Food and Bioprocess Technology

, Volume 10, Issue 12, pp 2208–2225

# 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

## Abstract

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.

## Keywords

Microbial inactivation Maximum inactivation rate Shoulder parameter Temperature effects

## Abbreviations

BF

Bias factor

CFU

Colony-forming unit

SHW

Standardised half width

SSE

Sum of squares of residuals

SST

Total sum of squares

## Nomenclature

$${A}_{{\mathrm{Arr}}_i}$$

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)

df

Degrees of freedom

D

Decimal reduction time (s; min)

e

Residual

Ea

Activation energy (J mol−1)

f(xi,θ)

Mathematical function with θ parameters, evaluated at x i

F

F distribution

k

Inactivation rate (s−1; min−1)

L

Lag or shoulder parameter (s; min)

n

Number of experimental points

nr

Number of replicates

N

p

Number of model parameters

R

Gas constant (8.314 J mol−1 K−1)

R2

Coefficient of determination

t

Time (s; min)

T

Temperature (K; °C)

x

Independent variable

y

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

z

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

z´

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

## Greek Symbols

α

Significance level (%)

ε

Experimental error

ρ

Correlation coefficient between parameters estimates

θ

Vector of model parameters

## Subscripts

aver

Average value

i

at observation i

inact

Inactivation

max

Maximum value

min

Minimum value

pred

Predicted by the model

pure

Related to pure error

r

at replicate r

ref

Reference value

res

Residual value

0

Initial value

## Notes

### Acknowledgments

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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## 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