Analysis Approaches Handling Both Symptomatic Severity and Frequency
Elodie L. Plan, Kristin E. Karlsson, Mats O. Karlsson
Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden
Background
Graded events analyses are often accompanied with a loss of information by not handling the true nature of the data. Pharmacodynamic outcomes commonly consist of symptoms that are defined as events happening at a certain point with a certain degree of severity. Pharmacometric modelling having substantially improved over the past two decades, the response rate (RR) approach is more and more replaced by the use of a cumulative logit model for longitudinal data. Lewis Sheiner introduced this population model in 1994[1] following an analgesic trial[2] and enabling the analysis of ordered categorical (OC) data. The state of the patient reported at regular time-points is adequately described with the OC model; however, spontaneous events happening at specific time-points involve data simplification, e.g. by utilizing the number of events or the maximal severity of events within equispaced time-intervals[3]. In order to pursue learning[4] and theory[5], analysis approaches handling both symptomatic severity and frequency are suggested and explored in this work.
Objectives
(i) To identify shortcomings of currently used approaches analyzing symptoms reported as graded on a severity scale,
(ii) To introduce new mixed-effects models retaining the original nature of data,
(iii) To illustrate benefits of the novel methods in terms of (a) data description, (b) drug effect assessment, (c) data simulation properties, (d) drug effect detection power, (e) real case analysis.
Methods
Repeated Time-To-Categorical Event model (RTTCE) model: The RTTCE model is based on a repeated time-to-event (RTTE) model describing the hazard for an event to occur. The hazard consists of a mixed-effects baseline parameter potentially affected by a function depending on time, and/or covariates, including the exposure. In order to capture the severity of the events that occur, in the same single step, the RTTE model is combined with an OC model. Cumulative probabilities of the different categories of severity are modelled on the logit scale.
Repeated Categorical Events per Time-interval (RCEpT) model: If reported data do not correspond to graded events at each occurrence, but rather only to maximal scores across time periods, they require the model to be adapted. The RCEpT model, built in the same fashion as the RTTCE one, but considering time-intervals of a defined length, is able to fit such data. Depending on whether the hazard is assumed to be varying or constant within time-intervals, the RTTE part follows an ordinary differential equation or its analytical solution, respectively. As records represent maxima over n number of events undergone during time-intervals, the discrete probability distribution of n enters the equation of the OC part. The expected number of occurrences λ entering the Poisson distribution function is the integrated hazard in the time-interval. The probability distribution of maximal severity score is a function of the OC sub-model and the frequency distribution given by the integrated RTTE sub-model.
Data: The RTTCE model was employed to simulate data mimicking a Phase IIa clinical trial. The design included 72 individuals equally allocated to placebo or one of the five drug treatment dose levels, 10, 50, 100, 200 or 400 mg. Observations, time and grade of the symptoms, were recorded with a 2-minute precision during 12 hours.
Study: Stochastic simulations and estimations (SSE) were performed 500 times to produce vectors of parameters subsequently used for computations and resimulations. SSEs were facilitated by a routine developed in PsN[6] running NONMEM VII[7] and enabling alternative models for the estimation step, RCEpT and OC in this case.
Results
(a) Objective function values displayed a systematic drop when analyzing summarized RTTCE data with an RCEpT compared to an OC model.
(b) Drug effect could be characterized on both the hazard of the events, through an Emax function, and the probabilities of their grades, with a linear function. Individual response distributions at dose levels excluded during estimation step were correctly retrieved, using the RTTCE and RCEpT models, but not the OC model.
(c) OC generated maximal grades per time-intervals, but RTTCE and RCEpT were able to reproduce realistic graded events. When computing summarized data, severity proportions were more accurately mimicking original data with simulations from RTTCE-type models than from OC model.
(d) Power observed with the novel models was substantially increased for the given study settings, thus a smaller sample size than initially considered was needed to detect the same treatment effect.
(e) Real data of spontaneous symptoms recorded as maximal grade per day were successfully analyzed with OC and RCEpT; the latter presented a better fit to the data.
Conclusions
Modeling graded symptoms by extensively summarizing the information originally contained in the data, results in a poor description of the events, an incomplete assessment of the drug effect, and a large sample size required. RTTCE-type models demonstrated multiple benefits, which include good population and individual predictions, appropriate simulations properties, and high power. Given that one of the main challenges in pharmacometrics is to adequately measure the effect of a drug[8], the novel methods presented above represent a step further, by enabling a two-dimension evaluation of the exposure-response relationship, which can be performed simultaneously, unlike previously done[9], and incorporate correlation.
References
[1] Sheiner, L.B. Clin Pharmacol Ther 1994.
[2] Sheiner, L.B. et al. Journal of the American Statistical Association 1997.
[3] Xie, R. et al. Clin Pharmacol Ther 2002.
[4] Sheiner, L.B. Clin Pharmacol Ther 1997.
[5] Sheiner, L.B. Clin Pharmacol Ther 1989.
[6] Lindbom, L. et al. Comput Methods Programs Biomed 2005.
[7] Beal, S.L., Sheiner L.B & Boeckmann, A.J. NONMEM Users Guides 1989-2006.
[8] Sheiner, L.B. et al. Clin Pharmacol Ther 2002.
[9] Kowalski, K.G. et al. J Pharmacokinet Pharmacodyn 2003.
