Parameter estimation for dynamic microbial inactivation: which model, which precision? K.D. Dolan a, b, * , V.P. Valdramidis c , D.K. Mishra a, d a Department of Biosystems & Agricultural Engineering, Michigan State University, East Lansing, MI 48824, USA b Department of Food Science & Human Nutrition, Michigan State University, East Lansing, MI 48824, USA c Department of Food Studies & Environmental Health, Faculty of Health Sciences, University of Malta, Msida 2080, Malta d Néstle Nutrition, Fremont, MI, USA article info Article history: Received 2 January 2012 Received in revised form 22 April 2012 Accepted 16 May 2012 Keywords: Microbial inactivation modeling Parameter estimation Non-isothermal Reference temperature Sublethal Sequential estimation abstract Ordinary least squares (OLS) one-step regression and the sequential procedure were applied to estimate the dynamic thermal microbial inactivation parameters of Escherichia coli K12 using the differential form of five different models. The best-performing models based on their statistical assessment were, in order: Geeraerd et al. sublethal (7 parameters), Geeraerd et al. stress adaptive (7 parameters); reduced Geeraerd et al. (6 parameters), Weibull (6 parameters), and the first-order model (5 parameters) all integrated with the secondary Bigelow model. The statistics used to evaluate the models were: lowest AIC c , minimum root mean square error (RMSE); distribution of residuals; asymptotic relative errors of parameters; scaled sensitivity coefficients; and sequential estimation. RMSE for the first-order model was more than twice that for Geeraerd et al. sublethal model, showing that the first-order model was inappropriate for these data. The optimum reference temperature (T ref ) for the secondary model (Bigelow type) was interpolated by estimating all other parameters for different fixed T ref values, and choosing T ref that minimized the correlation coefficient between AsymD ref and z. The advantage of finding the optimum T ref was that it minimized the relative error for AsymD ref . Scaled sensitivity coefficients of the Geeraerd et al. sublethal model revealed that a) none of the parameters was linearly correlated with others, and b) that the most easily estimated parameters were the three initial microbial concentrations logN(0), followed by AsymD ref , z, logC c (0), and sublethal b. The sequential method was also applied to estimate updated parameter values by successively adding each data point. Sequential results showed that each parameter reached a constant after w2.5 log reductions. These results show that a) parameters may be affected by rate of heating, and b) dynamic microbial inactivation parameters can be estimated accurately and precisely, directly from few experiments, potentially eliminating the need to apply isothermal parame- ters to dynamic industrial processes. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Transposition of results obtained from static to dynamic conditions has shown that adjustment of the initial mathematical structure is required (Bernaerts, Servaes, Kooyman, Versyck, & Van Impe, 2002; Dolan, 2003). A similar study has illustrated that inactivation model equations and their associated parameter values obtained under static acid stress conditions cannot be used directly for predicting inactivation under dynamic conditions limiting the value and reliability of the developed mathematical tools (Janssen et al., 2008). Dolan (2003) and Valdramidis, Geeraerd, Bernaerts, and Van Impe (2008) have also highlighted that even if the results are excellent by the use of isothermal inactivation param- eters one does not know the actual values of non-isothermal esti- mates. These observations pinpoint the importance of further studying parameter identification techniques under dynamic conditions representative of a realistic (processing) environment. Several microbial studies revealed that microbial adaptations are evident at different types of stressful environments (e.g., Skandamis, Stopforth, Yoon, Kendall, & Sofos, 2009; Valdramidis, Geeraerd, & Van Impe, 2007; Velliou et al., 2011). These environ- ments have an impact on the physiological state of the microor- ganisms and can result in an increase of their microbial resistance or a change to their adaptation time. Recent investigators tried to * Corresponding author. Department of Food Science & Human Nutrition, Mich- igan State University,135 Trout Food Science Building, East Lansing, MI 48824, USA. Tel.: þ1 517 355 8474x119; fax: þ1 517 353 8963. E-mail addresses: dolank@msu.edu (K.D. Dolan), vasilis.valdramdis@um.edu.mt (V.P. Valdramidis), Dharmendra.Mishra@rd.nestle.com (D.K. Mishra). Contents lists available at SciVerse ScienceDirect Food Control journal homepage: www.elsevier.com/locate/foodcont 0956-7135/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.foodcont.2012.05.042 Food Control 29 (2013) 401e408