Sample Size Determination and Statistical Hypothesis Testing for Core Centration in Press Coated Tablets

A novel statistical approach to evaluate the manufacturing quality of press coated tablets in terms of the centering of their core is presented. We also provide a formula to determine the necessary sample size. This approach is applied to real data.


Introduction
Dry-coated tablets provide an adequate and inexpensive technology for the development of controlled-release tablets.The combinations of different compositions between the core and the coat allows for a large variety of design of the release profile [1,2].It is important that the core is well centered inside of the tablet to make the tablet more light-stable, more water-stable or to preserve the release properties [2][3][4].The positioning of the core varies slightly from tablet to tablet during the manufacture process [5][6][7][8][9].A quality assessment process needs then to be established to guarantee that the centering of the core stays within certain specifications.Current methods used for evaluating core centration are either imprecise, where the core position is measured "by hand" [1,10], or expensive, where costly diagnostic equipment is required [3].An option to obtain well-centered cores is to use onestep dry-coated tablets [3,11] but the technology is rather new and not implemented everywhere.In this paper, we propose a relatively inexpensive and reliable methodology to monitor core centration during the manufacture of dry-coated tablets.The methodology is based on the cutting of a small sample of tablets (see Section 2), dyeing of the cut surface to distinguish the core from the coat, taking pictures of the cut surface with a digital camera, analysis of the pictures using image processing algorithms to extract core positioning parameters (see Section 3), and a statistical analysis of these data to provide core positioning information for the whole production batch and to compute the minimal sample size required to give a meaningful statistical answer.The goal of this paper is to present in details the statistical method that was devel-oped (see Section 4) and an example of application to real data (see Section 5).

The Core Centration Problem
The notations for the dimensions of the diameter and thickness of the tablet and of the core are given on Figure 1.
Generally, the exclusion criterion for deciding if the centering of a core is acceptable or not is if the minimum distance between the edge of the core and the border of the tablet is smaller to some fixed distance , let's say 1/10 th of min . We will assume that, for practical reasons (e.g. to prevent a breaking of the tablet), it is only possible to make one cut per tablet.The two types of cuts useful to measure the quality of core centration and easy to perform are the longitudinal cut and the transverse cut, as schematized on Figure 2.
Once several tablets have been cut, transversely or longitudinally, we measured various displacement quantities, as illustrated on Figure 3, using a modus operandi described in the next section.The needed measures of position and distance are:   O is the center of the tablet on the bottom surface of it, while O is the real center of the tablet;  C is the position of the center of the core in the tablet and H is its orthogonal projection on the rotational axis of symmetry of the tablet;  h is the distance (measured on the transverse cut) from H to the bottom surface of the tablet (along the axis of rotation), i.e. distance OH ;  r is the distance (measured on the longitudinal cut) from the center of the core C to the rotational axis of symmetry of the tablet;  that will be used in Section 4 to build the statistical test of core centration.
Moreover, note that we will suppose that vertical and horizontal displacements are independent, and we will also neglect tilt movements of the core, which is a realistic assumption as can be seen on Figure 5.

A Pattern Recognition Tool
The cut tablets are placed on a tray and two pictures are taken, one for the longitudinal cut (see Figure 4) and one for the transverse cut (see Figure 5).After dyeing, the core appears dark brown and the outer layer appears beige.Using Matlab and its Image Processing Toolbox, we developed algorithms for image recognition to identify the border of each tablet and each core.The necessary positioning parameters are then computed and used for  the statistical analysis.

The Statistical Formulation of the Question
Let be a random variable giving, for each observed tablet, the minimum distance between the edge of its core and the border of this tablet.Let 0 be a reference distance below which a tablet is declared unacceptable and let be the unknown probability that is above the threshold.
The objective is to significantly prove, namely with a small and controlled error risk (less than some fixed threshold α), that (in the population) p m0 > c = 1 -ε for a fixed and small given tolerance value  ; that is to say that our tablets are correctly manufactured with high probability.
We can formalize this problem in the statistical hypothesis jargon, using the so-called null and alternative hypotheses: Following the notations given in Section 2, we define , where we will  suppose that the random variables  and are independent.It is easy to see that are supposed to be independent, we have

The Statistic Used to Build the Test
Suppose we have a random sample of 1 2 tablets, where 1 (resp.2 ) tablets have been longitudinally (resp.transversely) cut.So we end up with the random samples , independent copies of and respectively.Based on these observations, we can estimate with the statistic where

Distribution of the Test Statistic
We have is a ran-  dom variable that can take values in 1 .In fact, one can show that it has a binomial distribution with so .Similarly, one can show that n F has a binomial distribution with .Thus the test statistic, , and whose distribution is the product of two independent binomial distributions The (unknown) probabilities of distribution are Let 1 and 2 be two independent random variables with distributions and respectively, and let , 0, , 0, , ; . ,

Critical Region and Statistical Decision
, we have from ( 5) For a pre-specified significance level  (0.05 or 0.1 are classical values), let be the largest (positive) value, called the critical value, such that: We will reject 0 H (i.e.we will show 1 H : our tablets are correctly manufactured) with a controlled type I error risk (probability of taking a wrong decision) as soon as (computed with our observations) will be greater than However, since h is unknown, we will substitute it with a value h given by an expert.This expert likely computed it from a preliminary study in a context where tablets were known (e.g. using more expensive and sophisticated techniques) to be correctly manufactured (i.e. ).

Determination of Sample Size
We define as a (fixed and known) effect size value greater than c.Under 0 1 , we have from (5) .The type II error risk : m H p     is defined using the following equation, where is the same value as before: It is often required that the power   1 of the test be at least equal to some fixed value , , n n (e.g.0.8).Now, using ( 6) and (7), it is possible to obtain a formula (possibly not in closed form) relating Two other approaches may be used to approximate the law L.We could resort to bootstrap methods (see [12]) to approximate it by say.Also, under some conditions we can approximate a binomial distribution with a Gaussian distribution.In this case, we could approximate L with the product of the preceding Gaussian distributions whose density is to be determined (knowing the joint density of the two Gaussians).
Here also, we will reject 0 H (i.e.we will show 1 H : our tablets are correctly manufactured) with a controlled probability of taking a wrong decision as soon as the observed value

A Real Data Application
We obtained samples of dry-coated tablets.The dimensions of these tablets are given in Table 1.We were able with a p-value equal to 0.03647.Thus, it is possible to conclude at level 5% that the tablets are correctly manufactured.Note that choosing the values 0   and , the sample size needed can be computed as being 2 × 29 = 58 tablets.0.9  

Conclusion
A statistical hypothesis test was developed to evaluate the manufacturing quality of dry-coated tablets, in terms of off-centering of their core.We also presented a formula to compute the sample size needed to get a fixed power.This research could be refined by taking into account the possible tilt movements of the core that have been neglected in this work.Also, sequential analysis and multiple testing problems could be investigated in this context.

Figure 1 .Figure 2 .
Figure 1.Diameter and thickness for the tablet and core.

Figure 3 .
Figure 3. Parameters of interest to determine core off-center.For symmetry reasons, the left and upper parts of the transverse cut may be omitted.

4 . 6 .
necessary sample size to use is thus obtained by solving in the two unknown values  Other Approaches to Approximate the Distribution L our observations) will fall too far in the upper low prob-