# Statistics of asbestos fibre mass determination by microscopy

INTRODUCTIONWhen total mass of asbestos fibres in a sample has to be determined by microscopy, a choice has to be made among several rules. Counting rules: If a fibre is not completely contained in the field of view, assign the statistical weight, r i , for any fibre overlapping the field of view: Rule R(12): r i =number of endpoints within field, divided by two. Rule R(1): r i =0 if lower endpoint in field, 1 if upper endpoint in field. Rule R(field): ratio of fibre length within field to total length. The volume W i of an individual fibre is calculated as W i =γ L i B i 2 , where L is length and B projected width of fibre. γ is a correction factor for cross section shape. Once W i (i=1 N) have been determined the expectation value of fibre volume W can be calculated using simple average value, E(W). If log-normality is assumed, maximum likelihood estimate, MLE(W), or unbiased minimum variance estimate can be used. Total fibre volume is calculated as W N. A simulation study was performed to determine the performance of these rules. Variability in density is not considered.METHODSFibre length and width was determined by phase contrast optical microscopy for a large number of samples prepared from different asbestos fibre types. The weights were recorded for each fibre. The data were pooled according to fibre type. Fibre volume determination was simulated by Bootstrap resampling from the pool of fibres (Efron and Tibshirani, 1993). Sampling was continued until the sum of the R(1) weights was equal to the preset number N. This would correspond to a given number of fields investigated. Since some of the observed fibres get r i =0, more fibres will be included using counting rules R(12) or R(field). This will reduce the variance of volume estimates based on R(12) and R(field) relative to estimates based on R(1), for the same time spent at the microscope.For counting until a given amount of matrix (determined without error) has been evaluated, the coefficient of variation CV of the total fibre volume is (Schneider and Holst, 1995) where E(M) is the amount of matrix scanned, λ is the expected number of particles per unit amount of matrix. This equation can be used to determine the minimum number of fibres that have to be sized, provided E(W 2 )/E(W) 2 can be estimated.For lath shaped fibres depositing on a surface it can be shown (Schneider, 1993) that even if they deposit without preferred orientation on a surface, γ≠1. It can be shown that γ e s t = exp[-σ 2 B (1 – ρ B H )], where exp(σ B ) is the geometric standard deviation of the individual fibre widths. ρ B H is the correlation between ln(B i ) and ln(H i ) and H i is fibre height. ρ B H cannot be determined from observing L and B, and it was assumed that ρ B H =0. If all fibres have square cross sections, ρ B H =1, and thus γ e s t =1.RESULTSW i were not log-normally distributed. Thus MLE and MLE c will give a biased estimate of the true total fibre volume. This was confirmed by the Bootstrap resampling. Thus only E(W) will be further considered. Fig. 1 shows the simulations of E(W) divided by the true volume for counting rule R(1). The relative standard deviation, RSD, was calculated for all three counting rules, for given N. The rule R(field) produced equal or smaller RSD than both R(12) and R(1) for given number of fields evaluated Thus, if cost is the determining factor the, the counting rule R(field) should be used. E(W 2 )/E(W) 2 is the key parameter determining CV. This is the theoretical equivalent to the common experience that a few large fibres determine the total fibre weight. Unfortunately, to obtain a reliable estimate N 100, as seen from Fig. 2.For amphiboles, γ e s t was on average 0.66. For cross sections measured by TEM the average was γ=0.68. For tremolite cleavage fragments, optical microscopy gave γ ranging from 0.3 to 0.5.While the lower limit of detection, LLD, for number of fibres in a matrix can be estimated using the Poisson distribution, there is no simple way to estimate LLD for fibre mass.Assume that the target performance of the analytical method is to determine the asbestos content by mass to within a factor of two of the true value, with 90% confidence. The simulations show that the minimum number of fibres that have to be counted, would range from about 15 to 100, for the range of sample types studied. Any experimental uncertainty would increase this number.CONCLUSIONSThe simulations have shown that estimators assuming a log normal distribution cannot be used and that the statistically determined correction factor for fibre cross section compares well with measurements of cross sections. The counting rule R(field) is superior to R(12) and R(1). No simple method for determining lower limit of detection of fibre volume can be given, unlike the case for determination of fibre number.

**First Author: ** Schneider T.

**Other Authors: ** Sethi, S.A., Davies, L.S.T., Burdett, G., Jørgensen, O.

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