SEARCH

SEARCH BY CITATION

Keywords:

  • kinetics;
  • nonlinear inactivation;
  • predictive microbiology;
  • thermal processing;
  • Weibull-Log logistic (WeLL) model

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Theoretical Background
  5. Procedure
  6. Demonstration of the Method with Actual Data
  7. Concluding Remarks
  8. Acknowledgments
  9. 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 Weibull-Log 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

  1. Top of page
  2. Abstract
  3. Introduction
  4. Theoretical Background
  5. Procedure
  6. Demonstration of the Method with Actual Data
  7. Concluding Remarks
  8. Acknowledgments
  9. References

Traditionally, the heat resistance parameters of microbial cells and bacterial spores have been determined from a series of isothermal semi-logarithmic 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 log-linear, is known as the “D-value”. It has also been assumed that this “D-value” has a log-linear 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 “D-value” 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 “D-value” versus temperature relationship is not always log-linear, in which case the original log-linear 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 D-value's reciprocal) and the absolute temperature (in °K) is log-linear. 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 log-linear or “1st-order” mortality kinetics has been extended to nonthermal inactivation, notably to ultra-high pressure processing (HPP). One can find long lists of microbial Dp and zp values, or kp 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 log-linear (“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):

  • image(1)

where S(t) is the survival ratio defined as N(t)/N0, N(t) and N0 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 high-pressure 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, nT(T) has only weak temperature dependence (van Boekel 2002, 2003), that is, n(T) ≈ constant =nT. In such a case, Eq. 1 can be reduced to the 2-parameter model:

  • image(2)

The exponent nT 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 nT > 1 the isothermal survival curve has downward concavity and when nT < 1 upward concavity. Log-linear inactivation, according to the model, is just a special case of Eq. 2 where nT= 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 ultra-high hydrostatic pressure, P, or a chemical preservative or disinfectant having concentration C. When the inactivation by these agents follows the Weibullian model, bT(T) and nT in Eq. 2 would be replaced by bP(P) and nP or bc(C) and nC, respectively.

The temperature dependence of bT(T) can be described by the log logistic model (for example, Corradini and others 2005; Pardey and others 2005; Peleg 2006):

  • image(3)

where Tc marks the onset of the lethal temperature regime and kT is the approximate rate (with respect to temperature) at which bT(T) rises when T >> Tc. (According to this model, at T << Tc, bT(T) ≈ 0 and at T >> Tc, bT(T) ≈k(TTc).)

A similar model can apply to high pressure or chemical preservation, in which case kT will be replaced by kP or kC and Tc by Pc or Cc, respectively (Peleg 2006). (Notice that the log-logistic model does not exclude the situation of Cc= 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:

  • image(4)

For ultra-high hydrostatic pressure preservation (HPP) it will become:

  • image(5)

and for chemical preservation:

  • image(6)

In either case, the organism's resistance to the lethal agent is characterized by 3 survival parameters, namely, nT, kT, and Tc, nP, kP, and Pc or nC, kC, and Cc 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 “iso-concentraton” log10S(t) compared with time data with Eq. 2 or one of its equivalents as a model, using nonlinear regression, will yield the bT(T), bP(P) or bC(C) value and the corresponding nT, nP, or nC. The other survival parameters; kT and Tc, kP, and Pc or kc and Cc can then be determined from the bT(T) compared with T, bP(P) compared with P or bC(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 ultra-high 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 high-temperature 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 Weibull-Log logistic (WeLL) survival model for isothermal inactivation of microorganisms and spores and (2) To demonstrate the method's potential applicability in ultra-high 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

  1. Top of page
  2. Abstract
  3. Introduction
  4. Theoretical Background
  5. Procedure
  6. Demonstration of the Method with Actual Data
  7. Concluding Remarks
  8. Acknowledgments
  9. References

Consider 3 ideal isothermal temperature profiles, T1, T2, and T3 starting at t= 0 and ending at tfinal1, tfinal2, and tfinal3, respectively, as shown in Figure 1. The corresponding logarithmic survival ratios at these points, determined experimentally, are log10S(tfinal1), log10S(tfinal2), and log10S(tfinal3). 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 log10S(0) = 0 (by definition) and that they end at the 3 experimentally determined final survival ratios log10S(tfinal1), log10S(tfinal2), and log10S(tfinal3). However, if the WeLL model with a fixed power nT 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:

  • image(7)
  • image(8)
  • image(9)
image

Figure 1—. Schematic view of how an organism or spore's 3 survival parameters can be extracted from 3 final survival ratios at the end of 3 isothermal treatments by the “3 end points method.” Note that the final points must lie on the corresponding survival curves (drawn as dotted lines), which are the solutions of the survival model equation. A similar argument applies to inactivation by HPP under isobaric conditions or by a chemical antimicrobial at constant concentration. (Also notice that if the inactivation can be characterized by 2 parameters only, 2 final survival ratios will suffice.)

Download figure to PowerPoint

These 3 simultaneous equations have 3 unknowns, namely, kT, Tc, and nT, 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 1st-order kinetics, as has been traditionally assumed, then the calculation would yield nT= 1. Or, if it is known a priori that nT= 1, then, theoretically, only 2 equations would be needed to determine the organism's 2 survival parameters, kT and Tc, 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 nT 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 nT is changed by small increments or decrements and the corresponding kT and Tc values become the solution of only 2 simultaneous equations. The values of kT and Tc so obtained together with that of the “temporary nT” 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 nT value and its corresponding kT and Tc produce a calculated survival ratio that matches the observed one within a specified tolerance, 0.01 log10 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://www-unix.oit.umass.edu/~aew2000/three_points_method/UHT_Appendices.html.

image

Figure 2—. Schematic view of how the incremental method can be used to extract survival parameters. Notice that as the assumed value of n increases, the prediction based on it and the corresponding values of the other 2 parameters will approach the correct (experimental) value.

Download figure to PowerPoint

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 nP, kP, and Pc or nC, kC, and Cc will replace nT, kT, and Tc, and P1, P2, and P3 or C1, C2, and C3, would replace T1, T2, and T3 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

  1. Top of page
  2. Abstract
  3. Introduction
  4. Theoretical Background
  5. Procedure
  6. Demonstration of the Method with Actual Data
  7. Concluding Remarks
  8. Acknowledgments
  9. 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 tfinal and corresponding log10S(tfinal) 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 nT value for each triplet of end points and records the corresponding values of kT and Tc. 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 nT, kT, and Tc values as its coefficients, against the reported experimental survival data.

The above-mentioned procedure was repeated with data on Escherichia coli inactivated under various levels of ultra-high 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 np, kp, and Pc and nc, kc, and cc were incorporated in their respective versions of the WeLL model equations, which were subsequently used to test and validate the method's applicability.

Demonstration of the Method with Actual Data

  1. Top of page
  2. Abstract
  3. Introduction
  4. Theoretical Background
  5. Procedure
  6. Demonstration of the Method with Actual Data
  7. Concluding Remarks
  8. Acknowledgments
  9. References

Isothermal inactivation

Clostridium botulinum spores The experimental survival data sets of C. botulinum spores are shown in Figure 3. They were determined at 7 constant temperatures in the range of 101 to 121 °C reported by Anderson and others (1996) and have exceptionally small scatter when compared with records from other sources. These data clearly indicate nonlinear inactivation kinetics at all 7 temperatures examined and noticeable “tailing”, that is, nT < 1 in terms of the Weibullian model (Eq. 1 or 2). To demonstrate the method's application and assess its performance, the database was divided into 7 subsets, two of them shown in Table 1. Each subset contained the final survival ratios of 6 out of the 7 isothermal treatments. These sets were further divided into several triplet temperature combinations (105, 109, and 121° C or 105, 113, and 119 °C, and so on—see the table) to calculate the magnitude of the survival parameters nT, kT, and Tc using the incremental procedure. As already explained, the procedure is based (see Figure 2) on increasing the value of nT in small steps and calculating the corresponding values of kT and Tc by solving the 2 simultaneous equations in which they appear (derived from the triplets 105, 109, and 121 °C or 105, 113, and 119 °C in the above-mentioned example, and so on). In each case, the intermediate values of kT and Tc, together with the temporary value of nT, were inserted back into the model to estimate the final survival ratio of the 3rd treatment (121 and 119 °C in the above-mentioned 2 examples) at each iteration. The procedure was halted when the resulting calculated final survival ratio matched the known value of the 3rd treatment within 0.01 log10 units, the arbitrarily predetermined tolerance. The final nT, kT, and Tc estimates obtained from 75 triplet combinations were then averaged. Their mean values—two are given in the table—were then used to generate the organism's whole survival curve at the 7th treatment, that is, the one whose final survival ratio had not been included in the parameters' determination. This procedure was repeated 7 times to cover all 7 temperatures. The resulting survival curves are shown as solid lines in Figure 3, together with the actual experimental survival ratios as reported by Anderson and others (1996). The averaged nT, kT, and Tc values of the 7 sets were themselves averaged to produce the parameters “global mean values,” that is, values representing these parameters' estimates when derived from all 7 data sets. These “global parameters” too were inserted into the WeLL model and used to produce the dashed curves shown in Figure 3. Unlike the solid curves, the dashed ones are not predictions in the strict sense of the term. This is because the final survival ratio of each of the treatments made a small but nevertheless real contribution to the calculated parameters' values. In principle, these “global means” could be used to estimate the organism's final survival ratios, or whole survival curves under other, new, treatments in the same medium at temperatures and temperature histories not yet tried. (Notice that, theoretically, with 7 data sets there are 210 triplet permutations of which half are the same in the context of this study. Because the equations' order has no effect on their solutions, the pairs 105 and 109 °C and 109 and 105 °C, when tested against 121 °C—see the first row in the table—are the same as far as the method is concerned. This still leaves 105 ways to choose the triplets. We actually examined 75 triplets and of these only 30 are listed in the table as an example.)

image

Figure 3—. Survival data of Clostridium botulinum spores inactivated at 7 constant temperatures (dots) and the predicted survival curves using averaged parameters obtained by the “3 end points method” (solid lines). Examples of calculated values using the various triplet combinations are given in Table 1. The dashed lines are the survival curves calculated with the “global parameters' values”—see table. The original experimental data are from Anderson and others (1996).

Download figure to PowerPoint

Table 1—.  The survival parameters of Clostridium botulinum spores calculated from experimental final survival ratio triplet combinations after isothermal treatments using the incremental (variable n) method and Eq. 4 as the inactivation model. The original data are from Anderson and others (1996).
Data combinationaTargetaTarget valuebObserved valueTested againstanTkT(°C−1)Tc(°C)
  1. aTemperature in °C of the final survival ratios pair taken for the iterative parameter calculation (1st column) in an attempt to match the final survival ratio at the “target temperature” given in the 2nd column.

  2. bThe “target value” is the final logarithmic survival ratio at the third, “target temperature,” given in the 2nd column, which was not used in the 2 equations solution.

105 and 109121−5.19−5.22 0.400.32103
105 and 113119−5.00−4.96 0.330.28101
105 and 117121−5.19−5.20 0.300.37101
105 and 119121−5.19−5.19 0.250.25 99
105 and 121119−5.00−5.00 0.280.26100
109 and 113105−3.81−4.28 0.580.27105
109 and 117113−5.59−5.581010.250.21 95
109 and 119113−5.59−5.58 0.250.21 96
109 and 121113−5.59−5.58 0.280.22 97
113 and 117119−5.00−5.00 0.350.31103
113 and 119109−5.31−5.32 0.250.21 96
113 and 121117−5.04−5.05 0.350.31103
117 and 119105−3.81−3.71 0.330.29102
117 and 121119−5.00−5.11 0.350.31103
119 and 121105−3.81−3.82 0.250.25 99
Mean    0.300.2799
101 and 109119−5.00−4.95 0.330.26100
101 and 113109−5.31−5.32 0.450.25103
101 and 117119−5.00−5.01 0.280.25 99
101 and 119121−5.19−5.19 0.220.20 96
101 and 121113−5.59−5.55 0.280.25 99
109 and 113101−3.01−3.06 0.430.25102
109 and 117101−3.01−3.00 0.330.26100
109 and 119101−3.01−2.911050.330.27100
109 and 121117−5.04−5.05 0.430.31103
113 and 117119−5.00−5.00 0.350.31103
113 and 119117−5.04−5.04 0.350.31103
113 and 121119−5.00−5.00 0.380.33104
117 and 119121−5.19−5.19 0.350.30103
117 and 121113−5.59−5.60 0.380.33104
119 and 121117−5.04−5.04 0.4 0.31103
Mean    0.350.28101
Global mean    0.300.26100

Comparison of the survival curves produced by the WeLL models into which the calculated mean nT, kT, and Tc values were incorporated with the experimental observation shows very good agreement in all 7 cases. As can be seen in Figure 3, the discrepancy between the predicted and observed values rarely exceeded 0.5 log10 units. This agreement confirms both the WeLL model's validity, including its underlying assumption that nT can be considered constant, and the procedure's applicability to the data at hand. It should also be added that Anderson and others' study had not been originally intended to test the model and hence no special effort was made to treat the final ratios (or counts) differently from the rest, which apparently has not affected the method's performance. All the above does not establish the model's uniqueness though. It is quite possible, at least theoretically, that alternative inactivation models might produce predictions of similar quality derived from different sets of survival parameters.

Salmonella and feline calicivirus Examples of experimental isothermal survival data of Salmonella reported by Newcomer (2004) and of a feline calicivirus published by Buckow and others (2008) are given in Figure 4. Both show a considerable scatter of the kind frequently found in the literature, reflecting on the difficulty to obtain highly reproducible counts in certain microbial systems. (At other temperatures, not shown, the scatter was even larger.) Not surprisingly, the method's “predictions,” based on either the “local” or “global” survival parameters, which are shown in the figure as solid and dashed curves, respectively, were not as good as those of the C. botulinum spores' inactivation patterns, which are shown in Figure 3. One reason for including Figure 4 in the article was to demonstrate that the semi-logarithmic survival curves' concavity, in both the Salmonella's cells and the virus's case, was clearly evident despite the noticeable experimental scatter. This suggests that the 1st trial of the method should be with nT≠1 and not with nT= 1 as in the traditional approach, where 1st-order inactivation kinetics is assumed. The other and more important reason was to suggest that especially in systems that tend to produce scattered survival data, such as when the medium is a real food for example, it might be more prudent to focus on the final survival ratios instead of attempting to determine the complete survival curves. In other words, accurate determination of the first and final counts might be a more effective way to obtain the targeted organism's survival parameters than trying to record the survival curves in their entirety. We hope that support for this proposition will come from other researchers after the method is tried with additional organisms.

image

Figure 4—. Examples of survival data of Salmonella typhimurium cells and feline calicivirus inactivated at 3 constant temperatures (dots) and the predicted survival curves using averaged parameters obtained by the “3 end points method” (solid lines). The dashed lines are the survival curves calculated with the same model but with the “global parameters.”

Download figure to PowerPoint

Isobaric inactivation

Five experimental survival data of E. coli ATCC 25922 treated by ultra-high pressure in the range of 200 to 400 MPa at 15 °C are shown in Figure 5. The results shown in the figure were reported by Koseki and Yamamoto (2007) who kindly shared with us their original data files to facilitate the analysis. To demonstrate the method's application and assess its performance with isothermal HPP data, the database was divided into 5 sets as shown in Table 2. Each set had the final survival ratios of 4 out of the 5 isobaric treatments. These sets were further divided into 6 triplet combinations (250, 300, and 350 MPa or 250, 350, and 300 MPa, and so on—see the first 2 rows in the table), covering all 30 theoretical possibilities. Each triplet was then used to calculate the magnitude of the survival parameters nP, kP, and Pc with the incremental method. It is based, as previously explained, on changing the value of nP in small increments and calculating the corresponding values of kP and Pc at each iteration. This was done, as before, by solving 2 simultaneous equations (derived from 250 and 300 MPa or 250 and 350 MPa in the above-mentioned example, and so on). The intermediate values of kP and Pc so calculated, together with that of nP, were then inserted back into model's equation to estimate the final survival ratio of the 3rd treatment at each iteration. The procedure was halted when the calculated test final survival ratio matched the known value of the 3rd treatment (350 and 300 MPa in the above-mentioned example) within the same predetermined tolerance. The final np, kp, and Pc estimates obtained from each of the 6 triplet combinations were then averaged. This was repeated 5 times, producing 5 averaged values of the survival parameters, one from each isobaric set. Four such mean values—see table—were then used to generate the organism's whole survival curve during the fifth treatment, that is, the one whose final survival ratio had not been included in the survival parameters' determination. The predicted survival curves generated with the WeLL model into which the averaged parameters had been inserted are presented as solid lines in Figure 5. They are shown superimposed on the actual experimental survival ratios that Koseki and Yamamoto (2007) reported. All the nP, kP, and Pc values listed in Table 2 were also averaged to produce their “global mean values.” As in the case of heat treatments previously discussed, the “global mean values” represent the survival parameters' estimates produced by the entire data set. They too were incorporated into the WeLL model to generate the dashed curves shown in the figure. Again, unlike the solid curves, the dashed ones are not predictions in the strict sense of the term because the final survival ratio of each of the treatments made a partial contribution to the calculated parameters' values. Nevertheless, these “global means” could be used to estimate the organism's final survival ratios, or whole isobaric survival curves in the same medium under pressure levels not yet tried.

image

Figure 5—. Survival data of Escherichia coli cells inactivated during 5 isobaric ultra-high pressure treatments at 15 °C (dots) and the predicted survival curves generated by the “3 end points method” with the averaged parameters (solid lines). The survival parameters' values calculated from the various triplet combinations are summarized in Table 2. The dashed lines are the survival curves calculated with the “global parameters' values”—see table. The original experimental data are from Koseki and Yamamoto (2007).

Download figure to PowerPoint

Table 2—.  The survival parameters of Escherichia coli ATCC 25922 calculated from experimental final survival ratio triplet combinations after isobaric treatments using the incremental (variable n) method and Eq. 5 as the inactivation model. The original data are from Koseki and Yamamoto (2007).
Data combinationaTargetaTarget  valuebObserved valuebTested againstanPkP(MPa−1)Pc(MPa)
  1. aPressure in MPa of the final survival ratios pair taken for the iterative parameter calculation (1st column) in an attempt to match the final survival ratio at the “target pressure” given in the 2nd column.

  2. bThe “target value” is the final logarithmic survival ratio at the third, “target pressure,” given in the 2nd column, which was not used in the 2 equations solution.

250 and 300350−6.28−6.26 0.6 0.02278
250 and 350300−5.53−5.54 0.6 0.02278
250 and 400350−6.28−6.03 0.2 0.03198
300 and 350400−7.98−7.952000.6 0.02278
300 and 400250−2.8 −3.02 0.1 0.01385
350 and 400300−5.53−5.51 0.250.02170
Mean    0.540.02234
200 and 300350−6.28−6.29 0.8 0.02322
200 and 350300−5.53−5.52 0.8 0.02322
200 and 400350−6.28−6.02 0.8 0.03199
300 and 350400−7.98−7.952500.250.02168
300 and 400350−6.28−6.29 0.250.02198
350 and 400200−1.39−1.28 0.3 0.02231
Mean    0.430.02240
200 and 250350−6.28−6.32 0.3 0.02218
200 and 350250−2.8 −2.82 0.250.03208
200 and 400350−6.28−6.02 0.2 0.03199
300 and 350200−1.39−1.403000.3 0.02218
300 and 400350−6.28−6.03 0.2 0.02198
350 and 400200−1.39−1.28 0.3 0.02231
Mean    0.260.02212
200 and 250300−5.53−5.58 0.450.02247
200 and 300250−2.8 −2.78 0.450.02247
200 and 400250−2.8 −2.78 0.2 0.03199
250 and 300200−1.39−1.413500.450.02247
250 and 400200−1.39−1.42 0.2 0.03198
350 and 400200−1.39−2.30 0.5 0.02242
Mean    0.380.02230
200 and 250350−6.28−6.32 0.3 0.02218
200 and 300250−2.8 −2.78 0.450.02247
200 and 350250−2.8 −2.82 0.250.03208
250 and 300200−1.39−1.414000.450.02246
250 and 350200−1.39−1.40 0.3 0.02218
300 and 350250−2.8 −2.78 0.6 0.02278
Mean    0.390.02236
Global mean    0.400.02234

Comparison of the reconstructed survival curves using the WeLL model with the actual ones shows general agreement within less than 0.5 log10 units in only 3 out of the 5 cases. The largest observed gap in the other 2 cases was 1 or 2 log10 units. The experiments reported by Koseki and Yamamoto (2007) had not been originally designed to test the model and hence no effort was made to improve the last points' accuracy by additional replicates, for example. Also, the number of data points, primarily dictated by technical and logistic considerations, was rather small and their scatter, as is common in such experiments, rather large. In light of these inevitable imperfections of the database, the observed level of agreement between predictions and observations tends to confirm the WeLL model's validity and the method's applicability. Although the method's performance was far from ideal, it could still yield 1st-order estimation of the survival parameters from the final survival ratios alone. Computer simulations have shown that the parameters' magnitude is highly sensitive to deviations in the final ratios used for their calculation (Peleg and others 2008). Here, this sensitivity is reflected in the parameters' range when determined with different triplet combinations as shown in Table 2. However, the model based on the averaged parameters was robust enough to be predictive, at least within the reported pressures' range.

Chemical inactivation

Experimental survival data sets of Pseudomonas aeruginosa treated with hydrogen peroxide in the range of 50 to 200 millimole per liter at ambient temperature, reported by Lambert and others (1999) are shown in Figure 6. To extract the survival parameters, they were divided into 4 subsets, each containing the final survival ratios of only 3 treatments. The result was an array of 12 different triplets, the theoretical maximum, as shown in Table 3. Each subset of 3 triplet combinations was used to estimate the corresponding nc, kc, and Cc values with the already described incremental procedure. As before, the 3 values of nc, kc, and Cc so calculated were averaged—see table—and the mean values were used to generate the entire survival curve of the 4th treatment whose final survival ratio had not been used in the parameter's calculation. The survival curves so created are shown as solid lines in Figure 6. As already explained, these curves are true predictions in the scientific sense and hence could be used for the model's validation. In all 4 trials, the match between the model's prediction and actual results was fairly close by microbial standards, that is, the gap between the experimental and predicted survival ratios was almost invariably within 0.5 log10 units or less.

image

Figure 6—. Survival data of Pseudomonas aeruginosa cells inactivated by hydrogen peroxide at 4 levels (dots) and the survival curves predicted by the “3 end points method” using averaged parameters (solid lines). The survival parameters' values calculated from the various triplet combinations are summarized in Table 3. The dashed lines are the survival curves calculated with the “global parameters' values”—see table. The original experimental data are from Lambert and others (1999).

Download figure to PowerPoint

Table 3—.  The survival parameters of Pseudomonas aeruginosa calculated from experimental final survival ratio triplet combinations after treatments with constant concentrations of hydrogen peroxide using the incremental (variable n) method and Eq. 6 as the inactivation model. The original data are from Lambert and others (1999).
Data combinationaTargetaTarget valuebObserved valuebTested againstanckc(L/mmol)Cc(mmol l−1)
  1. aConcentration in mmol/L of the final survival ratios pair taken for the iterative parameter calculation (1st column) in an attempt to match the final survival ratio at the “target concentration” given in the 2nd column.

  2. bThe “target value” is the final logarithmic survival ratio at the third, “target concentration,” given in the 2nd column, which was not used in the 2 equations solution.

100 and 150200−6.69−6.69 0.610.012191
100 and 200150−3.91−4.26500.610.012191
150 and 200100−2.58−2.59 0.610.012192
Mean    0.610.012191
50 and 150200−6.69−6.70 0.590.013183
50 and 200150−3.91−4.231000.590.013183
150 and 20050−1.45−1.49 0.590.012183
Mean    0.590.013183
50 and 100200−6.69−6.75 0.490.014142
50 and 200100−2.58−2.571500.500.013147
100 and 20050−1.45−1.48 0.520.013154
Mean    0.500.013148
50 and 100150−3.91−4.49 0.730.012227
50 and 150100−2.58−2.582000.730.012227
100 and 15050−1.45−1.45 0.730.012227
Mean    0.730.012227
Global mean    0.610.013187

Here too, the original experiments had not been designed to test the 3-point model. Thus, the possibility that one or two of the final ratios had a slight experimental error in it cannot be ruled out. As has been already mentioned, the magnitude of the survival parameters calculated by the 3-point's method is sensitive to deviations in the final survival ratios. Therefore, had there been substantial experimental errors in the final ratios that Lambert and others (1999) reported, then even superficial agreement between the predicted and reported survival curves would have been very improbable.

Table 3 also lists the “global mean” of the organism's survival parameters calculated by averaging the values obtained from all 4 treatments. These, as before, were used to generate the dashed survival curves shown in Figure 6. Again, strictly speaking, the dashed curves are not true predictions in the scientific sense because they were generated with parameters derived from data that included the particular treatment. However, as in the heat and pressure treatments, the global means can be considered as representing the result of the entire analysis, and in this capacity could have been used to estimate survival patterns in new disinfection treatments with hydrogen peroxide as an agent. Figure 6 demonstrates that the “predictions” constructed with the “global parameters” were not dramatically different from those produced with the “local” means. This might not have been the case had more accurate final survival ratios been available, especially if obtained in experiments especially designed for the purpose, that is, with the final survival ratios determined in more replicates.

Concluding Remarks

  1. Top of page
  2. Abstract
  3. Introduction
  4. Theoretical Background
  5. Procedure
  6. Demonstration of the Method with Actual Data
  7. Concluding Remarks
  8. Acknowledgments
  9. 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, ultra-high pressure, and chemical, having different characteristic time scales. The underlying assumption that the Weibullian-Log 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 log10 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 log-linear as the traditional 1st-order kinetic model demands. Had it been even close, the calculated WeLL model's concavity parameter (nT, nP, or nC) 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 semi-logarithmic 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 log-linear 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.
TreatmentOrganismMethod of calculationnTkT(°C−1)Tc(°C)Data source
HeatC. botulinum (spores)Three points0.300.26100Anderson and others (1996)
Regression0.330.28101 
HeatSalmonella entericaThree points0.310.14 64Newcomer (2004)
Regression0.420.15 71 
HeatFeline calicivirusThree points0.550.29 53Buckow and others (2008)
Regression0.500.30 53 
 
   nPkP(MPa−1)Pc(MPa) 
 
PressureE. coli ATCCThree points0.400.02237Koseki and Yamamoto (2007)
Regression0.32 0.022210 
 
   nCkC(L/mmol)Cc(mmol l−1) 
 
Hydrogen peroxidePseudomonas aeruginosaThree points0.610.013187Lambert and others (1999)
Regression0.690.011225 

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 3-points 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 “iso-concentration” 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 3-parameter 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.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. Theoretical Background
  5. Procedure
  6. Demonstration of the Method with Actual Data
  7. Concluding Remarks
  8. Acknowledgments
  9. References

The authors acknowledge the contribution of the Massachusetts Agricultural Experiment Station at Amherst and the New Jersey Experiment Station at New Brunswick. The authors also express their sincere thanks to Professor Koseki for sharing his group's experimental data on HPP inactivation.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Theoretical Background
  5. Procedure
  6. Demonstration of the Method with Actual Data
  7. Concluding Remarks
  8. Acknowledgments
  9. References
  • Alvarez I, Raso J, Sala FJ, Condon S. 2003. Inactivation of Yersinia enterocolitica by pulsed electric fields. Food Microbiol 20:691700.
  • Anderson WA, McClure PJ, Baird-Parker AC, Cole MB. 1996. The application of a log-logistic model to describe the thermal inactivation of Clostridium botulinum 213B at temperatures below 121.1 degrees C. J Appl Bacteriol 80:28390.
  • Augustin JC, Carlier V, Rozier J. 1998. Mathematical modeling of the heat resistance of Listeria monocytogenes. J Appl Microbiol 84:18591.
  • Buckow R, Isbarn S, Knorr D, Heinz V, Lehmacher A. 2008. Predictive model for inactivation of feline calicivirus, a norovirus surrogate, by heat and high hydrostatic pressure. Appl Environ Microbiol 74:10308.
  • Chen HQ, Hoover DG. 2003. Pressure inactivation kinetics of Yersinia enterocolitica ATCC 35669. Int J Food Microbiol 87:16171.
  • Corradini MG, Peleg M. 2004. Demonstration of the Weibull-Log logistic survival model's applicability to non isothermal inactivation of E. coli K12 MG1655. J Food Prot 67:261721.
  • Corradini MG, Normand MD, Peleg M. 2005. Calculating the efficacy of heat sterilization processes. J Food Engr 67:5969.
  • Corradini MG, Normand MD, Peleg M. 2008. Prediction of an organism's inactivation patterns from three final survival ratios determined at the end of three non-isothermal heat treatments. Int J Food Microbiol 126:98111.
  • [FDA] Food and Drug Administration. 2000. Kinetics of microbial inactivation for alternative food processing technologies. Available from: http://www.cfsan.fda.gov/~comm/ift-over.html. Accessed Oct 22, 2008.
  • Fernández A, Salmeron C, Fernández PS, Martínez A. 1999. Application of the frequency distribution model to describe the thermal inactivation of two strains of Bacillus cereus. Trends Food Sci Technol 10:15862.
  • Geeraerd AH, Valdramidis VP, Bernaerts K, Debevere J, Van Impe JF. 2004. Evaluating microbial inactivation models for thermal processing. In: RichardsonP, editor. Improving the thermal processing of foods. Cambridge , U.K. : Woodhead Publishing. p 42753.
  • Geeraerd AH, Valdramidis VP, Van Impe JF. 2005. GInaFit, a freeware tool to assess non-linear microbial survival curves. Int J Food Microbiol 102:95105.
  • Haas CN, Joffe J. 1994. Disinfection under dynamic conditions—modification of Hom model for decay. Environ Sci Technol 28:13679.
  • Haas CN, Joffe J, Anmangandla U, Jacangelo JG, Heath M. 1996. Water quality and disinfection kinetics. J Am Water Works Assoc 88:95103.
  • Heinz V, Knorr D. 1996. High pressure inactivation kinetics of Bacillus subtilis cells by a three-state-model considering distributed resistance mechanisms. Food Biotechnol 10:14961.
  • Hu XP, Mallikarjunan P, Koo J, Andrews LS, Jahncke ML. 2005. Comparison of kinetic models to describe high pressure and gamma irradiation used to inactivate Vibrio vulnificus and Vibrio parahaemolyticus prepared in buffer solution and in whole oysters. J Food Prot 68:2925.
  • Holdsworth SD. 1997. Thermal processing of packaged food. London , U.K. : Blackie Academic.
  • Holdsworth SD, Simpson R. 2007. Thermal processing of packaged foods. New York : Springer.
  • Jay JM. 1996. Modern food microbiology. New York : Chapman & Hall.
  • Koseki S, Yamamoto K. 2007. A novel approach to predicting microbial inactivation kinetics during high pressure processing. Int J Food Microbiol 116:27582.
  • Lambert RJW, Johnston MD, Simons EA. 1999. A kinetic study of the effect of hydrogen peroxide and peracetic acid against Staphylococcus aureus and Pseudomonas aeruginosa using Bioscreen disinfection method. J Appl Microbiol 87:7826.
  • Mafart P, Couvert O, Gaillard S, Leguerinel I. 2002. On calculating sterility in thermal preservation methods: application of the Weibull frequency distribution model. Int J Food Microbiol 72:10713.
  • Mattick KL, Jørgensen F, Wang P, Pound J, Vandeven MH, Ward LR, Legan JD, Lappin-Scott HM, Humphrey TJ. 2001. Effect of challenge temperature and solute type on heat tolerance of Salmonella serovars at low water activity. Appl Environ Microbiol 67:412836.
  • McKellarR, LuX. editors. 2004. Modeling microbial responses on foods. Boca Raton , Fla. : CRC Press.
  • Newcomer C. 2004. Modeling and risk assessment and the death kinetics of Salmonella. Masters thesis. New Brunswick , N.J. : Rutgers Univ.
  • Pardey KK, Schuchmann HP, Schubert H. 2005. Modelling the thermal inactivation of vegetative microorganisms. Chemie Ingenieur Technik 77:84152.
  • Peleg M. 2006. Advanced quantitative microbiology for food and biosystems: models for predicting growth and inactivation. Boca Raton , Fla. : CRC Press.
  • Peleg M, Cole MB. 1998. Reinterpretation of microbial survival curves. Crit Rev Food Sci Nutr 38:35380.
  • Peleg M, Normand MD, Corradini MG. 2005. Generating microbial survival curves during thermal processing in real time. J Appl Microbiol 98:40617.
  • Peleg M, Normand MD, Corradini MG, Van Asselt A, Ter Steeg PMC. 2008. Estimating the heat resistance parameters of bacterial spores from their survival ratios at the end of UHT and other heat treatments. Crit Rev Food Sci Nutr 48:63448.
  • Rodrigo D, Martinez A, Harte F, Barbosa-Canovas GV, Rodrigo M. 2001. Study of inactivation of Lactobacillus plantarum in orange-carrot juice by means of pulsed electric fields: comparison of inactivation kinetics models. J Food Prot 64:25963.
  • Valdramidis VP, Geeraerd AH, Bernaerts K, Van Impe JF. 2004. Dynamic versus static thermal inactivation: the necessity of validation some modeling and microbial hypotheses. Proc 9th Int Conf Eng & Food (ICEF 9). Montpellier , France . Societé de Chimie Industrielle, Paris , France . (CD-ROM). Paper 434.
  • Valdramidis VP, Geeraerd AH, Bernaerts K, Van Impe JF. 2006. Microbial dynamics versus mathematical model dynamics: the case of microbial heat resistance induction. Innov Food Sci Emerg Technol 7:807.
  • Van Boekel MAJS. 2002. On the use of the Weibull model to describe thermal inactivation of microbial vegetative cells. Int J Food Microbiol 74:13959.
  • Van Boekel MAJS. 2003. Alternate kinetics models for microbial survivor curves, and statistical interpretations, (Summary of a presentation at IFT Research Summit on Kinetic models for microbial survival). Food Technol 57:412.
  • Van Opstal I, Vanmuysen SCM, Wuytack EY, Masschalck B, Michiels CW. 2005. Inactivation of Escherichia coli by high hydrostatic pressure at different temperatures in buffer and carrot juice. Int J Food Microbiol 98:17991.