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International Journal of Food Microbiology 100 (2005) 323 – 333 www.elsevier.com/locate/ijfoodmicro Individual-based modelling of growth and migration of Salmonella enteritidis in hens’ eggs K. Grijspeerdta,*, J.-U. Kreftb, W. Messensa a Ministry of the Flemish Community, Centre of Agricultural Research Gent, Department of Animal Product Quality, Brusselsesteenweg 370, 9090 Melle, Belgium b Botanisches Institut der Universität Bonn, Abteilung Theoretische Biologie, Kirschallee 1, 53115 Bonn, Germany Received 24 September 2004; accepted 6 October 2004 Abstract An individual-based model (IbM) was developed to describe the growth and migration of Salmonella enteritidis in hens’ eggs. The Bacteria Simulator (BacSim) environment was used to implement the model; the bacteria are represented by spheres that grow by nutrient uptake and divide in two daughter cells upon exceeding a certain threshold volume. Motility of the Salmonella bacteria was described by a run and tumble mechanism. For the sake of simplicity, the bacteria were assumed to grow exponentially, an appropriate assumption for the initial phase of growth relevant for shelf-life predictions. Both albumen and yolk were assumed to be homogeneous. The impact of several model parameters (chemotaxis, growth rate, initial contamination numbers and bacterial swimming speed) was assessed by a sensitivity analysis. The results show that chemotaxis towards the yolk would have a strong effect on the time needed to reach the vitelline membrane, an aspect that future research should focus on. The contamination position had less impact on the time to reach the vitelline membrane. The simulation results illustrate the need for more detailed knowledge on the subject of bacterial migration in hens’ eggs. Our model can easily incorporate this knowledge when it becomes available. D 2004 Elsevier B.V. All rights reserved. Keywords: Individual-based model; Motility; Chemotaxis; Salmonella; Egg; Albumen; Yolk 1. Introduction Salmonella infections are one of the most widespread microbial infections (Mead et al., 1999). The * Corresponding author. Tel.: +32 9 2723012; fax: +32 9 2723001. E-mail addresses: [email protected] (K. Grijspeerdt)8 [email protected] (J.-U. Kreft). 0168-1605/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ijfoodmicro.2004.10.028 largest share is taken by Salmonella enteritidis infections, which accounted for 64% of total Salmonella infections in Belgium in 2001 (Ducoffre, 2003). S. enteritidis infections are mainly associated with consumption of shell eggs and egg dishes. Eggs can become contaminated by horizontal (bacterial penetration through the shell after laying) or vertical (contamination of the egg contents before the shell is formed) transmission. Upon vertical 324 K. Grijspeerdt et al. / International Journal of Food Microbiology 100 (2005) 323–333 transmission, both albumen and yolk may become infected. The predominant site of infection, however, remains unclear. It was generally believed that the majority of vertical contaminations occurred in the albumen (Humphrey et al., 1991), but recently there has been increasing evidence for contamination of the yolk (Gast et al., 2002), particularly its membrane (Gast and Holt, 2001b). Upon horizontal transmission, the organism crosses the shell and membranes and gains access to the albumen. The bacteria that succeed to migrate to the yolk encounter a growth-friendly environment, whereas albumen evolved to be an unfavourable growth medium (Hammack et al., 1993; Braun and Fehlhaber, 1995). It is the presence of inhibitory substances such as lysozyme, the shortage in available iron and the relatively high pH that make albumen an effective defence against bacteria (Baron et al., 1997). As a consequence, S. enteritidis will grow poorly or not at all in the albumen. If the initial contamination takes place in the albumen, the organisms will only thrive when they will have reached the yolk. It is not clear if this occurs by chance or by a chemotactic response to nutrients that diffuse out of the yolk. There are some reports on the migration of S. enteritidis in the albumen. After inoculating the albumen far from the yolk in situ, Gast and Holt (2000) found that significant growth required an inoculum dose of 150 cells and keeping temperature at 25 8C. With an inoculation dose of 15 cells, growth was not observed. However, the incubation time of only 2 days prevents definitive conclusions. In general, bacteria need to access the yolk before growth can start. Cogan et al. (2001) observed growth after 8 days at 20 8C in 7% of whole eggs inoculated in the albumen near the shell with as few as two cells. The fraction of contaminated eggs increased up to 50% when the initial inoculation level was increased to 2500 cells. The temperature also had a significant impact. Braun and Fehlhaber (1995) found that S. enteritidis can migrate from the albumen to the yolk in less than 1 day for 17% of the eggs inoculated with 10 cells/ml albumen and subsequent storage at 20 8C. After 4 weeks, 72% of the egg yolks were contaminated. Similar results were reported by Baker (1990), who observed contaminated yolks, albeit not frequently, during storage at 8 8C. It has to be noted that Braun and Fehlhaber (1995) used buffered peptone water for the S. enteritidis solution to be injected, which enhances bacterial growth in albumen. Using a rather high inoculation level (104 cells) in the albumen near the shell surface, growth was observed in the yolk after 2 days of storage at 26 8C (Hammack et al., 1993). Skipping the albumen layer, Gast and Holt (2001b) inoculated the vitelline membrane directly with S. enteritidis and found 6% positive yolk interiors after 6 h incubation at 25 8C, up to 100% positive after 24 h. At lower temperatures, the membrane was less frequently, but still significantly, penetrated. Upon vertical transmission, contamination occurs at very low initial numbers of only a few cells (Cogan et al., 2001). A fortiori, this will also be true for crossshell contamination, except for cracked eggshells. Traditional mass transport concepts to model S. enteritidis migration through the albumen, such as the growth-diffusion model presented by Grijspeerdt (2001), are based on a continuous representation of bacterial mass and are consequently not ideally suited to describe such low bacterial counts. A discrete approach appears to be the better choice here. Individual-based modelling (IbM) of bacterial growth has been successfully applied for describing colony and biofilm formation, and is applicable for the whole range of bacterial densities (Kreft et al., 1998, 2001). Because these types of model describe the individual cells, no global population-level assumptions are necessary, making them utilizable for a broad range of cases. An IbM was developed to simulate cell growth and movement within the egg. 2. The model The model is an extension of the BacSim model of the growth of bacterial colonies and biofilms (Kreft et al., 1998, 2001). Bacteria Simulator (BacSim) is an IbM where each bacterium is simulated as a sphere of variable volume in a continuous three-dimensional space. 2.1. Description of egg geometry Denys et al. (2003) compared four models describing the contours of an egg. The most flexible model was the four-parameter model of Carter (1974), which K. Grijspeerdt et al. / International Journal of Food Microbiology 100 (2005) 323–333 was applied to BacSim to define the eggshell. The yolk was assumed to be spherical, but can be positioned anywhere inside the egg. In general, the yolk will be situated close to the centre of the egg, as this is the most beneficial position to counter bacterial attacks. However, exceptions occur, especially with increasing egg age (Board et al., 1994). 2.2. The egg components Several regions can be defined inside the egg. The model in this paper simplifies the egg into three basic compartments: shell, albumen and yolk. For the simulations presented in this paper, the shell was considered to be an impermeable barrier for bacterial movement; hence, no bacteria can escape the egg. All simulations start from the premise of an initial contamination within the albumen. This contamination could be due to vertical or horizontal transmission, in the latter case assuming that the bacteria have penetrated the shell and shell membranes already. The albumen is assumed to be homogeneous, as is the yolk. Apart from viscosity differences, we are not aware of differences between the thick and thin albumen that may be relevant for bacterial growth or migration. From the viewpoint of bacterial motility, the viscosity of either type of albumen is relatively low justifying treating it as homogeneous. The vitelline membrane is not modelled as a compartment as it is assumed to have no extent, justified by its thickness of only 11–15 Am (AbouAshour and Edwards, 1970). Once a cell reaches the vitelline membrane, it sticks to this membrane for a certain period after which it enters the yolk. This is equivalent to supposing that the membrane holds back bacteria initially, but deteriorates after a period of time. It is left aside whether this is due to bacterial metabolism or aging effects. Daughter cells that are created during this hold-up period remain attached to the membrane as well. This approach is consistent with the time to membrane breakdown concept used in risk assessment studies (Baker et al., 1998; World Health Organization, 2003). Unfortunately, data on the yolk membrane breakdown time are very diverse (Gast and Holt, 2001a). Therefore, the time needed for the first cell to reach the vitelline membrane will be used as a reference point for most simulation examples. 325 2.3. Bacterial growth It is assumed that cells grow exponentially and do not die. During the time span of interest bacterial densities do not reach values high enough to cause significant deviation from exponential growth. For the sake of clarity, the lag phase is assumed to be absent, although the concept of IbM is very suited to include stochastic growth lag. Although considerable progress has been made in this field (McKellar and Knight, 2000; McKellar, 2001), information on lag phase distribution of individual bacteria remains uncertain, making it difficult to make a reasonable choice at this moment (Baranyi, 2002). For the same reason, death kinetics is not included in the model at this stage. The bacteria grow and divide into two daughter cells when the volume exceeds a certain threshold according to the cell division model of Donachie and Robinson (1987). The growth rate and cell size at division are varied as described in Kreft et al. (1998), making this growth model stochastic. This can be seen in Fig. 2 where a typical growth curve is shown. The cells clearly grow increasingly asynchronously. A minimum distance between the cells is maintained by a shoving mechanism, to prevent a closer packing of cells than physically possible (Kreft et al., 1998). Note that the importance of shoving is reduced when dealing with motile bacteria at low densities, as is the case here. 2.4. Bacterial motility Flagellated bacteria such as S. enteritidis swim by rotating their flagellar filaments (Blair, 1995; Purcell, 1997). When these flagella turn counterclockwise they form a synchronous bundle that pushes the bacterium forwards; the cell is said to drunT. When turning clockwise, the bundle comes apart and the cell moves in an erratic manner; the so-called dtumbleT mode (Berg, 2000). These two modes alternate and the cell executes a three-dimensional random walk. In the run mode, the cell would swim in a straight line were it not subject to rotational diffusion. In tumble mode, the cell reorients itself, albeit not completely randomly due to a forward bias. Tumbles are much shorter than runs, but they generate much larger changes in direction. The probability of switching modes is constant per unit time, and intervals between switching can thus be 326 K. Grijspeerdt et al. / International Journal of Food Microbiology 100 (2005) 323–333 described by a Poisson process. Swimming is superimposed on Brownian movement, which is several orders of magnitude slower and is therefore negligible. The bacterial swimming is modelled according to Berg (1993): ! In the run mode, each step (the displacement of a cell from one timestep to the next) is a vector two units long making an angle of 88 with the preceding vector but rotated around it by a random angle between 08 and 3608 (see Fig. 6.8 in Berg (1993)). ! After each run step, there is a probability of 0.1 that the cell switches to the tumble mode. ! In the tumble mode, each step is a vector one unit long making an angle of 808 with the preceding vector. ! After each tumble step, there is a probability of 0.3 that the cell reverts to the run mode. Note that this model of bacterial swimming was constructed based on experimental observations of Escherichia coli. As far as we know, there are no such data available for S. enteritidis, but the above model can be expected to be a reasonable approximation, since both bacteria are close relatives and use the same type of flagellar machinery to swim. 2.5. Chemotaxis Chemotaxis is the phenomenon that motile bacteria can bias their random walk in the presence of repellents or attractants (Armitage, 1999). In case of the latter, this is accomplished by reducing the probability of switching to the tumbling mode, making the straight runs longer (Berg, 1993, 2000). This mechanism is extremely fine-tuned, allowing chemotactic responses to changes in attractant concentration of less than 1% over a concentration range spanning five orders of magnitude (Stock, 1999). The tumbling frequency has been reported to decrease up to five-fold compared to the steady-state frequency (Staropoli and Alon, 2000) or to virtually zero (Maki et al., 2000). It has been postulated that there may be a leakage of nutrients from the yolk, certainly with increasing egg age (Lock et al., 1992; Humphrey and Whitehead, 1993), although the evidence to support this hypoth- esis is contradictory (Abou-Ashour and Edwards, 1970; Messens et al., 2004). Bacteria would then become attracted towards the yolk, reaching it sooner and thereby accelerating egg contamination. Chemotaxis towards these hypothetical attractants is implemented in the model by changing the tumbling probability linearly as a function of the swimming direction of the cells. When the cell swims straight to the yolk the tumbling probability is at the minimum value, when the cell swims parallel to the yolk it is at the default unstimulated value and when the cell swims away from the yolk it is at the maximum value. This is clearly an oversimplification. It is unlikely that a possible attractor gradient would be uniformly present in the egg. The possible influence of egg age and the associated deterioration of the yolk membrane are also not taken into account. Given the uncertainty about the possible leakage of yolk elements into the albumen, and the lack of information on their identity, it is clear that the model here can only serve to indicate the possible significance of chemotaxis. 2.6. Model parameters and simulation settings The simulated temperature was set to 20 8C, and the egg was supposed to be isothermal. This is a realistic situation in the EU where most shops store the eggs at room temperature since there is no legal requirement to cool eggs during storage. The model parameter values are summarised in Table 1. The bacterial swimming speed u has been measured by Schneider and Doetsch (1974) and Atsumi et al. (1996) for E. coli in solutions within and beyond the viscosity range found in albumen and yolk (1.7 10 3 to 4.310 3 Pa s in albumen and 4.810 2 to 0.31 Pa s in yolk, according to Scalzo Table 1 The model parameters and their default values used for the simulations Bacterial swimming speed in albumen Bacterial swimming speed in yolk Vitelline membrane hold-up time Egg length Egg width Yolk diameter Growth rate in albumen (20 8C) Growth rate in yolk (20 8C) 20 Am/s 4 Am/s 15 h 5.65 cm 4.15 cm 2.82 cm 0.1 1/h 0.75 1/h K. Grijspeerdt et al. / International Journal of Food Microbiology 100 (2005) 323–333 et al. (1999)). Within both viscosity ranges, the swimming speed of E. coli is relatively constant. Increasing the viscosity range from albumen to yolk, the swimming speed decreases approximately inversely proportional to viscosity. The hold-up time at the vitelline membrane was set at 15 h, in accordance with Gast and Holt (2001a). A typical egg geometry as measured in the lab was used. The growth parameters were obtained from growth curves of S. enteritidis at 20 8C in separated albumen and yolk, respectively (own results, data not shown). The cell volume was growth rate dependent as described in Kreft et al. (1998), and the minimal median volume was 0.410 15 l. Simulations were carried out using a time step of 1 min for simulating growth and an inner loop of about 0.1 s, depending on the actual swimming speed, was used for simulating the bacterial swimming. The results presented are averages of at least 100 repetitions if not mentioned otherwise. 3. Results 3.1. Time evolution An example of the time evolution of the bacterial distribution in the egg is shown in Fig. 1. To allow a 2-D visualisation of the 3-D cell locations, the cell positions are projected on the upper x–y plane, while preserving the distance to the egg centre. The initial contamination was one cell, located near the shell surface at a distance of 8.8 mm from the vitelline membrane. 327 The figure illustrates the simultaneous occurrence of bacterial motility and growth. Once the bacteria enter the yolk, the total bacterial number rises rapidly. An alternative view of the same phenomenon can be seen in the growth curve shown in Fig. 2. 3.2. Sensitivity analysis Several model parameters (chemotaxis, growth rate, initial number of cells and linear swimming speed) were varied individually within a certain range and the resulting model output was recorded. The specific growth rate and linear swimming speed were varied between 50% and +50% of their default values. For the latter, this is between the reported values of 10 and 30 Am/s for E. coli (Lowe et al., 1987). Growth rates in the albumen can vary considerably (Messens et al., 2004), but the range studied here is sufficiently wide to capture many potential cases. An increasing concentration of a chemoattractant reduces the tumbling probability from the unstimulated steady state value, causing the cells to swim longer in the same direction. Increasing the tumbling probability parameter above the unstimulated value would imply that the yolk acts as a repellent, which is implausible. Therefore, as a measure of chemotactic response towards an assumed yolk derived attractant, we varied the reduction of the tumbling frequency from the unstimulated value, in the range from 0% reduction (no chemotactic response) to 50% reduction. Finally, the initial number of cells was varied between 1 and 500. For all simulations, the initial contamination locus was the same as in Fig. 1. Fig. 1. S. enteritidis migration and growth starting from one initial cell. Indicated are the simulation time (t) in hours, and the total number of cells (n). 328 K. Grijspeerdt et al. / International Journal of Food Microbiology 100 (2005) 323–333 Fig. 2. Number of cells as function of time in the albumen, yolk and entire egg. Fig. 3 summarizes the sensitivity of the model response towards these parameters. Within the parameter ranges studied, it is clear that the presence of chemotaxis has the most profound impact on the time to reach the yolk. The initial number of cells is also very important at lower cell numbers, but for higher numbers, the effect levels off. Growth rate and bacterial swimming speed appear to be somewhat less influential. Fig. 3. Sensitivity analysis of the model responses towards parameter changes. The parameters varied are indicated on the y-axis together with the baseline values. The range between which the parameters were varied is shown in the scale below each bar. Chemotaxis and tumbling probability: a positive chemotactic response to an attractant leaking from the yolk would consists in reducing the tumbling probability (thereby increasing the duration of straight runs) from the steady-state probability of 0.1 in the absence of stimulation (temporal change in attractant concentration). A faster change of attractant concentration or a stronger response would cause a greater reduction of the tumbling probability. K. Grijspeerdt et al. / International Journal of Food Microbiology 100 (2005) 323–333 329 Fig. 4. Influence of initial contamination location, starting with one cell, on the time to reach the vitelline membrane. The linear trend is indicated. Fig. 5. Simultaneous influence of initial contamination distance from the vitelline membrane and tumbling probability reduction. 330 K. Grijspeerdt et al. / International Journal of Food Microbiology 100 (2005) 323–333 3.3. Influence of initial contamination location Fig. 4 shows the influence of initial contamination location on the time to reach the vitelline membrane. The variation is very large as would be expected from random walk movement, but there is a slightly positive correlation between the initial distance from the vitelline membrane and the required migration time. Addition of a chemotactic response towards the yolk diminishes the importance of the contamination position. This is illustrated by Fig. 5, where the simultaneous effect of initial distance and chemotaxis is shown. The data was smoothed with a first-order negative exponential filter to bring out the overall trend. 4. Discussion The newly developed IbM delivers interesting simulation results. Comparison of model simulations with experimental data is currently impossible, because the available data is very diverse, with time to reach the yolk ranging from less than a day to several weeks. This diversity points at the difficulty of in situ measurements in eggs, and the importance of using correct and standardised procedures. More efforts should be put in obtaining repeatable, reliable results. However, the simulations allow identifying some trends. In the absence of growth, the average time to traverse 8.8 mm of albumen by a cell (6 days) would probably be too long for the cell to survive in the hostile albumen environment. However, as soon as some growth and/or a limited chemotaxis towards the yolk are present, the required average migration time drops dramatically. There is a limited fraction of cases in which the vitelline membrane is reached faster, even in the absence of growth. In 1% of the replicate simulations, the time to reach the vitelline membrane is 12 h or less. These are the worst-case scenarios for a risk analysis. Whenever the bacteria have penetrated the yolk, the total number of cells increases rapidly (Fig. 2), justifying the use of the yolk membrane breakdown concept. However, it should be noted that the number of bacteria in the albumen should not be neglected in the framework of a risk assessment. Baker et al. (1998) mentioned an infectious dose as low as 10 cells when dealing with susceptible persons. The time to reach the vitelline membrane is significantly influenced by the initial contamination level when this is smaller than 100 cells. Cogan et al. (2001) illustrated that at such small inoculation levels, growth in albumen also occurs less frequently than when using larger inocula. This indicates that cells will suffer considerable damage by the defence mechanisms of the albumen and only a few cells will survive the initial phase after contamination. Note that for the results presented in Fig. 3 all cells were assumed to be in the same area, initially. When locations would be spread out more randomly, the average migration time would be lower. The contamination location plays a role, albeit less pronounced than chemotaxis (Fig. 3). There is no general consensus on the most important locus of contamination of eggs in the case of vertical transmission, both contaminations in the yolk and in the albumen have been reported (Gast et al., 2002). There is also evidence that a significant number of contaminations take place at the outside of the vitelline membrane (Gast and Holt, 2001a). When horizontal contamination occurs, it is obvious that the contamination of the egg internals starts in the albumen. The simulations with reduced tumbling switching probability pointed at the potential importance of chemotactic responses. Lock et al. (1992) found that bacteria migrate preferentially towards the yolk. In their set-up, the egg was broken first, so the question remains if the vitelline membrane was kept totally intact. A more rigorous implementation of chemotaxis would require more information on what is leaking out of the yolk, in what quantities, at what rate and where exactly. Information on the diffusion of leaked compounds into the albumen would also be required. The simulation of the diffusion of these compounds would carry a high computational cost but is straightforward (Kreft et al., 2001). Leaked components from the yolk could also enhance growth in the albumen (Humphrey and Whitehead, 1993), which in turn would lead to higher levels of cells in the albumen and a shorter time to reach the yolk. Swimming speed and growth rate appeared less influential on the model simulations within the range studied. Recently, Cogan et al. (2004) presented experimental evidence that motility is an important K. Grijspeerdt et al. / International Journal of Food Microbiology 100 (2005) 323–333 factor for growth of Salmonella enterica serovars in eggs because it allows the bacteria to reach the yolk and subsequently grow to high levels. These findings illustrate the importance of including bacterial motility for modelling bacterial behaviour in hens’ eggs. A series of assumptions were made in the model due to shortcomings in the current knowledge on bacterial migration in eggs. A major assumption is the homogeneity of the albumen. Albumen is a hostile environment to bacteria due to several reasons, but there is little knowledge about the distribution and timevarying nature of these factors in the albumen. The same applies to the possibility of nutrient leaking from the yolk. There is a lot of speculation on this subject, but no direct evidence is available. The common belief is that yolk leakage is absent in fresh eggs, and would only occur after about 3 weeks of storage as a result of alterations of the vitelline membrane. This conclusion was based on observations that S. enteritidis did not grow in the albumen during the first 3 weeks (Humphrey and Whitehead, 1993). Another explanation points at the ageing effect on the inhibitory properties of the albumen, although several authors have reported growth of S. enteritidis in fresh albumen (Schoeni et al., 1995; Baron et al., 1997; Cogan et al., 2001). Messens et al. (2004) did not detect an enhanced growth of Salmonella in albumen of older eggs, either separated from the yolk or in situ. Since bacterial levels in the simulations are far below asymptotic growth conditions, the assumption of exponential bacterial growth is justified. No lag phase was assumed for the simulations presented in this paper. IbMs are particularly suited to incorporate a stochastic lag phase concept as described by Baranyi (2002). Given the large uncertainties associated with stochastic lag, this was not yet added to the model. However, the trends that emerge from the simulation result can be expected to remain valid after inclusion of a lag phase. Since the egg was assumed to be isothermal, thermotaxis was not considered although swimming bacteria are likely to be susceptible to temperature gradients (Shi et al., 1993; Eisenbach, 2001). Thermotaxis could be an important factor when considering horizontal transmission just after lay. The warm egg cools off causing an underpressure in the egg, possibly sucking inwards bacteria present on the shell (Bruce and Drysdale, 1994). When the bacteria then swim 331 preferably towards the warmer parts of the cooling egg, i.e. the centre, the migration time could be significantly reduced. Furthermore, temperature gradients will also lead to convective flow (Denys et al., 2003), which superimposes on bacterial swimming. Possible interactions between the bacteria are not included in the current model. Cogan et al. (2001) have observed that S. enteritidis cells benefit from the presence of other cells and put forward two possible reasons. First, the combined enterochelin activity of a Salmonella population might outcompete ovotransferrin for iron. In this context, West and Buckling (2003) described a synergistic effect when communities of related bacteria produce siderophores, relating in higher growth rates. Second, the death of some cells in the albumen can allow others to utilise them as an energy source or a source of iron (cryptic growth). The presence of both mechanisms would lead to spreading patterns where the cells would be more clustered than in our simulations. An IbM would allow implementing interactions much more easily than continuous models. However, no evidence for the occurrence of these mechanisms in eggs could be found in the literature. The same applies for interaction between different microbial cultures. The results show that IbM is a promising technique to model the issue of bacterial migration in eggs. The main drawback of the methodology is the computational requirements. On a current desktop computer (Athlon XP 2000+, 1.5 GB RAM) it takes several days to reach a bacterial load of about 1.4106 cells/ egg (of which 104 cells are in the albumen) starting from one cell. Bacterial growth in eggs is characterised by a large variability. To allow convenient comparison of results, Clay and Board (1992) introduced the concept of generalised growth. Generalised growth is said to occur when the bacterial level reaches 104 cfu/ml (equivalent to about 35104 cfu/ albumen) or 106 cfu/ml (about 50106 cfu/egg) when inoculating separated albumen or whole eggs, respectively. Levels close to generalised growth can already be reached by IbM simulations. Extending simulations to higher bacterial numbers would not contribute much to the issue of food safety. Another approach when higher cell densities need to be simulated could be the combination of IbM and continuous methods. Once the IbM-simulated cell densities reach a certain threshold level, a continuous model could take over 332 K. Grijspeerdt et al. / International Journal of Food Microbiology 100 (2005) 323–333 using the final IbM output as initial state. A similar concept has been applied by McKellar (2001) when developing a continuous-discrete-continuous dynamic model to describe the lag phase of individual cells. Future developments of the model will include a refinement of the bacterial growth model with the inclusion of a lag phase and death kinetics and the introduction of more stochastic elements in both the bacterial growth and swimming models as well as in the egg model (membrane breakdown times, yolk location). 5. Conclusions IbM can be applied for modelling bacterial migration in eggs. Because of the low initial numbers of contamination, it is definitely better suited than continuous diffusion-type models to deal with this problem, and offers a level of flexibility and detail that is difficult to obtain by continuous methods. The model can already indicate some trends, but additional information on bacterial migration in eggs will have to be collected in order to further refine the model. Acknowledgements The authors are grateful for financial support from the European Community, project QLK5CT-2001-01606. 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