Pharmacometrics : Nonlinear mixed effect models in Statistics. Department of Statistics Ewha Womans University Eun-Kyung Lee
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1 Pharmacometrics : Nonlinear mixed effect models in Statistics Department of Statistics Ewha Womans University Eun-Kyung Lee 1
2 Clinical Trials Preclinical trial: animal study Phase I trial: First in human, small number of healthy v olunteers, dose finding Phase II/III trial: First in patient. Large number of patie nts. Prepare for FDA approval 2
3 Phase I clinical trial first in human small number of healthy volunteers, usually 10~20 Purpose find the right dose for human check toxicity After administration, collect the following information Cp: the drug concentration in blood Time: usually for 24 hours Effect: the effect of drug (eg. Blood pressure for hypertension drug) 3
4 Pharmacometrics PK PD
5 Schematics of PK Absorption Distribution Metabolism Elimination 5
6 Schematics of PK and PD PD Efficacy Toxicity PK Tissue (Effect site) Distribution Drug admin Absorption Blood Metabolism Excretion 6
7 Theophylline 7
8 One-comp model: Oral admin. with 1 st order absorp. K a K e Elimination X GI Dose X B K a : 1 st order constant absorption rate Data : C t, t, dose Parameters : K a, K e, V K e : elimination constant rate V : volume of distribution (theoretical volume that total amount of admin. drug would have to occupy to provide the same concentration; large V, more diluted in the blood) Cl : clearance
9 two-comp model
10 three-comp model
11 Population approach 1. Structural sub-model Overall trend using fixed-effect parameters 11
12 Population approach 2. Statistical sub-model 1) Intra-individual variability 12
13 Population approach 2) Inter-individual variability C ij : concentration of the ith patient and the jth timepoint K ai : absorption rate of the ith patient V i : volume of distribution of the ith patient K ei : elimination constant rate of the ith patient 13
14 Population approach 3. Covariance sub-model The relationship between covariate and model par ameter! Nonlinear mixed effect model with covariates 14
15 PK Model Individual level model Population level model y ij : measurement of i th subject at time t ij D i : dose t ij : time x i : covariates(weight, height, age, etc) θ : PK parameter(v, CL, Ka, etc) 15
16 Estimation Methods in NONMEM linear approximation approach First order method (FO) First order conditional method (FOCE) Laplacian method (Laplacian) EM algorithm based approach Iterative two stage (ITS) Important Sampling method (IMP) IMP assisted by mode a posteriori (IMPMAP) Stochastic approximation EM(SAEM) Bayesian approach 16
17 References for estimation methods Dempster, A. P., Laird N., and Rubin, D. B (1977) Maximum likelihood from incomplete data via the EM algorithm Laird, N. M. and Ware, J. H (1982) Random-effects models for longitudinal data Wu, C. F. (1983) On the convergence properties of the EM algorithm Lindstorm, M.J., and Bates, D. M(1988) Newton-Raphson and EM algorithms for linear mixed effects models for repeated-measures data Delyon, B., Laville, M., and Moulines, E.(1990) Convergence of a stochastic approximation version of the EM algorithm Lindstorm, M.J., and Bates, D.M(1990) Non-linear mixed-effects models for repeatedmeasures data Wolfinger, R. (1993) Laplace s approximation for nonlinear model Pinheiro, J.C. and Bates, D.M (1994) Approximations to the loglikelihood function in the nonlinear mixed effect model Davidian, M and Giltinan, D. M (1995) Nonlinear Models for Repeated Measurement Data Walker, S (1996) An EM algorithm for Nonlinear Random Effects Models Kuhn, E., and Lavielle, M. (2004) Coupling a stochastic approximation version of EM with a MCMC procedure Kuhn, E., and Lavielle, M.(2005) Maximum likelihood estimation in nonlinear mixed effects models Lavielle, M., and Meza, C. (2007) A parameter expansion version of the SAEM algorithm Meza, C., Jaffrezic, F., and Foulley, J-L (2007) REML estimation of variance parameters in nonlinear mixed effects models using the SAEM algorithm 17
18 Results from NONMEM fitting 1. Estimates of parameters PK/PD population parameter Inter-individual variablility Intra-individual variablility 2. Predictions PRED: predictive values without random effect ( when ) EBE: empirical Bayesian estimate of IPRED: individual level predicted values ( when ) 3. Residuals RES: residuals WRES: weighted residuals(for FO method) CWRES: conditional weighted residual (for FOCE method) 4. Covariates, TIME, DV, 18
19 Graphical Method: Prediction-based(1) 1. Prediction-based : PRED, IPRED, etc. PRED vs DV(obs) plot with line of identity (black line) lines between points for an individual ID values for outer edges of data Regression line (red line) IPRED vs DV plot Source: Intermediate NONMEM 7 Workshop, 2010, Seoul, Korea 19
20 Graphical Method: Residual-based(1) 2. Residual-based : RES, WRES, CWRES, etc. WRES (weighted residual) : use for FO method CWRES (conditional WRES) : use for FOCE method 20
21 Graphical Method: Residual-based(2) TIME vs. residual plot with lines between points for an individual ID values for outer edges of data line of y=0 (black line) lowess line line (red line) PRED vs. residual plot Covariate vs. residual plot Source: Intermediate NONMEM 7 Workshop, 2010, Seoul, Korea 21
22 Graphical Method: Residual-based(3) Estimation method vs. prediction/residual in NONMEM Estimation method PRED RES Weighted RES FO NPRED NRES NWRES FO INTER PREDI RESI WRESI FOCE CPRED CRES CWRES FOCE INTER CPREDI CRESI CWRESI MC based EPRED ERES ECWRES MC based INTER EPRED ERES EWRES MC based NPDE 22
23 Graphical Method: Residual-based(4) Source: Intermediate NONMEM 7 Workshop, 2010, Seoul, Korea 23
24 Graphical Method: Residual-based(5) Source: Intermediate NONMEM 7 Workshop, 2010, Seoul, Korea 24
25 Graphical Method: Residual-based(6) Source: Intermediate NONMEM 7 Workshop, 2010, Seoul, Korea 25
26 Shiny : Explore NONMEM data 26
27 Shiny : Explore NONMEM output 27
28 Shiny : Explore NONMEM output 28
29 Theophylline 29
30 Theophylline 30
31 Graphical Method: Simulation-based(1) Data Model : observed value, ith subject jth observation : observed time of ith subject jth observation : covariates of ith subject Estimate 31
32 Graphical Method: Simulation-based(2) Simulated data from model : estimates of Simulate data from 32
33 VPC : example Source: 33
34 VPC : Visual Predictive Checks (1) Step 1: Binning TIME data Make G intervals with Binning strategy Have similar amount of data in each bin No need binning if observation times are same between subjects 34
35 VPC : Visual Predictive Checks (2) Step 2: find PI of the original data Find 5%, 50%, 95% percentile of Also find median(50%percentile) of for each g=1,, G for each g=1,, G g med(tij) 5% percentile % percentile 95% percentile G * Prediction Interval(PI) of each bin is 5% percentile and 95% percentile 35
36 VPC : Visual Predictive Checks (3) Step 3: find PI of the simulation data Find 5%, 50%, 95% percentile of for each g=1,, G g 5% percentile % percentile 95% percentile G 36
37 VPC : Visual Predictive Checks (4) Step 4: find CI of 5%, 50%, and 95% percentile of the si mulation data (1) Find 5%, 50%, 95% percentile of for each g=1,, G and each k=1,,k 5% percentile 50% percentile 95% percentile k g 1 2 K k g 1 2 K k g 1 2 K G G G 37
38 VPC : Visual Predictive Checks (5) Step 4: find CI of 5%, 50%, and 95% percentile of the si mulation data (2) 5% percentile k 1 2 K g 1 2 G Find 2.5% and 97.5% percentile of CI of 5% percentile of the simulation data g 2.5% percentile % percentile G 38
39 VPC : Visual Predictive Checks (6) * Repeat this procedure for 50% and 95% percentile CI of 50% percentile of the simulation data g 2.5% percentile 1 2 G 97.5% percentile CI of 95% percentile of the simulation data g 2.5% percentile % percentile G 39
40 VPC : Visual Predictive Checks (7) Scatter VPC Make scatter plot of and with 3 lines - vs. - vs. - vs. Source: 40
41 VPC : Visual Predictive Checks (8) Percentile VPC Make plot with 6 lines - vs. - vs. - vs. - vs. - vs. - vs. Source: 41
42 VPC : Visual Predictive Checks (9) Confidence Interval VPC Make plot with 3 lines and 3 areas - vs. : line - vs. : line - vs. : line - vs. and : area - vs. and : area - vs. and : area 42
43 VPC : Visual Predictive Checks (10) Confidence Interval VPC Source: 43
44 VPC : Visual Predictive Checks (11) Handle covariate in VPC - Need to use stratification of covariate - For each strata, draw VPC plot - Pros - Allows subset of data/model to be inspected - Can increase resolution of model misspecification - Cons - Can dilute the signal - Multiple plot makes diagnostics complex - Use prediction correction 44
45 Prediction Correction (1) Step 1: Binning TIME/Covariate data Make T intervals with Make C intervals with Overall number of bins: G = T * C t T 1 T 2 T T c X 1 G 1 G 2 G T X 2 G T+1 X C G G 45
46 Prediction Correction (2) Step 2: Calculate pcvpc(prediction-corrected VPC) where : lower bound of yij from PRED : med(predij in bin g) : typical model prediction 46
47 Prediction Correction (3) Step 3: calculate pvcvpc(prediction- and variability-cor rected VPC) where 47
48 Prediction Correction (4) Step 4: plot pcvpc and pvcvpc - Combine all bins and draw one plot in the same manner of VPC - Use pcyij / pvcyij instead of Yij Source: AAPS Journal(2013), Vol.13(2)
49 Prediction Correction (4) VPC vs. pcvpc Source: AAPS Journal(2013), Vol.13(2)
50 QVPC/BVPC (1) Data : observed value, ith subject jth observation (including missing) : observed time of ith subject jth observation Rearranged Data 50
51 QVPC/BVPC(2) Simulated Data from model Bootstrap sample from data 51
52 QVPC/BVPC(3) Quantified VPC 1. Find 2. Calculate 3. For each t, draw parallel boxes or 52
53 QVPC/BVPC(3) Quantified VPC Source: J. Pharmacokinet Pharmacodyn(2008), Vol.35:
54 QVPC/BVPC(4) Bootstrap VPC 1. If, impute with 2. Find for each t and k k t K 54
55 QVPC/BVPC(5) Bootstrap VPC 3. Find 5%, 50%, 95% percentile of t 5% percentile % percentile 95% percentile 55
56 QVPC/BVPC(6) Bootstrap VPC 4. Draw scatter VPC plot and draw area with and draw line with and 56
57 Shiny : Visual Predictive Check 57
58 Shiny : Visual Predictive Check 58
59 Shiny : Visual Predictive Check 59
60 Shiny : Visual Predictive Check 60
61 Shiny : Visual Predictive Check 61
62 Shiny : Visual Predictive Check 62
63 fit4nm 63
64 fit4nm 64
65 fit4nm 65
66 fit4nm 66
67 fit4nm 67
68 fit4nm 68
69 fit4nm 69
70 fit4nm 70
71 asvpc: example (6) Bin-related weights Distance-related weights CI: 115 bins < 162 CI: 115 bins <
72 References Bergstrand, Hooker Wallin, and Karlsson (2011) Prediction-Corrected Visual Predictive Checks for Diagnosing Nonlinear Mixed-Effects Models, The AAPS Journal Vol 13(2) Brendel, K., Comets, E., Laffont, C., Laveille, C., and Mentre, F. (2006) Merics for External Model Evaluation with an Application to the Population Pharmacokinetics of Gliclazide, Pharmaceutical Researclh Vol 23(9) F. Mentre, and S. Escolano (2006) Prediction Discrepancies for the Evaluation of Nonlinear Mixed-Effects Models, Journal of Pharmacokinetics and Pharmacodynamics, 33(3), D. D. Wang, and S. Zhang (2012) Standardized Visual Predictive Check Versus Visual Predictive check for Model Evaluation, Journal of Clinical Pharmacology,52, P. R. Jadhav, and J. V.S. Gobburu (2005) A New Equivalence Based Metric for Predictive check to Qualify Mixed-Effects Models, The AAPS Journal, 7(3) E523-E531 E. Comets, K. Brendel, and F. Mentre (2008) Computing normalised prediction distribution errors to evaluate nonlinear mixed-effect models: The npde add-on package for R, Computer Methods and Programs in Biomedicine,90, MO Karlsson, and RM savic (2007) Diagnosing Model Diagnostics, Clinical Pharmacology and Therapeutics, 82(1), Y. Yano, S.L. Beal, and L. B. Sheiner (2001) Evaluating Pharmacokinetic/Pharmacodynamic Models Using the Posterior Predictive Check, Journal of Pharmacokinetics and Pharmacodynamics, 28(2), T.M. Post, J. I. Freijer, B. A. Ploeger, and M. Danhof (2008) Extensions to the Visual Predictive check to facilitate model performance evaluation, Journal of Pharmacokinetics and Pharmacodynamics, 35,
73 References X. Sun, K. Wu, and D. Cook (2011) PKgraph: An R package for graphically diagnosing population pharmacokinetic models, Computer methods and programs in biomedicine, 104, K. Ito and D. Murphy (2013) Application of ggplot2 to Pharmacometric Graphics, CPT: Pharmacometrics and systems Pharmacology, 2, e79 73
74 Questions 74
75 75
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