How to average microbial densities to characterize risk

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Wat. Res. Vol. 30, No. 4, pp. 1036-1038, 1996 Copyright © 1996 ElsevierScience Ltd Printed in Great Britain. All rights reserved 0043-1354/96 $15.00 + 0.00

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TECHNICAL NOTE HOW TO AVERAGE MICROBIAL DENSITIES TO CHARACTERIZE RISK C H A R L E S N. H A A S * LD Betz Professor of Environmental Engineering, Drexel University, Philadelphia, PA 19104, U.S.A. (First received 1 February 1995; accepted in revised form 1 October 1995)

Abstract~In the past, the geometric mean, or less frequently the arithmetic mean, has been used as a summary descriptor of microorganism densities in waters and other environmental media. It is shown in this paper that, if the interest is characterizing the risk from organisms that are described by either an exponential dose-response relationship, or a dose-response relationship that is linear at low doses, the appropriate summary descriptor is the arithmetic mean. Therefore, in future work, authors should be encouraged to report data on such organisms using this summary descriptor. Key words--microorganisms, statistics, risk, dose-response, infectivity

there is an associated risk, or probability, of an adverse occurrence---generally infection, denoted by p~. If the risk from individual exposures is an independent event, then the overall risk of infection from the cumulative exposure may be written as

INTRODUCTION The risk from oral ingestion of pathogenic organisms in drinking water, recreational water, food, or other vehicles, can be characterized by a dose-response relationship (Haas, 1983a,b; Gerba and Haas, 1988; Regli et al., 1991; Rose et al., 1991; Haas et al., 1993; Rose el al., 1993; Rose and Haas, 1994). The exponential or beta-Poisson models have been found to characterize all known pathogen oral dose-response relationships in humans. To assess risk in an actual system, where the microbial density may vary over time (e.g. day to day), it may be desirable to summarize density information. In the past, the arithmetic or geometric mean has been used for this purpose. However, a precise justification of the proper averaging procedure in the context of risk assessment has never, to the authors knowledge, appeared, although the averaging issue in more simplified contexts has appeared (Thomas, 1952). The purpose of this paper is to present a justification for the use of the arithmetic mean density as the proper summary characterization of risk from multiple exposures to a microbially contaminated medium.

Pz = l - I I (1 - p i ) ,

where p, is given as a function of the dose-response relationship, using the medium density existing during that particular exposure, m~. The average risk,/3, is defined as the per exposure risk that would, if it existed during each exposure, produce the same overall risk. This yields the following relationship pz = 1 - (1 - P)".

(2)

This can be inverted to yield /3 =

I

- (1 - pz)t'".

(3)

And thus, by combining (1) and (3), the average risk can be obtained as a function of the individual risk by

The average microbial density (rh) is defined as the concentration producing a single exposure risk/3. In other words, the same overall risk would result from exposure to the average microbial concentration as from the ensemble of microbial concentrations actually observed.

NOMENCLATURE The density (in number of organisms per unit volume) in a medium (drinking water, recreational water, food) on a given exposure (day, swimming exposure, meal) is given as mi where "i" denotes the exposure index. A series of "n" samples for each exposure (or a random sample of exposures) is taken. For each single exposure, *Author to whom all correspondence should be addressed.

(1)

i=l

DOSE-RESPONSE RELATIONSHIPS There are two major dose-response relationships that have been found to describe pathogen infectivity towards humans (Haas, 1983a,b). In the case of the exponential model, there is one parameter describing sensitivity, k, which represents the number of

1036

Technical Note microorganisms that must, on average, be ingested such that at least one survives to initiate an infection. The resulting equation is given by (5)

where V is the volume of medium ingested per exposure (or per day of exposure). The beta-Poisson model is derived from the exponential model with a heterogenous microbial survival probability distribution (Furumoto and Mickey, 1967a,b; Haas, 1983a,b), and may be written in terms of two dose-response parameters that characterize the interaction. The parameter a describes the degree of heterogeneity; as this increases, the exponential model is approached. The resulting equation is -

1037 AVERAGING

IN THE EXPONENTIAL

CASE

When the exponential dose-response model is obeyed, as is the case for human exposure to G i a r d i a and C r y p t o s p o r i d i u m , equation (5) can be substituted into equation (4) to define the appropriate average as follows 1 - exp(- -~)

= 1 - I~0 e x p ( - ~ ) 1

~'.

(8)

This can be simplified to yield the following rh = n

m,

(9)

i=1

And therefore, it is shown that the arithmetic mean of microbial densities is the appropriate metric to summarize results of a series of measurements for the purpose of developing a risk estimate.

(6) AVERAGING

One significant property of this relationship which has not been previously mentioned is that, at low doses (and risks) a linear relationship approximately characterizes the dose-response process. The linearized beta-Poisson model can be written as ~V p ~ ~-m.

IN THE GENERAL

LOW

DOSE

CASE

When the exponential model is not followed, derivation of the appropriate averaging scheme must proceed along a different line. First, from equation (4), if each of the individual risks is small ( < 0.01), then the product can be approximated by expanding the polynomial and neglecting second and higher order terms as follows

(7)

In general, the single exposure risk due to microbial exposures is quite low, and hence the approximation of equation 7 to (6) is quite good. This is shown in Fig. 1, from which it is concluded that the linearized beta-Poisson model is generally adequate to characterize the dose-response relationship under conditions of regulatory interest. It is noted, however, that the full beta-Poisson model (equation 6) must generally be used to model the dose-response relationship in actual experimental situations, since the infection proportion is most frequently in excess of 0.1.

k = 1 -

1 -

.

(10)

i

If the summation itself is also small, which will also generally be the case, then the fractional root can be approximated as well to yield ~b=l-[

1 - 1-n ,'~,P']

(11)

From this it is clear that ~b = n

P'"

(12)

i=l

0.1

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

i

.

. . ' " "'i ""

'

.'.

,;.'.,

AVERAGING

'-'-

0.o01

.-'..........................;~,,

S% i

Given equation (12), for the low-dose linearized beta poisson model, substitution of equation (7) in equation (12) clearly yields the following relationship

.........................

0.0001

................................................................................................

10-5

. . . . . . . .

0.01

{

0.1

,

-

Or.

BETA-POISSON

MODEL

O.Ol .....................................................#...................................~.....4,,~,...;i........................................ {~

FOR THE LINEARIZED

......

,

1

. . . . . . . .

10

Fig. 1. Maximum risk levels for specified relative errors in linearized approximation to beta-Poisson model.

-

--

-

n

= I

m,.

(13)

Therefore, as in the case of the exponential model, for the low dose linearized beta-Poisson model, the arithmetic average microorganism density is the proper summary statistic used to characterize risk.

1038

Technical Note DISCUSSION

From the above analysis, it is clear that the arithmetic average microorganism density should be used for a risk-based characterization of the microbial (pathogen) quality of an environmental medium. It should be recognized that the actual distribution of microorganisms in environmental samples may be highly skewed (Pipes et al., 1977; E1-Shaarawi et al., 1981; Haas and Heller, 1986), and therefore the proper (and precise) estimation of microorganism average density in environmental samples (and placing confidence limits on the average density) may require special methods. Further research, and importantly development of actual data sets, would be required to study this problem. However, 'inasmuch as it has become common for geometric means to be used for summary description of microbial prevalence, future works should be cautioned that the use of this statistic, while perhaps appropriate for merely describing central tendency, is inappropriate for characterizing risk. Parenthetically, it is noted however, that given the geometric mean and geometric standard deviation, the arithmetic average of a system can be determined, if the distributional form (e.g. log-normal) is known (EI-Shaarawi, 1989). It should also be noted that the development leading to equation (13) is quite general. For any dose response relationship that has linear low dose behavior, at low levels of risk (for each individual exposure, and for the overall exposure), the arithmetic mean concentration provides the appropriate summary statistic for characterizing risk. CONCLUSIONS It has been shown that the most appropriate summary descriptor for use in microbial risk assessment is the arithmetic mean, rather than the

geometric mean. Further work is required to determine the most efficient way to estimate this property from environmental data.

REFERENCES

EI-Shaarawi A. H. (1989) Inferences about the mean from censored water quality data. Wat. Resources Res. 25(4), 685-690. E1-Shaarawi A. H. et al. (1981) Bacterial density in water determined by Poisson or negative binomial distributions. Appl. environ. Microbiol. 41, 107. Furumoto W. A. and Mickey R. (1967a) A mathematical model for the infectivity~lilution curve of tobacco mosaic virus: experimental tests. Virology 32, 224. Furumoto, W. A. and R. Mickey (1967b) A mathematical model for the infectivity-dilution curve of tobacco mosaic virus: theoretical considerations. Virology 32, 216. Gerba C. P. and Haas C. N. (1988) Assessment of risks associated with enteric viruses in contaminated drinking water. A S T M Special Tech. Publ. 976, 489-494. Haas C. N. (1983a) Effect of effluent disinfection on risks of viral disease transmission via recreational exposure. J. Wat. Pollut. Control Fed. 55, 1111-1116. Haas C. N. (1983b) Estimation of risk due to low doses of microorganisms: a comparison of alternative methodologies. Am. J. Epidemiol. 118(4), 573-582. Haas C. N. and HeUer B. (1986) Statistics of enumerating total coliforms in water samples by membrane filter procedures. Wat. Res. 20, 525-530. Haas C. N. et al. (1993) Risk assessmentof virus in drinking water. Risk Analysis 13(5), 545-552. Pipes W. O. et al. (1977) Frequency distributions for coliform bacteria in water. J. Am. Wat. Wks Assoc. 69, 664. Regli S. et al. (1991) Modeling risk for pathogens in drinking water. J. Am. War. Wks Assoc. 83(11), 76-84. Rose J. B. and Haas C. N. (1994) Assessing and Controlling Risks o f Waterborne Protozoa, Ann. Conf. Am. War. Wks Assoc., New York. Rose J. B. et al. (1993) Waterborne pathogens: assessing health risks. Hlth Environ. Digest 7(3), 1-2. Rose J. B. et al. (1991) Risk assessment and the control of waterborne Giardiasis. Am. J. Publ. HIth 81, 709-713.

Thomas H. A. Jr (1952) On averaging results of coliform tests. Boston Soc. Civil Engrs J. 39, 253-270.