Modeling Sleep Data for a New Drug in Development using Markov Mixed-Effects Models

ABSTRACT

Purpose

To characterize the time-course of sleep in insomnia patients as well as placebo and concentration-effect relationships of two hypnotic compounds, PD 0200390 and zolpidem, using an accelerated model-building strategy based on mixed-effects Markov models.

Methods

Data were obtained in a phase II study with the drugs. Sleep stages were recorded during eight hours of sleep for two nights per treatment for the five treatments. First-order Markov models were developed for one transition at a time in a sequential manner; first a baseline model, followed by placebo and lastly the drug models. To accelerate the process, predefined models were selected based on a priori knowledge of sleep, including inter-subject and inter-occasion variability.

Results

Baseline sleep was described using piece-wise linear models, depending on time of night and duration of sleep stage. Placebo affected light sleep stages; drugs also affected slow-wave sleep. Administering PD 0200390 30 min earlier than standard dosing was shown through simulations to reduce latency to persistent sleep by 40%.

Conclusion

The proposed accelerated model-building strategy resulted in a model well describing sleep patterns of insomnia patients with and without treatments.

APPENDIX

I. Final parameter estimates for baseline sub-model with the estimated variabilities. Transition probabilities at each breakpoint are given in percent with the positioning of the breakpoint in minutes. One transition is described with a linear model, and for this model neither an internal breakpoint nor a transition probability for the internal breakpoint is given, as none was estimated. The stage time effect is given at each breakpoint with the positioning of the breakpoints in minutes. Stage time effect for the first breakpoint was by definition 1. First stage time breakpoint was by definition 0. Variability was modeled as additive on the logit scale. Variability was allowed to be different at the different breakpoints if supported. IS—initial sleeplessness, 0—wakefulness, 1—stage 1, 2—stage 2, 3—slow—wave sleep, 5—REM sleep

Transition		Baseline sub-model										Variability model
From stage	To stage	f(relative bedtime)						f(stage time)				Between Subject Variability			Between Occasion Variability
		Probability (%)			Breakpoint (min)			Stage time effect		Breakpoint (min)
		1st	2nd	3rd	1st	2nd	3rd	2nd	3rd	2nd	3rd	1st	2nd	3rd	1st	2nd	3rd
IS	1	0.01a	0.668	79.3	0a	5.23	300a	NA	NA	NA	NA	–	0.956	15.3	12.6	–	–
0	1	31.0	41.8	34.0	0a	304	480a	0.131	0.0001a	4.24	925a	0.138 × 0.134b	0.134	0.427	–	–	0.379
	5	0.01a	1.13	2.33	45a	74.9	480a	0.00615	0.0001a	1.22	295a	–	2.39	0.978 × 2.39b	–	0.501	–
1	0	21.6	8.79	12.7	0a	22.6	480a	0.745	0.108	0.500a	74a	0.202	–	0.128	–	0.143	–
	2	19.4	22.8	14.0	0a	252a	480a	1.62	0.0001a	0.596	74a	0.121c	0.121c	0.121c	0.113	0.304	−0.113d
	5	1.52	5.72	6.77	75a	159	480a	0.331	0.0001a	1.21	74a	–	–	0.283	3.98	0.643	–
2	0	2.80	3.37	4.7	0a	287	480a	0.417	1.02	11.0	91a	0.248	0.144	0.144b	–	–	–
	1	3.19	36.9	6.89	0a	454	480a	0.377	0.313	6.48	91a	–	0.192	–	0.316	–	–
	3	0.01a	2.42	0.0480	0a	29.4	480a	2.52	2.70	32.5	91a	1.38c	1.38c	1.38c	–	0.546	1.80
	5	0.01a	1.42	2.14	0a	73.0	480a	0.264	0.100a	9.30	91a	–	–	–	–	–	–
3	0e	0.627	NA	1.67	0a	NA	480a	0.854	5.85	0.500a	83a	–	1.18	–	–	–	–
	1	0.01a	0.747	1.95	0a	160a	480a	2.33	4.05	31.1	83a	16.7	0.0357 × 16.7d	–	–	–	–
	2	6.98	11.7	16.6	0a	120	480a	0.170	0.165	1.21	83a	1.09b	1.09	2.66 × 1.09b	–	–	–
5	0	1.69	1.73	2.34	0a	337	480a	1.87	2.92	6.00a	65a	0.451	0.190	0.190b	−	–	–
	1	2.34	1.17	1.48	0a	290	480a	2.89	0.0001a	41.9	65a	–	0.789	–	–	–	0.537
	2	0.01a	0.808	0.212	0a	37.0	480a	0.939	4.68	9.54	65a	0.438c	0.438c	0.438c	–	–	–

NA not applicable

^aFixed parameter

^bNo variance was estimated for this breakpoint as the random effect of this breakpoint is equal to the random effect of the internal breakpoint. A fixed effect is estimated to give the width of the variability at this breakpoint

^cSame variability for all three breakpoints

^dNo variance was estimated for this breakpoint as the random effect of this breakpoint is equal to the random effect of the first breakpoint, A fixed effect is estimated to give the width of this variability at this breakpoint

^eLinear model of relative bedtime

II. Final parameter estimates of the placebo, the PD 0200390 and zolpidem effect sub-models with the between-subject variability estimated for the effects of PD 0200390. Only one model included between-occasion variability estimated instead of between-subject variability. Two placebo sub-models were exponential with the estimated parameters half-life and intercept, and two were step models. All sub-models describing the effect of PD 0200390 were linear with the parameter slope, but two that were described used step models with the estimated parameters constant low dose and constant high dose. All effects of zolpidem were linear, with slope being the only estimated parameter. IS—initial sleeplessness, 0—wakefulness, 1—stage 1, 2—stage 2, 3—slow-wave sleep, 5—REM sleep

Transition		Placebo			PD 0200390				Zolpidem
From stage	To stage	Half-life (h)	Intercept	Constant	Slope (ml/μg)	Step, low dose	Step, high dose	Variability	Slope
IS	1	1.09	1a	NA	1.01b	NA	NA	–	5.22
0	1				0.704	NA	NA	0.500c	4.45
	5				NA	0.415	0.156	0.586	5.53
1	0	2.10	−0.605	NA	−0.786	NA	NA	0.409	−1.66
	5	NA	NA	0.326	0.291	NA	NA	0.430	–
2	0				−1.11	NA	NA	–	−3.02
	1				−1.04	NA	NA	–	–
3	0				−1.54	NA	NA	–	−6.62
	1				−1.20	NA	NA	–	−6.13
5	0				−0.765	NA	NA	0.281	−4.03
	1	NA	NA	0.263	NA	0	−0.334	–	–
	2				0.454	NA	NA	–	–

NA not applicable

^aParameter not estimated

^bParameter included at a significance level of p ≤ 0.10, instead of p ≤ 0.05 as the other parameters

^cThis variability estimate is between occasion variability

III. Example code for a transition.