Dose Response

Dose Response, Emax Curves and Drug Costs in Health Technology Assessment

Dose sits on both sides of a cost-effectiveness ratio. A higher dose usually costs more, while the extra benefit it buys shrinks as the curve flattens. That makes the shape of the curve, and the dose actually given and paid for, an economic question as much as a pharmacological one. Weight-based dosing, vial wastage and relative dose intensity turn dose into cost, and dose-ranging evidence and dose-response network meta-analysis feed health technology assessment. The term also has an epidemiological sense, in which exposure gradients drive public health models and support causal inference.

The Emax and sigmoid Emax models

Most dose-response analyses in drug development fit a curve that rises steeply at low doses and then levels off towards a ceiling. The sigmoid Emax model is the standard four-parameter form of that curve, and the simpler Emax model is its special case with a Hill coefficient of one. The expected response at a dose is given by the following equation.

$$E(D) = E_0 + \frac{E_{max} \, D^h}{ED_{50}^h + D^h}$$

where $E(D)$ is the expected response at dose $D$, $E_0$ is the response with no active drug (the placebo or baseline response), $E_{max}$ is the maximum effect the drug can add above $E_0$, $ED_{50}$ is the dose that produces half of $E_{max}$, and $h$ is the Hill coefficient, which sets how steeply the curve rises around $ED_{50}$. With $h$ equal to one the model reduces to the hyperbolic Emax model, $E(D) = E_0 + E_{max} D / (ED_{50} + D)$. Physiologically, $E_{max}$ is usually read as the effect reached when the drug saturates its receptors.

The share of the maximum effect depends only on the ratio of the dose to $ED_{50}$, which makes the curve easy to interpret.

$$\frac{E(D) - E_0}{E_{max}} = \frac{k^h}{1 + k^h}, \quad k = \frac{D}{ED_{50}}$$

where $k$ is the dose expressed as a multiple of $ED_{50}$ and the other terms are as defined above. With $h$ equal to one, doubling the dose from $ED_{50}$ raises the effect from 50% to 67% of the maximum; with $h$ equal to two, the same doubling raises it from 50% to 80%. The same form is known in pharmacology as the Hill equation, after A. V. Hill, who introduced it to describe the relationship between oxygen tension and the saturation of haemoglobin; he is not the Austin Bradford Hill of the causal viewpoints discussed below. Pharmacokinetics links the dose given to the concentration that drives the response, and in an exposure-response analysis a measured drug concentration takes the place of dose. Separate curves for benefit and harm underlie the therapeutic index.

Why dose is an economic quantity

For most medicines the acquisition cost rises in proportion to the amount given, while the effect follows the curve above. Once the dose is past the steepest part of the curve, which for a Hill coefficient above one lies below $ED_{50}$, each extra milligram costs the same but buys less health than the one before. The cost per extra unit of effect then rises with dose even when the average cost-effectiveness ratio still looks favourable, so choosing a dose is a marginal analysis problem, and the incremental cost-effectiveness ratio between adjacent doses is the quantity that matters in cost-effectiveness analysis.

Weight-based dosing, vials and wastage

When a dose is calculated from body weight or body surface area and the drug comes in single-use vials, the amount paid for is set by the vials opened rather than by the dose given. Leftover drug still has to be paid for when it is discarded, and body sizes rarely match vial content, so some drug is nearly always left over. Without vial sharing, the drug cost of one administration is as follows.

$$C_{admin} = \left\lceil \frac{D}{v} \right\rceil \times p_v$$

where $C_{admin}$ is the drug cost of one administration, $D$ is the prescribed dose in mg, $v$ is the content of one vial in mg, the brackets round up to the next whole vial, and $p_v$ is the price of one vial. With vial sharing, or payment per mg used, the cost falls to $D p_v / v$.

Vial size alone can change wastage substantially. One analysis of weight-based dosing using UK population data estimated that supplying pembrolizumab in 70 mg and 100 mg vials, in place of the 50 mg and 100 mg vials then available, would cut wastage from 13.3% to 8.7%. Dose banding, in which all patients with similar characteristics are allocated the same dose, is another departure from the dose the curve implies, since some patients then receive less intensive treatment, with a risk of reduced benefit that a model assuming banded doses should reflect.

Trial dose, licensed dose and the dose that is appraised

A cost-effectiveness model inherits its dose from trials run before the whole curve was known. The ICH E4 guideline on dose-response information records that drugs have often been marketed first at doses later recognised as excessive, well onto the plateau of the curve, and that no single dose is necessarily optimal for all patients.

In oncology the US Food and Drug Administration, through its Project Optimus initiative, has argued that dose selection inherited from cytotoxic chemotherapy too often leaves the doses of targeted therapies inadequately characterised before registration trials, so that a dose may give more toxicity without additional efficacy and require frequent dose reductions. The agency issued final guidance on dosage optimisation for oncology medicines in August 2024. For an economic evaluation a dose on the plateau adds cost and harm without adding benefit.

For England, NICE's manual (PMG36, section 2.2.6) rules out technology appraisal guidance for unlicensed or off-label use unless the Department of Health and Social Care indicates otherwise. The manual sets no rule on wastage. That rule sits in NICE's user guide for the company evidence submission template (PMG24, section 3.5.6), which states that drug wastage should be taken into account in calculating drug costs and that tablet packs are costed per pack unless evidence suggests there would be no wastage.

Relative dose intensity in oncology models

In cancer models the dose patients actually receive is often lower than the planned dose because of dose reductions, delays and missed doses. Relative dose intensity (RDI) is the ratio of the delivered dose intensity to the planned dose intensity, where dose intensity is the dose given per unit of time. Models use it to scale the drug cost of each cycle.

$$C_{drug} = C_{planned} \times RDI$$

where $C_{drug}$ is the modelled drug cost per treatment cycle, $C_{planned}$ is the cost of the full planned dose for that cycle and $RDI$ is the relative dose intensity, a proportion between 0 and 1.

A review of NICE appraisals of treatments for solid tumours, searched to 31 March 2024, found RDI or a dose delay factor in 63 of 265 appraisals. Across the appraisals reviewed, RDI ranged from 85% to 100%, methods varied widely, excluding the RDI adjustment generally raised the ICER by 5% to 10%, and committees frequently questioned the transparency of RDI calculations and the assumptions about delayed or missed doses and wastage. Two consistency checks follow from dose-response reasoning. Trial efficacy already reflects the doses received in the trial, so an RDI from the same trial lowers cost without changing effect, whereas an RDI from another source breaks that link. Under vial-based pricing a dose reduction saves money only when fewer vials are opened, so applying RDI to a per-mg cost can overstate the saving.

Dose-ranging evidence and dose-response network meta-analysis

Model-based meta-analysis (MBMA) pools trials through an explicit dose or time-course model, often an Emax model, so that treatments can be compared at doses never tested head to head. Network meta-analysis usually handles dose crudely. Doses of one agent are either lumped as if equally effective, which increases heterogeneity and the risk of inconsistency, or split into separate treatments, which lowers precision and can disconnect the network.

Mawdsley and colleagues proposed model-based network meta-analysis (MBNMA), which places a dose-response function inside a network meta-analysis while keeping comparisons within each trial randomised. Their framework allows $E_{max}$ and $ED_{50}$ to be exchangeable across agents in the same class, borrowing strength when one agent's dose range is too narrow; in their illustrative comparison of triptans for migraine, the limited dose range made $ED_{50}$ difficult to estimate reliably. For an economic model, MBNMA gives a relative effect at the dose actually costed, including a licensed dose that lies between two trial doses.

Exposure gradients in public health models and causal inference

In epidemiology the dose is a level of exposure and the response is a relative risk of disease, which public health economic models use to turn a change in exposure into changes in disease, costs and quality-adjusted life years; a model of alcohol pricing policies in England, for example, applied such risk functions to 47 illnesses. The shape matters. In a Global Burden of Disease meta-analysis the largest gains from physical activity occurred at lower levels, with diabetes risk falling a further 19% from 600 to 3,600 metabolic equivalent (MET) minutes a week but only 0.6% from 9,000 to 12,000. A biological gradient, the fifth of Bradford Hill's nine viewpoints, supports causal inference from an association, although he held that none of the viewpoints is required for it.

Worked example of cost per extra responder at three doses

The figures below are illustrative and describe a hypothetical drug. The response is the proportion of patients who respond, with $E_0 = 0.20$, $E_{max} = 0.40$, an $ED_{50}$ of 100 mg and $h = 2$. The drug costs £300 for a 100 mg single-use vial (£3 per mg), a course is 12 administrations, and other costs are left out so the arithmetic stays visible.

1. Response at each dose. The expected response at 150 mg is calculated first.

$$E(150) = 0.20 + 0.40 \times \frac{150^2}{100^2 + 150^2} = 0.20 + 0.40 \times \frac{22500}{32500} = 0.476923$$

where $E(150)$ is the expected proportion responding at 150 mg. The same calculation gives 0.40 at 100 mg, the $ED_{50}$, and 0.52 at 200 mg, where the fraction is 40000/50000 = 0.80. The extra responders compared with placebo are 0.20, 0.2769 and 0.32 per patient treated.

2. Drug cost per course with vial sharing. If leftover drug can be used for the next patient, cost is proportional to dose.

$$C(150) = 150 \times 3 \times 12 = 5400$$

where $C(150)$ is the drug cost in pounds of a course at 150 mg. Courses at 100 mg and 200 mg cost £3,600 and £7,200.

3. Average and incremental cost per extra responder. The average ratio divides each cost by the extra responders against placebo; the incremental ratio divides the change in cost by the change in responders between adjacent doses.

For the step from 150 mg to 200 mg, using the unrounded response at 150 mg, the calculation is as follows.

$$ICER_{200,150} = \frac{7200 - 5400}{0.52 - 0.476923} = \frac{1800}{0.043077} = 41786$$

where $ICER_{200,150}$ is the incremental drug cost in pounds per extra responder of moving from 150 mg to 200 mg. The average ratio rises only from £18,000 to £22,500, while the incremental ratio more than doubles, because each extra 50 mg costs the same £1,800 but adds fewer responders.

4. Drug cost without vial sharing. If leftover drug is discarded, 150 mg needs two 100 mg vials at every administration.

$$C_{admin}(150) = 2 \times 300 = 600$$

where $C_{admin}(150)$ is the drug cost in pounds of one 150 mg administration. A course then costs £7,200, the same as 200 mg, which uses the same two vials with nothing wasted. The 150 mg dose is weakly dominated and drops out, leaving 100 mg against 200 mg.

$$ICER_{200,100} = \frac{7200 - 3600}{0.52 - 0.40} = \frac{3600}{0.12} = 30000$$

where $ICER_{200,100}$ is the incremental drug cost in pounds per extra responder of moving from 100 mg to 200 mg.

5. Interpretation. With vial sharing, at an illustrative £25,000 per extra responder, 150 mg would be chosen, because the step to 200 mg costs £41,786. With wastage, the only step, from 100 mg to 200 mg, costs £30,000, so the decision falls back to 100 mg. The curve fixes the benefit side, but vial size and dispensing rules decide which doses are worth considering. A real evaluation would add adverse events, which have their own dose-response curve, and would measure benefit in quality-adjusted life years.

Where dose-response reasoning goes wrong in models

Several errors recur when dose-response evidence enters an economic model. Extrapolation of a fitted curve beyond the doses studied rests on the functional form rather than on data, and an $ED_{50}$ estimated from a narrow range is weakly identified, as the triptan example showed. A population-average curve does not apply to every patient, and a model that averages across weight or tolerability groups can hide subgroups for whom the dose sits on the plateau.

The dose received in practice is not randomised. Patients who tolerate treatment well receive more of it and may also have a better prognosis, so a gradient between received dose, or relative dose intensity, and survival in observational data can reflect confounding rather than an effect of dose. Uncertainty in the curve parameters, RDI and wastage belongs in probabilistic sensitivity analysis and scenario analysis rather than being fixed at point values.

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