Expected response at a dose under the sigmoid Emax model

Signature

E_D = E_0 + E_max * D^h / (ED_50^h + D^h)

Function

Dose-response curves and the dose-dependent drug cost behind incremental cost per extra responder

Computational function

• Computational function: dose ladder with Emax responses, shared and whole-vial course costs and incremental cost per extra responder

  Evaluates a ladder of doses: the sigmoid Emax response and extra responders over placebo at each dose, the course cost with vial sharing (cost in proportion to dose) and with whole single-use vials, the average cost per extra responder, and the incremental cost per extra responder along the cost-effectiveness frontier, with dominated doses left without a ratio. The inputs differ from the formula's: a vector of doses in any order, the vial size and price and the number of administrations per course.

  Inputs and outputs: dose: Doses compared, in any order. Unit: mg.; E0, Emax, ED50, h: Sigmoid Emax parameters (HE-FM-DOSE-001). Unit: proportion; proportion; mg; none.; vial_mg, vial_price: Content and price of one single-use vial. Unit: mg; pounds.; n_admin: Administrations per course. Unit: count.; response, extra: Expected response and extra responders over placebo. Unit: proportion.; cost: Drug cost per course, with sharing or with whole vials (HE-FM-DOSE-003). Unit: pounds.; average: Cost per extra responder against placebo. Unit: pounds per responder.; icer: Incremental cost per extra responder against the previous frontier dose, missing for a dominated dose. Unit: pounds per responder.

  Assumption: Placebo, at zero drug cost, is the starting point of the frontier; other costs are left out, and a dose is dominated when another dose or a blend of two gives more responders for no more cost (HE-FM-DOM-002, HE-FM-EXD-001).

  Worked example (Three doses with vial sharing, article example): With E_0 = 0.20, E_max = 0.40, ED_50 = 100 mg, h = 2, 100 mg vials at 300 pounds and 12 administrations, doses of 100, 150 and 200 mg cost 3,600, 5,400 and 7,200 pounds a course for 0.2000, 0.2769 and 0.3200 extra responders, with average ratios of 18,000, 19,500 and 22,500 pounds and incremental ratios of 18,000, 23,400 and 41,786 pounds, as in the article. dose = 100, 150, 200; E0 = 0.2; Emax = 0.4; ED50 = 100; h = 2; vial_mg = 100; vial_price = 300; n_admin = 12; icer = 18000, 23400, 41786

  Worked example (Same doses with whole vials): Without vial sharing 150 mg needs two vials and costs 7,200 pounds a course, the same as 200 mg, so it is dominated (average ratio 26,000 pounds, computed here for illustration) and the step from 100 mg to 200 mg costs 30,000 pounds per extra responder, as in the article. icer = 18000, dominated, 30000

  Excel: With doses in DoseGrid and the parameters named as in HE-FM-DOSE-001, =PlaceboResp+MaxEffect*DoseGrid^HillCoef/(ED50Dose^HillCoef+DoseGrid^HillCoef) returns the responses, =DoseGrid*VialPrice/VialMg*AdminsPerCourse the shared-vial costs and =ROUNDUP(DoseGrid/VialMg,0)*VialPrice*AdminsPerCourse the whole-vial costs; after sorting by cost and deleting dominated doses, with the frontier costs in FrontCost and extra responders in FrontExtra, =(FrontCost-VSTACK(0,DROP(FrontCost,-1)))/(FrontExtra-VSTACK(0,DROP(FrontExtra,-1))) returns the incremental ratios into FrontICER and =FrontCost/FrontExtra the average ratios into FrontAvg (Excel 365).

  R: dose_ladder <- function(dose, E0, Emax, ED50, h, vial_mg, vial_price, n_admin) { resp <- E0+Emax*dose^h/(ED50^h+dose^h); gain <- resp-E0; fr <- function(cost) { icer <- rep(NA, length(dose)); c0 <- 0; g0 <- 0; repeat { cand <- which(gain > g0); if (length(cand) == 0) break; ic <- (cost[cand]-c0)/(gain[cand]-g0); j <- cand[order(ic, -gain[cand])[1]]; icer[j] <- min(ic); c0 <- cost[j]; g0 <- gain[j] }; data.frame(dose = dose, response = resp, extra = gain, cost = cost, average = cost/gain, icer = icer) }; list(sharing = fr(dose*vial_price/vial_mg*n_admin), whole_vials = fr(ceiling(dose/vial_mg)*vial_price*n_admin)) } Base R only; dose_ladder(c(100, 150, 200), 0.20, 0.40, 100, 2, 100, 300, 12) returns both tables in input order.

  Python: def dose_ladder(dose, E0, Emax, ED50, h, vial_mg, vial_price, n_admin): resp = [E0+Emax*d**h/(ED50**h+d**h) for d in dose]; gain = [r-E0 for r in resp]; fr = lambda c, c0, g0: (lambda cand: [] if not cand else (lambda j: [(j, (c[j]-c0)/(gain[j]-g0))]+fr(c, c[j], gain[j]))(min(cand, key=lambda i: ((c[i]-c0)/(gain[i]-g0), -gain[i]))))([i for i in range(len(c)) if gain[i] > g0]); costs = {"sharing": [d*vial_price/vial_mg*n_admin for d in dose], "whole_vials": [math.ceil(d/vial_mg)*vial_price*n_admin for d in dose]}; return {k: {"dose": list(dose), "response": resp, "extra": gain, "cost": c, "average": [x/g for x, g in zip(c, gain)], "icer": [dict(fr(c, 0.0, 0.0)).get(i) for i in range(len(dose))]} for k, c in costs.items()} Needs import math; returns the same values as the R function, with None for a dominated dose.

  Test (Last frontier ratio no lower than its average ratio): Along a correct frontier the incremental ratios rise, so the last one is no lower than the average ratio of the most expensive frontier dose, which is a weighted mean of the ratios before it. Expected result: TRUE. FALSE shows the dominated 150 mg dose kept in the whole-vial frontier, which gives a last step from 150 mg to 200 mg of 0 pounds per extra responder against an average of 22,500. Excel check: =INDEX(FrontICER,COUNT(FrontICER))>=INDEX(FrontAvg,COUNT(FrontAvg))-1E-9*INDEX(FrontAvg,COUNT(FrontAvg))

  Common error (Ranking doses with per-mg costs when vials are single use): With vial sharing 150 mg is chosen at an illustrative 25,000 pounds per extra responder; with whole vials it is dominated and the choice falls back to 100 mg, so the dispensing rule decides which doses are worth considering.

  Source: Bornkamp B, Pinheiro J, Bretz F, et al. DoseFinding: Planning and Analyzing Dose Finding Experiments. R package version 1.4-1. 2025. Reference manual, help page drmodels (full text read). drmodels; Bach PB, Conti RM, Muller RJ, Schnorr GC, Saltz LB. Overspending driven by oversized single dose vials of cancer drugs. BMJ. 2016;352:i788. doi:10.1136/bmj.i788 (full text read). opening section; National Institute for Health and Care Excellence. Single technology appraisal and highly specialised technologies evaluation: user guide for company evidence submission template (PMG24). London: NICE; 2015, last updated 31 March 2026 (full text of the cost-effectiveness chapter read). Section 3.5.6.

  response = E0 + Emax * dose^h / (ED50^h + dose^h); extra = response - E0; cost_share = dose * vial_price / vial_mg * n_admin; cost_vial = ceiling(dose / vial_mg) * vial_price * n_admin; icer = (cost_j - cost_prev) / (extra_j - extra_prev) along the frontier

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Implementations

• Excel

  Sigmoid Emax response from named cells

  With DoseMg, PlaceboResp, MaxEffect, ED50Dose and HillCoef named, the formula returns the expected response, held in RespDose.

  =PlaceboResp+MaxEffect*DoseMg^HillCoef/(ED50Dose^HillCoef+DoseMg^HillCoef)

Assumptions

• Monotonic sigmoid curve fitted over the doses that matter

  The response rises monotonically with dose towards a plateau, and the curve's parameters come from trials covering the doses being compared; outside that range the result rests on the functional form, not on data.

• Population-average response on the sigmoid Emax curve

  E_D is the average response in the population the curve was fitted to; subgroups by weight or tolerability can sit on different parts of the curve.

Worked examples

• Response at 150 mg with ED50 100 mg and Hill coefficient 2

  With E_0 = 0.20, E_max = 0.40, ED_50 = 100 mg and h = 2, the response at 150 mg is 0.20 + 0.40 x 22,500 / 32,500, or 0.476923, as in the article.

  E_0 = 0.2; E_max = 0.4; D = 150; ED_50 = 100; h = 2; E_D = 0.476923

• Response at 200 mg with ED50 100 mg and Hill coefficient 2

  At 200 mg the fraction is 40,000 / 50,000 = 0.80, so the response is 0.52, as in the article.

  E_0 = 0.2; E_max = 0.4; D = 200; ED_50 = 100; h = 2; E_D = 0.52

• Response at 200 mg with a hyperbolic Emax curve

  With h = 1 the same 200 mg dose gives 0.20 + 0.40 x 200 / 300, or 0.4667 (computed here for illustration).

  E_0 = 0.2; E_max = 0.4; D = 200; ED_50 = 100; h = 1; E_D = 0.4667

Common errors

• Costing a dose that lies on the plateau

  The ICH E4 guideline records that drugs have often been marketed first at doses later recognised as excessive, well onto the plateau of the dose-response curve; a plateau dose adds cost and harm without adding benefit, as the step from 150 mg to 200 mg (41,786 pounds per extra responder) shows in the article.

• Estimating ED50 from a narrow dose range

  Mawdsley and colleagues found in their triptan example that, with a limited dose range, the ED50 parameter of an Emax model proved difficult to estimate reliably; they borrowed strength by treating Emax or ED50 as exchangeable across agents in a class.

• Judging a dose by its average cost per extra responder

  In the article the average ratio rises only from 18,000 to 22,500 pounds between 100 mg and 200 mg while the incremental ratio from 150 mg to 200 mg is 41,786 pounds; the choice of dose rests on the incremental ratio between adjacent doses.

Sources

• DoseFinding definition of the sigmoid Emax model

  Bornkamp B, Pinheiro J, Bretz F, et al. DoseFinding: Planning and Analyzing Dose Finding Experiments. R package version 1.4-1. 2025. Reference manual, help page drmodels (full text read). drmodels: the sigmoid Emax model is f(d) = E_0 + E_max d^h / (ED_50^h + d^h), an extension of the hyperbolic Emax model by a parameter h that determines the steepness of the curve at the ED50; e0 is the placebo effect, ed50 the dose giving half of the asymptotic maximum effect and h the Hill parameter.

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• Emax as receptor saturation and ED50 as the half-maximum dose in model-based network meta-analysis

  Mawdsley D, Bennetts M, Dias S, Boucher M, Welton NJ. Model-based network meta-analysis: a framework for evidence synthesis of clinical trial data. CPT: Pharmacometrics and Systems Pharmacology. 2016;5(8):393-401. doi:10.1002/psp4.12091 (full text read). Section on model-based meta-analysis: the Emax model is widely used to model drug effect as a function of dose; Emax corresponds, physiologically, to the drug saturating the body's receptors and ED50 is the dose at which half the maximum effect is reached. Triptan example: with a limited dose range the ED50 parameter proved difficult to estimate reliably, so models with exchangeable Emax or ED50 across agents were fitted.

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• Origin of the Hill equation

  Goutelle S, Maurin M, Rougier F, Barbaut X, Bourguignon L, Ducher M, Maire P. The Hill equation: a review of its capabilities in pharmacological modelling. Fundamental and Clinical Pharmacology. 2008;22(6):633-648. doi:10.1111/j.1472-8206.2008.00633.x (abstract read). Abstract: the Hill equation was first introduced by A.V. Hill to describe the equilibrium relationship between oxygen tension and the saturation of haemoglobin, and many pharmacokinetic-pharmacodynamic models use it to describe nonlinear dose-response relationships.

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• ICH E4 on doses marketed on the plateau

  International Conference on Harmonisation. Dose-Response Information to Support Drug Registration (ICH E4). Step 4 version, 10 March 1994 (full text read). Section I: drugs have often been initially marketed at what were later recognised as excessive doses, well onto the plateau of the dose-response curve for the desired effect, and any given dose provides a mixture of desirable and undesirable effects, with no single dose necessarily optimal for all patients.

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