Modeling Growth and Reduction of Microorganisms in Foods as Functions of Temperature and Time
Robert L. McMasters - Department of Mechanical Engineering
Virginia Military Institute
Ewen C. Todd - National Food Safety and Toxicology Center
Michigan State University
Food Safety Objectives (FSOs) are established in order to minimize the risk of foodborne illnesses to consumers, but these have not yet been incorporated into regulatory policy. A FSO states the maximum frequency and /or concentration of a microbiological hazard in a food at the time of consumption that provides an acceptable level of protection to the public and leads to a performance criterion for industry. However, in order to be implemented as a regulation, this criterion has to be achievable by the affected industry. The analysis performed as part of this work used existing models for growth of Listeria monocytogenes in conjunction with calculations of FSOs, to approximate the outcome of more than one introduction of the foodborne organism throughout the food processing path from the farm to the consumer.
Calculation of FSO
Logarithmic calculation based on single addition of contamination as originally proposed by ICMSF, :
Most models for the growth and reduction of foodborne illnesses are logarithmic in nature, which fits the nature of the growth of microorganisms, spanning many orders of magnitude. However, these logarithmic models are normally limited to a single introduction step and a single reduction step. The model presented as part of this research addresses more than one introduction of food contamination, each of which can be separated by a substantial amount of time . The advantage of treating the problem this way is the accommodation of multiple introductions of foodborne pathogens over a range of time durations and conditions. A new non-logarithmic calculation based on two additions of contamination is:
The logarithmic version of the above equation is:
Fundamental Assumptions for Listeria monocytogenes Growth
Using the Buchanan and Philips model from tryptose phosphate broth  our primary growth equation is
Where the variables used are:
L(t) = The base 10 log count of bacteria as a function of time: log(CFU/g)
Ho = The starting value of the base 10 log count of bacteria: log(CFU/g)
B = The growth rate in log(CFU/g)/h. The natural logarithm of B is defined in Reference  as
-8.7529 + 0.098T + 1.427P - 0.039S - 0.0188N - 0.00277T2 -0.1338P2 + 0.0003S2 + 0.000000334N2 + 0.0167TP - 0.000257TS -0.00000712TN + 0.00417PS + 0.00248PN - 0.000000197SN
C = The total amount of growth experienced as the bacteria count asymptotically approaches it's maximum value: log(CFU/g).
M = The time at which the bacterial growth rate is a maximum, expressed in hours. The natural logarithm of M is given in Reference  as 18.9022 - 0.1734T - 3.9898P + 0.0385S + 0.025N + 0.00312T2 + 0.3204P2 - 0.000121S2 - 0.00000048N2 -0.00801TP + 0.00011PS + 0.0000121TN - 0.00372PS - 0.00328PN - 0.00000225SN
N = Sodium nitrite concentration in μg/ml
P = pH (negative base 10 logarithm of the hydrogen ion concentration)
S = Sodium chloride concentration in g/l
T = Temperature in degrees C.
Figure 1: Simple growth Curves for L. monocytogenes at various temperatures, using the Buchanan
and Phillips model , with no salt or sodium nitrite content and pH of 7.
Figure 2: Growth curves for L. monocytogenes at 6oC with an introduction of 20 cfu/g. In the case of the "Single Addition Curve", the introduction took place all at once at time zero. In the case of the "Double Addition Curve" there were 10 cfu/g added at time zero and another 10 cfu/g added at time 200 hours. The end result is that the FSO (arbitrarily established at 100 CFU, or a logarithmic value of 2) is reached 43 hours earlier in the case of the two step addition, even though the net amount of the addition is the same as in the single step case.
Figure 3: A composite of outcomes for various time durations for a second addition of L. monocytogenes in comparison to a single addition. For these examples, the temperature was assumed to be that shown on the horizontal axis throughout the entire process. As in figure 2, the single addition case received one addition of 20 cfu/g and the two-step addition case received two introductions of L. monocytogenes, of 10 cfu/g each.
At the colder temperatures, the two-step addition cases reach the FSO in a shorter amount of time. At the higher temperatures, the later two-step addition cases actually take more time to reach the FSO because each of the two-step additions of the bacteria are 1/2 the single addition case and they take benefit of a later kill step. The combination of these factors can actually lead to a longer storage life.
- The time at which the introduction of the contamination takes place is critical in determining whether or not the product meets the FSO.
- This method accounts for increases in the potential for human illness, possibly stemming from growth of a microorganism introduced subsequent to cooking or processing, e.g., by the producer, during transportation, at retail or in the home after the package is opened, followed by a long storage period.
- In the example of L. monocytogenes in broth culture at 6oC, with an introduction of 20 cfu/g, the division of the contamination addition into two steps shortens the storage time 43 hours (from 411 hours to 368) before exceeding the FSO, arbitrarily set at 2 for this example.
- This method may represent a more realistic model for developing FSOs than the current ICMSF model and needs to be considered for setting policy.
 International Commission on Microbiological Specifications for Foods, Microorganisms in Foods 7, Kluwer Academic / Plenum Publishers, New York, (2002).
 R. McMasters and E. Todd, "Modeling Growth and Reduction of Microorganisms in Foods as Functions of Temperature and Time", Risk Analysis: An International Journal, Vol. 24, No. 2, pp. 209-214, 2004.
 R. Buchanan and J. Phillips, Response Surface Model for Predicting the Effects of Temperature, pH, Sodium Chloride Content, Sodium Nitrate Concentration and Atmosphere on the Growth of L. monocytogenes, Journal of Food Protection, Vol 53, May 1990.
Research support was provided by the U.S. Food and Drug Administration