls a balancing cut-off: the grade at which two stages reach capacity in the same year, so that neither is idle and neither is a bottleneck. There are three of these too, one for each pair of stages, and the six together are the only candidates the optimum can be. Lane's rule for choosing among them is mechanical: for each pair of stages, take the middle value of the two limiting grades and their balancing grade, and then choose among the three results. The consequence that matters is that the optimum is very often a balancing grade rather than a limiting one — it sits at the kink where two constraints meet, not at a smooth interior point where some marginal quantity is zero. A cut-off found by asking what the rock is worth will not find it. A cut-off found by asking which two stages should be made to fill at the same moment will.

the opportunity cost, and why the line falls as the mine ages

The last element is time. A mine is finite, and every tonne processed this year is a tonne that will not be processed later; the theory charges each unit of capacity used now with the value that the whole remaining operation forgoes by being one unit shorter. Lane writes this as an opportunity cost, a term added to the running costs in each of the limiting formulae, and it is proportional to the present value of everything still to come. Early in the life of the mine, when much remains, that value is large, the term is large, and the cut-off is high: the operation should take only the best rock, because the capacity it spends is dear. As the reserve is worked out the value of what remains falls, the term shrinks, and the optimum cut-off falls with it, until near the end the mine is treating rock it would once have dumped. This is the standard result of the theory and the standard shape of a planned cut-off policy: high in the early years, declining through the life of the mine. The rock at the bottom of the pit is no richer than it was; it is admitted because there is less future left to spend.

II.The Second Known Thing — the cutpoint that decides who is admitted

drug discovery · biomarkers and patient stratification

A drug that works better in some patients than others is developed, increasingly, by finding a measurable marker of the difference and enrolling only the patients above a line drawn on it. The marker is usually continuous — a percentage of tumour cells stained, a level in the blood, a score from a signature — and the line is the cutpoint. Above it a patient is marker-positive and admitted; below it, negative and not. The design is called enrichment, and its logic is not disputed: if the treatment effect rises with the marker, then a trial confined to high-marker patients needs fewer of them to show the effect, and a drug that would have failed in all comers can succeed in the enriched population and be labelled for it. Regulators have written guidance on it, and much of oncology is now organised around it.

how the field says a cutpoint is chosen

The literature on choosing the cutpoint is statistical. A cutpoint is placed where a receiver operating characteristic curve is best, or where the interaction between marker and treatment is largest, or where the p-value of a subgroup contrast is smallest, or by an optimal-design criterion that minimises the variance of the estimated threshold; more recent designs let the cutpoint adapt to the data as the trial proceeds. All of these methods have one thing in common: they treat the cutpoint as a property of the relation between the marker and the benefit, to be estimated from that relation as well as the data allow. The field's own account of why a cutpoint changes is, correspondingly, that the estimate changed — that a later trial saw benefit in patients a former cutpoint had excluded.

what the field also knows, and keeps in a different chapter

The field also knows that enrichment has a price in patients screened. Simon and Maitournam showed in 2004 that the efficiency of a targeted design depends on the prevalence of marker-positive patients and on how much of the effect survives in the negatives: a rare marker means many patients tested for each one enrolled, and the trial that needs the fewest randomised patients may need the most screened. This arithmetic is used to argue for or against enrichment as a design, and sometimes to weigh a screening cost, but it is kept apart from the choice of cutpoint, which remains a matter of biology. Nothing in the standard account makes the cutpoint depend on how many patients a programme can test in a month, on how many trial slots it has, on how large a labelled population it can sell to, or on how many years of exclusivity remain.

and yet the cutpoints move, and in one direction

Yet cutpoints do move, and they tend to move one way. Pembrolizumab was first approved in untreated lung cancer on a trial that admitted only patients whose tumours stained for PD-L1 on at least half their cells; a later trial lowered the line to one cell in a hundred, and the label followed, though by common reading the benefit in the newly admitted band was slight and the result was carried by the patients above the old line. Trastuzumab deruxtecan was developed for breast cancers with high HER2 expression, then shown active in cancers scored HER2-low, then in cancers scored lower still. Each move is explained, after the fact, as the arrival of new evidence about the marker. But the moves are almost always downwards, they come later in a drug's life rather than earlier, and the field has no theory that predicts them, only a series of trials that happened to look.

III.The Bridge

no mine plan cites a trial; no trial design cites a pit

The halls do not meet. Cut-off grade theory belongs to mining engineering and mineral economics; enrichment design belongs to biostatistics and regulatory science. No review spans them, and the coincidence of the word cut-off is lexical, not conceptual, since neither hall has borrowed anything from the other under that word or any other. Two neighbours must be set aside. It is not the choke censoring of an earlier specimen, in which an effort stops at the first of several limits it reaches and leaves the others untested; there the limits are fixed and the question is which one a passive process hits first, whereas here the threshold is a lever that is moved deliberately so that two limits are reached together. And it is not the border that moves with the question, in which a threshold on a continuous measure sits at different places for different observables; there the border belongs to the measure, whereas here it belongs to the operation and would move even if the measure and its observables stood still. What the halls share is a rule about where a sorting line falls when the sorted material passes through stages of fixed capacity.

X: the line belongs to the operation, not to the material

Call it the binding cut-off. Where a continuous grade sorts a stream of material into what is processed and what is set aside, and the processing runs through stages each of fixed capacity, the threshold that maximises the value of the whole operation is not a property of the material but of which stage binds. Four things follow. Each stage has its own limiting threshold, the grade at which the last unit admitted just pays for the capacity it consumes there, and these are different numbers for the same material. The operative threshold is either the limiting value of the stage that binds or a balancing value at which two stages fill together, so that the optimum sits typically at a kink between constraints and not at any point the material alone would pick out. The threshold carries an opportunity cost equal to the value the finite remaining life forgoes by spending a unit of capacity now, so it falls as that life shortens, and the same material is set aside early and admitted late without having changed. And the material contributes only its distribution over the grade — how much of it there is at each level — and nothing else about it enters the line. Read into the trial hall, each has a consequence the hall does not state. A biomarker cutpoint has three limiting values: one where the capacity to test patients binds, one where the capacity to treat them — sites, slots, drug supply, the months a readout can wait — binds, and one where the population a label can be sold to binds; and which of these is the cutpoint in force is a question about the programme, not about the marker. Where testing and treating balance, the positivity rate at the cutpoint is simply the ratio of enrolment capacity to screening throughput, a number that can be read from a screening diagram and has no biological content at all. The cutpoint carries the value of the exclusivity remaining and should fall as that runs out, which would make the downward drift of cutpoints over a drug's life a consequence of the calendar rather than of the marker. And the marker–benefit relation, which the field treats as the whole of the matter, enters only as the grade–tonnage curve enters a mine plan: it fixes how many patients there are at each level, and nothing more.

IV.The Question That Becomes Possible

neither discipline would ask this alone

The trial hall asks where on the marker the benefit begins and places the cutpoint there, treating every subsequent move of it as a correction of that estimate. The mining hall has no patients, but it settled more than fifty years ago that a sorting line through a graded stream is fixed by the stage that is full, sits usually where two stages fill together, and falls as the remaining life shortens, whatever the stream is made of. Neither hall asks whether the cutpoints actually chosen for a marker across the life of a drug are predicted by the capacities of the programme that chose them, separately from anything learnt about the marker in between.

the binding cut-off hypothesis

For a drug and a marker tested at more than one cutpoint across successive trials, is the cutpoint in force at each trial predicted by which capacity bound at the time — the positivity rate at the cutpoint matching the ratio of enrolment capacity to screening throughput where those two balance, and the cutpoint falling with the exclusivity remaining — independently of any change in the estimated marker–benefit gradient between the trials; and where a cutpoint has moved with no change in that gradient, has it moved in the direction the binding capacity predicts?

why it would matter

Because a cutpoint is read, by regulators, payers and physicians, as a statement about who benefits, and if it is in part a statement about how many patients a sponsor could test in a month and how many years of patent remained, then some of the patients on the wrong side of the line are there for reasons that have nothing to do with them. If the balancing rule holds, a programme can compute the cutpoint it is likely to end up with before any trial is run, from its own screening and enrolment numbers, and can see whether that line falls anywhere near where the biology would put it. If the opportunity-cost rule holds, the label broadening that now arrives as a sequence of surprises can be scheduled, and the trials that produce it can be planned rather than stumbled into. And if neither holds, the field will have shown that its cutpoints are what it says they are, which is worth knowing too.

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 needs no new trial; it needs the screening diagrams of trials already published and a coding nobody has yet done.

the honest status

Note the narrowness. The claim concerns cutpoints used for enrichment on markers assumed to rise with benefit, in trials run to support a label, and says nothing about prognostic markers, about stratification that admits everyone and merely balances the arms, or about markers that are categorical by nature. It does not say where a cutpoint should be, only what sets the one that is. It does not claim that the biology is irrelevant; the theory needs the grade–tonnage curve, and a marker with no gradient has no cutpoint to set. It is worth saying what does not transfer. A mine measures the grade of every tonne before deciding on it and can change its line every year; a trial fixes its line in a protocol and measures the marker once, and the line it fixes then binds the label for years. A mine's three capacities are engineered and known; a programme's are partly guessed and partly negotiated, and the market limit in particular is a forecast rather than a fact. The whole thing rests on one computation that a single group with the published trials could make in a season: whether the cutpoints a marker has actually carried are predicted better by the capacities of the programmes that chose them than by what those programmes knew about the marker. If they are not, this is an elaborate way of saying that trials are constrained by their budgets, and should be said so. Until that computation is made, 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.