neural networks; Gompertz model; Microbial growth; Multiple linear regression; Response ... of the biological nervous systems. A detailed description of CNNs ...
hr(emationalJoumal ofFoodMic&iobgy Food
International
Journal
of
Microbiology
34 (1997) 51-66
Computational neural networks for predictive microbiology II. Application to microbial growth Maha
N. Hajmeer”,
Imad A. Basheerb,
Yacoub
M. Najjarb,*
“Dt~purtma~tof’ Animal Sciences arld Industry, Kansas State Unisersity, Manhattan, KS 66506, USA bDepurtment of Civil Engineering, Kunscrs Starr Unicersity, Manhattun, KS 66506, USA Received
5 February
1996; revised
I8 June
1996; accepted
IO August
1996
Abstract
The growth of a specific microorganism on a certain food is influenced by a number of environmental factors such as temperature, pH, and salt concentration. Models that delineate the history of the growth of microorganisms are always subject to a considerable debate and scrutiny in the field of predictive microbiology. Regardless of its type, a growth model (e.g., modified Gompertz model) contains several parameters that vary depending on the microorganisms/food combination and the associated prevailing environmental conditions, The growth model parameters for a set of operating conditions are commonly determined from expressions developed via multiple linear regression. In the present study, a substitute for the nonlinear regression-based equations is developed using computational neural networks. Computational neural networks are applied herein on experimental data pertaining to the anaerobic growth of Shigellu,flexneri. Results have indicated that predictions by neural networks offer better agreement with experimental data as compared to predictions obtained via corresponding regression equations. Keywords: Computational neural networks; linear regression; Response surface models;
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author.
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+ 1 913 5325863;
0 1997 Elsevier 169-5
Gompertz model; Shigellu ,flexneri
fax:
+ 1 913 5327717
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reserved
Microbial
growth;
Multiple
1. Introduction The association of microorganisms with food is of major interest due to the various issues pertinent to safety and quality. The existence of microorganisms may indicate a potential food contamination which results in shortening the shelf life of the product. In addition, a hazard on public health will be plausible if the proliferating microorganisms are pathogenic. Food microbiologists are continuously concerned with conducting experimental investigations in an attempt to determine the extent of dependence of the microbial growth on factors related to the ecology or processing modes. As a consequence, models that relate the microbial growth of different types of microorganisms-causing-food deterioration to several environmental factors are of significant debate in the field of predictive microbiology. Regardless of their type, these predictive models are assumed to furnish rapid and accurate estimation of the microbial growth of a certain kind of organism on a specific food type under various prevailing environmental conditions. Predictive microbial growth models can be utilized: (i) to describe the conduct of microorganisms under a variety of physical and chemical conditions, (ii) to facilitate routes for the prediction of food shelf life and safety, (iii) to identify the critical points in the process of the production and distribution of the food, and (iv) to assist in optimizing the production chains (Zwietering et al., 1990). Consequently, predictive microbiology can be directed towards the product formulation and reformulation, process design. HACCP, time-temperature profiles. training and education, etc. (Walker, 1994). In the present study, an alternative approach to conventional methods of microbial growth prediction is proposed and analyzed. The suggested approach is based on the recent understanding of artificial computational neural networks (CNNs) as analogous mechanisms of the biological nervous systems. A detailed description of CNNs and in particular, backpropagation networks, was presented in the first paper of this series (Najjar et al., 199613). In the present paper, the microbial growth of microorganisms, with respect to the various operating parameters pertaining to food type and other environmental conditions, is outlined. Additionally, an application pertinent to the growth of Sh~~ellu,flexneri is investigated in order to address several issues pertinent to applicability of this computational tool in the area of predictive microbiology. The prediction accuracy of the developed neural network in regard to the generalization obtained is compared with the available response surface models derived by the stepwise multiple linear regression method. Additionally, the advantages of the adopted methodology over conventional methods of microbial growth modeling are addressed. Studies centered about the microbial growth constitute a considerable part of research conducted in the area of predictive microbiology. Most of those studies fdll within the category of modeling the time-based growth of certain microorganisms on specific food using different types of kinetic models. A list of the several bacterial growth models that are most commonly used in predictive microbiology are given in Zwietering et al. (1990).
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A predictive microbial growth model is an equation or a set of equations containing several parameters that need to be quantified for certain environmental (running) conditions and microorganism/food combination. For instance, the modified Gompertz model (Zaika et al., 1992) contains four parameters that may vary in relation to the running conditions on which the growth is occurring. Additionally, for the same running conditions, the Gompertz parameters will also vary depending upon the microorganism/food matrix employed in the study. The objective of the present paper is to outline a methodology based on neural networks that enables estimation of the growth model parameters from the knowledge of environmental running conditions without the need to: (i) conduct the lengthy type of microbial growth experiments, and (ii) determine model parameters by fitting (using a nonlinear optimization technique) the data to an assumed valid model. Response surface models are the most commonly used techniques for modeling. These models are simply multi-parameter regression equations that are usually developed through stepwise linear regression techniques driven by a forward selection or backward elimination method. Zaika et al. (1994) use backward elimination to develop necessary response surface models for the anaerobic growth of S. jlexneri as affected by the temperature, pH, NaCl and NaNO, concentrations. Using the same modeling technique, Buchannan and Bagi (1994), among others, developed response surface models for the growth of E. coli 0157:H7. Predictive modeling of microbial growth has been performed under various physical and chemical conditions for a wide variety of microorganisms. Examples of relevant microbial investigation include the work of Buchannan and Bagi (1994) Bhaduri et al. (1994) and Zaika et al. (1989, 1991, 1994). Additional studies involve McClure et al. (1994) Duffy et al. (1994) Davey and Daughtry (1995) and Buchanan and Golden (1995). A special double issue of the International Journal of Food Microbiology (Vol. 23, No. 3 and 4) has been entirely devoted for the microbial predictive modeling.
2. Microbial
growth parameters
As a kinetic model, the sigmoidal function is commonly employed to express the time-dependent growth of microorganisms. Among the various sigmoidal models discussed in the literature, the modified Gompertz equation was found to be the easiest to implement and statistically adequate to describe the growth of different types of microorganisms (see Zwietering et al., 1990). Additionally, the Gompertz parameters can be easily manipulated to determine several kinetic descriptors such as the exponential growth rate (EGR), lag phase duration (LPD), generation time (GT), and maximum population density (MPD) (Zaika et al., 1994). The determination of these kinetic descriptors may assist in understanding and rapidly comparing several microbial growth scenarios.
54
A4.h’. Hujmeer et ul. i Int. J. Food
The modified
Gompertz
equation
Microbiology
can be expressed
L(t) = A + C exp[ - exp( - B(r - M))]
34 (1997) 51-66
as (1)
where t(t) is the log count number of bacteria at time t, A denotes the initial level of bacteria [L(t = O)], C is the asymptotic amount of bacteria as t increases indefinitely [L(r = x)], A4 is the time corresponding to the maximum absolute growth rate, B is the relative growth rate at I = M, and e Y 2.718. As can be seen from Eq. (1) the several Gompertz growth model parameters (A, C, B, and hil) should be evaluated in order to determine the shape of the growth function and to quantify the various associated kinetic parameters (EGR, LPD, GT, MPD). For a specific food/microorganism, these growth model parameters vary in magnitude depending on the values of the operating environmental conditions. Most of the methods of evaluation of these parameters involve use of multiple linear regression methods that utilize the successive regression backward elimination (SRBE) technique (e.g., Zaika et al., 1994; Buchannan and Bagi, 1994). The obtained regression equations are commonly referred to as response surface models which relate the values of the growth model parameters (e.g., B and A4) to various environmental and ecological conditions such as those related to temperature, pH, salts concentration, etc. Most of the response surface models were found to be highly nonlinear involving a large number of terms embodying various combinations of the corresponding environmental variables. Some of these regression equations can be seen in Zaika et al. (1994) and Buchannan and Bagi (1994).
3. Application
on Shigella jlexnevi
In order to test the applicability of the CNN approach suggested in this paper, an example of the growth of S..flrxneri is discussed. Shigellosis has been associated with a wide range of food types (Satchel1 et al., 1990). In the United States, between 1973-1987, a total of 14 399 out of 108 906 (13%) cases involving bacteria were identified to be related to SIzigellu species (Bean and Griffin, 1990). However, it is assumed that these numbers are fractions of the actual number that occurs per year which has been estimated at 163 000 cases with an estimated associated cost of $63 million (Todd, 1989). The data of S..flexrwi growth was taken from an extensive experimental study by Zaika et al. (1994) which concentrates on the variation of growth of S. fkxneri as a function of temperature, pH, and concentration of sodium chloride (NaCl) and sodium nitrite (NaNO,). All growth experiments were conducted using brain heart infusion (BHI) media. The various experiments of Zaika et al. (1994) on S. j&xneri were selected using a factorial design that yielded a total of 124 different experiments. The ranges of the different environmental parameters were as follows: temperature: 12227°C; pH: 5.5-7.5; NaCl concentration: 0.50-4.0%; and NaNO, concentration: O-1000 ppm. It is to be noted that all the combinations studied by Zaika et al. (1994) were those that were known or observed to be suitable for the growth and survival of S. flexneri. The four Gompertz parameters were determined
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by Zaika et al. (1994) from the experimentally obtained growth data (i.e., L versus t) using the modified Gompertz expression Eq. (1) in conjunction with a multiple parameter regression program that employs a Gauss-Newton iteration procedure. The program fits the data to the Gompertz expression while minimizing the differences between observed and estimated bacteria count.
4. Developing CNN-based
predictive model
The data of Zaika et al. (1994) that contains 124 different combinations of environmental conditions was considered to develop the database for training the CNN. However, those data sets where no growth had occurred were eliminated from the present study. This has resulted in a possible database of an appreciable size of 83 sets. In order to train and test the accuracy of the CNN, the entire database was split into a training set and a validation set. For this purpose, the present database was divided into 66 training sets and a total of 17 sets (about 25%) were kept aside for testing the prediction accuracy of the developed CNN. To avoid any anticipated bias, the validation (testing) sets were randomly selected from the entire database using a method based on generating random numbers. The two parameters B and A4 were considered herein for modeling. Both A and C were found by Zaika et al. (1994) to take approximately constant values at 2.84 and 5.36, respectively. Moreover, the natural logarithm of both B and A4 was considered in developing the CNN. This has been carried out in order to compare with the corresponding correlations of Zaika et al. (1994) which involved logarithmic transformation. The architecture of the CNN was designed to contain four input parameters in the input layer and two output parameters in the output layer. The feed forward backpropagation algorithm (FFBP) discussed in Najjar et al. (1996a) was used to train the CNN with a momentum of CI= 0.45 and a learning rate, /I = 0.95. All runs were implemented on a mainframe computer using the FFBP algorithm encoded by the authors into a Fortran program. On average, the total running time for all training cycles with a single training phase of a particular network architecture did not exceed 3 min.
The number of training cycles (also called iterations or epochs) the network should be subjected to in order to reach the pre-assigned maximum tolerance depends on the specific problem analyzed. This is usually determined by a trial-anderror procedure by training a CNN with an arbitrary number of hidden nodes on the entire database. Typically, the number of training epochs is varied and the error (representing the difference between the target and predicted values for the entire database) is calculated. For the present study, a network with 20 hidden nodes was used to determine the appropriate number of training epochs. Fig. 1 depicts the variation of the mean of the absolute value of the relative error (MARE) for all the
56
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et (11. I ht.
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34 (1997)
51-66
training sets and the corresponding coefficient of determination, R2 (between the predicted and experimental values) as a function of the number of iterations. As can be seen from Fig. 1, the MARE and R2 for both Gompertz parameters--n(B) and Ln(M)-stabilize at 10000 iterations. This implies that any further increase in the number of iterations would not induce any reduction in the error (or increase in R2), nevertheless, a consequent substantial increase in the computation time and potential of overfitting can both be expected. Therefore, all runs of the CNNs were conducted using 10000 training cycles epochs. 4.2. Optimum number of hidden nodes The optimum number of hidden nodes was iteratively determined by developing several CNNs that vary only with the size of the hidden layer and simultaneously observing the change in both the indices of accuracy (i.e. MARE and R2). This was implemented by utilizing both the training and validation data sets. The effect of the number of hidden nodes (NH) on the accuracy of the CNN for prediction of the training data sets (i.e., the RECALL) is shown in Fig. 2(a) and Fig. 2(b) for Ln(B) and Ln(M), respectively. It can be seen from these two plots that both indices of accuracy have stabilized at NH = 20. However, the RECALL data cannot be used to finally decide on the optimum NH because of the problem of the potential occurrence of overfitting or memorization of the network. Virtually, MARE as low as 0.0 and R2 as high as 0.9999 can be achieved by testing the network to predict the training data on which it was trained. This can be done by increasing indefinitely, both the number of hidden nodes and the number of training cycles. Since any CNN is expected to generalize rather than to memorize, the 17 testing sets were also used to obtain the optimum NH. Fig. 2(c) and Fig. 2(d) depict the variation of MARE with NH for the 17 testing sets which have never
-1 .ooo
CNN 4-2-2
-0.960
2 B
-0.940
b
r? OI
0.900 5
10
15
20
25
30
35
40
45
50
Number of Iterations (Thousands)
Fig. I. The effect of number
of training
cycles on accuracy
of CNN
57
M.N. Hajmerr et al. /ht. J. Food Microbiology 34 (1997) 51-66
Number of Hldden Nodes (NH)
0
2
6 8 10 12 14 16 18 Number of Hidden Nodes (NH)
4
d 20.00CNN 4-20-2
0.004 0
2 I
4
20
22
MM)
6
8
10 I
12
14 ,
16 I
I8
20 ,
Number of Hidden Nodes (NH)
Fig. 2. (a,b) The effect of the number of hidden nodes on the accuracy of RECALL of the number of hidden nodes on the accuracy of PREDICTION data.
data. (c,d) The effect
been used in training. As can be observed, 20 hidden nodes were also found to give sufficiently low error for the present problem. A schematic of the architecture of the resulting optimized CNN is depicted in Fig. 3. In order to check for the memorization problem, both RECALL and PREDICTION (i.e., CNN prediction of the testing sets) can be compared using Fig. 2. As can be seen from the plots, the MARE for the RECALL data generally decreases with increasing NH, so does the MARE for the PREDICTION data.
5. The CNN prediction accuracy Using with the 4(a) and network
the optimized CNN (i.e., CNN 4-20-2), the RECALL data was compared experimental values of the Gompertz parameters, Ln(B) and Ln(M). Fig. Fig. 4(b) depict such a comparison. As can be seen from these figures, the was able to predict the 66 data sets on which it was trained with a high
accuracy. The developed CNN was able to predict Ln(B) with a MARE of 4.26% and R2 of 0.984, while it predicted Ln(M) with MARE of 3.70% and an R2 of 0.982. For the 17 testing data, Table 1 shows a comparison between experimental and as-predicted by the 4-20-2 CNN. Fig. 4(c and d) graphically illustrate the agreement between the predicted and the actual Gompertz parameters for the testing sets. A relatively good agreement was achieved with an estimated MARE of 12.37% and 10.94”/0 for Ln(B) and Ln(M), respectively. Also, the corresponding R’ for Ln(B) and Ln(M) for the testing data were calculated as 0.881 and 0.879. This may be considered as a high prediction capability of the developed CNN for sets it had never been trained on.
6. Comparison
with regression models
Zaika et al. ( 1994) presented linear multiple regression models for the determination of Ln(B) and Ln(A4) for the same data used in this paper. They considered second- and third-order correlations by utilizing several combinations of the input parameters (i.e., temperature, pH, NaCl, and NaNO, concentrations). The third-order regression equations were found to give closer agreement with experimental values of both Ln(B) and Ln(M). Using the third-order correlations of Zaika et al. (1994) the Gompertz parameters were calculated and compared to experimental values as plotted in Fig. 5(a) and Fig. 5(b) (for 83 data sets). The MARE was calculated as 11.56% and 8.62”/0 and the R’ as 0.904 and 0.893 for Ln(B) and Ln(M). respectively. It is worthy mentioning herein that Zaika et al. (1994) have
Fig. 3. The developed
S. fleltneri CNN
M.N.
Hajmeer
et al. I ht.
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34 (1997) 51-66
NN 4202 ECALL:
-3
-2
Measured Ln(B)
b 7 CNN
d ‘1 4-20-2
7 CNN
/
4-20-2
PREDICTION:
6. RECALL:
OG’#
7
Measured In(M)
Fig. 4. (a,b) The comparison between actual and CNN-predicted values comparison between actual and CNN-predicted values of PREDICTION
of RECALL data.
data.
(c,d) The
utilized the 83 data sets to construct their regression equations. Therefore, in order to conduct a fair comparison with the CNN prediction, a new CNN trained on the same 83 data sets was developed. As was found previously, a total of 20 hidden nodes was also found to generate good predictions for this new CNN. A better agreement between the measured and CNN-predicted data for the 83 cases was achieved as shown in Fig. 5(c) and Fig. 5(d). The present CNN was found to predict the 83 data sets with indices of accuracy as: MARE of 5.54% and 4.04% for Ln(B) and Ln(M), respectively, and a corresponding R2 of 0.977 and 0.983.
7. Sensitivity
analysis
Neural networks offer an interesting cance of the various input parameters
ability to test the sensitivity and signifito produce the output(s). This can be
60
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carried out by manipulating the connection weights and biases of the designed CNN using the method of weights partitioning described by Goh (1994). This method is superior to the conventional method of sensitivity analysis because of the less effort involved and its ease of implementation. The relative importance of the operating conditions; namely the temperature, pH and concentration of NaC1 and NaNO, in determining the values of both the Gompertz parameters, was studied using this approach. The CNN developed by training on the 66 data sets was analyzed. From such analysis, it was found that the temperature, pH and NaCl and NaNO, concentrations contribute to 32, 27, 21, and 20% of the influence on the output values of both Ln(B) and Ln(A4). In other words, while temperature seems to be the most influential for the growth of S. j?exncri, the other three operating conditions are of slightly lower significance. Therefore, high accuracy should be practiced in determining the various operating conditions as they could equally affect the output results. In order to test the effect of database size used for training the CNN on the sensitivity results, the CNN trained on the 83 data sets was also analyzed for sensitivity of the four input parameters. Similar results were also obtained; viz. 32% for temperature, 30% for pH, 21% for NaC1 concentration, and 17% for nitrite concentration.
Table I Comparison
between
Set #
and CNN-predicted
values for 17 testing
11 14 20 24 21 30 36 42 46 52 55 61 71 74 82 indecis
sets
Ln(M)
Ln(B)
8
Overall 0.879).
actual
Actual
CNN
Actual
CNN
-4.880 -4.351 -3.265 - 2.606 - 3.619 -3.154 - 3.344 - 1.710 - 2.478 - 2.328 - 1.835 - I.319 - 1.7s3 - 2.749 - I.172 -0.956 -2.278
-4.899 -4.720 - 3.000 - 2.850 - 3.220 -4.017 -3.919 ~ 1.735 - 2.455 -2.165 -2.010 - 2.053 - 1.782 - 2.969 - 1.082 - 0.943 - I.442
5.534 5.270 3.577 3.246 3.642 4. I49 4.437 1.973 2.89 I 2.779 2.268 3.092 1.985 3.978 I.758 1.601 3.383
5.960 5.630 3,098 3.410 3.437 5.060 5.750 2.050 2.963 2.744 2.448 2.6S2 2.236 4.060 2.077 1.511 2.455
of accuracy:
Ln(B):
(MARE,
R’) = (12.37’%1, 0.881):
Ln(M):
(MARE,
R’) = (10.94%,
M.N. Hajmeer et al. / ht. J. Food Microbiology 34 (1997) 51-66
a
O-
c
REGRESSION
O-
61
CNN 420.2 PREDICTION:
Measured
Ln(B)
REGRESSION
Measured
Ln(B)
3 Measured
4 Ln(M)
CNN 420-2
6-
PREDICTION:
1
08
0
i 0
Measured
Ln(M)
Fig. 5. (a,b) The comparison between entire database. (c,d) The comparison
1
2
5
6
actual and prediction by Zaika et al. (1994), correlation for the between actual and prediction by CNN for the entire database.
8.Discussion and conclusions A neural network for prediction of the modified Gompertz growth model parameters was developed. The anaerobic growth of S. &xneri was studied in this paper as a special case for testing the efficiency of neural networks in predicting the growth parameters. The neural networks developed herein were found to yield a better agreement with experimentally measured data as compared to data predicted by regression equations. A substantial increase in the values of R2 for both model parameters-O.977 and 0.983 for Ln(B) and Ln(M)-has been observed through using the developed CNN as compared to corresponding R2 of 0.904 and 0.893 with the regression models of Zaika et al. (1994). Unlike traditional methods of analyzing sensitivity, CNNs are also powerful tools for studying the relative significance of model input parameters. The temperature was found to be the most important factor to affect the growth rate of S. _ffexneri in an anaerobic environment.
For the developed CNN model--and Zaika et al. (1994) regression equations-only cases where microbial growth has been observed were used. Hence, one should be cautious to use these predictive models to estimate the growth parameters for a certain set of operating conditions. Zaika et al. (1994) observed that growth of S. flexneri was inhibited under the following operating conditions:
T (“C) 12 15 19
PH All 5.556.5 5.5
NaCl All All 4.0
(‘%I)
NaNO, All 50 0
(ppm)
In order to avoid obtaining erroneous results from the prediction models (CNN or regression) one should compare the running conditions with the above mentioned limiting values prior to use of these models. If the running conditions fall out of these ranges, the predictive models can be safely used. Furthermore. as is the case with all modeling techniques, it is not recommended to use the developed model for extrapolation. In other words, the user has to restrict use of the model within the limits on which it was developed. These limits are: temperature (12-37°C). initial pH (5.5-7.5) NaCl (0.554.0’%), and NaNO? (OPIOOO ppm). As a further development for the neural network approach discussed in this paper, the general microbial growth CNN suggested by Najjar et al. (1996b) should be investigated in more detail. The general growth CNN can furnish a universal model to study the growth of any type of microorganism in a wide variety of foodiecology conditions. While the ecology-related parameters can be easily identified, the foodand microorganism-type related parameters have to be precisely delineated. This might be regarded as a major, critical step in the development phase of any generalized microbial growth model.
9. Future work
Due to the apparent distinctive capability of CNNs, a more general neural model for predicting microbial growth, that deals with a wide variety of microorganisms and food systems, could be developed. Due to the countless number of microorganism/food matrices that may exist in the environment, a universal microbial growth model can furnish a model that has a wider domain of applicability. A study that aims at developing a general model for prediction of microbial growth for a variety of organism types and under a large number
MN.
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63
of combinations of environmental and ecological variables, and for a wide spectrum of foods (media) is lacking. The literature, on the other hand, has dealt with developing growth models (kinetic and empirical) for specific microorganisms growing on certain types of food. The various parameters for such a universal model are expected to exhibit ambiguous relationships among each other. The ability of CNNs to handle extremely nonlinear problems where parameters are highly interrelated renders such a technique promising for developing a general microbial growth model. A microbiological growth model (of any type) is assumed to be characterized by three different groups of data pertaining to: (i) food characteristics, (ii) microorganism type, and (iii) environmental conditions. The food characteristics category is designed to include parameters that distinguish one food type from another based on some factors pertaining to physical, chemical, and biological properties, For instance, physical properties may include the contents of protein, fat, carbohydrates, minerals, vitamins, water, etc. The microorganism type-related category is assumed to be function of the biochemical characteristics of the microorganisms under study. For instance, the DIFCO tabulated method of identifying microorganisms of the family Enterobacteriaceae (DIFCO, 1980) can be manipulated to represent such a category. A microorganism-characteristics vector is designed such that for two different microorganisms this vector cannot be identical. The last category, namely the environmental conditions-related category, is designed to contain many of the parameters relating to operating conditions under which growth is occurring. Depending on the factors that may affect growth, this category may include temperature, pH, sodium chloride content, sodium nitrite concentration, water activity, gaseous concentration, etc. While these three categories form the structure of the input layer of the general CNN model, the output vector of the CNN includes all constants that appear in the assumed applicable growth equation. For example, if the modified Gompertz model is assumed to be valid for a large number of microbial growth studies, the output layer can be designed to contain parameters such as ,4, C, B, and M which represent the constants of the modified Gompertz equation. A schematic of the suggested general microbiological growth CNN is shown in Fig. 6. 9.2. Puwly
rsperin7ental-bnsed
neural models
In the previous work, the developed neural models were based on experimental growth data that was assumed to follow the modified Gompertz equation. However, generally, growth may deviate substantially from the Gompertz assumptions. Bacterial growth typically follows more or less similar trends of exponential growth followed by stationary and death phases. Neural networks can be utilized to model experimental full growth curves without imposing, a priori, any simplifying assumptions. These networks can be directly trained on the obtained experimental data. In subsequent papers, the authors will apply
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CNNs to model the dynamic behavior of growth on several microorganisms. The belief in neural networks to be capable of modeling the real complex growth of microorganisms is attributed to several reasons, among which are: (i) Unlike regression modeling, neural networks impose no restrictionon the type of relationship governing the dependence of growth parameters on the various running conditions. Regression-based response surface models require stating the order of the model (e.g., second, third, or fourth order), while neural networks tend to implicitly match the input vector (e.g., running conditions) to the output vector (e.g., Gompertz parameters) for all examples available. This can eventually lead to the highest possible agreement between the experimental and model data. (ii) ‘Sequential’ or ‘dynamic’ neural networks (Najjar et al.. 1996a) can be designed to enable the prediction of time-based growth when trained on experimental growth data. These networks utilize, in addition to the running conditions, part of the input data representing the bacterial count of previous time intervals for prediction of bacterial counts at subsequent time steps. This modeling methodology can yield relatively continuous type of growth curves. After a dynamic CNN is trained on appropriate data. it can then be used to predict growth curves (with all phases) for new microbial growth cases without the need to conduct experimental investigation. Another type of microbial growth curve
Growth Model Parameters e,g. Gompertz Eq. : In B, In M
‘_
..,
,-
,’
i___
‘I
!_
:
HIDDEN
FOOD-CHARACTERISTICS RELATED PARAMETERS protein, fat, salt, water, etc. Fig. 6. An illustration
LAYER
MICROORGANlSM-TYPE RELATED PARAMETERS
ENVlRONMENTAL/ECOLOGYRELATED PARAMETERS
Shigella, Salmonella, etc.
T, pH, NaCI, NaN02, aw, gas, etc.
of a generalized
neural
n&work
for microbial
growth
prediction involves determination of ‘descretized’ curve by designing a neural network with an input vector, containing only the running conditions and an output vector including times corresponding to a number of bacterial growth levels expressed as percentage of the maximum potential bacterial count. For this methodology, the reader is referred to a similar study by Basheer and Najjar (1996).
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