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A case example: Building a population pharmacokinetic model for theophylline using NONMEM program.

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A case example: Building a population pharmacokinetic model for theophylline using NONMEM program. Pyry V litalo Orion Pharma 2.2.2010 * Pyry V litalo SSL Presentation – PowerPoint PPT presentation

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Title: A case example: Building a population pharmacokinetic model for theophylline using NONMEM program.


1
A case example Building a population
pharmacokinetic model for theophylline using
NONMEM program.
  • Pyry Välitalo
  • Orion Pharma

2
Population pharmacokineticsPharmacokinetics
using nonlinear mixed-effects models
  • Rather than model each individuals response
    separately, combine all the data in one model.
    The differences between individuals are described
    with random effects.
  • Benefit Possible to use data that would not be
    adequate for individual PK modeling (e.g. sparse
    sampling).
  • Benefit More data per model yields more power.
  • One model for each individual versus one model
    for all data
  • Benefit Easier to locate sources of
    between-subject variability?

3
The NONMEM program
  • NONMEM itself is a very general nonlinear
    mixed-effects regression program, but it comes
    with packages
  • PREDPP is a package of subroutines for most
    common problems.
  • NM-TRAN makes NONMEM input and output more
    user-friendly.
  • Main difference compared to SAS Designed
    spesifically to be used in population
    pharmacokinetics/pharmacodynamics.
  • Not as general as SAS.
  • Lots of canned solutions to typical
    pharmacokinetic problems.

4
Some nomenclature
  • Fixed effects Theta, ?. Mostly used to describe
    the structural model (e.g. clearance) and
    covariate relationships (e.g. effect of sex on
    clearance)
  • In general statistical nomenclature b
  • Random effects Eta, ?. The standard deviation of
    the random effects is ?. In population
    pharmacokinetics, random effects are most
    commonly used to describe between-subject
    variability.
  • In general statistical nomenclature bi
  • Residual variability Epsilon, e. The standard
    deviation of the residual variability is s.
  • The general statistical nomenclature is similar?

5
The dataset (theophylline)
  • Part of the data used in
  • Upton RA, Thiercelin JF, Guentert TW, Wallace SM,
    Powell JR, Sansom L, Riegelman S. Intraindividual
    variability in theophylline pharmacokinetics
    statistical verification in 39 of 60 healthy
    young adults. J Pharmacokinet Biopharm. 1982
    Apr10(2)123-34.
  • The dataset comes with NONMEM software.

6
The dataset
  • 12 individuals given a single dose of 3-6 mg/kg
    of oral theophylline.
  • 10 concentration measurements from 25 hours
  • A total of 120 observations
  • One continuous covariate Weight.

7
ADVAN2 subroutine The subroutine to be used
throughout this example.
  • ADVAN2 subroutine is used throughout the example.
    This subroutine defines an absorption compartment
    and a central compartment.
  • ADVAN2 requires the following to be specified
  • KA (absorption rate constant)
  • K (elimination rate constant)
  • How the observations are scaled (i.e. how do the
    drug amounts relate to concentrations)?
  • How the observation and individual prediction
    relate to each other, i.e. what type of residual
    error is used?
  • Everything else is optional.

8
Oral pharmacokinetic data and ADVAN2 subroutine
  • In its most basic form
  • At t0 A(1)Dose and A(2)0
  • dA(1)/dt-KAA(1) KA
  • dA(2)/dtKAA(1)-KA(2)

  • K
  • Define e.g. that
  • KATHETA(1) EXP(ETA(1))
  • KTHETA(2) EXP(ETA(2))
  • VTHETA(3) EXP(ETA(3)) Volume of distribution
  • YA(2)/V EPS(1) observation is the predicted
    concentration plus residual error

Depot(1)
Central(2)
9
Run1 The zero model
  • This control stream came with the dataset, and
    likely originates from year 1982
  • SUBROUTINES ADVAN2
  • KATHETA(1)ETA(1) absorption rate constant
  • KTHETA(2)ETA(2) elimination rate constant
  • CLTHETA(3)WTETA(3) clearance is scaled by
    weight
  • SCCL/K/WT scaling of concentration
    observations is done
  • by calculating VCL/K and scaling by weight
  • because dose is weight-adjusted.
  • THETA (.1,3,5) (.008,.08,.5) (.004,.04,.9) Fixe
    d effects estimates
  • OMEGA BLOCK(3) 6 .005 .0002 .3 .006 .4 Random
    effects variance-covariance
  • ERROR
  • YFEPS(1) additive error model
  • SIGMA .4 Residual error variance estimate
  • EST MAXEVAL450 PRINT5 First Order method

10
Run1 results
  • The parameter estimates make sense.
  • The parameter omega(K) near zero.
  • High relative standard errors on some random
    effects
  • Omega(Ka) 87
  • Covariance(Ka-CL) 370
  • Covariance(Ka-K) 270
  • Omega(K) 51

11
Run1 results Plasma concentrations and
predictions.
12
Final model Run20 (back-up slides include the
steps taken in model-building process)
  • SUBROUTINES ADVAN2
  • PK
  • D1THETA(6)EXP(ETA(3))
  • KATHETA(1)EXP(ETA(1))
  • VTHETA(2)EXP(ETA(2)THETA(5)) WT/70
  • CLTHETA(3)EXP(ETA(2)) WT/70
  • KCL/V
  • S2V
  • ERROR
  • IPREDF
  • IRESDV-IPRED
  • WTHETA(4) add error absorption phase
  • W2THETA(7) add error postabs phase
  • IWRESIRES/W
  • IF(TIME.GT.2) IWRESIRES/W2
  • YIPREDWEPS(1)
  • IF(TIME.GT.2) YIPREDW2EPS(1)

13
Aspects of the improved modelResidual error
model
  • SUBROUTINES ADVAN2
  • PK
  • D1THETA(6)EXP(ETA(3))
  • KATHETA(1)EXP(ETA(1))
  • VTHETA(2)EXP(ETA(2)THETA(5)) WT/70
  • CLTHETA(3)EXP(ETA(2)) WT/70
  • KCL/V
  • S2V
  • ERROR
  • IPREDF
  • IRESDV-IPRED
  • WTHETA(4) additive error defined
    with fixed effect. THETA(4) equals sd of res.var.
  • W2THETA(7) additive error defined with fixed
    effect. THETA(7) equals sd of res.var.
  • IWRESIRES/W IWRES are weighted with standard
    deviation of residual variability
  • IF(TIME.GT.2) IWRESIRES/W2 for
    post-absorption period
  • YIPREDWEPS(1) multiply by W because sd of
    residual variability is fixed to 1.
  • IF(TIME.GT.2) YIPREDW2EPS(1) for
    post-absorption period

14
Benefits of coding residual error model this way
  • We can obtain IWRES values as per definition
  • IWRES (observation IPRED)/s (e.g. Karlsson
    Savic 2005)
  • This is used to standardize the IWRES, so that
    they should have a standard deviation of 1 (and
    mean of 0).
  • Sometimes, using a fixed effect makes the model
    more stable than estimating a lot of random
    effects.
  • Regarding the use of separate residual errors for
    absorption and post-absorption phases This makes
    sense because the absorption phase typically has
    more noise than post-absorption phase (and IV
    data is yet more reliable).
  • Has been proposed in some articles, e.g.
  • Karlsson MO, Beal SL, Sheiner LB. J Pharmacokinet
    Biopharm. 1995 Dec23(6)651-72.
  • Chan PL, Weatherley B, McFadyen L. Br J Clin
    Pharmacol. 2008 Apr65 Suppl 176-85.

15
Aspects of the improved modelDistribution of
random effects, estimation method, implementing
weight as a covariate
  • SUBROUTINES ADVAN2 This is a predefined
    subroutine for 1-compartment model
  • with first order absorption
  • PK
  • D1THETA(6)EXP(ETA(3)) log-normal
    distribution
  • KATHETA(1)EXP(ETA(1)) log-normal
    distribution
  • VTHETA(2)EXP(ETA(2)THETA(5))
    WT/70 log-normal distribution, linear scaling
    by weight
  • CLTHETA(3)EXP(ETA(2)) WT/70 log-normal
    distribution, linear scaling by weight
  • KCL/V
  • S2V
  • ERROR
  • IPREDF
  • IRESDV-IPRED
  • WTHETA(4) add error
  • W2THETA(7) add err postabs
  • IWRESIRES/W
  • IF(TIME.GT.2) IWRESIRES/W2
  • YIPREDWEPS(1)
  • IF(TIME.GT.2) YIPREDW2EPS(1)

16
The practical difference between First Order and
First Order Conditional Estimation
  • First Order (FO) method expands around ETA0
  • Thus, the parameters are estimated for the mean
    response and the estimation could be described
    POPULATION AVERAGE.
  • First Order Conditional Estimation (FOCE) expands
    around the expected values of each random effect.
  • Thus, the parameters are estimated for the
    median subject and the estimation could be
    described SUBJECT SPESIFIC.
  • (On top of that, FOCE-I takes into account the
    interaction between random effects and residual
    error)

17
Implementing bodyweight in the model
  • Linear scaling of both clearance and volume of
    distribution by bodyweight.
  • Current trend, at least for pediatrics linear
    scaling for volume of distribution, nonlinear
    scaling for clearance (estimate an exponent for
    nonlinear scaling).
  • Probably not applicable to obese patients, etc.
  • Weight range in this data 55-86kg (median 70kg).
  • Probably not wide enough range to estimate
    nonlinearity.

18
Aspects of the improved modelAbsorption model
  • SUBROUTINES ADVAN2
  • PK
  • D1THETA(6)EXP(ETA(3)) D1duration of a
    hypothetical infusion into the depot
    compartment
  • KATHETA(1)EXP(ETA(1))
  • VTHETA(2)EXP(ETA(2)THETA(5)) WT/70
  • CLTHETA(3)EXP(ETA(2)) WT/70
  • KCL/V
  • S2V
  • ERROR
  • IPREDF
  • IRESDV-IPRED
  • WTHETA(4) add error
  • W2THETA(7) add err postabs
  • IWRESIRES/W
  • IF(TIME.GT.2) IWRESIRES/W2
  • YIPREDWEPS(1)
  • IF(TIME.GT.2) YIPREDW2EPS(1)

19
The absorption model Mixed 0- and 1st-order
absorption.
  • If (timeltD1) dA(1)/dtDose/D1 KAA(1)
  • If (timegtD1) dA(1)/dtKAA(1)
  • A lagtime model was also tried but the mixed 0-
    and 1st-order absorption proved better with the
    following benefits
  • Describes the data better (lower OFV)
  • Would make it possible to use proportional error
    model?
  • Not as radical a change point as lagtime
    possibly more stable than using lagtime?
  • Probably better still Transit compartmental
    absorption (was not tried in this case because of
    time limitations).
  • Savic RM, Jonker DM, Kerbusch T, Karlsson MO. J
    Pharmacokinet Pharmacodyn. 2007 Oct34(5)711-26.
    Epub 2007 Jul 26.

20
What was the additional effort worth?
  • Compared to the original run we have
  • More complicated absorption model.
  • More complicated residual error model .
  • (No random effect for elimination rate constant).
  • What did we benefit from this?
  • A better absorption model may also result in more
    accurate estimates of V/F and CL/F.
  • Since different magnitudes of residual error are
    identified for absorption and post-absorption
    phases, the model has better simulation
    properties.
  • The model overall describes the data better
    (lower residual error).

21
Original model parameter estimates versus final
model parameter estimates
Parameter Run3 Run20
OFV 111 (6 parameters) 1.84 (10 parameters)
KA (1/h) 1.6 (0.20) 2.8 (0.26)
VCL/K (l) 32 (0.045) 34 (0.038)
CL (l/h) 2.7 (0.079) 2.7 (0.074)
Omega(KA) 0.65 (0.53) 0.90 (0.44)
Omega(CL) 0.17 (0.30) 0.27 (0.56)
Sigma (mg/l) 0.71 (0.13) 0.66 (0.15) / 0.30 (0.11)
the numbers in parenthesis are relative standard
errors
22
Take-home messages
  • If you have problems making the model converge
  • Realize that non-continuous functions may be hard
    to integrate (e.g. lagtime for absorption)
  • Use the appropriate residual error model.
  • (In NONMEM start with a simple model and build
    from ground up)

23
Resources
  • NONMEM. Currently the golden standard in
    population pharmacokinetic modeling. Requires
    license.
  • http//www.icondevsolutions.com/nonmem.htm
  • Xpose An R package that helps in deciphering the
    output of NONMEM. Free.
  • http//xpose.sourceforge.net/
  • R A program needed by Xpose to operate. Free.
  • http//www.r-project.org/
  • Census. A helpful program for keeping record of
    NONMEM runs. Free.
  • http//census.sourceforge.net/
  • PsN (Perl-speaks-Nonmem). A collection of helpful
    Perl scripts for NONMEM that make life easier in
    a lot of ways. Free.
  • MONOLIX. Another population pharmacokinetic
    program. Free.
  • Advantages Shorter runtimes than in NONMEM,
    provides graphical output by itself.
  • Disadvantages Currently not as flexible as
    NONMEM. May be hard to make sense of the error
    messages encountered.
  • http//software.monolix.org/sdoms/software/

24
(Back-up slides)Run2 Modernize the original
run
  • Dataset modifications Input the exact dose that
    was given to each patient rather than a
    weight-scaled dose. Input weight as a covariate
    for every observation.
  • Rather than model additive random effects, model
    exponential random effects THETA(1)ETA(1)
    becomes THETA(1)EXP(ETA(1))
  • Because log-normal distributions very typical in
    pharmacokinetics

25
(Back-up slides)Run2 Modernize the original
run
  • Take away covariance matrix for now, but leave
    all the random effects parameters.
  • In NONMEM, unnecessary covariance structures can
    slow down the estimation and add to model
    instability.
  • Instead of First Order (FO) estimation method,
    use First Order Conditional Estimation with
    Interaction (FOCE-I) method In ESTIMATION
    block, METHOD1 INTER line is added
  • First Order method would expand around ETA0
    (Fixed effects estimated to describe Population
    Average)
  • First Order Conditional Estimation with
    Interaction expands around ?E(?). Thus, the
    fixed effects describe a typical subject rather
    than population average (Subject Spesific
    approximation).

26
(Back-up slides)Run2 results
  • OFV 111.985
  • Parameter estimate is near its lower boundary
    omega(K)
  • Left individual time-concentration plots.

Right Individual random effects KA versus CL.
No linear correlation seems to be present thus,
KA-CL covariance is not implemented.
27
(Back-up slides)Forward some runs
  • Run3 Remove random effect on K
  • OFV 111.983, large standard error found for
    omega(KA)
  • Run4 Based on run3. Remove random effect on KA
  • OFV 186.687. Even though there is difficulty
    estimating the random effect, it seems to be an
    important part of the model.
  • Run5 Based on run3. Implement additiveproportion
    al error.
  • OFV 105.320 , large standard error found for
    omega(KA)
  • Run6 Based on run3. Implement proportional
    residual error.
  • OFV 140.536, fairly large standard error found
    for omega(KA)
  • SUMMARY of this slide The random effect of K
    (the elimination rate) is not essential to the
    model. The additiveproportional residual error
    might improve the model a bit. However, more
    efforts will be spent on the residual error model
    later on thats why additiveproportional
    residual error is discontinued.

28
(Back-up slides)Run7 Based on run3. Estimate
KA, V and CL instead of KA, K and CL
  • BEFORE (run3)
  • KATHETA(1)EXP(ETA(1))
  • KTHETA(2)
  • CLTHETA(3)EXP(ETA(2))
  • VCL/K
  • AFTER MODIFICATION (run7)
  • KATHETA(1)EXP(ETA(1))
  • VTHETA(2)EXP(ETA(2))
  • CLTHETA(3)EXP(ETA(2))
  • KCL/V
  • In run3, it was found that a random effect is not
    needed for K. Since KCL/V, in run7 the same
    random effect is included for both V and CL.
    Results identical to run3 (OFV 111.983)

29
(Back-up slides)Runs 8-9 Still testing the
random effects
  • Run8 Take away random effect on V
  • OFV 130.861. Standard errors for parameter
    estimates increase somewhat.
  • Run9 Based on run7. Include a separate random
    effect for V and implement covariance for CL and
    V.
  • dOFV -5.4
  • According to the results, the correlation between
    the two random effects would be absolute,
    although the magnitude of these random effects
    differs. -gt Something fishy about
  • the dataset?
  • r covariance (ETA(CL)-ETA(V) )
  • (variance(ETA(CL))variance(ETA(V))

30
(Back-up slides)The covariance between var(CL)
and var(V)?
  • Run10 Reparameterize run9 Use the same random
    effect for both CL and V but scale with a fixed
    effect.
  • OFV 106.644. Results practically identical to
    run9.
  • The code in run9
  • VTHETA(2)EXP(ETA(3)) separate random
    effect for each
  • CLTHETA(3)EXP(ETA(2)) parameter.
  • OMEGA BLOCK(2) .2 .2 .2 var(CL), covar(CL,V),
    var(V)
  • And the code in run10
  • VTHETA(2)EXP(ETA(2)THETA(5)) scale by
    theta(5)
  • CLTHETA(3)EXP(ETA(2))
  • OMEGA .2 var(CL), used also for V.

31
(Back-up slides)Theophylline absorption modeling
  • Run11 Based on run10 Lagtime on
  • absorption? (1 parameter)
  • OFV 64.447. Improvement on some individual
  • plots.
  • Run12 Based on run11 Lagtime for absorption
  • with random effect? (1 parameter)
  • OFV 47.179. Some difficulties to make this run
  • work. A likely reason for difficulties is that
  • it is hard to integrate at timelagtime, as
    absorption starts instantaneously.
  • How to implement lagtime in a model? Write
  • ALAG1THETA(n) this would estimate lagtime for
    compartment 1.

32
(Back-up slides)Theophylline absorption modeling
  • Run13 Mixed 0- and 1st-order absorption
  • OFV 64.879
  • Run14 Based on run13. Mixed 0- and 1st-order
    absorption a random effect is included for
    0-order absorption rate.
  • 31.140. The standard error of var(KA) becomes
    lower. This seems the most reasonable solution.
  • How to implement mixed 0- and 1st-order
    absorption? Add a column RATE into the datafile
    and set it to -2 for every instance when AMTgt0
    (else .). Write in the modelfile
  • D1THETA(n) this would estimate the duration of
    a hypothetical
  • infusion into the absorption compartment

33
(Back-up slides)Absorption phase and residual
errors?
  • Oral dosing data is usually considered more
    noisy than intravenous dosing data. One reason
    for this is that despite the best absorption
    modeling efforts, drug absorption from GI tract
    can be erratic and/or unpredictable.
  • Thus, more variability can usually be observed in
    the absorption phase than in post-absorption
    phase.
  • There are instances when a time-dependent
    residual error has been assigned to differentiate
    between absorption phase and post-absorption
    phase, e.g.
  • IF(T.LE.2) YIPREDEPS(1) absorption phase,
    additive error
  • IF(T.GT.2) YIPREDEPS(2) post-abs phase,
    additive error
  • For example
  • Karlsson MO, Beal SL, Sheiner LB. J Pharmacokinet
    Biopharm. 1995 Dec23(6)651-72.
  • Chan PL, Weatherley B, McFadyen L. Br J Clin
    Pharmacol. 2008 Apr65 Suppl 176-85.

34
(Back-up slides) Runs 15-16 Test using
different residual errors for absorption and
post-absorption phases
  • Run15 Based on run10. Try setting the
    absorption-phase residual error to before 2 hours
    and post-absorption phase residual error to after
    2 hours.
  • Success OFV 49.023, the residual error of
    post-absorption phase drops to one-half of what
    it was previously.
  • Run16 Based on run15. Try using proportional
    residual error for post-absorption phase.
  • OFV 67.746. Does not improve the model.

35
(Back-up slides) Combine the time-dependent
residual error model with absorption model.
Implement weight as a covariate.
  • Run17 Based on run14. Combine mixed 0- and
    1st-order absorption and time-dependent residual
    error
  • OFV 19.804
  • Run18 Based on run17. Check if a random effect
    is still needed in the 0-order absorption rate.
  • OFV 31.530 and the standard error of omega(KA)
    increases.
  • Runs 19-22 Try different ways of scaling the
    volume of distribution and clearance by weight.
  • Run20 with linear scaling seems to work best
  • Conclusion Run20 shall be the final model.
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