BMS 123 · Common Core (Tronc commun)Section 6.2 · Dose–response, interactions and variability
Section 24 of 27
Quantal dose–response curves
Describe a quantal curve and use it to explain ED50, TD50, LD50 and the therapeutic index.
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Why this matters
A prescription is written for one person, but the evidence behind the dose comes from a population. Quantal curves are the bridge: they tell us what fraction of people respond, what fraction are harmed, and how far apart those two doses lie. The therapeutic index calculated from them is the single number most often used to describe how safe a drug is, and it explains why some medicines are given with routine monitoring while others are handed out with almost no follow-up.
What you need to know first
You need the difference between graded and quantal responses from Unit 6.1, and a basic idea of what a percentage is. Nothing else is assumed; ED50, TD50, LD50 and the therapeutic index are all defined here.
Words to know (5)
Quantal (all-or-none) response
An effect that either occurs or does not occur in a subject, so each subject is counted as a responder or a non-responder. Sleep produced by a hypnotic, or death after a toxic dose, are counted as quantal responses.
ED50 (median effective dose)
The dose at which 50% of the population shows the chosen therapeutic effect. If 50 of 100 patients improve at the same dose, that dose is the ED50 for that effect.
TD50 (median toxic dose)
The dose at which 50% of the population shows a defined toxic effect. The dose at which half of treated animals develop liver damage is that drug’s TD50 for hepatotoxicity.
LD50 (median lethal dose)
The dose that kills 50% of the subjects tested, used as the lethal end of the toxic dose range. LD50 is estimated in animal studies to guide the choice of a safe starting dose.
Therapeutic index (TI)
The ratio of the dose producing toxicity in 50% of subjects to the dose producing the therapeutic effect in 50%: TI = TD50 / ED50, or LD50 / ED50 when lethality is used. A drug with a TI near 1 has almost no gap between benefit and harm.
One idea at a timeIdea 1 / 5
In one sentence A quantal dose–response curve relates the dose to the percentage of subjects showing an all-or-none response, which may be therapeutic, toxic or lethal.
A quantal curve counts people, not effect size
A quantal dose–response curve relates the drug concentration to the percentage of subjects — patients, animals — responding with an all-or-none response. The response may be a therapeutic response, an adverse or toxic effect, or a lethal effect; the same drug tested against different criteria produces different quantal curves. The curve is built by giving a dose to each member of a group, deciding for each subject whether the criterion was met, and plotting the percentage of responders against the dose. Because each subject contributes a single yes or no, the curve rises from 0% to 100% instead of flattening at an Emax: the plateau at 100% means that everyone who can respond has responded.
For example
A hypnotic is tested in 100 volunteers at several doses, and sleep is recorded as present or absent. At a low dose 5 people sleep, at a middle dose 50 sleep, at a high dose 95 sleep: those points form a quantal curve.
Pause and check yourself
Why does a quantal curve not have the same plateau shape as a graded curve?
Show the answer
Because each subject gives only one yes or no answer, so the curve counts the proportion of responders and rises to 100% when everyone who can respond has responded, rather than rising to a tissue Emax.
Read detailed explanations
How quantal curves are actually built and what distorts them
The smooth S of a quantal curve is a summary of a staircase. In practice each animal or patient receives one dose, the criterion is scored as present or absent, and the proportion of responders at each dose is calculated; the proportion is then treated as if the population had a continuous threshold for the effect. That threshold varies between subjects, so the population curve is the accumulated distribution of individual thresholds, which is why the curve is S-shaped and why the gentle ends belong to unusually sensitive and unusually resistant subjects. The design of the study shapes the result in ways worth knowing. The criterion must be defined before the study and applied identically to everyone: “responded” might mean a 50% fall in pain score at one centre and a patient’s own statement of relief at another, and the two will not give the same ED50. The number and spacing of doses determine how well the median is estimated, and a study with only three widely spaced doses can only give a crude median. The species matters: an LD50 estimated in mice is a statement about mice, influenced by their metabolism, and its main use is to guide the design of safer human studies rather than to predict human lethality. Ethical and statistical practice has also moved away from treating LD50 as a routine number: modern toxicology prefers to estimate the dose that produces a defined toxic effect and to use fewer animals, because a median lethal dose tells us little about the mechanisms of harm that matter clinically. When you meet an ED50 or LD50 in a textbook or a paper, ask which criterion, which species and which dose spacing produced it.
Comparing two safety numbers: TI, margin of safety and standardised measures
The therapeutic index is the oldest and simplest safety number, but it has a weakness that becomes obvious when two drugs are compared. Because it is a ratio of two medians, it ignores the slopes of the curves. Two drugs can share the same TI yet behave differently: if the therapeutic and toxic curves of drug A are steep and far apart in the middle but overlap at the tails, a small dose change around the median is well tolerated while a large dose change can move a sensitive patient from unhelped to poisoned. If drug B has shallow, widely overlapping curves, its margin may be narrow at every dose even though its ratio is identical. This is why modern pharmacology uses measures that take variability into account. The margin of safety relates the toxic dose for the most sensitive 1% or 5% of subjects to the dose that is effective for 99% or 95%, which is precisely the comparison a clinician cares about because it asks whether the dose that helps nearly everyone can harm anyone at all. The standardised safety margin divides the difference between the TD50 and the ED50 by the ED50, expressing the gap as a proportion. A related idea from pharmacokinetics is the therapeutic range or window, the span of plasma concentrations within which most patients benefit and few are harmed; it is used with therapeutic drug monitoring when the dose–concentration relationship is unpredictable. Finally, the index has nothing to say about idiosyncratic reactions — anaphylaxis to penicillin, for example, happens in a tiny minority of patients and is unrelated to dose, so an enormously favourable TI does not make a drug allergy-proof.
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The quantal dose–response curve
When each subject gives only a yes or no answer, the curve is built from the percentage of responders, and the dose at which half the population responds is the ED50.
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01 / 5All-or-none answer
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All-or-none answer
In a quantal study each subject counts once: it responded or it did not. The response may be therapeutic (pain relieved), toxic (a rash or a fall in blood pressure) or lethal. The curve is built from the proportion of responders, not from the size of each response.
Schematic, not to scale. Real quantal curves are usually plotted against log dose; the spread and the height of the plateau depend on how strictly the responding criterion is defined.
Key points to remember
✓A quantal curve relates the dose to the percentage of subjects showing an all-or-none response, which may be therapeutic, toxic or lethal.
✓ED50 produces the therapeutic effect in 50% of the population, TD50 a toxic effect in 50%, and LD50 kills 50%.
✓The therapeutic index is TI = TD50 / ED50 or LD50 / ED50; values run from about 1 to more than 1000.
✓The higher the therapeutic index, the wider the margin of safety; the narrower the index, the more monitoring the drug requires.
✓These values describe populations, not individual thresholds, so they guide but cannot replace judgement for one patient.
Go deeper
Worked example
Calculating and interpreting a therapeutic index. A new anti-inflammatory drug is studied in 200 patients. Pain relief in 50% of them occurs at 40 mg, the agreed toxic effect (a rise in liver enzymes) occurs in 50% at 200 mg, and no deaths occurred in the study. A second drug, an anticancer agent, has an ED50 of 150 mg and a TD50 of 300 mg for bone-marrow suppression. Compare the two drugs and decide which needs closer monitoring.
Write down the ratio and use TD50 as the numerator, because the index compares the harmful dose to the useful dose: TI = TD50 / ED50 = 200 / 40 = 5 for the anti-inflammatory drug.
Repeat for the anticancer drug: TI = 300 / 150 = 2. The two indices are both low, but they are not the same, and the anticancer drug has the narrower margin.
Check what the numbers do not contain. The anti-inflammatory index used TD50 rather than LD50 because no deaths occurred; that is honest but means the number describes liver-enzyme toxicity, not lethality, and the two cannot be interchanged.
Decide the practical consequence. Both drugs need monitoring, and the anticancer drug needs the closer monitoring: a dose increase that improves the tumour response may at the same time move the patient into marrow suppression.
The therapeutic index converts two median doses into one safety statement: the larger the ratio, the wider the margin. It never replaces clinical judgement, because the index describes a population, names one toxicity and ignores every interaction the patient is actually exposed to.
Common misconception
Students often treat the therapeutic index as a property of the drug alone, like a melting point, and assume a high index means the drug cannot harm anyone. In fact the index depends on which toxicity and which therapeutic effect were chosen, on the population studied, and on the studies being comparable. Change the toxic endpoint and the number changes.
Check your reasoning
Explain why “this drug has a therapeutic index of 1000, so it is safe for every patient” is an unsound conclusion.
Reveal an answer
The index compares two medians in the population studied, using one named effect and one named toxicity. It says nothing about a highly sensitive individual, about allergies, about interactions with the patient’s other drugs, or about a different toxic effect that was not measured. Safety for a particular patient still requires history, dose adjustment and monitoring.
Where this comes from
BMS 123 General Pharmacology II lecture notes, Part 2 — Principles of Pharmacodynamics · Unit 3.4.2, Quantal dose-response curves: ED50, TD50, LD50 and therapeutic index
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About this course and its sources
Bilingual lessons organised from the supplied BMS 123 General Pharmacology II lecture notes (Principles of Pharmacokinetics and Principles of Pharmacodynamics). Diagrams are original educational schematics; pharmacology, nursing and French-language review are pending.