Seguridad Estructural Pág.55

Mathematics is the art of giving the same name to different things.

Jules Henry Poincaré Destruction and inhibition of microorganisms by chemical agents is an option frequently exercised in a variety of fields. Most notable examples are chemical preservation of foods and beverages and the disinfection of drinking water by chlorination or administration of other antimicrobials.

In medicine, the use of antibacterial (mostly antibiotics) and antifungal drugs is quite common and the same can be said about drugs that destroy or inhibit protists and worms. Antimicrobial agents of similar activities are also widely used in agriculture. Decontamination of space and equip-ment and by a gaseous chemical agent, regardless of whether the contam-inant is a naturally occurring microorganism or a deliberately introduced bacterium or spore (bioterrorism), is another example of chemical treat-ment that can have public safety implications.

There are numerous kinds of chemical antimicrobials and they are employed against a large variety of targets. The discussion in this chapter will focus on “traditional” microorganisms, primarily bacteria. However, the principles might be just as applicable to unicellular eukaryotes like yeast, or amoeba, except that the time scale might be quite different.

The chemical structure of antimicrobials and their physical properties also vary enormously, as does the mode of their activity. The way in which antimicrobials are administered and their effective dose also vary dramat-ically, depending on their chemical species, stability, and, among other things, whether they are in a solid, liquid, or gaseous state. The role of factors such as solubility, diffusivity and chemical affinity in the design of an effective treatment is well recognized and their importance cannot be overemphasized. The same can be said about the biochemical and biophysical interactions that render these chemical agents lethal to the targeted organism. Nevertheless, the focus of the following discussion will be on the inactivation kinetics, as expressed at the population level.

Thus, for the type of modeling addressed in this chapter, the main dis-tinction between the agents is whether their effective concentration, on the pertinent time scale, remains practically unchanged or dissipates appreciably.

The first class includes agents such as benzoates, sorbates, and the like, but also less stable agents like sulfite when administered in a massive dose, in which case an effective lethal concentration is maintained throughout the treatment. The second kind includes chemically unstable or volatile agents whose dissipating rate is high enough so that their effectiveness significantly diminishes during the treatment. The lethality of such agents may disappear altogether and they might need to be replenished so that the treatment will remain effective. For what follows, it will be assumed that in both cases the lethal effect is exclusively pro-duced by the agent’s presence and thus that its concentration’s profile alone determines the treated microbial population survival pattern. Obvi-ously, changes in temperature or pH, for example, must also affect the survival curve, a fact that has been long recognized.

Some existing models are intended to account for the combined effect of chemicals and heat on the logarithmic inactivation rate of microorgan-isms. Most of them are based on the incorporation of a temperature and concentration terms into an Arrhenius-type equation. One problem with such models, as has been mentioned in previous chapters, is that the concept of a time-independent logarithmic inactivation rate has a meaning in first-order mortality kinetics only, a kinetics that need not exist in reality (see below). However, even if microbial inactivation by a chemical agent did follow the first-order kinetics model, which is highly doubtful, any assumption regarding the nature of the combined temperature–concen-tration effect on the logarithmic rate constant, like that implied by the Arrhenius equation, for example, would need to be confirmed experimen-tally. Again, the confirmation should come through testing the predictive ability of the proposed model and not by demonstrating its fit.

Thus, expressing the logarithmic rate constant as the algebraic sum or product of two or more independent and interactive terms representing the concentration, the temperature and/or pH, etc. is based on the implicit assumption that these factors function independently or in another uni-versally prescribed manner dictated by the chosen model. It is not self-evident that either assumption must always be correct and therefore mod-els constructed in this way ought to be experimentally validated before they can be accepted for use. More on this issue will be discussed in Chapter 13.

For simplicity, we will only address situations in which the chemical inactivation occurs under isothermal or practically isothermal conditions, i.e., in the absence of synergistic or antagonistic effects that might be produced by other factors or agents.

Chemical Inactivation under “Constant” Agent Concentration

An example of published isothermal survival curves of Listeria monocyto-genes exposed to potassium sorbate at various concentrations is shown in Figure 5.1. They clearly demonstrate that the inactivation pattern was not log linear, as would have been expected from the first-order mortality kinetics. Moreover, all the semilogarithmic survival curves of the organ-ism when exposed to the sorbate had a noticeable downward concavity.

Such a pattern cannot be explained by an unnoticed loss of the sorbate.

If this had happened, the Listeria’s semilogarithmic survival curves would have to exhibit upward concavity (see below). Also, because the reported isothermal semilogarithmic thermal inactivation curves of Listeria are at most linear but more frequently concave upward, they could not provide, at least in this particular case, any qualitative or quantitative indication as to how the organism might respond to a specific chemical agent. All

FIGURE 5.1

Survival data of Listeria monocytogenes in the presence of potassium sorbate at different concentrations fitted with the Weibullian–power law model, with variable n(C) (Equation 5.1) and with a constant n (Equation 5.2). (The experimental data are from El-Shenawy, M.A.

and Marth, E.H., 1988, J. Food Prot., 51, 842–846.)

0 10 20 30 40 50 60 70

0

−1

−2

−3

−4

Log S(t)

n variable n fixed

0.25%

0.15%

0.10%

0.05%

Listeria monocytogenes

Time(days)

this suggests that unless there is a compelling reason to assume otherwise, the inactivation pattern of an organism must be determined experimen-tally; it cannot be assumed a priori that it follows the first-order kinetics.

The patterns shown in the figure could be described by the Weibul-lian–power law model as their thermal counterparts, albeit with a very different shape factor or power, i.e., n > 1 instead of the n ≤ 1 found for the isothermal heat inactivation curves. Expressed mathematically, the Weibullian model has the form:

log10S(t) = –b(C)t^n(C) (5.1) where b(C) and n(C) are concentration-dependent coefficients. In the case of a constant power, n(C) = n:

log10S(t) = –b(C)tⁿ (5.2) The actual fit of the model with constant and variable n(C) is shown as dashed and solid curves in Figure 5.1. Either way, the two versions of the Weibull model, like their equivalents in thermal activation, are consistent with the notion that the logarithmic inactivation rate depends not only on the agent’s concentration, but also on the exposure’s time. In the partic-ular case of the Listeria shown in Figure 5.1, the downward concavity of the semilogarithmic survival curves is an indication that a prolonged exposure to the sorbate sensitizes the survivors, presumably by accumu-lated damage to the cells. Here, too, a log linear survival curve, which traditionally has been treated as evidence of first-order kinetics, would be just a special case of the Weibullian model with n = 1.

Once the Weibullian model’s parameters have been determined exper-imentally, they can be used to determine the survival curve at any given agent concentration by interpolation. The same can be said about any alternative model, as long as the chemical agent can be assumed to be practically stable. The term ‘stable’ here requires clarification. For chem-ical agents to be effective, they need to interact with the organism and this may take very different forms. The interaction can be primarily phys-ical. For example, sugar at a concentration of above about 65% generates enough osmotic pressure to desiccate the cells. This osmotic preservation should not concern us here because, in most cases (notably in jam or marmalade preparation), heat and low pH are also involved and/or in the case of honey, the sugar concentration is so high that it is a major ingredient rather than a preservative only. The same can be said about alcoholic beverages, in which the alcohol plays more than a preservative role.

The interaction can be chemical, though, with the agent interfering with a crucial metabolic function of the microbial cell, for example. In such a case, the chemical agent is actually consumed, but the assumption is that the amount so lost does not significantly affect its overall concentration.

(For the case in which the concentration does change appreciably, see

“Microbial Inactivation with a Dissipating Chemical Agent” below.) Also, chemical preservatives, like sorbic, propionic, and benzoic acid, are known to be effective antimicrobials only in their undissociated form.

However, because these acids have very low water solubility, they are usually administered in their potassium, calcium, and sodium salts, respectively. These salts, especially the potassium sorbate and sodium benzoate, are effective preservatives only when the pH is low, which allows for their partial conversion into the respective acids.

Thus, there is a difference between the nominal concentration of the added salt and the effective concentration of the acid that should be kept in mind. In cured meats, there is a difference between the nominal and effective concentration of the nitrite that, among the added mixture of salts (the other two are sodium chloride and nitrate), is the real preserva-tive. As already stated, any upward concavity of a semilogarithmic sur-vival curve observed under a constant chemical preservative concentration can be in fact or at least partly due to the diminishing concentration of its effective form. This cannot be the case when the observed semilogarithmic survival curve has a clear downward concavity.

If the concentration of the effective form of the preservative indeed decreases somewhat with time, the true curvature of the semilogarithmic survival curve would be distorted to some extent.

However, whenever this factor plays a decisive role in shaping the inactivation pattern, the curve’s concavity direction would need to be inverted as a result. This creates an asymmetry between the interpretation of upward and downward concavities. Although the latter is a direct manifestation of the inactivation’s mode, the former can be the result of the diminishing effective concentration of the agent as well as of the early destruction of the sensitive or weak cells. This will become more clearly evident when we deal with chemical agents known to be unstable or volatile.

Microbial Inactivation with a Dissipating Chemical Agent Unstable or volatile chemicals are extensively used in the disinfection of water, equipment, packages, and foods. The most notable example is the

traditional chlorination of drinking water by perchlorates. More modern disinfectants (apart from ultraviolet light) include ozone generated in situ, hydrogen peroxide, and peracetic acid. Like the compounds produced in chlorination, these agents are volatile, very active chemically, and thus unstable. Therefore, unless continuously generated or replenished, their concentration will decrease with time and may reach an ineffective level before the required degree of inactivation has been accomplished.

When administered orally or by injection, antibiotics and other antimi-crobial drugs bear no chemical similarity to the agents used in water disinfection, of course. However, their effective concentration in the patient’s (or animal’s) body also diminishes with time and they too need to be replenished periodically or automatically by slow release. Thus, at least in principle, the same kind of mathematical model developed for assessing the inactivation kinetic of dissipating disinfectants can be just as applicable to the efficacy of antibiotics and other drugs in vivo. Obvi-ously, the time scale of the inactivation process, the model’s mathematical structure, and the magnitude of its parameters would be quite different.

However, the general kinetic considerations leading to the inactivation model’s derivation might be very similar despite the dissimilarities in these agent’s modes of activity.

A problem that arises in any attempt to model the effect of unstable chemical antimicrobials is that it is very difficult, if not utterly impossible, to obtain reliable microbial survival data under sets of constant agent con-centrations. (The problem is much more serious than in modeling the kinetics of thermal inactivation. This is because at least under certain circumstances, it is possible to record survival curves under conditions that are sufficiently close to ideal isothermal heat treatment. These are extremely useful in the establishment of the inactivation that are kinetics, despite their imperfec-tions.) Because of the inherent volatility and stability problem, any mention of constant agent concentration or isoconcentration in what follows will refer to hypothetical and not actual experimental conditions.

Traditional Models

When a microbial population is exposed to a lethal chemical agent with diminishing intensity, its semilogarithmic survival curve almost invari-ably has an upward concavity, i.e., it would exhibit notable “tailing.” This has been consistently observed in water treated with a variety of chemical disinfectants. The explanation of this finding is straightforward. As the agent’s concentration decreases so does its lethal effect. Therefore, the inactivation rate progressively diminishes; theoretically, when the agent’s concentration reaches a threshold level determined by the targeted organ-ism’s tolerance (which might be zero), the effect of the treatment vanishes

altogether. It was recognized very early that the lethal effect of a volatile agent or microorganisms depends not only on its initial concentration, but also on the exposure duration. Consequently, unlike traditional sur-vival curves that depict the number of surviving organisms or the sursur-vival ratio as a function of time, survival curves of organisms exposed to a dissipating chemical disinfectant are frequently presented in the form of a plot depicting the number of cells’ survival ratio as a function of a combined concentration–time parameter.

By far the most common combined parameter of this kind is the product of the momentary concentration, C(t), multiplied by the time, t, i.e., C(t)

⋅ t. The purpose of this modification of the time axis has been to compare experimental survival ratios obtained under conditions in which the agent dissipates at different rates. By plotting the survival ratio against a com-mon variable that accounts for the fact that the agent’s concentration varies with time, it has been hoped that the transient nature of the process would be captured. The difficulty with this approach is that the effects of time and concentration are assigned, arbitrarily, a universal reciprocal relation that probably does not exist in reality. To demonstrate the point, consider the following two hypothetical scenarios:

• An agent is applied at a constant concentration of 1000 ppm for 1 h, C(t) · t = 1000 ppm h, that is.

• The same agent is applied at a constant concentration of 5 ppm for 200 h, i.e., C(t) · t is also 1000 ppm h.

If the concept is correct, then the survival ratio at the end of the two treatments should be exactly the same, which is highly doubtful. If the agent’s concentration is not constant, but decreases with time, then accord-ing to the C(t) · t concept, the same lethality should be observed at any equivalent concentration–time combination regardless of the agent’s initial concentration and dissipation rate. Examination of published records of dis-infection experiments and computer simulations show that this would be a very unlikely occurrence. The same can be said about any alternative concentration–time combination, unless it can be experimentally demon-strated that it produces unique survival curves for any practical combi-nation of the agent’s initial concentration level and dissipation rate history.

Incorporating an Arrhenius type or a similar concentration term in the survival curve’s model equation would be equally problematic. This is because it requires the existence of a special preconceived relationship between the organisms’ response to different concentration–time combi-nations. Such a relationship needs to be established prior to the application of the model. If the existence of an Arrhenius-type relationship is consid-ered merely as an assumption, then its validity must be confirmed by

testing the model’s ability to predict experimental survival curves pro-duced in treatments that had not been used in the model’s formulation.

Alternative General Model

Consider a targeted organism whose hypothetical isoconcentration inac-tivation at the pertinent agent concentration range follows the Weibul-lian–power law model (Equation 5.1) with a constant power, n (Equation 5.2). The concentration independence of the power term, i.e., n(C) = n, is not a prerequisite for what follows and the purpose of using Equation 5.2 as the primary inactivation model is only to simplify the discussion. The same can be said about a number of alternative non-Weibullian models.

As in thermal processing, we assume that on the pertinent time scale the treatment is rigorous enough so that growth, adaptation, and recovery from injury do not occur. We also assume that the agent concentration’s dissipation history, or concentration profile (the C(t) vs. t relationship), can be expressed algebraically, although it will be shown later that this, too, is not a strict requirement. The reasoning that led to the development of the nonisothermal survival models for heat inactivation is also appli-cable here. A main difference is that, in contrast with heating where the process’s lethality progressively increases, the lethal intensity progres-sively decreases during a treatment with a dissipating disinfectant. (The analog of heating is a rise in the agent’s concentration while cooling is analogous to the agent’s dissipation.)

If the momentary logarithmic inactivation rate, dlog10S(t)/dt, under a continuously changing agent concentration is the rate that the agents’

momentary concentration produces at a time, t*, that corresponds to the momentary of the survival ratio as shown in Figure 5.2, we can write:

(Peleg et al., 2004):

(5.3) and

(5.4)

Combining the two equations results in the model rate equation:

(5.5)

If the power n is also concentration dependent, then n[C(t)] will replace n in the model equation. Although we do not have sets of experimental isoconcentration survival curves to determine b(C) and n or b(C) and n(C), if n(C) is thought to be concentration dependent too, we can still assume that b(C) will follow the log logistic model (Figure 5.3):

b(C) = ln {1 + exp[kc(C – Cc)]} (5.6)

FIGURE 5.2

Schematic view of the survival curve’s construction for an organism exposed to a dissipating chemical agent. Notice the assumption that the logarithmic inactivation rate is time depen-dent even under the (hypothetical) condition of constant agent concentration. (Courtesy of Dr. Maria G. Corradini.)

0

0 0 C₁ C₃ C₂

Time (arbitrary scale) C₇

C₇ C₆ C₅ C₄ C₃ C₂ C₁ C₆ C₅

C₄ Disinfectant’s concentration profile

Noniso-concentration survival curve Concentration Log S(t)

where C_c marks the concentration level at which lethality accelerates, and k_c the slope of b(C) vs. C at concentrations much higher than C_c. Like before, when C << C_c, b(C) ≈ 0 and when C >> Cc, b(C) ≈ kc(C – C_c) (see figure). Again, if the evidence is that the continuation of b(C) beyond C_c is nonlinear, the model can be amended by the addition of a power, m:

b(C) = ln {1 + exp [k_c(C – C_c)^m]} (5.7) Unlike in the application of this model to thermal inactivation, a situa-tion in which the agent has measurable lethality at any measurable con-centration cannot be ruled out. In such a case, C_c = 0 and Equation 5.6 and Equation 5.7 will be respectively reduced to:

b(C) = k_cC (5.8)

or

b(C) = kc′C^m (5.9)

Recall, though, that because we do not have experimental survival data obtained under constant agent concentration conditions, all the preceding

FIGURE 5.3

Schematic view of the log logistic concentration dependence of the rate parameter b(C).

Schematic view of the log logistic concentration dependence of the rate parameter b(C).