INTRODUCTION
Microbiological safety is one of the key tasks in
developing food production technology [1–4]. Most
existing solutions are based on microbial inactivation
via chemical or physical methods, as well as competitive
substitution of pathogenic and opportunistic pathogenic
microflora with probiotic bacteria. At the same time,
thermal sterilisation remains the most common way to
achieve the required level of microbiological safety.
The first stage in developing the thermal sterilisation
mode is traditionally the analysis of the microbial
death kinetics in the medium of the product for which
this mode is developed [5]. Microbiological studies are
carried out under isothermal conditions and repeated at
different temperatures. Then the experimental data are
approximated with a particular model [6–8]. Naturally,
the adequacy of the selected model to the experimental
data confirms the adequacy of ideas about the microbial
death kinetics in general and, in particular, regarding
extrapolation at concentrations of microorganisms down
to 1 CDU/g. Obtaining correct quantitative experimental
microbiological data for these concentrations is
associated with geometrically increasing resource
intensity and experimental error.
According to Whiting and Bushanan, all
mathematical models describing the microbial response
to the external negative effects can be divided into three
large groups – primary, secondary and tertiary models.
Thus, primary models approximate experimental data of
the microbial death kinetics under isothermal conditions
Research Article DOI: http://doi.org/10.21603/2308-4057-2019-2-348-363
Open Access Available online at http:jfrm.ru
Comparative evaluation of approaches to modelling kinetics
of microbial thermal death as in the case
of Alicyclobacillus acidoterrestris
Vladimir V. Kondratenko* , Mikhail T. Levshenko , Andrey N. Petrov ,
Tamara A. Pozdnyakova , Marina V. Trishkaneva
Russian Research Institute of Canning Technology, Vidnoye, Russia
* e-mail: nauka@vniitek.ru
Received June 05, 2019; Accepted in revised form July 26, 2019; Published October 21, 2019
Abstract: Microbial death kinetics modelling is an integral stage of developing the food thermal sterilisation regimes. At present,
a large number of models have been developed. Their properties are usually being accepted as adequate even beyond boundaries
of experimental microbiological data zone. The wide range of primary models existence implies the lack of universality of each
ones. This paper presents a comparative assessment of linear and nonlinear models of microbial death kinetics during the heat
treatment of the Alicyclobacillus acidoterrestris spore form. The research allowed finding that single-phase primary models (as
adjustable functions) are statistically acceptable for approximation of the experimental data: linear – the Bigelow’ the Bigelow
as modified by Arrhenius and the Whiting-Buchanan models; and nonlinear – the Weibull, the Fermi, the Kamau, the Membre
and the Augustin models. The analysis of them established a high degree of variability for extrapolative characteristics and, as a
result, a marked empirical character of adjustable functions, i.e. unsatisfactory convergence of results for different models. This
is presumably conditioned by the particularity and, in some cases, phenomenology of the functions themselves. Consequently,
there is no reason to believe that the heat treatment regimes, developed on the basis of any of these empirical models, are the most
effective. This analysis is the first link in arguing the necessity to initiate the research aimed at developing a new methodology
for determining the regimes of food thermal sterilisation based on analysis of the fundamental factors such as ones defined spore
germination activation and their resistance to external impact.
Keywords: Microorganisms, death kinetics, survival kinetics, sterilising effect, Alicyclobacillus acidoterrestris, model
Please cite this article in press as: Kondratenko VV, Levshenko MT, Petrov AN, Pozdnyakova TA, Trishkaneva MV.
Comparative evaluation of approaches to modelling kinetics of microbial thermal death as in the case of Alicyclobacillus
acidoterrestris. Foods and Raw Materials. 2019;7(2):348–363. DOI: http://doi.org/10.21603/2308-4057-2019-2-348-363.
Copyright © 2019, Kondratenko et al. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International
License (http://creativecommons.org/licenses/by/4.0/), allowing third parties to copy and redistribute the material in any medium or format and to remix,
transform, and build upon the material for any purpose, even commercially, provided the original work is properly cited and states its license.
Foods and Raw Materials, 2019, vol. 7, no. 2
E-ISSN 2310-9599
ISSN 2308-4057
349
Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363
as a function of processing duration. Secondary models
connect primary models, approximating them as a
function of additional factors, which include primarily
the temperature of processing, when developing
thermal sterilisation regimes. Tertiary models include
the software implementation of secondary models [9].
However, we believe, it would be more logical if the
adaptation of secondary models to non-isothermal
sterilisation regimes within specified boundary
conditions was referred to tertiary models.
The primary models of thermal inactivation of
microorganisms broke into development at the turn of
the XXth century. Then, basing on the analogy with the
first order chemical reaction, Chick, and later Bigelow
presented a model reflecting the kinetics of the microbial
death resulted from external adverse factors (Bigelow
model) as a linear kinetic model of the first order [10,
11]. Bigelow showed later that in general the kinetics
sought for can be adequately represented in semilogarithmic
coordinates [12]. Due to the simplicity of
form and further manipulations, this model has become
classic:
lg N = lg N0 − k ⋅τ (1)
where N is a trough concentration of microorganisms,
CFU/g;
0 N is an initial concentration of microorganisms,
CFU/g;
k is the rate of microbial death kinetics (often defined
as a constant), lg (CFU/g)·min–1 (time may be expressed
in s, h, etc.);
τ is a duration of processing, min (time may be
expressed in s, hour, etc.).
This model is based on the assumptions that all
cells of microorganisms have the same resistance to the
thermal impact in the processed product, and their death
kinetics complies with statistical regularities [9, 13]. As
a result, the death of each individual cell is considered
from the point of view of accidental inactivation of the
‘critical molecule’.
To avoid pure empiricalness, some researchers
attempted to adjust the Bigelow model. They expressed
the dependence of the microbial death rate on the
temperature of the process similarly to the dependence
of chemical kinetics on the activation energy according
to the Arrhenius equation [5]:
( )
0
0 exp
abs
k N E
R T T
 
= ⋅ − 
 ⋅ + 
(2)
where E0 is activation energy, J/mol;
R is the universal gas constant (8.3144598 J/mol·K);
Tabs is the absolute temperature of the triple point of
water (273.16 K);
T is the process temperature, °C.
However, Peleg et al. question the adequacy of this
approach in their work [14]. They argue this with the
cardinal difference of the microbial death kinetics from
the conventional chemical kinetics. The difference
manifests itself in the absence of microflora inactivation
under normal conditions and, therefore, in absence
of continuity of functional dependence of microbial
concentration on temperature over the entire range of its
determination.
Nowadays, there are at least four main types of
kinetics of microbial death as a result of heat treatment,
including linear (Fig. 1) [15].
Kinetics with a lag phase is one of the frequent
deviations of microbial death kinetics [16]. This type of
dependence approximates satisfactorily the Whiting-
Buchanan model proposed by Whiting and Buchanan
[17, 18]:
( )
0
0
lg ,
lg
lg ,
lag
lag lag
N
N
N k
τ τ
τ τ τ τ
 ≤ = 
− ⋅ − > 
(3)
where τlag is a lag phase duration.
Van Boekel carried out a large-scale study and
analysed more than 120 curves of microbial death
kinetics. He concluded that linear models described no
more than 5 percent of cases. It implies that this kinetics
is an exception rather than the rule [19]. Van Boekel
suggested that non-linear models should be used for
the most adequate description of kinetics. It should be
understood that nonlinearity of models is determined
primarily by their parameters, since the graphical
representation resulting from approximation can have
both nonlinear and linear views [16]. The simplest
nonlinear model, the Weibull model, is based on the idea
of statistical distribution of probability for the death of
cells and/or microbial spores under the adverse external
conditions as a result of their individual variability [20]:
0 lg lg
p
N N
τ
δ
= −    
 
(4)
where δ is a coefficient.
Figure 1 Dynamics of microbial death according to [15]. (A)
linear kinetics; (B) linear kinetics with a lag phase; (C) and (D)
nonlinear kinetics with ‘tails’; (E) and (F) sigmoidal kinetics
Population density, CFU/mL
Time, h
350
Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363
It is notable that at P > 1 the model represents a
convex curve, while at P < 1 the curve is concave.
When P = 1, the model becomes identical to the Bigelow
model. The coefficientδ becomes equivalent to the value
of D in the interpretation of the same model. It probably
gave Mafart a reason to adopt the coefficient δ as the
duration of the process required to reduce the microbial
concentration by one order [20].
Bhaduri et al. [21] were the first who showed that the
empirical modified Gompertz equation, the Gompertz
model, can effectively describe the thermal death
kinetics of Listeria monocytogenes:
{ ( ) } 0 lg N = lg N −C ⋅ exp −exp −b ⋅ τ −M  (5)
where C, b and M are coefficients.
A little earlier, Casolari [22] proposed a model that
satisfactorily approximates the microbiological data of
kinetics with ‘tails’, i.e. the first Casolari model:
0
lg lg lg 1
1 T
N N
b τ
 
= −    + ⋅ 
(6)
where bT is a coefficient.
He linked the microbial death to the critical
activity of water molecules when their energy exceeds
a threshold value E0, thus combining the probability
theory and Maxwell energy distribution:
( ) 2
2
0
0
2
a exp
T
H abs
b N E
M R T T
   ⋅ 
=   ⋅ −     ⋅ + 
(7)
where Na is Avogadro’s constant (6.022140857×1023,
mol–1);
H2 0 M is molar mass of water (18.01528 g/mol).
In the same work and later in [6, 23] was presented
the modification of this model, i.e. the second Casolari
model. It included a quadratic dependence on the
processing duration:
0 2
lg lg lg 1
1 T
N N
b τ
 
= +    + ⋅ 
(8)
In their turn, Daugthry et al. [24] proposed an
exponentially decreasing model, i.e the Daugthry
model. They justified its advantage over linear models
due to the approximation accuracy of the experimental
data on the death kinetics of Escherichia coli and
Staphylococcus aureus
( ) 0 lg lg exp d N = N − k ⋅τ ⋅ −λ ⋅τ (9)
where k is the initial rate of inactivation;
d λ
is the descending factor.
Data in [25, 26] demonstrate the expediency of a
logistic function, i.e. the Fermi model, for describing
the kinetics of microbial death limited by a number of
stressful factors:
( )
( ) 0
1 exp
lg lg lg
1 exp
lag
lag
b
N N
b
τ
τ τ
 + − ⋅  = +  
 +  ⋅ −  
(10)
where lag τ is a lag phase duration.
Cole et al. [27] proposed a four-factor logistic model,
i.e. the Cole model:
( )
0
0
0
lg
lg lg
4
1 exp
lg
lag
N N N
N
ω
σ τ τ
ω
−
= +
 ⋅ ⋅ − 
+  
 − 
(11)
where ω is the value of the lower asymptote of
microbial death kinetics, lg (CFU/g);
σ is the maximum rate of kinetics.
The model satisfactorily described the thermal
inactivation of Salmonella typhimurium, Cl. botulinum,
Salmonella enteritidis and E. coli.
Membre et al. proposed a modified logistic function,
the Membre model, to describe the thermal inactivation
of Salmonella typhimurium. They assumed that the
model could extrapolate on other microorganisms and
products [28]:
( ) ( ) 0 lg N = 1+ lg N − exp k ⋅τ (12)
Another kinetic model, the Kamau model, is
proposed by Kamau et al. [29] in relation to Listeria
monocytogenes and Staphylococcus aureus:
0 ( )
lg lg lg 2
1 exp d
N N
k τ
 
= +  
 + ⋅ 
(13)
where kd is a coefficient.
The complex logistic model, the Baranyi model,
was developed as an alternative to the Gompertz
model described by formula (5). It takes into account
the advantages of the Gompertz model and levels its
disadvantages. Xiong et al. [30] modified the Baranyi
model, which took the form:
{ ( ) ( ) }
0
lg lg 1 exp b b m T
N q q B
N
= + − ⋅ −μ ⋅ τ −  (14)
( )2
2 2
1 ln 3 arctg 2 3 arctg 1
3 2 3 3 T
r r T T r B
r r T T r
  +   ⋅ −    = ⋅  ⋅   + ⋅  + ⋅     − ⋅ +   ⋅   
( )2
2 2
1 ln 3 arctg 2 3 arctg 1
3 2 3 3 T
r r T T r B
r r T T r
  +   ⋅ −    = ⋅  ⋅   + ⋅  + ⋅     − ⋅ +   ⋅   
(15)
where qb is an indicator of the ‘tail’ of kinetics, reflecting
its manifestation or absence;
m μ is a maximum relative level of thermal
inactivation;
r is a lag parameter, numerically equal to half m μ ;
BT is a coefficient.
A distinctive feature of the Baranyi model is the
possibility to derive the kinetic model of the first order
on its basis.
Geeraerd et al. [23] proposed the Geeraerd model
as a complex approach to describing the complex
kinetics of thermal microbial inactivation. The model
takes into account the presence or absence of the ‘tail’
of the function, and also a lag phase in microorganisms
responding to the thermal effect:
( ) ( ) ( ( ) lg lg 0 lg exp
lg 10 10 10 exp
1 exp 1 res res N N N lag
lag
k
N k
k τ
τ
τ  ⋅ = + − ⋅ − ⋅ ⋅   +  ⋅ −  ⋅
351
Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363
) ( ) ( )
( ) ( )
lg 0 lg exp
10 10 exp
1 exp 1 exp
res N N lag
lag
k
k
k k
τ
τ
τ τ
 ⋅ 
− ⋅ − ⋅ ⋅  
 +  ⋅ −  ⋅ − ⋅ 
(16)
The analysis of these models shows that in many
cases logistic functions are mathematical abstractions
that describe experimental microbiological data more
or less satisfactorily. The Augustin model proposed
in [31] and originally intended to describe the kinetics
of thermal inactivation of Listeria monocytogenes is no
exception
0 2
lg N lg N lg 1 exp lg m
s
  τ −  = −  +     
(17)
where m and s are coefficients.
However, due to the commonality of this model with
other logistic functions, this model may be ostensibly
applicable to the description of thermal inactivation
kinetics of other microorganisms.
The Fernandez model stands apart. It was proposed
in [32] and based on the statistical approach to
estimating the density of distribution:
lg 1 exp ( )N = a−b ⋅b ⋅τ b− ⋅ − τ a b   
(18)
A new empirical nonlinear model was developed, i.e.
the Chiruta model [33]:
( )2
0
lg N 1 exp a b ln c ln
N
= −  + ⋅ τ + ⋅ τ  (19)
where a, b and c are coefficients.
The model is a modified polynomial function with all
properties characteristic of approximation functions of
this class. The properties are satisfactory interpolation
and sensitivity to approximated data. Extrapolation of
this model is possible, but it must be carried out with
caution and mandatory experimental validation.
Analysing the existing array of the experimental
microbiological data, Cerf [34] was one of the first
to who suggested that the deviation of kinetics from
linear (in semilogarithmic coordinates) is most likely a
consequence of simultaneous presence of at least two
microbial subpopulations with different resistance to
external negative effects in a genetically homogeneous
population. The deviation may also be caused by
artefacts (generated by a set of perturbation factors that
are not taken into account, or are not levelled in the
formulation and execution of studies). Figure 2 presents
a graphical interpretation of this approach.
In relation to thermal inactivation kinetics, the result
of this conclusion is the use of two-phase models, taking
into account the contribution of each subpopulation
to the integral response. Models of this kind were
developed by Geeraerd et al [23], Kamau et al [29],
Xiong et al [30], Cerf [34], Whiting and Buchanan [35],
and Coroller et al [36].
The Cerf model is described by the formula:
( ) ( ) ( ) 0 1 2 lg N = lg N + lg  f ⋅ exp −k ⋅τ + 1− f ⋅ exp −k ⋅τ 
( ) ( ) ( ) 0 1 2 lg N = lg N + lg  f ⋅ exp −k ⋅τ + 1− f ⋅ exp −k ⋅τ  (20)
where f is a share of the first subpopulation in the test
culture;
1 and 2 are indices of coefficients belonging to the
subpopulation.
The Kamau model is described by the formula:
( )
( )
0 ( )
1 2
2 2 1 lg lg lg
1 exp 1 exp
f f N N
b τ b τ
 ⋅ ⋅ − 
= +  + 
 + ⋅ + ⋅ 
(21)
The Whiting-Buchanan model:
( )
( )
( ) ( ( )
1 2
0
1 2
1 exp 1 1 exp
lg lg lg
1 exp 1 exp
lag lag lag
f b f b
N N
b b
τ τ
τ τ τ τ
 ⋅  + − ⋅  − ⋅  + − ⋅ = +  +  +  ⋅ −  +  ⋅ − ( )
( )
( ) ( )
( )
1 2
0
1 2
1 exp 1 1 exp
lg lg lg
1 exp 1 exp
lag lag
lag lag
f b f b
N N
b b
τ τ
τ τ τ τ
 ⋅  + − ⋅  − ⋅  + − ⋅   = +  + 
 +  ⋅ −  +  ⋅ −  
(22)
The Coroller model:
( )
1 2
1 2
0 lg lg lg 10 1 10
p p
N N f f
τ τ
δ δ
   
−  − 
   
 
= +  ⋅ + − ⋅ 
 
(23)
The Xiong model:
0
0,
lg l
( ),
lag
lag
N
N f
τ τ
τ τ τ
 ≤
=  > 
(24)
where
{ ( ) ( ) ( ) } 1 2 ( ) lg exp 1 exp lag lag f τ = f ⋅ −k ⋅ τ −τ + − f ⋅ −k ⋅ τ −τ 
{ ( ) ( ) ( ) } 1 2 ( ) lg exp 1 exp lag lag f τ = f ⋅ −k ⋅ τ −τ + − f ⋅ −k ⋅ τ −τ  (25)
The Geeraerd model:
( ) ( ) ( ) ( 0 1 2
lg lg lg exp 1 exp
1 exp N N f k f k
τ τ
 = +  ⋅ − ⋅ + − ⋅ − ⋅ ⋅  +  Figure 2 Graphical interpretation of the subpopulation
approach to the microbial death kinetics during heat treatment
(adapted from [36])
0
1
2
3
4
5
6
0 2 4 6 8 10 12 14 16 18
log10(N)
Time, h
I subpopulation II subpopulation Superposition
log10(N0)
p > 1
δ1 δ2
α
(1)
(2)
(3)
(1) –
0
1
2
3
4
5
6
0 2 4 6 8 10 12 14 16 18
log10(N)
Time, h
I subpopulation II subpopulation Superposition
log10(N0)
p > 1
δ1 δ2
α
0
1
2
3
4
5
6
0 2 4 6 8 10 12 14 16 18
log10(N)
Time, h
I subpopulation II subpopulation Superposition
log10(N0)
p > 1
δ1 δ2
α
0
1
2
3
4
5
6
0 2 4 6 8 10 12 14 16 18
log10(N)
Time, h
I subpopulation II subpopulation Superposition
log10(N0)
p > 1
δ1 δ2
α
(2) – (3) – s
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Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363
( ) ( ) ( ) ( )
( ) ( )
1
1 2
1 1
exp
exp 1 exp
1 exp 1 exp lag
k
k f k
k k
τ
τ τ
τ τ
⋅  − ⋅ + − ⋅ − ⋅ ⋅ 
+  ⋅ −  ⋅ − ⋅ 
(26)
Theoretically, there may be a significant difference
in the concentration of subpopulations (10 or more
times). As a result, the calculation of the share of the first
subpopulation f may be complicated due to its small
correction from unity. To solve this problem, Geeraerd
proposed to substitute the indicator f with its more
variable form:
lg
1
f
f
α
 
=   −
(27)
Then:
10
1 10
f
α
α =
+ (28)
A wide variety of primary models implies the lack
of universality of each of them. It turns the application
of such models into their adjusting to specific
experimental data of microbial thermal inactivation
kinetics. At the same time, it is still unclear whether
the results of extrapolating these models outside the
scope of determining the experimental values of input
and output factors are adequate. The adequacy can
be partly justified with the arguments by the authors
of the models. However, as the authors of the models
suggested, the arguments are also very empirical.
Accordingly, the issue of adequate applicability
or non-applicability of empirical approaches to
determining thermal sterilisation regimes requires
comparative assessment of models and modes predicted
with the help of the approaches. The convergence of
their results as well as compliance with the existing
trends of the development of food technologies and
requirements for quality and safety of food products
should be taken into consideration.
Thus, the aim of this work was to analyse the
convergence of existing approaches to modelling the
microbial death kinetics during heat treatment. To
achieve this goal, we solved the following tasks:
– an analytical review of existing approaches to
modelling the microbial death kinetics during the heat
treatment;
– a comparative evaluation of (a) primary kinetic models
of microbial spore death; (b) secondary kinetic models
of microbial spore death; and (c) models describing rate
dynamics of microbial death kinetics.
STUDY OBJECTS AND METHODS
The research focused on the following objects:
the spore form of the guaiacol-positive strain of
Alicyclobacillus acidoterrestris RNCIM V-1008 from
the centre for culture collection of microorganisms, the
Laboratory of Quality and Food Safety of the Russian
Research Institute of Canning Technology.
The spore suspension of A. acidoterrestris was
prepared according to [37]. For this purpose, the culture
from the collection was activated by means of double
or triple relocation to the liquid nutrient enrichment
medium, i.e. the YSG medium (HiMedia Laboratories
Pvt. Ltd., India). Subsequently, the actively growing daily
culture was planted into Petri dishes with pre-prepared
BAT-agar (HiMedia Laboratories Pvt. Ltd., India).
For this purpose, 0.1–0.2 cm3 of the cell culture fluid
was evenly distributed on the surface of the medium
with a spatula. The platings were thermostated at 40°C
for 96 h. To detect the spores, the native sample was
studied with phase contrast microscopy, using the Zeiss
Axioscope microscope, equipped with Canon PC 1200
camera and original AxioVision Rel.4.8 software. The
culture contained light refractive shiny spores. Their
amount was not less than 70% compared to the total
number of cells. The spores produced on a solid nutrient
medium were washed off with a phosphate buffer (0.1M
aqueous solution of phosphate buffer, pH 6.98), according
to [38], approximately 10–15 cm3 solution per 75 cm2
surface. The spores were separated from the medium
by centrifuging the culture fluid at 275 g for 30 min.
Washing and centrifuging were repeated several times.
The washed sediment was suspended in the medium
of concentrated apple juice. The resulting suspension
had a spore concentration of not less than 107 CFU/g. To
inactivate the remaining vegetative cells, the suspension
was heated at 80°C for 10 min. The concentration of
spores in the suspension was determined by plating
appropriate dilutions on BAT-agar within Petri dishes. The
obtained suspension was used to determine the parameters
of thermal stability in the concentrated apple juice (ACJ).
The capillary method was used to determine
the parameters of thermal stability of the spores in
the studied juice. For this purpose, the medium was
contaminated by applying the spore suspension in
sterile conditions. The capillaries were thin-walled glass
tubes, 75 mm long, outer diameter of 3 mm. The spore
suspension was injected into capillaries by 0.1 cm3. Each
capillary contained spores at a concentration of 5.31 lg
(CFU/g). The filled capillaries were warmed up in the
circulating thermostat series LOIP LT-311 (Russia) in the
glycerine medium at 100°C and over and the aqueous
medium at temperatures below 100°C.
The contaminated samples were thermostated in
capillaries at 90 and 95°C for 420 s, 100°C for 300 s,
and 105°C for 150 s. The trough concentration of the
surviving spores was established after 0, 120, 240,
360 and 420 s for 90 and 95°C; after 0, 60, 120, 180,
240, and 300 s for 100°C, and after 0, 30, 60, 90, 120,
150 s for 105°С. The trough concentration of the
surviving spores was determined according to [37] by
direct inoculation method. The samples of 1 cm3 were
analysed using YSG-agars as dense nutrient media. The
initial processing of the inoculation results was carried
out according to [39]. All microbiological studies were
carried out in four-fold repetition, rejecting statistically
unreliable data.
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To determine the dynamics of heating the ACJ in
the least heated zone, the sterile medium was placed
in a glass jar of 100 cm3 with a twist-off lid and a
thermocouple element fixed by a vertical axis of the jar,
15 mm off the inside surface of the bottom. Hermetically
sealed jars with ACJ were thermostated at 90, 95, 100
and 105°C.
The temperature was measured in the least heated
point at 30-second intervals for 42 min. After that, the
jars were cooled in a water tank at 15°C for 10 min. The
automatic multichannel thermometer CTF9008 (Ellab
A/S) connected to the thermocouples was used for
temperature control during the heat treatment. To reduce
statistical error, each experiment was carried out in a
three-fold repetition, rejecting statistically unreliable data.
Mathematical processing and modelling was carried
out using a spreadsheet processor Microsoft Excel 2010
(Microsoft Corporation) with installed add-ons ‘Data
Analysis’, ‘Solution Search’ and ‘Parameter Selection’,
as well as specialised software — TableCurve 2D v.5.01
(SYSTAT Software Inc.) and Wolfram Mathematica 10.4
(Wolfram Research Inc.).
Approximation of the experimental data was
carried out under the following parameters of
TableCurve 2D Fitting Controls option of TableCurve
2D v.5.01: linear approximation was by Singular Value
Decomposition; the level of robustness (stability) of
nonlinear approximation was high (Pearson VII Lim);
minimisation by natural logarithm of the square root of
the sum ‘1 + squared remainder’.
RESULTS AND DISCUSSION
Comparative evaluation of primary kinetic
models of the microbial spore death. The literary
data review showed that at present a basic provision
underlying any approach to determining the thermal
sterilisation regimes for canned products is experimental
determination of the microbial death kinetics in the
analysed product with the subsequent approximation
and extrapolation of the obtained model. Therefore,
the initial criterion for assessing the adequacy of
the particular primary model application to describe
experimentally fixed kinetics is the convergence of
approximating (averaging) and interpolating properties
(corresponding to the numerical values in the nodes, i.e.
experimental points).
In the first approximation, this criterion is
numerically equivalent to the determination coefficient,
i.e. the square of the correlation coefficient. However,
not all models can be calculated directly. Thus, in this
study, linearising transformations were carried out
previously for a number of models, i.e. the Kamau
model, the Membre model and the Augustin model:
– the Kamau model
( ) 0 2
ln ln 1
K N k
N
τ
 ⋅  =  −  = ⋅
 
(29)
– the Membre model
( )
0
ln ln lg
10
M N k
N
τ
  
= −   = ⋅   ⋅ 
(30)
– the Augustin model
( ) 0
2
ln ln 1 lg
A N m
N s
  τ − =  −  =
 
(31)
The lower threshold of the determination coefficient
of 0.9 was adopted as a boundary condition determining
the applicability of the primary model for approximation
of the experimental data.
Thus, out of the described set of primary models,
only seven models complied with the experimental data
of survival kinetics of A. acidoterrestris. There were
two linear models (the Bigelow model and the Whiting-
Buchanan model) and five nonlinear models (the Weibull
model, the Fermi model, the Kamau model, the Membre
model and the Augustin model). In addition, in order to
expand the potential of the primary models used for the
Bigelow model, the activation energy of the microbial
spore death was calculated in each of the temperature
options according to Arrhenius. That indirectly
increased the number of analysed models. At the same
time, both the Bigellow model and its modification by
Arrhenius in its primary form were actually identical.
The analysis of the study results showed a somewhat
larger aggregate (for all temperature variants) adequacy
of nonlinear models at experimental data approximation
(Table 1). Thus, the determination coefficient did not fall
below 0.965 for all temperature variations in nonlinear
models. On the other hand, this coefficient decreased
up to 0.936 and 0.956 for the Bigelow model and the
Whiting-Buchanan model, respectively, in linear models.
The non-linear Membre model was the only exception,
comparable in aggregate adequacy to the Whiting-
Buchanan model.
It is noteworthy that the activation energy of the
microbial death, corresponding to the Bigellow model
as modified by Arrhenius, was not constant. When the
processing temperature was increasing, this value was
monotonously decreasing, which presumably confirms
the fact that many factors impact microbial resistance
to external adverse conditions. When the temperature
increases within the range of values corresponding
to proteins denaturation, the number of such factors
inevitably decreases. As a result, less energy is required
to reach the target effect of the microbial death.
Table 1 presents characteristic indicators and
determination coefficients corresponding to linear and
non-linear models.
Table 1 demonstrates the heterogeneity of the
approximation efficiency for any of the selected models
within the temperature values in the experiment
variants. It should be noted for most nonlinear models
(except the Augustin model) that the approximation
adequacy decreases when the linearity of experimentally
determined kinetics increases and vice versa. The latter
was established by the determination coefficient increase
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for linear models. It proves a low universality of the
primary (in fact – adjustable) models. The consequence
is an increase in the deviation of model-predicted values
of the surviving microbial spore concentrations when
extrapolating from the experimental data, intuitively
expected on the basis of visual estimation (Figs. 3 and 4).
Thus, at 105°C, extrapolating both linear and most
non-linear models towards increasing the duration
showed lower heat treatment efficiency than it was
intuitively assumed from the visual estimation of
experimental data. It is true even for the Augustin
model, approximating experimental data as formally
adequately as possible. Conclusions were formed on
the basis of modelling microbial death kinetics under
thermal influence and its subsequent extrapolation. They
were expressed in the form of the ratio of temperature
and duration of treatment to achieve a given sterilising
effect. Though, as a result of the above said, the
conclusions will be inevitably overrated against the true
state of affairs.
Table 1 Characteristic indicators and determination coefficients corresponding to linear and nonlinear models
Temperature,
°C
Linear models
Bigelow model Bigelow model modified
by Arrhenius
Whiting model
k, lg (CFU/g) /s D, min r2 E0, J/mol k, lg (CFU/g) /s τlag, s r2
90 1.089×10–3 15.31 0.9585 57 745 1.136×10–3 13.904 0.9611
95 2.964×10–3 5.62 0.9876 55 183 2.985×10–3 2.85×10–11 0.9875
100 5.560×10–3 3.00 0.9932 53 980 5.591×10–3 1.226 0.9933
105 14.291×10–3 1.17 0.9355 51 889 16.357×10–3 13.041 0.9564
Non-linear models
Weibull model Membre model Kamau model
p δ, s r2 k, s–1 r2 k, s–1 r2
90 1.17 797.976 0.9722 9.14×10–4 0.9748 3.802×10–3 0.9764
95 340.534 0.9711 20.31×10–4 0.9526 8.663×10–3 0.9652
100 187.498 0.9853 35.44×10–4 0.9651 15.848×10–3 0.9841
105 74.419 0.9686 82.25×10–4 0.9960 38.245×10–3 0.9698
Fermi model Augustin model
b, s–1 τlag, s r2 m, ln (s) s, [lg (s)]1/2 r2
90 5.045×10–3 182.908 0.9806 2.432 0.561 0.9957
95 8.701×10–3 3.30×10–11 0.9649 1.965 0.496 0.9980
100 15.834×10–3 2.02×10–11 0.9822 1.789 0.436 0.9975
105 51.767×10–3 39.110 0.9949 1.596 0.356 0.9881
Figure 3 Approximation of experimental data of microbial
death kinetics of A. acidoterrestris during heat treatment
(t = 105°C) by using linear models
Figure 4 Approximation of the experimental data of microbial
survival kinetics of A. acidoterrestris during heat treatment
(t = 105°C) by using nonlinear models
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
0 30 60 90 120 150 180 210
Concentration, log10 (CFU/g)
Time, s
Weibull Fermi Kamau
Membre Augustin
3.5
4.0
4.5
5.0
5.5
6.0
Concentration, log10 (CFU/g)
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
0 30 60 90 120 150 180 210
Concentration, log10 (CFU/g)
Time, s
Weibull Fermi Kamau
Membre Augustin
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
0 20 40 60 80 100 120 140 160 180 200 220
Concentration, log10 (CFU/g)
Time, s
Whiting-Buchanan Bigelow Experiment
(1)
(2)
(3)
(5)
(4)
(1) (2) (3)
(4) (5)
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Thus, finding the primary model satisfying all
variants of experimentally established kinetics even in
the field of determining variable independent factors is a
non-trivial problem. This problem appears in conditions
of empiricism and simplification, and in many cases
phenomenology of both existing primary models and
approaches to their development. This conclusion
is compounded by the ambiguity of extrapolating
properties of the models. However, the whole further
algorithm of determining the sterilisation regimes was
built on these properties.
Comparative evaluation of secondary kinetic
models of the microbial spore death kinetics. The
numerical values of coefficients for primary microbial
death models are different when calculated as a result
of approximation of experimental data at different
temperatures of heat treatment. It suggests some
dependence of these values on temperature. The format
of primary models has a certain logic though simplified.
Unlike them, the functional dependencies of these
models coefficients on the processing temperature
are exclusively adjustable functions. These functions
approximate an array of numerical values most
efficiently and extrapolate logically. They extend the
value of the independent factor in any direction beyond
the scope of determining experimental values.
Thus, the coefficients of linear primary models
depend on temperature as follows:
– the Bigelow and Whiting-Buchanan models:
lg k a T
b
= − + (32)
– the Bigelow model:
lg D a T
b
= − (33)
– the Whiting-Buchanan model:
lag lg a b exp exp T c T c 1
d d
τ
  −  −  = + ⋅ −  −  − +     
(34)
– the Bigelow model as modified by Arrhenius:
0 lg E = −a +T ⋅b (35)
This approach resulted in differences in functional
description of secondary models: the Bigelow model
and the Bigelow model as modified by Arrhenius. The
coefficient k carried the function of the kinetic rate
of the thermal microbial death and in the first case it
depended on temperature linearly (in semilogarithmic
coordinates). Conversely, in the second case, this
dependence acquired nonlinearity due to the complex of
activation energy and the processing temperature. The
activation energy itself had a linear dependence on the
processing temperature in semi-logarithmic coordinates:
0 ( )
exp 10
a T b
abs
k N
R T T
 − + ⋅ 
= ⋅ − 
 ⋅ + 
(36)
The coefficients of nonlinear primary models
depended on temperature as follows:
– the Weibull model:
lg a T
b
δ = − (37)
– the Fermi model:
lgb a T
b
= − + (38)
( ) ln 1 2 lg exp 1
2 lag
T c
a b
d
τ
  ⋅ −   = + ⋅ − ⋅   
   
(39)
– the Kamau model and the Membre model:
lg k a T
b
= − + (40)
– the Augustin model:
lgm a T
b
= − (41)
lg s a T
b
= − + (42)
The coefficients are functionally expressed through
formulae (32)–(42). In general, they allowed obtaining
more informative – secondary – analogues for each of
the analysed primary models (Figs. 5–8).
It is mandatory to bear in mind that primary
models represent some degree of approximation of the
functional idea of the microbial death dynamics as a
result of thermal effect at the specified temperature.
In its turn, the functional dependence of coefficients
on temperature allows determining their values in the
process of inter- and/or extrapolation with a certain
degree of approximation as well. Piling up with the
error generated by the primary model, it determines the
total degree of approximation to the experimental data
in determining the temperature and processing time.
This also sets some uncertainty in further extrapolation.
The consequence of this conclusion presents itself in
the degree to which secondary models comply with the
experimental data on the basis of which these models
were obtained.
The models were superimposed on the array of
experimental microbiological data in Figs. 5–8. Colour
variations indicated the consistency of experimental
data, taking into account their errors, and the model.
The green cubes show experimental data that exceeded
the concentration of the surviving microbial spores
calculated on the basis of the model. Black triangles
show the data having values below the calculated ones.
Blue spheres indicate values the same as the model
within the error range of experimental data.
The analysis of the convergence of experimental
data and models showed that there is a poor convergence
with the kinetics at 95°C in the case of linear primary
models. In addition, the secondary models – the
Whiting-Buchanan model (Fig. 5b) and the Bigelow
model as modified Arrhenius (Fig. 6a) – showed a
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greater progression in the microbial death kinetics
at 100°C compared to the kinetics as determined by
the experimental data. However, these same models
showed unsatisfactory extrapolative properties at 105°C
towards overrating the concentration of the surviving
microorganisms.
There was even a greater heterogeneity in the case of
secondary models based on nonlinear primary models.
Thus, the secondary model based on the Fermi model
(Fig. 7a) has the visually worst approximation properties
and, as a consequence, extrapolative characteristics.
This model overestimates the calculated concentration
of the surviving microorganisms for the most part in
comparison with the experimental values. The Weibull
model (Fig. 6b) is characterised by almost the same
disadvantages as those noted for the Whiting-Buchanan
model and the Bigelow model modified by Arrhenius.
The same statement, but to a lesser extent, can
be applied to the secondary models, i.e. the Kamau
model (Fig. 7b) and the Membre model (Fig. 8a). Of
all secondary models, the Augustin model was the
most appropriate in terms of convergence with the
experimental data. However, it was also characterised
by overestimating the calculated values of surviving
microbial concentration at the treatment temperature of
95°C, as well as overestimating the calculated values at
extrapolation at the temperature of 105°C.
However, the graphical analysis of the secondary
models showed that each of them was characterised
(b)
Figure 6 Secondary linear model and nonlinear model of microbial death A. acidoterrestris during heat treatment. (a) Bigelow model modified by Arrhenius, (b) Concentration, log10 Concentration, log10 (CFU/g) (CFU/g)
Time, s
Temperature, oC
(a) (b)
Figure 5 Secondary linear models of microbial death of A. acidoterrestris during heat treatment. (a) Bigelow model, (b) Whiting-
Buchanan model
(a)
(b)
Figure 5 Secondary linear models of microbial death of A. acidoterrestris during heat treatment.
(a) Bigelow model, (b) Whiting-Buchanan model.
(a)
Temperature, oC
Temperature, oC
Concentration, log10 (CFU/g)
Time, s
Time, s
Time, s
Concentration, log10 (CFU/g)
Concentration, log10 (CFU/g)
Temperature, oC
(a)
(b)
Figure 5 Secondary linear models of microbial death of A. acidoterrestris during heat treatment.
(a) Bigelow model, (b) Whiting-Buchanan model.
(a)
Temperature, oC
Temperature, oC
Concentration, log10 (CFU/g)
Time, s
Time, s
Time, s
Concentration, log10 (CFU/g)
Temperature, oC
(a) (b)
Figure 6 Secondary linear model and nonlinear model of microbial death A. acidoterrestris during heat treatment. (a) Bigelow
model modified by Arrhenius, (b) Weibull model
(a)
(b)
Figure 5 Secondary linear models of microbial death of A. acidoterrestris during heat treatment.
(a) Bigelow model, (b) Whiting-Buchanan model.
(a)
Temperature, oC
Temperature, oC
Concentration, log10 (CFU/g)
Time, s
Time, s
Time, s
Concentration, log10 (CFU/g)
Concentration, log10 (CFU/g)
Temperature, oC
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by the continuous area in the ‘duration – temperature’
coordinates. It was true for this area that a trough
concentration of surviving microbial spores reached the
value, not exceeding a given threshold. This fitted well
into the formal logic that ‘a higher temperature value
corresponds to a shorter processing time’. In Figs. 5–8,
the trough concentration of surviving microbial spores
of 10–2 CFU/g was randomly chosen as such threshold
value. It corresponded to the sterilising effect of
reducing microbial concentrations from the initial value
by more than seven orders.
Comparative evaluation of rate dynamics models
for microbial death kinetics. The graphical display
of the secondary models was characterised by external
homogeneity (if connection to the experimental data
was removed). However, the key factor for the overall
microbial death kinetics was the change rate indicator of
concentration of the surviving microorganisms after the
heat treatment. The process rate was defined as a value
derived from the kinetics of the analysed index (in this
case, the concentration of microorganisms).
Therefore, if there was a functional dependence
reproducing the kinetics of the analysed index, the
rate could be defined as the first time derivative. If the
dynamics of microbial concentration increase was
negative, which occurred during heat treatment, the
rate value was also be negative. However, for greater
convenience and clarity without levelling adequacy,
(a) (b)
Figure 7 Secondary nonlinear models of microbial death of A. acidoterrestris during heat treatment (a) the Fermi model,
(b) Kamau model
(b)
linear model and nonlinear model of microbial death A. acidoterrestris during heat treatment. (a) Bigelow model modified by Arrhenius, (b) Weibull model.
(a)
(b)
Figure 7 Secondary nonlinear models of microbial death of A. acidoterrestris during heat treatment
(a) the Fermi model, (b) Kamau model.
Concentration, log10 (CFU/g) Concentration, log10 (CFU/g)
Time, s
Time, s
Time, s
Temperature, oC
Temperature, oC
Temperature, oC
(a)
(b)
Figure 7 Secondary nonlinear models of microbial death of A. acidoterrestris during heat treatment
(a) the Fermi model, (b) Kamau model.
Concentration, log10 (CFU/g)
Time, s
Time, s
Temperature, oC
Temperature, oC
(a) (b)
Figure 8 Secondary nonlinear models of microbial death of A. acidoterrestris during heat treatment (a) Membre model,
(b) Augustin model
(a)
Concentration, log10 (CFU/g) Concentration, log10 (CFU/g)
Time, s
Temperature, oC
(a)
b)
Figure 8 Secondary nonlinear models of microbial death of A. acidoterrestris during heat treatment
(a) Membre model, (b) Augustin model.
Concentration, log10 (CFU/g)
Concentration, log10 (CFU/g)
(CFU/g)/s
Time, s
Time, sec
Temperature, oC
oC
(a)
Concentration, log10 (CFU/g) Time, s
Temperature,
oC
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we used the rate value with the opposite sign in further
calculations and representations. Thus, the rate of
microbial death kinetics corresponding to the secondary
models could be expressed as the following:
– the Bigelow and Whiting-Buchanan model:
10
a T
υ b − + = (43)
where υ is the rate of microbial death kinetics,
lg (CFU/g) /s;
– the Bigelow model modified by Arrhenius:
0 ( )
exp 10
a T b
abs
N
R T T
υ
 − ⋅ 
= ⋅ − 
 ⋅ + 
(44)
– the Weibull model:
1 10
T p a
υ p τ τ b − − + = ⋅ ⋅ ⋅   
 
(45)
– the Kamau model:
( )
10
1 exp 10 ln 10
a T
b
a T
b
υ
τ
− +
 +  − ⋅ − +  ⋅      
=
(46)
– the Fermi model:
( ) ( )
1
10
1 exp 10 10 10 ln 10
b
b
b lag
b lag lag
a T
b
a b T a T b b τ
τ τ τ
υ
− +
 
− + − ⋅ ⋅  
    +  − ⋅ ⋅
  
=
  
⋅
(47)
where b a is the coefficient corresponding to the
coefficient a in formula (38);
b b is the coefficient corresponding to the coefficient
b in formula (38);
lag aτ is the coefficient corresponding to the coefficient
a in formula (39);
lag bτ is the coefficient corresponding to the coefficient
b in formula (39);
– the Membre model:
10 exp 10
a T a T
υ b τ b − + − + ⋅ = ⋅   
  (48)
– the Augustin model:
( )
( )
( )
( ) ( )
2 2 2 2
2 2 2
ln
10 exp 10 10
ln 10
ln
1 exp 10 10 ln 10
ln 10
s s m
s s m
s m
s m
a T a T a T
b b b
a T a T
b b
τ
τ
υ
τ
⋅ ⋅
− ⋅ + − ⋅ + −
⋅
− + −
  
⋅   −    
     +   − ⋅ ⋅      
= (49)
where m a is the coefficient corresponding to the
coefficient a in formula (41);
m b is the coefficient corresponding to the coefficient
b in formula (41);
s a is the coefficient corresponding to the coefficient
a in formula (42);
s b is the coefficient corresponding to the coefficient
b in formula (42).
Models of the rate dynamics for microbial death
kinetics during the heat treatment are featured in
Figs. 9–12.
Analysis of formulae (43)–(49) and their graphical
representation showed that all linear models (Figs. 9
and 10a) were invariant in terms of the heat treatment
duration, while nonlinear models (Figs. 10b, 11, 12)
included the time component.
The rate growth intensity of microbial death kinetics
differed visually almost twice even in externally
similar linear models, such as the Bigelow model and
the Bigelow model modified by Arrhenius. It occurred
within graphically represented area of temperature
determination of the heat treatment (80–130°C).
Some nonlinear models (the Weibull model, the
Fermi model and the Kamau model) demonstrated a
pronounced effect of treatment duration on the rate
(a) (b)
Figure 9 Model of the rate dynamics for microbial death kinetics of A. acidoterrestris during heat treatment.(a) Bigelow model,
(b) the Bigelow model modified by Arrhenius
(b)
Figure 9 Model of the rate dynamics for microbial death kinetics of A. acidoterrestris during heat treatment.(a) Bigelow model, (b) the Bigelow model modified Rate, log10 (CFU/g)/s 10 (CFU/g)/s
Time, s
Temperature, oC
(a)
(b)
Figure 8 Secondary nonlinear models of microbial death of A. acidoterrestris during heat treatment
(a) Membre model, (b) Augustin model.
(a)
Concentration, log10 (CFU/g) Concentration, log10 (CFU/g)
Rate, log10 (CFU/g)/s
Time, s
Time, sec
Time, s
Temperature, oC
Temperature,
oC
Temperature, oC
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of microbial death kinetics only at the initial stages
of processing at temperatures over 100°C. For others
(the Membre model), the rate was determined by
the pronounced effect of both factors on the entire
graphically represented area of determination. In the
Augustin model, the rate dynamics decreased to zero by
increasing both temperature and the processing duration.
Presumably, these differences did not have
any profound fundamental effect due to the initial
representation of the primary models in question.
The models were used for further constructions as
adjustable (empirical) functions that were not bound
to fundamental aspects, i.e. molecular and possibly
supermolecular mechanisms, which directly determine
the spore resistance to thermal treatment.
The dependence of the rate dynamics for microbial
death kinetics on the processing duration for nonlinear
models results in the need to determine the formal
starting moment of the heat treatment. This moment
should serve the starting point for measuring the real
duration in order to determine the actual values of the
sterilising effect for each temperature value obtained
during the experimental heating. Indeed, the initial
(starting) concentration of microbial spores in the
product in the real conditions of thermal sterilisation
was significantly (by 5–8 orders) lower than their initial
(a) (b)
Figure 10 Model of the rate dynamics for microbial death kinetics of A. acidoterrestris during heat treatment.
(a) Whiting-Buchanan model, (b) Weibull model
(b)
rate dynamics for microbial death kinetics of A. acidoterrestris during heat treatment.(a) Bigelow model, (b) the Bigelow model modified by Arrhenius.
(a)
(b)
of the rate dynamics for microbial death kinetics of A. acidoterrestris during heat treatment. (a) Whiting-Buchanan model, (b) Weibull model.
Rate, log Rate, log10 (CFU/g)/s 10 (CFU/g)/s
Time, s
Time, s
Time, s
Temperature, oC
Temperature, oC
Temperature, oC
(a)
(b)
Figure 10 Model of the rate dynamics for microbial death kinetics of A. acidoterrestris during heat treatment. (a) Whiting-Buchanan model, (b) Weibull Rate, log10 (CFU/g)/s Time, s
Time, s
Temperature, oC
Temperature, oC
(a) (b)
Figure 11 Model of rate dynamics for microbial death kinetics of A. acidoterrestris during heat treatment. (a) Fermi model,
(b) Kamau model
(a)
Rate, log10 (CFU/g)/s Rate, log10 (CFU/g)/s
Time, s
Temperature, oC
(a)
(b)
Figure 11 Model of rate dynamics for microbial death kinetics of A. acidoterrestris during heat treatment. (a) Fermi model, (b) Kamau model.
log10 (CFU/g)/s Rate, log10 (CFU/g)/s Rate, log10 (CFU/g)/s
Time, s
Time, s
Temperature, oC
Temperature, oC
360
Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363
concentration in microbiological experiment of heating
the contaminated product. Then, in general terms, the
conditions for achieving the starting microbial spore
concentration and the final concentration achieved at
an arbitrary time point under the known functional
dependences which approximated microbiological data
satisfactorily, we got the following form:
( )
( )
0
0
lg lg ,
lg lg ,
st st N N f T
N N f T
τ
τ
 = − 
 = −
(50)
where Nst is a starting microbial spores concentration in
the product, CFU/g;
τst is a theoretical duration of the heat treatment to
achieve the starting microbial concentration according to
the selected model (starting treatment duration), s.
In this case, at an arbitrary time point, the sterilising
effect n was defined as:
n lg lg ( , ) ( , ) st st = N − N = f τ T − f τ T (51)
The obvious consequence of formula (51) was the
conclusion that reaching n ≥ 0 required fulfilling the
condition τ ≥ τst.
To simplify the calculations in this study, the starting
microbial spore concentration was assumed equal to
1 CFU/g. Then lg st N = 0. In this case n = − f (τ ,T ). This
conclusion could be used in the calculation of sterilising
effects at each temperature value. For this, the calculated
sterilisation duration for each given temperature value at
the real moment of determination, adjusted to τst, must
be substituted into the formula of the rate of microbial
death kinetics.
The analysis of secondary models showed that
for a given Nst value τst was a function of the process
temperature. However, due to the peculiarities of
formulas describing secondary models, not each of
them could have an explicit form of dependence. In
this regard, the ratios {T, τst} in the area of temperature
determination from 80 to 130°C were numerically
determined for each model. Then they were
approximated with the following functions:
– the Bigelow model, the Weibull model, the Kamau
model and the Membre model:
lg( ) st τ = a − b ⋅T (52)
– the Bigelow model modified by Arrhenius:
lg( ) exp( ) st τ = a + b ⋅ −c ⋅T (53)
– the Whiting-Buchanan model and the Fermi model:
( ) lg 2 2 st  τ  = a + b ⋅T + c ⋅T (54)
– the Augustin model:
lg( ) exp st
a b T
c
τ = + ⋅  − 
 
(55)
Figure 13 shows graphical representations
of functional dependences of τst on temperature,
corresponding to both linear and non-linear models.
The visual analysis showed a relatively high degree
of variability of this dependence for nonlinear models. It
indicated the pronounced variability in the dynamics of
extrapolative properties for the different models studied
due to the expressed empirical character of the adjustable
functions. In its turn, this variability was presumably
conditioned by the particularity and, in some cases, the
phenomenological nature of the functions themselves,
while there was no adequate connection with the
fundamental mechanism of microbial spore inactivation.
In addition, this variability must inevitably lead to the
significant variability of the final regimes of thermal
sterilisation. The regimes could be determined by
(a) (b)
Figure 12 Model of rate dynamics for microbial death kinetics of A. acidoterrestris during heat treatment. (a) Membre model,
(b) Augustin model
(b)
Model of rate dynamics for microbial death kinetics of A. acidoterrestris during heat treatment. (a) Fermi model, (b) Kamau model.
(a)
Rate, log10 (CFU/g)/s Time, s
Time, s
Temperature, oC
Temperature, oC
(b)
Figure 12 Model of rate dynamics for microbial death kinetics of A. acidoterrestris during heat treatment. (a) Membre model, (b) Augustin model.
Rate, log10 (CFU/g)/s
Time, s Temperature, oC
361
Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363
applying the analysed models to the same array of initial
microbiological and thermophysical data.
CONCLUSION
According to the results of the research and
subsequent analysis, the single-phase primary models
(adjustable functions) are statistically acceptable
(at r2 > 0.9) for approximation of experimental data
on the kinetics of thermal death of Alicyclobacillus
acidoterrestris spores. The models were linear, namely
the Bigelow model, the Bigelow model modified by
Arrhenius, and the Whiting-Buchanan model, and
nonlinear, such as the Weibull model, the Weibull
model modified by Malfart et al., the Fermi model, the
Kamau model, the Membre model, and the Augustin
model. At the same time, nonlinear models approximate
experimental microbiological data on death kinetics of
microbial spores during the heat treatment statistically
more adequately.
The unsatisfactory convergence of extrapolation
results and dynamics caused by the rate models and
the temperature coefficient was shown for the first
time. In other words, the expressed empirical use of
(a) (b)
Figure 13 Dependence of starting treatment duration required for concentration of surviving A. acidoterrestris to reach Nst on
sterilisation temperature for linear (a) and nonlinear (b) models
adjustable functions was established analytically.
This was presumably conditioned by the particularity
and, in some cases, phenomenology of the functions
themselves. Other causes were the lack of the criteria
for the unambiguous choice of the original model and
the absence of adequate connection with a fundamental
mechanism of microbial spore inactivation based on the
targeted blocking of the system of spore germination
initialisation in combination with the conditions of the
environment.
Consequently, there is no reason to believe that heat
treatment regimes based on these empirical models
were the most effective and provided a maximum
sterilising effect at a minimum heat load. Thus, the
analysis was the first link in arguing the necessity to
initiate the research aimed at developing a methodology
for determining the regimes of thermal sterilisation for
food products including the analysis of the fundamental
factors of spore germination activation and their
resistance to external impact.
CONFLICT OF INTEREST
The authors state that there is no conflict of interest.

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