Uncertainty Evaluation of Human Risk Analysis (HRA) of Chemicals by Multiple Exposure Routes

2.1. Substances

The description of the industrial site and data is reported in the literature.⁽¹⁷⁾ The three contaminant substances with carcinogenic potency were considered, i.e., 1,1 dichloroethene, tetrachloroethene, and trichloroethene; they are indexed in the following by i= A, B, or C, respectively. Slope factor and parameters of statistical distribution of concentration values for each substance are reported in Table I.

2.2. Cancer Risk Estimation

A risk equation for a point estimation is assumed to be an expression of all independent variables, namely: concentration at the exposure point (C_ep); toxicological parameter (toxicity value, TV), i.e., slope factor (SF) for carcinogenic; and rate of exposure E.⁽²⁾

(1)

The risk evaluation in this article considers the direct ingestion of contaminated water without any other attenuation factor and the inhalation of vapors of the contaminant substances during a shower.

An application of the general expression of risk in Equation (1) for single substance ingestion by drinking water is:⁽²⁾

(2)

Where C_w is the substance concentration in water (mg/L), SF_o is the oral slope factor for ingestion of the substance (kg d/mg), IR_w is the water ingestion daily rate (kg/d), EF_w is the exposure frequency to water intake from wells (d/yr), ED is the exposure duration (yr), BW is the body weight (kg), and AT is the averaging time (yr·d/yr) over which risk is evaluated.

An application of the general expression of R (1) for inhalation of vapors is:⁽²⁾

(3)

Where C_a is the substance concentration in air into the shower (mg/m³), SF_i is the slope factor for inhalation of the substance (kg d/mg), IR_a is the air inhalation rate (m³/min), SD is the daily shower duration (min/d), EF_s is the exposure frequency to shower (d/yr), ED, BW, and AT are the same as in Equation (2).

The concentration of the substance in air into the shower was calculated by the model proposed by Foster and Chrostowski⁽¹⁸⁾ for a volatile organic compound (VOC):

(4)

Where C_a and C_w are substance concentration in air into shower (mg/m³) and in water (mg/L); FR is the water flow rate (L/min); V_s is the shower volume (m³); AE_s is air exchange rate in shower (m³/m³ min); t is the falling time of the water drop in shower (s); d is the water drop diameter (m); φ is the shape factor value for drop (φ= 6 for spherical drops); K_aL is the global mass transfer coefficient of VOC through the air/water interface (cm/h); the number 360 arrange h with s and cm with mm.

K_aL can be calculated by:⁽¹⁸⁾

(5)

Where R is the ideal gas constant (m³atm/mol K); T_a and T_ws are the absolute temperature of air and water in shower (K); H_VOC is the Henry's constant of the volatile organic compound (m³atm/mol); and k_g,H2O are the mass transfer coefficient of CO₂ in water and H₂O in air at 20°C (cm/h); MW are the molecular weight (g/mol); CF is the temperature correction that depends on T_ws, given in K, as it follows:

(6)

The total cancer risk is the sum of oral and inhalation risk of each substance:⁽²⁾

(7)

Where o and i refer to oral ingestion and inhalation, respectively, and A, B, and C refer to the three substances as reported in Table I.

Combining Equations (2) to (7) applied to each substance the following system of equations result:

(8)

The model reported in Equation (8) is strongly nonlinear; moreover, temperature affects the most variables that are thus correlated.

2.3. GUM Procedure for Evaluating Uncertainty

The uncertainty characterizing the value of risk is evaluated by applying the procedure proposed by GUM⁽¹⁴⁾ for combined standard uncertainty for the discussed input quantities. Their correlation is at first neglected and the procedure for uncorrelated was thus applied. More specifically, “r” is the best estimate of the value of total risk R calculated by Equation (8) and u(r), the standard uncertainty associated with “r,” is a combination of the standard uncertainties of the input quantities.

In analytical form:

(9)

Where x_j is the jth variable, ∂R/∂x_i the partial derivative of risk by the jth variable, u indicates standard uncertainties, and c indicates the sensitivity coefficients, i.e., ∂R/∂x_i calculated at the current value of the each parameter.

As an example, Equation (9) applied to Equation (2) gives:

(10)

Where capital letters indicate variables and small letters indicate values.

Equation (9) is used to evaluate the combined uncertainty of risk u_c(r_o). Sensitivity coefficients are limited to first-order Taylor series when nonlinearity of risk expression is considered negligible, i.e., sensitivity coefficients are the partial derivatives of Equation (2) calculated at the point at which risk is evaluated.

As an example, the sensitivity coefficient of BW in Equation (10) is reported:

(11)

When nonlinearity of the exposure model is significant, higher-order terms in the Taylor series expansion must be included in the expression of uncertainty.⁽¹⁴⁾

When health and safety of humans are involved, GUM⁽¹⁴⁾ prescribes to state not the uncertainty associated with the value of the measurand, but the so-called expanded uncertainty (U), which provides the boundaries of a coverage interval that is expected to encompass the value with a stated probability.

Extensive experience with and full knowledge of the uses to which an estimation result will be used can facilitate the selection of the proper value of the coverage factor. The coverage probability is, as mentioned above, subject to political decisions. However, the GUM and many regulations require a coverage probability of 0.95. This implies for a Gaussian that the so-called coverage factor, i.e., the ratio between U and u(y), has the value 1.96 or, rounded, 2. If the probability density function (PDF) is not symmetric (see GUM⁽¹⁴⁾ G 5.3), one can introduce U₋ and U₊ or simply state the boundaries of the shortest coverage interval. U₋ and U₊ may be calculated as the difference between 95th percentiles of distribution and (best) estimate of the value. In the standard framework of GUM, one has reason to assume—except for certain cases—that the PDF for the measurand is a Gaussian. In this case, the coverage interval is symmetric about the expectation that is taken as (best) estimate of the value, and the expanded uncertainty is the half width of that coverage interval.

The GUM procedure permits to create a “budget of uncertainty” that underlines the sources of uncertainty for the risk and ranks the sources based on their sensitivity coefficients. The budget is a very useful tool to point out how to reduce uncertainty, how to determinate the boundary conditions for the reduction, and which are the relevant variables that affect uncertainty.

The budget of uncertainty is a table that summarizes in each row the parameters of a variable (refer to Tables IV–VI). Parameters are: acronym (X), units ([X]), value (x), uncertainty at value x (u (x)), relative uncertainty at value x (u (x)/x), sensitivity coefficient calculated at values x (c (x)), quadratic term (u² (x) c2 (x)), criticism. The last row shows the parameters of the combined quantity, i.e., risk (R). X and u(x) (excluded last row) are the independent values; all the other values in the table are calculated from them.

Relative uncertainty is calculated as the ratio between standard uncertainty u(x) and the value x for each variable; sensitivity coefficients may be determined by analytical or numerical sensitivity analysis and they are calculated at the values x; quadratic terms are calculated from sensitivity coefficient c(x) and standard uncertainty u(x).

Risk value r is calculated from values x by Equation (1). The quadratic term of risk is calculated as the sum of quadratic terms u²(x)c²(x), i.e., by Equation (9). Uncertainty of risk is calculated as the square root of the quadratic term of risk.

Criticism is calculated as the ratio between the quadratic term of the variable and the maximum quadratic term of the summation.

(12)

Criticism mirrors and visualizes the relative impact of the variables in order to rank the contribution of the variables to the uncertainty. It is possible to state a threshold value for criticism under which variables have a negligible contribution to uncertainty, e.g., 0.1 can be considered as the threshold value.

The budget of uncertainty includes and explicates all the calculations used. A single sheet of Excell^® was the only thing need to perform calculations.

2.4. Information About Independent Variables

In order to calculate risk and its uncertainty for each independent variable it is necessary to recover information. Table II lists the 37 independent variables that result from the system of Equation (8).

To implement RME risk evaluation a reasonable value must be stated for each independent variable on the basis of an approximate reasoning about the role of the variable. The more the model is complex, the less it is possible to arrange a believable reasoning about all the variables. Generally, the 95th percentile, the mode, the median, or the mean are considered as reasonable values.⁽¹⁹⁾

EPA⁽¹⁹⁾ suggests values for BW, ED, EF_w, AT, SF_o, and SF_i for RME. The RME value for contaminants concentration may be stated at the 95th percentile of the statistical distribution.⁽¹⁹⁾ Smith suggested, without any explicit justification, RME values lower than the 95th percentile and greater than the mean for IR_w, IR_a, C_w, and RME values and equal to the mean for AE_s, SD, V_s, EF_s, d, FR_ws, and T_ws. No suggestions are available for the other variables, mean values were stated as RME values according to the approach reported by Smith.⁽¹⁷⁾ Nevertheless, caution would suggest a safeguard value greater than the mean for FR_ws and T_ws. RME values and type are reported in Table II.

To implement Monte Carlo risk evaluation, a statistical distribution must be stated for each independent variable. Distribution may account for the sole variability (as it was considered by Smith⁽¹⁷⁾) or to the total uncertainty about the true value of the variable. This article deals mainly with variability because of a lack of information on the other sources of uncertainty. The statistical distributions used in this article to account for variability are the same as those used by Smith⁽¹⁷⁾ and they are listed in Table II.

To implement GUM procedure a representative value and an uncertainty must be stated for each independent variable. The uncertainty u(x) of each variable must be evaluated from all the information available on the estimation of their values x using Type A or Type B evaluation of standard uncertainty.⁽¹⁴⁾ All kinds of information may be used to implement a Type B evaluation of uncertainty. At least an expert evaluation about the way by which the representative value is calculated, estimated, or measured, and an expert evaluation about the possible variability of the representative value should be expressed.

Table III reports the equations used to calculate the representative value and the uncertainty from statistical distribution parameters. GUM⁽²⁶⁾ and the new development in metrology⁽³¹⁾ do not give any indication to treat lognormal distribution. Because of the properties of the asymmetric distributions, the median was chosen as a representative value of lognormal distribution⁽²⁷⁾ and σ_L of lognormal distribution was chosen as relative uncertainty.

Table III. Properties of Statistical Distributions and Type B Evaluation of Uncertainty

Distribution	Constant	Uniform	Triangular	Normal	Lognormal*
Acronym	C(K)	U(h; L)	TR(h; mo; L)	N(M, SD)	LN(M;SD)
Value x	x = m = K	x = M = m = (h+l)/2		x = M = m =μN
Uncertainty u(x)	u = 0			u =σN
X95	K			M + z95SD
Confidence of a threshold XT	0 for XT<K
	1 for XT>K	0 for XT<l	0 for XT<l
		1 for XT>h	1 for XT>h
Special properties				μN= M
				σN= SD

• Parameters of distribution: z₉₅= 1.645; M = mean; m = median; SD = standard deviation; GM = geometric mean; GSD = geometric standard deviation; h = highest value; l = lowest value; mo = mode = highest frequency value.
• *Not treated by GUM.

Tables IV, V, and VI list the representative values and the standard uncertainty for all the independent variables for RME, variability, and total uncertainty, respectively.

3.1. Uncertainty of Deterministic Estimation

RME of SF_o, SF_i, EF_w, EF_s, and ED are safeguard values; they are affected by uncertainty due to caution. RME is a mean value for the other variables. Deeper information about the estimation of RME values is not available; thus some assumption must be stated in order to evaluate uncertainty.

RME safeguard values may be considered the 95th percentile of probability distribution of the variable considered.⁽¹⁷⁾ Moreover, a PDF has to be assumed; the uniform distribution is the simplest one. Similar reasoning may apply to C_w and AT.

For safeguard values, the Type B evaluation reported in GUM⁽¹⁴⁾ Clause 4.3.7 of uncertainty was calculated:

(13)

BW is considered as an example RME mean value; the reason to estimate 70 kg is not known, but a uniform distribution in the range 30–120 kg with a 95% confidence may be representative of the evaluation. Type B evaluation⁽¹⁴⁾ for BW is:

(14)

If different information is available about RME values estimation, a different set of values may be calculated, but a negligible difference is expected. Similar reasoning applies to all the variables. Table IV shows the results of uncertainty estimation for RME.

By means of the equations in Table III, mean and standard deviation can be calculated from the value and the uncertainty; a lognormal distribution LN(1.53 10⁻³; 4.78 10⁻³) describes the distribution of risk. Risk results are lower than RME at 95% confidence, greater than 10⁻⁶ at 99.992% confidence, and lower than 10⁻⁴ at 16% confidence.

For the symmetric distributions mode, median and mean have the same value, but for the asymmetric distributions (as lognormal) mode, median, and mean have different values and each of them may be considered representative of the value of the risk; mean was chosen to report the results as the most representative value in the contest of risk analysis.

In the GUM procedure, one expects usually that the PDF for the output quantity tends to be a Gaussian; in this case, the coverage factor can be read off tables, and is 1.96 for a coverage probability of 0.95; this is rounded to 2.0. Nevertheless, GUM suggests to choose other values for specific applications (as for strong asymmetric distributions). In the present case, in order to reach the 95% confidence, the cover factor k was stated at 6.0. Risk estimation result may be reported⁽¹⁴⁾ as r= 1.5 10⁻³+ 4.4 10⁻³. Criticism is higher than 10% for 12 variables (reported in bold in Table IV); these variables have importance in determining uncertainty. It is important to underline that only substance A (1,1 dichloroethene) has a role in determining uncertainty.

3.2. Uncertainty of Statistical Estimation

3.3. Comparison: GUM vs. Monte Carlo

Figs. 1 and 2 show cumulative distribution function and probability density function, respectively, calculated by both the GUM procedure and Monte Carlo simulation.

The GUM procedure results, stating a lognormal distribution for risk, show a good fitting of Monte Carlo simulation results. The difference between 95th percentiles of risk is lower than 10%, while the difference between medians is lower than 2%. GUM gives the lowest values, but owing the large uncertainty the values of Monte Carlo and GUM may be considered the same value.

The nonlinearity of the risk model does not invalidate the applicability of a first-order Taylor series approximation. Nevertheless, if fitting is not good for strong nonlinearity in the model, the second-order Taylor series approximation may be used for approximation. Also, the approximation of neglecting correlation between variables does not affect results. Nevertheless, GUM suggests the way to overcome this kind of trouble.

This comparison confirms the validity of the choice of lognormal distribution for risk.