Improved pharmacometric model building techniques172497/FULLTEXT01.pdfplines such as clinical pharmacology, statistics, computer science, classical pharmacology, etc. Indeed, pharmacometrics

ACTA

UNIVERSITATIS

UPSALIENSIS

UPPSALA

2008

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Pharmacy 80

Improved pharmacometric modelbuilding techniques

RADOJKA SAVIC

ISSN 1651-6192ISBN 978-91-554-7275-7urn:nbn:se:uu:diva-9272

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To my mum and dad

This thesis is based on following papers: I Savic RM, Jonker DM, Kerbusch T and Karlsson MO. Imple-

mentation of a Transit Compartment Model for Describing Drug Absorption in Pharmacokinetic Studies. J Pharmacokin Pharma-codyn 34(5): 711-726, 2007

II Østerberg O, Savic RM, Karlsson MO, Simonsson USH, Nør-gaard JP, Walle JV and Agersø H. Pharmacokinetics of Desmo-pressin Administrated as an Oral Lyophilisate Dosage Form in Children With Primary Nocturnal Enuresis and Healthy Adults. J Clin Pharmacol. 46: 1204-11, 2006

III Wilkins JJ, Savic RM, Karlsson MO, Langdon G, McIlleron H,

Pillai GC, Smith PJ and Simonsson USH. Population Pharma-cokinetics of Rifampin in Pulmonary Tuberculosis Patients In-cluding a Semi-mechanistic Model to Describe Variable Absorp-tion. Antimicrob Agents Chemother. 52(6): 2138-48, 2008

IV Savic RM, Kjellsson MC, Karlsson MO. Evaluation of the Non-parametric Estimation Method in NONMEM VI. (Submitted)

V Baverel PG, Savic RM, Wilkins JJ and Karlsson MO. Evaluation

of the Nonparametric Estimation Method in NONMEM VI: Ap-plication to Real Data. (Submitted)

VI Savic RM and Karlsson MO. Evaluation of an Extended Grid

Method for Estimation Using Nonparametric Distributions. (In manuscript)

VII Petersson KJF, Hanze E, Savic RM and Karlsson MO. Semi-

parametric Distributions with Estimated Shape Parameters. (In manuscript)

VIII Karlsson MO and Savic RM. Diagnosing Model Diagnostics.

Clin Pharmacol Ther, 82:17-20, 2007

IX Savic RM and Karlsson MO. Importance of Shrinkage in Em-pirical Bayes Estimates for Diagnostics: Problems and Solutions. (Submitted)

Reprints were made with permission from the publisher.

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Contents

1.� Introduction .........................................................................................11�1.1.� Pharmacometrics ........................................................................11�1.2.� Non-linear mixed effect models .................................................12�1.3.� Model building techniques .........................................................13�

1.3.1.� Structural model development ...............................................14�1.3.2.� Stochastic model development ..............................................16�1.3.3.� Model diagnostics ..................................................................18�1.3.4.� Other important model building considerations ....................21�

2.� Aims.....................................................................................................22�

3.� Methods ...............................................................................................23�3.1.� Improvement of structural model development..........................23�

3.1.1.� Description of the transit compartment model and its derivation .............................................................................................23�3.1.2.� Extension for multiple dosing events.....................................26�3.1.3.� Improvement with the Stirling approximation.......................27�3.1.4.� Evaluation of the absorption model .......................................28�

3.2.� Improvement of stochastic model building ................................29�3.2.1.� Description of the novel nonparametric estimation method ..29�3.2.2.� Description of the extended grid method...............................32�3.2.3.� Description of the semiparametric method ............................34�3.2.4.� Evaluations of the methods....................................................36�

3.3.� Improvement of model diagnostics ............................................40�3.3.1.� PRED-based diagnostics........................................................40�3.3.2.� RES-based diagnostics...........................................................40�3.3.3.� EBE-based diagnostics ..........................................................41�

4.� Results .................................................................................................44�4.1.� Structural model development (Papers I-III)..............................44�4.2.� Stochastic model development (Papers IV-VII).........................52�

4.2.1.� Performance of the nonparametric estimation method (Papers IV-V) 52�4.2.2.� Performance of the extended grid method (Paper VI) ...........60�4.2.3.� Performance of the semiparametric method (Paper VII) .......64�

4.3.� Model diagnostics (Papers VIII-IX) ...........................................67�4.3.1.� PRED-based diagnostics........................................................67�

4.3.2.� RES-based diagnostics...........................................................68�4.3.3.� EBE-based diagnostics ..........................................................69�

5.� Discussion............................................................................................77�5.1.� A novel structural model for absorption.....................................77�5.2.� Stochastic model development ...................................................79�5.3.� Model diagnostics.......................................................................82�

6.� Conclusions .........................................................................................85�

7.� Acknowledgements..............................................................................87�

8.� References ...........................................................................................91�

Abbreviations

ARB Absolute relative bias CHMP Committee for Medicinal Products for Human Use CL Clearance CRCL Creatinine clearance DV Dependent variable EBE Empirical Bayes estimate EC50 Drug concentration producing half maximal effect EM Expectation-Maximization Emax Maximal effect F Bioavailability FDA Food and Drug Administration FO First order method FO-NONP Nonparametric method preceded by FO FOCE First order conditional estimation method FOCE-NONP Nonparametric method preceded by FOCE GAM Generalized additive model HT Heavy tailed IIV Inter-individual variability IOV Inter-occasional variability IPRED Individual prediction IWRES Individual weighted residual k Elimination rate constant ka Absorption rate constant ktr Transit rate constant Lik Likelihood under the kth possible model for individual i MAE Mean absolute error MC Monte Carlo ME Mean error MTT Mean transit time n Number of transit compartments NMVI Software NONMEM VI NONP Nonparametric NPAG Nonparametric adaptive grid NPC Numerical predictive check NPDE Normalized prediction distribution error NPEM Nonparametric expectation maximization

NPML Nonparametric maximum likelihood NPOFV Nonparametric objective function value OFV Objective function value PI Prediction interval pk Population probability of belonging to mixture k PD Pharmacodynamics PK Pharmacokinetics PRED Population prediction REE Relative estimation error RES Residual RPE Relative percentile error RV Residual variability SCM Step-wise covariate model building Sh� Eta-shrinkage Sh� Epsilon-shrinkage t Time tad Time after dose tdos Time of the dose tlag Lag time parameter WRES Weighted residual � Difference between individual prediction and observation

(residual error) �i Difference between population and individual parameter

estimate � Fixed-effect parameter (typical value) � Standard deviation of �s � Variance-covariance matrix of �s � Standard deviation of �s

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1. Introduction

1.1. Pharmacometrics Pharmacometrics is a scientific discipline that has evolved in the last 30 years. Most often this discipline has been defined as “research focusing on non-linear mixed effects models, which describes response-time profiles observed in clinical trials with a focus on determining sources of variability within a studied population” (1). A more recent definition describes phar-macometrics as “the science of developing and applying mathematical and statistical models to characterize understand and predict a drug’s pharma-cokinetics (PK), pharmacodynamics (PD) and biomarker-outcome behavior” (2). An even broader definition of the field has been recently suggested which defines pharmacometrics as “that branch of science concerned with mathematical models of biology, pharmacology, disease and physiology used to describe and quantify interactions between xenobiotics and patients, including beneficial effects and side effects resultant from such interfaces” (1). The latest definition aims to accentuate the relationship with other disci-plines such as clinical pharmacology, statistics, computer science, classical pharmacology, etc. Indeed, pharmacometrics is a discipline which integrates principles from several quantitative science disciplines with a major focus on developing predictive models of drug and disease. These models are utilized in many different ways, but probably the most important use of the models is to determine the optimum dose, schedule and treatment duration of novel therapies, thereby streamlining and supporting decision-making during drug development and potentially bringing novel therapies to patients with unmet need faster (3).

Pharmacometrics has great potential to impact on a multitude of global

health aspects via enabling data-driven decision-making based on elegant synthesis of available knowledge. The importance of pharmacometrics has been recognized by regulatory agencies, the pharmaceutical industry and academia. Consequently, (i) pharmaceutical industry is incorporating model-based analysis and data synthesis as an essential part of the drug-

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development process (4-8), (ii) regulatory agencies are accepting, requesting and even performing their own pharmacometric analyses to support/address issues arising during the regulatory assessment process (9-11), and (iii) sev-eral graduate programs have been or are in the process of being established at different academic sites (1, 12-14). Pharmacometrics as a science in itself is very dynamic and has been evolving rapidly. As evidence of the prolifera-tion and wider acceptance of the discipline, the publication record of papers focusing on this discipline has grown from 32 in the 1980s, to 259 in the 90s, and 588 in the last 8 years (Pubmed search performed using “NONMEM” as the search term in the cited time interval). Therefore, the pharmacometric science is focused on developing the appropriate models based on all avail-able knowledge as well as the model building techniques to do so.

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1.2. Non-linear mixed effect models Mathematical models represent a set of mathematical relationships equa-tions) describing the system of interest. These models can be used to explore the structure and behavior of the system (under a range of different condi-tions). In pharmacometrics, systems of interest involve the change of drug concentration over time in human body (effect of body on drug, i.e., drug pharmacokinetics, PK), the relationship between drug exposure and drug response, the dynamics of drug response over time (effect of drug on body, i.e., drug pharmacodynamics, PD) and the dynamics of disease itself, so-called disease progression. These relationships are most often described by nonlinear functions, thus these models are referred to as nonlinear models.

Pharmacometric models are built to describe the experimental data col-lected in clinical trials from both healthy volunteers and patients. Examples of collected data include the measurement of the plasma concentration over time, or measurement of a biomarker characterizing drug effect or disease state over time. Inferences from the data are drawn and summarized in terms of estimated model parameters, such as drug clearance (CL) or maximal drug effect (Emax). Because there are no two identical human bodies, an im-portant feature for this type of data is that different sets of model parameter estimates are expected to characterize physiological processes across indi-viduals. This brings another important component of the pharmacometric model, variability between individuals, which is manifest in the model pa-rameters describing the process. Thus, parameters of the model are treated as distributions, rather than single values. The mean of such a distribution is considered the typical parameter value while the distribution is estimated

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from the data and it describes the distribution of individual parameters in the studied population (2, 3, 15-17).

Part of the observed variability can be explained by some known factors or covariates, such as body-weight, genotype, creatinine clearance (CRCL), etc. However a major part of the variability between individuals that cannot be explained simply by any known covariate is deemed to be random. Simul-taneous estimation of the mean profile (typical parameter) as well as the random variability in referred to mixed effect analysis. Although there are other types of methods that can be used to deal with this class of models, non-linear mixed effect analysis is by far the most commonly used and will be referred to throughout this thesis.

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1.3. Model building techniques The focus of pharmacometric science is to find an adequate model which would meet all predefined requirements. It has been widely discussed in literature that one never can find a true model, because the biological system is too complex. However, a modeler can still develop a useful model which could help to explain the data, to understand the underlying system and to be able to predict response of the system in different circumstances (18). The process of finding the optimal model is defined as model building or “mod-eling”. Once an experiment is performed and data are collected, cleaned and explored graphically, the model building process follows as an iterative pro-cedure involving four major steps (19):

1. Model definition 2. Model fit 3. Model diagnostics. 4. Model evaluation

In the pharmacometric framework, model development is comprised of

the definition of three major model components: the structural, stochastic and covariate model, and most often these components are introduced se-quentially. Early in the process of the model development, an adequate struc-tural model, which can explain the mean data response, is determined. Next, an appropriate stochastic model, which involves variability quantification in model parameters as well as identification of possible correlations between model parameters, is developed. Finally, a covariate model, which aims at finding plausible relationships between model parameters and measured

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covariates, is introduced. At each step, the model is fitted to the data and evaluated for adequacy using visual model diagnostics and derivation of goodness of fit statistics. At each modeling milestone, and most importantly once a final model has been determined, the model needs to be validated through different techniques to assess fulfillment of predefined criteria for model acceptance. The model-building approach described above is infor-mally accepted by practicing modelers (20, 21). Of central interest in this work is a formal evaluation and discussion of the three components central to pharmacometrics, the development of the structural model, the stochastic model and model diagnostics.

1.3.1. Structural model development A structural model aims to describe the central tendency of the data. There is a certain range of general models that can be applied to each studied case. For example, one-, two-, three- compartment disposition models with ex-travascular input following first-, zero-order or combined input kinetics are available to describe the pharmacokinetics of a drug. In the case of drug pharmacodynamics, Emax, sigmoidal Emax, effect compartment model, or indirect response models are often used to describe the data. Although, these models contain parameters with useful physiological meaning, such as CL (rate of elimination in relation to plasma concentration of drug) or EC50 (concentration of drug which would produce half maximal effect), these models are considered as empirical models, because they do not provide further insight into the underlying physiology/pathophysiology and are gen-erally applicable across drug classes. However, these models can still be very useful for their intended purposes (2, 3).

In the effort towards a mechanistic description of the system, compo-

nents specific to the a) drug itself (e.g. specific release profile, drug binding, drug distribution etc), b) drug pharmacology (affinity to receptor, binding kinetics), and c) the underlying disease and physiology (e.g. cell aging, regu-latory feedback mechanisms need to be introduced. Clearly, in such a case, the model building approach and the structural model itself would be case-specific and it is more difficult to give general recommendations on how to build such models. The modeler’s path and the complexity of the model, in this case, would be guided and/or limited by several factors, among them the most important being the data available, and, the goal of the analysis.

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1.3.1.1. Absorption structural model One of the more difficult tasks for a modeler is to find appropriate structural description of drug absorption. Drug absorption is a complex process de-pendent both on physicochemical properties of the drug, the biopharmaceu-tical properties of the drug formulation and the physiological processes in-volved. Biopharmaceutical properties of the drug formulation determine processes such as disintegration of the drug-delivery system, dissolution and release from the dosage form. Physiological processes governing drug ab-sorption include gut motility, pH, gastric emptying rate and metabolism in the gut wall. These processes determine the rate, extent, and the length of time before the drug appears in systemic circulation.

Physiology-based absorption pharmacokinetic (PK) models have been developed that account for physicochemical properties (such as dissolution rate or the pH dependence of drug solubility of the drug) as well as for the complex physiological processes (such as the metabolism in gut or liver and drug transit to the absorptive surface) involved in the drug absorption. These mechanistic models require extensive prior knowledge, such as information about the absorptive surface area, the rate of gastric emptying, drug concen-tration in the lumen, enzyme abundance in the gastrointestinal wall and liver and liver blood flow (22, 23). This information is not usually available, pre-venting the routine application of physiologic models in drug absorption estimation.

Data available for developing absorption model are collected from plasma subsequent to drug passage through these multiple barriers and is usually limited to a few samples, hence the difficulty in describing this process accu-rately. Thus, the traditional models used to describe the absorption process are simple and include a parameter describing the absorption rate (first or zero order absorption rate constant) and usually a lag time parameter charac-terizing any potential absorption delay. Occasionally, Weibull or inverse-Gaussian empirical functions are introduced to deal with erratic absorption profiles (24-26). Even though the importance of characterizing the absorp-tion delay via lag time has been widely applied, a lag time often fails to give an adequate description of the absorption delay. This is probably due to fact that a lag time postpones the dose administration, which also leads to an abrupt increase in the absorption rate from zero to a maximal value. This is not a physiological approach and therefore it is not surprising if such a model fails to describe the system adequately. Additionally, the discontinu-ous nature of the resulting concentration-time profile (so-called a change-point model) may cause difficulties in finding the optimal parameter value using available optimization algorithms.

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In general, the range of alternative structural absorption models is lim-ited. Furthermore, experimental limitations restrict availability of more abundant data to fully characterize drug absorption. Therefore, there is a need for an alternative, generally applicable, absorption model, capable of describing various drug absorption profiles and yet still simple enough to be estimable from the available data.

1.3.2. Stochastic model development One of the major challenges and aims of the pharmacometric analysis is to identify and quantify variability in model parameters. An erroneously speci-fied stochastic model may translate into poor simulation properties of the model, bias in parameter estimates, error in hypothesis testing or failure to understand the system (27, 28). For example, it has been shown that neglect-ing inter-occasion variability, leads to biased parameter estimates (29, 30).

A portion of variability in observed data can be explained in terms of

available covariates, while the remaining part is treated as random variabil-ity. Random variability captures the majority of observed variability in clini-cal data, thus it is important to quantify it. Despite the label “random”, it is not completely random but is simply variability not directly explained in physiological terms.

The sources of variability can be multiple. Random variability potentially

encapsulates variability due to covariates not measured, and/or suggests that the model is too crude to fully explain underlying physiological processes. Most often, at least two levels of variability are defined and quantified in mixed effect analysis: variability between different individuals – inter-individual variability (IIV) and residual variability (RV), which represents all remaining noise in the data after other sources of variability are taken into account. Residual variability, includes noise due to measurement error, erro-neous data records (such as time when sample was taken or the actual amount of drug administered), changes in individual biology over time, or error which arises due to model misspecification. Often, if the drug was studied at different study occasions, variability between these occasions may also be quantified (inter-occasion variability IOV). Examples of incorporat-ing fourth and fifth levels i.e. referring to inter-study and inter-center vari-ability, respectively (31), of random effects have been demonstrated. Clearly, there are many possible sources of variability in the PKPD data and it is important both to separate them and to quantify them correctly.

Variability estimation is not a straight forward statistical problem and several different approaches exist.

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1.3.2.1. Parametric approach Parametric methods assume that the random part of variability arises from a specific parametric distribution, most often the normal or a transformation of the normal distribution (32). Parametric methods are widely used in phar-macometric analysis and they have been previously discussed intensively (2, 15, 16, 32). Assumption of log-normality in the distribution of model pa-rameters facilitates and simplifies the estimation procedure to a great extent. No matter what is the size of the data set, parameter variability is always summarized in terms of two parameters, central tendency and variance of the distribution. The mean value refers to the typical parameter estimate while the variance relates to the parameter describing inter-individual (IIV) vari-ability in the studied population.

In conjunction with the parametric assumption, the maximum likelihood

estimator is most often used for the parameter estimation as it is efficient. Even though the normality assumption is often relevant (33), it is still an assumption and does not necessarily need to hold true. The simplest example of violation of such an assumption is the presence of enzyme polymorphism which leads to an existence of different subpopulations. Each subpopulation has its own parameter distribution which results in a multimodal distribution in CL for the studied population. Another simple example of deviation from normality is the presence of outliers, which can change the shape of the pa-rameter distribution from normal to heavy tailed.

The true parameter distribution does not need to reflect any known shape.

Making an erroneous assumption may lead to problems such as overestima-tion of the variability, parameter bias, hidden model misspecification or poor simulation and predictive properties of the model (16). Therefore, another class of methods has been developed and which are less restrictive with re-spect to the assumed shape of the distribution.

1.3.2.2. Nonparametric approach Nonparametric methods are another class of methods which are at the other extreme compared to parametric methods. They make no assumption about the distribution shape, but only define the parameter space. Development of these methods and algorithms started almost 30 years ago with the work of Laird and Lindsey who studied properties of the maximum likelihood esti-mator when no assumption is made on distribution of random effects (34, 35).The first algorithm introduced in the pharmacometric area, the nonpara-metric maximum likelihood method (NPML) developed by Mallet, demon-strated that nonlinear mixed effects modeling could be performed without any assumptions regarding the distribution shape leaving the distribution function completely unrestricted (36). A similar algorithm described by

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Schumitzky et al., called nonparametric EM algorithm (NPEM), was in fact a generalization of the previously developed expectation-maximization (EM) algorithm (37). The more computationally efficient version of this algorithm, the nonparametric adaptive grid (NPAG) has since been introduced also as a separate software (38). These methods have proven to be powerful in esti-mating underlying distribution shapes (39-41). However their use is associ-ated with certain drawbacks, which include increased computation time, no imprecision measurements, and the impossibility to estimate residual vari-ability, which has limited their usage to a certain extent (42).

1.3.2.3. Semiparametric approach An alternative path that is in between the parametric and nonparametric es-timation methods is the semiparametric approach, which enables transforma-tion of parameter to any distribution shape. These transformations may be more complex where additional parameter(s), so called shape parameters, are introduced and estimated along with other model parameters. These esti-mated distributions do not fully relax the assumptions about distribution shapes but allow a wider range of distributions to be adequately described. Despite historically being named “semiparametric” (16, 43) these methods still retain the advantages of parametric over nonparametric methods. Simi-lar approach, but without estimated shape parameters, is used often even in the parametric framework, where specific parameter distributions are re-quired. For example a simple log-normal transformation is almost always used to constrain parameters to be positive. Also, a logit transformation is used for parameters that have theoretical upper and lower bounds, for exam-ple fractions or bioavailability parameters. The first semi-parametric trans-formation was developed and implemented in the NLMIX software by Davidian and Gallant in 1993 (43). This method uses polynomials as trans-formation functions. Another developed method uses spline functions (44, 45). Neither the polynomial nor the spline transformations appear to have been used in published papers outside the citations above. Likely reasons for this are instability, complex implementation and lack of supported software.

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1.3.3. Model diagnostics Selection of the appropriate modeling approach for a given data can be aided by model diagnostics. As described earlier, the model development is a se-quential and iterative process of building several submodels – the structural, covariate and stochastic model. Model diagnostics are therefore an essential part of model development. They serve both as an aid to compare alternative models and to assess model appropriateness. Throughout the model building process, assessing model appropriateness includes evaluating numerous as-

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pect of a model, such as predictive properties, simulation properties, model robustness, etc (46, 47).

An informally agreed upon set of diagnostics is often used for each new

model iteration, while another set is used for covariate model building, and yet a third and more comprehensive set of diagnostics is used prior to accep-tance and characterization of the final model. As can be expected, a modeler may chose to use one set of diagnostics to assist their own modeling exercise only to be prompted to use another set of diagnostics depending on whom they are communicating with about the model. Thus model diagnostic ap-proaches may differ depending on whether a model is being presented to colleagues from the field, regulatory authorities, clinical team or people from other areas, such as business or marketing.

Furthermore, model diagnostics (graphical and/or numerical) seldom

provide a modeler with definitive answers. Thus it is important that a phar-macometrician can interpret the information obtained correctly. Also, a pharmacometrician needs to be sufficiently knowledgeable in order to ap-proach model diagnostics critically, to be aware of each diagnostic’s as-sumptions, strengths and weaknesses. Model diagnostics may be roughly divided into graphical and numerical diagnostics.

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1.3.3.1. Graphical diagnostics Graphical diagnostics are considered a powerful and intuitive visual method to be used not only by pharmacometricians, but also as a communication tool. The US FDA (Food and Drug Administration) and the European CHMP (Committee for Medicinal Products for Human use) guidances’ ex-plicitly mention the utility of graphical diagnostics (48, 49). Furthermore, graphical displays are extensively used during the model building process and are considered as essential tools for data visualization, inspection of model adequacy and assumption testing. The importance and power of this type of diagnostics has been discussed and demonstrated by others (46). Several tools have been developed to assist a modeller to produce these types of diagnostics with increasing ease (50). Graphical diagnostics are appealing in their simplicity as a modeller simply evaluates if graphical out-put resembles an expected pattern or not. However, even though human eye has very fine perception of visual trends, there is a great variation in how different modellers would draw conclusions from different figures. Lastly, the shortcomings as well as assumptions that graphical diagnostics are based on have seldom been discussed.

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1.3.3.2. Numerical diagnostics Numerical model diagnostics may also be used alongside graphical diagnos-tics. These types of diagnostics include quantities such as goodness-of-fit statistics, parameter estimates, uncertainty in parameter estimates, and diag-nostics for detection of influential individuals.

A maximum likelihood quantification that data are arising from the speci-

fied model is summarized as a single number, the objective function value OFV, which is approximately proportional to -2 times the log likelihood under the assumption that the model is the true model and that the errors are normally distributed (16, 51, 52). If two nested models are to be compared, a formal statistical test, the likelihood ratio test may be performed based on these values in order to determine the statistically significantly better model. However, a modeler needs to be aware that this test may show a higher type 1 error rate, (i.e. the risk of getting false positive results in certain circum-stances), especially when data are rich in terms of the number of observa-tions per individual in conjunction with the first order (FO) estimation algo-rithm or when small data sets are used with any estimation algorithm (27, 53-55).

The primary goal of all optimization algorithms is to provide model pa-rameter estimates, and the role of the pharmacometrician is to check that these estimates are physiologically plausible, the model is stable and reliably converges to similar parameter estimates even with perturbed initial esti-mates, and that final parameter estimates do not coincide with parameter space boundaries. Additionally, uncertainty estimates provide a modeler with the additional information about the parameter. The uncertainty may be assessed as a classical standard error, or it can be described using a confi-dence interval estimate obtained via computer intensive methods, such as log-likelihood profiling or a bootstrap procedure (56). These are often used to diagnose possible model overfit. Certain diagnostic procedures, such as bootstrap or case-deletion diagnostics, are used to assess model robustness, while influential individuals may be detected either using case deletion diag-nostics or individual contribution to likelihood (individual OFV) (57, 58).

Clearly, numerical diagnostics are useful for many different purposes,

however they are seldom useful to assess whether a model can adequately describe the observed data in an absolute sense.

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1.3.4. Other important model building considerations So far, concepts around building the structural model, the stochastic model and model diagnostics have been introduced. The model building process also includes other important parts which won’t be discussed in greater detail in this thesis. These will be briefly mentioned below.

Covariate model building is a part of model development where poten-

tially important (predictive) covariates as well as the relationship between covariate and parameter are described. If identified, these covariates may help decide upon dosing scheme and schedule as well as to serve as guidance for individualized dosing. Several techniques have been developed to build covariate models, with step-wise covariate model building (SCM) and gen-eralize additive model (GAM), being probably those used most often. This topic has been discussed in a great detail in work by others (59-62). Some details regarding graphical inspection of covariate relationships as well as use of individual parameter estimates in GAM will be touched upon in this thesis.

Model qualification is a process of formal assessment of if a model serves

its purpose. As there are many different purposes for a model, the range of model qualification tools is also broad and constantly developing. The most frequently used model validation tools include different types of predictive checks (posterior, numerical and visual predictive checks) (28, 63-65), non-parametric bootstrap techniques, case-deletion techniques and normalized prediction distribution error (NPDE) (66). In this work, some of these tech-niques will be used and touched upon. In particular the numerical predictive check will be discussed in greater detail.

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2. Aims

The major aim of this thesis is to advance the approaches used in phar-macometrics by introducing novel model components and methods for ap-plication in essential parts of model building procedure: development of a structural model, development of a stochastic model and model diagnostics. Specific aims of this thesis are:

1. Contribute to structural model development:

� Develop an alternative to the lag model for describing delays in ab-sorption

� Evaluate this alternative model, the transit compartment model us-ing real case studies, including studies with sparse sampling design and studies of a compound with variable absorption

2. Contribute to stochastic model development:

� To evaluate a novel nonparametric method for estimation of pa-rameter distribution

� To develop and evaluate a method which address shortcomings of the proposed nonparametric method

� To develop a semiparametric approach for dealing with deviations from parametric assumptions

3. Contribute to model diagnostics:

� To critically discuss the shortcomings of the most commonly used model diagnostics

� To systematically evaluate model diagnostics dependent upon indi-vidual parameter estimates

� To provide information and tools by which modelers can judge in-formation value in some common diagnostics.

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3. Methods

3.1. Improvement of structural model development 3.1.1. Description of the transit compartment model and its

derivation The transit compartment model is a suitable alternative to the traditional lag time model for description of both delayed outcomes in pharmacokinetics as well as in pharmacodynamics. In a transit compartment model the delay in an observable event arises as a consequence of the drug transit through a chain of intermediate compartments before reaching the observation com-partment (67-69). This model is shown schematically in Figure 1, where the absorption delay is described by the passage of drug through a series of tran-sit compartments with a single transfer rate constant, ktr (Equation 1). The rate of change of the amount of drug in the nth compartment is given by:

� � ntrntrn akak

dtda

�� 1 (1)

In Equation 1, dan /dt denotes the rate of change of substance a in com-

partment n, an is the drug amount in the nth compartment, ktr is the transit rate constant from the nth-1 compartment to the nth compartment and n is the number of transit compartments.

Figure1. Schematic view and mathematical description of drug flow through a chain of transit compartments.

24

A commonly used model-building strategy for time delayed phenomena is to manually hard code all the transit compartments separately. This approach requires multiple stepwise addition/subtraction of transit compartments one at a time. This is a time consuming process, especially when the optimum number of transit compartments is high. Furthermore, once hard-coded the number of transit compartments is fixed for the entire population and cannot be varied between individuals. Numerical estimation of the optimal number of transit compartments would address both these concerns.

In order to numerically estimate the optimal number of transit compart-

ments, a system of n differential equations as shown in Equation 1 describ-ing the drug transit from the input through to the final transit compartment needs to be solved, and an analytical solution for drug amount in the last compartment of the transit compartment chain derived. This analytical solu-tion is given by the function:

tktrn

trn e

ntkDoseFta ��

��!

)()( (2)

In Equation 2, F is the drug bioavailability and n! is the factorial function

with argument n. To compute this solution numerically, Stirling’s approxi-mation is employed to compute n! (Equation 3).

nn enn �� 5.02! (3)

Transfer of drug from the ultimate presystemic transit compartment to the central compartment is via an absorption compartment in which the disap-pearance of drug is described by the rate constant ka. The rate of change of drug amount in the absorption compartment (dAa/dt) is given by:

� �aann

tktrntr

tr Aken

etkkFDosedt

dAa��

��

��

��

5.02 (4)

Stirling’s approximation is a continuous function of n, which allows im-plementation of this model in software for non-linear mixed effects model-ling (NONMEM). Furthermore, it also enables estimation of a non-integer number of transit compartments n. To prevent the numerical difficulties aris-ing from rapid increments in function values with an increase in n, the trans-formation shown in Equation 5 may be ued.

25

� �

aaen

etkkFDoseAke

dtdAa nn

tktrntr

tr

��

��

��

�� )2

ln( 5.0

(5)

Thus, in the final implementation of the model, the parameters defined are n, number of transit compartments, ktr, transit rate constant and the mean transit time (MTT), which represents the average time spent by drug mole-cules traveling from the first transit compartment to the absorption com-partment. The relationship between these three parameters is shown in Equa-tion 6. Because of this relationship, only two parameters need to be esti-mated while the third parameter is derived.

MTTnktr

1��

(6)

The concentration-time profile obtained with the transit compartment model has a smoother initial increase in the concentration-time curve as a consequence of the gradually increasing absorption rate, as shown in Figures 2 and 3. This is in contrast to the abrupt on/off absorption profile that results when the lag time model is used and in which an abrupt switch in the absorp-tion rate from 0 to a constant value occurs at tlag.

LAG vs TRANSIT

18

20

14

16

on

TRANSITLAG

Absorption

8

10

12

ncen

trat

i Absorption

Disposition

4

6con Disposition

0

2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

time

Figure 2. Simulated concentration-time profile depictions using lag-time (dotted line) and transit compartment (solid line) models, respectively) when k=0.5 1/h, ka=1.7 1/h, Dose=30mg, n=5 and MTT (or tlag) =1h

26

100

(%)

sorp

tion

(

50

ulat

ive

abs

Lag modeln=2000

Cum

u

1n=20n=2

n=200

00n 2

Absorption delay (h)1 20

Absorption delay (h) Figure 3. Profiles of the cumulative amount of drug reaching the absorption com-partment simulated using the lag-time model (tlag=1h) and several transit compart-ment models, which differ in the number of transit compartments (2, 20, 200, 2000) when Dose=100 mg and MTT=1h

3.1.2. Extension for multiple dosing events The solution for the transit compartment model presented in previous para-graph is applicable only for the single dose data, since the nature of the dif-ferential equations allows the initial conditions of the system (the dose ad-ministered) to be set at time 0 only and does not allow for the introduction of new drug boluses into the system at later time points. In order to apply this model to a repeated-dose design, modifications are required. Time t in Equa-tion 4, the actual time at which a given sample is drawn, can be changed to time after dose (tad) by subtracting the time of the last dose (tdos) from t Equation 7). In this way, each dosing event is allowed to occur at time 0, meeting the requirements of the original model and thus allowing the transit model to accommodate multiple-dosing schemes.

� �aann

tdostktrntr

tr Aken

etdostkkFDosedt

dAa��

��

��

��

5.0

)(

2(

(7)

27

This type of implementation assumes that the entire bioavailable dose reaches the absorption compartment, i.e. leaves the delay chain, before the next dose is given. This is usually a valid assumption for the majority of dosing schedules.

3.1.3. Improvement with the Stirling approximation Stirling’s approximation has been shown previously to be precise for values of n higher than 2 (approximation error is less than 1% for n > 2). If n is smaller than 2, the approximation error increases and propagates into the entire function. If n is less than 2 an improved version of the approximation, shown in Equation 8 needs to be used. Comparison of the true function with the original approximation and the improved approximation is shown in Figure 4.

)12

11(2! 5.0

nenn nn

�� (8)

2.0

ion

1.5

ma

func

t

1.0o ga

mm

0 5mat

ion

to

0.5

App

roxi

m

Stirling approximationTrue gamma functionImproved Stirling approximation

0 0 0 5 1 0 1 5 2 0

0.0

A

0.0 0.5 1.0 1.5 2.0

n Figure 4: Approximations to gamma function when n < 2

28

3.1.4. Evaluation of the absorption model Implementation of the Stirling’s approximation in the transit compartment model was evaluated in 3 different circumstances: (i) via application to al-ready published datasets, (ii) via application using sparse data sets collected in children (desmopressin), and (iii) via application to a dataset which ex-hibit variable absorption (rifampin). Model performance was compared to the traditional lag time model. All analyses were performed using the NONMEM software version VI (15).

3.1.4.1. Published data sets Available published/soon to be published datasets from pharmacokinetic studies with four different compounds: glibenclamide, moxonidine, fu-rosemide and amiloride were analyzed. PK analyses of the glibenclamide and moxonidine data sets have been published previously (70-72) while the manuscripts reporting PK analyses of furosemide and amiloride are in prepa-ration (Frick et al, in preparation) (73). Details about these studies can be found in referenced papers as well as in the final publication based on this work. The major objective was to investigate whether the transit compart-ment model could offer any improvements over final models implemented for these datasets.

3.1.4.2. Sparse data sets in children This analysis was performed on sparsely sampled pediatric dataset for des-mopressin. In total, 139 observations were collected from 72 children, mak-ing this dataset sparse and therefore a substantial challenge to model, espe-cially in the absorption phase (60 observations sampled in first 2 hours, i.e., on average less than 1 observation per child available for estimation of ab-sorption parameters). Details of the study are available in the final publica-tion based on this work (74).

3.1.4.3. Data set exhibiting variable absorption (rifampin) In order to evaluate the performance of the proposed transit compartment model on data that exhibit variable absorption, the model was evaluated us-ing a rifampin dataset collected from pulmonary tuberculosis patients who showed extremely variable pharmacokinetics. Rifampin treatment is a chronic treatment, which also means that this example was suitable for test-ing the extension of the transit compartment model to fit data from multiple dosing schedules. In total 2913 samples were collected from 261 patients receiving rifampin for at least 10 days. Details of the study are available in the final publication based on this work (75).

29

3.2. Improvement of stochastic model building This part of the thesis deals with three novel methods available for develop-ment of stochastic models when the major assumption of normality does not hold.

�

3.2.1. Description of the novel nonparametric estimation method

Motivation: A novel method (available in the recently released program NONMEM VI (15)) has been introduced into pharmacometric community for nonparametric distribution estimation. This method bridges the widely-used parametric statistical methods with nonparametric methods, which is an attractive compromise because it borrows strengths from both these ap-proaches. In the novel method, the parameter distributions are approximated by a discrete probability density functions at a number of parameter values (support points) equal to or less than the number of individuals in the data set, in a similar manner as previously presented for the nonparametric meth-ods. However, the support points are obtained via parametric estimation of the empirical Bayes estimates (EBE’s). By simplifying this step, the novel method avoids issues related to the large dimension optimization problem faced previously by pharmacometricians, while still providing them with the entire probability estimate of the parameter distribution. Description: The nonparametric estimation method in NONMEM VI (NMVI) is a maximum likelihood method which consists of two steps. The first step involves the estimation of the discrete locations at which the non-parametric parameter distribution is to be evaluated and the second step es-timates the population probability associated with each support point. Sup-port point estimates are set at the estimates of individual parameters (i.e. empirical Bayes estimates – EBEs) in parametric settings. Thus any method available in NONMEM (FO in conjunction with a POSTHOC step, FOCE or LAPLACE) may be used to estimate EBEs, implying that the assumption of normality is influencing the initial support point estimation. Subsequently, each support point is estimated in a multidimensional space with the number of dimensions equaling the number of parameter distributions to be esti-mated, and it is defined with a vector of model parameters unique for each individual. Thus, the maximum number of support points is restricted to the number of individuals in the dataset, similar to other nonparametric soft-ware. Once support points are estimated, second step- joint probability esti-mation is implemented.

30

The nonparametric method in NONMEM operates as a large mixture model with the number of mixtures/submodels equal to the number of unique vectors of individual parameters in the data set. Each submodel is determined by the set of individual parameters and each parameter distribu-tion within the submodel has the mean that is equivalent to each individual parameter estimate with zero variance. Thus, each mixture may be viewed as only one point in a multidimensional space where the number of dimensions is determined by the number of model parameters. The probability of data belonging to each mixture is estimated as in any other mixture model and this probability actually defines the entire joint nonparametric probability of the data given the model. Once estimated, the joint probability may be used to derive, for example, the cumulative marginal probability distribution for each parameter, which is defined at a number of discrete locations. The like-lihood for individual i under a nonparametric model, as in the case of the mixture model, is given by

��

�r

kikki LpL

1 (9) where Lik is the likelihood under the kth possible model for individual i, pk is the probability of belonging to mixture k and r is the number of submodels and which is equal to number of individuals n. The total likelihood of all data is the sum of individual likelihoods and it is output from NONMEM as nonparametric objective function value (NPOFV).

The output from the nonparametric estimation includes estimates of pa-

rameter distributions, summarized as a collection of parameter values (sup-port points) and associated probabilities. These nonparametric distributions can be statistically summarized in terms of reporting mean, median, vari-ance, covariance, distribution percentiles, etc. Visually, they are most often summarized as spiky plots which can be further smoothed (Figure 5) or transformed into a cumulative density plots (Figure 6) in order to facilitate the comparison with other distributions, for example, the normal distribution (Figures 5, 6 and 7).

31

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Individual parameter estimates

0.0

0.2

0.4

0.6

0.8

1.0C

umul

ativ

e pr

obab

ility

den

sity

func

tion

Nonparametric distributionNormal distribution

-2 -1 0 1 2

-0.01

0.01

0.03

0.05

Nonparametric distributionSmoothed nonparametric distributionNormal distribution

32

2tio

n

1

etric

dis

tribu

0

ed n

onpa

ram

-1Est

imat

e

-3 -2 -1 0 1 2 3-2

Normal Distribution Figure 5-7. Useful graphical diagnostics for inspection of nonparametric distribu-tions versus the normal distribution, illustrated using a single example with an un-derlying heavy tailed true distribution visualized via: 5) a cumulative probability density function, 6) a nonparametric distribution smoothed with a Kernel smoother, and 7) a classical QQ plot

3.2.2. Description of the extended grid method Motivation: Empirical Bayes estimates (EBEs), which are used as support points as described above, have shortcomings and are particularly impacted by the so-called shrinkage phenomenon in certain circumstances (76). There-fore they may not always provide a suitable range of support points. Fur-thermore, it is well known that nonparametric methods suffer major draw-backs when the number of subjects in the datasets is low because the distri-bution is estimated at the maximal number of support points, equal to num-ber of individuals in the dataset. In this circumstance, when the nonparametric grid is not optimal, the discrete nature of nonparametric methods becomes questionable both in terms of precise parameter estimation and for simulation purposes of novel scenarios, since simulations from the model only can represent as many unique individuals as were present in the original study. This was the motivation for development of a novel method which can provide a modeler with an extended nonparametric grid which would also result in an improved estimate of parameter distribution.

33

Description: In the extended grid method, the nonparametric grid is enriched by addition of new support points. These are user-defined, and in the default version of the method, they are simulated from the parametric distribution with the mean and variance estimated in the preceding parametric analysis. Similar to other simulation methods, points of support can be generated us-ing inflated/deflated variances, by simulations from other distributions (e.g. uniform), or sampling from the parameter space after dividing it into equally probable subspaces, etc. The extended grid method does not have a required upper limit for the number of support points and it also may or may not keep the EBEs from the preceding parametric step in the nonparametric grid. Once additional points of support are introduced, the entire nonparametric joint probability density function is estimated based on the new extended grid. This method leads to a nonparametric distribution estimated at the lar-ger number of support points than individuals in the datasets (Figure 8).

0 50 100 150 200 250

Volume of distribution

0.0

0.2

0.4

0.6

0.8

1.0

Cum

ulat

ive

prob

abili

ty fu

nctio

n

True distributionExtended grid NP distributionDefault NP distribution

Figure 8. A visual comparison of true, estimated default, and extended-grid non-parametric distributions of data sets containing 30 individuals

34

3.2.3. Description of the semiparametric method Motivation: Often, though a modeler may notice certain deviations from normality, they do not instantly move into the nonparametric framework. There are different reasons for this, most often because these methods are not widely used, or access to software is not available or modelers are not familiar with the methodology itself. The first choice is probably to stay within parametric framework, but possibly to handle these issues differently. The present range of alternatives are rather few but include application of logit transformation or application of the mixture model, even though with the latter, a modeler is already moving into the field where two models are not easily comparable any longer – the likelihood ratio test does not hold (77). Therefore the motivation arose to develop a transformation, similar to a logit or log transformation, but with the possibility of approximating addi-tional shapes. The major idea behind these novel transformations introduced here is that they are easy to implement in the existing software within para-metric framework and yet, capable of handling range of different underlying shapes, for a cost of one or two additional shape parameters.

Description: The goal of this investigation was to develop and evaluate pa-rameter distribution transformations with estimated shape parameters. The basic principle behind this investigation was that a default randomly distrib-uted variable �i could be transformed, using a characteristic mathematical function, into the transformed variable �i,transformed. The chosen mathematical function contains the shape parameter which is estimated from the data thereby allowing the observed data to inform the actual parameter distribu-tion shape. By doing so, parameter distribution is estimated from the data under less rigid distribution assumptions, to allow that it fits the underlying data in best possible way.

When selecting suitable transformations, certain predefined criteria were

sought: (i) the ability of the transformation to collapse into the identity trans-formation at some values for the shape parameter(s) which would also en-able hypothesis testing whether the transformation inferred a statistically significant improvement in fit, (ii) enabling that the typical parameter value before the transformation (�i =0) would remain the same after the transfor-mation �I,transformed=0, and (iii) to retain approximately, the correlation struc-ture among random effects before and after transformation by assuring that the rank orders of �i and �i,transformed are the same.

Three transformations, denoted as logit, box-cox, and heavy-tailed (HT),

were investigated.

35

The logit transformation uses mathematical Equation 10 to transform a normal distribution into a left- or right-skewed, spiky, wide or bimodal dis-tribution (Figure 9). Two shape parameters are introduced, �1 which governs the skewness of the distribution and �2, which determines the width. These values need to have boundaries; for �1 these are between 0 and 1 and for �2, the parameter space is constrained to be positive. When � is small, �2 large and �1 is 0.5, the logit transformation collapses into the identity transforma-tion, resulting in a normal distribution of �i,transformed. (10)

The box-cox transformation uses Equation 11 to transform a normal dis-tribution into a left- or right-skewed distribution (Figure 9). One shape pa-rameter is used, �1, which can take any value from the entire parameter space except 0, at which point the function is not defined. Therefore, as this pa-rameter approaches 0, the box-cox transformation approaches the identity distribution.

1

_1

(( ) 1)i

i Transformede �

�

�� (11)

The heavy tailed transformation uses Equation 12 to transform normally

distributed variable into a heavy tailed using one parameter which deter-mines how “heavy” the distribution tails are (Figure 9). Additionally, when the shape parameter takes negative values, a symmetric bimodal shape can be obtained. Similar to the box-cox transformation, this function is also non-continuous at �1 = 0 and approaches the identity distribution as �1 goes to 0. Therefore, the parameter needs no boundaries but it could be restricted to a positive value to get only heavy-tailed distributions.

1

_i Transformed i i�

� � (12)

� �1

1

11 1

(1 )1

_ 1 21iLOGLOG i

i Transformed e e�

�

� �

� ��

��

��

��

� ��

� ��

36

Den

sity

0.0

0.5

1.0

1.5

2.0

−1 0 1

Logit

−1 0 1

Box−Cox

−1 0 1

Heavy tailed[1st parameter;(2nd);Standard deviation]

[0.1;12;1]

[0.2;2;4]

[0.3;6;0.5]

[0.5;1]

[1;1]

[2;1]

[0.2;1]

[1;1]

[2;1]

Figure 9. Examples of normal distribution transformations. The shape parameter values used to create the distributions and the standard deviation of the original distribution are indicated.

3.2.4. Evaluations of the methods �

3.2.4.1. MC simulations Nonparametric and parameter distribution transformations methods were evaluated via Monte-Carlo simulation studies (100 replicates) to assess their statistical properties. The objective was to assess the capability of the meth-ods to estimate precisely the entire parameter distributions, specifically those that are non-normal. Additionally, both small and large data sets were stud-ied (30-50 and 200 individuals respectively), as well as sparse and rich data-sets in terms of number of observations (1 and 1.5 observations per random effect respectively). All Monte-Carlo simulation studies had similar general step-wise settings: � Data were simulated from a predefined model (one compartment iv bolus

model or one compartment model with first order absorption), and study design with predefined non-log normal parameter distribution shapes

� These data were analyzed using both the current standard and proposed novel methods, including the true model, which was used to generate the datasets

� Results were collected in terms of estimates of entire parameter distribu-tions

� Results were evaluated so that estimated parameter values at pre-defined distribution percentiles were compared with the true parameter values at the same percentiles and relative percentile error (RPE) was assessed (also defined as relative estimation error (REE))

37

� Results were summarized and presented both numerically, in terms of bias and imprecision values and graphically, as box plots.

Brief specific settings for each studied method follow. Detailed simula-

tion settings are presented in corresponding publications.

Default nonparametric estimation method. Three different scenarios with respect to distribution shapes were studied: log-normal, bimodal and heavy tailed. Different combinations of bimodal distributions were also in-vestigated. Both, large (N=200) and small (N=50) datasets were investi-gated. Also, rich (1.5 observation / random effect and/or low residual vari-ability) and sparse (1 observation / random effect and/or high residual vari-ability) data were studied. Support points were estimated, using both the first order (FO) and the first order conditional estimation (FOCE) methods.

Extended grid method. The evaluation of this method was focused on studying small rich data sets (N=30 with 1.5 observation/random effect) and sparse data sets (200 individuals with 1 observation/random effect). The underlying parameter distribution was heavy-tailed. Introduction of different numbers of additional points into the nonparametric grid (200, 400, 800 and 1600) was studied. Also, for each case, generation of additional points of support by simulation from the normal distribution with mean 0 and variance equal to 1, 1.5 and 2 times the estimated parametric variance was explored.

Semiparametric method. The evaluation of the method focused on study-ing 500 extensively sampled individuals, 2.3 observations / random effect) and sparse (50 individuals with 1 observation / random effect) data with skewed underlying parameter distributions. Skewness was created by simu-lating the parameter distribution using a mixture of 4 subpopulations with different mean and variance values. In this work, the type one error rate was also assessed in order to interpret the likelihood ratio test.

��

3.2.4.2. Real data Performance of all methods was also evaluated using actual clinical data. A remark needs to be made here with respect to the methodology chosen in this part of the work. An obstacle with the real data sets exists in the sense that the true distribution is unknown which makes it difficult to judge whether a method is performing better or worse. Thus, different approaches were used and these are explained briefly below.

38

Default nonparametric estimation method Even though the true distribution of parameters can never been known, there are certain properties of the model which are dependent on how well the estimated parameter distributions resemble the true distribution. These model properties include simulation and predictive properties. Under the assumption that the structural model is not misspecified, simulation and pre-diction properties of the model will to a large extent depend on how well the stochastic mode has been defined, i.e. how well parameter distribution and correlations between them are estimated. Thus, in order to compare different methods, internal and external simulation or prediction properties of the model developed with these methods can be evaluated and compared. In order to evaluate external prediction properties of the model, i.e. how well model can predict future studies and patients, data from these future studies and patients are needed. Clearly, there is little possibility to obtain such data, restricting the evaluation of the models external predictive properties. How-ever, in the evaluation of internal prediction properties of the model, via visual and numerical predictive checks, there is no need for generation of new data, thus this methodology presents an alternative possibility. Specifi-cally, the numerical predictive check was chosen as a quantitative technique for this exercise.

The numerical predictive check (NPC) characterizes model appropriate-

ness by means of prediction intervals (PIs) derived by simulation from final model parameters and parameter distribution estimates. Based on simulated data, prediction intervals are created for each unique observation, so that the simulated observations, usually 1000 of them, corresponding to that particu-lar observation are ranked, and prediction intervals are read off. For exam-ple, the 90% PI would take the 50th and 950th values as the upper and lower prediction interval values. In this manner, 5% of the simulated data is below and 5% is above the 90% prediction interval. Theoretically, if the model is correct, the probability of finding the actual observation outside these predic-tion intervals is the same – a 5% chance that observation will be above or below the prediction interval. Overall, when all observations are evaluated, 10% of all data is expected to be outside the 90% PIs. Numerical predictive checks were performed to compare two major approaches for parameter distribution estimation – parametric and nonparametric using FO and FOCE estimation methods. In total, 4 methods were compared, FO, FOCE, non-parametric preceded by FO (FO-NONP) and nonparametric preceded by FOCE (FOCE-NONP).

Datasets and models used came from 25 completed analyses, previously

developed using the parametric approach. All these models were considered final (29, 60, 70, 71, 74, 78-94). Overall, a wide range of PK (16) and PD

39

(14), sparsely and intensively sampled (3-45 per individual), with various number of subjects (8-637), simple to complex models (stochastic model complexity of 1-9 random effects) were investigated.

Once results were obtained, the number of actual observations was com-

pared with the number of expected observations for each studied prediction interval for each studied case; results were summarized in terms of mean error (ME) and mean absolute error (MAE) as an indicator of imprecision and bias respectively. Lastly, two-tailed t-tests for paired samples were used with a significance level of 5% to compare studied methods (FO vs. FO-NONP; FOCE vs. FOCE-NONP; FO-NONP vs. FOCE; FO-NONP vs. FOCE-NONP) with regard to MAE and ME (95).

Extended grid method The desmopressin data set introduced earlier in this thesis as an example of the sparse data set used to evaluate the transit compartment model, was also used in this study to further evaluate the extended grid method. As men-tioned earlier, these data were sparse, with 139 observations coming from 72 children. The model contained random effects on five parameters (CL, V, Ka, mean transit time (MTT) and number of transit compartments (n) which made it extremely sparse (0.28 observations per random effect) and thus a good example to evaluate the extended grid method. The shrinkage extent was substantial, suggesting rather poor performance of the default nonpara-metric method. The final parameter estimates from the (parametric) model were considered as the true parameters. One hundred datasets were simu-lated from the model using the original sparse study design in children. Non-parametric distributions for all 5 parameters were estimated using the default nonparametric method in NONMEM as well as the extended grid method under 3 different conditions (200, 500 and 1000 additional support points) to enrich the nonparametric grid. Additionally, for each parameter distribution, 90% prediction intervals (PIs) based on 100 nonparametric distributions estimated using the extended grid were constructed and compared to both the true distribution and to the default nonparametric distribution from the origi-nal data.

Semiparametric method The possibility of making model selection based on likelihood ratio test made this case easier to evaluate with real data.

Thirty final models were evaluated for possible improvement in goodness

of fit (significant drop in OFV) after applying three different transformations

40

of the normal distribution. The transformations were added to one random effect at a time; if the inclusion of a transformation was found to be signifi-cant (p<0.05), the transformation was included on more distributions within the same model. Each transformation was tested separately and different transformations were not applied to the same model simultaneously.

�

3.3. Improvement of model diagnostics �

3.3.1. PRED-based diagnostics This diagnostic is often used as being simple and intuitive. A common way of displaying this diagnostic is as a plot of observations versus population predictions. A line of identity, and sometimes also a regression line, is in-cluded to illustrate how well the observations and predictions agree. Some common assumptions are made about how this plot should appear if the model is correct: (i) plotted data should be evenly spread around line of iden-tity, and (ii) the regression line should be superimposed on the identity line.

Here we investigate relevance of these assumptions by creating a simple

example when data are simulated and fitted with the same model and this type of diagnostics is explored to evaluate the relevance of previously men-tioned assumptions. The studied cases include: (i) a PK model with data omission due to censoring, (ii) a dose adaptation study (titration to suitable response), and (iii) sigmoidal Emax model with covariance between Emax and EC50.

3.3.2. RES-based diagnostics This type of diagnostic includes different type of residuals, based on popula-tion prediction (RES=observations – PRED), based on individual prediction (IWRES) or residual known as WRES, which is standard output from soft-ware, such as NONMEM. Often, WRES is used for detection of model mis-specification. Recently, a shortcoming of WRES has been revealed which is that this residual is always computed using the FO method, even when the FOCE method is used. Consequently, a new residual was introduced, the conditional weighted residual (CWRES) which is computed based on the FOCE method (96). The usefulness of these two residuals in the presence of the correct model was investigated. Examples included simulations, fitting

41

and creating diagnostics based on a sigmoid-Emax model, a PK model with Michaelis-Menten elimination, and the transit compartment model.

3.3.3. EBE-based diagnostics Diagnostics based on individual parameter estimated are widely used among modelers and are quite simple to interpret. In guidance’s from both the US FDA (Food and Drug Administration) and the European CHMP (Committee for Medicinal Products for Human use), graphical diagnostics based on indi-vidual parameter estimates are specifically mentioned (48, 49). However, if these diagnostics are to be trusted, so should the underlying individual pa-rameter estimates which are used for their derivation.

The individual parameter estimates are estimated using Bayesian method-

ology, they are generally referred to as empirical Bayes estimates (EBEs) as they represent estimates informed both by a prior population parameter and residual error distribution and the individual data. At one extreme, with no observations available, the patient will be regarded as a typical patient. At the other extreme, when data for an individual goes towards infinity, the prior will have marginal impact; in between these extremes, both factors will contribute and there will be a tendency for individual estimates to be closer to the population mean than the true individual parameter value (Figure 10 and 11).

No data

Sparse dataSparse data

ω

Ri h d

TRUE parameter

Rich data

0ηi

η distribution Figure 10. Concept of ��shrinkage. The estimate of �i (broken blue line) will be dependent on the actual data available. The true �i value is indicated with the red line.

42

ctio

nns

ity F

unab

ility

Den

Ω decrease

Prob

a

Post Hoc η values Figure 11. Impact of data information content on POSTHOC � distribution.

We define the phenomenon of EBE variance shrinking towards the popu-

lation mean as the � – shrinkage (sh�). Similarly, with respect to the IWRES distribution - with informative data IWRES will approach a normal distribu-tion with zero-mean and unit variance, and vice versa, and as data quantity diminishes the IWRES distribution will shrink towards zero. We define this phenomenon as (epsilon) �-shrinkage (sh�). Both, � – and �- shrinkage can be quantified using Equation 13 and 14 respectively.

�

)(1 EBESDsh ��

(13)

)(1 IWRESSDsh �� (14)

The consequences of shrinkage for EBE-based diagnostics were explored

using PK (one compartment, first/zero order absorption) and PD (Emax, indi-rect effect) models. Datasets were simulated from these models and EBEs were estimated in NONMEM based on the true or a misspecified model. If

43

EBEs were estimated using the true model, diagnostics were supposed to confirm expected results with no unexpected trends in graphs. When EBEs were estimated using the misspecified model, diagnostics should reveal the model misspecification. Dataset informativeness was altered with respect to the number of observations (nobs) and the residual error magnitude. The rela-tionship between diagnostics and shrinkage extent was assessed both qualita-tively and quantitatively. Exploration of diagnostics was limited to those mostly widely used, namely: � EBE distribution shape which is used for evaluation of the assumption of

normality

� ETABAR outcome which is a p value obtained from the hypothesis test that speaks to if the mean value of EBEs is significantly different from 0

� parameter vs parameter plots, often used for investigation of potential correlations between parameters

� parameter vs covariate plots, often used for inspection of potential rela-

tionships between parameters and covariates � dependent variable vs IPRED plot, commonly used for detection of struc-

tural model misspecification � IWRES vs IPRED plot, commonly used for assessment of the appropri-

ateness of the residual error model

The relationship between �- and �-shrinkage was also assessed.

44

4. Results

4.1. Structural model development (Papers I-III). The transit model was successfully implemented in a non-linear mixed effect framework and applied to all investigated data sets. In the first comparison, when the transit model was applied to already published models, the transit model was superior to the traditional lag time model, both statistically, in terms of a significant improvement in the fit (�OFV of up to -483 units), and visually, with more accurate descriptions of concentration-time profiles, especially in the absorption phase and around the concentration peak (Fig-ures 12-13). For disposition parameters, CL and V, the population parameter estimates obtained with the transit model were similar to those from former analyses; however more pronounced differences were seen for absorption pharmacokinetic parameters (ka and tlag / MTT). The estimated number of transit compartments for glibenclamide, furosemide, amiloride and moxonidine were 22.9, 20.1, 8.15 and 7.17, respectively. Also, maximum likelihood profiles for parameter n were the same whether they were ob-tained via hard-coding of transit compartments or via implementation of the Stirling approximation, which confirms that both methods lead to similar results (Figure 14). Close inspection of Figure 14 highlights that the uncer-tainty profile of this parameter is asymmetrical.

45

4001.2

Glibenclamide Moxonidine

300

ion

(ng/

ml)

0 6

0.8

1.0

tion

(ng/

ml)

Glibenclamide Moxonidine

100

200

conc

entr

ati

0.2

0.4

0.6

conc

entr

at

0 2 4 0 2 4

Time (h)

0

0 2 4 6 8 0 2 4 6 8

Time (h)

0.0

12

2000

)

Time (h) ( )

Amiloride Furosemide

8

atio

n (n

g/m

l)

1000

1500

atio

n (n

g/m

l)4

conc

entr

a

500co

ncen

tra

0 2 4 6 0 2 4 6

Time (h)

0

0 2 4 6 0 2 4 6

Time (h)

0

Figure 12. A comparison of individual predictions obtained with the lag time model (thin grey line) and the transit compartment (thick black line) models, for 2 repre-sentative profiles of each drug. Observations are represented with an open circle.

0Amiloride

-200

Amiloride

Furosemidesignificant OFV dropof 3.84 (p=0.05)

400FV -400

Δ O

F

Glibenclamide

-600

-800 Moxonidine

NO LAG LAG TRANSIT TRANSIT -IIV(N) Figure 13. Improvement in the goodness-of-fit with the transit compartment model for all investigated compounds. NO LAG – model without a lag time, LAG – model with a lag time, TRANSIT – transit compartment model with estimated IIV in MTT if possible, TRANSIT-IIV (N) – transit compartment model with estimated IIV in number of transit compartments

46

0 5 10 15 20 25 30

Number of transit compartments

1710

1750

1790

1830

OF

V

Hard coded transit compartmentsEstimation of the transit compartments

Figure 14. Maximum likelihood profile for number of transit compartments obtained via hard coding and estimation of transit compartments.

In a second evaluation, the transit model was also selected as be-ing the superior model when describing desmopressin pharmacokinet-ics based on sparse data collected in children. Because desmopressin absorption was very rapid and the estimated number of transit com-partments was low (1.18), an improved version of the Stirling ap-proximation was required in order to minimize the approximation er-ror. The transit model also allowed detection of differences in absorp-tion between pediatric and adult populations, when the combined model was developed.

In a third evaluation, the rifampin data were best described by the transit

compartment model. The model showed substantial improvement over the other tested absorption models, such as lag time model or sequential zero- and first- order absorption process, which produced poorer fits and resulted in both larger unexplained variability in absorption parameters and higher residual variability (Table 1), The transit model was found to be flexible enough to fit a wide range of typical and atypical absorption profiles (Figure 15). Also, as this data came from multiple dosing schedules, the solution of the transit model for multiple dose data was successfully implemented.

Abs

orpt

ion

Mod

el

�OFV

(df)

C

L/F

(IIV

) V

/F (I

IV)

ka (I

IV)

� add

� e

xp

Firs

t-ord

er

0 (0

) 19

.4 (0

.321

) 52

.8 (0

.427

) 1.

61 (2

.39)

0.

256

0.30

2

Firs

t-ord

er a

nd la

g tim

e -8

3.02

6 (1

) 20

.2 (0

.387

) 53

.4 (0

.415

) 1.

64 (0

.689

) 0.

409

0.25

5

Sequ

entia

l zer

o-

and

first

-ord

er

-270

.703

(2)

19.2

(0.3

14)

51.5

(0.3

84)

1.23

(0.7

87)

0.25

4 0.

259

Tran

sit c

ompa

rt-m

ents

-3

91.9

39 (3

) 19

.2 (0

.279

) 53

.2 (0

.188

) 1.

15 (0

.439

) 0.

0923

0.

222

Tab

le I.

Com

paris

on o

f par

amet

er e

stim

ates

and

obj

ectiv

e fu

nctio

n va

lues

from

com

petin

g ab

sorp

tion

mod

els.

OFV

= c

hang

e in

NO

NM

EM

obje

ctiv

e fu

nctio

n va

lue;

df =

deg

rees

of f

reed

om; C

L/F

= ap

pare

nt c

lear

ance

, L·h

-1; V

/F =

app

aren

t vol

ume

of d

istri

butio

n, L

; ka =

abs

orpt

ion

rate

con

stan

t, h-1

; �ad

d = a

dditi

ve re

sidu

al e

rror

, mg·

L-1; �

ccv =

pro

porti

onal

resi

dual

err

or; I

IV =

inte

rindi

vidu

al v

aria

bilit

y (v

aria

nce)

.

48

0 2 4 6 8 0 2 4 6 8

0 2 4 6 8

0

5

10

15

200

5

10

15

20

0

5

10

15

20

Time after dose (h)

Rifa

mpi

n co

ncen

trat

ion

(mg/

L)

Poor fit Typical fit Good fit

Typicalindividuals

LowAUC0-∞

Atypicalabsorption

Figure 15. Plots of the observations (open circles), individual predictions (solid lines), and population predictions (dotted lines) from the final rifampin model, as used to illustrate goodness-of-fit for different classes of individual (typical individu-als, individuals with low RIF exposure, and individuals with atypical absorption profiles). “Poor fit,” “Typical fit,” and “Best fit” denote goodness-of-fit, as catego-rized by inspection of IWRES and WRES

While evaluating the transit model, several observations were made,which

are an interesting prelude to the next results section. Both, the population parameter distribution and uncertainty distribution of parameter n appeared to be skewed. Figure 16 shows the distribution of n in examples of rich data sets, where shrinkage is presumed not to interfere with the results. Moreover, skewness of the uncertainty distribution became apparent while likelihood profiling this parameter to obtain confidence intervals for glibenclimide, moxonidine, amiloride and furosemide examples (Figure 14 and Table 2).

Inspection of WRES plots for the transit compartment model on moxonidine data indicated the presence of remaining model misspecification even though model fit was substantially improved, both in terms of decrease in OFV and visual description of the observed data (Figure 17). Also, the DV vs IPRED plot showed perfect agreement (perfect-fit phenomenon) in the desmopressin example, both when traditional lag time and transit com-partment models were fitted suggesting similar performance for both models (Figure 18).

49

Compound Estimate 95% confidence intervals

Glibenclimide 22.9 10.80 - 64.80 Moxonidine 8.17 4.66 - 8.88 Amiloride 8.13 2.42 - 21.21

Furosemide 20.1 8.60 - 144.57

Table II: Confidence intervals obtained via log-likelihood profiling for parameter n for four studied compounds.

0 0.5 1 1.5 2 2.5

0 0.5 1 1.5 2 2.5

ηn

0.0

0.5

1.0

1.5

2.0

2.5

IIV IOV

Figure 16. Empirical Bayes estimate distribution of parameter n with an underlying assumption that the random variable �n is normally distributed.

50

Moxonidine, transit compartment model

Time after dose

Wei

ghte

d re

sidu

als

−10

−5

0

5

0 2 4 6 8

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Figure 17. WRES indicating model misspecification after applying the transit com-partment model on moxonidine data, despite the substantial improvement in the goodness of fit (�OFV = 483) compared to the lag- time model

Lag

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e m

od

el

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vidu

al p

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re 1

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s IPR

ED p

lot f

or tw

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odel

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e “p

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52

4.2. Stochastic model development (Papers IV-VII) 4.2.1. Performance of the nonparametric estimation method

(Papers IV-V) 4.2.1.1. Evaluation via MC simulation (Paper IV)

Overall, the novel nonparametric estimation method could identify non-normal parameter distributions and correct bias in parameter estimates ob-served when applying the true models with the FO estimation method.

For all default simulated scenarios (Nsubjects=200, nobs=1.5/random ef-

fect/individual, underlying distributions = log-normal, bimodal and heavy tailed), the estimated nonparametric parameter distribution matched the true parameter distribution with an absolute value of relative bias (ARB) of less than 4 % at all evaluated percentiles, regardless of the method chosen for support point estimation (FO or FOCE) (Figure 19-22). The results obtained with the nonparametric method were as good as the results obtained with the true model fitted with the FOCE method. If the true model was fitted with the FO method then the nonparametric results were an improvement over results obtained with the true model. There was even an improvement over the fit with the true model (mixture model for the true bimodal distribution and parametric model for true log-normal distribution) when FO was the method of estimation.

53

10 25 50 75 90 10 25 50 75 90 10 25 50 75 90 10 25 50 75 90

Percentiles

-10

0

10

20

30

Rel

ativ

e es

timat

ion

erro

r (%

)P_FO Nonp_FO P_FOCE Nonp_FOCE

Log-normal data

10 25 50 75 90 10 25 50 75 90 10 25 50 75 90 10 25 50 75 90 10 25 50 75 90 10 25 50 75 90

Percentiles

-50

0

50

100

Rel

ativ

e es

timat

ion

erro

r (%

)

P_FO Mix_FO Nonp_FO P_FOCE Mix_FOCE Nonp_FOCE

Quadro-modal data

54

10 25 50 75 90 10 25 50 75 90 10 25 50 75 90 10 25 50 75 90 10 25 50 75 90 10 25 50 75 90

Percentiles

-50

0

50

100

Rel

ativ

e es

timat

ion

erro

r (%

)P_FO Mix_FO Nonp_FO P_FOCE Mix_FOCE Nonp_FOCE

Trimodal data

10 25 50 75 90 10 25 50 75 90 10 25 50 75 90 10 25 50 75 90 10 25 50 75 90 10 25 50 75 90

Percentiles

-20

0

20

40

60

Rel

ativ

e es

timat

ion

erro

r (%

)

P_FO Mix_FO Nonp_FO P_FOCE Mix_FOCE Nonp_FOCE

Heavy-tailed data

Figure 19-22 . Distribution of REE (n=100) at 5 evaluated distribution percentiles (10th, 25th, 50th, 75th and 90th) using different estimation methods for 4 studied sce-narios. Estimation methods are noted as P_FO – parametric method with FO, P_FOCE – parametric method with FOCE, Mix_FO – mixture method with FO, Mix_FOCE – mixture methods with FOCE, Nonp_FO – nonparametric method preceded with FO and Nonp_FOCE - nonparametric method preceded with FOCE. In Figure 22, the scale was narrowed down for the purpose of clarity, thus results from P_FO are not fully visible.

55

When small datasets were analyzed (Nsubjects=50, nobs=1.5/random ef-fect/individual, underlying distributions = log-normal, bimodal and heavy tailed), results from all methods were less precise. However the NONP method still showed either an improvement in bias or similar performance compared to the true model in all instances.

When sparse data sets were studied (Nsubjects=50, nobs=1/random ef-

fect/individual, underlying distributions = log-normal, bimodal and heavy tailed) the NONP method still showed an improvement or performed simi-larly compared to the true model for almost all scenarios, except when the true distribution was bimodal and the method to assess points of support was FO. This was mainly due to inability of the FO method to separate different sources of variability. In this particular case, unexplained model variability was mainly expressed as high residual variability. The estimate of residual variability from the FO estimation showed both high bias and imprecision with a median value of 34 % (14.5 – 48.9 % for the 5th and 95th percentiles respectively), where the true value was 10%.

4.2.1.2. Evaluation using real data (Paper V) Overall, less imprecision and less bias were observed with nonparametric methods than with parametric methods in terms of the outcome of the nu-merical predictive check in this exercise. Across the 25 models that were evaluated, t-tests revealed that imprecision and bias were significantly lower (p<0.05) with FOCE-NONP than with FOCE for half of the NPC outcomes investigated. Improvements were even more pronounced with FO-NONP in comparison with FO. This indicates improved simulation properties of the model when the parameters are estimated with the nonparametric estimation method.

An example is displayed in Table 3, which shows the values of low, high and total outlier percentages of the 90%PI obtained for each of the 25 mod-els and after running FO and FO-NONP methods. Similar tables were cre-ated for all studied methods (FOCE and FOCE-NONP) and prediction inter-vals (50% and 90% PIs). Each numerical predictive check performed was based on 1000 simulations. By inspecting these tables, a general trend was observed that results from the nonparametric simulations were closer to ex-pected results than those from parametric simulations.

56

FO FO-NONP NPC: 90% Prediction Inter-val

Percentages of outliers 90% U 90% L 90% T 90% U 90% L 90% T

PK1 - Cladribine 7.08% 2.99% 10.07% 6.44% 5.17% 11.62%

PK2 - Pefloxacin 5.93% 6.82% 12.76% 5.64% 5.34% 10.98%

PK3 - Glibenclamide lag time 0.35% 7.32% 7.67% 3.14% 9.41% 12.54%

PK4 - Glibenclamide transit 0.35% 6.97% 7.32% 2.44% 9.06% 11.50% PK5 - Desmopressin joint

transit 0.20% 0.78% 0.98% 2.34% 2.54% 4.88%

PK6 - Levosimendan 3.62% 4.46% 8.08% 4.46% 5.01% 9.47%

PK7 - Antibody atm-027 0.24% 0.48% 0.73% 3.39% 1.21% 4.60%

PK8 - Moxonidine lag time 8.41% 2.35% 10.76% 6.36% 3.03% 9.39%

PK9 - Moxonidine transit 6.56% 4.40% 10.96% 5.97% 5.09% 11.06%

PK10 - Gefitinib 1.84% 3.40% 5.25% 4.40% 3.55% 7.94%

PK11 - Melagatran 4.01% 7.45% 11.46% 3.78% 7.01% 10.79%

PK12 - Pirazinamide 4.56% 2.88% 7.44% 4.17% 3.49% 7.66%

PK13 - Prazosin 2.25% 6.31% 8.57% 3.83% 7.33% 11.16%

PK14 - Antibody X 7.51% 2.68% 10.20% 7.87% 3.04% 10.91%

PK15 - Voriconazole 1.18% 11.46% 12.64% 5.34% 4.47% 9.81%

PK16 - Tobramycin 4.04% 5.28% 9.32% 5.28% 6.52% 11.80%

PD1 - Cladribine 7.62% 0.88% 8.50% 5.28% 7.92% 13.20%

PD2 - Docetaxel 3.80% 1.80% 5.60% 6.05% 1.32% 7.37%

PD3 - Levodopa 5.05% 7.29% 12.34% 4.35% 7.05% 11.40%

PD4 - Moxonidine SBP 6.13% 4.33% 10.45% 5.92% 4.74% 10.66%

PD5 - Moxonidine HR 2.37% 7.46% 9.83% 4.84% 5.25% 10.08%

PD6 - Moxonidine NA 7.21% 3.69% 10.90% 6.16% 5.74% 11.90%

PD7 - Tezaglitazar FPG 3.07% 4.21% 7.29% 4.93% 3.54% 8.48%

PD8 - Tezaglitazar HbA1c 5.55% 3.95% 9.50% 4.70% 5.10% 9.80%

PD9 - Digoxin 3.05% 2.41% 5.46% 2.92% 6.10% 9.02%

Table 3. Observed total (90% T), upper (90% U), and lower (90% L) outlier per-centages of the 90% prediction interval obtained for each model using the paramet-ric estimation (FO) and nonparametric estimation (FO-NONP) methods with NONMEM VI, respectively.

57

The mean absolute errors (MAEs) and mean errors (MEs) computed for FO, FO-NONP, FOCE and FOCE-NONP and for all the outcomes of the NPC (including the above, below and total percentages of outliers of the 90% and 50% PIs) are presented in Table 4 and 5, as well as the p-values of the different t-tests performed during the statistical analyses.

Tabl

e 4.

Mea

n ab

solu

te e

rror

s (M

AEs

) of t

he re

sults

obt

aine

d af

ter r

unni

ng a

num

eric

al p

redi

ctiv

e ch

eck

for e

ach

of th

e 25

mod

els w

ith 4

di

ffer

ent e

stim

atio

n m

etho

ds. P

rese

nted

are

out

lier p

erce

ntag

es o

f tot

al (T

), up

per (

U) a

nd lo

wer

(L) o

bser

vatio

ns o

f the

90%

PI a

nd o

f the

50

% P

I. Th

e ou

tcom

es o

f the

stat

istic

al t-

test

s per

form

ed a

re a

lso

sum

mar

ized

in th

is ta

ble.

The

ast

eris

ks in

dica

te p

<0.0

5.

p-va

lue

p-va

lue

p-va

lue

p-va

lue

FO v

s FO

-NO

NP

FOC

E v

s FO

CE

-NO

NP

FO-N

ON

P vs

FO

CE

FO-N

ON

P vs

FO

CE

-NO

NP

90%

U

0.02

30.

011

0.02

10.

012

0.0

00 *

0.0

02 *

0.0

01 *

0.70

8

90%

L

0.02

20.

016

0.01

30.

011

0.07

80.

298

0.23

60.

053

90%

T0.

024

0.01

60.

021

0.01

6 0

.020

*0.

114

0.17

60.

918

50%

U0.

056

0.01

70.

033

0.02

0 0

.000

* 0

.014

* 0

.000

*0.

513

50%

L

0.05

20.

026

0.03

50.

025

0.0

14 *

0.12

00.

230

0.84

2

50%

T

0.04

60.

025

0.05

10.

025

0.0

07 *

0.0

00 *

0.0

01 *

0.92

5

FOC

E

FOC

E-N

ON

P M

AE

sFO

FO

-NO

NP

Tabl

e 5.

Mea

n er

rors

(ME)

of t

he re

sults

obt

aine

d af

ter r

unni

ng a

num

eric

al p

redi

ctiv

e ch

eck

for e

ach

of th

e 25

mod

els w

ith 4

diff

eren

t est

ima-

tion

met

hods

. Pre

sent

ed a

re o

utlie

r per

cent

ages

of t

otal

(T),

uppe

r (U

) and

low

er (L

) obs

erva

tions

of t

he 9

0% P

I and

of t

he 5

0% P

I. Th

e ou

t-co

mes

of t

he st

atis

tical

t-te

sts p

erfo

rmed

are

als

o su

mm

ariz

ed in

this

tabl

e. T

he a

ster

isks

indi

cate

p<0

.05.

p-va

lue

p-va

lue

p-va

lue

p-va

lue

FO v

s FO

-NO

NP

FOC

E v

s FO

CE

-NO

NP

FO-N

ON

P vs

FO

CE

FO-N

ON

P vs

FO

CE

-NO

NP

90%

U

0.00

90.

002

0.00

60.

002

0.0

46 *

0.22

80.

280

0.89

4

90%

L

0.00

5-0

.001

0.00

4-0

.002

0.19

3 0

.010

*0.

240

0.78

1

90%

T0.

014

0.00

10.

010

-0.0

01 0

.003

* 0

.006

*0.

093

0.67

0

50%

U0.

024

0.00

30.

010

0.00

90.

117

0.94

10.

347

0.26

4

50%

L

0.00

20.

012

0.02

10.

006

0.52

0 0

.030

*0.

337

0.30

8

50%

T

0.02

60.

015

0.03

00.

015

0.28

00.

072

0.09

90.

992

FO

FO-N

ON

PFO

CE

FO

CE

-NO

NP

MEs

60

4.2.2. Performance of the extended grid method (Paper VI) 4.2.2.1. Evaluation using MC simulations

For both the small and rich data sets, the parameter distributions estimated using the extended grid method showed good agreement with the true pa-rameter distributions examined visually by comparison of their cumulative density functions (Figure 23).

Scenario A, a small data set

1.0

Scenario A, a small data set

0.8

nctio

n

0.6

babi

lity

fun

0.4

lativ

e pr

ob

True distributionExtended grid NP distributionDefault NP distribution

0.2

Cum

ul

0 50 100 150 200 250

0.0

0 50 100 150 200 250

Volume of distribution

Scenario B, a sparse data set

1.0

0.8

func

tion

0.6

obab

ility

f

0.4

ulat

ive

pro

True distribution

0.2

Cum

u True distributionExtended grid NP distributionDefault NP distribution

0.0

0 50 100 150 200 250

Volume of distribution Figure 23. A visual comparison of distributions for volume of distribution estimated using a default nonparametric method and extended grid method with the true distri-bution in case of small and sparse data set. A randomly chosen example is shown for both scenarios.

61

In the case of the small data sets, bias was slightly larger for the extended grid method, but surprisingly low for both methods (less than 10%). How-ever, imprecision was decreased with the extended grid method (Figure 24). The parameter distribution estimated on the extended grid was also smoother compared to the default nonparametric distribution.

In the case of the sparse data sets, bias was slightly lower for the ex-tended grid method while precision was substantially improved with this method compared to the default method. Addition of 400, 800 and 1600 support points in the extended grid improved results even further with de-creased bias and similar variance compared to default addition of 200 sup-port points (Figure 25). Using inflated variances to generate additional points of support did not further improve results for any of the studied cases.

40

default extended

40

rror

(%)

20

entil

e er

0

ve p

erce

-20

Rel

ativ

5 10 20 25 30 50 70 75 80 90 95 5 10 20 25 30 50 70 75 80 90 95

-40

5 10 20 25 30 50 70 75 80 90 95 5 10 20 25 30 50 70 75 80 90 95

Percentiles

default extended_200 extended_400 extended_800 extended_1600

40

20

rror

(%)

0rcen

tile

erel

ativ

e pe

-20Re

-40

5 10 20 25 30 50 70 75 80 90 95 5 10 20 25 30 50 70 75 80 90 95 5 10 20 25 30 50 70 75 80 90 95 5 10 20 25 30 50 70 75 80 90 95 5 10 20 25 30 50 70 75 80 90 95

Percentiles Figure 24-25. RPEs (n=100) at 11 evaluated distribution percentiles (5th, 10th, 20th, 25th, 30th, 50th, 70th, 75th , 80th, 90th and 95th) using default nonparametric estimation method in NONMEM (“default”) and extended grid method (”extend”) for small data scenario (Figure 24) and sparse sample scenario (Figure 25).

62

4.2.2.2. Evaluation using Real data The default nonparametric method performed poorly in this case, especially for the estimation of the absorption parameter distributions. The extended grid method improved results substantially for all parameter distributions, both in terms of decreased bias and variances of relative percentile errors. Also, prediction intervals derived based on the distribution estimated on the extended grid showed good agreement with the true parameter distributions. This was not the case with the default nonparametric parameter distribution estimated using the true model and the original data set (Figure 26).

10

True

dis

tribu

tion

Def

ault

NP

dis

tribu

tion

ηC

Ldi

strib

utio

n

0.8

1.0

ensity

90%

PI -

ext

ende

d gr

id N

P d

istri

butio

nm

edia

n - e

xten

ded

grid

NP

dis

tribu

tion

0.6

bability d 0.4

tive prob 0.2

Cumulat

06

01

04

09

0.0

-0.6

-0.1

0.4

0.9

ηC

L

True

dis

tribu

tion

Df

ltN

Pdi

tib

ti

ηV

dist

ribut

ion

08

1.0

ensity

Def

ault

NP

dist

ribut

ion

90%

PI -

ext

ende

d gr

id N

P d

istri

butio

nm

edia

n - e

xten

ded

grid

NP

dist

ribut

ion

0.6

0.8

ability de 0.4

ive proba 0.2

Cumulat

07

02

03

08

0.0

C

-0.7

-0.2

0.3

0.8

ηV

ηka

dist

ribut

ion

1.0

y

True

dis

tribu

tion

Def

ault

NP

dist

ribut

ion

90%

PI -

ext

ende

d gr

id N

P di

strib

utio

nm

edia

nex

tend

edgr

idN

Pdi

strib

utio

n

0.8

y density

med

ian

- ext

ende

dgr

idN

Pdi

strib

utio

n

0.6

obability 0.4

ative pro 0.2

Cumula 0.0

-3-2

-10

12

ηka

η MTT

dist

ribut

ion

1.0

True

dis

tribu

tion

Def

ault

NP

dist

ribut

ion

90%

PIex

tend

edgr

idN

Pdi

strib

utio

n

η MTT

dist

ribut

ion

0.8

density90

%PI

- ex

tend

edgr

idN

Pdi

strib

utio

nm

edia

n - e

xten

ded

grid

NP

dist

ribut

ion

0.6

abilityd 0.4

ve prob 0.2

umulativ 0.0

C

-0.7

-0.2

0.3

0.8

ηM

TT

ηN

Ndi

strib

utio

n

1.0

ty

True

dis

tribu

tion

Def

ault

NP

dis

tribu

tion

90%

PI -

ext

ende

d gr

id N

P d

istri

butio

nm

edia

n-e

xten

ded

grid

NP

dist

ribut

ion

0.8

y densit

med

ian

exte

nded

grid

NP

dist

ribut

ion

0.6

obability 0.4

ative pro 0.2

Cumula 0.0

C

-4-2

02

4η

NN

Figu

re 2

6. S

pars

e pa

edia

tric

exam

ple:

90

% p

redi

ctio

n in

terv

als (

blue

line

) and

a m

edia

n (b

lack

line

) for

eac

h of

5 p

aram

eter

dis

tribu

tions

cr

eate

d ba

sed

on 1

00 e

stim

ated

non

para

met

ric d

istri

butio

ns u

sing

the

exte

nded

grid

met

hod

(100

0 ad

ditio

nal s

uppo

rt po

ints

), an

d its

com

pari-

son

to th

e tru

e (g

rey

line)

and

def

ault

nonp

aram

etric

(red

line

) dis

tribu

tion

estim

ated

with

the

orig

inal

real

dat

a, re

spec

tivel

y.

64

4.2.3. Performance of the semiparametric method (Paper VII)

4.2.3.1. Evaluation using MC simulation In the simulation study, the inclusion of an estimated transformation that allows a skewed distribution (logit and box-cox) provided a better fit to the data, both in terms of better estimation of underlying parameter distributions and significant drops in OFV, for both rich and sparse data (Figure 27, 28). Inclusion of the transformation also improved the models’ simulation per-formances – this can be seen in Figure 29 where a visual predictive check for an individual data set is shown. In this example, misspecification of the pa-rameter distribution by using a fixed log-normal distribution leads to a visi-bly poorer agreement with observations compared to a model based on box-cox transformations. Of note, including a transformation that can adapt to the skewed distribution corrected for this overprediction.

Clearance

Den

sity

0.02

0.04

0.06

20 40 60 80

True MixtureLognormalBox−coxMixture

Figure 27. Randomly selected example of a comparison of the true and estimated distributions using mixture, parametric, and box-cox transformation models.

65

Relative percentile error and Difference in OFV, 500 individuals, 7 observations per individual

Percentiles

Rel

ativ

e er

ror

−0.4

−0.2

0.0

0.2

0.4

10 20 30 40 50 60 70 80 90

�

�

�

�

�

�

�

�

�

�

� �

�

�

�

�

Normal

10 20 30 40 50 60 70 80 90

��

�

� �

�

��

�

��

��

� ��

��

Logit

10 20 30 40 50 60 70 80 90

��

��

�

�

��

�

��

�

� ��

�

� ��

��

Box−Cox

10 20 30 40 50 60 70 80 90

�

�

�

��

�

�

�

�

�

�

�

�

�

�

��

�

�

��

�

�

Heavy tailed

10 20 30 40 50 60 70 80 90

��

��

�

�

�

��

� � �

�

Mixture (true)

Difference in OFV

dOFV

−150

−100

−50

0

50

100Normal

�

Logit

�

Box−Cox

�

��

Heavy tailed

�

�

�

��

Mixture (true)

Figure 28. Relative errors of investigated percentiles and differences in OFV be-tween fixed lognormal distribution (Normal) and other models for rich data with 500 individuals and 7 observations per individual.

Time

Con

cent

ratio

n

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

2 4 6 8 10 12 14

Observed medianSimulated median (95% CI)Observed 95% PISimulated 95% PI (95% CI)

Time

Con

cent

ratio

n

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

2 4 6 8 10 12 14

Observed medianSimulated median (95% CI)Observed 95% PISimulated 95% PI (95% CI)

Figure 29. Visual predictive checks for models with a misspecified lognormal clear-ance distribution (left) and after inclusion of a box-cox transformation with esti-mated shape parameters (right). 95 percentiles and median of observations are com-pared to the 95% confidence intervals of simulated 95 % prediction intervals (PI) and median.

4.2.3.2. Evaluation using Real data Parameter transformations appeared to be useful when added into existing models with the real data. A significant drop in OFV was seen in 20, 19 and

66

22 of the models (out of 30) for the logit, box-cox and HT transformations, respectively (Figures 30 and 31).

PK

1

PK

2

PK

3

PK

4

PK

5

PK

6

PK

7

PK

8

PK

9

PK

10

PK

11

PK

12

PK

13

PK

14

PK

15

PK

16

PK

17

PK

18

PK

19

PK

20

PK

21

Dec

reas

e in

obj

ectiv

e fu

nctio

n va

lue

0

10

20

30

40

50

60LogitBox−CoxHeavy tailed

Figure 30. Improvement in goodness-of-fit expressed as a decrease in OFV after implementation of distribution transformations in all real PK models. The dotted line represents a significance level of 0.05 for inclusion of one model parameter.

PD

1

PD

2

PD

3

PD

4

PD

5

PD

6

PD

7

PD

8

PD

9

Dec

reas

e in

obj

ectiv

e fu

nctio

n va

lue

0

50

100

150

200

250

LogitBox−CoxHeavy tailed

Figure 31. Improvement in goodness-of-fit expressed as a decrease in OFV after implementation of distribution transformations in all real PD models. The dotted line represents a significance level of 0.05 for inclusion of one model parameter.

67

4.3. Model diagnostics (Papers VIII-IX) 4.3.1. PRED-based diagnostics

PRED-based diagnostics, i.e. a plot of the observations versus population predictions (PRED) generated from the true model, for three studied condi-tions is shown in Figure 32. These results indicate that the commonly made assumptions such as equal spread of the data around line of identity and overlap of the regression line and the identity line do not hold true in these cases. Thus, it is likely that the true models would be rejected, if these as-sumptions are not questioned.

Imax model with high variability in EC50

Population predictions

Obs

erva

tions

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

●●

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Imax model with high variability in EC50, BASE and Emax


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tions

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One compartment model with a first order absorption and a lag time


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tions

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Figure 32. Observations versus population predictions when observations are simu-lated with the same model as is used to calculate population predictions. The black line is the line of identity, and the red line is a linear regression line.

68

4.3.2. RES-based diagnostics The comparison of two the residuals, WRES and CWRES, derived based on the fit of the true model to the data is shown in Figure 33. Clearly, if WRES is to be trusted, all the true models are likely to be rejected, as in all exam-ples shown, WRES indicated trends in the plots. When CWRES was applied, this trend disappeared or remained in the sigmoidal Emax model and the Michaelis Menten model respectively indicating better properties of CWRES, but with apparent limitations too.

Figure 33. Conditional weighted residuals (CWRES) and weighted residuals (WRES) versus independent variable plots when both CWRES and WRES were calculated from the correct models.

69

While evaluating the transit compartment model on the moxonidine data set, the two residuals showed similar properties. Even though the model fit showed clear structural improvement after applying the transit compartment model, WRES plots indicated a model misspecification trend. With CWRES, this trend disappeared (Figure 34).

Moxonidine, transit compartment model

Time after dose

Res

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Figure 34. CWRES indicates the disappearance of the WRES-indicated model mis-specification trend identified earlier using the transit compartment model on the moxonidine data set

4.3.3. EBE-based diagnostics In general, in the absence of shrinkage, all EBE-based diagnostics are essen-tial and powerful model evaluation tools. These diagnostics separate differ-ent sources of variability, thus graphs become easier to interpret. However, in the presence of shrinkage, commonly used EBE-based diagnostics begin to lose their informativeness and could potentially become misleading.

The presence of �-shrinkage may have several consequences. In addition to shrinkage, the EBE distribution may also be asymmetric, which could cause ETABAR to be significantly different from 0, even if the model is correctly specified. An example of this can be seen in Figure 35 where dis-tribution of ETABARs obtained after fitting the true model (100 samples) using several study designs in which the times of the first observations was

70

varied, is shown. For �ka, when the first sample is taken early enough, i.e. when datum is informative on this parameter, ETABAR is close to zero, as expected. If the first sample is collected at a later time, this sample is not particularly informative for individuals that have fast absorption because their absorption process is essentially complete by that time, but it will still be informative for individuals that have slow absorption; thus shrinkage is expected to occur mainly for individuals with high ka values, which leads to asymmetric shrinkage resulting in ETABAR values significantly different from zero.

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When EBEs are used for inspections for parameter correlations, they can

indicate a correlation when there is none (Figure 36, upper panel). Con-versely, correlations would sometimes not been detected by EBEs when they in fact are present (Figure 36, lower panel). Commonly induced correlations include those between �EC50 ~ �Emax and �ka ~ �V. Upper and lower left corner panels of Figures 36 represent the true parameter relationships, while other

71

panels show relationships based on EBEs estimated under the true model for different study designs. The quantitative relationship shown in Figure 37 indicates that both phenomena, induced and hidden correlations, become apparent when shrinkage is higher than 20-30 %.

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73

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75

One consequence of �-shrinkage is that the power of IPREDs to detect model misspecification is reduced. This is shown in Figure 41 where a mis-specified model was fitted to the data generated with another model design with varying degrees of informativeness. Here, the IPRED vs DV plot is expected to reveal model misspecification, however with increased shrink-age, it fails to do so. This is consequence of the so-called “perfect-fit phe-nomenon” where the individual prediction is shrinking towards the actual observation and eventually all IPREDs would be lined up on the line of iden-tity.

a.) Linear Emax model fitted to data simulated with a sigmoidal Emax model (Nindividuals=200)

b.) One-compartment disposition PK model fitted to data simulated with a two-compartment model (Nindividuals=100)

c.) First order absorption PK model fitted to data simulated with a transit compartment absorption model (Nindividuals=100)

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Figure 41. Observed versus individual predictions for three different structural model misspecifications at varying degrees of information in data, expressed through the �-shrinkage value.

76

Similarly, the power of IWRES to detect residual model misspecification diminishes with increase in � -shrinkage. Figure 42 shows IWRES diagnos-tics produced based on a misspecified model (proportional residual error only) fitted to the data simulated with the model that had combined error model (additive + proportional residual error).

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77

5. Discussion

5.1. A novel structural model for absorption One of the aims of this doctoral thesis was to develop and evaluate a novel structural model for absorption. An adequate absorption model is essential for studying drug absorption properties and may be of importance in the development of pharmacodynamic models on the basis of predicted concen-tration-time profiles. Probably the most commonly used model for oral drug administration in pharmacometric analysis is first order absorption model with a lag time. This model has three serious drawbacks. First, the lag time model predicts a sudden increase in plasma concentration, due to a sudden increase in absorption rate from zero to a maximal value. This is not likely to be a realistic physiological description of the drug absorption process, thus leading to often poor description of the absorption phase. Second, it is diffi-cult to estimate a variability term in this parameter, as the partial derivative of the function with respect to the lag time, necessary for variability estima-tion, is not defined at the lag time value itself. This finding was confirmed in our first example in Paper I. Third, the lag time model often produces pa-rameter estimates biased by the observation times. This was observed in our first analysis, where the estimated tlag values for furosemide and glibenclim-ide were 0.5 h and 0.3 h, respectively. These values were very close to the time of the first observation after dosing (0.5 h for furosemide and 0.333 for glibenclamide).

This body of work introduces a transit compartment model, which enables

numerical estimation of the optimal number of transit compartments neces-sary for the full description of drug-absorption delay. The transit model seems to be able to address above-mentioned drawbacks observed when using the lag time model. Furthermore, the transit model also provides better physiological description of the absorption profiles and results in statistically significant improvement in model fit.

The transit model describes the potential delay in absorption as drug tran-

sit through a chain of identical compartments that are linked to the central compartment by a first-order absorption process. The same principle has been applied in the population PK model developed by Rousseau et al., for orally administered cyclosporine, where the absorption is described by a

78

linear chain of 5 compartments placed upstream of the central compartment and connected by a single exiting rate constant (68). The concept of transit compartments has also been used in population pharmacoki-netic/pharmacodynamic (PK/PD) analysis for modeling delayed PD re-sponses (69) . An example is the semi-physiological model for myelosup-pression developed by Friberg et al., which successfully used a chain of transit compartments to mimic the different cell stages within the bone mar-row in order to model the time course of leukocytes after varying schedules of anticancer drug (67).

In the transit compartment models that have been published so far, the op-

timal number of transit compartments is assessed by stepwise addition of one compartment at a time. The analytical solution derived for the transit model describes the absorption delay and is given by the gamma distribution func-tion. Thus, the absorption profile obtained by the transit model should be distinguished from the absorption profile modeled by means of the Erlang distribution function, which is a special case of the gamma distribution func-tion when n is constrained to an integer number. The Erlang function used by Rousseau et al., is equivalent to the step-wise addition approach (68). The usage of the Erlang distribution function, as a discrete function, requires manual optimization of number of transit compartments whereas the transit model described in this thesis is able to determine the optimum number of the pre-systemic compartments by computation. This offers three advan-tages. First, the manual optimization of transit compartments is a time-requiring procedure especially when the optimal number of transit compart-ments is high. With numerical estimation the time required for the analysis is shortened. Second, the transit model allows estimation of both, a low and high number of transit compartments; the latter is not possible with manual optimization, since NONMEM allows maximally up to 20 compartments to be coded within the control stream. When the number of transit compart-ments needed is low, an improved version of Stirling approximation has been suggested in order to correct for bias and stabilizes the system due to approximation error of the original Stirling approximation. Lastly, the transit model can be extended with additional parameters, such as IIV and covari-ance terms for n.

The original implementation of the model was derived for and applied to

single dose data, however an extension of the model for application to multi-ple dose data was developed when such data became available for analysis. This extension was developed under the assumption that the entire ingested dose has reached the absorption site before the next dose is administered. Even though this assumption is reasonable to make for most dosing sched-ules, it needs to be verified prior to model application. Ingestion of multiple tablets in a short time period, diseases which may lead to delayed gastric

79

emptying etc may be circumstances when the application of this model would not be appropriate. The transit model was evaluated on both rich (moxonidine, glibenclamide, rifampin) and sparse data sets (furosemide, amiloride, desmopressin). Considerable improvements in goodness of fit were observed with the rich data sets. However, when sparse data were ana-lyzed, the improvement in the fit was not as pronounced in terms of OFV drop, but still statistically significant and visible in goodness-of-fit plots. Additionally, the transit model was more stable and allowed estimation of the lag phase length and IIV in MTT, the latter being not possible with the lag time model. This suggests that the transit model may also perform better in comparison to the lag time model when analyzing sparse data.

5.2. Stochastic model development In order to improve stochastic model building methodology, we introduced three novel methods for building stochastic models for cases when the as-sumption of normal distribution of random variability was not appropriate. Methods were evaluated both via extensive simulation studies and via appli-cation to real data sets.

The nonparametric estimation method introduced in NONMEM VI per-

formed well in the majority of the studied cases. It was capable of precisely estimating the underlying distribution with very low bias and imprecision. Other algorithms, such as NPML, NPEM and NPAG, solve the complex problem of estimation n optimal support points for each parameter, n being number of individuals, by using different algorithms and techniques. In con-trast, the support points in NONMEM are the empirical Bayes estimates (EBEs) of individual parameters. Thus, the support point estimation in the nonparametric method in NONMEM is a parametric problem, and so is eas-ier and faster to solve. While in most cases this novel approach appears as an advantage, there are circumstances when it could be a disadvantage. The EBEs are a compromise between the data for an individual and the popula-tion parameter distribution. As data per individual are reduced, the reliance on the population distribution becomes stronger, which was introduced ear-lier in this thesis as a shrinkage phenomenon. Thus, the performance of the nonparametric estimation method will be dependent on how well the EBEs are estimated. If there is no substantial shrinkage, the nonparametric method will perform well. Conversely, if shrinkage is substantial, the available range of support points will not be sufficient for precise parameter distribution estimation, thus the nonparametric method will perform less well. This is a general phenomenon where precise estimation of parameter variability is challenging when individual data are sparse and noisy, not only for non-

80

parametric methods, but also for less numerically demanding parametric methods.

An additional challenge for nonparametric methods is to precisely esti-

mate the entire probability density function from such data. On a similar note, like all statistical methods, the nonparametric method becomes less precise when datasets are small in terms of number of individuals. In this case, the nonparametric grid will be sparsely populated as the number of support points has to be equal to the number of individuals. These two cir-cumstances were a motivation for development of a novel method, the ex-tended grid method, which had the aim of improving estimation properties in these two extreme cases. Indeed, with yet another way of obtaining support points, the extended grid method offered further improvement in parameter distribution estimation when compared with the default method. The results were more precise the greater the number of introduced support points in the nonparametric grid, although, introduction of additional support points above 400 per random effect resulted in only marginal improvements.

It was shown that simple simulations from the normal parametric distribu-

tion were sufficient to generate these additional support points which cov-ered the parameter space sufficiently well. In order to generate additional support points, simulations from the normal distribution using a mean and variance estimated in preceding parametric analysis were performed, which seemed to be satisfactory. The majority of parametric methods are adequate when quantifying the variance of parameter distribution. However, this ap-proach can be questioned if the parametric estimate of variance is too low or strong covariances indicate perfect correlations. This was also observed spo-radically in our analyses. If that is the case, the variance used to simulate support points needs to be inflated. In our case, inflation of variances did not help to further improve the results– it rather slightly biased the results. Also, the nonparametric OFV, which is the measurement of the exact likelihood, increased. There is little experience of using nonparametric OFV for model building decisions. Rival nonparametric models are seldom nested so the likelihood ratio test based on OFV cannot be used. For non-nested models with different number of parameters, where each support point represents a parameter, criteria such as the Akaike Criterion may possibly be useful, but no experience to support such use exists to date. However, for two models which have the same number of parameters, a comparison based on OFV seems appropriate with a lower OFV indicating a better fit.

Both the nonparametric and extended grid methods, exhibited a good per-

formance when evaluated on the real data. The default nonparametric method was capable of improving simulation properties of the model, even though there was little room for improvement as these models were already

81

developed and considered as final, implying already satisfactory simulation properties. The improvement was observed irrespective of how much shrinkage there was in the original analysis. Indeed, there was a varying degree of shrinkage present in each real data example which may have influ-enced nonparametric distribution estimation, as well as simulations from such a distribution. However, the results were still an improvement over classical parametric simulations. If nonparametric distributions were esti-mated with the extended grid method, it is likely that results would be im-proved even further, as this method is not influenced by shrinkage and there-fore has a better performance.

Despite that nonparametric methods have been available for a long time,

they are not used nearly as often as parametric methods. There are several reasons for this including a) long run times, b) no methodology to separate different sources of variability, c) lack of methods for imprecision measure-ment, d) inconsistency in input data formatting to commonly used formats for other platforms, e) lack of a formal way for performing model compari-son, and f) no direct approaches for covariate model building.

The methods presented in this thesis, as well as some recent develop-

ments of the methodology in the field, may hopefully help change this trend based on following facts: (i) the nonparametric method based on EBEs is likely to be faster because it does not involve nonparametric estimation of support points. This was also confirmed in study of Antic et al where the method in NONMEM appeared to be the fastest one among nonparametric methods (97). (ii) A novel way of specifying residual error model has been proposed which involves estimation of entire (nonparametric) distribution of the residual variability. This method seemed to be an improvement over traditional methods of fixing the residual error to the assay error value or more recent method of quantifying this error with the parametric software (42, 98). (iii) A novel method for uncertainty estimation has been developed which offers several advantages (99). In addition to quantifying the uncer-tainty, the NONMEM nonparametric method may assist a modeler in select-ing the correct shape for parameter distribution. It also can be used in order to incorporate uncertainty in prospective simulations from nonparametric models. (iv) A novel method for covariate model building from the non-parametric distribution has been proposed based on weighted GAM or weighted linear regression which seems to perform as well as the parametric methods for covariate selection (100), and (v) the presence of these methods in NONMEM, which is the most widely used software for non-linear mixed effect analysis and quite flexible with respect to data and model specifica-tion, is likely to encourage all modelers to use these methods more often. Also, since the first step of the analysis is a parametric step, all knowledge from this step can be utilized in subsequent nonparametric analysis which

82

nicely links the knowledge from both methodologies. These methods could also aid in the model building procedure, as a simple diagnostic tool for in-spection of parameter distribution shape as well as a tool for optimizing the variance-covariance matrix.

If the modeler would still like to stay within a parametric framework, but

still has the problem of parameter distribution deviation from the assumed (log-normal) shape, then the semi-parametric transformations presented herein are an attractive alternative. These transformations are easy to imple-ment and they require the simple addition of one or two additional parame-ters. In the examples presented, the addition of shape parameters appeared to be powerful in describing the true parameter distribution. They also offered an improvement in fit in more than 2/3 of the models evaluated, which also implies that there were some parameter distributions which were better de-scribed with different shape than the assumed log-normal distribution in the majority of these models. This also can be taken as strong evidence that the assumption of normally distributed random effects often is not valid and therefore it is relevant and important to evaluate it.

5.3. Model diagnostics �

Commonly used diagnostics and their associated assumptions were evalu-ated in order to critically discuss their advantages and disadvantages. PRED-based diagnostics were designated least generalizable as the pattern dis-played by a correct model was situation dependent. The spread of the data around line of identity depends on the residual variability and interindividual variability, range of observed values, data censoring, dose adaptation etc. Thus, the commonly made assumption that this spread should be uniform does not always hold true. Similarly, numerous conditions may lead to the situation where the regression line does not overlap with the line of identity, and most importantly, the regression procedure itself does not take into ac-count that data are coming from different individuals and that parameter variability enters nonlinearly into the model.

Residual-based diagnostics are also flawed. In particular, WRES-based

diagnostics, the residual used as an indicator for structural model building, appeared not to be optimal since it may indicate model misspecification even if there truly is none. This is particularly the case for highly nonlinear mod-els, similar to models investigated (Michaelis-Menten model, transit com-partment absorption model, sigmoidal Emax model). CWRES-based diag-

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nostics appeared to have better properties in these circumstances, although they too may show sometimes limitations. This is simply because the ap-proximations these residuals are based on, FO and FOCE may be too crude to approximate a highly nonlinear likelihood function.

Diagnostics based on individual parameter estimates, although very useful

and intuitive in the presence of rich data, appeared to be less and less infor-mative with decreased information content per individual, potentially be-coming misleading. If shrinkage is not considered, these diagnostics may support acceptance of the misspecified structural, stochastic, residual and covariate models. One interesting finding was that EBEs may, in presence of shrinkage, indicate false relationships or hide true relationships, when used for covariate screening. This is of special interest when EBEs are used for selection of covariates which will be further tested in the model for their significance. If only certain parameters are screened for covariates, it may happen that EBEs would indicate false parameter-covariate relationships which may even turn out to be significant when tested directly in the model, if the covariate was truly related to another parameter (e.g., V in our exam-ple). If, for example V was not screened for significant covariates, but only ka, a modeler may include a false covariate in the model. Similarly, the co-variate relationship may be hidden when EBEs are used for screening, which also may erroneously guide a modeler not to even test important covariates. The shrinkage phenomenon is a function of both data richness in terms of number of observations per individual and study design. While � - shrinkage depends equally on both, � - shrinkage depends more on the data richness and it becomes obvious/substantial only when number of observations is equal or less than number of random effects per individual.

Even though, �- and � - shrinkage are positively correlated, the concept of

the individual prediction shrinking towards the actual observations is often counterintuitive, as one would probably expect individual prediction (IPRED) to shrink towards the typical individual prediction (PRED) in the same manner as the individual parameter is shrinking towards the typical individual parameter value. Indeed, this would be the case with extreme shrinkage. However, this is most often not the case as shrinkage usually takes a value between two extremes (no shrinkage and absolute shrinkage). If the number of random effects is larger than number of observations, which is the condition when substantial � - shrinkage occurs, one would expect that the collection of estimated individual parameters will be still flexible enough to fit the individual model straight through the observations. This is also referred to as an overfit phenomenon. Often, when this occurs, a modeler is guided to reduce the model so that fewer random effects are estimated. The important point is that even if data are not sufficiently abundant to lead to precise individual parameter estimates, available data could still be adequate

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for estimation of a population variance. It is probably more plausible to in-clude variability in the model, rather than to state that all subjects have the same parameter value.

Shrinkage of 20-30% is usually sufficient enough to render the EBE-

based diagnostics essentially of no value. A display of the degree of shrink-age in conjunction with the diagnostic graph is essential for a relevant inter-pretation of the pattern in the plot.When shrinkage is high, other diagnostics and more direct population model estimation need to be employed in model building and evaluation.

�

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6. Conclusions

The aim this thesis was to improve model building techniques used in the pharmacometric field. Contributions were made with respect to three essen-tial model development components – the structural and stochastic model building, and model diagnostics.

A novel absorption model, the transit compartment model, was devel-

oped. This model offers great flexibility for modeling the drug absorption process even when data are sparse. In comparison with traditional absorption models, this model offer several advantages: the continuous nature of the model is reflected in a gradual (and not abrupt) increase in the absorption rate which better approximates physiological processes, the ability of the model to estimate the optimal number of transit compartments numerically, favorable computational properties for likelihood optimization due to the absence of a change-point, and finally, the stability and flexibility of the model allows further extension of the absorption model by including inter-individual variation on all absorption parameters. The model has been suc-cessfully applied to describe pharmacokinetics of desmopressin, using both sparse children and rich adult data. The pharmacokinetics of rifampin in South African tuberculosis patients was described with the proposed model. In addition to being the only model capable of fitting the erratic and variable absorption profiles, the model was also extended to fit multiple dose data.

Contribution to the stochastic model building procedure was made by de-

velopment of three novel methods for parameter distribution estimation. The nonparametric method as available in NONMEM VI has good estimation properties. It can identify non-normal distribution shapes, it can correct the bias seen with the FO method and results in improved simulation properties of the model. The performance of the method is diminished when data are sparse and from few subjects, thus resulting in a too narrow and sparse range of support points. In these cases, an extended grid method, which provides the nonparametric grid with wider and denser range of support points, is an attractive alternative. This method has improved estimation properties over the default nonparametric method and it can precisely estimate the entire distribution function even when a default method is not performing that well.

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Semiparametric distributions with an estimated shape parameter are a fur-ther attractive alternative for dealing with parameter distributions with non-known shapes, where a modeler prefers to stay within the parametric frame-work. Three transformations have been suggested which can approximate skewed, spiky, heavy-tailed and broad shape. Inclusion of these transforma-tions into real models caused significant improvement in two thirds of tested models implying that in model building, more often than not, the parameter distribution assumptions are violated.

Contribution to model diagnostics was made by identifying the weak-

nesses of the most commonly used model diagnostics. For the diagnostics derived on typical individual values, no generalizations can be made about the expected pattern of the plot. WRES may be too crude to identify struc-tural model misspecification when models are nonlinear. CWRES shows better properties, but has limitations too. In order to get idea of what a plot should look like, a reference plot can be created by simulating data from the models and deriving the same model diagnostics, but based on simulated data. The quality of all diagnostics based on individual parameter estimates is dependent on the data richness. At shrinkage values of 20-30%, this diag-nostics becomes of no value, thus it is important to report the shrinkage magnitude. A way around the issue of shrinkage is to perform more model testing and to use other types of diagnostics, for example those that are simu-lation-based.

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7. Acknowledgements

There have been many people that made the work included and excluded in this thesis, and my entire life as a PhD student extremely enjoyable. There-fore, I have many people to thank. My supervisor, Prof. Mats Karlsson, for five extraordinary years full of chal-lenge, great work, brilliant ideas and fun! I appreciate immensely how you have taught me to think scientifically in my efforts to become a good scien-tist. Thank you for all the knowledge, opportunities, and most importantly for being an exceptional human being whom I greatly respect. My second supervisor Dr. Siv Jönsson, it was such a comforting thought knowing that your door was always open and I could come to you asking for anything at any time. Thank you for the great support during all these years. Prof. Margareta Hammarlund-Udenaes, the head of the division - Thank you for making me feel very welcome at the department and for always hav-ing encouraging words for me and my work. Dr. Catharina Svensson, the head of the UGSBR school - Thank you for believing in me and giving me an opportunity to join a fantastic research school in Sweden, which has set me on a new path in life. Dr. Daniël Jonker, my first supervisor on the transit compartment project- Thank you for opening the pharmacometrics door for me, for teaching me all first steps in NONMEM and for having great patience with me during my first year! My undergraduate pharmacokinetic teacher from Belgrade, Prof. Branislava Miljkovi� - Thank you for making this field extremely interesting for me. I also would like to thank you for your great support when moving to Sweden. Thank you to all the senior people at the department, Dr. Andy Hooker, for being a great colleague while teaching the NONMEM VI course; Dr. Lena Friberg and Dr.Ulrika Simonsson for always showing an interest in my

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work and advices; Dr. Niclas Jonsson, for always adding that little gold nugget to my work. All my master students: Paul Baverel for all your work on Paper V and for a great and fun time while supervising you! Klas Petersson, for all your work on paper VII especially in the last weeks of writing this thesis. It was so easy to work with you and I’m so happy that both of you stayed at the division as graduate students. Moa Grahm, for all your work on heavy-tailed transfor-mation and for a good time. I hope all of you learned from me, I definitely did learn from you- thank you! Mia Kjellsson, my co-author on Paper IV- for being such a great person to work with and to learn from. Justin Wilkins, my co-author on Paper III and Paper V-for being an extraor-dinary colleague and even better friend. Grant Langdon-for collaborating on Paper III, and for all witty comments and the many laughs we had together. Thank you to all my co-authors – Dr. Thomas Kerbusch on Paper I, for a great collaboration and for being always supportive of my work and having a nice word for me; a Danish team, Dr. Ole Østerberg, Dr. Jens-Peter Nør-gaard, Dr. Johan Vande Walle, Dr. Thomas Senderovitz and Dr. Henrik Agersø, for a trust in the transit model from the very beginning and a nice collaboration on Paper II, the South African team, Dr. Helen McIlleron, Dr. Goonaseelan (Colin) Pillai, and Dr. Peter J Smith for a great collaboration on the Paper III. All participants of the NONMEM VI courses for your great questions and discussions on all the topics. Prof. Nick Holford, for discussion on the transit model and multiple dose solution and a great input on my work while in Paris; Prof. France Mentré, for the enjoyable discussions on nonparametric methods and for accepting me as a Postdoc; Dr. Emmanuelle Comets for your valuable input on my work while in Paris. I’m looking forward to work with both of you! I would also like to thank other scientists for showing an interest in my research, supporting me in different ways, which motivated and encouraged me to do even better: Dr. Rik Schoemaker, Dr. Toufigh Gordi, Dr. Marie Sandström, Prof. Steve Duffull, Dr. Pete Bonate. Thank you to The Swedish Academy of Pharmaceutical Society (Apotekar-societeten) for my Postdoc scholarship.

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Dr. Janet Wade, thank you for proof-reading my thesis and for always en-couraging and supporting me. Dr. Peter Milligan, thank you for making possible my internship in Pfizer and for the great experience that I gained there. Pontus - Thank you for the much needed technical support during all my projects, and Magnus for your help with the final manuscripts, and Kjell for all your computer-related help Thank you Shasha!

My wonderful friend and roommate, Elodie for being such a trustful and fun person. Thank you for all support, laughs and understanding! My other wonderful friend, Angelica for being such a lovely friend with such a good heart! Thank you for all your help regarding anything I needed, for all your patience while teaching/practicing Swedish with me and all fun we had while traveling to many conferences. Joe, for proof-reading my thesis, for applying all my research work and all the fun we had while traveling. Anubha and Doaa, thank you for all your friendship, support and for being such lovely persons to work with and talk to.

My great friends, Joy, thank you for all long talks we had, Kristin C, for a being a great friend and a colleague, Anthe for much fun, Samuel for all good talks and Jan-Stefan for all our shared loud and crazy laughs. All present friends from the division, Hanna, Kristin, Bettan, Jörgen, Emma H., Stefanie, Guangli, Martin, Rocio, Joakim, Dominik, Kajsa, Brigitte, Johan W. and all the others for creating a great atmosphere to work in. All my former colleagues and friends, Poppi, for being a great roommate; Jakob, for “living” with me all these years; MaMa for all fun and laughs while in Pfizer; Anja for all long talks and nice words during late hours while you were working on your thesis; Lasse, Anders, Emma B., Petra, Bengt, Stina, you all gave me an excellent example how to prepare and de-fend a thesis. All visitors that I had pleasure to collaborate with: Eva H., Edwin, Dong-Seok, Alistair, Rory, Alex P.

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My fabulous friends across the earth, Ana (London), Ankica (New York), Minja (Amsterdam), Jelena (Dresden), Ema (Belgrade), Danijela (Ljubl-jana) and Dijana (Australia) for staying my friends all these years despite the long distance, for supporting and believing in me and keeping the silly side of me in shape. Kundayi, Chido, Tabitha and Ngozi, for wii tournaments while writing my thesis that kept me energized and smiling. My friends, Brent, for your help with gamma function and all the fun; Ulli for a wonderful time during my first year in Sweden Damir, for all support while moving to Sweden and for all good and relaxing times. Hvala za sve Dachi!

My brother, Saša, for all your extraordinary support over the last four years. I appreciate immensely what you did for me! Debeli, hvala na podršci i na svemu! My mum, for letting me chase my dream despite all the difficult times we have been through. Maki, ti si moja Ruža! I love you mum! My dad, for all that you taught me in our short time together, for planting in me a true love for knowledge and a hunger to strive for the best while still staying simple, which made me become what I am today. I hope this thesis would make you proud.

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80. R. Bruno, B. Lacarelle, V. Cosson, J.W. Mandema, Y. Le Roux, G. Montay, A. Durand, M. Ballereau, and A. M.. Evaluation of Bayesian estimation in comparison to NONMEM for population pharmacokinetic data. . J Pharmacokinet Biopharm 20:653-669 (1992).

81. L. Brynne, J.L. McNay, H.G. Schaefer, K. Swedberg, C.G. Wiltse, and M.O. Karlsson. Pharmacodynamic models for the cardiovascular effects of moxonidine in patients with congestive heart failure. . Br J Clin Pharmacol 51:35-43 (2001).

82. L.E. Friberg, A. Henningsson, H. Maas, L. Nguyen, and M.O. Karlsson. Model of chemotherapy-induced myelosuppression with parameter consistency across drugs. . J Clin Oncol 20:4713-4721 (2002).

83. B. Hamren, E. Bjork, M. Sunzel, and M. Karlsson. Models for Plasma Glucose, HbA1c, and Hemoglobin Interrelationships in Patients with Type 2 Diabetes Following Tesaglitazar Treatment. . Clin Pharmacol Ther (2008).

84. B. Hamren, H. Ericsson, O. Samuelsson, and M.O. Karlsson. Mechanistic modelling of tesaglitazar pharmacokinetic data in subjects with various degrees of renal function--evidence of interconversion. Br J Clin Pharmacol 65:855-863 (2008).

85. G. Hempel, M.O. Karlsson, D.P. de Alwis, N. Toublanc, J.L. McNay, and H.G. Schaefer. Population pharmacokinetic-pharmacodynamic modeling of moxonidine using 24-hour ambulatory blood pressure measurements. . Clin Pharmacol Ther. 64:622-635 (1998).

86. B. Hornestam, M. Jerling, M.O. Karlsson, and P. Held. Intravenously administered digoxin in patients with acute atrial fibrillation: a population pharmacokinetic/pharmacodynamic analysis based on the Digitalis in Acute Atrial Fibrillation trial. . Eur J Clin Pharmacol 58:747-755 (2003).

87. J. Li, M.O. Karlsson, J. Brahmer, A. Spitz, M. Zhao, M. Hidalgo, and S.D. Baker. CYP3A phenotyping approach to predict systemic exposure to EGFR tyrosine kinase inhibitors. . J Natl Cancer Inst. 98:1714-1723 (2006).

88. S. Lindemalm, J. Liliemark, A. Gruber, S. Eriksson, M.O. Karlsson, Y. Wang, and F. Albertioni. Comparison of cytotoxicity of 2-chloro- 2'-arabino-fluoro-2'-deoxyadenosine (clofarabine) with cladribine in mononuclear cells from patients with acute myeloid and chronic lymphocytic leukemia. . Haematologica. 88:324-332 (2003).

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89. S. Lindemalm, R.M. Savic, M.O. Karlsson, G. Juliusson, J. Liliemark, and F. Albertioni. Application of population pharmacokinetics to cladribine. BMC Pharmacology. 5: (2005).

90. Savic R.M., Jonker D.M., Kerbusch T., and Karlsson M.O. Evaluation of a transit compartment model versus a lag time model for describing drug absorption delay. In P.A. [www.page-meeting.org/?abstract=513] (ed.), 2004.

91. I.F. Troconiz, T.H. Naukkarinen, H.M. Ruottinen, U.K. Rinne, A. Gordin, and M.O. Karlsson. Population pharmacodynamic modeling of levodopa in patients with Parkinson's disease receiving entacapone. Clin Pharmacol Ther 64:106-116 (1998).

92. J.J. Wilkins, G. Langdon, H. McIlleron, G.C. Pillai, P.J. Smith, and U.S. Simonsson. Variability in the population pharmacokinetics of pyrazinamide in South African tuberculosis patients. . Eur J Clin Pharmacol 62:727-735 (2006).

93. E.N. Jonsson, S. Antila, L. McFadyen, L. Lehtonen, and M.O. Karlsson. Population pharmacokinetics of levosimendan in patients with congestive heart failure. . Br J Clin Pharmacol. 55: (2003).

94. P.H. Zingmark, C. Edenius, and M.O. Karlsson. Pharmacokinetic/pharmacodynamic models for the depletion of Vbeta5.2/5.3 T cells by the monoclonal antibody ATM-027 in patients with multiple sclerosis, as measured by FACS. Br J Clin Pharmacol 58:378-389 (2004).

95. L.B. Sheiner and S.L. Beal. Some suggestions for measuring predictive performance. J Pharmacokinet Biopharm. 9: (1981).

96. A.C. Hooker, C.E. Staatz, and M.O. Karlsson. Conditional Weighted Residuals (CWRES): A Model Diagnostic for the FOCE Method. Pharm Res (2007).

97. J. Antic, C. Laffont, and D. Concordet. Some Stochastic Algorithms For The Smooth Non Parametric (SNP) Estimator, PAGE, 2008.

98. R.M. Savic, M.C. Kjellsson, and M.O. Karlsson. Evaluation of the nonparametric estimation method in NONMEM VI beta, PAGE 15 (2006) Abstr 937 [www.page-meeting.org/?abstract=937], 2006.

99. R.M. Savic, P.G. Baverel, and M.O. Karlsson. A novel bootstrap method for obtaining uncertainty measurement around the nonparametric distribution., PAGE 17, 2008.

100. P.G. Baverel, R.M. Savic, and K. M.O. Covariate Model Building Method for Nonparametric Estimation Method in NONMEM VI: Application to Simulated Data, PAGE, Marseille, 2008.

Acta Universitatis UpsaliensisDigital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Pharmacy 80

Editor: The Dean of the Faculty of Pharmacy

A doctoral dissertation from the Faculty of Pharmacy, UppsalaUniversity, is usually a summary of a number of papers. A fewcopies of the complete dissertation are kept at major Swedishresearch libraries, while the summary alone is distributedinternationally through the series Digital ComprehensiveSummaries of Uppsala Dissertations from the Faculty ofPharmacy. (Prior to January, 2005, the series was publishedunder the title “Comprehensive Summaries of UppsalaDissertations from the Faculty of Pharmacy”.)

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Improved pharmacometric model building techniques172497/FULLTEXT01.pdfplines such as clinical pharmacology, statistics, computer science, classical pharmacology, etc. Indeed, pharmacometrics