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.
\[ \Delta \mathrm{QTc}_{i j k} = \left( \theta_1 + \eta_{1, i} \right) + \theta_2 \mathrm{TRT}_j + \left( \theta_3 + \eta_{3, i} \right) C_{i j k} + \theta_4 \mathrm{TIME}_j + \theta_5 \left( \mathrm{QTc}_{i, j=0} - \overline{\mathrm{QTc}_0} \right) \tag{1}\]
[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
- \(C_{i j k}\): 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.