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Computational Challenges of PK/PD NLME

Computational Challenges of PK/PD NLME . Bob Leary Pharsight Corporation. Computational challenge #1 – make execution time reasonable. Many PK/PD NLME software packages - NONMEM (with many choices for methods) is by far the most popular, but not necessarily always the most appropriate

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Computational Challenges of PK/PD NLME

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  1. Computational Challenges of PK/PD NLME Bob Leary Pharsight Corporation © Tripos, L.P. All Rights Reserved

  2. Computational challenge #1 – make execution time reasonable • Many PK/PD NLME software packages - NONMEM (with many choices for methods) is by far the most popular, but not necessarily always the most appropriate • All methods are to some degree computationally intensive – execution time can be a limiting factor, even for a single run • Many types of analyses require multiple runs (bootstrap, covariate search, likelihood profiling, etc. – execution time constraints can be severe). © Tripos, L.P. All Rights Reserved Slide 2

  3. Execution time, cont’d • There are trades-offs between accuracy/statistical quality and speed: FO vs FOCE vs MCPEM/SAEM/NPAG • Technology (parallel computing) can help a lot, but algorithmic improvements are at least equally important (SAEM, MCPEM vs. FOCE) © Tripos, L.P. All Rights Reserved Slide 3

  4. © Tripos, L.P. All Rights Reserved Slide 4

  5. NPAG Outperforms NPEM • CPU HRS MB LOG -LIK • NPEM: 2037 10000 -433.1 • NPAG: 0.5 6 -425.0 © Tripos, L.P. All Rights Reserved Slide 5

  6. Computational Challenges #2 - #4 • PK/PD NLME models and data are complex and computationally demanding, probably much more so that most other NLME application areas. Special purpose software is needed. • Many of the methods are complex, not well documented, approximate, not easily understood by the user base, and at least somewhat fragile • Software is relatively difficult to learn and use © Tripos, L.P. All Rights Reserved Slide 6

  7. A chronology of events in development of NLME • 1972 – Sheiner, Rosenberg, Melmon paper (FO) • 1977 – NONMEM group established at UCSF • (L. Sheiner and S. Beal) • 1979 – First NONMEM FO program appears • 1986 – First nonparametric method NPML (A. Mallet) • 1990 – First FOCE method (Lindstrom/Bates) • 1990 – First Bayesian method (Gelfand/Smith – Bugs and PKBugs) © Tripos, L.P. All Rights Reserved Slide 7

  8. Chronology, cont’d • 1991 - NPEM nonparametric method (Schumitzky) • 1992 – First PAGE meeting (63 participants, 500+ in 2010) • 1993 - First Laplacian method - enables general LL models (Wolfinger) • 1999 – FDA Guidance for POP PK • 2004 –2005 EM methods (SAEM, MCPEM, PEM) , Lyon inter-method comparison exercises, MONOLIX • 2007 – EMEA guidelines for POP PK • 2009 – NONMEM SAEM/MCPEM/Bayesian, Pharsight PHOENIX © Tripos, L.P. All Rights Reserved Slide 8

  9. Some PK/PD software • NONMEM (L. Sheiner and S. Beal, UCSF 1979 – to date) • -primarily parametric modeling, although has primitive NP method • -classical approximate likelihood methods (FO, FOCE, FOCEI, Laplacian) • -’new’ accurate likelihood EM methods (SAEM and MCPEM) (2009) • -Bayesian methods (2009) • USC*PACK (R. Jelliffe, USC/LAPK et al., 1993-to date) • -nonparametric (NPEM, NPAG) (A. Schumitzky, R. Leary) • -individual dosing optimization – multiple model control (D. Bayard) © Tripos, L.P. All Rights Reserved Slide 9

  10. PK/PD software, cont’d • Monolix (INSERM, 2005 - to date) - SAEM (Stochastic Approximation Expectation Maximization) • Adapt/S-Adapt (USC/BMSR, D. D’Argenio, R. Bauer, 1989-to date) MCPEM (Monte Carlo Parametric Expectation Maximization) + Bayesian • PHOENIX (Pharsight, 2009 – to date) classical NM methods + AGQ + SAEM + QMCPEM + NPAG + WinNonLin single subject and NCA modeling • BUGS, WinBUGS – (1999 to date) – Bayesian • S+ NLME, R NLME, SAS PROC-NLMIXED can be used, but not well suited for PK/PD © Tripos, L.P. All Rights Reserved Slide 10

  11. PK/PD Software User Base • WinNonLin (Single Subject, NCA): 6000 (3000 academic, 3000 commercial) • NONMEM (Population NLME): 1500 • Commercial demand for experienced users exceeds supply © Tripos, L.P. All Rights Reserved Slide 11

  12. FDA Guidance for Industry, 1999 • Population PK analysis is concerned with identifying and quantifying the random [random effects] and nonrandom [covariate effects] variability in the PK behavior of the patient population • About 25% of recent submissions at time of writing included a ‘population’ analysis • Magnitude of random variability is particularly important because the safety and efficacy of a drug is affected. • Mentions Standard Two Stage and NLME modeling as possible methods © Tripos, L.P. All Rights Reserved Slide 12

  13. EMEA Guidelines 2007 • NLME Pop PK analysis appears to be mandatory, or at least expected • No mention of STS • Extensive specification of model validation diagnostics and validation techniques (CWRES, predictive checks, etc.) • Notes FDA Guidance is from 1999 and • “The FDAguidance should be read bearing in mind that it was written in 1999 and that population pharmacokinetics is an evolving science” © Tripos, L.P. All Rights Reserved Slide 13

  14. Obligatory ODE section © Tripos, L.P. All Rights Reserved Slide 14

  15. ODE Considerations Most PK models are dynamical systems that can be described by ordinary differential equations (ODEs) ODEs often need to be solved numerically (many PK/PD software packages use ODEPACK, a library of ODE solvers developed by A. Hindmarsh at LLNL) If system is linear and homogeneous with constant coefficients, the matrix exponential can be used Some special cases (1, 2, and 3-compartment models) are best handled by built-in closed form solutions. Special handling capabilities are built in to the software for lag times, bioavailability, etc. © Tripos, L.P. All Rights Reserved Slide 15

  16. A Simple PK Model as ODE : 1-Compartment IV Bolus © Tripos, L.P. All Rights Reserved Slide 16

  17. IV Bolus closed form solution © Tripos, L.P. All Rights Reserved Slide 17

  18. Multiple Doses: Use superposition if model ODE is linear Covariate models with time varying covariates pose additional complications – suppose K=tvK(1+(coef)(SCR-SCR0)) © Tripos, L.P. All Rights Reserved Slide 18

  19. 1-Comp first order absorption extra-vascular dosing © Tripos, L.P. All Rights Reserved Slide 19

  20. 1-Comp first order absorption extra-vascular dose solution © Tripos, L.P. All Rights Reserved Slide 20

  21. 1 compartment 0-order (IV) dosing ODE © Tripos, L.P. All Rights Reserved Slide 21

  22. General N-compartment model : 0 and 1st order dosing © Tripos, L.P. All Rights Reserved Slide 22

  23. Nonlinear cases must be solved numerically with ODE solvers (ODEPACK) © Tripos, L.P. All Rights Reserved Slide 23

  24. ODE ‘solver’ order of preference/speed • Closed form (1, 2, 3 compartment, 0 and 1st order dosing) • Matrix Exponential (Linear, constant coefficient) • Non-stiff numerical ODE solver (Runge-Kutta, Adams) • Stiff ODE solver (Gear BDF) Node execs = (Niter_out)(Nfix+Nran)(Nsub)(Niter_in)(Nran)(Ntime) (100)(10)(1000)(20)(5)(10) = 1,000,000,000 © Tripos, L.P. All Rights Reserved Slide 24

  25. End of ODE section, Start of methods section © Tripos, L.P. All Rights Reserved Slide 25

  26. Simple (single subject) regression Model • PK Model • Data • Residual Error Model © Tripos, L.P. All Rights Reserved Slide 26

  27. Extended least squares objective function © Tripos, L.P. All Rights Reserved Slide 27

  28. Computational challenge : minimize • Nonlinear, nonconvex, • But no likelihood approximations are necessary in single subject case • Unconstrained (can add bound constraints if desired) • No exploitable structure • Use general purpose unconstrained quasi-Newton method UNCMIN from TOMS is 99+% reliable, but may encounter problems with multiple minima • - © Tripos, L.P. All Rights Reserved Slide 28

  29. Regression model to estimate V and K © Tripos, L.P. All Rights Reserved Slide 29

  30. A simple population PK model: IV Bolus cont’d © Tripos, L.P. All Rights Reserved Slide 30

  31. Population Likelihood function © Tripos, L.P. All Rights Reserved Slide 31

  32. Li cannot be evaluated analytically – how to proceed? • Numerical quadrature - adaptive Gaussian quadrature, Monte Carlo integration , quasi-Monte Carlo integration – very slow, dimensionality problems • Laplace approximation – FO, FOCE, Laplace (Y. Wang, 2006) • Use a method that does not require integration (SAEM,PEM, MCPEM, Bayesian methods, nonparametric methods) © Tripos, L.P. All Rights Reserved Slide 32

  33. Laplacian Approximation (FO, FOCE, Laplacian) © Tripos, L.P. All Rights Reserved Slide 33

  34. Joint log likelihood J(q,s2,W,h) and Laplacian, FOCE, and FO approximations © Tripos, L.P. All Rights Reserved Slide 34

  35. Conditional methods (FOCE, Laplace) require nested optimizations to find mode of J, FO does not • Each top level evaluation of • requires Nsub mode-finding optimizations of • Total number of innter optimizations = (Neval)(Nsub) - can • easily reach 100,000 or more, leading to a reliability problem © Tripos, L.P. All Rights Reserved Slide 35

  36. Lyon 2004-2005 ‘bake-off’ of NLME methods © Tripos, L.P. All Rights Reserved Slide 36

  37. © Tripos, L.P. All Rights Reserved Slide 37

  38. STATISTICAL EFFICIENCIES © Tripos, L.P. All Rights Reserved Slide 38

  39. Approximate likelihoods can destroy statistical efficiency © Tripos, L.P. All Rights Reserved Slide 39

  40. SAEM, MCPEM, NPEM/NPAG © Tripos, L.P. All Rights Reserved Slide 40

  41. The ideal case –Vi and Ki can be observed Parametric estimators Nonparametric histogram © Tripos, L.P. All Rights Reserved Slide 41

  42. The real case: Vi and Ki are not directly observable • We only have time profiles of drug plasma concentrations • Measurement and dosing protocols are not uniform over different individuals • At best, we can get estimates • by solving a regression model © Tripos, L.P. All Rights Reserved Slide 42

  43. Standard Two-Stage Method Vi and Ki are estimated by simple nonlinear regression methods Parametric estimators Nonparametric histogram © Tripos, L.P. All Rights Reserved Slide 43

  44. MCPEM and SAEM are Monte Carlo versions of STS © Tripos, L.P. All Rights Reserved Slide 44

  45. NPEM and NAG: Many PK/PD populations have sub-populations that would be missed by parametric techniques B – Best normal approximation to population distribution A - True two-parameter population distribution © Tripos, L.P. All Rights Reserved Slide 45

  46. NPEM and NPAG • Assign an unknown probability (or probability density value) pj to each grid point • Grid the relevant portion of the (V,K) with grid points (Vj,Kj) • Estimate probabilities pj by maximizing the (exact) nonparametric log likelihood © Tripos, L.P. All Rights Reserved Slide 46

  47. NPEM vs NPAG • NPEM uses a fixed, static grid and and EM algorithm to solve optimization problem (no formal numerical optimization) for the probabilities pj • NPAG uses an adaptive grid (multiple iterations) and a convex special purpose primal-dual algorithm to optimize the log likelihood • A later extension of NPAG incorporated a d-optimal design criterion based on the dual solution that enables candidate new grid points to be tested very rapidly for potential for improving the likelihood • Final optimal nonparametric distribution is discrete with at most Nsub support points. © Tripos, L.P. All Rights Reserved Slide 47

  48. NPAG results format looks like ideal case of direct observation © Tripos, L.P. All Rights Reserved Slide 48

  49. PHX NPAG vs FOCE for bimodal distribution of Ke values © Tripos, L.P. All Rights Reserved Slide 49

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