Maximum Cumulative Ratio

Price and Han (2011) propose the Maximum Cumulative Ratio (\(\operatorname{MCR}\)) which is defined as the ratio of the cumulative exposure received by an individual on an intake day to the largest exposure received from a single substance:

\[\operatorname{MCR} = {\text{Cumulative exposure}\over\text{Maximum exposure}}\]

This \(\operatorname{MCR}\) statistic is also picked up as a practical device in a recent JRC report, Bopp et al. (2015), to investigate cumulative exposure. If \(\operatorname{MCR}\) is large, it is important to consider cumulative effects. If \(\operatorname{MCR}\) is close to 1, the individual exposure will not be much different from a single-substance assessment. The \(\operatorname{MCR}\) can therefore be interpreted as the degree to which the risk of being exposed is underestimated by not performing a cumulative risk assessment.

The \(\operatorname{MCR}\) statistic is implemented in MCRA for both the acute risk and the chronic risk cases. In the acute risk case the short-term (single-day) exposures are used. For the chronic case long-term individual exposures (estimated by aggregating over the available survey days of each individual) are used.

Table 112 shows an artificial example how the \(\operatorname{MCR}\) is calculated in the acute risk case. First the cumulative exposure per day is calculated by cumulating the exposure of each substance multiplied by the relative potency factors (\(\operatorname{RPF}\)). Then, for each day, the cumulative exposure (in equivalents of the reference substance) is divided by the maximum exposure of a single substance on that day. The last column shows the \(\operatorname{MCR}\) values, with the substance with the highest exposure in parenthesis. The \(\operatorname{MCR}\) has a value of 1 or close to 1 for mixtures where the exposure is dominated by one substance (e.g. day 1, substance B). When all substances have approximately equal exposure (e.g. day 3) the \(\operatorname{MCR}\) value is equal or close to the number of substances, here 4. Day 2 represents an intermediate case. The \(\operatorname{MCR}\) suggest that for exposure days (or persons) with \(\operatorname{MCR}\) values close to 1, the need for a cumulative risk assessment is low.

Table 112 Maximum Cumulative Ratios

	Substance A	Substance B	Substance C	Substance D	total exposure	ratio
day 1	0.01	0.99	0	0	1	1.01 (B)
day 2	0.1	0.2	0.3	0.4	1	2.50 (D)
day 3	0.25	0.25	0.24	0.26	1	3.99 (D)

In the example, all days have equal values for total exposure. For real data, total exposure will vary. It is obviously of interest to know if the \(\operatorname{MCR}\) is high or low at those days (or individuals) where the total exposure is highest.

In Figure 51, French steatosis data (39 substances, 4079 persons) are used to calculate the chronic exposure matrix. For each individual the \(\operatorname{MCR}\) is calculated and plotted against the total exposure. The different colours are used to identify the single substances with maximum exposure. From the original 39 substances, 10 different substances have the largest exposures. For the total exposure and \(\operatorname{MCR}\), the \(p_{5}\), \(p_{50}\) and \(p_{95}\) percentiles are indicated with the black line segments. The red line indicates the ratio with value 5. The dashed green lines indicate the \(p_{95}\) percentiles for the \(\operatorname{MCR}\) value for different ranges of the total exposure.

Figure 51 Maximum Cumulative Ratios vs total exposure

The plot shows that \(\operatorname{MCR}\) values with Imazalil as risk driving substance (purple) are predominantly found in the lower part of the plot for relatively high values of the total exposure. A second finding is that \(\operatorname{MCR}\) values decline when total exposure increases. This implies that cumulative exposure for most individuals is driven by multiple substances. At the right site of the plot, individuals are found with high exposure. Because \(\operatorname{MCR}\) values tend to be lower here, higher exposures are received from one predominant substance and not because many substances are above the average level. For those individuals a cumulative risk assessment has less value.

Because Figure 51 can be very dense, in Figure 52, 95% confidence regions representing bivariate lognormal distributions of \(\operatorname{MCR}\) and total exposure are plotted. The latter figure facilitates interpretation of the first figure. Note that substances with just one or two observations cannot be plotted in this display (substances with 2 observations are represented by a line).

Figure 52 Bivariate distributions \(\operatorname{MCR}\) vs total exposure

In Figure 53 and Figure 54 scattered \(\operatorname{MCR}\) distributions for the total and upper tail (here 37%) that drive the cumulative exposure are shown. The red line indicates the \(\operatorname{MCR}\) threshold, 1.5. The black lines represent the regression lines \(\operatorname{MCR}\) vs ln(Cumulative exposure) for each tail. Substances with an exposure contribution less than 15% are not displayed.

Figure 53 Using \(\operatorname{MCR}\) to identify substances that drive cumulative exposures, scatter distributions (total).

Figure 54 Using \(\operatorname{MCR}\) to identify substances that drive cumulative exposures, scatter distributions (upper tail 37%).

In Table 113 contributions to tail exposures at various percentile are shown. Column \(\operatorname{MCR} = 1\) shows the percentage of tail exposure due to individual(day)s with a single substance. Column \(1 < \operatorname{MCR} ≤ 2\) shows the percentage of tail exposure due to individual(day)s with multiple substances, but the \(\operatorname{MCR} ≤\) 2. Column \(\operatorname{MCR} > 2\) shows the percentage of tail exposure due to individual(day)s with multiple substances with \(\operatorname{MCR} > 2\).

Table 113 Maximum Cumulative Ratio summary

Tail %	% with MCR = 1	Substances	% with 1 < MCR<=2	Substances	% with MCR > 2	Substances
37	20.6	Difeno, Tebu	73.7	Difeno, Tebu	5.7	Difeno, Tebu
50	19.2	Difeno, Tebu	75.6	Difeno, Tebu	5.2	Difeno, Tebu
90	16.3	Difeno, Tebu	78.8	Difeno, Tebu	5.0	Difeno, Tebu
95	15.0	Difeno, Tebu	82.5	Difeno, Tebu	2.5	Difeno, Tebu
99	25.0	Difeno	75.0	Difeno, Tebu Propi	0.0

For \(\operatorname{MCR}\) settings, see exposure mixture settings.