The standard of visual cleanliness is commonly applied to the evaluation of surface contamination. Numerous published studies have examined the visually clean standard as a means of verifying cleaning effectiveness in pharmaceutical manufacturing, and methods for the quantitation of visible residue limits (VRLs) have been provided. Current methods for establishing VRLs are not statistically justifiable, however. The author proposes a method for estimating VRLs based on logistic regression.
Visually clean (VC), a term that refers to inspection with the naked eye, is a common cleanliness standard employed for evaluating surface contamination and cleaning in high-technology manufacturing, including that of pharmaceuticals, where surface cleaning is of utmost importance. The importance of the VC standard for pharmaceutical manufacturing is evident in the following facts:
In the pharmaceutical industry, cleaning is defined as limiting contamination to a level below practical, achievable, justifiable, and verifiable limits. CV is the documented evidence of cleanliness.
Common bases for establishing CV acceptance limits, as described in literature and in regulatory guidance documents, include the following (6, 7):
Because the VC standard is relevant to many technological areas, tremendous efforts have been devoted to defining and devising novel and efficient ways to develop justifiable and quantifiable VRLs for monitoring and validating cleaning procedures. Table II lists some definitions of the VC standard from the literature. The VC standard and VRLs are based on the following common principles:
Although the VC standard may be highly subjective, personnel have successfully quantified VRLs by establishing well-controlled experiments and programs. For industries other than pharmaceuticals, variables and parameters associated with the VC standard (e.g., viewing distance and light intensity) have been quantified and well documented (8).
The most popular method, henceforth referred to as the current method, for determining VRLs in the pharmaceutical industry involves spiking the selected material surface with known amounts of residue at concentrations of about 0–10 μg/cm2. Trained inspectors then examine the surfaces under controlled viewing conditions (e.g., light, viewing angle, and viewing distance) for the presence of residue (9–11). The lowest level of residue that is detected is then considered the VRL for that particular residue. The only drawback with the method is that it is not statistically justifiable and, hence, not scientifically definable. The primary objective of this article is to establish a method for setting scientifically and statistically justifiable VRLs and to provide a meaningful definition of the VC standard.
Statistical limitations of the current method
One statistical limitation of the current method is that the VRL is determined based on observed data without describing a relationship between observations and the experimental parameters. Suppose that in a VC verification study, a residue is spiked at levels of 0, 0.5, 1.0, 2.0, 3.0, and 4.0 μg/cm2. If all four inspectors detect residue at 2.0 μg/cm2 and only three inspectors detect residue at 1.0 μg/cm2, then the VRL would be 2.0 μg/cm2, assuming that the residue levels between 1.0 and 2.0 μg/cm2 would not be detected by all inspectors. The VRL is inappropriate because a residue level between 1.0 and 2.0 μg/cm2 could possibly have been detected by all inspectors.
To predict the number of observers that would detect residue at levels other than those spiked (e.g., 1.5 μg/cm2), the observed data must be incorporated into a reasonable model that describes a relationship between an outcome and a set of independent variables. The results obtained from spiking studies for verifying the VC criterion are binary (i.e., only two values are possible) rather than continuous. The regulatory guidelines and the available literature do not explain how to establish a modeling procedure, based on these discrete responses, that could be used to derive VRLs, however.
The objective of VC verification studies is to prove that the VC criterion would ensure cleanliness if implemented in the manufacturing setting. During VC verification, spiking studies are performed and inspectors state whether they can detect residue visually under controlled viewing conditions. VC verification studies thus provide a basis for the establishment of VRLs and the determination of appropriate viewing conditions. The lowest concentration of residue that is visually detected by all the observers is then used as VRL. In the current method, VRL could be defined mathematically as the lowest residue concentration for which the ratio of the number of observers able to detect the residue to the total number of observers is equal to 1. As discussed earlier, any knowledge about the outcome in future situations could not be obtained from the observed data unless the data were fitted with the most conservative model that explains the data.
One of the most common examples of modeling is the linear-regression technique. However, linear regression is not suitable for binary data. If we represent the binary responses "Yes" and "No" with values of 1 and 0, respectively, then the mean is the proportion of cases with a value of 1 and can be interpreted as proportion or probability of detection. Although the proportions and probabilities cannot exceed 1 or fall below 0, fitting the data with linear regression could give predicted values of the response variable above 1 and below 0. Clearly, linear regression is not appropriate when the data must lie between 0 and 1 because predictions from the model are not similarly constrained. Other problems that arise when fitting binary data with linear regression are that the variance of the error term is not constant and that the error term is not normally distributed.
Although the relationship between observed probability of detection and the residue concentration is nonlinear, a generalized linear modeling technique can be applied to these data. The logistic-regression technique fits the observed data with a linear model, the parameters for which are estimated using the maximum likelihood technique. Next, logistic regression transforms this linear model into a nonlinear logistic curve also known as an S-shaped or sigmoid curve. Logistic regression can therefore be seen as the conversion of a linear model into a nonlinear model that is naturally suited to the description of a binary response variable (13). The link function, commonly known as logit (the logarithm of odds), is used for converting the linear model to nonlinear logistic model and vice versa.
in which P(Y =1) represents the predicted probability of response being equal to 1 (i.e., the predicted probability of detection); e is the exponent function; β0, β1, β2, ... βk are coefficients estimated from the data (obtained using the method of maximum likelihood); x1, x2, ... xk are independent variables; and k is the number of independent variables.
For the data presented in Table III, which involves only one independent variable (i.e., residue concentration), logit = β0 + β1x1. Once a meaningful relationship is defined between spiked residue concentration and probability of detection, VRL could easily be obtained from the regression model.
Results and discussion
These point estimates provided a framework for evaluating the reliability of the logistic model. The logistic models and associated point estimates could be considered reliable if the observed probability of detection was found to be consistent with the predicted probability of detection. If one assumes 0.999 to be approximately equal to 1, then the VRL for the given residue should be 2.921 μg/cm2, which is larger than the one obtained with the current method (see Table VI). Thus, based on the logistic-regression model, the residue concentration at 2.921 μg/cm2 is predicted to be detected by all the observers with a 95% confidence interval of 2.266–4.761. Because the probability of detection increases significantly with an increase in spiked-residue concentration, setting higher acceptance criteria would give larger VRLs. Table VI shows that as the acceptance criterion approaches 1, the relationship requires a larger change in the explanatory variable to have the same effect as a smaller change in the explanatory variable at the middle of the curve. For example, a change in the predicted probability of detection from 0.9 to 0.99 requires a larger change in residue concentration (i.e., a change of 0.674 μg/cm2) than does a change in the probability from 0.5 to 0.6 (i.e., a change of 0.114 μg/cm2). Similarly, the confidence interval for these VRLs would tend to be wider as the acceptance criterion increases. Logistic regression may provide a much larger VRL than the current method. However, manufacturers may achieve lower VRLs by adjusting the acceptance criterion.
Logistic regression, as previously described, can be generalized to incorporate more than one explanatory variable, which may be continuous or categorical. However, care should be taken when interpreting and reporting results from multiple logistic-regression models. To correctly interpret the results from a multiple logistic-regression analysis and arrive at meaningful conclusions, appropriate steps must be taken to incorporate statistical interaction or curvilinear effects properly (e.g., including additional x1 × x2 or polynomial terms such as x12 in the systemic component of the model) (13). If the logistic coefficient for the product or polynomial term is not statistically significant, then the interaction or curvilinear effect is not statistically significant. One problem that may arise while modeling multiple explanatory variables is that sometimes the value of one or more independent variables may raise the probability of the dependent variable close to 1, therefore the effects of other variables cannot have much influence. In that case, such variables should be excluded from the model or individual VRLs should be determined for the most appropriate viewing conditio
Logistic regression was demonstrated to be a better approach than the current method for estimating accurate and statistically justifiable VRLs based on discrete responses. It has the advantage of always making biologically meaningful predictions and, in most cases, its predictions closely reflect observations. Logistic regression should be used to determine VRLs rather than current method. Because the model may provide a much larger VRL than the current method does, the quality of visual inspection, in terms of the discrete response, can be improved by properly controlling the experimental variables and defining the acceptance criterion for the estimation of VRL.
Based on the modeling procedure, VRL can be defined as a scientifically justifiable residue concentration that, when viewed with the unaided eye, as measured by a specific method, would be detected by the observers with a predefined acceptance criterion. Once established, the VRL could then be used for CV and routine monitoring purposes. It would be appropriate to define the VC criterion as the absence of all particulate and nonparticulate contaminants above the established VRL from the surface when viewed with the unaided eye under preverified viewing conditions.
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