Biological and Physical Bases for Deviations

Equation (7.3) would predict a straight line (either increasing or decreasing) of the logarithm of microorganism concentration versus time. However, under many circumstances, this is not the case. Causes of these deviations will be discussed specifically for decay processes and growth processes. For some processes, particularly removal or reduction by physical processes (e.g., filtration), a first-order law is appropriate.

So the equation for microorganism concentration versus distance estimated from the data is

(7.5)

We would like N to be equal to 0.1/l; therefore, ln(0.1) = − 2.302. Substituting this into Equation (7.5), we find that z = (−2.302 − 4.47/−0.0105) = 639m.

General Forms of Decay and Reasons for Nonlinearity

A baseline for modeling microbial decay has long been the concept that the rate of decay is a first-order process in remaining viable microorganism concentrations (e.g., [5]). This leads to functional forms as in Equations (7.2) and (7.3) (with the sign of the rate being negative). In the context of decay promoted by disinfection, this formulation was advanced as early as Chick [6].

When a disinfectant is added, Watson [7] proposed that the “k” in Chick’s Law be replaced by a power function in disinfection concentration, that is, k →  kCⁿ. This dependency has been found to be frequently valid, even when there are deviations from the exponential nature of Chick’s Law. The combination of this power law dependency on disinfection concentration with Chick’s Law is generally referred to as the Chick–Watson Law. In differential form,

(7.6)

which when integrated yields

(7.7)

Chick herself, and numerous later researchers, noted that a plot of viable microorganism numbers or concentrations was not always linear on a semilogarithmic plot. Rather, a variety of forms, including “shoulders,” “tails,” and S-shaped or inverse S-shaped curves, can be seen, as illustrated in Figure 7.2.

There are several potential mechanisms that could explain deviations from Chick’s Law:

1. For chemical agents, there may be decay of the agent, that is, C diminishes as a function of time. In this case, the differential equation of Equation (7.6) must be integrated using C = f(t) (e.g., first-order decay). In this case, the integrated form as shown in Equation (7.7) is inappropriate. Rather, a decay-corrected relationship such as derived in [9] must be used.
   
2. Microorganisms being disinfected may exist in “clumps”—stable aggregates containing multiple infectious units. In order for a “clump” to be completely inactivated, each single infectious unit must be inactivated—thereby requiring multiple “hits” to inactivate each clump [11]. For fixed clump size, this multiple hit approach results in a disinfection curve with a shoulder.
   
3. There may be intrinsically distinct subpopulations that have different sensitivities to inactivation. For example, if a broad assay, such as heterotrophic plate count or total coliform, is used to measure decay, there are different populations in each of these assays with different sensitivities. In principle, if the survivors from a decay experiment are isolated and subcultured, the resultant population should possess enhanced resistance. This experiment has not been done frequently, but when attempted, no resistance enhancement has been observed [12].
   
4. Even in a population of only one species of microorganism, there may be different physiological states of individuals in the population. These states may result in individual cells having different susceptibilities, even though they represent genetically identical progeny.
   
5. In a real system, there may be physical factors that temporarily (or permanently) render a portion of otherwise susceptible microorganisms less accessible to disinfectant. Examples of these factors include entrapment in solids [13–15] or adsorption to container walls.
   
6. Disinfectants may cause multiple reactions with microorganisms. In addition to the inactivation step, in some circumstances, disinfectants could cause sublethal changes, which result in reduced sensitivity (hardening) to further disinfection [16].

A number of authors have proposed modifications of Chick’s Law in which general nonlinear behaviors can be described [17–20]. These can be summarized in the following manner, by analogy to Equations (7.2) and (7.3) [21]:

(7.8)

(7.9)

If x = 1 and m = 0, Equation (7.9) simplifies to the Chick–Watson relationship. Interestingly, if x is different than 1, there is a dependency of inactivation efficiency (i.e., the combination of C and t that leads to a given value of N/N₀) on N₀. This has been observed experimentally in several studies in which the initial microorganism concentration was varied systematically [18, 22].

Spontaneous/Endogenous

Microorganisms will have a spontaneous rate of decay even in the absence of the introduction of inhibitory or antagonistic agents. Fundamentally, since microorganisms are highly organized systems, that is, regions of lower entropy, in the absence of continual utilization of energy, they are expected to decay, that is, result in increase of entropy. The rate of decay will depend on environmental conditions and the type and physiological state of the microorganism. For example, in both air and water (as well as other moist environments, such as foods), the survival of organisms is a function of temperature and water activity/relative humidity [23, 24]. These dependencies can be used in order to accelerate microbial decay, such as by high temperature—as discussed further in the succeeding text.

Chemical Agents

There are numerous chemical agents that can be used to accelerate the inactivation of microorganisms. It is impossible to give an exhaustive list, but such agents include halogens and halogen compounds, ozone, heavy metals, alcohols, aldehydes (e.g., formaldehyde), synthetic antimicrobials, etc. Regardless of the agent used, the process of inactivation can be described by equations in the prior section—taking into account any chemical agent decay that may occur.

The inactivation rate constants, for example, k in Equation (7.9), may be functions of environmental conditions that modulate the effectiveness of chemicals in inactivating organisms. For example, in the case of chlorine and hypochlorites, the presence of ammonia nitrogen and changes in pH [25] are well known to alter intrinsic effectiveness against a wide variety of microorganisms.

In the application of chemical disinfectants, the disinfectant concentration may diminish due to reaction with other environmental components (or for highly reactive disinfectants, such as ozone, by rapid spontaneous decay). Some of the diminution may be virtually instantaneous; however, a slower decay process may occur. In the presence of such diminution, the differential equation describing disinfectant decay must be integrated along with the differential equation describing inactivation (e.g., Eq. (7.6)).

For example, if there is no immediate demand, but instead, the decay can be described by the following first-order process:

(7.10)

where k^* is the first-order rate constant for disinfectant decay, then Equations (7.6) and (7.10) must be solved simultaneously. First, we note that Equation (7.10) can be integrated directly to yield

(7.11)

Now, Equation (7.11) can be substituted into Equation (7.6) to produce

(7.12)

This system has the following analytical integral—which, if plotted, would exhibit “tailing off” behavior (one explanation for such nonlinear inactivation curves):

(7.13)

Analogous modifications of the Hom [9] and other disinfection models for demand can be computed.

Thermally Induced

The use of elevated temperature to reduce microorganisms is a well-known and ancient process. In particular, in food safety, elevation of temperature (during cooking) is an efficient way to reduce spoilage and pathogenic organisms. Conventionally, the reduction of target microorganisms at a constant elevated temperature with time is logarithmic and can be written in the following form [26]:

(7.14)

The parameter “D” is the decimal reduction time, that is, the time to reduce microorganism concentration by a factor of 10.

“D” is a function of temperature—once the temperature is reached promoting inactivation, then the dependency on temperature can be written as

(7.15)

and thus “z” can be interpreted as the increase in temperature that reduces the decimal reduction time by a factor of 10. Given “D” and “z” for a particular pathogen–food combination, the surviving microorganism population can be computed in the presence of inactivation.

As with chemical disinfection, with thermal inactivation, nonlinear curves (i.e., not linear on a semilogarithmic plot of survivors vs. time) can arise [27]. One way to treat such abberations is by the use of power functions of time in the form introduced by [28] justified by a Weibull distribution:

(7.16)

While Equation (7.16) can explain “shoulders” and “tailing off,” the parameters α and β may both be functions of temperature, although the analyses of Reference [28] suggest that in many cases β could be regarded as independent of temperature and that α had significant temperature dependence.

Ionizing and Nonionizing Radiation

Radiation can serve to inactivate microorganisms. Radiation may be supplied by sources of electromagnetic waves or from the disintegration of radioactive isotopes. The relevant forms of radiation that cause inactivation are those in the higher-frequency (shorter wavelength) visible spectrum and above—as shown in Figure 7.4.

The distinction between ionizing and nonionizing radiation is based on the ability of electromagnetic waves greater than a certain frequency (below a certain wavelength) to cause electrons to be dissociated from molecules. UV radiation at the commonly used wavelengths of approximately 254 nm can cause mild chemical alterations in biological molecules, but not ionization per se; however, higher-frequency UV (shorter wavelength) as well as X-rays and the various forms of radioactive decay are capable of causing ionization.

Nonionizing radiation (including 254 nm UV) primarily acts by virtue of direct interaction with susceptible molecules in microorganisms, such as nucleic acids or proteins. Shorter-wavelength, ionizing, radiation acts by virtue of both direct interaction as well as the formation of reactive chemical intermediates (such as free radicals) that themselves can be antimicrobial in nature.

For nonionizing radiation, a first approach to assess the kinetics of microbial inactivation is to use the equations presented earlier, for example, Equation (7.8) and its simplifications, with the incident light intensity, I, replacing the disinfectant concentration, C. Units of I are typically in power per unit area, for example, W/m² of the effective wavelength(s). It is further often assumed that the dependency of inactivation rate is linearly proportional to this intensity; in other words, n = 1.

For ionizing radiation, there may be multiple mechanisms of inactivation. The radiation itself may directly damage biomolecules. In addition, the radiation may produce by-products (e.g., see [30]), which themselves can exert antimicrobial activity. Thus, for ionizing radiation, a detailed chemical model of free radical production and inhibition is required, and the net inactivation rate is an addition of the rate of inactivation via a direct and an indirect mechanism.

With radiation (either ionizing or nonionizing), passage through a medium (air, water, food, etc.) may result in the loss of intensity with distance traveled. This decline, which is a function of wavelength as well as concentration of materials in solution, obeys a logarithmic decline relationship—termed Beer’s Law [31]. Hence, the presence of a large degree of absorbing material at the active wavelengths for inactivation will result in different energy doses at different distances from the source(s) of radiation, in the absence of vigorous mixing. This stratification can impede effectiveness of UV disinfection processes [32].

Predation and Antagonism

Environments into which pathogens are released, especially if they are moist (water, soil, food, etc.) can also harbor biological antagonists, which may include predatory or parasitic organisms. Examples can include viruses (phage) and protozoans. In addition, bacteria, for example, Bdellovibrio, can parasitize other bacteria [33, 34]. The rate at which pathogens can be decreased by this mechanism is a function of both the pathogen and the predator/parasite concentration.

If there is both a predation loss mechanism and other loss mechanisms, then the rates can be regarded as occurring simultaneously. Then, an overall relationship for the decline in an infectious agent might be written as

(7.17)

where r_g is the growth rate (if it exists), discussed in the following section; r_c is the decay rate due to the occurrence of any chemical disinfectants (as well as an intrinsic decay rate); r_rad is the rate of inactivation due to radiation; and r_pred is the rate of inactivation due to the predator or parasite. In a predator–prey system, in order to be able to predict the change in prey numbers, the rates of growth and decay of predators must also be modeled. Approaches to this are discussed later in this chapter.

If an environment contains predators, then materials or factors that act more substantively against predators than prey may result in a net increase in the prey (e.g., pathogenic microorganisms) [35]. This appears to have been observed in a full-scale study in Chicago, where cessation of disinfection apparently resulted in a diminution of indicator organisms far downstream—presumably because disinfectant residuals had been highly inhibitory toward indigenous protozoan predators [36].