A pharmaceutical company operating in the U.S. had encountered frequent release failures of its top selling product. In addition, it had set up a fairly common stability limit (1), which triggered investigations when a stability point differed by more than 5% from its previous measurement. As a result:
- This organization lost several million dollars in the profit that this product was supposed to generate;
- the company experienced several supply chain interruptions, which complicated its relationships with some of its main customers; and
- there were frequent inconclusive stability failure investigations, all of which consumed funds, time and organizational resources.
Management assembled a cross-disciplinary team with the relevant functional departments. I was hired to manage this team and provide quality engineering skills to the organization. Quality engineering is the science of understanding and controlling variation.
Understanding the fundamentals of variation (2, 3)
In a given specification, the total observed variability is the sum of the variability of the assay plus the variability of the process. Mathematically, this can be expressed as:
σ2observed = σ2process + σ2assay
where σ2observed is the observed variance, σ2process is the variance of the process, and σ2assay is the variance of the assay.
The variance of the assay is given by:
σ2assay = σ2repeatability + σ2reproducibility
where σ2repeatability is the precision, or the variance obtained when the same assay is performed by the same analyst, in the same equipment, during consecutive measurements, during the same day. σ2repeatability is the variation between different analysts performing the same assay, in the same equipment, at different days. Both of these parameters are normally determined as part of the assay validation.
It is good practice to set the release specification range at a minimum of 8 σs of the observed variance. The measurement error is six times σ2assay and it should be less than 10% of the specification range. In addition, it is recommended that the ratio of the assay to the observed variance be less than 10%.
σ2assay can be reduced by increasing the number of measurements by using the central limit theorem of statistics, which states that in any population, the variance of means of samples from that population will be the total population variance divided by the
where n is the number of measurements.
Note that, when the sample size increases, an assay will produce means closer to the true mean, and the spread of the assay results will be narrower.
The capability of a process to meet its specifications can be measured with a term called process capability, which in mathematical terms is expressed as:
where USL and LSL are the upper and lower spec limits, respectively.
σ is the observed standard deviation (from the process and from the assay).
The higher the Cpk values the better. A Cpk of 1 means that the process is barely meeting its specifications and there is no room for variability beyond random variation. A Cpk below 1 means the process is not meeting its specifications, and that out of spec results will be expected. A Cpk above 1 indicates that the process has room for some variability beyond random variation.
Solving the problem of release failures
Release data from approximately 30 lots was collected to calculate the process average and the observed variability. This allowed the calculations of the Cpks with the old specifications. This product has several actives, hence one Cpk per active was calculated. Not surprisingly, several of the Cpks were below 1, which indicated that the specifications were tighter than the normal and random observed variation from both the assays and the process. The solution was then to increase specification ranges of the actives with the lowest Cpks.
The definition of process capability given before suggests two ways to increase the Cpk. One could increase the range of the specifications (LSL and USL), or one could reduce the observed variance. The first option is preferred because it is cheaper and faster. Sometimes there are market- or customer-related limitations on how wide the specification ranges need to be. For example, there are some toxic drugs for which the nurses must know the drug concentration within very narrow ranges to be able to give exact doses to the patients. Or, medical reasons could dictate that drug doses do not exceed certain limits. That was not the case with this product. Therefore, with the approval of the organization's Medical Affairs department, the specification ranges were increased to +/- four standard deviations from the process averages. By definition, this increased the Cpks to 1.33.
The option to increase Cpk through reduction in the observed variability is far more complicated, lengthy and expensive. Assay variability can be calculated from the assay validation reports. Observed variance can be calculated from the release data. Then, from the first formula provided in this article, the process variance can be deducted as the difference between the observed variance and the assay variance. Depending on which is one is relatively bigger, efforts could then be oriented toward reducing either the assay or the process variances.