Use of modified Richards model to predict isothermal and non-isothermal microbial growth

(1)

h ttp : / / w w w . b j m i c r o b i o l . c o m . b r /

Food

Microbiology

Use

of

modiﬁed

Richards

model

to

predict

isothermal

and

non-isothermal

microbial

growth

Jhony

Tiago

Teleken

a

_,

_Alessandro

_Cazonatto

_Galvão

b

_,

_Weber

_da

_Silva

_Robazza

b,∗

a_Universidade_Federal_de_Santa_Catarina,_Departamento_de_Engenharia_Química_e_Engenharia_de_Alimentos,_{Florianópolis,}_SC,_Brazil b_Universidade_do_Estado_de_Santa_Catarina,_Departamento_de_Engenharia_de_Alimentos_e_Engenharia_Química,_Pinhalzinho,_SC,_Brazil

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received22December2016 Accepted18January2018 Availableonline7March2018 AssociateEditor:ElaineDeMartinis

Keywords:

Richardsmodel Predictivemicrobiology Non-isothermalregimens

a

b

s

t

r

a

c

t

Mathematicalmodelsareoftenusedtopredictmicrobialgrowthinfoodproducts.An impor-tantclassofthesemodelsinvolvestheadaptationofclassicalsigmoidfunctions,suchas theGompertzandlogisticfunctions.Thisstudyaimedtovalidatetheuseofthemodified Richardsmodelinvarioussituations,whichhavenotpreviouslybeentested.Themodel wasobtainedthroughsolvingasystemoftwodifferentialequationsandcouldbeapplied tobothisothermalandnon-isothermalenvironments.Totestandvalidatethismodel,we usedpublisheddatasetscontainingdataforthegrowthofPseudomonasspp.infish prod-ucts.Theresultsobtainedafterfittingthemodelshowedthatitcouldbeeffectivelyusedto describeandpredictthePseudomonasgrowthcurvesundervarioustemperatureregimens. However,theinfluenceoftheshapeparameteronthegrowthcurveisanissuethatneeds furtherevaluation.

Introduction

Thecontrol ofthe processing, packaging,distribution, and storageconditionsofmanyfoodproductsisofmajor impor-tancefortheassuranceoffoodsafety.Themicrobialstability of food is affected by several environmental factors such as temperature, pH, and water activity. These factors can undergosigniﬁcantchangesthroughouttheproductionchain, andfoodmaybecomepotentiallyunsafeduetothepresence ofspoilage and/orgrowthofpathogenic microorganisms.1,2

Consequently,the concentration ofmicroorganismsshould becontinuouslymonitoredduringthetransportationofraw products,until theyreach their ﬁnal destination. However,

∗ _{Corresponding}_author.

E-mail:[email protected](W.S.Robazza).

this task is virtually impossible to accomplish because

enumerating microorganisms during different stages of

food processing using traditional approaches is slow and expensive.3,4 _Therefore,_classical_microbial_evaluation,

espe-ciallythatofperishablefoods,isoflimitedvalueforpredictive purposessincethesefoodsareconsumedorsoldbeforethe resultsofmicrobiologicaltestsbecomeavailable.

Theuseofequationsandmathematicalmodelstopredict microbial growth has been adopted in predictive microbi-ology. This quantitative area enables users to objectively estimatetheeffectofprocessingandstorageoperationson themicrobiologicalsafetyandqualityoffood.5_The_main

mod-els commonlyappliedintheabove-mentionedﬁeldare the

modiﬁed Gompertz model and the Baranyi model.6–10 _The

https://doi.org/10.1016/j.bjm.2018.01.005

(2)

Baranyimodelisthoughttoprovideabettergoodness-of-ﬁt thanthemodiﬁedGompertzequation.10,11_It_should_also_be

mentionedthatthemodiﬁedGompertzmodellacksa

mech-anisticbasisandthattheBaranyimodelprovidesabiological interpretationforthedurationofthelagphase.12

In addition to the above-mentioned issues, when one

addressesquantitativemicrobialriskassessment,itis

impor-tant to develop microbial growth models that take into

account the inﬂuence of environmental changes on the

behavior of selected pathogens along the food production

and supply chain. In this respect, temperature changes

during different stages of the production chain process

are determinant factors affecting pathogen growth. Thus, the predictive models able to take into account the

inﬂu-ence of temperature changes on microbial growth may

contribute to the prediction of a food’s shelf life and/or riskassessment whendealing withspoilage or pathogenic microorganisms.4,9,13

Themainpurposeofthisstudywastovalidatea mathe-maticalmodeldescribingmicrobialgrowthinfoodproducts underconditionsthathadnotbeenpreviouslytested.Totest theabilityofthemodeltocorrelatewithexperimentaldata, six different isothermal datasets and four non-isothermal

datasets pertaining to Pseudomonas spp. growth in ﬁsh

productswere selected from the literature.The adjustable parametersobtainedwereusedtoestimatethekinetic

prop-erties, suchas the maximumspeciﬁc growth rate and the

durationofthelagphaseofthemicroorganismunder inves-tigation.

Materials

and

methods

Modeldescription

Inthisstudy,themodifiedRichardsmodelwastestedfor tem-peratureprofilesthat hadnotbeenpreviouslyevaluated. It hasalreadybeenshownthatthemodelcanbederivedfroma systemoftwodifferentialequations.14_The_first_equation

con-sistsofthemostelementarymodelbuildingblockdescribing microbialevolution15_and_can_be_written_as_follows:

dN(t)

dt =(N)N(t) (1)

whereN(t)representsthemicrobialloadand(N)corresponds tothespeciﬁcgrowthrateofthemicroorganism.

Theseconddifferentialequationisrelatedtothedensity dependenceofthespeciﬁcgrowthrate.Themodelisbasedon theusualassumptionthat(N)mustbeadecreasing func-tion of the population, due to the accumulation of waste metaboliteswithinthefoodenvironment.Itisassumed,as representedbyEq.(2),thatthisdecreasefollowsapowerlaw:

d(N)

dN =−˛N

m ₍₂₎

where˛isapositiveparameter,whichisassumedtodepend ononlytheenvironmentalconditionsandnotonthe partic-ularmicroorganismunderstudy.Theparametermisashape parameterandhasnobiologicalmeaning.

Isothermalenvironment

Theanalyticalsolutionofthesystemofequationscomposed ofEqs. (1) and (2) atthe initialconditions ofN(0)=N0 and (N0)=0 inanisothermalenvironmentisgivenbythe fol-lowingequation14_: N(t)=

⎧

⎪

⎨

⎪

⎩

Nf

⎧

⎪

⎨

⎪

⎩

Nm+1_f −Nm+1₀ Nm+1₀

e −t

0(m+1)Nm+1_f Nm+1_f −Nm+1₀

+1

⎫

⎪

⎬

⎪

⎭

−(1/(m+1)) ,m/=−1 N0e ln(Nf /N0)

1−e − t0 ln(Nf/N0)

, m=−1 (3)

Eq.(3)expressesthemicrobialload,N(t),asafunctionof theinitialpopulation,N0,theinitialspeciﬁcgrowthrate,0, theshapeparameter,m,andtheﬁnalpopulation,Nf.Itshould

benotedthat,ifm=−1,theequationissimilartoa reparame-terizedversionoftheGompertzmodelandifm=0,itissimilar toareparameterizedversionofthelogisticequation.Inthis study, the population, N(t), was replaced bylogN(t), which is similar tothe modiﬁed Gompertz equation. Eq.(3) with

m /= −1isareparameterizedversionoftheRichardsgrowth model,19_and_for_predictive_microbiology_purposes,_it_is_known

asamodiﬁedRichardsmodel.

TheapplicationofEq.(3)isusefulforstatic environmen-talconditions(e.g.,constanttemperature).ToobtainEq.(3),

the parameter ˛ was replaced by a function of the other

parametersofthemodel,14_as_can_be_observed_in_Eq.₍₄₎_._This

procedureholdsonlyforanisothermalenvironment. There-fore,thisparameterisnotusedtoﬁtthemodeltobacterial growthinanisothermalenvironment.

˛=

⎧

⎪

⎨

⎪

⎩

0(m+1)Nm+1_f

Nm+1 f −Nm+10

, m /= −1 0 ln(Nf)−ln(N0), m=−1 (4) Non-isothermalenvironment

Forsituationsthatdonotcorrespondtoisothermalgrowth, Eqs.(1)and(2)shouldbeused.Dynamicconditions,suchas an increaseor decrease of thetemperature, require that a

secondarymodel expressingthedependenceofthegrowth

rate on the temperature be used. In the current study, it wasselected theRatkowsky’ssquarerootmodelbecause it is largely adopted and validated in the ﬁeld of predictive microbiology.16,17 _Because_the _other _parameters ₍

0 and ˛) arenotfoundinothermodels,anotherempiricalsecondary modelsfortheseparameterswereincluded,andtheresulting differentialequationwasnumericallysolved.

Biologicalparameters

In the present study, the following biological parameters were evaluated: the net logarithmic growth, A, the maxi-mumspeciﬁcgrowthrate,max,andthedurationofthelag phase,.TheparametersAandmax canbeobtainedusing

(3)

Eqs.(5)and(6).Itshouldbementionedthatmaxisestimated throughthecalculationoftheinﬂectionpointofthegrowth curve.Thedurationofthelagphasewasestimatedusingthe procedureadoptedbyBuchananandCygnarowicz.18

A=log(Nf)−log(N0) (5) max= 0·(m+1)·Nm+2_f Nm+1_f −Nm+1₀ ·(m+2) (−m−2)/(m+1) ₍₆₎ Experimentaldata

Thebacterialgrowthdatawerecollectedfromtheliterature20

and extracted from published curves using GetData Graph

Digitizer2.24software(www.getdata-graph-digitizer.com).Six isothermal(0,2,5,8,10,and15◦C)andfournon-isothermal datasetscontainingPseudomonasspp.growthdatainﬁshmeat wereselected.Thenon-isothermaltemperatureproﬁlesused inthisstudycorrespondtosuddenshiftsintemperatureand canbeexpressedbyfunctionswithif-statements,asfollows:

T(t)=

⎧

⎪

⎨

⎪

⎩

T1 if t<t1 T2 if t1≤t<t2 T3 if t2≤t<t3 .. . ... ... Tn if tn−1≤t<tn (7) Numericalmethods

Fittingoftheprimaryandsecondarymodelstothe experimen-talmicrobialgrowthdatawasperformedusingthefitfunction ofthecurvefittingtoolboxavailableintheMatlabR2011b soft-ware,version7.13(MathWorks,Natick,MA,USA).Thistoolbox usesthenonlinearleastsquaresmethodandthetrust-region reflectiveofNewton’smethod.Theinitialestimateforeach parameterofthemodelwasobtainedthroughavisual inspec-tionofthecurves.

Thenumericalsolutionofthedifferentialequationsforthe non-isothermalgrowthcurveswasobtainedwiththeode23 functionavailableinMatlab.Theode23functionisa

general-purposealgorithmbasedonthe Runge–Kuttamethod.This

methodcanautomaticallyadjustthecomputationalstepsize toachieveadesiredaccuracywhennumericallysolving ordi-narydifferentialequations.

Statisticalanalysis

Theperformanceofthemodelwasevaluatedusingrootmean squareerror,RMSE(Eq. (8)),and thecoefﬁcient of determi-nation,R2_(Eq.₍₉₎_)._In_these_expressions,_N

expstandsforthe experimentaldata,Ncalstandsfortheresultpredictedbythe model,Ntrepresentsthetotalnumberofexperimentalpoints,

andkcorrespondstothenumberofparametersinthemodel.

RMSE=

1 Nt−k Nt

n=1

Nexp(n)−Ncal(n)

2 (8)

Fig.1– FitofEq.(5)tothedataforPseudomonasspp.growth inﬁshobtainedat()0◦C,()2◦C,()5◦C,(♦)8◦C,(+) 10◦C,and(䊉)15◦C. R2=1−

Nt n=1

Nexp(n)−Ncal(n)

2

Nt n=1

Nexp(n)−_N1 t

Nt n=1Ncal(n)

2 (9)

Acomparisonbetweentheexperimentalvaluesand the

responses predicted by the non-isothermal mathematical

modelwascarriedoutusingtheRMSEandthebias(BF)and accuracyfactors(AF),whichare givenbyEqs.(10)and(11), respectively21_: BF=10 1 Nt

_Nt n=1log

Ncal(n) Nexp(n)

(10) AF=10 1 Nt

_Nt n=1

log

Ncal(n) Nexp(n)

(11)

Results

Isothermalgrowth

In the present study, the primary model given by Eq. (3)

wasﬁttedtotheexperimentaldataobtainedatsixdifferent temperatures.ThecurvesobtainedareshowninFig.1.The residualsobtainedafterﬁttingthemodelareplottedinFig.2

and are shown tobe approximately independentand

nor-mallydistributed.However,itwasobservedformostofthe temperatures studiedthat theinitiallog countwasslightly underestimatedrelativetotheexperimentalresults.

Table 1 shows the numerical values estimated for the modelparameterswithinthe95%confidencelimit,aswellas theRMSEandR2_values._These_values_were_obtained_after fit-tingthemathematicalmodeltotheexperimentaldataforthe sixtemperaturesinvestigated.ThefitofthemodifiedRichards model tothe Pseudomonasspp.experimentaldataprovided asgoodaperformanceasthatofthefour-parameterlogistic modelusedintheoriginalresearch,20_with_an_R2_greater_than 0.983andanRMSElowerthan0.226logofcolonyformingunits (CFU)/g,asshowninTable1.

Someauthorshavereportedtheuseofotherempiricaland semi-mechanistic modelsto ﬁtthe data obtainedfor Pseu-domonas spp. growingin ﬁsh.Corradini andPeleg22 _used_a

(4)

Fig.2–PlotoftheresidualsobtainedaftertheﬁtofEq.(3)

totheexperimentaldatageneratedat()0◦C,()2◦C,() 5◦C,(♦)8◦C,(+)10◦C,and(䊉)15◦C.

modiﬁedversionofthelogisticequationtomodelthesame experimentaldatasetsthatwereusedhere.Theyconsidered

Y(t)=log[N(t)/N0]andincludedinthelogisticequationa con-stantthat satisﬁes thefollowingconditions: att=0, Y(t)=0 and as t→∞, Y(t)=constant. The empirical growth mod-elsadoptedinthestudiesofKoutsoumanis20_and_Corradini

andPeleg22_were_different_{modiﬁcations}_of_the_classical

Ver-hulstmodel.Lee23_showed_that_the_{semi-mechanistic}_Baranyi

model and the Huang model also provided an adequate

descriptionofPseudomonasspp.growth.Bothmodelsinclude anadaptationfunctionthatdescribesthedurationofthelag phaseintheelementarydifferentialequation(Eq.(1)).Inthe Baranyimodel,theadaptationfunctionquantiﬁesthe physio-logicalstateofthemicrobialcells,12_while_in_the_Huang_model

theadaptationfunctionisonlyanempiricalexpression (logis-ticfunction).24_Robazza_et_al.25_also_used_a_{semi-mechanistic}

modelbasedontheCentralLimitTheoremtomodelthesame experimentaldata.Thismodelprovidesalinkbetweenthe physiologicalstateofindividualcellsandthespeciﬁcgrowth rateofthepopulation.

Someotherauthorshavealsomodeledthegrowthof Pseu-domonasspp.inpoultry9,26 _and_pork_meat.2_The_Baranyi_and

modiﬁedGompertzmodelswereﬁttedtothegrowthdataof

Pseudomonasspp.inpoultrymeatobtainedunderﬁve isother-malconditions.9_Both_models_showed_{goodness-of-ﬁt}_results

closetowhatwasexpectedforallofthedatasetsadopted, withtheRMSEvalueslowerthan0.1477andthepseudo-R2 valuesgreaterthan0.9779.Lietal.2_studied_Pseudomonas_spp.

growthinporkmeatatsixisothermalconditionsand eval-uatedthreeprimarymodelsfordataﬁttingtoselectthebest

Table2–Summaryofthebiologicalparameters estimatedforeachtemperaturestudied.

T(◦C) A(log10CFU/g) max(h−1) (h)

0 5.59 0.024 126.577 2 5.23 0.030 74.424 5 5.34 0.044 33.699 8 5.03 0.063 28.690 10 4.42 0.072 8.302 15 4.10 0.111 4.503

one.ThemodiﬁedGompertz,BaranyiandHuangmodelswere usedinthestudy,butnosinglemodelcouldprovidea consis-tently preferablegoodness-of-ﬁtmeasureforallthegrowth data.Lietal.26_showed_that_the_Baranyi_model_was_more

suit-ablethantheGompertzandHuangmodelsforﬁttingthedata forPseudomonasspp.growth inchicken productsunderﬁve differentstoragetemperatures.

AsshowninTable1,theshapeparametermisstatistically lessstablethantheotherparameters.Theresultssuggestthat this parametershouldbeﬁxedpriortoﬁttingthemodelto thedata.Table2summarizestheresultsobtainedforthese parametersforeachtemperatureadopted.

FromtheresultspresentedinTable2,itcanbeobserved thatthevaluesobtainedfortheparameters,mainlymaxand ,aredifferentfromthoseobtainedinpreviousresearchusing thesame datasets.20 _This_result_is_not_surprising,_since_the

twostudiesuseddifferentmathematicalmodelsand differ-entproceduresforevaluationoftheparameters.Theseresults canalsoexplaintheoverestimationofthedurationofthelag phase.ItcanbeconfirmedbyvisualinspectionofFig.1thatthe resultspresentedinTable2donotreproducetheactual val-uesofthelagphaseduration.However,themajordrawback ofthemodelisthemparameter.Sinceitcannotbeestimated withconfidence,smallvariationsinthisparametercan signifi-cantlychangethevaluesoftheremainingparameters.Despite thegoodperformanceofthemodel,whichisreflectedinthe coefficientofdeterminationvaluespresentedinTable1,there maybemanycombinationsofthemodelparametersthatcan providesimilarfits.AscanbeseenfromTable1,evenforthe smallrangeoftemperaturesadopted(from0◦Cto15◦C)inthe presentstudy,itwasobtainedalargevariationinthevaluesof theparameterm(over100%,rangingfrom3.204to6.607).Thus, itisdifficulttoestablishasinglevalueaprioritothis param-eter.Somecommonlyusedmodels,liketheBaranyi model, arbitrarilyfixesthisvalueasbeingequalto1.7,8_However,_from

theresultspresentedinTable1,ithardlycanbesaidthatthis value isthe bestoption forthe datasets analyzed. There-fore, it wasopted not toestablish asinglevalueto m and

Table1–Summaryofthemodelparametersandstatisticalindicesobtainedforisothermalgrowthforeachtemperature studied. T(◦C) Nf N0 m 0 RMSE R2 0 8.849±0.339 2.960±0.189 6.607±6.443 0.0041±0.0006 0.1554 0.9963 2 7.954±0.193 2.726±0.213 5.504±5.111 0.0059±0.0010 0.1449 0.9962 5 8.301±0.281 2.934±0.240 3.204±3.361 0.0096±0.0022 0.1706 0.9941 8 8.027±0.204 2.993±0.251 5.496±7.034 0.0123±0.0028 0.1638 0.9949 10 8.078±0.275 3.659±0.247 4.880±6.660 0.0143±0.0040 0.1919 0.9891 15 8.110±0.332 4.013±0.393 5.958±11.212 0.0209±0.0088 0.2262 0.9834

(5)

Fig.3–ObservedandpredictedgrowthcurvesofPseudomonasspp.undernon-isothermalconditions:(A)temperature profile1,(B)temperatureprofile2,(C)temperatureprofile3and(D)temperatureprofile4.

ﬁttheresultingequationtoexperimentaldata.Instead,the focusconsistedinevaluatingthebehaviorofminarangeof conditionsasacasestudyforasystematicuseofthemodel infuturestudies.

Non-isothermalgrowth

Toevaluatenon-isothermalgrowth,itwasconsideredthat˛ and0 couldbeexpressedasfunctionsofthetemperature inEq.(2). Thus,the solution ofthis equation providesthe followingresult: =0(T)− ˛(T) m+1

Nm+1₋_Nm+1 0 Nm+1_f

(12)

Insertion of Eq. (12) into Eq. (1), with the temperature proﬁle considered in terms of ˛(T) and 0(T), provides a differentialequationthatmustbenumericallysolvedto gen-eratepredictionsfornon-isothermalgrowth.Thetemperature dependenceoftheseparameterswasobtainedbyﬁtting lin-ear(0)andsquare-root(˛)modelstothedatafromTable1. Theseexpressionswereempiricallyselectedandhaveno spe-cialmeaning.TheresultscanbeexpressedbyEqs.(13)and (14):

0(T)=0.0011T+0.0038, R2=0.9953 (13)

˛(T)=0.0139T+0.1630, R2₌_0.9515 ₍₁₄₎

Theother parameters ofthe model (m and Nf) did not

signiﬁcantlychangewithtemperature.Thus,themean

val-ues of m and Nf were determined to be 5.275 and 8.220,

Table3–Summaryofthestatisticalindicesobtainedfor eachnon-isothermaltemperatureproﬁlestudied.

Temperatureproﬁle RMSE BF AF

Proﬁle1 0.4504 1.0433 1.0547

Proﬁle2 0.9232 1.1065 1.1065

Proﬁle3 0.4343 1.0424 1.0579

Proﬁle4 0.4372 1.0372 1.0541

respectively.Sincetheseresultswere includedinthe conﬁ-denceintervalsoftheseparametersforallthetemperatures studied,itwasassumedthattheycouldbesafelyusedforthe ﬁttingprocess.Inotherwords,theassumptionisthatunder non-isothermalconditionsthemomentarygrowthrate coin-cideswiththeisothermalgrowthrateatthesametemperature atatimethatcorrespondstothemomentarygrowthlevel,as discussedbyCorradiniandPeleg.22

Fig.3showsthepredictionssuppliedbythemodelforthe non-isothermalconditions.Thedotsrepresentexperimental data,thecurvesrepresenttheﬁtsobtainedbythemodel,and thegraylinescorrespondtothetemperatureproﬁles.

Table 3 presents the results obtained forthe statistical indicesadoptedinthepresentstudy.

Discussion

Gospavicetal.9_observed_a_good_agreement_in_the_maximum

speciﬁc growth rates between the modiﬁed Gompertz and

Baranyimodels,butthedurationofthelagphasedidnotagree aswell.Theseauthorsexplainedthedifferencebasedonthe confidenceintervalscomputedforthemodelparameters.For themodifiedGompertzmodel,theconfidenceintervalsforthe

(6)

parameterswerelargerthanthoseobtainedfortheBaranyi model,indicatingthatthelagphaseparametersobtainedby theBaranyimodelweredeterminedwithahigherconﬁdence

than the ones obtainedby the modiﬁed Gompertz model.

Regardingthe modelevaluated inthepresent study,itcan beobservedfromtheresultspresentedinTable1,itcanbe observedthattheparameterswithadirectbiological interpre-tation,N0andNfaremorestableandcanbeestimatedwith

higherconﬁdence.Theparameter0isrelatedtotheinitial speciﬁcgrowthratewhichisclosetozeroatthelagphase.It canbeseenthatthisparameterislessstableforhigher tem-peratureswhichcanbeascribedtothesmallerdurationofthe lagphase.

FromanalysisofthedatapresentedinTable3,itcanbe con-cludedthat,ingeneral,themodiﬁedRichardsmodelshowed goodagreementwiththeexperimentaldata,asdemonstrated bytheBF valuesrangingfrom 1.0372to1.1065andthe AF valuesrangingfrom1.0541to1.1065.Comparedtothe logis-ticmodelusedforthesamedatasetsinpreviousresearch,20

theperformanceofthemodiﬁedRichardsmodelwasbetter,20

sincethestatisticalindiceswere,ingeneral,closertothose inthepresentstudy.Basedonthepredictionparameters(BF andAF),theperformanceofthemodiﬁedRichardsmodelwas similartothatoftheBaranyiand Huangsemi-mechanistic models23_and_Central_Limit_Theorem_model.25

TheBaranyimodelfurnishedBFvaluesthatrangedfrom 1.023to1.083andAFvaluesfrom1.053to1.087,whilethe

HuangmodelshowedBFvaluesfrom 1.009to1.081andAF

valuesthatrangedfrom1.044to1.089.Themodelbasedon theCentralLimitTheoremfurnishedBFvaluesthatranged from0.99to1.13andAFvaluesfrom1.04to1.13.Theseranges ofBFandAFvaluesobtainedbythenon-isothermalmodels areconsideredacceptable.Itwasdemonstratedthatthe mod-iﬁedRichardsmodelhadaslightdiscrepancycomparedwith theexperimentaldata,but onlyforProﬁle2.Similarresults wereobtainedforothermodelsusingthesameexperimental dataset.20,23,25

Bruckneretal.4 _developed_a_dynamic_model_by

combin-ingthemodiﬁedGompertzmodelastheprimarymodeland

theArrheniusequationasthesecondarymodeltopredictthe growthofPseudomonasspp.infreshporkand freshpoultry meats.Forthevalidationofthemodelundernon-isothermal conditions,ninedynamictemperaturescenarioswereused.In thisstudy,BFrangedfrom0.87to1.01forporkandfrom0.89to 1.04forpoultry,andAFvariedbetween1.05and1.24forpork andbetween1.05and1.16forpoultry.Koutsoumanisetal.27

developedandvalidated amicrobialmodelforPseudomonas

growthingroundmeat(beefandpork)underdynamic temper-atureconditions.TheBaranyimodelwasusedtopredictfood spoilageandthegrowthofthebacteriumunderfour differ-entﬂuctuatingtemperatureproﬁles,withtemperatureshifts from0to20◦C.Theperformanceofthemodelwasevaluated bythepercentageofrelativeerrors(%RE)and93.3%ofthe pseudomonadpredictionsfellwithinthe−10to10%REzone forallthescenariostested,whilenonefelloutsidethe−20to 20%REzone.27

Thedifferentpredictive abilities ofthe modelsfor non-isothermalproﬁlesare relatedtothe goodness-of-ﬁtofthe modelparameterswiththetemperatureandtothebehavior ofbacteriawhensubjectedtoenvironmentalstress.13,27 _The

lackofunderstandingofthephysiologyofbacteriawhen sub-jectedtostressconditions,suchastemperatureshifts,isstill anobstacle toimprovingthe growthmodels andobtaining betterpredictions.

Conclusions

Inthepresentstudy,amodifiedversionoftheRichardsgrowth modelwasevaluatedfordescribingmicrobialgrowthinfood products.Themodeliscomposedoffouradjustable parame-tersandprovidedaccuratefitsforisothermalenvironments.In addition,itwasvalidatedfornon-isothermalenvironments.It consistsinadifferentapproachtodescribemicrobialgrowth inanon-isothermalenvironment,sinceittakesintoaccount the influenceofthe shapeparameteronthe growthcurve. Sincethemodelcanbeconsideredareparameterizationof theRichardsgrowthmodelthatencompassesbothlogisticand Gompertzgrowthmodels,itcanbeexpectedthatthemodel, ingeneral,wouldprovideanaccuratefittothe experimen-taldata.However,thereisashapeparameterincludedinthe mathematicalequationsthatdoesnothaveabiological mean-ingandexhibitsanirregularbehaviorinnonlinearregression. Therefore,furtherworkisneededtostudythepropertiesof the model and the behavior ofits parameters inorder for themodeltobeappliedconfidentlyandsafelyinpredictive microbiology.

Conﬂicts

of

interest

Theauthorsdeclarenoconﬂictsofinterest.

Acknowledgment

WewouldliketothanktheConselhoNacionalde Desenvolvi-mento CientíﬁcoeTecnológico(CNPq)forﬁnancialsupport (Process484037/2013-7).

r

e

f

e

r

e

n

c

e

s

1.ZwieteringMH,deKoosJT,HasenackBE,deWittJC,Van’t RietK.Modelingofbacterialgrowthasafunctionof temperature.ApplEnvironMicrobiol.1991;57:1094–1101.

2.LiM,NiuH,ZhoG,etal.Analysisofmathematicalmodelsof

Pseudomonasspp.growthinpallet-packageporkstoredat

differenttemperatures.MeatSci.2013;93:855–864.

3.LebertI,LebertA.Quantitativepredictionofmicrobial behaviourduringfoodprocessingusinganintegrated modelingapproach:areview.IntJRefrig.2006;29:968–984.

4.BrucknerS,AlbrechtA,PetersenB,KreyenschmidtJ.A predictiveshelflifemodelasatoolfortheimprovementof qualitymanagementinporkandpoultrychains.Food

Control.2013;29:451–460.

5.McDonaldK,SunDW.Predictivefoodmicrobiologyforthe meatindustry:areview.IntJFoodMicrobiol.1999;52:1–27.

6.GibsonAM,BratchellN,RobertsTA.Predictingmicrobial growth:growthresponsesofSalmonellaeinalaboratory mediumasaffectedbypH,sodiumchlorideandstorage temperature.IntJFoodMicrobiol.1988;6:155–178.

(7)

7. BaranyiJ,RobertsTA,McClureP.Anon-autonomous differentialequationtomodelbacterialgrowth.Food

Microbiol.1993;10:43–59.

8. BaranyiJ,RobertsTA.Adynamicapproachtopredicting bacterialgrowthinfood.IntJFoodMicrobiol.1994;23:277–294.

9. GospavicR,KreyenshmidtJ,BrucknerS,PopovV,HaqueN. Mathematicalmodellingforpredictingthegrowthof

Pseudomonasspp.inpoultryundervariabletemperature

conditions.IntJFoodMicrobiol.2008;127:290–297.

10.ZhouK,GuiM,LiP,XingS,CuiT,PengZ.Effectofcombined functionoftemperatureandwateractivityonthegrowthof

Vibrioharveyi.BrazJMicrobiol.2012;43:1365–1375.

11.BuchananRL,WhitingRC,DamertWC.Whenissimplegood enough:acomparisonoftheGompertz,Baranyi,and three-phaselinearmodelsforﬁttingbacterialgrowthcurves.

FoodMicrobiol.1997;14:313–326.

12.BaranyiJ,RobertsTA.Mathematicsofpredictivefood microbiology.IntJFoodMicrobiol.1995;26:119–218.

13.LonghiDA,DalcantonF,AragãoGMF,CarcioﬁBAM,Laurindo JB.Assessingthepredictionabilityofdifferentmathematical modelsforthegrowthofLactobacillusplantarumunder non-isothermalconditions.JTheorBiol.2013;21:88–96.

14.TelekenJT,RobazzaWS,GomesGA.Mathematicalmodeling ofmicrobialgrowthinmilk.CienTecAlim.2011;31:891–896.

15.VanImpeJF,PoschetF,GeeraerdAH,VereeckenKM.Towards anovelclassofpredictivemicrobialgrowthmodels.IntJFood

Microbiol.2005;100:97–105.

16.RatkowskyDA,LowryRK,McMeekinTA,StokesAN, ChandlerRE.Modelforbacterialculturegrowthrate throughouttheentirebiokinetictemperaturerange.J

Bacteriol.1983;154:1222–1226.

17.RatkowskyDA,OlleyJ,McMeekinTA,BallA.Relationship betweentemperatureandgrowthrateofbacterialcultures.J

Bacteriol.1982;149:1–5.

18.BuchananRL,CygnarowiczML.Amathematicalapproach towarddeﬁningandcalculatingthedurationofthelag phase.FoodMicrobiol.1990;7:237–240.

19.RichardsFJ.Aﬂexiblegrowthfunctionforempiricaluse.J ExpBot.1959;10:290–301.

20.KoutsounamisK.Predictivemodelingofshelflifeofﬁsh undernonisothermalconditions.ApplEnvironMicrobiol.

2001;67:1821–1829.

21.RossT.Indicesforperformanceevaluationofpredictive modelsinfoodmicrobiology.JApplBacteriol.1996;81:501–508.

22.CorradiniMG,PelegM.Estimatingnon-isothermalbacterial growthinfoodsfromisothermalexperimentaldata.JAppl

Microbiol.2005;99:187–200.

23.LeeDS.Comparisonoftwomechanisticmicrobialgrowth modelstoestimateshelflifeofperishablefoodpackage underdynamictemperatureconditions.MathProblEng.

2014:1–7.

24.HuangL.GrowthkineticsofListeriamonocytogenesinbroth andbeeffrankfurters–determinationoflagphaseduration andexponentialgrowthrateunderisothermalconditions.J FoodSci.2008;73:235–242.

25.RobazzaWS,TelekenJT,GalvaoAC,MiorelliS,StolfDO. Applicationofamodelbasedonthecentrallimittheoremto predictgrowthofPseudomonasspp.inﬁshmeat.Food

BioprocessTechnol.2017;10:1685–1694.

26.LiM,LiY,HuangX,ZhoG,TianW.Evaluatinggrowthmodels

ofPseudomonasspp.inseasonedpreparedchickenstoredat

differenttemperaturesbytheprincipalcomponentanalysis (PCA).FoodMicrobiol.2014;40:41–47.

27.KoutsoumanisK,StamatiouA,SkandamisP,NychasGJE. Developmentofamicrobialmodelforcombinedeffectof temperatureandpHonspoilageofgroundmeat,and validationofthemodelunderdynamictemperature conditions.ApplEnvironMicrobiol.2006;72:124–134.