Model probability of detecting a pathogen in food using the Poisson Distribution.

Despite the global use of HACCP systems and a legal requirement for the use of HACCP in many jurisdictions' food poisoning remains an endemic problem and large numbers of people continue to be hospitalised, die and as a result companies either face substantial legal costs and / or in many cases are forced to cease trading.

While the use of HACCP systems significantly reduces the need for microbiological end point testing of foods, sampling schemes and microbial analysis have important roles in system validation and quality assurance.

This raises an issue concerning the adequacy of sampling schemes and microbial analysis in commercial food manufacture.

In September 2015 the US Centers for Disease Control and Prevention reported on a multistate outbreak of listeriosis allegedly caused by Mediterranean-type soft cheeses. Some 30 people were affected, twenty-eight people were hospitalized and three deaths were reported. However, listeria were not isolated from the cheeses produced by the manufacturer concerned.

The rejection of infant milk powders contaminated with *Cronobacter sakazakii * by Chinese regulatory authorities has clearly shown significant limitations in "approved" sampling schemes operated by major companies in both Europe and the US.

In sampling methods we ideally want to obtain a high level of certainty that we can detect pathogens and other organisms. Normally we want to work with a 95% level of confidence.

So, how reliable are sampling schemes for detecting microorganisms in foods and how many samples are required to give meaningful results when validating a new process?

Let's say we want to calculate the probability of detecting a pathogen or fault causing microorganism present at a concentration of 2 organisms per kilogram in a food and we assume that the organism is uniformly distributed- frequently not the case -to make the mathematics easy and that 10 random samples of 25g are taken for analysis. Note, the use of 10 random samples of 25 grams to test a complete production batch of some foods is not uncommon.

To calculate the probability we first need to calculate the total number of potential sample lots, this is obtained by dividing 1000 (grams in a kilogram) by 25 (weight of sample). This gives 40.

We know that there are organisms in 2 sample lots.

So there is no pathogen in 38 samples, or in 38/40. We can calculate the probability of there being no pathogen in the product by finding the power of this fraction using the number of samples as the power (38/40)^{10 }. This gives a probability of 0.599 or 60% of not finding the organism. The probability of a pathogen being present is 1-0.599 or about 40%.

### This low probability of detection is some way from the desirable 95% confidence level that is frequently used when working with pathogens and spoilage organisms!

The above example is indicative of a pathogen being distributed homogeneously and is close to an ideal situation. A more practical way of describing bacterial populations in foods is to mathematically describe how they are distributed. A number of distribution models are used including the binomial, Poisson and log-normal distributions.

The Poisson Distribution is particular useful in working with homogeneous populations. With this distribution, the probability of finding at least one sample positive for a pathogen can be described as p=1-e^^{ -(c.n.s ) }, where p is the probability (%), e is a mathematical constant and is the base of natural logarithms with a value of 2.7183 to 4 decimals, c is the concentration of bacteria per gram or mL, n is the number of samples and s is the sample weight or volume. in addition to the concentration of bacteria, in situations where bacteria are distributed homogenously, it is the total mass of sample (n x s) rather than the number of samples that influences the probability. With other distributions the number of samples, in addition to sample masses or volumes, is important.

I have programmed a calculator using the Poisson Distribution to explore the relationship between the number of microorganisms actually present in a food, the number of samples taken and the weight or volume of the samples on the probability of detection. For example, by keeping the number of samples, and the sample weight constant the effect of contaminant numbers on the probability of detection can be investigated, try 10 CFU/kg. By doing this, it is apparent that the sampling regime required to give a high degree of confidence in detection is dependent on the total mass or volume of sample. As the total mass or volume of samples increase it becomes easier to detect pathogens and fault-causing microorganisms with higher levels of confidence. It should be apparent in this type of situation you could take fewer samples of larger weight or volume and obtain the same level of assurance. However, ideal situations like this are much less common than many people realise.

In this case we have also assumed that we can detect the organism being investigated. Modern PCR techniques can give almost 100% detection in many cases, however, this is not true for traditional microbiological plating methods. So the odds of detection will be even lower!

Contaminants are frequently not distributed uniformly in food. The spatial distribution of bacteria in food is being increasingly studied and is not well understood. Fractionation and mixing effects in particular can markedly affect the distribution of microorganisms. The detachment, or sloughing, of cells from biofilms on e.g. pipelines, heat exchangers, stainless steel surfaces in general can generate significant clusters of bacteria periodically during production processes. Some internationally approved sampling systems were not designed to detect these periodic microbial 'spikes' and modification of sampling systems are required to allow for these probabilities. I will add further content dealing with sampling in relation to the more typical heterogeneous distribution of pathogens and fault causing organisms in foods.

## Model probability of detecting a pathogen in food.

Recommended references on sampling

Hoelzer, K. and Pouillot, R. (2013).Practical considerations for the interpretation of microbial testing results based on small numbers of samples. Foodborne Pathogens and Disease. **10**, 907-915

Zwietering, M.H. and den Besten, H.M.W. (2016). Microbial testing in food safety: effect of specificity and sensitivity on sampling plans — how does the OC curve move. Current Opinion in Food Science. **12,** 42-51.

## Acknowledgements

The author acknowledges helpful discussions with Dr David Kilpatrick, Dr Colin Weatherup and Mr Tony Duffy. All views expressed are those of the author as are any errors in this article.

**How to cite this article**

Mullan, W.M.A. (2015).
[On-line]. Available from: __https://www.dairyscience.info/index.php/food-model/275-sampling.html .__ Accessed: 2 April, 2020.
Updated March 2019