12.1 Dose-Response Relationships: Threshold vs Non-Threshold, Linear vs Non-Linear Models
Key Takeaways
- Quantal data are all-or-none incidences (death, tumor, malformation); continuous data are graded measurements (body weight, ALT). LD50, LC50, ED50, and EC50 are midpoint potencies, not chronic points of departure.
- A frequency plot of individual thresholds is often roughly bell-shaped; the cumulative plot is the sigmoid whose 50% point is the LD50 or ED50. Slope, plateau (Emax), and midpoint are separate geometric properties.
- Most noncancer apical effects are treated as threshold (nonlinear) processes. EPA 2005 cancer guidelines use default linear no-threshold extrapolation for agents with a mutagenic mode of action.
- Hormesis is a J-shaped or inverted-U low-dose stimulation with high-dose inhibition; endocrine non-monotonic curves change slope sign with dose. Neither pattern is an automatic default for chemical reference values.
- Therapeutic index is TD50/ED50 (example: 200/20 = 10). Essential nutrients follow a U-shape (deficiency plus excess). PBPK models replace administered mg/kg with internal metrics such as AUC, Cmax, or amount metabolized.
Why dose–response is Domain III.C
Handbook III.C.1 asks whether you can describe how a biological system changes as dose or concentration changes, then defend a model shape for that endpoint and mode of action. Independent OpenExamPrep teaching in this section covers quantal versus continuous data, potency midpoints (LD50, LC50, ED50, EC50), curve geometry (slope, plateau, midpoint), frequency versus cumulative plots, hormesis and other non-monotonic shapes, the usual threshold treatment of most noncancer effects, the U.S. Environmental Protection Agency (EPA) 2005 Guidelines for Carcinogen Risk Assessment default linear no-threshold (LNT) treatment of mutagenic carcinogens, the therapeutic index, the U-shaped curve for essential nutrients, and why physiologically based pharmacokinetic (PBPK) models replace administered dose with an internal metric. This material is not an ABT product and does not claim official approval, review, or partnership with ABT or EPA.
Domain III.C is 9% of the examination. Items here punish mixing an acute lethality screen with a chronic cancer assessment, or treating every sigmoid as proof that a biological threshold equals the study NOAEL.
Quantal versus continuous responses
Quantal (dichotomous) data are all-or-none for each animal or person: dead or alive, tumor present or absent, malformation yes or no. The group response is an incidence (3/10, 30%). Likelihood is binomial. Extra risk and added risk live on this scale (section 12.2).
Continuous data are graded measurements: terminal body weight, serum alanine aminotransferase (ALT), red-cell acetylcholinesterase activity, fetal weight. Each animal contributes a number. Modeling uses means and variances. A common benchmark is a change of one control standard deviation (1 SD), not a 10% incidence.
The same study can contain both: hepatocellular necrosis scored present/absent (quantal) and ALT in U/L (continuous). Do not convert a mean 12% ALT rise into an “LD12.” Do not treat a 50% drop in a continuous enzyme activity as an LD50 unless death was the endpoint.
LD50, LC50, ED50, EC50
These are midpoint potency numbers. They locate the curve on the dose axis. They are not safety limits, not chronic points of departure, and not first-in-human starting doses.
| Symbol | Quantity | Typical setting | What 50% means |
|---|---|---|---|
| LD50 | Median lethal dose | Acute oral, dermal, or injection lethality | Dose estimated to kill 50% of the tested animals (quantal death) |
| LC50 | Median lethal concentration | Acute inhalation; aquatic lethality | Air or water concentration killing 50% |
| ED50 | Median effective dose | Pharmacology; some toxicology | Dose producing 50% of Emax, or affecting 50% of subjects, depending on how the endpoint was defined |
| EC50 | Median effective concentration | In vitro assays; aquatic sublethal tests | Concentration at the midpoint of a concentration–response curve |
Worked midpoint. Ten rats per group receive a single oral gavage. Deaths: 0/10 at 50 mg/kg, 2/10 at 100, 5/10 at 200, 8/10 at 400, 10/10 at 800. Incidence crosses 50% at 200 mg/kg, so the LD50 is near 200 mg/kg. That number does not tell you the NOAEL of a 90-day study, the BMDL10 of a cancer bioassay, or a therapeutic starting dose. It only locates acute lethality.
Slope, plateau, and midpoint
A sigmoid (logistic, probit, Hill) has three geometric jobs that items like to separate:
- Midpoint — LD50/ED50/EC50: where the curve is located on the x-axis.
- Plateau (Emax) — the maximum response. Quantal incidence cannot exceed 100%. Continuous inhibition may plateau below 100% if residual activity remains.
- Slope (Hill coefficient n) — how fast the curve rises between about 20% and 80% of Emax. A steep slope means a small dose increment moves the population from few responders to most responders (some ion-channel toxins, some acute lethality curves). A shallow slope means effects accumulate over a wide dose range (many chronic organ-weight changes).
Teaching form of the Hill equation:
E = Emax × C^n / (EC50^n + C^n)
If n = 1, the rise is gradual. If n = 4, the curve is almost a step. Two chemicals can share an ED50 of 10 mg/kg and still differ enormously in how dangerous a two-fold dosing error is—that difference is slope, not the midpoint.
Linear versus nonlinear in risk-assessment language is about the low-dose region, not about whether a laboratory sigmoid looks curved on arithmetic paper. A probit fit to log-dose is a nonlinear function of dose and still becomes a straight line through the origin in an EPA mutagenic-cancer extrapolation below the PoD. A Michaelis–Menten metabolic saturation is nonlinear and can produce a threshold-like apical effect once repair is exceeded. Name the region and the endpoint before you name the model.
Common families, conceptually (not a software tutorial): probit (cumulative normal of log dose), logit (log-odds linear in log dose), multistage polynomials used historically in cancer assessment, Weibull, and for continuous data power, Hill, and exponential models. Model shopping is section 12.2; this section is shape literacy.
Frequency versus cumulative curves
Individual animals do not all share one threshold. A frequency plot (histogram) of the dose at which each animal first responds is often roughly bell-shaped. The cumulative curve adds those animals and is the familiar sigmoid of probit analysis.
Worked frequency table. Ten mice have unique individual lethal thresholds (mg/kg): 12, 14, 15, 16, 18, 20, 22, 24, 28, 40. Sum = 209, so the mean is 20.9 mg/kg. The median of the ten values is the average of the 5th and 6th, (18 + 20) / 2 = 19 mg/kg.
| Dose bin (mg/kg) | New deaths in bin (frequency) | Cumulative deaths | Cumulative % |
|---|---|---|---|
| 10–14 | 2 | 2 | 20% |
| 15–19 | 3 | 5 | 50% |
| 20–24 | 3 | 8 | 80% |
| 25–40 | 2 | 10 | 100% |
Frequency peaks in the middle two bins (3 + 3). Cumulative percent is 20, 50, 80, 100—the sigmoid. The LD50 sits where cumulative incidence is 50%, here in the 15–19 mg/kg bin, near the median of 19, not at the 40 mg/kg outlier and not automatically at the histogram’s visual pile-up if the distribution is skewed. Confusing the frequency mode with the cumulative median is a high-yield error.
Threshold, linear no-threshold, and other nonlinear shapes
Threshold (nonlinear, a dose below which the adverse effect does not appear above background). For most noncancer apical effects—necrosis after a repairable insult, cholinesterase inhibition that must exhaust reserve, renal failure after tubular compensation is spent—homeostasis implies a region of no incremental adversity in the tested population. Risk assessment may still apply uncertainty factors below the PoD (Chapter 13). That arithmetic is not a claim that the biological threshold equals the NOAEL.
Linear no-threshold. In EPA 2005 cancer practice, agents with a mutagenic mode of action are assessed with default linear extrapolation: extra cancer risk is treated as proportional to dose at low dose, with no dose assigned zero incremental risk. The PoD (often a BMDL10) is connected to the origin; the slope of that line is the cancer slope factor in section 12.3. EPA does not force LNT on every carcinogen. If a well-supported nonlinear mode of action is accepted—for example cytotoxicity with regenerative proliferation as the driver of tumors—a threshold-style reference value may be derived instead. The examination tests whether you know the mutagenic-carcinogen default, not whether you can relitigate radiation epidemiology.
Nonlinear without a sharp threshold. Saturation of clearance, receptor occupancy approaching 100%, and glutathione depletion produce curves that steepen or flatten once a pathway saturates. That is still nonlinear. It is not automatically hormesis.
Hormesis and endocrine non-monotonicity
Hormesis is a J-shaped or inverted-U pattern: a low-dose change in the opposite direction from the high-dose toxic effect (low-dose growth stimulation, high-dose inhibition). It is a hypothesis to test with data, not a default for IRIS files or ICH M3 packages.
Non-monotonic dose–response (NMDR) is broader: the slope changes sign at least once. Endocrine-active agents (some receptor agonists and antagonists) are the teaching example, because receptor occupancy can stimulate at low occupancy and then desensitize, recruit different cofactors, or give way to ordinary cytotoxicity at high concentration. A reporter assay that peaks at 0.1 nM and returns toward baseline at 10 nM is NMDR. Killing the cells at 10 µM is high-dose toxicity, not proof of hormesis.
Design consequence: a study that only tests 100, 300, and 1000 mg/kg may miss a low-dose endocrine peak. That is a dose-placement problem, not a license to draw a U-shape through two noisy means.
Therapeutic index and the nutrient U-shape
Therapeutic index (TI) in classical pharmacology is TD50 / ED50 (sometimes LD50 / ED50). If the median toxic dose is 200 mg/kg and the median effective dose is 20 mg/kg, TI = 200 / 20 = 10. A TI of 10 is more forgiving of pharmacokinetic variability than a TI of 2. TI uses population midpoints. It is not an RfD, not a CSF, and not a substitute for organ-specific NOAELs on a 90-day study.
Essential nutrients (vitamin A, copper, selenium, fluoride) produce a U-shaped health curve: deficiency disease at too little, toxicity at too much, with an adequate-intake trough in between. Treating selenium like a mutagenic carcinogen with LNT would “protect” people into deficiency cardiomyopathy. Treating vitamin A deficiency as proof that high-dose retinoids are harmless would miss teratogenicity. The U-shape is about homeostatic requirement plus excess toxicity, not a claim that every industrial chemical is a nutrient.
PBPK outputs as the x-axis
Administered mg/kg/day is a convenient external dose. When species differ in ventilation, protein binding, or metabolic clearance, the same external dose is not the same internal dose. PBPK models predict time courses of parent and metabolite in blood and tissues. Dose–response is then plotted against:
- Area under the curve (AUC) in venous blood or the target tissue, when cumulative exposure drives the effect
- Peak concentration (Cmax) when the peak drives the effect (some acute CNS effects; some teratogenic peaks)
- Amount metabolized when a reactive metabolite is the toxicant (amount of NAPQI, not milligrams of acetaminophen in the bottle)
Using AUC as the metric can remove apparent species differences that were only clearance differences. Using Cmax when AUC is the true driver, or the reverse, is a misspecified dose metric. PBPK does not invent a threshold; it changes the x-axis on which threshold or linear models are fitted. Section 12.3 returns to PBPK as an interspecies tool for first-in-human dose.
Scenario
A 90-day rat study of a liver enzyme inducer shows a shallow continuous increase in relative liver weight (plateau not reached), no necrosis, and a quantal incidence of thyroid follicular hypertrophy of 0/10, 0/10, 2/10, 8/10 across four doses. The liver-weight curve is continuous and shallow; the thyroid finding is quantal and steep. An Ames-positive impurity in the same program is assessed for cancer with the mutagenic LNT default, not with the thyroid threshold story. A PBPK model that reports thyroid AUC of the parent, not gavage mg/kg, is the right x-axis if clearance differs between rat and human. A second file is selenium in feed: both the deficiency arm and the excess arm raise adverse findings—the U-shape—not an invitation to apply a cancer CSF to essential-nutrient deficiency.
Traps
- Using LD50 as a chronic PoD or as an FIH dose.
- Calling every laboratory sigmoid a biological threshold equal to the NOAEL.
- Applying LNT to essential-nutrient deficiency, or applying a nutrient U-shape to a mutagenic industrial carcinogen without evidence.
- Reading a frequency histogram’s peak as the ED50 without looking at the cumulative median.
- Plotting administered dose when the toxicant is a metabolite AUC.
- Treating hormesis as the required default for endocrine or cancer assessment.
Which statement correctly contrasts quantal and continuous dose–response data and the meaning of an LD50?
Under EPA 2005 cancer assessment practice, which default low-dose shape is used for an agent with a mutagenic mode of action?
A nuclear-receptor agonist shows reporter stimulation at 0.1 nM, a return toward baseline at 10 nM, and cytotoxicity at 10 µM. Which description fits the low- and mid-dose pattern?