Fuzzy-based Fugacity Model for Propagating Uncertainty in Assessing Swimmer Exposures to

Disinfection Byproducts

Roberta Dyck, Rehan Sadiq School of Engineering

University of British Columbia Okanagan Kelowna, BC, Canada

Manuel Rodriguez, Sabrina Simard École supérieure d’aménagement du territoire

Université Laval Québec, QC, Canada

Robert Tardif Département de santé environnementale et santé au travail

Université de Montréal Montréal, QC, Canada

Abstract— Aleatory and epistemic uncertainty are widely accepted as unavoidable components of human health risk assessment. Model uncertainty and variability in exposure factors contribute to the overall uncertainty in the assessed risk. Probabilistic methods are commonly used to characterize these uncertainties. Exposures to disinfection byproducts have been linked to cancer as well as reproductive and respiratory health effects. Swimmers are exposed to disinfection byproducts mainly through inhalation and dermal contact and to a lesser extent through ingestion. A fugacity model has previously been used to estimate inhalation, dermal contact and ingestion exposures for swimmers. There is no consensus in literature regarding the most applicable models to use to estimate these parameters. Uncertainties related to differences between these models have been addressed here using fuzzy set theory. Possibilistic treatment of these uncertainties provides a helpful alternative to arbitrarily chosen models or probability distributions. A case study using disinfection byproduct concentrations from Italian swimming pools is presented.

Keywords- Swimming pools, exposure assessment, Level III fugacity model, disinfection byproduct (DBPs), fuzzy sets

I. INTRODUCTION It is important to understand the risks associated with

exposure to disinfection byproducts (DBPs) in indoor swimming pools due to potential health effects that swimmers and staff could experience. The purpose of this study is to explore fuzzy techniques for addressing uncertainties involved in modeling exposures to DBPs in the swimming pool environment.

A. DBPs in indoor swimming pools

Swimming pools must be disinfected to protect the public from waterborne illnesses caused by viruses, bacteria, parasites and fungus [1]. DBPs are formed when organics present in

water react with the disinfectants. The presence of DBPs in drinking water and in air and water of swimming pools has been linked to potential health effects such as cancer [2], respiratory effects [3] and reproductive effects [4]. Over 100 different DBPs have been detected in swimming pools [5]. The most prevalent DBPs in water are trihalomethanes (THMs), and haloacetic acids (HAAs). Although DBPs are linked to potentially negative health effects and are found in swimming pools, they are not consistently required to be measured and few guidelines exist for their control.

Swimmers are exposed to DBPs through three main exposure routes: ingestion, inhalation and dermal contact [6]. Understanding and managing swimming pool exposures to DBPs and associated risks is important to protect human health. To that end, several models have been used previously to estimate exposures to DBPs in swimming pools including SWIMODEL [7]; a multi-layer exposure model [8]; and a fugacity exposure model [9].

B. Uncertainty

For human health risk assessment, aleatory uncertainty can be present in concentrations of environmental samples and in exposure factors (e.g. exposure duration and frequency, receptor weight, inhalation and ingestion rates). In risk assessment, epistemic uncertainty can occur in models that attempt to describe processes for which information may be incomplete or conflicting. This uncertainty must be included in any exposure model and properly characterized to allow the most accurate interpretation of the model results.

Monte Carlo simulations are a common probabilistic tool for propagating uncertainties in environmental modeling [10]. While this method may be suitable for variability in exposure factors and concentrations, other model parameters (e.g. partitioning properties) lack either the data or the precision to

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use a pdf for their characterization. Another method is needed to represent the uncertainties within the model itself and the vagueness that can exist in the model parameters.

C. Non-probabilistic methods

Due to the difficulty in addressing epistemic uncertainty through stochastic methods, fuzzy set theory has been increasingly used to address uncertainty in civil and environmental engineering applications. The objective of this paper is to present fuzzy-based environmental modeling applied to DBP exposures in indoor swimming pools.

Fuzzy set theory was derived by Zadeh [11] as a way to represent the uncertainties in vague or imprecise information. It allows the use of qualitative knowledge and linguistic descriptions in numerical analysis and reasoning. It is especially well adapted to decision making, and incorporating expert opinion or linguistic variables [12]. Fuzzy sets, fuzzy numbers and fuzzy arithmetic have been well described elsewhere [13–16], so only the application to DBPs in swimming pools will be considered here.

II. FUGACITY-BASED MODEL FOR DBPS IN SWIMMING POOLS

Exposure to THMs in swimming pools occurs through three exposure routes: inhalation, dermal contact, and to a lesser extent ingestion [6]. Previous studies have found inhalation to be an important exposure route for swimmers and swimming pool attendants [17], [18]. Therefore, it is important to have a clear understanding of the concentrations of THMs in the air of the pool environment as well as in the water in the pool.

A. Fugacity model A fugacity-based model has been developed to estimate

exposures to THMs in indoor swimming pools [9]. Fugacity is defined as the escaping tendency of a chemical to leave one medium in preference for another [19]. At the low concentrations expected in the swimming pool environment, a linear relationship exists between concentration (C), fugacity (f) and fugacity capacity (Z)

C = f·Z (1)

The use of fugacity in contaminant fate modeling allows simultaneous consideration of all the media and processes occurring in the environment. Intermedia transport and advection of the contaminant in and out of the “evaluative environment” are included in mass balance equations which relate the concentrations in the different media in the environment [20]. Four levels of fugacity models have been proposed to handle different equilibrium and dynamic conditions. Level III model, discussed below and used by Dyck et al. [9], deals with steady state, non-equilibrium conditions with flow in and out of the system occurring.

The characterization of the environment and exposure scenario are described by Dyck et al. [9]. The physico-chemical properties for the contaminant of concern must be characterized for inclusion in the model using chemical handbooks and literature [21]. The fugacity capacities (Z) are calculated from

the physico-chemical properties. Intermedia transport is then calculated using transfer equations provided by Mackay and Yuen [22]. Mass balance equations were used to calculate concentration of THMs in the air and swimmers.

B. Sources of uncertainty in the model Variability (aleatory uncertainty) can exist in exposure

models for exposure factors such as swimming frequency, swimming duration, swimmer weight and surface area, ingestion rates and inhalation rates. These are parameters for which sufficient data usually exist to approximate a pdf to represent the expected value and the associated uncertainty. Other uncertainties exist as epistemic uncertainty. These could be due to uncertainty in the model or in model parameters or variables such as the mass transfer coefficients. Because the aleatory uncertainty can be represented using probabilistic methods, only the epistemic uncertainty which originates from the model itself is considered here.

For the use of the model the concentration in water (C2) was available. The fugacity capacities (Z) were calculated as crisp numbers from physico-chemical properties. The flow of air and people out of the environment were reported during sampling of the water. The mass transport between the water and skin were calculated using an equation given by McKone and Howd [23]. The parameter D12, representing transport between water and air, is the most uncertain.

(2)

where

• kG is the air side mass transfer coefficient,

• A12 is the surface area of the water,

• Z1 and Z2 are the fugacity capacities for the air and water compartments, respectively; and

• kL is the water side mass transfer coefficient.

There were no sources for conclusive equations for the parameters kG and kL. Guo and Roache [24] listed seven different equations for kG [25–32]. Three sources were also found for equations for kL [24], [33], [34].

For the purposes of probabilistic modeling, Dyck et al. [9] used Monte Carlo simulations to propagate the uncertainty in the model. A pdf was estimated for the parameter kG using the seven equations given. The results of the Monte Carlo simulations are discussed in the next section.

C. Results of probabilistic model

Fantuzzi et al. [35] published results of water and air sampling at swimming pools in Modena, Italy. These results were used to validate the probabilistic fugacity-based model by Dyck et al. [9]. In Table I, the measured air and water concentrations of chloroform are presented for three swimming pools along with the results of the probabilistic model.

The water and air concentrations from the pools in Modena, Italy presented by Fantuzzi et al. [35] were given as crisp

The research is supported by UBC Internal funding. The authors gratefullyacknowledge the input and assistance of Guglielmina Fantuzzi, Elena Righi and Gabriella Aggazzotti for sharing data and insight into operation of pool facilities in Italy.

values, therefore only the modeled air concentrations include a representation of the uncertainty (standard deviation).

TABLE I. RESULTS OF PROBABILISTIC MODEL

Modena Pool

Number [9]

Measured Concentrations

Modeled Air Concentrations

Water (µg/L) Air (µ/m3) Mean

(µ/m3) Standard Deviation

1 26 58.6 87.7 37.8

3 18.7 67.7 50.8 18.1

5 6.1 19 16.9 6.3

III. FUZZY-BASED MODEL One limitation encountered in the probabilistic model by

Dyck et al. [9] is the need for sufficient data to generate a pdf. The mass transport coefficients kL and kG were derived from three and seven equations, respectively. Because none of the equations directly addressed the environmental characteristics of an indoor swimming pool, all of the equations were considered equally likely to be correct. Rather than choose one equation or attempt to fit a pdf to the seven calculated results, here we use a fuzzy-based fugacity model to calculate the mass transport coefficients (kG and kL), mass transfer term (D12), fugacity in air (f1) and finally concentration of chloroform in the air. Chloroform was chosen for this assessment because it is both commonly occurring and extensively studied [36].

A. Membership functions

For the transport coefficient kG, a trapezoidal membership function was defined. The seven values calculated (from the formulas listed by Guo and Roache [24]) were considered equally likely and so form the top of the trapezoid. The minimum and maximum values were calculated as theoretical limits based on physico-chemical properties of chloroform and the facility characteristics such as air recirculation rates.

For the transport coefficient kL, a triangular membership function was defined. The three values calculated (Table II) represent the minimum, most likely and maximum values.

TABLE II. CALCULATED VALUES OF KG AND KL

Pool 1 Pool 3 Pool 5 Source of Equation

kG cm/hr

27.0 25.7 25.1 [25], [26]

51.0 48.5 47.5 [27] [28]

76.5 59.2 66.6 [29] [28]

26.9 25.6 25.1 [30] [28]

26.6 26.6 26.6 [31] [32]

73.7 63.0 66.1 [26]

43.8 34.5 38.2 [28] [30] Pool 1 Pool 3 Pool 5 Source of Equation

kL cm/hr

0.58 0.58 0.58 [24]

12.1 12.1 12.1 [33]

61.1 46.2 41.5 [34]

B. DSW algorithm

The functions for D12, f1 and C1 were evaluated using the DSW algorithm [37]. The algorithm uses α-cuts and interval analysis to generate fuzzy output for the functions. There are four basic steps [16]:

• Select a value α in the membership functions (the same for each variable)

• Find the intervals in the input membership functions that correspond to this α.

• Using interval operations, compute the interval for the output membership function for the selected α-cut level.

• Repeat these steps for different values of α to build a fuzzy result.

For this analysis, we chose α values of 0, 0.25, 0.5, 0.75 and 1.0. Using the α-cuts on each membership function (for kG and kL) as shown in Fig. 1, we obtain intervals (left and right) as shown in Table III.

Next we use the interval values in the equation for D12 (2) as shown in an example using α=0 for Modena Pool 1:

The intervals for α0 for kG and kL in Modena Pool 1 are kGα =0 = [20, 99] and kLα =0 = [0.58, 61.1]. So we evaluate (2) as: 1120, 99 10.58, 61.1

Figure 1. α-cuts on kG and kL fuzzy membership functions (Modena Pool 1)

TABLE III. INTERVAL VALUES FOR KG AND KL

Pool 1 Pool 3 Pool 5

kG (cm/hr)

α-cut levels left right left right left right

0 20 99 20 99 20 99

0.25 21.7 93.4 21.4 90.0 21.3 90.9

0.5 23.3 87.7 22.8 81.0 22.5 82.8

0.75 25.0 82.1 24.2 72.0 23.8 74.7

1 26.6 76.5 25.6 63.0 25.1 66.6

kL (cm/hr)

α-cut levels left right left right left right

0 0.58 61.1 0.58 46.2 0.58 41.5

0.25 3.47 48.9 3.47 37.7 3.47 34.2

0.5 6.36 36.6 6.36 29.2 6.36 26.8

0.75 9.25 24.4 9.25 20.6 9.25 19.5

1 12.14 12.1 12.14 12.1 12.14 12.1

The parameters A12, Z1 and Z2 are crisp numbers. Fuzzy arithmetic operations are then applied [16]. By using the interval values in the equation for D12, we get values for the left and right end of the intervals for kG and kL, for a total of four possible results for each α-cut level, as shown in Table IV. The minimum and maximum values of D12 for each α-cut are highlighted. These intervals can be considered as a fuzzy number because our function meets the requirements of being convex, increasing.

In a similar fashion, we use the fuzzy results of mass transport, D12, to calculate fugacity, (f1) and use those results to calculate the concentration of chloroform in the air (C1) using equations presented by Dyck et al. [9].

TABLE IV. VALUES OF D12 FOR INTERVALS OF KL AND KG FOR FIVE Α-CUTS

Pool 1 Pool 3 Pool 5

α-cut kG left kG right kG left kG right kG left kG right

0 kL left 3.92E-03 4.48E-03 3.77E-03 4.31E-03 3.14E-03 3.59E-03

kL right 2.39E-02 9.93E-02 2.27E-02 8.96E-02 1.88E-02 7.26E-02

0.25 kL left 1.37E-02 2.24E-02 1.31E-02 2.14E-02 1.09E-02 1.79E-02

kL right 2.55E-02 9.03E-02 2.38E-02 7.91E-02 1.95E-02 6.46E-02

0.5 kL left 1.86E-02 3.48E-02 1.76E-02 3.26E-02 1.46E-02 2.74E-02

kL right 2.67E-02 8.02E-02 2.46E-02 6.82E-02 2.01E-02 5.62E-02

0.75 kL left 2.20E-02 4.31E-02 2.07E-02 3.91E-02 1.71E-02 3.32E-02

kL right 2.71E-02 6.75E-02 2.47E-02 5.63E-02 2.01E-02 4.70E-02

1 kL left 2.49E-02 4.83E-02 2.33E-02 4.20E-02 1.91E-02 3.60E-02

kL right 2.49E-02 4.83E-02 2.33E-02 4.20E-02 1.91E-02 3.60E-02

minimum values

maximum values

A. Fuzzy model results

The fuzzy output results for f1 are presented in Table V. The fuzzy output results for C1 are shown in Fig. 2.

The results of the probabilistic model using Monte Carlo simulations are presented as pdfs in Fig. 2. The three plots represent the three pools in Modena, Italy for which air and water concentration data were available. They were normalized (so the maximum value was one) for comparison to the fuzzy model results [38]. The results of the fuzzy model are presented on the same plots. The actual measured values of chloroform air concentration are presented in Fig. 2 as a dashed vertical line. There was no information on error or uncertainty included with the measurement data, therefore this is a crisp number, which stays the same for every level of likelihood and is represented as a line.

TABLE V. FUZZY OUTPUT RESULTS FOR FUGACITY, F1

α-cut Pool 1 Pool 3 Pool 5

0 left 1.32E-04 1.41E-04 3.29E-05

right 3.23E-03 3.18E-03 7.32E-04

0.25 left 4.62E-04 4.88E-04 1.14E-04

right 2.95E-03 2.83E-03 6.54E-04

0.5 left 6.24E-04 6.54E-04 1.52E-04

right 2.63E-03 2.45E-03 5.72E-04

0.75 left 7.39E-04 7.67E-04 1.77E-04

right 2.22E-03 2.04E-03 4.80E-04

1 left 8.33E-04 8.59E-04 1.98E-04

right 1.60E-03 1.53E-03 3.70E-04

Figure 2. Results of fuzzy model, probabilistic model and measured air

concentrations

IV. DISCUSSION In Fig. 2 we see that the fuzzy model fits well with the

measured value for pool 1 and pool 3, but not as well for pool 5. The fuzzy model results more closely resemble the probabilistic model results for Modena Pool 1 and Modena Pool 5. The fuzzy-based model provided the most likely value with similar accuracy to the probabilistic model using Monte Carlo simulations.

In Fig. 2, for Modena Pool 1 and Modena Pool 5, the pdf for the air concentrations from the probabilistic model extends beyond the fuzzy model results. For the fuzzy model, a maximum value is specified to generate the fuzzy membership functions for the model input parameters. For the probabilistic model, the pdf is generated from all possible values of the model input parameters being included with no maximum to truncate the pdf on the right side. The probabilistic model may not have extended beyond the fuzzy model results if we had used a maximum value for the model parameters, or if a

distribution other than lognormal were used for the model input parameters

One major benefit to using the fuzzy numbers for uncertain model parameters is that fuzzy numbers may provide a more realistic estimate of the parameter (and the associated uncertainty) than can be generated by using a pdf with too few data points, or by arbitrarily choosing one of the formulae to represent the mass transport coefficients. For DBPs in swimming pools, this allows better characterization of the air concentration and related uncertainty. Another benefit to using fuzzy model parameters is the decreased computing complexity involved [39]. The interval operations involved in fuzzy arithmetic are generally easy to use and don’t require specialized risk analysis software or numerous model simulations to generate a result.

One disadvantage to using fuzzy-based models is that some engineers still find fuzzy-based methods unfamiliar [40]. Also, while they are easy to work with, triangular or trapezoidal membership functions may be over-simplifications of the actual membership of input parameters. The minimum and maximum values are still somewhat arbitrarily chosen by the analyst, when in fact those limits could themselves be considered fuzzy which would add another layer of complexity to the analysis.

In exposure assessment and risk assessment, there are some model parameters for which risk assessors have a clear understanding and abundant data. In these cases, probabilistic methods can be useful in addressing aleatory uncertainty (eg. the natural variability in population height, swimming duration, and body weight). Several environmental models have been reported [14], [39] which couple probabilistic methods for aleatory uncertainty (in input parameters such as site characteristics) with fuzzy-based methods for the epistemic uncertainty (uncertainty in our understanding of the model itself). That approach may be useful for exposure and risk assessment for DBPs in swimming pools and is recommended for further investigation.

V. SUMMARY AND CONCLUSIONS Both epistemic and aleatory uncertainty can be present

in contaminant fate, exposure and risk assessment models. While probability theory can address variability in exposure factors or contaminant concentrations; model uncertainty is epistemic and better address by fuzzy-based models. A fuzzy-based fugacity model was applied to assessing concentrations of DBPs in exposures to swimmers. The mass transport coefficients were assigned triangular and trapezoidal fuzzy membership functions. The DSW algorithm was used to evaluate the mass balance equations in the fugacity model in order to generate fuzzy output results for mass transport, fugacity and concentration of chloroform in air for three swimming pools in Modena, Italy. The fuzzy-based model predicted the air concentration with similar results to the probabilistic model (which used Monte Carlo simulations). The fuzzy model input may better express our understanding (or lack of understanding) of the model than pdfs generated with insufficient data. We recommend to use the air concentrations from the model for exposure and risk analysis and potentially use a combined approach with

probabilistic methods addressing the aleatory uncertainty and fuzzy-based methods addressing the epistemic uncertainty. Indeed, fuzzy-based environmental models require less computational complexity while providing a better representation of our true understanding of a model when data are lacking or vague.

ACKNOWLEDGEMENT

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