Models in food technology, at the simplest level, are equations showing the relationship between two or more variables.  Dunn (1986) defined a mathematical model of a process as a 'system of equations whose solution, given specified input data, is representative of the response to a corresponding set of inputs'.

Commercially, cold-filled acidic pickles, sauces (e.g. salad cream, mayonnaise) and food dressings are preserved, and their microbiological safety assured, by the use of acetic acid, salt (NaCl) and sugar. This article provides an overview of a preservation model and access to the model to enable the effect of sauce components and pH to be investigated.

 The Comite´ des Industries des Mayonnaises et Sauces Condimentaires de la Communaute´ Économique Européenne (CIMSCEE) has provided guidance on a safety value, Σs, for a microbiologically safe product preserved using acetic acid (Anonymous, 1993). A safe product has been defined as one which is so formulated that when an inoculum of viable cells of  Escherichia coli is  added to the product this is reduced by 3 log cycles in less than 72 h. Products exhibiting this level of antibacterial activity have a CIMSCEE safety value (Σs) of greater than 63. Σs is calculated using equation 1:

 Σs =15.75 (1 - ɑ) (total acetic acid %) + 3.08 (salt %) + (hexose %) + 0.5 (disaccharide %) + 40 (4-pH).

The value (1 - ɑ) represents the proportion of acetic acid which is undissociated, this is dependent on a number of factors but principally on the pH of the aqueous phase of the product; using the pK_a (4.76 at 25°C) for acetic acid and the pH of the aqueous phase, (1 - ɑ) can be calculated using the Henderson–Hasselbalch equation, equation 2, where pK_a is the negative logarithm of the ionization constant (K) of an acid, this is the pH of the solution in which half of the acid molecules are ionised. Note that the pK_a value is not an absolute one; it will vary with the ionic strength of a solution so salt addition for example will change the value. 

Equation 2,


 Equation 2 can be simplified and rearranged to give:

Commercial cheese correctly manufactured with pasteurised milk and lactic starter cultures has a well deserved reputation as a nutritious and safe product. However, under certain circumstances cheese may support the growth of food poisoning bacteria or serve as a ‘vehicle’ for their transmission.

Four pathogens are of particular significance, Listeria monocytogenes, Salmonella species, enteropathogenic Escherichia coli and Staphylococcus aureus.  Listeria monocytogenes, the causal agent of listeriosis, is arguably the most significant of this group.

L. monocytogenes is particularly significant since it can grow / survive for long periods in cheese and cause serious illness leading to death; the death rate arising from listeriosis can exceed 30%. It can also induce abortion in humans and its ability to cross the placenta, and access the brain makes it a particularly dangerous pathogen.

 This article provides an introduction to the binary and ordinal logistic regression models developed by Bolton and Frank (1999) for predicting the probability of L. monocytogenes growing in cheese after 42 days storage at 10°C.

Characteristics of Listeria monocytogenes

L. monocytogenes is a Gram-positive, non-sporing bacterium that can grow in high salt environments (up to 10 % sodium chloride), and over a wide pH (5.0-9.6) and temperature range (< 3° – 45°C); it can grow aerobically and microaerophilically ( Bajard et al., 1996; Pearson and Marth, 1990).

While the organism is relatively sensitive to heat there has been considerable debate regarding its sensitivity to pasteurisation.  While   D _71-72°C  values have generally been reported as < 4s, a D_72°C value of 4.6 ± 0.5 s has been reported (Bunning et al., 1992) for heat-shocked cells.  Assuming a D_72°C value of 5s and raw milk containing 1000 (a high value) CFU / ml, pasteurisation at 72°C for 15s would be predicted to result in around 1 CFU / ml surviving (see http://www.dairyscience.info/newcalculators/listeria-d.asp) emphasising the critical importance of ensuring  only low concentrations of this pathogen in raw milk.

The calculator converts temperature readings to lethal rates, plots the lethal rates against time, and determines F values for a heat process. The area under the curve is determined using the trapezoid rule. Accurate F determinations for most thermal processes can be obtained. In general the more values, the more accurate the value for F will be.

The lethal effect of high temperatures on microorganisms is dependent on both temperature and holding time.

Because microorganisms in foods are exposed to lethal temperatures as they reach the target processing temperature and during cooling, it is necessary to calculate the cumulative effect of heat on microbial destruction during both heating and cooling as well at the holding time at the target temperature.

The logarithmic reduction in time required to kill the same number of microorganisms as temperature is increased has been well described.  This can be expressed by calculating lethal rate.

The lethal rate is a dimensionless number and can be calculated using equation 1 (Stobo, 1973).

Equation 1, Lethal rate = 10 ^(T-Tr)/z  where T is the temperature, in Celsius, at which the lethal rate is calculated and Tr is the reference temperature at which the equivalent lethal effect is compared. The z-value measured in °C is the reciprocal of the slope of the thermal death curve for the target microorganism or spore; 10° C is the value frequently used in F_o calculations performed on low acid foods.

Use of this equation can be illustrated using the following example. Calculate the lethal rate at 110° C compared to that at 121.11° C (Tr), given that the most heat resistant organism present has a Z value of 10° C.

Lethal rate= 10 ^{(110-121.11)/10}

 = 0.077.

Lethal rates when plotted against process time can be used to calculate the F value of a thermal process.  The F value can be defined as the time taken to reduce initial microbial numbers, at a specified temperature, by a particular value, normally a multiple of the D-value for the target organism.

Note the original work on themobacteriology used a reference temperature of 250° F. This is equivalent to 121.11°C.

Tr will vary depending on whether Fo is being calculated or whether a pasteurisation process or other heat treatment is being assessed. A Tr value of 121.11° C is used in the determination of Fo. If F₇₀ or other F value is required then Tr can be set at 70° C or other temperature.

Tr and the Z-value can be varied by the user. There is also a facility to perform a lethality calculation using an uploaded CSV file. Basic instructions are provided in the help file. The application can be tested using internal data and you can also upload a data file for analysis of a process.

The lethal rate calculator was validated using a Microsoft Excel spreadsheet  and following consultation with site users this is now available for download.

The spreadsheet converts temperature to lethal rate and plots lethal rate and temperature against time. The trapezoid rule is used to calculate the area under the curve and depending on the reference temperature chosen Fo or other F value can be calculated. A graph showing lethal rate and temperature against time is also plotted. The download contains a Microsoft Excel spreadsheet and a document in PDF form. The PDF file explains the basis of lethal rate calculation and also how to use Microsoft Excel to calculate the area under a curve using the trapezoid rule. All the cell formulas are unlocked.

Click here to use the calculator

 
How to cite this article

Mullan, W.M.A. (2007). [On-line]. Available from: http://www.dairyscience.info/food-model.html . Accessed: 8 September, 2010.