In the thesis two different approaches to assess homogeneity and factors that may influence homogeneity have been introduced.
The first approach described in Appendix E leads to a model of the best ob-tainable blend and content uniformity derived from the distribution of particle radii. However, the best obtainable homogeneity is a ’theoretical’ limit that holds for all batches with the same distribution of particle radii, and therefore this approach does not lead to information on the actual homogeneity of a given batch. The second approach described in Appendix A introduces two methods to assess the homogeneity of a specific batch.
3.2.1 The effect of particle size distribution
Particle size distributions are often seen to be skewed and it has been shown in Appendix E that this feature affects the distribution of content in blend and tablet samples.
For a log-normal distribution of particle diameters, the resulting distribution of particle mass (volume) is also a log-normal distribution. It is found that skewness and excess (heavy-tailedness) of the distribution of particle radii is amplified when transformed to particle mass. The larger the coefficient of variation in the distribution of particle radii the more pronounced the amplifi-cation. The relation between the coefficient of variation for particle mass and the coefficient of variation for particle radii is given in a table.
Beside the variation in particle mass the variation in dose content is affected by variation in the number of particle in a sample. For a homogeneous blend with a random scattering of particles over the blend it is demonstrated that for a given distribution of particle sizes the variation in the distribution of absolute doses is proportional to the average number of particles in the samples (tablets).
Further, it is shown that the larger the average number of particles in the sample the closer the distribution of content in the samples is to a normal-distribution.
3.2. METHODS TO ASSESS HOMOGENEITY AND FACTORS THAT MAY
INFLUENCE HOMOGENEITY 23
For spherical particles an explicit relation between the variation in the relative doses and the mean and the coefficient of variation in the distribution of particle radii is given.
3.2.2 Assessment of homogeneity in specific batches Two methods are introduced in Appendix A to assess homogeneity and factors that may influence homogeneity in a specific blend. The two methods are based on respectively Generalized Linear Models (GENMOD) and General Linear Models (GLM). General Linear Models can be used to assess differences in mean content in respectively layers and areas within a layer. Generalized Lin-ear Models are here used to assess differences in variance. More specific to assess whether the size of the small scale variation is constant throughout the batch.
In practical applications a Generalized Linear Model should be applied first to assess if the small scale variation/variation between replicates is constant throughout the blend. If this is not so, it should be accounted for in the General Linear Model.
GENMOD
In Appendix A a Generalized Linear Model is used to assess the influence of layers on the small scale variation. For samples simulated from a hierarchi-cal model with three layers, four areas within each layer, and three replicates within each area the following was found: For the 5% level test the standard deviation between replicates within an area has to be 4.5 times larger in one layer than the standard deviation corresponding to replicates within an area in another layer for the effect of layers to be declared significant with a probabil-ity of at least 0.95.
However, depending on the experimental design and the assumptions made, the method presented in this section can also be used to assess factors influencing sample error (variation). For example the method can be used to test if one sampling thief leads to larger variation between replicate samples than another thief.
24 CHAPTER 3. RESULTS AND DISCUSSION
Using the thief leading to the smallest variation between replicates will reduce the risk of incorrectly rejecting a batch because of suspicion of inhomogeneity.
Other external sources of variation (as e.g. the sampling procedure) may have a similar effect on the small scale variation. Examples of such analysis are given in Appendix F.
GLM
When a Generalized Linear Model has been applied a General Linear Model can be applied specially to assess the medium and the large scale variation as well as factors that may influence these types of variation. In case the variation between replicates is found not to be constant throughout the blend, this should be corrected for in the analysis by introducing appropriate weights
Under the assumption that the variation between replicate samples is indepen-dent of the layer and that there is no interaction between the factors in the model, two statistical methods to describe blend homogeneity have been in-vestigated.
The first statistical method (using the aggregated model) corresponds to an
’aggregated’ definition of homogeneity in the sense that large and medium scale variation in the batch is assessed as a whole.
The other statistical method (using the hierarchical model) corresponds to a homogeneity definition with two different criteria; one explicitly regarding the large scale variation and the other explicitly regarding the medium scale varia-tion.
The analysis showed that the two methods are approximately equally good at detecting inhomogeneity. That is, an analysis according to the aggregated model can be used to detect inhomogeneity even in situations with large scale variation (variation between layers).
The most important difference between the two types of analysis is that when inhomogeneity is declared according to the aggregated analysis the result does not reveal whether this inhomogeneity is due to large or medium scale variation in the batch. However, the hierarchical model explicitly assesses respectively large and medium scale variation.
3.2. METHODS TO ASSESS HOMOGENEITY AND FACTORS THAT MAY
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The power of the respective tests as a function of the standard deviation corre-sponding to respectively variation between layers and variation between areas within a layer are shown in Figure 3 to Figure 5 in Appendix A. The standard deviation corresponding to respectively variation between layers and variation between areas within a layer are measured relatively to the standard deviation corresponding to replicates.
Finally, for the given design (and forσrep = 1) the power of the test of a thief effect was shown in Figure 6 in Appendix A. The test of the thief effect is independent ofσlayerandσarea,hi.
With the given design (and forσrep = 1) a difference greater than 1.5 in mean content in samples from two different thieves will be detected with a probabil-ity of at least 0.95.
3.2.3 Example
As an example of how the robustness and power of a test can be used to evaluate the test result the test of the small scale variation in Appendix F is discussed in relation to the results from Appendix A.
As the resulting design in Appendix F is neither balanced nor identical to the design from which the robustness and power are assessed, the example should be seen as a guidance on the type of considerations to make when evaluating test results.
Variation between replicate samples
In Appendix F the variation between replicates, σrep2 , tends to be larger the lower the layer the samples are sampled from. However, the tendency is not significant.
The estimated variance components,σ2rep, are given in the Table 3.1. The esti-mates in the table are multiplied with 1000 compared to the results in Case F.
The reason is that in Appendix A the unit is%LC rather than fraction LC.
26 CHAPTER 3. RESULTS AND DISCUSSION
Batch 1 Batch 1
Thief 1 2 3 1 2 3
σ2rep,top 0 0 1 0 3 8
σ2rep,bottom 1 7 17 4 35 98
Table 3.1: Estimates of variance components from Appendix F.
It is seen from the three cases in the table whereσrep,top2 6= 0that the estimate ofσrep,bottom2 is between 10 and 20 times as large as σrep,top2 . At first a dif-ference that large may be expected to be found significant. However, for the design in Appendix A it was showed thatσrepin one layer should be more than 4.5 times as large asσrepin an other layer for the test to show significance with a high probability. This corresponds to one variance estimate being4.52 ≈20 times larger than the smallest. This result is valid for the balanced design with 12 pairs of replicates in each batch and analysed with a model with one factor (layer).
The design in Case F is not balanced, it has only 6 pairs of replicates in each batch and two factors included in the model (layer and thief). Hence, with an estimate of σ2rep,bottom being 10 to 20 times as large as σrep,top2 it is not surprising that the test shows no significance - however, the estimated differ-ence betweenσ2rep,topand σrep,bottom2 may still be real and give practical and valuable insight to the sampling process.