Food and Bioprocess Technology

, Volume 10, Issue 12, pp 2208–2225 | Cite as

Mathematical Models for Prediction of Temperature Effects on Kinetic Parameters of Microorganisms’ Inactivation: Tools for Model Comparison and Adequacy in Data Fitting

  • Maria M. Gil
  • Fátima A. Miller
  • Teresa R. S. Brandão
  • Cristina L. M. Silva
Original Paper


Microbial inactivation often follows a sigmoidal kinetic behaviour, with an initial lag phase, followed by a maximum inactivation rate period and tending to a final asymptotic value. Mathematically, such tendencies may be described by using primary kinetic models (Gompertz based model is one example) that describe microbial survival throughout processing time when stressing conditions are applied. The parameters of kinetic models are directly affected by temperature. Despite the number of mathematical equations used to describe the dependence of the kinetic parameters on temperature (so-called secondary models), there is a lack of studies regarding model comparison and adequacy in data fitting. This work provides a review of mathematical models that describe the temperature dependence of kinetic parameters related to microbial thermal inactivation. Regression analysis schemes and tests seeking model comparison are presented. A case study is included to provide guidance for the assessment of secondary model adequacy and regression analyses procedures. When modelling temperature effects on sigmoidal inactivation kinetics of microorganisms, one should be aware about the regression methodology applied. The most adequate models according to the two-step regression methodology may not be the best selection if a global fit is applied.


Microbial inactivation Maximum inactivation rate Shoulder parameter Temperature effects 



Bias factor


Colony-forming unit


Standardised half width


Sum of squares of residuals


Total sum of squares


\( {A}_{{\mathrm{Arr}}_i} \)

Parameter i of the Arrhenius model that relates L with T (i = 1, 2)

\( {A}_{{\mathrm{Dav}}_i} \)

Parameter i of the Davey model that relates L with T (i = 1, 2, 3)

\( {A}_{{\mathrm{Ratk}}_i} \)

Parameter i of the Ratkowsky modified model that relates L with T (i = 1, 2)

\( {A}_{{\mathrm{WLF}}_i} \)

Parameter i of the Williams-Landel-Ferry model that relates L with T (i = 1, 2, 3)

\( {A}_{{\mathrm{Hip}}_i} \)

Parameter i of the hyperbole model that relates L with T (i = 1, 2, 3)

\( {C}_{{\mathrm{pel}}_i} \)

Parameter i of the Peleg model that relates k max with T (i = 1, 2, 3)

\( {C}_{{\mathrm{Ratk}}_i} \)

Parameter i of the Ratkowsky model that relates k max with T (i = 1, 2)

\( {C}_{{\mathrm{RatkM}}_i} \)

Parameter i of the Ratkowsky modified model that relates k max with T (i = 1, 2)

\( {C}_{{\mathrm{WLF}}_i} \)

Parameter i of the Williams-Landel-Ferry model that relates k max with T (i = 1, 2, 3)


Degrees of freedom


Decimal reduction time (s; min)




Activation energy (J mol−1)


Mathematical function with θ parameters, evaluated at x i


F distribution


Inactivation rate (s−1; min−1)


Lag or shoulder parameter (s; min)


Number of experimental points


Number of replicates


Microbial load (CFU mL−1)


Number of model parameters


Gas constant (8.314 J mol−1 K−1)


Coefficient of determination


Time (s; min)


Temperature (K; °C)


Independent variable


Dependent response variable; logarithm of microbial loads normalised in relation to initial values


Temperature change required to change D value by a factor of 10 (°C)


Temperature change required to change L parameter by a factor of 10 (°C)

Greek Symbols


Significance level (%)


Experimental error


Correlation coefficient between parameters estimates


Vector of model parameters



Adjusted value


Average value


at observation i




Maximum value


Minimum value


Predicted by the model


Related to pure error


at replicate r


Reference value


Residual value


Initial value



Fátima A. Miller and Teresa R.S. Brandão gratefully acknowledge Fundação para a Ciência e Tecnologia (FCT) and Fundo Social Europeu (FSE) for the financial support through the Post-Doctoral grants SFRH/BPD/65041/2009 and SFRH/BPD/101179/2014, respectively. This study had the support of FCT, through the strategic project UID/MAR/04292/2013 granted to MARE and through project UID/Multi/50016/2013 granted to CBQF.


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Maria M. Gil
    • 1
  • Fátima A. Miller
    • 2
  • Teresa R. S. Brandão
    • 2
  • Cristina L. M. Silva
    • 2
  1. 1.MARE—Marine and Environmental Sciences Centre, ESTMPenichePortugal
  2. 2.CBQF—Centro de Biotecnologia e Química Fina, Escola Superior de BiotecnologiaCentro Regional do Porto da Universidade Católica PortuguesaPortoPortugal

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