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R VI WS
Shelf-life prediction: theory and application
Bin Fu and Theodore P. Labuza*
Although most food companies have a distinct interest in the shelf-life of their food products, little has been done to determine true shelf-life as a function of variable environmental conditions. Research efforts are generally aimed at improving quality with respect to a longer shelf-life. For true shelf-life prediction, especially for refrigerated foods, knowledge of predictive microbiology is needed. Models that describe influences of temperature and water activity on microbial deteriora- tion are corn criteria of r P ared and the square-root model is found to be the best based on the and mean square error. Appropriate approaches are presented for prediction of remaining quality or shelf-life of foods undergoing fluctuating temperature conditions. Finally, the Weibull hazard analysis technique for determination of shelf-life is introduced.
Keywords: Shelf-life; prediction; microbial growth; model; Weibull hazard analysis
INTRODUCTION
The shelf-life of a food is the time period for the product to become unacceptable from sensory, nutritional or safety perspectives. The criterion level corresponding to the end of shelf-life is determined by any legal requirement, e.g. the 80/80 rule for nutri- tional loss in natural foods, as well as consumer taste attributes, marketing distribution requirements and cost. From the viewpoint of the food industry, shelf-life is based on the extent of quality loss in a food that the food company will allow prior to product consumption. At that point, the food may still be organoleptically acceptable. For consumers, the end of shelf-life is the time when the food no longer has an acceptable taste. Realizing that one can never satisfy all consumers all of the time and that food systems, along with their mechanisms of deterioration, are inherently complex, a universal definition of shelf-life is thus virtually impos- sible to establish. Despite this, shelf-life can be deter- mined, and subsequently predicted, for individual food
Department
of
Food
Science and Nutrition, University of Minnesota, 1334 Eckles Avenue, St Paul, MN 55108, USA. *To whom correspondence should be addressed. Presented at the International Food Technology Exposition and Confer- ence (IFTEC), 15-18 November 1992, The Hague, The Netherlands
0956-7135/931030125-09 0 1993 B’utterwotth-Heinemann Ltd
products based on some primary mode of deteriora- tion. Microbiological decay is one of the major modes of food deterioration, especially for fre$r or minimally processed refrigerated products. Microorganisms may cause spoilage of foods or result in foodborne diseases. Processors should conduct shelf-life tests to determine when spoilage occurs and should verify the effective- ness of the various barriers to pathogenic growth, using appropriate scientific studies to evaluate potential risk of their food products (NFPA, 1988). Spoiled products, even if no health hazard is present, are considered adulterated in the USA under Section 402(a)(l) and 402(a)(3) of the Food, Drug and iCosmetic Act. Accurate shelf-life prediction is an impbrtant aspect of food science, not only to corporations but to govern- ments and the general public as well. A premature loss of shelf-life can lead to a loss of consumer confidence and of revenues to the food manufackurer. Shelf-life testing also allows the company to m nimize costs in formulation and packaging. Open
I
at ng (e.g. sell-by date and best-if-used-by date) of foods must be based on some type of shelf-life testing as well. With the new EC requirements, all foods must have some type of open date. To predict the shelf-life of refrigerated foods based on microbial growth, the concept of ‘predictive micro- biology’ must be employed (e.g. Buchanan, 1993). Growth of microorganisms are first studied under
Food Control 1993 Volume 4 Number 3
125
Shelf-life prediction: 6. Fu and T. P. Labuza
several different but fixed conditions. An appropriate growth curve
model e.g. Gompertz
model) can be applied to the data (Zwietering er
al.,
1990), from which growth parameters (such as specific growth rate and lag time) are derived. Then another model is employed to determine the effects of compositional and environmental factors on the growth. The combination of the above two steps can be used to predict the microbial level and thus the shelf-life and safety of perishable foods under commercial distribution con- ditions. Models with one or two variables are practic- ally useful, even though multivariant response surface models (Buchanan
et al.,
1989; Gibson and Roberts, 1989; Buchanan and Phillips, 1990) can also be used to predict the growth of microorganisms in response to food formulation, packaging and environmental fluc- tuations, as long as the limits of the model are not exceeded. This latter approach is especially useful in the stage of new product development. The purpose of this paper is to present the mathe- matical models that are available to describe quantita- tive effects of temperature and water activity
a,,,),
with the emphasis on temperature, on microbial growth parameters, particularly specific growth rate and lag time and to demonstrate the applicability of these models in predicting shelf-life of foods under real world conditions. Since sensory evaluation is often used in evaluating quality deterioration of perishable foods, the Weibull hazard analysis technique is introduced for shelf-life determination. Nutrient degradation reac- tions, such as the browning reaction, vitamin C loss and lipid oxidation, which are important to many other foods (Labuza, 1984) will not be discussed.
MICROBIAL GROWTH AS A FUNCTION OF TEMPERATURE AND WATER ACTIVITY
There are many temperature-dependent models to describe microbial growth in the literature (Zwietering
et al.,
1991; Labuza
et
al.,
1992; Buchanan, 1993), however, only a few are applicable for practical shelf- life prediction. Based on Bayesian statistical analysis (also known as Ockham’s Razor rule), the fewer number of parameters in a model, the closer to reality based on physical principles is the equation (Jefferys and Berger, 1992). Thus a higher order model may give a very good fit but then eliminates the ability to determine the effects of various factors on the precision of the measurement. Therefore, only simple models with two or three parameters will be discussed below with a comparison based on
r2
and mean square error (MSE).
Arrhenius model
Since microbial growth is a biochemical process, it is expected that, for a certain temperature range, the Arrhenius law would follow. Thus the temperature dependence of the growth rate can be characterized by an overall activation energy if all other ecological factors are kept constant. This two-parameter function takes the form: where
k
is the specific growth rate determined from the growth curve,
A
is the collision factor,
T
is the absolute temperature (K), R is the universal gas constant (8.314 J mol-’ K-‘) and EA (J/mol) is called the activa- tion energy, which is a measure of the temperature sensitivity of the growth rate. The value of
EA
can be determined from the slope of the plot of In
k
versus l/T. Successful applications of Equation (1) for predictive microbiology are available in the literature for many different organisms if the temperature range used is limited (Labuza
et al.,
1992). Significant deviation from linearity for the plot of In
k
versus l/T has also been noted by Ratkowsky
et
al.
(1982) if too large a temperature range is used. The Arrhenius relationship can also be applied to model the temperature depend- ence of the lag phase, which would be critical for prediction of the shelf-life under variable temperature conditions where there is an initial low microbial load. The inverse of the lag time (l/t,, i.e. lag rate) is used to make the Arrhenius plot. Kinetic parameters for the exponential and lag phase are given in
Table I
for the data of Fu
et
al.
(1991). Davey (1989) modified the Arrhenius equation for predicting the combined effect of temperature and
a,
on the growth rate of bacteria which took the form: In
k = CO + C,IT+ C2/T2 + C, a, + C, aw2 4
where C,, to C, are the five coefficients to be deter- mined by multiple-linear regression. It generally gives good fit due to added terms. This equation has also been used for lag time data (Davey, 1991). When
a,
is kept nearly constant, which could be true for most fresh foods since it would require a significant moisture content change to alter the
a,
significantly, Equation (2) can be simplified to account only for the effect of temperature.
Table I
shows that the Davey model gives a higher r2 and lower MSE than the Arrhenius model for both growth phases.
Square root model
Ratkowsky
et al.
(1982) proposed a simple two- parameter empirical equation for the temperature dependence of microbial growth up to the optimum temperature
T,,,)
as:
V/k= b T- Tmin) 3)
where
k
is the specific growth rate as before,
b
is the slope of the regression line of ti versus temperature,
T
is the test temperature (in either “C or K) and
Tmin
is the notional microbial growth temperature where the regression line cuts the temperature axis at fi = 0. This equation generally gives a better linear fit from the
Tmin
up
to and including the
Topt.
Ratkowsky
et al.
(1982, 1983) showed that the equation accurately described the growth rate data of many organisms. However, since the actual
Tmin
for growth may occur several degrees above the extra- polated values, the growth predicted near the lower extreme of growth temperature based on the square root equation, could be more than would actually occur. The kinetic parameters of the square root model for
Pseudomonas fragi
growth are provided in
Table I.
As seen it has a slightly larger r2 value and significantly lower MSE for both phases, than does the Arrhenius; 126
Food Control 1993 Volume
4
Number 3
Shelf-life prediction: B. Fu and T. P. Labuza
Table 1
Model equations and kinetic parameters determined for
Pseudomonas frugi
in a simulated milk” Arrhenius model Davey model Square root model Linear model Exponential model Exponential phase Equation Ink=30.10- 8.90 x 103/T) Parameters InA = 30.10
EA =
73.9OkJlmol
r2
0.984
0.999
0.998 0.976 0.977 MSEb 0.00260
0.00013
0.00007 0.00144 0.00420
Lag
phase Equation In l/t,) = 29.90- 9.17 x 103/T) Parameters In
A
= 29.90 EA = 76.20 kJ/mol
r2
0.963 0.994 0.993 0.980 0.956
MSE 12.79
2.79 2.53 175.72 13.99 Ink= -160.97+
(99.92 x 10”IT) - (15.48 x w/T*) c,= -160.97 C, = 99.92 x 10’ c*= -15.48X10h
In l/t,) = -237.75 + 143.24 x 103/T)- 2168x lob/T*) C, = -237.75 C, = 143.24 x lo3 C, = 21.68 x lo6
d/k =
O.O306 T+
k-
-0.0315 1 - 7.85) l.17T)
k = 0.088
e”.“” T
b =
0.0306 h-“‘“C a = -0.0315 h-’
k,, =+ .088
h-’
Tmi. =
-7.85”C c= -1.17”C
s=o.llo”c~-’
l/v/t, = 0.0172 T+ l/r, = -0.0093 1 - 7.65) I .22
T)
l/t, = 0.026 e”.“3 ’
b =
0.0172 h-“*“C a = -0.0093 h-’
k,-O.O26h-’
Tmi. = -
7.65”C c = - I .22”C’
s=o.113”c-’
“Based on the data
of
Fu et
al.
(1991) bMSE = Z[(observed - predicted) ]
and similar r* but lower MSE than does the three- on r2 value but the parameter Davey model. data. McMeekin et
al.
(1987) modified the basic square root equation to incorporate
a,:
xponential model
d/k = Cv a,,,- MINa,) T- Tmin) 4)
where C is a constant and MINa, is the theoretical minimum
a,
for growth of the organism. This equation may be used for predicting the growth rate at any combination of
a,
and
T
as long as they are in the test range (McMeekin et
al.,
1987). If the temperature range of concern is no more than 20 to 30 degrees, then a simple plot of thei specific growth rate on semilog graph paper versus temperature (instead of inverse absolute temperature) also gives a straight line. The exponential model has the following form:
k = kO
exp
ST) 6)
Linear model
Spencer and Baines (1964) proposed a linear model based on the research of microbial growth on white fish. They postulated that the effect of temperature on the rate of spoilage of fish stored at a constant temperature between -1 and 25°C was found to be approximately linear and could be expressed in the form:
k=a l+cT) 5)
where
k
is the specific growth rate at storage tempera- ture
T “C), a
is the rate at 0°C and c is the temperature coefficient (UC). Thus a plot of the growth rate
k
versus
T
gives a straight line. Such a response would be expected if both the temperature range and the
EA
were small and if the organism was able to grow at a few degrees below 0°C. Jorgensen et
al.
(1988) reported that the shelf-life of iced whole cod can be predicted using a linear model but not that of vacuum-packed fillets because of the greater variability of bacterial activity in packaged fish. The parameters for the same data from Fu et
al.
(1991) are shown in
Table I.
As seen, it fits fairly well based MSE is very high for the lag phase where
k
is the specific rate at
T
in “C, and
k,,
is the specific rate at O”C, s is the slope of a plot of Ink versus
T.
This model is also applicable tb lag phase data or shelf-life data (Labuza, 1984). As s own in
Table 2,
the exponential model has essentially t % e lowest r2 for both phases, the highest MSE for the exponential phase and the second to highest MSE for
;
he lag phase. Overall speaking, the square root model best fits both the lag and exponential phase data of p.
fragi
growth. Qro
and other models
Q o
is defined as the temperature isensitivity of a reaction.It is the increase in reaction (or growth) rate or the decrease in shelf life for a~ 10°C increase in temperature. It has been used to predict quality or nutrient losses for many foods and potency degradation for drugs. The Qio is usually assumed’constant over a narrow range of temperature. L eporte, Qio values for microbial growth under refrigeration conditions range from 2 to 10. The relationship of Qio~ with previously discussed models and the Qio values for
P. fragi
growth (Fu et
al.,
1991) are shown in
Tab@ 2.
As seen, a different temperature model can lead to a different (2 i. value at the same reference temperature.
Food Control 1993
Volume 4
Number
3 127
Shelf-life prediction: 6. Fu and T. P. Labuza
Table 2
The relationship of Q,(, with other models and Q,,) values for P.
fragi
growth Q,,) for P.
fragi
growth” Model type Q,,) expression Lag phase Exponential phase Arrhenius Davey Square root Linear exp lOC, T(T+ 10) - 10&(2T+ 10) T*(T+ lo)2 >
(T+
10 - T,i,)’ (T- T,i,)*
IOC
1+-
l cT
3.2 3.1 3.7 3.4 3.5 3.4 4.1 4.2 Exponential exp (10s) 3.1 3.2 “Based on the parameters in
Table I
and calculated at
T =
4°C
SHELF LIFE PREDICTION IN THE REAL WORLD
General consideration
Under the ideal conditions, the level of microbial growth or shelf-life of a food based on microbial growth can be easily predicted by using the following equations: N = No exp [k(t - tL)] (7a) ts = In (N,IN,)Ik + tL (7b) where No is the initial microbial load, N is the microbial level at time t, N, is the microbial level corresponding to the end of shelf-life,
t,
is the shelf-life at that constant temperature,
k
and
tL
are the specific growth rate and lag time at a constant storage temperature, which can be predicted from an appropriate model, such as the square root model as discussed earlier. For example, Griffiths and Phillips (1988) were able to predict the shelf-life of pasteurized milk at different constant storage temperatures using the square root model for microbial growth developed from several other con- stant temperatures ranging from 2-14°C. Unfortunately, the reality is not that simple. Shelf- life depends on the initial level of contamination of the product. However, the initial microbial load is usually not well controlled in real foods. With pasteurized milk, Maxcy and Wallen (1983) noted considerable variation in initial levels of contamination which was responsible for differences in product shelf-life. This presents a serious problem for shelf-life prediction, but may be overcome by specifying a maximum initial level of the contamination organisms of concern. A rela- tively stable initial No value can be set by employing the hazard analysis and critical control point (HACCP) system and good manufacturing practice (GMP) pro- cedures and taking into account the shelf-life required for a particular market. No more than 10 pseudo- monads per ml was deemed to be appropriate for pasteurized milk (Chandler and McMeekin, 1989). In addition, the variety of species that make up the initial population is not always known. If two types of microbes or strains with different Q,, values are responsible for the quality loss in a food, the one 128
Food Control 1993 Volume 4 Number 3
with the higher Q,u may predominate at higher temperatures and the other with the lower Q,, may predominate at the lower storage condition, confound- ing the prediction. Spoilage at different temperatures could be attributed to the growth of different groups of bacteria and could be influenced by package type. On the other hand, a product-dependent factory flora may be established, specifically selected/adapted to growth in the conditions prevailing in the food or factory environment. Such a flora may modify the spoilage characteristic and shelf-life of a product. Another consideration is the correlation of microbial level with the quality or shelf-life. In general, a bacterial level of 106-10’ cfu/unit (Ns) indicates the end of shelf-life. However, in some products, the number of microbes may not be a valid indicator for shelf-life. In these cases, the quality deterioration caused by micro- bial growth may be better evaluated by other indices, e.g. organoleptic quality change. If a toxin-producing pathogen is concerned, then the shortest lag time before the toxin can be detected may be used as the end of shelf-life (Baker and Genigeorgis, 1990). To predict the growth under variable time- temperature conditions, researchers usually run kinetic experiments at several constant temperatures to obtain kinetic growth parameters for the organism. Because of the possible shift of dominant microorganisms and the minimum-optimum-maximum temperature behaviour, the choice of the temperature range for such a study is quite limited. Suggested values for refriger- ated foods are 4-15°C with controls stored at just above the freezing point. One more concern is the history effect, which is defined as the situation in which the actual growth rate that is measured after a temperature shift is signifi- cantly different from that predicted by a model under- going the same temperature shift. Several varying temperature studies of microbial growth showed some history effects (Ng
et al.,
1962; Shaw, 1967; Fu
et al.,
1991), yet others did not find any history effect within the temperature range studied (Langeveld and Cuperus, 1980; Simpson
et al.,
1989). If there is a history effect, the prediction for microbial growth rate and lag time using any model may significantly deviate from the real growth. In the following discussion, we assume there is no history effect or that the effect is negligible.

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