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Detection probability of Campylobacter
Food Control 21 (2010) 247–252
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Food Control journal homepage: www.elsevier.com/locate/foodcont
Detection probability of Campylobacter E.G. Evers a,*, J. Post b, F.F. Putirulan b, F.J. van der Wal b a b
National Institute for Public Health and the Environment, P.O. Box 1, 3720 BA Bilthoven, The Netherlands Central Veterinary Institute of Wageningen UR, P.O. Box 65, 8200 AB Lelystad, The Netherlands
a r t i c l e
i n f o
Article history: Received 12 December 2008 Received in revised form 27 May 2009 Accepted 2 June 2009
Keywords: Detection probability Food processing Pathogen concentration Scheduling
a b s t r a c t A rapid presence/absence test for Campylobacter in chicken faeces is being evaluated to support the scheduling of highly contaminated broiler flocks as a measure to reduce public health risks [Nauta, M. J., & Havelaar, A. H. (2008). Risk-based standards for Campylobacter in the broiler meat chain. Food Control, 19, 372–381]. Although the presence/absence test is still under development, an example data set of test results is analysed to illustrate the benefit of the detection probability concept. The detection probability of Campylobacter increases with the logarithm of the Campylobacter concentration in faeces according to an S-shaped curve which stretches about 2–3 log units. The detection probability is 50% at a Campylobacter concentration of 7.4 106 cfu/g. The uncertainty in the detection probability is 32% at the most for a 90% confidence interval. This type of information allows for realistic calculations on the Campylobacter status of different food processing paths after splitting. Usable quantitative estimates on detection probability await a data set of test results from a test that is ready for use or has similar properties. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction In many cases, analysis of the level of microbiological contamination of food, faeces, water, etc., only consists of testing samples for absence or presence of the species of interest whereas concentrations are not measured. This can be inevitable, due to the low concentrations present, or concentrations are not considered relevant by the researcher or the legislator. In such presence/absence tests, detection of a micro-organism is related to its concentration in the investigated matrix. This relation is usually described with the established concept of the detection limit. The detection limit stands for a number or concentration below which the micro-organism will not be detected and above which it will always or in a certain part of the tests be detected, as set by the sensitivity of the test. The concept of detection limit is found to be applied using either of two possible approaches. For the first approach, no experimental results are needed. It is simply stated that the detection limit is equal to one micro-organism in the tested sample size, which is then usually converted into a concentration. The second approach is to determine it experimentally, e.g. by preparing a 1:10 dilution series of a sample with known concentration in duplicate and determining presence/absence. The dilution with the lowest concentration in which the micro-organism is detected is considered as the detection limit. * Corresponding author. E-mail address: [email protected] (E.G. Evers). 0956-7135/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.foodcont.2009.06.004
In our view, the concept of detection limit as described above is theoretically unrealistic. We cannot think of a realistic theoretical concept that results in a concentration–detection relation in which the micro-organism will never be detected below a certain concentration and will always or in a certain percentage of cases be detected above this concentration. We think it is more realistic to consider this type of experiments in terms of a probability process. It is assumed that there is a probability of detection which increases with increasing concentration or number of microorganisms. This probability is equal to 0 and 1 if the concentration or number of micro-organisms is equal to 0 and infinity, respectively. The objective of this research is to present a simple model that describes the relation between the concentration of a micro-organism in a matrix and the probability of detection in a sample. This relation uses the sample size and the probability of detection of one micro-organism in a sample as parameters. Elaboration of the detection probability concept will lead to a better understanding of the processes that play a role during presence/absence measurements. The resulting concentration–detection relation also has important practical usefulness. It shows microbiologists that there is considerable uncertainty associated with the conclusions that can be drawn on concentrations when based only on presence/absence measurements, and it gives a quantification of this uncertainty. More specific examples are the possibility to calculate the number of presence/absence measurements necessary to detect a specified minimum concentration of a micro-organism, or the removal of the underestimation that currently exists in MPN (Most
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Probable Number) tables. The detection probability concept can also be useful in the field of risk assessment calculations. This is especially the case when the public health effect of measures, in which food processing paths are scheduled based on the results of presence/absence measurements, is to be calculated (Nauta and Havelaar, 2008). The measurements of Campylobacter in chicken faeces that were used as an example to illustrate the method were collected to support exactly this type of calculations. As this presence/absence method is still under development, the specific quantitative results presented here cannot be used for the implementation of measures.
We calculate the detection probability D by calculating 1 minus the probability of no detection. The probability of no detection (1D) equals the probability of absence of micro-organisms in the sample (1P) added to the sum of the probabilities of no detection for each possible number of cfu in the sample. Slight rearrangement results in the formula for the detection probability D:
2. Materials and methods
where z is a given number of micro-organisms in the sample. Eq. (4) is equal to
D¼P ¼1
1 P
z¼0
D ¼ 1 eka
2.1.1. Presence/absence test In order to determine presence or absence of a micro-organism in a matrix, a sample is taken from this matrix. The concentration of the micro-organism in the matrix is equal to k (which may be 0). We assume that the micro-organism is randomly distributed in the matrix where the samples are taken from. This implies that the number of cfu in a sample is Poisson distributed with mean ka, with
ka ¼ ka
ð1Þ
where a is the size of the sample in grams. For detection to occur, one or more cfu of the micro-organism has to be present in the sample. The probability that this occurs, the presence probability P, equals:
P ¼ 1 Poið0jka Þ
ð2Þ
where Poi is the Poisson distribution. Given that a micro-organism is present in the sample, its presence has to be detected. We assume that every single cfu of a certain micro-organism in the sample has the same probability of being detected given a certain experimental set-up. This probability is the parameter d. It will vary between types of micro-organisms and experimental set-ups. When a certain number of cfu n is present in the sample, the probability Dn that the micro-organism will be detected is equal to 1 minus the probability that none of the n cfu’s will be detected, using a binomial distribution (Bin):
Dn ¼ 1 Binð0jn; dÞ
ð3Þ
The parameter d and the probability Dn are somewhat theoretical entities. The detection probability D of a micro-organism in a sample in relation to the concentration k in the matrix is of more practical use. The Poisson and Binomial variability indicated above (Eqs. (2) and (3)) have to be incorporated in the estimation of D.
Table 1 Symbols used in the detection probability model. Symbol
Definition
k k0 ka kc a c P
Concentration of a micro-organism in the matrix in cfu/g Original undiluted value of k Mean number of a micro-organism in the absence/presence sample in cfu Mean number of a micro-organism in the plate count sample in cfu Absence/presence sample size in gram Plate count sample size in gram The presence probability of one or more cfu of a micro-organism in the sample The detection probability of one cfu of a micro-organism in the sample The detection probability of n cfu of a micro-organism in the sample The detection probability of a micro-organism in the sample in relation to k
d Dn D
ð4Þ
Poiðzjka Þ Binð0jz; dÞ
2.1. Model For the used symbols, we refer to Table 1.
Poiðzjka Þ Binð0jz; dÞ
z¼1 1 P
1 X ½ka ð1 dÞz z! z¼0
ð5Þ
and using that for a Poisson distribution with mean ka(1d) it holds that 1 X ½ka ð1 dÞz eka ð1dÞ ¼1 z! z¼0
ð6Þ
Eq. (5) can be simplified to
D ¼ 1 edka
ð7Þ
This shows an analogy with the dose–response working field as this formula is identical to the formula for the exponential dose– response relation (see e.g. Teunis & Havelaar, 2000). 2.1.2. Plate counts The concentration k of a micro-organism is determined with plate counts. We assume that the number of colonies on a plate is Poisson distributed with mean kc, with
kc ¼ kc
ð8Þ
where c is the size of the sample in grams. 2.2. Experimental set-up 2.2.1. Sample preparation Chicken faeces was collected from cages and was checked for the absence of Campylobacter using a selective enrichment broth as described in the ISO procedure 10272 (Anonymous, 2006a). Campylobacter-negative chicken faeces was stored frozen in aliquots. A Campylobacter-negative chicken faeces suspension of 0.167 g/ml was prepared by adding 0.2 g faeces to 1 ml borate buffer (100 mM, pH 8.8), and used for spiking with Campylobacter. In order to do this, a Campylobacter jejuni strain was grown overnight at 37 °C under microaerobic conditions in HI (Heart Infusion broth; Becton Dickinson) in an orbital shaker. The culture was diluted 1/ 100 into 50 ml HI and cultured for 5 h. The bacteria were pelleted (30 min, 4120g) and resuspended to a concentration of 2 1010 cfu/ml in HI, assuming that an optical density of 0.1 at 600 nm corresponds with 2 109 cfu/ml. The actual number of viable bacteria was determined afterwards (see below). The resulting Campylobacter suspension was used to spike chicken faeces to 1.3 1010 cfu/g faeces (0.15 g faeces/ml), which was taken as H, the highest concentration tested. With the spiked faeces a dilution series was prepared by repeated dilution into Campylobacter-negative chicken faeces (1 in 3.16), resulting in a series of spiked faeces samples ranging from H to 0.00032H. The spiked faeces samples were split in two and used for the presence/absence test or for enumeration with plate counts.
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2.2.2. Presence/absence test A prototype Campylobacter on site test was used to detect Campylobacter. The spiked faeces samples were prepared for the test by removal of large particles using a syringe filter, and subsequent boiling for half an hour. The test is based on the principle of lateral flow and consisted of a nitrocelluloses strip of 5 mm wide (Prima 40, Whatman) with a piece of filter paper attached to the top as absorbent pad. On the strip 0.4 lg polyclonal antibody was applied with a non-contact quantitative dispenser (ZX1000 dispense platform with BioJet Quanti 3000 dispensers; Biodot Ltd., Chichester, UK). The polyclonal antibody solution was prepared by protein-G purification of serum obtained from rabbits that had been immunised with Campylobacter antigens. Tests were run by first letting 100 ll sample flow into the strip from the bottom, followed by 100 ll buffer. For detection of bound antigens 100 ll of carbonconjugated polyclonal antibody (Van Amerongen et al., 1993) was used for flow through, followed again by 100 ll buffer. The tests were performed in tenfold and the results were visually scored positive or negative. Previously, sensitivity and specificity of this test in terms of detection of strains and species was investigated (Van der Wal, unpublished). It was found that 59 of 60 tested strains of C. jejuni were detected and 20 of 20 strains of Campylobacter coli. Further, Clostridia, Lactobacillus and a number of other species of bacteria that occur in chicken intestines and litter gave no signal in the presence/absence test. 2.2.3. Plate counts Enumeration was performed in triplicate according to the direct plate count method described in the ISO procedure 10272 (Anonymous, 2006b).
with
kc;j ¼ ej bj k0 c
ð10Þ
and
ka;j ¼ bj k0 a
ð11Þ
where i stands for the replicate number of plate counts and j for the dilution, J = 8 and I = 3 in the present experiment (see Table 2), xij stands for the measured number of cfu on a plate, kc,j (Eq. (10)) is the mean number of cfu on plate for dilution j, yj is the measured number of positives by the presence/absence test, r = 10 is the number of replicates of the presence/absence test, Dj is the detection probability D in the presence/absence test for dilution j where the mean number in the sample is ka,j (Eq. (11)) and is given by Eq. (4). bj is the dilution factor. This e.g. equals 0.32 for the concentration ‘0.32H’ (see Tables 2 and 3). ej is the dilution that had to be made to arrive at a countable number on the plate (see Table 2). We chose the following uninformed priors p1 and p2 for k0 and d, respectively:
1 k0 p2 ðdÞ ¼ 1
p1 ðk0 Þ ¼
ð12Þ ð13Þ
The posterior distribution f is proportional to the product of the likelihood function L and the prior distributions p1 and p2:
f ðd; k0 jxij ; yj Þ / p1 ðkÞp2 ðdÞLðxij ; yj jd; k0 Þ
ð14Þ
ð9Þ
Point estimates for k0 and d were obtained from a simultaneous fit of Eq. (14) to the data, giving the highest value of f (see Tables 2 and 3). For the main part, the Poisson distribution had to be approximated by the Normal distribution, due to the high numbers of cfu involved. In order to obtain posterior uncertainty distributions for d and k0, first the ranges of d and k0 for which the value of f is not neglectable were determined by pilot numerical work. Then, marginal distributions of the simultaneous posterior distribution of p and k0 were determined. The relevant ranges for d and k0 were found to be (2 106 2 105) and (1.1 109 1.4 109 cfu/g). For the posterior distribution of k0, we took for each of a series of 101 values for k0 (1.1, 1.103, 1.106. . . 1.4 109 cfu/g), 1000 random samples for d from a Uniform (2 106, 2 105) distribution. The 1000 posterior values were summed for each value of k0. Analogously, for the posterior distribution of d, we took for each of a series of 101 values for d (2, 2.18, 2.36. . . 20 106), 1000 random samples for k0 from a Uniform (1.1 109, 1.4 109 cfu/g) distribution. The 1000 posterior values were summed for each value of d. In the relation between the concentration k and the detection probability D the uncertainty in the detection probability d has to be taken into account. To incorporate this uncertainty, we used
Table 2 Observed (replicates 1–3) and fitted (kc,j) plate counts of Campylobacter. ‘H’ stands for the highest concentration tested. ‘Dilution’ stands for the dilution factor ej that was applied to arrive at a countable number on the plate.
Table 3 Experimental and fitting results for the fast test. The second column gives the experimental results, where ten replicates were assessed per dilution. The positive and negative control were observed to be positive and negative, respectively. The third and fourth column gives fitted estimates for the detection probability Dj and the no. of cfu in a sample analysed in the fast test (ka,j).
Concentration
Dilution
Replicate 1
Replicate 2
Replicate 3
kc,j
Concentration
No. of positives/total no.
Dj
ka,j
H 0.32 H 0.1 H 0.032 H 0.01 H 0.0032 H 0.001 H 0.00032 H
106 106 105 105 104 104 104 103
31 45 99 42 160 35 21 130
25 28 83 36 130 32 37 230
37 51 57 51 75 32 31 270
183 59 183 59 183 59 18 59
H 0.32H 0.1H 0.032H 0.01H 0.0032H 0.001H 0.00032H
1 1 1 1 0.7 0.3 0 0
1.00 1.00 1.00 0.975 0.683 0.308 0.109 0.0361
1.83 107 5.86 106 1.83 106 5.86 105 1.83 105 5.86 104 1.83 104 5.86 103
2.3. Statistics For the calculations Microsoft Excel 2002 SP3 (Microsoft Corporation, 1985–2001) with @Risk 4.5.1 – Professional Edition Add-in (Palisade Corporation, 2002) was used. We use Bayesian statistics to fit the model to the data (see e.g. Vose, 2000). In order to calculate the posterior distribution f, we need prior distributions for k0 (the original undiluted concentration k in the matrix) and d, and a formula for the likelihood function L. The likelihood function L incorporates the observations from the presence/absence test and the plate counts (see Tables 2 and 3). The dilution error made in obtaining lower concentrations k from k0 is neglected. The likelihood function L, in general the product of probabilities of all observations as a function of parameter values, equals for the present experiments
Lðxij ; yj jd; k0 Þ ¼
J Y I Y
Poiðxij jkc;j ÞBinðyj jr; Dj Þ
j¼1 i¼1
E.G. Evers et al. / Food Control 21 (2010) 247–252
a General distribution based on the 101 values of the marginal posterior distribution for d as described above, and on an assumed posterior value of 0 for d = 0 or 2.1 106. We took for each of a multiplicative series of 301 values for ka (1.50 103, 1.56 103, 1.62 103,. . .,1.50 108 cfu), 1000 random samples from the General distribution for d. These values were inserted into Eq. (4) to obtain the corresponding values for the detection probability D. For each value of ka, corresponding to values for k according to Eq. (1) 5% and 95% percentile values of D were determined. The best estimates for D for each of the 301 k values were calculated using the best fitting point estimate for d. The best estimates for Dj were also obtained by Eq. (4) using the best fitting point estimate for d and for k0 through Eq. (11).
0.25
0.2
Probability density
250
0.15
0.1
0.05
0 0
2
4
6
8
10
12
14
16
18
20
d, x10-6
3. Results
Fig. 2. Uncertainty distribution of d.
4. Discussion Faeces samples found positive by classical microbiological analysis were always found positive also by the presence/absence method (Van der Wal, unpublished). An important reason for the
20
Probability density
16
12
8
4
0 1.1
1.15
1.2
1.25
1.3
λ 0, x109 cfu/g Fig. 1. Uncertainty distribution of k0.
1.35
1.4
1
detection probability/fraction detection
The plate count and presence/absence test measurements are presented in Tables 2 and 3, respectively. The experimental setup implies that a = 0.015 g and c = 0.15 g. Fitting of the model to the measurements results in the best point estimate for the concentration k0 of 1.22 109 cfu/g. Linked to this (Eq. (10)) are estimates for the number of cfu of Campylobacter in the samples for plate counts kc,j, which are also given in Table 2. Comparing observed and estimated plate counts shows deviations, especially at H, 0.1H and 0.00032H. The best point estimate for d is 6.27 106. This results in values for Dj as presented in Table 3, which agree well with the observations. Table 3 also shows the no. of cfu of Campylobacter in a fast test sample, ka,j, for the different concentrations tested. Figs. 1 and 2 give the uncertainty distributions of k0 and d. It must be stressed that these are distributions of uncertainty and not of variability as it is assumed that k0 and d are constants. The value of k0 appears to be known quite precisely, given this dataset of 24 plate counts, and the model assumptions. The 90% interval is 1.19–1.25 109 cfu/g. The value of d is more uncertain; the 90% interval is 4.1–10.3 106. Fig. 3 shows the relationship between the concentration in the matrix k and the probability of detection D, including the uncertainty. The 5–95% interval in this figure is based on the uncertainty of the detection probability d. D = 0.5 for k = 7.4 106 = 106.9 cfu/g. The maximum width of the 5–95% interval for D is 0.32, at k = 9.6 106 = 107.0 cfu/g.
0.8
0.6
0.4
Detection probability D
0.2
Fraction detection 5% 95%
0 5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
λ, 10log cfu/g Fig. 3. The estimated probability of detection D and the observed fraction of detection, both as a function of the concentration in the matrix k (10log cfu/g).
presence/absence method being still under development is the large difference in method sensitivity between naturally and artificially contaminated chicken faeces, using plate counts as a reference. When Campylobacter harvested from a plate after 48 h of growth is added to Campylobacter-negative faeces, about 108 cfu/ g are required for detection, whereas this test can detect Campylobacter in naturally contaminated faeces that contains about 105 cfu/g. A possible explanation for this might be that naturally contaminated faeces contains a high concentration of dead bacteria of which the antigens are still available. This is supported by the fact that in artificially contaminated faeces the intensity of the signal increases as older cultures are used for spiking. An alternative explanation might be that Campylobacter expresses the proteins that are used as target for this test at a higher level in its natural habitat than under laboratory conditions. The experimental results show an increase in detection probability with concentration and not a step function. This contradicts the concept of a detection limit: no detection below a certain concentration, always or in a certain fraction of cases detection above this concentration. It agrees with the detection probability concept. As is to be expected, there is no perfect fit with the model. This familiar phenomenon in modeling can be ascribed to the model being simplified or wrong, or to an unknown source of variation that might turn out to be experimentally uncontrollable. Using such a variation source one can state that there is a theoretical detection limit that varies, resulting in the observations of this experiment. However, as stated in the introduction, there is no the-
E.G. Evers et al. / Food Control 21 (2010) 247–252
oretical basis for the concept of a detection limit and this is an important argument to use the theoretical concept of detection probability. As for many models, it will be difficult to prove experimentally the correctness of the detection probability concept in the broad sense and even more so of a specific detection probability model. The model uses two important assumptions that may prove to be wrong in all or a part of the cases. The first assumption is that of random distribution of micro-organisms in the batch where the sample is taken from. It might be that the micro-organism is under- or overdispersed in the batch. This can be described by variability in k, as was developed in Reinders, De Jonge, and Evers (2003), Reinders, De Jonge, and Evers (2004), but this was not done in the simple model used here. When counts and presence/absence are determined in the same sample, it is only the uncertainty of k that is not well estimated in case of under/overdispersion, but the estimate of d is not affected. When using different samples, the estimate of d will be affected as well. In the experimental set-up there is a 1:3.2 dilution step of the faeces. This has identical consequences, except that it is then not clear whether the over/underdispersion occurs in the original matrix, in the dilution step, or both. The second assumption is that the detection probability d is identical for every single cfu of a micro-organism in the sample given a certain experimental set-up. It might vary between individual cfu’s, but this will not be measurable. It might also vary between measurements, e.g. as done here to determine the relationship between concentration and detection probability. When this is uncontrollable variation in the test, e.g. in the molecular techniques used, it can be described by a variability distribution for d, which when fitted to the data gives an estimate of the size of this variability. It might also be that through further research the source of this variation becomes clear and this will lead to better understanding of the processes that occur in the presence/absence test, extension of the model and a better estimate for d. The usual approach in microbiological research to determine a detection limit is to make a 1:10 dilution series in singular. The highest concentration in which the micro-organism is detected is called the detection limit. Even for the practically inclined microbiologist, who is not interested in the theoretical model elaborated here, this paper presents important messages. To start with, a negative test result does not imply with great certainty that the concentration of the micro-organism is lower than the detection limit. Conversely, a positive test result does not imply that it is higher. Of course the probability that a very low concentration leads to a positive result is low. A maybe even more import message deals with the approach used to determine a detection limit. The exact location of the detection limit might be not that important in a part of the cases, so using a 1:3.2 dilution series might be not so relevant. However, a singular dilution series really gives too uncertain results to use the resulting detection limit again and again in interpreting measurements. One of the dilutions might be located near the 50% detection probability point and this then could lead to an error of a factor 10 in the detection limit. It is therefore advised to measure these dilution series in multiplicate. Measurement in quadruplicate is sufficient because then the probability to be around the 50% point without noticing this is reasonably small (0.54). Presence/absence tests are also used for setting standards in food safety regulations. For instance in the Dutch Warenwetbesluit ‘Bereiding en behandeling van levensmiddelen’ (preparation and handling of food), it is stated that ‘pathogenic micro-organisms must be absent in food in amounts that can be harmful for public health’ which is implemented for Salmonella and Campylobacter by demanding that it is not detectable in 25 g or ml. This approach uses the detection limit concept as a starting point, where a con-
251
centration lower than 0.04 cfu/g (1 in 25 g) is considered as a negligible risk for public health. Apart from that the reasoning behind this value is not clear, this paper shows that presence/absence measurements have limitations in drawing conclusions on concentrations, as these measurements are to be considered as the result of a probability process. It is very well possible that a concentration higher than 0.04 cfu/g leads to a negative test result (or a lower concentration to a positive result). Further, these standards are used for enforcement. This creates an unfair situation in the concentration range around the 50% detection probability point, as e.g. different producers of meat with pathogens in the same concentration are facing a random process of detection and therewith a warning or fine or otherwise. When a maximum acceptable concentration of a micro-organism, say km, has been determined, the detection probability concept can be used to define the set-up (i.e., the number of replicates m) of presence/absence measurements to be used for routine surveillance, thus replacing the laborious measurements of concentrations. For example, in this way standards for food safety as discussed in the previous section can be improved. To achieve this, the relation between detection probability D and concentration k must be experimentally determined in a very precise way. Then the model (Eqs. (4) and (8)) is fitted to the experimental data. Finally, we estimate the required number of replicates m of negative presence/absence results such that the probability that the concentration in the matrix is higher than km is smaller than a certain number, e.g. 0.05. This is the solution to the equation:
ð1 Dm Þm 6 0:05 where Dm is the detection probability at concentration km. Using m replicates will then give a reliable routine surveillance. When the concentration of a micro-organism in a matrix is too low to be determined by direct counts, or when there is too much interfering competitive flora, researchers often turn to the MPN (most probable number) method to still obtain estimates for the concentration. According to this method, absence/presence is determined in a number of replicates for a number of dilutions. With MPN tables (see e.g. De Man, 1983) the observations can be translated into a concentration. These tables are based on a simple model for the Binomially distributed number of positives where being positive is Poisson distributed. It is assumed that every single cfu of a micro-organism has a 100% probability of being detected. This is not realistic. The value of d is for the immunological test in this article shown to be 6.27 106. In general, also when incubation to allow for growth is part of the experimental procedure, it will be smaller than 1. This implies that the values in MPN tables are generally an underestimation of the real concentration. MPN estimates can be improved by estimating d experimentally (which is labour-intensive and micro-organism specific) and introducing the parameter d in the MPN model. Acknowledgements We want to thank Maarten Nauta (National Institute for Public Health and the Environment) for his contribution to the modeling work and Aart van Amerongen and Jan Wichers (Institute of the Agrotechnology and Food Sciences Group, Wageningen University and Research Centre) for advice on lateral flow immuno-assays. References Anonymous (2006a). ISO 10272–1. Microbiology of food and animal feeding stuffs – Horizontal method for detection and enumeration of Campylobacter spp. – Part 1: Detection method. Geneve, Switzerland: International Organisation for Standardisation. Anonymous (2006b). ISO 10272–2. Microbiology of food and animal feeding stuffs – Horizontal method for detection and enumeration of Campylobacter spp. – Part 2:
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Colony-count technique. Geneve, Switzerland: International Organisation for Standardisation. De Man, J. C. (1983). MPN tables, corrected. European Journal of Applied Biotechnology, 17, 301–305. Nauta, M. J., & Havelaar, A. H. (2008). Risk-based standards for Campylobacter in the broiler meat chain. Food Control, 19, 372–381. Reinders, R. D., De Jonge, R., & Evers, E. G. (2003). A statistical method to determine whether micro-organisms are randomly distributed in a food matrix, applied to coliforms and Escherichia coli O157 in minced beef. Food Microbiology, 20(3), 297–303. Reinders, R. D., De Jonge, R., & Evers, E. G. (2004). Corrigendum to: ‘‘A statistical method to determine whether micro-organisms are randomly distributed in a
food matrix, applied to coliforms and Escherichia coli O157 in minced beef” [Food Microbiol. 2003 20(3) 297–303]. Food Microbiology, 21(6), 819. Teunis, P. F. M., & Havelaar, A. H. (2000). The beta poisson dose–response model is not a single-hit model. Risk Analysis, 20(4), 513–520. Van Amerongen, A., Wichers, J. H., Berendsen, L. B., Timmermans, A. J., Keizer, G. D., Van Doorn, A. W., et al. (1993). Colloidal carbon particles as a new label for rapid immunochemical test methods: Quantitative computer image analysis of results. Journal of Biotechnology, 30(2), 185–195. Vose, D. (2000). Risk analysis, a quantitative guide (2nd ed.). Chichester, England: John Wiley & Sons, Ltd.
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Detection probability of Campylobacter E.G. Evers a,*, J. Post b, F.F. Putirulan b, F.J. van der Wal b a b
National Institute for Public Health and the Environment, P.O. Box 1, 3720 BA Bilthoven, The Netherlands Central Veterinary Institute of Wageningen UR, P.O. Box 65, 8200 AB Lelystad, The Netherlands
a r t i c l e
i n f o
Article history: Received 12 December 2008 Received in revised form 27 May 2009 Accepted 2 June 2009
Keywords: Detection probability Food processing Pathogen concentration Scheduling
a b s t r a c t A rapid presence/absence test for Campylobacter in chicken faeces is being evaluated to support the scheduling of highly contaminated broiler flocks as a measure to reduce public health risks [Nauta, M. J., & Havelaar, A. H. (2008). Risk-based standards for Campylobacter in the broiler meat chain. Food Control, 19, 372–381]. Although the presence/absence test is still under development, an example data set of test results is analysed to illustrate the benefit of the detection probability concept. The detection probability of Campylobacter increases with the logarithm of the Campylobacter concentration in faeces according to an S-shaped curve which stretches about 2–3 log units. The detection probability is 50% at a Campylobacter concentration of 7.4 106 cfu/g. The uncertainty in the detection probability is 32% at the most for a 90% confidence interval. This type of information allows for realistic calculations on the Campylobacter status of different food processing paths after splitting. Usable quantitative estimates on detection probability await a data set of test results from a test that is ready for use or has similar properties. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction In many cases, analysis of the level of microbiological contamination of food, faeces, water, etc., only consists of testing samples for absence or presence of the species of interest whereas concentrations are not measured. This can be inevitable, due to the low concentrations present, or concentrations are not considered relevant by the researcher or the legislator. In such presence/absence tests, detection of a micro-organism is related to its concentration in the investigated matrix. This relation is usually described with the established concept of the detection limit. The detection limit stands for a number or concentration below which the micro-organism will not be detected and above which it will always or in a certain part of the tests be detected, as set by the sensitivity of the test. The concept of detection limit is found to be applied using either of two possible approaches. For the first approach, no experimental results are needed. It is simply stated that the detection limit is equal to one micro-organism in the tested sample size, which is then usually converted into a concentration. The second approach is to determine it experimentally, e.g. by preparing a 1:10 dilution series of a sample with known concentration in duplicate and determining presence/absence. The dilution with the lowest concentration in which the micro-organism is detected is considered as the detection limit. * Corresponding author. E-mail address: [email protected] (E.G. Evers). 0956-7135/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.foodcont.2009.06.004
In our view, the concept of detection limit as described above is theoretically unrealistic. We cannot think of a realistic theoretical concept that results in a concentration–detection relation in which the micro-organism will never be detected below a certain concentration and will always or in a certain percentage of cases be detected above this concentration. We think it is more realistic to consider this type of experiments in terms of a probability process. It is assumed that there is a probability of detection which increases with increasing concentration or number of microorganisms. This probability is equal to 0 and 1 if the concentration or number of micro-organisms is equal to 0 and infinity, respectively. The objective of this research is to present a simple model that describes the relation between the concentration of a micro-organism in a matrix and the probability of detection in a sample. This relation uses the sample size and the probability of detection of one micro-organism in a sample as parameters. Elaboration of the detection probability concept will lead to a better understanding of the processes that play a role during presence/absence measurements. The resulting concentration–detection relation also has important practical usefulness. It shows microbiologists that there is considerable uncertainty associated with the conclusions that can be drawn on concentrations when based only on presence/absence measurements, and it gives a quantification of this uncertainty. More specific examples are the possibility to calculate the number of presence/absence measurements necessary to detect a specified minimum concentration of a micro-organism, or the removal of the underestimation that currently exists in MPN (Most
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Probable Number) tables. The detection probability concept can also be useful in the field of risk assessment calculations. This is especially the case when the public health effect of measures, in which food processing paths are scheduled based on the results of presence/absence measurements, is to be calculated (Nauta and Havelaar, 2008). The measurements of Campylobacter in chicken faeces that were used as an example to illustrate the method were collected to support exactly this type of calculations. As this presence/absence method is still under development, the specific quantitative results presented here cannot be used for the implementation of measures.
We calculate the detection probability D by calculating 1 minus the probability of no detection. The probability of no detection (1D) equals the probability of absence of micro-organisms in the sample (1P) added to the sum of the probabilities of no detection for each possible number of cfu in the sample. Slight rearrangement results in the formula for the detection probability D:
2. Materials and methods
where z is a given number of micro-organisms in the sample. Eq. (4) is equal to
D¼P ¼1
1 P
z¼0
D ¼ 1 eka
2.1.1. Presence/absence test In order to determine presence or absence of a micro-organism in a matrix, a sample is taken from this matrix. The concentration of the micro-organism in the matrix is equal to k (which may be 0). We assume that the micro-organism is randomly distributed in the matrix where the samples are taken from. This implies that the number of cfu in a sample is Poisson distributed with mean ka, with
ka ¼ ka
ð1Þ
where a is the size of the sample in grams. For detection to occur, one or more cfu of the micro-organism has to be present in the sample. The probability that this occurs, the presence probability P, equals:
P ¼ 1 Poið0jka Þ
ð2Þ
where Poi is the Poisson distribution. Given that a micro-organism is present in the sample, its presence has to be detected. We assume that every single cfu of a certain micro-organism in the sample has the same probability of being detected given a certain experimental set-up. This probability is the parameter d. It will vary between types of micro-organisms and experimental set-ups. When a certain number of cfu n is present in the sample, the probability Dn that the micro-organism will be detected is equal to 1 minus the probability that none of the n cfu’s will be detected, using a binomial distribution (Bin):
Dn ¼ 1 Binð0jn; dÞ
ð3Þ
The parameter d and the probability Dn are somewhat theoretical entities. The detection probability D of a micro-organism in a sample in relation to the concentration k in the matrix is of more practical use. The Poisson and Binomial variability indicated above (Eqs. (2) and (3)) have to be incorporated in the estimation of D.
Table 1 Symbols used in the detection probability model. Symbol
Definition
k k0 ka kc a c P
Concentration of a micro-organism in the matrix in cfu/g Original undiluted value of k Mean number of a micro-organism in the absence/presence sample in cfu Mean number of a micro-organism in the plate count sample in cfu Absence/presence sample size in gram Plate count sample size in gram The presence probability of one or more cfu of a micro-organism in the sample The detection probability of one cfu of a micro-organism in the sample The detection probability of n cfu of a micro-organism in the sample The detection probability of a micro-organism in the sample in relation to k
d Dn D
ð4Þ
Poiðzjka Þ Binð0jz; dÞ
2.1. Model For the used symbols, we refer to Table 1.
Poiðzjka Þ Binð0jz; dÞ
z¼1 1 P
1 X ½ka ð1 dÞz z! z¼0
ð5Þ
and using that for a Poisson distribution with mean ka(1d) it holds that 1 X ½ka ð1 dÞz eka ð1dÞ ¼1 z! z¼0
ð6Þ
Eq. (5) can be simplified to
D ¼ 1 edka
ð7Þ
This shows an analogy with the dose–response working field as this formula is identical to the formula for the exponential dose– response relation (see e.g. Teunis & Havelaar, 2000). 2.1.2. Plate counts The concentration k of a micro-organism is determined with plate counts. We assume that the number of colonies on a plate is Poisson distributed with mean kc, with
kc ¼ kc
ð8Þ
where c is the size of the sample in grams. 2.2. Experimental set-up 2.2.1. Sample preparation Chicken faeces was collected from cages and was checked for the absence of Campylobacter using a selective enrichment broth as described in the ISO procedure 10272 (Anonymous, 2006a). Campylobacter-negative chicken faeces was stored frozen in aliquots. A Campylobacter-negative chicken faeces suspension of 0.167 g/ml was prepared by adding 0.2 g faeces to 1 ml borate buffer (100 mM, pH 8.8), and used for spiking with Campylobacter. In order to do this, a Campylobacter jejuni strain was grown overnight at 37 °C under microaerobic conditions in HI (Heart Infusion broth; Becton Dickinson) in an orbital shaker. The culture was diluted 1/ 100 into 50 ml HI and cultured for 5 h. The bacteria were pelleted (30 min, 4120g) and resuspended to a concentration of 2 1010 cfu/ml in HI, assuming that an optical density of 0.1 at 600 nm corresponds with 2 109 cfu/ml. The actual number of viable bacteria was determined afterwards (see below). The resulting Campylobacter suspension was used to spike chicken faeces to 1.3 1010 cfu/g faeces (0.15 g faeces/ml), which was taken as H, the highest concentration tested. With the spiked faeces a dilution series was prepared by repeated dilution into Campylobacter-negative chicken faeces (1 in 3.16), resulting in a series of spiked faeces samples ranging from H to 0.00032H. The spiked faeces samples were split in two and used for the presence/absence test or for enumeration with plate counts.
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2.2.2. Presence/absence test A prototype Campylobacter on site test was used to detect Campylobacter. The spiked faeces samples were prepared for the test by removal of large particles using a syringe filter, and subsequent boiling for half an hour. The test is based on the principle of lateral flow and consisted of a nitrocelluloses strip of 5 mm wide (Prima 40, Whatman) with a piece of filter paper attached to the top as absorbent pad. On the strip 0.4 lg polyclonal antibody was applied with a non-contact quantitative dispenser (ZX1000 dispense platform with BioJet Quanti 3000 dispensers; Biodot Ltd., Chichester, UK). The polyclonal antibody solution was prepared by protein-G purification of serum obtained from rabbits that had been immunised with Campylobacter antigens. Tests were run by first letting 100 ll sample flow into the strip from the bottom, followed by 100 ll buffer. For detection of bound antigens 100 ll of carbonconjugated polyclonal antibody (Van Amerongen et al., 1993) was used for flow through, followed again by 100 ll buffer. The tests were performed in tenfold and the results were visually scored positive or negative. Previously, sensitivity and specificity of this test in terms of detection of strains and species was investigated (Van der Wal, unpublished). It was found that 59 of 60 tested strains of C. jejuni were detected and 20 of 20 strains of Campylobacter coli. Further, Clostridia, Lactobacillus and a number of other species of bacteria that occur in chicken intestines and litter gave no signal in the presence/absence test. 2.2.3. Plate counts Enumeration was performed in triplicate according to the direct plate count method described in the ISO procedure 10272 (Anonymous, 2006b).
with
kc;j ¼ ej bj k0 c
ð10Þ
and
ka;j ¼ bj k0 a
ð11Þ
where i stands for the replicate number of plate counts and j for the dilution, J = 8 and I = 3 in the present experiment (see Table 2), xij stands for the measured number of cfu on a plate, kc,j (Eq. (10)) is the mean number of cfu on plate for dilution j, yj is the measured number of positives by the presence/absence test, r = 10 is the number of replicates of the presence/absence test, Dj is the detection probability D in the presence/absence test for dilution j where the mean number in the sample is ka,j (Eq. (11)) and is given by Eq. (4). bj is the dilution factor. This e.g. equals 0.32 for the concentration ‘0.32H’ (see Tables 2 and 3). ej is the dilution that had to be made to arrive at a countable number on the plate (see Table 2). We chose the following uninformed priors p1 and p2 for k0 and d, respectively:
1 k0 p2 ðdÞ ¼ 1
p1 ðk0 Þ ¼
ð12Þ ð13Þ
The posterior distribution f is proportional to the product of the likelihood function L and the prior distributions p1 and p2:
f ðd; k0 jxij ; yj Þ / p1 ðkÞp2 ðdÞLðxij ; yj jd; k0 Þ
ð14Þ
ð9Þ
Point estimates for k0 and d were obtained from a simultaneous fit of Eq. (14) to the data, giving the highest value of f (see Tables 2 and 3). For the main part, the Poisson distribution had to be approximated by the Normal distribution, due to the high numbers of cfu involved. In order to obtain posterior uncertainty distributions for d and k0, first the ranges of d and k0 for which the value of f is not neglectable were determined by pilot numerical work. Then, marginal distributions of the simultaneous posterior distribution of p and k0 were determined. The relevant ranges for d and k0 were found to be (2 106 2 105) and (1.1 109 1.4 109 cfu/g). For the posterior distribution of k0, we took for each of a series of 101 values for k0 (1.1, 1.103, 1.106. . . 1.4 109 cfu/g), 1000 random samples for d from a Uniform (2 106, 2 105) distribution. The 1000 posterior values were summed for each value of k0. Analogously, for the posterior distribution of d, we took for each of a series of 101 values for d (2, 2.18, 2.36. . . 20 106), 1000 random samples for k0 from a Uniform (1.1 109, 1.4 109 cfu/g) distribution. The 1000 posterior values were summed for each value of d. In the relation between the concentration k and the detection probability D the uncertainty in the detection probability d has to be taken into account. To incorporate this uncertainty, we used
Table 2 Observed (replicates 1–3) and fitted (kc,j) plate counts of Campylobacter. ‘H’ stands for the highest concentration tested. ‘Dilution’ stands for the dilution factor ej that was applied to arrive at a countable number on the plate.
Table 3 Experimental and fitting results for the fast test. The second column gives the experimental results, where ten replicates were assessed per dilution. The positive and negative control were observed to be positive and negative, respectively. The third and fourth column gives fitted estimates for the detection probability Dj and the no. of cfu in a sample analysed in the fast test (ka,j).
Concentration
Dilution
Replicate 1
Replicate 2
Replicate 3
kc,j
Concentration
No. of positives/total no.
Dj
ka,j
H 0.32 H 0.1 H 0.032 H 0.01 H 0.0032 H 0.001 H 0.00032 H
106 106 105 105 104 104 104 103
31 45 99 42 160 35 21 130
25 28 83 36 130 32 37 230
37 51 57 51 75 32 31 270
183 59 183 59 183 59 18 59
H 0.32H 0.1H 0.032H 0.01H 0.0032H 0.001H 0.00032H
1 1 1 1 0.7 0.3 0 0
1.00 1.00 1.00 0.975 0.683 0.308 0.109 0.0361
1.83 107 5.86 106 1.83 106 5.86 105 1.83 105 5.86 104 1.83 104 5.86 103
2.3. Statistics For the calculations Microsoft Excel 2002 SP3 (Microsoft Corporation, 1985–2001) with @Risk 4.5.1 – Professional Edition Add-in (Palisade Corporation, 2002) was used. We use Bayesian statistics to fit the model to the data (see e.g. Vose, 2000). In order to calculate the posterior distribution f, we need prior distributions for k0 (the original undiluted concentration k in the matrix) and d, and a formula for the likelihood function L. The likelihood function L incorporates the observations from the presence/absence test and the plate counts (see Tables 2 and 3). The dilution error made in obtaining lower concentrations k from k0 is neglected. The likelihood function L, in general the product of probabilities of all observations as a function of parameter values, equals for the present experiments
Lðxij ; yj jd; k0 Þ ¼
J Y I Y
Poiðxij jkc;j ÞBinðyj jr; Dj Þ
j¼1 i¼1
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a General distribution based on the 101 values of the marginal posterior distribution for d as described above, and on an assumed posterior value of 0 for d = 0 or 2.1 106. We took for each of a multiplicative series of 301 values for ka (1.50 103, 1.56 103, 1.62 103,. . .,1.50 108 cfu), 1000 random samples from the General distribution for d. These values were inserted into Eq. (4) to obtain the corresponding values for the detection probability D. For each value of ka, corresponding to values for k according to Eq. (1) 5% and 95% percentile values of D were determined. The best estimates for D for each of the 301 k values were calculated using the best fitting point estimate for d. The best estimates for Dj were also obtained by Eq. (4) using the best fitting point estimate for d and for k0 through Eq. (11).
0.25
0.2
Probability density
250
0.15
0.1
0.05
0 0
2
4
6
8
10
12
14
16
18
20
d, x10-6
3. Results
Fig. 2. Uncertainty distribution of d.
4. Discussion Faeces samples found positive by classical microbiological analysis were always found positive also by the presence/absence method (Van der Wal, unpublished). An important reason for the
20
Probability density
16
12
8
4
0 1.1
1.15
1.2
1.25
1.3
λ 0, x109 cfu/g Fig. 1. Uncertainty distribution of k0.
1.35
1.4
1
detection probability/fraction detection
The plate count and presence/absence test measurements are presented in Tables 2 and 3, respectively. The experimental setup implies that a = 0.015 g and c = 0.15 g. Fitting of the model to the measurements results in the best point estimate for the concentration k0 of 1.22 109 cfu/g. Linked to this (Eq. (10)) are estimates for the number of cfu of Campylobacter in the samples for plate counts kc,j, which are also given in Table 2. Comparing observed and estimated plate counts shows deviations, especially at H, 0.1H and 0.00032H. The best point estimate for d is 6.27 106. This results in values for Dj as presented in Table 3, which agree well with the observations. Table 3 also shows the no. of cfu of Campylobacter in a fast test sample, ka,j, for the different concentrations tested. Figs. 1 and 2 give the uncertainty distributions of k0 and d. It must be stressed that these are distributions of uncertainty and not of variability as it is assumed that k0 and d are constants. The value of k0 appears to be known quite precisely, given this dataset of 24 plate counts, and the model assumptions. The 90% interval is 1.19–1.25 109 cfu/g. The value of d is more uncertain; the 90% interval is 4.1–10.3 106. Fig. 3 shows the relationship between the concentration in the matrix k and the probability of detection D, including the uncertainty. The 5–95% interval in this figure is based on the uncertainty of the detection probability d. D = 0.5 for k = 7.4 106 = 106.9 cfu/g. The maximum width of the 5–95% interval for D is 0.32, at k = 9.6 106 = 107.0 cfu/g.
0.8
0.6
0.4
Detection probability D
0.2
Fraction detection 5% 95%
0 5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
λ, 10log cfu/g Fig. 3. The estimated probability of detection D and the observed fraction of detection, both as a function of the concentration in the matrix k (10log cfu/g).
presence/absence method being still under development is the large difference in method sensitivity between naturally and artificially contaminated chicken faeces, using plate counts as a reference. When Campylobacter harvested from a plate after 48 h of growth is added to Campylobacter-negative faeces, about 108 cfu/ g are required for detection, whereas this test can detect Campylobacter in naturally contaminated faeces that contains about 105 cfu/g. A possible explanation for this might be that naturally contaminated faeces contains a high concentration of dead bacteria of which the antigens are still available. This is supported by the fact that in artificially contaminated faeces the intensity of the signal increases as older cultures are used for spiking. An alternative explanation might be that Campylobacter expresses the proteins that are used as target for this test at a higher level in its natural habitat than under laboratory conditions. The experimental results show an increase in detection probability with concentration and not a step function. This contradicts the concept of a detection limit: no detection below a certain concentration, always or in a certain fraction of cases detection above this concentration. It agrees with the detection probability concept. As is to be expected, there is no perfect fit with the model. This familiar phenomenon in modeling can be ascribed to the model being simplified or wrong, or to an unknown source of variation that might turn out to be experimentally uncontrollable. Using such a variation source one can state that there is a theoretical detection limit that varies, resulting in the observations of this experiment. However, as stated in the introduction, there is no the-
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oretical basis for the concept of a detection limit and this is an important argument to use the theoretical concept of detection probability. As for many models, it will be difficult to prove experimentally the correctness of the detection probability concept in the broad sense and even more so of a specific detection probability model. The model uses two important assumptions that may prove to be wrong in all or a part of the cases. The first assumption is that of random distribution of micro-organisms in the batch where the sample is taken from. It might be that the micro-organism is under- or overdispersed in the batch. This can be described by variability in k, as was developed in Reinders, De Jonge, and Evers (2003), Reinders, De Jonge, and Evers (2004), but this was not done in the simple model used here. When counts and presence/absence are determined in the same sample, it is only the uncertainty of k that is not well estimated in case of under/overdispersion, but the estimate of d is not affected. When using different samples, the estimate of d will be affected as well. In the experimental set-up there is a 1:3.2 dilution step of the faeces. This has identical consequences, except that it is then not clear whether the over/underdispersion occurs in the original matrix, in the dilution step, or both. The second assumption is that the detection probability d is identical for every single cfu of a micro-organism in the sample given a certain experimental set-up. It might vary between individual cfu’s, but this will not be measurable. It might also vary between measurements, e.g. as done here to determine the relationship between concentration and detection probability. When this is uncontrollable variation in the test, e.g. in the molecular techniques used, it can be described by a variability distribution for d, which when fitted to the data gives an estimate of the size of this variability. It might also be that through further research the source of this variation becomes clear and this will lead to better understanding of the processes that occur in the presence/absence test, extension of the model and a better estimate for d. The usual approach in microbiological research to determine a detection limit is to make a 1:10 dilution series in singular. The highest concentration in which the micro-organism is detected is called the detection limit. Even for the practically inclined microbiologist, who is not interested in the theoretical model elaborated here, this paper presents important messages. To start with, a negative test result does not imply with great certainty that the concentration of the micro-organism is lower than the detection limit. Conversely, a positive test result does not imply that it is higher. Of course the probability that a very low concentration leads to a positive result is low. A maybe even more import message deals with the approach used to determine a detection limit. The exact location of the detection limit might be not that important in a part of the cases, so using a 1:3.2 dilution series might be not so relevant. However, a singular dilution series really gives too uncertain results to use the resulting detection limit again and again in interpreting measurements. One of the dilutions might be located near the 50% detection probability point and this then could lead to an error of a factor 10 in the detection limit. It is therefore advised to measure these dilution series in multiplicate. Measurement in quadruplicate is sufficient because then the probability to be around the 50% point without noticing this is reasonably small (0.54). Presence/absence tests are also used for setting standards in food safety regulations. For instance in the Dutch Warenwetbesluit ‘Bereiding en behandeling van levensmiddelen’ (preparation and handling of food), it is stated that ‘pathogenic micro-organisms must be absent in food in amounts that can be harmful for public health’ which is implemented for Salmonella and Campylobacter by demanding that it is not detectable in 25 g or ml. This approach uses the detection limit concept as a starting point, where a con-
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centration lower than 0.04 cfu/g (1 in 25 g) is considered as a negligible risk for public health. Apart from that the reasoning behind this value is not clear, this paper shows that presence/absence measurements have limitations in drawing conclusions on concentrations, as these measurements are to be considered as the result of a probability process. It is very well possible that a concentration higher than 0.04 cfu/g leads to a negative test result (or a lower concentration to a positive result). Further, these standards are used for enforcement. This creates an unfair situation in the concentration range around the 50% detection probability point, as e.g. different producers of meat with pathogens in the same concentration are facing a random process of detection and therewith a warning or fine or otherwise. When a maximum acceptable concentration of a micro-organism, say km, has been determined, the detection probability concept can be used to define the set-up (i.e., the number of replicates m) of presence/absence measurements to be used for routine surveillance, thus replacing the laborious measurements of concentrations. For example, in this way standards for food safety as discussed in the previous section can be improved. To achieve this, the relation between detection probability D and concentration k must be experimentally determined in a very precise way. Then the model (Eqs. (4) and (8)) is fitted to the experimental data. Finally, we estimate the required number of replicates m of negative presence/absence results such that the probability that the concentration in the matrix is higher than km is smaller than a certain number, e.g. 0.05. This is the solution to the equation:
ð1 Dm Þm 6 0:05 where Dm is the detection probability at concentration km. Using m replicates will then give a reliable routine surveillance. When the concentration of a micro-organism in a matrix is too low to be determined by direct counts, or when there is too much interfering competitive flora, researchers often turn to the MPN (most probable number) method to still obtain estimates for the concentration. According to this method, absence/presence is determined in a number of replicates for a number of dilutions. With MPN tables (see e.g. De Man, 1983) the observations can be translated into a concentration. These tables are based on a simple model for the Binomially distributed number of positives where being positive is Poisson distributed. It is assumed that every single cfu of a micro-organism has a 100% probability of being detected. This is not realistic. The value of d is for the immunological test in this article shown to be 6.27 106. In general, also when incubation to allow for growth is part of the experimental procedure, it will be smaller than 1. This implies that the values in MPN tables are generally an underestimation of the real concentration. MPN estimates can be improved by estimating d experimentally (which is labour-intensive and micro-organism specific) and introducing the parameter d in the MPN model. Acknowledgements We want to thank Maarten Nauta (National Institute for Public Health and the Environment) for his contribution to the modeling work and Aart van Amerongen and Jan Wichers (Institute of the Agrotechnology and Food Sciences Group, Wageningen University and Research Centre) for advice on lateral flow immuno-assays. References Anonymous (2006a). ISO 10272–1. Microbiology of food and animal feeding stuffs – Horizontal method for detection and enumeration of Campylobacter spp. – Part 1: Detection method. Geneve, Switzerland: International Organisation for Standardisation. Anonymous (2006b). ISO 10272–2. Microbiology of food and animal feeding stuffs – Horizontal method for detection and enumeration of Campylobacter spp. – Part 2:
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