Optimal Design for Model Discrimination and Parameter Estimation for Itraconazole Population...

DOI: 10.1007/s10928-005-0026-2Journal of Pharmacokinetics and Pharmacodynamics, Vol. 32, Nos. 3–4, August 2005 (© 2005)

Optimal Design for Model Discrimination andParameter Estimation for Itraconazole PopulationPharmacokinetics in Cystic Fibrosis Patients

Timothy H. Waterhouse,1,∗ Stefanie Redmann,2 Stephen B. Duffull,2

and John A. Eccleston1

Received March 21, 2005—Final June 2, 2005

Optimal sampling times are found for a study in which one of the primary purposes is todevelop a model of the pharmacokinetics of itraconazole in patients with cystic fibrosis forboth capsule and solution doses. The optimal design is expected to produce reliable estimatesof population parameters for two different structural PK models. Data collected at these sam-pling times are also expected to provide the researchers with sufficient information to reason-ably discriminate between the two competing structural models.

KEY WORDS: itraconazole; optimal design; model discrimination; pharmacokinetic model-ling; population approach.

INTRODUCTION

Background and Significance

Allergic bronchopulmonary aspergillosis (ABPA) is a complication ofcystic fibrosis (CF), occurring in approximately 10% of patients mostlyafter the age of 6 years (1). It presents as a hypersensitive reaction toAspergillus fumigatus (AF) antigens in some CF patients colonised withAF in the lower respiratory tract (2) and requires treatment. Itraconaz-ole has a broad spectrum as an oral antifungal drug. It is active againstmost fungi including Aspergillus sp. A study has shown that patients

1School of Physical Sciences, The University of Queensland, Australia.2School of Pharmacy, The University of Queensland, Australia.∗To whom correspondence should be addressed. Department of Mathematics, The Univer-sity of Queensland, Brisbane, QLD 4072, Australia; e-mail: [email protected]

521

1567-567X/05/0800-0521/0 © 2005 Springer Science+Business Media, Inc.

522 Waterhouse, Redmann, Duffull, and Eccleston

achieved an improvement in the inflammatory response when itraconazolewas given in combination with systemic glucocorticosteroids (3).

Although relatively little is presently known about the pharmacoki-netics of itraconazole in patients with CF, there is evidence which sug-gests that it is different to the PK of the drug in non-CF patients (4–6).It is also known that there is often large variability in the pharmaco-kinetic parameters (e.g. absorption rate constant, volume of distribution,and clearance) of drugs administered to CF patients (7,8). It has beenobserved that high doses of itraconazole are required in CF patients forthe treatment of ABPA (2).

Itraconazole (Fig. 1) is a highly lipophilic drug (log octanol-waterpartition coefficient = 5.66, pH 8.1), and is considered ‘insoluble’ in water(less than 5 µg/ml) and has a molecular weight of 705.64 (9). It is aweak base and is ionised only at low pH (pka = 3.7) (10). Itraconaz-ole is widely distributed in the body, with a reported volume of distri-bution of 10.7 l/kg following intravenous administration (9). Up to 99.8%of the drug is bound to human plasma proteins. Itraconazole is exten-sively metabolised by the hepatic cytochrome P450 isoenzymes and has atleast 30 metabolites, each representing less than 1–5% of the dose. Themain metabolite is considered to be hydroxyitraconazole (also shown in

N

NNN N

O O

N

N

N

CH2

Cl

Cl

CH2

O

O

HCH3

CH3

Itraconazole

N

NNN N

OO

N

N

CH2

Cl

Cl

O

O

HCH3

CH3

OH

Hydroxyitraconazole

Fig. 1. Chemical structures of itraconazole and hydroxyitraconazole.

Optimal Design for Model Discrimination and Parameter Estimation 523

Fig. 1), which also exhibits antifungal activity against a variety of species(9,11,12).

Several studies have shown that the absorption of the capsuleformulation is enhanced by a low gastric pH, and taking the capsules afterfood or with an acidic beverage (e.g. cola) can improve the absorption(9,13,14).

The absorption of the oral solution is reported not to be influencedgreatly by the degree of gastric acidity (15), and the absorption is morerapid in the fasting state (16–19). It has been shown in healthy volunteersthat the oral availability of itraconazole oral solution taken under theseconditions can be up to 60% higher than that of capsules (15,17,20).

The pharmacokinetics of itraconazole and its main metabolite hasbeen shown to exhibit the behaviour of two linked compartmental models.Figure 2 shows the general structure of the models, including the rates ofmass transfer between each compartment. The amounts of itraconazole inthe gut, central and peripheral compartments are given by A1, A2 and A3,respectively; the amount of its metabolite (hydroxyitraconazole) is denotedby A4. The elimination from the central compartment to a compoundother than hydroxyitraconazole (shown in Fig. 2) has been described by

Fig. 2. Compartmental model of the pharmacokinetics of itraconazole and hydroxyitraconazole.


some as linear (elimination rate F20C L2/V2)(10), and by others as non-linear (elimination rate F20Vmax/(Km + A2/V2)) (9,14,19,21).

Structural Models

The differential equations describing the two models (linear and non-linear) from Fig. 2 are shown in Eqs. (1–8).

Model 1. The linear model:

d A1

dt= −F12ka A1 − F14ka A1, (1)

d A2

dt= F12ka A1 + Q

V3A3 − Q

V2A2 − F24

C L24

V2A2 − F20

C L2

V2A2, (2)

d A3

dt= Q

V2A2 − Q

V3A3, (3)

d A4

dt= F14ka A1 + F24

C L24

V2A2 − C L4

V4A4. (4)

Model 2. The nonlinear model:

d A1

dt= −F12ka A1 − F14ka A1, (5)

d A2

dt= F12ka A1 + Q

V3A3 − Q

V2A2 − F24

C L24

V2A2 − F20

Vmax

Km + A2/V2A2,

(6)

d A3

dt= Q

V2A2 − Q

V3A3, (7)

d A4

dt= F14ka A1 + F24

C L24

V2A2 − C L4

V4A4. (8)


Note that since the differential equations describe the rates of changeof the amount of drug in each compartment (rather than the concentra-tion), the last term in Eq. (6) describing the Michaelis–Menten eliminationis slightly different to the form which may be familiar to the reader. Thedivisor still involves the concentration (i.e. A2/V2), but we multiply by A2to give the rate of change in the amount. The units of Vmax and Km arestill of the form ‘mass/(volume × time)’ and ‘mass/volume’, respectively. Inboth sets of differential equations, the initial conditions are

A1(0) = dose, A2(0) = A3(0) = A4(0) = 0. (9)

Aim

This paper describes the development of an optimal design in realtime for a population pharmacokinetic study of itraconazole when givenas capsules and solution in adult patients with CF.

THEORY

We are interested in modelling the behaviour over time of the amountof the parent drug in the central compartment and its metabolite (A2 andA4), and hence their concentrations (C2 = A2/V2 and C4 = A4/V4).

It is assumed that the fractions F12, F14, F24 and F20 are known fixedconstants and that their values are available from previous experiments.The parameters to be estimated which are common to both models aretherefore the volumes V2, V3 and V4; and the clearances C L4, C L24 andQ. The linear model has one additional parameter, C L2, while the non-linear model also has parameters Vmax and Km. Considering now that weare interested in the pharmacokinetics of itraconazole in both capsule andsolution form, there are actually four models under consideration: the lin-ear model for capsules, the linear model for solutions, the nonlinear modelfor capsules and the nonlinear model for solutions. The models for cap-sules and solutions have the same structure, they only differ in terms oftheir parameter values. Hence there are four sets of parameter values:

β1,c = (k1,ca , V 1,c

2 , V 1,c3 , V 1,c

4 , C L1,c4 , C L1,c

24 , Q1,c, C L1,c2 )T , (10)

β2,c = (k2,ca , V 2,c

2 , V 2,c3 , V 2,c

4 , C L2,c4 , C L2,c

24 , Q2,c, V 2,cmax, K 2,c

m )T , (11)


β1,s = (k1,sa , V 1,s

2 , V 1,s3 , V 1,s

4 , C L1,s4 , C L1,s

24 , Q1,s, C L1,s2 )T , (12)

β2,s = (k2,sa , V 2,s

2 , V 2,s3 , V 2,s

4 , C L2,s4 , C L2,s

24 , Q2,s, V 2,smax, K 2,s

m )T , (13)

where for βu, f the superscripts u and f indicate the structural modelu(u = 1 for linear, u = 2 for nonlinear) and the form of dose f ( f = cfor capsule, f = s for solution), and “T” denotes the vector/matrix trans-pose.

For the time being, suppose that only one structural model and onlyone form of dose is under consideration. The superscripts u and f aredropped, and the p-vector of parameters is denoted by β. Let θi be thevector of parameter values for the ith individual. Then for a vector of ni

sampling times ξi = (ti1, . . . , tini )T , suppose that the ni -vector of observa-

tions is modelled by

yi = η(θi , ξi ) + εi . (14)

Employing a nonlinear mixed effects model, it is assumed that eachof the parameters varies randomly between individuals. Let β be the vec-tor of fixed effects and let bi be the vector of random effects for individ-ual i . So the random effects are written as either θi = β + bi (for additiverandom effects) or θi = β exp(bi ) (for exponential random effects). Thusthe structural model can be written in an alternative form, η(β, bi , ξi ). Itis assumed that bi ∼ N (0,�), where � = diag(ω1, . . . , ωp). It is alsoassumed that, as in Retout and Mentre (22), the variances of the normallydistributed error terms εi (conditional on the random effects) have addi-tive and proportional components, i.e. εi |bi ∼ N (0, diag(σA + σPη(θi , ξi )

2)for some σA and σP . Denote the entire vector of model parameters by = (βT , ω1, . . . , ωp, σA, σP )T .

Optimal Design

Model discrimination aside (momentarily), the objective of the exper-iment is to obtain estimates of the relevant parameters with the great-est possible efficiency. For this purpose, the Fisher information matrix isemployed. For a population design = (ξ1, . . . , ξN )T this is given by

MF (,) = E

(−∂2l(; y)

∂∂T

), (15)

where l(; y) is the log-likelihood of the vector of observations y =(y1, . . . , yN )T for the population parameters . The population design is


limited to Q groups, such that the Nq subjects of the qth group each havethe same set of nq sampling times ξq . So we have

MF (,) =Q∑

q=1

Nq MF (, ξq). (16)

The Cramer–Rao Lower Bound states that the variance-covariancematrix of any unbiased estimator of is bounded below by M−1

F (,). Itfollows that any population design which maximises |MF (,)| minimisesthe volume of the joint asymptotic confidence ellipsoid of the parameters.Such a design is called D-optimal. See Atkinson and Donev (23) for anintroduction to D-optimal design.

For an elementary design ξ , Retout and Mentre (22) have shown thatby using a first-order Taylor series expansion of η(θ, ξ) around the expec-tation of the random effects, an approximation to the Fisher informationmatrix for an elementary design ξ is given by

MF (, ξ) ≈ 12

[A(E, S) C(E, S)

CT (E, S) B(E, S)

], (17)

where

(A(E, S))mn = 2∂ ET

∂βmS−1 ∂ E

∂βn, m, n = 1, . . . , p, (18)

(B(E, S))mn = tr(

∂S

∂λmS−1 ∂S

∂λnS−1

), m, n = 1, . . . , p + 2, (19)

(C(E, S))mn = tr(

∂S

∂λmS−1 ∂S

∂βnS−1

), m = 1, . . . , p + 2, n = 1, . . . , p.

(20)

Here E and S represent the expectation and variance of the observations,respectively:

E = E(y) ≈ η(β, 0, ξ), (21)

S = S(y) ≈[∂η(β, 0, ξ)

∂bT

]�

[∂η(β, 0, ξ)

∂bT

]T

+ diag(σA + σPη(β, 0, ξ))2.

(22)

We assume that S is independent of β, as in Retout et al. (24). We alsomake the assumption that the fixed and random effects are uncorrelated,i.e. that C(E, S) is a matrix of zeros.


Parameter Estimation in Multiresponse Situations

Consider for a moment that the observations are in fact modelled bya simpler fixed effects model with multiple response types. Suppose thatthe outcomes of an experiment on a single subject with r simultaneousresponses can be modelled as

y(a)i = η(a)(θ, ti ) + ε

(a)i (a = 1, . . . , r; i = 1, . . . , n), (23)

where the θ = (θ1, . . . , θp)T is now a p-vector of fixed effects parame-

ters; E(ε(a)i ) = 0; E(ε

(a)i ε

(b)j ) = 0 for i �= j; E((ε

(a)i )2) = σ 2

a = σaa ;

and E(ε(a)i ε

(b)i ) = ρabσaσb = σab for a �= b. So the covariance matrix of

responses for the i th sampling time is given by

� = {σab}a,b=1,...,r =

σ11 σ12 · · · σ1rσ21 σ22 · · · σ21r...

.... . .

...σr1 σr21 · · · σrr

(24)

where σab = σba . Let �−1 = {σ ab}a,b=1,...,r .Again, denote the elementary design by ξ = (t1, . . . , tn)T . Draper and

Hunter (25) have shown that the information matrix for multiple responsesis given by

MF (θ, ξ) =r∑

a=1

r∑b=1

σ ab FaT Fb, (25)

where

Fa =[

∂η(a)u (θ, t1)

∂θ, . . . ,

∂η(a)u (θ, tn)

∂θ

], a = 1, . . . , r. (26)

The D-optimal design then maximises the determinant of MF (θ, ξ) asgiven in Eq. (25).

It follows that if the information matrix in Eq. (25) is simplified to

MF (θ, ξ) =r∑

a=1

r∑b=1

FaT Fb, (27)

i.e. if it is assumed that the σ ab are all equal, then this is equivalent tomaximising |FT F |,


where

F =r∑

a=1

Fa, (28)

=[

∂∑r

a=1 η(a)(θ, t1)

∂θ, . . . ,

∂∑r

a=1 η(a)(θ, tn)

∂θ

]. (29)

This will greatly simplify computation, as all that is required for the crite-rion is to sum the r responses in the model, and then calculate the Fisherinformation matrix as in the single response case. While this assumptionmay not be realistic for this study, to best of the authors’ knowledge thereare no published values for the covariance of itraconazole and hydrox-yitraconazole, so this algebraically convenient (and hence computationallysimple) simplification is reasonable in these circumstances.

This idea is extended to the more complex nonlinear mixed effectsmodels with heteroscedastic error. To accomplish this, the Fisher infor-mation matrix described in Eqs. (17–22) is computed, replacing η(β, 0, ξ)with

∑ra=1 η(a)(β, 0, ξ). In the itraconazole example, the only responses of

interest are the concentrations in the 2nd and 4th compartments. Hencethe sum is only over a = 2, 4.

Model Discrimination

There are two competing structural models to describe the pharmaco-kinetics of itraconazole. The first involves linear elimination of the parentdrug, the second involves a nonlinear Michaelis–Menten process. Atkinsonand Fedorov (26) introduced the T -optimality criterion for model discrim-ination, which could be applicable here. However, T -optimality is quiteexpensive computationally, and is not efficient for parameter estimation(see Waterhouse et al. (27) for examples). A viable alternative is to usethe product of the determinants of the information matrices, raised tothe power of the reciprocal of the number of population parameters (orequivalently, the product of efficiencies, as suggested by Atkinson and Cox(28)). Waterhouse et al. (27) also suggest that in some circumstances thismethod can produce designs which are useful for both goals: parameterestimation and model discrimination. This is demonstrated by the simula-tions in Section ‘Design Evaluation’.

Combining Capsule and Solution Responses

As mentioned in the section Structural Models there are four mod-els under consideration. We are, however, only interested in discriminating


between the linear and nonlinear structural models. To do this, a capsuledose is administered, followed by a solution dose at a time after whichit is assumed that there is no residual amount of the capsule dose. It isalso assumed that there is no effect of the order of the doses. An elemen-tary design ξ includes sampling times for the capsule (ξ c) and solution(ξ s). The responses of subject i to both doses are combined to give theoverall response. So the parameters for capsules and solutions for a givenstructural model must also be combined, i.e. for u = 1 and 2 (linear andnonlinear models, respectively), θu = ((θu,c)T , (θu,s)T )T . In the context ofnonlinear mixed effects models with additive random effects, we now havethe vector of parameters in model u for subject i given by

θui = ((θu,c

i )T , (θu,si )T )T , (30)

= ((βu,c + bu,ci )T , (βu,s + bu,s

i )T )T , (31)

= βu + bui , (32)

where βu = ((βu,c)T , (βu,s)T )T and bui = ((bu,c

i )T , (bu,si )T )T , with the

random effects for dose form f ( f = c, s) for each subject being nor-mally distributed with zero mean and with covariance matrix �u, f =diag(ω

u, f1 , . . . , ω

u, fpu, f ). The combined covariance matrix for model u is then

�u = diag(�u,c,�u,s). Equations (31) and (32) may be adjusted for expo-nential random effects by replacing β + bi with β exp(bi ) (for any super-scripts on β and bi ).

So the entire vector of population parameters under model u is then

u = ((u,c)T , (u,s)T )T (33)

with

u, f = ((βu, f )T , ωu, f1 , . . . , ω

u, fpu, f , σ

u, fA , σ

u, fP )T . (34)

In this notation, the combined ath response (for compartment a,where a = 2 or 4) for capsule and solution can be written

η(a)u (βu, bu

i , ξi ) = (η(a)u (βu,c, bu,c

i , ξ ci )T , η(a)

u (βu,s, bu,si , ξ s

i )T )T . (35)

Final Criterion

Hence the overall product criterion to be maximised is given by

DP (1, 2, ) = |M1F (1, )|1/p1 |M2

F (2, )|1/p2 , (36)


where

MuF (u, ) =

Q∑q=1

Nq MuF (u, ξq), u = 1, 2. (37)

u is the vector of population parameters for the uth model. For the uthmodel, the elementary information matrix is given by

MuF (u, ξq) ≈ 1

2

[A(Eu, Su) C(Eu, Su)

CT (Eu, Su) B(Eu, Su)

], (38)

with A(·, ·), B(·, ·) and C(·, ·) as described in Eqs. (18–19), and

Eu = Eu(y) ≈∑

a=2,4

η(a)u (βu, 0, ξq), (39)

Su = Su(y) ≈[

∂∑

a=2,4 η(a)u (βu, 0, ξq)

∂bT

]�u

[∂

∑a=2,4 η

(a)u (βu, 0, ξq)

∂bT

]T

+ diag

σa + σp

∑a=2,4

η(a)u (βu, 0, ξq)

, (40)

where the sums are over the responses for compartments 2 and 4.

METHODS

In this section we outline the constraints of the sampling design, pro-cedures used in selecting appropriate parameter and constant values, anumber of simplifications to the models (to fit design constraints), anda brief overview of computational methods used to optimise the designcriterion.

Design Constraints

A number of limitations on the experimental design were imposedfor reasons of time and expense. A maximum of 30 patients were to beenrolled and studied on two occasions. The number of blood samples wasalso constrained to be four samples per occasion. A single 200 mg dose ofthe drug in capsules was to be given to each patient on the first occasion,followed by a single 200 mg dose in solution on the second occasion.

A time constraint of approximately 2 months was placed on thedesign of the experiment, from exploration of the literature to computa-tion of the final design.


Patients

Thirty adults with CF who are admitted to hospital for treatment of achest exacerbation will receive two doses of itraconazole (one 200 mg dosein capsules and one 200 mg dose as an oral solution) for this study.

The participating patients are divided randomly into three groups,with each group being assigned one of three elementary sampling designs.

Accordingly patients will be required to take two Sporonax� Cap-sules as one dose (200 mg itraconazole), which is a standard adult dose.Four blood samples (0.5 ml) will be then drawn by finger prick accordingto the optimum sampling times of the design.

After a wash-out period of at least three days (after the initial dose),the same dose of 200 mg (20 ml of 100 mg/10 ml solution) itraconazolewould be taken as Sporonax� Oral Solution. Four blood samples (0.5 mlwill again be drawn by finger prick according to the optimum samplingtimes of the design.

Simplification of Models

The design optimisation problem under consideration is quite compu-tationally intensive. Any justifiable reduction in the number of parametersto be estimated (which leads to a reduction in the dimension of the Fisherinformation matrix) is greatly beneficial.

We believe that the different forms of dose will effect the input pro-cess only. So rather than treat the model parameters for capsule and solu-tion doses as being entirely separate, as in Eqs. (10–13), we suppose thatthere are only two sets of parameters to be estimated, the first for thelinear model and the second for the nonlinear model:

β1 = (k1a, V 1

2 , V 13 , V 1

4 , C L14, C L1

24, Q1, C L12)

T , (41)

β2 = (k2a, V 2

2 , V 23 , V 2

4 , C L24, C L2

24, Q2, V 2max, K 2

m)T . (42)

The parameters kua represent the absorption rate for capsules (i.e. they are

strictly ku,ca ). The distinction between absorption rates is instead addressed

by a multiplicative factor Fka = 0.9452/0.4159 ≈ 2.273, the ratio of theabsorption rate for solutions and the absorption rate for capsules (not tobe estimated). So the response for the solution under model u is given byusing the absorption rate Fka ku

a . The smaller of the two absorption ratesis used for this design, as it will be the most difficult to estimate.

In addition to the computational intensity of the problem, the designconstraints (outlined in Section ‘Design Constraints’) mean that not all ofthe parameters (of which there is still a large number) will be estimable.


Some preliminary investigation of expected standard errors of the esti-mates (given by the square roots of the diagonals of the inverse of theFisher information matrices) show that, under both linear and nonlin-ear models, the parameters C L24, Q and V3 will have very impreciseestimates under the constraints of the sampling design. Further to this,Km is not estimable with any precision under the nonlinear model. Thesefour parameters are therefore assumed to be fixed for the purposes ofthis design. They are included in the calculation of the Fisher informationmatrices, but their corresponding rows and columns are deleted for thecalculation of the optimality criterion. This method produces subtly differ-ent information matrices to those calculated by leaving the parametersout of the calculation altogether. The differences in the methods are dueto the approximations used in calculating the matrices. For a ‘true’ infor-mation matrix, the inclusion of other parameters should not effect theexisting elements of the matrix. For example, E

(− ∂2l(;y)

∂βi ∂β jT

)should not

be effected by whether the derivatives with respect to βk are includedelsewhere in the matrix. In retrospect, these fixed parameters should havebeen left out of the calculation of the information matrix, but as thedifference between the two methods is minimal, this was not considereda major issue.

To further simplify computation, the residual amounts of drug andmetabolite in each compartment at the time of the second dose wasassumed to be insignificant. This was found to have a negligible effect onthe criterion.

Parameter and Constant Values

Finding the D-optimal design for any model which is nonlinear inits parameters requires prior specification of the model parameters. Suchdesigns are known as ‘locally’ D-optimal.

The parameter values (and values of other constants) used in the con-struction of our design were taken from a variety of sources. To find thelocal optimum design, parameters which are common to both linear andnonlinear elimination models (e.g. V2, C L4) are given the same value forboth model types, despite our treatment of the parameters as separateentities when constructing the design criterion. For example, V 1

2 and V 22

are not the same population parameters, but for convenience we assignthem the same value in the construction of the optimum design.

A recently published population pharmacokinetic study byKoks et al. (29) gave us most of the parameter values we needed. In par-ticular, the clearance of itraconazole from the central compartment and its


volume of distribution in the central compartment, and therefore the elim-ination rate constant from the central compartment (k20), but also the dis-tribution rate constants (k23, k32) and the formation rate constant (k24).

These authors did not report any nonlinear elimination of the drug.But in several other pharmacokinetic studies nonlinear pharmacokineticbehaviour was shown for itraconazole (9,14,15,19). We consequently col-lected information about the Michaelis–Menten kinetic parameters(km and Vmax) from other sources (14).

Since we are planning to study the relative bioavailability of the oralsolution compared to the capsule formation of itraconazole in patientswith CF, we had to estimate the different absorbtion rate constants (ks

aand kc

a) as well. This was done using

ka = loge 2t1/2, absorbtion

, (43)

where t1/2, absorbtion is estimated by tmax/3. The times of peak plasmaconcentration for itraconazole in the oral solution and the capsule formu-lation from the drug information web page of the Food and Drug Admin-istration (FDA) (30) were used.

As no further information on PK parameters was available, theremaining model parameters were estimated by ‘best guess’. Table Ishows, along with all previously mentioned parameters and constants,the values chosen for the elimination rate constant, the volume of distri-bution, and the clearance of the metabolite (k40, V4, and C L4, respec-tively). The best guess values of these parameters were chosen as thosethat produced a simulated concentration-time curve that was closelycomparable to profiles provided by the authors of other clinical pharma-cokinetic studies (9,32). Deterministic simulations were performed usingMATLAB (these are shown in Fig. 3).

Residual variability was assumed to have both additive and propor-tional components, with σA = 5 × 10−3 mg/l (half of the lowest concentra-tion detectible by the assay) and σP = 0.15.

Between subject variability was assumed to be lognormal with vari-ance 0.1, i.e. θi = β exp(bi ) with bi ∼ N (0, 0.1Ip) for all parameters, whereIp is the p × p identity matrix.

Computational Methods

A set of MATLAB routines, POPT� Version 2.0, was used to calcu-late the optimal population design, optimising the criterion given in Sec-tion ‘Final Criterion’. This software allows the user to find D-optimal


Table I. Parameter and Constant Values Used to Calculated the Optimal Design

Variable Value Source

ksa = absorption rate constant, solution 0.945 h−1 (A)

kca = absorption rate constant, capsule 0.416 h−1 (A)

Fka = absorption rate multiplicative factor 2.27 = ksa/kc

ak20 = elimination rate constant from the 7.6 × 10−2 h−1 (B)

central compartment, linear modelk23 = distribution rate constant 0.126 h−1 (B)k32 = distribution rate constant 0.15 h−1 (B)k24 = formation rate constant 2.93 × 10−3 h−1 (B)k40 = elimination rate constant for hydroxy-itraconazole 7.6 × 10−2 h−1 = k20C L2 = clearance of itraconazole 27.9 l/h (B)C L4 = clearance of the metabolite 1.75 l/h = k40V4Q = inter-compartmental clearance 46 l/h = k23V2Vmax = theoretical maximum rate of the process 9.54 ng/(ml h) (C)Km = Michaelis–Menten constant 329 ng/ml (C)V2 = volume of central compartment 365 l (B)V4 = volume of metabolite compartment 23 l ≈ 1

16 V2V3 = volume in peripheral compartment 307 l = Q/k32C L24 = clearance of itraconazole by 1.07 l/h = k24V2

metabolism to hydroxyitraconazoleF12 = fraction of parent in the gut absolute bioavailability 0.55 (A)F14 = fraction of metabolite after first pass metabolism 4.3 × 10−2 (D)F24 = fraction of parent converted to metabolite 0.5 (E)F20 = fraction of parent eliminated 0.5 = 1 − F24

References: (A) = Janssen Pharmaceutica Products, L.P. (30); (B) = Koks et al. (29); (C) =Barone et al. (14); (D) = Janssen Pharmaceutica Products, L.P. (31); (E) = best guess derivedfrom MATLAB simulation.

sampling times by specifying the number of blood samples, the upper andlower boundaries for the samples, the dose interval for taking blood sam-ples, and the number of groups in the study. A paper by Duffull et al.,Consideration of designs for population pharmacokinetic studies (man-uscript submitted for publication), gives further description of the soft-ware used. POPT� was modified to incorporate the product criterion andnumerical solutions to the differential equations.

Simulated Annealing

POPT� employs an algorithm based on simulated annealing forcontinuous variables to maximise the criterion in each case. Our imple-mentation is based on the algorithm of Corana et al. (33) which appliessimulated annealing to optimisation over continuous spaces. Also see


0 12 24 36 480

100

200

300

400

500

Time (hr)

Con

cent

ratio

n (n

g/m

L pl

asm

a)

(a)

0 12 24 36 480

100

200

300

400

500

Time (hr)

(b)

Fig. 3. Concentration–time profiles of itraconazole (solid line) and hydroxyitraconazole(dashed line) after a 200 mg solution dose of itraconazole, with (a) linear elimination of thedrug; and (b) nonlinear elimination of the drug. The values of the various parameters andconstants used to produce the plots are given in Table I.

Goffe et al. (34) for examples of its use. We give a brief overview of theprocedure here.

Consider the design as an n-vector with elements i . At the kthiteration of the algorithm, each element of (i.e. each of the samplingtimes) is perturbed in sequence. The sizes of the perturbations are definedby a vector s = {si } of step sizes. The new value of the design variable i

is taken from a uniform distribution on the interval i ± vi . The vi areadjusted at specified intervals so that, on average, around half of the newdesigns are rejected.

For each new design, its criterion DP (1, 2, ) (described in Eq. (36))is calculated and compared to the criterion for the current ‘best’ design ∗.Designs with higher criteria are always accepted, designs with lower crite-ria are accepted or rejected according to the Metropolis criterion, i.e. theacceptance probability is


exp

[DP (1, 2, ) − DP (1, 2, ∗)

Tk

](44)

for the current temperature Tk . The temperature is lowered at specifiedintervals by geometric cooling, Tk+1 = αTk (α = 0.75 was used here tospeed up run times). The initial temperature was set at 1000.

The algorithm stops when the average step size has reached a suitablevalue, i.e. the stopping criterion is

∑i si/n < ε, for a pre-specified small

tolerance ε. This differs slightly from the algorithm of Corana et al. (33),who instead rely on the criterion reaching a stable value, regardless of thestep sizes. Sufficiently small step sizes ensure that the criterion does in factreach a stable value.

Numerical Solutions to Differential Equations

In the case of the nonlinear model, there does not exist a closed formsolution to Eqs. (5–8), so we cannot analytically isolate the responses ofinterest, A2 and A4. Instead numerical solutions to the equations are used.

For this purpose, we used MATLAB’s built-in ODE solver ode45,which is applicable to nonstiff problems and has a medium order of accu-racy. A relative tolerance (RelTol) of 10−4 was used in these calculations.

Numerical Derivatives

Once the solutions to Eqs. (5–8) are found, we require their partialderivatives with respect to each parameter (as in Eq. (40)) . These deriva-tives were approximated numerically using the central difference method,

∂ f (x)

∂x≈ f (x(1 + h1) + h2) − f (x(1 − h1) − h2)

2(xh1 + h2)(45)

for a suitably small h1 and h2. This method is preferred over forwardand backward difference methods as the error associated with the centraldifference method is proportional to the square of the step size (xh1 + h2)rather than the step size itself. This leads to more stable approximationsto the derivatives.

Since we are using numerical approximations to both the ODE solu-tions and the partial derivatives, it is important that the ODE solutionsare more precise than the numerical derivatives. For example, if the ODEsolutions are correct to the 4th decimal place, we must choose h1 andh2 such that the changes in f (·) must be larger than 10−4, otherwise theresults may be inaccurate. An alternative solution would be to use the‘direct method’ (see Atkinson and Bogacka (35) for an example of its use).


RESULTS

Optimal Design

The set of optimal sampling times ∗ are given in Table II. The timeshave been rounded to the nearest minute. As the sampling times have beenchosen from a continuous interval, these schedules are difficult to imple-ment in practice. Indeed, in some cases they are actually impossible, withseveral blood samples being taken concurrently. It is therefore necessaryto extend the population design to a sequence of intervals, or windows,in which the samples may be taken. We chose sampling windows of 3 hr(smaller for the earlier sampling times), roughly centered around the opti-mal times. These windows are also shown in Table II.

The efficiency of a population design in this case is given by

DP (1, 2, )

DP (1, 2, ∗), (46)

where ∗ is the optimal population design. Any deviation from the opti-mal sampling times will obviously lead to a decrease in efficiency. The sam-pling windows were evaluated by taking 1000 designs at random from allthe given intervals and calculating their efficiencies. The 1000 designs were

Table II. Optimal Population Design ∗ for the Population Pharmacokinetics Study ofItraconazole with 10 Patients in Each of the Three Elementary Design Groups

Capsule Solution

Elementary Sampling Elementary SamplingGroup Design Window Design Window(q) Nq ξc

q (hrs:mins) (hrs) ξ sq (hrs:mins) (hrs)

1:14 0.1 → 3.0 0:17 0.1 → 1.08:56 7.0 → 10.0 3:55 3.0 → 3.5

1 10 25:49 24.0 → 27.0 3:56 3.5 → 4.051:45 50.0 → 53.0 3:56 4.0 → 4.5

6:13 50.0 → 8.0 0:18 0.1 → 1.09:50 8.0 → 11.0 4:06 3.0 → 4.0

2 10 29:29 28.0 → 29.5 4:06 4.0 → 5.029:29 29.5 → 31.0 72:00 69.0 → 72.0

8:08 7.0 → 10.0 0:17 0.1 → 1.028:00 26.5 → 29.5 4:22 3.0 → 6.0

3 10 72:00 69.0 → 70.5 27:08 26.0 → 29.072:00 70.5 → 72.0 72:00 69.0 → 72.0


0.85 0.9 0.95 10

50

100

150

200

250

300

350

Fig. 4. Histogram of efficiencies of 1000 simulations of sampling times taken uniformlywithin all of the sampling windows described in Table II.

selected by taking a set of sampling times from uniform distributions onall of the sampling windows, i.e. these are joint (not marginal) samplingwindows. Figure 4 shows the distribution of these efficiencies. The averageefficiency was about 0.95, with 96.2% of the designs having an efficiency of0.90 or greater. The ‘worst’ design found had an efficiency of about 0.85.The sampling windows were deemed acceptable in light of these results, anddid not need to be shortened to give any further increase in efficiency.

Design Evaluation

Model Discrimination

The optimal design was evaluated in terms of its ability to selectwhether the ‘true’ underlying model involves linear or nonlinear elimina-tion from the parent compartment.

To do this, we simulated 100 sets of data under each model (linearelimination and nonlinear elimination) using the parameter values given inTable I and the optimal design provided in Table II. The models were fittedto each set of data using NONMEM (version 5 with the G77 FORTRAN


compiler, using FOCE & INTERACTION), and the ‘best’ model was chosenempirically as the one with the smallest objective function (which is propor-tional to minus twice the log-likelihood). Recall that C L24, Q and V3 areconsidered to be fixed under both the linear and nonlinear models, and thatKm is considered fixed in the nonlinear elimination model.

After simulating under the linear model, the linear model was chosencorrectly 74% of the time. After simulating under the nonlinear model, thenonlinear model was chosen correctly 100% of the time. For all estimationruns, the NONMEM run was reported to have converged successfully.

These results are considered to be reasonable, especially consideringthat at low concentrations, the difference between linear andMichaelis–Menten elimination is minimal.

Standard Errors

The results from these simulations were also used to calculate stan-dard errors of the parameters estimates. Empirical estimates of the stan-dard errors were obtained from the standard deviation of the estimatedparameter values from the 100 simulated data sets. This was performedonly in those cases where the same model was used for simulation andestimation. These are given in Table III for both the linear and nonlinearmodels. We can compare these standard errors to those predicted by theinformation matrix used to calculate the optimal design. These ‘expected’standard errors are given by the square roots of the diagonals of theinverse of each Fisher information matrix. These are given in Table III.

The comparison of expected and simulated standard errors revealquite satisfactory results, considering the difficulty in estimating betweensubject variability and absorption rate constants. The only parameter witha significant difference between the expected and estimated standard errorsis ka . The absorption rate constant is notoriously difficult to estimate inPK models, so it is not unexpected that the estimates from the simulationswere so variable.

Parameter Estimation

To assess the benefit of the optimal population design with respect toparameter estimation, two additional designs were calculated: one is opti-mal in terms of the linear model alone, and the other optimal in terms ofthe nonlinear model alone. The optimal design for model u, denoted by∗

u , was found by maximising the criterion

|MuF (u, )|1/pu , (47)

using the same methods as described in Section ‘Computational Methods’.


Table III. Standard Errors (Expressed as Coefficients of Variation) for the Population Meanand Between Subject Variability for Parameters Estimated Under the Linear and NonlinearModels

Model parameter

Linear elimination of parent ka V2 V4 C L4 C L2

Population meanExpected CV (%) 8.39 16.39 10.64 10.65 8.57Estimated CV (%) 54.34 18.71 18.20 10.76 8.32

Between subject variabilityExpected CV (%) 41.22 82.17 51.57 45.58 39.95Estimated CV (%) 121.00 114.61 80.85 68.30 89.80

Nonlinear elimination of parent ka V2 V4 C L4 Vmax

Population meanExpected CV (%) 8.19 15.55 11.26 12.51 27.84Estimated CV (%) 25.25 10.30 9.85 10.58 33.44

Between subject variabilityExpected CV (%) 40.81 49.04 49.28 61.62 197.73Estimated CV (%) 111.00 76.10 83.00 81.10

The expected standard errors are calculated from the Fisher information matrix, the estimatedstandard errors are the standard deviations of parameter estimates from 100 NONMEM sim-ulation-estimation runs. The dash indicates that normality was rejected by the Kolmogorov–Smirnov test (the data was highly positively skewed), in which case calculation of standarderrors is meaningless.

The efficiency of any population design with respect to model u isdefined (in a similar fashion to Eq. (46)) to be the ratio

Effu(,∗u) = |Mu

F (u, )|1/pu

|MuF (u, ∗

u)|1/pu, (48)

where ∗ is the product optimal design given in Table II.For the product optimal design ∗, the criteria given by Eq. (47) were

calculated under the linear and nonlinear models separately. The valuesfound were 37.591 and 115.1, respectively. The criteria for designs ∗

1 and∗

2, which are optimal under the linear and nonlinear model, respectively,are 38.993 and 120.01. Hence the efficiencies of ∗ in terms of the linearand nonlinear models are:

Eff1(∗, ∗

1) = 37.59138.993

= 0.9640

and


Eff2(∗, ∗

2) = 115.1120.01

= 0.9591.

This indicates that the product optimal design is suitably efficient at esti-mating the parameters of both models.

DISCUSSION

The practical example of the itraconazole study is still underway atthe time of writing. The authors are currently waiting on results from theactual trial.

For comparison, an empirical ‘rich’ sampling design, in which 14samples are taken over almost the entire dosing interval (72 hr) is given by(0.01, 0.5, 1, 1.5, 2, 4, 6, 8, 12, 18, 24, 36, 48, 70). In this case, again in astudy involving thirty patients, all parameters in question would be estima-ble. The relative standard error of all fixed effects would all be less than20%, in most cases less than 10%.

The construction of the optimal design was subject to several con-straints. A timeline of approximately 2 months was given to developthe entire design, from collating prior information on itraconazole andresearching applicable methods, to computing the final design (a proce-dure which took over 7 days in itself). Future research in this area includesthe consideration of further methodology (such as the Direct Method(35)). Under the given time constraints, computation time was a majorissue, with the final design computation taking over a week to run (ona 2.4 GHz Pentium 4 with 1 GB of RAM). Cost limitations of the studyand limitations imposed by the relevant ethics committee also limited thepossible number of patients and the number of samples per patient, whichfurther constrained the design in terms of the number of parameters whichare estimable.

Several ideas were used in this design which are novel to the field ofexperimental design for population pharmacokinetic studies. These includethe use of the D-optimality product criterion for model discrimination andthe incorporation of multiple responses into the Fisher information matrix(a problem previously considered in different contexts (36,37)).

The results of the power tests and simulations show that the useof this product criterion has created a design which is efficient in termsof both parameter estimation and model discrimination. Whether theindividual D-optimal designs were also adequate for model discriminationwas not investigated for this design. However, we have seen evidence (27)that suggests that this is not true for nonlinear models.


The standard errors predicted by the inverse of the Fisher informationmatrix generally match those estimated by the simulations. We believe thatthese results show that the errors introduced by the approximations madeby Retout and Mentre (22) are mostly inconsequential. Also, the assump-tions about the covariance structure of the two responses in the modelseem to be justified by these results.

The sparse sampling design presented in this paper gives a simple andcost-effective data collection regimen which should provide acceptable esti-mates of the parameters of interest.

ACKNOWLEDGMENTS

The author acknowledge helpful discussions with Ivelina Gueorguieva,Andrew Hooker, and France Mentre.

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