Models for Predicting the Combined Effect of

Models for Predicting the Combined Effect of Environmental Process Factors on the Exponential and Lag Phases of Bacterial Growth - Development and Application and an Unexpected Correlation K. R. Davey Food Technology Research Group Department of Chemical Engineering Adelaide University Adelaide 5005 Australia

The bacterial spoilage of foods and the growth of pathogens are of major importance to community health and safety and to the food process industries. The food industries are generally a nation’s largest manufacturing industry and among the most stable. In recent years mathematical models have emerged to offer the potential for predictive detection, and longer-term, process control. Future efficient processing, storage and handling (including distribution) of foods will depend heavily on the development of mathematical models and these will therefore have a central place in planning and design. Longer term, models are necessary to optimisation and computer control - superseding current empirical (rule of thumb) design and limited analyses. The mathematical models necessary for food processing operations and storage cannot simply be copied from models elaborated in chemical engineering and traditional biotechnology. The reasons that this emerging new field of microbiological process modelling, needs to build its own store of tools for understanding the underlying kinetics are reviewed and the potential for practical applications from this multidisciplinary field is discussed. The Davey Linear-Arrhenius (DL-A) model is reviewed in detail and the evolution of ideas is traced in its development and applications. An unexpected correlation emerges that is presented. Its application to some 93 years of published and independent growth data is highlighted. Findings suggest the prospect of a new qualitative law based on the notion of an environmental constant. The DL-A model can reliably predict the effect of an environmental process factor (such as T, aw and pH) on bacterial growth in the lag and growth phase from just one empirically determined coefficient. INTRODUCTION Worldwide, the food industries are generally a nation’s largest manufacturing sector, and importantly, among the most stable (Davey 1993a). From a process view, heat and mass transfer aspects appear well understood. Concurrently there has been in recent years a very rapid rise in mathematical techniques and computer technologies that promise process optimisation and the ability to control in real time (Davey 1992a, b). What is missing however is a quantitative understanding of the kinetics of bacterial growth, death (inactivation) and survival - especially as affected by combined environmental (process) factors such as temperature (T), pH, water activity (aw) and others, for e.g., the availability of oxygen. This important gap in knowledge and lack of a rigorous mathematical description severely limits development (Davey 1993a, b). We have been very interested over some 25 years in attempts to fill in this knowledge gap. The rewards in terms of industry competitiveness, reliable food quality and safety assurance are of course huge. However food-microorganism interactions are intriguingly complex. This has meant that mathematical models and techniques from the chemical or bioreactor literature cannot simply be copied. There is a pressing need to build a store of specific resource tools for the food industry. Our early experience gained in investigating food-microorganism interactions highlighted the need for a true multi-disciplinary approach. In 1994 an opportunity was seized to form the Food Technology Research Group (FTRG) at Adelaide University to continue the research. FTRG involves the departments of Statistics (now Biometrics SA), Mathematics, Molecular Biosciences (formerly Microbiology & Immunology) and Chemical Engineering. Members cooperate in research, teaching, graduate supervision and consultancy and collaborate with specialist researchers worldwide. We are interested in kinetics of bacterial growth, sterilisations and disinfections (bacterial death), especially as affected by combined environmental factors (Davey, Lin & Wood 1978, Davey 1982, Davey 1989a, b, Davey 1991, Davey 1993a, b, c, Coker, Davey & Kristall 1993, Davey, Hall & Thomas 1995, Davey & Daughtry 1995, Daughtry, Davey & King 1995, Gay, Cerf & Davey 1996, Cerf, Davey & Sadoudi 1996, Chiruta, Davey & Thomas 1997a, b, c, Davey 1994, Daughtry, Davey & King 1997, Daughtry et al 1997, Davey 1999, Davey & Thomas 2000). In this paper a model synthesised in the course of our work for kinetics of bacterial growth is reviewed in context and the emergence and slow recognition of an unexpected correlation for the complex food-

6th World Congress of Chemical Engineering Melbourne, Australia 23-27 September 2001

microorganism interaction is highlighted. Practical application is demonstrated and speculative conclusions about its surprising meaning are reviewed. One finding is a contradiction to widely promoted ideas of synergism in reactions. Another is the suggestion of a possible new qualitative (micro)biological law. All symbols used are listed in the Notation. DEVELOPMENT OF MICROBIOLOGICAL PROCESS MODELLING Given the size and importance worldwide of the food industries, and the driving need for a specialised resource in modelling kinetics of bacterial growth and survival, a new research field has emerged. Mathematical modeling, a technique familiar to chemical engineers, was seen to offer potential for predictive detection as an alternative to costly microbiological diagnostic techniques. The first international conference in this new field was in 1992 (Anon. 1992). Australian researchers carried out much of the early work (McMeekin et al 1993). Microbiologists, who after all generate the necessary microbiological data, initially dominated the field. The field has emerged however as a more cooperative one involving statisticians (who are most useful for economic design of experiments), mathematicians (who are very useful in helping to look at model structure), computer specialists and biochemical engineers in the modelling of the process. To reflect this and better recognise the process involved in broadening the field the descriptor microbiological process modelling was suggested (Davey 1993b). Currently, there are a number of research groups, these are in: Australia, UK, France, USA and Holland. The three earliest were in Hobart, CSIRO-Adelaide University and at Unilever (England). The Hobart and CSIRO-Adelaide University involved chemical engineers at conception. The reader should note that there is strong potential for a leadership role by (bio)chemical engineers because of a strong background in mathematics, and emphasis on process, together with adequate microbiology. At Adelaide University we have been interested in demonstrating the overall process with integration of bacterial kinetics with, for example, aspects of process rheology and flow hydrodynamics and heat and mass transfer. FOODS AND THE BIOREACTOR LITERATURE For foods, models elaborated in chemical engineering and traditional biotechnology cannot simply be copied from the bioreactor literature. The food-microorganism interaction is very complex. There has been a recognition that there is a real need to build a store of dedicated resource tools (Chiruta, Davey & Thomas 1996, J. Baranyi pers. comm.). In summary, this is because: • With foods, the aim is to minimise (or prevent) growth of cells, rather than optimise as in bioreactors • The concentration of cells in foods is considerably lower than in bioreactors (typically>106 cell mL-1). Methods validated at high concentration (e.g. turbidity, biomass or conductance), therefore cannot be applied directly to foods • In foods, the lag phase is of great importance, whereas in bioreactors this is less important (Monod’s equation, for example, loses significance with food handling because substrate limitation is rare) • The physical environment of foods is less accurately known than in bioreactors. Methods that necessarily involve simplifying assumptions and many empirical elements are required for foods. MODEL FORMULATION OF BACTERIAL GROWTH A look at a bacterial growth curve reveals that there are at least four identifiable phases: the lag phase, growth phase, exponential phase, stationary phase, and death or decline phase (Stanier, Doudoroff & Adelerg 1972). Growth starts from a zero rate that accelerates with time to a maximal value resulting in the lag phase. The lag phase can vary considerably and does not always occur - indeed in bioreactor fermentation one does not want any lag time - the aim is to get straight into biomass production. In the growth or exponential phase the bacteria divide regularly with daughter cells behaving in an identical manner. This gives rise to the exponential increase observed in cell numbers. The stationary phase is reached when the growth rate decreases to zero. There are no further cell divisions. In the final phase the death of cells dominate and an exponential decrease in viable cell numbers is observed. Model Formulation Model formulation has been largely concerned with the growth and lag phase. The growth phase of bacteria is important to foods as possible food spoilers or pathogens. The lag phase is important to modelling foodborne pathogens where delaying the initiation of growth by maximising the lag is the primary consideration (Davey 1999). Of particular interest are predictive models for the combined effect of a number of environmental factors, especially, temperature (T), pH and water activity (aw). Because temperature is of primary importance to

6th World Congress of Chemical Engineering Melbourne, Australia 23-27 September 2001

bacterial cell behaviour, model formulation usually starts with it. In growth of bacteria in foods, the effect of T has been widely modelled using the Arrhenius equation, where the original activation energy (ΔE) has been replaced by μ - a temperature characteristic term. However, Ratkowsky, a chemical engineer, working with the Hobart group demonstrated that this is a poor descriptor to published experimental data (Ratkowsky et al 1982). Arrhenius plots of lnk vs 1/T showed curves, not straight lines as would be expected (Figure 1). Inspection of Fig. 1 reveals clearly the curvature in these data for five bacteria and one fungi.

Figure 1 Arrhenius plot of five bacteria and a fungi (Reproduced from Ratkowsky et al 1982)

Recognising this, a number of researchers have used broken curves with various values of the temperature characteristic for different portions of the growth curve. This approach led to a rather complex statement of things. Figure 3 shows the model of Schoolfield, Sharpe and Magnuson (1981) for Escherchia coli in glucose-minimal medium. Intuitively, this is unsatisfactory (Davey 1989b). How many straight-line portions of the curve can be treated in this way?

Figure 2 Arrhenius plot of growth rate of Escherchia coli in glucose-minimal medium (Reproduced from Schoolfield, Sharpe & Magnuson 1981)

Model Categorisation Davey (1993a) proposed a broad categorisation of model approaches, namely: Schoolfield non-Arrhenius, Square-Root - a rediscovery of the Belehradek form - (Belehradek 1926, McMeekin et al 1993, Davey 1999), Polynomial and a Linear Arrhenius - so called Davey Linear-Arrhenius (DL-A) equation (McMeekin et al 1993, Whiting & Buchanan 1994, ICMSF 1996, Holdsworth 1997, Gardner & Peel 1998).

6th World Congress of Chemical Engineering Melbourne, Australia 23-27 September 2001

To give an express meaning to model development Davey (1992a) proposed a standard terminology, including the term model. This was based broadly on statistical terminology (widely familiar to biochemical engineers). Baranyi and Roberts (1992) pointed out that if adopted as a convention it would make the field a more exact discipline. A number of categorisations and comparisons between models have been made (McMeekin et al 1993, Davey 1989b, Adair, Kilsby & Whittall 1989, Buchanan 1991, Ratkowsky et al 1991, Davey 1999). The Perfect Model The perfect model of course is the one that represents reality. However such a model would have the same drawbacks as a map as large and as detailed as the city it represents – a map depicting every park, every street, every building and every map of the city. The nth-order polynomial form seems to be a bit like such a map, Figure 3.

Figure 3 Polynomial cubic model for predicting the effects of combined T, pH, and salt (NaCl and NaNO3) on the aerobic growth of Listeria monocytogenes Scott A, for use with the Gompertz Model: Log (Nt) = A + C exp(-B(t-M)) (Reproduced from McMeekin et al 1993)

Clearly this very specificity would appear to defeat its primary purpose, namely, to generalise and to abstract. A drawback is that the form lacks generality. A different kinetic function is required for each foodmicroorganism. This would seem to imply each behaves differently. Further, it is impossible to interpret the coefficients of the model when there are so many. Nature appears to form patterns. We should be looking for these patterns to a universal form. Models should simplify as much as they mimic the real world. What therefore are acceptable criteria for an adequate model for complex food-microorganism interactions? CRITERIA FOR AN ADEQUATE MODEL The criteria for an adequate model for bacterial growth kinetics as affected by combined environmental factors must include (Davey 1993b, McMeekin et al 1993, Davey & Daughtry 1995): • Accuracy of prediction against observed data • Ease of synthesis and of use • Relative complexity (parsimony). The model should contain the minimum number of justifiable terms. From a process point of view, the model should, in addition, be of a form that can be readily integrated with other equations to form an overall process model • Universality i.e. a genralised form applicable to a wide range of food-microrganism interactions • Potential for physiological interpretation and significance of coefficients. The percent variance accounted for (%V) has been adopted in our work as one of two elements for a stringent test of a model’s accuracy of prediction against observed data. The %V is given by:  (1 − r 2 )(n − 1)  (1) %V = 1 − × 100 (n − N T − 1)  

It is a more stringent test of goodness of fit than either the correlation coefficient (r2) or Mean Square Error (MSE), particularly, when there are few data and the model form has a large number of terms (Davey 1993a, b, Davey & Cerf 1996, Davey 1999, Daughtry et al 1997). Importantly, the %V permits significant comparison of models forms with different numbers of terms. The use of r2 has been widely criticised (Ratkowsky 1990, Davey 1989b) and has little significance if the model form is non-linear. Further, r2 can be misleading when there are few data and thereby a high value is obtained. MSE has been shown to be an inappropriate measure of goodness of fit in some instances (Davey 1999). It is claimed to be only reliable if the residuals are normally distributed (McMeekin et al 1993). The second element adopted in the test of a model’s accuracy of prediction against observed data is an appraisal of residual plots (predicted value vs observed value). Adequate model parameterisation and a

6th World Congress of Chemical Engineering Melbourne, Australia 23-27 September 2001

guide to complexity of the model form can be estimated from residual plots. Diagnosis sheds light on model fits and the influence of inherent assumptions (Weisberg 1985). THE LINEAR-ARRHENIUS MODEL The DL-A model fulfils these important criteria – as is illustrated in the following. The last criterion, the potential for physiological interpretation of the coefficients is important, although often overlooked by researchers, is developed separately in discussion. In general form, it is given as (Davey 1994): j

ln k = C 0 + ∑ (C 2 i −1V + C 2 iV i 2 )

(2)

i =1

For predicting the combined effect of two environmental factors, say, T (as 1/T) and aw, on bacterial growth, the DL-A is seen from equation (2) to be given by: (3) ln k = C0 + C1 / T + C2 / T 2 + C3aw + C4 aw2 Applied to the lag phase of growth with T as the sole environmental factor the model is given by:

ln(

1 ) = C0 + C1 / T + C 2 / T 2 lagtime

(4)

where it is the reciprocal of the lag time that is modelled to maintain consistent units (of time-1). The DL-A model is said to be Linear-Arrhenius because it is Arrhenius formulated to account for observed curvature in published data. In statistical terminology it is a linear model (i.e. a plot of lnk vs 1/T2 gives a straight line). The model is additive, and in common with Arrhenius, has as yet, no theoretical foundation. The model is readily obtained using regression analyses by unsophisticated users. Because the DL-A is quadratic in form an optimum value of growth (or lag) can be found for each environmental factor (Davey 1994, Davey & Daughtry 1995). Applications Table 1 presents independently published experimental data for the effect of T only, and also, for the combined effect of T-aw together with the fit of the DL-A model. These data are summarised from Davey (1989a), Davey & Daughtry (1995) and Daughtry, Davey & King (1997) and Davey & Daughtry (pers. comm.). As is revealed by the high values of %V the model gives a very good fit to these data. The data of Table 1 includes a wide range of bacterial types: a Gram negative bacterium, Gram positive, rods, cocci, a silage bacterium and a spore former in a wide range of foods including meat, UHT milk and laboratory media. The data range over some 93 years with 12 independent researchers, laboratories and methods. There is the suggestion therefore of a universality of the DL-A form in Table 1. The hypothesis that the model might also fit the lag phase was tested. Any model simulation of food handling that did not include the lag phase would be overly conservative, largely defeating the purposes of the models. In the chilling of meat, for example, some 4-6 h might elapse in the lag phase before initiation of growth. Table 2 summarises the fit of the DL-A model to a range of independent lag data with T as the sole environmental factor. The values of %V show a very high degree of accuracy of fit. Table 3 illustrates the goodness of fit by comparing the predicted and observed values of the lag time to the data of Mackey and Kerridge (1987) for Salmonella spp. in minced beef. Over the T range 10-350C the DL-A model is seen to give a very good fit. A careful search of the literature highlighted that there were fewer published data for the lag phase than for the growth phase. The effect of pH on growth is highly significant (Davey 1994, Davey & Daughtry 1995). Table 4 summarises the fit of the DL-A model to independent data for the rate coefficient for the growth phase as effected by combined T-pH in three acidulants. The model is seen from the table to give an excellent fit to these independent data. The fit of the DL-A model to the experimental data of Gibson, Bratchell & Roberts (1988) for the combined effect of 3-environmental factors, namely, T-aw-pH, on Salmonella spp. is presented in Table 5. The table shows, through the value of %V = 97.6%, a very good fit to these independent data. Overview A retrospective overview of applications of the DL-A model underscores: • High values of %V that illustrate a high degree of fit and accuracy of prediction • Model coefficients which are all significant with no significant interaction terms (e.g. T*aw, T*aw2) • The sign on the model coefficients is consistent across all the available data • It is unlikely the model could be improved as there appeared no structure in residual plots • With only 2-terms for each environmental factor the model could be said to fulfil the criterion of parsimony (especially as compared with Polynomial forms)

6th World Congress of Chemical Engineering Melbourne, Australia 23-27 September 2001

•

The model coefficients can be easily obtained by relatively unsophisticated users with linear regression on a PC or hand-held calculator (this contrasts with the Square Root model of McMeekin et al 1993, see Davey 1999 for a discussion) • The model has fitted a very wide range of food-microorganism interactions without exception over some 93 years • The simplicity of the model form permits ready integration with other equations to describe an overall process • The model form is apparently universal. The model therefore is a powerful and convenient predictive tool. The smooth nature of the model surface suggests limited extrapolation of predictions could be reliably carried out (Davey 1993a, 1994, Daughtry & Davey 1995). Table 1 Fit of the DL-A model to independent data for the rate coefficient for the growth phase: Lnk = C0 + C1/T + C2/T2 Lnk = C0 + C1/T + C2/T2 + C3aw + C4aw2 Microorganism

%V

n

aw

-C1/C2

-C3/C4

12 15 20 10

T (0C) 10-40.8 -1-30 8-46 8-45

Bacillus coli Pseudomonas Escherichia coli Pseudomonas aeruginosa Pediococcus cerevisiae Microbacterium thermosphactum Clostridium botulinum Aerobacter aerogenes Salmonella typhimurium Staphylococcus xylosus Pseudomonas spp. Yersinia enterocolitica Listeria monocytogenes

98.8 99.0 98.8 96.1

0.960-0.993 -

0.0063 0.0067 0.0063 0.0064

1.97 -

Barber 1908 Scott 1937 Ingraham 1958 Ingraham 1958

92.2

16

24-42

0.944–0.998

0.0065

1.99

Lanigan 1963

98.0

18

0-25

0.930–0.990

0.0066

1.96

Brownlie 1969

97.9

15

15-30

0.970–0.995

0.0057

1.99

Ohye, Christian & Scott 1967

94.0

18

15-37

0.950–0.975

0.0065

1.97

96.9

28

9-30

0.94–0.98

0.0065

1.99

97.2

185

7-27

0.848–0.996

0.0066

1.99

Casolari, Spotti & Castelvetri 1979 Broughall, Anslow & Kilsby 1983 McMeekin et al 1987

99.2

15

4.4-28.1

-

0.0063

-

Chandler 1988

99.8

10

2.8-24.1

-

0.0066

-

Adams, Little & Easter 1991

99.6

18

0-30.6

-

0.0066

-

Grau & Vanderlinde 1993

Reference

Table 2 Fit of the DL-A model to independent data for the lag phase with T as the sole factor: Ln(1/lagtime) = C0 + C1/T + C2/T2 Microorganism-food Coliforms in blended meat Escherichia coli SF in blended meat Salmonella typhimurium in blended meat Salmonella in minced meat Pseudomonas spp. Gram – ve spoilage flora in raw meat Staphylococcus aureus in pastry Mixed salmonellae Escherichia coli O157: H7

%V

n 8 8

T (0C) 8.2 - 40 8.2 - 40

98.9 99.2

0.0064 0.0065

Smith 1985 Smith 1985

96.0

7

10 - 40

0.0064

Smith 1985

99.7 99.5 98.3

6 15 10

10 - 35 4.4 - 28.1 -2 - 30

0.0066 0.0064 0.0063

Mackey & Kerridge 1987 Chandler 1988 Adair, Kilsby & Whittall 1989

95.0

13

10.6 - 45.6

0.0064

Adair, Kilsby & Whittall 1989

96.5 99.2

5 13

10 - 30 10 - 42

0.0065 0.0064

A Gibson (pers. comm.) R Buchanan (pers. comm.)

-C1/C2

Reference

6th World Congress of Chemical Engineering Melbourne, Australia 23-27 September 2001

Temperature (0C)

10 15 20 25 30 35

Lag Time (h) Observed Predicted

40.6 10.8 4.2 2.3 2.3 2.6

41.3 10.4 4.1 2.5 2.2 2.6

Table 3 Comparison of the observed with predicted data of Mackey & Kerridge (1987) for the lag time of Salmonella in minced beef (%V=99.7) Ln(1/lagtime) = -623.4 + 3.7667x105/T - 5.697x107/T2

Table 4 Fit of the DL-A model to independent data for the rate coefficient for the growth phase: Lnk = C0 + C1/T + C2/T2 + C5pH + C6pH2 Microorganism Yersinia enterocolitica Yersinia enterocolitica Listeria monocytogenes Escherichia coli O157:H7

Acid

%V

n

pH

-C1/C2

-C5/C6

Reference

48

T (0C) 1 - 24

H2SO4

98.6

4.5-6.5

0.0066

12.9

97.8

34

2 - 24

5.0-6.5

0.0062

13.5

HCl

95.7

25

5 - 37

4.5-7.5

0.0065

13.5

HCl

96.6

25

10 - 42

4.5-8.5

0.0063

14.0

Adams, Little & Easter 1991 Adams, Little & Easter 1991 R Buchanan (pers. comm.) R Buchanan (pers. comm.)

Lactic

Table 5 Fit of the DL-A model to independent data for the rate coefficient for the growth phase: Lnk = C0 + C1/T + C2/T2 + C3aw + C4aw2 + C5pH + C6pH2 Microorganism

%V

n

Salmonella spp.

97.6

66

T (0C) 10-30

aw

pH

-C1/C2

-C3/C4

-C5/C6

Reference

0.7–4.56*

5.63-6.77

0.0066

1.98

12.8

Gibson, Bratchell & Roberts 1988

* aw is expressed as salt concentration (see Davey & Daughtry 1995)

AN UNEXPECTED CORRELATION A curious observation to emerge in our work is that the value of the ratio of the two coefficients (C2i-1/C2i) for each environmental factor is remarkably constant. This ratio will have been seen to be given in each of the Tables 1, 2, 4 & 5. The value of this ratio for each of the three environmental factors across all data is summarised in Table 6, where for example, for T (expressed as 1/T) the value across all data is 0.00643. This ratio is tentatively called the environmental constant and given the symbol αVi. Given the fact of unrelated researchers together with a very wide range of different bacteria spanning 93 years of published data, and range of laboratory techniques, it is all the more curious that it should appear constant. This finding means however that the DL-A model can now be written as: j

ln k = C 0 + ∑ (C iV i (1 + α V iV i )

(6)

i =1

If, as it appears, αVi is constant over the range of values of interest of the environmental (process) factors (T-aw-pH & others e.g. atmosphere) for a wide range of bacteria in foods, then from equation (6) it is clear that the effect of each and all environmental factors can be predicted on the basis of only one empirically determined coefficient. This of course is as economically simple as a model can be and may hint at the prospect of some sort of qualitative law based on the notion of the environmental constant. Although we have been aware of the emergence of the constancy of the ratio for some time (McMeekin et al 1993, Davey pers. comm.), attempts to explain it have proved more difficult. The structure of the model suggests that each of the environmental factors acts independently of each other on cell growth, in both the growth and lag phases (J Baranyi pers. comm.). This suggests some quantitative support for the hurdle theory of Leister (1985) but would appear to challenge notions of synergism in cell responses to environmental factors. The excellent agreement between model simplicity and accurate predictions to extensive and independent data implies strong potential for physiological interpretation of the model coefficients. To speculate, it

6th World Congress of Chemical Engineering Melbourne, Australia 23-27 September 2001

appears that there is only one bacterial cell response to each of the environmental (process) factors. The appearance of the environmental constant, i.e. αVi in the model, might suggest a form of a qualitative law pointing to a particular cell element. The errors introduced in prediction by its use can be demonstrated from values in Table 6 to be significantly smaller than that introduced for e.g. from a standard plate count of viable cells of about ±15%. Environmental Factor (Vi)

1/T aw pH

Environmental Constant Overall Mean

αVi ± sd 0.006427 (0.00002) 1.980 (0.011952) 13.34 (0.49)

Table 6 Value of the environmental constant (αVi) derived from independent data for environmental factors (Vi): 1/T, aw and pH

It is not yet known if anything can be deduced from a ranking of the values of αVi shown in Table 6, i.e. approximately 156, 13 & 2, for respectively, T, pH and aw. However, the notion that only one cell response should be apparent for each of the process factors, strongly suggests that all data could be represented on an arbitrarily chosen co-ordinate. (We are in fact currently investigating this exciting line of thought with an arbitrarily chosen co-ordinate where the intercept is the value of, for e.g. T, that corresponds to one percent of the optimum growth rate i.e. T1%). The reader should appreciate that the generation of the data needed for reliable modelling requires the resources of a large group of researchers. The experimental determination of adequate data represents a “ … monumental amount of work … indicative of … time consuming, painstaking effort“ (McMeekin et al 1993). The unavoidable cost associated with generating data is undoubtedly a significant factor contributing worldwide to increasingly restricted access. Much data remains commercial in confidence1. CONCLUSIONS 1. Predictions from the Linear Arrhenius (DL-A) model give excellent agreement with extensive, independent and unrelated data over some 93 years for a wide range of food-microorganism systems. 2. The model can be readily derived from experimental data and integrated with a range of equations to describe a unit operation or process in foods handling. 3. The excellent agreement between the DL-A model predictions and published data implies potential for physiological interpretation of model coefficients. 4. The mathematical structure of the model suggests each of the environmental (process) factors act independently of the others on cell growth and survival in the lag phase. 5. The effect of a particular environmental factor on bacterial growth, in either the growth phase or lag phase, can be predicted from just one empirically determined coefficient. NOMENCLATURE aw A B Ci,j k M n Nt

1

exposure (process) water activity model coefficient, Table 3 relative growth rate at M (Gompertz equation coefficient), Table 3 coefficients of Davey Linear-Arrhenius (DL-A) model, Equation (2) rate coefficient for microorganism growth time at which absolute growth rate is maximal (Gompertz equation coefficient), Table 3 number of observations in the data set number of viable cells at time = t, (Table 3)

NT number of terms in a model r2 correlation coefficient t exposure (process) time, Table 3 T exposure (process) temperature Vi,j environmental (process) factors e.g. 1/T, aw, pH %V percent variance accounted for, Equation (1) Greek symbols αVi environmental constant ΔE Arrhenius chemical activation energy μ Arrhenius temperature characteristic term for microorganism growth

We wish to record an indebtedness to the generosity of Dr Angela Gibson, CSIRO, Sydney for unrestricted access to the data for Salmonella spp. and to Dr Bob Buchanan, Department of Health & Human Services, USFDA, Washington for access to Escherichia coli O157: H7 data.

6th World Congress of Chemical Engineering Melbourne, Australia 23-27 September 2001

REFERENCES Adair C, Kilsby D C and Whittall P T 1989 Comparison of the Schoolfield (non-linearArrhenius) model and the square root model for predicting bacterial growth in foods. Food Microbiology 6, 7-17. Adams M R, Little C L and Easter M C 1991 Modelling the effect of pH, acidulant and temperature on the growth rate of Yersinia enterocolitica. Journal of Applied Bacteriology 71, 65-71. Anon. 1992 1st International Conference on the Application of Predictive Microbiology and Computer Modeling Techniques to the Food Industry, Tampa, Florida, USA, April 12-15. Baranyi J and Roberts T A 1992 A terminology for models in predictive microbiology – a reply to K R Davey. Food Microbiology 9, 353-356. Barber M A 1908 The rate of multiplication of Bacillus coli at different temperatures. Cited by Buchanan R E & Fulmer E I 1930 In: Physiology and Biochemistry of Bacteria Vol. 2, Williams & Williams, Baltimore, p. 66 ff. Belehradek J 1926 Influence of temperature on biological processes. 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