INTRODUCTIONMicrobiological safety is one of the key tasks indeveloping food production technology [1–4]. Mostexisting solutions are based on microbial inactivationvia chemical or physical methods, as well as competitivesubstitution of pathogenic and opportunistic pathogenicmicroflora with probiotic bacteria. At the same time,thermal sterilisation remains the most common way toachieve the required level of microbiological safety.The first stage in developing the thermal sterilisationmode is traditionally the analysis of the microbialdeath kinetics in the medium of the product for whichthis mode is developed [5]. Microbiological studies arecarried out under isothermal conditions and repeated atdifferent temperatures. Then the experimental data areapproximated with a particular model [6–8]. Naturally,the adequacy of the selected model to the experimentaldata confirms the adequacy of ideas about the microbialdeath kinetics in general and, in particular, regardingextrapolation at concentrations of microorganisms downto 1 CDU/g. Obtaining correct quantitative experimentalmicrobiological data for these concentrations isassociated with geometrically increasing resourceintensity and experimental error.According to Whiting and Bushanan, allmathematical models describing the microbial responseto the external negative effects can be divided into threelarge groups – primary, secondary and tertiary models.Thus, primary models approximate experimental data ofthe microbial death kinetics under isothermal conditionsResearch Article DOI: http://doi.org/10.21603/2308-4057-2019-2-348-363Open Access Available online at http:jfrm.ruComparative evaluation of approaches to modelling kineticsof microbial thermal death as in the caseof Alicyclobacillus acidoterrestrisVladimir V. Kondratenko* , Mikhail T. Levshenko , Andrey N. Petrov ,Tamara A. Pozdnyakova , Marina V. TrishkanevaRussian Research Institute of Canning Technology, Vidnoye, Russia* e-mail: nauka@vniitek.ruReceived June 05, 2019; Accepted in revised form July 26, 2019; Published October 21, 2019Abstract: 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 boundariesof experimental microbiological data zone. The wide range of primary models existence implies the lack of universality of eachones. This paper presents a comparative assessment of linear and nonlinear models of microbial death kinetics during the heattreatment of the Alicyclobacillus acidoterrestris spore form. The research allowed finding that single-phase primary models (asadjustable functions) are statistically acceptable for approximation of the experimental data: linear – the Bigelow’ the Bigelowas modified by Arrhenius and the Whiting-Buchanan models; and nonlinear – the Weibull, the Fermi, the Kamau, the Membreand the Augustin models. The analysis of them established a high degree of variability for extrapolative characteristics and, as aresult, a marked empirical character of adjustable functions, i.e. unsatisfactory convergence of results for different models. Thisis 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 mosteffective. This analysis is the first link in arguing the necessity to initiate the research aimed at developing a new methodologyfor determining the regimes of food thermal sterilisation based on analysis of the fundamental factors such as ones defined sporegermination activation and their resistance to external impact.Keywords: Microorganisms, death kinetics, survival kinetics, sterilising effect, Alicyclobacillus acidoterrestris, modelPlease 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 Alicyclobacillusacidoterrestris. 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 InternationalLicense (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. 2E-ISSN 2310-9599ISSN 2308-4057349Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363as a function of processing duration. Secondary modelsconnect primary models, approximating them as afunction of additional factors, which include primarilythe temperature of processing, when developingthermal sterilisation regimes. Tertiary models includethe software implementation of secondary models [9].However, we believe, it would be more logical if theadaptation of secondary models to non-isothermalsterilisation regimes within specified boundaryconditions was referred to tertiary models.The primary models of thermal inactivation ofmicroorganisms broke into development at the turn ofthe XXth century. Then, basing on the analogy with thefirst order chemical reaction, Chick, and later Bigelowpresented a model reflecting the kinetics of the microbialdeath resulted from external adverse factors (Bigelowmodel) as a linear kinetic model of the first order [10,11]. Bigelow showed later that in general the kineticssought for can be adequately represented in semilogarithmiccoordinates [12]. Due to the simplicity ofform and further manipulations, this model has becomeclassic: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 definedas a constant), lg (CFU/g)·min–1 (time may be expressedin s, h, etc.);τ is a duration of processing, min (time may beexpressed in s, hour, etc.).This model is based on the assumptions that allcells of microorganisms have the same resistance to thethermal impact in the processed product, and their deathkinetics complies with statistical regularities [9, 13]. Asa result, the death of each individual cell is consideredfrom the point of view of accidental inactivation of the‘critical molecule’.To avoid pure empiricalness, some researchersattempted to adjust the Bigelow model. They expressedthe dependence of the microbial death rate on thetemperature of the process similarly to the dependenceof chemical kinetics on the activation energy accordingto the Arrhenius equation [5]:( )00 expabsk N ER 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 ofwater (273.16 K);T is the process temperature, °C.However, Peleg et al. question the adequacy of thisapproach in their work [14]. They argue this with thecardinal difference of the microbial death kinetics fromthe conventional chemical kinetics. The differencemanifests itself in the absence of microflora inactivationunder normal conditions and, therefore, in absenceof continuity of functional dependence of microbialconcentration on temperature over the entire range of itsdetermination.Nowadays, there are at least four main types ofkinetics of microbial death as a result of heat treatment,including linear (Fig. 1) [15].Kinetics with a lag phase is one of the frequentdeviations of microbial death kinetics [16]. This type ofdependence approximates satisfactorily the Whiting-Buchanan model proposed by Whiting and Buchanan[17, 18]:( )00lg ,lglg ,laglag lagNNN kτ ττ τ τ τ ≤ = − ⋅ − > (3)where τlag is a lag phase duration.Van Boekel carried out a large-scale study andanalysed more than 120 curves of microbial deathkinetics. He concluded that linear models described nomore than 5 percent of cases. It implies that this kineticsis an exception rather than the rule [19]. Van Boekelsuggested that non-linear models should be used forthe most adequate description of kinetics. It should beunderstood that nonlinearity of models is determinedprimarily by their parameters, since the graphicalrepresentation resulting from approximation can haveboth nonlinear and linear views [16]. The simplestnonlinear model, the Weibull model, is based on the ideaof statistical distribution of probability for the death ofcells and/or microbial spores under the adverse externalconditions as a result of their individual variability [20]:0 lg lgpN Nτδ= −     (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 kineticsPopulation density, CFU/mLTime, h350Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363It is notable that at P > 1 the model represents aconvex curve, while at P < 1 the curve is concave.When P = 1, the model becomes identical to the Bigelowmodel. The coefficientδ becomes equivalent to the valueof D in the interpretation of the same model. It probablygave Mafart a reason to adopt the coefficient δ as theduration of the process required to reduce the microbialconcentration by one order [20].Bhaduri et al. [21] were the first who showed that theempirical modified Gompertz equation, the Gompertzmodel, can effectively describe the thermal deathkinetics 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 thatsatisfactorily approximates the microbiological data ofkinetics with ‘tails’, i.e. the first Casolari model:0lg lg lg 11 TN Nb τ = −    + ⋅ (6)where bT is a coefficient.He linked the microbial death to the criticalactivity of water molecules when their energy exceedsa threshold value E0, thus combining the probabilitytheory and Maxwell energy distribution:( ) 22002a expTH absb N EM 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 presentedthe modification of this model, i.e. the second Casolarimodel. It included a quadratic dependence on theprocessing duration:0 2lg lg lg 11 TN Nb τ = +    + ⋅ (8)In their turn, Daugthry et al. [24] proposed anexponentially decreasing model, i.e the Daugthrymodel. They justified its advantage over linear modelsdue to the approximation accuracy of the experimentaldata on the death kinetics of Escherichia coli andStaphylococcus aureus( ) 0 lg lg exp d N = N − k ⋅τ ⋅ −λ ⋅τ (9)where k is the initial rate of inactivation;d λis the descending factor.Data in [25, 26] demonstrate the expediency of alogistic function, i.e. the Fermi model, for describingthe kinetics of microbial death limited by a number ofstressful factors:( )( ) 01 explg lg lg1 explaglagbN Nbττ 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:( )000lglg lg41 explglagN N NNωσ τ τω−= + ⋅ ⋅ − +   − (11)where ω is the value of the lower asymptote ofmicrobial death kinetics, lg (CFU/g);σ is the maximum rate of kinetics.The model satisfactorily described the thermalinactivation 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 inactivationof Salmonella typhimurium. They assumed that themodel could extrapolate on other microorganisms andproducts [28]:( ) ( ) 0 lg N = 1+ lg N − exp k ⋅τ (12)Another kinetic model, the Kamau model, isproposed by Kamau et al. [29] in relation to Listeriamonocytogenes and Staphylococcus aureus:0 ( )lg lg lg 21 exp dN Nk τ = +   + ⋅ (13)where kd is a coefficient.The complex logistic model, the Baranyi model,was developed as an alternative to the Gompertzmodel described by formula (5). It takes into accountthe advantages of the Gompertz model and levels itsdisadvantages. Xiong et al. [30] modified the Baranyimodel, which took the form:{ ( ) ( ) }0lg lg 1 exp b b m TN q q BN= + − ⋅ −μ ⋅ τ −  (14)( )22 21 ln 3 arctg 2 3 arctg 13 2 3 3 Tr r T T r Br r T T r  +   ⋅ −    = ⋅  ⋅   + ⋅  + ⋅     − ⋅ +   ⋅   ( )22 21 ln 3 arctg 2 3 arctg 13 2 3 3 Tr r T T r Br r T T r  +   ⋅ −    = ⋅  ⋅   + ⋅  + ⋅     − ⋅ +   ⋅   (15)where qb is an indicator of the ‘tail’ of kinetics, reflectingits manifestation or absence;m μ is a maximum relative level of thermalinactivation;r is a lag parameter, numerically equal to half m μ ;BT is a coefficient.A distinctive feature of the Baranyi model is thepossibility to derive the kinetic model of the first orderon its basis.Geeraerd et al. [23] proposed the Geeraerd modelas a complex approach to describing the complexkinetics of thermal microbial inactivation. The modeltakes into account the presence or absence of the ‘tail’of the function, and also a lag phase in microorganismsresponding to the thermal effect:( ) ( ) ( ( ) lg lg 0 lg explg 10 10 10 exp1 exp 1 res res N N N laglagkN kk τττ  ⋅ = + − ⋅ − ⋅ ⋅   +  ⋅ −  ⋅351Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363) ( ) ( )( ) ( )lg 0 lg exp10 10 exp1 exp 1 expres N N laglagkkk kτττ τ ⋅ − ⋅ − ⋅ ⋅   +  ⋅ −  ⋅ − ⋅ (16)The analysis of these models shows that in manycases logistic functions are mathematical abstractionsthat describe experimental microbiological data moreor less satisfactorily. The Augustin model proposedin [31] and originally intended to describe the kineticsof thermal inactivation of Listeria monocytogenes is noexception0 2lg N lg N lg 1 exp lg ms  τ −  = −  +     (17)where m and s are coefficients.However, due to the commonality of this model withother logistic functions, this model may be ostensiblyapplicable to the description of thermal inactivationkinetics of other microorganisms.The Fernandez model stands apart. It was proposedin [32] and based on the statistical approach toestimating 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]:( )20lg N 1 exp a b ln c lnN= −  + ⋅ τ + ⋅ τ  (19)where a, b and c are coefficients.The model is a modified polynomial function with allproperties characteristic of approximation functions ofthis class. The properties are satisfactory interpolationand sensitivity to approximated data. Extrapolation ofthis model is possible, but it must be carried out withcaution and mandatory experimental validation.Analysing the existing array of the experimentalmicrobiological data, Cerf [34] was one of the firstto who suggested that the deviation of kinetics fromlinear (in semilogarithmic coordinates) is most likely aconsequence of simultaneous presence of at least twomicrobial subpopulations with different resistance toexternal negative effects in a genetically homogeneouspopulation. The deviation may also be caused byartefacts (generated by a set of perturbation factors thatare not taken into account, or are not levelled in theformulation and execution of studies). Figure 2 presentsa graphical interpretation of this approach.In relation to thermal inactivation kinetics, the resultof this conclusion is the use of two-phase models, takinginto account the contribution of each subpopulationto the integral response. Models of this kind weredeveloped 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 testculture;1 and 2 are indices of coefficients belonging to thesubpopulation.The Kamau model is described by the formula:( )( )0 ( )1 22 2 1 lg lg lg1 exp 1 expf f N Nb τ b τ ⋅ ⋅ − = +  +  + ⋅ + ⋅ (21)The Whiting-Buchanan model:( )( )( ) ( ( )1 201 21 exp 1 1 explg lg lg1 exp 1 explag lag lagf b f bN Nb bτ ττ τ τ τ ⋅  + − ⋅  − ⋅  + − ⋅ = +  +  +  ⋅ −  +  ⋅ − ( )( )( ) ( )( )1 201 21 exp 1 1 explg lg lg1 exp 1 explag laglag lagf b f bN Nb bτ ττ τ τ τ ⋅  + − ⋅  − ⋅  + − ⋅   = +  +  +  ⋅ −  +  ⋅ −  (22)The Coroller model:( )1 21 20 lg lg lg 10 1 10p pN N f fτ τδ δ   −  −     = +  ⋅ + − ⋅  (23)The Xiong model:00,lg l( ),laglagNN fτ ττ 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 2lg lg lg exp 1 exp1 exp N N f k f kτ τ = +  ⋅ − ⋅ + − ⋅ − ⋅ ⋅  +  Figure 2 Graphical interpretation of the subpopulationapproach to the microbial death kinetics during heat treatment(adapted from [36])01234560 2 4 6 8 10 12 14 16 18log10(N)Time, hI subpopulation II subpopulation Superpositionlog10(N0)p > 1δ1 δ2α(1)(2)(3)(1) –01234560 2 4 6 8 10 12 14 16 18log10(N)Time, hI subpopulation II subpopulation Superpositionlog10(N0)p > 1δ1 δ2α01234560 2 4 6 8 10 12 14 16 18log10(N)Time, hI subpopulation II subpopulation Superpositionlog10(N0)p > 1δ1 δ2α01234560 2 4 6 8 10 12 14 16 18log10(N)Time, hI subpopulation II subpopulation Superpositionlog10(N0)p > 1δ1 δ2α(2) – (3) – s352Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363( ) ( ) ( ) ( )( ) ( )11 21 1expexp 1 exp1 exp 1 exp lagkk f kk kττ ττ τ⋅  − ⋅ + − ⋅ − ⋅ ⋅ +  ⋅ −  ⋅ − ⋅ (26)Theoretically, there may be a significant differencein the concentration of subpopulations (10 or moretimes). As a result, the calculation of the share of the firstsubpopulation f may be complicated due to its smallcorrection from unity. To solve this problem, Geeraerdproposed to substitute the indicator f with its morevariable form:lg1ffα =   −(27)Then:101 10fαα =+ (28)A wide variety of primary models implies the lackof universality of each of them. It turns the applicationof such models into their adjusting to specificexperimental data of microbial thermal inactivationkinetics. At the same time, it is still unclear whetherthe results of extrapolating these models outside thescope of determining the experimental values of inputand output factors are adequate. The adequacy canbe partly justified with the arguments by the authorsof the models. However, as the authors of the modelssuggested, the arguments are also very empirical.Accordingly, the issue of adequate applicabilityor non-applicability of empirical approaches todetermining thermal sterilisation regimes requirescomparative assessment of models and modes predictedwith the help of the approaches. The convergence oftheir results as well as compliance with the existingtrends of the development of food technologies andrequirements for quality and safety of food productsshould be taken into consideration.Thus, the aim of this work was to analyse theconvergence of existing approaches to modelling themicrobial death kinetics during heat treatment. Toachieve this goal, we solved the following tasks:– an analytical review of existing approaches tomodelling the microbial death kinetics during the heattreatment;– a comparative evaluation of (a) primary kinetic modelsof microbial spore death; (b) secondary kinetic modelsof microbial spore death; and (c) models describing ratedynamics of microbial death kinetics.STUDY OBJECTS AND METHODSThe research focused on the following objects:the spore form of the guaiacol-positive strain ofAlicyclobacillus acidoterrestris RNCIM V-1008 fromthe centre for culture collection of microorganisms, theLaboratory of Quality and Food Safety of the RussianResearch Institute of Canning Technology.The spore suspension of A. acidoterrestris wasprepared according to [37]. For this purpose, the culturefrom the collection was activated by means of doubleor triple relocation to the liquid nutrient enrichmentmedium, i.e. the YSG medium (HiMedia LaboratoriesPvt. Ltd., India). Subsequently, the actively growing dailyculture was planted into Petri dishes with pre-preparedBAT-agar (HiMedia Laboratories Pvt. Ltd., India).For this purpose, 0.1–0.2 cm3 of the cell culture fluidwas evenly distributed on the surface of the mediumwith a spatula. The platings were thermostated at 40°Cfor 96 h. To detect the spores, the native sample wasstudied with phase contrast microscopy, using the ZeissAxioscope microscope, equipped with Canon PC 1200camera and original AxioVision Rel.4.8 software. Theculture contained light refractive shiny spores. Theiramount was not less than 70% compared to the totalnumber of cells. The spores produced on a solid nutrientmedium were washed off with a phosphate buffer (0.1Maqueous solution of phosphate buffer, pH 6.98), accordingto [38], approximately 10–15 cm3 solution per 75 cm2surface. The spores were separated from the mediumby centrifuging the culture fluid at 275 g for 30 min.Washing and centrifuging were repeated several times.The washed sediment was suspended in the mediumof concentrated apple juice. The resulting suspensionhad a spore concentration of not less than 107 CFU/g. Toinactivate the remaining vegetative cells, the suspensionwas heated at 80°C for 10 min. The concentration ofspores in the suspension was determined by platingappropriate dilutions on BAT-agar within Petri dishes. Theobtained suspension was used to determine the parametersof thermal stability in the concentrated apple juice (ACJ).The capillary method was used to determinethe parameters of thermal stability of the spores inthe studied juice. For this purpose, the medium wascontaminated by applying the spore suspension insterile conditions. The capillaries were thin-walled glasstubes, 75 mm long, outer diameter of 3 mm. The sporesuspension was injected into capillaries by 0.1 cm3. Eachcapillary contained spores at a concentration of 5.31 lg(CFU/g). The filled capillaries were warmed up in thecirculating thermostat series LOIP LT-311 (Russia) in theglycerine medium at 100°C and over and the aqueousmedium at temperatures below 100°C.The contaminated samples were thermostated incapillaries at 90 and 95°C for 420 s, 100°C for 300 s,and 105°C for 150 s. The trough concentration of thesurviving 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 thesurviving spores was determined according to [37] bydirect inoculation method. The samples of 1 cm3 wereanalysed using YSG-agars as dense nutrient media. Theinitial processing of the inoculation results was carriedout according to [39]. All microbiological studies werecarried out in four-fold repetition, rejecting statisticallyunreliable data.353Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363To determine the dynamics of heating the ACJ inthe least heated zone, the sterile medium was placedin a glass jar of 100 cm3 with a twist-off lid and athermocouple element fixed by a vertical axis of the jar,15 mm off the inside surface of the bottom. Hermeticallysealed jars with ACJ were thermostated at 90, 95, 100and 105°C.The temperature was measured in the least heatedpoint at 30-second intervals for 42 min. After that, thejars were cooled in a water tank at 15°C for 10 min. Theautomatic multichannel thermometer CTF9008 (EllabA/S) connected to the thermocouples was used fortemperature control during the heat treatment. To reducestatistical error, each experiment was carried out in athree-fold repetition, rejecting statistically unreliable data.Mathematical processing and modelling was carriedout using a spreadsheet processor Microsoft Excel 2010(Microsoft Corporation) with installed add-ons ‘DataAnalysis’, ‘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 wascarried out under the following parameters ofTableCurve 2D Fitting Controls option of TableCurve2D v.5.01: linear approximation was by Singular ValueDecomposition; the level of robustness (stability) ofnonlinear approximation was high (Pearson VII Lim);minimisation by natural logarithm of the square root ofthe sum ‘1 + squared remainder’.RESULTS AND DISCUSSIONComparative evaluation of primary kineticmodels of the microbial spore death. The literarydata review showed that at present a basic provisionunderlying any approach to determining the thermalsterilisation regimes for canned products is experimentaldetermination of the microbial death kinetics in theanalysed product with the subsequent approximationand extrapolation of the obtained model. Therefore,the initial criterion for assessing the adequacy ofthe particular primary model application to describeexperimentally fixed kinetics is the convergence ofapproximating (averaging) and interpolating properties(corresponding to the numerical values in the nodes, i.e.experimental points).In the first approximation, this criterion isnumerically equivalent to the determination coefficient,i.e. the square of the correlation coefficient. However,not all models can be calculated directly. Thus, in thisstudy, linearising transformations were carried outpreviously for a number of models, i.e. the Kamaumodel, the Membre model and the Augustin model:– the Kamau model( ) 0 2ln ln 1K N kNτ ⋅  =  −  = ⋅ (29)– the Membre model( )0ln ln lg10M N kNτ  = −   = ⋅   ⋅ (30)– the Augustin model( ) 02ln ln 1 lgA N mN s  τ − =  −  = (31)The lower threshold of the determination coefficientof 0.9 was adopted as a boundary condition determiningthe applicability of the primary model for approximationof the experimental data.Thus, out of the described set of primary models,only seven models complied with the experimental dataof survival kinetics of A. acidoterrestris. There weretwo linear models (the Bigelow model and the Whiting-Buchanan model) and five nonlinear models (the Weibullmodel, the Fermi model, the Kamau model, the Membremodel and the Augustin model). In addition, in order toexpand the potential of the primary models used for theBigelow model, the activation energy of the microbialspore death was calculated in each of the temperatureoptions according to Arrhenius. That indirectlyincreased the number of analysed models. At the sametime, both the Bigellow model and its modification byArrhenius in its primary form were actually identical.The analysis of the study results showed a somewhatlarger aggregate (for all temperature variants) adequacyof nonlinear models at experimental data approximation(Table 1). Thus, the determination coefficient did not fallbelow 0.965 for all temperature variations in nonlinearmodels. On the other hand, this coefficient decreasedup to 0.936 and 0.956 for the Bigelow model and theWhiting-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 themicrobial death, corresponding to the Bigellow modelas modified by Arrhenius, was not constant. When theprocessing temperature was increasing, this value wasmonotonously decreasing, which presumably confirmsthe fact that many factors impact microbial resistanceto external adverse conditions. When the temperatureincreases within the range of values correspondingto proteins denaturation, the number of such factorsinevitably decreases. As a result, less energy is requiredto reach the target effect of the microbial death.Table 1 presents characteristic indicators anddetermination coefficients corresponding to linear andnon-linear models.Table 1 demonstrates the heterogeneity of theapproximation efficiency for any of the selected modelswithin the temperature values in the experimentvariants. It should be noted for most nonlinear models(except the Augustin model) that the approximationadequacy decreases when the linearity of experimentallydetermined kinetics increases and vice versa. The latterwas established by the determination coefficient increase354Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363for linear models. It proves a low universality of theprimary (in fact – adjustable) models. The consequenceis an increase in the deviation of model-predicted valuesof the surviving microbial spore concentrations whenextrapolating from the experimental data, intuitivelyexpected on the basis of visual estimation (Figs. 3 and 4).Thus, at 105°C, extrapolating both linear and mostnon-linear models towards increasing the durationshowed lower heat treatment efficiency than it wasintuitively assumed from the visual estimation ofexperimental data. It is true even for the Augustinmodel, approximating experimental data as formallyadequately as possible. Conclusions were formed onthe basis of modelling microbial death kinetics underthermal influence and its subsequent extrapolation. Theywere expressed in the form of the ratio of temperatureand duration of treatment to achieve a given sterilisingeffect. Though, as a result of the above said, theconclusions will be inevitably overrated against the truestate of affairs.Table 1 Characteristic indicators and determination coefficients corresponding to linear and nonlinear modelsTemperature,°CLinear modelsBigelow model Bigelow model modifiedby ArrheniusWhiting modelk, lg (CFU/g) /s D, min r2 E0, J/mol k, lg (CFU/g) /s τlag, s r290 1.089×10–3 15.31 0.9585 57 745 1.136×10–3 13.904 0.961195 2.964×10–3 5.62 0.9876 55 183 2.985×10–3 2.85×10–11 0.9875100 5.560×10–3 3.00 0.9932 53 980 5.591×10–3 1.226 0.9933105 14.291×10–3 1.17 0.9355 51 889 16.357×10–3 13.041 0.9564Non-linear modelsWeibull model Membre model Kamau modelp δ, s r2 k, s–1 r2 k, s–1 r290 1.17 797.976 0.9722 9.14×10–4 0.9748 3.802×10–3 0.976495 340.534 0.9711 20.31×10–4 0.9526 8.663×10–3 0.9652100 187.498 0.9853 35.44×10–4 0.9651 15.848×10–3 0.9841105 74.419 0.9686 82.25×10–4 0.9960 38.245×10–3 0.9698Fermi model Augustin modelb, s–1 τlag, s r2 m, ln (s) s, [lg (s)]1/2 r290 5.045×10–3 182.908 0.9806 2.432 0.561 0.995795 8.701×10–3 3.30×10–11 0.9649 1.965 0.496 0.9980100 15.834×10–3 2.02×10–11 0.9822 1.789 0.436 0.9975105 51.767×10–3 39.110 0.9949 1.596 0.356 0.9881Figure 3 Approximation of experimental data of microbialdeath kinetics of A. acidoterrestris during heat treatment(t = 105°C) by using linear modelsFigure 4 Approximation of the experimental data of microbialsurvival kinetics of A. acidoterrestris during heat treatment(t = 105°C) by using nonlinear models2.02.53.03.54.04.55.05.50 30 60 90 120 150 180 210Concentration, log10 (CFU/g)Time, sWeibull Fermi KamauMembre Augustin3.54.04.55.05.56.0Concentration, log10 (CFU/g)2.02.53.03.54.04.55.05.50 30 60 90 120 150 180 210Concentration, log10 (CFU/g)Time, sWeibull Fermi KamauMembre Augustin2.02.53.03.54.04.55.05.56.00 20 40 60 80 100 120 140 160 180 200 220Concentration, log10 (CFU/g)Time, sWhiting-Buchanan Bigelow Experiment(1)(2)(3)(5)(4)(1) (2) (3)(4) (5)355Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363Thus, finding the primary model satisfying allvariants of experimentally established kinetics even inthe field of determining variable independent factors is anon-trivial problem. This problem appears in conditionsof empiricism and simplification, and in many casesphenomenology of both existing primary models andapproaches to their development. This conclusionis compounded by the ambiguity of extrapolatingproperties of the models. However, the whole furtheralgorithm of determining the sterilisation regimes wasbuilt on these properties.Comparative evaluation of secondary kineticmodels of the microbial spore death kinetics. Thenumerical values of coefficients for primary microbialdeath models are different when calculated as a resultof approximation of experimental data at differenttemperatures of heat treatment. It suggests somedependence of these values on temperature. The formatof primary models has a certain logic though simplified.Unlike them, the functional dependencies of thesemodels coefficients on the processing temperatureare exclusively adjustable functions. These functionsapproximate an array of numerical values mostefficiently and extrapolate logically. They extend thevalue of the independent factor in any direction beyondthe scope of determining experimental values.Thus, the coefficients of linear primary modelsdepend on temperature as follows:– the Bigelow and Whiting-Buchanan models:lg k a Tb= − + (32)– the Bigelow model:lg D a Tb= − (33)– the Whiting-Buchanan model:lag lg a b exp exp T c T c 1d dτ  −  −  = + ⋅ −  −  − +     (34)– the Bigelow model as modified by Arrhenius:0 lg E = −a +T ⋅b (35)This approach resulted in differences in functionaldescription of secondary models: the Bigelow modeland the Bigelow model as modified by Arrhenius. Thecoefficient k carried the function of the kinetic rateof the thermal microbial death and in the first case itdepended on temperature linearly (in semilogarithmiccoordinates). Conversely, in the second case, thisdependence acquired nonlinearity due to the complex ofactivation energy and the processing temperature. Theactivation energy itself had a linear dependence on theprocessing temperature in semi-logarithmic coordinates:0 ( )exp 10a T babsk NR T T − + ⋅ = ⋅ −  ⋅ + (36)The coefficients of nonlinear primary modelsdepended on temperature as follows:– the Weibull model:lg a Tbδ = − (37)– the Fermi model:lgb a Tb= − + (38)( ) ln 1 2 lg exp 12 lagT ca bdτ  ⋅ −   = + ⋅ − ⋅      (39)– the Kamau model and the Membre model:lg k a Tb= − + (40)– the Augustin model:lgm a Tb= − (41)lg s a Tb= − + (42)The coefficients are functionally expressed throughformulae (32)–(42). In general, they allowed obtainingmore informative – secondary – analogues for each ofthe analysed primary models (Figs. 5–8).It is mandatory to bear in mind that primarymodels represent some degree of approximation of thefunctional idea of the microbial death dynamics as aresult of thermal effect at the specified temperature.In its turn, the functional dependence of coefficientson temperature allows determining their values in theprocess of inter- and/or extrapolation with a certaindegree of approximation as well. Piling up with theerror generated by the primary model, it determines thetotal degree of approximation to the experimental datain determining the temperature and processing time.This also sets some uncertainty in further extrapolation.The consequence of this conclusion presents itself inthe degree to which secondary models comply with theexperimental data on the basis of which these modelswere obtained.The models were superimposed on the array ofexperimental microbiological data in Figs. 5–8. Colourvariations indicated the consistency of experimentaldata, taking into account their errors, and the model.The green cubes show experimental data that exceededthe concentration of the surviving microbial sporescalculated on the basis of the model. Black trianglesshow the data having values below the calculated ones.Blue spheres indicate values the same as the modelwithin the error range of experimental data.The analysis of the convergence of experimentaldata and models showed that there is a poor convergencewith the kinetics at 95°C in the case of linear primarymodels. In addition, the secondary models – theWhiting-Buchanan model (Fig. 5b) and the Bigelowmodel as modified Arrhenius (Fig. 6a) – showed a356Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363greater progression in the microbial death kineticsat 100°C compared to the kinetics as determined bythe experimental data. However, these same modelsshowed unsatisfactory extrapolative properties at 105°Ctowards overrating the concentration of the survivingmicroorganisms.There was even a greater heterogeneity in the case ofsecondary models based on nonlinear primary models.Thus, the secondary model based on the Fermi model(Fig. 7a) has the visually worst approximation propertiesand, as a consequence, extrapolative characteristics.This model overestimates the calculated concentrationof the surviving microorganisms for the most part incomparison with the experimental values. The Weibullmodel (Fig. 6b) is characterised by almost the samedisadvantages as those noted for the Whiting-Buchananmodel and the Bigelow model modified by Arrhenius.The same statement, but to a lesser extent, canbe applied to the secondary models, i.e. the Kamaumodel (Fig. 7b) and the Membre model (Fig. 8a). Ofall secondary models, the Augustin model was themost appropriate in terms of convergence with theexperimental data. However, it was also characterisedby overestimating the calculated values of survivingmicrobial concentration at the treatment temperature of95°C, as well as overestimating the calculated values atextrapolation at the temperature of 105°C.However, the graphical analysis of the secondarymodels 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, sTemperature, 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, oCTemperature, oCConcentration, log10 (CFU/g)Time, sTime, sTime, sConcentration, 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, oCTemperature, oCConcentration, log10 (CFU/g)Time, sTime, sTime, sConcentration, log10 (CFU/g)Temperature, oC(a) (b)Figure 6 Secondary linear model and nonlinear model of microbial death A. acidoterrestris during heat treatment. (a) Bigelowmodel 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, oCTemperature, oCConcentration, log10 (CFU/g)Time, sTime, sTime, sConcentration, log10 (CFU/g)Concentration, log10 (CFU/g)Temperature, oC357Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363by the continuous area in the ‘duration – temperature’coordinates. It was true for this area that a troughconcentration of surviving microbial spores reached thevalue, not exceeding a given threshold. This fitted wellinto the formal logic that ‘a higher temperature valuecorresponds to a shorter processing time’. In Figs. 5–8,the trough concentration of surviving microbial sporesof 10–2 CFU/g was randomly chosen as such thresholdvalue. It corresponded to the sterilising effect ofreducing microbial concentrations from the initial valueby more than seven orders.Comparative evaluation of rate dynamics modelsfor microbial death kinetics. The graphical displayof the secondary models was characterised by externalhomogeneity (if connection to the experimental datawas removed). However, the key factor for the overallmicrobial death kinetics was the change rate indicator ofconcentration of the surviving microorganisms after theheat treatment. The process rate was defined as a valuederived from the kinetics of the analysed index (in thiscase, the concentration of microorganisms).Therefore, if there was a functional dependencereproducing the kinetics of the analysed index, therate could be defined as the first time derivative. If thedynamics of microbial concentration increase wasnegative, which occurred during heat treatment, therate value was also be negative. However, for greaterconvenience 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, sTime, sTime, sTemperature, oCTemperature, oCTemperature, 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, sTime, sTemperature, oCTemperature, 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, sTemperature, 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)/sTime, sTime, secTemperature, oCoC(a)Concentration, log10 (CFU/g) Time, sTemperature,oC358Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363we used the rate value with the opposite sign in furthercalculations and representations. Thus, the rate ofmicrobial death kinetics corresponding to the secondarymodels could be expressed as the following:– the Bigelow and Whiting-Buchanan model:10a Tυ b − + = (43)where υ is the rate of microbial death kinetics,lg (CFU/g) /s;– the Bigelow model modified by Arrhenius:0 ( )exp 10a T babsNR T Tυ − ⋅ = ⋅ −  ⋅ + (44)– the Weibull model:1 10T p aυ p τ τ b − − + = ⋅ ⋅ ⋅    (45)– the Kamau model:( )101 exp 10 ln 10a Tba Tbυτ− + +  − ⋅ − +  ⋅      =(46)– the Fermi model:( ) ( )1101 exp 10 10 10 ln 10bbb lagb lag laga Tba b T a T b b ττ τ τυ− + − + − ⋅ ⋅      +  − ⋅ ⋅  =  ⋅(47)where b a is the coefficient corresponding to thecoefficient a in formula (38);b b is the coefficient corresponding to the coefficientb in formula (38);lag aτ is the coefficient corresponding to the coefficienta in formula (39);lag bτ is the coefficient corresponding to the coefficientb in formula (39);– the Membre model:10 exp 10a T a Tυ b τ b − + − + ⋅ = ⋅     (48)– the Augustin model:( )( )( )( ) ( )2 2 2 22 2 2ln10 exp 10 10ln 10ln1 exp 10 10 ln 10ln 10s s ms s ms ms ma T a T a Tb b ba T a Tb bττυτ⋅ ⋅− ⋅ + − ⋅ + −⋅− + −  ⋅   −         +   − ⋅ ⋅      = (49)where m a is the coefficient corresponding to thecoefficient a in formula (41);m b is the coefficient corresponding to the coefficientb in formula (41);s a is the coefficient corresponding to the coefficienta in formula (42);s b is the coefficient corresponding to the coefficientb in formula (42).Models of the rate dynamics for microbial deathkinetics during the heat treatment are featured inFigs. 9–12.Analysis of formulae (43)–(49) and their graphicalrepresentation showed that all linear models (Figs. 9and 10a) were invariant in terms of the heat treatmentduration, while nonlinear models (Figs. 10b, 11, 12)included the time component.The rate growth intensity of microbial death kineticsdiffered visually almost twice even in externallysimilar linear models, such as the Bigelow model andthe Bigelow model modified by Arrhenius. It occurredwithin graphically represented area of temperaturedetermination of the heat treatment (80–130°C).Some nonlinear models (the Weibull model, theFermi model and the Kamau model) demonstrated apronounced 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)/sTime, sTemperature, 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)/sTime, sTime, secTime, sTemperature, oCTemperature,oCTemperature, oC359Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363of microbial death kinetics only at the initial stagesof processing at temperatures over 100°C. For others(the Membre model), the rate was determined bythe pronounced effect of both factors on the entiregraphically represented area of determination. In theAugustin model, the rate dynamics decreased to zero byincreasing both temperature and the processing duration.Presumably, these differences did not haveany profound fundamental effect due to the initialrepresentation of the primary models in question.The models were used for further constructions asadjustable (empirical) functions that were not boundto fundamental aspects, i.e. molecular and possiblysupermolecular mechanisms, which directly determinethe spore resistance to thermal treatment.The dependence of the rate dynamics for microbialdeath kinetics on the processing duration for nonlinearmodels results in the need to determine the formalstarting moment of the heat treatment. This momentshould serve the starting point for measuring the realduration in order to determine the actual values of thesterilising effect for each temperature value obtainedduring the experimental heating. Indeed, the initial(starting) concentration of microbial spores in theproduct in the real conditions of thermal sterilisationwas 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)/sTime, sTime, sTime, sTemperature, oCTemperature, oCTemperature, 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, sTime, sTemperature, oCTemperature, 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)/sTime, sTemperature, 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)/sTime, sTime, sTemperature, oCTemperature, oC360Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363concentration in microbiological experiment of heatingthe contaminated product. Then, in general terms, theconditions for achieving the starting microbial sporeconcentration and the final concentration achieved atan arbitrary time point under the known functionaldependences which approximated microbiological datasatisfactorily, we got the following form:( )( )00lg lg ,lg lg ,st st N N f TN N f Tττ = −  = −(50)where Nst is a starting microbial spores concentration inthe product, CFU/g;τst is a theoretical duration of the heat treatment toachieve the starting microbial concentration according tothe selected model (starting treatment duration), s.In this case, at an arbitrary time point, the sterilisingeffect 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 theconclusion that reaching n ≥ 0 required fulfilling thecondition τ ≥ τst.To simplify the calculations in this study, the startingmicrobial spore concentration was assumed equal to1 CFU/g. Then lg st N = 0. In this case n = − f (τ ,T ). Thisconclusion could be used in the calculation of sterilisingeffects at each temperature value. For this, the calculatedsterilisation duration for each given temperature value atthe real moment of determination, adjusted to τst, mustbe substituted into the formula of the rate of microbialdeath kinetics.The analysis of secondary models showed thatfor a given Nst value τst was a function of the processtemperature. However, due to the peculiarities offormulas describing secondary models, not each ofthem could have an explicit form of dependence. Inthis regard, the ratios {T, τst} in the area of temperaturedetermination from 80 to 130°C were numericallydetermined for each model. Then they wereapproximated with the following functions:– the Bigelow model, the Weibull model, the Kamaumodel 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 sta b Tcτ = + ⋅  −  (55)Figure 13 shows graphical representationsof functional dependences of τst on temperature,corresponding to both linear and non-linear models.The visual analysis showed a relatively high degreeof variability of this dependence for nonlinear models. Itindicated the pronounced variability in the dynamics ofextrapolative properties for the different models studieddue to the expressed empirical character of the adjustablefunctions. In its turn, this variability was presumablyconditioned by the particularity and, in some cases, thephenomenological nature of the functions themselves,while there was no adequate connection with thefundamental mechanism of microbial spore inactivation.In addition, this variability must inevitably lead to thesignificant variability of the final regimes of thermalsterilisation. 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, sTime, sTemperature, oCTemperature, 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)/sTime, s Temperature, oC361Kondratenko V.V. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 348–363applying the analysed models to the same array of initialmicrobiological and thermophysical data.CONCLUSIONAccording to the results of the research andsubsequent analysis, the single-phase primary models(adjustable functions) are statistically acceptable(at r2 > 0.9) for approximation of experimental dataon the kinetics of thermal death of Alicyclobacillusacidoterrestris spores. The models were linear, namelythe Bigelow model, the Bigelow model modified byArrhenius, and the Whiting-Buchanan model, andnonlinear, such as the Weibull model, the Weibullmodel modified by Malfart et al., the Fermi model, theKamau model, the Membre model, and the Augustinmodel. At the same time, nonlinear models approximateexperimental microbiological data on death kinetics ofmicrobial spores during the heat treatment statisticallymore adequately.The unsatisfactory convergence of extrapolationresults and dynamics caused by the rate models andthe temperature coefficient was shown for the firsttime. 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 onsterilisation temperature for linear (a) and nonlinear (b) modelsadjustable functions was established analytically.This was presumably conditioned by the particularityand, in some cases, phenomenology of the functionsthemselves. Other causes were the lack of the criteriafor the unambiguous choice of the original model andthe absence of adequate connection with a fundamentalmechanism of microbial spore inactivation based on thetargeted blocking of the system of spore germinationinitialisation in combination with the conditions of theenvironment.Consequently, there is no reason to believe that heattreatment regimes based on these empirical modelswere the most effective and provided a maximumsterilising effect at a minimum heat load. Thus, theanalysis was the first link in arguing the necessity toinitiate the research aimed at developing a methodologyfor determining the regimes of thermal sterilisation forfood products including the analysis of the fundamentalfactors of spore germination activation and theirresistance to external impact.CONFLICT OF INTERESTThe authors state that there is no conflict of interest.