## Mathematical Models

Roark (1972) developed the following model to describe the microbiological contamination of spacecraft. Its principles are applicable to any form of contamination, including the manufacture of sterile pharmaceutical products. Number of contaminants after time t = A . X(t) . ^(t) . q-, where

A = the surface area exposed to contamination. The larger the area the greater the probability of microbiological contamination X(t)= the deposition rate of microbial contaminants on the item of surface A. The symbol (t) describes the elapsed time in which the item is exposed to the contamination potential. V(t) = the removal rate of microbial contaminants through physical means or death. The symbol (t) describes elapsed time. The proportion of survivors within a microbial population diminishes as a function of time. q- = the number and manner in which microbial contaminants may be present in the contaminating environment whether as individual bacteria (- = 1) or as groups or clumps borne on nonviable physical particles. The symbol -represents the number of microbial contaminants (0, 1, 2, n) that may be present in any one clump. There is extensive microbiological evidence to indicate that in nature many viable airborne microorganisms are attached to larger, nonviable particles such as skin flakes, dust, lint. The size and composition of these larger particles influences the ease or difficulty with which the particles will settle out of air and also the ease or difficulty of their physical removal from contaminated items (e.g., due to their very small size, discrete microorganisms are very difficult to remove, but conversely, they are not protected from desiccation by any extrinsic factors).

The mathematical complexity of this model is unimportant. It is consistent with observations, and identifies the factors that must be resolved in order to describe the contamination processes. None of the functions are resolved in this model to the point where it could be applied to pharmaceutical manufacturing.

Bradley et al. (1991) studied contamination in a containment room, where they established uniform, stable concentrations of 104, 106 and 107 discrete airborne spores per cubic millimetre (mm3) of Bacillus subtilis var niger. The test system was a blow-fill-seal machine filling Tryptone Soy Broth (TSB) at a fixed rate into plastic ampoules.

They demonstrated a regular relationship between the logarithm of the fraction of product units contaminated and of the spore challenge concentration in air. The team claimed that, by extrapolation of this relationship, they could substantiate a sterility assurance level (SAL) for practical operating conditions in actual airborne contamination. They stressed that the observed relationship was specific to the experimental conditions.

The regularity of the form of the relationship described by Bradley et al. (1991) is observationally important as an unconfused, assumption-free description of how airborne contamination relates to product contamination. It is, however, only a measure of the contamination frequency.

In terms of Roark's (1972) model it only describes the deposition rate X(t) and does not extrapolate to SAL. It does not take contamination by contact into account (perhaps understandably, in that the blow-fill-seal process affords little opportunity for contamination by contact than other aseptic filling processes).

By using a TSB recovery system, the findings disregard the product-specific die-off of microorganisms in filled units.

Within its description of deposition rate, the result only accounts for contamination by discrete microorganisms. Natural patterns of contamination from airborne sources would be much more complex, and should consider the size and nature of nonviable clumps.

The general form of the log-log relationship in these studies could form the basis of a more complex model for what Bernuzzi et al. (1997) described as the "background" contamination from airborne sources, which operates throughout an aseptic filling process.