Evaluation of High Pressure Processing Kinetic Models for Microbial
Inactivation Using Standard Statistical Tools and Information Theory
Criteria, and the Development of Generic Time-Pressure Functions for
Process Design
Serment-Moreno, V., Fuentes, C., Barbosa-Cánovas, G., Torres, J. A., &
Welti-Chanes, J. (2015). Evaluation of High Pressure Processing Kinetic Models
for Microbial Inactivation Using Standard Statistical Tools and Information Theory
Criteria, and the Development of Generic Time-Pressure Functions for Process
Design. Food and Bioprocess Technology, 8(6), 1244-1257.
doi:10.1007/s11947-015-1488-x
10.1007/s11947-015-1488-x
Springer
Accepted Manuscript
http://cdss.library.oregonstate.edu/sa-termsofuse
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Evaluation of high pressure processing kinetic models for microbial
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inactivation using standard statistical tools and information theory
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criteria, and the development of generic time-pressure functions for
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process design
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Author: Vinicio Serment-Moreno
Affiliation: Escuela de Ingenierías y Ciencias, Tecnológico de Monterrey, Av. Eugenio
Garza Sada 2501 Sur, Col. Tecnológico, 64849, Monterrey, NL, México
E-mail address: vsermentm@gmail.com
Author: Claudio Fuentes
Affiliation: Department of Statistics, Oregon State University, 54 Kidder Hall, Corvallis,
OR 97331, USA
Phone: 1-541-737-3587
Fax: 1-541-737-3489
E-mail address: fuentesc@stat.oregonstate.edu
Author: Gustavo Barbosa-Cánovas
Affiliation: Center for Nonthermal Processing of Food, Washington State University,
Pullman, WA 99164-6120, U. S. A.
Phone: 1-509-335-6188
Fax: 1-509-335-2722
E-mail address: barbosa@wsu.edu
Corresponding Author: José Antonio Torres
Affiliation: Food Process Engineering Group, Department of Food Science and
Technology, Oregon State University, 100 Wiegand Hall, Corvallis, OR 97331, U. S. A.
Phone: 1-541-737-4757
Fax: 1-541-737-1877
E-mail address: j_antonio.torres@oregonstate.edu
Corresponding Author: Jorge Welti-Chanes
Affiliation: Escuela de Ingenierías y Ciencias, Tecnológico de Monterrey, Av. Eugenio
Garza Sada 2501 Sur, Col. Tecnológico, 64849, Monterrey, NL, México
Phone: +52-81-8358-2000 ext. 4923
Fax: +52-81-8358-4293
E-mail address: jwelti@itesm.mx
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Abstract
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Generic kinetic models for microbial inactivation by high pressure processing (HPP) would accelerate the
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development of commercial applications. The aim of this work was to develop a generic model obtained
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by fitting peer-reviewed microbial inactivation data (124 kinetic curves) to first order kinetics (LKM),
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Weibull (WBLL), and Gompertz (GMPZ) primary and secondary models. Standard statistics (coefficient of
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determination (R ), variance, residuals plots, experimental vs predicted plots) and information theory
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criteria (Akaike Information Criteria, AIC; Akaike differences, ∆AICi, Bayesian Information Criteria)
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determined their goodness of fit. Standard statistics showed no differences between WBLL and GMPZ,
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whereas information theory criteria identified WBLL as the best model (lowest AICi value, 61.3% of
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cases). LKM performed poorly according to all statistics (e.g., ∆AICi > 10, 58.1% of cases). The
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dispersion of model parameters prevented the derivation of a secondary model for the whole dataset, but
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clear trends and sufficient data (56 kinetic curves) were found to develop one for milk. A secondary WBLL
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model (b’ = 0.056-2.230, n = 0.758-0.403; 150-600 MPa) was the best alternative (AICi = 183.8). A GMPZ
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model yielded similar predictions, but registered ∆AICi = 19.3 reflecting its larger number of parameters (p
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= 8). Selecting datasets with pressure holding times of commercial interest (t ≤ 10 min) yielded different
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parameter estimates for the generic WBLL model (b’ = 0.079-1.859, n = 1.340-0.557; 300-600 MPa). In
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conclusion, information theory criteria complemented standard statistics, and the simpler WBLL
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secondary model (p = 4) provided a product-specific time-pressure function of industrial relevance.
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Keywords: High pressure processing (HPP); microbial inactivation kinetics; first order kinetics model;
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Weibull model; Gompertz model; Akaike Information Criteria (AIC); information theory criteria
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1. Introduction
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1.1 High pressure processing microbial inactivation kinetic models
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High Pressure Processing (HPP) can deliver safe food products while preserving most of their nutritional
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and sensorial properties (Mújica-Paz et al. 2011; Heinz and Buckow 2009). Consumer acceptance of
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HPP food products keeps raising and by 2018 they are expected to represent a US$12,000 million
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market, while sales of HPP equipment will reach US$600 million (Spinner 2014). Still, the availability of
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microbial and enzymatic kinetic information remains well behind the standardized databases and
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calculation methodologies available for traditional food technologies such as thermal processing
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(Serment-Moreno et al. 2014). A mathematical model predicting microbial inactivation as a function of
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time (primary model) is an essential tool when designing HPP experiments and industrial processes.
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Furthermore, the application of a primary model can be extended if a secondary model describing the
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pressure dependence of the primary model parameters is available too.
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Santillana Farakos and Zwietering (2011) presented a methodology for the prediction of microbial
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inactivation rates as a function of pressure. Decimal reduction times (D) and z-values were estimated by
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assuming a first order inactivation rate (Eq. 1) for microbial inactivation by HPP treatments. Under this
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assumption, plots of microbial survival fraction (log10 N/N0) versus time should generate straight lines with
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a slope proportional to the inactivation rate constant (k). However, numerous authors report that linearity
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is rarely observed and that non-linear equations should be employed to model HPP microbial inactivation
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data (Bermúdez-Aguirre and Barbosa-Cánovas 2011; Campanella and Peleg 2001; Corradini et al. 2009;
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Wilson et al. 2008; Serment-Moreno et al. 2014).
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log10
N
= −k ⋅ t
N0
(1)
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The Weibull distribution used to predict the failure time of mechanical and electrical systems has been
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applied to describe microbial inactivation. In this application, the cumulative probability of the Weibull
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distribution determines the time at which microbial cells fail to survive (Eq. 2). The cumulative distribution
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function of the Weibull distribution has been widely used to describe non-linear microbial inactivation
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kinetics due to its simplicity, flexibility and semi-empirical nature. The parameter b’ is equivalent to the
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inactivation rate constant to the -nth power. The exponent n defines the concavity of the curve and the
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type of microbial population, classified as resistant (downward concavity; n > 1), low-resistant (upward
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concavity; n < 1), or homogeneous (straight line; n = 1) (Serment-Moreno et al. 2014; van Boekel 2002;
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see Chapters 2 and 6, Peleg 2006).
n
log10
N
N
1
1
t
= −b'⋅t n
=−
⋅   ; b' =
; log10
n
N0
N0
2.303  b 
2.303 ⋅ b
(2)
3
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Secondary models for the Weibull parameters include a logistic-exponential expression for parameter b’,
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indicating that the inactivation rate constant will be close to zero until a critical pressure level is reached
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(Pc; Eq. 3). If the pressure level is increased beyond Pc, b’ will increase at a rate wP. An exponential
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decay model has been used to predict pressure effects on the Weibull parameter n (Eq. 4), where the
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dimensionless parameter d0 is the value of n at pressures P approaching 0, and d1 is the rate at which n
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exponentially decays (Doona and Feeherry 2007; Peleg et al. 2002; see Chapter 6, Peleg 2006; Doona et
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al. 2007). Pilavtepe-Çelik et al. (2009) proposed a Bigelow-type model to describe the pressure
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dependence of parameter b’ (Eq. 5), where Pref is the reference pressure, b’ref is the Weibull rate constant
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at Pref, and zP is the amount of pressure needed to increase b’ by 10. Alternative Weibull secondary
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models mainly consist of empirical equations aiming to linearize the effects of pressure on the parameters
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b’ and n (Carreño et al. 2011; Chen and Hoover 2003b; Pilavtepe-Çelik et al. 2009; Serment-Moreno et
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al. 2014).
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b' (P ) = ln{1 + exp[wP ⋅ (P − PC )]}
(3)
n(P ) = d 0 ⋅ exp(− d1 ⋅ P )
(4)
b' (P ) = bref ⋅ 10
 P − Pref
−

zP





(5)
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Other alternatives for non-linear HPP microbial inactivation curves include those capable of describing
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concavity changes such as the log-logistic (Eq. 6) and the modified Gompertz models (Eq. 7) (Chen and
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Hoover 2003b; Guan et al. 2005; Saucedo-Reyes et al. 2009). However, the log-logistic model has an
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unfortunate mistake when applying the chain rule in its theoretical derivation that leads into the
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misinterpretation of its parameters (Serment-Moreno et al. 2014; Serment-Moreno et al. 2015). Another
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inconvenience of the log-logistic model (Eq. 6) is the need to approximate log10 (t = 0) with an arbitrary
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value which were chosen as -1 (Cole et al. 1993) or -6 (Chen and Hoover 2003b). More recently,
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Serment-Moreno et al. (2015) tested log10 (t = 0) from -1 to -9 to evaluate its effect on the non-linear
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regression, and reported that the log-logistic model parameter estimates varied for different
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approximation values for log10 (t = 0).
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log 10
N
=
N0
A
 4Ω

⋅ (t − log 10 t )
1 + exp 
 A

log 10 t 0 ≈ −1 or − 6
−
A
 4Ω

⋅ (t − log 10 t 0 )
1 + exp 
 A

(6)
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The modified Gompertz model has been rarely used to model the microbial inactivation kinetics by HPP
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treatments even though biological meaning can be assigned to its parameters. Parameter A represents
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the difference between the lower and upper asymptotes of microbial survival curves (log10 N/N0 vs time),
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µmax is the maximum inactivation rate, and parameter λ is the inflection point, or the time at which the
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linear portion of the curve starts. An extensive and recent literature review showed that no secondary
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model is available to describe the relationship of the log-logistic or Gompertz model parameters with
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pressure (Serment-Moreno et al. 2014).
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log10

N
 m ⋅ exp(1)

= A ⋅ exp− exp  max
⋅ (l − t ) + 1 
N0
A



(7)
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1.2 Information theory criteria for model selection
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Statistical quantities based on squared differences such as the coefficient of determination R , the mean
141
squared error (MSE), and variance estimates ( σˆ ) are widely used for the evaluation of mathematical
142
models. The coefficient of determination R alone is useful to obtain a quick assessment of a model
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performance but it is an insufficient criterion to discriminate between different models (Breiman 2001;
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Johnson and Omland 2004; van Boekel 2008). Mathematical functions that yield the highest R , and the
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2
lowest MSE or σˆ are usually chosen as the model best describing the experimental data (Chen and
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Hoover 2003a; Guan et al. 2006). However, Spiess and Neusmeyer (2010) reported that R alone is a
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poor discriminator stating that similar non-linear models displayed differences only after the third or fourth
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decimal place of the computed R values. The authors also suggested that R is better suited for linear
149
regression, and discourages R to determine the goodness of fit of nonlinear models (Spiess and
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Neumeyer 2010). Additionally, the sole use of statistical quantities based on the squared differences may
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be misleading if the model consistently over- or under- estimates the observed values, for which residuals
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plots and experimental vs plots are recommended to detect consistent model deviations (Ross 1996).
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2
2
2
2
2
2
2
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On the other hand, information theory criteria measuring the information lost when an empirical probability
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distribution given by a proposed model is used to approximate the underlying “true” distribution of a given
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dataset, provide a powerful alternative for model selection (Burnham and Anderson 2002; Stoica and
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Selén 2004). Despite its widespread application in environmental science and industrial engineering,
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information theory criteria have been rarely used for model comparisons in food science (Albert and
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Mafart 2005; Koseki et al. 2009; López et al. 2004; van Boekel 2008; Reyns et al. 2000; Coroller et al.
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2006). The Akaike Information Criteria (AIC) favors the accuracy of the model through the maximum log-
161
likelihood estimate (ℓ; Eq. 8) while penalizing the indiscriminate use of extra parameters (p; Eq. 8) when
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assessing goodness of fit among nested or non-nested models. If the underlying distribution of the
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sample data is a normal distribution, ℓ can be calculated from the data sample size (n) and the variance
5
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estimate ( σˆ ). The smaller the computed AICi value, the better that model (i = 1, 2, … M models) can
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describe a particular dataset.
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AICi = −2 + 2 p;  = −
n
ln σˆ 2
2
(8)
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The parameter AICi is likely to change with differences in the sample size and variability of dataset even
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when the models compared remain unchanged. Therefore, several authors recommend using instead the
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model AICi differences (∆AICi) calculated using the minimum AICi as a reference (Eq. 9). Furthermore,
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∆AICi values can be normalized via the log-likelihood term of AICi (Eq. 8-11) to obtain Akaike weights
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(wAICi) generating a 0-1 relative scale to help in visualizing Akaike analysis differences. As a general rule
173
of thumb (Table 1), ∆AICi values in the 0-2 range yield exp(-½ ∆AICi) values around 1.0-0.3 (Eq. 10),
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suggesting that the experimental data supports the model fit. In the case of larger ∆AICi values, Eq. 10
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generates lower values indicating that the current experimental data does not support the model, with
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∆AICi ≥ 10 having little or no weight on the normalization. For example, assuming that the hypothetical
177
model 1 has the minimum AICi value, then ∆AIC1 = 0 (Eq. 9) and exp(-½ ∆AIC1) = 1 (Eq. 10; Table 1). For
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model 2, if it is assumed that ∆AIC2 = AIC2 - AIC1 = 2 then exp(-½ ∆AIC2) = 0.368. The denominator of Eq.
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11 (exp[-½ ∆AIC2] + exp[-½ ∆AIC2]) would yield 1.000 + 0.368 = 1.368, and the corresponding Akaike
180
weights as wAIC1 = 0.731 and wAIC2 = 0.269, for which both models are considered to adequately describe
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the current experimental dataset. As the value of ∆AIC2 increases, wAIC2 tends to zero and the likelihood
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that the current dataset supports model 2 is more questionable. Another way to interpret wAICi is to
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consider wAICi as conditional probabilities according to a Bayesian framework briefly mentioned in the next
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paragraph. In this setting, the lower the value of wAICi, the less probable that the current dataset supports
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model i (Burnham and Anderson 2002; Johnson and Omland 2004; Wagenmakers and Farrell 2004;
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Burnham and Anderson 2004).
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∆AICi = AICi − AIC min

 1
 = n ln σ 2 ∝ exp − ∆AIC i 

 2
w AICi =
(9)
(10)
exp(− 12 ∆AIC i )
M
∑ exp(−
i =1
1
2
∆AIC i )
(11)
188
189
INSERT TABLE 1
190
6
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The Bayesian Information Criteria (BICi) is another alternative for model selection (Eq. 12). As in the case
192
of ΔAICi values, models yielding the lowest ΔBICi value (Eq. 13) describe best the experimental data
193
collected. The Bayesian Information Criteria is not derived from information theory criteria such as the
194
AICi, but from a Bayesian approach that assumes that one of the tested models is true. Akaike weights
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can be related to the BICi probabilities as shown in Eq. (14).
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BICi = −2 ln  + p ln n
(12)
∆BICi = BICi − BIC min
(13)
pi = Pr[g i | data ] =
exp(− 12 ∆BICi )
M
∑ exp(−
i =1
1
2
∆BICi )
=
exp(− 12 ∆AICi )
M
∑ exp(−
i =1
1
2
∆AICi )
= w AICi
(14)
197
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In this work, the possibility of developing a generic pressure-time mathematical expression capable of
199
describing HPP microbial inactivation kinetics was explored, expecting that the resultant model will
200
provide a starting point towards the standardization of HPP kinetic data. The first order kinetics, Weibull
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and Gompertz models were fit to HPP microbial inactivation data reported in peer-reviewed sources. The
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goodness of fit of kinetic models was evaluated using standard statistics (R , mean squared error values
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(MSE), residuals plots, experimental vs. model prediction value plots) and information theory criteria
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(maximum log-likelihood estimate ℓ, Akaike Information Criteria AICi, Bayesian Information Criteria BICi).
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Furthermore, secondary models describing the pressure effect on the primary model parameters were
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analyzed and proposed as an attempt to establish a much needed model predicting pressure effects on
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microbial inactivation rates.
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2. Methodology
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2.1 High pressure processing microbial inactivation data and model fit
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An exhaustive review of references reporting microbial, chemical and enzymatic HPP kinetics was
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performed to generate the kinetic database of this study, by selecting kinetic curves with ≥4 data points
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(including t = 0). The most extensive database (Table 2) was found for HPP kinetic data of aerobic
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microorganisms at ambient temperature (20-25°C), and low acidity media (pH > 4.5), totaling 124
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microbial survival curves (974 pressure-time microbial counts sets). All microbial inactivation data was
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manually digitized using Engauge (V4.1, www.freecode.com/projects/engauge), a freeware tool that has
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been used in other studies for the digitalization and analysis of previously published scientific data
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(Treseder 2008; Beauchemin et al. 2007). Data fit to the first order kinetics (LKM, Eq. 1), Weibull (WBLL,
219
Eq. 2) and modified Gompertz (GMPZ, Eq. 7) models were performed with the MATLAB “fit” function
220
(Version R2012b, www.mathworks.com/ products/matlab) using the Levenberg-Marquardt algorithm.
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INSERT TABLE 2
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2.2 Statistical analysis
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Standard statistical analysis included residuals analysis, that is the inspection of the differences between
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experimental (yobs) and predicted (yfit) values (Eq. 15), coefficient of determination (R ; Eq. 16), mean
227
squared error values (MSE; Eq. 17), and variance estimates ( σ ; Eq. 18). The theoretical information
228
criteria that complemented this statistical analysis of goodness of fit required the following approach.
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Under standard assumptions and if the model is correctly specified, the residuals (Eq. 15) should be
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distributed with a population mean µ equal 0, and a variance estimate that depends on the proposed
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model parameters obtained through non-linear regression (Eq. 18). Therefore, the maximum log-
232
likelihood estimate ℓ is calculated with Eq. 19, AICi values are determined with Eq. 20, and ∆AICi values
233
with Eq. 9 (Burnham and Anderson 2002; Stoica and Selén 2004). It should be noted that the variance
234
estimate is also considered a parameter in the calculation of AICi values, thus the number of parameters
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(p) for each model is: (a) LKM, p = 2 (k,
236
).
2
2



σ 2 ); (b) WBLL, p = 3 (b’, n, σ 2 ); (c) GMPZ, p = 4 (A, µmax, λ, σ 2
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residuals = y obs − y fit
(15)
SST SSE
∑ ( y − yˆ )
R =
−
= 1−
2
SST SST
∑ (y − y)
2
2
∑ (y
MSE =
− y fit )
2
obs
n− p
sˆ 2 =
=−
∑ (y
(16)
− y fit )
(17)
2
obs
(18)
n
n
ln σˆ 2
2
AICi = −2 + 2 p = n ln σˆ 2 + 2 p
(19)
(20)
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239
2.3 Development of a secondary model for HPP microbial inactivation kinetics in milk
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First, the datasets used in this study describing the kinetics for the microbial inactivation by HPP were
241
grouped according to the type of media (milk, juice, buffer, other) and microbial population (Listeria,
242
Escherichia, mesophiles) to explore possible common microbial inactivation patterns. Next, the following
243
equations were evaluated as secondary model alternatives: (a) Eyring model (Eq. 21) for the LKM model
244
parameter k (Katsaros et al. 2010); (b) logistic-exponential (Eq. 3) and exponential decay (Eq. 4)
245
equations for the WBLL model parameters b’ and n, respectively (Peleg 2006); (c) Bigelow type model
246
(Eq. 5) and exponential decay (Eq. 4) equations for the WBLL model parameters b’ and n; (d) an
8
247
exponential sigmoidal equation (Eq. 22) for parameter A, an exponential decay equation (Eq. 23) for the
248
parameter µmax; and an exponential decay equation (Eq. 24) for the parameter λ in the GMPZ model
249
proposed in this study. Finally, the parameters obtained for the secondary model equations were used as
250
initial values to perform a one-step non-linear regression analysis in which primary and secondary models
251
were integrated into one equation.
252
 ∆V ≠ ⋅ (P − Pref )
k (P ) = k 0 ⋅ exp −

R ⋅T


(21)
A(P ) = q1 ⋅ exp[− exp(q 2 − P ⋅ q 3 )]
(22)
m max (P ) = f1 ⋅ exp(− f 2 ⋅ P )
(23)
λ (P ) = g1 ⋅ exp(− g 2 ⋅ P )
(24)
253
254
3. Results and Discussion
255
3.1 Goodness of fit analysis of primary models with standard statistical criteria
256
The WBLL and GMPZ models gave the best data fit yielding R ≥ 0.9 on 90% of the 124 kinetic curves
257
tested in this study. The comparison of R and
258
apparent differences as summarized in Table 3. In the case of MSE, the WBLL registered slightly higher
259
and more disperse values (0.477±0.876) than the GMPZ model (0.369±0.548; Table 3), although it is
260
hard to tell if these differences are relevant due to the lack of a scale or defined interval for MSE values
261
as is the case of* the [0, 1] interval available for R . As expected, the LKM model greatly underperformed
262
consistently yielding high MSE values, low R < 0.9, and consequently high variance estimates, resulting
263
in substantial differences when compared to the WBLL and GMPZ models (Table 3).
2
2

σ2
values of the WBLL and GMPZ equations showed no
2
2
264
265
INSERT TABLE 3
266
2
267
LKM was the only model to register R < 0 (-0.382, -1.189), which occurred after fitting LKM to E. coli
268
inactivation data (200 and 250 MPa) reported in Yamamoto et al. (2005). Nevertheless, the coefficient of
269
determination should never give negative values. In this case, R < 0 occurs because the deviation of
270
model prediction is higher than the variability measured by the total sum of squares (Eq. 16), reflecting a
271
very deficient model choice and the inadequacy of R as a statistical criteria for model selection (Spiess
272
and Neumeyer 2010). The microbial survival curves of E. coli yielding R < 0 presented a very steep drop
273
of microbial counts (3-5 log reductions) in very short times (t ~ 1.0-1.5 min), and slower inactivation rates
274
afterwards that resulted in clear nonlinear curves with upward concavities (data not shown). These
275
microbial survival curve shapes caused extremely high LKM prediction deviations since the intercept (a0)
2
2
2
9
276
was fixed as a0 = 0 to force LKM fit through the origin (0, 0). Strictly, LKM prediction should be forced
277
through the origin because the integration constant (N0) is already included when microbial counts are
278
normalized as log10 (N/N0) (Serment-Moreno et al. 2014). If a0 is obtained with the regression, a0 ≠ 0
279
increases the goodness of fit as expected, yielding R = 0.615 (200 MPa) and 0.479 (250 MPa). However,
280
the equation obtained with a0 ≠ 0 is no longer a first order kinetic model. Furthermore, the resultant
281
mathematical expression becomes an empirical first order polynomial model with two parameters that
282
describes a behavior rarely observed in HPP microbial inactivation kinetics.
2
283
284
The analysis of plots of predicted vs. experimental values also showed non-significant differences
285
between the WBLL and GMPZ models (Figure 1b,c). Although both yielded R > 0.98 with slope m
286
ranging 0.991-0.993, the WBLL model showed a straighter and less disperse line (Figure 1b). Even LKM
287
values were acceptable yielding R = 0.881 and m = 0.934 (Figure 1a); however, the predictions clearly
288
deviate from the linear trend due to high overestimations observed for x-axis values in the -7 to -4 range
289
which is more evident in residuals plot (Figure 1d). The residuals for the other two models showed similar
290
90% confidence intervals (-0.45 to 0.47, WBLL; -0.54 to 0.58, GMPZ), and inaccurate predictions with
291
high positive residual values for the experimental microbial log reductions near zero (Figure 1e,f).
292
Moreover, the GMPZ residuals plot displayed an important deviation for values lower than -7 in the x-axis
293
in addition to an oscillating trend (Figure 1f).
2
2
294
295
INSERT FIGURE 1
296
297
Most GMPZ model sub-estimations occurred for microbial survival curves with two or more concavity
298
changes (DAT07, DAT08, DAT09, DAT11, and DAT17). In these cases, the asymptotic GMPZ model
299
parameter A, and the weight of microbial log count measurements prior to a last concavity change near
300
the end of the microbial survival data, do not allow predictions to reach lower values. All standard
301
statistical criteria confirmed the need to discard the use of the LKM approach, but only the subjective
302
residuals plot analysis suggested a slight superiority of the WBLL over the GMPZ model as the most
303
reliable alternative to describe the 28 microbial inactivation datasets used in this study. The GMPZ model
304
has been rarely considered as a model alternative to describe data on microbial inactivation by HPP
305
treatments. The widely preferred log-logistic model was not included in the comparisons presented in this
306
study for the reasons previously stated in section 1.1 (Serment-Moreno et al. 201X). The WBLL model
307
has been reported previously as yielding better estimates than the GMPZ model. However, in these -
308
reports, MSE and R values were the preferred statistical criteria for model comparisons (Chen and
309
Hoover 2003a, b; Guan et al. 2005; Guan et al. 2006; Koseki et al. 2009; Yamamoto et al. 2005).
2
310
311
3.2 Goodness of fit of primary models assessed with information theory criteria
10
312
When used to assess HPP microbial inactivation kinetics, AICi values varied greatly from -46.9 to
313
25.5, -54.1 to 9.1, and -178.9 to 8.9 for the LKM, WBLL, and GMPZ model, respectively. As described in
314
section 1.2, this behavior was expected since the first term of the AICi factor, the maximum log-likelihood
315
estimate (ℓ) calculated through the dataset variance, depends largely on the population size and
316
dispersion. The WBLL model yielded the lowest ∆AICi, i.e., values of 0 and ≤ 2 for 76 (61.3%) and 98
317
(79%), respectively, of the 124 kinetic curves analyzed (Table 4). The GMPZ model varied widely
318
registering ∆AICi values of 0 and ≤ 2 in 27.4 and 37.9% of the curves analyzed, while for LKM more than
319
half (50.8%) of the ∆AICi values exceeded 4. As expected the LKM model performed poorly and
320
information theory criteria just confirmed the results observed with standard statistics. While yielding the
321
lowest ∆AICi values on 11.3% of the curves analyzed, the LKM reached values > 10 in 58.1% of them
322
(Table 4), suggesting that this particular model may not be so reliable to predict HPP microbial
323
inactivation kinetics. The conclusions obtained using the BICi criteria agreed with the AICi for all tested
324
models, thus the results are not reported.
325
326
INSERT TABLE 4
327
328
3.3 HPP microbial inactivation kinetic model in milk
329
Primary model parameters showed wide scattering when plots as a function of pressure were prepared
330
for the complete microbial inactivation data available for this study (data not shown). Therefore, microbial
331
inactivation data was classified according to the media and microorganism involved in each dataset.
332
Kinetic data for microorganisms in milk (see Table 2) provided the largest amount of information with 56
333
kinetic curves, and showed identifiable trends of pressure effects on primary model parameters despite
334
wide differences in the type of microorganisms involved (data not shown). The Eyring, logistic-
335
exponential, Bigelow-exponential, sigmoidal and exponential secondary models listed in section 2.3
336
showed good agreement when correlating the corresponding primary kinetic model parameters with the
337
pressure level of the microbial inactivation treatment used. However, when the secondary model
338
parameters were used to predict the log microbial counts for the corresponding pressure-time
339
combinations (n = 429), the LKM model severely underperformed by scoring the largest variance estimate
340
( σ = 3.37), and ∆AICi = 345.8 (Table 5).
2
341
342
These results differ with previous work by Santillana Farakos and Zwietering (2011) who found good
343
agreement when using the Bigelow model to describe pressure and temperature effects on LKM
344
parameters. These authors fitted the Bigelow model to decimal reduction time (D) values reported in
345
peer-reviewed sources, and generated a different dataset from the one used in this study. However, when
346
D values were not included in the published reports used in their work, the LKM regression included only
347
the linear portion of the microbial survival curve even when concave trends were observed (Santillana
348
Farakos and Zwietering 2011).
11
349
350
INSERT TABLE 5
351
352
Standard statistical criteria showed a similar goodness of fit when describing microbial inactivation using
353
WBLL and GMPZ models parameters as a function of pressure (Table 5). However, a one-step non-linear
354
regression could not be performed when incorporating Eq. 22-24 into the primary GMPZ model (Eq. 7)
355
due to the complexity of the resulting mathematical expression. The secondary models for pressure
356
effects on GMPZ parameters registered a variance estimate
357
observed values with slope m = 0.930, considering the 429 pressure-time kinetic data combinations
358
available for milk. The 90% confidence residuals interval was (-2.13, 2.26; Table 5) but values were not
359
evenly distributed as only positive residuals occurred for x-axis values between [-1, 0], and only negative
360
residuals were observed for x-axis values [-9, -7] (data not shown). The maximum inactivation rate and
361
the inflection point tended to follow exponential trends when plotted against pressure.

σ 2 = 1.55 (Table 5) and a fit of predicted vs.
362
363
The asymptote difference parameter (A) showed the greatest dispersion of all three parameters. The
364
divergence might be attributed to microbial log reductions achieved during the come up time (CUT). Even
365
when reports used in this study presented CUT values (see Table 2; DAT 01-02, DAT10, DAT12-16,
366
DAT18-19, DAT26-28), traditional normalization of microbial inactivation data (log10 N/N0 or log10 N/NCUT)
367
to linearize the kinetic data does not take into account the CUT reduction effect as exemplified with data
368
reported by Ahn et al. (2007). Normalizing the spore inactivation data as log10 N/NCUT does not consider
369
the 2-3 log reductions during CUT at 700 MPa/105°C, or the 4-6 log reductions at 700 MPa/121°C (see
370
Figures 1b and 1d presented by Ahn et al. 2007). Conversely, if the microbial inactivation data is
371
normalized as log10 N/N0 the spore survival curves will present a significantly steeper slope, theoretically
372
resulting in a higher b’ value for the WBLL model (Eq. 2), or a larger A parameter in the case of the
373
GMPZ model (Eq. 7). Still, 69.2% of the milk HPP microbial inactivation data fell within the interval defined
374
by the GMPZ model prediction ± standard deviation (yfit ± σ ; Figure 2). The 300-400 MPa pressure range
375
registered 14.2% of the cases in which the experimental data remained outside the yfit ± σ interval, and a
376
similar percentage occurred when the pressure level exceeded 400 MPa (14.7%). These deviations may
377
be attributed to microbial physiological factors or biochemical changes occurring in milk at pressure
378
exceeding 200 MPa (Considine et al. 2007; Huppertz et al. 2006).


379
380
This work represents the first time that secondary models are developed for sigmoidal kinetics for
381
microbial inactivation by HPP. Erkmen (2009) used Arrhenius and square root models as secondary
382
models, but only evaluated the temperature effect for Salmonella typhimurium inactivation in tryptic soy
383
broth (200-350 MPa; 15-45°C). The use of the expressions here proposed could be extended to the
384
analysis of sigmoidal curves observed in other microbial inactivation alternatives including conventional
385
thermal food processing. Regarding other sigmoidal secondary models, Serment-Moreno et al. (201X)
12
386
analyzed the possibility of developing a secondary model for the log-logistic model (Eq. 6), but the
387
dependence of the log-logistic model parameters on pressure could not be determined.
388
389
INSERT FIGURE 2
390
391
The logistic-exponential WBLL model obtained after a one-step non-linear regression yielded higher than
392
expected predicted values caused by microbial inactivation data following sigmoidal curves. Predicted
393
survival counts were comparable to those obtained when using the secondary GMPZ model (Figure 3),
394
although the WBLL model clearly registered the highest goodness of fit described by the AICi and ∆AICi
395
values reported in Table 5. The comparison of the variance estimate using a log scale via the log-
396
likelihood estimate and the parsimony term of the AICi value (Eq. 20) played an important role when
397
addressing WBLL and GMPZ model differences, as variance estimates differed just by 0.05 units (Table
398
5).
399
400
INSERT FIGURE 3
401
402
The WBLL model displayed upward concavity at all tested pressure levels as denoted by predicted n
403
values <1 and decreasing exponentially from n = 0.758 at 150 MPa to n = 0.403 at 600 MPa. The value of
404
n under isothermal conditions remained relatively unchanged (|n(P1) – n(P2)| < 0.1) for narrow pressure
405
intervals (~100 MPa) as reported in other studies (Table 6). Nevertheless, precautions must be taken with
406
this assumption, since currently there is no evidence that non-apparent differences among n values will
407
result in non-significant differences of the predicted microbial counts which are more transcendent. The
408
WBLL inactivation rate constant increased from b’ = 0.056 to b’ = 2.230 min
409
range. Both parameters (b’, n) displayed good agreement with the microbial pressure resistance
410
described by the Weibull secondary model (Eq. 2-4, van Boekel 2002; see Chapter 6, Peleg 2006;
411
Serment-Moreno et al. 2014).
-n
for the same pressure
412
413
INSERT TABLE 6
414
415
The Bigelow model predicting the pressure dependence of b’ (Eq. 5) yielded zP = 290 MPa (Table 5),
416
which is similar to zP = 298 MPa for Listeria spp. Reported by Santillana Farakos and Zwietering (2011).
417
Nevertheless these results are not equivalent, since zP obtained by these authors corresponds to the first
418
order kinetic model fit of Listeria spp. data in various media such as beef, pork, milk or buffers. The
419
Bigelow model yielded b’ref = 0.0481 at Pref = 117.4 MPa (Table 5), which is close to b’ = 0.056 (100 MPa)
420
predicted with the logistic-exponential model. The parameters describing the pressure dependence of n
421
(d0, d1; Eq. 4), and the estimated variances also remained close when comparing the logistic-exponential
422
and Bigelow-exponential WBLL models (Table 5). Despite this apparent similarities, the logistic-
13
423
exponential model clearly resulted a better option registering ℓ = -86.9, while the Bigelow-exponential
424
model yielded ℓ = -111.9 and a wider, non-symmetric residuals interval (Table 5). The Bigelow-
425
exponential model also uses one more parameter than the logistic-exponential model, resulting in a large
426
∆AICi = 51.9 (Table 5).
427
428
Doona et al. (2008) used the logistic-exponential Weibull secondary model (Eq. 3-4) to analyze E. coli
429
inactivation in a whey protein model (214-439 MPa; 30-50°C). The low MSE values (0.003-0.932) and
430
visual inspection of the model fit showed a good agreement between experimental values and model
431
predictions. The concavity of the curves changed when the pressure level applied increased, showing
432
upward concavity at the higher pressure levels with the exponent decreasing from n = 3.37 (214 MPa) to
433
n = 0.70 (439 MPa) at 30°C. The Weibull rate constant increased from b’ = 2.70x10 to b’ = 0.482 min
434
for the same pressure-temperature conditions (214-439 MPa; 30°C). The parameter estimates obtained
435
by Doona et al. (2008) diverged from values obtained in this study using the generic WBLL model for
436
HPP microbial inactivation data in milk at 25°C. Different experimental factors can affect non-linear
437
regression results including the pressure-temperature range, media treated, and microbial flora type likely
438
being the most influential ones.
-8
-n
439
440
Most importantly, the results in the current study imply that a relatively simple equation such as the
441
logistic-exponential WBLL four parameter secondary model might be sufficient to deliver an effective
442
mathematical tool for the experimental and industrial design of HPP treatments when inactivation kinetic
443
data of similar microorganism types in a common media is grouped together. In spite of the regular use of
444
the WBLL model in thermal food processing (Halder et al. 2007; Peleg 2002; van Boekel 2002; Peleg
445
2006), except for the work by Doona et al. (2008), no other secondary model applications to model HPP
446
inactivation kinetics were found in published work. Although alternative Weibull secondary models have
447
been proposed, these mathematical expressions have been tested only with the datasets used to derive
448
them (Chen and Hoover 2003b; Buzrul et al. 2008; Pilavtepe-Çelik et al. 2009; Serment-Moreno et al.
449
2014).
450
451
3.3 HPP microbial inactivation kinetic model in milk with short processing times (<10 min)
452
Microbial inactivation data for extremely prolonged HPP treatments is of questionable relevance to
453
industrial applications of this technology. In addition, longer times increased the prevalence of sigmoidal
454
curves in the milk data used in this study. Therefore, the one-step non-linear regression with the
455
secondary logistic-exponential Weibull model was repeated by analyzing only microbial kinetic curves
456
with n ≥ 3 data points and treatment times below 10 min. The pressure range for the remaining 12 kinetic
457
curves (n = 73 pressure-time microbial data combinations) covered the 300-600 MPa range (DAT08-09,
458
DAT11, DAT13-14, and DAT24-25). Although this dataset is very limited, the results showed important
459
differences when compared to the complete milk kinetic data used in Section 3.3. The parameters b’ and
14
-n
460
n ranged from 0.079 to 1.859 min and 1.340 to 0.557, respectively, for the 300-600 MPa range whereas
461
the complete milk dataset yielded b’ = 0.056 to 2.230 min and n = 0.758 to 0.403 for the 150-600 MPa
462
region. The secondary model parameters wP = 0.0139 MPa and Pc = 478.7 MPa remained close to the
463
values reported in Table 5, whereas the parameter d0, representing the value of the Weibull exponent (n;
464
Eq. 2) at low pressure levels, dramatically increased from d0 = 0.9358 (Table 5) to d0 = 3.224.
465
Consequently, the concavity of the survival curves changed from downward to upward concavity as the
466
pressure level increased, displaying linear trends in the 400-450 MPa region, while the Weibull model
467
obtained for the complete milk dataset only showed upward concavity curves. Despite the limited amount
468
of kinetic data, these results suggest that an inappropriate choice of the HPP experiment duration can
469
result in estimates of kinetic parameters that do not describe well HPP treatment conditions of commercial
470
interest.
-n
-1
471
472
4. Conclusions
473
This study provided a much needed comparison for three of the most widely used microbial inactivation
474
kinetic models. It showed also that the sole use of standard statistical tools (R , MSE, residuals plots, and
475
fit of experimental vs. model prediction plots) to select a microbial inactivation model may be insufficient.
476
Standard statistics could not distinguish whether the WBLL or the GMPZ model were the most adequate
477
primary HPP microbial inactivation model. Further analysis with statistical parameters based on
478
information theory (maximum log-likelihood estimate, ℓ; Akaike Information Criteria, AICi; Akaike
479
differences, ∆AICi) indicated that the WBLL outperformed the LKM and the GMPZ primary models on 76
480
of the 124 kinetic curves analyzed. In spite of its historical importance in food processing, LKM is not a
481
reliable alternative to model HPP microbial inactivation kinetics since it performed poorly during primary
482
and secondary modeling. Akaike differences were ≥10 in 58.1% of the kinetic curves analyzed for primary
483
modeling, whereas the secondary Eyring model registered ∆AICi = 345.8 when compared to the logistic-
484
exponential Weibull secondary model. Most importantly, this study showed that information theory criteria
485
adequately complemented the use of standard statistics, and highlighted the importance of considering
486
several statistical parameters to evaluate the goodness of fit of mathematical models.
2
487
488
Although a global model for HPP inactivation could not be developed, this study suggests that generic
489
models are possible when limited to the HPP inactivation of similar types of microorganisms and media,
490
which would be useful for HPP design. The logistic-exponential WBLL and empirical GMPZ secondary
491
models derived for microbial inactivation kinetics in milk had equal goodness of fit according to standard
492
statistics. However, the logistic-exponential WBLL secondary model registered the lowest AICi value
493
(181.8), and a high difference when compared to the large AICi value of the GMPZ model (∆AICi = 19.3)
494
or the Bigelow-exponential WBLL model which performed poorly (∆AICi = 51.9). The simplicity of the
495
logistic-exponential Weibull model was an advantage, since only 4 parameters plus the variance estimate
496
are required by the flexible, easy to use logistic-exponential and exponential decrease functions used in
15
497
the secondary WBLL model. The empirical equations defining the secondary GMPZ model (8
498
parameters) also predicted accurately microbial log counts at different pressure-time combinations, but
499
they were deemed to be impractical because their complexity prevented a one-step non-linear regression.
500
These empirical equations are the only ones available for HPP microbial inactivation data following a
501
sigmoidal behavior and their application could be extended to analyze the kinetics of microbial
502
inactivation for other food processing technologies.
503
504
Finally, an important concern raised by this study is that much of the research on microbial inactivation by
505
HPP treatment does not match commercial processing conditions. When a microorganism survives a low
506
pressure level, researchers consistently extend treatment times far beyond values with commercial
507
viability. These long treatment times often result in microbial inactivation curves characterized by multiple
508
concavity changes that may interfere with the estimation of “practical” kinetic parameter for more realistic
509
HPP treatment conditions. A non-linear regression analysis considering only microbial data for t ≤ 10 min
510
(12 kinetic curves; n = 73 pressure-time microbial data combinations) supports the recommendation of
511
focusing on short pressure holding times. The estimated WBLL model parameters and the shape of the
512
microbial survival curves clearly differed from those obtained when long treatment times were included.
513
All resources are limited and they should be employed to generate microbial inactivation data at an
514
appropriate pressure level to deliver significant microbial inactivation in a reasonable time (preferably less
515
than 10 min) that complies with the needs of industrial HPP food processors.
516
517
Acknowledgments
518
Authors Vinicio Serment-Moreno and Jorge Welti-Chanes acknowledge the support from the Tecnológico
519
de Monterrey (Research chair funds CAT 200 and CDB081), and México’s CONACYT Scholarship
520
Program. This project was also supported by Formula Grants no. 2011-31200-06041 and 2012-31200-
521
06041 from the USDA National Institute of Food and Agriculture.
522
16
523
Nomenclature
A
AICi
AICmin
b’
b'ref
d0
d1
fi; i=1,2
gi; i=1,2
HPP
k
k0
ℓ
MSE
N
N0
n
P
Pc
Pref
p
qi; i=1,2,3
2
R
SSE
SST
T
t
wAICi
wP
yfit
yobs
zP
-1
Asymptote difference, Gompertz model, log-logistic model; log10 cfu ml
Akaike Information Criterion for model i
Minimum AICi value among the models tested
-n
Slope parameter, Weibull model; min
-n
Slope parameter at a reference pressure; Bigelow type Weibull secondary model; min
Parameter n at atmospheric pressure, Weibull secondary model; dimensionless
-1
Exponential pressure dependence of parameter n, Weibull secondary model; MPa
Parameters of empirical secondary Gompertz model
Parameters of empirical secondary Gompertz model
High Pressure Processing
-1
Inactivation rate constant, first order kinetics model; min
-1
Inactivation rate constant at a reference pressure, Eyring model; min
Maximum log-likelihood estimate
Mean squared error
-1
-1
Microbial population; cfu g , cfu ml
-1
-1
Microbial population prior to the high pressure treatment; cfu g , cfu ml
Number of observed data points
Exponent, Weibull model
Pressure; MPa
Critical pressure parameter, Weibull log-logistic secondary model; MPa
Reference pressure, Eyring model, Bigelow type Weibull secondary model ; MPa
Number of estimated parameters, Akaike Information Criterion
Parameters of empirical secondary Gompertz model
Coefficient of determination
Error sum of squares; coefficient of determination
Total sum of squares; coefficient of determination
Temperature; °C
Time; min
Akaike weight factor
-1
Exponential pressure dependence of parameter b’, Weibull secondary model; MPa
Microbial counts with primary or secondary HPP inactivation models
Experimental microbial counts for literature reported HPP treatments
Pressure resistance factor; Bigelow type Weibull secondary model; MPa
524
525
Greek symbols
α
ΔAICi
≠
∆V
λ
Ω
δ
μmax
σˆ
σ̂ 2
τ
Significance levels for statistical analysis
AICi differences among the tested models by taking AICmin as reference
3
-1
Activation volume change, Eyring model; cm mol
Inflection or starting point of the linear portion of sigmoidal inactivation curves,
Gompertz and log-logistic models; min
-1
Maximum inactivation rate, log-logistic model; log cfu min
-1
Slope parameter, log-logistic model; decimal microbial reductions (log time units)
-1
-1
Maximum inactivation rate, Gompertz model; log10 (cfu ml ) min
Standard deviation
Variance estimate
Log time for maximum inactivation rate in “vitalistic” log-logistic model; log10 min
526
17
527
Table 1. Exponentiation of
Akaike differences (∆AICi)
∆AICi
0
1
2
3
4
5
7
10
exp(-½ ∆AICi)
1.000
0.607
0.368
0.223
0.135
0.082
0.030
0.007
528
18
529
Table 2. References reporting microbial inactivation by high pressure processing (HPP) of low acid foods at ambient temperature (2025°C)
Label
T(°C)
P(MPa)
Microorganism
Media
pH
Number of
microbial
survival curves
DAT01
DAT02
DAT03
20
20
25
150-350
150-350
200-300
Milk
Milk
Ewe milk
NR
NR
6.7
4
4
3
DAT04
25
200-300
Ewe milk
6.7
3
DAT05
DAT06
DAT07
DAT08
DAT09
DAT10
DAT11
DAT12
DAT13
DAT14
DAT15
DAT16
DAT17
DAT18
DAT19
DAT20
25
25
22
22
25
21
25
25
25
25
25
25
25
20
20
25
200-500
200-500
300-450
350-500
400-500
250-400
400-600
200-700
300-600
300-600
300-600
300-600
350-600
200-500
300-600
100-300
Listeria monocytogenes
Natural microflora
Escherichia coli
Pseudomonas
fluorescens
Lactobacillus helveticus
Staphylococcus aureus
Yersinia enterocolitica
Yersinia enterocolitica
Listeria monocytogenes
Mesophiles
Listeria monocytogenes
Listeria monocytogenes
Mesophiles
Listeria monocytogenes
Mesophiles
Listeria monocytogenes
Salmonella typhimurium
Escherichia coli
Escherichia coli
Escherichia coli
Ewe milk
Ewe milk
Phosphate buffer
Milk
Milk
Milk
Milk
BHIB
Milk
Milk
Peach juice
Peach juice
Milk
HEPES-KOH buffer
Carrot juice
Phosphate buffer
6.7
6.7
7.0
6.7
6.7
NR
6.7
6.7
6.7
6.7
5.2
5.2
NR
7.0
6.6
6.8
5
5
4
4
2
4
3
6
3
3
3
3
6
7
7
6
DAT21
20
200-400
Escherichia coli
Peptone water
NR
5
DAT22
DAT23
DAT24
DAT25
DAT26
DAT27
DAT28
23
23
22
22
25
25
25
350-550
450-600
400-600
400-600
250-400
250-400
200-350
Staphylococcus aureus
Staphylococcus aureus
Escherichia coli
Listeria innocua
Escherichia coli
Listeria monocytogenes
Salmonella typhimurium
Phosphate buffer
Ham
Whole milk
Whole milk
Fish slurry
Fish slurry
Tryptone soy broth
6.7
6.7
6.6
6.6
NR
NR
7.0
6
6
5
5
4
4
4
Reference
(Mussa et al. 1999)
(Mussa et al. 1999)
(Gervilla et al. 1999a)
(Gervilla et al. 1999a)
(Gervilla et al. 1999b)
(Gervilla et al. 1999b)
(Chen and Hoover 2003b)
(Chen and Hoover 2003b)
(Chen & Hoover, 2003a)
(Pandey et al. 2002)
(Chen and Hoover 2004)
(Dogan and Erkmen 2004)
(Dogan and Erkmen 2004)
(Dogan and Erkmen 2004)
(Dogan and Erkmen 2004)
(Dogan and Erkmen 2004)
(Guan et al. 2005)
(Van Opstal et al. 2005)
(Van Opstal et al. 2005)
(Yamamoto et al. 2005)
(Koseki and Yamamoto
2007)
(Tassou et al. 2007)
(Tassou et al. 2007)
(Buzrul et al. 2008)
(Buzrul et al. 2008)
(Ramaswamy et al. 2008)
(Ramaswamy et al. 2008)
(Erkmen 2009)
530
531
19
2
Table 3. Coefficient of determination (R ), mean
squared error (MSE) and variance estimate ( σˆ ) for
the first order kinetics (LKM), Weibull (WBLL) and
Gompertz (GMPZ) model describing HPP microbial
inactivation at ambient temperatures (20-25°C)
2
2
σˆ 2
LKM
mean
sd
min
max
R
0.789
0.294
-1.189
0.997
MSE
7.969
13.652
0.005
97.618
0.925
1.644
0.001
10.846
WBLL
mean
sd
min
max
0.965
0.050
0.708
0.999
0.477
0.876
-5
0.987x10
5.616
0.110
0.232
-5
4.894x10
1.871
GMPZ
mean
sd
min
max
0.965
0.037
0.822
1.000
0.369
0.548
0.000
2.709
0.110
0.165
0.000
1.156
532
20
533
Table 4. Akaike difference (∆AICi) distribution for the first order
kinetics (LKM), Weibull (WBLL) and Gompertz (GMPZ) primary
inactivation models describing HPP microbial inactivation
kinetics in milk at ambient temperature (20-25°C)
Criteria
LKM
WBLL
GMPZ
∆AICi = 0
11.3%
61.3%
27.4%
∆AICi ≤ 2
19.4%
79.0%
37.9%
2 < ∆AICi ≤ 4
4.8%
5.7%
11.3%
4 < ∆AICi ≤ 10
17.7%
9.7%
25.8%
∆AICi > 10
58.1%
5.7%
25.0%
21
534
535
536
Table 5. Statistical summary of secondary HPP microbial inactivation models in milk at ambient temperature (20-25°C).
(1)
(1)
(1)
LKM
WBLL
WBLL
k(P)
Parameter
k0
0.0255
≠
-∆V
Pref
Residuals
(d)
range
b'(P)
-19.65
wP
Pc
(a)
n(P)
0.0111
d0
d1
408.7
0.9358
0.0014
b'(P)
bref
Pref
zP
240.3
(b)
0.0481
117.4
290
n(P)
d0
d1
0.9022
0.0010
GMPZ
A(P)
p1
p2
p3
Λ(P)
µmax(P)
f1
-7.795
f2
2.111
-0.017
-3
6.174x10
g1
429.8
g2
-0.0174
-3
8.996x10
-3.30 to 2.00
-1.92 to 1.96
-2.16 to 1.78
-2.13 to 2.26
3.37
1.50
1.68
1.55
AICi
529.6
183.8
235.7
203.1
∆AICi
345.8
0
51.9
19.3
(1) Results obtained from a one-step non-linear regression incorporating the secondary model into the primary model as one equation; (a) logisticexponential model; (b) Bigelow type-exponential model; (d) 90% confidence interval of residuals
22
Table 6. Parameter n average values for different pressure ranges.
Microorganism
Various
Pressure range
Pressure
(MPa)
levels
0.661±0.038
200-300
3
0.574±0.033
300-400
3
Generic Weibull
0.499±0.029
400-500
3
secondary milk model of
0.434±0.025
500-600
3
current work
0.467±0.046
400-600
5
0.477
300-400
3
Milk
0.583
400-500
3
Milk
0.483
400-600
5
Chen and Hoover (2004)
400-600
5
Buzrul et al. (2008)
Media
Milk
Phosphate
Y. eneterocolitica
L. monocytogenes
buffer
E. coli
L. innocua
Milk
n
0.780
0.790
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23
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690
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691
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693
29
694
6. List of Tables
695
696
Table 1.
697
Exponentiation of Akaike differences (∆AICi).
698
699
Table 2.
700
References reporting microbial inactivation by high pressure processing (HPP) at ambient temperature
701
(20-25°C) and low acidity foods.
702
703
Table 3.
704
Coefficient of determination (R ), mean squared error (MSE) and variance estimate ( σˆ 2 ) for the first order
705
kinetics (LKM), Weibull (WBLL) and Gompertz (GMPZ) models describing HPP microbial inactivation at
706
ambient temperatures (20-25°C).
2
707
708
Table 4.
709
Akaike difference (∆AICi) distribution of the first order kinetics (LKM), Weibull (WBLL) and Gompertz
710
(GMPZ) primary inactivation models describing milk HPP microbial inactivation kinetics at ambient
711
temperature (20-25°C).
712
713
Table 5.
714
Statistical summary of secondary HPP microbial inactivation models in milk at ambient temperature (20-
715
25°C).
716
717
30
718
7. List of Figures
719
720
Figure 1.
721
Model vs. reported data plot and residuals plots for HPP primary inactivation models for microbial
722
inactivation at ambient temperature (20-25°C) and low acidity foods.
723
724
Figure 2.
725
Empirical secondary Gompertz model fit (solid line), secondary model standard deviation (dotted line),
726
and reported microbial inactivation data (color symbols) at different pressure-time combinations for milk
727
HPP treatments at ambient temperature (20-25°C).
728
729
Figure 3.
730
Logistic-exponential secondary Weibull model fit (solid line), secondary model standard deviation (dotted
731
line), and reported microbial inactivation data (color symbols) at different pressure-time combinations for
732
milk HPP treatments at ambient temperature (20-25°C).
733
31
734
735
32
736
737
33
738
739
34
740
er
35