This chapter considers the changes in the number of survivors (individual microorganisms capable of reproduction) with time. It is often applied to industrial disinfection, pasteurization, and sterilization (terminal or commercial sterilization) processes aimed at assuring safety, stability, and quality of pharmaceutical parenteral drugs and devices and of shelf-stable food products. These concepts can be studied using a variety of individual microorganism types or even mixed populations. But the emphasis of this chapter is focused on the inactivation of bacterial spores because they are highly resistant to antimicrobial agents and also have stability properties that support their use as biological indicators to verify the efficacy of sterilization processes (see chapter 65). Development, improvement, and troubleshooting of sterilization processes benefit from a good understanding of related inactivation kinetics.^{1},^{2} When a validated mathematical model is available, the related tasks may be supported more efficiently than using a purely empirical trial-and-error approach because the time, money, and other resources are significantly reduced. Indeed, in the food industry, a simulation package is well accepted by the regulatory and process authorities for simulation of moist heat-based (F_{0}; see equation 10 and chapters 11 and 28) processes.
Bacterial spore structure, formation, and significant transformations are well understood^{3} (see chapter 3). The significant transformations concerning sterilization are dormancy, activation, inactivation, clumping,^{4} and injury. Figure 7.1 presents a system diagram that includes these transformations and the corresponding subpopulations. “D” is applied to the inactivation transformation of the different subpopulations, and the inverted triangles represent the inactivated individuals corresponding to the different subpopulations. System analysis from population dynamics is used here to work on the corresponding models at the conceptual and mathematical levels. Dormancy is the transformation of the bacteria into its corresponding spore that renders it resistant to antimicrobial processes (see chapter 3), including moist heat, and microscopically birefringent. Activation is the transformation of dormant bacterial spores that enables them to potentially germinate into an actively dividing cell that can produce colony-forming units (CFUs) on culturing. Activated spores lose their high heat resistance and birefringence (looking opaque in the phase contrast microscope, discussed in the following text). Inactivation is a transformation of microorganisms that renders them unable to multiply and therefore produce CFUs. Clumping of bacterial spores (and other microorganisms) is where they form groups or clumps, but due to this affinity may only produce one CFU per clump of cells on cultivation. Finally, injury renders them initially incapable of producing a CFU, unless modified growth conditions (such as additional nutrients, lower incubation temperature, enzymatic treatments, extended incubation time, etc) are applied or occur. Other transformations have been conceptually described^{5} but have not yet been described mathematically well enough for practical applications.
Survival curves, which are generally semilog plots of the ratio of the concentration of surviving organisms to the initial number versus time, are commonly used to present inactivation data and to interpret the inactivation kinetics of microorganisms. Typical curves may be linear but are frequently observed to be nonlinear. Figure 7.2 presents some microbial inactivation curve types that may be seen in practice due to the presence of the different subpopulations and transformations described earlier. For instance, activated spores or vegetative cells will lead to an initial sharp drop in survivors because activated spores have a significantly lower resistance to moist heat that is similar to the resistance of vegetative cells. In addition, the activation transformation leads to a negative exponential curve and will lead to a hump in the inactivation curve that may be significant at relatively lower temperatures and other conditions. These survivorship/inactivation curves may be described by the validated models available. Despite this, it is important to design products and processes that minimize the complexity of the description of the kinetics of inactivation without introducing significant error in practical terms. For instance, a more resistant subpopulation that may occur at occluded interfaces between glass and rubber stoppers may be eliminated by supersaturating the stoppers with water before sterilization, for example (see chapter 28). Figure 7.3 gives examples of how dormant bacterial spores change when viewed under phase-contrast microscopy. Dormant spores show birefringence and are heat resistant, whereas bacterial spores that have broken dormancy look opaque and have lost their high heat resistance. The bacterial spore resistance properties were recently reviewed.^{6}
FIGURE 7.1 System diagram for bacterial spores undergoing sterilization. This system includes inactivation, activation, injury, and the presence of a less resistant subpopulation. The gamma (γ) symbols represent the rate for these subpopulations (undergoing a transformation ie, injury, activation, death, etc.); mathematically, they are the time derivatives of the density (individuals per unit volume or unit mass; colony-forming units per gram, etc.) of the corresponding subpopulation undergoing a transformation.
Conceptually, the inactivation of microorganisms under the effect of a lethal agent follows well-defined patterns, with the rate of inactivation diminishing with time in a mostly ordered fashion. The exponential nature of inactivation prevents the goal of zero survivorship from being realistic (infinite time would be required). Thus, low levels of survivorship have been defined and validated practically as the goals of related industrial sterilization or disinfection (including pasteurization) processes, such as terminal sterilization for the medical device and pharmaceutical industry and commercial sterilization for the low acid-canned foods in the food industry. This chapter reviews procedures that can be used to assist in the efficient development and validation of such processes. The complexity and diversity of industrial sterilization processes precludes the selection of a single approach to describe mathematically the inactivation of microorganisms. In addition, different lethal agents may attack different components of the targeted microorganisms (see chapter 5). For instance, DNA, enzymes, or other subsystems required for germination and growth may be targeted and can affect the results. Finally, study of the kinetics of microbial inactivation caused by some lethal agents such as radiation may benefit from a non-homogeneous kinetic approach (see chapter 29). As an illustration, Figure 7.4 presents the damage caused by an electric field in a piece of wood. Clearly, if electroporation was intended as the lethal process, some regions would receive high-intensity treatments, whereas most of the material would receive negligible treatments. Therefore, the model used to describe the corresponding kinetics of inactivation should be the simplest model that describes the inactivation process properly. Models with corresponding degrees of complexity are further discussed in this chapter. Moist heat (including ultrahigh pressure), dry heat, chemical, and radiation sterilization processes are reviewed in detail, due to their practical importance. Application of mathematical models to such practical problems is currently enabled using mathematical software in various computer systems. Use of mathematical programs eliminates the need to use traditional procedures (eg, semilog paper). Examples presented here as illustrations correspond to worksheets developed using MathCAD versions 14 or 15 (MathSoft Engineering and Education Inc, Cambridge, MA). The worksheets presented have been carefully developed, but no warranty is given regarding their application. Computerized systems must be validated properly in each instance per the corresponding regulatory requirements or guidelines.
FIGURE 7.2 Examples of survivorship curves of bacterial spores undergoing sterilization that are commonly found in industrial practice.
FIGURE 7.3 Bacterial spores seen through the phase-contrast microscope showing the birefringence of dormant spores (left) that coincides with high spore heat resistance, and opaque spores (right) that have germinated and lost their resistance to heat.
FIGURE 7.4 Fractal nature of high-voltage electric field into wood as an illustration of lethal agents where nonhomogeneous kinetics may be useful (ie, with radiation).
MATHEMATICAL MODELING
In 1910, Chick^{7} showed that microbial inactivation by moist heat resembled a first-order chemical reaction and that the temperature dependency of the corresponding rate constant was well described by Arrhenius law (note that “Acronyms and Abbreviations” section is provided at the end of this chapter for reference)
where, N is the concentration of microorganisms at time t, and k is the pseudo-first order inactivation rate constant, and t is time. The rate of inactivation (minus sign) was found to be proportional to the number of survivors (N). For isothermal lethal conditions, the solution is
The last expression has the form of a straight line when the dependent variable is the logarithm of the number of viable spores. Using decimal logarithms equation (equation 2) is
The decimal reduction time (D) is the time required by the inactivation curve to go through an order of magnitude (for instance, from 1 000 000 to 100 000 CFUs/mL or a 1 log_{10} reduction of the population). Therefore, for isothermal conditions (square-wave heating), the time required to reach a desired reduction in the population of microorganisms can be estimated by multiplying the number of orders of magnitude by the decimal reduction time. The effect of temperature on the decimal reduction time is normally described using the parameter z, or the number of degrees of temperature required to change the decimal reduction time by a factor of 10. The Arrhenius law has been shown to be a better descriptor, but the temperature range of moist heat sterilization processes is small enough that both equations are expected to work reasonably well.^{8} In addition, the decimal reduction time is often used to define the effect of the menstruum composition on the resistance of the test organisms. Figure 7.5 presents the application of the decimal reduction time obtained from corresponding thermal destruction time studies to define which of two solutions should be used as a master solution for routine qualification tests when z-values are significantly different.
In 1908, Chick^{9} also explored chemical disinfection processes and found that the kinetics of inactivation could be described as a first-order chemical reaction and that the rate of inactivation was proportional to the number of survivors times the concentration to a power n. Thus, when the concentration of the lethal agent is to be considered, the model is
FIGURE 7.5 Use of the Bigelow (D_{250}, z) model to select a solution that imparts a higher resistance to bacterial spores within a temperature range for intended sterilization processes. This worksheet example performs the calculations to decide which solution will impart a higher heat resistance to a biological challenge (BI) when z-values differ significantly in the solutions (based on inspection of the decimal reduction time, D, and temperature curves). The data for D at 250°F and z-values for each solution are entered and analyzed between 245°F and 260°F (the temperature range of practical interest).
In this case, for instance for gas sterilization using ethylene oxide (EO), the decimal reduction time will include the kinetic constant k and the concentration to the power n.
The power n is often near one, and the rate of inactivation will be twice as fast if the concentration is doubled. There are cases such as the disinfection using phenolic compounds when n is near three.^{10} This means that if the concentration is twice as large, the rate of inactivation will be 8 times faster. This theory may be limited within certain ranges of concentrations depending on the antimicrobial chemical under investigation and associated process conditions (eg, temperature, humidity, formulation, pH). Table 7.1 presents published values for exponent n corresponding to disinfection processes using chlorine, chlorine dioxide, or ozone as lethal agents.^{11}Table 7.2 presents results of the application of these kinetic models to spore inactivation by EO in transient processes showing the accuracy of this approach and its potential usefulness to simulate practical applications^{12} (see chapter 31 on EO sterilization).
TABLE 7.1 Coefficient of dilution (n) for free chlorine, chlorine dioxide, and ozone^{a}
In general, the decimal reduction time defines the rate of microbial inactivation.
There are a series of factors that have been found to have a significant effect on the value of the decimal reduction time such as temperature (T), water activity (Aw), pressure (P, at very high values), pH, radiation dose, etc. The decimal reduction time (D), or its natural logarithm [ln(D)] is therefore a function of a series of variables
Correspondingly, the spore log reduction (SLR) may be estimated for transient lethal treatments using the following formula:
Use of this formula is illustrated in detail in Figure 7.6, as an example of the effect of Aw in a process.
The differential of the decimal reduction time as function of the mentioned series of variables is
(Note: The notation is standard mathematical notation for partial derivatives of functions.)
Application of the Gauss law for the propagation of error enables us to define the corresponding variance
Therefore, the kinetic mathematical differential and its statistical variances are closely linked concepts.
Moist and Dry Heat
The ordered nature of the inactivation using thermal methods may be understood based on the idea that the corresponding molecular transformations require that the molecules surpass the energetic barrier represented by the activation energy. The Boltzmann distribution of the speed of the molecules for a certain molecular structure depends only on the absolute temperature. Figure 7.7 shows an example of the effect of absolute temperature on the shape of the corresponding frequency distribution curves for several absolute temperature values. Therefore, the fraction that will reach the energy needed to overcome the energetic barrier remains constant (for instance, 90%); thus, after D minutes, the inactivation transformation is expected to diminish the survivors by the same factor (for instance, 90%) and so on. In addition, the fraction that reaches the higher speed increases with temperature leading to a corresponding reduction in the value of the decimal reduction time. This interpretation has also been verified for condensed systems (eg, DNA suspensions).^{13},^{14},^{15}
FIGURE 7.6 Empirical description of the effect of water activity on the decimal reduction time. The function needed to calculate the spore log reduction corresponding to a sterilization process with variable Aw is included.
FIGURE 7.7 Distribution of the speed of molecules as a function of absolute temperature.
Estimation of the survivors to a moist or dry heat process may be performed using the following formula. The integration is to be performed numerically
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