2.1 Environmental Health

Human footprints on the nature significantly deteriorate the environmental quality of air, water, and soil, pose a great threat to human health and reduce human life expectancy. Taking air pollution as an example, the primary and secondary ambient air standards of particulate matter of diameter 2.5 µm (PM_2.5) in the United States are 12 and 15 µg/m³, respectively (US EPA, 2016). The corresponding standards of PM_2.5 in China are set at 15 and 30 µg/m³, respectively (China MEP, 2016). However, the average concentration of PM_2.5 for 10 provinces in China was greater than 55 (µg/m³) in 2015. According to the data from the Administration of Energy of China, 64% of primary energy in China is from coal. China with 1.4 billion population burns about 47% of the world’s total coal consumption. The annual average PM_2.5 for the top 10 most polluted provinces in China is at least twice as much as the secondary standard of 35 µg/m³ as shown in Figure 2.1.

In China, mortality rates due to environmental damage are significantly greater than natural mortality rates. Lung, liver, and stomach cancers are the top three mortality causes in China. Epidemic studies indicate that lung cancer is associated with chemical smog due to coal burning, liver cancer with drinking water and strong liquor, and stomach cancers with contaminated food due to soil pollution. About 90% of the 161 cities whose air quality was monitored in 2014 were below Chinese official standards according to the National Bureau of Statistics of China (NBSC). More than half of China's surface water is so polluted that it cannot be used as drinking water resources. As a result, about 4000 people died every day in China due to coal burning as the likely principal cause. Due to its severity, air pollution is the leading cause of lung cancer and other respiratory infections. Figure 2.2 shows the percentage of leading causes of death in the world. In China, the most common causes of death are (i) lung cancer with 1.59 million deaths, (ii) liver cancer with 745 000 deaths, (iii) stomach cancer with 723 000 deaths, (iv) colorectal cancer with 694 000 deaths, (v) breast cancer with 521 000 deaths, and (vi) esophageal cancer with 400 000 deaths.

Worldwide, environmental pollution is also the leading cause of cancer and contributes to major premature mortality. WHO (2014) estimated that there were 3.1, 1.6, and 1.5 million people who died from lower respiratory infections, trachea bronchus lung cancers, and diarrhea diseases in the world as shown in Figures 2.2 and 2.3. There were approximately 14 million new cases, and the number of new cases is expected to rise by about 70% over the next two decades (WHO, 2014). In 2012, ischemic heart disease, stroke, and chronic obstructive pulmonary disease caused 7.4, 6.7, and 3.1 million deaths, respectively (NBSC, 2012).

Figure 2.4 shows that the leading causes of mortality due to malignant tumor is 168 and 159 deaths per 100 000 population in urban and rural China, respectively, which is about 28 and 25% of the leading causes of death in urban and rural China.

Further evidence of the environmental health issue can be seen from Figures 2.5 and 2.6. Zhao et al. (2010) showed that lung, liver, stomach, esophageal, and colorectal cancer were 32, 19, 18, 9, and 8% in urban China, while these data were 24, 23, 23, 15, and 5% in rural China, respectively, as shown in Figure 2.5.

To monitor the progress of protecting human health, the United Nation uses an improved sanitation facility as a metric. It is defined as piped sewerage, septic tanks, and pit latrines with slabs or composting toilets not shared with other households. From 1990 to 2015, the coverage of improved sanitation at these facilities rose from 54% to around 68% globally as shown in Figure 2.7. This number, however, missed the Millennium Development Goal (MDG) target by 9%. In 2015, about 946 million people were still practicing open defecation worldwide (WHO, 2016).

To achieve the Sustainable Development Goals (SDG) target, cities all over the world need to reduce open defecation, promote handwashing, and improve management and treatment of faucal wastes from both collected sewer and on‐site facilities. For drinking water, the UN uses pipe water on premise and public standpipes, boreholes, protected wells, springs, and rainwater as indicators for improved drinking water. In 2015, there were 6.6 billion people using improved drinking water, while 0.7 billion still used an unimproved drinking water source or surface water (WHO, 2016), as shown in Figure 2.8.

About one‐quarter of improved sources are contaminated with feces and approximately 1.8 billion people drink water containing such contamination (WHO, 2016). Therefore, only 68% in urban areas and 20% in rural areas have truly safe drinking water. Unsafe drinking water causes liver cancer due to disinfection by‐products (DBPs) and other contaminants such as antibiotics in drinking water.

2.2 Environmental Standards

The US EPA establishes environmental standards using health risk assessment (HRA) according to carcinogenic or noncarcinogenic chemicals. For chemicals that are known or expected to cause adverse health effects, the EPA established an enforceable maximum contaminant level (MCL) or nonenforceable maximum contaminant level goal (MCLG). The Safe Drinking Water Act (SDWA) proclaims standards and health advisories (HAs) for DBPs. HRA quantifies factors such as adsorption during ingestion, pharmacokinetics, mutagenicity, reproductive and developmental effects, and carcinogenicity (Pontius, 1990). The MCLs are federally enforceable limits for contaminants in drinking water established as the national primary drinking water regulations (NPDWRs). The secondary MCLs are established under the SDWA to protect public welfare such as odor, taste, and appearance. The HAs for drinking water contaminants are levels considered to be without appreciable health risk for specific durations of exposure and are not legally enforceable. Similarly, the MCLGs are nonenforceable health goals, which are to be set at levels at which no known or anticipated adverse effects on the health of persons occur with an adequate margin of safety. Table 2.1 is the description of the SDWA standards and HA categories.

To assess an individual’s risk, bioassays are converted to estimate human risk based on human exposure. The dose–response curves from animal tests were used to determine the equivalent human dose–response curve. The MCLG is set using a three‐category approach dependent upon the evidence of carcinogenicity. If the evidence of carcinogenicity is strong, the MCLG is set to zero. If the evidence of carcinogenicity is limited, one of two methods is utilized to calculate the MCLG depending on the toxicity data available. For every chemical compound that has an MCLG, an MCL or treatment technique must be determined. Considering the cost of remediation, the MCL is based on the best available technology (BAT) and set close to the MCLG. If BAT cannot achieve zero, MCL cannot equal to zero. When the MCL is within a risk of 10⁻⁴ to 10⁻⁶, the MCLG is set to zero.

In the past, reference dose (RfD) expressed in units of milligrams per kilogram of body weight per day (mg/kg/day) was used. The RfD is based on lifetime exposure level at which there is no significant risk to humans. It is calculated by dividing the no observed adverse effect level (NOAEL) or the lowest observed adverse effect level (LOAEL) by an uncertainty factor. The uncertainty factor accounts for differences between human and animal and differences within the human population and varies from 10 to 1000 depending upon the toxicity data available. RfD is determined using the following equation:

(2.1)

Since an extremely high uncertainty factor is used, major limitations of RfDs are the following: (i) it is limited to one of the doses in the study and is dependent on study design, (ii) it does not account for variability in the estimate of the dose–response, (iii) it does not account for the slope of the dose–response curve, and (iv) it cannot be applied when there is no NOAEL, except through the application of an uncertainty factor (Crump, 1984; Kimmel and Gaylor, 1988).

To overcome the high uncertainty factor of the RfD, the EPA developed benchmark dose (BMD) methods by fitting mathematical models to dose–response data and to select a BMD associated with a predetermined benchmark response (BMR), such as a 10% increase in the incidence of a particular lesion or a 10% decrease in body weight gain. Results from all models include a reiteration of the model formula and model run options chosen by the user, goodness‐of‐fit information, the BMD, and the estimate of the lower‐bound confidence limit on the BMD (BMDL). The benchmark dose software (BMDS 2.6) by the US EPA (2016) has the nested models, parameter standard error reporting, and parameter initialization for continuous models. BMDS 2.6 contains thirty different models appropriate for the analysis of dichotomous (quantal) data, continuous data, nested developmental toxicology data, multiple tumor analysis, and concentration–time data. Typical models used in the software are shown in Table 2.2.

The SDWA proclaims standards and HAs for DBPs (US EPA, 1997). However, setting the MCLs and MCLGs involves a lot of uncertainty, because discrepancies may exist for lifetime and longer‐term exposure HAs due to conservative policies. For example, the uncertainty factor may vary from 5 to 5000 when the lifetime health advisory concentration is estimated (U.S. EPA, 1996). Human HRA and environments risk assessment (ERA) are used to develop both ambient and discharge standards by the US EPA. Risk assessment is a systematic approach to characterize the nature and magnitude of the risks associated with environmental or health hazards.

Since chlorination is the major disinfection process in the United States, regulation of DBP concentration in drinking water is one of the major challenges faced by the US EPA. Major human and financial resources have been devoted to identify, monitor, assess, and regulate the human health effect of DBPs. As a result, HRA guidelines of DBPs were developed to protect the public from both biological and chemical risks. For example, the US EPA developed the Stage 2 DBPR aiming to reduce peak DBP concentrations in the distribution system. When a water treatment plant (WTP) assesses its disinfection strategy, both the disinfectant effectiveness against the target pathogen and the DBPs formed as a result of the disinfectant must be considered in the decision‐making process. Since no toxicity test could be performed on humans, animal test data are used with quantified uncertainty and variation involved. During HRA of DBP, information for accurate evaluation of DBP risk may not be complete, and the uncertainty factor in the assessment may be quite large (Cothern et al., 1986). For example, for total trihalomethanes (TTHM) and five haloacetic acids (HAA5), the drinking water standards were set at 80 and 60 µg/l as locational running annual average (LRAA) by the EPA (2006), respectively. If there was no specific toxicity information available, the MCLG was set up based upon a quantitative structure–activity relationship (QSAR) study. For example, the MCLG of 1,1‐dichloroethylene and cis‐1,2‐dichloroethylene were developed using QSAR approach.

In addition to pathogenic bacteria and viruses, Cryptosporidium is one of the new public health concerns for the EPA. The EPA established the Long Term 2 Enhanced Surface Water Treatment Rule (LT2ESWTR) based on Cryptosporidium concentrations in source water and current treatment practices. Four bins were recommended corresponding to additional treatment requirements for filtered WTPs as shown in Table 2.3. WTPs with average Cryptosporidium concentrations less than 0.075 oocysts per liter (oocysts/l) are placed in bin 1 where no additional treatment is required. For concentrations of 0.075 oocysts/l or more, treatment beyond the existing processes is required. The additional treatment required for each bin, specified in terms of log removal, depends on the type of treatment that the WTP already uses. In setting the biological standards, the term “log” means the order of magnitude reduction in concentration; e.g. 2‐log removal equals a 99% reduction, 3‐log removal equals a 99.9% reduction, and 4‐log removal equals a 99.99% reduction. Giardia and virus need 3‐log and 4‐log removal and/or inactivation, respectively.

To reduce biological and chemical risks simultaneously, UV disinfection appeared to be the best option for drinking water and treated wastewater effluent. Unlike chemical disinfectants, UV leaves no residual that can be monitored to determine UV dose and inactivation credit. To earn disinfection credits, however, a relationship between the required UV dose and these parameters must be established and then monitored at a WTP to ensure sufficient disinfection of microbial pathogens. The UV dose depends on the UV intensity (measured by UV sensors), the flow rate, and the UV transmittance (UVT). The US EPA (2006) recommended UV dose requirements (mJ/cm²) in Table 2.4.

2.3 Health Risk Assessment

Human health risk assessment (HRA) quantifies factors such as adsorption during ingestion, pharmacokinetics, mutagenicity, reproductive and developmental effects, and carcinogenicity (Pontius, 1990). It is to quantify the likelihood that adverse human health effects may occur or are occurring as a result of exposure to one or more stressors (US EPA, 1992). To reduce environmental health mortality, HRA is used to detect issues, identify hazards, characterize hazards, assess exposure, and manage and communicate risk. By using HRA, the EPA developed an Integrated Risk Information System (IRIS) that contains the human health effect databases of chemicals that may result from exposure to various chemicals in the environment. The IRIS 10⁻⁶ risk level is the contaminant concentration (in µg/l) in drinking water that would yield no greater than an additional risk of one in a million (10⁻⁶) after a lifetime of drinking that water. The acute 10‐day values apply specifically to acute toxic effects on children but are expected to be protective for adults. For noncarcinogenic chemicals, this value is typically the same as the MCLG. The chronic (lifetime) values for cancer are set at a level that should yield no greater than an additional 10⁻⁶ risk over a lifetime exposure. According to the IRIS, the EPA cancer risk is classified in six different categories: (i) H is carcinogenic to humans, (ii) L is likely to be carcinogenic to humans, (iii) L/N is likely to be carcinogenic above a specified dose, (iv) S is suggestive evidence of carcinogenic potential, (v) I is inadequate information to assess carcinogenic potential, and (vi) N is not likely to be carcinogenic to humans.

2.3.2 Dose–Response Curves

A dose–response relationship describes how severity of adverse health effects (the responses) is related to the exposure amount to a chemical compound. The measured response usually increases as the dose increases. To establish drinking water standards, dose–response curves of animal toxicity tests are used to establish toxicity slopes or BMD and to document the dose–response relationship over the range of observed doses. With animal toxicity data, the health risk is extrapolated beyond the lower range of available observed data until the dose level begins to be adverse to human health. The shape of the dose–response relationship curve depends on the chemical, the kind of response, incidence of disease, and death. However, there are major gaps during the extrapolation from animal to human and from high dose to low dose to predict human health risk. Great uncertainty could be introduced during these extrapolations through either nonlinear or linear dose–response models for different modes of action. For example, animal tumor data is based on the development of the dose–response parameter as effect dose (ED) at 10% (ED₁₀) or lethal dose at 10% (LD₁₀). The actual dose heavily depends upon the exposure routes and animals. The EPA lists the general LD₅₀ of different chemical compounds with great variations in Table 2.7. For dioxin (TCDD), the LD₅₀ is as low as 0.001 mg/kg. For inhalation toxicity test, the LD₅₀ for rats and mice are 293 and 137 mg/kg, respectively. Table 2.7 shows LD₅₀ for different chemicals varied thousands of times.

Chemical	LD50 (mg/kg)	Chemical	LD50 (with route and animal)
Ethyl alcohol	10 000	Caffeine	620 mg/kg – oral mouse
Sodium chloride	4 000		192 mg/kg – oral rat
			105 mg/kg – i.v. rat
			68 mg/kg – i.v. mouse
Ferrous sulfate	1 500	Chlorine (LC50)	293 ppm/1 h – rat
			137 ppm/1 h – mouse
Morphine sulfate	900	THC (from marijuana)	175 mg/kg – i.v. mouse
Strychnine sulfate	150		155 mg/kg – i.v. rabbit
			100 mg/kg – i.v. dog
Nicotine	1	Mercury(I) chloride	210 mg/kg – oral rat
			8 mg/kg – i.v. mouse
Black widow	0.55	Mercury(II) chloride	37 mg/kg – oral rat
			10 mg/kg – oral mouse
Curare	0.50	Arsenic acid (V oxidation state)	48 mg/kg – oral rat
Rattlesnake	0.24	Arsenic trioxide (III oxidation state)	20 mg/kg – oral rat
Dioxin (TCDD)	0.001	Dimethylarsenic acid (methylated arsenic form used as a cotton defoliant)	700 mg/kg – oral rat

2.3.2.1 Nonlinear Dose–Response Assessment

In nonlinear dose–response assessment, the threshold of toxicity is where the effects (or their precursors) begin to occur. The no observed adverse effect level (NOAEL) is the highest exposure level at which no statistically or biologically significant increases are seen in the frequency or severity of adverse effect between the exposed population and its control population. Different mathematical models were used to establish the bench mark dose (BMD) or benchmark dose lower confidence limit (BMDL) in the range from 1 to 10% depending on toxicity tests. The BMDL is a statistical lower confidence limit on the dose that produces the selected response. The lowest observed adverse effect level (LOAEL), NOAEL, or BMDL is used as the point of departure for extrapolation to lower doses. The EPA (2012) developed a guideline on using dose–response modeling to obtain BMD, i.e. dose levels corresponding to specific response levels, near the low end of the observable effect as shown in Figure 2.9. The fraction of animals affected in each group is indicated by the points with the error bars of 95% confidence intervals. In developing the BMD, the EPA requires the following statistical information of the process: (i) rationale, (ii) estimation procedure, (iii) estimates of model parameters, (iv) goodness of fit such as log‐likelihood and Akaike Information Criterion (AIC), and (v) standardized residuals. The US EPA (2012) provided excellent examples to illustrate some important aspects of computing benchmark doses (BMDs) and BMDLs from simple datasets and endpoints using EPA’s BMDS package.

Parameter	Maximum likelihood estimates (MLEs)	Standard error
Background	0.12	0.132665
Beta1	0.00930036	0.141898
Beta2	0.00925286	0.0246904

Parameter	Maximum likelihood estimates (MLEs)	Standard error
Background	0.111488	0.111488
Beta1	0.120556	0.120556

Dose	Estimated probability	Expected number responding	Observed number responding	Group size	Scaled residual
0.0000	0.1115	5.574	6	50	0.086
2.8300	0.2417	11.842	10	49	−0.205
5.6700	0.3531	17.657	19	50	0.118

2.3.2.2 Linear Dose–Response Assessment

For carcinogens, if “mode of action” information is insufficient, then linear extrapolation is typically used as the default approach for dose–response assessment. A straight line is drawn from the point of departure for the observed data (typically the BMDL) to the origin (where there are zero dose and zero response). The slope of this straight line is referred to as the slope factor (SF) or cancer SF that is used to estimate risk at exposure levels. When linear dose–response is used to assess cancer risk, excess lifetime cancer risk is calculated as follows:

(2.2)

Total cancer risk is calculated by adding the individual cancer risks for each pollutant in each pathway of concern (i.e. inhalation, ingestion, and dermal absorption) by using reasonable maximum exposure (RME) and then adding together the risk for all pathways.

2.3.3 Exposure Assessment

Exposure assessment estimates the magnitude, frequency, and duration of human exposure to an agent in the environment or estimates future exposures for an agent that has not yet been released. Exposed concentration can be measured at the point of contact (the outer boundary of the body), estimated by separately evaluating the exposure concentration and the time through different scenarios, or reconstructed through internal indicators (biomarkers, body burden, and excretion levels) after the exposure. Table 2.12 lists the variables used in the US EPA Guidelines for Exposure Assessment (1992) for HRA.

Parameter	Definition	Default – child	Default – adult
TRL	Target risk level (unitless)	10−6	10−6
BW	Body weight (kg)	15	70
AT	Averaging time (year)	70	70
SFABS	Absorbed cancer slope factor (mg/kg/day)−1	Chemical specific	Chemical specific
ED	Exposure duration (year)	6	30
EV	Event frequency (events/day)	1	1
EF	Exposure frequency (days/year)	350	350
FA	Fraction absorbed (unitless)	Chemical specific	Chemical specific
tevent‐RME	Event duration (h)	1 (bathing)	0.58 (showering)
SA	Surface area (cm2)	6 600	18 000
Kp	Permeability coefficient (cm/h)	Chemical specific	Chemical specific
ABSGI	Absorption fraction (unitless)	Chemical specific	Chemical specific
τevent	Lag time per event (h)	Chemical specific	Chemical specific
SFo	Oral cancer slope factor (mg/kg/day)	Chemical specific	Chemical specific
t*	Time to reach steady state (h)	Chemical specific	Chemical specific
DAD	Dermal absorbed dose (mg/kg/day)	Site specific	Site specific
ADevent	Absorbed dose per event (mg/cm2/event)	Site specific	Site specific
B	Dimensionless ratio of the permeability coefficient of a compound through the stratum corneum relative to its permeability coefficient across the viable epidermis (ve) (dimensionless)	Chemical specific	Chemical specific

The internal dose via the dermal route, µg/kg bw/day, can be calculated as follows:

(2.3)

where

• 
• 
• 
• 
• 
• 
• 
• 

2.3.3.1 Cancer Screening Calculation for Dermal Contaminants in Water

For a given cancer risk level at 10⁻⁶, the following equations can be used to estimate dermal absorbed dose (DAD) in mg/kg/day:

• For cancer risk
  (2.4)
• For hazard quotient
  (2.5)
• Evaluate AD_event:
  (2.6)
• Evaluate permissible water concentration, C_w: For organics
  (2.7)
  (2.8)

• For inorganics
  (2.9)

Example 2.3 illustrates the steps used to calculate the cleanup level from dermal exposure to compounds in water given an acceptable risk of 10⁻⁶. The default scenarios used in the calculations are (i) the adult 30‐year exposure and (ii) an age‐adjusted 30‐year exposure incorporating a child bathing for 1 h/event (RME value), once a day, 350 days/year for 6 years and an adult showering at 35 min/event (RME value), once a day, 350 days/year for 24 years. The general equations could be applied to any compound, and the example gives the calculation for one compound in water with a cancer risk of 10⁻⁶.

2.3.3.2 Noncancer Screening Calculation for Contaminants in Residential Soil

The following equations are provided by the EPA in the exposure assessment process. The scenario to be evaluated is residential soil. Equations (2.10), (2.11), and (2.12) are used for calculating the soil concentration, C_soil:

Child or adult:

(2.10)

Age adjusted:

(2.11)

The age‐adjusted, body‐part‐weighted dermal factor is as presented in equation

(2.12)

For toxicity assessment, cancer SF can be derived based on absorbed dose:

(2.13)

while RfD can be expressed as follows based on absorbed dose:

(2.14)

2.4 QSAR Analysis in HRA

The US EPA permits and uses quantitative structure and activity relationship (QSAR) principles to classify and prioritize DBPs because more than 600 DBPs have been identified and cataloged by the US EPA but only small fraction of them have been studied on toxicity quantification. QSAR can be used to predict the toxicity of a specific DBP or molecular property of DBP such as log P, which reflects hydrophobicity of a chemical compound. Experts use the principles of mechanism‐based structure and activity relationship (SAR) relative to known carcinogens, and mechanisms include structural analogy to known carcinogens, toxicokinetic and toxicodynamic factors, potency indicators for a structural analog, short‐term test data, and metabolic activation. Therefore, QSAR analysis has its unique role in setting drinking water standards.

QSAR analysis is based upon a simple fact that similar chemical structures are expected to exhibit similar chemical behavior. With tens of thousands of chemical structures to assess and many different empirical tests, QSAR is often used in strategic screening in product development through HI and risk assessment. Given sufficient knowledge on structurally or functionally related compounds, QSAR may be used for screening well‐defined biological, toxicological, or pharmacological endpoints of interest and associated kinetic characteristics such as absorption, distribution, metabolism, and excretion. In QSAR analysis, chemical structure is quantitatively correlated with a well‐defined process, such as biological activity, chemical reactivity, or toxicity of a chemical compound. The US EPA uses QSAR analysis to prioritize DBPs using three general types of predictive models:

1. Mathematical models such as SARs and QSARs use descriptors and mathematical relationships to derive predictions. Examples of SAR/QSAR models are ECOSAR and select modules contained in EPISuite™ like WSK_OWWIN.
2. Fragment‐based models such as the BioWin module within EPISuite evaluate the features of molecular fragments present on the molecule to make predictions.
3. Expert systems, like OncoLogic™, use rule‐based decision trees to mimic an expert’s judgment. Other expert systems utilize artificial neural networks and molecular models.

There are hundreds of chemicals that have been identified as DBPs, a large proportion of which fall into the general category of halogenated DBPs. Due to evidence of carcinogenicity and developmental effects, halogenated DBPs are of particular interest. Two major molecular descriptors can be used in predicting the toxicity of DBP. One is log P and the other is the number of carbon and halogen atoms. For QSAR analysis, halogenated DBPs are classified into eight classes as illustrated in Figure 2.14.

According to the molecular structure of DBPs, Pierotti (1999) developed a DBP database containing more than one thousand DBP compounds for the eight classes of DBPs. The database includes more than 20 different physical and chemical characteristics for each DBP compound. Major variables are chemical name, address, subclass, CAS number, SMILES, disinfection process, molecular formula and structure, molecular weight, log P, cLog P, log P reference, melting point (°C), boiling point (°C), vapor pressure (mm of Hg), solubility (mol/l), E_HOMO, E_LUMO, surface area, MCLG (mg/l), MCL (mg/l), 10‐day exposure for a 10 kg child (mg/l), long‐term exposure for 10 kg child (mg/l), long‐term exposure for a 70 kg adult (mg/l), mg/l at 10⁻⁴ cancer risk, cancer group, SDWA reference, and toxicological effect. The database is used in developing QSAR models in the following section to illustrate statistical methods such as regression, outlier detection, quantification of uncertainty and sensitivity, and validation of QSAR models.

2.4.1 Multiple Linear Regression (MLR)

Multiple linear regression (MLR) is a common algorithm in deriving QSAR models. It relates the dependent variable y to a number of independent (predictor) variables, x_j, by using a linear equation as follows:

(2.15)

where

• ŷ = calculated dependent variable
• x_j = predictor variable
• b_j = regression coefficient

To assess goodness of fit quantified by the correlation coefficient of multiple determination, (R²) is calculated by Equation (2.17). R² estimates the proportion of the variation of y that is explained by regression equation (Massart et al., 1997). If there is a perfect fit, R² = 1, while if there is no linear relationship between the dependent and independent variables, R² = 0:

(2.16)

where

• 
• 
• 
• 
• 
• 

In predicting the toxicity of a chemical compound, two important molecular descriptors are hydrophobicity such as log P and electronic properties such as the energy of the lowest unoccupied molecular orbitals (E_LUMO). Log P reflects the hydrophobicity of molecules, which often correlates well with the bioactivity of chemicals (Leo and Hansch, 1999). The logarithm of the partition coefficient (log P), also referred to as log K_ow, describes the distribution of a compound between organic (usually n‐octanol) and water phases by the following equation:

(2.17)

where

• 
• 

If log P is greater than zero (0), the compound has a greater solubility in the organic phase; if log P is less than zero (0), the compound has a greater solubility in the aqueous phase. Chen et al. (2016) reported that log P can be predicted accurately by the E_LUMO, the number of carbon (N_C), and the number of chlorine (N_Cl). A general MLR model is to predict log P is

(2.18)

To develop a robust QSAR model, outliers have to be detected to improve the correlation coefficiency of the QSAR model. The Hotelling test and the associated leverage statistics can be used to detect outliers. The leverage of a chemical provides a measure of the distance of the chemical from the centroid of its active set. Chemicals in the active set have leverage values between 0 and 1. A warning leverage (h^*), defined in Equation (2.19) (Eriksson et al., 2003), is a critical value to cut off the outliers from the dataset:

(2.19)

where

• 
• 
• 

The William plot has a double‐ordinate Cartesian plot of cross‐validation residuals (first ordinate), standard residuals (second ordinate), and leverage (Hat diagonal: abscissa) values (h), which defined the domain of applicability of the model as a squared area within ±2 band for residuals and a leverage threshold, h^*.

Figure 2.16 is a box plot that shows the contribution of each variable such as E_LUMO, N_Cl, and N_C to the standardized coefficients after the outliers were removed. It suggests that N_C contributes to log P most positively, while E_LUMO contributes to log P negatively to a lesser extent. The number of Cl and N_Cl contributes to log P positively but less than both N_C and E_LUMO.

Figure 2.17 shows that all the predicted log P values are within the boundary of 95% confidence of the observed log P values.

2.5 Quantification of Uncertainty

All the models have intrinsic uncertainty that has to be quantified to communicate to the public about risks. Constraints, uncertainties, and assumptions having an impact on the risk assessment should be explicitly considered at each step in the risk assessment. Therefore, the quantitative description of uncertainty in HRA, carrying capacity, and climate change are critical for policy makers and the general public. For example, the IPCC Fifth Assessment (2015) adopted the following technical terms in dealing with uncertainty due to the uncertainties in quantifying all consequences of different emissions:

• Confidence in the validity of a finding, based on the type, amount, quality, and consistency of evidence (e.g. mechanistic understanding, theory, data, models, expert judgment) and the degree of agreement. Confidence is expressed qualitatively.
• Quantified measures of uncertainty in a finding expressed probabilistically (based on statistical analysis of observations or model results or expert judgment).

For the policy makers, the IPCC adopted the following terms for quantitative description purposes to express the assessed likelihood of an outcome or a result. Seven different descriptive terms are assigned to seven different likelihood probability ranges as in Table 2.19.

To quantify different uncertainty, the US EPA identifies scenario, parameter, and model uncertainty. In each category of uncertainty, the sources of uncertainty are identified with a specific example in Table 2.20.

Furthermore, the US EPA lists different quantification methods of uncertainty with examples in Table 2.21.

2.5.1 Quantification of QSAR Model’s Uncertainty

For a QSAR model in which r is function of j variables X₁, X₂, …, X_j:

(2.22)

The uncertainty due to errors in variable can be expressed as follows (Coleman and Steele, 1989):

(2.23)

(2.24)

When the above uncertainty equation is applied to log P as function r, while E_LUMO, N_Cl, and N_C are the three variables, the uncertainty equation of log P can be expressed in Equations 2.25 and 2.26:

(2.25)

(2.26)

When the data reduction expression is very complex and the task of computing the partial derivatives in the above equations is extremely laborious, Monte Carlo simulation (MCS) that is the most efficient quantification of uncertainty according to distribution of independent variables can be used (Cox et al., 2001). MCS is used as an example to quantify the uncertainty of the predicted log P using E_LUMO, N_Cl, and N_C as three independent variables.

2.5.2 Monte Carlo Simulation

Monte Carlo Simulation (MCS) has been extensively used for quantifying uncertainty of linear equations (Tang et al., 2009). Estimating propagation of error distributions by MCS is based on theoretical principles and supports a fully consistent and transferable estimation of measurement uncertainty. In general case, a linear model equation is to measure output Z indirectly obtained from the input variables X₁, X₂, …, X_n by a functional relationship F:

(2.27)

The knowledge about the values that may be reasonably attributed to quantities X_i, considered as continuous random variables, is expressed by their probability distribution function (PDF), , within the corresponding domain. An expectation or best estimate for the value of X_i E(X_i) and the uncertainty associated with this value, μ(X_i), assimilated to the standard deviation σ(X_i), are obtained from the PDF:

(2.28)

(2.29)

If the equation is linearized by means of a Taylor expansion about the point, the estimated uncertainty of the measured output μ(Z) is from the input μ = (μ₁, μ₂, …, μ_n) while the second‐ and higher‐order terms are neglected, the estimated uncertainty of μ(Z) from the input variables is calculated as follows:

(2.30)

Taking μ(Z) as the values F(μ) = F(μ₁, μ₂, …, μ_n) leads to the well‐known law of propagation of uncertainty:

(2.31)

In the Guide to the Expression of Uncertainty in Measurement (GUM, 1993), which is the internationally accepted master document for the evaluation of uncertainty, the combined standard uncertainty μ(Z) is evaluated from the standard uncertainty of the input variables, μ(X_i), and the covariances between correlated ones, cov(X_i, X_j), if all the input variables are independent, (X_i, X_j) = 0. The essence of the MCS is to simulate sampling to a target population with a given expectation μ_z and variance σ²(Z). A random sample of size M is obtained from the simulation of M independent and identically distributed random variables Z₁, Z₂, …, Z_M. According to the central limit theorem (Martinez, 2002), the distribution of the sum is approximately normal, with expectation M_µz and variance M_σ2:

(2.32)

According to the rule of “two sigmas,” the probability

(2.33)

By dividing Equation (2.33) by M on both sides, the expression becomes

(2.34)

By substituting into Equation (2.34), the expression becomes

(2.35)

This relationship is the foundation of the MCS because it establishes the rule to evaluate the error of set as . The sample variance may be best estimated using σ²:

(2.36)

After defining the coverage probability, P, the confidence interval for the result is evaluated as , where extremes correspond to the 2.5 and 97.5% percentiles of the sorted Z values. When the skewness value of the Z forecast discrete distribution is near zero, the confidence interval becomes symmetric and expanded uncertainty U(Z) is estimated by Equation (2.38):

(2.37)

MCS is a powerful tool in quantifying uncertainty in addition to experimental, analytical, or other numerical methods. MCS protocol can replace point estimates with random variables drawn from probability density functions. To perform MCS, a computer program is used to generate the pseudorandom numbers to simulate the values of the variables within a given PDF. Several commercial software programs such as Crystal Ball, LabVIEW, @RISK, and Analytica are the most popular for this purpose. There are three major steps if Crystal Ball by Oracle (2016) is used to carry out the MCS. First, a probability distribution is incorporated into a spreadsheet cell, and each time the spreadsheet is recalculated, a new value of the random variable is selected from the distribution and used for calculations. Second, the entire simulation is run at least 10 000 times to satisfy the required high number of trials. Each time new values of the random variables are selected, a new estimate of the final target is generated. Third, the results of simulations are summarized in a user‐friendly interface such as a table and a figure.

A step‐by‐step procedure for uncertainty assessment of regression QSAR model for halogenated DBPs is illustrated in Figure 2.21. The first step is the compilation of the DBP data. Three variables such as E_LUMO, the number of chorine (N_Cl), and the number of carbon (N_C) were used. The characterization of the probability distributions is carried out by statistic software SAS 9.4 to obtain the data average and standard deviation. All these parameters are fed into the MCS that gives the results in a probability distribution around a mean value that is used to carry out a detailed sensitivity analysis. A sensitivity analysis is applied to identify which parameters had the most impact on the predicted log P. If small modifications of one parameter characterized by a probability distribution strongly influenced the final result, it may be concluded that the sensitivity of the variable is very high. Sensitivity is crucial in determining what variables are the most important in predicting a dependent outcome. This can be analyzed by displaying the sensitivity as a percentage of the contribution from each parameter to the variance of the final result. Crystal Ball presents contribution in terms of percentage for each independent variable with the sum of percentage contribution to 100%. A general procedure in the uncertainty quantification of the predicted log P of a class DBP through MCS is outlined as follows:

1. Selection of significant sources of uncertainty. Molecular descriptors E_LUMO, N_Cl, N_C, the intercept b₀, and the coefficiencies b₁, b₂, and b₃ are the significant sources contributing to the model’s uncertainty.
2. Identification of the probability density function corresponding to the uncertainty sources selected. All uncertainty sources, e.g., E_LUMO, N_Cl, N_C, b₀, b₁, b₂, and b₃ are analyzed by Crystal Ball to obtain its best fit probability distribution functions. The results of the probability functions for each variable are then used in the best fit MLR equation to predict log P.
3. Selection of the number M of Monte Carlo Simulation trials. For example, MCS was carried out 10 000 times in order to have a sufficiently high number of trials.
4. Simulation of M samples {X_i1, X_i2, …, X_in,} for each X_i (E_LUMO, N_Cl, N_C, b₀, b₁, b₂, and b₃) uncertainty source, which was considered a random variable with a probability density function p(E_LUMO), p(N_Cl), p(N_C), p(b₀), p(b₁), p(b₂), p(b₃).
5. Computation of the M results {Z₁, Z₂, …, Z_n} by applying Equation (2.27) or (2.37) to M samples for each variable X_i (E_LUMO, N_Cl, N_C, b₀, b₁, b₂, and b₃).
6. Analysis and interpretation of Monte Carlo Simulation results.

Example 2.9 presents the MCS results produced by Crystal Ball software. The methodology can be used for the quantification of uncertainties of any regression curve in general.

2.5.4 Sensitivity Analysis by Monte Carlo Simulation

The influence of different variables on the outcome of model’s prediction can be visually presented through sensitivity analysis in Crystal Ball. Table 2.25 summarizes the sensitivity results for each DBP class. Figure 2.26 demonstrates that N_Cl is the most influential molecular descriptor in predicting log P of all DBP classes, except halogenated alkane.

No.	Compound class	ELUMO (%)	NCl (%)	NC (%)
1	Halogenated alkane	−16.0	0.8	83.2
2	Halogenated alkene	−44.0	53.9	2.1
3	Halogenated aromatic	0.6	90.6	8.7
4	Halogenated aldehyde	−7.3	64.9	27.8
5	Halogenated ketone	0.0	93.0	7.0
6	Halogenated carboxylic acid	36.5	55.6	7.9

Table 2.25 shows the major difference between sensitivities of single bond compounds, such as halogenated alkanes. Log P of chlorinated alkane changes significantly with the length of the carbon chain, N_C. However, for all the other classes of DBPs containing unsaturated π bond, log P will change mostly with the number of chlorine, N_Cl. Log P reflects the amount of hydrogen bonding between the chemical compound and the hydrogen in the water molecule in the absence of oxygen in the chlorinated alkanes. It appears that when chlorine attaches to carbon, it only has influence on the electron cloud of the attached carbon and such electronic interaction is tightly bonded; therefore, the number of chlorine on the chlorinated alkane carbon chain does not significantly influence the strength of hydrogen bonds. However, with DBP compounds containing unsaturated π bonds, the chlorine atom would attract electron cloud to itself since most compounds containing unsaturated bonds may have a resonance structure through which the influence of chlorine will be transmitted. As a result, the strength of the hydrogen bond between the chemical compound and water molecule would be significantly influenced by the number of chlorine, N_Cl. Therefore, log P is more sensitive to the number of chlorine.

2.6 Exercise

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