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Food Microbiology and Safety

An Artificial Neural Network Approach to Escherichia Coli O157:H7 Growth Estimation R.M. GARCÍA-GIMENO, C. HERVÁS-MARTÍNEZ, E. BARCO-ALCALÁ, G. ZURERA-COSANO, AND E. SANZ-TAPIA

Introduction

M

ODELS RELATING THE GROWTH OF MICROORGANISMS TO A

number of environmental factors are of considerable importance in the field of predictive microbiology. Food microbiologists are constantly concerned to determine, experimentally, the extent to which microbial growth depends on factors relating to ecology or processing modes (Hajmeer and others 1997). The models most widely used to describe relationships between combinations of factors and growth parameters are based on three main types: the Arrhenius equation, the square root model (Bèlehrádek model), and the Response Surface Model or linear regression ( Whiting and Buchanan 1994). Recently, a number of new models have been introduced, some involving the application of artificial neural networks (ANN), which can estimate kinetic parameters from the growth curve (primary model) and also from factors affecting microbial growth (secondary model) (Hajmeer and others 1997; Geeraerd and others 1998; Hervás and others 2001b; Jeyamkondan and others 2001; García and others 2002). An ANN is a highly interconnected network, consisting of many simple processing elements and capable of performing massive parallel computations for data processing; it is inspired by the elementary principles of the nervous system. A detailed description of artificial neural networks is provided by Najjar and others (1997). Unlike regression modeling, ANN imposes no restrictions on the type of relationship governing the dependence of growth parameters on the various running conditions (Hajmeer and others 1997; Jeyamkondan and others 2001). For these authors, regressionbased response surface models require that the order of the model be stated (that is, second, third, or fourth order), while ANNs tend implicitly to match the input vector (that is running conditions) to the output vector (that is Gompertz parameters). For Geeraerd and others (1998), ANNs constitute a low-complexity nonlinear modeling technique, capable of accurately describing experimental data in the field of secondary models in predictive microbiology. For these authors, the complexity of the ANN model in a specific application can be adapted, taking into account the general trend and the number of data points. As experimental results on the influence of other environmental factors become available, ANN models can be extended simply by adding more neu© 2003 Institute of Food Technologists Further reproduction prohibited without permission

rons and/or layers. For Jeyamkondan and others (2001), ANNs-due to their inherent interconnected structure-are powerful tools for modeling a system in which different processes are highly interrelated, and can be used to effectively model microbial interactions. Network architecture can be designed using a genetic algorithm with a selection procedure and crossing/mutation operators that introduce some diversity to the algorithm (Hervás and others 1998, 2000, 2001a, 2001b). The fitness function used in the genetic algorithm has 2 objectives, (1) to minimize the residual sum of squares, and (2) to reduce the unnecessary parameters. The 2nd term penalizes more complex models (models with a larger number of parameters) as compared with less complex models offering the same generalization capacity. Some techniques, collectively termed “pruning,” reduce the network size by modifying not only the connection parameters (weights) but also the network structure during training, beginning with a network design with an excessive number of nodes and gradually eliminating the unnecessary nodes or connections (Le Cun and others 1990; Hassibi and Stork 1993). Escherichia coli O157:H7 is considered an emerging pathogen; it is an infectious agent whose incidence in humans has increased dramatically over the last 20 y. The first case of E. coli O157:H7 was recorded in the United States in 1982 by Riley and others (1983) in two outbreaks of hemorrhagic colitis affecting people who had eaten at fast-food restaurants. It has subsequently been involved in numerous outbreaks, most of them in the U.S.A., Canada, U.K., and Japan (Carter and others 1987; Sharp and others 1994). The incidence of E. coli O157: H7 in Spain is not high; the last outbreak was confirmed in Barcelona in September 2000, following consumption of undercooked sausages, and a large number of scholarship students were affected. Several models have been constructed for E. coli O157: H7, but none have considered the 5 factors studied here. The aim of the present study was to analyze the potential enhancement of predictive microbiology afforded by ANN, optimized by means of genetic algorithms, for the study of Escherichia coli O157:H7 growth at different temperatures, pH, salt, and nitrite concentrations and in aerobic/anaerobic conditions and compare the results with those obtained using the Response Surface Model. Vol. 68, Nr. 2, 2003—JOURNAL OF FOOD SCIENCE

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ABSTRACT: Artificial neural networks (ANN) was evaluated and compared with Response Surface Model (RSM) results using growth response data for E.coli O157:H7 as affected by 5 variables: pH, sodium chloride, and nitrite concentrations, temperature, and aerobic/anaerobic conditions. The best ANN obtained, where the 2 kinetic parameters, growth rate and lag-time, were estimated jointly, contained 17 parameters and displayed a slightly lower Standard Error of Prediction (% SEP) than those obtained with RSM. Mathematical lag-time validation with additional data gave a lower %SEP for ANN (18%) than for RSM (27%), although growth-rate values were the same (22%). ANN thus should provide the innovative possibility of obtaining a single predictive model for the estimation of several kinetic parameters. Keywords: microbial growth, Escherichia coli O157:H7, artificial neural network modeling, genetic algorithm

ANN for Escherichia coli O157:H7 . . . Table 1—Average replicate values for observed and estimated growth rate (Gro, GrANN, GrRSM) and lag-time (Lago, LagANN, LagRSM) of E. coli O157:H7 by Artificial Neural Networks (ANN) and Response Surface Model(RSM) T(°C)

pH

NaCl(%)

9 12 12 12 12 12 12 13 13 13 13 13 13 13 15 15 15 15 15 15 15.3 18 18 18 18 18 18 18 21 21 21 21 21

6.5 5.5 5.5 6.5 6.5 7.5 7.5 5 5 6 6 6 6 6 6.5 6.5 6.5 6.5 6.5 8.5 6.5 5.5 5.5 5.5 7.5 7.5 7.5 7.5 6.5 6.5 6.5 6.5 6.5

2 2 2 2 4 2 2 3 3 3 3 5 5 5 0 4 4 4 6 4 6 2 2 6 2 2 6 6 0 2 4 4 6

NaNO2(ppm) 100 50 150 50 100 50 150 70 116.8 70 116.8 70 116.8 163.5 100 0 100 200 93.5 100 186.9 50 150 50 50 150 50 150 50 100 50 100 100

Gro (SD)(h-1) 0.0054 0.0361 0.0137 0.0167 0.0239 0.0222 0.0245 0.0106 0.0086 0.0338 0.0303 0.0223 0.0153 0.0106 0.0674 0.0394 0.0429 0.0331 0.0407 0.0533 0.0330 0.1040 0.0479 0.0160 0.1422 0.1635 0.0753 0.0734 0.2044 0.1779 0.1419 0.1169 0.0728

(0.0019) (0.0043) (0.0042) (0.0048) (0.1230) (0.0071) (0.0077) (0.0007) (0.0022) (0.0072) (0.0079) (0.0058) (0.0070) (0.0048) (0.0239) (0.0071) (0.0095) (0.0040) (0.0316) (0.0225) (0.0091) (0.0159) (0.0048) (0.0080) (0.0504) (0.0462) (0.0142) (0.0120) (0.0165) (0.0141) (0.0083) (0.0483) (0.0198)

GrANN (h-1) 0.0001 0.0194 0.0185 0.0236 0.0139 0.0248 0.0316 0.0148 0.0104 0.0262 0.0266 0.0134 0.0129 0.0111 0.0915 0.0432 0.0461 0.0414 0.0287 0.0571 0.0252 0.0865 0.0526 0.0284 0.1387 0.1443 0.0816 0.0822 0.2188 0.1609 0.1363 0.1198 0.0812

GrRSM (h-1)

Lago (SD)(h)

LagANN (h)

0.0063 0.0342 0.0202 0.0322 0.0312 0.0162 0.0422 0.0254 0.0132 0.0404 0.0376 0.0278 0.0250 0.0222 0.0845 0.0569 0.0569 0.0569 0.0312 0.0670 0.0324 0.0942 0.0682 0.0230 0.1482 0.1622 0.0770 0.0910 0.2298 0.1864 0.1470 0.1410 0.0876

56.19 (3.17) 46.93 (5.82) 65.31 (9.08) 37.74 (1.88) 102.92 (7.55) 34.62 (4.33) 36.09 (2.25) 45.78 (10.36) 138.05 (22.41) 30.17 (3.01) 32.18 (3.42) 161.92 (28.35) 126.84 (17.38) 133.94 (26.26) 19.80 (2.10 47.43 (2.43) 54.01 (1.85) 52.80 (2.56 209.49 (24.89) 80.32 (6.48) 229.56 (32.25) 16.09 (1.11) 16.66 (0.67) 76.58 (7.40) 15.20 (0.94) 15.85 (0.65) 92.80 (4.68) 91.46 (10.09) 7.40 (0.28) 11.15 (0.32) 23.88 (0.67) 24.22 (1.99) 72.89 (8.34)

60.92 43.55 45.41 43.36 97.31 43.87 44.3 51.88 54.21 50.72 52.09 141.5 149.8 159 22.58 53.19 56.92 63.13 195.6 59.74 207.9 25.84 34.31 97.38 14.29 13.94 97.65 111.1 –1.02 11.79 19.43 23.86 58.44

Lag

RSM(h)

65.23 40.52 56.64 35.78 79.19 42.32 35.88 60.40 76.32 43.84 49.28 113.59 132.56 154.65 16.76 40.86 48.19 56.83 157.46 70.32 188.98 15.95 22.29 103.68 16.66 14.12 108.30 107.76 7.44 10.11 19.74 21.44 71.17

SD: Standard deviation

Material and Methods

tions, selected randomly from the ranges indicated, were employed for model validation.

Experimental design


A complete star factorial design was performed with the following intervals for the conditions studied: temperature (9 to 21 °C); pH (4.5 to 8.5), NaCl (0 to 8%); NaNO2 concentration (0 to 200 ppm) in both aerobic and anaerobic conditions. Additional condi-

Preparation of inoculum Escherichia coli O157:H7 strain CECT 4076 (Spanish Collection of Culture Types, Valencia, Spain) was grown separately in 50 mL of tryptone soya broth (TSB, Oxoid) and subcultured on 3 successive days. The 3rd subculture was grown for 18 h at 37 °C as far as the stationary phase. The number of microorganisms present in the culture was calculated from the absorbance value using a previously obtained calibration line. Subsequently, the necessary dilutions were made in TSB to obtain a total of 106 CFU/mL per well. A calibration line (Figure 1) was performed to determine with some degree of accuracy the number of cells injected into the samples, by taking 3 previous calibrations made using Bioscreen C (Labsystems, Finland), with readings at 600 nm under an optimal temperature condition of 37 °C, and the E. coli O157:H7 strain CECT 4076. For this purpose, double serial dilutions were made in TSB at different initial microorganism concentrations. Four wells were filled with 250 ␮L of each dilution and only those with optical density (OD) values greater than 0.3 were accepted. At the same time, the inoculum was controlled by counting on Tryptone Soy Agar (TSA, Oxoid) and incubated at 37 °C for 24 h.

Sample preparation Figure 1—Calibration line, optical density (OD) as compared with cell concentration (log10 N CFU/mL) of E.coli O157:H7 640

JOURNAL OF FOOD SCIENCE—Vol. 68, Nr. 2, 2003

The different medium conditions were achieved by adding the corresponding concentration of sodium chloride (Oxoid) (0 to 8%) to 100 ␮L of TSB (Oxoid); after complete dissolution, pH (4.5 to 8.5)

ANN for Escherichia coli O157:H7 . . . Table 2—Average replicate values for observed and estimated growth rate (Gro, GrANN, GrRSM) and lag-time (Lago, LagANN, LagRSM) of E. coli O157:H7 by Artificial Neural Networks (ANN) and Response Surface Model(RSM) for additional data randomly selected for the mathematical validation. T(°C)

pH

NaCl (%)

13 15 15 15.3 15.3 16 17.5 17.5 17.5 17.5 18 18 19.8 21 21 21

6 6.5 6.5 6.5 6.5 5.5 6 6 7 7 6.5 6.5 6.5 6.5 6.5 6.5

5 4 6 4 6 4 5 5 5 5 2 6 6 0 2 6

NaNO2 (ppm) 116.8 100 140.2 140.2 140.2 93.4 116.8 163.5 116.8 163.5 50 50 140.2 100 50 50

Gro (SD) (h–1) 0.0148 0.0440 0.0365 0.0605 0.0502 0.0206 0.0433 0.0340 0.1136 0.1025 0.1521 0.0467 0.0506 0.1982 0.1767 0.0789

(0.0103) (0.0118) (0.0243) (0.0066) (0.0320) (0.0028) (0.0089) (0.0060) (0.0219) (0.0180) (0.0234) (0.0102) (0.0120) (0.0152) (0.0300) (0.0142)

GrANN (h–1) GrRSM (h–1) 0.0129 0.0461 0.0271 0.0489 0.0296 0.0338 0.0448 0.0309 0.078 0.0743 0.1221 0.0644 0.0603 0.2034 0.1767 0.0986

0.0250 0.0569 0.0311 0.0599 0.0327 0.0374 0.0478 0.0408 0.0851 0.0875 0.1282 0.0569 0.0690 0.2238 0.1924 0.0935

Lago (SD) (h) 127.57 56.50 211.70 69.64 231.69 60.49 51.36 53.32 53.50 54.05 15.57 80.30 80.31 7.66 10.78 61.76

(11.21) (3.26) (46.73) (8.90) (32.87) (36.46) (1.92) (3.17) (2.26) (1.43) (0.43) (5.96) (10.40) (0.54) (0.29) (3.84)

Lag

ANN

(h) Lag

149.82 56.92 208.07 56.15 194.82 51.87 64.95 70.96 60.63 64.62 17.93 95.12 78.06 2.83 7.82 51.62

RSM

(h)

132.56 48.19 176.54 49.16 168.55 51.68 66.38 77.44 57.41 59.59 14.08 91.56 90.56 7.46 9.68 62.96

SD: Standard deviation

was adjusted with HCl (5 M) and NaOH (5 M). After sterilization, selected NaNO2 (0 to 200 ppm) concentrations were added after previous filtration by cellulose nitrate membrane filters (0.2 ␮m, Whatman, England). All wells were topped up with 200 ␮L of medium and 50 ␮L of E. coli O157:H7 inoculum. Six replicates of each selected condition were performed and 3 blank samples were run (Table 1). All selected conditions were repeated, but wells were topped up with 250 mL of sterile paraffin to create anaerobic conditions.

Growth curve fitting The DMFit curve fitting Program designed by Baranyi (1998) was used for the optical density (O.D.) data fit, applying the modified Gompertz function (Mc Clure and others, 1993) and Baranyi function (Baranyi and Roberts, 1995) for the estimation of growth rate and lag-time. Both models were compared using the following equation, to select that of least variance:

where (s) the transference function for the neurons in the hidden layer was sigmoid, and (l) the transference function for the units in the output layer was lineal; “output” here corresponded to the kinetic parameters estimated: lag –time and growth rate The genetic algorithm used searched the best net by least square methodology in the selection procedure and crossing/mutation operators (procedures which introduce some diversity into the algorithm), looking for the minimum weights in the pruning phase. The whole process was repeated 20 times (runs) and the best net was chosen, that is the net with lowest error and number of connections, and good generalization capacity (Hervás and others 1998, 2000, 2001a, 2001b). The error criterion chosen was the Standard Error of Prediction (SEP) expressed as a percentage, which has the advantage of being dimensionless and is obtained by the following expression:

(1)

where yobs: observed absorbance value at time t and ypred: predicted absorbance value for the model at time t. To test the significance of variance, a t test was performed using Statistics for Windows (Microsoft, Redmond, Wash., U.S.A.).

Artificial neural network model Seventy percent (396) of the growth curves were taken for training or building of the model (Table 1) and 30% (192) were used for generalization processes, also termed mathematical validations (Table 2) following the methodology described by Hervás and others (2001b). The initial architecture of the ANN was designed to contain five neurons in the input layer, associated with environmental factors (temperature, pH, nitrite, and NaCl concentration and aerobic / anaerobic conditions), one hidden layer with 4 initial neurons and 1 output layer with 1 neuron associated with the kinetic parameter to be estimated, growth rate or lag-time, and another ANN with the same architecture but with 2 neurons in the output layer, for the simultaneous estimation of both kinetic parameters. A sigmoid function for the hidden layer and a linear function for the output layer were used for transference functions, where the nomenclature of the network architecture was as follows:

(2)

where obs: observed value; pred: predicted value; and mean obs: mean of observed values. The algorithms were implemented in C programming language and the training process was carried out using a Silicon Graphics Origin 2000(Calif., U.S.A.). Variables (temperature, pH, % NaCl, NaNO2 , growth rate, and lag-time) were scaled [0.1, 0.9] according to their different measurement ranges to avoid saturation problems in the sigmoid transference functions. The new scaled variables were named as t*, p*, c*, n*, g*, and l* respectively, and were obtained as follows:

(3)

where x* is the scaled variable value, X the original value, Xmin and Xmax the minimum and maximum value taken by the considered variable. Vol. 68, Nr. 2, 2003—JOURNAL OF FOOD SCIENCE

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VAR= (yobs – ypred)2/(yobs – meanyobs)

“nr of input neurons” : “nr hidden neurons” s : “nr of output neurons” l

ANN for Escherichia coli O157:H7 . . . The output values of the node hi in the hidden layer were obtained using a sigmoid function such as

(4) where wi,n are the connections’ weights or parameters values between the input and hidden layers. The output values of g* and l* were estimated using a linear function such as

Response surface model As well as for ANN, the kinetic parameters obtained from the primary model were used for Response Surface Analysis on the same 396 time-optical density curves (Table 1) and validated afterwards as ANN using 192 additional curve data (Table 2). The independent variables considered were temperature, pH, salt and sodium nitrite concentrations, and aerobic/anaerobic conditions, and the dependent variables were growth rate (Gr), and lag-time (lag). For estimation of the parameters of the fitting function and additional statistical analysis, SPSS version 9.0 (SPSS Inc., Ill., USA) software was used, considering the Levenberg-Marquardt algorithm as suitable for optimization of the error function.

Mathematical validation or generalization where wg,n are the connections weights or parameters values between the output and hidden layers. These values were descaled by: (7)

where Y* is the scaled variable value, Y the original value, Ymin and Ymax the minimum and maximum value taken by the considered variable.

Once the best model had been obtained by the training or modeling process, it was tested against a new data set, obtained in a similar manner and designated “generalization data set”, to evaluate the predictive capacity of the proposed model. This may be considered equivalent to what other predictive microbiology researchers have termed “validation”(Geeraerd and others 1998). Generalization capacity was evaluated by means of the Standard Error of Prediction percentage (% SEP), which is a relative standard deviation from mean prediction values and has the advantage, compared to other error measures, of not depending on the magnitude of the measurements.

Food Microbiology and Safety Figure 2—Graphic expression of the Artificial Neural Network model obtained of E.coli O157:H7. 642


ANN for Escherichia coli O157:H7 . . . Table 3—Standard error of predictions (%SEP), RMSE and number of parameters obtained by Artificial Neural Networks (ANN) and Response Surface Models (RSM) estimations of growth rate (Gr) and lag-time (Lag) of E. coli O157:H7

Gr ANN (h–1) Lag ANN (h) Gr RSM (h–1) Lag RSM (h)

Based on individual data Model Validation %SEP RMSE %SEP

RMSE

Based on average of data Model Validation %SEP RMSE %SEP

35.29 35.45 38.86 35.51

0.0244 20.69 0.0238 25.74

16.90 29.57 21.93 30.79

0.0211 23.04 0.0226 23.74

32.00 27.00 31.14 33.59

0.0098 19.77 0.0128 20.59

22.24 18.30 20.98 27.10

RMSE

no parameters

0.0170 14.03 0.0160 20.77

17 11 9

RMSE: Residual Mean Square Error

Results and Discussion

T

HE OPTICAL DENSITY VALUES OBTAINED BY BIOSCREEN C MEASURE-

ments were adjusted by DMFit (Baranyi 1998) as described in the Material and Methods section, obtaining a total of 588 curves, 294 for aerobic conditions and 294 for anaerobic conditions. The Baranyi and Roberts equation (Baranyi and Roberts 1995) was chosen as best describing the growth of E. coli O157:H7. Selection of the most suitable growth curve model is of great importance, since much of the observed disparity between different models for the same microorganisms could be caused by the adjustment equation chosen. Several studies had demonstrated, for example, that the generation time estimated from the de Gompertz equation and that estimated by the traditional method of calculating the tangent at the maximum point of the slope of the growth curve are quite different (Withing and Cygnarowicz-Provost 1992; Baranyi and others 1993; Ross 1993; Daalgard and others 1994). No significant differences (P > 0.05) were observed for the kinetic parameters (growth rate and lag-time) in aerobic and anaerobic conditions, so this variable was eliminated from models. Perfect adaptation of E. coli to anaerobic conditions has been described by Buchanan and Klawiter (1992); other authors have reported that if this microorganism is heat treated, the viable cells that are able to survive will recover more easily in anaerobic than in aerobic conditions (Bromberg and others 1998; George and others 1998; Semanchek and Golden 1998). Three different types of nets were designed: the 1st was set up to predict only growth rate values, the 2nd predicted only lag-time values, and the 3rd was designed to predict growth rate and lagtime jointly. After genetic algorithm processing, the final architectures were 4:3s:1l for growth rate, with 11 connections or parameters, 30.73 %SEP; 4:3s:1l for lag-time, with 11 connections and 25.76% SEP and 4:3l: 2s for growth rate and lag-time jointly, with a training or model SEP of 35.29% for Gr and 35.45% for lag (Table 3), with 17 connections, that is less than the sum of parameters for the 2 models separately. The next step was to ascertain the statistical significance of the differences observed between the estimation errors of the different models using a t test, and assuming that the SEP values obtained had a normal distribution. This analysis showed that there was no statistically significant effect due to network architecture (P > 0.05) so the 4:3l:2s model was chosen (Figure 2), due to the advantage of estimating both parameters simultaneously with the same model, a possibility not offered by RSM. When the same errors were calculated from the average values of replicates, results were different, as shown in Table 3: SEPs for the model or training set were 16.90 and 29.57% for growth and lag respectively. Authors do not usually specify in their reports whether they calculate model error from the average data for individual replicates; as Table 3 shows, the results may be quite different. Only a

few papers address the application of ANN to prediction of kinetic parameters for food microorganisms; an added difficulty, when comparing the results obtained here with those reported elsewhere, is that errors are all expressed in different ways, thus hindering comparison. Some authors report smaller errors, 0.71 to 0.12 and 4 to 9 RMSE values for generation time and lag time, respectively (Jeyamkondan and others 2001), but do not specify the architecture or complexity of the model; moreover, only 2 variables are considered, pH and temperature. Geeraerd and others (1998) describe a similar number of parameters for individual ANNs, with a mean square error of 0.02284 for growth rate. The ANN study by Hajmeer and others (1997) involved four variables with very low error values of around 4% mean absolute relative error but using a very complex model with a large number of parameters. Hervás and others (2001b) report a SEP of around 4 to 9% depending on whether the net was pruned or not. Acuña and others (1998) report mean error indices of 15 to 30, although modeling only three variables. The model equations are as following: For growth rate:

For lag-time:


Another way of studying the validation of a model is by calculating the accuracy and bias factors described by Ross (1996), and using graphic representations of observed versus estimated values.

where

Only those terms that were statistically significant were considered in the RSM model. Different transformations of the data, such as the application of logarithms, were also studied so as to improve the model. The equations for the best models were: Gr = 0.1913 – 0.0354 × T + 0.0316 × NaCl – 0.0010 × NaNO2 + 0.0060 × T × pH – 0.0023 × T × NaCl – 0.00002 × T x NaNO2 + 0.0002 × pH × NaNO2 + 0.0006 × T2 – 0.0070 × pH2 – 0.010 × NaCl2 (15) Vol. 68, Nr. 2, 2003—JOURNAL OF FOOD SCIENCE

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ANN for Escherichia coli O157:H7 . . . Table 4—Bias and accuracy factors and RMSE estimated for the growth rate (Gr) and lag-time (Lag) of Artificial Neural Networks ANN) and Response Surface Model (RSM) with experimental additional data

The results show a 2nd-order Response Surface Model since the 3rd-order model displayed regression coefficients whose differences were not significant (P > 0.05). As Table 3 shows, % SEPs of the model or training set for growth rate and lag time estimated by RSM were 38.86 and 35.51, respectively; that is very similar values. When the same errors were calculated from the average values of the replicates, results were different (Table 3): 21.93 and 30.79% respectively. Variables considered in constructing E. coli O157:H7 growth models exerted varying influences on microorganism development. Temperature, and to a lesser extent concentrations of NaCl and pH, were the factors most affecting the kinetic parameters, growth rate and lag-time. The concentration of NaNO2 did not significantly influence lag-time, but did affect growth rate. No significant differences were detected for E. coli O157:H7 growth under aerobic or anaerobic conditions. The model was validated using additional data at different experimental conditions not included in the model; the average values of the 6 replicates are shown in Table 2. When the model was validated against other data with different medium conditions from those used for the elaboration of data, the error—although remaining within the range for the model—was generally higher due to interpolation; however, this enhanced the robustness of the model.

The calculated SEP of these validation or generalization sets is shown in Table 3, where in general RSM errors were greater than those recorded with ANN. The model yielded greater errors in conditions approaching inhibition of microorganism growth. Generalization or validation was repeated excluding conditions involving over 6% of NaCl and 100 ppm of NaNO2, from the remaining 156 growth curves, estimated SEP fell to 24.04% and 19.59% for Gr and lag respectively. Therefore, model users wishing to obtain predictions for these suboptimal conditions should bear in mind that the error will be higher, especially for lag-time. After validating both models and examining their generalization capacity, they were compared in terms of RMSE, %SEP, model complexity, advantages and disadvantages. Errors for lag time were lower in ANN than in RSM (Table 3). Estimations of growth rate were similar in both models. Other authors report a better estimation with ANN than with RSM (Hajmeer and others 1997; Hervás and others 2001b; García and others 2002). Another factor requiring study is the number of parameters, or complexity, of models. Hajmeer and others (1997) reported on an ANN for Shigella flexneri with smaller error values (4 to 12% mean absolute relative error) but with considerable complexity (142 parameters). Some researchers do not approve the use of ANNs for estimation of growth parameters, due to their complexity. However, thanks to genetic algorithm pruning, ANNs have been shown to be even simpler than regression (Hervás and others 2001b; García 2002). In the present study, ANN required 17 parameters for joint estimation of both variables, compared with 11 parameters needed for the estimation of Gr plus 9 parameters required for estimating lag in the RSM.(Table 3). One advantage of ANN is that it imposes no restrictions on the type of relationship governing the dependence of the growth parameters on the various running conditions (Hajmeer and others 1997; Geeraerd and others 1998). Regression-based RSM requires the order of the model to be stated (that is, 2nd, 3rd, or 4th order), while ANN tends to implicitly match the input vector (that is growth conditions) to the output vector (that is kinetic parameters). A further advantage of ANN is that it allows the inclusion of nogrowth data and, as this study shows, it can be performed with one, two or more output or estimated parameters.

Figure 3—Observed against predicted growth rate values for E.coli O157:H7 with additional experiment data. The diagonal line is the line of identity.

Figure 4—Observed against predicted lag-time values for E.coli O157:H7 with additional experiment data. The diagonal line is the line of identity.

Bias Factor GrANN (h–1) LagANN (h) GrRSM (h–1) LagRSM (h)

0.95 0.95 1.09 0.98

Accuracy Factor 1.25 1.24 1.27 1.17

Ln(lag) = 11.1286 – 0.2574 × T – 1.7525 × pH + 0.0163 × NaNO2 – 0.0025 × pH × NaNO2 + 0.0004 × NaCl × NaNO2 + 0.0034 × T2 + 0.1461 × pH2 + 0.056 × NaCl2 R2 = 0.92 (16)

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ANN for Escherichia coli O157:H7 . . .

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Hajmeer M, Basheer I, Najjar Y. 1997. Computational neural networks for predictive microbiology II. Application to microbial growth. Int J Food Microbiol 34:51-66. Hassibi B, Stork D. 1993. Second order derivatives for a network pruning: Optimal brain surgeon. In: Hanson SJ, Cowan JD, Giles CL, Editors. Advances in neural information-processing systems 5. San Mateo, Calif.: Morgan Kaufmann. p 164-71. Hervás C, Ventura S, Silva M, Pérez D. 1998. Computational neural networks for resolving nonlinear multicomponent systems based on chemiluminescence methods. J Chem Inform Comp Sci 38:1119-24. Hervás C, Algar JA, Silva M. 2000. Correction for temperature variations in kinetic-based determinations with pruning computational neural networks by using genetic algorithms. J Chem Inform Comp Sci 40:724-31. Hervás C, Toledo R, Silva M. 2001a. Use of pruned computational neural networks for processing the response of oscillating chemical reaction with a view to analyzing nonlinear multicomponent mixtures. J Chem Inform Comput Sci 41:1083-92. Hervás C, Zurera G, García RM, Martínez JA. 2001b. Optimisation of computational neural network for its application to the prediction of microbial growth in foods. Food Sci Technol Int 7:159-63. Jeyamkondan S, Jayas DS, Holley RA. 2001. Microbial growth modeling with artificial neural networks. Int J Food Microbiol 64:343-54. Le Cun Y, Denker J, Solla S. 1990. Optimal brain damage. In: Touretzky D.S. (Ed.). Advances in neural information-processing systems. Vol. 2. San Mateo, Calif.: Morgan Kaufmann. p 598-605. Mc Clure PJ, Baranyi J, Boogard E, Kelly TM, Roberts TA. 1993. A predictive model for the combined effect of pH, sodium chloride and storage temperature on the growth of Brochothrix thermosphacta. Int J Food Microbiol 19:161-78. Najjar Y, Basheer I, Hajmeer M. 1997. Computational neural networks for predictive microbiology. I. Methodology. Int J Food Microbiol 34:27-49. Riley LW, Remis RS, Helgerson SD, McGee HB, Wells JG, Davis BR, Herbert RJ, Olcott ES, Johnson LM, Hargrett NT, Blake PA, Cohen ML. 1983. Hemorragic colitis associated with a rare Escherichia coli serotype. N Engl J Med 308:6815. Ross T. 1993. A philosophy for the development of kinetic models in predictive microbiology [DPhil Thesis]. Hobart, Australia: Univ. of Tasmania. Ross T. 1996. Indices for performance evaluation of predictive models in food microbiology. J Appl Bacteriol 81:501-8. Semanchek JJ, Golden DA. 1998. Influence of growth temperature on inactivation and injury of Escherichia coli O157:H7 by heat, acid, and freezing. J Food Prot 61:395-401. Sharp JC, Coia JE, Curnow J, Reilly WJ. 1994. Escherichia coli O157 infections in Scotland. J Med Microbiol 40:3-9. Whiting R, Buchanan R. 1994. Microbial modeling. Food Technol 48:113-20. Whiting R, Cygnarowicz-Provost M. 1992. A quantitative model for bacterial growth and decline. Food Microbiol 9:269-77. MS 20020421, Submitted 7/12/02, Revised 8/28/02, Accepted 10/1/02, Received 10/10/02 This work has been partly financed by the CICYT ALI98-0676-C02-01 and ALI98-0676-CO202, and Research Group AGR-170 and TIC 148 AYRNA. The authors thank Dr. Jôszef Baranyi for kindly providing his program.

Authors García-Gimeno, Barco-Alcalá, and Zurera-Cosano are with the Dept. of Food Science and Technology, Univ. of Córdoba, Córdoba, Spain. Author Hervás-Martínez is with the Dept. of Computer Science and Numerical Analysis, Univ. of Córdoba, Córdoba, Spain. Author Sanz-Tapia is with the Campus Univ. de Rabanales, Córdoba, Spain. Direct inquiries to author García-Gimeno (E-mail: [email protected]).


The major disadvantage of ANN’s performance is that it is more complicated to apply, and requires both an expert staff and a much longer run-time. Another way of comparing or studying the validation of a model is that proposed by Ross (1996), by accuracy and bias factors and also the graphic representation of observed compared with predicted values. Table 4 shows that both calculated indices, accuracy, and bias factors and yield acceptable values, very close to one. Growth rate is slightly on the dangerous side (bias 0.95) using ANN, since the prediction should be greater than the observed value (that is value higher than 1), but the deviation is very small. This is more readily appreciable in Figures 3 and 4, which show very little scattering from the identity line. ANN has proved to be a useful tool for the estimation of E. coli O157:H7 kinetic parameters, with less estimation error than RSM, for a similar complexity, and with the great advantage of having both kinetic parameters in just one model, which is an important advance in traditional predictive modeling.

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