“I occasionally might have some reasonably good ideas, but they are nothing compared to the brilliance of nature”

Till min familj

List of Papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Nyberg J., Karlsson M.O., Hooker A.C. (2009) Simultaneous

optimal experimental design on dose and sample times. Journal of Pharmacokinetics and Pharmacodynamics, 36(2):125–145

II Hennig S., Nyberg J., Hooker A.C., Karlsson M.O. (2009) Trial treatment length optimization with an emphasis on disease progression studies. Journal of Clinical Pharmacology, 49(3):323–335

III Hennig S., Nyberg J., Fanta S., Backman J.T., Hoppu K.,

Hooker A.C., Karlsson M.O. (2011) Application of the Optimal Design Approach to Improve a Pretransplant Drug Dose Finding Design for Ciclosporin. Journal of Clinical Pharmacology, [Epub ahead of print]

IV Silber H.E., Nyberg J., Hooker A.C., Karlsson M.O. (2009) Optimization of the intravenous glucose tolerance test in T2DM patients using optimal experimental design. Journal of Pharma-cokinetics and Pharmacodynamics, 36(3):281–295

V Sjögren E., Nyberg J., Magnusson M.O., Lennernäs H., Hooker A.C., Bredberg U. (2011) Optimal Experimental Design for Assessment of Enzyme Kinetics in a Drug Discovery Screening Environment. Drug Metabolism and Disposition, 39(5):858–863

VI Nyberg J., Höglund R., Bergstrand M., Karlsson M.O., Hooker A.C., Serial correlation in optimal design for nonlinear mixed effects models. [Submitted]

VII Nyberg J., Svensson A., Karlsson M.O., Hooker A.C. Optimal design in nonlinear mixed effects models with discrete type data including Categorical, Count, Dropout and Markov models. [In manuscript]

VIII Nyberg J., Ueckert S., Strömberg E., Hennig S., Karlsson M.O., Hooker A.C., PopED - An extended, parallelized, population optimal design tool. [Submitted]

Reprints were made with permission from the respective publishers.

Contents

Introduction ................................................................................................... 13 Pharmacometrics ...................................................................................... 13 Optimal experimental design in general ................................................... 14 Population modeling ................................................................................ 14 Nonlinear mixed effects models ............................................................... 15

Nonlinear mixed effects models for continuous data .......................... 16 Nonlinear mixed effect models for discrete data ................................. 17 Population likelihood and parameter estimation ................................. 18

Fisher Information Matrix ........................................................................ 21 Maximizing the Fisher information matrix .......................................... 22

Applied population optimal design .......................................................... 24 Clinical trial simulation versus optimal design ........................................ 25

Aims .............................................................................................................. 26

Methods ........................................................................................................ 27 Further development of the population FIM ............................................ 27

Linearizing the residual model ............................................................ 27 Including intrinsic variability .............................................................. 29 Improved linearization method ............................................................ 30 Calculating simulation-based Fisher Information Matrices ................. 31

Finding the optimal design ....................................................................... 32 Simultaneous versus sequential optimization ...................................... 32 Group size optimization ....................................................................... 33 Design criteria ...................................................................................... 34 Optimized design variables .................................................................. 35

Evaluating designs - models and methods ............................................... 36 Simulation- and estimation-based diagnostics ..................................... 37

Applied optimal design ............................................................................ 38 Drug compound screening experiments (Paper V) .............................. 38 Intravenous glucose tolerance test (Paper IV) ..................................... 39 Therapeutic drug monitoring of ciclosporin (Paper III) ...................... 40

The optimal design software PopED (paper VIII) ................................... 41 Software architecture ........................................................................... 42 Efficiency translation ........................................................................... 42 Using PopED in parallel ...................................................................... 42

Results and Discussion ................................................................................. 44 Performance of the extended population FIM .......................................... 44

Linearization around the mode ............................................................ 44 Linearization with serial correlation .................................................... 45 Simulation-based Fisher Information Matrix ...................................... 47

Optimization using the FIM ..................................................................... 49 Simultaneous versus sequential optimization ...................................... 49 Group size optimization and efficiency translation ............................. 50 Optimization using other design variables........................................... 51

Applied optimal design ............................................................................ 53 Optimizing in vitro drug compound screening experiments ............... 53 Optimization of the intravenous glucose tolerance test ....................... 56 Optimization of a drug monitoring design for ciclosporin .................. 59

Optimal design software tool PopED ....................................................... 60 Parallel implementation ....................................................................... 61

Conclusions ................................................................................................... 63

Populärvetenskaplig sammanfattning ........................................................... 64

Acknowledgements ....................................................................................... 66

References ..................................................................................................... 69

Abbreviations and Symbols

CTS Clinical trial simulation D Dropout predictions - changes the number of individuals in

the design DD Dropout predictions - changes the number of individuals in

the design and gives additional information on disease param-eters

DP Disease progression DS Ds-optimal design where the autocorrelation parameter is

treated as uninteresting EBE Empirical Bayes estimates (the mode of the joint density) ED Expectation D-optimal design EID Expectation optimal design of determinant of inverse Fisher

information matrix EXP Exponential decay model for intrinsic clearance FIM Fisher information matrix FA Fast approximation group size optimization FIXAR Optimal design for all parameters except the autocorrelation

parameter in the Fisher information matrix FO First order FOCE First order conditional estimation FOCEM First order conditional estimation around the median mode GLM Generalized linear model GLMM Generalized linear mixed model G-OD General optimal design GS Global search group size optimization GUI Graphical user interface ID Dropout predictions - changes the number of individuals in

the design, gives additional information on disease parame-ters, and the dropout parameters are included in the Fisher in-formation matrix

IIV Inter-individual variability IOV Inter-occasion variability IV Intravenous administration of drug IVGTT Intravenous glucose tolerance test JM Job manager, distributing parallel jobs LAP Laplace method based population likelihood

LMEM Linear mixed effects model MC Monte Carlo based population likelihood ML Maximum likelihood MLE Maximum likelihood estimate MM Michaelis-Menten model (nonlinear elimination) MPI Message-passing interface ND No dropout is included in the optimization NLMEM Nonlinear mixed-effects model NOAR Optimal design without present autocorrelation NOFIX Optimal design with all parameters in the Fisher information

matrix OD Optimal design ODE Ordinary differential equation OFV Objective function value, in estimation often -2 log likelihood

and in optimization the criterion value OMPI Open MPI project for message passing PD Pharmacodynamics PK Pharmacokinetics RMSE Root mean squared error RSE Relative standard error RUV Residual unexplained variability SAEA Simulation with autocorrelation followed by estimation in-

cluding autocorrelation SAEN Simulation with autocorrelation followed by estimation with-

out autocorrelation SAEM Stochastic approximation expectation maximization SD Standard deviation SDE Stochastic differential equation SNEA Simulation without autocorrelation followed by estimation

including autocorrelation SNEN Simulation without autocorrelation followed by estimation

without autocorrelation SRD Standard rich design SSD Standard sparse design STD-D Standard design STS Standard two-stage modeling approach TAD Time after dose TDM Therapeutic drug monitoring XML Extensible markup language Population median (typical value, fixed effect) Individual deviation from typical value Difference from individual parameter between occasions Differences between individual predictions and observations All population parameters

Population covariance matrix for Population covariance matrix for Population covariance matrix for

13

Introduction

The cost of releasing a new drug on the market is increasing rapidly and was recently estimated to be over one billion dollars [1]. The increasing cost is well known and recognized by regulators [2] and external experts [3], as well as in academia [1, 4-6]. The reasons for the increasing cost and low success rates can differ between drugs, but possible causes include higher ethical demands, lack of effect, safety issues, reduced profits, conflicts of interest, and inadequate study design. It is clear that correct decisions need to be made earlier in the process and the information gain need to be maxim-ized throughout the drug development phases. Doing this will increase the probability that the development of ineffective drugs will be stopped early and increase understanding of successful drugs, and will therefore lower the overall costs of drug development.

Pharmacometrics Pharmacometrics is a scientific discipline that has the potential to deal with some of the issues in drug development. This field combines biostatistics with pharmacology and is “the science of developing and applying mathe-matical and statistical models to characterize, understand and predict a drug’s pharmacokinetics, pharmacodynamics and biomarker-outcome be-havior” [7]. Several examples of the benefits of pharmacometrics over tradi-tional statistical analysis have been published in the literature [8, 9]. Pharmacometric models can describe [10]

Pharmacokinetics (PK) – “what the body does to the drug”. Pharmacodynamics (PD) – “what the drug does to the body”.

For instance, a PK model could describe the concentration-time profile of a drug after administration, and a PD model could show how the effect of a drug changes with time. In pharmacometrics, it is common to simultaneously model as many components as possible to describe the variability in the data. For example, data from a long term disease progression trial could be mod-eled using a PK model that includes the disease status (a PD model), where some patients might drop out due to their disease status (dropout model)

14

[11]. Once a pharmacometric model is developed, it can be used to make statistical inferences, test new designs in a clinical trial simulation setting [12], individualize treatments [13], or find optimal designs [14].

Optimal experimental design in general It was recognized early in the 20th century that the information obtained in an experiment changes with the design of the experiment [15]. It was R.A. Fisher who discovered that this information is asymptotically linked to the Fisher Information Matrix (FIM) [16]. Later, in the 1940s, Rao [16] and Cramer [17] independently stated that the uncertainty of any unbiased esti-mator is bounded from below by the inverse of the FIM:

(1)

This asymptotic behavior formed the Cramer-Rao inequality and is a funda-mental part of optimal design (OD) theory. Equation (1) shows that maxim-izing the FIM with respect to the design variables q will result in the most informative (optimal) study setup for estimating the parameters as pre-cisely as possible.

Population modeling In drug treatment and drug development it is very common to perform longi-tudinal (over time) repeated measurement studies. Often, these studies in-volve multiple subjects such as patients, animals, molecules etc., and a natu-ral way to model such data is to use a model that describes the population of subjects. The simplest way to model repeated subject data is to pool the data and model it as one entity, i.e. naïve pooling [18]. The problem with this approach is that only the population mean will be modeled and therefore no information about the differences between subjects is obtained. This method is also sensitive to unbalanced designs, i.e. where the design differs among subjects. This is because a subject with rich data is more influential than a subject with sparse data, even though data from the subject with sparse data might be more similar to the mean of the population.

Another population approach is to use the standard two-stage (STS) method. In this method, the individual data are modeled separately and summary statistics, based on the subject parameters, are then used to calcu-late the population characteristics [18-20]. The STS approach produces un-biased estimates for rich data but will bias the parameters when the subject data are sparse. Moreover, this approach does not separate the subject vari-

1 ˆ,qFIM COV

ˆ

15

ance from the residual variance and therefore does not capture the true un-derlying biological variability. Furthermore, if a subject has very sparse data, a different individual model with fewer parameters might be necessary and hence it can be hard to calculate summary statistics over parameters with different definitions.

A third approach is to pool the data and use all the data to estimate the population parameters by adjusting for the correlations between subject data [18, 20]. This approach is called nonlinear mixed effects modeling (NLMEM), where the population parameters are estimated directly. After-wards (or simultaneously), the subject parameters can be derived from the subject data and the population parameters. This method has several ad-vantages over the other two approaches, for instance it is possible to use sparse data for one subject within the same structural model. Moreover, sparse data will not bias the population parameters; instead, the individual parameter estimates will shrink towards the population median parameters (i.e. the typical value or fixed effect). Nowadays when a population model is discussed, it is usually an NLMEM.

Nonlinear mixed effects models The term mixed effects refers to the simultaneous modeling of both fixed (population parameters) and random (variance of the population and individ-ual data) effects. An NLMEM typically describes the inter-individual varia-bility (IIV), the inter-occasion variability (IOV) [21] and the residual unex-plained variability (RUV). The RUV contains both the assay error from the method of analyzing the data and other sources of variability such as errors in the recorded dose and sampling history as well as model misspecifica-tions; hence the term “unexplained”. Other levels of variability can also be incorporated: inter-study variability [22] or intrinsic variability (serial corre-lation) between observations [23], which can arise from e.g. structural model misspecification. It is straight-forward to include several sources of variabil-ity in the model building. However, in general, increasing the levels of vari-ability will increase the computational burden when fitting the model to the observed data.

NLMEM can address continuous data, such as plasma concentration measurements, as well as discrete data, such as the number of seizures, dropouts from a study, etc. The parameters included in the model and the definition of the NLMEM will differ depending on whether the data are con-tinuous or discrete.

16

Nonlinear mixed effects models for continuous data

A multi-response continuous NLMEM is defined as

(2)

where g() is a vector function describing the individual parameters, the co-variates vector ai, and the discrete variables vector xi of individual i. The individual parameters deviate from the population typical value vector by the IIV [ i~N(0, )] and the IOV [ i~N(0, )]. ti is a vector with independent variables (e.g. sample times) of size ni (number of independent variables), h() is the vector function describing the residual error model of size ni, where

i~N(0, ), and f() (size ni) describes the model for individual i. The matrices , , and are covariance matrices describing the correlations between the

individual parameters ( , ) and between the residual error parameters ( ). The model and residual error model can be further split into different re-sponses by

(3)

where tij is the ith individual vector of the independent variables in response j. For example, f1 could be a PK model with an additive residual error model h1, f2 could be a PD model with a proportional residual error model h2, and so on.

In the setting described in equation (2) with a general function g(), the pa-rameters can have any transformed parameter distribution, e.g. normal addi-tive + or proportional (1+ ), log-normal +e , etc. The residual error model h() is also general and can be homoscedastic (additive) i, heterosce-dastic (proportional) fi* i, or combinations of both, etc. As mentioned earlier, intrinsic variability could be present in the data. This can be included in the general definition in equation (2) by stating that the residual errors are serially correlated by e.g. the autoregressive model of order one (AR(1)) [20, 23, 24]. That is

f ,g , , , , h ,g , , , , ,i i i i i i i i i i i iy t a x t a x

1 1 1

1 1 1 1

f ,

f ,g

f ,

h , ,

h ,g(),

h , ,

i

i

j ij j

i i

i i

j ij j ij

t g

t

t g

t g

t

t g

17

(4)

where C(j,k) represents the correlation between the independent variables j and k. t½ is the parameter determining the half-life of the correlation. In this model, the correlation strength diminishes exponentially with the difference between tj and tk and is exactly 1 if the independent variables are equal. Since tj and tk often represent sample times, the residual correlation will be weaker if these are widely spread. Another possible implementation of serial correlation in NLMEM is to use stochastic differential equations (SDEs) [25, 26]. This approach includes statistical functionality for e.g. the AR(1) model and easily allows for the use of other correlation structures. However, it is not straightforward to implement and solve SDEs within an NLMEM framework. The parameters that describe the population (population parameters) can be combined into a vector incorporating the population median, IIV, IOV and RUV: =[ , , , ]. Note that the covariance matrices are symmetric and hence only the lower triangular parts (including the diagonal) are parameters, i.e. n*(n+1)/2 population parameters from a matrix of size n*n).

Nonlinear mixed effect models for discrete data The NLMEM for discrete data is a probabilistic model where the probability of observing an individual response yi, given the parameters, is modeled as

(5)

where pi() is the individual probability (likelihood) function. All the other parameters and variables are described in the previous section. A popular approach to modeling discrete data is to use a generalized linear model (GLM) [27] where a link function describes how to connect the probabilistic model to a discrete response. The work by McGullagh and Nelder for GLMs has been extended to include random effects and is known as generalized linear mixed models (GLMM) [28, 29]. As with continuous data, it is possi-ble to include intrinsic variability in discrete data models, e.g. an autoregres-sive model [29] or an equicorrelation structure [28]. In pharmacometric modeling, Markov elements have been used to acknowledge correlations between states or observations. For example, the importance of including a Markovian predictor for a model of disease progression in acute stroke pa-tients has been investigated [30]. Markov elements have also been used in a model for sleep states [31] and when modeling count data of epilepsy sei-zures [32, 33].

ln 2

½( , )j kt t

tj k eC

p | ,g , , , ,i i i i i i iy t a x

18

Population likelihood and parameter estimation In order to estimate the population parameters , it is necessary to include a function defining how likely the data are, given the parameters, i.e. the like-lihood function. A maximum likelihood (ML) approach is commonly used in NLMEM, and the population likelihood L is maximized (or else the -2 log L is minimized) with respect to the population parameters

(6)

where li() is the individual likelihood, p() is the probability of the individual deviations from the fixed effects, given the population parameters in and

, and n is the number of individuals in the population. The product

is referred to as the joint density and the integral is the marginal likelihood or the individual contribution to the population likelihood. Figure 1 illustrates an example of a marginal likelihood and joint density in one dimension, i.e. estimation of one population parameter. The marginal likeli-hood cannot be calculated for NLMEM with a closed form solution. The reason for this is that the model is nonlinear with respect to the individual deviations i and i. However, there are several ways to approximate the marginal likelihood (or linearize the model with respect to i and i).

Numerous regression and statistical software are available for incorporat-ing NLMEM; for example, NONMEM [34], Monolix [35], R (nlme pack-age) [36], SAS (NLMIXED procedure) [37], etc. The first software that im-plemented NLMEM (and the most widely used) is NONMEM, which was initially developed by Beal and Sheiner [34]. NONMEM also has the most approximation and estimation methods for NLMEM and can be used with methods where the data are either assumed to belong to a certain distribution (parametric) or are without this assumption (nonparametric). Some of the parametric approximation methods available in NONMEM are briefly de-scribed below.

First order (FO) method This method, which was originally implemented in the NLMEM software NONMEM [38], utilizes a linearization of the model around the most com-mon individual value, i.e. i, i=0. The marginal likelihood is then calculated, assuming Gaussian distribution of the data with the mean and variance of the linearized model. Exactly the same marginal likelihood can be calculated using a first order Laplace (integral) approximation to the marginal likeli-hood around the typical values while approximating the hessian matrix of the individual likelihood li() by the individual FIM based on the second moment of the typical individual score function (see the Fisher Information Matrix

1

argmin 2 ln , , | , , | , d dn

i ii

l y p

il p

19

section below) [39]. This approximation is, in comparison to other approxi-mations, fast to use but has been proven to produce biased estimates [40, 41] and will not in general work well when the inter- and intra-individual varia-bility is large. Besides, this approximation is not suitable for probabilistic models, used with discrete data, because the normality assumption will be too inaccurate.

Figure 1. Approximation methods of the marginal likelihood. The left panel shows the joint density with the maximum likelihood estimate (MLE) from the FO approx-imation. In the middle panel, the marginal likelihood is plotted for different parame-ter values. The right panel shows a typical PK profile using the MLE from different approximations.

First order conditional estimation (FOCE) method The FOCE method was first introduced by Lindstrom and Bates [42] who linearized the model around the mode of the joint density which can be obtained by maximizing

(7)

As in the FO method, the marginal likelihood is approximated by assuming Gaussian distribution of the data with a mean and variance which are linear in . This method, which uses a slightly different optimization algorithm to find the maximum likelihood estimates (MLE), was later implemented in NONMEM. As with the FO method, an approximation of the marginal like-lihood can be achieved by using the Laplace integral method around the mode and further approximating the hessian of the individual likelihood by

-0.6 -0.4 -0.2 0 0.20

0.2

0.4

0.6

0.8

1

join

t den

sity

0 0.5 1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

conc

entr

atio

n

1 1.5 2 2.5 30.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

population param

mar

gina

l lik

elih

ood

TrueLaplaceFOCEFO

ˆ ˆ,

,

ˆ ˆ, argmax , , | , , | , i il y p

ˆ ˆ,

20

using the individual FIM based on the second moment of the individual score function [39]. However, even this improved approximation is in gen-eral too poor to use with models for discrete data.

Laplace method This method was named as a tribute to Pierre-Simon Laplace, whose integral method is used in FO, FOCE and the Laplace method. Here, the approxima-tion of the marginal likelihood is not achieved using direct linearization of the model; instead, the first order Laplace integral method is used. As with the FOCE method, the mode is used when approximating the marginal like-lihood and the hessian of the individual likelihood is calculated. Because the hessian matrix is used, this method is more computer intensive but it will in theory be closer to the true underlying marginal likelihood, simply because it uses a higher order approximation than FOCE, see Figure 1. For discrete data, the Laplace method works well in many situations but less well than the more accurate Adaptive Gaussian Quadrature method implemented in SAS [37] within the NLMIXED procedure [33, 43].

Stochastic and Monte Carlo-based methods The marginal likelihood Li can be calculated without any linearization by using the Monte Carlo integration [44]

(8)

where k and k are samples from their respective Gaussian probability dis-tributions and n is the number of Monte Carlo samples. A problem with this method is that the convergence rate will be very poor for problems with many dimensions (parameters), if the variance of the sampling distribution is large (high IIV, IOV) or when the typical values contribute little to the inte-gral (the mode of the joint density is very different from the most probable individual samples). This method has nonetheless provided accurate and precise estimates, despite long run times, according to the pharmacometrics literature [45, 46].

In the past two years, NONMEM updates have included several new al-gorithms built on more sophisticated Monte Carlo techniques: the Monte Carlo importance sampling method (IMP), the stochastic approximation expectation algorithm (SAEM) [47], and others.

The SAEM algorithm for NLMEM was originally implemented in Mono-lix [35] and is an extension of previous EM algorithms [48]. Briefly, the method simulates individuals using a Markov Chain and then approximates the expected likelihood using a stochastic approach (the E-step). The param-eters are then updated to maximize the likelihood (M-step). The algorithm

1

1, , | ,

n

i i i k kk

L l yn

21

iteratively repeats the E- and M-steps until convergence to produce the max-imum likelihood estimates.

The IMP method also uses the E- and M-steps but uses a Monte Carlo in-tegration (E-step) with sampling from a distribution that approximates the joint density (a more accurate but slower method of computing the joint den-sity compared to SAEM).

Fisher Information Matrix The FIM can be used to calculate the expected precision of the unknown parameters in an upcoming experiment (equation 1). The FIM is defined as the expectation over the second moment of the score function (i.e. the partial derivative of the log likelihood with respect to the parameters) [49]

(9)

However, the first moment of the score is zero for the maximum likelihood estimates and, because of weak regularity conditions, the FIM can equally be defined as the variance of the score [49]

(10)

One consequence of the Cramer-Rao bound, equation (1), is that the infor-mation in an experiment is dependent on the design of the experiment and hence maximizing the FIM with respect to the design will give the best pos-sible precision in a future study, i.e. the OD.

For linear models without random effects, the FIM can be calculated ana-lytically and will not be dependent on the unknown parameters [50]. If the model is nonlinear without random effects, the FIM will be dependent on the unknown parameters of the upcoming experiment but can still be ex-pressed in a closed form. However, if the model is nonlinear with nonlinear random effects, the FIM must be approximated because the marginal likeli-hood is not generally solvable (see section population likelihood).

Mentré et al. [51] defined the FIM for NLMEM with continuous data by linearizing the model (using the FO approximation) to a linear mixed effects model (LMEM). They also used an LMEM closed form expression for the FIM, assuming that the data are normally distributed. Initially, this FIM was derived for a homoscedastic residual error model, assuming that the residual

2

, logq E LFIM

2

2, logq E LFIM

22

variance parameter was known. This work was further limited in that the IIV distributions were assumed to be normal or lognormal. The FIM was later extended to heteroscedastic residual error models, including the unknown residual parameters [14], to other IIV distributions by linearizing around the parameter function g() [52], and with extra interaction terms in the FO line-arization of proportional residual errors [53]. Other extensions to the origi-nally derived FIM for NLMEM have utilized the covariances between the fixed effects and random effects [53-55], referred to as the full FIM (as op-posed to the reduced FIM where the covariances are not acknowledged).

Breslow and Clayton [29] derived a FIM for GLMM based on probabilis-tic models for data where a link function is known. This derivation was based on a second order Taylor linearization of the GLMM and did not in-clude covariances between fixed and random effects. However, a full FIM was expressed for certain Gaussian random effect structures [56]. One draw-back associated with these expressions is that they involve conditional ex-pectations that in general need to be calculated by numeric integration. An-other full FIM was derived for a logit model [57], although this derivation is not generalizable to other model families.

Maximizing the Fisher information matrix The inverse of the FIM can be interpreted as the expected covariance in an upcoming study, with the matrix dimensions equal to the number of un-known parameters. Optimizing the future experiment aims to minimize this matrix or, equally, to maximize the FIM. In order to use standard methods for optimization, a scalar function (criterion) of the FIM must be defined. In OD theory, a vast number of criteria are available. The most common one is the (local) D-optimal design criterion, where D stands for the determinant of the FIM [50]. Maximizing this criterion minimizes the volume (or area in 2 dimensions, figure 2) of the hyperbolic confidence ellipsoid in the unknown parameter space. Another commonly used criterion is the A criterion, which minimizes the sum of the variances (sum of the axis) of the confidence ellip-soid, i.e. the trace of the inverse FIM, figure 2. [50]. An alternative local design approach is to only consider a subset of the parameters and to opti-mize those. The most common criterion in this family is the Ds-optimal de-sign [50], which minimizes the volume of the confidence ellipsoid for the interesting parameters but still acknowledges the correlation with the unin-teresting parameters.

When working with nonlinear models, the unknown parameters must be known to calculate the FIM. To get around this, robust design (global de-sign), where a parameter distribution is assumed known instead of a point value, can be introduced. These types of designs have been proven to spread the support points in the OD and hence are more robust against different parameter values [50]. Again, several criteria are available: ED OD [58],

23

which calculates the average D criterion over the parameter distribution; EID design [59], which calculates the average of the determinant of the inverse FIM over the parameter distribution; etc. In local ODs, where the parameter values are assumed known, the minimization of the inverse of the FIM will be equal to maximizing the FIM; this is not the case for global ODs where the OD using the EID criterion is different from the OD using the ED crite-rion [59]. Calculating average ODs (ED, EID) can have the disadvantage that some parameter values might be more influential on the design than others.

Another possible robust approach that is less sensitive to influential pa-rameter values is to consider average efficiency ODs [60], where e.g. a local D-optimal design is found for all the values of the parameter distribution (a discrete set of values). Optimizing with the average efficiency criterion then calculates the design that will lower the overall efficiency calculated with the D-optimal design for each parameter value. This approach is in general very robust but is also computer intensive due to the search of the local OD for each parameter value.

One alternative to the average designs is to use a robust criterion that con-siders the worst possible scenario (e.g. a function over the parameter space like the minimal D-optimal design [61]) and optimize over that criterion, a so-called minimax design. Again, this approach demands several calcula-tions of local ODs and is therefore computer intensive.

When the prior information of the parameters is of high quality, a Bayesi-an OD is a suitable approach. In population OD, it can e.g. be used to opti-mize individual treatments for Bayesian estimation by optimizing on the individual level, given a population prior [62].

In general, the robust designs do not have analytic solutions and must be calculated with numerical integration, which makes them computationally slow.

24

Figure 2. Confidence ellipses. The left panel shows the confidence ellipse of local criteria: D criterion= 1* 2, A criterion= 1+ 2. The right panel illustrates 100 confi-dence ellipses with parameters sampled from a log normal distribution ( 1) and a normal distribution ( 2). The minimax criterion corresponds to the minimum |FIM-1| and the EID criterion is approximated by E[FIM( ) -1].

Applied population optimal design OD theory relies on a well-defined solid mathematical foundation but the final purpose of the theory is to design experiments. Since the first derivation of the NLMEM FIM (population FIM) [51] almost 15 years ago, numerous examples of applied population ODs have been published. For example, Bayesian OD was used to optimize sample times for the individual treat-ments of nortriptyline given a population prior [62, 63]. The sample times, subject assignments to different elementary designs, and discrete doses of ivabradine were also optimized using a PD model in two Phase I studies. This was done to illustrate that a much sparser trial design could achieve similar efficiency to that obtained using the original design [64]. Another application optimized the sample times for a study to build a PK model for administration of itraconazole to patients with cystic fibrosis [65]. Here, the focus of the OD was discrimination between two suggested PK models, as well as having a design that estimates parameters with good precision. The criterion used was an extension of the T-optimal design criterion introduced by Atkinsson and Fedorov [66]. Other work used cost or constraint OD, which demonstrates optimization around a more realistic design region or utility function [51, 67-70], e.g. where the total cost of a study is restricted. Other methodological publications discussed the benefits and implementa-

Global criteria

Local criteria

11-SD(

1)

2-SD(

2)

2+SD(

2)

1+SD(

1)

1

2 2

tr(FIM-1)1/2=(SD(1)2+SD(

2)2)1/2

21/2

11/2

- D-criterion. . EID-criterion+ Minimax-criterion

25

tion of robust designs for population models [55, 71, 72]. Clinically more relevant designs using design windows where the OD is altered to become more feasible while remaining adequate have also been demonstrated [73-76].

In most of the NLMEM OD applications, the sampling times or group as-signments have been optimized. However, there are no limitations within OD theory with regard to the inclusion of other design variables like the start and stop times of studies, durations of infusion, order of multiple doses, co-variates, duration of wash-out periods, etc. In fact, ultimately all design vari-ables, both continuous and discrete, can be optimized simultaneously.

Clinical trial simulation versus optimal design The outcomes of a new study can also be predicted by using clinical trial simulation (CTS) techniques [77, 78]. In one CTS setting that is comparable with OD, the data are simulated with a given model, parameters and number of fixed designs, the model is re-estimated using the different designs, and some measurement of the performance of the simulated trials, e.g. bias and precision, power, etc., is calculated. CTS is a powerful tool because it can handle different linked models for simulating data, and it can easily calculate power with type I error correction for bias, test performance of estimation methods, etc. However, CTS is a very computer-intensive method (in gen-eral up to 1000 simulation datasets with re-estimations are used for each design investigated), and therefore only a limited number of candidate de-signs can be tested. This is a drawback if many variables are to be optimized and if the performance of the trial is sensitive with respect to the design pa-rameters. In these types of scenarios, OD is favorable because it allows for optimization of multiple continuous design variables as well as multiple discrete variables, and it is in general faster than CTS. Also, recent advances in population OD allow for optimization of power [79, 80], utilizing methods without linearization, suitable for non-identifiable models [81] and using adaptive designs [82, 83]. Nevertheless, CTS has proven its use [12, 77, 78] and will be an important component of designing studies whenever OD methods are not applicable or to verify the ODs.

26

Aims

The overall aim of this thesis was to develop tools and adopt methodology to aid designing optimal experiments in drug development and drug treatment. The specific aims were to: Develop methodology to optimize with more accurate methods, to ena-

ble optimization of discrete data models and models with intrinsic varia-bility;

Show the benefits and availability of simultaneously and sequentially optimizing less common design variables such as trial start and stop times, dose order, infusion start and stop times and duration, etc.;

Show applied examples where OD can be used in early drug develop-ment as well as in late phases of drug development and drug treatment;

Develop software to improve the ease of use of NLMEM OD and that

can be used to run complex run-time intensive models in parallel.

27

Methods

Further development of the population FIM All the work in this thesis used extended linearization of the residual error model. In paper VIII, an extension of the FOCE linearization [54] was de-rived, while paper VI used an FO linearization including serially correlated residual errors, and paper VII used a simulation-based FIM with either a Laplace method or a Monte Carlo integration technique to calculate the mar-ginal likelihood. Linearizing the residual model An extended linearization of the population FIM was derived in order to implement more advanced residual error structures. The first extension of the FIM was to linearize equation 2 using the FO method:

(11)

where o is the number of occasions for individual i and

(12)

(13)

, ,

,

, ,1

f ,g , , , , h ,g , , , , ,

f ,g , 0, 0, ,

,g ,g ,

i i i i m i i i i i m i i i

i i i m i i

o

i i i i i i i m i i mm

y t a x t a x

t a x

t t t gL H T

,1: :, 0

f,g ,g

dim dim

i i m o

T

i i ii

i i

t t

y

L

,1: :, , 0

h,g , ,g ,

dim dim

i i m o i

T

i i i i ii

i i

t t

y

H

28

(14)

The other parameters and functions are defined in equation 2. The occasion linearization is slightly different from previous derivations because a sum is used instead of increasing the occasion matrix size [54]. When a model is linear in the random effects (LMEM) the standard derivation of the popula-tion FIM for individual i with a design qi can be used [14, 51, 54, 55, 84]:

(15)

where

(16)

(17)

(18)

and the matrices M1,M2 and M3 are defined as

(19)

(20)

,

,, , 0

f,g ,g

dim dim

i i m

T

i m i ii m

i

t t

y

T

1 2 2 3

2 3 3

, , , ,1,

2 , , ,

i i i i i i

i i

i i i i i

y yq

y y

A M M V C M M VFIM

C M M V B M V

1

1 1

1 2 1 1( ) ( )2 2

2, ,

tr

, 1,...,dim

Ti i i

i i i n mi i i i

yy

y y

m n

M V MA M M V

M V M V

1 1( ) ( )3 3 3, tr

, 1,...,dim , ,

m ni i i i i iy y y

m n

B M V M V M V

1 1( )2 3 3 2, , tr

1,...,dim

1,...,dim , ,

nmi i i i i i iy y y

m

n

C M M V M V M V

1

E, dim( ) dimi

i i

yyM

2

2 , dim dimii i

vec yy

VM

29

(21)

where dim([ , , ]) denotes the number of population parameters to esti-mate in the covariance matrices. The expectation and variance of the linear-ized model are:

(22)

(23)

The expression in equation 15 is the FIM for individual i, but often the indi-viduals have the same design and can therefore be lumped together in an elementary design (design group). Doing this will finally yield the popula-tion FIM:

(24)

where gi is the number of individuals in the elementary design i and Ng is the number of design groups. This expression of the population FIM, or modifi-cations of it, were used throughout this thesis.

Including intrinsic variability Serially correlated residual errors were included in the expression for the FIM (paper VI) by updating the linearized variance with a correlation matrix C

(25)

of size (dim(yi) x dim(yi)) where the element j,k in the matrix C e.g. corre-sponds to the AR(1) model, equation 4. The implementation was investigat-ed using different scenarios: NOAR – no included autocorrelation, FIXAR – autocorrelation included in the variance but excluded from the FIM, NOFIX

2

3 , dim dim , ,, ,

ii i

vec yy

VM

E f ,g , dimi i iy t y

, ,1

dim dim

oT T T

i i i i m i m i im

i i

y diag

y y

V L L T T H H

1

, ,gN

i i ii

q g qFIM FIM

, ,1

oT T T

i i i i m i m i im

y diagV L L T T H H C

30

– all parameters in the FIM including the AR parameter, and DS – autocor-relation parameter treated as uninteresting. The AR parameter value might be difficult to ascertain before an experiment is run and therefore a robust design (ED-optimal) was also investigated. Moreover, the performances of the designs were evaluated in a simulation study with four different scenari-os: 1) SAEA – simulate with autocorrelation, re-estimate with a model in-cluding autocorrelation; 2) SAEN – simulate with autocorrelation and re-estimate without the AR parameter; 3) SNEA – simulate without AR but use an AR parameter in re-estimation; and 4) SNEN – simulate and re-estimate without autocorrelation.

Improved linearization method When estimating parameters, the FOCE linearization [42] is superior to the FO linearization. This is because the FOCE method linearizes around the mode of the joint density. In fact, it is necessary to use this type of lineariza-tion, both in estimation and OD, if there is a power relationship between the fixed effect and the random effect, e.g. a Box-Cox transformation [85] of the parameter. An extended linearization for OD has been proposed before [54], where the model was linearized around a sample from the IIV distribution [ i~N(0, )]. This was repeated many times and the expectation of the FIM over the different samples was used. A drawback with this method is that the model is not linearized around the mode and hence will not mimic the FOCE method used in estimation. An obvious reason for not using the mode is that individual data are needed when calculating the mode (see equation 7, where the individual likelihood is data-dependent). A slightly different approach, based on the expected data for an individual i, was used to develop the FO-CEMode linearization (paper VIII) with E[yi] and V[yi] defined as

(26)

(27)

, ,

, ,

, ,ˆ ˆ1

ˆ ˆf ,g , , , ,

ˆ ˆi i i m i m

i i i i i m i m i i

o

i i i m i mm

E y t a x

L T

, , , ,

ˆ ˆ

, ,ˆ ˆ1

, , 0 , , 0

i i i i

i m i m i m i m

T

i i i

oT

i m i mm

T

y

diag

V L L

T T

H H

31

where is the expected mode of the joint density given by equation 7 and the data are replaced by the median of the individual data

(28)

Given this expectation and variance (equations 26 and 27), which are linear in the random effects, the FIM can be calculated by equation 15. The OD and predictive performance of this linearization was investigated (paper VIII) using a sigmoidal E-max model, where the doses in two dose groups were optimized. The predictive performance of the FIM using this extended linearization was also investigated in a simulation study.

Calculating simulation-based Fisher Information Matrices A simulation-based FIM was developed (paper VII) to demonstrate optimi-zation of more complicated discrete data models. The idea was to calculate the expectation over data using a Monte Carlo approach over several indi-vidual marginal likelihoods. Two different approaches to calculate the mar-ginal likelihood were used: A Monte Carlo integration over the joint density (equation 8) where data

were simulated according to the underlying individual data distribution (e.g. Poisson, Binomial, Weibull, etc.)

A Laplace method [39] to approximate the marginal likelihood

(29)

Again, the individual data were simulated according to the underlying indi-vidual distribution and the marginal likelihood was approximated around the mode of the joint density. Given the two different marginal likelihood approximations, the FIM can be calculated from no simulated datasets:

(30)

where Ng is the number of elementary designs, gi is the number of individu-als in the elementary design i, Lpop,j is the population likelihood (sum of all

ˆ ˆ,

, , , , ,i i i i i iE y f t g a x

2

ˆ

ln1ˆ ˆ2 ln 2 ln ln ln

2T i

i i i i T

lL l -1 -1

2

, ,21 1

1ln |

g oN n

pop j pred ji jo

L yn

FIM

32

individual marginal log likelihoods) for dataset j and ypred,j is a dataset con-sisting of gi individuals simulated with the design in the elementary design i . If equation 8 is used to calculate the marginal likelihood, the simulation-based FIM does not involve any linearization. On the other hand, if the La-place method is used, the assumption is that the joint density can be approx-imated by Gaussian distribution. To calculate this FIM, for any underlying NLMEM with discrete data, knowledge of the properties of the underlying distributions is not required. However, data need to be generated with a proper simulation model. Paper VII demonstrated the simulation-based FIM with four different models: 1) a dichotomous model, 2) a count data model, 3) a multi-response continuous disease progression model with a dropout model, and 4) a dichotomous model with correlated states using Markov elements. The models were used (when possible) to compare the closed form GLMM FIM [29] with the simulation-based FIM. In the disease progression model, different scenarios were investigated with either a constant hazard survival model, where the number of dropouts is not dependent on the dis-ease status, or with a disease status-dependent survival model, where the number of dropouts increases with a worsening disease status. The scenarios investigated were: the dropouts were ignored (ND), affecting only the num-ber of individuals in the study (D); the dropout parameters were excluded from the FIM but additional information about the disease progression pa-rameters from the dropout model were included (DD); or the FIM included both dropouts and disease progression parameters and the additional infor-mation about the disease progression parameters (ID). The simulation-based FIM had to be used in both the DD and ID scenarios due to the information from the dropout model.

Finding the optimal design One of the major obstacles when working with ODs is the actual optimiza-tion. This is not surprising because the criterion surface may be locally dis-continuous (discrete design variables), with many dimensions (multiple de-sign variables). It is also hard to judge beforehand where the OD is located and therefore a local search (e.g. gradient-based methods) will not, in gen-eral, find the global (optimal) solution. Instead a global search over the en-tire design space is needed.

Simultaneous versus sequential optimization In paper I, the optimization of multiple continuous design variables, e.g. doses and sample times, is discussed. There are several ways this can be performed:

33

Simultaneous optimization, i.e. doses and sample times optimized at the same time (T,D).

Sequential optimization, i.e. optimization of doses first with fixed sam-ple times, then the optimal doses used to optimize the sample times (D|T). Alternatively, sample times can be optimized first with fixed dos-es, then the optimal sample times used to optimize the doses (T|D).

Iterative sequential optimization where either (D|T) or (T|D) are used but are repeated until there is no difference in the criterion value between it-erations.

All of these approaches can be used with other design variables but, in the examples presented (paper I), continuous doses and sample times were used.

Group size optimization Another important task in population studies is to optimize the distribution of a fixed number of individuals between elementary designs (design groups). For example, the task might be to distribute individuals between a low-dose group, an intermediate-dose group and a group receiving a high dose. This type of optimization is not new; it has, for example, been utilized with the Fedorov-Wynn algorithm [79, 86, 87]. However, the Fedorov-Wynn algorithm is most suitable for continuous designs where fractions of the total number of individuals in each elementary design are optimized. This method can lead to ODs with fractions of individuals in an elementary design (obvi-ously not physically possible). Another approach when performing group size optimization is to take advantage of the population FIM properties. The group size algorithm can be formulated as (paper VIII):

(31)

where maxstud and minstud are the maximum and minimum allowed number of subjects in the study, mini and maxi are the minimum and maximum allowed number of subjects in design group i, Ng is the number of elementary de-signs, gi is the number of subjects in elementary design i, and FIM1,i is the

1,1

1

,

arg max subject to

min max

min max

g

i

g

N

i i ii

gN

stud i studi

i i i

criterion g q

g

g

FIM

34

Fisher Information Matrix for one subject in design group i. A closer look at equation 31 reveals that the FIM only needs to be calculated once per design group, given that the other design variables qi and the parameters are fixed (or have fixed distributions). This implicitly makes the algorithm sequential with respect to optimization of other design variables. If a few elementary designs with a few subjects are optimized, an extensive search over all com-binations of group assignments is possible, i.e. a global search (GS). Howev-er, if the number of combinations of group assignments becomes too large, an approximated global optimization can be used; in this work, this is an algorithm referred to as the fast approximation (FA). In the FA algorithm, the subjects are iteratively assigned to the most informative elementary de-sign, i.e. the design group that increases the criterion the most. A group size optimization is demonstrated in paper VIII using an E-max model with mul-tiple dose groups.

Design criteria Several OD criteria were used in the papers covered by this thesis, all of which have different pros and cons. The most commonly used criterion was the D-optimal criterion for local ODs but global criteria were also used, see table 1. The D, Ds, A, ED and EDs criteria are well described in the literature [50, 87] but the criteria/methods used in papers III and V are less common.

Table 1. Design criteria used in the thesis

Criteria Papers

Local designs D criterion I, IIIa, VI, VII Ds criterion IIIa, IV, VI A criterion VIII Global designs ED criterion I, II, VI

EDs criterion III

Average D-efficiency V a A global approach over the individual parameter space but not over the population parame-

ters.

Although standard criteria were used (D, DS) in paper III, they were calcu-lated over an average individual FIM combined with a prior from the popu-lation parameters

(32) MAP iE -1FIM FIM

35

where the average is taken over the entire population of individual parame-ters i and the matrix -1 is a fixed prior, i.e. the IIV from the population model. FIMMAP indicates that this is the Fisher information matrix for the estimator used for maximum a posteriori estimation which is suitable for e.g. therapeutic drug monitoring [13, 88, 89]. Optimizing the FIMMAP will give an optimal individualized design in an average sense where the information given by the population model is still acknowledged.

In paper V, an average criterion over a discrete parameter distribution was considered. However, a standard approach like the ED criterion could not be used because the parameter values in the distribution were very different from one another and an expectation over determinants, given extremely different parameter values, would outweigh the most informative parameter sets. Instead, a penalized (or normalized) ED criterion was used where, for each set of potential parameter values, the determinant was penalized by the D criterion from the D-optimal design for those potential parameters. Fur-thermore, the criterion only included the parameter sets that had an expected relative standard error (RSE; see equation 34) less than 50% with each pa-rameter specific D-optimal design. The criterion can be formulated as:

(33)

where qi* is the OD for the parameter set i, p is the number of parameters in

the model and n is the number of parameter sets in the discrete distribution. The OD under this criterion is identical to the OD using the average efficien-cy approach [60] except for the RSE restrictions.

Optimized design variables When optimizing a study, any design variable that is possible to change in clinic can be included in the optimization. In this work, several design varia-bles that are not commonly included in an OD have been considered. The reasons for not including such variables could include that they are clinically unfeasible, but also that it is computationally harder and that the necessary

1/

*1

*

i

,arg max

,

with

arg max ,

subject to

RSE , 50%

i

i

pn

i

qi i

i i iq

q

q

q q

F

FIM

FIM

FIM

36

features are not available in most of the standard NLMEM OD tools. The complete list of design variables investigated in the different papers is pre-sented in table 2.

Table 2. Design variables optimized or evaluated

Design variables Paper

Continuous Discrete

Dosesab and sampling timesab I

Start timeb and stop timeb of study Samples per individualsa II Dosesb and sampling timesb, time of second doseb, infusion durationb

Administration ordera, number of samples per dosea

III

Dosesab, startab and stop timeab of infusion, infu-sion durationab, sampling timesa

Exclusion of labeled glucosea IV

Concentrations and sampling timesb V Sampling timesb VI Start timeb and stop timeb of study Dosesb VII Dosesa, start time of studyb, fraction of individu-als in elementary designsb

Group sizesa, dosesb VIII

a The design variable was optimized sequentially with respect to other design variables. b The design variable was optimized simultaneously with other design variables.

Paper II demonstrates an example where a disease progression (DP) model was used to optimize the start and stop times of the treatment of a drug. This was done with three disease progression models with different drug effects: a symptomatic effect, a disease modifying (protective) effect, and a combina-tion of both symptomatic and protective effects. A forth scenario was also investigated, where all three models were combined into a multi-response model where each response corresponded to one of the models described above. The hypothesized trial involved one design group with 200 individu-als. 13 observation times were fixed, equally spaced between 0-12 time units, and a global criterion (ED criterion) was used, assuming that the natu-ral disease progression parameter was less certain than other parameters because of the lack of a placebo arm. This was implemented with a normal distribution around that parameter, with 15% uncertainty.

Evaluating designs - models and methods In most of the included papers in this thesis, common PK-PD models were used to demonstrate the performance of the design and to compare the pre-dicted precision with the empirical precision from a simulation study. Only the models in papers III-V were developed (previously by others [90-92]) from observed data. Table 3 lists the models used in this thesis.

37

Simulation- and estimation-based diagnostics Often when a design is optimized, a confirmation diagnostic is performed (papers II, IV, V, VI, VII and VIII). This is done with a reference design (usually the OD) to see whether the empirical precision is similar to the pre-cision predicted by the FIM with the reference design. The empirical preci-sion is computed using simulations and re-estimations, which also allows for investigations of the bias, given the reference design. This can be important, because the OD theory addresses parameter precision and not parameter bias. Typically a simulation and re-estimation procedure is performed with at least n=100 (most often n=500 or n=1000) datasets simulated and then re-estimated. After the re-estimation, summary statistics are calculated. In this thesis, the NLMEM software NONMEM [34] was used in all simulations and re-estimations with the FO, FOCE or Laplace methods with or without interactions between the residual error model and the individual parameters.

Table 3. Models used to evaluate and optimize designs

Model type Models Papers

PK 1 compartment, IV admin-istration, nonlinear elimina-tion

I, V

1 compartment, IV admin-istration, linear elimination

VI, V

3 compartments, oral or infusion administration, linear elimination

III

7 ODEsa divided into 3 sub-models of insulin, glucose and hot glucose

IV

PD Linear disease progression

(with and without dropouts) with slow- or immediate-onset of drug effect

II,VII, VIII

Logistic regression VII Poisson model VII Logistic regression with Markov elements

VII

Emax model VIII PK-PD 1 compartment, IV PK with

linear elimination linked with an Emax model

I

1 compartment, oral PK with linear elimination linked with an Emax model

I

a ODE = ordinary differential equation.

38

Three different summary statistical methods were used (all relative to the true parameters used to simulate data): the relative standard error (RSE), the mean relative error (MRE) and the root mean squared error (RMSE)

(34)

(35)

(36)

where n is the number of simulated and re-estimated datasets. The RSE measures the precision of the parameters, the MRE represents the size of the bias and the RMSE is a combination of both bias and precision. For an unbi-ased estimator, the RSE, according to equation 1, is asymptotically propor-tionally linked to the square root of the diagonal inverse of the FIM, the MRE is 0 and the RMSE is equal to the RSE.

Applied optimal design All of the methodology and examples in this thesis can be used to optimize applied models for drug treatment or drug development. Three of the models in this thesis were developed with real data, with the primary aim of the OD of these models to either reduce an existing design or to develop a more in-formative design without increasing the number of samples or number of subjects in the study.

Drug compound screening experiments (Paper V) A Michaelis-Menten (MM) model with nonlinear elimination or an exponen-tial decay model (EXP) [93] was used to estimate the maximum velocity of the metabolic reaction Vmax and the Michaelis constant Km or, alternatively, to estimate the intrinsic clearance CLint (where CLint=Vmax/Km) of drug com-pounds in a screening experiment. These parameters describe the metabolic stability of the compound and are important when deciding whether it will be progressed in the drug development process. Screening experiments are highly standardized and a common design (STD-D) is to use a start concen-tration of 1 M and sample every 10 minutes for a 40 minute experiment.

ˆest

true

SDRSE

,1

1 ˆn

est i trueitrue

MREn

2

,1

1 1 ˆn

est i trueitrue

RMSEn

39

The samples are taken in triplicate, i.e. 15 samples per compound. Only the EXP model with the CLint parameter can be considered for this type of de-sign because the enzyme kinetics are often linear at such low concentrations. However, allowing the design to change the start concentration C0 enables use of the MM model. An OD was computed to allow usage of the MM model. The design space for the proposed screening experiment assumed incubation with 15 replicates of each compound with an initial concentration C0 of between 0.01 and 100 M, and a sample time between 0 and 40 min. A discrete distribution of 76 published Vmax and Km values from different com-pounds [94] was used to improve the robustness of the design against differ-ent parameter values. The robust criterion used is explained in equation 33. Compounds with an expected precision, under the D-optimal design, corre-sponding to a RSE (equation 34) of CLint > 50% were excluded from the discrete distribution and this resulted in a total of 52 compounds in the opti-mization. The design, initial concentrations C0, and sample times were opti-mized and labeled G-OD. A pragmatic design more suitable for the laborato-ry (OD), adjusted from the G-OD, was also evaluated. After the optimiza-tion, a simulation study was executed to check the performance of the OD compared to the standard design (STD-D) with both the MM and EXP mod-els. The simulation study (n=500) was executed for each compound, model and design.

Intravenous glucose tolerance test (Paper IV) Another applied example used a model for the intravenous glucose tolerance test (IVGTT) [90]. This model is a semi-physiological model of a complex system (insulin-glucose regulation) and is therefore quite intricate and in-volves several linked ordinary differential equations (ODEs). The model is able to describe several parts of this physiological system and could there-fore be useful when optimizing studies for new anti-diabetic treatments. The model consists of 7 linked ODEs, divided into three subsystems, a glucose model, an insulin model and a hot (labeled) glucose model. The sub-models are linearly or nonlinearly interconnected and several feedback loops exist in the system. During an IVGTT, concentrations for all sub-models were col-lected and a standard rich design (SRD) was used to take 30-34 samples/sub-model over 240 minutes. To get reasonable run times, this design was initial-ly reduced to 10 samples/sub-model, i.e. the standard sparse design (SSD). The SSD was optimized with the Ds-criterion (excluding residual error pa-rameters) and included clinical restrictions of the glucose concentrations. The SSD was optimized with respect to several different scenarios (see table 2) and compared to the SRD and SSD.

40

Therapeutic drug monitoring of ciclosporin (Paper III) A population model was developed from all the pediatric renal transplant patients in Finland [92]. The data were collected during 1998-2005 from 162 patients who received an infusion (IV) of ciclosporin and 89 who received an oral dose (PO); 77 of these patients received both IV and PO ciclosporin. The sampling design was intensive: 12 samples within 28 h for IV and 10 samples within 24 hours for PO, on different occasions. The aim of the OD was to reduce the design so as to give IV and PO doses on the same day, within 8 hours of each other, and to have a maximum of three samples per individual. This while maintaining enough information about the individual parameters so that they could be used for post-treatment dose predictions and therapeutic drug monitoring (TDM). The FIMMAP (equation 32) was used to optimize individual parameter estimations, given the population model. Sev-eral design variables where optimized (see table 2) with the Ds- or D-criteria over the FIMMAP. Three different scenarios were investigated: assuming that the individual clearance CL and bioavailability F were the parameters of interest, assuming that all individual parameters except the individual resid-ual error parameters (MAP 6) were of interest, or assuming that all individu-al parameters (MAP All) were of interest. For the CL, F and MAP 6 scenari-os, the parameters excluded were treated as uninteresting parameters in the Ds-criterion. A covariate (weight) was included in the model to alometrically scale the clearance and volume parameters. The average of the FIMMAP was calculated for all the individual parameters (from the population model) who received both doses (i.e. 77 patients), figure 3. The individual weights were sampled, together with the individual parameters from the parameter distri-bution, to keep any correlations (due to model misspecification) between individual parameters and the weights. After optimization, the new designs were compared with the original design with respect to expected parameter precision as well as D and Ds efficiencies.

41

Figure 3. The figure illustrates the 77 patients who received ciclosporin both IV (first) and PO (second), given the discrete individual parameter distribution. In the figure, both doses are within an 8-hour time interval with the second dose adminis-tered 4 hours after the first. However, the real data were collected on different occa-sions for IV and PO and over a longer sampling period (28 h and 24 h).

The optimal design software PopED (paper VIII) All of the optimizations and design evaluations in this work were imple-mented in the NLMEM OD tool PopED (Population Experimental Design). The tool was initially developed by Foracchia et al. [55] but was adapted to MATLABTM and FreeMat [95] with a new graphical user interface (GUI) developed in C# .NET. The aim of the tool is not only to implement the methodology and optimize models as presented in this thesis but also to make the OD methods available to population modelers in general. Another purpose of the software is to have tools for pre- and post-processing of the OD, e.g. for calculating sampling windows to make designs more clinically relevant, translating efficiency to a more meaningful value, such as the num-ber of study subjects needed, plotting criterion surfaces of the design varia-bles to visualize the sensitivity of the design, etc. Several of the models used in this thesis have very long run times due to e.g. numerous parameters, ODEs, robust criteria, more advanced linearizations, FIM based on simula-tions, etc. A natural way to reduce the run time for these models is to paral-

0 2 4 6 80

1

2

3

4

5

6

7

8

977 discrete patient profiles

Time (h)

Cic

los

po

rin

co

nc

en

tra

tio

n (

ln[m

cg

/L])

42

lelize the tool to take advantage of multi-core processors as well as computer clusters.

Software architecture The structure of the software consists of two different modules: the script library (written in MATLABTM/FreeMat) and the GUI. The script library is used to calculate the FIM and to optimize using the FIM, while the GUI defines the models, parameters, design restrictions, etc. The GUI also con-tains a model library for common models and examples, demonstrating how to carry out more advanced ODs. The basic idea behind the separation of the GUI and the script library is that any of these modules can be substituted, e.g. to a module written in another programming language. To allow for this, a standardized protocol for communication between the modules was devel-oped using the extensible markup language (XML).

Efficiency translation Efficiency is a widely used measurement when comparing two designs, e.g. q1 and q2. The expression for efficiency is dependent on the criterion used but it can be interpreted as the additional design “effort” that is needed for the less informative design q1 to be as efficient as the more informative de-sign q2. In population modeling, design “effort” is not easily interpreted be-cause the “effort” could be to include more samples per subject, more sub-jects, more cohorts, etc. However, a population consists of subjects and it is therefore natural to relate the efficiency measurement to the additional (or fewer) number of subjects needed to achieve a given amount of information. The group size algorithm was used (equation 31) to optimize the group as-signment and also to find the worst possible scenario (efficiency) given a number of subjects. This can also be done for a number of different subjects to get a closed set (graph) of possible efficiencies versus subjects. This is a tool available in PopED; an example of the tool for the A and D criteria was implemented using an Emax model with three dose groups.

Using PopED in parallel A parallel implementation of the software PopED was developed using a Message Passing Interface (MPI) system or the built-in Parallel Computing Toolbox [96] (PCT) in MATLABTM. For the PCT implementation, the paral-lel computing is done automatically within the toolbox. With the MPI im-plementation, the message handling and the calculation of the FIM must be implemented separately. Briefly, this was done by 1) compiling the FIM calculation into a C-shared library with the Matlab Runtime Compiler, 2) linking the C-shared library with an MPI implementation that handles inter-

43

process communication, using a C++ compiler. These two steps created an executable that could distribute jobs and evaluate a number of different de-signs in parallel by running the executable on different computer nodes or cores. The executable works both as a job manager (JM) distributing jobs and a worker calculating the FIMs. The parallel implementation was evalu-ated using a model with long and short run times over 4, 8 and 34 processing units. This was done in two different scenarios: 1) an inactive JM, only dis-tributing jobs, and 2) an active JM, distributing jobs and calculating the FIMs. The parallel performance was calculated as the parallel efficiency relative to an ideal parallel program where message passing and process initialization takes no time

(37)

where exp is the experienced speed increase when executing in parallel

(38)

where tserial is the time for executing the calculation on one processing unit and tparallel is the parallel executing time. max is the maximal possible speed increase, i.e. [3, 7, 33] for an inactive JM and [4, 8, 34] for an active JM in the example above with 4, 8 or 34 processing units.

exp

maxrelEff

expserial

parallel

t

t

44

Results and Discussion

Performance of the extended population FIM

Linearization around the mode The expected RSE of the parameters from the FIM when using the lineariza-tion around typical values (FO), individual samples (FOCE) and the ex-pected mode (FOCEM) are presented in table 4. The results show that the linearization around the expected mode predicts the empirical precision from NONMEM better than the other linearization methods. The sum of the abso-lute RSE differences from the empirical precisions is 27.4% for FIMFO, 36.7 % for FIMFOCE and only 11.0% for FIMFOCEM. The design surfaces also dif-fer between the linearization methods (paper VIII). The OD for FIMFO is a dose1 that is not equal to dose2 while, in the FIMFOCE and FIMFOCEM, dose1=dose2 (paper VIII). The design is also predicted to be less sensitive around the OD with FIMFOCEM than with FIMFOCE.

Table 4. Evaluations with different FIM approximations

Methoda RSE(%)

EC50 2EC50

2

FIMFO 2.64 14.50 10.00

FIMFOCE b

1.86 18.50 9.46

FIMFOCEM b

2.52 18.60 9.43

NONMEM c 2.54 17.16 9.26

a Emax model with (30, 30) design b 1000 individual samples used c 1000 sim/est with FOCE method

Individual shrinkage can occur when calculating the mode in estimation [97], i.e. the individual deviations ( , ) from the typical values shrink to-wards zero. Shrinkage can also occur when using the expected mode. How-ever, the FIMFOCEM will, in general, experience more built-in shrinkage than that present during estimation, figure 4. This is because the mode [empirical bayes estimates (EBEs)] in estimation will depend on the data and the popu-lation parameters used to calculate the EBEs in estimation are the maximum

45

likelihood estimates given the data (not the values used to simulate the data). The MLEs in estimation are naturally spread due to the uncertainty and bias of the estimator. When calculating expected EBEs in the FIMFOCEM method, with the data from the individual median, the population parameters are as-sumed to be known MLEs and will therefore shrink even more than the es-timated EBEs. In spite of these inaccuracies, in the example shown, which has massive shrinkage, the FIMFOCEM still performs better than the other methods.

Figure 4. Simulated individual values versus Empirical Bayes Estimates (EBEs) estimated during parameter estimation. The grey stars represent the expected EBEs calculated from the median data of the simulated individual values. The white dotted line is a smoothed line of the estimated EBEs.

Linearization with serial correlation When using serial correlation between the residuals, the clustered sample times seen in OD without serial correlation spread out, figure 5. The NOAR example has two replicated support points. The DS, FIXAR and NOFIX designs, which incorporate serial correlation, all spread out the clustered support points and have four support points instead of two. The OD in figure 5 was also used to calculate the empirical performance (RMSE) in a simula-tion study. The results, presented in figure 6, show that the designs without autocorrelation (NOAR or NOARb) perform slightly better for both the SNEN and the SNEA scenarios (serial correlation not present in the simulat-ed data). However, when serial correlation is present in the data but not in-cluded in the estimation (SAEN), the optimal designs including AR are much more informative than NOARb.

46

Figure 5. Four sample D-optimal designs using the different scenarios. The line represents the typical value profile of the model. Note that the y-axis is shifted so that the concentration is 0 at the NOAR y-value.

Surprisingly, the SAEA approach (simulate and estimate with AR) is slightly better with the NOARb design (AR is excluded when calculating the OD). This result could be explained by the different approximation methods used when optimizing (FO) and estimating (FOCE with interaction). To confirm this hypothesis, a simulation study using the FO method was also performed and the results showed that the DS, FIXAR and NOFIX outperformed the NOARb design when the FO method was used in estimation (paper VI).

The most realistic scenario is probably the SAEN approach (simulate with AR and estimate without AR) because the data (by nature) are likely to be correlated but the correlation is rarely incorporated in estimation. In this scenario, accounting for the serial correlation massively improves the esti-mation performance (RMSE). Support points for NLMEM ODs de-coalesce if serial correlation is included in the model used to compute the design. Here, an optimal design for an NLMEM was combined with an analytic expression for the AR(1) model. This approach is a fast way of incorporating serial correlation, compared to methods that incorporate SDE to describe serial correlation, since SDE must be implemented in terms of a differential equation system and is thus slower to optimize [70, 98]. Another possibility is to use the exchange algorithm when optimizing with serial correlation [99]; however, this algorithm is de-rived from continuous design theory which assumes a convex optimization problem. This is not the case when including serial correlation due to the possibility of singular information matrices and it is therefore not straight-forward to use the exchange algorithm.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

NOFIX

FIXAR

DS

NOAR

1

3

5

7

9

11

13

con

cen

tra

tion

(m

g/L

)

time (h)

22

47

Figure 6. The RMSE from the different AR simulations/estimations scenarios given the optimal designs.

Simulation-based Fisher Information Matrix The ODs with the simulation-based FIMs (FIMMC, FIMLAP) were different from those with the analytic approximation to the FIM (FIMAnalytic) for the dichotomous model, figure 7. However, the differences were small and the D criterion surface looks very similar for the different FIM approximations. Similar results, i.e. small differences between the methods, were obtained for the count model (paper VII).

0

10

20

30

40

50

60

70

80

90RMSE(%) SAEA

RM

SE

(%)

CL Vt½ 2

CL2V

2prop

0

50

100

150

200

250

300

350RMSE(%) SAEN

RM

SE

(%)

CL V2CL

2V

2prop

0

10

20

30

40

50

60

70

80

90RMSE(%) SNEA

RM

SE

(%)

CL Vt½ 2

CL2V

2prop

0

10

20

30

40

50

60

70

80

90RMSE(%) SNEN

RM

SE

(%)

CL V2CL

2V

2prop

NOFIX =153.50%

FIX =152.57%

DS =153.50%

NOARb =147.45%

NOFIX =143.68%

FIX =143.85%

DS =143.69%

NOARb =127.37%

NOFIX =137.99%

FIX =136.04%

DS =138.00%

NOARb =120.56%

NOAR =121.55%

NOFIX =123.54%

FIX =126.32%

DS =123.57%

NOARb =665.82%

48

Figure 7. The D-design surfaces using different methods when calculating the FIM. There is more information in the lighter areas and less in the darker areas. The D-optimal designs for FIMMC and FIMLAP indicate a dose of 0 and 0.5 units while, in FIMAnalytic, the D-optimal design indicates one dose of 0 and another dose of 0.46 units.

Including dropouts changed the start and stop times of the disease progres-sion study, figure 8. None of the scenarios including dropouts had an OD similar to the no dropout (ND) scenario. Using the constant hazard dropout model, where the number of dropouts is not dependent on the disease status, shifts the designs from a start time at 0 (ND) to a start time at 4 (D, ID). A similar design is obtained when the number of dropouts depends on the dis-ease status but no additional information from observing dropouts is includ-ed in the FIM (D). When additional information about the disease status parameters are obtained by observing the dropouts (DD, ID), the design shifts to a start time of 1 and a stop time of 9. In these scenarios, the level of information in the designs is also more sensitive to the start and stop times of the study.

Although analytic expressions for survival models exist for NLMEM FIMs using e.g. frailty models [100], these expressions are not generalizable when it comes to including random effects in the survival model. In contrast, the method presented in this thesis, with a higher computational cost, can include any regular survival model connected with other responses such as disease progression. As seen in the results (figure 8), dropouts clearly change the OD and are therefore important to acknowledge when designing a trial. This has already been addressed when estimating pharmacometric models using mixed effects disease progression models [101] and it is therefore nat-ural to include dropout when designing experiments with OD techniques. This example is specific with a time-to-dropout model but can be extended to e.g. repeated time-to-event models or other types of time-to-event models within the same methodology.

49

Figure 8. Disease progression model with different dropout scenarios. A white ring indicates the optimal design.

Two FIMs based on simulations were developed (paper VII) and imple-mented in PopED (paper VIII). Figure 7 shows that the simulation-based FIMs are similar to the standard analytic approximation of the FIM and seem to perform well in the complete design region. For the dichotomous model, the analytic expression (which is much faster to compute) can be used instead of the simulation-based FIMs. However, when more complex models are used it might be hard to derive a proper link and variance func-tion so as to be able to use the FIMAnalytic. Instead, the FIMLAP or FIMMC could be used, since the only knowledge needed is to be able to simulate data from the individual probability model. Furthermore, if the joint density (equation 6) is far from a Gaussian distribution, FIMMC without normality assumptions can be used.

Optimization using the FIM

Simultaneous versus sequential optimization In figure 9, it is evident that the OD differs for different optimization meth-ods. The sample times are the same in all the methods for PK sampling, but the sequential approaches T|D and D|T have different PD samples as well as different optimal doses.

0 5 100

2

4

6

8

10

12No dropout (ND)

0 5 100

2

4

6

8

10

12Constant hazard (D)

0 5 100

2

4

6

8

10

12Constant hazard (ID)

0 5 100

2

4

6

8

10

12DP dependent hazard (D)

0 5 100

2

4

6

8

10

12DP dependent hazard (DD)

0 5 100

2

4

6

8

10

12DP dependent hazard (ID)S

tart

tim

e

Stop time

50

Figure 9. The optimal designs for five elementary designs using different optimiza-tion methods. The PK optimal sampling times are seen to the left. The PD sampling times are seen in the middle and the optimal doses are to the right. Two PK samples were available per group, which gives a total of 10 PK samples per approach. Simi-larly, three PD samples were taken per group, which gives a total of 15 PD samples per approach. Finally, one dose was given per group, which gives a total of 5 doses per approach.

Another example, figure 10, shows that the increase in RSE per parameter can be as high as 14% for the worst case scenario when using the dose-first (D|T) approach. The results are not surprising because the design space is restricted when using sequential optimization and hence local optima might be found instead of global optima. In the examples in paper I, the worst sce-nario was always the dose-first (D|T) scenario. This is because the design space with respect to the dose is less influential than the sample times, in this case. For this reason, the sequential approach with the sample times first (T|D) is preferable to (D|T) in these examples but not in general. Paper I also illustrates that it is possible to optimize several continuous design variables, which might be important when designing a new study.

Figure 10. The increase in relative standard error as % per parameter of the sequen-tial approaches D|T, T|D to the simultaneous (T,D) approach.

Group size optimization and efficiency translation Figure 11 shows an efficiency translation curve using the GS and FA algo-rithms, equation 31. The A efficiencies are the same for both GS and FA.

0 0.2 0.4 0.6 0.8 1

PK times (h)

5 5

5 5

5 5

0 0.2 0.4 0.6 0.8 1

PD times (h)

4 2 2 7

6 9

4 1 10

0 1 2 3 4 5

Dose (mg)

3 2

2 2 1

2 2 1T,D

T|D

D|T

0

5

10

1 group 2 groups 3 groups 4 groups 5 groups

Incr

ease

in

RS

E/p

aram

eter

(%

)

3a D|T

3a T|D

51

The D efficiency differs for the low D efficiency line between GS and FA (up to ~370 individuals), with the largest difference in ~71 subjects at an efficiency of ~82%. The speed difference between GS and FA is significant; the optimization time for the reference design in figure 11 is ~1 second with the FA algorithm and ~1000 seconds with the GS algorithm.

Figure 11. D and A efficiency translation and group size optimization using the Global Search (GS) or the Fast Approximation (FA) method. The reference design (100% eff), using 250 individuals, is marked with a red star. The black dotted line represents the efficiency when using the same proportions of group sizes as the reference design [80, 100, 70]. The A and D ODs using 400 individuals are marked as black rings.

The group size optimization algorithms, FA and GS, provide a fast way to optimize group assignments of subjects between elementary designs, with only one evaluation of the FIM for each elementary design. The efficiency translation plot, using the group size optimization algorithm, is a useful tool for translating efficiencies into numbers of subjects, where the worst and best case (OD) scenarios are presented. The tool can also be used as an indi-cation of the importance of group size optimization for a given model. The tool is available in the OD tool PopED (paper VIII).

Optimization using other design variables In paper II, optimization of design variables other than the standard (doses, sample times, group sizes or group patterns) was demonstrated. In this paper, the sample times were fixed and the trial start and stop times were opti-mized. As seen in figure 12, the optimal start and stop times vary with the

100 200 300 400 500 600 700 800 9000

50

100

150

200

250

300

350

400

Effi

cien

cy [%

]

Number of Individuals

Same proportionsD-eff FAD-eff GSA-eff GS,FAReference design

52

model used. The symptomatic-effect model is the least sensitive design while the protective-effect model has a more narrow informative design re-gion.

Figure 12. The ED criterion surface for the different models versus start and stop times of the study. Whiter areas indicate more information.

The importance of study duration versus the number of samples per individ-ual is seen in Figure 13. The figure uses the protective and symptomatic model and shows that it is more important to increase the study duration than to include more samples per individual, in this example. Paper II shows the importance of optimizing other design parameters, in this case the start and stop times of a disease progression trial. The optimization used a global criterion (ED) which enables optimization with uncertainty around parameters. The paper also shows that it could be more important to increase the study duration than to add more samples per individual in order to be able to capture the disease progression parameters.

53

Figure 13. The log ED criterion surface of different study durations versus the num-ber of samples per individual for the protective & symptomatic model.

Applied optimal design

Optimizing in vitro drug compound screening experiments In paper V, a drug screening experiment design was optimized using an MM model instead of the usual EXP model. The standard design STD-D, the optimal design G-OD and the adjusted optimal design OD are presented in table 5. The penalized ED criterion value (OFV) is also shown in the table. If the design is as informative as the D-optimal design for each compound, the OFV will be equal to the number of included compounds, i.e. 52. The G-OD has an OFV of 28, which means that the general design is on average slightly more than half as informative as the compound-specific designs. This can be compared to the STD-D design, which is not suitable for estimating with the MM model and, hence, has an OFV of 0.011 which roughly corresponds to an average of 0.02% of the information in the individual compound designs.

05

1015

2025

0

5

10

15

20

25

0

5

10

15

20

25

30

35

Samples/Individuals

Length of Study Period versus Samples per Individual

Study Length

Lo

g(d

et[F

IM])

54

Table 5. The standard design (STD-D), optimal design (G-OD) and adjusted optimal design (OD) for the MM model.

Sample STD-D, OFV=0.011 G-OD, OFV=28 OD, OFV=17

C0 ( M) Time (min) C0 ( M) Time (min) C0 ( M) Time (min)

1 1 0 0.427 40 0.45 40 2 1 0 0.427 40 0.45 40 3 1 0 0.427 40 0.45 40 4 1 10 0.427 40 2.5 5 5 1 10 0.433 40 2.5 40 6 1 10 0.436 40 18 5 7 1 20 2.38 40 18 40 8 1 20 17.6 40 30 5 9 1 20 30.3 40 30 40 10 1 30 62.1 40 60 40 11 1 30 92.5 40 90 40 12 1 30 93.0 40 90 40 13 1 40 100 40 100 40 14 1 40 100 40 100 40 15 1 40 100 40 100 40

The performance of the different designs when estimating either the CLint parameter using the EXP model or the Vmax, Km parameters using the MM model and then calculating CLint is presented in figure 14. The results show that the OD is favorable in all examples (99%) for the precision (RSE) of the CLint parameter. Although the OD adds some additional bias (RMSE) for compounds with an elimination half-life of ~70-500 minutes, it is still as good or better in 78% of the compounds.

55

Figure 14. The upper panel shows the lowest RMSE for the CLint parameter with either MM or the EXP model with the optimal adjusted design (OD) and the stand-ard design (STD-D). A circle indicates that both Vmax and Km could be estimated with an RMSE < 30%. The lower panel shows the lowest RSE for CLint with both OD and STD-D.

In figure 14, the best performing model (lowest RSE and lowest RMSE) for fitting the data to either the EXP or MM was used. In reality, the parameters are unknown and it is therefore hard to discriminate between the different models. However, it is possible to fit both models to the data and look at the usual model discrimination measures, for example the Akaike criteria, the log likelihood ratio criteria or the observed FIM (FIM given a dataset).

56

Moreover, simultaneously fitting both models will not significantly increase the screening run times because the fitting is in general very fast.

Paper V shows the benefits, i.e. better precision and bias and the ability to estimate nonlinear enzyme kinetics, of using OD in the early stages of drug development, i.e. drug compound screening experiments. Optimization of the intravenous glucose tolerance test The results from the optimization of the three responses in the IVGTT are seen in figure 15. The SRD (full), SSD (reduced) and OD (samples only) are shown together with the prediction interval from the simultaneous optimiza-tion of the glucose and insulin doses. Not surprisingly, the optimal doses are higher than the SSD doses.

Figure 15. Simulated concentration-time prediction intervals (1000 simulated indi-viduals) of glucose, insulin and hot glucose using standard doses of glucose and insulin (solid lines) and the doses given when optimizing on both glucose dose and insulin dose (dotted lines). The optimal sampling scheme when optimizing the sam-ple times only, the reduced design without optimization and the full design without optimization are presented in the lower panel. The sampling time at 240 min for the optimal sampling scheme is replicated.

The results of the optimizations are presented in Table 6, where the opti-mized design parameters are presented along with the efficiency calculated from either the SRD or the SSD and the mean predicted RSE from the FIM. If only one type of design parameter is considered, optimizing the sample times is the most informative, followed by the insulin dose. The optimization scenarios of the SSD gave the same (or higher) efficiency as the SRD in three scenarios, even though the numbers of samples were reduced to a third

0

500

1000

1500

2000

0

500

1000

1500

Con

cent

ratio

ns (

mg/

DL)

0

100

200

300

400

0 50 100 150 200 250

Full

Reduced

Optimal

Time (min)

90% PI - Standard design

90% PI - Optimal design

Glucose

Insulin

Hot Glucose

57

of those in the SRD. The three most informative optimization scenarios were: 1) insulin, glucose doses simultaneously using the SSD, 2) insulin dose at the optimal sample times (sequentially) and 3) sample times. The efficiency was almost the same as the non-optimized SSD scenario when optimizing sample times restricted to within 2 hours. The results also show that including labeled glucose is important. Excluding labeled glucose re-duced the efficiency of the SSD by almost 30%. The efficiency was less than that of the SRD (which is used in clinics) in most of the optimization scenar-ios investigated. However, the OD uses only a third of the samples and the predicted mean RSE for those scenarios was still quite low and similar to the SRD predicted mean RSE. It might therefore still be valid to use one of the optimized SSD designs with lower efficiency than the SRD.

The optimization of the IVGTT demonstrates that OD can be used to op-timize complex semi-physiological NLMEMs. This applied example opti-mized sample times as well as other design variables simultaneously or se-quentially and shows that the efficiency of a design used clinically can be preserved even with a reduction of the number of sample times to a third of those in the original design. The efficiency can even be increased compared to the original design if the glucose and insulin doses are optimized, still using a reduced number of samples.

Tab

le 6

. Opt

imiz

atio

n sc

enar

ios

wit

h th

e IV

GT

T m

odel

Op

t. p

aram

eter

O

pt.

inte

rval

G

luc.

dos

e (g

/kg)

a In

s. d

ose

(mU

/kg)

In

f.

star

t

Inf.

st

op

Sam

ple

tim

esb

Eff

icie

ncy

(%

) (S

SD

)c E

ffic

ien

cy

(%)

(SR

D)d

Exp

ecte

d m

ean

R

SE

(%

)

Sta

ndar

d ri

ch d

esig

n -

0.3

50

20

25

SR

D

150

100

22.5

Sta

ndar

d re

d. d

esig

n -

0.3

50

20

25

SS

D

100

66.8

26

.1

Insu

lin

dose

1-

300

mU

/kg

0.3

88.4

20

25

S

SD

12

6.4

84.4

21

.5

Sta

rt in

suli

n in

f.

0-23

5 m

in

0.3

50

28.6

+

5 S

SD

10

6.7

71.3

25

.3

Sto

p of

insu

lin

inf.

25

-240

min

0.

3 50

20

34

.8

SS

D

101.

9 68

.1

25.8

Sta

rt, s

top,

dos

e of

in

suli

n in

fusi

on

1-30

0 m

U/k

g,

0.1-

240

min

, 0.

1-24

0 m

in

0.3

83.9

28

.9

31.5

S

SD

13

4.1

89.6

21

.1

Sto

p, d

ose

of in

suli

n in

fusi

on

1-30

0 m

U/k

g,

25-2

40 m

in

0.3

86.9

20

32

.8

SS

D

129.

3 86

.3

21.3

Glu

cose

dos

e 0.

1-3

g/kg

0.

81

50

20

25

SS

D

114.

6 76

.5

28.0

G

luco

se d

ose,

insu

lin

dose

0.

1-10

mg/

kg,

1-10

00 m

U/k

g 1.

42

151

20

25

SS

D

166.

6 11

1.2

20.9

10 s

ampl

es

0-24

0 m

in

0.3

50

20

25

0, 2

.1, 9

.6, 2

0, 2

3.2,

37.5

, 56

.7, 1

12, 2

40, 2

40

150.

7 10

0.6

19.5

10 s

ampl

es, 2

hou

rs

0-12

0 m

in

0. 3

50

20

25

0,

2.1

, 2.1

, 11.

7, 2

3.7,

36,

55

.2, 7

2, 1

20, 1

20

99.6

66

.5

32.8

Exc

lude

hot

-

0.3

50

20

25

SS

D

72.9

48

.7

78.8

Insu

lin

dose

wit

h O

D

sam

ple

tim

es

1-30

0 m

U/k

g 0.

3 95

.2

20

25

0, 2

.1, 9

.6, 2

0, 2

3.2,

37.5

, 56

.7, 1

12, 2

40, 2

40

155.

6 10

3.9

19.1

a B

old

num

bers

indi

cate

opt

imiz

ed d

esig

n pa

ram

eter

b S

SD

indi

cate

s st

anda

rd s

pars

e de

sign

wit

h th

e fo

llow

ing

sam

ples

tim

es: 0

, 2, 1

0, 1

5, 3

0, 4

5, 7

0, 1

00, 1

50, 2

40 m

in

c Eff

icie

ncy

calc

ulat

ed b

ased

on

the

Ds-

crit

erio

n of

FIM

com

pare

d to

the

SS

D

d Eff

icie

ncy

calc

ulat

ed b

ased

on

the

Ds-

crit

erio

n of

FIM

com

pare

d to

the

SR

D

59

Optimization of a drug monitoring design for ciclosporin The expected precision (SD) of the parameters, given the simultaneous OD variables (sample times, doses, infusion duration and time of second dose), are presented in table 7. The results indicate that giving the intravenous dose before the oral dose (IVPO) is slightly less informative than POIV. The effi-ciency is around 50% for all scenarios compared to the standard design. Since CL and F are used to calculate steady-state concentrations, they are perhaps the most interesting parameters. The uncertainty (SD) of these pa-rameters decreases from (0.17, 0.23) to (0.10, 0.11) in the best-case scenario (CL & F design). This can be compared to (0.07, 0.08) with the standard rich design including the prior.

Table 7. Expected precision and efficiency of the optimized designs

IVPO POIV

Prior CL & F MAP 6 MAP ALL CL & F MAP 6 MAP ALL

Efficiency (%)a 49.5 44.6 44.9 50.5 49.4 49.2 -

SD

CL 0.10 0.09 0.09 0.11 0.13 0.13 0.17 V2 Q4 V4 0.33 0.33 0.33 0.38 0.36 0.36 0.40 ka 0.19 0.19 0.20 0.18 0.18 0.18 0.35 Fb 0.11 0.12 0.12 0.11 0.12 0.12 0.23 V3 0.41 0.41 0.41 0.40 0.37 0.37 0.42 Q3 0.30 0.30 0.30 0.31 0.27 0.27 0.32 RUV IV 0.30 0.30 0.30 0.30 0.30 0.30 0.43

RUV PO 0.32 0.32 0.32 0.32 0.32 0.32 0.52 a Efficiency compared to the standard (rich) design with priors b SDF~F*(1-F)*SDEBE(F)

The optimal reduced designs for the CL & F scenarios are presented in table 8. The total dose for IV and PO always reached the dose restriction, i.e. a total dose of 10 mg/kg. The duration of the infusion is doubled for the POIV scenario but the infusion rate (0.74 mg/[kg*h]) is similar for both IVPO and POIV. The second dose is given one hour later for the POIV than for the IVPO and both designs use the full allowed sampling interval (up to 8 hours). For the pre-kidney transplant dose of ciclosporin, the number of samples was reduced from 22 to 6. The POIV design is slightly more favorable than the IVPO design because similar fasting conditions to those in the original de-sign could be used before the oral dose. Previously, the intravenous dose and the oral dose were given on two different occasions but the OD restricted the time of dosage to both within 8 hours on the same day. The 8 hour restriction was used to ensure that the same nurse gives the treatment and that the chil-dren are not hospitalized overnight. Even with these strong restrictions, the efficiency of the OD is 50% compared to the original design, still with a minor precision loss in the parameters, especially for the most important CL

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and F parameters. The ODs were calculated with covariate data from the entire existing population of transplantation children in Finland (from 1998 to 2005) and are therefore likely to reliably represent new renal patients. The ODs were clinically adjusted by sampling and dose windows (paper III) which provided a clinically feasible design with an efficiency reduction within 10%. The designs presented are suitable for clinical therapeutic drug monitoring and will be less demanding for both patients and medical staff.

Table 8. Optimal designs when CL & F are treated as parameters of interest

Design IVPO POIV

Original design CL & F CL & F

Sampling times IV h (TADa) 1.92, 3.60, 3.60 0.01, 3.12, 3.20 0, 2, 4, 5, 6, 7, 8, 10,

13, 15, 20, 28

Sampling times PO h (TADa) 0.70, 3.82, 4.38 0.64, 1.12, 4.64 0, 1, 2, 3, 4, 6, 9, 12,

16, 24

Dose IV (mg/kg) 0.78 1.70 3

Dose PO (mg/kg) 9.22 8.30 10

IV Infusion duration (h) 1.06 2.27 4

Time of second dose (h) 3.62 4.64 Another occasion a Time after dose (TAD) is reported, time after infusion = TAD-Infusion duration.

Optimal design software tool PopED The graphical interface of the tool, which is written in C# .NET Framework 2.0, is presented in figure 16. The GUI communicates with the calculation engine, written in MATLABTM/FreeMat [95], via XML files. The GUI ena-bles usage of PopED and NLMEM optimal design without extensive knowledge of programming or OD theory. All settings enabled via the GUI can also be entered directly in the PopED engine (matlab scripts). A stand-ardized wizard function is available, which takes the user, step-by-step, through all the design and model settings needed to build up the model and design. The wizard also enables the use of models from a pre-specified li-brary.

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Figure 16. The PopED GUI main window.

The PopED GUI also enables surface plots of the design criterion surface (figures 7, 8 and 12) and calculation of efficiency translations (figure 11). The PopED software is open-source and freely available [102].

Parallel implementation The parallelization of PopED (paper VIII) was successfully implemented using the Open MPI [103] project (OMPI) or the parallel toolbox in MATLABTM, Parallel Computing Toolbox © (PCT) [96]. The performance of the OMPI parallelization was investigated with evaluations of 231 de-signs; the results are presented in Figure 17. For a model with long run times, the achieved parallel efficiency was in general high (>80%) but when the message passing is intense, due to short run times, the efficiency de-creases. This is especially evident when the number of computing units in-creases (efficiency < 20% for 34 cores) because the number of messages to be sent between the computing units increases rapidly.

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Figure 17. The efficiency of the parallel implementation of the FIM calculations. The plot is presented with two types of job managers (JMs): i) an active JM, doing FIM calculations or ii) an inactive JM, only distributing jobs, both evaluating with the long (Mlong) and the short (Mshort) run time model.

Parallel implementation is an important part of this thesis because the run times of ODs with complex models or robust designs could be in the order of months. It is possible to reduce the computing times into the order of days, and hence make the OD calculations more usable in practice, if a computer cluster with many computing units is used.

0

20

40

60

80

100

4 cores 8 cores 34 coresRela

tive

effic

ienc

y (%

)Achieved parallel efficiency

Mshort inactive JM Mshort active JM

Mlong inactive JM Mlong active JM

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Conclusions

This thesis has advanced the practical application of OD using nonlinear mixed effects models. Several methodological points in population OD were addressed: (1) optimi-zation over multiple continuous or discrete design variables, (2) the use of robust designs when needed, (3) a faster group size optimization algorithm, (4) interpretation of efficiency, (5) optimizing both continuous and discrete data, (6) improving FIM calculations, both through linearization and by using simulation-based FIMs and (7) the investigation of models with serial correlations. Additionally, in three projects presented here, some of these methodologies were applied to real world problems: (I) optimization of a glucose-insulin provocation study, (II) optimization of an enzyme kinetic study in early drug development and (III) optimization of an individual dose-finding study for pre-transplant treatment with ciclosporin. In examples (I) and (III), the de-signs were massively reduced, although they remained sufficiently informa-tive, to less demanding designs for patients and with a potential cost saving. In example (II), the designs were able to pick up nonlinear enzyme kinetics for drug compounds, which is important for decisions on whether the com-pounds can be further used in the drug development process. All of the methodology and examples in this thesis were developed and im-plemented in the PopED software. This software has been made freely avail-able in order to make the methodology in this thesis accessible to others. A parallelization of the software was also demonstrated; this enables optimiza-tion of more realistic run time-intensive models.

This thesis has shown with applied examples as well as methodology that studies in drug development and drug treatment can be optimized with non-linear mixed-effects OD. This provides a tool than can lower the cost and increase the overall efficiency in drug development and drug treatment. More importantly, it has the potential to decrease human suffering and the number of animals sacrificed in drug development and treatment.

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Populärvetenskaplig sammanfattning

Kostnaden för att utveckla ett nytt läkemedel har under de senaste årtiondena skjutit i höjden. Anledningarna till detta är många och skiljer sig mellan olika läkemedel men vad som är säkert är att det finns ett behov att effektivi-sera läkemedelsutvecklingen och att lägga ner läkemedel som inte är tillräck-ligt lönsamma eller inte har önskvärd effekt tidigare i utvecklingsprocessen. Att utveckla ett läkemedel tar oftast lång tid och flertalet studier om läke-medlets egenskaper genomförs på försöksdjur, på friska frivilliga och hos patienter. Målet med studierna skiljer sig beroende på hur långt i utveckl-ingsprocessen läkemedlet har kommit men exempelvis behövs studier om hur höga doser av läkemedlet som skall tas, hur ofta det skall tas och om läkemedlet är skadligt för vissa patientgrupper. När en studie genomförs så är det under mycket kontrollerade former med en förutbestämd s.k. studiede-sign. Denna design är utformad för att kunna besvara syftet med studien och typiska designval är; Hur många och vilka doser skall ges? Vid vilken tid-punkt skall man mäta effekten och blodkoncentrationen av läkemedlet? När i studien ska läkemedelsbehandlingen börja och hur lång ska den vara? etc. Dessa designval görs utifrån tidigare erfarenheter av läkemedlet eller lik-nande läkemedel, ofta genom att använda en modell för hur läkemedlet för-väntas uppföra sig.

En möjlig teknik för att effektivisera designvalen i en studie och därmed effektivisera läkemedelsutveckling är att använda ”optimal design” (OD) d.v.s. bästa tänkbara design. OD går ut på att beräkna den design som ger maximal information om läkemedlet i en kommande studie under antagandet att modellen någorlunda speglar verkligheten. Tidigt i utvecklingen av ett läkemedel kan modellen vara väldigt osäker men i och med att fler studier genomförts så är det möjlig att uppdatera modellen och således få en modell som blir bättre och bättre på att förutspå hur läkemedlet fungerar. Det finns många sätt att uppdatera modellen men förenklat så anpassas modellen till att på bästa sätt förklara vad som observerats i en studie. Det finns också olika typer av modeller men vissa typer beskriver inte bara hur varje individ reagerar på ett läkemedel utan också hur en grupp reagerar, en så kallad po-pulation. Forskning har visat att populationsmodeller kan vara bättre på att förutspå effektiviteten i en studie än individuella modeller, men de är också svårare att beräkna och oftast mer komplicerade.

Denna avhandling handlar om att utveckla metoder och visa på hur OD kan användas i både läkemedelsutveckling och läkemedelsbehandling, ex-

65

empelvis vid individuell dosering genom att beräkna den bästa designen givet en populationsmodell. Mer konkret har detta visats genom att testa mer okonventionella designval, såsom studielängd, start och stopptid för en stu-die m.fl. Avhandlingen visar också på möjligheterna att designa studier där populationsmodellen hanterar diskreta data, exempelvis antal epilepsianfall, patientbortfall från en studie, om patienten överlever eller inte, etc.

Denna avhandling tar också upp tre exempel på studier som används idag där designen har förbättrats till att t.ex. endast använda en tredjedel av ob-servationerna, att studien genomförs under kortare tid och att designen möj-liggör att mer invecklade modeller kan användas.

Alla exempel och metoder i denna avhandling har utvecklats i programva-ran PopED som är fritt tillgänglig med öppen källkod.

Sammanfattningsvis visar denna avhandling att OD kan användas med såväl fysiologiska populationsmodeller som med modeller för läkemedels-molekyler och kan därigenom effektivisera stora delar av läkemedelsutveckl-ingsprocessen. Således kan kostnaden för att utveckla ett läkemedel reduce-ras och läkemedelsbehandlingar kan bli effektivare och mindre mödosamma för patienter.

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Acknowledgements

This thesis was carried out at the Department of Pharmaceutical Biosciences, Faculty of Pharmacy, Uppsala University, Sweden. I would like to thank all people who have contributed to this thesis and helped me during my past years as a research student. My main supervisor Associate Prof. Andrew Hooker for accepting me as a PhD-student and teaching me about OD. You are always curious when I present my crazy ideas and trying to put things into a scientific perspective. Also thanks for keeping me on track, pushing me to write papers and for your hard effort teaching me in scientific writing. It has been a pleasure to work and share ideas with you during the past years. Visst har vi haft kul?! My co-supervisor Prof. Mats Karlsson for coming up with new ideas and seeing practical applications in a field which is filled with Greek letters and technicalities. It takes a really talented scientist to see these opportunities. Prof. Margareta Hammarlund-Udenaes and Prof. Sven Björkman for creat-ing a stimulating working environment at the department. Magnus Jansson for helping me with the manuscript papers in this thesis. From now on, I will never ever write any inline equations, it’s a promise!

My project collaboration partners - Sebastian Ueckert for co-authoring paper VIII, for proof reading this thesis; your suggestions improved a lot, and for putting up with my knock on the door and “Do you have a minute for a quick discussion?” without kicking me out half an hour later. Erik Sjögren för att du försökte lära mig om enzymkinetik (papper V) och andra obegrip-liga saker. OD är enkelt det är bara det att det ibland inte fungerar som man vill! Mats Magnusson – för värdefull input på papper V. Hanna Silber – för att du är så fantastiskt enkel att jobba med (papper IV), dessutom en hejare på att skriva. Vi saknar dig i UBS. Stefanie Hennig – I really enjoyed our collaboration (paper II, III & VIII). Please continue keeping me updated via Picassa and I hope we get a chance to work together again in the future. Ni andra medförfattare till artiklar inkluderade i denna avhandling; Samuel Fanta, Hans Lennernäs, Ulf Bredberg, Janne Backman och Kalle Hoppu.

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My master students: Joakim Ringblom (jag kallar dig min x-jobbare även om du inte formellt var det), Richard Höglund (papper VI), Anna Svensson (papper VII) & Eric Strömberg (papper VIII) – ni var alla grymma! Förlåt om jag hade en miljard andra saker att göra samtidigt som att handleda er och för att jag förmodligen visade er en massa obegripliga formler utan att förklara dem ordentligt!

My co-workers in other projects, especially: Angelica, Rocio, Camille, Peter Gennemark, Alexander Danis and Warwick Tucker. Thanks to all former and present PhD-students, post docs, researchers, sen-iors, administrators and teachers at the department – You all make this de-partment a great place to work at with your knowledge and enthusiasm. Especially thanks to: Vi som började våra utbildningar ungefär samtidigt – Klas, för våra ändlösa samtal under sena nätter på olika konferenser och för din förmåga att hålla ut LÄNGE på fester så att vi inte behöver känna oss så gamla och tråkiga. Mar-tin, för att du är så generös, både med att dela med dig av kunskap och att bjuda på öl. För ditt medförfattarskap av papper VI. Jag hoppas våra samtal om forskning inte slutar här! Emma, för att man kan fråga dig om allt och du alltid har koll på läget, synd bara att du inte disputerade innan mig! Men då får jag väl åtminstone en chans att bjuda tillbaka. Elodie, for sharing a few bottles of wine and a cheese or two at Lipari ;) and Kinder-surprising me on my birthday. Paul B, for our great tête-à-têtes in the student’s room. Mina rummisar – Bettan, för att du stod ut med mig som rummis och dessu-tom banade väg till min avhandling genom att dela med dig av material och erfarenheter av vad som måste göras, och när. My new rummis Niels, for providing me with chocolate, bringing fika and talking to me whenever I needed a break from this thesis. My other rummisar – Doa, Dominik, Robert, Ulrika, Agneta & Ron; sorry that my phone was always ringing.

Jakob som inte bara handledde mig som x-jobbare utan även kom ihåg att ringa mig en vårdag 2006 för att fråga mig om jag var intresserad av att dok-torera, det är mycket tack vare dig som jag skriver denna mening… Kaffeautomaten i personalrummet i C2-korridoren, utan dig hade denna avhandling inte varit möjlig! Jörgen – för att du alltid frågar hur man mår och hittar på upptåg, det är per-soner som du som gör det så trevligt att arbeta på avdelningen.

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Min familj och mina vänner, speciellt: TPPD-gruppen med respektive (Erik & Stina, Oskar & Frida, Stefan & So-fia, Gustav). Nu när jag är klar ska jag bli mer aktiv, jag lovar. Till Pontus (Mästarnas mästare) & Carolina, Cecilia & Henrik, Robban & Camilla, Erik A - för att vi har så kul när vi träffas. Mina svärföräldrar (Per & Gunilla) – För att ni alltid ställer upp. Jag kan inte tänka mig att det finns några bättre morföräldrar åt lilla H och svärföräldrar åt mig. Hanna & Tom, Emma för att ni alltid fått mig att känna mig som en del i familjen. Noah för att du är en så stolt kusin. Morbror Torbjörn & Lena, Samuel & Hanna, Philip, faster Marie och far-bror Per, mormor Ebba – fortsätt att bo nära så att vi kan träffas ofta. Min kära Mor & Far (eller ska jag säga farmor och farfar) – för att ni alltid har dörrarna öppna och bjuder in till middagar. Hos er är man alltid väl-kommen, det är ovärderligt! Mina underbara syskon med respektive – Chrille & Erika, snälla bli klara med huset snart, nu måste vi börja med vårt, C – jag blir troligtvis den första riktiga doktorn ;) Mirjam & Jesper, tack för att ni försett H med en kusin, snart har även X en jämnårig kusin Josefine & Kalle för att ni är så unga så att ni kan lära mig coola uttryck (och känna mig gammal), vi ses på fekken. Edwin & Malva – för att ni tvingat mig att få frisk luft varenda dag, oavsett väder. X – Om några veckor är du här om du inte redan är det (helst inte den 9/12). Jag ser fram emot att få tillbringa tid med dig, du är efterlängtad. Min älskade Hedvig - aldrig hade jag trott att du skulle innebära så mycket jobb. Tur att det kompenseras med så oändligt mycket mer lycka! Sophia – för att du lärt mig ett av dina favorituttryck ”Bakom varje fram-gångsrik man står en förvånad kvinna” som jag skulle vilja omformulera till: ”Bakom min avhandling står mitt livs stora kärlek”. Jag älskar dig och ser fram emot resten av våra liv tillsammans

Joakim

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