a program for the optimum design of pharmacokinetic, pharmacodynamic, drug metabolism and...

Computer Methods and Programs in Biomedicine (2005) 78, 237—249

A program for the optimum design ofpharmacokinetic, pharmacodynamic, drugmetabolism and drug—drug interaction models

Gordon Grahama,∗, Ivelina Gueorguievab, Kelly Dickensb

a Novartis Pharma AG, Lichtstrasse 35, Basel 4052, Switzerlandb Centre for Applied Pharmacokinetic Research, School of Pharmacy and Pharmaceutical Sciences,University of Manchester, Oxford Road, Manchester, M13 9PL, UK

Received 18 May 2004; received in revised form 27 January 2005; accepted 10 February 2005

KEYWORDSOptimum design;Pharmacokinetics;Pharmacodynamics;Drug metabolism;Drug—drug interactions

Summary Planning any experiment includes issues such as how many samples areto be taken and their location given some predictor variable. Often a model is usedto explain these data; hence including this formally in the design will be benefi-cial for any subsequent parameter estimation and modelling. A number of criteriafor model oriented experiments, which maximise the information content of thecollected data are available. In this paper we present a program, Optdes, to inves-tigate the optimal design of pharmacokinetic, pharmacodynamic, drug metabolismand drug—drug interaction models. Using the developed software the location ofeither a predetermined number of design points (exact designs) or together withthe proportion of samples at each point (continuous designs) can be determined.Local as well as Bayesian designs can be optimised by either D- or A-optimality cri-teria. Although often the optimal design cannot be applied for practical reasons,alternative designs can be readily evaluated.© 2005 Elsevier Ireland Ltd. All rights reserved.

1. Introduction

There is an abundance of information in the liter-ature describing the design of model-oriented ex-periments. This often falls under the title of ‘alpha-betic’ optimum design because many of the designcriteria are named after letters of the Latin alpha-bet (see for example [1—3] for a discussion of differ-

ent criteria). A design criterion for parameter esti-mation usually requires the maximisation of somefunction of the Fisher information matrix. Modelbased optimum design has previously been appliedto clinical pharmacology problems in a theoreticalsetting but the methodology has rarely been usedin practice. Early examples include the selectionof sampling times for pharmacokinetic experiments

* Corresponding author. Tel.: +41 61E-mail address: gordon.graham@p

(G. Graham).

[4—6], determining doses for pharmacodynamic ex-

0169-2607/$ — see front matter © 200doi:10.1016/j.cmpb.2005.02.005

324 2897.harma.novartis.com

periments, and substrate concentrations for drugmetabolism studies [7]. More general theoreticalwork has been reported by Walter and Pronzato [8].

5 Elsevier Ireland Ltd. All rights reserved.

238 G. Graham et al.

Software is available for the application of optimumdesign methodology to pharmacokinetic problems.For example, ADAPT II [9] allows the user to con-sider D and c-optimum designs but only for exactdesigns, i.e. designs where the number of samplingtimes is predetermined and only the locations aresought. In the area of population pharmacokinet-ics, optimum design has received greater exposureof late, particularly for the evaluation of study de-signs [10] as well as the optimisation of popula-tion pharmacokinetic designs [11]. These programsare specifically for (nonlinear) mixed effects mod-els and have been used in several example scenarios[12—14]. However there seems to be a reluctanceto adopt such methodology in practice. This couldbe due to a lack of widespread knowledge about themethodology, a belief that the approach is too com-plex or gives unrealistic and logistically infeasibledesigns.

Optdes was developed for the purpose of de-signing pharmacokinetic, pharmacodynamic, drugmetabolism and drug—drug interaction experi-ments. Pharmacokineticists designing these exper-iments may not be aware of all the relevant sta-tistical tools that could be utilised in their design

be collected with n being the number of observa-tions. These response data are collected at prede-fined design points, denoted by (x1, x2, . . ., xn).These design points (also called predictors or inde-pendent variables in a modelling context) do notnecessarily have to be distinct values, for exam-ple xi = xj for i �= j, i and j taking values in the set{1, 2, . . ., n}. The data are to be analysed using amodel of the form yi = f(�, xi) + εi, i = 1, . . ., n wherei denotes the ith observation, f is the structuralmodel, � the p× 1 vector of model parameters, andεi = yi − f(�, xi) is the residual term. A standard ap-proach to estimating parameters is to minimise thesum of squared residuals with respect to the modelparameters. This method of parameter estimationis known as least squares (LS) estimation and thecriterion function can be written as

LS(�) =n∑i=1

ε2i =n∑i=1

(yi − f(�, xi))2

The best fit of the model to the data is obtainedwhen � = � is the value that minimises LS(�) (in thiscase when ∂LS(�)/∂�|�=� = 0) for which there arem[

caamttEld

L

wthlwohztfhmft

work. This program was developed for model basedoptimum design to be explored by such scientistsand is also intended to help demonstrate that thismethodology has a role to play. It can also be con-sidered as a precursor to the more complex designsinvolved in population pharmacokinetics and phar-macodynamics and as a stepping stone to the use ofother optimum design programs. Optdes is a standalone program working under Windows.

In Section 2 an introduction to optimum exper-imental design and analysis methodology is pre-sented. The structure of Optdes and an outline ofits utilisation are presented in Section 3. Two exam-ples are described in Section 4: a one compartmentfirst order absorption model first presented in [4],and a triazolam metabolite formation study. A dis-cussion is given in Section 5.

2. Optimum experimental designmethodology

In this section, we describe themethodology behindthe design of model-oriented experiments. First theframework for modelling experimental data is de-scribed.

2.1. Modelling framework

An experiment is to be implemented where the re-sponse data denoted by y = (y1, y2, . . ., yn) is to

any well established optimisation methods (see15] for a text on numerical optimisation methods).An alternative approach to model fitting is to

onsider the distribution of the response dataround the mean model, f(�, x). The commonestpproach to use is maximum likelihood (ML) esti-ation. Frequently the assumption is made thathe data are independently and identically dis-ributed (i.i.d.) as a normal random variable, where(Yi) = f(�, xi) and var(Yi) = �2, for i = 1, 2, . . ., n. Theikelihood is obtained as the product of each point’sistributional contribution

(�; y) =n∏i=1

q(yi|�)

here L is the likelihood and q the probability dis-ribution function used in constructing the likeli-ood. To obtain estimates of the parameters, theikelihood or log-likelihood, is first differentiatedith respect to the structural model parameters (tobtain the score function). At the maximum likeli-ood estimate, �, the score function is a vector oferos because the maximum likelihood estimate ishe maximum of the likelihood surface and there-ore will have gradient zero (assuming the likeli-ood is differentiably continuous). LS and ML esti-ation will result in the same parameter estimatesor i.i.d. normal random variables since they arehe same criterion up to an additive constant.

A program for the optimum design of pharmacokinetic, pharmacodynamic 239

For a general model, f(�, x), the inverse of theFisher information matrix is the asymptotic lowerbound of the parameter variance—covariance ma-trix. The Fisher informationmatrix is determined bydifferentiating the negative log-likelihood functiontwice with respect to the model parameters,

Iobs(�) =[− ∂2l

∂�T∂�

]

This form is known as the observed Fisher infor-mation matrix because it can still be a function ofthe observed response data. The expected Fisherinformation matrix is obtained by replacing y byf(�, x) and results in a matrix dependent on the ob-served data through the maximum likelihood esti-mates only. To obtain the variance—covariance ma-trix, the inverse of the expected Fisher informationmatrix is calculated, var(�) = I−1(�). An alternativeapproach to calculating the Fisher information ma-trix is by the following identity in the case wherethe residuals are i.i.d. normal.

−[

∂2l

∂�j∂�k

]=[∂f(�, x)∂�j

]˙−1(jk)

[∂f(�, x)∂�k

]T,

w

actnThs

2

TspismmrrDpp

posed such as G-optimum design, which minimisesthe prediction error. The general equivalence the-orem [3] shows that D- and G-optimum design areequivalent under certain conditions.

2.3. D- and A-optimum experimental design

Assume we are concerned with the data analysisframework described in the previous section. Thetwo design criteria we are considering are definedas follows:

D-optimality: Maximise the determinant of theexpected Fisher information matrix, �D = arg{max(�D(�, �))} where �D(�, �) = |I(�, �)|. The de-sign variable is denoted by � to distinguish it fromother independent variables in a modelling frame-work, � is the design space, |.| denotes the deter-minant of a matrix and � denotes a general designcriterion function. The D-optimum design criterionminimises the confidence region associated withthe parameter estimates.A-optimality: Minimise the trace of thevariance—covariance matrix, �A = arg{min� ∈�(�A(�, �))} where �A(�, �) = trace(I−1(�, �)).

2

Tin

Tl(gtdd

siaFtpmlf

j, k = 1, ..., p

here the Jacobian is given by

∂f(�, x)∂�j

=[∂f(�, x1)∂�j

∂f(�, x2)∂�j

· · · ∂f(�, xn)∂�j

]T,

j = 1, ..., p

nd ˙ = �2In×n is the n× n residual variance—ovariance matrix. Another common assumption forhe likelihood function is that the errors are log-ormally distributed around the geometric mean.his type of likelihood assumption results in aeterogeneous variance model (on the originalcale).

.2. Model oriented optimum design

o make the analysis efficient the study designhould be specific to the data analysis that will beerformed. One possible design approach is to min-mise the uncertainty associated with estimatingome characteristic of the model. Examples includeinimising the standard errors of the parameters,inimising the volume of the parameter confidenceegion or minimising the uncertainty of a predictedesponse. The criteria chosen for this program are- and A-optimum design. These criteria are appro-riate when interest lies in the estimation of modelarameters. Other design criteria have been pro-

.4. Example: linear model

he D-optimum design of a linear regression models calculated as follows. The linear model in matrixotation is specified as

y = X� + ε, ε ∼ N(0,˙),

X =

1 �1

......

1 �n

T

, � =[�1

�2

], ε =

ε1

...εn

, ˙ = �2In

he maximum likelihood estimates (or equiva-ently least squares estimates) are given by � =XT˙−1X)−1XT˙−1y and the information matrix isiven by I(�)=XT˙−1X. It is not difficult to seehat the information in the parameter estimatesepends on the design through the selection of theesign variable, �i.If the interval on which the design variable is

ampled is given by [xl, xu] then the optimum designs that which takes an equal number of observationst the extremes of the interval. This is displayed inig. 1 along with a graph showing the influence ofhe number of observations at the optimum designoints on the confidence interval around the linearodel. There is a substantial amount of work pub-ished on the D-optimum design of linear models,or example [16—19].


Fig. 1 The left panel shows the logarithm of the D-optimum criterion surface vs. the two distinct design points. Theright panel shows the confidence intervals around the linear model for an increasing number of observations (8, 16and 24 divided equally between the two optimal design points).

For the linear model, the optimum design is inde-pendent of the parameter values. When the modelis nonlinear then the situation is less straightfor-ward. In this case the information matrix is a func-tion of the parameters and when optimising the D-or A-criterion, typically an analytical solution willnot be available. This means that initial parameterestimates are required so that the Fisher informa-tion matrix becomes a function of the design vari-able alone.

2.5. Exact versus continuous optimumdesign

So far, the way in which we have considered the op-timum design problem is that the number of mea-surements, n, has been predefined and the designpoint positions have then been optimised. This ap-proach is known as exact design because only theposition of each point is to be determined. An alter-native approach is continuous design where ratherthan defining the number of design points, the max-imum number of distinct points is chosen and aweight associated with each of the optimum design

that if a large amount of data were to be collected,then the data should be collected at the optimumdesign points �i with the number of observationstaken at each design point approximately given byωi/ω0. As well as the design points, the weights arealso optimised.

2.6. Bayesian optimum design

When we are concerned about the credibility of theinitial parameter values and their effect on the de-sign, then it is possible to incorporate uncertaintyin our knowledge of the parameter values into theexperimental design. The uncertainty in the esti-mates can be specified in terms of probability den-sity functions. The inclusion of the uncertainty isachieved by weighting the design criterion with re-spect to the prior distribution on the parametersas∫� ∈�

� (�, �)p(�) d�

where p(�) is the prior density of themodel parame-ters and � is the parameter space. The effect of in-tpbcimWttBi

points is found. In this situation the information ma-trix can be written as I(�, �) = ∑k

i=1ωiI(�, �i) suchthat ωi ≥ 0 for every i∈ {1, 2, ..., k}, ∑k

i=1ωi = ω0and k is the number of distinct design points. Theweight, ωi, can take any value on the interval [0,ω0]. This means that if ω0 represents the number ofmeasurements to be taken during the experiment,then ωi/ω0 is the fraction of observations taken atthe design point, �i. This allows for the possibilityof a noninteger number of observations at each de-sign point. Obviously this is not feasible but means

egrating the design criterion over a distribution ofarameter values is that the optimum design pointsecome spread out rather than being replicated atertain design points [20]. Bayesian optimum designs related to Bayesian modelling by the fact that theodel parameters are random variables a priori.hen data have been collected we derive a pos-erior distribution rather than a point estimate forhe model parameters given the observed data byayes theorem, p(�|y) =p(y|�)p(�)/p(y). In Optdes,t is assumed that all data analyses for parameter


estimation will employ least squares or maximumlikelihood methods. These asymptotic Bayesian de-signs could however be used for a Bayesian analysisif it is known that the posterior distribution can beapproximated by a normal distribution.

Bayesian D-optimum design is defined as∫� ∈�

log |I(�, �)|p(�) d�

and Bayesian A-optimum design is defined as∫� ∈�

trace(I(�, �)−1)p(�) d�

The Bayesian D-optimum design criterion is some-times referred to as a posteriori information (API)optimum design. The API criterion was adopted asthe form of Bayesian D-optimality in this programbecause of the derivation from a fully Bayesian de-sign approach [21].

Bayesian A and D-optimality can be combinedwith exact and continuous design criteria. Contin-uous Bayesian D-optimum design is defined as

∫ ∣∣ k∣∣

UaodataeC{d

∫

wpptmf

2

Fm

the optimum design will change as the parameterspecification varies. Thus it is necessary to performsensitivity analysis to assess how efficient the de-sign will be to misspecification of the initial param-eter estimates or distributions. Efficiency can beused to assess the loss of information for a partic-ular design compared to the optimum design. Theefficiency for D-optimum designs is calculated as

EffD =(

|I(�, �U)||I(�, �D)|

)1/p

and for A-optimum designs is calculated as

EffA = trace(var(�, �A))trace(var(�, �U))

where �U is an alternative design. For D-efficiency,the ratio of the determinant of Fisher informationmatrices is normalised by the pth root (the numberof structural model parameters). The efficiency isbetween 0 and 1 as long as the optimum design hasbeen found. One useful function of the efficiency isthe gain function defined as gain = 100(1− Eff)/Eff[ntos1avtdt

a

E

w

Fa

E

wlt

� ∈�log

∣∣∣∣∑i=1

ωiI(�, �i)∣∣∣∣p(�) d�

sually these integrals are analytically intractablend require some approximation. Quadraturemeth-ds have been used for Bayesian logistic optimumesign [21]. Discretisation of the prior distributionllows the integral to be replaced by a summa-ion over the parameter space [5]. The integral canlso be approximated by Monte Carlo techniques asmployed by Zucchi and Atkinson [22]. The Montearlo approach samples a set of parameter values�(1), . . ., �(h)} from p(�) and then approximates theesign criterion as follows

� ∈�� (�, �)p(�) d� ≈ 1

h

h∑j=1

� (�(j), �)

here h is the number of random samples. In thisrogram the Monte Carlo approximation is em-loyed and the number of sampled parameter vec-ors is set to 1500. Three distributions are accom-odated in Optdes: the normal; log-normal and uni-orm distributions.

.7. Efficiency

or any design criterion the initial parameter esti-ates or prior distribution chosen is crucial because

6]. This expresses the percentage increase in theumber of observations for the alternative designo have the same normalised criterion value as theptimum design. For example, the D-optimal de-ign for a linear model on the design space [−1,] is to take equal observations at −1 and 1. Anlternative design with an equal number of obser-ations at −0.5 and 0.5 is 50% efficient. To attainhe same amount of information as the D-optimumesign, twice as many observations are required forhe design at −0.5 and 0.5.For Bayesian D-optimum designs the efficiency is

pproximated by

ffDB ≈(

|I(�, �)||I(�, �D)|

)1/p

here I(�, �) = 1h

h∑j=1

M(�(j), �).

or Bayesian A-optimum designs the efficiency ispproximated by

ffAB ≈ trace(var(�, �A))trace(var(�, �))

here var(�, �) = (I(�, �))−1. Similar efficiencies for

ocal and Bayesian design can be computed for con-inuous designs.


3. The program

Optdes was developed and programmed in MatlabVersion 6.1 [27]. Object oriented features wereutilised, which allowed the program to be compiledand created as a stand-alone, Windows-based pro-gram with a graphical user interface. The programtogether with a user manual can be obtained fromthe authors.

3.1. Models

3.1.1. Structural modelsThe program incorporates a range of pharma-cokinetic, pharmacodynamic, drug metabolism anddrug—drug interaction structural models as listed inTable 1. Also included are three linear models andthe power model. For the pharmacokinetic models,single and multiple dosing can be selected. In allcases, a single design variable can be optimised.However for the drug—drug interaction models, in-hibitor and substrate concentrations need to be se-lected/optimised. In this case the inhibitor concen-trations are held fixed while the substrate concen-trations are optimised or the substrate concentra-

proportional error model yi = f(�, xi)(1+ εi), theerrors are proportional to the mean response. Acombined additive and proportional error modelis given by yi = f(�, xi)(1+ ε1i)+ ε2i where ε1i ∼N(0, �21) and ε2i ∼ N(0, �22).

3.2. Optimisation

Optdes includes five algorithms for optimisation.These include the simplex algorithm first describedin [23] with the implementation used here givenin [24]. Simulated annealing and adaptive randomsearch were implemented as described in Refs.[25,26], respectively. A hybrid simulated annealing-simplex algorithm is also implemented. This algo-rithm works by firstly using simulated annealing un-til a certain temperature is reach and then switch-ing to the simplex method. This enables firstly theneighbourhood of the optimum design to be foundand then the simplex can be used as it is quickerthan simulated annealing. The final algorithm is anexchange algorithm, similar to that described byFedorov [1]. At each iteration of the exchange al-gorithm, each design point in turn is replaced by ancTnboptgfattd

etab

ral

selin

tions are held fixed while the inhibitor concentra-tions are optimised. The models are hard coded aspart of the software.

3.1.2. Error modelsAs well as the structural model, the error modelis also a determining factor for the optimal de-sign. Three error models are included in Optdes: anadditive error model; a proportional error model;combined additive and proportional error model.The assumption for the additive error model, yi =f(�, xi)+ εi, is that all errors are normally dis-tributed with mean zero and variance, �2. For the

Table 1 Pharmacokinetic, pharmacodynamic, drug mOptdes

Pharmacokinetic Pharmacodynamic/gene

• One cmpt iv bolus • Emax• One cmpt first order abs • Sigmoid Emax• One cmpt zeroth order abs • Emax with baseline• One cmpt inf • Sigmoid Emax with ba• Two cmpt iv bolus• Two cmpt first order abs • Linear• Two cmpt zeroth order abs • Quadratic• Two cmpt inf • Cubic

• Power• Three cmpt iv bolus

ew design point that is guaranteed to increase theriterion until the some tolerance level is reached.he search space is discretised so that only a finiteumber of design points are considered. As a resultoth the efficiency in terms of time and accuracyf the resulting D-optimal sampling times are de-endent on the pre-specified step size. Apart fromhe exchange algorithm, all of the procedures areeneral-purpose optimisers that can be used for anyunctional optimisation problem. None of the abovelgorithms guarantee convergence to the true op-imum design, so several optimisers were includedo allow users to assess differences in the optimisedesigns.

olism and drug—drug interactions model available in

Drug metabolism Drug—drug interaction

• Michaelis—Menten • Competitive Inhibition• Hill • Noncompetitive Inhibition• One site linear • Uncompetitive Inhibition

e • Two site • Mixed Inhibition


3.3. User defined designs

Evaluation of designs can be performed if optimisa-tion is not required. This may be desirable in thecase where a set of candidate designs has beenselected and a comparison of their properties issought.

4. Examples

4.1. One compartment first orderabsorption model

This example was first considered in [5]. It concernsthe selection of plasma sampling times for an indi-vidual whose parameter values are uncertain. Thestructural model is assumed to be a one compart-ment first order absorption model parameterisedas

C(t) = Dka

V(ka − ke)(e−ket − e−kat)

atbOprvswtfwPo

medasapmCc Ta

ble2

Optimaldesign

pointsandruntimes

fortheonecompartmentfirstorderabsorption

model

No.

ofdesign

points

Bayesian

D-optimalsamplingtimes

Time(m

in)

30.49

(0.43)

2.3(2.3)

7.5(7.5)

9.02

40.37

(0.35)

1.4(1.9)

3.8(3.9)

9.0(9.4)

14.43

50.30

(0.30)

0.94

(0.91)

2.2(2.2)

4.9(4.9)

9.8(9.8)

23.58

60.27

(0.29)

0.70

(0.69)

1.5(1.6)

3.0(3.1)

5.5(6.0)

10.1(10.4)

38.02

70.26

(0.27)

0.55

(0.56)

1.2(1.3)

2.0(2.2)

3.8(3.8)

6.1(6.6)

10.4(10.6)

61.55

80.25

(0.26)

0.45

(0.52)

0.95

(1.0)

1.6(1.9)

2.6(2.8)

4.5(4.8)

6.5(7.2)10.7(10.9)

100.8

90.25

(0.25)

0.36

(0.51)

0.86

(0.81)

1.2(1.5)

2.2(2.6)

2.9(3.2)

5.3(6.0)

6.5(7.5)11.0(11.3)

135.7

100.25

(0.23)

0.30

(0.55)

0.78

(0.55)

0.92

(1.5)

1.8(1.6)

2.3(3.2)

3.7(3.8)

5.6(7.2)

7.0(7.5)11.2

(11.7)

544.4

Entriesinbracketsaretheresultsfrom

D’Argenio(1990).

nd the new subject’s pharmacokinetic parame-ers are assumed to be realisations from the distri-ution

log(kai)log(kei)log(Vi)

∼ N

log(2)log(0.25)log(15)

,1 0 00 1/2 1/120 1/12 1/3

ptdes does not include covariance terms in therior density so cov(kei , Vi) was set to zero. Theesidual error model is assumed to be additive withariance 0.5. Optimisation was performed using theimplex method. The Bayesian D-optimum designsere determined for a range of sampling times andhe results are displayed in Table 2. The run timesor determining the optimum design are also givenhen run on a Windows 98 operating system with aentium II 450MHz processor, 128MB RAM and 15Gbf hard drive space.The sampling times are approximately in agree-ent with those obtained by D’Argenio. How-ver they were not expected to be the sameue to the omission of the intersubject covari-nce between ke and V, a different parameteri-ation of the model is used in Optdes (ka, Cl, V)nd a different algorithm for integrating over therior distribution (D’Argenio used a discretisationethod). The run-times are long due to the Montearlo method of evaluating the Bayesian designriterion.


4.2. Optimum design for triazolammetabolism in human cryopreservedhepatocytes

4.2.1. IntroductionA range of drug metabolism in vitro systems suchas rat liver microsomes and hepatocytes, and hu-man liver microsomes have been studied to assesstheir predictiveness of in vivo hepatic clearance forthe benzodiazepines. Preliminary studies in humanliver microsomes showed large variability in intrin-sic clearance, Clint. Further work has been plannedto use human liver cryopreserved hepatocytes. Thisis a particularly expensive system to work with andthere was a desire to limit the number of substrateconcentrations used. Triazolamwas initially studiedand the results are presented here. Previous stud-ies in microsomes showed solubility problems forhigher substrate concentrations so care was neededin determining the optimum substrate concentra-tions.

The empirical approach usually taken to designdrug metabolism studies is to use a 1/5 to 5-foldrange around the Km value for adequate estima-

tion of the model parameters. It is generally be-lieved a minimum of eight substrate concentrationsis necessary and anything up to 30 concentrationsmay be needed for more complex enzyme kineticmodels.

This study considers the design of a triazolam(TZ) metabolite formation experiment. There aretwo main metabolites, 1′-hydroxy-triazolam (1′-OHTZ) and 4-hydroxy-triazolam (4-OH TZ) are de-scribed by Michaelis—Menten and sigmoidal kinet-ics, respectively. Previous metabolite formationstudies in 12 human liver microsome preparationsgave the results for 1′-OH TZ and 4-OH TZ with themean curves shown in Fig. 2.

4.2.2. Objectives of the experimentThe purpose of the experiment was to study theformation properties of the major metabolites oftriazolam in cryopreserved human liver hepato-cytes and ultimately to assess the prediction ofin vivo hepatic clearance. The requirement of thisexperiment is therefore to produce data that willlead to the efficient parameter estimation of theMichaelis—Menten or Hill equation parameters.

Fig. 2 Kinetic profiles for the 12 human livers from microsomTZ (right), respectively. Corresponding line styles in each plot

e preparations corresponding to 1′-OH TZ (left) and 4-OHrepresent the same individual.


4.2.3. Constraints on the designThe design region: In previous microsome experi-ments, the range of substrate concentrations usedwas 1—500�M. For the human cryo-preserved hep-atocytes an upper bound of 250�M was preferred.For the metabolite 1′-OH TZ this range seemedadequate but for 4-OH TZ, it was expected therewould be problems in estimating Vmax and Km. Thisconstraint was imposed because higher substrateconcentrations require a stronger solvent for thedrug to remain in solution. These solvents are notideal for any in vitro system as they can inhibit theP450 activity [28—30].Number of substrate concentrations: For the mi-crosome study, either 9 or 10 substrate concen-trations were used. Cryo-preserved human hepa-tocytes are expensive and there was a desire tolimit the number of distinct substrate concentra-tions to 5 or 6.Number of replicates: Each substrate concentra-tion is performed in duplicate.Reference (empirical) designs: The empirical de-sign that was proposed includes six substrate con-centrations at 1, 5, 10, 50, 100 and 250�M. If thedesign region were extended then a 500�M con-

4T

v

wsltilmtbsb

4TmVfKm

Fig. 3 Exact local D-optimal designs for 1′-OH TZ(dashed line) and 4-OH TZ (solid line).

concentrations are required to render the modelidentifiable. Endrenyi [7] gives a detailed descrip-tion of the exact local D-optimum design for theMichaelis—Menten model. One important featureof this design is that it is sensitive to the upperbound of the design region. The D-optimum designis usually found at a substrate concentration cor-responding to the Km and the other correspondingto Vmax if the upper bound is sufficiently greaterthan Km. If the upper bound is not much higherthan Km then the D-optimum design is given bythe highest possible substrate concentration and�2 = Km�max/(2Km + �max). Assuming the upperlimit, �max = 250�M, the D-optimum design (ob-tained by Optdes and in agreement with [7]) isgiven by 49.5 and 250�M. The efficiency of the de-sign where concentrations are at the Km and upperbound is 89% compared to the D-optimum design. Ifthe upper bound of the design region were 500�M,then the D-optimum design is 61.7 and 500�M. Thedesign with a substrate concentration at the Kmand 500�M would be 97% efficient when comparedto the D-optimum design.

For the Hill equation, that is 4-OH TZ, threedistinct substrate concentrations are required foritaVcpt

tsF

u

centration would be used in addition to, or insteadof 250�M.Error model: The error model for each of the 12livers was assumed to be normal with the residualstandard deviation equal to 0.07 nmol/minM cells.

.2.4. The modelhe Hill equation can be written as

= VmaxS&

K&m + S&

here the Michaelis—Menten model is obtained byetting & = 1. In theory, this particular design prob-em produces a multivariate response, in this casehe velocity of the two major metabolites. Since its known in advance that the data for the metabo-ites will be analysed separately, we assume theetabolites can be considered separately, evenhough we are considering a single design that wille used for both metabolites. A compromise de-ign between the designs of each metabolite wille used.

.2.5. Exact local D-optimum designhe initial parameter estimates (arithmeticeans of the two livers) for 1′-OH TZ weremax = 0.53 nmol/minM cells and Km = 82.02�M andor 4-OH TZ were Vmax = 2.09 nmol/minM cells,m = 409�M and & = 1.57. For the Michaelis—Mentenodel, that is 1′-OH TZ, two distinct substrate

dentifiability. If the upper bound is much largerhan the Km then the expected design is 210, 795�Mnd a substrate concentration corresponding tomax. The D-optimum design given the design regiononstraints is 64.9, 168.5 and 250�M. When com-ared to the expected design with no constraints,he D-efficiency is 2.6%.Now assume that there are six substrate concen-

rations, each in duplicate, and the reference con-traints apply, the D-optimum designs are shown inig. 3.It can be seen from Fig. 3 that for 4-OH TZ, the

pper concentration bound is not adequately high


Fig. 4 Plot of Bayesian D-optimal substrate concentrations for triazolam metabolites 1′-OH TZ (dashed line) and 4-OHTZ (solid line). Left panel is for an upper bound of 250�M and the right panel is for an upper bound of 500�M.

to achieve Vmax (2.09). This is borne out by theefficiency of the reference design (34.6%). For 1′-OH TZ, the expected percent standard errors forVmax and Km are 17% and 43%, respectively and aparameter estimation correlation of 0.91 for theD-optimum design. This is not a desirable featureof a good design. This correlation is induced bythe upper bound not being adequately high. To as-sess the effect of the upper bound, a duplicatesubstrate concentration at 500�M was added toboth the reference design and the optimum de-sign. The percent standard errors for Vmax and Kmare 12% and 37%, respectively. The correlation be-tween the parameter estimates is now given by0.875. If the D-optimum design was determinedwith an upper bound of 500�M then the designreplicates the substrate concentrations at 61.8 and500�M, which gives a parameter correlation of0.795.

4.2.6. Exact Bayesian D-optimum designIf we assume a log-normal distribution on theparameters for both metabolites, the geometricmean (approximate coefficient of variation) for1′-OH TZ were Vmax = 0.33 nmol/minM cells (122)

and Km = 69.62�M (61) and for 4-OH TZ wereVmax = 1 nmol/minM cells (158), Km = 368�M (49)and & = 1.57 (5). The Bayesian D-optimum design isshown in Fig. 4. In this case the upper bound is as-sumed to be 250�M.

To assess the effect of the upper bound, theBayesian D-optimum design was re-run with a500�M upper bound. The results are also shown inFig. 4.

Based on the local and Bayesian D-optimum de-signs, four compromise design were selected, eachwith an upper limit of 500�M since it was clear thatan upper limit of 250�M was not adequate for 4-OHTZ. The information gain of these compromise de-signs was compared to the reference design witheither a 250�M or 500�M upper substrate concen-tration for 1′-OH TZ. The designs and efficienciesare given in Table 3.

The D-efficiency of the design with an upperbound of 500�M compared to the reference de-sign with a 250�M upper bound results in valuesbetween 9.26 and 11.09. This means that for thereference design to have the same D-optimum cri-terion value, approximately ten times more dataat the reference design points would need to be


Table 3 Information gain for compromise designs when compared to the reference design with either an upperbound of 250�M or 500�M. Results are for 1′-OH TZ

Design (substrate concentrations) Information gain(1, 5, 10, 50, 100, 250)a

Information gain(1, 5, 10, 50, 100, 500)

5 10 50 100 250 500 9.26 1.8210 30 60 100 300 500 10.18 2.1710 40 80 150 300 500 10.97 2.145 50 70 150 300 500 11.09 2.11

a Reference design.

Fig. 5 Data and regression lines for 1′-OH TZ and 4-OH TZ.

collected. When the reference design has an upperlimit of 500�M the loss of efficiency was not so se-vere. In this case approximately twice asmanymea-surements would need to be taken for the refer-ence design as compared to the compromise design.Similar results were obtained for 4-OH TZ and forthe Bayesian A-optimum design criterion for bothmetabolites. The design 10, 40, 80, 150, 300 and500�M was selected as it has good characteristicsfor both metabolites and for all the design criteriaconsidered.

4.2.7. Results from the experimentThe results of the metabolite formation experimentin human cryopreserved hepatocytes are displayedin Fig. 5. Regression analysis for 1′-OH TZ and 4-OH TZ resulted in Vmax estimates of 0.072 and 0.23with percent standard errors of 3.5% and 9%, re-spectively. The corresponding Km mean estimateswere 21.55 and 282.63 with 15.5% and 15% percentstandard errors, respectively. It is clear that 1′-OHTZ attains Vmax but 4-OH TZ does not. Also solubil-ity problems occurred at the highest concentrationand measurements at 500�M could not be used forparameter estimation. However the standard errorso2Tc

than those used at the design stage (& was esti-mated to be one).

5. Discussion

The use of Optdes for the design of pharma-cokinetic, pharmacodynamic, drug metabolism anddrug—drug interaction experiments is in the mainfor exploratory purposes and for experimentalistsin these areas coming to model oriented optimumdesign for the first time. The conditions specifiedfor the use of this program enables a wide classof designs to be considered. However there aremany types of designs that are beyond the scopeof this program that are frequently met in phar-macology. These include designs for model discrim-ination; models with multivariate responses; non-normal probability models and mixed effects mod-els. These settings are naturally more complex,however, even for the relatively simple models con-sidered in this program complex issues are still in-volved. This program presents the basic theory ofmodel based optimum design and provides a plat-form from which more complex designs can be in-vdsm

f the parameters for bothmetabolites are less than0%, which can be considered satisfactory. For 4-OHZ, a Michaelis—Menten model was found to be ac-urate and all the estimated parameters were less

estigated. To facilitate the optimisation proce-ures, especially when Bayesian designs are con-idered, the analytical solution to the informationatrix rather than the numerical solution was used.


This approach places limitations on the flexibility ofthe software regarding the addition of new models.It is envisaged that future versions will allow usersto specify their own models in analytical as well asdifferential equation form.

Optdes uses a range of design criteria thatare often considered for parameter estimation. D-optimality is frequently considered in practice. Ex-act and continuous design methodology can alsobe applied as well as incorporating uncertaintythrough the use of Bayesian optimality criteria. Apossibility for running several different optimisa-tion algorithms is provided in the program. Ourchoice of algorithms was guided by their availabil-ity as coded routines and the desire to assess con-vergence to the optimum design by the use of sev-eral optimisation methods with different search-ing properties. There is no uniformly optimal algo-rithm, which would perform consistently well in allcases. Ideally, if time and resources permit severaloptimisers should be used for the maximisation ofany design and if similar maxima are obtained theuser can be confident that the global maximum hasbeen located.

There are many situations in which the design

pharmacodynamic model, J. Pharmacokinet. Biopharma-ceut. 23 (1995) 101—125.

[7] L. Endrenyi, Kinetic Data Analysis. Design and Analysis ofEnzyme and Pharmacokinetic Experiments, in: Design of ex-periments for estimating enzyme and pharmacokinetic pa-rameters, Plenum, New York, 1981, pp. 137—167.

[8] E. Walter, L. Pronzato, Identification of Parametric Modelsfrom Experimental Data, Springer, Paris, 1987.

[9] D.Z. D’Argenio, A. Schumitzky, ADAPT II User’s Guide,Biomedical Simulations Resource, University of California,Los Angeles, 1998.

[10] S. Retout, S. Duffull, F. Mentre, Development and im-plementation of the population Fisher information ma-trix for the evaluation of population pharmacokinetic de-signs, Comput. Meth. Programs Biomed. 65 (2001) 141—151.

[11] M. Tod, J.M. Rocchisani, Comparison of, ED, EID, and API cri-teria for the robust optimisation of sampling times in phar-macokinetics, J. Pharmacokinet. Biopharmaceut. 25 (1997)515—537.

[12] M. Tod, F. Metnre, Y. Merle, A. Mallet, Robust optimal designfor the estimation of hyperparameters in population phar-macokinetics, J. Pharmacokinet. Biopharmaceut. 26 (1998)689—716.

[13] S.B. Duffull, F. Mentre, L. Aarons, Optimal design of a pop-ulation pharmacodynamic experiment for ivabradine, Phar-maceut. Res. 18 (2000) 83—89.

[14] S. Retout, F. Mentre, R. Bruno, Fisher information matrixfor non-linear mixed-effects models: Evaluation and appli-cation for optimal design of enoxaparin population pharma-

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criteria presented are applicable and occur fre-quently in different phases of drug research anddevelopment. This program also fills some of thesoftware gaps in the application of optimum designmethodology, such as Bayesian designs. The two ap-plications show how the program can be appliedto real scenarios. The one compartment first or-der absorption model shows how incorporating thevariability between subjects in designing a samplingschedule results in a more robust design. This isalso exemplified for the enzyme kinetic example fortriazolam where the experimental design selectedwas actually implemented and the resulting data al-lowed the model parameters to be estimated well.

References

[1] V.V. Fedorov, Theory of Optimal Experiments, AcademicPress, New York, 1972.

[2] S.D. Silvey, Optimum Design, Chapman and Hall, London,1980.

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[4] D. D’Argenio, Optimal sampling times for pharmacokineticexperiments, J. Pharmacokinet. Biopharmaceut. 9 (1981)739—756.

[5] D. D’Argenio, Incorporating prior parameter uncertainty inthe design of sampling schedules for pharmacokinetic pa-rameter estimation experiments, Math. Biosci. 99 (1990)105—118.

[6] Y. Merle, F. Mentre, Bayesian design criteria: computation,comparison, and application to a pharmacokinetic and a

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24] J.C. Lagarias, J.A. Reeds, M.H. Wright, P.E. Wright, Conver-gence of the Nelder-Mead simplex method in low dimen-sions, SIAM J. Optimisation 9 (1998) 112—147.

25] A. Corana, M. Marchesi, C. Martini, S. Ridela, ACM Trans.Math. Software 13 (1987) 262—280.

26] S.F. Masri, G.A. Bekey, F.B. Safford, A global optimisationalgorithm using adaptive random search, Appl. Math. Com-put. 7 (1980) 353—375.

27] Matlab User’s Guide Version 6.1 (The MathWorks, Natick,MA, 2001).

28] W.F. Busby Jr., J.M. Ackerman, C.L. Crespi, Effect ofmethanol, ethanol, dimethyl sulfoxide, and acetonitrile onin vitro activities of cDNA-expressed human cytochromesP-450, Drug Metab. Dispos. 29 (1999) 249—266.


[29] J. Easterbrook, C. Lu, Y. Sakai, A.P. Li, Effects of organicsolvents on the activities of cytochrome P450 isoforms,UDP-dependent glucuronyl transferase, and phenol sulfo-transferase in human hepatocytes, Drug Metab. Dispos. 29(2001) 141—144.

[30] D. Hickman, J.-P. Wang, Y. Wang, J.D. Unadkat, Evaluationof the selectivity of the in vitro probes and suitabilityof organic solvents for the measurement of humancytochrome P450 monooxygenase activities, Drug Metab.Dispos. 26 (1998) 207—215.

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