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Dynamic kinetic analysis of growth of Listeria monocytogenes in pasteurized cow milk

Open ArchivePublished:January 14, 2021DOI:https://doi.org/10.3168/jds.2020-19442

      ABSTRACT

      The objective of this study was to develop a dynamic model for predicting the growth of Listeria monocytogenes in pasteurized cow milk under fluctuating temperature conditions during storage and temperature abuse. Six dynamic temperature profiles that simulated random fluctuation patterns were designed to change arbitrarily between 4 and 30°C. The growth data collected from 3 independent temperature profiles were used to determine the kinetic parameters and construct a growth model combining the primary and secondary models using a 1-step dynamic analysis method. The results showed that the estimated minimum growth temperature and maximum cell concentration were 0.6 ± 0.2°C and 7.8 ± 0.1 log cfu/mL (mean ± standard error), with the root mean square error (RMSE) only 0.3 log cfu/mL for model development. The model and the associated kinetic parameters were validated using the data collected under both dynamic and isothermal conditions, which were not used for model development, to verify the accuracy of prediction. The RMSE of prediction was approximately 0.3 log cfu/mL for fluctuating temperature profiles, and it was between 0.2 and 1.1 log cfu/mL under certain isothermal temperatures (2–30°C). The resulting model and kinetic parameters were further validated using 3 growth curves at 4, 7, and 10°C arbitrarily selected from ComBase (www.combase.cc). The RMSE of prediction was 0.8, 0.4, and 0.5 log cfu/mL, respectively, for these curves. The validation results indicated the predictive model was reasonably accurate, with relatively small RMSE. The model was then used to simulate the growth of L. monocytogenes under a variety of continuous and square-wave temperature profiles to demonstrate its potential application. The results of this study showed that the model developed in this study can be used to predict the growth of L. monocytogenes in contaminated milk during storage.

      Key words

      INTRODUCTION

      Listeria monocytogenes, a serious foodborne gram-positive pathogenic microorganism, can cause sporadic but severe infection in certain susceptible populations, such as pregnant women, patients with weakened immune systems, and seniors aged 65 yr or older (
      • Vázquez-Boland J.A.
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      Listeria pathogenesis and molecular virulence determinants.
      ;
      • van Nassau T.J.
      • Lenz C.A.
      • Scherzinger A.S.
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      Combination of endolysins and high pressure to inactivate Listeria monocytogenes.
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      • Mammel M.K.
      • Datta A.R.
      Global transcriptomic response of Listeria monocytogenes during growth on cantaloupe slices.
      ). According to the US Centers for Diseases Control and Prevention (
      • CDC
      Listeriosis (Listeria infection).
      ), at least 1,600 cases of listeriosis are reported each year in the United States. According to the European Center for Disease Prevention and Control (ECDC), 2,502 confirmed cases were reported in Europe in 2017 (
      • ECDC (European Centre for Disease Prevention and Control)
      Listeriosis - annual epidemiological report for 2017.
      ). The presence of L. monocytogenes in ready-to-eat products is a particular concern because it can survive and multiply even at refrigeration temperatures commonly used to control pathogens in foods (4–10°C;
      • WHO (World Health Organization)
      Risk assessment of Listeria monocytogenes in ready-to-eat foods: Technical report.
      ;
      • Ryser E.
      Pathogens in Milk| Listeria monocytogenes.
      ). Various ready-to-eat products have been reported to be associated with listeriosis outbreaks, such as cheeses (
      • Linnan M.J.
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      • Goulet V.
      • May S.
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      • Plikaytis B.D.
      • Fannin S.L.
      • Kleks A.
      • Broome C.V.
      Epidemic listeriosis associated with Mexican-style cheese.
      ;
      • EFSA and ECDC (European Food Safety Authority and European Centre for Disease Prevention and Control)
      The European Union summary report on trends and sources of zoonoses, zoonotic agents and food-borne outbreaks in 2015.
      ;
      • CDC
      Multistate outbreak of listeriosis linked to soft raw milk cheese made by Vulto Creamery.
      ), deli ham (
      • CDC
      Outbreak of Listeria infections linked to deli ham.
      ), ice cream (
      • CDC
      Multistate outbreak of listeriosis linked to blue bell creameries products.
      ) and other dairy products (
      • CDC
      Multistate outbreak of Listeriosis linked to Roos Foods dairy products.
      ), and fresh-cut fruits and vegetables (
      • Ho J.L.
      • Shands K.N.
      • Friedland G.
      • Eckind P.
      • Fraser D.W.
      An outbreak of type 4b Listeria monocytogenes infection involving patients from eight Boston hospitals.
      ;
      • CDC
      Multistate outbreak of listeriosis linked to frozen vegetables.
      ;
      • Garner D.
      • Kathariou S.
      Fresh produce - Associated listeriosis outbreaks, sources of concern, teachable moments, and insights.
      ).
      Cow milk is a nutritious and valuable dairy product, especially for infants and children. It is also a suitable growth medium for pathogenic microorganisms (
      • Oliver S.P.
      • Jayarao B.M.
      • Almeida R.A.
      Foodborne pathogens in milk and the dairy farm environment: Food safety and public health implications.
      ) and has been identified as one of the main food vectors for L. monocytogenes transmission (
      • Jordan K.
      • Hunt K.
      • Dalmasso M.
      Listeria monocytogenes in milk products.
      ). While pasteurized milk is consumed in almost every part of the world, L. monocytogenes outbreaks associated with the consumption of pasteurized milk products have been reported previously. For example, 7 infant and 42 adult cases of listeriosis were epidemiologically linked to consumption of pasteurized milk in Massachusetts in the United States in 1983 (
      • Ryser E.
      Pathogens in Milk| Listeria monocytogenes.
      ), and 66 cases of gastroenteritis and fever were directly traced to consumption of pasteurized chocolate milk in Illinois (
      • Dalton C.B.
      • Austin C.C.
      • Sobel J.
      • Hayes P.S.
      • Bibb W.F.
      • Graves L.M.
      • Swaminathan B.
      • Proctor M.E.
      • Griffin P.M.
      An outbreak of gastroenteritis and fever due to Listeria monocytogenes in milk.
      ), which was most likely caused by cross-contamination due to contact with contaminated equipment and processing environment (
      • Ferreira V.
      • Wiedmann M.
      • Teixeira P.
      • Stasiewicz M.J.
      Listeria monocytogenes persistence in food-associated environments: Epidemiology, strain characteristics, and implications for public health.
      ;
      • Meloni D.
      • Consolati S.G.
      • Mazza R.
      • Mureddu A.
      • Fois F.
      • Piras F.
      • Mazzette R.
      Presence and molecular characterization of the major serovars of Listeria monocytogenes in ten Sardinian fermented sausage processing plants.
      ;
      • Ortiz S.
      • López V.
      • Martínez-Suárez J.V.
      Control of Listeria monocytogenes contamination in an Iberian pork processing plant and selection of benzalkonium chloride-resistant strains.
      ).
      Although a few studies have investigated the growth kinetics of L. monocytogenes and developed predictive models in cheese and other dairy products (
      • Lobacz A.
      • Kowalik J.
      • Tarczynska A.
      Modeling the growth of Listeria monocytogenes in mold-ripened cheeses.
      ;
      • Tiwari U.
      • Walsh D.
      • Rivas L.
      • Jordan K.
      • Duffy G.
      Modelling the interaction of storage temperature, pH, and water activity on the growth behavior of Listeria monocytogenes in raw and pasteurized semi-soft rind washed milk cheese during storage following ripening.
      ;
      • Salazar J.K.
      • Bathija V.M.
      • Carstens C.K.
      • Narula S.S.
      • Shazer A.
      • Stewart D.
      • Tortorello M.L.
      Listeria monocytogenes growth kinetics in milkshakes made from naturally and artificially contaminated ice cream.
      ;
      • Thomas M.
      • Tiwari R.
      • Mishra A.
      Predictive model of Listeria monocytogenes growth in queso fresco.
      ;
      • Tirloni E.
      • Stella S.
      • Bernardi C.
      • Dalgaard P.
      • Rosshaug P.S.
      Predicting growth of Listeria monocytogenes in fresh ricotta.
      ), these models were developed using the traditional 2-step approach from the growth curves observed under isothermal conditions.
      • Xanthiakos K.
      • Simos D.
      • Angelidis A.
      • Nychas G.E.
      • Koutsoumanis K.
      Dynamic modeling of Listeria monocytogenes growth in pasteurized milk.
      attempted to model dynamic growth of L. monocytogenes in pasteurized milk, but the models were also developed using the traditional 2-step approach using isothermal growth curves obtained at temperatures below 16°C. Although the 2-step approach can be used for kinetic parameters estimation and model development, it is more time-consuming and labor-intensive when compared with 1-step dynamic analysis (
      • Versyck K.J.
      • Bernaerts K.
      • Geeraerd A.H.
      • Van Impe J.F.
      Introducing optimal experimental design in predictive modeling: A motivating example.
      ;
      • Dolan K.D.
      • Valdramidis V.P.
      • Mishra D.K.
      Parameter estimation for dynamic microbial inactivation: Which model, which precision?.
      ;
      • Huang L.
      Dynamic determination of kinetic parameters, computer simulation, and probabilistic analysis of growth of Clostridium perfringens in cooked beef during cooling.
      ). The 1-step dynamic analysis method can directly construct a combined model (primary + secondary) for predicting the growth of pathogens under both dynamic and isothermal conditions (
      • Versyck K.J.
      • Bernaerts K.
      • Geeraerd A.H.
      • Van Impe J.F.
      Introducing optimal experimental design in predictive modeling: A motivating example.
      • Huang L.
      Dynamic determination of kinetic parameters, computer simulation, and probabilistic analysis of growth of Clostridium perfringens in cooked beef during cooling.
      ,
      • Huang L.
      Dynamic kinetic analysis of growth of Listeria monocytogenes in a simulated comminuted, non-cured cooked pork product.
      ;
      • Huang L.
      • Vinyard B.T.
      Direct dynamic kinetic analysis and computer simulation of growth of Clostridium perfringens in cooked turkey during cooling.
      ;
      • Huang L.
      • Hwang C.A.
      Dynamic analysis of growth of Salmonella enteritidis in liquid egg whites.
      ;
      • Li M.
      • Huang L.
      • Yuan Q.
      Growth and survival of Salmonella Paratyphi A in roasted marinated chicken during refrigerated storage: Effect of temperature abuse and computer simulation for cold chain management.
      ;
      • Li M.
      • Huang L.
      • Zhu Y.
      • Wei Q.
      Growth of Clostridium perfringens in roasted chicken and braised beef during cooling - One-step dynamic analysis and modeling.
      ). Therefore, the objective of this study was to use the 1-step dynamic analysis method to construct and validate a mathematical model to predict the growth behavior of L. monocytogenes in milk at dynamic temperatures ranging from 4 to 30°C with a goal to use it for food-safety decision making and risk assessment in the future. The model validation process was conducted using the independent experimental data collected in this study and external data retrieved from ComBase (www.combase.cc). Moreover, the potential application of the generated model was evaluated by simulating the growth of L. monocytogenes in milk exposed to temperature fluctuation and abuse conditions.

      MATERIALS AND METHODS

      Bacterial Cultures and Preparation

      Four strains of L. monocytogenes, including CICC 21632 (serotype 7, human feces), CICC 21633 (serotype 1/2a, poultry), CICC 21635 (serotype 4b, human), and CICC 21639 (serotype 1/2a), were obtained from the China Center of Industrial Culture Collection (CICC, Beijing, China). The L. monocytogenes strains were induced into resistance to rifampicin (Rif) at 100 mg/L for easy separation from background microorganisms in the samples (
      • Fang T.
      • Liu Y.H.
      • Huang L.H.
      Growth kinetics of Listeria monocytogenes and spoilage microorganisms in fresh-cut cantaloupe.
      ). These Rif-resistant strains were stored on tryptic soy agar (TSA; BD/Difco Laboratories, Sparks, MD) containing 100 mg/L of Rif (TSA/Rif) at 4°C. Before the experiment, a loop of each strain was transferred to 10 mL of brain heart infusion broth (BD/Difco Laboratories) containing 100 mg/L of Rif. The cultures were incubated on an orbital shaker at 37°C for 18 to 19 h. After incubation, the cultures were centrifuged for 15 min at 2,500 × g at 4°C, and the pellets were washed twice with 0.1% sterile peptone water (PW). After washing, each pellet was resuspended in 5 mL of PW. An equal amount of each strain was mixed to form a 4-strain cocktail of L. monocytogenes. The concentration of L. monocytogenes in the cocktail was around 109 cfu/mL, which was then properly diluted in PW to obtain an inoculum of ∼105 cfu/mL.

      Sample Preparation and Inoculation

      The HTST-pasteurized milk was purchased from a local supermarket (Fuzhou, China) and stored it in a refrigerator (∼4°C) overnight. Before experiments, the presence of background bacteria in the sample was checked by direct plating on TSA plates (without enrichment) and PALCAM Listeria selective agar plates (Guangdong HuanKai Microbial Sci. & Tech. Co., Guangzhou, China), and then samples were incubated at 37°C overnight. No colony was observed on the plates after overnight incubation, suggesting that the milk samples were properly processed and that there was no detectable native L. monocytogenes in the milk. The milk was divided (10 mL) into sterile culture tubes, and each tube was inoculated with a 100-μL aliquot of the L. monocytogenes cocktail such that the initial inoculation level was approximately 102 cfu/mL in the samples. The inoculated samples were vortexed to properly mix the milk with the bacterial cells.

      Growth Studies and Enumeration

      The inoculated samples were transferred to a precision programmable incubator [model KB-115 (E3.1), Binder GmBH, Tuttlingen, Germany]. Six dynamic temperature profiles (DT_A, DT_B, DT_C, DT_D, DT_E, and DT_F) were programmed to simulate random temperature changes during storage and distribution, particularly during temperature abuse. The dynamically changing temperature ranged from 4 to 30°C, and the storage time ranged from 120 to 314 h. Six dynamic studies were independently conducted to collect the growth data of L. monocytogenes in milk samples under each dynamic temperature profile. The growth data obtained under temperature profiles of DT_A, DT_B, and DT_C were analyzed simultaneously to develop growth models and determine the kinetic parameters associated with the growth of L. monocytogenes in pasteurized milk. The other 3 independent dynamic growth curves under temperature profiles of DT_D, DT_E, and DT_F were used to validate the growth models. In addition, isothermal growth studies of L. monocytogenes in pasteurized milk were also conducted at 2, 4, 8, 12, 16, 20, 25, and 30°C for validation of the developed growth model. Two independent growth experiments per temperature (2 replicates) were performed. The samples were retrieved regularly to enumerate the counts of L. monocytogenes in milk.
      To enumerate L. monocytogenes counts, an aliquot (0.1 mL or 1 mL) from each sample was withdrawn, properly diluted with PW, and then plated in duplicates, either directly or after serial dilutions, onto TSA/Rif plates, which effectively suppressed the background microorganisms, allowing accurate enumeration of Rif-resistant L. monocytogenes. The plates were incubated at 37°C for 48 h. The L. monocytogenes colonies on each agar plate were counted and converted to log cfu/mL (base 10) of sample. During data analysis, the L. monocytogenes counts in log cfu/mL were converted to ln cfu/mL by multiplication with 2.303.

      Growth Models

      The 2-compartment growth model (Eq. 13) reported by
      • Huang L.
      Dynamic determination of kinetic parameters, computer simulation, and probabilistic analysis of growth of Clostridium perfringens in cooked beef during cooling.
      was used to simulate the growth of L. monocytogenes. The changes of the lag and dividing cells were individually described by Eq. 1 and 2, and the total counts of L. monocytogenes cells (N) were calculated by the sum of both lag state and dividing cells using Eq. (3) as follows:
      dNLdt=-γμmaxNL,
      [1]


      dNDdt=γμmaxNL+μmaxND(1+NL+NDNmax),
      [2]


      N=NL+ND,
      [3]


      with initial conditions as N = NL, ND = 0, at t = 0, where NL, ND, and N are the numbers of cells in the lag phase, exponential growth phase, and the total population (cfu/mL); Nmax is the maximum cell concentration, or the carrying capacity of the system (cfu/mL), which is labeled as Ymax after being converted to the nature logarithm of Nmax. The natural logarithm of N is Y, which can be calculated from the log cfu/mL of bacterial counts; μmax is the specific growth rate of the dividing cells (ln cfu/mL per hour, or hour−1), and γ is a coefficient that defines the conversion of cells from the lag state to the dividing state and determines the duration of the lag phase.
      To describe the effect of temperature on the specific rate of dividing cells, suboptimal Huang square-root model was used (Eq. 4;
      • Huang L.
      • Hwang C.A.
      • Phillips J.
      Evaluating the effect of temperature on microbial growth rate - the Ratkowsky and a Bělehrádek-type models.
      ). In Eq. 4, a is a regression coefficient, T is the storage temperature (°C), and Tmin is the minimum growth temperature (°C).
      μmax=a(T-Tmin)0.75.
      [4]


      Determination of Kinetic Parameters and Data Analysis

      The 1-step dynamic analysis methodology reported by
      • Huang L.
      Dynamic determination of kinetic parameters, computer simulation, and probabilistic analysis of growth of Clostridium perfringens in cooked beef during cooling.
      ,
      • Huang L.
      Dynamic kinetic analysis of growth of Listeria monocytogenes in a simulated comminuted, non-cured cooked pork product.
      ) and
      • Li M.
      • Huang L.
      • Yuan Q.
      Growth and survival of Salmonella Paratyphi A in roasted marinated chicken during refrigerated storage: Effect of temperature abuse and computer simulation for cold chain management.
      was used to solve the ordinary differential equations (Eq. 13) for determining the kinetic parameters. Numerical analysis was implemented using MATLAB 2018 (The MathWorks Inc., Natick, MA). The ordinary differential equations were solved using a fourth-order Runge-Kutta method, and a numerical optimization technique was used to search for optimized kinetic parameters (γ, a, Tmin, and Ymax) such that the difference between the calculated and observed values [Y, or ln (N)] of bacterial growth in the dynamic temperature profiles of DT_A, DT_B, and DT_C was minimized. During numerical optimization, the Fitnlm procedure based on the Levenberg-Marquardt nonlinear least squares algorithm in MATLAB Statistics and Machine Learning Toolbox was used.

      Model Validation

      To examine the accuracy of the resulting models, the remaining 3 independent growth curves obtained under dynamic temperature profiles (DT_D, DT_E, and DT_F) that were not previously used in model development and determination of kinetic parameters were used to validate the models. The models were also validated using isothermal growth curves of L. monocytogenes observed at 2, 4, 8, 12, 16, 20, 25, and 30°C. During validation, the kinetic parameters obtained from dynamic analysis were applied in the models to calculate the growth of L. monocytogenes under different temperature conditions. In addition, 3 isothermal growth curves of L. monocytogenes in milk observed at 4°C [record # J270_Lm; strain: Scott A; food name: skim milk (pH = 6.7),
      • Larson A.E.
      • Yu R.R.
      • Lee O.A.
      • Price S.
      • Haas G.J.
      • Johnson E.A.
      Antimicrobial activity of hop extracts against Listeria monocytogenes in media and in food.
      ], 7°C [record # L168_4; mixed strains: CRA499, 1576, 1579; food name: milk (pH = 7)], and 10°C [record # L168_1; mixed strains: CRA499, 1576, 1579; food name: milk (pH = 7)], arbitrarily retrieved from ComBase, were used to validate the models.
      The square root of an unbiased estimator of the variance (RMSE; Eq. 5) was calculated to evaluate the accuracy of the model. The residual errors were analyzed using @RISK 7.6 Industrial (Palisade Corp., Ithaca, NY) to determine the distribution pattern. The RMSE was determined as follows:
      RMSE=i=1n(yi-yˆi)2n-p,
      [5]


      where yi is the measured experimental bacterial counts (log cfu/mL) at time point i of each sample collection, yˆi is the corresponding bacterial counts calculated by the developed model (log cfu/mL), n is the number of observed points, and p is the number of parameters in the model.

      Model Application and Simulation of Growth Under Temperature Fluctuation

      Temperature fluctuation and potential abuse is inevitable during distribution and storage of food. Therefore, once the models and kinetic parameters were validated, they were used to simulate the growth of L. monocytogenes in milk exposed to temperature fluctuation and abuse conditions. The growth was simulated under square-wave and sine-wave temperature profiles fluctuating between 2 and 4°C, and 2 and 10°C with 3 different cycle times (24, 12, and 4 h). The temperature fluctuation between 2 and 4°C was used to mimic normal storage temperature, and the fluctuation between 2 and 10°C was used to simulate temperature abuse conditions. The total storage time was 150 h, or 6.25 d. These hypothetical temperature profiles were designed to show the capability and flexibility of the resulting model for simulating complex changes in temperature during normal storage and temperature abuse. The initial population of L. monocytogenes was set to 2 log cfu/mL.

      RESULTS AND DISCUSSION

      Determination of Kinetic Parameters and Numerical Optimization

      Figure 1 shows the growth curves of L. monocytogenes observed under 3 dynamic temperature profiles (DT_A, DT_B, and DT_C). These profiles were designed to vary from 4 to 30°C, which covered the range that milk may experience during cold-chain logistics. These 3 growth curves were used together to estimate all the kinetic parameters (γ, a, Tmin, and Ymax) using 1-step dynamic analysis, which converged easily during numerical analysis. Table 1 lists the kinetic parameters estimated from these curves.
      Figure thumbnail gr1
      Figure 1Growth of Listeria monocytogenes under dynamic temperature condition and mathematical modeling. Smooth curves are the predictive model plots. Dashed lines are the temperature profiles (DT_A, DT_B, and DT_C).
      Table 1Estimates of kinetic parameters for describing growth of Listeria monocytogenes in milk
      Parameter
      Tmin = minimum temperature; Ymax = maximum cell concentration; RMSE = root mean square error.
      Predictive valuesSEt-valueP-value95% confidence limit
      LowerUpper
      γ0.970.118.671.98 × 10−130.751.20
      a0.060.00235.604.94 × 10−540.060.07
      Tmin (°C)0.60.22.90.4 × 10−20.21.1
      Ymax (ln cfu/mL)18.0
      7.8 log cfu/mL.
      0.365.51.8 × 10−7617.518.6
      RMSE (log cfu/mL)0.3
      Number of observations92
      Error degrees of freedom88
      1 Tmin = minimum temperature; Ymax = maximum cell concentration; RMSE = root mean square error.
      2 7.8 log cfu/mL.
      According to Table 1, the estimated values of γ, a, Tmin, and Ymax were 0.97, 0.06, 0.6°C, and 18.0 ln cfu/mL (or 7.8 log cfu/mL), respectively, with very low P-values (<0.05). It is worth mentioning that the estimated minimum growth temperature was 0.6 ± 0.2°C (mean ± SE), which is very close to the typical value (−0.4°C) for L. monocytogenes (
      • Food and Agriculture Organization (FAO)
      Report of the FAO expert consultation on the trade impact of Listeria in fish products.
      ), and is well within the range reported in the literature. The RMSE of optimization was only 0.3 log cfu/mL, suggesting that the estimated kinetic parameters can be used to describe the growth of L. monocytogenes in pasteurized milk.
      The effect of temperature on the specific growth rate (μmax) was based on the Huang square-root model (Eq. 4) to estimate a and Tmin (Figure 2). In Figure 2, the μmax obtained in this study is also compared with previous studies (
      • Noviyanti F.
      • Shimizu S.
      • Hosotani Y.
      • Koseki S.
      • Inatsu Y.
      • Kawasaki S.
      Predictive growth model of Listeria monocytogenes under fluctuating temperature conditions in pasteurized milk by using real-time polymerase chain reaction.
      ;
      • Rosenow E.M.
      • Marth E.H.
      Growth of Listeria monocytogenes in skim, whole and chocolate milk, and in whipping cream during incubation at 4, 8, 13, 21, and 35°C.
      ;
      • Xanthiakos K.
      • Simos D.
      • Angelidis A.
      • Nychas G.E.
      • Koutsoumanis K.
      Dynamic modeling of Listeria monocytogenes growth in pasteurized milk.
      ).
      • Rosenow E.M.
      • Marth E.H.
      Growth of Listeria monocytogenes in skim, whole and chocolate milk, and in whipping cream during incubation at 4, 8, 13, 21, and 35°C.
      investigated the growth of L. monocytogenes in skim, whole, and chocolate milk, though neither primary model nor secondary model was developed. The generation time (GT) of L. monocytogenes at different constant temperatures was determined and converted to μmax through μmax=ln2/GT for comparison with the current result. Both
      • Xanthiakos K.
      • Simos D.
      • Angelidis A.
      • Nychas G.E.
      • Koutsoumanis K.
      Dynamic modeling of Listeria monocytogenes growth in pasteurized milk.
      and
      • Noviyanti F.
      • Shimizu S.
      • Hosotani Y.
      • Koseki S.
      • Inatsu Y.
      • Kawasaki S.
      Predictive growth model of Listeria monocytogenes under fluctuating temperature conditions in pasteurized milk by using real-time polymerase chain reaction.
      studied the growth kinetics of L. monocytogenes in pasteurized milk using the traditional 2-step method.
      • Xanthiakos K.
      • Simos D.
      • Angelidis A.
      • Nychas G.E.
      • Koutsoumanis K.
      Dynamic modeling of Listeria monocytogenes growth in pasteurized milk.
      used the Baranyi model (
      • Baranyi J.
      • Roberts T.A.
      A dynamic approach to predicting bacterial growth in food.
      ) as the primary model and the Ratkowsky square-root model (
      • Ratkowsky D.A.
      • Lowry R.K.
      • McMeekin T.A.
      • Stokes A.N.
      • Chandler R.E.
      Model for bacterial culture growth rate through the entire biokinetic temperature range.
      ) as the secondary model to fit the growth curves obtained between 1.5 and 16°C. Although
      • Noviyanti F.
      • Shimizu S.
      • Hosotani Y.
      • Koseki S.
      • Inatsu Y.
      • Kawasaki S.
      Predictive growth model of Listeria monocytogenes under fluctuating temperature conditions in pasteurized milk by using real-time polymerase chain reaction.
      used the same modeling approach reported in
      • Xanthiakos K.
      • Simos D.
      • Angelidis A.
      • Nychas G.E.
      • Koutsoumanis K.
      Dynamic modeling of Listeria monocytogenes growth in pasteurized milk.
      , the bacterial counts were obtained by the real-time PCR. Judging from Figure 2, it is apparent that the specific growth rate of L. monocytogenes calculated from the model developed in our study is in close agreement with those reported by
      • Noviyanti F.
      • Shimizu S.
      • Hosotani Y.
      • Koseki S.
      • Inatsu Y.
      • Kawasaki S.
      Predictive growth model of Listeria monocytogenes under fluctuating temperature conditions in pasteurized milk by using real-time polymerase chain reaction.
      ,
      • Rosenow E.M.
      • Marth E.H.
      Growth of Listeria monocytogenes in skim, whole and chocolate milk, and in whipping cream during incubation at 4, 8, 13, 21, and 35°C.
      , and
      • Xanthiakos K.
      • Simos D.
      • Angelidis A.
      • Nychas G.E.
      • Koutsoumanis K.
      Dynamic modeling of Listeria monocytogenes growth in pasteurized milk.
      at temperatures above about 4°C. The difference becomes apparent and more significant at temperatures below 4°C. This difference is caused by the choice of the secondary model. In the linear model for square root for μmax in the Ratkowsky square-root model, or μmax=a(T-T0) the parameter T0 is the nominal minimum temperature; this is often lower than the true minimum temperature (Tmin), which may allow the growth of L. monocytogenes growth, due to the model structure (Rosso, 1993;
      • Dantigny P.
      • Molin P.
      Influence of the modeling approach on the estimation of the minimum temperature for growth in Bělehrádek-type models.
      ;
      • Huang L.
      • Hwang C.A.
      • Phillips J.
      Evaluating the effect of temperature on microbial growth rate - the Ratkowsky and a Bělehrádek-type models.
      ). It should be noted that the nominal minimum temperature in the Ratkowsky square-root model may not be the true biological minimum growth temperature.
      Figure thumbnail gr2
      Figure 2Comparison of the maximum specific growth rate (μmax) of Listeria monocytogenes in the present study with other published models (
      • Rosenow E.M.
      • Marth E.H.
      Growth of Listeria monocytogenes in skim, whole and chocolate milk, and in whipping cream during incubation at 4, 8, 13, 21, and 35°C.
      ;
      • Xanthiakos K.
      • Simos D.
      • Angelidis A.
      • Nychas G.E.
      • Koutsoumanis K.
      Dynamic modeling of Listeria monocytogenes growth in pasteurized milk.
      ;
      • Noviyanti F.
      • Shimizu S.
      • Hosotani Y.
      • Koseki S.
      • Inatsu Y.
      • Kawasaki S.
      Predictive growth model of Listeria monocytogenes under fluctuating temperature conditions in pasteurized milk by using real-time polymerase chain reaction.
      ) for estimating the μmax of L. monocytogenes in milk.
      • Augustin J.C.
      • Zuliani V.
      • Cornu M.
      • Guillier L.
      Growth rate and growth probability of Listeria monocytogenes in dairy, meat and seafood products in suboptimal conditions.
      evaluated the growth rate of L. monocytogenes collected from microbiological media, dairy, meat, and seafood products in suboptimal conditions using growth rate data published in the literature. The authors used both the cardinal temperatures model (model 1;
      • Rosso L.
      • Lobry J.R.
      • Flandrois J.P.
      An unexpected correlation between cardinal temperatures of microbial growth highlighted by a new model.
      ) and the Ratkowsky square-root model (model 2) to estimate the minimum growth temperature. If without considering other environmental effects, such as pH, aw, CO2, and phenol concentration, which was the case for this study, the minimum temperature (Tmin) estimated from cardinal temperatures model is −0.95 ± 1.49°C (mean ± SD). For the Ratkowsky square-root model, the nominal minimum growth temperature (T0) is −1.66 ± 1.33°C (mean ± SD). The mean T0 is about 0.7°C lower than Tmin estimated by the cardinal temperatures model. The T0 value is −2.32°C in
      • Xanthiakos K.
      • Simos D.
      • Angelidis A.
      • Nychas G.E.
      • Koutsoumanis K.
      Dynamic modeling of Listeria monocytogenes growth in pasteurized milk.
      and −2.8°C in
      • Noviyanti F.
      • Shimizu S.
      • Hosotani Y.
      • Koseki S.
      • Inatsu Y.
      • Kawasaki S.
      Predictive growth model of Listeria monocytogenes under fluctuating temperature conditions in pasteurized milk by using real-time polymerase chain reaction.
      . According to
      • Rosso L.
      • Lobry J.R.
      • Flandrois J.P.
      An unexpected correlation between cardinal temperatures of microbial growth highlighted by a new model.
      , the precision of estimation is ± 5°C for Tmin.
      • Junttila J.R.
      • Niemela S.I.
      • Hirn J.
      Minimum growth temperatures of Listeria monocytogenes and non-haemolytic Listeria.
      examined the minimum growth temperature of 100 strains of Listeria, including 78 strains of L. monocytogenes isolated from animals and 22 from nonhemolytic strains, using a flooding technique at temperatures between −1.6 and 14.5°C. They reported that the mean minimum growth temperature was 1.1 ± 0.3°C for L. monocytogenes and 1.7 ± 0.5°C for nonhemolytic strains of Listeria.
      • Huang L.
      • Hwang C.A.
      • Phillips J.
      Evaluating the effect of temperature on microbial growth rate - the Ratkowsky and a Bělehrádek-type models.
      demonstrated that the use of the Huang square-root model may produce an estimate of Tmin close to the biological minimum growth temperature. On the hand, the nominal minimum growth temperature may overpredict the growth due to estimation of the minimum growth temperature (lower than the biological minimum growth temperature), which may give an impression that the bacteria may grow at the temperatures below the typical biological minimum growth temperature. The Tmin value of 0.6°C obtained in our study is closer to the mean minimum growth temperature (1.1°C) reported by
      • Junttila J.R.
      • Niemela S.I.
      • Hirn J.
      Minimum growth temperatures of Listeria monocytogenes and non-haemolytic Listeria.
      and the value (−0.4°C) recommended by Food and Agriculture Organization (
      • Food and Agriculture Organization (FAO)
      Report of the FAO expert consultation on the trade impact of Listeria in fish products.
      ).

      Validation of Growth Model With Dynamic Temperature Profiles

      The other 3 different dynamic temperature profiles (DT_D, DT_E, and DT_F), which were not used for estimating the kinetic parameters in the previous section, were applied to check the accuracy of the resulting predictive models. Figure 3 compares the dynamic growth curves of L. monocytogenes with the values calculated by using the predictive models and the estimated parameters with the experimental observations for the different profiles of DT_D, DT_E, and DT_F. The comparison shows that the predicted growth values are close to the experimental observations, suggesting that the predictive models can predict the growth of L. monocytogenes under dynamic conditions. The RMSE of the predictions for profiles DT_D, DT_E, and DT_F were 0.3, 0.3, and 0.4 log cfu/mL, respectively. The residual errors can be best described by a normal distribution, with mean of 0.1 log cfu/mL and standard deviation of 0.3 log cfu/mL (Figure 4A). For the predictions, 92.2% of the residual errors were between −0.5 and 0.5 log cfu/mL, suggesting that the residual errors are well within the normal range of experimental errors. Figure 4B shows the standardized residuals, which was obtained by dividing the residual with the standard deviation of the residuals. In Figure 4B, none of the standardized residuals were outside of ± 1.96 (2 SD) in profile DT_D, and only 1 in profile DT_E and 2 in profile DT_F were outside of this range. This suggested that only 3 observations in all 56 data points were likely to be larger than normal errors of prediction.
      Figure thumbnail gr3
      Figure 3Validation of the predictive model using dynamic temperature profiles (DT_D, DT_E, and DT_F).
      Figure thumbnail gr4
      Figure 4Residual errors (ε) distribution for dynamic growth validation. (A) Prediction versus observation; (B) standardized residuals (residual/SD) for dynamic profiles of DT_D, DT_E, and DT_F.

      Validation of Growth Model With Isothermal Temperature Profiles

      Figure 5 depicts the experimental growth data of L. monocytogenes in pasteurized milk incubated at 2, 4, 8, 12, 16, 20, 25, and 30°C, as well as the growth curves predicted by the developed predictive model. Overall, the predictive growth curves match well with the experimental observations at all the tested temperatures except at 4 and 12°C. At 4°C, the growth of L. monocytogenes was accurately predicted by the model when the incubation time was <300 h; however, the model overpredicted after about 300 h (12.5 d). Compared with other growth curves, these errors were most likely caused by some experimental errors at these 2 temperatures. For 4 and 12°C, the RMSE of prediction was about 1.1 to 1.2 log cfu/mL. At the other isothermal temperatures (2, 8, 16, 20, 25, and 30°C), the predictions generally agreed with the observed data, with RMSE values of 0.3, 0.9, 0.5, 0.3, 0.2, and 0.3 log cfu/mL, respectively. The RMSE of prediction of all temperatures (including 4 and 12°C) was 0.6 log cfu/mL. Therefore, the predictive model built in this study is suitable for predicting the isothermal growth of L. monocytogenes in pasteurized milk. The residual errors can be best described by a Laplace distribution (location = 0.1 log cfu/mL, scale factor = 0.6 log cfu/mL), with 71.1% of the residual errors between −0.5 and 0.5 log cfu/mL for the predictions (Figure 6). The errors of prediction were also within the normal range of experimental errors. With a Laplace distribution, the probability of observing ε decreases exponentially as |ε| increases. For isothermal data, the calculation of standardized residuals showed that, out of 201 data points (both replicates), only 18 data points were outside the 2 standard deviation range, including 8 data points at 4°C (4 for each replicate), 3 at 8°C (second replicate), and 7 at 12°C (3 for replicate 1 and 4 for replicate 2), suggesting that only 9% of the prediction errors may be likely larger than the remaining predictions, due to some abnormal experimental error.
      Figure thumbnail gr5
      Figure 5Growth of Listeria monocytogenes in milk at different constant temperatures (2–30°C). Dashed line = temperature; circles = growth data; solid line = predicted growth curve; empty circles and triangles = data collected in experiments (replicates 1 and 2, respectively).
      Figure thumbnail gr6
      Figure 6Residual error (ε) distribution for isothermal growth validation.
      The models were further validated with 3 growth curves obtained from ComBase (Figure 7). These data served the purpose of external validation, as the data were independently collected by other researchers. Again, the predicted results also agreed with the published data. The RMSE of prediction was 0.8, 0.4, and 0.5 log cfu/mL, respectively, at 4, 7, and 10°C, suggesting that the developed models are practically suitable for predicting the growth of L. monocytogenes in milk. While the models and kinetic parameters had been validated using laboratory-collected dynamic and isothermal growth curves, as well as the third-party growth curves from ComBase, it is noteworthy to point out that this study was based on a cocktail of L. monocytogenes. Because of this, the model would predict the behavior of the fast-growing strain in the cocktail, thus representing the worst-case prediction for the strains used in this study. It does not explain the variability in the growth behavior among the strains. The users would need to check the accuracy of prediction when different strains are involved.
      Figure thumbnail gr7
      Figure 7Model application to published growth data of Listeria monocytogenes at 4 (record # J270_Lm), 7 (record # I168_4), and 10°C (record # L168_1) using data from ComBase (www.combase.cc).

      Model Application and Simulation of Growth Under Temperature Fluctuation

      A review reported by
      • James C.
      • Onarinde B.A.
      • James S.J.
      The use and performance of household refrigerators: A review.
      clearly suggested that many household refrigerators throughout the world are maintained and operated at temperatures higher than recommended temperatures, and many households are not aware of the recommended refrigeration temperature range. Depending on the country, the mean temperatures in the household refrigerators range from 3.5 to 9.3°C, and the maximum temperature can range from 10 to 37°C. The weighted mean temperatures of all mean minimum, mean, and maximum temperatures are reported as −1.5, 6.1, and 16.1°C, respectively. This report clearly suggests the potential problem of storage temperature abuse in household refrigerators. Another survey in 5 European countries showed that the minimum and maximum temperatures of the refrigerators of households with consumers vulnerable to L. monocytogenes are 3.7 and 8.2°C in France, 1.1 and 9.2°C in the United Kingdom, 3.2 and 9.1°C in Portugal, 1.8 and 12.2°C in Romania, and 1.2 and 8.8°C in Norway (
      • Dumitra'cu L.
      • Nicolau A.I.
      • Neagu C.
      • Didier P.
      • Maître I.
      • Nguyen-The C.
      • Skuland S.E.
      • Møretrø T.
      • Langsrud S.
      • Truninger M.
      • Teixeira P.
      • Ferreira V.
      • Martens L.
      • Borda D.
      Time-temperature profiles and Listeria monocytogenes presence in refrigerators from households with vulnerable consumers.
      ). Also, according to a survey of refrigerators in France, the minimum, average, and maximum temperatures are 0.9, 6.6, and 11.4°C, respectively (
      • Laguerre O.
      • Derens E.
      • Palagos B.
      Study of domestic refrigerator temperature and analysis of factors affecting temperature: A French survey.
      ).
      With the models validated, they were used to simulate the bacterial growth under fluctuating temperature profiles mimicking normal storage (2–4°C) and temperature abuse conditions (2–10°C). These hypothetical temperature profiles were designed to show that computer simulation could be used to predict the potential growth of L. monocytogenes under a wide range of temperature fluctuation, especially for situations where the temperature control of household refrigerators may malfunction. The results shown in Figures 8A1, B1, and C1 simulate the change in the population of L. monocytogenes under sine-wave profiles between 2 and 4°C with cycle times of 24 h (A1), 12 h (B1), and 4 h (C1), respectively. Figures 8A2, B2, and C2 show the simulation of bacterial growth under square-wave temperature profiles. The simulation results shown in Figure 8 clearly demonstrate that, regardless of the shape (sine wave or square wave) and cycle time, L. monocytogenes in milk stored at 2 to 4°C would grow slowly and increase by approximately 0.7 log cfu/mL at the end of 150 h storage due to psychrotrophic nature of this microorganism.
      Figure thumbnail gr8
      Figure 8Simulation of bacterial growth under fluctuating temperatures (Temp.) between 2 and 4°C. A1, B1, and C1: sine-wave, with cycle time = 24, 12, and 4 h, respectively. A2, B2, and C2: square-wave, with cycle time = 24, 12, and 4 h, respectively.
      Temperature abuse during refrigerated storage presents an increased risk to public health. Figure 9 shows the simulation of bacterial growth under temperature abuse conditions when milk is stored under 2 to 10°C. Figures 9A1, B1, and C1 demonstrate the growth of L. monocytogenes under sine-wave temperature profiles, and Figures 9A2, B2, and C2 depict the growth under square-wave temperature profiles, with cycle times of 24 h (A), 12 h (B), and 4 h (C), respectively. Figure 9 shows that, regardless of the shapes (sine wave or square wave) and cycle time, the population of L. monocytogenes in milk stored under such conditions would increase by approximately 3.3 to 3.5 log cfu/mL at the end of 150 h of storage, suggesting that temperature abuse may present a potential risk of human listeriosis. Therefore, it is very important to control temperature abuse to prevent the growth of L. monocytogenes.
      Figure thumbnail gr9
      Figure 9Simulation of bacterial growth under fluctuating temperatures (Temp.) between 2 and 10°C. A1, B1, and C1: sine-wave, with cycle time = 24, 12, and 4 h, respectively. A2, B2, and C2: square-wave, with cycle time = 24, 12, and 4 h, respectively.

      CONCLUSIONS

      This study constructed a dynamic mathematical model for the growth of L. monocytogenes in pasteurized milk using a 1-step dynamic method based on the experimental growth data. The kinetic parameter (γ, a, Tmin, and Ymax) values for describing the growth of L. monocytogenes estimated and optimized from the observed growth points were 0.97, 0.06, 0.6°C, and 18.0 ln cfu/mL (or 7.8 log cfu/mL). In addition, the predictive model developed in this study was verified under both dynamic and isothermal conditions, using observed growth data. The results of analysis demonstrated that the developed dynamic model can be used to predict the growth of L. monocytogenes in pasteurized milk not only under dynamic conditions, but also under isothermal conditions. This study also demonstrated that it is very important to prevent temperature abuse during refrigerated storage, as the simulation results show that the population of L. monocytogenes in contaminated milk may increase to a dangerously high level, potentially presenting a serious risk for consumers. This study also demonstrated that predictive modeling and the model developed herein are useful for predicting the growth of L. monocytogenes in pasteurized milk during storage throughout the cold chain and for cold-chain management, and may potentially be used for risk assessment.

      ACKNOWLEDGMENTS

      This work was financially supported by the National Natural Science Foundation of China (NSFC 31601393, 31401597, Beijing, China), the National and Science Foundation of Fujian Province (2018J01696, Fuzhou, Fujian, China), Fujian Agricultural and Forestry University (KXb16012A, Fuzhou, Fujian, China), and Fujian Education Department (JAT160147, Fuzhou, Fujian, China). Mention of trade names or commercial products in this publication is solely for the purpose of providing specific information and does not imply recommendation or endorsement by the US Department of Agriculture. USDA is an equal opportunity employer and provider. The authors have no conflicts of interest.

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