On the comparison between the performance of the Weibull/log-logistic and the log-Linear/Arrhenius survival models in sterility calculations

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  2. On the comparison between the performance of the Weibull/log-logistic and the log-Linear/Arrhenius survival models in sterility calculations
  3. References

The authors of a recent article published in the Journal of Food Science entitled “Uncertainty in thermal process calculation due to variability in first-order and Weibull kinetic parameters” have concluded that “due to the large effect of variability in the input parameters on the final log reduction, the effort to move toward more accurate kinetic models needs to be weighed against inclusion of variability” (Halder and others 2007). They have reached this conclusion after Monte Carlo simulations of non-isothermal survival curves have shown that “with the same percent variability in [the] kinetic parameters, uncertainty in the final log reduction for [the] Weibull kinetics was smaller or equal to that for first-order kinetics”. But while this observation might be correct, the conclusion that it should be a major consideration in whether to use non-linear kinetics in sterility calculations is not.

When the isothermal survival curve of a microorganism or spore is not log-linear, forcing a straight line through the curved data will introduce a systematic error, which can result in an error of several orders of magnitudes in the final survival ratio's estimate - see Figure 5 in Halder and others' paper. Thus, if the targeted organism or spore's isothermal semi logarithmic survival curves are concave upward, especially if they exhibit a flat “tail” (see Pardey and others 2005), the use of the ‘first-order kinetics’ can introduce a false sense of safety. On the other hand, if the isothermal semi logarithmic survival curves have downward concavity, significant over-processing will result, with unnecessary destruction of nutrients, inferior product quality and an increased risk of thermally produced harmful compounds (see Arnoldi 2001).

Apart from the attempt to grant legitimacy to passing a straight line through curved data, an unacceptable or discouraged practice in any other scientific field, the way the two survival models have been compared also raises some questions. The Weibullian model's b(T) and n(T) are not the equivalents of the D and z values in the conventional model. Even if n(T)=constant (different from one), then when b(T) has log-logistic temperature dependence (Peleg 2006), the number of survival parameters is three not two. [This is why the Weibullian/log-logistic and similar non-linear survival models can predict the inactivation patterns of organisms or spores that have log-linear isothermal survival curves as well as those that do not.]

The Arrhenius equation, its popularity in food research not withstanding, has several serious flaws when applied to microbial survival. Cells and spores are not gas molecules, for which the model has been originally developed. Their heat resistance in a given medium changes not only with temperature, as the Arrhenius equation requires, but also with the exposure time at that temperature. Also, only rarely does any inactivation rate parameter, based on either linear or nonlinear kinetics, vary by several orders of magnitudes in the pertinent lethal temperature regime, and therefore there is no need for their logarithmic conversion. The same can be said about the temperature scale's compression through the transformation to degrees K reciprocals. It too serves no useful purpose. And as to the supposedly temperature-independent “energy of activation”, what is a ‘mole of bacterial spores’? Some 200,000 metric tons or so as a simple calculation will show? And what does the Universal Gas Constant have to do with the biophysical/biochemical processes that cause a cell's death or a spore's inactivation?[For more on the problems created by the use of the Arrhenius equation in microbiology, see Peleg 2006.] There has been an attempt to anchor the first order inactivation kinetics to the “Maxwell-Boltzman distribution”, which too had been originally developed for elastically colliding gas molecules at thermal equilibrium. Equilibrium conditions simply do not exist in cells or spores undergoing heat inactivation, isothermal or non-isothermal.

The real major consideration should be whether in the 21st century universities in the United States and around the world continue to teach a defunct theory of microbial survival kinetics and whether the food and pharmaceutical industries continue to use sterility criteria based on passing straight lines through curved data.

Uncertainties in the survival parameters magnitudes will always exist and simulation of the kind reported by Halder and others (2007) can be a useful tool to assess their safety implications. The same can be said about uncertainties and variations in the temperature histories of the individual cans. Both should be used to establish rational safety factors in thermal processing. [The idea to investigate the potential role of non-linear kinetics in the safety of heat-processed foods, including that of uncertainties in their parameters' magnitude, was proposed about 10 years ago. It was rated by the Food Safety Program of the USDA-NRICGP as ‘unworthy of support even if [plenty] of funds were available’!]

The issue before the Food Science community is whether the non-linear characters of microbial inactivation should be acknowledged and taken into account in sterility calculations, now and in the future. Halder and others' suggestion that nonlinear semi logarithmic survival curves might be a result of experimental errors and artifacts (Halder and others 2007) is based on circular reasoning, akin to the classic ‘begging the question fallacy’. The ‘argument’ can be reversed. One can equally claim that those who have reported first-order kinetics inactivation did not detect the semi-logarithmic survival curves' curvature because of experimental errors and artifacts. [Or maybe bias has also played a role?].

Free software based on the Weibullian/log-logistic model to do sterility calculations (with MS Excel®), are already available on the web – see: The posted programs allow the user to generate the temperature profile or paste his or her own time-temperature data and modify the targeted organism's survival parameters. [This also enables the user to assess the role of uncertainties of various kinds, including in the survival parameters.] Each program can generate the survival ratio in real time and it calculates, automatically, the equivalent time at any chosen reference temperature like the standard 121.1C (250F) – see Corradini and others 2006 and Peleg 2006. Using these programs is no more complicated than to calculate the dubious ‘F0 value’. [Although rarely acknowledged, the ‘F0 value’ cannot be directly translated into a survival ratio if either the isothermal survival curves are not all log-linear or even if only the ‘log k vs. T’ (not vs. 1/T) is non-linear, as the traditional methods require – see Datta 1993 and Peleg 2006.] Canned foods are safe to eat because they are grossly over-processed, not because the ‘first order kinetic theory’ is correct or the Arrhenius equation applicable (Peleg 2006). With the wide availability of non-linear regression programs, there is no reason to continue to use the log-linear inactivation model when the microorganisms and spores repeatedly tell us that it is invalid (see van Boekel 2002, Fernandez and others 1999, Periago and others 2004, Pardey and others 2005, for example, and many others. Just open Google Scholar under ‘Weibull microbial inactivation’ and see for yourself). With today's computer capabilities and user-friendly software based on non-linear kinetics, there is no justification to use methods known to be incorrect. It is therefore surprising that there is so much opposition to the introduction of new methods to calculate sterility not only from a conservative industry but also from respectable academic institutions as Halder and others' paper demonstrates.


  1. Top of page
  2. On the comparison between the performance of the Weibull/log-logistic and the log-Linear/Arrhenius survival models in sterility calculations
  3. References
  • Arnoldi A. 2001. Thermal processing and food quality: analysis and control. In: RichardsonP, Editor. Thermal Technologies in Food Processing. Woodhead CRC, Cambridge , U.K.
  • Corradini MG, Normand MD, Peleg M. 2006. On expressing the equivalence of non-isothermal and isothermal heat sterilization processes. J Sci Food Agric 86:78592.
  • Datta AK. 1993. Error estimates for approximate kinetic parameters used in the food literature. J Food Eng 18:18199.
  • Fernandez A, Salmeron C, Fernandez PS, Martinez A. 1999. Application of the frequency distribution model to describe the thermal inactivation of two strains of Bacillus cereus. Trends Food Sci Tech 10:15862.
  • Halder A, Datta AK, and Geedipalli SSR. 2007. Uncertainty in Thermal Process Calculations due to Variability in First-Order and Weibull Kinetic Parameters. J Food Sci 72:E15567.
  • Pardey KK, Schuchmann HP, and Schubert H. 2005. Modeling the thermal inactivation of vegetative microorganisms [in German]. Chem Ingen Technik 77:84151.
  • Peleg M. 2006. Advanced Quantitative Microbiology for Food and Biosystems: Models for Predicting Growth and Inactivation. CRC Press, Boca Raton , Fla.
  • Periago PM, Van Zuijlen A, Fernandez PS, Klapwijk PM, Ter Steeg PF, Corradini MG, and Peleg M. 2004. Estimation of the non-isothermal inactivation patterns of Bacillus sporothermodurans IC4 spores in soups from their isothermal survival data. Intl J Food Microbiol 95:20518.
  • Van Boekel MAJS. 2002. On the use of the Weibull model to describe thermal inactivation of microbial vegetative cells. Intl J Food Microbiol 74:13959.