Abstract
 Top of page
 Abstract
 Introduction
 Theoretical Background
 Procedure
 Demonstration of the Method with Actual Data
 Concluding Remarks
 Acknowledgments
 References
ABSTRACT: Theoretically, if an organism's resistance can be characterized by 3 survival parameters, they can be found by solving 3 simultaneous equations that relate the final survival ratio to the lethal agent's intensity. (For 2 resistance parameters, 2 equations will suffice.) In practice, the inevitable experimental scatter would distort the results of such a calculation or render the method unworkable. Averaging the results obtained with more than 3 final survival ratio triplet combinations, determined in four or more treatments, can remove this impediment. This can be confirmed by the ability of a kinetic inactivation model derived from the averaged parameters to predict survival patterns under conditions not employed in their determination, as demonstrated with published isothermal survival data of Clostridium botulinum spores, isobaric data of Escherichia coli under HPP, and Pseudomonas exposed to hydrogen peroxide. Both the method and the underlying assumption that the inactivation followed a WeibullLog logistic (WeLL) kinetics were confirmed in this way, indicating that when an appropriate survival model is available, it is possible to predict the entire inactivation curves from several experimental final survival ratios alone. Where applicable, the method could simplify the experimental procedure and lower the cost of microbial resistance determinations. In principle, the methodology can be extended to deteriorative chemical reactions if they too can be characterized by 2 or 3 kinetic parameters.
Introduction
 Top of page
 Abstract
 Introduction
 Theoretical Background
 Procedure
 Demonstration of the Method with Actual Data
 Concluding Remarks
 Acknowledgments
 References
Traditionally, the heat resistance parameters of microbial cells and bacterial spores have been determined from a series of isothermal semilogarithmic survival curves recorded at several lethal temperatures. The reciprocal of the slope of the log survival ratio compared with time plot, which has been assumed to be loglinear, is known as the “Dvalue”. It has also been assumed that this “Dvalue” has a loglinear temperature dependence characterized by its own slope, which is known as “z” (for example, Jay 1996; Holdsworth 1997; Holdsworth and Simpson 2007). Thus, according to the classic theory of microbial inactivation, knowing an organism's “Dvalue” at a reference temperature and its “z” is sufficient to construct this organism's isothermal or “dynamic” survival curves in the particular medium where these parameters had been determined. (The restriction also applies to the organism's growth stage and to treatments that do not allow for adaptation, issues that will not concern us here.) It had been observed fairly early that the “Dvalue” versus temperature relationship is not always loglinear, in which case the original loglinear model was replaced by the Arrhenius equation. The Arrhenius model is based on the assumption that the relationship between the exponential rate constant k (the Dvalue's reciprocal) and the absolute temperature (in °K) is loglinear. With this model too, an organism's heat resistance is characterized by 2 parameters only, namely, k at a reference temperature and the inactivation's “energy of activation”.
The classic theory of microbial loglinear or “1storder” mortality kinetics has been extended to nonthermal inactivation, notably to ultrahigh pressure processing (HPP). One can find long lists of microbial D_{p} and z_{p} values, or k_{p} and “activation energies” defined in terms of pressure, for example, instead of temperature—see FDA (2000). Despite the widespread acceptance of these theories, there is now accumulated evidence that microbial survival only rarely follows the loglinear (“first order kinetics”) model and that there is no reason that it should (for example, Peleg and Cole 1998; Fernández and others 1999; Mafart and others 2002; van Boekel 2002; Peleg 2006). What is known as “nonlinear inactivation” has been described by a variety of mathematical models, of various degrees of complexity (for example, Anderson and others 1996; Augustin and others 1998; Corradini and Peleg 2004; Geeraerd and others 2004, 2005; McKellar and Lu 2004; Valdramidis and others 2004, 2006). Among these, perhaps the simplest and most intuitive is the Weibullian model (Peleg and Cole 1998):
 (1)
where S(t) is the survival ratio defined as N(t)/N_{0}, N(t) and N_{0} being the momentary (instantaneous) and initial count, respectively, and b(T) and n(T) temperature dependent coefficients. Equation 1 emerges when the inactivation or mortality events have a Weibull temporal distribution (Peleg and Cole 1998; van Boekel 2002), which is quite common in failure phenomena. Therefore, it should not come as a surprise that this flexible mathematical model could fit many microbial survival curves (for example, Fernández and others 1999; Mafart and others 2002; van Boekel 2002). For the same reason, the Weibullian model was also found to fit survival data under ultra highpressure processing (for example, Heinz and Knorr 1996; Chen and Hoover 2003; Hu and others 2005; Van Opstal and others 2005; Koseki and Yamamoto 2007; Buckow and others 2008), chemical disinfection (for example, Haas and Joffe 1994; Haas and others 1996), and even pulsed electric fields (for example, Rodrigo and others 2001; Alvarez and others 2003).
In most cases, n_{T}(T) has only weak temperature dependence (van Boekel 2002, 2003), that is, n(T) ≈ constant =n_{T}. In such a case, Eq. 1 can be reduced to the 2parameter model:
 (2)
The exponent n_{T} can be determined by averaging, for example, or by searching for the value that will result in the best fit overall. According to the Weibullian model, when n_{T} > 1 the isothermal survival curve has downward concavity and when n_{T} < 1 upward concavity. Loglinear inactivation, according to the model, is just a special case of Eq. 2 where n_{T}= 1. Therefore, all that follows will apply to the traditional models of survival as well as to the Weibullian. The same can be said about inactivation by means of a nonthermal agent like ultrahigh hydrostatic pressure, P, or a chemical preservative or disinfectant having concentration C. When the inactivation by these agents follows the Weibullian model, b_{T}(T) and n_{T} in Eq. 2 would be replaced by b_{P}(P) and n_{P} or b_{c}(C) and n_{C}, respectively.
The temperature dependence of b_{T}(T) can be described by the log logistic model (for example, Corradini and others 2005; Pardey and others 2005; Peleg 2006):
 (3)
where T_{c} marks the onset of the lethal temperature regime and k_{T} is the approximate rate (with respect to temperature) at which b_{T}(T) rises when T >> T_{c}. (According to this model, at T << T_{c}, b_{T}(T) ≈ 0 and at T >> T_{c}, b_{T}(T) ≈k(T–T_{c}).)
A similar model can apply to high pressure or chemical preservation, in which case k_{T} will be replaced by k_{P} or k_{C} and T_{c} by P_{c} or C_{c}, respectively (Peleg 2006). (Notice that the loglogistic model does not exclude the situation of C_{c}= 0, in which case the chemical agent is lethal to the targeted organism at any concentration.)
Combining Eq. 2 and 3 yields the isothermal inactivation model:
 (4)
For ultrahigh hydrostatic pressure preservation (HPP) it will become:
 (5)
and for chemical preservation:
 (6)
In either case, the organism's resistance to the lethal agent is characterized by 3 survival parameters, namely, n_{T}, k_{T}, and T_{c}, n_{P}, k_{P}, and P_{c} or n_{C}, k_{C}, and C_{c} instead of the conventional two.
These survival parameters can be determined from experimental inactivation curves recorded at several constant temperatures, pressures, or chemical agent concentrations. Fitting the experimental isothermal, isobaric, or “isoconcentraton” log_{10}S(t) compared with time data with Eq. 2 or one of its equivalents as a model, using nonlinear regression, will yield the b_{T}(T), b_{P}(P) or b_{C}(C) value and the corresponding n_{T}, n_{P}, or n_{C}. The other survival parameters; k_{T} and T_{c}, k_{P}, and P_{c} or k_{c} and C_{c} can then be determined from the b_{T}(T) compared with T, b_{P}(P) compared with P or b_{C}(C) compared with C, respectively, again by nonlinear regression but with Eq. 4, 5, or 6 as a model.
Once determined in this manner, these survival parameters can be inserted into a rate model equation (Corradini and Peleg 2004; Pardey and others 2005; Peleg and others 2005; Peleg 2006), which can be used to predict the inactivation patterns under dynamic conditions. Agreement between this model's predictions and experimental dynamic inactivation data will validate the model and the assumptions on which its derivation is based.
At this point, a word of caution is in order. Application of ultrahigh hydrostatic pressure also produces adiabatic heating. Consequently, special arrangements are required to generate survival data under different pressures but the same temperature—see subsequently. Similarly, maintaining a constant concentration of a volatile disinfectant requires its constant generation or supplementation under controlled conditions. What follows will only address this kind of experiment.
In certain systems, even determination of an isothermal survival curve is difficult for technical reasons. A case in point is hightemperature processing where withdrawing a sample during the short treatments is hardly an option. There are situations where logistic considerations concerning labor and expense might limit the number of counts. In HPP, withdrawing a sample without affecting the pressure is a problem in itself, forcing the researcher to repeat the whole process with different holding times operating several pressurized vessels simultaneously. In all such circumstances, determination of the survival parameters without recording the whole survival curve would greatly simplify the experimental procedure and could bring considerable monetary savings.
The possibility to determine an organism's thermal resistance parameters from the final survival ratios of several heat treatments has recently been studied and its practicality demonstrated (Corradini and others 2008; Peleg and others 2008). The initial investigation focused on heat inactivation by UHT in which obtaining reliable isothermal survival data is a serious problem. Therefore, the idea was to retrieve the Weibullian survival parameters from the final survival ratios determined after three or more nonisothermal (dynamic) treatments. But because under dynamic conditions the inactivation kinetics is governed by a differential (rate) equation, extraction of the survival parameters from the data required an elaborate and difficult mathematical procedure (Peleg and others 2008). In systems where generation of isothermal temperature profiles is not a technical issue, the survival parameters calculation from the final survival ratios can be simplified considerably. The objectives of this study were: (1) To develop and test the simplified version of the procedure based on the WeibullLog logistic (WeLL) survival model for isothermal inactivation of microorganisms and spores and (2) To demonstrate the method's potential applicability in ultrahigh pressure (HPP) inactivation studies and in chemical preservation under constant agent concentration, where the microbial inactivation most probably follows similar kinetics, albeit characterized by different survival parameters.
Theoretical Background
 Top of page
 Abstract
 Introduction
 Theoretical Background
 Procedure
 Demonstration of the Method with Actual Data
 Concluding Remarks
 Acknowledgments
 References
Consider 3 ideal isothermal temperature profiles, T_{1}, T_{2}, and T_{3} starting at t= 0 and ending at t_{final1}, t_{final2}, and t_{final3}, respectively, as shown in Figure 1. The corresponding logarithmic survival ratios at these points, determined experimentally, are log_{10}S(t_{final1}), log_{10}S(t_{final2}), and log_{10}S(t_{final3}). The actual temperatures are unimportant as long as they all fall within the clearly lethal range and are not too close to each other. The exposure duration is also unimportant, except that it has to be long enough to reduce the number of cells or spores by several orders of magnitude. Also shown in Figure 1 is that the actual survival curves produced by the 3 heat treatments are initially unknown. All we do know about them is that they start at log_{10}S(0) = 0 (by definition) and that they end at the 3 experimentally determined final survival ratios log_{10}S(t_{final1}), log_{10}S(t_{final2}), and log_{10}S(t_{final3}). However, if the WeLL model with a fixed power n_{T} is valid, then these 3 final survival ratios ought to lie on the 3 corresponding isothermal survival curves characterized by this model. (These yet unknown curves are shown schematically as dotted lines in the figure.) Mathematically, all this translates into the following equations:
 (7)
 (8)
 (9)
These 3 simultaneous equations have 3 unknowns, namely, k_{T}, T_{c}, and n_{T}, the organism or spore's survival parameters in the particular medium. Although nonlinear equations, they can be easily solved numerically by Mathematica^{®} (Wolfram Research, Champaign, Ill., U.S.A.), the program used in this study, and by almost any advanced mathematical software. Thus, at least in principle, the survival parameters can be extracted from 3 (isothermal) final survival ratios alone, provided that the underlying model, the WeLL model in our case, is indeed applicable. The model's validity can be tested and established by using the survival parameters calculated in this manner to predict the final survival ratios at temperatures not used in their determination and then comparing the predicted values with those determined experimentally. If the assumed underlying model is inappropriate for the particular organism or spore's survival, there will either be no solution to the 3 simultaneous equations or one or more of the resulting parameters will have no physical meaning, being negative or complex, for example. In other words, the method provides its own criterion of whether it is applicable to the experimental data at hand.
If an organism or bacterial spore's inactivation indeed follows 1storder kinetics, as has been traditionally assumed, then the calculation would yield n_{T}= 1. Or, if it is known a priori that n_{T}= 1, then, theoretically, only 2 equations would be needed to determine the organism's 2 survival parameters, k_{T} and T_{c}, or their equivalents in the conventional models. The same will be true for any alternative 3 or 2 parameter survival model, except that Eq. 7 to 9 would have to be reconstructed accordingly.
In practice, because microbial counts almost always have a scatter, the experimentally determined final survival ratios, or at least one of them, can deviate somewhat from their “correct values.” The result would be somewhat distorted survival parameters or, in case of a substantial error or errors, an inoperable method. The latter would be manifested in either totally unrealistic or absurd survival parameters (for example, negative or complex) or failure of the numerical solution to converge. These situations can be avoided in 2 ways (Corradini and others 2008):
 1
By fixing the magnitude of n_{T} to a value (or values) deemed reasonable by a preliminary analysis thus reducing the number of simultaneous equations and calculated survival parameters to two, or
 2
By calculating the survival parameters not from a single set of 3 final survival ratios but from several pair or triplet ratio combinations determined at four or more constant temperatures and averaging the results.
The 1st option, although by far the most convenient, is less desirable because of the arbitrary element involved. However, its simplicity can be exploited by its incorporation into an iterative calculation procedure. The survival parameters extraction is more efficient when n_{T} is changed by small increments or decrements and the corresponding k_{T} and T_{c} values become the solution of only 2 simultaneous equations. The values of k_{T} and T_{c} so obtained together with that of the “temporary n_{T}” are then inserted into the model's equation, which is used to predict the final survival ratio at the 3rd temperature, as shown schematically in Figure 2. The procedure is repeated until the n_{T} value and its corresponding k_{T} and T_{c} produce a calculated survival ratio that matches the observed one within a specified tolerance, 0.01 log_{10} units say, as shown in the figure. For more on the incremental method, see Peleg and others (2008) and Corradini and others (2008). The 1st publication also provides the URL of the 4 Mathematica 6^{®} programs, which were used to do the calculations. They are labeled Appendix A, B, C, and D and can be freely downloaded at http://wwwunix.oit.umass.edu/~aew2000/three_points_method/UHT_Appendices.html.
In principle, the described “3 end points method” can be applied to nonthermal microbial inactivation under a constant agent's intensity, except that the WeLL model equations would have different coefficients. For HPP or chemical preservation, the triplet n_{P}, k_{P}, and P_{c} or n_{C}, k_{C}, and C_{c} will replace n_{T}, k_{T}, and T_{c}, and P_{1}, P_{2}, and P_{3} or C_{1}, C_{2}, and C_{3}, would replace T_{1}, T_{2}, and T_{3} in Eq. 7 to 9. And, as before, the model's ability to predict the outcome of treatments not used in its derivation would serve as its validation criterion.
Procedure
 Top of page
 Abstract
 Introduction
 Theoretical Background
 Procedure
 Demonstration of the Method with Actual Data
 Concluding Remarks
 Acknowledgments
 References
Published isothermal inactivation data on Clostridium botulinum spores (Anderson and others 1996) served as the primary database for the 1st part of the work. For demonstration it was supplemented by a set of published feline calicivirus inactivation data (Buckow and others 2008) and a set of data on Salmonella cells taken from Newcomer (2004). The experimental procedures are described in detail in the original publications. (The one used by Newcomer [2004] can also be found in Mattick and others [2001])
The isothermal survival ratio compared with time data sets were digitized and their end points recorded and saved separately. Various triplet combinations of the t_{final} and corresponding log_{10}S(t_{final}) values, taken at different temperatures, were then used to calculate the survival parameters. This was done by solving Eq. 7 to 9 using the incremental method, the principle of which is demonstrated in Figure 2. When this procedure is employed, as already mentioned, the program itself identifies the best n_{T} value for each triplet of end points and records the corresponding values of k_{T} and T_{c}. The 3 survival parameters so obtained were then averaged and their means recorded. They are reported side by side with the predicted survival ratios calculated by the program and the actual experimental values. The method's performance was also tested by comparing the entire survival curves generated with the WeLL model (Eq. 4), having the mean n_{T}, k_{T}, and T_{c} values as its coefficients, against the reported experimental survival data.
The abovementioned procedure was repeated with data on Escherichia coli inactivated under various levels of ultrahigh pressure (HPP) at 15 °C published by Koseki and Yamamoto (2007) and on Pseudomonas aeruginosa inactivated by hydrogen peroxide at various concentrations reported by Lambert and others (1999). Here too the averaged values of the parameters n_{p}, k_{p}, and P_{c} and n_{c}, k_{c}, and c_{c} were incorporated in their respective versions of the WeLL model equations, which were subsequently used to test and validate the method's applicability.
Concluding Remarks
 Top of page
 Abstract
 Introduction
 Theoretical Background
 Procedure
 Demonstration of the Method with Actual Data
 Concluding Remarks
 Acknowledgments
 References
The applicability of the method to estimate survival parameters from the final ratios of three or more treatments has been demonstrated in very different organisms and with 3 very different inactivation modes, thermal, ultrahigh pressure, and chemical, having different characteristic time scales. The underlying assumption that the WeibullianLog logistic (WeLL) model can effectively describe the inactivation kinetics in all 3 processes has also been confirmed by this model's repeated ability to predict entire survival curves, which had not been used in its parameters' derivation. In most cases, the predictions were within 0.5 log_{10} units of the actual survival ratios, that is, within the experimental scatter range frequently found in published survival data. None of the studies whose results were used in the survival parameters' calculations had been originally designed to test the WeLL model or to validate the described calculation procedure. Consequently, the authors of the original research that had produced the experimental data made no attempt to improve the final survival ratios' accuracy. Yet, as Table 4 shows, the survival parameters calculated by the “3 end points method” were fairly close to those obtained from the complete experimental record using nonlinear regression. That the agreement has been observed in all 3 treatments suggests that the “3 end points method” can be robust enough to extract an organism's survival parameters from a considerably smaller experimental database than that required by conventional procedures. In all the cases, discussed in this study, the inactivation was clearly not loglinear as the traditional 1storder kinetic model demands. Had it been even close, the calculated WeLL model's concavity parameter (n_{T}, n_{P}, or n_{C}) would have been approximately one regardless of whether it was calculated by the “3 end points method” or regression. Instead, the calculated values were invariably in the range of 0.3 to 0.7 as shown in the tables. The semilogarithmic survival curves' pronounced upper concavity, which implies “tailing,” is of course also evident in the appearance of the experimental data themselves, as seen in Figure 3 to 6. Nevertheless, if and when the inactivation or any other degradation process, see subsequently, does follow the loglinear model, the procedure would be simplified considerably by testing ratio pairs instead of triplets.
Table 4—. Comparison of the WeLL model's survival parameters calculated by the “3 points method” and by regression using all the data in the experimental survival curves.Treatment  Organism  Method of calculation  n_{T}  k_{T}(°C^{−1})  T_{c}(°C)  Data source 

Heat  C. botulinum (spores)  Three points  0.30  0.26  100  Anderson and others (1996) 
Regression  0.33  0.28  101  
Heat  Salmonella enterica  Three points  0.31  0.14  64  Newcomer (2004) 
Regression  0.42  0.15  71  
Heat  Feline calicivirus  Three points  0.55  0.29  53  Buckow and others (2008) 
Regression  0.50  0.30  53  

   n_{P}  k_{P}(MPa^{−1})  P_{c}(MPa)  

Pressure  E. coli ATCC  Three points  0.40  0.02  237  Koseki and Yamamoto (2007) 
Regression  0.32  0.022  210  

   n_{C}  k_{C}(L/mmol)  C_{c}(mmol l^{−1})  

Hydrogen peroxide  Pseudomonas aeruginosa  Three points  0.61  0.013  187  Lambert and others (1999) 
Regression  0.69  0.011  225  
Replacing the traditional set of complete survival curves by a set of final survival ratios might only have a clear advantage whenever extracting a sample during the process is technically difficult or where counting the targeted organism at sufficiently short intervals during the treatments is a costly option. The savings accrued by avoiding the need to determine entire survival curves experimentally might allow more reliable determination of the final survival ratios through added replicates and with it the organism or spore's survival parameters.
Since there is nothing in the general concept that limits the 3points method's applicability to microbial inactivation under constant conditions, it can be extended to dynamic processes whenever creating ideal or next to ideal isothermal, isobaric, or “isoconcentration” conditions is not feasible. Such an extension, though, will translate into a much more elaborate mathematical procedure that requires special programming and/or the use of advanced mathematical software such as Mathematica^{®} (Corradini and others 2008; Peleg and others 2008). The described principle can also be extended to chemical degradation, such as vitamin loss during thermal processing, provided that a single 2 or 3parameter model can characterize its kinetics in the pertinent temperature range. The same can be said about chemical synthesis of unwanted (or, in some food preparation processes, wanted) compounds such as the products of the Maillard reaction (Peleg and others 2008). But here too, the model's validity would have to be first established by testing its predictions against known final concentrations or concentration ratios determined under conditions that have not been used in its parameters' calculation.
The study's main objectives, as stated, have been to propose and evaluate a concept, not to determine the inactivation characteristics of any particular microorganism or microorganisms. We hope that this study will encourage others to test the calculation method's applicability with data especially generated for the purpose. Future research might also evaluate the merits of alternative kinetic models and compare their performance in the characterization and prediction of survival patterns and the outcome of chemical degradation or synthesis reactions.