Till Agnes och Lovisa

Papers discussed

This thesis is based on the following papers, which will be referred to in the text by the Roman numerals assigned below.

I Zingmark PH, Ekblom M, Odergren T, Ashwood T, Lyden P, Karlsson MO and Jonsson EN. Population pharmacokinetics of clomethiazole and its effect on the natural course of sedation in acute stroke patients. British Journal of Clinical Pharmacology,2003;56(2):173-183.

II Zingmark PH, Edenius C and Karlsson MO. Pharmacoki-netic/Pharmacodynamic models for monoclonal antibody actions on the number of target cells and receptors as measured by FACS. British Journal of Clinical Pharmacology, 2004; 58(4):378-389.

III Zingmark PH, Kågedal M and Karlsson MO. Modelling a spon-taneously reported side effect by use of a Markov mixed-effects model. Accepted for publication in Journal of Pharmacokinetics and Pharmacodynamics.

IV Zingmark PH*, Kjellsson MC*, Jonsson EN and Karlsson MO. Comparing the differential drug effect model to the proportional odds model. Manuscript.

* Zingmark PH and Kjellsson MC contributed equally to this work.

Reprints were made with kind permission of Springer Science and Business Media and Blackwell Publishing

Contents

Abbreviations..................................................................................................9

Introduction...................................................................................................11Population PK/PD modelling ...................................................................12

Non-linear mixed effects models.........................................................12Clinical trial simulations......................................................................14

Categorical data analysis ..........................................................................14Models for ordered categorical responses ...........................................15Goodness of fit.....................................................................................15Commonly used scales ........................................................................15

Aim ...............................................................................................................17

Methods ........................................................................................................18Clinical studies .........................................................................................18

Stroke studies.......................................................................................18Multiple sclerosis studies.....................................................................19Side effect study ..................................................................................21

Data analysis ............................................................................................23Pharmacokinetics.................................................................................23PK-PD models .....................................................................................23Proportional odds model......................................................................23Markov model......................................................................................24Differential drug effect model .............................................................25Software...............................................................................................29

Results...........................................................................................................30Proportional odds model ..........................................................................30

Stroke studies.......................................................................................30Multiple Sclerosis studies ....................................................................34

Markov model ..........................................................................................37Differential drug effect model..................................................................41

Simulation study ..................................................................................41Application to real data........................................................................41

Discussion.....................................................................................................46Perspectives..............................................................................................49

Conclusions...................................................................................................51

Populärvetenskaplig sammanfattning ...........................................................52

Acknowledgements.......................................................................................53

References.....................................................................................................55

Abbreviations

AUC Area under the plasma concentration time curve AUCt Cumulative area under the plasma concentration

time curve until time t AUC50 Area under the plasma concentration time curve

giving half of maximum effect BASE0 Baseline at time 0 BASEt Baseline at time t C Concentration Ce Concentration in effect compartment CTol Concentration in tolerance compartment CL Clearance CLASS Clomethiazole Acute Stroke Study CNS Central nervous system CT Computer tomography C50 Concentration giving half of the differential drug

effectD Drug effect EC50 Concentration giving half of the maximum effect Emax Maximum effect FACS Fluorescence activated cell sorter fdiff Differential drug effect FO First order estimation method FOCE First order conditional estimation method FOCE INTER First order conditional estimation method with

interactionlx Logit LAPLACE Laplacian estimation method LC-MS Liquid chromatography-Mass spectrometry MS Multiple sclerosis NIH-SS National Institute of Health Stroke Scale OFV Objective function value

OFV Difference in objective function value P Probability pi Individual probability PPC Posterior predictive check PD Pharmacodynamic

PK Pharmacokinetic PT Probability of sedation as a function of time SD Standard deviation TC50 Concentration giving half of the maximum toler-

anceTCR V 5.2/5.3 T-cell receptor t-PA Tissue Plasminogen Activator UV Ultra violet WT Body weight Difference between observation and individual

prediction Concentration giving half of the differential drug

effect Difference between population and individual pa-

rameter Typical value of a parameter, fixed effects parame-

ter Change in drug effect introduced by the differential

drug effect 2 Variance of the s2 Variance of the s

11

Introduction

In drug development clinical trials are designed to investigate whether a new treatment is safe and has the desired effect on the disease in the target patient population. The trials have pre-specified measurements to enable these evaluations. Usually, the pre-specified endpoints are evaluated with a rather simple statistical test where the patients who received active treatment are compared to the patients who received placebo or another comparative treatment. The evaluations can also be done with the help of a more complex mathematical model that describe the observed endpoint over time and its dependence of the different study variables. Statistical comparisons can then be performed within the model (1). Categorical endpoints are commonly used in drug development and the outcomes measures are usually ranked, i.e. the measurements have an order. The severity of a side effect is generally described as mild, moderate or severe and different ranking scales are often used as efficacy measures. These can be of different types; a patient partici-pating in a trial can be judged to a responder or a non-responder to the treat-ment given, a binary outcome, or the outcome can be measured on an or-dered categorical scale with several levels, for example pain scores (2-4) or sedation scales (5-7).

The purpose with pharmacokinetic-pharmacodynamic (PK/PD) modelling is to quantitatively describe the time course of the plasma concentrations of a drug and its relationship to the effect measured (8-10) with the goal of find-ing the optimal dosing regimen in the target patient population. PK/PD mod-els are valuable tools for understanding the time course of drug action and are used in all stages of drug development (11-15) and are also utilized in regulatory decision-making (16-21). The models can be used for predictive purposes when designing new studies or for intra- or extrapolations of the data to situations not yet studied (22-24). Although extensive statistical lit-erature exists in the field of categorical data analysis (for example (25)), analyses of repeated measures categorical data in non-linear mixed effects models is a relatively new field. The purpose of this thesis is to present PD models to be used for analyzing ordered categorical data encountered in drug development.

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Population PK/PD modelling The aim with population PK/PD analyses is to describe the PK, i.e. parame-ters such as clearance and volume of distribution, and the PD, i.e. parameters such as Emax and EC50, in a specific population. Previously, this was done using the two-stage method where pharmacokinetic or pharmacodynamic parameters first were established in each individual and then summarized by means and standard deviations over the population studied (26; 27). That approach requires extensive sampling in all individuals in the study popula-tion. Today population PK/PD analyses are mostly done using non-linear mixed effects modelling, where means and variances of the model parame-ters are estimated simultaneously from all individuals making it possible to use sparse sampling schemes. The term mixed refers to the use of both fixed effects that characterize the typical individual (the mean) and random effects that characterize the variability components of the data (28). The random components are usually residual variability and inter-individual variability, but depending on the study design it might also be possible to estimate a third level of variability, intra-individual variability which for practical rea-sons is estimated as inter-occasion variability (29).

Non-linear mixed effects models The models have the following general structure. The jth observation in an individual i (yij) can be described by:

ijiijij Pxfy ),( (1)

where f(…) is the individual prediction described by a non-linear or linear function with parameter vector Pi and independent variables xij that for a pharmacokinetic model usually are dose and time and for a pharmacody-namic model are concentration and time. ij is a random effect and the values of ij are assumed to be normally distributed with the variance of 2. ij is describing the discrepancy between the individual prediction and the obser-vation and is referred to as the residual error. Other models are also possible to use (30), for example models were the deviation is proportional to the function. Residual errors usually reflects analytical assay errors or model misspecifications. The other source of variability, inter-individual variabil-ity, is due to differences between the individuals in the study population. It isincorporated in the models in the following way:

kikkip (2)

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where pki is the kth individual parameter in Pi and k is the typical value of pk. The difference between the individual parameter and the typical value is described by the random effect ki, which is assumed to be normally distrib-uted with the estimated variance k

2. For biological systems, this distribution usually is log-normally distributed and the parameterization of the model is then:

kiep kki (3)

The typical value of a parameter can also be dependent on individual patient factors, covariates, such as body weight or kidney function. In such cases the covariates are included in the function and thereby explains part of the inter-individual variability in the parameter.

Estimation methods Several software packages are available for population PK/PD analyses (31). All packages are based on hierarchical nonlinear mixed-effects models but uses different estimation methods. The parametric maximum likelihood is used in NONMEM (Globomax Coorporation) (32), WinNonMix (Pharsight Coorporation) (33), NLMIXED within SAS (SAS Institute) (34) and NLME within S-plus (Insightful Coorporation) (35). Non-parametric maximum likelihood is used within NPLM (36) and NPEM (37) and Bayesian methods are used within BUGS (38). For population PK/PD modelling, NONMEM is the most commonly used software.

The parametric maximum likelihood estimation methods used in NON-MEM estimates the model parameters by maximizing the likelihood of the data given the model. This is done by minimization of the extended least squares objective function value, which is proportional to – 2 log likelihood of the data (39). The computation of the likelihood requires the evaluation of a multiple integral, which usually cannot be solved exactly. The different estimation methods within NONMEM use different approximations of the integral evaluation. The first order method, FO, uses the first order Taylor series linearization of the prediction based on the model parameters. The derivative of the function is then evaluated around the typical value of the population, ( i = 0). The conditional estimation methods, FOCE and LAPLACE, instead evaluate the function around the conditional Empirical Bayes estimates of the model parameters. The FOCE method uses the first order Taylor series linearization of the function and the Laplacian estimation method uses the second order Taylor series linearization of the function (32).

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Clinical trial simulations Clinical trial simulations refers to the structured use of PK/PD models for designing and predicting the outcome of future clinical trials (40-42). These simulations enable evaluation of different study factors before running the actual trial (43). For example it is possible to study the impact of different study designs, e.g. parallel group vs. cross-over designs, but also the impact of different sampling strategies, selection of different study populations, impact of between and within patient variability but also of the uncertainty of the different model parameters used. The models can also be extended to include economic aspects of the investigated treatment (44; 45). This means that far more situations and scenarios can be investigated before running the actual clinical trial. The use of models for simulations of complex systems are widely used in areas such as telecom and airplane construction but are to an increasingly extent also being used in drug development. Although not the primary aim of this thesis work, the models developed from the phase I data for the monoclonal antibody ATM-027 in paper II was successfully used to predict the outcome of the phase II clinical trial that was designed with a rather complex adaptive study design.

Categorical data analysis A wide variety of approaches to analyze categorical data exist and the method of choice depends on the nature of the data and what the aim with the analysis is. Categorical variables can be divided into two primary types of scales, nominal and ordinal (25). Variables having categories without a natural ordering such as race or gender are called nominal whereas categori-cal variables having a natural order are called ordinal. Most of the categori-cal endpoints used in drug development are ordinal and are the focus of this thesis. It is also possible distinguish between marginal and conditional mod-els for analyses of the data. Marginal models, for example the Generalized Estimation Equations, describe population-averaged effects, whereas condi-tional models describe subject-specific effects that also can be used for in-terpretation on the population level. A mixed-effect model is an example of a conditional model. Marginal models are used for comparisons between groups and are for example used frequently in epidemiological studies (25). The categorical PD data collected in clinical trials are most often recorded repeatedly over time within each individual in the study. When analyzing this type of data it is important to consider that observations over time are coming from the same individual (4). Therefore all analyses in this thesis were done using conditional random-effects models as implemented in the software NONMEM with the Laplacian estimation method.

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Models for ordered categorical responses Logistic regression models are frequently used in medical statistics. The models estimate the probability of the data to be in a certain category. The most popular model for ordered categorical data is a generalization of the logistic regression model to a multi-category response (46). This model uses the logit function as a link function to link the observations to the cumulative probabilities of the model parameters. The model assumes an identical effect of the predictors for each cumulative probability and is therefore often re-ferred to as the proportional odds model (47). Depending on the shape of the response curve, it is possible to use the model with another link function than the logit, for example the probit or the complementary log-log function. It is also possible to allow for separate effects of some, but not all, logits and use the so called partial proportional odds model (48).

The proportional odds model was introduced in the field of PK/PD model-ling by Lewis Sheiner in 1994 (2), where he used the proportional odds model with subject-specific random effects to analyze a 4-degree pain scale. The details of the model are presented in the methods section and this model was the starting point for the thesis work.

Goodness of fit Goodness of fit for models for ordered categorical PD data is different from ordinary PD models where various forms of residual plots are important goodness of fit tools (49). Since the models estimates the probability of a score at a certain time point, not a prediction of the score itself, residuals are not available for evaluating the goodness of fit. During the model building process changes in the objective function value calculated by NONMEM can be evaluated using a likelihood ratio test. Simulations from the model are also used for model evaluation, and a thorough simulation exercise such as the posterior predictive check (50) is probably the best way to actually be able to investigate several aspects of the model performance.

Commonly used scalesThe most commonly used categorical scale in drug development is the rank-ing of adverse events, which is used in almost every clinical study per-formed. The adverse events collected are ranked as mild, moderate or severe. Mixed effect modelling with data from ordered categorical scales has been reported for a wide variety of therapeutic areas. The endpoints analyzed are in most cases different ranking scales. The published analyses are to date: pain scores (2; 51), pain relief scores (3; 4), dry mouth severity (52), nicotine craving scores (53), severity of allergic rhinitis (54), severity of adverse

16

bleeding events (55), grade of neutropenia (56), oncology toxicity (57), ad-verse event severity (58), change in thrombus size (59), grades of diarrhoea (60) and different sedation scales (5-7). There is also one example of PK/PD modelling with a non-ordered categorical scale, a nominal scale, using a transition model to evaluate transitions between different sleep stages after administration of temazepam (61). The published examples illustrate that models for categorical PD data are useful in a wide variety of diseases and therapeutic areas.

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Aim

The aim of the thesis was to develop predictive models for ordered categori-cal pharmacodynamic data encountered in drug development.

Three types of data were considered: sedation data, T-cell data and side-effect data. The aims with the work on sedation data from patients suffering from an acute stroke were to evaluate the heterogeneity in the disease by quantifying both the disease and the drug effects on sedation but also to evaluate the heterogeneity in the sedation scale used. The aims with the work on the T-cell data were to develop models for both the continuous and cate-gorical data collected. The aim with the side-effect data was to develop models for a spontaneously reported event.

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Methods

Clinical studiesStroke studies Clomethiazole has been shown to have neuroprotective effects in rat, gerbil, and marmoset models of cerebral ischemia (62-67). These findings sug-gested that clomethiazole might prevent neuronal degeneration following ischemic stroke and the drug therefore underwent clinical development as a potential treatment. Three randomized, double blind, placebo-controlled phase III studies were conducted and have been published in detail (68-70). Patients were recruited to the different studies based on the results of a CT scan. Patients who had an acute ischemic stroke and a combination of limb weakness, higher cortical dysfunction and visual field disturbance were en-rolled in the CLASS-I study, patients with an intracerebral haemorrhage were enrolled in the CLASS-H study and ischemic stroke patients treated with t-PA were enrolled in the CLASS-T study. In total 1600 patients (1200 for CLASS-I, 200 for CLASS-H and CLASS-T respectively) from 166 cen-ters in USA and Canada were to be included in the studies. To be eligible for inclusion the patients had to have a clinical diagnosis of acute stroke within twelve hours after onset of stroke. Sedation is an important aspect of the disease and since clomethiazole had known sedative properties it was impor-tant to monitor and quantify the relations between the treatment and seda-tion.

TreatmentClomethiazole edisilate was given as a three-phase iv-infusion over 24 hours, 6 mg/kg over 0.25 hours, 31 mg/kg from 0.25 to 8 hours and 31 mg/kg from 8 to 24 hours. The aim of the treatment was to achieve a target steady state plasma concentration of clomethiazole above 10 µmol/L.

Blood sampling Blood samples for pharmacokinetic assessment were drawn at 15 minutes, at any time between 1 and 2 hours and at the end of the infusion or 24 hours whichever occurred first. If a dose adjustment was needed (see below), an extra sample was drawn at the first time the infusion was interrupted. To obtain more pharmacokinetic information, the above sampling times were

19

changed for patients included in the late stage of the CLASS-I study. The new sampling times were any time between 2 and 10 hours, any time be-tween 10 and 23 hours and any time between 26 and 36 hours after the start of the infusion. The determination of clomethiazole concentrations in plasma was made by reversed phase liquid chromatography with UV detection.

Sampling design for the sedation score data Sedation was monitored throughout the entire treatment period. The degree of sedation was measured on a discrete ordinal scale with six levels (Table 1). If the patient became excessively sedated, defined as a score of 4 or above on the sedation scale, the infusion was stopped and optionally re-sumed at half of the previous rate as soon as the subject scored 3 or less on the sedation scale. The sedation was measured at least every 3 hours up to 24 hours after the start of infusion, but if a score of 3 or greater was observed, sedation was checked approximately every 15 minutes.

Table 1, The sedation scale.

1 = Fully awake

2 = Drowsy but answers when spoken to

3 = Answers slowly when spoken to

4 = Reacts when spoken to but does not answer

5 = Reacts only to pain

6 = Does not react to pain

Multiple sclerosis studies Autoreactive T-cells are implicated in the pathogenesis of several autoim-mune conditions e.g. rheumatoid arthritis, psoriasis and multiple sclerosis (MS) (71; 72). A humanized antibody, ATM-027, specific for a subgroup of T-cells (V 5.2/5.3+ T-cells) thought to be associated with MS, was produced with the aim of developing an effective first line treatment for this disease. One phase I and one phase II study in MS patients were performed to evalu-ate the safety and tolerability of ATM-027 and also its effects on the target V 5.2/5.3+ T-cells.

Phase I study This was a single centre, open, non-comparative individual dose escalation study in patients with relapsing-remitting or secondary-progressive MS.

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After approval from the local ethics committee at Karolinska Institutet, Stockholm, Sweden, 14 patients, 7 females and 7 males were treated with ATM-027. The study consisted of four parts: a run-in period of 4 weeks when assessments were performed on three occasions, a treatment period of 4-19 days, a follow-up period of 8 weeks (with assessments conducted 1, 2, 4 and 8 weeks after last dose) and a long-term follow-up period of up to 18 months (with assessments conducted at 4, 6, 7, 10, 12 and 18 months after last dose). ATM-027 was administered as a constant rate infusion over 30 minutes. There were initially six different dose levels in the study, 0.03 (pa-tients 1-6 only), 0.3, 3, 30, 100 and 300 mg. Provided it was safe, the dose was escalated every 72 h until the V 5.2/5.3+ T-cells were depleted (defined as < 0.5% V 5.2/5.3+ T-cells of the total CD3 positive T-cell population in peripheral blood 24 hours after drug administration). Details of the individ-ual dosing are found in (73).

Blood samples for the analysis of ATM-027 were taken at the end of the run-in period, at the end of infusion, and at each follow up visit up to and including 8 months. Blood samples for flow cytometry were drawn 4 weeks, 1-2 weeks and 24 h before treatment, at 2, 24 and 72 h after start of each infusion and at each follow-up visit up to and including 18 months.

Phase II study This was a randomized, double-blind, placebo-controlled, individual FACS-controlled dose titration, multi-centre study in relapsing-remitting MS pa-tients conducted at six centers in the Netherlands, United Kingdom and Sweden (74). After approval from the local ethics committees, 59 patients were enrolled in the study; 47 patients were given ATM-027 and 12 patients received placebo. The study started with a 4-month run-in period, followed by a 6-month treatment period with monthly administration of ATM-027 or placebo, clinical and immunological assessments. A 7-month follow up pe-riod ended the study and included three additional clinical and immunologi-cal assessments.

The aim of the dosing regimen used was to keep the total number of V 5.2/5.3+ T-cells at < 1.2 % of the total CD3 positive T-cell population and the T-cell receptor (TCR) density on the remaining target cells of < 20% of normal (TCR-low). Target cells with a TCR density of 20-100% of normal were denoted TCR-high. All patients received 6 monthly injections and the patients randomized to ATM-027 received a starting dose of 6 mg. Subse-quent dosing were based on both the levels of target cells and their TCR density 3 days before the next dose was scheduled. If the total levels of V 5.2/5.3+ T-cells was > 1.2 % or the levels of V 5.2/5.3+ T-cells with TCR-high density was > 0.3%, the patient was to receive another dose of ATM-027. If not, patients received placebo. A secondary aim of the dose

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regimen was to find a dose that depleted target cells for more then 4 weeks. Thus, if the starting dose of 6 mg did not result in target cell depletion for 4 weeks, the dose was increased to 15 mg at the next injection. A final dose adjustment up to 30 mg was permitted, but a dose was never allowed to be lowered.

Analysis of ATM-027, V 5.2/5.3+ T-cells and receptor expression Blood samples for the analysis of ATM-027 were taken before and within 1 hour after administration of each dose, at the visit 14 days after the first and the second dose and at all follow up visits. Blood samples for flow cytome-try were drawn monthly during the run in period, 2 days before administra-tion of each dose, at the visit 14 days after the first and the second dose and at all follow up visits. Concentrations of ATM-027 in plasma were assayed by a sandwich immunofluorescent assay. The target V 5.2/5.3+ T-cell sub-sets were analyzed by two-colour flow cytometry in blood. The results are presented as the proportion of target T-cells within the CD3+ T-cell popula-tion.

Categorization of the receptor expression The fact that not only the numbers of target T-cells but also the receptor expression on the T-cell surface, TCR density, was affected by ATM-027 was an unexpected finding in the phase I study (73). This was first presented by the bioanalyst as a trichotomized variable: dim, intermediate and bright and the classification was a subjective classification by the analyst. This categorization was first used in the modelling analysis. Later it was realized that it was possible to calculate a continuous figure between 0 and 1, corre-sponding to the proportion V 5.2/5.3 receptors of the total number of recep-tors on the V 5.2/5.3+ T-cell surface. The different categories of the receptor expression in the phase I study corresponded approximately to; dim: < 0.25, intermediate: 0.25 < X < 0.35 and bright: > 0.35. Since the classification was subjective, a few observations differed from that. In the phase II study this was changed to first determine the receptor density and then categorize it with a new definition: a receptor density < 0.2 was defined as low, and a receptor density > 0.2 was defined as high.

Side effect study In the clinical development of a new drug a specific CNS side effect was encountered and its occurrence appeared to be related to the plasma concen-tration of the drug. The side effect occurred suddenly and was described as a specific CNS feeling or sensation. The sensation disappeared as suddenly as it appeared. It was uncomfortable but was not judged to constitute any safety concern. To investigate the concentration-response characteristics of this

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side effect a clinical study aiming at provoking the side effect by use of dif-ferent infusion rates was conducted.

Study design This was a single centre, double blind, and randomized crossover study in 12 healthy male volunteers. The drug was infused intravenously on four differ-ent occasions per subject. In the first open session all subjects were given an intravenous infusion of the drug in order to be familiarized with the specific CNS-symptoms caused by the drug, to obtain individual PK information and to exclude possible subjects not reporting the side effect. In sessions 2-4 the subjects reporting symptoms in the first session and willing to continue the study were randomized to anyone of the six possible sequences of three infu-sion rate levels in a double blind crossover fashion. The rate levels were: low (L), intermediate (I) and high (H). The infusion was stopped on all occasions when the specific side effect of moderate intensity had occurred. The infu-sion rates were chosen to mimic absorption profiles of different oral formu-lations where the low infusion rate was similar to an extended release formu-lation, the high infusion rate was similar to an oral aqueous solution and the intermediate infusion rate was in between the other.

In order to obtain similar unbound drug concentration versus time profiles in all subjects at the different drug input rates, the infusion rates were indi-vidually adjusted to ensure a predicted unbound plasma concentration of the drug after 4 hours of infusion at the low rate level was as close to 2.87 nmol/L as possible. The concentration level of 2.87 nmol/L was chosen since 80% of subjects in previous studies reported CNS-symptoms on that level or below and in this study it was necessary to get concentrations high enough to actually be able to study the side effect.

The infusion was to be stopped immediately after CNS-symptoms of moderate or severe intensity was reported. The maximal total duration of the infusion including placebo infusion was 5 hours. The time point of start of active drug infusion was single blinded in all sessions. The infusion rate was not blinded in the first open session and double blinded in sessions 2-4. The self-reported occurrence of the specific CNS side effect was recorded during all study sessions, but the analysis only concerns the side effect data from sessions 2-4. For each reported event, the time and type of change in severity was stated. The side effects were classified as: 0 = no side effect, 1 = mild side effect and 2 = moderate or severe side effect. All subjects started their treatment on the observation 0.

BioanalysisPlasma samples for determination of drug concentrations were drawn at 5, 10, 15, 30 min and at 1, 2, 3, 4 hours and at 4h 10 min after start of infusions

23

in session 2-4. Additional samples were taken in all sessions at the stop of infusions and at 10, 20, 30 min and at 1, 2, 4, 6, and 8 hours after the end of the infusions. Concentrations of the drug in plasma and in ultra filtrate of plasma (free concentration of the drug) were determined by use of LC-MS-MS (liquid chromatography-mass spectrometry-mass spectrometry).

Data analysis PharmacokineticsIndividual pharmacokinetic parameters were estimated in paper I-III. In the phase I study in paper II and in paper III, the individual pharmacokinetic parameters were estimated using the software WinNonLin, version 2.1 (Pharsight Corporation). In paper I and in the phase II study in paper II, in-dividual pharmacokinetic parameters were estimated using a non-linear mixed-effects modelling approach as implemented in the software NON-MEM, version V, level 1.1 where the individual pharmacokinetic parameters were generated using the POSTHOC option.

PK-PD models In all PD analyses fixed individual pharmacokinetic parameters were used in the model building process to generate predicted plasma concentrations at the time points for the PD observations. In paper I, the final model was a step model only dependent on whether clomethiazole or placebo were given. In the phase I study in paper II a link-model (75) between concentrations and the continuous measurements of the receptor densities was used to account for the observed time-delay. The link model was also used in paper III to-gether with a model for the tolerance development seen. The tolerance was modelled as a hypothetical competitive antagonist in a tolerance compart-ment (76; 77).

Proportional odds model The proportional odds model was used in all papers; for the analyses (paper I and II) and for comparisons with other models (paper III and IV). When applying the proportional odds model to the data, the probability of an event is analysed. If Yi=(Yi1,Yi2,…,Yin) is the vector of categorical response for the ith individual, then the probability that Yit is greater than or equal to the score m has the following general structure:

,it i i m i if P Y m logit p f (4)

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in which,

log1

ii

i

plogit pp

(5)

giving,

,

,1

m i i

m i i

f

it i feP Y m

e (6)

The function f[x] denotes the logit transform of a probability, which is used to ensure the probability to be between zero and one. The fm,i is a function of baseline conditions, possible drug and placebo effects. i is a normally dis-tributed, zero mean random variable with the variance 2 describing the interindividual variability.

m

jjim placebodrugf

2, (7)

where j specifies the baseline probabilities of the categories. The intercept parameter for the lowest category needs not be estimated since the cumula-tive probability of event score being larger than or equal to the lowest cate-gory is by definition one. The drug and placebo parameters can have any shape and different forms of these parameters were evaluated in the different analyses presented.

Markov model The Markov model was used in the side effect analysis (paper III). This model estimates the probabilities of a having a certain score conditioned on the preceding observation. The Markov model can be viewed as a propor-tional odds model where the probabilities are dependent on the preceding stage through a first-order Markov element. The model was implemented on the three-level side effect classification. Let S denote an observed score. The logits, lx, of the probabilities that S 1 or S =2, given the preceding observa-tion (pre = 0, 1 or 2), was described by:

25

Dbbprel

Dbprel

Dbbprel

Dbprel

Dbbprel

Dbprel

S

S

S

S

S

S

652

51

432

31

212

11

2

2

1

1

0

0

(8)

Where the bxs are baseline fixed effects parameters of the model and D is the drug effect. Different models for D can be applied as appropriate. The prob-abilities corresponding to the logits in Eq 1 are given by:

xl

xl

x e1ePC (9)

The actual probabilities, px, of observing a particular score is given by: pS=0 = 1-PCS 1, pS=1 = PCS 1-PCS=2, pS=2 = PCS=2. Incorporation of different normally distributed random effects , with mean zero and estimated vari-ance 2, was evaluated. Implementation was as additive distributions for baseline parameters (b1, b3 and/or b5) and exponential distributions for con-centration-related PD parameters in the drug part of the model.

The side effect data was represented as observations of severity stage on an equal-spaced time grid, with observations frequent enough to adequately represent the data. A three-minute interval between observations was se-lected as this was sufficiently frequent that in no interval more than one tran-sition between severity stages occurred. Further, a three-minute interval was considered sufficiently fine for all practical use of the model. In the evalua-tion of model robustness, data sets were also developed in a similar manner but with intervals of one and six minutes, respectively. All data were ana-lyzed with both the Markov model and the proportional odds model.

Differential drug effect model One important feature of the proportional odds model is that it assumes that the effect size of the predictors is the same on all categories (47). This as-sumption is valid when the categorical data represents a categorization of a continuous scale, but it may however not hold true when analyzing a ranking scale. For example, it is possible to think of situations where a drug affects the lower part of a ranking scale different from the higher part of the ranking scale, especially when the ranking scale measures a complex or composed

26

endpoint. Modelling with different effect sizes on different parts of a cate-gorical scale could be accomplished by assuming a non-symmetric response curve and then use a different link function than the logit, for example the complementary log-log function or the probit function (25). A modification of the proportional odds model that also allows for different effect sizes has also been presented, the partial proportional odds model (48). This model is model is hierarchical to the proportional odds model, allows for different effect sizes of the predictor on the different observed scores, but does not assume another distribution than the proportional odds model do. However, as the name of the model implies, parts of the model should have odds that are proportional to each other, i.e. some of the categories should have the same effect of the predictor. An extension of the partial proportional odds model introduced in paper IV allows all categories to have different effects of the predictor if the data indicates this, while still being hierarchical to the proportional odds model. This model is called the differential drug effect model and is implemented by the function fdiff, which is multiplied with the drug effect model in the general equation:

,

2

m

m i j diff

j

f drug f +placebo . (10)

fdiff specifies the differential drug effect and is for the proportional odds model equal to 1 for all categories, whereas for the differential drug effect model as implemented here it follows the equation:

31

j

j

m

diff

j

efe

(11)

in which j specifies the change in drug effect introduced by the differential drug effect. The differential drug effect for the lowest category modelled is by definition 1 so that this category can be used as a reference category for the differential drug effect of the other categories. If all j’s are large, fdiff for all categories will be 1 and the differential drug effect model collapses into the proportional odds model, hence they are nested. The partial proportional odds model, which is a special case of the differential drug effect model, is achieved if at least one category has a fdiff=1 and at least one category has a fdiff=0. Figure 1 illustrates of the different forms of the function fdiff. The logits for a 4-category scale with different sizes of fdiff are plotted versus a predictor.

27

-15

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Figure 1. Illustration of the different forms of the function fdiff . The logits for a 4-category scale with different sizes of fdiff are plotted versus a predictor.

Non-Proportional Odds (fdiff1=0.5, fdiff2=0.5)

Partial Proportional Odds (fdiff1=1, fdiff2=0)

Proportional Odds (fdiff1=1, fdiff2=1)

28

Implementing the differential drug effect as a logit transform is done to ensure the effect to be between zero and one, while the estimated parameter,

j is allowed to vary between - and . However, the null hypothesis, the proportional odds model, places the parameter on the boundary of the pa-rameter space of the alternative hypothesis, the differential drug effect model. Consequently, the parameter is only allowed to move in one direc-tion, from to - , making an addition of a parameter in the differential drug effect model not to constitute a full degree of freedom. This phenomenon has also been observed for variance parameters (78-80). Wälhby et al (2002) (80) showed that one added variance parameter constitutes approximately ½ degree of freedom and this was also used for the differential drug effect model. However, fdiff can also be parameterized in other ways. If a signifi-cant fdiff, parameterized as described above, is found during the model devel-opment, its concentration dependence can be evaluated.

Paper IV investigates the performance of the differential drug effect model by a simulation study and by applying the model to data previously analyzed by the proportional odds model.

Simulation study The Type I error rate for the differential drug effect was investigated with a Monte Carlo simulation study. The study was designed as follows: 1000 datasets consisting of 1000 patients, evenly divided into 4 dose groups (pla-cebo, 7.5, 15 and 30 units of drug) with 4 observations per patient: 1 baseline observations and 3 observations when treatment was given, were simulated using the proportional odds model. The placebo effect was a step model and the drug effect was linear to the dose and all measurements were on a 4-category scale, starting at 0. At baseline, the frequency of reporting any of the categories was approximately 25% with an increase for category 3 to 50% when the highest dose, 30 units of drug, was given. The data was esti-mated using the reduced model, the proportional odds model, and the full model, the differential drug effect model, and the differences in objective function value (OFV) between the models were evaluated. The full model differed from the reduced model by two additional parameters.

Application to real data To assess possible improvements with the differential drug effect model, three previously published clinical studies were re-analyzed with the use of the differential drug effect model instead of the proportional odds model. The results from the re-analyses were compared with the previous results, thus comparing the proportional odds model to the differential drug effect model by comparing the changes in OFV. The model was applied to the sedation data from the stroke study, the phase I data from the multiple scle-rosis study and to an analysis of a 5-category diarrhoea scale where both the

29

effect of the drug and 2 metabolites were evaluated. That study was per-formed in 104 patients where all patients had measurements of the drug and one metabolite and 51 patients had also measurements of the other metabo-lite. The diarrhoea scale was measured 1-2 times per individual. The drug effect was linearly related to the AUC for the substance and for its metabo-lites. More details are found in Xie et al (2002) (60).

SoftwareAll PD analyses were done using a non-linear mixed effects approach as implemented in the NONMEM software version V, level 1.1 (Globomax Corporation) and version VI with the Laplacian estimation method and the Likelihood option. Model discrimination was based on goodness of fit plots, simulations and changes in NONMEM's objective function value. Xpose (Uppsala University) (81) was used for data checkout, graphics and other diagnostic techniques, in order to assist the model building. This program was run in an S-PLUS environment (Insightful Corporation, version 2000 for Windows). The software Excel (Microsoft Corporation) was used for data handling and graphics. The software Perl was used for data handling and simulations.

30

Results

Proportional odds model Stroke studies In total 1546 patients of the planned 1600 patients from all three studies were available for the analysis. 1545 of these patients had recorded sedation score measurements and were included in the sedation analysis. Several pa-tients had dose-interruptions as directed by the protocol when reaching a sedation score of 4 or greater (43% of those treated with clomethiazole, 6% of those treated with placebo). The pharmacokinetic evaluation was done on the 774 patients on active treatment.

Model for the time course of stroke induced sedation As the degree of sedation in the placebo patients seemed to change during the observation period, it was necessary to develop a model for this time dependence. In the final model the time dependence was included as an ex-ponential model and defined as:

t321T eP (12)

where 1, 2 and 3 are parameters of the model and t is the time after the start of the first infusion.

During the placebo model development it was noted that the values of 1were bi-modally distributed (Figure 2). Further investigation of the individu-als with their 1 in the lower modal of the distribution revealed that they had primarily sedation scores of 1. In other words, these individuals seemed to be less sensitive to the stroke induced sedation. To accommodate this phe-nomenon, a mixture model in b2 was used. In a mixture model, the model is defined so that two or more different, distinct, states are allowed. During the estimation of the model, the parameters of the different states are estimated together with the proportions of the likelihood that can be ascribed each state. The inclusion of the mixture in b2 significantly improved the fit and the classification of individuals into sensitive and non-sensitive to stroke in-duced sedation agreed well with Figure 2. The proportion of non-sedation sensitive patients in the placebo data set was estimated to be 0.18. Covariate

31

effects were evaluated on the term PT. Stroke severity was found to be the most important predictor of sedation. The relationship was found to be non-linear and sedation increased by 0.041 per NIH point for scores below 16 (the median value) and with 0.0095 per NIH point for scores above 16. Age was also found to increase sedation in a non-linear way. Sedation increased by 0.3 % per year below an age of 74 (the median value) and by 1.6 % per year above 74.

Figure 2. Distribution of 1 from a model for the placebo data not including a mix-ture for b2.

The model for clomethiazole induced sedation From Figure 3 it is quite clear that patients treated with clomethiazole on average reach a higher level of sedation than placebo patients. To accommo-date this fact, a drug effect parameter D was added to the logits. The logits now contain terms for both the stroke-induced sedation and clomethiazole induced sedation, where D is zero for placebo patients. Various forms of D were investigated. D could either be a single parameter, i.e. a step effect, or a function of the predicted clomethiazole concentrations from the final PK model. The functions tried in the latter case were the Emax model, the sigmoid Emax model, an exponential model or a power model, in models without and with delay between the concentrations and the sedation (effect-compartment model).

Based on the objective function value, graphics and parameter estimates it was concluded that a step model appropriately described the effect of clome-thiazole on sedation. No improvement over this model could be gained by taking the actual clomethiazole levels into account. For example, in an Emax

0

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model, EC50 was estimated to be as low as 0.25 µmol/L with no improve-ment in the objective function value compared to the step model. Covariate effects were then evaluated on the term D and weight were found to influ-ence the clomethiazole induced sedation. The clomethiazole induced seda-tion increased linearly with weight. The covariate was modelled as a slope intercept model, centered around the median values:

))75WT(1()WT(cõv WT|3cov3cov3 (13)

Clomethiazole induced sedation increased by 0.7 % per kg increase in body weight.

Final Pharmacodynamic Model The final model for stroke and clomethiazole induced sedation was a logistic regression model with the logits given by

1654326

154325

14324

1323

122

DPbbbbblDPbbbbl

DPbbblDPbbl

DPbl

TS

TS

TS

TS

TS

(14)

where D is zero for placebo patients and D = D*(1+cov3) for clomethiazole treated patients, where cov3 included covariate effects for weight, and PT =

1- 2e- 3t*(1+cov1)*(1+cov2), where cov1 included covariate effects for NIH and cov2 included covariate effects for age. The observed and predicted se-dation score frequencies from the final pharmacodynamic model for both placebo and clomethiazole treated patients are displayed in Figure 3. The final model parameters and comparisons of this analysis with the use of the differential drug effect model are presented in the differential drug effect model section.

33

Clomethiazole treated patients

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Figure 3. Goodness of fit plots for the final pharmacodynamic model showing the sedation scores over 10 time intervals with equal number of observations in each category. The upper panel shows the observed data for the clomethiazole and pla-cebo treatment groups respectively and the bottom panel shows the predicted seda-tion scores from the final pharmacodynamic model.

34

Multiple Sclerosis studies Models for TCR density The proportional odds model was used for the analysis of the categorical TCR data. The final drug effect models for both the phase I and for the phase II data were ordinary Emax-models. For the phase II data, essentially any low value of parameter b1 would describe data equally well. This corresponds to the probability of a non-high observation in the absence of drug being negli-gible. The value of this parameter was fixed to the value obtained for phase I data, to facilitate comparison of other parameters. Simulated proportions agreed well with observed for both the phase I and phase II models.

The final drug effect models for both phase I and phase II continuous TCR density data were sigmoidal Emax models with exponential interindi-vidual variabilities on baseline and EC50. The residual error models were for the phase I data a combined proportional and additive error model and for the phase II data a proportional model with exponential interindividual vari-ability were used. No significant covariate relationships were found. The results are very similar between the phase I and phase II models, the largest difference was seen in the shape of the Emax-curve, the Hill factor. It was in the phase I model estimated to 1.48 and in the phase II model to 0.705.

Figure 4 shows the individual EC50 values from the categorical and con-tinuous TCR models for the phase I and phase II data respectively, plotted against each other. The absolute values of EC50 can not be compared since the EC50 values from the categorical models are estimated on an logit scale, but the correlation between the models shows that the main trend in the data are preserved both for the trichotomized phase I data and also for the di-chotomized phase II data, even though some phase II subjects have some-what higher EC50 values when estimated from the dichotomized data. The categorical EC50 values for the phase I and phase II data were different be-cause of the different way the TCR expression was categorized.

35

0 100 200 300 400EC50 categorical model

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Figure 4. Comparisons between the individual EC50 values for the continuous and categorical receptor expression models for the phase I and phase II studies respec-tively. Please note that the categorical EC50 values from the phase I and phase II data are different because of the different way the TCR expression was categorized.

36

Models for the number of target T-cells It was observed in the long term follow-up period in the phase I study that the V 5.2/5.3+ T-cells never, at least not during the 18 months the study was carried on, returned to the levels observed before any drug was given. Ap-parently ATM-027 caused a very long acting reduction in the number of V 5.2/5.3+ T-cells. In the modelling this was taken care of by an empirical model which starts with a baseline before any drug administration, and dur-ing drug treatment a new baseline is established. The model used was pa-rameterized as:

))C/(*C1(*BASE))AUC/(A*AUC1(*BASEBASE

BASE

50max

50max0

10

ECEEAUC

t

ttt

i

(15)

where BASE0 is the baseline levels of V 5.2/5.3+ T-cells and the cumulative AUC of ATM-027, AUCt, is driving the establishment of the new baseline, BASEt, similar to an Emax-model. The actual level of V 5.2/5.3+ T-cells is also dependent on the actual concentration of ATM-027 and its effect is in-cluded in the model as an ordinary Emax-model, where C is the concentration of ATM-027. The model used to estimate the number of target T-cells was the same for both phase I and phase II.

The estimation of the drug effect differed slightly between phase I and phase II data. The parameter Amax, was similar in both studies and the new baseline was established on 25.7 % and 28.9 % of the baseline for the phase I and phase II data respectively, suggesting that the new baseline is established on approximately the same level irrespective of the amount ATM-027 given. The parameters Emax and EC50, which are accounting for the effects of each dose on the overall levels of target T-cells, differs more between the studies. Looking at an ATM-027 concentration of 100 µg/L and a cumulative AUC of 200 000 µg*h/L, the phase I parameters predicts a level of 0.88 % V 5.2/5.3+ T-cells of the total CD3+ T cell population, whereas the phase II parameters predicts a level of 1.48 %. The effects seen from the single doses of ATM-027 in the phase I study were estimated to be greater and also to occur at lower concentrations compared with the model prediction of the phase II data. The parameterization of the model with cumulative AUC driv-ing the establishment of the new baseline together with the actual concentra-tions driving the predicted cell number, could lead to correlations between the parameters. This was however not the case. When evaluation the correla-tions by looking at the correlation matrix yield by NONMEM, the highest correlation between two parameters were estimated to 0.8. When estimating the model with a full omega block, a clear overparameterisation of the

37

model, the highest correlation between two omegas was 0.63, indicating that no parameter correlations was overlooked.

Simulation of phase II trial Based on the continuous TCR expression and the cell count models from phase I, the phase II study was simulated. After obtaining the actual phase II results comparisons with the simulated studies were made. This evaluation was done with regard to both the predicted vs. actual observations of the two different PD variables, and also by looking at how they combined could predict the outcome of the adaptive design used in the phase II study. For example, the observed vs. predicted results from day 26, the continuous TCR expression had an observed value of 0.18 and the model predicted (mean [SD] of 10 studies) was 0.20 [0.09]. The observed value of the total number of V 5.2/5.3+ T-cells was 1.05 % and the model predicted (mean [SD] of 10 studies) was 0.69 % [0.73]. Apparently, the simulation model was doing well with regard to predicting the TCR expression, but the number of V 5.2/5.3+

T-cells was slightly underpredicted. The cells were coming back somewhat faster after a dose than the phase I model predicted. This is in agreement with the differences seen in the parameter estimates when comparing the phase I and phase II results from the model for the number of T-cells showed in the previous section. The observed and predicted highest doses given to each patient showed that the model in general predicted the used dosing regimen well.

Markov model The observed numbers of transitions between the side effect scores in paper III were: from 0 -> 1: 24, from 0 -> 2: 11, from 1 -> 2: 23, from 2 -> 1: 1, from 2 -> 0: 32 and from 1 -> 0: 2. In no instance did the same type of tran-sition occur twice in any individual. The data indicated both a time-delay between concentrations and effects and tolerance development. Therefore the final model consisted of an effect-compartment model (link-model) (13) with a tolerance compartment where tolerance was modelled as a hypotheti-cal competitive antagonist (14). Different shapes of the effect model were also evaluated, e.g. the linear effect model, the Emax model and the sigmoidal Emax model. In the final model, the drug effect D had the form of:

CeeTCCEC

eTCCECED

i

i

Tol

Tolpre

5050

50502,1,0max

/1

/1 (16)

where Emax is the maximal drug effect, EC50 is the concentration in the effect compartment producing 50% of the maximal effect, CTol is the concentration

38

in the tolerance compartment, TC50 is the concentration in the tolerance compartment which doubles EC50 and Ce is the concentration in the effect compartment. The final model used three different Emax for the preceding observation being 0, 1 or 2. All individuals started all sessions with a side effect score of 0 and as the probability of scoring anything else in the ab-sence of drug was very low, the parameter b1 was fixed to a low value.

To evaluate the predictive properties of the Markov model, a posterior predictive check (PPC) (12) was performed where 100 data sets were simu-lated from the final model. For example, looking at the average number of predicted transitions from the 100 simulated datasets: from 0 -> 1: 26, from 0 -> 2: 11, from 1 -> 2: 25, from 2 -> 1: 1, from 2 -> 0: 35 and from 1 -> 0: 1, the numbers of transitions were very close to the observed data. Average observed times to the first transition (0 -> 1 or 0 -> 2) were for the different treatments in the study: low infusion rate, 0.79h, intermediate infusion rate 0.58h and for the high infusion rate 0.36h. Average times to first transition from the 100 datasets in the PPC were for the low infusion rate: 0.80h, for the intermediate infusion rate: 0.54h and for the high infusion rate: 0.34h.

The difference between the Markov model and the proportional odds model was investigated by fitting both types of models to different datasets with observations each minute, every 3rd minute and every 6th minute. From the final models posterior predictive checks were performed where 100 data-sets were simulated from the final models. From the results the difference between the models is evident. Both models predict well the time to first transition and the concentration at the first transition, although the propor-tional odds model fails somewhat in this respect for the data with more fre-quent observations (Table 2). The prediction failure of the proportional odds model becomes even more evident when looking at the total number of pre-dicted transitions (Table 3), where it predicts markedly too many transitions. Further, it is also clear that predictions will depend on the frequency by which the observed data are represented. The Markov model on the other hand predicts well compared to the observed data and predictions are inde-pendent on the observation frequency representation of the data set. Not only the predictions, but also the parameter estimates (with intercepts, bX, cor-rected for observation frequency) and parameter uncertainty for the final Markov model was similar for the three data sets. 95% confidence intervals were determined by likelihood profiling for one well estimated parameter, EC50, and one poor estimated parameter kto for the Markov model on differ-ent datasets. The estimated confidence intervals were for EC50 (0.04; 0.12), (0.05; 0.11) and (0.05; 0.12) and for kto (7E-10; 1.1), (7E-12; 1.2) and (1E-15; 1.1) for the 1 min, 3 min and 6 min datasets respectively.

Table 2. Results from the comparative posterior predictive checks for the proportional odds model and the Markov model on different datasetswith observations each minute, every 3rd minute and every 6th minute. The results shown are the average and (10th and 90th percentiles) from 100 simulated datasets.

Proportional odds model Markov model

1 min 3 min 6 min 1 min 3 min 6 min Observed

data

Time to first positive transition (h)

Low infusion rate 0.63 (0.51; 0.76) 0.72 (0.59; 0.90) 0.79 (0.66; 0.95) 0.81 (0.66; 1.0) 0.80 (0.64; 0.95) 0.83 (0.66; 1.1) 0.79

Medium infusion rate 0.41 (0.35; 0.46) 0.47 (0.40; 0.54) 0.52 (0.45; 0.60) 0.54 (0.45; 0.62) 0.54 (0.46; 0.64) 0.54 (0.45; 0.63) 0.58

High infusion rate 0.27 (0.24; 0.30) 0.30 (0.27; 0.35) 0.35 (0.31; 0.39) 0.34 (0.29; 0.39) 0.34 (0.30; 0.38) 0.35 (0.31; 0.39) 0.36

Free concentration at first transition (nmol/L)

Low infusion rate 1.0 (0.92; 1.3) 1.1 (1.0; 1.3) 1.2 (1.1; 1.4) 1.4 (1.2; 1.5) 1.4 (1.2; 1.6) 1.4 (1.2; 1.6) 1.1

Medium infusion rate 1.3 (1.2; 1.4) 1.4 (1.3; 1.6) 1.6 (1.3; 1.7) 1.6 (1.4; 1.8) 1.6 (1.5; 1.8) 1.6 (1.4; 1.8) 1.7

High infusion rate 1.9 (1.7; 2.1) 2.1 (1.9; 2.3) 2.3 (2.1; 2.5) 2.3 (2.0; 2.5) 2.3 (2.1; 2.5) 2.3 (2.0; 2.5) 2.4

Table 3. Results from the comparative posterior predictive checks for the proportional odds model and the Markov model on different datasets with observations each minute, every 3rd minute and every 6th minute. The results shown are the average and (10th and 90th percentiles) from 100 simulated datasets.

Proportional odds model Markov model

Number of transitions 1 min 3 min 6 min 1 min 3 min 6 min

Observed data

0-1 362 (324; 403) 137 (120; 152) 76 (65; 85) 26 (21; 30) 26 (21; 30) 26 (23; 30) 24

0-2 289 (249; 335) 117 (102; 133) 72 (62; 82) 11 (9; 14) 11 (8; 15) 11 (8; 15) 11

1-2 178 (133; 223) 79 (64; 95) 48 (39; 59) 25 (21; 29) 25 (20; 29) 25 (21; 29) 23

2-1 178 (133; 222) 78 (62; 94) 48 (38; 57) 1 (0; 3) 1 (0; 3) 1 (0; 2) 1

2-0 289 (242; 338) 117 (98; 135) 71 (60; 81) 35 (32; 38) 35 (31; 39) 35 (32; 38) 32

1-0 362 (230; 402) 135 (119; 150) 76 (66; 85) 1 (0; 3) 1 (0; 2) 1 (0; 3) 2

41

Differential drug effect model Simulation study The Type I error rate was evaluated in paper IV by looking at the distribu-tion of the differences in NONMEMs objective function value ( OFV) be-tween the differential drug effect model and the proportional odds model for the 1000 simulated datasets. When assuming a 2-distribution of the OFV values, the percent of the datasets showing a difference of more than 3.84 and 6.63 represents an estimate of the Type I error rate corresponding to expected p-value of 5% and 1%, respectively. The results of the simulation study shows that the Type I error rate is as expected; 4.9% of the analyses of the simulated datasets showed a significant drop in OFV at a 5% signifi-cance level and the corresponding value at a 1% significance level was 0.84%. The differential drug effect model appears not to be indicated as measured by the OFV when it should not offer any improvements of the model. When analyzing the data using the proportional odds model, 998 runs minimized successfully with a covariance step, hence calculations of the type I error rate were independent of if the calculations were based on suc-cessful runs or all runs.

Application to real data Results of the implementation of the differential drug effect model to the previously analyzed clinical studies showed no statistical improvements for the 3-category T-cell data and for the 5-category diarrhoea. For the 6-category sedation data, however, the differential drug effect model provided a statistical significant model improvement. The original analysis with a proportional odds model gave a step-effect for the drug effect on sedation, i.e. the placebo group had one effect and the actively treated group had an-other effect. Implementation of the differential drug effect model as de-scribed in the methods section gave significant model improvement. That result encouraged further evaluation of the shape of the differential drug effect and concentration dependent models for the differential drug effect were implemented as follows:

m

j jjjdiff conc

concconcf3

1 (17)

Where j is the differential drug effect with boundaries 0 -> 1 and j is the C50 effect of the differential drug effect, bound to be positive. In the sedation

42

example, two of the possible differential effects were estimated and in one of these a C50 was estimated. In the other estimated differential effect, was set to 1000 to generate a linear dependence with concentration; hence the full equation was as follows:

5436

435

555

3444

3

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1

diffdiffdiffdiff

diffdiffdiff

diffdiff

diff

ffff

ffconc

concf

fconc

concf

f

(18)

This approach provided further model improvement and the final model thus had a linear concentration effect on the probability of scoring 4 and an Emax-like shape of the probability of scoring 5 on the scale while the differential drug effect on the other probabilities were independent of concentration. The parameter estimates from the proportional odds model, the differential drug effect model and the differential drug effect model with concentration de-pendence are shown in Table 4 and show that the parameter estimates when applying the different parameterizations of the differential drug effect does not alter the parameter estimates of the basic model estimated with the pro-portional odds model.

43

Table 4. Parameter estimates (Relative Standard Error) for the proportional odds model, the differential drug effect model and the differential drug effect model de-pendent on concentrations on the 6-category sedation scale used in acute stroke patients.

Parameter Proportional Odds Differential Drug effect

Differential Drug effect dependent

on concentrations

Baseline2 -4.99 (6.89)

-9.12 (7.79)

-4.85 (7.71)

-8.80 (8.36)

-4.88 (9.02)

-8.84 (10.2)

Baseline3 -3.14 (2.43) -3.15 (2.41) -3.15 (2.42)

Baseline4 -1.56 (3.85) -1.42 (9.30) -1.43 (11.7)

Baseline5 -1.86 (5.48) -1.18 (15.7) -1.17 (15.9)

Baseline6 -2.07 (13.2) -1.17 (24.5) -1.18 (25.0)

Drug 2.49 (5.70) 2.60 (6.31) 2.61 (6.67)

Mixed 0.199 (30.8) 0.232 (32.1) 0.227 (38.3)

Plc1 5.12 (4.94) 5.03 (4.97) 5.03 (5.51)

Plc2 1.70 (5.52) 1.70 (5.54) 1.70 (5.52)

Plc3 0.304 (12.8) 0.304 (11.5) 0.303 (13.7)

3 - 1* 1*

4 - 0.94 (34.8) 0.94† (7.96)

5 - 0.65 (57.8) 0.443 (41.3)

6 - 0* 0*

5 - - 13.4 (8.36)

2 3.69 (12.8) 3.44 (14.1) 3.49 (15.3) * Parameter fixed for estimation

† Linear model with concentrations

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The predictive performances of the different models are shown in Figure 5 where the empirical cumulative probabilities of the different scores are shown for the different models and for the placebo and drug treated groups respectively. The improvement achieved by the differential drug effect model lies in the prediction of the higher scores on the scale as shown in the lower panel in Figure 5. Since the proportional odds model adds the same effect of treatment on all scores it overpredicts the probability of scoring a 6 (the highest number on the scale) for patients on active treatment but also underpredicts the probability of scoring a 6 for the placebo group.

As seen from the observed data, drug treatment did not give rise to more observations of 6 compared to placebo, rather the contrary, and the differen-tial drug effect model predicts the observed data better than the proportional odds model both for the drug treated and placebo treated groups. For the sedation data, the differential drug effect model did not only give a statistical significant model improvement; it appeared also to have a better predictive performance.

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Figure 5. The predictive performances of the proportional odds model and the dif-ferential drug effect model for the 6-category sedation data. Observed data are shown together with the empirical cumulative probabilities of simulated data of the different models for the placebo and drug treated groups respectively. The im-provement achieved by the differential drug effect model lies in the prediction of the higher scores on the scale as shown in lower panel in the figure.

Simulated sedation data

46

Discussion

The aim of the thesis was to develop models for ordered categorical PD data encountered in drug development. This has been done by analysis of seda-tion data from a phase III stroke program, analysis of T-cell data from a phase I and II multiple sclerosis program and by analysis of side effect data from a phase I study. The models used were the traditionally used propor-tional odds model, but 2 new models were also developed and applied. The Markov model for analysis of a spontaneously reported event and the differ-ential drug effect model that enables analyses of data where the size of the drug effect is different on different parts of an ordinal scale. These examples show that categorical data encountered in clinical trials with different design and different categorical endpoints could successfully be incorporated in PK/PD models. The models presented can also be applied to analyses of other ordinal scales than those presented in this thesis.

The sedation data from the clomethiazole stroke trials were analyzed with the proportional odds model. Sedation was a problem that needed monitoring in the clinic, since both the drug but also the disease itself causes sedation and excessive sedation led to dose interruptions. It was therefore important to understand the time course of both the natural course of sedation in stroke patients and the effect of clomethiazole induced sedation. The PK of clome-thiazole had previously not been described in stroke patients and therefore a large population PK analysis with covariate evaluation was an important part of the analysis. The heterogeneity in the stroke population with respect to sedation was in the PD analysis accounted for by use of a mixture model. The analysis with the proportional odds model could describe the sedation data in stroke patients and its relations to drug treatment, time after start of treatment, stroke severity at admission, age and body weight. The analysis showed no concentration-effect relationship for clomethiazole. However, later it was possible to find concentration-effect relationships for some of the scores by use of the differential drug effect model. For the aim of finding a concentration-effect relationship, a design with different fixed doses result-ing in a larger spread of the plasma concentrations would have been better than the used regimen with weight adjusted doses aiming for a certain target concentration. The model was evaluated by simulating back the original data but a more thorough model evaluation with a PPC would have been desir-able. This was not possible to accomplish given the study design, with a dose

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interruption at a score of 4 and the subjective choice from the investigator to restart the infusion when sedation decreased. A PPC would have required several assumptions regarding the subjective part of the design and would have made the results from such an evaluation dependent on them.

It is very uncommon that the same endpoint is collected with both con-tinuous and categorical measurements; usually categorization of a continu-ous scale is done before designing a study and only the amount of data fal-ling into the different categories are collected. The ATM-027 data was an exception to that rule and the same endpoint was actually available both as categorical and continuous data. During the clinical development of ATM-027 as treatment for MS, the T-cell receptor density data were first presented as a subjective categorical classification by the bioanalyst and incorporated in the PK/PD-models but later it was realized that the data also could be made available as continuous measurements. This gave the possibility to study what impact the information loss by categorization of the scale actu-ally means when predicting the phase II trial. However, since the categoriza-tion in the phase II trial was different from phase I both in terms of changing a trichotomous variable to a dichotomous but also by altering the cut-points for the classifications, simulation of phase II given the phase I categorical model could not be accomplished. The phase II trial was then only simulated from the continuous phase I receptor number and receptor density models, but the simulation model could simulate the rather complicated adaptive dosing regimen used with results very close to the observed data in the phase II trial. The proportional odds model used for the categorical receptor ex-pression data could adequately describe the observed classifications in both the phase I and phase II trials.

Self-reported data are sometimes encountered in drug development but modelling of these data are seldom seen. The side effect data presented here may also have been analyzed by use of a time-to-event approach since the exact time of each event was known. Such an analysis has to my knowledge not been presented for a graded event, but it might be possible to develop different time-to-event models for all observed events, in this case the 6 dif-ferent transitions between the side effect scores. However, that model would be rather cumbersome to use. If interpreting all data as transitions between stages, a full transition model could have been used. That approach will also require the development of 6 different models for the observed transition between side-effect scores and in this case very little information was avail-able for some transitions, for example only one transition between 2 -> 1. A transition model would also be much more cumbersome to use than the Markov model presented here. The Markov model can also be applied to other situations than those with spontaneously reported data. Studies with frequent observations is likely to have correlations between the observations

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and to ignore them and use the proportional odds model could lead to the development of an erroneous model with bad predictive performance. As shown in paper III, the sampling frequency is important for the number of transitions predicted by the proportional odds model whereas the Markov model predicts the same number of transitions independent of the sampling frequency in the dataset.

The differential drug effect model enables different effect sizes of the drug to be applied to different parts of an ordered categorical scale and could be used for analyses of heterogeneous scales or in other situations when the observed data not does represent a categorization of a continuous scale. It is a hierarchical expansion of the proportional odds model and it appears to have the desired property of not being indicated where the drug effect is truly proportional on the entire measured scale as shown by the simulation study. The differential drug effect do not have to be a fixed effect, it can be a function of different predictors, such as concentration, and the predictors can act differently on different parts of the analyzed scale. Applied to data simu-lated from the proportional odds model, the differential drug effect model showed no improvement over the proportional odds model. When applied to real data, the differential drug effect model did not show any improvements for the 3-category T-cell data nor for the 5-category diarrhoea data. The T-cell data was a categorization of a continuous scale and it is for such data the proportional odds model once was derived and the diarrhoea data appears also to be homogenous enough to mimic a continuous scale. The reason for the good performance of the differential drug effect model on the sedation data probably lies in the nature of the scale. Sedation is a complex outcome and the severe degrees of sedation, score 5 = reacts only to pain and score 6 = does not react to pain, could be caused by other physiological mechanisms than the lighter degrees of sedation and the scale appears not to be a catego-rization of a continuous scale. In similar situations, the differential drug ef-fect model should be considered during the model building process. The presented implementation of the differential drug effect model is rather pragmatic approach to the problem of not having a proportional drug effect on the entire analyzed scale; it is however also possible to use methods for more formal testing of whether the proportional odds assumption is correct (82).

In general, categorization of a continuous scale means throwing away in-formation (83). It is therefore surprisingly often that is done in order to use a simple statistical test of the endpoint. It is understandable to take this ap-proach for explaining complex study results in a simple way, but during drug development those approaches can lead to ignorance about important aspects of drug action. If a categorical scale needs to be used, it is wise to use all information possible to get from it. That is not always the case as shown by

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an example from migraine therapy. All migraine trials measure pain relief on a 4-level categorical scale: 1 = no pain, 2 = mild pain, 3 = moderate pain and 4 = severe pain. Even if this scale is rather rough itself, it is common practice to evaluate if a patient, who initially had the score of 3 or 4 that is required for entering the study, reaches 1 or 2 on the scale. The patient is then judged to be a responder to the treatment. The scale is being dichotomized and the number of responders and non-responders is being compared between the different treatments in the study (84). By applying some of the approaches presented in this thesis, more information of the concentration-effect rela-tionships of the studied drug is likely to be gained and could lead to devel-opment of more informative study designs and better dosing regimens.

PerspectivesThe use of PK/PD modelling is increasing in drug development and the models are to a larger extent being used by drug companies in discussions with drug regulating authorities, leading to the use of PK/PD models for situations not previously explored. Analysis of ordered categorical data is a relatively new area within PK/PD modelling and more aspects of these mod-els will be explored over the coming years. The models are likely to be ap-plied to new therapeutic areas and new scales, but improvements of the cur-rently used models will probably also take place. Transition models have so far only been used in 2 applications (61; 85) and will be explored further. Combinations of these models with time-to-event models or count-models could also be a field for future research. Different estimation techniques are also likely to be adopted. The use of the Laplacian likelihood method within NONMEM has been shown to lead to bias in the parameter estimates for highly skewed data, whereas the Gaussian quadrature method within the software SAS seems to be more robust (86). However, a method to eliminate the problem with biased estimates within NONMEM, the back-step method, has been proposed (87). Another interesting field for future research is to incorporate models for categorical PD data in analyses within a Bayesian framework.

When PK/PD models for ordered categorical data are being used more frequently, improvements of the use of these models can be done. Experi-ments could be designed using optimal design theories to optimize the esti-mation of the model parameters. Nestorov et al (51) have published the first study where optimal design points for a proportional odds model were evaluated. The analysis of ordered categorical PD data might be a more suit-able field for optimal design theory than regular PK/PD modelling with con-tinuous endpoints. When analyzing ordered categorical PD data, fewer model options are available which makes model discrimination easier. There

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is no residual error model and interindividual variability is in most cases only applied to the baseline parameters. The choice of model could therefore be done in advance more easily than a traditional PK/PD analysis. Taken together, this would favor the use of models for categorical data in optimal design experiments.

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Conclusions

The examples presented in this thesis show that categorical data obtained in clinical trials with different design and different categorical endpoints suc-cessfully could be incorporated in PK/PD models. The models presented can also be applied to analyses of other ordinal scales than those presented.

The proportional odds model could describe the sedation data in stroke pa-tients and its relations to drug treatment, time after start of treatment, stroke severity at admission, age and body weight. Further a mixture model was used to account for the baseline-heterogeneity in the patient population with respect to the sedation following stroke.

The proportional odds model could adequately predict the different categori-zations of the V 5.2/5.3 receptor expressions and the effects of categorizing a continuous scale were evaluated.

A new model, the Markov model, was developed for the spontaneously re-ported side effect and was shown to be superior to the proportional odds model for this type of data. The model can also be used in other situations when dependence between observations exists.

A new model, the differential drug effect model, was developed to enable evaluations of heterogeneous scales. It was shown to have the property of not being indicated where it is not necessary and to provide model improve-ments when the analyzed scale not is a categorization of a continuous scale.

The work has also gained insight in other aspects of the analyzed data in-cluding:

A PK model for clomethiazole in acute stroke patients. A model for the natural time course of sedation in stroke patients. A PK model for the monoclonal antibody ATM-027 in multiple sclerosis patients.A PD model for the time-course of the depletion of V 5.2/5.3+ T-cells in MS patients. The models developed from the phase I MS data were successfully used for prospective prediction of the outcome of the phase II ATM-027 trial with an adaptive study design.

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Populärvetenskaplig sammanfattning

Under klinisk utveckling av nya läkemedel designas kliniska studier för att svara på frågor om det nya läkemedlet är säkert och har den önskade effek-ten på sjukdomen i den patientgrupp som ska behandlas. Många av de mät-ningar som görs i studierna utförs med olika skalor som rangordnar effekter eller en biverkan. Farmakokinetiska/farmakodynamiska modeller är matema-tiska modeller som användes för att beskriva tidsförloppet av plasmakon-centrationerna av ett läkemedel samt sambandet mellan koncentrationer och de effekter läkemedlet utövar på kroppen. Dessa modeller är viktiga under läkemedelsutvecklingen för förståelsen av hur de läkemedel som utvecklas beter sig i kroppen och de används också i diskussioner med myndigheter som godkänner nya läkemedelsbehandlingar. De farmakodynamiska data som används i modellerna beror på vilken effekt som studeras och är van-ligtvis mätningar av någon kontinuerlig effekt, exempelvis blodtryck eller en hormonnivå. I denna avhandling har farmakodynamiska modeller för data som är olika typer av rankingskalor, och som också kallas för ordnade kate-goriska variabler, studerats. Den mest frekvent använda modellen för sådana analyser kallas för den proportionella odds modellen. Den modellen använ-des för analys av sedationspoäng i patienter som fått en akut stroke samt för en analys av receptoruttrycket på en T-cell i studier på Multipel Skleros pati-enter. En vidareutveckling av denna modell användes för att analysera en spontant rapporterad biverkan och en ny modell utvecklades också för skalor som är heterogena eller där läkemedelseffekten är olika stor för olika delar av skalan. De nya modellerna jämfördes med den proportionella odds model-len och de verkar innebära förbättringar i de studerade situationerna genom att bättre kunna beskriva de observerade data. De nya modellerna kan an-vändas för analyser av andra sorters rankingskalor och möjliggör adekvata analyser av ordnade kategoriska farmakodynamiska variabler i fler situatio-ner än de som hittills studerats.

53

Acknowledgements

The thesis work was carried out at the department of Clinical Pharmacology, AstraZeneca R&D Södertälje, Sweden, and at the Division of Pharmacoki-netics and Drug Therapy, Department of Pharmaceutical Biosciences, Fac-ulty of Pharmacy, Uppsala University, Sweden.

I would like to express my sincere gratitude to all who have helped and en-couraged me over the years. Especially I would like to thank:

Professor Mats Karlsson, my supervisor, for your brilliance, never-ending enthusiasm and your willingness to share your deep knowledge. I am very grateful for having been taught pharmacometrics by you and I really hope that we will continue to have some kind of collaboration in the future as well.

Associate professor Niclas Jonsson, my co-supervisor, for your engagement, knowledge and your much valued practical support during my first years as a PhD-student, I have really enjoyed working with you. I have also had a lot of fun during our nighttime discussions and I am glad that you still want to play squash against me, despite a total lack of victories…

Dr. Gunilla Osswald, for employing me and together with Dr. Marcus Jer-ling giving me the opportunity to be an external PhD-student at AstraZeneca. I am very grateful that you have fulfilled your initial commitment, despite numerous re-organisations, project cancellations and other hurdles in our ever-changing environment.

Professor Margareta Hammarlund-Udenaes for providing nice working facilities and always welcoming me at the division.

All co-authors of my papers: Marianne Ekblom, Tomas Odergren, Tim Ash-wood, Patrick Lyden, Carlotte Edenius, Matts Kågedal and Maria Kjellssonfor your contributions. Especially I would like to thank Maria Kjellsson for sharing the differential drug effect model paper and for your hard work to meet my tight deadlines during your late-time pregnancy.

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All heads of the different departments I have been working in: Marianne Ekblom, KG Jostell and Marie Gårdmark, for letting me continue the started work. Especially I would like to thank KG Jostell for your support, encour-agement and for reading through the different manuscripts, the thesis and always providing valuable comments.

My PK-colleagues at AstraZeneca: Marcus Björnsson, Yi-Fang Cheng, Ur-ban Fagerholm, Gunilla Huledal and Matts Kågedal for numerous interest-ing discussions. Especially I would like to thank Matts Kågedal for all our interesting chats by the white-board and Marcus Björnsson for all valuable discussions about everything from NONMEM coding to sports and childcare and for your help with perl-programming.

All other colleagues at former Experimental Medicine, especially Annie,Björn, Stig, Ann and Ewy for a lot of fun discussions and gossiping!

Bengt Hamrén for sharing the good and bad things of being an external PhD-student and for your contributions to the pharmacometric field within AZ.

All past and present colleagues at the Division of Pharmacokinetics and Drug Therapy (Anja, Karin, Siv, Lena, Ulrika W, Ulrika S, Lasse, Mama, Jörgen, Emma, Jacob, Rikard, Elisabeth, Anders, Hanna, Justin, Fredrik, Mia, Andy, Kristin, Johan and all the rest of you), thank you for always wel-coming me at the department and for lots of fun during parties and travels!

Jag vill också tacka all släkt och alla vänner som har stött mig under dokto-randarbetet:

Per för att stunderna med H-båten innebär en fantastiskt skön avkoppling från allt annat! Johan & Malin för trevliga minisemestrar, Johan & Ida, Anna & Martin, Gia & Fredrik, Marcus & Erika för trevliga middagar och andra upptåg, Johan och Jacob för roliga seglingar.

Femkampsgänget: Pelle, Johan, Stefan och Henrik för ständiga utmaningar. Jag vann!

Per, Johan, Martin, Tobias och Carl för spelglädjen på torsdagskvällarna!

Mamma Anna-Lenah, pappa Per-Anders och lillebror Carl för att ni alltid finns där med stöd och omtanke.

Mina älskade flickor Agnes och Lovisa, för all kärlek och omtanke och för all kraft ni ger mig genom att bara vara. Tack för att ni har stått ut med mig trots all tid jag ägnat åt denna avhandling istället för att vara med er.

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