C-QTc hysteresis check: a signed loop area on model residuals

1. The SAP step this formalizes
2. Statistic
3. Why the model matters
4. Operating characteristics
5. Diurnal confound and the Day -1 check
6. SAP text
7. What it does not do
8. Provenance and credit

The SAP step this formalizes

Concentration-QTc (C-QTc) analysis under ICH E14 Q&A (R3) (Q5.1) fits a prespecified model of ddQTcF (the change from baseline in QTcF minus the time-matched placebo mean) on plasma concentration, usually a linear mixed-effects model (Garnett et al. 2018). That model assumes the effect follows concentration with no delay. Before fitting it, the SAP checks for hysteresis: an effect that lags concentration, from distribution to the site of action, a metabolite or slow channel kinetics.

The check is usually visual. The registry SAPs listed under credit assess it with joint plots of the mean ddQTcF and mean concentration by time and a "hysteresis plot", and consider other models if "the curve of the hysteresis plot shows a counterclockwise loop" and the time of the peak effect is more than 1 hour from tmax. The CiPA Phase 1 SAP adds a formal one-sided Wilcoxon test at 1% alongside the same 1 hour rule. This page replaces the visual judgment with one number per participant and a test whose size and power are documented on synthetic data, and keeps the tmax gap beside it.

Statistic

One synthetic trial (32 active and 8 placebo participants, seed 20261025) through hysteresis_check(), the function the mock shells mirror:

Why the model matters

The closed-path area of any straight line (x, a + b x) is exactly 0, and a participant-level constant adds no area. So under a linear f the residual area equals the raw area of ddQTcF itself, whatever the fitted slope and intercepts. A curved direct effect is different. Sampled at 12 times, the path up and the path down pass through different points of the curve and the closing segment cuts a chord, so the raw path encloses area with no delay at all: 1.40 ms for the noise-free typical participant under the Emax model, against -0.002 ms on a 0.01 h grid and 2.34 ms for a real delay (linear, ke0 = 0.7 per h).

With noise, 32 active and 8 placebo participants, the raw-area test rejects in 0.296 (MCSE 0.014) of trials with an Emax direct effect and no delay. On residuals from the Emax fit the rate is 0.036 (MCSE 0.006), and 0.039 two-sided. The CiPA-style linearity check (a quadratic term tested two-sided at 5%, then the Emax model kept when its AIC is lower) picks Emax in 0.997 of these trials and 0.139 of the linear ones, and the area on residuals from the f it selects rejects in 0.036 (Emax truth) and 0.037 (linear truth); the MCSE of these rates is at most 0.016. When the Emax effect is also delayed (ke0 = 0.7 per h), both areas keep their power: 0.999 raw and 0.986 on Emax residuals. The residual area is conditional on the prespecified form: it removes the curvature the model describes, not curvature it misses.

F2. Example loops. Solid: the mean over 200 simulated active participants, corrected by the mean of 200 placebo participants (more than a trial has, so that the expected shape is visible), in time order (arrows; whiskers are plus or minus one standard error; the dotted segment closes the path from 24 h back to 0 h). Dashed: the noise-free typical participant (ka 1.2 per h, ke 0.15 per h). A single noisy participant's loop is mostly noise here (residual SD 5 ms against a peak effect of about 6 ms under the linear model); the test pools the signed areas across participants. Standalone files: a, b, c, d.

Operating characteristics

Synthetic known-answer design (32 scenarios; 1,000 replicates each, so the MCSE of a rate near 0.05 is 0.007; scenario k uses seed 20261025 + k, listed in T1 and in the CSV). 32 active participants and a pooled placebo group of 8 (for example, four cohorts of 8 active and 2 placebo participants), sampled at 0, 0.5, 1, 1.5, 2, 3, 4, 6, 8, 12, 16 and 24 h. One-compartment oral PK with ka = 1.2 exp(N(0, 0.3²)) per h, ke = 0.15 exp(N(0, 0.3²)) per h and D/V = 2; an effect compartment dCe/dt = ke0 (C - Ce), solved analytically. Every participant's dQTcF carries a common daily rhythm, 4 sin(2π(t + 18)/24) ms, and N(0, 5²) noise; active participants add a drug effect of 4 ms × Ce, or 20 Ce / (Ce + 0.4) ms for the Emax variant. ddQTcF is active dQTcF minus the time-matched placebo mean, as in the mock shells. One-sided alpha 0.05.

• KA1, size. Linear model, direct effect: the placebo-adjusted t-test rejects in 0.046 (MCSE 0.007); two-sided, 0.050; the parametric bootstrap, 0.048. With 16 and 32 placebo participants it rejects in 0.053 and 0.055. The one-sample sign-flip test on the same areas rejects in 0.236, 0.184 and 0.113: too often whenever the placebo mean is shared. In the crossover variant, where each participant's ddQTcF uses their own placebo period, the sign-flip test is exact and rejects in 0.048 (MCSE 0.007); two-sided, 0.056.
• KA2, curvature. Emax, no delay: raw area 0.296, Emax-residual area 0.036, bootstrap from the Emax fit 0.052 (MCSE at most 0.016).
• KA3, power. Linear model with delay, 32 active and 8 placebo participants: 0.254 at ke0 = 2 per h (equilibration half-life 0.35 h), 0.489 (MCSE 0.016) at 0.7 per h (1 h) and 0.551 at 0.35 per h (2 h). Power rises little beyond ke0 = 0.7 per h (F1), because a still slower effect site makes the effect itself smaller and flatter. The placebo group sets most of the noise: at ke0 = 0.7 per h power is 0.707 with 16 placebo participants, 0.876 with 32, and 0.876 in the crossover variant with the sign-flip test (MCSE at most 0.016).
• KA4, one SAD cohort. 6 active participants against the pooled placebo group of 8: size 0.048 (MCSE 0.007), against 0.084 for the one-sample sign-flip test, and power only 0.315 at ke0 = 0.7 per h and 0.352 at 0.35 per h. A cohort this size cannot rule a delay in or out; pooling the active cohorts, as the C-QTc model does, gives the 32-participant rows.
• KA5, tolerance. E = 4 C - 3 M with dM/dt = 0.35 (C - M) produces clockwise loops (noise-free area -1.86 ms). The one-sided test rejects in 0.001 (MCSE 0.001) and the two-sided test in 0.262 (MCSE 0.014).
• The tmax gap rule fires in 0.123 (individual peaks) and 0.165 (mean profile) of trials with no delay, and in 0.634 and 0.579 at ke0 = 0.7 per h, where the area test rejects in 0.489 (MCSE 0.016). In a single SAD cohort of 6 with no delay it fires in 0.297 and 0.277. It is reported beside the test, not used as the decision.

32 active + 8 placebo 32 active + 32 placebo 6 active (one SAD cohort) + 8 placebo

Diurnal confound and the Day -1 check

QTc has a daily rhythm. The time-matched placebo correction removes the part that active and placebo participants share: every simulated trial carries the common 4 ms rhythm above, and it never reaches ddQTcF (the tests check this to 1e-12). A time pattern the active participants do not share with the placebo group stays in ddQTcF, for example when a cohort's ECGs fall at different clock times from those of the placebo participants it is compared with. It can fake a loop or hide one, depending on where its peak falls relative to tmax. NC1 adds 3 sin(2π(t + phi)/24) ms to the active participants only, on Day 1 and Day -1, under the linear direct-effect model: the one-sided test rejects in 0.165 (MCSE 0.012) of trials at phi = 0 h, 0.701 at phi = 18 h and 0.000 at phi = 6 h (a clockwise artifact; two-sided 0.603), against 0.046 (MCSE 0.007) without it. This is why the SAP requires a time-matched placebo correction from participants studied alongside the active ones.

NC2, the Day -1 check. Pair each active participant's Day -1 time-matched QTcF (no drug, centered within the participant, minus the placebo group's centered Day -1 mean, as on Day 1) with that participant's Day 1 x_j, compute the same signed area and test it the same way, two-sided. With no drug on Day -1, any loop it shows comes from timing alone. Without a residual pattern it keeps nominal size: 0.050 (MCSE 0.007). A one-sample sign-flip test of the same areas rejects in 0.360, too often for the same reason as on Day 1. It is a weak detector. It flags the pattern in 0.564 of trials at phi = 18 h, where the Day 1 one-sided test rejects in 0.701, and in 0.576 at phi = 6 h. At phi = 0 h the primary one-sided Day 1 test is inflated to 0.165 (MCSE 0.012), 3.3 times nominal, but the Day -1 check flags the pattern in only 0.092 (MCSE 0.009) of trials, so it misses most trials of that kind; at phi = 12 h it flags 0.112. A clean Day -1 check does not rule a residual time pattern out.

SAP text

"Before the C-QTc analysis, delay between concentration and ddQTcF will be assessed by the mean signed area of each participant's concentration-ddQTcF loop, computed on residuals from the prespecified direct-effect model with concentrations scaled to each participant's Cmax. Because ddQTcF subtracts the same time-matched placebo mean from every active participant, the mean area will be tested with a t-test whose standard error adds the variance from the placebo mean (from each placebo participant's signed area, about the placebo mean, against the active participants' mean scaled concentration profile), with Welch-Satterthwaite degrees of freedom; one-sided alpha set in the SAP. If each participant's ddQTcF is corrected by their own placebo period, a participant-level sign-flip test will be used instead (exact for 20 or fewer participants per dose, otherwise 9,999 random sign vectors). The tmax gap will be reported alongside, and a Day -1 pseudo-loop check will be reported. Evidence of delay leads to consideration of an effect-compartment model."

A significant result prompts an effect-compartment model (Sheiner et al. 1979) or a metabolite explanation. It does not replace the C-QTc analysis. A non-significant result in a study with a small placebo group is weak evidence of no delay (KA3, KA4).

Mock shells that report it, in the TLF libraries (pseudocode; the SAS twins are not executed, no license):

• sad/f_cqtc_hysteresis.R · .sasFigure 14.3.5.4: mean ddQTcF against mean concentration in time order, one panel per dose
• sad/t_cqtc_hysteresis.R · .sasTable 14.3.5.5: n active and placebo, A-bar (95% CI), placebo-adjusted p values, tmax gap, model for f, Day -1 check
• mad/f_cqtc_hysteresis.R · .sasFigure 14.3.6.4: the same, for each profile day
• mad/t_cqtc_hysteresis.R · .sasTable 14.3.6.5: by dose and profile day

What it does not do

• It does not replace the prespecified C-QTc model, its estimate of the effect at the geometric mean Cmax, or effect-compartment modeling. It decides only whether those need a delay term.
• It is conditional on the form of f. It removes curvature that the prespecified model describes (linear or Emax here); curvature the model misses can still look like a loop, as the raw area does under Emax.
• Its size is not exact. Every active participant's ddQTcF shares the time-matched placebo mean, so the A_i are dependent and the test is a Welch-type t-test whose size was checked only on these synthetic designs: 0.046 with 32 active and 8 placebo participants and 0.048 with 6 and 8 (MCSE 0.007). The participant-level sign-flip test is exact only when each participant's ddQTcF uses their own placebo profile; applied to pooled-placebo ddQTcF it rejected in 0.236 of trials with no delay. With an estimated Emax curve the test was slightly conservative, at 0.036.
• Its power is set mostly by the placebo group. With 8 placebo participants the placebo-mean term is most of the variance of A-bar (the two groups' areas have a similar spread, so with a quarter as many placebo participants it is about four fifths): power at ke0 = 0.7 per h was 0.489, against 0.876 with 32 placebo participants.
• It does not estimate the delay. The area grows with the effect size and depends on the sampling times around tmax; it is a test, not a ke0 estimate.
• The synthetic concentrations have no assay error, every participant shares the nominal grid, placebo and active participants have the same noise SD, and multiple-dose steady state was not simulated. Those were not studied.
• A residual time pattern, a period effect or any other time pattern left in ddQTcF can produce a loop with no drug delay. The Day -1 check detects such a pattern only if Day -1 is processed the same way as Day 1, and even then it often misses it (NC2).
• Nothing here is validated or a regulatory deliverable. The R and SAS shells were written by the same author (no independent double programming), the SAS is not executed, and every dataset is synthetic.

Provenance and credit

Established prior art, credited first.

• ICH E14 (May 2005), E14 Q&A (R3) (10 Dec 2015; Q5.1 on concentration-response modeling) and E14/S7B Q&As (Step 4, 21 Feb 2022).
• Garnett C, Bonate PL, Dang Q, et al. Scientific white paper on concentration-QTc modeling. J Pharmacokinet Pharmacodyn 2018;45(3):383-397, doi:10.1007/s10928-017-9558-5; correction 45(3):399, doi:10.1007/s10928-017-9565-6.
• Sheiner LB, Stanski DR, Vozeh S, Miller RD, Ham J. Simultaneous modeling of pharmacokinetics and pharmacodynamics: application to d-tubocurarine. Clin Pharmacol Ther 1979;25(3):358-371, doi:10.1002/cpt1979253358 (the effect compartment).
• Louizos C, Yáñez JA, Forrest ML, Davies NM. Understanding the hysteresis loop conundrum in pharmacokinetic/pharmacodynamic relationships. J Pharm Pharm Sci 2014;17(1):34-91, doi:10.18433/j3gp53 (review).
• Zhong Y. Development of hysteresis analysis as a model-independent approach to assess temporal dissociation in pharmacokinetics and pharmacodynamics. PhD dissertation, Washington State University, 2016, hdl:2376/12054 (model-independent loop metrics, including the loop centroid).
• Upton RN, Doolette DJ. A quantitative alternative to the hysteresis plot for measurement of drug transit time. J Pharmacol Toxicol Methods 2002;47(1):45-51, doi:10.1016/s1056-8719(02)00207-1 (a model-independent area method).
• Wang J, Li W. Test hysteresis in pharmacokinetic/pharmacodynamic relationship with mixed-effect models: an instrumental model approach. J Biopharm Stat 2014;24(2):326-343, doi:10.1080/10543406.2013.859149 (a formal model-based hysteresis test).
• Cyclicity analysis, where the signed area between two series measures which leads: Baryshnikov Y, Schlafly E. Cyclicity in multivariate time series and applications to functional MRI data. 2016 IEEE 55th Conference on Decision and Control, 1625-1630, doi:10.1109/CDC.2016.7798498; Giusti C, Lee D. Iterated integrals and population time series analysis. arXiv:1811.03558 (2018).
• Welch BL. The generalization of "Student's" problem when several different population variances are involved. Biometrika 1947;34(1-2):28-35, doi:10.1093/biomet/34.1-2.28; Satterthwaite FE. An approximate distribution of estimates of variance components. Biometrics Bulletin 1946;2(6):110-114, doi:10.2307/3002019 (the degrees of freedom).
• CiPA Phase 1 ECG biomarker validation study, NCT03070470, statistical analysis plan (March 2017), section 5.2.1: a one-sided one-sample Wilcoxon test at the 1% level combined with a time gap of more than 1 hour between tmax and the largest mean effect.
• Registry statistical analysis plans for dedicated QTc studies, NCT06215521 (section 6.2.1) and NCT04818229 (section 6.1.1): hysteresis judged from the hysteresis plot (a counterclockwise loop), with a delay of more than 1 hour between the peak ddQTcF and tmax among the conditions.

What is new here is small: the area computed on residuals from the prespecified direct-effect model, so the check is conditional on the form the SAP already names; a test whose standard error allows for the time-matched placebo mean that every active participant's ddQTcF shares, with the exact sign-flip test kept for designs where nothing is shared; documented operating characteristics, including where the raw area fails under a curved direct effect, how much a one-sample test overstates significance with a pooled placebo, and how far the placebo group limits power; and the Day -1 confound check, with its weakness shown.

Provenance note. The framing, splitting a lag-1 cross-moment into symmetric and antisymmetric parts with a time-reversal null, follows my author draft Anticipating Network Contagions: When Directedness Outperforms Connectedness (with the editor at Applied Network Science; author draft 5 July 2026; github.com/patrickiben/directed-asymmetry-contagion).

Rerun it

Rscript cqtc_hysteresis/R/run_oc.R        # from the repository root; about 3 minutes; rewrites out/ and this page
Rscript cqtc_hysteresis/tests/test_area.R # geometry, exactness, the placebo-mean identity, known answers and the shell gates

• R/sim.Rdata-generating model, with the pooled placebo group
• R/area.Rarea, fits, placebo-adjusted t, sign-flip and bootstrap tests, tmax gap
• R/run_oc.Rscenarios, seeds and outputs
• tests/test_area.Rtests
• out/oc_results.csvscenario, n active and placebo, seed, method, rejection rate, MCSE, exact 95% interval
• out/noise_free_areas.csvnoise-free typical-participant areas

Run with R version 4.6.0 (2026-04-24), base R plus nlme (only for the random-intercept comparison in the tests). Two consecutive runs wrote byte-identical files. The direct-effect fits estimate the participant intercepts as fixed effects (profile least squares); because a constant has no area, the intercepts never change A_i, and the tests compare the Emax result with an nlme random-intercept fit.