Microbial QuantificationENVR 421Mark D. SobseySpringMicrobial Quantification• Determining microbial concentrations and loads specimens and samples: critical information for:– Ecology and natural history; monitoring & surveillance• Analysis of vehicles, vectors, reservoirs, etc.– Measuring exposures from environmental media– Quantifying human health risks of exposures– Estimating human infectivity and dose-response– For risk assessment and risk management• Survival, persistence and proliferation in environment• Effect of treatment processes; prevention and control• Determine if samples meet regulatory and other requirements for microbial quality• Based on fundamental statistical principles of measuring concentrations of discrete objects• Requires consideration of sources of non-homogeniety, variability and uncertaintyQuantification and Distributions of Microorganisms in Hosts and Environmental Media• Based on fundamental statistical principles of measuring concentrations of discrete objects• Requires consideration of sources of non-homogeniety, variability and uncertaintyMean or MedianRangeConcentrationCumulative Frequency Distribution of Number of Organisms in Unit Volume of SampleNo. Organismsµ = Meanσ = Standard DeviationProbability Density FunctionThe magnitude of variability of an estimated mean concentration (e.g., of microbes in a sample) depends on the number of observationsFrequency Distributions:Different ExamplesEstimating Microbe Concentrations from Quantal Data: Relationship Between Dilution and % Positive Sample VolumesDose or Dilution (log scale)% Pos.100050Dose-response relationships:Often based on few data pointsOften sigmoidalDifficult to estimate mid-pointDifficult to extrapolate to low dose or dilution63The microbe concentration in a unit volume of sample that gives a positive result 63% of the time when analyzed repeatedly corresponds to a mean concentration of 1 microbe per unit volume in the bulk sampleProbits and Their Application to Estimating 50% Infectious DoseAssume population response is "normally distributed", i.e., a Gaussian distributionUseful for “straightening” plots and facilitate extrapolation.Cumulative dose-responses as % are often sigmoidal and not linear.The most accurate data are near the mid-point, which is the average or mean or 50% point.How does one extrapolate to the extremes where there are no experimental data?• To avoid positive and negative values (and be consistent with the expression of cumulative frequency distributions and dose-response data) the mean value of a normal probability function is assigned a probit value of 5– consistent with the 50% response point in a cumulative frequency distribution or a dose-response relationshipRelationship Between Dilution and Percentage of Positive Sample VolumesDose or Dilution (log scale)Probits159Log-Normal Distribution% Responses expressed as probits:Can linearize the dose-responseFacilitates extrapolationProbits and Their Use for Log-Normally Distributed Data• Express ordinate (Y-axis) in multiples of the standard deviation, i.e., "normal equivalent deviates.”• Hence, probits are unit values assigned to standard deviations• 1 standard deviation = 1 probit unit• 2 standard deviations = 2 probits units• …etc.%ResponseNormalEquivalentDeviate(Std. Dev.)Probit99.9 +3 897.7 +2 784.0 +1 650 0 516 -1 42.3 -2 30.1 -3 2* ±±±±1 SD (16-84%) incl. 67% of pop.* ±±±±2 SD (2.3-97.7%) incl. 95% of pop.* ±±±±3 SD (0.1-99.9%) incl. 99.7% of pop.Dose-Response Relationship Based on % Response and Probits as a Function of Dose on a Log ScaleEstimating Microbial Concentrations from QuantalData: the Poisson Distribution and the Most Probable Number (MPN)• Poisson distribution describes “low probability” or rare events– such as, the probability that an inoculated culture of broth will or will not contain 1 or more bacteria that will grow.• If large numbers of replicate tubes are inoculated with large numbers of closely-spaced dilutions of a sample containing microbes, a sigmoidal dose-response curve for % positive tubes per dilution is likely to be generated.Estimating Microbial Concentrations from Quantal Data: Relationship Between Dilution, % of Positive Sample Volumes and The MPN Unit of ConcentrationDose or Dilution% Pos.100050• With increasing sample dilution, fewer and fewer culture tubes are positive• For the sample dilution that contains on average 1 microbe per volume inoculated into a culture tube, what %age of culture tubes would be positive?• According to the Poisson distribution, 63% would be positive• So, the “unit” of the Poisson distribution estimated by the Most Probable Number is the inoculumvolume that contains on average 1 organism and gives 63% positive cultures631 organismRandom (Poisson) Distributions of Organisms• Organisms are randomly distributed• P(x = N) = [(µV)N/N!]e(-µV)Where:N = number of organisms u = mean density of organisms “true concentration”V = volume of sampleP = probabilityPoisson Distribution Example• If the “true” mean number of microbes in a sample is 1 per ml, what is the probability that a given 1 ml sample will contain 0, 1, 2,…..n microbes per ml?• P(x = N) = [(µV)N/N!]e(- µV)• So, • P(0) = [(1x1)0/0!]e-1x1 = (1/1)e(-1) = 1e-1= 0.37• P(1) = [(1x1)1/1!]e-1x1 = (1/1)e-1 = 1e(-1)= 0.37• P(2) = [(1x1)2/2!]e-1x1 = (1/2)e-1 = 0.5e(-1)= 0.185Random (Poisson) Distributions of Organisms• The mean (µ) is equal to the variance (s2):µ = s2• As an approximation:x-bar = s2• and as an approximation the standard error, s, equals the square root of (x-bar/n):s = (x-bar/n)1/2where n = number of samples• If n = 1, this becomes the standard deviationS.D. = ±x1/2as an approximation, the 95% confidence interval is± 2(S.D.)Standard Error (of the Mean) of Poisson CountsAccording to Poisson distribution, the standard error of the mean (SE) for n number of assays is given by:(x/n)1/2The 95% confidence limit of the mean count for n number of assays in then approximated by:95% CL = x ± 2 (x-/n)1/2If the number of replicate assays, n, is small, then the 95% confidence limits should be adjusted by substituting the appropriate t-value for n-1 degrees of freedom in place of the number “2” in the equation aboveStandard Deviation of Poisson CountsStandard deviation (s):s = x1/2Example: if x = 64 (e.g., colony counts on a plate)s = 64 1/2= ±895% confidence