Abstract
According to their structure, predictive models can be primary, secondary, or tertiary. This classification mainly depends on the final purpose and type of prediction generated. There has been a significant evolution in the past few years toward better understanding of microbial behavior in foods. Therefore, models that describe the biological process of microbial growth and inactivation have been subsequently developed. Also, fitting methods for linear and nonlinear regression together with goodness-of-fit indexes give us useful information about how the model is able to explain the observed data. Finally, models cannot be applied if a validation process is not previously accomplished, which typically consists of confirming the predictions experimentally by using any quantitative method. In this chapter, a comprehensive review of the most popular validation methods is provided.
An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-1-4614-5520-2_8
An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-1-4614-5520-2_8
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References
Abdi H (2007) The method of least squares. In: Salkind N (ed) Encyclopedia of measurements and statistics. Sage, London
Akaike H (1974) A new look at the statistical model identification. IEEE T Automat Contr 19:716–723
Almeida JS (2002) Predictive non-linear modeling of complex data by artificial neural networks. Curr Opin Biotechnol 13:72–76
Augustin JC, Carlier V (2000a) Modelling the growth rate of Listeria monocytogenes with a multiplicative type model including interactions between environmental factors. Int J Food Microbiol 56:53–70. doi:10.1016/S0168-1605(00)00224-5
Augustin JC, Carlier V (2000b) Mathematical modelling of the growth rate and lag time for Listeria monocytogenes. Int J Food Microbiol 56:29–51. doi:10.1016/S0168-1605(00)00223-3
Baranyi J (1992) Letters to the editor: A note on reparameterization of bacterial growth curves. Food Microbiol 9:169–171. doi:10.1016/0740-0020(92)80024-X
Baranyi J, Pin C (2001) A parallel study on bacteria growth and inactivation. J Theor Biol 210:327–336. doi:10.1006/jtbi.2001.2312
Baranyi J, Roberts TA (1994) A dynamic approach to predicting bacterial growth in food. Int J Food Microbiol 23:277–294. doi:10.1016/0168-1605(94)90157-0
Baranyi J, Roberts TA, McClure P (1993) A non-autonomous differential equation to model bacterial growth. Food Microbiol 10:43–59. doi:10.1006/fmic.1993.1005
Baranyi J, Robinson TP, Kaloti A, Mackey BM (1995) Predicting growth of Brochothrix thermosphacta at changing temperature. Int J Food Microbiol 27:61–75. doi:10.1016/0168-1605(94)00154-X
Baranyi J, Pin C, Ross T (1999) Validating and comparing predictive models. Int J Food Microbiol 48:159–166. doi:10.1016/S0168-1605(99)00035-5
Basheer I, Hajmeer M (2000) Artificial neural networks: fundamentals, computing, design and application. J Microbiol Methods 43:3–31. doi:10.1016/S0167-7012(00)00201-3
Blackburn CW, Curtis LM, Humpheson L, Billon C, McClure PJ (1997) Development of thermal inactivation models for Salmonella enteritidis and Escherichia coli O157:H7 with temperature, pH and NaCl as controlling factors. Int J Food Microbiol 38:31–44. doi:10.1016/S0168-1605(97)00085-8
Buchanan RL, Golden MH (1995) Model for the non-thermal inactivation of Listeria monocytogenes in a reduced oxygen environment. Food Microbiol 12:203–212. doi:10.1016/S0740-0020(95)80099-9
Buchanan RL, Klawitter LA (1991) Effect of temperature history on the growth of Listeria monocytogenes Scott A at refrigeration temperatures. Int J Food Microbiol 12:235–246.http://dx.doi.org/10.1016/0168-1605(91)90074-Y
Buchanan RL, Whiting RC, Damert WC (1997) When is simple good enough: a comparison of the Gompertz, Baranyi, and three-phase linear models for fitting bacterial growth curves. Food Microbiol 14:313–326. doi:10.1006/fmic.1997.0125
Campos DT, Marks BP, Powell MR, Tamplin ML (2005) Quantifying the robustness of a broth-based Escherichia coli O157: H7 growth model in ground beef. J Food Prot 68:2301–2309
Cerf O, Davey KR, Sadoudi AK (1996) Thermal inactivation of bacteria. A new predictive model for the combined effect of three environmental factors: temperature, pH and water activity. Food Res Int 29:219–226. doi:10.1016/0963-9969(96)00039-7
Chatterjee S, Hadi AS (2006) The problem of correlated errors. Regression analysis by example. Wiley, New York, pp 197–219
Dalgaard P, Ross T, Kamperman L, Neumeyer K, McMeekin TA (1994) Estimation of bacterial growth rates from turbidimetric and viable count data. Int J Food Microbiol 23:391–404. doi:10.1016/0168-1605(94)90165-1
Davey KR (1993) Linear-Arrhenius models for bacterial growth and death and vitamin denaturations. J Ind Microbiol 12:172–179. doi:10.1007/BF01584187
Delignette-Müller ML, Cornu M, Pouillot R, Denis JB (2006) Use of Bayesian modelling in risk assessment: application to growth of Listeria monocytogenes and food flora in cold-smoked salmon. Int J Food Microbiol 106:195–208. doi:10.1016/j.ijfoodmicro.2005.06.021
Devlieghere F, Geeraerd H, Versyck KJ, Vandewaetere B, Van Impe J, Debevere J (2001) Growth of Listeria monocytogenes in modified atmosphere packed cooked meat products: a predictive model. Food Microbiol 18:53–66. doi:10.1006/fmic.2000.0378
Dym CL (2004) Principles of mathematical modeling. Elsevier Academic Press, London, 4
Fernández PS, Ocio MJ, Rodrigo F, Rodrigo M, Martínez A (1996) Mathematical model for the combined effect of temperature and pH on the thermal resistance of Bacillus stearothermophilus and Clostridium sporogenes spores. Int J Food Microbiol 32:225–233. doi:10.1016/0168-1605(96)01118-X
Fernandez PS, George SM, Sills CC, Peck MW (1997) Predictive model of the effect of CO2, pH, temperature and NaCl on the growth of Listeria monocytogenes. Int J Food Microbiol 37:37–45. doi:10.1016/S0168-1605(97)00043-3
García-Gimeno RM, Hervás C, de Silóniz MI (2002) Improving artificial neural networks with a pruning methodology and genetic algorithms for their application in microbial growth prediction in food. Int J Food Microbiol 72:19–30. doi:10.1016/S0168-1605(01)00608-0
Garthright WE (1997) The three-phase linear model of bacterial growth: a response. Food Microbiol 14:395–397. doi:10.1006/fmic.1996.9997
Geeraerd AH, Herremans CHML, Herremans ML, Cenes C, Van Impe JF (1998) Application of artificial neural networks as a non linear technique to describe bacterial growth in chilled food products. Int J Food Microbiol 44:49–68. doi:10.1016/S0168-1605(98)00127-5
Geeraerd AH, Herremans CH, Van Impe JF (2000) Structural model requirements to describe microbial inactivation during a mild heat treatment. Int J Food Microbiol 59:185–209. doi:10.1016/S0168-1605(00)00362-7
Geeraerd AH, Valdramidis VP, Devlieghere F, Bernaert H, Debevere J, Van Impe JF (2004) Development of a novel approach for secondary modelling in predictive microbiology: incorporation of microbiological knowledge in black box polynomial modelling. Int J Food Microbiol 91:229–244. doi:10.1016/S0168-1605(03)00388-X
Geeraerd AH, Valdramidis VP, Van Impe JF (2005) GInaFiT, a freeware tool to assess non-log-linear microbial survivor curves. Int J Food Microbiol 102:95–105. doi:10.1016/j.ijfoodmicro.2004.11.038
Gibson AM, Bartchetll N, Roberts TA (1987) The effect of sodium chloride and temperature on the rate and extent of growth of Clostridium botulinum type A in pasteurised pork slurry. J Appl Bacteriol 62:479–490. doi:10.1111/j.1365-2672.1987.tb02680.x
Gibson A, Bratchell N, Roberts T (1988) Predicting microbial growth: growth responses of Salmonellae in a laboratory medium as affected by pH, sodium chloride and storage temperature. Int J Food Microbiol 6:155–178. doi:10.1016/0168-1605(88)90051-7
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. doi:10.1016/S0168-1605(96)01169-5
Hervás C, Zurera G, García-Gimeno RM, Martinez J (2001) Optimization of computational neural network for its application to the prediction of microbial growth in foods. Food Sci Technol Int 7:159–163. doi:10.1106/6Q2A-8D7R-JHJU-T7F6
Hervás-Martínez C, García-Gimeno RM, Martínez-Estudillo AC, Martínez-Estudillo FJ, Zurera-Cosano G (2006) Improving microbial growth prediction by Product Unit Neural Networks. J Food Sci 71(2):31–38. doi:10.1111/j.1365-2621.2006.tb08904.x
Hills BP, Mackey BM (1995) Multicompartment kinetic models for injury, resuscitation induced lag and growth in bacterial-cell populations. J Theor Biol 12:333–346. doi:10.1016/S0740-0020(95)80114-6
Huang L, Hwang A, Phillips J (2011) Effect of temperature on microbial growth rate-mathematical analysis: the Arrhenius and Eyring–Polanyi connections. J Food Sci 76:553–560. doi:10.1111/j.1750-3841.2011.02377.x
Jeyamkondan S, Jayas DS, Holley RA (2001) Microbial growth modelling with artificial neural networks. Int J Food Microbiol 64:343–354. doi:10.1016/S0168-1605(00)00483-9
Juneja JK, Marmer BS, Phillips JG, Miller AJ (1995) Influence of the intrinsic properties of food on thermal inactivation of spores of nonproteolytic Clostridium botulinum: development of a predictive model. J Food Saf 15:349–364. doi:10.1111/j.1745-4565.1995.tb00145.x
Karadavut U, Palta Ç, Kökten K, Bakoğlu A (2010) Comparative study on some non-linear growth models for describing leaf growth of maize. Int J Agric Biol 12:227–230
Koutsoumanis K (2001) Predictive modeling of the shelf life of fish under nonisothermal conditions. Appl Environ Microbiol 67:1821–1829. doi:10.1128/AEM.67.4.1821-1829.2001
Le Marc Y, Huchet V, Bourgeois CM, Guyonnet JP, Mafart P, Thuault D (2002) Modelling the growth kinetics of Listeria as a function of temperature, pH and organic acid concentration. Int J Food Microbiol 73:219–237. doi:10.1016/S0168-1605(01)00640-7
Lebert I, Robles-Olvera V, Lebert A (2000) Application of polynomial models to predict growth of mixed cultures of Pseudomonas spp. and Listeria in meat. Int J Food Microbiol 61:27–39. doi:10.1016/S0168-1605(00)00359-7
Lee SH, Hou CL (2002) An art-based construction of rbf networks. IEEE Trans Neural Netw 13(6):1308–1321
Leguérinel I, Mafart P (1998) Model for combined effects of temperature, pH and water activity on thermal inactivation of Bacillus cereus spores. J Food Sci 63:887–889. doi:10.1111/j.1365-2621.1998.tb17920.x
Mafart P, Leguérinel I (1998) Modeling combined effects of temperature and pH on heat resistance of spores by a linear-Bigelow equation. J Food Sci 63:6–8. doi:10.1111/j.1365-2621.1998.tb15662.x
McClure 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 19:161–178. doi:10.1016/0168-1605(93)90074-Q
McClure PJ, Beaumont AL, Sutherland JP, Roberts TA (1997) Predictive modelling of growth of Listeria monocytogenes. The effects on growth of NaCl, pH, storage temperature and NaNO. Int J Food Microbiol 34:221–232. doi:10.1016/S0168-1605(96)01193-2
McKellar RC (2001) Development of a dynamic continuous-discrete-continuous model describing the lag phase of individual bacterial cells. J Appl Microbiol 90:407–413. doi:10.1046/j.1365-2672.2001.01258.x
McKellar RC, Knight KP (2000) A combined discrete-continuous model describing the lag phase of Listeria monocytogenes. Int J Food Microbiol 54:171–180. doi:10.1016/S0168-1605(99)00204-4
McKellar RC, Lu X (2004) Modelling microbial responses in food, CRC Series in Contemporary Food Science. CRC, London. ISBN 0-8493-1237-X
McKellar RC, Butler G, Stanich K (1997) Modelling the influence of temperature on the recovery of Listeria monocytogenes from heat injury. Food Microbiol 14:617–625. doi:10.1006/fmic.1997.0124
McMeekin TA, Olley J, Ross T, Ratkowsky DA (1993a) Predictive microbiology: theory and application. Research Studies Press, Taunton
McMeekin TA, Olley J, Ratkowsky DA, Ross T (2002) Predictive microbiology: towards the interface and beyond. Int J Food Microbiol 73:395–407. doi:10.1016/S0168-1605(01)00663-8
Membré JM, Ross T, McMeekin TA (1999) Behaviour of Listeria monocytogenes under combined chilling processes. Lett Appl Microbiol 28:216–220. doi:10.1046/j.1365-2672.1999.00499.x
Miller FA, Ramos B, Gil MM, Brandao TRS, Teixeira P, Silva CLM (2009) Influence of pH, type of acid and recovery media on the thermal inactivation of Listeria innocua. Int J Food Microbiol 133:121–128. doi:10.1016/j.ijfoodmicro.2009.05.007
Nerbrink E, Borch E, Blom H, Nesbakken T (1999) A model based on absorbance data on the growth rate of Listeria monocytogenes and including the effects of pH, NaCl, Na-lactate and Na-acetate. Int J Food Microbiol 47:99–109. doi:10.1016/S0168-1605(99)00021-5
Pin C, Baranyi J, de Fernando GG (2000) Predictive model for the growth of Yersinia enterocolitica under modified atmospheres. J Appl Microbiol 88:521–530. doi:10.1046/j.1365-2672.2000.00991.x
Pin C, Avendaño-Pérez G, Cosciani E, Gómez N, Gounadakic A, Nychas G, Skandamis P, Barker G (2011) Modelling Salmonella concentration throughout the pork supply chain by considering growth and survival in fluctuating conditions of temperature, pH and aw. Int J Food Microbiol 145:S96–S102. doi:0.1016/j.ijfoodmicro.2010.09.025
Pouillot R, Albert I, Cornu M, Denis JB (2003) Estimation of uncertainty and variability in bacterial growth using Bayesian inference. Application to Listeria monocytogenes. Int J Food Microbiol 81:87–104. doi:10.1016/S0168-1605(02)00192-7
Presser KA, Ratkowsky DA, Ross T (1997) Modelling the growth rate of Escherichia coli as a function of pH and lactic acid concentration. Appl Environ Microbiol 63:2355–2360
Psomas AN, Nychas GJ, Haroutounian SA, Skandamis PN (2011) Development and validation of a tertiary simulation model for predicting the growth of the food microorganisms under dynamic and static temperature conditions. Comput Electron Agric 76:119–129. doi:10.1016/j.compag.2011.01.013
Ratkowsky DA (ed) (1983) Nonlinear regression modeling: a unified practical approach. Dekker, New York
Ratkowsky DA (2004) Model fitting and uncertainty. In: McKellar RC, Lu X (eds) Modelling microbial responses in foods. CRC Press, Boca Raton, pp 191–195
Ratkowsky DA, Olley J, McMeekin TA, Ball A (1982) Relationship between temperature and growth rates of bacterial cultures. J Bacteriol 149:1–5
Ratkowsky DA, Lowry RK, McMeekin TA, Stokes AN, Chandler RE (1983) Model for bacterial culture growth rate throughout the entire biokinetic temperature range. J Bacteriol 154:1222–1226
Reichart O (1994) Modeling the destruction of Escherichia coli on the base of reaction kinetics. Int J Food Microbiol 23:449–465. doi:10.1016/0168-1605(94)90169-4
Robinson TP, Ocio MJ, Kaloti A, Mackey BM (1998) The effect of the growth environment on the lag phase of Listeria monocytogenes. Int J Food Microbiol 44:83–92. doi:10.1016/S0168-1605(98)00120-2
Ross T (1996) Indice of performance evaluation of predictive models in food microbiology. J Appl Bacteriol 81:501–508. doi:10.1111/j.1365-2672.1996.tb03539.x
Ross T, Dalgaard P, Tienungoon S (2000) Predictive modelling of the growth and survival of Listeria in fishery products. Int J Food Microbiol 62:231–245. doi:10.1016/S0168-1605(00)00340-8
Ross T, Ratkowsky DA, Mellefont LA, McMeekin TA (2003) Modelling the effects of temperature, water activity, pH and lactic acid concentration on the growth rate of Escherichia coli. Int J Food Microbiol 82:33–43. doi:10.1016/S0168-1605(02)00252-0
Rosso L, Lobry JR, Bajard S, Flandrois JP (1995) Convenient model to describe the combined effects of temperature and pH on microbial growth. Appl Environ Microbiol 61:610–616
Rosso L, Bajard S, Flandrois JP, Lahellec C, Fournaud J, Veit P (1996) Differential growth of Listeria monocytogenes at 4° and 8°C: consequences for the shelf life of chilled products. J Food Prot 59:944–949
Schepers A, Thibault J, Lacroix C (2000) Comparison of simple neural networks and nonlinear regression models for descriptive modeling of Lactobacillus helveticus growth in pH-controlled batch cultures. Enzyme Microb Technol 26:431–445. doi:10.1016/S0141-0229(99)00183-0
Shadbolt C, Ross T, McMeekin TA (2001) Differentiation of the effects of lethal pH and water activity: food safety implications. Lett Appl Microbiol 32:99–102. doi:10.1046/j.1472-765x.2001.00862.x
Silva AR, Sant’Ana AS, Massaguer PR (2010) Modelling the lag time and growth rate of Aspergillus section Nigri IOC 4573 in mango nectar as a function of temperature and pH. J Appl Microbiol 109:1105–1116. doi:10.1111/j.1365-2672.2010.04803.x
Smyth GK, El-shaarawi AH, Piegorsch WW (2002) Nonlinear regression. Environmetrics 3:1405–1411
Stringer M, George SM, Peck MW (2000) Thermal inactivation of Escherichia coli O157:H7. Symp Ser Soc Appl Microbiol 29:79S–89S
Stumbo CR, Purohit KS, Ramakrishnan TV (1975) Thermal process lethality guide for low acid foods in metal containers. J Food Sci 40:1316–1323. doi:10.1111/j.1365-2621.1975.tb01080.x
Sutherland JP, Bayliss AJ (1994) Predictive modelling of growth of Yersinia enterocolitica: the effects of temperature, pH and sodium chloride. Int J Food Microbial 21:197–215. doi:10.1016/0168-1605(94)00082-H
Sutherland JP, Bayliss AJ, Roberts TA (1994) Predictive modelling of growth of Staphylococcus aureus: the effects of temperature, pH and sodium chloride. Int J Food Microbiol 21:217–236. doi:10.1016/0168-1605(94)90029-9
te Giffel MC, Zwietering MH (1999) Validation of predictive models describing the growth of Listeria monocytogenes. Int J Food Microbiol 46:135–149. doi:10.1016/S0168-1605(98)00189-5
Valero A, Hervás C, García-Gimeno RM, Zurera G (2007) Product unit neural network models for predicting the growth limits of Listeria monocytogenes. Food Microbiol 24:452–464. doi:10.1016/j.fm.2006.10.002
Van Asselt E, Zwietering MH (2006) A systematic approach to determine global thermal inactivation parameters for various food pathogens. Int J Food Microbiol 107:73–82. doi:10.1016/j.ijfoodmicro.2005.08.014
Van Boekel MAJS (2002) On the use of the Weibull model to describe thermal inactivation of microbial vegetative cells. Int J Food Microbiol 74:139–159. doi:10.1016/S0168-1605(01)00742-5
Whiting RC (1993) Modeling bacterial survival in unfavourable environments. J Ind Microbiol 12:240–246. doi:10.1007/BF01584196
Whiting RC, Cygnarowicz-Provost M (1992) A quantitative model for bacterial growth and decline. Food Microbiol 9:269–277. doi:10.1016/0740-0020(92)80036-4
Wijtzes T, McClure PJ, Zwietering MH, Roberts TA (1993) Modelling bacterial growth of Listeria monocytogenes as a function of water activity, pH and temperature. Int J Food Microbiol 18:139–149. doi:10.1016/0168-1605(93)90218-6
Willocx F, Mercier M, Hendrickx M, Tobback P (1993) Modelling the influence of temperature and carbon dioxide upon the growth of Pseudomonas fluorescens. Food Microbiol 10:159–173. doi:10.1006/fmic.1993.1016
Zurera G, García-Gimeno RM, Rodríguez-Pérez MR, Hervás C (2004) Performance of response surface model for prediction of Leuconostoc mesenteroides growth parameters under different experimental conditions. Food Cont 17:429–438. doi:10.1016/j.foodcont.2005.02.003
Zwietering MH, Jongenburger I, Rombouts FM, Van’t Riet D (1990) Modelling of the bacterial growth curve. App Environ Microbiol 56:1876–1881
Zwietering MH, Witjzes T, de Wit JC, Van’t Riet K (1992) A decision support system for prediction of the microbial spoilage in foods. J Ind Microbiol 12:324–329. doi:10.1007/BF01584209
Zwietering MH, de Wit JC, Cuppers HG, van’t Riet K (1994) Modeling of bacterial growth with shifts in temperature. Appl Environ Microbiol 60:204–213
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Pérez-Rodríguez, F., Valero, A. (2013). Predictive Models: Foundation, Types, and Development. In: Predictive Microbiology in Foods. SpringerBriefs in Food, Health, and Nutrition, vol 5. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5520-2_3
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