Transcript Model-Based adaptive optimal design in pharmacometrics

Influence of the size of the cohorts in
adaptive design for nonlinear mixed effect
models: an evaluation by simulation for a
pharmacokinetic (PK) and pharmacodynamic
(PD) model in oncology
Giulia Lestini, Cyrielle Dumont, France Mentré
IAME UMR 1137, INSERM, University Paris Diderot, Paris, France
PODE 2014
September 11, 2014
1
Outline
•
•
•
•
•
•
Context
Objectives
Methods
Simulation Study
Results
Conlusion and Perspectives
2
Context: Optimal design in NLMEM
• Choosing a good design for a planned study is essential
– Number of patients
– Number of sampling times for each patient
– Sampling times (allocation in time)
• Optimal design depends on prior information (model and
parameters)
• D-optimality criterion
– Local Designs
– Robust designs
Atkinson, Optimum Experimental Designs. (1995)
Dodds et al., J Pharmacokinet Pharmacodyn. (2005)
Pronzato and Walter, Math Biosci. (1988)
3
Context: Adaptive design
• AD: clinical trial designs that use accumulating
information to decide how to modify predefined aspects
of the study
– Areas of interest: predicting clinical data; Phase 1 studies
– ADs are useful to provide some flexibility but were rarely used
for NLMEM
• Two-stage designs could be more efficient than fully
adaptive design (not yet tested in NLMEM)
• Dumont et al. implemented two-stage AD in NLMEM
• AD questions:
– How many adaptations? (e.g stages)
– How many patients in each cohort? (i.e. cohorts size)
Foo et al., Pharm Res. (2012)
Mentré et al., CPT Pharmacometrics Syst Pharmacol. (2013)
Fedorov et al., Stat Med. (2012)
Dumont et al., Commun Stat. (2014)
4
Objectives
1. To compare by simulation one and two-stage
designs using a PKPD model in oncology
2. To study the influence of the size of each cohort in
two-stage designs
3. To test extensions of two-stage adaptive design as
three- and five-stage adaptive designs
5
Methods: Basic mixed effect model
• Individual model (one continuous response)
yi =𝑓(𝜙𝑖 , ξ𝑖 ) + 𝜀𝑖 vector of ni observations
•
•
•
•
•
ξ𝑖 : individual sampling times tij j=1, … ni
𝜙𝑖 : individual parameters (size p)
𝑓: nonlinear function defining the structural model
𝜀𝑖 : gaussian zero mean random error
var (𝜀𝑖 ) = (𝜎𝑖𝑛𝑡𝑒𝑟 + 𝜎𝑠𝑙𝑜𝑝𝑒 𝑓(𝜙𝑖 ,ξ𝑖 ))2 combined error model
• Random-effects model
– 𝜙𝑖 = 𝛽 × exp(𝑏𝑖 )
– 𝑏𝑖 ~𝑁 0, Ω
here Ω diagonal: 𝜔𝑘2 = Var(𝑏𝑖𝑘 )
• Population parameters: Ψ (size P)
– 𝛽 (fixed effects)
– unknowns in Ω (variance of random effects)
– 𝜎𝑖𝑛𝑡𝑒𝑟 and/or 𝜎𝑠𝑙𝑜𝑝𝑒 (error model parameters)
6
Methods: Basic population design
• Assumption
– N individuals i
– same elementary design ξ in all N patients (ξ𝑖 = ξ) with
𝑡1 , … , 𝑡n sampling times
– ntot= N × n
• n = number of samples for each individual
• Population design
Ξ = ξ, N
7
Methods : Fisher Information Matrix (FIM)
• Elementary FIM: 𝑀𝐹 Ψ, ξ = 𝐸
−𝜕2 𝐿(𝑦;Ψ)
𝜕Ψ𝜕Ψ𝑇
• no analytical expression for FIM  FO approximation
• Population Fisher Information Matrix for one group design
𝑀𝐹 Ψ, Ξ = 𝑁 × 𝑀𝐹 Ψ, ξ
• 𝑀𝐹 is implemented in the R function « PFIM »
• In PFIM 4.0 (April 2014) it is possible to include prior information on
FIM for two-stage design
Mentré et al., Biometrika (1997)
Bazzoli et al., Comput Methods Programs Biomed. (2010)
Mentré et al., PAGE Abstr 3032 (2014)
Dumont et al., Commun Stat. (2014)
www.pfim.biostat.fr
8
Method: K-stage Adaptive Design
COHORT 1: N1
Model M
Initial parameters Ψ1
Design
Optimisation
Design ξ1
...
COHORT k: N𝑘
COHORT K: NK
Model M
Ψ𝑘−1
Model M
ΨK−1
Design
Optimisation
Design ξ𝑘
...
Data Y1 + ⋯ +
Y𝑘
Data
Y1
Estimation
Ψ1 (from Y1 )
...
Estimation
Ψ𝑘 (from Y1 , … , Y𝑘 ,
N = N1 + ⋯ + N𝑘 )
...
...
...
Design
Optimisation
Design ξK
Data
Y1 + ⋯ Y𝑘 + ⋯ + YK
Estimation
ΨK (from Y1 , Y2 , … , YK ,
N = N1 + … + N𝑘 + ⋯ + NK )
1st stage:
from a priori Ψ1 , find 𝜉1 that maximizes determinant of
kth
MF(Ψ1 , N1 𝜉 ) = N1 MF(Ψ1 , 𝜉)
stage:
using estimated Ψ𝑘−1 , find 𝜉𝑘 that maximizes determinant of
MF(Ψ𝑘−1 ,N1 𝜉1 +...+N𝑘 𝜉) = N1 MF(Ψ𝑘−1 ,𝜉1 ) +…+N𝑘−1 MF(Ψ𝑘−1 ,𝜉𝑘−1 )+N𝑘 MF(Ψ𝑘−1 , 𝜉)
9
Simulation Study: PKPD Model
• 2 responses model, developed for a novel oral
transforming growth factor β (TGF − β)
PK: concentration
𝑓𝑃𝐾 (𝜙, 𝑡) =
𝑘=
𝐷𝑂𝑆𝐸 𝑘𝑎
𝑉
𝑘𝑎 −𝑘
𝑒
−𝑘𝑡
−𝑒
−𝑘𝑎 𝑡
ka
Concentration
,
CL/V
𝐶𝐿
𝑉
Parameters: 𝑘𝑎 , 𝑉, 𝐶𝐿
PD: relative inhibition of TGF-β
𝑑𝑓𝑃𝐷 (𝜙, 𝑡)
𝑑𝑡
𝐼𝑚𝑎𝑥 = 1
=
𝐼
∙𝑓
(𝜙, 𝑡)
𝑘𝑜𝑢𝑡 𝑚𝑎𝑥 𝑃𝐾
𝑓𝑃𝐾 (𝜙, 𝑡)+𝐼𝐶50
− 𝑘𝑜𝑢𝑡 ∙ 𝑓𝑃𝐷 (𝜙, 𝑡),
ksyn
Effect
kout
Parameters: 𝑘𝑜𝑢𝑡 , 𝐼𝐶50
Gueorguieva et al., Comput Methods Programs Biomed. (2007)
Gueorguieva et al., Br J Clin Pharmacol. (2014)
Bueno et al., Eur J Cancer. (2008)
10
Simulation Study: Parameters
Prior (Ψ0)
True (Ψ*)
𝜔𝑘2𝑎
2
100
40
0
2
100
10
0
𝜎𝑖𝑛𝑡𝑒𝑟,𝑃𝐾
𝜎𝑠𝑙𝑜𝑝𝑒,𝑃𝐾
0.49
0.49
0
0.2
0.49
0.49
0
0.2
PD Parameters
Prior (Ψ0)
True (Ψ*)
𝑘𝑜𝑢𝑡 h−1
𝐼𝐶50 mg/L
𝜔𝑘2𝑜𝑢𝑡
2
0.3
0.49
0.2
0.3
0.49
𝜎𝑖𝑛𝑡𝑒𝑟,𝑃𝐷
𝜎𝑠𝑙𝑜𝑝𝑒,𝑃𝐷
0.49
0.2
0
0.49
0.2
0
PK Parameters
𝑘𝑎 h−1
𝑉 L
𝐶𝐿 Lh−1
𝜔𝑉2
2
𝜔𝐶𝐿
2
𝜔𝐼𝐶50
11
Simulation Study: Evaluated designs
• N=50
One-stage designs
• Rich design, n=6 sampling times: 𝜉𝑟𝑖𝑐ℎ = (0.1, 0.5, 1.5, 4, 6, 12)
• 2 optimal designs, n=3 sampling times among the 6 of 𝜉𝑟𝑖𝑐ℎ :
– 𝜉0 = 𝜉0𝑃𝐾 = 0.1, 4, 12 ; 𝜉0𝑃𝐷 = 0.5, 1.5, 4 (D-optimal for Ψ0)
– 𝜉∗ = 𝜉∗𝑃𝐾 = 0.1, 4, 12 ; 𝜉0𝑃𝐷 = 4, 6, 12 (D-optimal for Ψ*)
– mixed design 𝜉0∗ (N1=25 patients with 𝜉0 ; N2=25 patients with 𝜉∗ )
Two-stage designs
• Balanced: 𝜉25−25 (N1=N2=25)
• Various sizes for cohorts 1 and 2: 𝜉10−40 , 𝜉15−35 , 𝜉35−15 , 𝜉40−10
Three-stage designs
• Small size for cohorts 1 (N1=10): 𝜉10−20−20 , 𝜉10−10−30
Five-stage design
𝜉10−10−10−10−10 (N1=N2=N3=N4=N5=10)
12
Simulation Study: Clinical Trial Simulation
• 100 data sets simulated with parameters Ψ* and design 𝜉𝑟𝑖𝑐ℎ
– For the designs to be evaluated were kept only the corresponding sampling
times
Parameter estimation: SAEM algorithm in MONOLIX 4.3
– 5 chains, initial estimates: Ψ0
• Comparison of one-, two-,three- and five- stage designs from 100
estimated Ψ1 , Ψ2 , Ψ3 , Ψ5 :
– Relative Estimation Error (REE)
– Relative Bias (RB)
– Relative Root Mean Squared Error (RRMSE)
13
Results: 1-stage vs 2-stage balanced design
• Relative Estimation Error (REE) for PK parameters Ka and CL
Ka
𝜉𝑟𝑖𝑐ℎ
RB
0.4
𝜉∗
0.9
𝜉0
1.0
CL
𝜉0∗
1.1
𝜉25−25
0.5
𝜉𝑟𝑖𝑐ℎ
RB
1.6
𝜉0
𝜉∗
1.7
1.7
𝜉0∗
1.7
𝜉25−25
1.8
14
Results: 1-stage vs 2-stage balanced design
• Relative Estimation Error (REE) for PD parameters Kout and IC50
Kout
𝜉𝑟𝑖𝑐ℎ
RB
0.6
𝜉∗
2.6
𝜉0
34.2
IC50
𝜉0∗
3.2
𝜉25−25
3.7
𝜉𝑟𝑖𝑐ℎ
RB
-0.5
𝜉∗
-0.3
𝜉0
53.1
𝜉25−25
𝜉0∗
0
1.5
15
Results: 1-stage vs 2-stage balanced design
• Relative Root Mean Squared Error (RRMSE) for PD parameters
𝜉∗
𝜉0
𝜉0∗
𝜉25−25
16
Results: 1-stage vs 2-stage balanced design
• Relative Root Mean Squared Error (RRMSE) for PD parameters
𝜉∗
𝜉0
𝜉0∗
* RRMSEs standardized to 𝜉∗ (best 1-stage design)
𝜉25−25
17
Results: Cohort size influence in 2-stage design
• Relative Root Mean Squared Error (RRMSE) for PD parameters
𝜉10−40
𝜉15−35
𝜉25−25
𝜉35−15
𝜉40−10
18
Results: Cohort size influence in 2-stage design
• Relative Root Mean Squared Error (RRMSE) for PD parameters
𝜉10−40
𝜉15−35
𝜉25−25
𝜉35−15
* RRMSEs standardized to 𝜉∗ (best 1-stage design)
𝜉40−10
19
Results: 2-stage vs 3- and 5-stage adaptive designs
• Relative Root Mean Squared Error (RRMSE) for PD parameters
𝜉10−40
𝜉10−20−20
𝜉10−10−30 𝜉10−10−10−10−10
20
Results: 2-stage vs 3- and 5-stage adaptive designs
• Relative Root Mean Squared Error (RRMSE) for PD parameters
𝜉10−40
𝜉10−20−20
𝜉10−10−30 𝜉10−10−10−10−10
* RRMSEs standardized to 𝜉∗ (best 1-stage design)
21
Results: 2-stage vs 3- and 5-stage adaptive designs
• Relative Root Mean Squared Error (RRMSE) for PD parameters
𝜉10−40
𝜉10−20−20
𝜉10−10−30 𝜉10−10−10−10−10
* RRMSEs standardized to 𝜉∗ (best 1-stage design)
𝜉25−25
22
Conclusions
1. With the balanced two-stage design 𝜉25−25
– results are very close to those of 𝜉∗ and are much better than those of 𝜉0
2. The balanced 𝜉25−25 was the best two-stage design compared to
unbalanced cohort size, especially if the second cohort was of
small size
3. In case of small first cohort, more adaptive steps are needed, but
these designs are more complex to implement
• Perspectives:
– Use robust approach for first stage
– Expand the approach for dose-finding
– Perform other studies
23
Thank you for your attention !
The research leading to these results has received support from the
Innovative Medicines Initiative Joint Undertaking under grant
agreement n° 115156, resources of which are composed of financial
contributions from the European Union's Seventh Framework
Programme (FP7/2007-2013) and EFPIA companies’ in kind
contribution. The DDMoRe project is also financially supported by
contributions from Academic and SME partners
24
Back up
25
Results: Cohort size influence in 2-stage design
• Relative Estimation Error (REE) for PK parameters Ka and CL
Ka
CL
𝜉10−40 𝜉15−35 𝜉25−25 𝜉35−15 𝜉40−10
RB
0.5
0.8
0.5
0.7
0.9
𝜉10−40 𝜉15−35 𝜉25−25 𝜉35−15 𝜉40−10
RB
1.7
1.7
1.8
1.8
1.8
26
Results: Cohort size influence in 2-stage design
• Relative Estimation Error (REE) for PD parameters Kout and IC50
RB
Kout
IC50
𝜉10−40 𝜉15−35 𝜉25−25 𝜉35−15 𝜉40−10
𝜉10−40 𝜉15−35 𝜉25−25 𝜉35−15 𝜉40−10
5.9
5.0
3.7
8.9
10.4
RB
8.3
5.3
1.5
8.8
12.7
27
Results: number of different elementary designs (ndesigns)
and number of datasets with 𝜉 = 𝜉∗ (ndatasets ) in two-, three- and
five- stage design
Designs
Two-stage
ξ10−40
ξ15−35
ξ25−25
ξ35−15
ξ40−10
Three-stage
ξ10−20−20
ξ10−10−30
Five-stage
ξ10−10−10−10−10
2nd Stage
3rd Stage
4th Stage
5th Stage
ndesigns n∗datasets ndesigns n∗datasets ndesigns n∗datasets ndesigns n∗datasets
12
8
6
6
6
24
35
49
47
45
12
12
27
28
5
6
71
61
12
28
7
60
4
69
4
76
28