A combined discrete–continuous model describing the lag phase of ...

172 R.C. McKellar, K. Knight / International Journal of Food Microbiology 54 (2000) 171 –180
function; however, there are some limitations to the Some of the fundamentals of this approach have
use of this type of function. For example, it can be been discussed by McMeekin et al. (1993). These
shown mathematically that the Gompertz rate is authors and others emphasized the limits of this
always the maximum rate and occurs at an arbitrary method, the most severe of which is the fact that OD
point of inflection (Garthright, 1991, 1997). The methods are comparative only, and cannot be used to
Gompertz equation tends to overestimate m as it fits predict viable counts unless some attempt at calia
sigmoidal curve to a straight line. In addition, the l bration is made (McMeekin et al., 1993; Baranyi and
calculated with the Gompertz is always at a defined Roberts, 1995). McClure et al. (1993) used a simple
point relative to the upper and lower asymptotes quadratic equation to relate OD to viable counts.
(Garthright, 1991, 1997). Thus, empirical equations Dalgaard et al. (1994) used two equivalent methods
have a limited ability to enhance our knowledge for calibration: one in which stationary phase cells
concerning the physiological stages of bacterial were diluted to the appropriate OD, and the other in
adaptation to new environment and subsequent which samples for OD and viable count were taken
growth. during growth. Predicted generation times were
It has been suggested that connecting the behavior lower with viable count data (Dalgaard et al., 1994),
of a single cell to that of the whole population is the and this factor has been taken into account in later
next stage in developing a more mechanistic ap- studies (Miles et al., 1997). Similar methods have
proach to predictive food microbiology (Baranyi, been used to relate turbidimetric and viable count
1997). While there have been some attempts to data (Chorin et al., 1997).
develop mechanistic models for bacterial growth In some studies, the Gompertz equation was fitted
(Baranyi and Roberts, 1994, 1995; Hills and Wright, directly to OD data; however, no data was available
1994; Hills and Mackey, 1995), these models tend to
7
at below the minimum detectable OD (ca. 10 cfu/
view bacteria as a homogeneous population, and ml) thus the estimates for l and m should be
there have been few attempts to model bacterial questioned (Hudson, 1994; Hudson and Mott, 1994).
adaptation and growth on the basis of single cells. A form of calibration was achieved by relating l
Recently, a model was proposed in which the using OD measurements to that determined with
bacterial population was divided into non-growing viable counts by a regression equation (Hudson and
and growing cells (McKellar, 1997). This model was Mott, 1994). McMeekin et al. (1993) have discussed
expressed in the form of differential equations, and the correct way to fit the Gompertz function to %
the behavior of the two types of cells was modeled transmittance data, and this method has been used to
independently. Buchanan et al. (1997) have proposed calculate generation times (Neumeyer et al., 1997).
a model which takes into account the variation in Other studies have been carried out without any
adaptation (or lag) time of individual cells. Simula- apparent calibration (Huchet et al., 1995). l values
tions with this model gave rise to ‘‘traditional’’ have been estimated from OD data by extrapolation
growth curves; however, these authors did not pro- of the exponential portion of the curve back to the
vide experimental evidence for their model. More initial cell numbers (Breand et al., 1997); however,
recently, Baranyi and Pin (1999) and Baranyi (1998) this method may be inaccurate since the growth rate
have proposed a lag model based on behavior of estimated from the OD data may be lower than that
individual cells.
obtained during the period of maximum growth
Construction of models using viable count data is (McMeekin et al., 1993).
time consuming and expensive, and researchers have Interestingly, the time to detection (t
d) approach
explored other, more rapid, methods for accumulat- has not been used to any great extent. The td
for a
ing sufficient data for modeling. One of the simplest turbidimetric instrument can be defined as the time
method is the use of optical density (OD), where required for an initial measurable increase in OD.
growth can be related to the increase in turbidity of a The difference between td
for serial two-fold dilu-
bacterial culture. This method lends itself particu- tions gives the doubling time, from which the m can
larly well to automation, and a number of studies be determined (Cuppers and Smelt, 1993). The l can
have used automated turbidmetric instruments such be calculated subsequently by the difference between
as the Bioscreen (McClure et al., 1993; Huchet et al., the predicted td
based on the m, and the observed td
1995). (Cuppers and Smelt, 1993). This method was used to