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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

Computational Challenges of PK/PD NLME

Bob Leary

Pharsight Corporation

© Tripos, L.P. All Rights Reserved

computational challenge 1 make execution time reasonable
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

execution time cont d
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

npag outperforms npem
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

computational challenges 2 4
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

a chronology of events in development of nlme
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

chronology cont d
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

some pk pd software
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

pk pd software cont d
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

pk pd software user base
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

fda guidance for industry 1999
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

emea guidelines 2007
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

obligatory ode section
Obligatory ODE section

© Tripos, L.P. All Rights Reserved Slide 14

ode considerations
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

a simple pk model as ode 1 compartment iv bolus
A Simple PK Model as ODE : 1-Compartment IV Bolus

© Tripos, L.P. All Rights Reserved Slide 16

iv bolus closed form solution
IV Bolus closed form solution

© Tripos, L.P. All Rights Reserved Slide 17

multiple doses use superposition if model ode is linear
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

1 comp first order absorption extra vascular dosing
1-Comp first order absorption extra-vascular dosing

© Tripos, L.P. All Rights Reserved Slide 19

1 comp first order absorption extra vascular dose solution
1-Comp first order absorption extra-vascular dose solution

© Tripos, L.P. All Rights Reserved Slide 20

1 compartment 0 order iv dosing ode
1 compartment 0-order (IV) dosing ODE

© Tripos, L.P. All Rights Reserved Slide 21

general n compartment model 0 and 1 st order dosing
General N-compartment model : 0 and 1st order dosing

© Tripos, L.P. All Rights Reserved Slide 22

nonlinear cases must be solved numerically with ode solvers odepack
Nonlinear cases must be solved numerically with ODE solvers (ODEPACK)

© Tripos, L.P. All Rights Reserved Slide 23

ode solver order of preference speed
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

end of ode section start of methods section
End of ODE section, Start of methods section

© Tripos, L.P. All Rights Reserved Slide 25

simple single subject regression model
Simple (single subject) regression Model
  • PK Model
  • Data
  • Residual Error Model

© Tripos, L.P. All Rights Reserved Slide 26

extended least squares objective function
Extended least squares objective function

© Tripos, L.P. All Rights Reserved Slide 27

computational challenge minimize
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

regression model to estimate v and k
Regression model to estimate V and K

© Tripos, L.P. All Rights Reserved Slide 29

a simple population pk model iv bolus cont d
A simple population PK model: IV Bolus cont’d

© Tripos, L.P. All Rights Reserved Slide 30

population likelihood function
Population Likelihood function

© Tripos, L.P. All Rights Reserved Slide 31

l i cannot be evaluated analytically how to proceed
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

laplacian approximation fo foce laplacian
Laplacian Approximation (FO, FOCE, Laplacian)

© Tripos, L.P. All Rights Reserved Slide 33

joint log likelihood j q s 2 w h and laplacian foce and fo approximations
Joint log likelihood J(q,s2,W,h) and Laplacian, FOCE, and FO approximations

© Tripos, L.P. All Rights Reserved Slide 34

conditional methods foce laplace require nested optimizations to find mode of j fo does not
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

lyon 2004 2005 bake off of nlme methods
Lyon 2004-2005 ‘bake-off’ of NLME methods

© Tripos, L.P. All Rights Reserved Slide 36

statistical efficiencies
STATISTICAL EFFICIENCIES

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

approximate likelihoods can destroy statistical efficiency
Approximate likelihoods can destroy statistical efficiency

© Tripos, L.P. All Rights Reserved Slide 39

saem mcpem npem npag
SAEM, MCPEM, NPEM/NPAG

© Tripos, L.P. All Rights Reserved Slide 40

the ideal case v i and k i can be observed
The ideal case –Vi and Ki can be observed

Parametric estimators Nonparametric histogram

© Tripos, L.P. All Rights Reserved Slide 41

the real case v i and k i are not directly observable
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

standard two stage method v i and k i are estimated by simple nonlinear regression methods
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

mcpem and saem are monte carlo versions of sts
MCPEM and SAEM are Monte Carlo versions of STS

© Tripos, L.P. All Rights Reserved Slide 44

slide45

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

npem and npag
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

npem vs npag
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

npag results format looks like ideal case of direct observation
NPAG results format looks like ideal case of direct observation

© Tripos, L.P. All Rights Reserved Slide 48

phx npag vs foce for bimodal distribution of ke values
PHX NPAG vs FOCE for bimodal distribution of Ke values

© Tripos, L.P. All Rights Reserved Slide 49