Aspects on tolerance limit estimation - some common approaches and flexible modeling

Abstract

In a dose finding study the aim is to attain a safe and efficient drug therapy in a certain population. By a dose finding study we generally refer to a study where the dosage is successively adjusted after analyzing the responding concentration in e.g. the blood. Measurements of interest are e.g. area under curve, time to peak, peak value, time to zero and tolerance limits. The estimates obtained from the pharmacokinetical data are compared with the known safety and efficiency profiles of the drug, and the dosage may be adjusted to attain a satisfactory result. Different estimation approaches have to be used depending on the safety and efficiency profiles of the drug. When estimating tolerance limits there are roughly two different approaches, one intended for drugs with severe side effects and one intended for drugs with harmless side effects. However, sometimes the approaches are used as interchangeable resulting in misleading estimates. It is also important to consider the data utilization. A variety of estimation techniques are used in pharmacokinetics. Some of the proposed estimators use cross sectional data and the more advanced ones use the longitudinal structure of the data to different extents. The benefits of the latter are often considerable, especially for small sample sizes common in pharmacokinetics. For crude data where we can not assume fixed regression parameters flexible regression models tend to be superior.

In Report I two different estimation approaches for tolerance limits are considered: the conservative approach intended for drugs with severe side effects and the closeness approach intended for drugs with harmless side effects. It is obvious, but rarely discussed, that the conservative approach tends to result in less efficient drug therapies than the closeness approach. In the case with severe side effects this disadvantage may be well motivated by safety aspects. However, this approach is sometimes also proposed for drugs with harmless side effects. In Report I it is shown in a simulation study that the conservative approach can be considerably less efficient and some examples are given. Both parametric and non-parametric cross sectional estimators are used in the study.

In many applications data is based on repeated observations over time. In Report II this longitudinal structure of the data is taken into consideration. A flexible model which after linearization has a random intercept and a fixed slope is proposed. The model is flexible in the sense that the time dependence is modeled by tP where t is the time value and p is an unknown parameter. A two-step estimation approach is proposed where p is estimated in the first step, and the rest of the parameters are estimated in the second step by using standard regression techniques. The effects of first estimating p upon the rest of the estimators are demonstrated by a simulation study. It is concluded that bias and precision of estimators of the variance components and the dependent variable remain quite unaffected by the two-step procedure, as well as bias of the slope and intercept parameters. However, the precision of the latter estimators may be poor for small values of p, provided that the variance of the measurement errors is large. ii