J Pharmacokinet Pharmacodyn (2014) 41:553–569 DOI 10.1007/s10928-014-9373-1
ORIGINAL PAPER
Population model of longitudinal FEV1 data in asthmatics: meta-analysis and predictability of placebo response Eleonora Marostica • Alberto Russu • Shuying Yang • Giuseppe De Nicolao • Stefano Zamuner • Misba Beerahee
Received: 21 November 2013 / Accepted: 6 August 2014 / Published online: 15 August 2014 Ó Springer Science+Business Media New York 2014
Abstract Asthma is an obstructive lung disease where the mechanism of disease progression is not fully understood hence motivating the use of empirical models to describe the evolution of the patient’s health state. With reference to placebo response, measured in terms of FEV1 (Forced Expiratory Volume in 1 s), a range of empirical models taken from the literature were compared at a single trial level. In particular, eleven GSK trials lasting 12 weeks in mild-tomoderate asthma were used for the modelling of longitudinal placebo responses. Then, the chosen exponential model was used to carry out an individual participant data meta-analysis on eleven trials. A covariate analysis was also performed to find relevant covariates in asthma to be accounted for in the meta-analysis model. Age, gender, and height were found statistically significant (e.g. the taller the patients the higher the FEV1, the older the patients the lower the FEV1, and females have lower FEV1). By truncating each trial at week 4, the predictive properties of the meta-analysis model were also investigated, showing its ability to predict long-term FEV1 response from truncated trials. Summarizing, the study suggests that: (i) the exponential model effectively describes the placebo response; (ii) the meta-analysis approach may prove helpful to simulate new trials as well as E. Marostica (&) A. Russu G. De Nicolao Department of Industrial and Information Engineering, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy e-mail: [email protected] Present Address: A. Russu Janssen Research and Development, Janssen Pharmaceutical Companies of Johnson&Johnson, Beerse, Belgium S. Yang S. Zamuner M. Beerahee Clinical Pharmacology Modelling and Simulation, GlaxoSmithKline, Stockley Park, UK
to reduce trial duration in view of its predictive properties; (iii) the inclusion of available covariates within the metaanalysis model provides a reduction of the inter-individual variability. Keywords Asthma Population approach Metaanalysis Predictability analysis Covariate analysis
Introduction Asthma, a chronic long-term disease that affects airways [1], is a complex and multi-factorial disease whose causes are not fully understood. Patients’ health state is usually assessed by the Forced Expiratory Volume in the first second (FEV1) which is a relevant parameter related to the pulmonary function obtained through a non-invasive spirometry test [2–6]. As for other diseases, the physio-pathological knowledge of asthma is insufficient to suggest mechanistic models of disease progression. Rather, a range of empirical models of longitudinal type are investigated, similar to those adopted for the empirical modelling of other diseases such as in psychiatry [7–12]. In particular, the placebo effect in patients affected by asthma has been widely discussed, as for other diseases such as pain, depression, and Parkinson [13–16]. In order to gain a better understanding of the placebo response, moving from the analysis of a single trial to the joint analysis of multiple trials might bring a definite advantage. Indeed, meta-analyses are being increasingly adopted across several therapeutic areas [17–21], although to our knowledge the only meta-analysis regarding placebo response in asthma is due to Wang and co-workers that elaborated the mean values of FEV1 obtained in 34 trials
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from the literature using an exponential model [22]. Herein, we perform an individual participant data (IPD) population meta-analysis of eleven trials considering not only the mean placebo responses of each trial but also all the individual FEV1 profiles. The eleven trials, retrieved from the GSK repository, were conducted in the North America, including United States and Canada, as part of the development of fluticasone propionate (FP) or FP/ Salmeterol. Only subjects who were randomized in the placebo group were analysed. The present work has four main objectives. First of all, an extensive comparison at single trial level is carried out to determine the best longitudinal model among seven candidate empirical models (linear, polynomial, Bateman, Weibull-and-linear, Emax, Hill, and exponential). The second objective is to obtain a statistical characterization of inter-trial variability by running an IPD meta-analysis considering simultaneously all the individual data from the eleven trials. The third objective is to perform a predictability analysis on the meta-analysis model to see whether it can be used to predict long-term outcomes from shortterm trials. The last main objective is to investigate whether the inclusion of covariate effects may improve the meta-analysis model, in order to attempt more efficient clinical trial designs. In particular, the inclusion of covariate effects in the model could potentially explain part of the random effects hence improving clinical trial efficiency in term of sample size requirement. Moreover, the design of a clinical trial may be improved for instance by optimizing duration and schedule of visits.
consent before starting the clinical trial. The clinical response was measured through the Forced Expiratory Volume in the first second (FEV1), that represents the amount of air that can be exhaled from the lungs in the first second of a forced exhalation. A spirometry testing was used to assess the lung function by measuring the FEV1. The sampling schedule was different for each subject in the different trials. Hence, a different number of samples per subject were collected. In particular, a maximum of 9 observations per subject were available. Overall, 6272 FEV1 placebo data were sampled from 1,151 patients. For each clinical trial, demographic covariates such as age, gender, body weight, height, and race were available. Moreover, asthma duration, smoking status, and concurrent asthma medication were considered. Further, the year of the beginning of the study was considered as a covariate. In particular, age, body weight, height, and the year of the beginning of each study (see Table 1) are continuous covariates, while gender, race, smoking status, asthma duration, and concurrent asthma medication were available in categorical format. More details about the continuous and categorical covariates are reported in Tables 2 and 3, respectively. All the clinical trials were comparable in terms of baseline FEV1 values. Concerning the duration of asthma, comparable values were available across the trials. In particular, these are the percentage of patients in each trial having asthma duration (i) lower than 5 years: from 9 to 18 %, (ii) greater than 5 years and lower than 10 years: from 12 to 27 %, and (iii) greater than 10 years: from 63 to 74 %.
Materials Methods In the present work, longitudinal data from eleven clinical trials in inhaled corticosteroids (ICS) naı¨ve patients diagnosed with mild-to-moderate asthma were considered (http://www.gsk-clinicalstudyregister.com/). In Table 1 a summary of the eleven clinical trials is reported. In particular, data from the placebo arm of the following trials were adopted to assess the performance of the proposed model:
Since several clinical trials were available, a two-step approach was adopted. First, all the trials were analysed separately (single-trial analysis). Once the best model was selected through the single-trial analysis, data from all the trials were merged and the best model was identified using all individual data (IPD meta-analysis). The population paradigm [23] was adopted to perform the analyses.
–
Software implementation
– –
two Phase II randomized, double-blind, doubledummy, parallel-group, placebo-controlled trials, six Phase III and one Phase IIIb randomized, doubleblind, parallel-group, placebo-controlled trials, two Phase IV double-blind, double-dummy, parallelgroup, placebo-controlled trials.
All the clinical trials lasted 12 weeks and placebo was given once or twice daily depending on the trial. The studies considered were approved by the Investigational Review Board (IRB) and all the patients gave informed
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Parameter estimation was carried out through software WinBUGS 1.4.3 [24]. Data management and graphical output were performed through R 2.13.1 [25]. Software R was also used to perform the covariate analysis. In particular, the function cor.test was used to perform the correlation test and obtain the p-values for the continuous covariates, while aov and TukeyHSD functions provided the p-values for the categorical covariates through the
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Table 1 Summary of the clinical trials Trial
Phase
Years
# Subjects
Table 2 Summary of the continuous covariates # FEV1 data
# Median (min, max) data/ subject
Covariate
Median (min, max)
Age (years)
30 (12, 78)
Height (cm)
170 (138, 199) 76 (32, 151)
1-FAP30008
III
2000–2001
100
554
6 (1, 7)
Weight (kg)
2-FLD302
III
1992–1993
84
669
9 (2, 9)
3-FLD304
III
1993–1994
40
301
9 (1, 9)
Minimum, median, and maximum values are reported for each covariate
4-FLI202
II
1991–1992
73
417
5 (1, 9)
5-FLI302
III
1991–1992
87
495
6 (1, 9)
6-FLTA2003
II
1995–1996
73
523
9 (1, 9)
7-FLTA3020
III
1997–1998
43
269
8 (1, 8)
8-FLTA4030
IV
1997–1998
115
748
7 (1, 7)
9-FLTA4031
IV
1997–1998
114
731
7 (1, 7)
10-FPD40009
IIIb
2000–2001
210
965
5 (1, 6)
11-SAS30022
III
2001–2003
212
600
3 (1, 3)
Table 3 Summary of the categorical covariates Covariate Gender Race
ANOVA test. The TukeyHSD function refers to the Tukey’s Honest Significant Difference method that assesses the differences between the mean values of the compared groups defined in terms of the covariate values.
Smoking status
Asthma duration
Single-trial analysis Concurrent asthma medication
Methods In order to define the most suitable model to describe the available data, a comparison was carried out by identifying a range of empirical models taken from the literature at a single trial level. According to a population framework, the index j in the following models refers to the subject of each trial, that is j ¼ 1; . . .; Nsubj , where Nsubj is the number of subjects in the trial. Prior distributions for each parameter are reported for each of the following models, assessed on each trial. In particular, the same log-normal prior distribution was assumed for the baseline, that is logAj N ðlA ; sA Þ; lA Nð1; 10Þ; sA C1 ð1; 0:01Þ. 1.
2.
3.
FEV1j ðtÞ ¼ Aj þ Bj t þ Cj t2
ð2Þ
where parameters Aj and Bj have the same meaning as in the linear model (1) as well as the same prior
Male
586 (51 %)
Female
565 (49 %)
White
933 (81 %)
African American
123 (11 %)
Asian
18 (2 %)
Hispanic
60 (5 %)
Other Never smoked
17 (1 %) 896 (81 %)
Former smoker
205 (18 %)
Current smoker
7 (1 %)
\5 years
146 (13 %)
5 years to \10 years
192 (17 %)
10 years
813 (70 %)
No
474 (41 %)
Yes
677 (59 %)
distributions, while Cj is the quadratic coefficient normally distributed as Cj NðlC ; sC Þ; lC N ð0:5; 1Þ; sC C1 ð1; 0:01Þ. Bateman [7, 8, 10]: FEV1j ðtÞ ¼ Aj þ Bj ðekon j t ekrec j t Þ
ð1Þ
where Aj is the baseline value of the response and Bj represents the slope. In particular, Bj NðlB ; sB Þ; lB Nð1; 1Þ; sB C1 ð1; 0:01Þ. Polynomial [7, 10]:
# Subjects ( %)
For the study FLTA3020 the covariate smoking status was not available
Linear [10]: FEV1j ðtÞ ¼ Aj þ Bj t
Value
4.
ð3Þ
where Aj is the baseline, Bj NðlB ; sB Þ; lB N ð0:1; 1Þ; sB C1 ð1; 0:5Þ modulates the amplitude of the response, konj controls the onset of the treatment effect and krecj regulates the recovery of the health state towards the baseline. The same log-normal prior distribution was assumed for both konj and krecj as logPj NðlP ; sP Þ; lP Nð0:5; 2Þ; sP ð1; 0:01Þ, where Pj represents konj and krecj . Weibull ? Linear [7, 8, 11]: FEV1j ðtÞ ¼ Aj e
ðt t Þbj dj
þ hrecj t
ð4Þ
where Aj is the baseline, td j and bj account for the amelioration of the health state with respect to the
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Table 4 Comparison of the different models on the different trials: DIC values Trial
Linear
Polynomial
Bateman
Weibull?Linear
Emax
Hill
Exponential
FAP30008
39.6
36.5
18.1
23.3
159.0
105.7
2172.1
FLD302
365.2
357.2
360.2
302.5
241.0
232.0
234.7
FLD304
153.0
156.5
152.6
101.8
53.8
102.7
37.8
FLI202
104.8
34.4
157.4
83.5
57.8
80.6
32.2
FLI302
455.5
385.2
419.9
319.7
300.3
285.1
304.2
FLTA2003
322.9
273.5
377.7
256.3
175.9
237.9
183.6
FLTA3020
199.5
227.3
186.5
164.6
192.4
184.8
153.0
FLTA4030
586.1
643.4
618.0
549.0
456.9
480.5
394.4
FLTA4031
341.6
372.2
364.8
279.3
206.8
203.5
197.5
FPD40009
233.8
247.9
131.6
138.8
1.2
112.5
60.8
SAS30022
418.8
143.6
–
–
218.5
–
210.7
5.
baseline. In particular, logbj Nðlb ; sb Þ; lb N logtd j Nðltd ; std Þ; ltd N ð0:5; 10Þ; sb ð1; 0:01Þ, ð4; 10Þ; std ð1; 0:01Þ, and loghrecj Nðlhrec ; shrec Þ; lhrec Nð0:01; 5Þ; shrec ð1; 0:01Þ Emax [12]: FEV1j ðtÞ ¼ Aj þ EMAXj t=ðET50j þ tÞ
6.
where Aj is the baseline, EMAXj defines the maximum effects, and ET50j is the time relative to the 50 % of the maximum effect. The parameter EMAXj is characterized by the same prior distributions considered for the baseline, while logET50j NðlET50 ; sET50 Þ; lET50 Nð0; 1Þ; sET50 Nð1; 0:1Þ. Hill [12]: c
FEV1j ðtÞ ¼ Aj þ EMAXj tcj =ðET50j j þ tcj Þ
7.
ð5Þ
ð6Þ
where Aj ; EMAXj , and ET50j have the same meaning as in the Emax model (5), while cj is the Hill coefficient determining the steepness of the sigmoid-shaped profile. Same prior distributions as for the Emax model were considered. Moreover, cj has the same prior as ET50j . Exponential: FEV1j ðtÞ ¼ Aj þ Gj ð1 e
Tt
j
Þ
Results
ð7Þ
where Aj is the baseline, Gj is the gain and Tj is the time constant. In particular, the gain represents the asymptotic improvement of the patient’s health state, and the time constant determines the time needed to reach the steady-state. Note that, the smaller T the faster the response. Log-normal prior distributions were assumed for all the parameters. In particular the baseline and the gain are characterized by the same distribution, whereas logTj NðlT ; sT Þ; lT Nð0; 1Þ; sT C1 ð1; 0:1Þ.
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Parameter estimation of all the population models was performed through WinBUGS [24] carrying out the comparison in terms of the Deviance Information Criterion (DIC) [26], an index automatically provided by WinBUGS to select optimal model complexity. In particular, the model providing the highest number of wins on the clinical trials in terms of DIC was selected as the best model. Moreover, established distributions were used for the priors. Also, for each model parameter reasonably flat priors were defined. To make an example, the prior for lT assumes that the typical value of the time constant T lies in ð1:96 102 ; 1:96 102 Þ days with probability 0.95. Since the settling-time of the exponential model is about 4T, the adopted prior is very flat as it encompasses responses showing practically instantaneous onsets and ones whose transient ends after more than 2 years. The predictive performance of the model was assessed through the Visual Predictive Checks (VPCs) [27]. Since heterogeneous sampling schedules were adopted for each subject in each trial, the binning procedure was used to perform the VPCs. In particular, about the same number of observations were included in each bin.
Considering each trial separately, the seven models selected from the literature were compared in terms of DIC values which are reported in Table 4. The exponential model yielded the best performance in six out of eleven trials. Moreover, concerning trial SAS30022, Bateman, Weibull?Linear, and Hill models could not be identified because only a maximum of three data per subject were available. Based on this analysis, the exponential model was selected for the subsequent IPD meta-analysis as it resulted to be the best model most frequently (six case out of
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CV% of A
TV of A 3.0 40
2.8
35
CV%
2.6
TV (L)
Fig. 1 Box-plots of the posterior distributions of the typical value TV (on the left) and of the percent coefficient of inter-individual variation CV% (on the right) for each parameter in each clinical trial
557
2.4 2.2
30 25 20
2.0
15
1.8 1
2
3
4
5
6
7
8
9
10 11
1
2
3
4
5
Trial
6
7
8
9
10 11
8
9
10 11
Trial
TV of G
CV% of G 500
1.5
1.0
CV%
TV (L)
400
0.5
300 200 100 0
1
2
3
4
5
6
7
8
9
10 11
1
3
4
5
6
7
Trial
TV of T
CV% of T 10^150
1000
100
10^100
CV%
TV (days)
2
Trial
10
10^50 1
0.1
10^0 1
2
3
4
5
6
Trial
eleven). As seen from the posterior distributions (Fig. 1), satisfactory parameter estimates were obtained for all the parameters in all the clinical trials. Nevertheless, it was found that the gain G had quite a high inter-individual variability, and the time constant T was characterized by a high inter-individual variability within the trials. This can be due to the variability of response shapes observed in the data. In particular, there are non-responder patients who have a flat profile (i.e. G close to 0), while there are responder patients who may have a very fast (i.e. low T) or a slow (i.e. high T) response. Figure 1 shows the box-
7
8
9 10 11
1
2
3
4
5
6
7
8
9 10 11
Trial
plots of the posterior distributions of the typical value (left column) and of the percent coefficient (right column) of inter-individual variation of each parameter for each trial. The exponential model was able to describe well both the typical curve and the individual profiles of each clinical trial. Moreover, the inter-individual variability (assessed by means of VPCs) of each study was explained well. An example of individual profiles is shown in Fig. 2 (top panel), while VPCs of a representative trial are shown in Fig. 3.
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Single−trial Analysis 0
20
40
60
80 100
0
20
40
60
80 100
Subject 834
Subject 933
Subject 1033
Subject 1078
Subject 1150
Subject 1190
Subject 1205
Subject 1213
5 4 3 2
FEV1 (L)
Fig. 2 Individual fittings obtained with the single-trial analysis (top panel) and with the mixture meta-analysis (bottom panel) for a representative subset of patients belonging to trial 2 (FLD302). Solid line individual profile; grey dashed lines 90 % credible intervals; black dashed line typical curve. Note that, the overall typical curve is shown in the bottom panel
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1
5 4 3 2 1
0
20
40
60
80 100
0
20
40
60
80 100
Time (days)
Meta−Analysis 0
20
40
60
80 100
0
20
40
60
80 100
Subject 834
Subject 933
Subject 1033
Subject 1078
Subject 1150
Subject 1190
Subject 1205
Subject 1213
5 4 3
FEV1 (L)
2 1
5 4 3 2 1
0
20
40
60
80 100
0
20
40
60
80 100
Time (days)
Meta-analysis Methods Based on the results obtained through the single-trial analysis step, the exponential model was extended in order to perform a population model-based IPD meta-analysis. In order to better address the inter-individual variability characterizing each trial, it was assumed to identify two possible sub-populations, i.e. responder and non-responder subjects. Moreover, the meta-analysis was performed to assess differences among the trials and estimate the inter-trial variability. In particular, each model parameter is characterized by a subject-related term, that accounts for the differences between patients, and a study-
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related term which explains differences between the clinical trials. As in the single-trial analysis, WinBUGS was used to implement the model. In the following, a description of the population model is reported in detail. Since a mixture model was assumed, the identification of responder and non-responder subjects was performed. In particular, in addition to the parameters A; G, and T, a parameter relative to the probability of being responder or non-responder was considered. More in detail, the mixture meta-analysis model was defined as: FEV1i ðtÞ ¼ Ai þ bi ðGi ð1 et=Ti ÞÞ
ð8Þ
where i ¼ 1; . . .; N denotes the i-th patient, N is the total number of patients of all the trials, and bi BernoulliðpÞ. In
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found that mA ¼ 0:25; mG ¼ 2:73, and mT ¼ 1:64, while tA ¼ tG ¼ tT ¼ 2:30. The marginalized log-normal priors for the overall individual parameters Ai ; Gi , and Ti can be obtained
Single−trial Analysis
5
ðlogAi jlA ; qA ; sA Þ NðlA ; qA þ sA Þ
FEV1 (L)
4
ðlogGi jlG ; qG ; sG Þ NðlG ; qG þ sG Þ
ð13Þ
3
ðlogTi jlT ; qT ; sT Þ NðlT ; qT þ sT Þ 2
It can be demonstrated that the variance of the prior distribution for each individual parameter Ai ; Gi , and Ti is given by the sum of the variance of the subject-related term and the one of the study-related term, considering the logdomain. More in detail, considering Pi the generic individual parameter, based on the definition of meta-analysis model:
1
0
20
40
60
80
Time (days)
Fig. 3 Visual Predictive Checks of FEV1 data for a representative trial (2-FLD302) obtained with the single-trial analysis
logPi ¼ logPsubji þ logPstudyk
particular, the probability p is described through a Betað1; 1Þ distribution. Therefore, bi ¼ 1 identifies a responder, i.e.:
Since there is no correlation, the variance of logPi is: Var ½logPi ¼ Var logPsubji þ Var logPstudyk ¼ sP þ qP
FEV1i ðtÞ ¼ Ai þ Gi ð1 et=Ti Þ
FEV1i ðtÞ ¼ Ai
ð10Þ
The individual parameters were defined as follows:
As in the single-trial analysis, the measurement error was assumed to be normally distributed with variance defined through the inverse gamma prior C1 ð1; 0:001Þ. Predictability analysis
1
ðlogAi jAstudy k ; sA Þ NðlogAstudy k ; sA Þ
sA C ð1; 0:01Þ
ðlogGi jGstudy k ; sG Þ NðlogGstudy k ; sG Þ
sG C1 ð1; 0:01Þ
ðlogTi jTstudy k ; sT Þ NðlogTstudy k ; sT Þ
ð15Þ
ð9Þ
While, if bi ¼ 0, the model for the non-responders is:
sT C1 ð1; 0:1Þ
ð11Þ where k ¼ 1; . . .; 11 identifies the clinical trial. Further, Astudy ; Gstudy and Tstudy are the study-related terms for each parameter. Moreover, log-normal prior distributions were assumed for the individual parameters. The associated variance was specified by means of inverse gamma distribution ðC1 Þ. More specifically, the priors related to the study-effect were
The proposed mixture meta-analysis model was applied to perform a predictability analysis. Its aim was to assess whether the model could be used to predict long-term outcomes from short-term trials. Therefore, the following procedure was carried out: 1. 2.
One at a time, each trial was truncated at week 4 removing subsequent data. Parameter estimation of the mixture meta-analysis model was performed considering all the data up to 12 weeks of the remaining ten trials and the data until week 4 of the truncated one. In particular, besides A; G, and T, also the probability of being responders or
ðlogAstudy k jlA ; qA Þ NðlA ; qA Þ
lA NðmA ; tA Þ
qA C1 ð1; 0:01Þ
ðlogGstudy k jlG ; qG Þ NðlG ; qG Þ
lG NðmG ; tG Þ
qG C1 ð1; 0:01Þ
ðlogTstudy k jlT ; qT Þ NðlT ; qT Þ
ð14Þ
lT NðmT ; tT Þ
The values of the means mA ; mG ; mT , and variances tA ; tG and tT were calculated from the data. In particular, it was
ð12Þ
1
qT C ð1; 0:1Þ
non-responders, given by p, as well as the parameter b were estimated.
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3.
4.
5.
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The prediction Root Mean Square Error (RMSE pred) was calculated based on the difference between the removed and the predicted FEV1. For each truncated trial, the prediction RMSE was then compared to the RMSE based on the difference between the observed FEV1 data and those predicted using the complete datasets of the eleven trials. Moreover, two commonly used metrics were calculated for each truncated trial: the median of the absolute value of the percent prediction errors (PE %) and the absolute value of the median of PE % [28]. Typical and individual profiles, and VPCs of both complete and truncated trials were computed over the 12 weeks. This procedure was repeated for all the eleven trials.
2.
3.
Covariate analysis Once the mixture meta-analysis model was identified on all the clinical trials, a further analysis was performed in order to detect statistically significant covariates which may affect the patient’s health state. For the purpose of establishing a base model to be used for the covariate analysis, responder and nonresponder subjects were identified using the posterior estimates of individual parameters bi obtained through model (8). In particular, the ith subject was considered as a responder if pðbi ¼ 1jFEV1Þ 0:5, or a non-responder otherwise. Model (8) was then re-identified by assuming bi ¼ 1 for responders and bi ¼ 0 for non-responders. Such model was therefore used as the base model for the covariate analysis. Since the covariate smoking status was not available for study FLTA3020, the covariate analysis was carried out considering the remaining ten trials (to ensure consistency, the meta-analysis model was re-identified on these ten trials). Moreover, the covariate concurrent asthma medication was not used because all the patients belonging to 8 clinical trials had this covariate equal to 1. Therefore, age, height, weight, gender, race, smoking status, asthma duration, and year of the beginning of the study were tested. The following steps were performed to carry out the covariate analysis: 1.
Preliminary selection: a correlation test was performed between each continuous covariate and the model parameters A; G, and T for each individual. As for the categorical covariates, the ANOVA test [29] was performed to assess if the mean values of the parameters A; G, and T in the groups identified by the categorical covariates were significantly different from each other. For both continuous and categorical covariates the parameter estimates were considered in the logarithmic domain. Note that, the possible correlation between each covariate and the parameters gain and time constant was considered only for responder subjects.
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4.
Single-covariate analysis: based on step 1, a subset of covariates was selected. Each of them was included, one at a time, in the meta-analysis model modulating the parameter for which a significant correlation had been found. If a covariate had been found to be significant on two or all parameters, all the possible combinations of models were assessed. For instance, if covariate COVi had been found significant for parameter A and G, three models were investigated: A depending on COVi ; G depending on COVi , and both A and G depending on COVi . The best model for each covariate was selected through the DIC. Full model: the statistically significant covariates (i.e. the ones yielding lower DIC values with respect to the meta-analysis model without covariates) selected through the single-covariate analysis were included in the meta-analysis model in order to define the full model. Backward procedure: once the full model was identified on all the data from the ten trials, a backward approach was carried out. Each covariate was removed one at a time. The models so obtained were identified in order to obtain a DIC value. This process ended when no lower DIC can be found, with respect to the full model.
In terms of mathematical formulation, the meta-analysis model accounting for a covariate COVi modulating a generic parameter P is: logPi ¼ logPsubj i þ logPstudy k þ logPk ðCOVi COV k Þ ð16Þ where i ¼ 1; . . .; N; k ¼ 1; . . .; 11, and COV k is the average value of the covariate within the kth study. Above, logPk is a study parameter, with log-normal, flat prior distribution. Moreover, parameter P could be the baseline A, the gain G, or the time constant T. Note that if the covariate is binary, then COV k coincides with the proportion of subjects such that COVi ¼ 1 within the kth study. Results The exponential model was identified after merging data from all the trials. The mixture meta-analysis model provided satisfactory parameter estimates. In particular, the identification of the responder and non-responder subpopulations produced a shrinking of the posterior distributions of the inter-individual variability of the gain and the time constant. The summary of the posteriors of the parameter estimates obtained with the meta-analysis model are reported in Table 5. The exponential model yielded also good performance when applied in the meta-analysis context. The proposed
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Table 5 Posterior summaries of the parameter estimates of the mixture meta-analysis model Parameter
Mean
SD
2.5 %
Median
97.5 %
Table 6 Number and percentage of responders identified for each clinical trial according to the meta-analysis model (# Responders MA (%)) and for each truncated trial according to the predictability analysis (# Responders PA (%))
TV of A
2.39
0.048
2.30
2.39
2.49
Trial
# Subjects
TV of G
0.66
0.033
0.6
0.66
0.73
# Responders MA (%)
# Responders PA (%)
1-FAP30008
100
15 (15 %)
18 (18 %)
2-FLD302
84
40 (48 %)
84 (100 %)
TV of T
7.28
1.13
5.17
7.23
9.57
CV% of Asubj
22.8
0.53
21.8
22.8
23.9
CV% of Astudy
5.98
1.44
3.86
5.73
9.40
3-FLD304
40
31 (78 %)
35 (88 %)
CV% of Gsubj
38.8
3.36
32.2
38.8
45.4
4-FLI202
73
26 (36 %)
52 (71 %)
CV% of Gstudy
10.38
3.96
5.11
9.62
20.4
5-FLI302
87
28 (32 %)
56 (64 %)
CV% of Tsubj
164
43.7
100.2
156.2
269.3
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73
35 (48 %)
73 (100 %)
CV% of Tstudy
31.3
13.9
15.8
27.9
66.02
7-FLTA3020
43
13 (30 %)
13 (30 %)
Variance of
0.061
0.0013
0.059
0.061
0.064
8-FLTA4030
115
58 (50 %)
61 (53 %)
9-FLTA4031
114
37 (32 %)
24 (21 %)
10-FPD40009 11-SAS30022
210 212
10 (5 %) 45 (21 %)
3 (1 %) 15 (7 %)
TV typical value, CV % percent coefficient of inter-individual or intertrial variation
model was able to describe both the typical curve of the whole meta-analysis and the typical curve of each clinical trial. Note that, since the meta-analysis approach is applied, the typical curve of each specific study is obtained through the parameter estimates provided by the identification of the meta-analysis model. Moreover, good individual fittings were obtained and they were similar to the ones provided by the single-trial analysis. An example of a subset of individual profiles is shown in Fig. 2 (bottom panel). The mixture meta-analysis model was able to identify the responders and non-responders sub-populations. More in details, the percentage of responders identified for each clinical trial according to the meta-analysis model ranged from 5 % (Trial FPD40009) to 78 % (Trial FLD304) (see Table 6). The estimated overall percentage of responders was 29 %. Further, the predictive performances of the mixture meta-analysis model were assessed in terms of VPCs. In particular, stratified VPCs on the two identified sub-populations (responders and non-responders) were performed. The proposed mixture model was able to explain well both the inter-individual and the inter-trial variability for responders and non-responders considering all the data (Fig. 4, bottom panels). Also, stratified VPCs on the two sub-populations were done for each of the specific trial. The model could describe the inter-individual variability in each trial properly. An example of VPCs for a representative trial is shown in Fig. 4 (top panels).
A column with the total number of subjects for each trial is also reported
week 4 onward based on the complete observed data. This is due to the increased uncertainty consequent to the removal of data in the next 8 weeks. In particular, an increase of the RMSE ranging from 22 % (trial 7-FLTA3020 in Table 1) to 54 % (trial 11-SAS30022 in Table 1), was obtained (see Fig. 5). In order to better assess the predictive performance of the mixture meta-analysis model, two commonly used metrics were calculated for each truncated trial: the median of the absolute value of the percent prediction errors (PE %) and the absolute value of the median of PE %. The worst values obtained were 8.8 and 4.18 % for the median of the absolute value of PE % and the absolute value of the median of PE %, respectively. Such values were both lower than the correspondent commonly used thresholds of 30 and 15 %, respectively [30–32]. In particular, the model was able to predict well both the individual profiles and the inter-individual variability. A subset of individual profiles obtained from truncated observations of representative patients is shown in Fig. 6. Moreover, VPCs of the sub-populations characterized by a sufficient number of subjects for the truncated trials are shown in Fig. 7. Further, Table 6 reports the number and percentage of responders identified through the predictability analysis in each truncated trial.
Predictability analysis Covariate analysis Based on the mixture meta-analysis model, the predictability analysis was performed as described in ‘‘Methods’’ section. For each truncated trial, the prediction RMSE was compared to the RMSE using all the data. Figure 5 shows that the prediction RMSE is higher than the RMSE from
Since the covariate smoking status was not available for the study FLTA3020, the covariate analysis was performed based on the parameter estimates provided by the metaanalysis model identified on the other ten trials. Parameter
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Fig. 4 Top panels Visual Predictive Checks of FEV1 data relative to non-responders (left panel) and responders (right panel) of a singletrial (2-FLD302) obtained with the mixture meta-analysis model.
Bottom panels VPCs obtained with the mixture meta-analysis for nonresponder patients (left panel) and responder patients (right panel)
estimates of the meta-analysis model identified on ten trials are reported in Table 7 (top). First, a preliminary selection of a subset of statistically significant covariates was performed through the correlation test between each covariate and the estimates of A, G, and T provided by the meta-analysis model. For the continuous covariates, the correlation test suggested that age, height, weight, gender, race, and smoking status had to be included in the model modulating the baseline (p-value \0.05, see Table 8), while height, weight, and gender modulated the gain. Since height and weight were correlated, the body mass index (bmi) was calculated. It was found that bmi had a statistically significant correlation with the baseline. For categorical covariates, the ANOVA test was performed. For all parameters, statistically significant differences were found between males and females. Also, based on the TukeyHSD test, the estimates of the baseline turned out to be statistically different between the White and the African American races. No
significant p-value was found considering all the other possible pairs of races. Based on the low number of subjects in the race groups different from the White one (see Table 3), it was decided to treat the race as a binary covariate, where the two categories were White and nonWhite. When considering the race as a binary covariate, a significant correlation was found for the baseline (p-value = 0.001). Furthermore, the estimate of A was statistically different between the former smoker and the patients who never smoked. Since only 7 subjects over 1151 were current smoker and given that no statistically significant difference was found between former smoker and current smoker, the two groups were merged and a binary covariate smoker/no-smoker was considered. Hence, the ANOVA test was performed between the introduced binary covariate and the three model parameters A; G, and T, and statistically significant p-values were found for the baseline. Finally, no significant correlation was found between the asthma duration and the model parameters. Table 8 reports
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which may modulate more than one parameter (e.g. for the covariate height, three candidate models were investigated: a first one where height affects the baseline, a second one where it affects the gain, and a third one accounting for a simultaneous effect of height on both baseline and gain). According to this procedure, it was found that the models yielding lower DIC values with respect to the metaanalysis model without covariates were the ones having:
0.2
– – – 0.1
RMSE (L)
0.3
RMSE RMSE pred
These three models were combined together to define the full model, where the baseline was modulated by age, height, and gender whereas the gain was modulated by height and gender. After the identification of the full model, the backward procedure was applied, by removing age, height, and gender one at a time. Moreover, the covariates height and gender were removed on both baseline and gain as well as on either baseline or gain separately. The final meta-analysis model accounted for age, height, and gender affecting the baseline and gender modulating the gain. More details on the DIC values are reported in Table 9. This final model provided satisfactory posterior distributions. In particular, the posterior distributions of the typical values of the three parameters A, G, and T were similar to the ones obtained with the meta-analysis model identified on ten trials. Moreover, compared to the estimates yielded by the meta-analysis model on ten trials, the inclusion of covariates modulating the baseline and the gain provided lower values for the percent coefficient of the inter-individual variation of both the two parameters.
0.0
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Fig. 5 Bar-plot of the Root Mean Square Error for each truncated trial calculated when the data after week 4 were removed from model identification (RMSE pred) and when they were included (RMSE). Trial FLTA3020 (trial 7 in Table 1) is the most predictable. The percent increase of the RMSE when predicting the truncated data ranges from 22 % (trial 7-FLTA3020 in Table 1) to 54 % (trial 11 SAS30022 in Table 1)
all the p-values obtained through the correlation test between the parameter estimates and each covariate. Based on the p-values obtained, the meta-analysis models accounting for the relevant covariates were developed to perform the single-covariate analysis (see ‘‘Methods’’ section). Note that, in this phase of the covariate analysis, each model can be affected only by one covariate
Predictability Analysis 0
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the baseline modulated by age, the baseline and the gain modulated by height, the baseline and the gain modulated by gender.
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b Fig. 7 Predictability analysis: Visual Predictive Checks of FEV1 data
for truncated trials obtained with the mixture meta-analysis model. VPCs not shown for trial 7-FLTA3020 because of the limited number of subjects available (13 responders and 30 non-responders, see Table 6). Solid lines observed 2.5, 50 and 97.5 percentiles; grey bands 95 % simulated prediction intervals; filled circles observations available for the model identification; open circles observations removed from the truncated trial
Table 9 DIC values of the meta-analysis model identified on 10 trials :(i) without covariates (MA), (ii) with the covariate age modulating the baseline (Age on A), (iii) with the covariate height modulating the baseline and the gain (Height on A and G), (iv) with the covariate gender modulating the baseline and the gain (Gender on A and G), (v) full model (FULL), (vi) final model (FINAL) Model
DIC
D
MA
1,454.3
Table 7 Posterior summaries of the parameter estimates of the mixture meta-analysis model identified on ten trials
Age on A
1,438.0
16.3
Height on A and G
1,419.3
35
Parameter
Gender on A and G
1,438.7
15.6
FULL
1,380.9
73.4
FINAL
1,376.6
77.7
Mean
SD
2.5 %
Median
97.5 %
No covariates TV of A
2.39
0.06
2.28
2.39
2.51
TV of G
0.64
0.03
0.58
0.64
0.71
TV of T
7.52
1.20
5.29
7.49
10.0
CV% of Asubj
22.8
0.53
21.8
22.8
23.8
CV% of Astudy
6.85
1.70
4.29
6.54
11.2
CV% of Gsubj
40.4
3.40
34.1
40.4
47.2
CV% of Gstudy
10.2
3.90
5.05
9.40
19.6
CV% of Tsubj
150.7
34.3
102.0
144.9
234.6
CV% of Tstudy
29.1
11.6
14.4
26.4
59.8
Variance of
0.06
0.0013
0.058
0.06
0.063
TV of A
2.42
0.06
2.32
2.42
2.53
TV of G TV of T
0.62 7.63
0.03 1.18
0.56 5.48
0.62 7.59
0.69 10.1
CV% of Asubj
13.4
0.34
12.7
13.4
14.1
CV% of Astudy
6.76
1.70
4.33
6.48
10.9
CV% of Gsubj
38.1
3.23
32.1
37.9
44.8
CV% of Gstudy
10.1
3.56
5.04
9.43
18.7
CV% of Tsubj
171.6
45.8
100.8
165.8
271.4
CV% of Tstudy
29.5
10.6
15.5
27.5
55.3
Variance of
0.06
0.0013
0.058
0.06
0.063
With covariates
TV typical value, CV% percent coefficient of inter-individual or intertrial variation
Table 8 p-values of the correlation test performed between the parameter estimates obtained from the identification of the metaanalysis model on ten trials and each covariate Covariate
A
G
T
Age
\ 1016
0:12
0:60
Height
\ 1016
106
0:16 0:42
8
Weight
10
0:005
Bmi
105
0:35
0:72
Gender
1016
104
0:69
Race
0:001
0:17
0:72
Smoking status
106
0:75
0:84
Asthma duration Study year
0:24 0:96
0:37 0:35
0:87 0:75
Moreover, the difference with respect to the model without covariates is reported in column D
Again, the inter-trial variability was lower than the interindividual one for each parameter, as already observed for the meta-analysis model. Posterior summaries are reported in Table 7 (bottom). Figure 8 shows the box-plots of the posterior distributions of the parameters relative to the covariates for each clinical trial. Considering the baseline parameters, AHeight was estimated greater than 1, suggesting that the taller the patient the higher the FEV1. Moreover, it was found that the older the patients the lower the FEV1. Finally, females are characterized by lower FEV1. In addition, similar values of the covariate parameters were obtained in each clinical trial. Figure 9 shows VPCs stratified on the covariate gender for the responder and non-responder populations. Each of the ten trials was simulated using the observed values of the individual continuous covariates age and height. In Fig. 8 it can be seen that in the female group (both responders and non-responders) the median and the upper percentiles appeared to be over-estimated, whereas in the male responder group the median and the lower percentiles are slightly under-estimated. This appears to be related to the baseline parameter (e.g. the upper percentile for the females is over-estimated already at t = 0). In particular, such issue may be possibly due to the different magnitudes of inter-individual variability characterizing the baseline for males and females. It is interesting to note that similar inter-individual variability is observed for responders and non-responders within the same gender.
Discussion Asthma is a complex and multi-factorial disease and the mechanistic knowledge of the underlying physiological processes is still scarce. Therefore, when limited physiopathological knowledge is available, the use of empirical
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Fig. 8 Box-plots of the posterior distributions obtained for the parameters relative to the covariates height, age, and gender for each clinical trial
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Height
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models offers a way to mathematically model placebo response. In this work, several empirical models taken from the literature were investigated on placebo responses collected in eleven asthma trials. The linear, polynomial, Bateman, Weibull-and-linear, Emax, and Hill empirical models are already successfully used in several therapeutic areas, such as pain and psychiatry [7–12]. Based on the single-trial analysis, the exponential model was selected as the best model since it provided the highest number of wins on the clinical trials available in terms of Deviance Information Criterion (DIC) value. In this work, this criterion was adopted to select the best model, however another possibility would have been to consider the sum of the DIC values provided by each model across the eleven trials. The exponential model had already been used to describe the disease progression of asthma in placebo treated subjects by Wang et al. [22] who only analysed mean responses, amounting to 34 longitudinal profiles, one
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for each trial. In contrast, in the present study a population analysis is carried out using individual data (1,151 subjects in total). The model being proposed is characterized by three parameters: the FEV1 starting value (baseline), the difference between the FEV1 value achieved at steadystate and the baseline (gain), and a parameter proportional to the time needed to reach the steady-state (time constant). This seemed to be the minimal set of parameters needed to capture all the essential features of placebo response. When the trials were analysed separately, overall parameter estimation provided satisfactory posterior distributions. Only the inter-individual variability of the time constant was more difficult to estimate. This was due to the wide range of possible individual profiles present in the data. In particular, there were patients that exhibited either very fast onset of placebo responses or very slow responses. Subsequent to the single-trial analysis, a model-based meta-analysis was performed. In particular, in order to
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Fig. 9 Covariate analysis: Visual Predictive Checks of FEV1 data for non-responder (left panels) and responder (right panels) patients stratified on the covariate gender. Solid lines observed 2.5, 50 and 97.5 percentiles; grey bands 95 % simulated prediction intervals
better explain the inter-individual variability, two possible sub-populations were assumed and identified (i.e. responder and non-responder patients). The IPD meta-analysis model proposed accounted for a trial effect on all the three parameters (baseline, gain, and time constant). The metaanalysis model provided narrower posterior distributions of both typical values and percent coefficients of inter-individual variation with respect to the single-trial analysis. Moreover, it was found that the inter-individual variability was greater than the inter-trial variability. The mixture meta-analysis model was able to identify 29 % of responders considering all the data from all the eleven
trials. Moreover, no statistically significant correlation between covariates and being responder was found. This finding motivated a predictability analysis at the meta-analysis level. The predictability of the mixture metaanalysis model was investigated and showed that the model was able to predict long-term outcomes (until week 12) based on trials truncated at week 4. The results of the predictability analysis obtained in this study suggests that placebo response in asthma can be predicted using shorter trials. Statistically significant covariates were identified, such as age, height, and gender which were able to explain part
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of the inter-individual variability characterizing both the baseline and the gain. Finally, some of the useful outcomes of this meta-analysis of placebo response in asthma include (i) more efficient design of future trials, given the population longitudinal analysis of the FEV1 response, (ii) potential reduction in the trial duration while maximising placebo response, and (iii) potential lack of effect of patient covariates studied on the placebo model parameters. A common debate for drug development teams regards the duration of efficacy studies early in clinical development prior to confirmatory studies. This meta-analysis provides some encouraging insight into the potential to undertake short term trials to predict placebo response that would be observed in longer term trials. It will be worth verifying whether this historical analysis outcome is confirmed in future studies in asthma. A natural extension of this work would be to develop a general longitudinal model accounting for both placebo and treatment effects. The points to be investigated would largely coincide with those studied herein: the possibility of describing drug response with a simple exponential model, the statistical significance and clinical relevance of covariates, and, finally, the predictability of long-term outcomes based on shorter observation periods.
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Acknowledgments The authors thank colleagues in the Respiratory Therapeutic Area in particular Shuyen Ho and Anna Ellsworth who supported preparation of the database for this work.
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References
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1. American Lung Association (2007) Trends in asthma morbidity and mortality. American Lung Association Epidemiology & Statistics Unit Research and Program Services 2. Miller MR, Hankinson J, Brusasco V et al (2005) Standardisation of spirometry. Eur Respir J 26:319–338. doi:10.1183/09031936. 05.00034805 3. Pierce R (2005) Spirometry: an essential clinical measurement. Aust Family Phys 34:535–539 4. American Thoracic Society (1991) Lung function testing: selection of reference values and interpretative strategies. Am Rev Respir Dis 144:1202–1218 5. Wagner NL, Beckett WS, Steinberg R (2006) Using spirometry results in occupational medicine and research: common errors and good practice in statistical analysis and reporting. Indian J Occup Environ Med 10:5–10. doi:10.4103/0019-5278.22888 6. David PJ, Pierce R (2008) Spirometry handbook, 3rd edn., Spirometry: the measurement and interpretation of ventilatory function in clinical practiceNational Asthma Campaign, Melbourne, pp 1–24 7. Gomeni R, Merlo-Pich E (2006) Bayesian modelling and ROC analysis to predict placebo responders using clinical score measured in the initial weeks of treatment in depression trials. Br J Clin Pharmacol 63:595–613. doi:10.1111/j.1365-2125.2006. 02815.x 8. Nucci G, Gomeni R, Poggesi I (2009) Model-based approaches to increase efficiency of drug development in schizophrenia: a can’t
123
19.
20.
21.
22.
23.
24.
miss opportunity. Expert Opin Drug Discov 4:837–856. doi:10. 1517/17460440903036073 Holford N, Li J, Benincosa L, Birath M (2002) Population disease progress models for the time course of HAMD score in depressed patients receiving placebo in anti-depressant clinical trials. Population Approach Group in Europe (PAGE) 11th Meeting, Abstract 311. http://www.page-meeting.org/?abstract=311 Reddy VP, Kozielska M, Johnson M, Vermeulen A, de Greef R, Liu J, Groothuis GMM, Danhof M, Proost JH (2011) Structural models describing placebo treatment effects in schizophrenia and other neuropsychiatric disorders. Clin Pharmacokinet 50:429–450. doi:10.2165/11590590-000000000-00000 Russu A, Marostica E, De Nicolao G, Hooker AC, Poggesi I, Gomeni R, Zamuner S (2012) Joint modeling of efficacy, dropout, and tolerability in flexible-dose trials: a case study in depression. Clin Pharmacol Ther 91:863–871. doi:10.1038/clpt. 2011.322 Atkinson AJ, Huang SM, Lertora JJL, Markey SP (2012) Principles of clinical pharmacology. Academic Press, London Shang EY, Gibbs MA, Landen JW, Krams M, Russell T, Denman NG, Mould DR (2009) Evaluation of structural models to describe the effect of placebo upon the time course of major depressive disorder. J Pharmacokinet Pharmacodyn 36:63–80. doi:10.1007/s10928-009-9110-3 Matre D, Casey KL, Knardahl S (2006) Placebo-induced changes in spinal cord pain processing. J Neurosci 26:559–563. doi:10. 1523/JNEUROSCI.4218-05.2006 Mayberg HS, Silva JA, Brannan SK, Tekell JL, Mahurin RK, McGinnis S, Jerabek PA (2002) The functional neuroanatomy of the placebo effect. Am J Psychiatry 159:728–737 de la Fuente-Fernandez R, Ruth TJ, Sossi V, Schulzer M, Calne DB, Stoessl AJ (2001) Expectation and dopamine release: mechanism of the placebo effect in Parkinson’s disease. Science 293:1164–1166. doi:10.1126/science.1060937 Delahoy P, Thompson S, Marschner IC (2010) Pregabalin versus gabapentin in partial epilepsy: a meta-analysis of dose–response relationships. BMC Neurol 10:104. doi:10.1186/1471-2377-10104 Houk BE, Bello CL, Kang D, Amantea M (2009) A population pharmacokinetic meta-analysis of sunitinib malate (SU11248) and its primary metabolite (SU12662) in healthy volunteers and oncology patients. Clin Cancer Res 15:2497–2506. doi:10.1158/ 1078-0432.ccr-08-1893 Chan PL, Weatherley B, McFadyen L (2008) A population pharmacokinetic meta-analysis of maraviroc in healthy volunteers and asymptomatic HIV-infected subjects. Br J Clin Pharmacol 65:76–85. doi:10.1111/j.1365-2125.2008.03139.x Ito K, Ahadieh S, Corrigan B, French J, Fullerton T, Tensfeld T (2010) Disease progression meta-analysis model in Alzheimer’s disease. Alzheimers Dement 6:39–53. doi:10.1016/j.jalz.2009.05. 665 Forey BA, Thornton AJ, Lee PN (2011) Systematic review with meta-analysis of the epidemiological evidence relating smoking to COPD, chronic bronchitis and emphysema. BMC Pulm Med 11:36. doi:10.1186/1471-2466-11-36 Wang X, Shang D, Ribbing J, Ren Y, Deng C, Zhou T, Guo F, Lu W (2012) Placebo effect model in asthma clinical studies: longitudinal meta-analysis of forced expiratory volume in 1 second. Eur J Clin Pharmacol 68:1157–1166. doi:10.1007/s00228-0121245-2 Rowland M, Sheiner LB, Steimer JL (1985) Variability in drug therapy: description, estimation and control. Raven Press, New York Lunn DJ, Thomas A, Best N, Spiegelhalter DJ (2000) WinBUG: a Bayesian modeling framework: concepts, structure and extensibility. Stat Comput 10:325–337
J Pharmacokinet Pharmacodyn (2014) 41:553–569 25. Development Core Team R (2011) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna 26. Spiegelhalter DJ, Best NG, Carlin BP, van der Linde A (2002) Bayesian measures of model complexity and fit. J R Statist Soc B 64(Part 4):583–639. doi:10.1111/1467-9868.00353 27. Karlsson M, Holford N (2008) A tutorial on Visual Predictive Checks. Population Approach Group in Europe (PAGE) 17th Meeting, Abstract 1434. http://www.page-meeting.org/default. asp?abstract=1434 28. Sheiner LB, Beal SL (1981) Some suggestions for measuring predictive performance. J Pharmacokinet Biopharm 9:503–512 29. Fisher RA (1925) Statistical methods for research workers. Oliver & Boyd, Edinburgh
569 30. Ette EL, Williams PJ (2013) Pharmacometrics: the science of quantitative pharmacology. Wiley, Hoboken 31. Barraclough KA, Isbel NM, Kirkpatrick CM, Lee KJ, Taylor PJ, Johnson DW, Campbell SB, Leary DR, Staatz CE (2011) Evaluation of limited sampling methods for estimation of tacrolimus exposure in adult kidney transplant recipients. Br J Clin Pharmacol 71:207–223. doi:10.1111/j.1365-2125.2010.03815.x 32. Sasakawa T, Masui K, Kazama T (2013) Iwasaki H. Comparison of Five Parameter Sets. American Society of Anesthesiologists, Performance of Compartmental Pharmacokinetic Model for Rocuronium
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ORIGINAL PAPER
Population model of longitudinal FEV1 data in asthmatics: meta-analysis and predictability of placebo response Eleonora Marostica • Alberto Russu • Shuying Yang • Giuseppe De Nicolao • Stefano Zamuner • Misba Beerahee
Received: 21 November 2013 / Accepted: 6 August 2014 / Published online: 15 August 2014 Ó Springer Science+Business Media New York 2014
Abstract Asthma is an obstructive lung disease where the mechanism of disease progression is not fully understood hence motivating the use of empirical models to describe the evolution of the patient’s health state. With reference to placebo response, measured in terms of FEV1 (Forced Expiratory Volume in 1 s), a range of empirical models taken from the literature were compared at a single trial level. In particular, eleven GSK trials lasting 12 weeks in mild-tomoderate asthma were used for the modelling of longitudinal placebo responses. Then, the chosen exponential model was used to carry out an individual participant data meta-analysis on eleven trials. A covariate analysis was also performed to find relevant covariates in asthma to be accounted for in the meta-analysis model. Age, gender, and height were found statistically significant (e.g. the taller the patients the higher the FEV1, the older the patients the lower the FEV1, and females have lower FEV1). By truncating each trial at week 4, the predictive properties of the meta-analysis model were also investigated, showing its ability to predict long-term FEV1 response from truncated trials. Summarizing, the study suggests that: (i) the exponential model effectively describes the placebo response; (ii) the meta-analysis approach may prove helpful to simulate new trials as well as E. Marostica (&) A. Russu G. De Nicolao Department of Industrial and Information Engineering, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy e-mail: [email protected] Present Address: A. Russu Janssen Research and Development, Janssen Pharmaceutical Companies of Johnson&Johnson, Beerse, Belgium S. Yang S. Zamuner M. Beerahee Clinical Pharmacology Modelling and Simulation, GlaxoSmithKline, Stockley Park, UK
to reduce trial duration in view of its predictive properties; (iii) the inclusion of available covariates within the metaanalysis model provides a reduction of the inter-individual variability. Keywords Asthma Population approach Metaanalysis Predictability analysis Covariate analysis
Introduction Asthma, a chronic long-term disease that affects airways [1], is a complex and multi-factorial disease whose causes are not fully understood. Patients’ health state is usually assessed by the Forced Expiratory Volume in the first second (FEV1) which is a relevant parameter related to the pulmonary function obtained through a non-invasive spirometry test [2–6]. As for other diseases, the physio-pathological knowledge of asthma is insufficient to suggest mechanistic models of disease progression. Rather, a range of empirical models of longitudinal type are investigated, similar to those adopted for the empirical modelling of other diseases such as in psychiatry [7–12]. In particular, the placebo effect in patients affected by asthma has been widely discussed, as for other diseases such as pain, depression, and Parkinson [13–16]. In order to gain a better understanding of the placebo response, moving from the analysis of a single trial to the joint analysis of multiple trials might bring a definite advantage. Indeed, meta-analyses are being increasingly adopted across several therapeutic areas [17–21], although to our knowledge the only meta-analysis regarding placebo response in asthma is due to Wang and co-workers that elaborated the mean values of FEV1 obtained in 34 trials
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from the literature using an exponential model [22]. Herein, we perform an individual participant data (IPD) population meta-analysis of eleven trials considering not only the mean placebo responses of each trial but also all the individual FEV1 profiles. The eleven trials, retrieved from the GSK repository, were conducted in the North America, including United States and Canada, as part of the development of fluticasone propionate (FP) or FP/ Salmeterol. Only subjects who were randomized in the placebo group were analysed. The present work has four main objectives. First of all, an extensive comparison at single trial level is carried out to determine the best longitudinal model among seven candidate empirical models (linear, polynomial, Bateman, Weibull-and-linear, Emax, Hill, and exponential). The second objective is to obtain a statistical characterization of inter-trial variability by running an IPD meta-analysis considering simultaneously all the individual data from the eleven trials. The third objective is to perform a predictability analysis on the meta-analysis model to see whether it can be used to predict long-term outcomes from shortterm trials. The last main objective is to investigate whether the inclusion of covariate effects may improve the meta-analysis model, in order to attempt more efficient clinical trial designs. In particular, the inclusion of covariate effects in the model could potentially explain part of the random effects hence improving clinical trial efficiency in term of sample size requirement. Moreover, the design of a clinical trial may be improved for instance by optimizing duration and schedule of visits.
consent before starting the clinical trial. The clinical response was measured through the Forced Expiratory Volume in the first second (FEV1), that represents the amount of air that can be exhaled from the lungs in the first second of a forced exhalation. A spirometry testing was used to assess the lung function by measuring the FEV1. The sampling schedule was different for each subject in the different trials. Hence, a different number of samples per subject were collected. In particular, a maximum of 9 observations per subject were available. Overall, 6272 FEV1 placebo data were sampled from 1,151 patients. For each clinical trial, demographic covariates such as age, gender, body weight, height, and race were available. Moreover, asthma duration, smoking status, and concurrent asthma medication were considered. Further, the year of the beginning of the study was considered as a covariate. In particular, age, body weight, height, and the year of the beginning of each study (see Table 1) are continuous covariates, while gender, race, smoking status, asthma duration, and concurrent asthma medication were available in categorical format. More details about the continuous and categorical covariates are reported in Tables 2 and 3, respectively. All the clinical trials were comparable in terms of baseline FEV1 values. Concerning the duration of asthma, comparable values were available across the trials. In particular, these are the percentage of patients in each trial having asthma duration (i) lower than 5 years: from 9 to 18 %, (ii) greater than 5 years and lower than 10 years: from 12 to 27 %, and (iii) greater than 10 years: from 63 to 74 %.
Materials Methods In the present work, longitudinal data from eleven clinical trials in inhaled corticosteroids (ICS) naı¨ve patients diagnosed with mild-to-moderate asthma were considered (http://www.gsk-clinicalstudyregister.com/). In Table 1 a summary of the eleven clinical trials is reported. In particular, data from the placebo arm of the following trials were adopted to assess the performance of the proposed model:
Since several clinical trials were available, a two-step approach was adopted. First, all the trials were analysed separately (single-trial analysis). Once the best model was selected through the single-trial analysis, data from all the trials were merged and the best model was identified using all individual data (IPD meta-analysis). The population paradigm [23] was adopted to perform the analyses.
–
Software implementation
– –
two Phase II randomized, double-blind, doubledummy, parallel-group, placebo-controlled trials, six Phase III and one Phase IIIb randomized, doubleblind, parallel-group, placebo-controlled trials, two Phase IV double-blind, double-dummy, parallelgroup, placebo-controlled trials.
All the clinical trials lasted 12 weeks and placebo was given once or twice daily depending on the trial. The studies considered were approved by the Investigational Review Board (IRB) and all the patients gave informed
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Parameter estimation was carried out through software WinBUGS 1.4.3 [24]. Data management and graphical output were performed through R 2.13.1 [25]. Software R was also used to perform the covariate analysis. In particular, the function cor.test was used to perform the correlation test and obtain the p-values for the continuous covariates, while aov and TukeyHSD functions provided the p-values for the categorical covariates through the
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Table 1 Summary of the clinical trials Trial
Phase
Years
# Subjects
Table 2 Summary of the continuous covariates # FEV1 data
# Median (min, max) data/ subject
Covariate
Median (min, max)
Age (years)
30 (12, 78)
Height (cm)
170 (138, 199) 76 (32, 151)
1-FAP30008
III
2000–2001
100
554
6 (1, 7)
Weight (kg)
2-FLD302
III
1992–1993
84
669
9 (2, 9)
3-FLD304
III
1993–1994
40
301
9 (1, 9)
Minimum, median, and maximum values are reported for each covariate
4-FLI202
II
1991–1992
73
417
5 (1, 9)
5-FLI302
III
1991–1992
87
495
6 (1, 9)
6-FLTA2003
II
1995–1996
73
523
9 (1, 9)
7-FLTA3020
III
1997–1998
43
269
8 (1, 8)
8-FLTA4030
IV
1997–1998
115
748
7 (1, 7)
9-FLTA4031
IV
1997–1998
114
731
7 (1, 7)
10-FPD40009
IIIb
2000–2001
210
965
5 (1, 6)
11-SAS30022
III
2001–2003
212
600
3 (1, 3)
Table 3 Summary of the categorical covariates Covariate Gender Race
ANOVA test. The TukeyHSD function refers to the Tukey’s Honest Significant Difference method that assesses the differences between the mean values of the compared groups defined in terms of the covariate values.
Smoking status
Asthma duration
Single-trial analysis Concurrent asthma medication
Methods In order to define the most suitable model to describe the available data, a comparison was carried out by identifying a range of empirical models taken from the literature at a single trial level. According to a population framework, the index j in the following models refers to the subject of each trial, that is j ¼ 1; . . .; Nsubj , where Nsubj is the number of subjects in the trial. Prior distributions for each parameter are reported for each of the following models, assessed on each trial. In particular, the same log-normal prior distribution was assumed for the baseline, that is logAj N ðlA ; sA Þ; lA Nð1; 10Þ; sA C1 ð1; 0:01Þ. 1.
2.
3.
FEV1j ðtÞ ¼ Aj þ Bj t þ Cj t2
ð2Þ
where parameters Aj and Bj have the same meaning as in the linear model (1) as well as the same prior
Male
586 (51 %)
Female
565 (49 %)
White
933 (81 %)
African American
123 (11 %)
Asian
18 (2 %)
Hispanic
60 (5 %)
Other Never smoked
17 (1 %) 896 (81 %)
Former smoker
205 (18 %)
Current smoker
7 (1 %)
\5 years
146 (13 %)
5 years to \10 years
192 (17 %)
10 years
813 (70 %)
No
474 (41 %)
Yes
677 (59 %)
distributions, while Cj is the quadratic coefficient normally distributed as Cj NðlC ; sC Þ; lC N ð0:5; 1Þ; sC C1 ð1; 0:01Þ. Bateman [7, 8, 10]: FEV1j ðtÞ ¼ Aj þ Bj ðekon j t ekrec j t Þ
ð1Þ
where Aj is the baseline value of the response and Bj represents the slope. In particular, Bj NðlB ; sB Þ; lB Nð1; 1Þ; sB C1 ð1; 0:01Þ. Polynomial [7, 10]:
# Subjects ( %)
For the study FLTA3020 the covariate smoking status was not available
Linear [10]: FEV1j ðtÞ ¼ Aj þ Bj t
Value
4.
ð3Þ
where Aj is the baseline, Bj NðlB ; sB Þ; lB N ð0:1; 1Þ; sB C1 ð1; 0:5Þ modulates the amplitude of the response, konj controls the onset of the treatment effect and krecj regulates the recovery of the health state towards the baseline. The same log-normal prior distribution was assumed for both konj and krecj as logPj NðlP ; sP Þ; lP Nð0:5; 2Þ; sP ð1; 0:01Þ, where Pj represents konj and krecj . Weibull ? Linear [7, 8, 11]: FEV1j ðtÞ ¼ Aj e
ðt t Þbj dj
þ hrecj t
ð4Þ
where Aj is the baseline, td j and bj account for the amelioration of the health state with respect to the
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Table 4 Comparison of the different models on the different trials: DIC values Trial
Linear
Polynomial
Bateman
Weibull?Linear
Emax
Hill
Exponential
FAP30008
39.6
36.5
18.1
23.3
159.0
105.7
2172.1
FLD302
365.2
357.2
360.2
302.5
241.0
232.0
234.7
FLD304
153.0
156.5
152.6
101.8
53.8
102.7
37.8
FLI202
104.8
34.4
157.4
83.5
57.8
80.6
32.2
FLI302
455.5
385.2
419.9
319.7
300.3
285.1
304.2
FLTA2003
322.9
273.5
377.7
256.3
175.9
237.9
183.6
FLTA3020
199.5
227.3
186.5
164.6
192.4
184.8
153.0
FLTA4030
586.1
643.4
618.0
549.0
456.9
480.5
394.4
FLTA4031
341.6
372.2
364.8
279.3
206.8
203.5
197.5
FPD40009
233.8
247.9
131.6
138.8
1.2
112.5
60.8
SAS30022
418.8
143.6
–
–
218.5
–
210.7
5.
baseline. In particular, logbj Nðlb ; sb Þ; lb N logtd j Nðltd ; std Þ; ltd N ð0:5; 10Þ; sb ð1; 0:01Þ, ð4; 10Þ; std ð1; 0:01Þ, and loghrecj Nðlhrec ; shrec Þ; lhrec Nð0:01; 5Þ; shrec ð1; 0:01Þ Emax [12]: FEV1j ðtÞ ¼ Aj þ EMAXj t=ðET50j þ tÞ
6.
where Aj is the baseline, EMAXj defines the maximum effects, and ET50j is the time relative to the 50 % of the maximum effect. The parameter EMAXj is characterized by the same prior distributions considered for the baseline, while logET50j NðlET50 ; sET50 Þ; lET50 Nð0; 1Þ; sET50 Nð1; 0:1Þ. Hill [12]: c
FEV1j ðtÞ ¼ Aj þ EMAXj tcj =ðET50j j þ tcj Þ
7.
ð5Þ
ð6Þ
where Aj ; EMAXj , and ET50j have the same meaning as in the Emax model (5), while cj is the Hill coefficient determining the steepness of the sigmoid-shaped profile. Same prior distributions as for the Emax model were considered. Moreover, cj has the same prior as ET50j . Exponential: FEV1j ðtÞ ¼ Aj þ Gj ð1 e
Tt
j
Þ
Results
ð7Þ
where Aj is the baseline, Gj is the gain and Tj is the time constant. In particular, the gain represents the asymptotic improvement of the patient’s health state, and the time constant determines the time needed to reach the steady-state. Note that, the smaller T the faster the response. Log-normal prior distributions were assumed for all the parameters. In particular the baseline and the gain are characterized by the same distribution, whereas logTj NðlT ; sT Þ; lT Nð0; 1Þ; sT C1 ð1; 0:1Þ.
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Parameter estimation of all the population models was performed through WinBUGS [24] carrying out the comparison in terms of the Deviance Information Criterion (DIC) [26], an index automatically provided by WinBUGS to select optimal model complexity. In particular, the model providing the highest number of wins on the clinical trials in terms of DIC was selected as the best model. Moreover, established distributions were used for the priors. Also, for each model parameter reasonably flat priors were defined. To make an example, the prior for lT assumes that the typical value of the time constant T lies in ð1:96 102 ; 1:96 102 Þ days with probability 0.95. Since the settling-time of the exponential model is about 4T, the adopted prior is very flat as it encompasses responses showing practically instantaneous onsets and ones whose transient ends after more than 2 years. The predictive performance of the model was assessed through the Visual Predictive Checks (VPCs) [27]. Since heterogeneous sampling schedules were adopted for each subject in each trial, the binning procedure was used to perform the VPCs. In particular, about the same number of observations were included in each bin.
Considering each trial separately, the seven models selected from the literature were compared in terms of DIC values which are reported in Table 4. The exponential model yielded the best performance in six out of eleven trials. Moreover, concerning trial SAS30022, Bateman, Weibull?Linear, and Hill models could not be identified because only a maximum of three data per subject were available. Based on this analysis, the exponential model was selected for the subsequent IPD meta-analysis as it resulted to be the best model most frequently (six case out of
J Pharmacokinet Pharmacodyn (2014) 41:553–569
CV% of A
TV of A 3.0 40
2.8
35
CV%
2.6
TV (L)
Fig. 1 Box-plots of the posterior distributions of the typical value TV (on the left) and of the percent coefficient of inter-individual variation CV% (on the right) for each parameter in each clinical trial
557
2.4 2.2
30 25 20
2.0
15
1.8 1
2
3
4
5
6
7
8
9
10 11
1
2
3
4
5
Trial
6
7
8
9
10 11
8
9
10 11
Trial
TV of G
CV% of G 500
1.5
1.0
CV%
TV (L)
400
0.5
300 200 100 0
1
2
3
4
5
6
7
8
9
10 11
1
3
4
5
6
7
Trial
TV of T
CV% of T 10^150
1000
100
10^100
CV%
TV (days)
2
Trial
10
10^50 1
0.1
10^0 1
2
3
4
5
6
Trial
eleven). As seen from the posterior distributions (Fig. 1), satisfactory parameter estimates were obtained for all the parameters in all the clinical trials. Nevertheless, it was found that the gain G had quite a high inter-individual variability, and the time constant T was characterized by a high inter-individual variability within the trials. This can be due to the variability of response shapes observed in the data. In particular, there are non-responder patients who have a flat profile (i.e. G close to 0), while there are responder patients who may have a very fast (i.e. low T) or a slow (i.e. high T) response. Figure 1 shows the box-
7
8
9 10 11
1
2
3
4
5
6
7
8
9 10 11
Trial
plots of the posterior distributions of the typical value (left column) and of the percent coefficient (right column) of inter-individual variation of each parameter for each trial. The exponential model was able to describe well both the typical curve and the individual profiles of each clinical trial. Moreover, the inter-individual variability (assessed by means of VPCs) of each study was explained well. An example of individual profiles is shown in Fig. 2 (top panel), while VPCs of a representative trial are shown in Fig. 3.
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Single−trial Analysis 0
20
40
60
80 100
0
20
40
60
80 100
Subject 834
Subject 933
Subject 1033
Subject 1078
Subject 1150
Subject 1190
Subject 1205
Subject 1213
5 4 3 2
FEV1 (L)
Fig. 2 Individual fittings obtained with the single-trial analysis (top panel) and with the mixture meta-analysis (bottom panel) for a representative subset of patients belonging to trial 2 (FLD302). Solid line individual profile; grey dashed lines 90 % credible intervals; black dashed line typical curve. Note that, the overall typical curve is shown in the bottom panel
J Pharmacokinet Pharmacodyn (2014) 41:553–569
1
5 4 3 2 1
0
20
40
60
80 100
0
20
40
60
80 100
Time (days)
Meta−Analysis 0
20
40
60
80 100
0
20
40
60
80 100
Subject 834
Subject 933
Subject 1033
Subject 1078
Subject 1150
Subject 1190
Subject 1205
Subject 1213
5 4 3
FEV1 (L)
2 1
5 4 3 2 1
0
20
40
60
80 100
0
20
40
60
80 100
Time (days)
Meta-analysis Methods Based on the results obtained through the single-trial analysis step, the exponential model was extended in order to perform a population model-based IPD meta-analysis. In order to better address the inter-individual variability characterizing each trial, it was assumed to identify two possible sub-populations, i.e. responder and non-responder subjects. Moreover, the meta-analysis was performed to assess differences among the trials and estimate the inter-trial variability. In particular, each model parameter is characterized by a subject-related term, that accounts for the differences between patients, and a study-
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related term which explains differences between the clinical trials. As in the single-trial analysis, WinBUGS was used to implement the model. In the following, a description of the population model is reported in detail. Since a mixture model was assumed, the identification of responder and non-responder subjects was performed. In particular, in addition to the parameters A; G, and T, a parameter relative to the probability of being responder or non-responder was considered. More in detail, the mixture meta-analysis model was defined as: FEV1i ðtÞ ¼ Ai þ bi ðGi ð1 et=Ti ÞÞ
ð8Þ
where i ¼ 1; . . .; N denotes the i-th patient, N is the total number of patients of all the trials, and bi BernoulliðpÞ. In
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found that mA ¼ 0:25; mG ¼ 2:73, and mT ¼ 1:64, while tA ¼ tG ¼ tT ¼ 2:30. The marginalized log-normal priors for the overall individual parameters Ai ; Gi , and Ti can be obtained
Single−trial Analysis
5
ðlogAi jlA ; qA ; sA Þ NðlA ; qA þ sA Þ
FEV1 (L)
4
ðlogGi jlG ; qG ; sG Þ NðlG ; qG þ sG Þ
ð13Þ
3
ðlogTi jlT ; qT ; sT Þ NðlT ; qT þ sT Þ 2
It can be demonstrated that the variance of the prior distribution for each individual parameter Ai ; Gi , and Ti is given by the sum of the variance of the subject-related term and the one of the study-related term, considering the logdomain. More in detail, considering Pi the generic individual parameter, based on the definition of meta-analysis model:
1
0
20
40
60
80
Time (days)
Fig. 3 Visual Predictive Checks of FEV1 data for a representative trial (2-FLD302) obtained with the single-trial analysis
logPi ¼ logPsubji þ logPstudyk
particular, the probability p is described through a Betað1; 1Þ distribution. Therefore, bi ¼ 1 identifies a responder, i.e.:
Since there is no correlation, the variance of logPi is: Var ½logPi ¼ Var logPsubji þ Var logPstudyk ¼ sP þ qP
FEV1i ðtÞ ¼ Ai þ Gi ð1 et=Ti Þ
FEV1i ðtÞ ¼ Ai
ð10Þ
The individual parameters were defined as follows:
As in the single-trial analysis, the measurement error was assumed to be normally distributed with variance defined through the inverse gamma prior C1 ð1; 0:001Þ. Predictability analysis
1
ðlogAi jAstudy k ; sA Þ NðlogAstudy k ; sA Þ
sA C ð1; 0:01Þ
ðlogGi jGstudy k ; sG Þ NðlogGstudy k ; sG Þ
sG C1 ð1; 0:01Þ
ðlogTi jTstudy k ; sT Þ NðlogTstudy k ; sT Þ
ð15Þ
ð9Þ
While, if bi ¼ 0, the model for the non-responders is:
sT C1 ð1; 0:1Þ
ð11Þ where k ¼ 1; . . .; 11 identifies the clinical trial. Further, Astudy ; Gstudy and Tstudy are the study-related terms for each parameter. Moreover, log-normal prior distributions were assumed for the individual parameters. The associated variance was specified by means of inverse gamma distribution ðC1 Þ. More specifically, the priors related to the study-effect were
The proposed mixture meta-analysis model was applied to perform a predictability analysis. Its aim was to assess whether the model could be used to predict long-term outcomes from short-term trials. Therefore, the following procedure was carried out: 1. 2.
One at a time, each trial was truncated at week 4 removing subsequent data. Parameter estimation of the mixture meta-analysis model was performed considering all the data up to 12 weeks of the remaining ten trials and the data until week 4 of the truncated one. In particular, besides A; G, and T, also the probability of being responders or
ðlogAstudy k jlA ; qA Þ NðlA ; qA Þ
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qA C1 ð1; 0:01Þ
ðlogGstudy k jlG ; qG Þ NðlG ; qG Þ
lG NðmG ; tG Þ
qG C1 ð1; 0:01Þ
ðlogTstudy k jlT ; qT Þ NðlT ; qT Þ
ð14Þ
lT NðmT ; tT Þ
The values of the means mA ; mG ; mT , and variances tA ; tG and tT were calculated from the data. In particular, it was
ð12Þ
1
qT C ð1; 0:1Þ
non-responders, given by p, as well as the parameter b were estimated.
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The prediction Root Mean Square Error (RMSE pred) was calculated based on the difference between the removed and the predicted FEV1. For each truncated trial, the prediction RMSE was then compared to the RMSE based on the difference between the observed FEV1 data and those predicted using the complete datasets of the eleven trials. Moreover, two commonly used metrics were calculated for each truncated trial: the median of the absolute value of the percent prediction errors (PE %) and the absolute value of the median of PE % [28]. Typical and individual profiles, and VPCs of both complete and truncated trials were computed over the 12 weeks. This procedure was repeated for all the eleven trials.
2.
3.
Covariate analysis Once the mixture meta-analysis model was identified on all the clinical trials, a further analysis was performed in order to detect statistically significant covariates which may affect the patient’s health state. For the purpose of establishing a base model to be used for the covariate analysis, responder and nonresponder subjects were identified using the posterior estimates of individual parameters bi obtained through model (8). In particular, the ith subject was considered as a responder if pðbi ¼ 1jFEV1Þ 0:5, or a non-responder otherwise. Model (8) was then re-identified by assuming bi ¼ 1 for responders and bi ¼ 0 for non-responders. Such model was therefore used as the base model for the covariate analysis. Since the covariate smoking status was not available for study FLTA3020, the covariate analysis was carried out considering the remaining ten trials (to ensure consistency, the meta-analysis model was re-identified on these ten trials). Moreover, the covariate concurrent asthma medication was not used because all the patients belonging to 8 clinical trials had this covariate equal to 1. Therefore, age, height, weight, gender, race, smoking status, asthma duration, and year of the beginning of the study were tested. The following steps were performed to carry out the covariate analysis: 1.
Preliminary selection: a correlation test was performed between each continuous covariate and the model parameters A; G, and T for each individual. As for the categorical covariates, the ANOVA test [29] was performed to assess if the mean values of the parameters A; G, and T in the groups identified by the categorical covariates were significantly different from each other. For both continuous and categorical covariates the parameter estimates were considered in the logarithmic domain. Note that, the possible correlation between each covariate and the parameters gain and time constant was considered only for responder subjects.
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Single-covariate analysis: based on step 1, a subset of covariates was selected. Each of them was included, one at a time, in the meta-analysis model modulating the parameter for which a significant correlation had been found. If a covariate had been found to be significant on two or all parameters, all the possible combinations of models were assessed. For instance, if covariate COVi had been found significant for parameter A and G, three models were investigated: A depending on COVi ; G depending on COVi , and both A and G depending on COVi . The best model for each covariate was selected through the DIC. Full model: the statistically significant covariates (i.e. the ones yielding lower DIC values with respect to the meta-analysis model without covariates) selected through the single-covariate analysis were included in the meta-analysis model in order to define the full model. Backward procedure: once the full model was identified on all the data from the ten trials, a backward approach was carried out. Each covariate was removed one at a time. The models so obtained were identified in order to obtain a DIC value. This process ended when no lower DIC can be found, with respect to the full model.
In terms of mathematical formulation, the meta-analysis model accounting for a covariate COVi modulating a generic parameter P is: logPi ¼ logPsubj i þ logPstudy k þ logPk ðCOVi COV k Þ ð16Þ where i ¼ 1; . . .; N; k ¼ 1; . . .; 11, and COV k is the average value of the covariate within the kth study. Above, logPk is a study parameter, with log-normal, flat prior distribution. Moreover, parameter P could be the baseline A, the gain G, or the time constant T. Note that if the covariate is binary, then COV k coincides with the proportion of subjects such that COVi ¼ 1 within the kth study. Results The exponential model was identified after merging data from all the trials. The mixture meta-analysis model provided satisfactory parameter estimates. In particular, the identification of the responder and non-responder subpopulations produced a shrinking of the posterior distributions of the inter-individual variability of the gain and the time constant. The summary of the posteriors of the parameter estimates obtained with the meta-analysis model are reported in Table 5. The exponential model yielded also good performance when applied in the meta-analysis context. The proposed
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Table 5 Posterior summaries of the parameter estimates of the mixture meta-analysis model Parameter
Mean
SD
2.5 %
Median
97.5 %
Table 6 Number and percentage of responders identified for each clinical trial according to the meta-analysis model (# Responders MA (%)) and for each truncated trial according to the predictability analysis (# Responders PA (%))
TV of A
2.39
0.048
2.30
2.39
2.49
Trial
# Subjects
TV of G
0.66
0.033
0.6
0.66
0.73
# Responders MA (%)
# Responders PA (%)
1-FAP30008
100
15 (15 %)
18 (18 %)
2-FLD302
84
40 (48 %)
84 (100 %)
TV of T
7.28
1.13
5.17
7.23
9.57
CV% of Asubj
22.8
0.53
21.8
22.8
23.9
CV% of Astudy
5.98
1.44
3.86
5.73
9.40
3-FLD304
40
31 (78 %)
35 (88 %)
CV% of Gsubj
38.8
3.36
32.2
38.8
45.4
4-FLI202
73
26 (36 %)
52 (71 %)
CV% of Gstudy
10.38
3.96
5.11
9.62
20.4
5-FLI302
87
28 (32 %)
56 (64 %)
CV% of Tsubj
164
43.7
100.2
156.2
269.3
6-FLTA2003
73
35 (48 %)
73 (100 %)
CV% of Tstudy
31.3
13.9
15.8
27.9
66.02
7-FLTA3020
43
13 (30 %)
13 (30 %)
Variance of
0.061
0.0013
0.059
0.061
0.064
8-FLTA4030
115
58 (50 %)
61 (53 %)
9-FLTA4031
114
37 (32 %)
24 (21 %)
10-FPD40009 11-SAS30022
210 212
10 (5 %) 45 (21 %)
3 (1 %) 15 (7 %)
TV typical value, CV % percent coefficient of inter-individual or intertrial variation
model was able to describe both the typical curve of the whole meta-analysis and the typical curve of each clinical trial. Note that, since the meta-analysis approach is applied, the typical curve of each specific study is obtained through the parameter estimates provided by the identification of the meta-analysis model. Moreover, good individual fittings were obtained and they were similar to the ones provided by the single-trial analysis. An example of a subset of individual profiles is shown in Fig. 2 (bottom panel). The mixture meta-analysis model was able to identify the responders and non-responders sub-populations. More in details, the percentage of responders identified for each clinical trial according to the meta-analysis model ranged from 5 % (Trial FPD40009) to 78 % (Trial FLD304) (see Table 6). The estimated overall percentage of responders was 29 %. Further, the predictive performances of the mixture meta-analysis model were assessed in terms of VPCs. In particular, stratified VPCs on the two identified sub-populations (responders and non-responders) were performed. The proposed mixture model was able to explain well both the inter-individual and the inter-trial variability for responders and non-responders considering all the data (Fig. 4, bottom panels). Also, stratified VPCs on the two sub-populations were done for each of the specific trial. The model could describe the inter-individual variability in each trial properly. An example of VPCs for a representative trial is shown in Fig. 4 (top panels).
A column with the total number of subjects for each trial is also reported
week 4 onward based on the complete observed data. This is due to the increased uncertainty consequent to the removal of data in the next 8 weeks. In particular, an increase of the RMSE ranging from 22 % (trial 7-FLTA3020 in Table 1) to 54 % (trial 11-SAS30022 in Table 1), was obtained (see Fig. 5). In order to better assess the predictive performance of the mixture meta-analysis model, two commonly used metrics were calculated for each truncated trial: the median of the absolute value of the percent prediction errors (PE %) and the absolute value of the median of PE %. The worst values obtained were 8.8 and 4.18 % for the median of the absolute value of PE % and the absolute value of the median of PE %, respectively. Such values were both lower than the correspondent commonly used thresholds of 30 and 15 %, respectively [30–32]. In particular, the model was able to predict well both the individual profiles and the inter-individual variability. A subset of individual profiles obtained from truncated observations of representative patients is shown in Fig. 6. Moreover, VPCs of the sub-populations characterized by a sufficient number of subjects for the truncated trials are shown in Fig. 7. Further, Table 6 reports the number and percentage of responders identified through the predictability analysis in each truncated trial.
Predictability analysis Covariate analysis Based on the mixture meta-analysis model, the predictability analysis was performed as described in ‘‘Methods’’ section. For each truncated trial, the prediction RMSE was compared to the RMSE using all the data. Figure 5 shows that the prediction RMSE is higher than the RMSE from
Since the covariate smoking status was not available for the study FLTA3020, the covariate analysis was performed based on the parameter estimates provided by the metaanalysis model identified on the other ten trials. Parameter
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Fig. 4 Top panels Visual Predictive Checks of FEV1 data relative to non-responders (left panel) and responders (right panel) of a singletrial (2-FLD302) obtained with the mixture meta-analysis model.
Bottom panels VPCs obtained with the mixture meta-analysis for nonresponder patients (left panel) and responder patients (right panel)
estimates of the meta-analysis model identified on ten trials are reported in Table 7 (top). First, a preliminary selection of a subset of statistically significant covariates was performed through the correlation test between each covariate and the estimates of A, G, and T provided by the meta-analysis model. For the continuous covariates, the correlation test suggested that age, height, weight, gender, race, and smoking status had to be included in the model modulating the baseline (p-value \0.05, see Table 8), while height, weight, and gender modulated the gain. Since height and weight were correlated, the body mass index (bmi) was calculated. It was found that bmi had a statistically significant correlation with the baseline. For categorical covariates, the ANOVA test was performed. For all parameters, statistically significant differences were found between males and females. Also, based on the TukeyHSD test, the estimates of the baseline turned out to be statistically different between the White and the African American races. No
significant p-value was found considering all the other possible pairs of races. Based on the low number of subjects in the race groups different from the White one (see Table 3), it was decided to treat the race as a binary covariate, where the two categories were White and nonWhite. When considering the race as a binary covariate, a significant correlation was found for the baseline (p-value = 0.001). Furthermore, the estimate of A was statistically different between the former smoker and the patients who never smoked. Since only 7 subjects over 1151 were current smoker and given that no statistically significant difference was found between former smoker and current smoker, the two groups were merged and a binary covariate smoker/no-smoker was considered. Hence, the ANOVA test was performed between the introduced binary covariate and the three model parameters A; G, and T, and statistically significant p-values were found for the baseline. Finally, no significant correlation was found between the asthma duration and the model parameters. Table 8 reports
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which may modulate more than one parameter (e.g. for the covariate height, three candidate models were investigated: a first one where height affects the baseline, a second one where it affects the gain, and a third one accounting for a simultaneous effect of height on both baseline and gain). According to this procedure, it was found that the models yielding lower DIC values with respect to the metaanalysis model without covariates were the ones having:
0.2
– – – 0.1
RMSE (L)
0.3
RMSE RMSE pred
These three models were combined together to define the full model, where the baseline was modulated by age, height, and gender whereas the gain was modulated by height and gender. After the identification of the full model, the backward procedure was applied, by removing age, height, and gender one at a time. Moreover, the covariates height and gender were removed on both baseline and gain as well as on either baseline or gain separately. The final meta-analysis model accounted for age, height, and gender affecting the baseline and gender modulating the gain. More details on the DIC values are reported in Table 9. This final model provided satisfactory posterior distributions. In particular, the posterior distributions of the typical values of the three parameters A, G, and T were similar to the ones obtained with the meta-analysis model identified on ten trials. Moreover, compared to the estimates yielded by the meta-analysis model on ten trials, the inclusion of covariates modulating the baseline and the gain provided lower values for the percent coefficient of the inter-individual variation of both the two parameters.
0.0
1
2
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11
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Fig. 5 Bar-plot of the Root Mean Square Error for each truncated trial calculated when the data after week 4 were removed from model identification (RMSE pred) and when they were included (RMSE). Trial FLTA3020 (trial 7 in Table 1) is the most predictable. The percent increase of the RMSE when predicting the truncated data ranges from 22 % (trial 7-FLTA3020 in Table 1) to 54 % (trial 11 SAS30022 in Table 1)
all the p-values obtained through the correlation test between the parameter estimates and each covariate. Based on the p-values obtained, the meta-analysis models accounting for the relevant covariates were developed to perform the single-covariate analysis (see ‘‘Methods’’ section). Note that, in this phase of the covariate analysis, each model can be affected only by one covariate
Predictability Analysis 0
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Fig. 6 Predictability analysis: predicted individual profiles of a subset of representative patients belonging to trial 2 (FLD302). Observations (filled circles) before week 4 (vertical grey line) were available for the model identification, while the ones (open circles) after week 4 were removed. Solid line individual curve obtained by truncating the data; dashed line individual curve obtained when all the data are available for model identification
the baseline modulated by age, the baseline and the gain modulated by height, the baseline and the gain modulated by gender.
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b Fig. 7 Predictability analysis: Visual Predictive Checks of FEV1 data
for truncated trials obtained with the mixture meta-analysis model. VPCs not shown for trial 7-FLTA3020 because of the limited number of subjects available (13 responders and 30 non-responders, see Table 6). Solid lines observed 2.5, 50 and 97.5 percentiles; grey bands 95 % simulated prediction intervals; filled circles observations available for the model identification; open circles observations removed from the truncated trial
Table 9 DIC values of the meta-analysis model identified on 10 trials :(i) without covariates (MA), (ii) with the covariate age modulating the baseline (Age on A), (iii) with the covariate height modulating the baseline and the gain (Height on A and G), (iv) with the covariate gender modulating the baseline and the gain (Gender on A and G), (v) full model (FULL), (vi) final model (FINAL) Model
DIC
D
MA
1,454.3
Table 7 Posterior summaries of the parameter estimates of the mixture meta-analysis model identified on ten trials
Age on A
1,438.0
16.3
Height on A and G
1,419.3
35
Parameter
Gender on A and G
1,438.7
15.6
FULL
1,380.9
73.4
FINAL
1,376.6
77.7
Mean
SD
2.5 %
Median
97.5 %
No covariates TV of A
2.39
0.06
2.28
2.39
2.51
TV of G
0.64
0.03
0.58
0.64
0.71
TV of T
7.52
1.20
5.29
7.49
10.0
CV% of Asubj
22.8
0.53
21.8
22.8
23.8
CV% of Astudy
6.85
1.70
4.29
6.54
11.2
CV% of Gsubj
40.4
3.40
34.1
40.4
47.2
CV% of Gstudy
10.2
3.90
5.05
9.40
19.6
CV% of Tsubj
150.7
34.3
102.0
144.9
234.6
CV% of Tstudy
29.1
11.6
14.4
26.4
59.8
Variance of
0.06
0.0013
0.058
0.06
0.063
TV of A
2.42
0.06
2.32
2.42
2.53
TV of G TV of T
0.62 7.63
0.03 1.18
0.56 5.48
0.62 7.59
0.69 10.1
CV% of Asubj
13.4
0.34
12.7
13.4
14.1
CV% of Astudy
6.76
1.70
4.33
6.48
10.9
CV% of Gsubj
38.1
3.23
32.1
37.9
44.8
CV% of Gstudy
10.1
3.56
5.04
9.43
18.7
CV% of Tsubj
171.6
45.8
100.8
165.8
271.4
CV% of Tstudy
29.5
10.6
15.5
27.5
55.3
Variance of
0.06
0.0013
0.058
0.06
0.063
With covariates
TV typical value, CV% percent coefficient of inter-individual or intertrial variation
Table 8 p-values of the correlation test performed between the parameter estimates obtained from the identification of the metaanalysis model on ten trials and each covariate Covariate
A
G
T
Age
\ 1016
0:12
0:60
Height
\ 1016
106
0:16 0:42
8
Weight
10
0:005
Bmi
105
0:35
0:72
Gender
1016
104
0:69
Race
0:001
0:17
0:72
Smoking status
106
0:75
0:84
Asthma duration Study year
0:24 0:96
0:37 0:35
0:87 0:75
Moreover, the difference with respect to the model without covariates is reported in column D
Again, the inter-trial variability was lower than the interindividual one for each parameter, as already observed for the meta-analysis model. Posterior summaries are reported in Table 7 (bottom). Figure 8 shows the box-plots of the posterior distributions of the parameters relative to the covariates for each clinical trial. Considering the baseline parameters, AHeight was estimated greater than 1, suggesting that the taller the patient the higher the FEV1. Moreover, it was found that the older the patients the lower the FEV1. Finally, females are characterized by lower FEV1. In addition, similar values of the covariate parameters were obtained in each clinical trial. Figure 9 shows VPCs stratified on the covariate gender for the responder and non-responder populations. Each of the ten trials was simulated using the observed values of the individual continuous covariates age and height. In Fig. 8 it can be seen that in the female group (both responders and non-responders) the median and the upper percentiles appeared to be over-estimated, whereas in the male responder group the median and the lower percentiles are slightly under-estimated. This appears to be related to the baseline parameter (e.g. the upper percentile for the females is over-estimated already at t = 0). In particular, such issue may be possibly due to the different magnitudes of inter-individual variability characterizing the baseline for males and females. It is interesting to note that similar inter-individual variability is observed for responders and non-responders within the same gender.
Discussion Asthma is a complex and multi-factorial disease and the mechanistic knowledge of the underlying physiological processes is still scarce. Therefore, when limited physiopathological knowledge is available, the use of empirical
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Fig. 8 Box-plots of the posterior distributions obtained for the parameters relative to the covariates height, age, and gender for each clinical trial
Age
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A Height
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models offers a way to mathematically model placebo response. In this work, several empirical models taken from the literature were investigated on placebo responses collected in eleven asthma trials. The linear, polynomial, Bateman, Weibull-and-linear, Emax, and Hill empirical models are already successfully used in several therapeutic areas, such as pain and psychiatry [7–12]. Based on the single-trial analysis, the exponential model was selected as the best model since it provided the highest number of wins on the clinical trials available in terms of Deviance Information Criterion (DIC) value. In this work, this criterion was adopted to select the best model, however another possibility would have been to consider the sum of the DIC values provided by each model across the eleven trials. The exponential model had already been used to describe the disease progression of asthma in placebo treated subjects by Wang et al. [22] who only analysed mean responses, amounting to 34 longitudinal profiles, one
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for each trial. In contrast, in the present study a population analysis is carried out using individual data (1,151 subjects in total). The model being proposed is characterized by three parameters: the FEV1 starting value (baseline), the difference between the FEV1 value achieved at steadystate and the baseline (gain), and a parameter proportional to the time needed to reach the steady-state (time constant). This seemed to be the minimal set of parameters needed to capture all the essential features of placebo response. When the trials were analysed separately, overall parameter estimation provided satisfactory posterior distributions. Only the inter-individual variability of the time constant was more difficult to estimate. This was due to the wide range of possible individual profiles present in the data. In particular, there were patients that exhibited either very fast onset of placebo responses or very slow responses. Subsequent to the single-trial analysis, a model-based meta-analysis was performed. In particular, in order to
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Fig. 9 Covariate analysis: Visual Predictive Checks of FEV1 data for non-responder (left panels) and responder (right panels) patients stratified on the covariate gender. Solid lines observed 2.5, 50 and 97.5 percentiles; grey bands 95 % simulated prediction intervals
better explain the inter-individual variability, two possible sub-populations were assumed and identified (i.e. responder and non-responder patients). The IPD meta-analysis model proposed accounted for a trial effect on all the three parameters (baseline, gain, and time constant). The metaanalysis model provided narrower posterior distributions of both typical values and percent coefficients of inter-individual variation with respect to the single-trial analysis. Moreover, it was found that the inter-individual variability was greater than the inter-trial variability. The mixture meta-analysis model was able to identify 29 % of responders considering all the data from all the eleven
trials. Moreover, no statistically significant correlation between covariates and being responder was found. This finding motivated a predictability analysis at the meta-analysis level. The predictability of the mixture metaanalysis model was investigated and showed that the model was able to predict long-term outcomes (until week 12) based on trials truncated at week 4. The results of the predictability analysis obtained in this study suggests that placebo response in asthma can be predicted using shorter trials. Statistically significant covariates were identified, such as age, height, and gender which were able to explain part
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of the inter-individual variability characterizing both the baseline and the gain. Finally, some of the useful outcomes of this meta-analysis of placebo response in asthma include (i) more efficient design of future trials, given the population longitudinal analysis of the FEV1 response, (ii) potential reduction in the trial duration while maximising placebo response, and (iii) potential lack of effect of patient covariates studied on the placebo model parameters. A common debate for drug development teams regards the duration of efficacy studies early in clinical development prior to confirmatory studies. This meta-analysis provides some encouraging insight into the potential to undertake short term trials to predict placebo response that would be observed in longer term trials. It will be worth verifying whether this historical analysis outcome is confirmed in future studies in asthma. A natural extension of this work would be to develop a general longitudinal model accounting for both placebo and treatment effects. The points to be investigated would largely coincide with those studied herein: the possibility of describing drug response with a simple exponential model, the statistical significance and clinical relevance of covariates, and, finally, the predictability of long-term outcomes based on shorter observation periods.
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Acknowledgments The authors thank colleagues in the Respiratory Therapeutic Area in particular Shuyen Ho and Anna Ellsworth who supported preparation of the database for this work.
17.
References
18.
1. American Lung Association (2007) Trends in asthma morbidity and mortality. American Lung Association Epidemiology & Statistics Unit Research and Program Services 2. Miller MR, Hankinson J, Brusasco V et al (2005) Standardisation of spirometry. Eur Respir J 26:319–338. doi:10.1183/09031936. 05.00034805 3. Pierce R (2005) Spirometry: an essential clinical measurement. Aust Family Phys 34:535–539 4. American Thoracic Society (1991) Lung function testing: selection of reference values and interpretative strategies. Am Rev Respir Dis 144:1202–1218 5. Wagner NL, Beckett WS, Steinberg R (2006) Using spirometry results in occupational medicine and research: common errors and good practice in statistical analysis and reporting. Indian J Occup Environ Med 10:5–10. doi:10.4103/0019-5278.22888 6. David PJ, Pierce R (2008) Spirometry handbook, 3rd edn., Spirometry: the measurement and interpretation of ventilatory function in clinical practiceNational Asthma Campaign, Melbourne, pp 1–24 7. Gomeni R, Merlo-Pich E (2006) Bayesian modelling and ROC analysis to predict placebo responders using clinical score measured in the initial weeks of treatment in depression trials. Br J Clin Pharmacol 63:595–613. doi:10.1111/j.1365-2125.2006. 02815.x 8. Nucci G, Gomeni R, Poggesi I (2009) Model-based approaches to increase efficiency of drug development in schizophrenia: a can’t
123
19.
20.
21.
22.
23.
24.
miss opportunity. Expert Opin Drug Discov 4:837–856. doi:10. 1517/17460440903036073 Holford N, Li J, Benincosa L, Birath M (2002) Population disease progress models for the time course of HAMD score in depressed patients receiving placebo in anti-depressant clinical trials. Population Approach Group in Europe (PAGE) 11th Meeting, Abstract 311. http://www.page-meeting.org/?abstract=311 Reddy VP, Kozielska M, Johnson M, Vermeulen A, de Greef R, Liu J, Groothuis GMM, Danhof M, Proost JH (2011) Structural models describing placebo treatment effects in schizophrenia and other neuropsychiatric disorders. Clin Pharmacokinet 50:429–450. doi:10.2165/11590590-000000000-00000 Russu A, Marostica E, De Nicolao G, Hooker AC, Poggesi I, Gomeni R, Zamuner S (2012) Joint modeling of efficacy, dropout, and tolerability in flexible-dose trials: a case study in depression. Clin Pharmacol Ther 91:863–871. doi:10.1038/clpt. 2011.322 Atkinson AJ, Huang SM, Lertora JJL, Markey SP (2012) Principles of clinical pharmacology. Academic Press, London Shang EY, Gibbs MA, Landen JW, Krams M, Russell T, Denman NG, Mould DR (2009) Evaluation of structural models to describe the effect of placebo upon the time course of major depressive disorder. J Pharmacokinet Pharmacodyn 36:63–80. doi:10.1007/s10928-009-9110-3 Matre D, Casey KL, Knardahl S (2006) Placebo-induced changes in spinal cord pain processing. J Neurosci 26:559–563. doi:10. 1523/JNEUROSCI.4218-05.2006 Mayberg HS, Silva JA, Brannan SK, Tekell JL, Mahurin RK, McGinnis S, Jerabek PA (2002) The functional neuroanatomy of the placebo effect. Am J Psychiatry 159:728–737 de la Fuente-Fernandez R, Ruth TJ, Sossi V, Schulzer M, Calne DB, Stoessl AJ (2001) Expectation and dopamine release: mechanism of the placebo effect in Parkinson’s disease. Science 293:1164–1166. doi:10.1126/science.1060937 Delahoy P, Thompson S, Marschner IC (2010) Pregabalin versus gabapentin in partial epilepsy: a meta-analysis of dose–response relationships. BMC Neurol 10:104. doi:10.1186/1471-2377-10104 Houk BE, Bello CL, Kang D, Amantea M (2009) A population pharmacokinetic meta-analysis of sunitinib malate (SU11248) and its primary metabolite (SU12662) in healthy volunteers and oncology patients. Clin Cancer Res 15:2497–2506. doi:10.1158/ 1078-0432.ccr-08-1893 Chan PL, Weatherley B, McFadyen L (2008) A population pharmacokinetic meta-analysis of maraviroc in healthy volunteers and asymptomatic HIV-infected subjects. Br J Clin Pharmacol 65:76–85. doi:10.1111/j.1365-2125.2008.03139.x Ito K, Ahadieh S, Corrigan B, French J, Fullerton T, Tensfeld T (2010) Disease progression meta-analysis model in Alzheimer’s disease. Alzheimers Dement 6:39–53. doi:10.1016/j.jalz.2009.05. 665 Forey BA, Thornton AJ, Lee PN (2011) Systematic review with meta-analysis of the epidemiological evidence relating smoking to COPD, chronic bronchitis and emphysema. BMC Pulm Med 11:36. doi:10.1186/1471-2466-11-36 Wang X, Shang D, Ribbing J, Ren Y, Deng C, Zhou T, Guo F, Lu W (2012) Placebo effect model in asthma clinical studies: longitudinal meta-analysis of forced expiratory volume in 1 second. Eur J Clin Pharmacol 68:1157–1166. doi:10.1007/s00228-0121245-2 Rowland M, Sheiner LB, Steimer JL (1985) Variability in drug therapy: description, estimation and control. Raven Press, New York Lunn DJ, Thomas A, Best N, Spiegelhalter DJ (2000) WinBUG: a Bayesian modeling framework: concepts, structure and extensibility. Stat Comput 10:325–337
J Pharmacokinet Pharmacodyn (2014) 41:553–569 25. Development Core Team R (2011) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna 26. Spiegelhalter DJ, Best NG, Carlin BP, van der Linde A (2002) Bayesian measures of model complexity and fit. J R Statist Soc B 64(Part 4):583–639. doi:10.1111/1467-9868.00353 27. Karlsson M, Holford N (2008) A tutorial on Visual Predictive Checks. Population Approach Group in Europe (PAGE) 17th Meeting, Abstract 1434. http://www.page-meeting.org/default. asp?abstract=1434 28. Sheiner LB, Beal SL (1981) Some suggestions for measuring predictive performance. J Pharmacokinet Biopharm 9:503–512 29. Fisher RA (1925) Statistical methods for research workers. Oliver & Boyd, Edinburgh
569 30. Ette EL, Williams PJ (2013) Pharmacometrics: the science of quantitative pharmacology. Wiley, Hoboken 31. Barraclough KA, Isbel NM, Kirkpatrick CM, Lee KJ, Taylor PJ, Johnson DW, Campbell SB, Leary DR, Staatz CE (2011) Evaluation of limited sampling methods for estimation of tacrolimus exposure in adult kidney transplant recipients. Br J Clin Pharmacol 71:207–223. doi:10.1111/j.1365-2125.2010.03815.x 32. Sasakawa T, Masui K, Kazama T (2013) Iwasaki H. Comparison of Five Parameter Sets. American Society of Anesthesiologists, Performance of Compartmental Pharmacokinetic Model for Rocuronium
123
