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

R.C. McKellar, K. Knight / International Journal of Food Microbiology 54 (2000) 171 –180 175
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Fig. 2. Discrete–continuous model designed with ModelMaker to simulate td
and standard deviation of individual cell lag times (S.D.
L)
based on kinetic parameters derived from Bioscreen data.
pre-defined time; and component event blocks are variation in individual cell lag times, t
L, whereas
square, and are activated in response to other com- S.D. refers to the variation between wells observed
ponents in the model.
with any defined number of cells per well.
When the model run is initiated, the AssignLag The model described in Fig. 2 was extended to
block (an independent event block activated at t
0)
allow the simulation of a complete growth curve.
uses a random number generator based on a trun- The discrete adaptation function was retained, and
cated (positive values only) normal distribution with combined with the HPM to provide the continuous
mean tL and S.D.
L
from Table 1 to assign tL
values growth function (Fig. 4). In this model, the Adaptato
each of up to 64 cells. These values are stored in tion block moves one cell from the NonGrowing to
the defined value Triggers block. The Adaptation the Growing compartment at each of the Trigger
block reads these values, and adds a single cell to the times. This preserves the initial number of cells in
Growing compartment at each time corresponding to the model. The blocks to calculate the log of the cell
an individual cell t
L. Once in the Growing compart- numbers are provided to assist in the visualization of
ment, cells start growing immediately according to a the growth curve.
logistic equation (McKellar, 1997):
The output of this model for a total of 64 cells is
shown in Fig. 5. Note that values of the the Growing
dG G compartment are not shown where the number of
] 5 Gm S1 2 ]] D (4)
dx N max
cells was zero. These results show that the simulated
values for the NonGrowing cells decreased as the
where G is the number of cells in the Growing numbers of Growing cells increased. When the total
compartment and Nmax
is the maximum population
density.
The LogGrowing block calculates the log cfu, and
at each time increment the component event Monitor
block tests to see if the value of LogGrowing has
6
exceeded the detection limit (defined as 3.510
cfu/well). The defined value TimetoDetection block
holds the calculated t
d.
The model in Fig. 2 was used to simulate values
for td
corresponding to up to 32 cells per well. The
simulated S.D. values were derived from a total of
20 simulations for each of 1, 2, 4, 8, 16 or 32 cells
per well. Fig. 3 shows that both the simulated td
and
S.D. values are in close agreement with the ex- Fig. 3. Comparison of t
d
(s) and S.D.
L
(h) from experimental
perimental findings. Note that S.D. refers to the data (open symbols) or simulated data (closed symbols).
L