Concentration-QT (C-QT) modeling

Under construction

Data

The default dependent variable (DV) is ΔQTcF.

Modeling

The pre-specified linear mixed effect (LME) C-QTc model (Equation 1) is used as the primary analysis.

$$\begin{array}{}\text{(1)}& \mathrm{\Delta }{\mathrm{QTc}}_{ijk}=({\theta }_1+{\eta }_{1,i})+{\theta }_2{\mathrm{TRT}}_j+({\theta }_3+{\eta }_{3,i})C_{ijk}+{\theta }_4{\mathrm{TIME}}_j+{\theta }_5({\mathrm{QTc}}_{i,j=0}-\overline{{\mathrm{QTc}}_0})\end{array}$$

[1]

• ${\theta }_1$: Population mean intercept wihtout drug effect
• ${\theta }_2$: Treatment effect (1=drug, 0=placebo)
• ${\theta }_3$: Slope (Continuous covariate. Parent or metabolite concentrations)
• ${\theta }_4$: Mean QTc change from baseline at TIME = k for placebo
• ${\theta }_5$: Baseline QTc effect
• $C_{ijk}$: Concentration for subject i in treatment j and time k;
• ${\mathrm{QTc}}_{i,j=0}$: Baseline QTc
• $\overline{{\mathrm{QTc}}_0}$: Mean of all the baselines

Evaluation

Plot	Model assumption tested	What to evaluate	Model impact
Exploratory plots
Time course of HR stratified by dose	No drug effect on HR	Consistency of change from baseline HR ΔHR with time, dose and treatment	If dose- or concentration-dependent effects on HR are observed, the relationship between QT and RR may differ between on- and off-treatment, impacting the QT correction differently between the two conditions This could potentially violate the assumption that the applied QTc correction is an adequate heart rate correction method
QTc versus RR intervals	QTc is independent of HR for drug-free and/or placebo treatments	Linear regression line should show the lack of relationship between QTc and RR intervals Range of HR are similar off- and on-drug	Individual correction factor is potentially poorly estimated due to narrow range of RR intervals within each subject which could bias the C-QTc model
Time course of mean concentrations and mean ΔQTc, ΔΔQTc intervals	Explore direct effect assumption Evaluate PK/PD hysteresis	Shape of PK- and QTc-time profiles, e.g., time course of effect, time of peak, return to baseline Magnitude of variability in PK and QTc	High inter-subject variability in ΔQTc can mask signal in mean curves-this is important in small-sized studies
C-ΔQTc	Evaluate linearity and heterogeneity assumptions between exposure and QTc across doses and studies	Shape of C-QTc relationship Magnitude of ΔQTc over observed concentration range Concentration range covers worse case clinical exposure scenario	Model-independent observations are not corrected for covariates and might therefore not appear to match model prediction Confounding factors not accounted for Heterogeneity between doses/trials
Goodness of fit plots
Model predicted versus observed ΔQTc	Model specification is adequate.	Model and observed values should fall around the line of unity without evidence of systematic bias. Loess smooth line with 95% CI should include the unity line over range of values	Systematic bias indicates model misspecification. For example, model predictions will be negatively biased at high values when PK/PD hysteresis is ignored and model predictions will be positively biased at high values when a linear model is applied to nonlinear data
Quantile-Quantile plot of residuals	Residuals follow normal distribution with mean of zero	Residuals should fall on the line of unity	Heavy tails indicate model misspecification. The plot does not indicate source of misspecification
Concentrations versus residuals Baseline QTc versus residuals	Model covariates are adequate	Residuals should be randomly scattered around zero The 95% CI of the loess line should include zero	Bias in residuals indicates model misspecification. A residual plot should be made for each model parameter
Time versus residuals
Active treatment versus residuals
Quantiles of concentrations and ΔQTc overlaid with slope of final model	Drug effect model is adequate	The concentration-QTc relationship obtained from final model should describe the observed data	Any systematic differences between the modeled versus observed data indicates model misspecification

References

[1]

Garnett C, Bonate PL, Dang Q, Ferber G, Huang D, Liu J, et al. Scientific white paper on concentration-QTc modeling. J Pharmacokinet Pharmacodyn 2018;45:383–97. https://doi.org/10.1007/s10928-017-9558-5.