P. LAFAYE DE MICHEAUX, V. LEMAIRE

270

Figure 2. Schematization of longitudinal and transverse cut.

Figure 3. Parameters of interest to determine core off-cen-

ter. For symmetry reasons, the left and upper parts of the

transverse cut may be omitted.

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 tab-

let and

is its orthogonal projection on the rota-

tional axis of symmetry of the tablet;

h is the distance (measured on the transverse cut)

from

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;

The distances

:2haa he and

:2d r that will be used in Section

4 to build the statistical test of core centration.

bbrD

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 realis-

tic assumption as can be seen on Figure 5.

3. 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

Figure 4. Picture of longitu d inally cut tablets.

Figure 5. Picture of transv ers ely cut tablets.

the statistical analysis.

4. A Statistical Hypothesis Test

4.1. 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

Μ

m

m

0

m be the unknown probability

that is above the threshold.

0

Μpm

Μm0

The objective is to significantly prove, namely with a

small and controlled error risk (less than some fixed thresh-

old α), that (in the population) pm0 > c = 1 – ε for a fixed

and small given tolerance value

following system:

0

0

.

k

k

;

;

kEk

kEk

ˆ

L

*

L

4.6. Other Approaches to Approximate the

Distribution L

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 Gaus-

sian distribution. In this case, we could approximate L

with the product of the preceding Gaussian distribu-

tions whose density is to be determined (knowing the joint

density of the two Gaussians).

Here also, we will reject 0

(i.e. we will show 1

:

our tablets are correctly manufactured) with a controlled

probability of taking a wrong decision as soon as the

observed value 0

12, of 0

12 (computed with

our observations) will fall too far in the upper low prob-

ability regions of or .

ˆm obs

p

*

L

nn ˆm

nnp

ˆ

L

5. A Real Data Application

We obtained samples of dry-coated tablets. The dimen-

sions of these tablets are given in Table 1 . We were able

Table 1. Dimensions of the tablets.

Type 1 Type 2 Type 3

D 9.53 mm 10.32 mm 12.7 mm

E 5.1 mm 6.53 mm 6.05 mm

d 6.35 mm 7.14 mm 7.94 mm

e 2.8 mm 4.56 mm 5.06 mm

to obtain the following i and measurements from

58 tablets of Type 1, from which 1 were cut lon-

gitudinally and

hi

r

28n

30

n2 were cut transversely:

hi: 2.42413 2.43201 2.50639 2.38244 2.28380 2.40008

2.49024 2.36265 2.35915 2.45703 2.48598 2.42386

2.43107 2.44298 2.41016 2.41928 2.43686 2.29511

2.41397 2.52839 2.37148 2.46547 2.52098 2.46419

2.28514 2.27086 2.52790 2.40166 2.28546 2.39130;

ri: 0.32041 0.31734 0.22026 0.37408 0.21759 0.52229

0.35651 0.23147 0.33771 0.33981 0.28513 0.25708

0.23791 0.09140 0.26171 0.30145 0.34680 0.25996

0.31784 0.47770 0.06728 0.08803 0.14386 0.26170

0.14298 0.13447 0.19959 0.16734.

We used the R software, version 2.11.0 (2010-04-22)

[13] and our code is available from the first author. We

performed the test using the (reasonable) values α = 0.05,

0.73c

, h0.94p

00.89m

ˆ728

nn p

0.2

and . We obtained the

following results: 0

12 ,m obs 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 tab-

lets.

0.9

6. 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 for-

mula to compute the sample size needed to get a fixed

power. This research could be refined by taking into ac-

count 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.

7. Acknowledgements

The authors would like to thank the company which pro-

vided the samples of tablets.

REFERENCES

[1] P. A. Kulkarni, M. Jaiswal, S. Jawale, S. V. Shirolkar and

P. V. Kasture, “Development and Evaluation of Press

Coated Tablets for Chronopharmaceutical Drug Delivery

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