© tripos, l.p. all rights reserved computational challenges of pk/pd nlme bob leary pharsight...

© Tripos, L.P. All Rights Reserved

Computational Challenges of PK/PD NLME

Bob Leary

Pharsight Corporation

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


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)


103

104

105

106

107

108

109

-800

-750

-700

-650

-600

-550

-500

-450

-400Pipericillin model convergence with grid size

Number of grid points

Log

likel

ihoo

d


NPAG Outperforms NPEM

CPU HRS MB LOG -LIK

NPEM: 2037 10000 -433.1

NPAG: 0.5 6 -425.0


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


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)


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


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)


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


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


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

FDA Guidance for Industry, 1999


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 FDA guidance should be read bearing in mind that it was written in 1999 and that population pharmacokinetics is an evolving science”


Obligatory ODE section


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.


A Simple PK Model as ODE : 1-Compartment IV Bolus

/

( ) ( ) /

(0)

dA dt K A

C t A t V

A Dose


IV Bolus closed form solution

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time t

plas

ma

conc

entr

atio

n

t1/2=0.46

( )Kte

C t DV


Multiple Doses: Use superposition if model ODE is linear

1

01

( )0 1

1

( )( ) ,0

( ) ( )( ) ,

Kt

K t TKt

Dose eC t t T

V

Dose e Dose eC t t T

V

Covariate models with time varying covariates pose additionalcomplications – suppose K=tvK(1+(coef)(SCR-SCR0))


1-Comp first order absorption extra-vascular dosing

1 12 1

2 12 1 22 2

2

1

12

22

/

/

( ) ( ) /

(0)

1st order absorption rate constant

elimination rate constant

d A dt k A

d A dt k A k A

C t A t V

A Dose

k

k


1-Comp first order absorption extra-vascular dose solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.091-compartment extravascular first order aborption

time

conc

22 12 )12

12 22

( ) (( )

k t k tD kC t e e

V k k


1 compartment 0-order (IV) dosing ODE

1

2 12 1 22 2

1

2

2

12

22

/ 0

/

(0) 1

(0) 0

( ) ( ) /

IV infusion rate

elimination rate constant

d A dt

d A dt k A k A

A

A

C t A t V

k

k


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

[ ]

2

/

(0) (dosing at t=0)

General solution using matrix exponential

( ) (0)

[ ]1 ...

2!Accurate, fast, and reliable software libraries for matrix exponentials exist

and outper

ij

Kt

M

d A dt K A

A f

A t e A

Me M

form numerical ODE solvers


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

max

Michaelis-Menten elimination

/

( )( )

(0)

m

VdA dt A

K A

A tC t

VA Dose


ODE ‘solver’ order of preference/speed

1. Closed form (1, 2, 3 compartment, 0 and 1st order dosing)

2. Matrix Exponential (Linear, constant coefficient)

3. Non-stiff numerical ODE solver (Runge-Kutta, Adams)

4. 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


End of ODE section, Start of methods section


Simple (single subject) regression Model•PK Model

•Data

•Residual Error Model

( )Kte

C t DV

Concentration profile: ( , ( ) ), 1,..,j obs j obst C t j N

2

( ) ( ) ( )

~ (0, )

obsC t C t C t

N


Extended least squares objective function

2 2

22 2

2 21

( , , ) 2 ln( ( , , ))

( ( ) / )ln( / ) )

( / )

j

j

j

KtNobsKtobs j

Ktj

ELS V K l V K const

C t De VDe V

De V


Computational challenge : minimize2( , , )ELS V K

•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

-


Regression model to estimate V and K

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.610

-1

100

time t

C(t

)

V = 1/C(0)

slope = -K

log (V) = -log (C) - Kt


A simple population PK model: IV Bolus cont’d

2

( ) or

( )

( , ) (0, )

: ( ( ), , ), 1, , 1,

parameters to be fit:

fixed effects: ,

residual error:

population covariance elements: , ,

V V

K

i

tvlV

V K

obs ij ij i obs

VV VV KV

V tvV e V e

K tvK e

N

Data C t t D i Nsub j N

tvV tvK


Population Likelihood function

1

2

2

2

log log( )

( , , , )

( , , | , ) ( , | ) )

( | , , , )

Nsub

ii

i

i V K V K V K

L L

L tvV tvK

l tvV tvK h d d

J tvV tvK d


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)


Laplacian Approximation (FO, FOCE, Laplacian)

( ) ' ( )

2

/2

mode

2mode2

( )

( ) (2 ) / det( )

( )

( )

H

d

J Ae

J d A H

A J

JH


Joint log likelihood J() and Laplacian, FOCE, and FO approximations

-2 -1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

eta

Joint likelihood and Laplace, FOCE, FO approximations

FO J(eta)

Laplace

FOCE


Conditional methods (FOCE, Laplace) require nested optimizations to find mode of J, FO does notEach 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

1

log log( )Nsub

ii

L L

2( | , , , )J tvV tvK d


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


STATISTICAL EFFICIENCIES


Approximate likelihoods can destroy statistical efficiency

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

2

4

6

8

10

12

14

16

histogram (blue) of NONMEM FOestimators

histogram (white) of PEM estimators


SAEM, MCPEM, NPEM/NPAG


The ideal case –Vi and Ki can be observed

Parametric estimators Nonparametric histogram

{( , ), 1/ }i i iF V K p N

1/ 2

2

1

1ˆ ˆ( )

( 1)

N

V i Vi

VN

0.5 1 1.5 20

10

20

30

40

50

60

70

80

V

frequency

1

1ˆ

N

V ii

VN


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

ˆ ˆ,i iV K


Standard Two-Stage Method Vi and Ki are estimated by simple nonlinear regression methods

Parametric estimators Nonparametric histogram

{( , ), 1/ }i i iF V K p N

1/2

2

1

1ˆ ˆ( )

( 1)

N

V i Vi

VN

0.5 1 1.5 20

10

20

30

40

50

60

70

80

V

frequency

1

1ˆ

N

V ii

VN


MCPEM and SAEM are Monte Carlo versions of STS

2

2

1. inpute ( , ), 1, for each subject i

by drawing random samples from the

(unnormalized) posterior:

( , ) ~ ( | , , , )

( : ~ 500, SAEM: ~ 1)

2. Compute from inputed C(t) a

ik ik

V K i

ik

V K k Nsamp

J tvV tvK

MCPEM Nsamp Nsamp

2

nd data

3. Compute updated , , , values

from STS formulas - no numerical optimization is necessary

tvV tvK


NPEM and NAG: Many PK/PD populations have sub-populations that would be missed by parametric techniques

A - True two-parameter populationdistribution

B – Best normal approximation to population distribution


NPEM and NPAG

1. Assign an unknown probability (or probability density value) pj to each grid point

2. Grid the relevant portion of the (V,K) with grid points (Vj,Kj)

3. Estimate probabilities pj by maximizing the (exact)

nonparametric log likelihood

1

log log( )

0, 1

Nk

NP j iji j

j jj

L p l

p p


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.


NPAG results format looks like ideal case of direct observation

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

V

K


PHX NPAG vs FOCE for bimodal distribution of Ke values

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

Simulated (true) Ke values

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

Post-hoc estmate of eta Ke

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

Nonparameteric mean eta Ke - optimized support points

© tripos, l.p. all rights reserved computational challenges of pk/pd nlme bob leary pharsight...

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