Risk-based principles and incompleteness theorems for linear dose-response extrapolation for carcinogenic chemicals

Chemosphere 247 (2020) 125934

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Risk-based principles and incompleteness theorems for linear dose-response extrapolation for carcinogenic chemicals Zijian Li School of Public Health (Shenzhen), Sun Yat-sen University, Guangdong, 510275, China

h i g h l i g h t s

g r a p h i c a l a b s t r a c t


Risk-based principles and theorems were introduced to extrapolate linear models.
The dose-based static and hazard risks were defined to derive the cancer slope factors.
The safe ranges of the cancer slope factors for 708 carcinogenic chemicals were simulated.
Individual variability significantly impacted the selection of linear models.
Linear extrapolation might not be practical for a homogenous population.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 November 2019 Received in revised form 9 January 2020 Accepted 15 January 2020 Available online 16 January 2020

To conduct better health risk assessments, this study introduced two risk-based principles and a series of line-lognormal-intersection theorems that helped derive the safe ranges of the cancer slope factors (CSFs) for 708 carcinogenic chemicals. The extrapolated linear dose-response relationships presented in this study can ensure safety with respect to both static and dose-based instantaneous risks compared to the lognormal dose-response model. The theorems proved that the maximum static and dose-based hazard risk ratios of a lognormal curve and a linear model are independent of a chemical’s toxicity (the effect dose that corresponds to a 50% response, or ED50), where the selected linear extrapolation (m value) and the individual variability (s) of the responses to carcinogenic chemicals are two determining factors. The theorems also indicated that individual variability determines the range of m if the acceptable risk ratios were regulated. When s was 1.36 (i.e., the 50th percentile of the individual variability’s lognormal distribution), the safe range of m was derived as [11.22, 21.46] (i.e., from ED11.22 to ED21.46); if the 95th percentile of the s lognormal distribution was used, the safe range of m was [1.13, 4.57] (i.e., from ED1.13 to ED4.57). This study also showed that for a relatively homogenous population (i.e., s is relatively small) that has similar characteristics, the linear dose-response extrapolation method might not be completely effective due to the shape shift of the lognormal curve that draws the static risk of the extrapolated linear model away from the lognormal model. © 2020 Elsevier Ltd. All rights reserved.

Handling editor: A. Gies Keywords: Cancer risk assessment Low-dose extrapolation Dose-response modeling Static and instantaneous risk

1. Introduction Once humans entered the industrial era, tons of synthetic chemicals were made and released into the environment. These E-mail address: [email protected]. https://doi.org/10.1016/j.chemosphere.2020.125934 0045-6535/© 2020 Elsevier Ltd. All rights reserved.

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Z. Li / Chemosphere 247 (2020) 125934

chemicals such as pesticides, Polycyclic Aromatic Hydrocarbons (PAHs), Polychlorinated Biphenyls (PCBs), petroleum products and so on can enter the human body through all kinds of exposure pathways (Sharma et al., 2009; Ghanbari and Moradi, 2017; Godoy et al., 2019; Martin et al., 2018; Lui et al., 2017; Dong et al., 2019; Luo et al., 2019; Gao et al., 2019; Helmfrid et al., 2019; Fantke et al., 2012, 2019; Das et al., 2020), of which the oral route always dominates the overall exposure and it includes the ingestion of chemical contaminated soil, water, and food (Jennings, 2010; Irvine et al., 2014; Li, 2018; Valcke et al., 2017). Most of these chemicals are toxic to humans, especially carcinogenic chemicals, even when people ingest them at very low doses, which might be due to evolution. That is, our biological systems do not naturally filter our exposure to synthetic chemicals. People from worldwide countries are suffering from cancers resulting from exposure to carcinogenic chemicals, especially when it is related to their daily lives (Wogan et al., 2004), which significantly increases the medical burden and hinders the sustainable development of our society (Lin et al., 2018; Madia et al., 2019). Therefore, assessing human health risks and properly setting regulations are significant to understanding and quantifying the toxicity for carcinogenic chemicals. The individual cancer incidence risk that is induced by carcinogenic chemicals is determined by many complexed factors, including enzyme’s metabolic activity, cell detoxification, the DNA Repair mechanism, and so on (Hattis and Barlow, 1996; Koual et al., 2019). This individual variability in the responses to carcinogenic chemicals usually results in a lognormal distribution of the cancer risks among a population (Hattis and Barlow, 1996; National Research Council, 2009; Saltzman, 1987, 2001), which is commonly plotted as a lognormal dose-response relationship and is derived from animal testing studies. The lognormal doseresponse curve is important for health risk assessments and indicates that higher cancer incidence rates disproportionately occur at higher doses and are determined by multiple independent random variables (Saltzman, 1987, 2001). Except for some highdose exposure incidents, e.g., occupational medical accidents, people are usually exposed to toxic chemicals at low doses; for example, the cancer risks are within 106 and 104, for which the low dose extrapolation from the dose-response curve is always applied to facilitate and simplify the life cycle assessment process (Olin et al., 1995; USEPA, 2005; Crettaz et al., 2002; Cogliano et al., 2012). In addition, some environmental agencies such as USEPA adopted the benchmark dose (BMD10) or lower bound effect dose (LED10) method for the regulatory process, which is estimated from the 95% lower confidence limit (Crettaz et al., 2002; USEPA, 1996a, 1996b, 2005). This estimate is also a relatively conservative compared to other low-dose extrapolation models, such as the multistage, Weibull, multi-hit, and probit models (Stara and Erdreich, 1984; Saltzman, 1987). In addition to low dose extrapolation, the ED10 (Effective Dose) method is commonly applied in the life cycle assessment studies. This method is derived by selecting the maximum likelihood estimate of the dose that makes 10% of the population have cancer, which is relatively independent of the model selection and does not apply the statistical upper bounds that could distort the relative rankings (USEPA, 1989; Cogliano, 1996; Cogliano et al., 2012). Other linear extrapolation methods such as ED01 and ED05 have also been derived from chronic low dose animal tests (Cranmer, 1981; Kodell et al., 1983; National Research Council, 2006), and even linear extrapolation using higher tumor (malignant) incidences such as ED20 and ED40 has been widely applied (Schneider et al., 2002). The linear extrapolation method can be applied as an empirical formula. This approach is practical and has been widely used for risk assessment, e.g., derivation of the cancer slope factors (CSFs), especially for the ED10 method on the life cycle assessment or the

BMD10 method on the guideline regulation (Crettaz et al., 2002; USEPA, 1996a, 1996b; Slob, 1999; Warren et al., 2004). Compared to a lognormal model, the extrapolated linear dose-response relationship has an intrinsic slope at the origin point of the coordinates, which immediately introduces a significant instantaneous risk, even when the dose of a chemical is zero. This study defined the dose-based instantaneous risk, or the dose-based hazard, as the chance of the occurrence of cancer as a function of the exposure dose, which should approach zero when a population is exposed to a very low dose of a chemical because there should be only a small portion of people who are very sensitive to the carcinogen in a population due to their relatively vulnerable anticancer biological systems. In addition, the dose-based hazards for cancer risks would dramatically increase as the exposure dose increases because of the large portion of people who will have increased tumor responses to carcinogenic chemicals at some certain dose intervals. The linear extrapolation method might ignore this dynamic hazard risk, which could be further magnified by the inaccurate measurement of the exposure dose during the animal-human toxicology data transaction and the risk assessment processes, including inaccurate experiments, the uncertainty in intra- and inter-species analysis, etc. In addition to the hazard risk, the estimated health risk, or the static risk, from linear extrapolation could also cause large inaccuracies, although in most cases, linear extrapolation overestimates the real risk that is allowed due to the precautionary principle. However, the estimated static risk from the linear extrapolation should not be too much higher than the real risk; otherwise, it can affect risk assessments. So far, the linear extrapolation approach, especially for the ED10 method, has been widely applied, but no studies have been performed to further discuss it from both static and hazard risk perspectives. To solve this issue, this study introduced two risk-based principles and a series of the line-lognormalintersection theorems to determine a reliable linear dose-response relationship, based on which the safe ranges of the CSFs for carcinogenic chemicals were derived. Hopefully, the methods and results that are provided by this study could help to better manage cancer risks and conduct more accurate risk assessments. 2. Modeling framework In this study, a lognormal model was applied to fit the human cancer dose-response curves for carcinogenic chemicals that were based on animal experimental data and human-lifetime-exposure derivations (Huijbregts et al., 2005). Fig. 1 illustrates the modeling framework that is introduced for the linear dose-response extrapolation of carcinogenic chemicals. The limitations of the linear extrapolation were illustrated from the static risk and the dosebased hazard risk perspectives. The static risk, or health risk, is the probability that cancer occurs in the human population, which is also equal to the y-axis values. The dose-based hazard risk, or instantaneous risk, indicates the likelihood of cancer occurring due to the lifetime-ingested doses of carcinogenic chemicals. Mathematically, the linear extrapolation creates a difference between the linearly extrapolated risk and the lognormal-based risk, which could significantly overestimate or underestimate the real risk. This study regulated and restricted these risk differentials based on two main principles and further quantified them by introducing a series of line-lognormal-intersection theorems. Furthermore, the safe range for the linear dose-response extrapolation, i.e., the CSFs, were derived for carcinogenic chemicals. 2.1. Dose-based hazard risk functions In this study, the dose-based survival and hazard functions for carcinogenic chemicals were applied. In them, the cancer incidence

Z. Li / Chemosphere 247 (2020) 125934

3

Fig. 1. Modeling framework for the linear dose-response extrapolation of carcinogenic chemicals.

among a human population was a function of the lifetime ingestion of carcinogenic chemicals. It should be noted that this approach could also be applied for daily-dose-based toxicological studies, such as the original animal test data, and this study adopted the lifetime-adjusted dose to conduct the human health risk assessment, especially since carcinogenic chemicals are typically evaluated using the lifetime exposure. Therefore, x in Eq (1). denotes the continued survival dose (kg) of the chemicals that are ingested over a lifetime using a probability density function (PDF) f ðxÞ and a cumulative distribution function (CDF) FðxÞ.

FðxÞ ¼ PðX  xÞ

(1)

Then, the dose-based survival function could be expressed as follows:

SðxÞ ¼ PðX > xÞ ¼ 1  FðxÞ

dx/0

Pðx < X  x þ dxÞ f ðxÞ ¼ dx SðxÞ

ðx HðyÞdy

(4)

0

2.2. Linear dose-response relationship extrapolation The general form of the linear low-dose extrapolation that is commonly applied to approximate the dose-response relationship, e.g., ED10, can be expressed as follows:

bEDm% ¼

m0 0:01m ¼ EDm  ED0 EDm

HðxÞ ; 0 < x  EDm Hb ðxÞ

(6)

where Hb ðxÞ is the hazard function for the extrapolated linear doseresponse relationships of chemicals. Likewise, the ratio function for the sum of instantaneous risks, RL ðxÞ, could be expressed as follows:

ðx RL ðxÞ ¼

LðxÞ ¼ ð x0 Lb ðxÞ 0

HðyÞdy ; 0 < x  EDm Hb ðyÞdy

(7)

where L denotes the cumulative risks.

(3)

To further evaluate and compare the sum of the instantaneous risks, i.e., the cumulative risks, between the simulated curve and the linear low-dose response, the following cumulative hazard function was applied.

LðxÞ ¼

RH ðx; mÞ ¼

(2)

where SðxÞ denotes the probability that someone survives beyond that chemical dose, i.e., they do not suffer from cancer due to their lifetime ingestion. Then, the dose-based hazard function HðxÞ, i.e., the dose-based instantaneous risks for cancer effects, can be expressed as follows:

HðxÞ ¼ lim

for m% of the selected population. This study extrapolated the linear relationships and bEDm s for chemicals to further compare the dose-based static and instantaneous risks using the experimentbased model (i.e., a lognormal model). In it, the instantaneous risk ratio function, RH ðx; mÞ, was introduced as follows:

(5)

where bEDm (kg1) is the lifetime-adjusted human cancer slope factor that is derived from the effect dose (EDm ) that causes cancer

2.3. Linear extrapolation from a lognormal model This study applied the lognormal model to fit the cancer doseresponse curve for the human population as X  LNðm; s2 Þ, where m is the mean of the natural logarithms of the lognormally distributed dose values, i.e., lnED50. The ED50 values of the lifetime ingestion of carcinogenic chemicals were taken from Huijbregts et al. (2005). s is the standard deviation of the natural logarithms of the dose values, which indicates the variability of individual responses to carcinogens. This individual variability can be attributed to gender, age, genetic expression, health conditions, etc. Due to the toxicological data limit for specific carcinogens, only the variability in human susceptibility for general carcinogenic effects was estimated (Huijbregts et al., 2005). A central estimated s was simulated as 1.36 (i.e., 0.59 for the decadic logarithms) with the 5th to 95th percentiles as [0.85, 2.10] (i.e., [0.37, 0.91] for the decadic logarithms) due to the variability in individual metabolism, genetic activities, detoxification, pharmacokinetic factors, and uptake parameters (Hattis et al., 1986, 1996, 1997). Then, the dose-based survival SLN ðxjm; sÞ, hazard HLN ðxjm; sÞ, and cumulative hazard LLN ðxjm; sÞ functions for a lognormal model were expressed as follows:

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SLN ðxjm; sÞ ¼ 1  F



lnðxÞ  m

 (8)

s

  m 4 lnðxÞ s   HLN ðxjm; sÞ ¼  m sx 1  F lnðxÞ s    lnðxÞ  m LLN ðxjm; sÞ ¼  ln 1  F

s

 (9)



Hb ðx; mÞ ¼

f ðx; mÞ ¼ Sðx; mÞ

0:01m

EDm 1

0:01  ¼ EDm m  0:01x x

(11)

(12)

0:01m EDm

    0:01m x þC Lb ðx; mÞ ¼  ln 1  EDm



ln 1 

exp½mþsF

ð0:01mÞ

(18)

0:01m 1 exp½mþsF ð0:01mÞ

(10)

where F($) and 4($) are the CDF and PDF of the standard normal distribution, respectively. Likewise, the dose-based survival Sb ðx; mÞ, hazard Hb ðx; mÞ, and cumulative hazard Lb ðx; mÞ functions based on the selected EDm value were derived as follows:

 0:01m x; 0 < m < 100; 0 < x  EDm Sb ðx; mÞ ¼ 1  EDm

LLN ðxjm; sÞ ¼ Lb ðx; mÞ    m ln 1  F lnðxÞ s   0:01m x þ 1

RL ðx; mjm; sÞ ¼

(13)

3. Principles and incompleteness theorems 3.1. Principles for linear dose-response extrapolation The foundation of the two principles of the linear dose-response extrapolation that is introduced in this study adopts the precautionary approach. Based on the precautionary approach, the first principle (Principle 1) is that the estimated static cancer risks from the linear dose-response extrapolation should be always larger than the experiment-based risks, i.e., the lognormal model in this study. In addition, the approximated linear risks should not significantly overestimate the cancer risks, which indicate that the maximum static risk differential between the linear extrapolation and the lognormal model should be within an acceptable range. Therefore, Principle 1 can be expressed as follows. Principle 1:

In Eq. (13), C is an arbitrary constant for the indefinite integral, which could be solved by mathematically setting x as zero.

MinfFb ðxÞ  FLN ðxÞg  0; 0 < x  EDm

(19)

    0:01m 0 þ C ¼ ln 1  EDm

MaxfFb ðxÞ  FLN ðxÞg  DA FLN ðxÞ; 0 < x  EDm

(20)

0:01m

0:01m  ¼ EDm 0 1  0:01m EDm EDm

(14)

Principle 1 can also be simplified and combined as follows:

Therefore, the cumulative hazard function for the linear doseresponse function was expressed as follows:

    0:01m 0:01m x þ Lb ðx; mÞ ¼  ln 1  EDm EDm

(15)

For the lognormal dose-response curve, EDm can be expressed using m based on the inverse function for a lognormal CDF as follows:

h i EDm ¼ exp m þ sF1 ð0:01mÞ ; 0 < m < 100

(16)

where F1($) denotes the percent point function of the standard normal distribution. To compare the instantaneous cancer risks that were extrapolated by the linear model and the lognormal model, the dose-based hazard risk ratios, including the RH ðx; mÞ and RL ðx; mÞ that were derived from Eqs. (6) and (7), respectively, were expressed as follows:

  1 m 4 lnðxÞ s B C H ðxÞ  C  ¼B RH ðx; mjm; sÞ ¼ LN @ A Hb ðx; mÞ m sx 1  F lnðxÞ s

Fb ðxÞ   1 þ DA ; 0 < x  EDm FLN ðxÞ

(17)

(21)

where Min{$} and Max{$} denote the minimum and maximum value functions, respectively, and DA is the acceptable differential factor. For example, if DA is set as 1.0, i.e., 100% exceedance, it means that the maximum static risk ratio of the linear extrapolation to the lognormal model should be less or equal to 2.0. In addition to the static risk ratio, the second principle (Principle 2) was introduced based on the instantaneous risk ratio that was defined in Eq (17). to describe the dynamic risk of the dose-based dimension. Compared to the first principle, the estimated instantaneous risk for the linear dose-response extrapolation could be a little bit lower than the experiment-based risks since it indicates a potential value that should still be within the acceptable range. Therefore, Principle 2 could be expressed using RH ðxÞ as follows. Principle 2:

RH ðxÞ ¼

0

    exp m þ sF1 ð0:01mÞ x 0:01m

1

HLN ðxÞ  1 þ RA ; 0 < x  EDm Hb ðx; mÞ

(22)

where RA is the selected acceptable exceedance ratio factor for the hazard ratio. For example, if RA is set as 0.50, i.e., 50% exceedance, it means that the maximum hazard ratio of the lognormal model and the linear extrapolation should be less than or equal to 1.50, which indicates that the linear extrapolation model should not significantly underestimate the instantaneous risk.

Z. Li / Chemosphere 247 (2020) 125934

3.2. Theorems for linear-lognormal intersection 3.2.1. Theorem 1 The boundary conditions for principle 1, i.e., to obtain m, could be solved using the line-lognormal CDF intersection theorems. The F ðxÞ

lower boundary condition problem (FLNb ðxÞ  1) could be solved using Theorem 1 Theorem 1. A straight line function Fb ðxÞ that passes through the origin point of the coordinates intersects the real-valued continuous

5

   0:01m exp m  s2 EDm rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    pffiffiffiffiffiffiffi 2s3 F1 ð0:01mÞ  2s2 ln 0:01ms 2p þ s4     1 þ DA FLN ED1st m ! where FLN ðED1st m Þ ¼ F

lnðED1st m Þm

s

. Then,

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    pffiffiffiffiffiffiffi 2sF1 ð0:01mÞ  2 ln 0:01ms 2p þ s2 0:01m exp  s2  sF1 ð0:01mÞ  s ffi  1 þ DA   rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 F  2sF ð0:01mÞ  2 ln 0:01ms 2p þ s2  s

lognormal ðm; s2 ÞCDF FLN ðxÞat the point ðEDm ; 0:01mÞ. For any F ðxÞ

EDm 2ð0; mode, it makes Fb ðxÞ  FLN ðxÞ, i.e., FLNb ðxÞ  1, where x2 ð0; EDm Þ. The proof of Theorem 1 was provided in Appendix I. Therefore, based on Theorem 1, the range of m could be preliminarily determined as follows.

0 < m  100F

 

 ln exp m  s2  m

s

(25)

(26)

3.2.3. Theorem 3 Based on Theorems 1 and 2, Theorem 3 can help to further determine the range of m. Theorem 3. A straight line function Fb ðxÞ that passes through the origin point of the coordinates intersects the real-valued continuous lognormal ðm; s2 ÞCDF FLN ðxÞat the point ðEDm ;0:01mÞ. The maximum ðxÞ value for the ratio of their hazard functions, i.e., RH ðxÞ ¼ HHLN ðx;mÞ, exists b

¼ 100FðsÞ

(23)

when x ¼ EDm and that maximum value is independent of m, i.e., lnðED50 Þ. The proof of Theorem 3 was provided in Appendix III. Therefore, based on Theorem 3, the range of m could be further determined as follows:

 4 F1 ð0:01mÞ  1 þ RA 0:01ms

(27)

3.2.2. Theorem 2 The upper boundary condition problem, i.e.,

Fb ðxÞ FLN ðxÞ

 ð1 þ DA Þ,

can be solved using Theorem 2. Theorem 2. A straight line function Fb ðxÞ that passes through the origin point of the coordinates intersects the real-valued continuous lognormal ðm; s2 ÞCDF FLN ðxÞat the point ðEDm ;0:01mÞ. The maximum value of Fb ðxÞ  FLN ðxÞexists at the first critical point where F ’LN ðxÞ ¼ F ’b ðxÞ ¼ 0:01m EDm and x2ð0; EDm Þ. Definition: The first critical point is the point ðED1st m ; 0:01mÞ on

1st 1st FLN ðxÞ where F ’LN ðxÞ ¼ F ’b ðxÞ ¼ 0:01m EDm and EDm < mode, i.e., EDm <

expðm  s2 Þ. The proof of Theorem 2 was provided in Appendix II. Therefore, based on Theorem 2, the range of m could be further determined as follows. At the first critical point, we obtain the following:

    1 þ DA FLN ED1st Fb ED1st m m where ED1st m could be solved using Eq. (25) below. Then,

(24)

3.2.4. Theorem 4 Theorems 1e3 could be used to extrapolate the linear doseresponse relationships based on the static and instantaneous risk analysis. It should be noted that theorems 1e3 based on Principles 1 and 2 are incomplete for optimizing the linear dose-response extrapolation, although it is very convenient to choose the mode of the lognormal model to narrow down the range of m. Theorem 4, i.e., the complementary theorem for Theorems 1e3, could be applied to help completely describe Principles 1 and 2 by sacrificing the simplicity. However, when the simulated range of m due to choosing the mode according to Theorem 1 is not compatible with Theorems 2 and 3 or is too narrow, then the complementary theorem should be used to further determine m’s range. Theorem 4. (complementary theorem): A straight line function Fb ðxÞthat passes through the origin point of the coordinates intersects with the real-valued continuous lognormal ðm; s2 ÞCDF FLN ðxÞat the second critical point of ðED2nd m ; 0:01mÞ. 1. For any x2ð0; ED2nd m Þ, it makes Fb ðxÞ > FLN ðxÞ, i.e.,

FLN ðxÞ Fb ðxÞ < 1.

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Z. Li / Chemosphere 247 (2020) 125934

2nd 2 2. ED2nd m > mode, i.e., EDm > expðm  s Þ.

Definition: The on FLN ðxÞ where

second critical point is the point ðED2nd m ; 0:01mÞ 2nd F ’LN ðxÞ ¼ F ’b ðxÞ ¼ 0:01m and ED > mode, i.e., m ED2nd m

2 ED2nd m > expðm  s Þ The proof of Theorem 4 was provided in Appendix IV.

Corollary. A straight line function Fb ðxÞthat passes through the origin point of the coordinates intersects with the real-valued continuous lognormal ðm; s2 ÞCDF FLN ðxÞ. At the first 2nd ðED1st m ; 0:01mÞand the second ðEDm ; 0:01mÞcritical points on the

FLN ðxÞcurve where

F ’LN ðxÞ

’

¼ F b ðxÞ, the probabilities PðX 

ED1st m Þand

3.3.1. Step 1 Based on Theorem 1, the mode point of the lognormal doseresponse CDF model was used as the initial screen to obtain the maximum m, which was expressed as follows:

 

 ln exp m  s2  m ¼ 100FðsÞ 0 < m  100F

s

3.3.2. Step 2 Based on Theorem 2, the maximum static risk differential at the first critical point was used to narrow down the range of m as follows:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    pffiffiffiffiffiffiffi 2sF1 ð0:01mÞ  2 ln 0:01ms 2p þ s2 0:01m exp  s2  sF1 ð0:01mÞ  s ffi  1 þ DA   rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 2 F  2sF ð0:01mÞ  2 ln 0:01ms 2p þ s  s

PðX  ED2nd m Þ are independent of m, i.e., lnED50 . The proof of Corollary was provided in Appendix IV. Then, based on Theorem 4, the maximum m, complementary to Theorem 1, could be determined as follows:

 4 F1 ð0:01mÞ ¼1 1:36  102 m

(28)

Theorem 4 could be used to find the theoretically maximum EDm (i.e., the second critical point or tangent point) for the linear dose-response extrapolation that could ensure that the estimated static health risk would higher than the experimental-based risk (i.e., the lognormal model) according to Principle 1. To avoid the complex tangent function, the mode of the lognormal distribution can be applied to help quantify EDm for the linear extrapolation based on the incompleteness theorems because the mode is usually close to ED2nd m , as shown in the result section, and the mode of a lognormal distribution is present where the peak probability happens. Therefore, although it is mathematically incomplete and imperfect, implementing the mode as one of the boundary conditions can maintain the simplicity and typicality.

(29)

(30)

where DA was determined in the results and discussion section (e.g., DA  1). 3.3.3. Step 3 Based on Theorem 3, the maximum hazard risk ratio at the intersection point of the lognormal CDF and linear model was used to further narrow down the range of m as follows:

 4 F1 ð0:01mÞ  1 þ RA 0:01ms

(31)

where RA was determined in the results and discussion section (e.g., RA  0:25). 3.3.4. Step 4 (if necessary) Based on Theorem 4, the second critical point (i.e., tangent point) could be used as a complementary method to re-determine the maximum m as follows.

 4 F1 ð0:01mÞ ¼1 1:36  102 m

(32)

3.3. Derivation for the safe range of m Based on the principles and theorems above, the safe range of m was derived by the following steps.

3.3.5. Integration of Steps 1e4 Steps 1e4 could be integrated as a group of inequality functions as follows:

4 F1 ð0:01mÞ



 8 mmax ∽Ið1; if T13T2⊎3; 0; if T1IT2⊎3Þ½m ¼ 100FðsÞ þ ð1  IÞ ¼ 1 ; T1⊎T4 > 0:01ms > > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > >     p ffiffiffiffiffiffiffi > > > 2sF1 ð0:01mÞ  2 ln 0:01ms 2p þ s2 < 0:01m exp  s2  sF1 ð0:01mÞ  s m ffi  1 þ DA ; T2  rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  > pffiffiffiffiffiffiffi > 1 2 > > F  2sF ð0:01mÞ  2 ln 0:01ms 2p þ s  s > > > > > :

4 F1 ð0:01mÞ  1 þ RA ; T3 0:01ms

(33)

Z. Li / Chemosphere 247 (2020) 125934

where “∽” denotes m’s value or range, which could be further derived using the following equations including inequalities. I($) denotes the indicator function. T is the Theorem that is introduced in this study. “T13T2⊎3” means that the computed maximum m (mmax) based on Theorem 1 is compatible with Theorems 2 and 3; therefore, when an incompatibility emerges, i.e., T1IT2⊎3, Theorem 4 will be applied to derive mmax. It should be noted that if the derived mmax using Theorem 1 for the mode of a lognormal distribution is too small, even though it is compatible with Theorems 2 and 3, Theorem 4 could also be used to expand the range of m. Then, the range of CSF for carcinogenic chemicals could be expressed as follows:

CSF ¼

0:01m 0:01m ¼   EDm exp m þ sF1 ð0:01mÞ

(34)

4. Model applications and discussion 4.1. Impact of the selected linear relationship on hazard ratios To demonstrate the impact of the selected linear relationship (m), i.e., linear dose-response extrapolation, on the dose-based hazard risk (instantaneous) ratio of the experiment-based human model and the linear approximation, i.e., RH ðx; mjm; sÞ, the lognormal dose-response model of DDT for human health was applied with parameters (natural logarithm) 3.61 (m) and 1.36 (s) (Huijbregts et al., 2005; Hattis, 1997). The predicted model could be expressed as follows:

Fig. 2. Dose-based hazard (instantaneous) risk ratio RH Hb DDT ðx; mÞ plotted as a function of ln x and m.

DDT ðx; mj3:61; 1:36Þ

7

  1 4 lnðxÞ3:61 1:36 B C B  C  DDT ðx; mj3:61; 1:36Þ ¼ @ A 1:36x 1  F lnðxÞ3:61 1:36 0

RH

 

   exp 3:61 þ 1:36F1 ð0:01mÞ x 0:01m

(35)

Fig. 2 illustrated the hazard risk ratio for DDT as a function of x and m ð1  m  25Þ, which presented a nearly half-curve surface (0 < x  14:80). When 0 < m < 5:28, the simulated RH DDT values at the high end of the dose distribution, i.e., x/EDm , are higher than 1.50 over the intervals, and by decreasing this range of m, the linear extrapolation could increase the hazard ratio of the linear model to the lognormal model by up to 1.50 or higher. The simulated RH DDT values at the high end of the distribution decrease as m increases because the multiplicative independent variables that respond to DDT among a population magnify the potential cancer effects along the dose axis, which would affect the lognormal dose-response model that increase the dose-based hazard risk when m is larger. Meanwhile, the linear extrapolation approximation has a relatively flat change for the dose-based hazard risk, which assumes a continuous cancer response to carcinogenic chemicals as a function of one or more independent variables. This phenomenon demonstrates that the linear extrapolation that is defined for very low dose of DDT, i.e., less than ED5.28, could seriously underestimate the dose-based instantaneous risk, especially when the measurement inaccuracies could be magnified by the large dose-based hazard ratio. It must be noted that the simulation in Eq (35). was based on an s of 1.36 (natural logarithm), i.e., the central estimate from the human population (Hattis, 1997). When 5:28  m < 11:22, the simulated RH DDT values near the

for DDT of the experiment-based human model HLN

DDT ðxj3:61; 1:36Þ

to the linear approximation

8

Z. Li / Chemosphere 247 (2020) 125934

high values of the dose interval could be higher than 1.25 over the intervals, indicating a 25% hazard risk exceedance of the experiment-based lognormal dose-response model to the linear model. This range of m includes the commonly used ED10 linear extrapolation method, i.e., m ¼ 10, which indicates that the ED10 method could underestimate the dose-based instantaneous cancer risk by at least 25%, although this usually happens at doses that are near the high end of the interval. As m increases over its interval, the extrapolated linear dose-response relationships, i.e., bEDm ¼ 0:01m, EDm

tend to diminish the instantaneous risk ratios. The simulated

RH DDT ðx; mj3:61; 1:36Þ values for DDT also indicated that when x approaches zero, i.e., at very low doses, the hazard ratio will approach zero due to the intrinsic slope of the linear extrapolation. For example, based on the common ED10 extrapolation method, RH DDT is 1.29 at the intersection point (6.48, 0.10), but that value dramatically decreases to 0.55 when x ¼ 1.0 kg and to 0.015 when x ¼ 0.1 kg. Additionally, the corresponding m (i.e., 8.69) for the mode of the DDT dose distribution is within this range, and this means that even the derivation of the linear dose-response relationship at the maximum of the DDT-dose-response PDF could ignore the instantaneous risk exceedance. At this point, since the mode-derived linear extrapolation could not effectively manage the instantaneous risk due to the incompleteness of Theorem 1, i.e., the derived range of m based on Theorem 1 is incompatible with Theorem 3, further determination of the range of m using Theorem 4 (the second critical point) is required. If the linear extrapolation is derived at the mode point (5.83, 8.69  102), i.e., the human lifetime cancer slope factor (CSF) for DDT is 1.49  102 kg1, the maximum dose-based hazard ratio at the mode (i.e., x ¼ 1.49  102) is 1.34, and approximately 36.88% (i.e., 3:68  x  5:83) of the whole length of the interval for the mode-based linear extrapolation causes hazard ratios of over 1.25. The mode of the

Fig. 3. Dose-based cumulative hazard risk ratio RL Lb DDT ðx; mÞ plotted as a function of ln x and m.

DDT ðx; mj3:61; 1:36Þ

lognormal DDT dose-response model indicates that the lifetime cancer incidence appears most often for a human population; therefore, the dose-based hazard for the lognormal model near the mode should be attentively addressed using a toxicology study. When the selected EDm passes the second critical point (12.64, 21.46  102), i.e., the tangent and intersection point of the linear and lognormal models, the dose-based hazard risk ratios for DDT at the maximum (i.e., the intersection point according to Theorem 3) begin to be lower than 1.0 because at the second critical point, the slopes of the linear and lognormal models are equal, and when m is larger than 21.46,

HLN ðxÞ Hb ðx;mÞ

becomes a downward-concave function

over x  12:64. Although the instantaneous risk is well controlled by selecting EDm over the second critical point, the static health risk that is calculated from the linear model could exceed the lognormal model based on Theorem 2. To further evaluate whether the hazard functions and the corresponding risk ratios for DDT are changing for different doses, the dose-based cumulative hazard risk ratio of LLN ðxj3:61; 1:36Þ to Lb ðx; mÞ was derived as follows:

RL ðx; mj3:61; 1:36Þ ¼    ln 1  F lnðxÞ3:61 1:36     0:01m x þ ln 1  1 exp½3:61þ1:36F

ð0:01mÞ

0:01m 1 exp½3:61þ1:36F ð0:01mÞ

(36) For the overall selected linear approximations, the simulated RL ðx; mj3:61; 1:36Þ values are lower than 1.0 over its intervals, indicating that the overall changing hazard of the DDT dose for the lognormal model is lower than that for the linear extrapolation. This finding can be further illustrated by setting m to 1.0 in Fig. 3,

for DDT of the experiment-based human model LLN

DDT ðxj3:61; 1:36Þ

to the linear approximation

Z. Li / Chemosphere 247 (2020) 125934

which showed that the dose-based hazards are increasing over the DDT dose for both the lognormal and linear models, and the average increase of the hazard for the linear extrapolation was greater, though the increasing hazard rate for the lognormal model is higher over the interval ð0 < x  EDm Þ, which can be further demonstrated by the increasing cumulative ratio, i.e., RL DDT ðx; mj3:61; 1:36Þ: The phenomenon of the increasing hazard risk for different doses can be further discovered when m increases, i.e., RL ðx; mj3:61; 1:36Þ increases as m increases over the selected interval. For example, when m ¼ 5.0 (EDm ¼ 3.95 kg), the maximum value of RL ðx; 5:0j3:61; 1:36Þ is 0.80 at the intersection point, and this maximum value goes up to 0.87 at the intersection point when m reaches 10.0, i.e., the ED10 linear dose-response extrapolation. Compared to hazard functions where RH DDT ðx; mj3:61; 1:36Þ decreases as m increases over the selected interval, the increasing cumulative hazard ratios of the lognormal and linear models when m increases indicated that the hazard for the lognormal model increases faster than the linear model as the DDT dose increases. 4.2. Impact of chemical toxicity on hazard ratios To explore the impact of a chemical’s toxicity on hazard ratios, the current ED10 extrapolation was applied. The dose-based chemical toxicity on a human population could be quantified by using the lognormal parameters: m, i.e., lnðED50 Þ; and s, i.e., the individual variability in the responses to carcinogens. The lnðED50 Þ values for a total of 708 carcinogenic chemicals were taken from Huijbregts et al. (2005), which followed a lognormal distribution ~ LN(1.77, 3.09) (Supplementary Information) and the 5th to 95th percentile range is [-3.32, 6.85]. The 5th to 95th percentile of s (natural logarithm) is [0.85, 2.10]. It must be noted that due to the toxicological data limit for specific carcinogens (Huijbregts et al., 2005), only the general variability and uncertainty for chemicals were considered. Thus, it was assumed that m and s are mutually independent, and the s for carcinogens is only dependent on the selected human populations (Huijbregts et al., 2005). Based on Theorem 3, the RH ðx; mjm; sÞ of HLN ðxjm; sÞ to Hb ðx; mÞ is its maximum at the intersection point (EDm ; 0:01m), i.e., (ED10 ; 0:1), which could help to evaluate whether the current ED10 method could effectively manage the instantaneous risks; therefore, in this section, RH ðED10 ; 10jm; sÞ was applied to discuss the impacts of m and s on hazard ratios. Fig. 4 illustrates RH ðED10 ; 10jm; sÞ and RL ðED10 ; 10jm; sÞ as

9

functions of m and s at the intersection point (ED10 ; 10). Based on Theorem 3, RH ðEDm ; mjm; sÞ is a rational function of s and independent of m, which decreases as s increases, which indicates that the ED10 linear extrapolation seems to be safer in terms of the hazard risks when the interindividual variability in a population’s responses to carcinogenic chemicals is large, regardless of the specific chemicals. The simulated RH ðED10 ; 10jm; sÞ values at the intersection point are higher than 1.0 when s < 1:33, and those values are higher than 1.25, i.e., 25% exceedance, when s < 1:18. When the 5th percentile of the human variability to carcinogens, i.e., s ¼ 0:85, is applied, the ED10 linear dose-response extrapolation for carcinogenic chemicals will have the maximum instantaneous risk ratio of 2.43, which is over two times higher than the experimental-based lognormal model. This phenomenon could be explained by the shape of the lognormal model. As the shape parameter (s) decreases, the shape of a lognormal CDF becomes more centralized, which flattens the shape of the initial stage for the model. This flattening makes the slope of the linear extrapolation very small at low dose values (e.g., ED10) compared to that of the lognormal model at the intersection point, which means that if the interindividual variability in the responses to carcinogens is small, e.g., a relatively homogeneous population, the ED10 linear extrapolation method might significantly underestimate the real hazard risk. On the other hand, for a relatively heterogeneous population with some different characteristics, a large interindividual variability to carcinogenic chemicals could steepen the initial shape of the lognormal dose-response model and increase the hazard risk for the extrapolated linear model at low dose values. Therefore, from the instantaneous risk perspective, the selection of the linear extrapolation for carcinogenic chemicals should consider the individual variability, i.e., the study population. As opposed to RH ðEDm ; mjm; sÞ, RL ðEDm ; mjm; sÞ at the intersection point is dependent on both m and s because the dose-based cumulative hazard ratio compares how the hazard risks for the linear extrapolation and the lognormal model change due to doses, of which LLN ðEDm jm; sÞ is determined by the lognormal parameters that could not be eliminated by the linear model. As shown in Fig. 4, RL ðED10 ; 10jm; sÞ decreases when s increases due to the multiplicative factor of the percent point function value at 0.1, which is consistent with RH ðEDm ; mjm; sÞ because of the negative F1 ð0:1Þ value. On the other hand, RL ðED10 ; 10jm; sÞ is an increasing function of m, i.e., lnED50, over its interval. As the lnED50 value for carcinogenic chemicals increases, the RL ðED10 ; 10jm; sÞ values that are

Fig. 4. Dose-based hazard RH ðEDm ; mjm; sÞ and cumulative hazard risk RL ðEDm ; mjm; sÞ ratios as functions of m, i, e., lnðED50 Þ; and s at the intersection point (ED10 ; 0:1) for carcinogenic chemicals.

10

Z. Li / Chemosphere 247 (2020) 125934

simulated for the ED10 linear extrapolation approach 1.0, which can be explained by comparing the different location parameters of the lognormal models that have the same shape parameter. If a lognormal model has a relatively large location parameter, i.e., a population trends to significantly respond to a carcinogenic chemical at a high dose, the lognormal dose-response curve shifts to high dose values and makes the slope extrapolated ED10 linear model flatter, which results in a relatively low cumulative hazard that is simulated by the linear model along the interval. When the lnED50 value is low, i.e., the lognormal dose-response curve shifts to low dose values, the LLN ðED10 jm; sÞ of a lognormal model is far less than the Lb ðED10 ; 0:1Þ of a linear model because of the initial slope that the linear extrapolation possesses at low-dose values. To further quantify the impacts of m and s on RL ðED10 ; 10jm; sÞ values, Monte Carlo sensitivity analysis (XLSTAT software, Paris, France) was applied based on the simulated lognormal distributions for m, i.e., LN  ð1:77; 3:09Þ, and s, i.e., LN  ð0:31; 0:28Þ. Here, m contributes to approximately 97.95%, indicating that the lnðED50 Þ value of carcinogenic chemicals impacts the outcome the most. 4.3. Linear extrapolation for carcinogenic chemicals According to Theorems 1e4, the maximum static risk ratio of Fb ðxÞ to FLN ðxÞ at the first critical point and the maximum hazard (instantaneous) risk ratio of HLN ðxÞ to Hb ðxÞ at the intersection point helped to derive the safe range of m for the linear extrapolation with the slope of 0:01m EDm . Fig. 5b illustrates the maximum static risk ratio of Fb ðxÞ to FLN ðxÞ at the first critical point (ED1st m ; 0:01m) as a function of m and s, which is independent of m according to Eq (23). Fig. 5b could provide the safe range of m for linear extrapolation

with respect to the static risk ratio (or differential), which was based on the 5th to 95th percentiles of the individual variability distribution. When DA ¼ 1 and RA ¼ 0:25, it means that the acceptable maximum static risk ratio, i.e.,

Fb ðED1st m Þ , FLN ðED1st m Þ

is set as 2.0, and

the acceptable maximum instantaneous risk ratio at the intersecðEDm Þ A tion point, i.e., HHLN ðEDm ;mÞ, is set as 1.25. It should be noted that D and b

RA could be arbitrarily selected. Here, a DA of 2.0, i.e., 100% exceedance, for static health risk was applied because at a low dose, the extrapolated linear dose-response relationship has an intrinsic slope, which would immediately draw a risk distance away from the lognormal model. Additionally, the extrapolated static risk was always higher than the experiment-based lognormal model; therefore, the 100% exceedance of the experiment-based model was considered to be acceptable. On the other hand, since the linear extrapolation always underestimates the instantaneous cancer risk at the intersection point, which should be precautionarily controlled, 25% below the experiment-based model was applied to optimize and control the hazard risk. Fig. 5c illustrates that the maximum dose-based hazard risk ratio

HLN ðxÞ Hb ðxÞ

at the intersection

point is a decreasing function of both m and s over their intervals, which could be further explained by the slope ratio of the lognormal CDF and linear models at the intersection point. A lower m makes the linear extrapolation flatter, and a lower s value could boost the slope of the lognormal CDF model more than the extrapolated linear one at the intersection point. Therefore, a relatively homogenous population has large dose-based hazard risks for carcinogenic chemicals at lower dose values, and the extrapolated linear dose-response relationship might significantly underestimate the instantaneous risks. On the other hand, for a

F ðxÞ

Fig. 5. (a) Relationship between mmax and population variability (s) for carcinogenic chemicals based on Theorems 1 and 4, (b) maximum static risk ratio FLNb ðxÞ at the first critical ðxÞ point plotted as a function of m and s, (c) maximum dose-based hazard (instantaneous) risk ratio HHLNðxÞ at the intersection point plotted as a function of m and s, and (d) derived b ranges of the cancer slope factors (CSFs) for carcinogenic chemicals plotted as a cumulative distribution function (CDF) based on two selected s values.

Z. Li / Chemosphere 247 (2020) 125934

relatively heterogenous population that has large individual variability to carcinogens, the extrapolated linear model at very low dose values (i.e., m is pretty small) could also significantly underestimate the instantaneous risks by flatting the linear slope. As s decreases, the maximum static risk ratio increases. For example, when s is equal to 0.85, i.e., 5th percentile of the individual variability distribution (lognormal), the minimum

Fb ðED1st m Þ FLN ðED1st m Þ

is

2.39, which is larger than the selected acceptable guideline in this study of m ¼ 46.84. Therefore, based on the acceptable DA that is selected in this study, the linear extrapolation method might not be suitable for carcinogenic chemicals due to the variability of 0.85 (natural logarithm) in the individual responses in the population. When s is equal to 1.36 (i.e., 50th percentile of the distribution), the range of m according to the selected DA (Theorem 2) is determined to be [4.10, 58.50], which makes the maximum static risk ratio less or equal to 2.0. When s is 2.10 (95th percentile of the distribution), Fb ðED1st m Þ FLN ðED1st m Þ

is less than 2.0 over the interval m 2ð0; 64:27. However,

when m is too high, the values of

Fb ðxÞ FLN ðxÞ

over the interval x > ED2nd m ,

i.e., the tangent point, will be less than 1.0, which is incompatible with the precautionary principle and requires Theorems 1 and 4 to further restrict the range of m. As s approaches zero, i.e., a homogenous population with same characteristics, the shape of the lognormal CDF moves towards to the central tendency (i.e., lnED50 ) and makes the initial part of the FLN ðxÞ flatter, which makes the slope of the extrapolated linear model relatively steep and draws the distance away from the lognormal CDF. When s < 1:07, the simulated maximum static risk ratios at the first critical point are larger than 2.0 for all ms in its interval. Therefore, the linear extrapolation method could overestimate the health risk too much when the individual variability in the responses to carcinogenic chemicals is small. This phenomenon will become more obvious if the selected population has higher homogeneity. Based on the simulated range of m, the acceptable linear extrapolation for carcinogenic chemicals could be identified and expressed as the range of CSF. Fig. 5d illustrates two groups of the acceptable CSF range clusters for 708 carcinogenic chemicals, which were

11

computed using the s values of 1.36 and 2.10, respectively. When s ¼ 1:36 (i.e., 50th percentile of the individual variability lognormal distribution), the maximum value of m based on Theorem 1 is 8.69, which is incompatible with the derived minimum value of m (i.e., 11.22) according to Theorems 2 and 3. This incompatibility means that the linear extrapolation at the modevalue based point that a carcinogenic response appears most often for a human population could not effectively control the introduced dose-based static or instantaneous risks. Therefore, using the second critical point that is completed by Theorem 4, the range of m for carcinogenic chemicals with the individual variability of 1.36 (natural logarithm) is determined to be [11.22, 21.46], based on which the range for the linear dose-response model can be extrapolated from ED11.22 to ED21.46. This linear extrapolation range can ensure the safety for both dose-based static and instantaneous risks. When the 95th percentile of the individual variability distribution is applied (i.e., s ¼ 2:10), the derived safe ranges of m based on the incompleteness theorems and the complementary theorems are [1.13, 1.79] and [1.13, 4.57], respectively, which indicates that for a relatively heterogeneous population that has large individual variability to carcinogenic chemicals, the extrapolated linear dose-response model can protect human health at very lowdose values. That is, the upper limit is ED4.57, which is smaller than that for the common ED10 extrapolation method for the CSF estimation. It should be noted that this study did not try to prove that the current linear extrapolation method, especially that for ED10, was wrong since at low values of the dose interval the dose-based static and hazard risk ratios could be within the acceptable range. However, to ensure the safety for the entire dose interval, the linear extrapolation method based on the theorems that are introduced by this study is suggested. 5. Model limitations and future studies This study formulated the linear extrapolation method for carcinogenic chemicals by introducing static and instantaneous risk principles. Some limitations about the modeling framework should be addressed. This study applied the lognormal dose-response relationship, whereas other models such as Weibull, multistage,

Fig. 6. Cancer slope factors (CSFs) (log-transformed) by Theorems 2e4 and s ¼ 1:36 compared to USEPA oral slope factors (OSFs) (log-transformed).

12

Z. Li / Chemosphere 247 (2020) 125934

probit, logit, and so on could also be used to fit the experimental data. Different dose-response models might result in significant variances in the low-dose extrapolation, although these models could statistically fit the data. For example, Fig. 6 illustrated the derived CSFs by Theorems 2e4 as compared to USEPA’s oral slope factors (OSFs) (USEPA, 2019) which were transformed by humanlifetime-exposure derivations. For these 160 carcinogenic chemicals, the average ratio of USEPA’s OSFs to CSFs ranges from 30.49 (min) to 32.74 (max), indicating that the CSFs derived by this study were generally lower than current regulatory OSFs. This is due to the selection of models and toxic endpoints. For example, USEPA’s Integrated Risk Information System (IRIS) applied the linearized multistage procedure and hepatocellular carcinomas in rats to derive the OSF for DDT (USEPA, 1988), which is 6.50  101 kg1 (i.e., 3.40  101 mg kg1d1). The CSFs for DDT based on by Theorems 2e4 and the central estimate from the human population (s ¼ 1:36) ranges from 1.58  102 to 1.70  102. In addition, USEPA used the statistical upper confidence bounds, while this study directly applied the dose-response curves, which further deviates the simulated results. Besides the difference to current regulatory OSFs, this study needs to consider the mechanism of carcinogenesis for specific chemicals and background cancer risks in future research. Starr and Swenberg (2013, 2016) introduced a novel bottom-up method to estimate low-dose cancer risks without requiring high-dose toxicity data. Their novel approach can significantly promote the low-dose cancer risk estimation because it only requires the background risk information, which will minimize the limitations of animal tests and model selections. In addition, Starr and Swenberg’s novel approach can help to validate current linear doseresponse extrapolations as an independent check. Thus, the lognormal-based linear extrapolation method in this study should be validated and incorporated with Starr and Swenberg’s bottomup method once their method is generalized to more carcinogenic chemicals. Also, risk principles and the related theorems should be developed for other dose-response models in future studies.

6. Conclusions This study introduced two principles for extrapolating linear dose-response relationships based on a lognormal model for carcinogenic chemicals, which included dose-based static and hazard (instantaneous) risks. To quantify the acceptable range of the derived CSF for carcinogenic chemicals, incompleteness and complementary theorems were introduced and proved. The study concluded that the individual variability in a population’s responses to human carcinogens significantly impacted the range of m, and using the EDm that was calculated based on the toxicity parameters of the chemicals could finally determine the acceptable CSF range. It was also concluded that for a relatively homogenous population that has similar characteristics, the linear doseresponVDse extrapolation method might not be completely effective due to the shape shift of the lognormal CDF towards the central tendency, which could steepen the slope of the extrapolated linear model and significantly overestimate the static risk. Additionally, the results were compared to the common ED10 linear extrapolation method and they indicated that the ED10 method might not be completely able to manage the dose-based static or hazard risks, even though the ED10 method could be used for the low-dose interval. Hopefully, the theorems and principles that were introduced by this study could help toxicologists to better quantify and extrapolate cancer risks for human health risk assessments.

Declaration of competing interest The author has declared that is no conflict of interest in this paper. CRediT authorship contribution statement Zijian Li: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing - original draft, Writing - review & editing. Acknowledgements This study was supported by Sun Yat-sen University (grant 58000e18841211). The author highly appreciates Mark Huijbregts and Dale Hattis for the discussion about the interindividual variability. Also, the author gratefully acknowledges the anonymous reviewers for their careful reading of the manuscript and their valuable comments. Appendix A. Proof of Theorem 1

Theorem. 1 A straight line function Fb ðxÞthat passes through the origin point of the coordinates intersects the real-valued continuous lognormal ðm; s2 ÞCDF FLN ðxÞat the point ðEDm ; 0:01mÞ. For any EDm 2ð0; mode, it makes Fb ðxÞ  FLN ðxÞ, i.e.,

Fb ðxÞ FLN ðxÞ

 1, where x2ð0; EDm Þ.

Proof: q FLN ðxÞ denotes the real-valued continuous lognormal CDF curve with parameters m and s2 .

∴F ’LN ðxÞ ¼

∴F ’’LN ðxÞ ¼

" # 1 ðln x  mÞ2 pffiffiffiffiffiffiffi exp  2s2 xs 2p

(A1)

" #!  1 ðln x  mÞ2 ln x  m pffiffiffiffiffiffiffi exp   1  s2 2s2 x2 s 2p (A2)

where F ’LN ðxÞ and F ’’LN ðxÞ denote the first and second derivatives of FLN ðxÞ, respectively.

" # 1 ðln x  mÞ2 pffiffiffiffiffiffiffi exp  q >0 2s2 x2 s 2p

(A3)

∴ To make F ’’LN ðxÞ > 0;

ln x  m   1>0 2

s

(A4)

then,

 x < exp m  s2

(A5)

where expðm s2 Þ is the mode of the lognormal ðm; s2 Þ. Therefore, FLN ðxÞ is concave up, i.e., a convex-downward function, over its interval ð0; modeÞ: According to the definition of the convexdownward function, for any x1 ; x2 2ð0; EDm Þ and every l2½0; 1, one has

Z. Li / Chemosphere 247 (2020) 125934

FLN ðlx1 þ ð1  lÞx2 Þ  lFLN ðx1 Þ þ ð1  lÞFLN ðx2 Þ qF ’LN ðxÞ ¼

(A6)

" # 1 ðln x  mÞ2 pffiffiffiffiffiffiffi exp  >0 2s2 xs 2p

(A7)

Then, based on Theorem 1, lim G’ ðxÞ could be expressed as follows: x/0

lim G’ ðxÞ ¼ lim F ’LN ðxÞ  lim F ’b ðxÞ x/0 x/0

x/0

¼ lim

Fb ðxÞ FLN ðxÞ  lim x x/0 x

¼ lim

FLN ðxÞ  Fb ðxÞ <0 x

x/0

∴FLN ðxÞ < FLN ðEDm Þ; x2ð0; EDm Þ

(A8)

then,

x/0

FLN ðlx1 þ ð1  lÞx2 Þ < lFLN ðx1 Þ þ ð1  lÞFLN ðEDm Þ

(A9)

13

therefore,

lim G’ ðxÞ < 0

then,

(A18)

x/0

lim ½FLN ðlx1 þ ð1  lÞx2 Þ < lim ½lFLN ðx1 Þ þ ð1  lÞFLN ðEDm Þ

x1 /0

x1 /0

(A10) lim ½FLN ð0 þ ð1  lÞx2 Þ < lim ½lFLN ð0Þ þ lim ½ð1  lÞFLN ðEDm Þ

x1 /0

x1 /0

lim F ’LN ðxÞ < lim F ’b ðxÞ ¼

x/0

x/0

(A12)

q FLN ðxÞ intersects with Fb ðxÞ at the point of ðEDm ; 0:01mÞ.

 ∴FLN ðEDm Þ ¼ Fb ðEDm Þ ¼ EDm

0:01m EDm

 (A13)

(A19)

x/EDm

lim G’ ðxÞ ¼ lim F ’LN ðxÞ  lim F ’b ðxÞ

x/EDm

FLN ðð1  lÞx2 Þ < ð1  lÞFLN ðEDm Þ

0:01m EDm

Additionally, lim G’ ðxÞ could be expressed as follows:

x1 /0

(A11)

x/EDm

x/EDm

Fb ðEDm Þ  Fb ðxÞ FLN ðEDm Þ  FLN ðxÞ  lim EDm  x EDm  x x/EDm x/EDm

¼ lim ¼ lim

x/0

(A20)

Fb ðxÞ  FLN ðxÞ >0 EDm  x

therefore,

then,

FLN ðð1  lÞx2 Þ < ð1  lÞFb ðEDm Þ

(A14)

FLN ðð1  lÞx2 Þ < FLN ðð1  lÞEDm Þ < ð1  lÞFb ðEDm Þ   0:01m ¼ ð1  lÞEDm EDm

(A15)

then, ð1 lÞEDm 2ð0; EDm Þ can be replaced by x2ð0; EDm Þ to obtain the general form as follows:

 FLN ðxÞ <

(A17)

 0:01m x ¼ Fb ðxÞ EDm

(A16)

∴FLN ðxÞ < Fb ðxÞ and at the intersection point FLN ðxÞ ¼ Fb ðxÞ.

lim G’ ðxÞ ¼ > 0

lim F ’LN ðxÞ > lim F ’b ðxÞ ¼

x/EDm

x/EDm

0:01m EDm

therefore,      ’ 0:01m > 0 lim F ’LN ðxÞ 0:01m F ðxÞ  G’ ðxÞ2 < 0; lim LN EDm EDm x/EDm

x/0

(A22)

is

monotonically increasing over its interval x2ð0; EDm Þ. GðxÞ has the minimum value GðxÞmin when G’ ðxÞ ¼ 0, i.e.,

G’ ðxÞ ¼ F ’LN ðxÞ  F ’b ðxÞ ¼ 0

(A23)

0:01m EDm

(A24)

F ’LN ðxÞ ¼

Appendix B. Proof of Theorem 2

(A21)

x/EDm

therefore, when F ’LN ðxÞ ¼ 0:01m EDm , FLN ðxÞ  Fb ðxÞis at its minimum value Theorem. 2 A straight line function Fb ðxÞthat passes through the origin point of the coordinates intersects the real-valued continuous lognormal ðm; s2 Þ CDF FLN ðxÞat the point ðEDm ; 0:01mÞ. The maximum value of

(negative), which could be further expressed as follows:

Fb ðxÞ  FLN ðxÞexists at the first critical point where F ’LN ðxÞ ¼ F ’b ðxÞ ¼ 0:01m and x2ð0; ED Þ. m EDm

Fb ðxÞ  FLN ðxÞ ¼ △max

Definition: The first critical point is the point ðED1st m ; 0:01mÞ on

1st 1st FLN ðxÞwhere F ’LN ðxÞ ¼ F ’b ðxÞ ¼ 0:01m EDm and EDm < mode, i.e., EDm <

expðm  s2 Þ Proof: Let

GðxÞ ¼

FLN ðxÞ 

Fb ðxÞ.

Then,

G’’ ðxÞ ¼ F ’’LN ðxÞ  F ’’b ðxÞ ¼ F ’’LN ðxÞ > 0according to Eq. (A2)-(A5) in

Theorem 1. Therefore, G’ ðxÞ is a monotonically increasing function on its interval x2ð0; EDm Þ.

FLN ðxÞ  Fb ðxÞ ¼  △max

△max > 0

(A25) (A26)

2 Proof for ED1st m < expðm  s Þ:

q F ’LN ðxÞis a monotonically increasing function over its interval

x 2ð0; expðm s2 ÞÞaccording to the proof in Theorem 2.

     Additionally;qF ’LN exp ms2 >F ’b exp ms2 ¼F ’LN ED1st m  2 ∴ED1st m
14

Z. Li / Chemosphere 247 (2020) 125934

Appendix C. Proof of Theorem 3

Appendix D. Proof of Theorem 4 and Corollary

Theorem. 3 A straight line function Fb ðxÞ that passes through the origin point of the coordinates intersects the real-valued continuous lognormal ðm; s2 ÞCDF FLN ðxÞat the point ðEDm ; 0:01mÞ. The maximum value for ðxÞ the ratio of their hazard functions, i.e., RH ðxÞ ¼ HHLN ðx;mÞ, exists when x ¼ b

EDm and that maximum value is independent of m, i.e., lnðED50 Þ. Proof: According to Eqs (2). and (3), RH ðxÞ could be further expressed as follows. F ’LN ðxÞ   BF ’LN ðxÞC 1  Fb ðxÞ HLN ðxÞ 1FLN ðxÞ C ¼ ’ RH ðxÞ ¼ ¼B @ 0:01m A 1  F ðxÞ ; 0 < x F b ðxÞ Hb ðx; mÞ LN ED 1Fb ðxÞ

ðm; s2 Þ CDF FLN ðxÞat the second critical point of ðED2nd m ; 0:01mÞ. 3 For any x2ð0; ED2nd m Þ, it makes Fb ðxÞ > FLN ðxÞ, i.e., 2nd 2 4 ED2nd m > mode, i.e., EDm > expðm  s Þ.

Definition: The second critical point is the point ðED2nd m ; 0:01mÞon FLN ðxÞ

m

G’ ðxÞ ¼

 EDm (A27) F ’LN ðxÞis

q a monotonically increasing function over its interval x2 ð0; EDm Þ according to the proof in Theorem 2. 0 1 BF ’ ðxÞC LN C is at its maximum when x ¼ EDm . ∴B @ 0:01m A EDm

exists when x ¼ EDm .

At the intersection point (EDm ;0:01m), RH ðxÞcould be expressed as follows:

  lnðEDm Þ  m 4 H ðEDm Þ F ’ ðEDm Þ ¼ ¼ ’ LN RH ðEDm ; mjm; sÞ ¼ LN Hb ðEDm ; mÞ F ðEDm ; mÞ b

4 ¼

  lnðEDm Þ  m

F ’LN ðxÞ ¼ F ’b ðxÞ ¼ 0:01m 2nd and

ED2nd m > mode,

EDm

i.e.,

" # 1 ðln x  mÞ2 0:01m pffiffiffiffiffiffiffi exp   EDm 2s2 xs 2p

(A30)

h i EDm ¼ exp m þ sF1 ð0:01mÞ ; 0 < m < 100 Let G’ ðxÞ ¼ 0. Then,

" # 1 ðln x  mÞ2 0:01m G ðxÞ ¼ pffiffiffiffiffiffiffi exp    ¼0 2 2 s xs 2p exp m þ sF1 ð0:01mÞ (A31) then,

"

Therefore, the maximum value for the ratio of their hazard functions, i.e., RH ðxÞ ¼

where

’

q Fb ðxÞ  FLN ðxÞover its interval x2ð0; EDm Þaccording to the proof in Theorem 2.   1F ðxÞ ∴ 1FLNb ðxÞ  1 is at its maximum value of 1 when x ¼ EDm . HLN ðxÞ Hb ðx;mÞ,

FLN ðxÞ Fb ðxÞ < 1.

2 ED2nd m > expðm  s Þ Proof: According to Theorem 2,

1

0

Theorem 4 (complementary theorem): A straight line function Fb ðxÞthat passes through the origin point of the coordinates intersects with the real-valued continuous lognormal

mÞ exp  ðln2x s2

# 2

 pffiffiffiffiffiffiffi ¼ 0:01m 2p x s   exp  m  sF1 ð0:01mÞ

(A32)

then,

"

s

ðln x  mÞ2 þ m þ sF1 ð0:01mÞ 2s2  pffiffiffiffiffiffiffi ¼ ln 0:01ms 2p x

EDm s 0:01m EDm

#!

ln exp 

(A33)

then,

s

0:01ms (A28)

m þ sF1 ð0:01mÞ ¼

 pffiffiffiffiffiffiffi ðln x  mÞ2 þ lnx þ ln 0:01ms 2p 2 2s (A34)

then,

h i qEDm ¼ exp m þ sF1 ð0:01mÞ

∴RH ðEDm ; mjm; sÞ ¼

i ln expm þ sF1 ð0:01mÞ  m 4

s

0:01ms

¼



4 F1 ð0:01mÞ 0:01ms

(A29)

Z. Li / Chemosphere 247 (2020) 125934

 pffiffiffiffiffiffiffi 2ms2 þ 2s3 F1 ð0:01mÞ  m2  2s2 ln 0:01ms 2p 2  2   þ s2  m ¼ ðlnxÞ2 þ 2s2  2m lnx þ s2  m

15

2nd ðED1st m ; 0:01mÞand the second ðEDm ; 0:01mÞcritical points on the

(A35)

FLN ðxÞcurve where F ’LN ðxÞ ¼ F ’b ðxÞ, the probabilities PðX  ED1st m Þand 2nd PðX  EDm Þare independent of m, i.e., lnED50 . Proof:

then,

2s F 3

According to Eq. (A40) and (A41) in Theorem 4, PðX  ED1st m Þ and 1

2   pffiffiffiffiffiffiffi ð0:01mÞ  2s ln 0:01ms 2p þ s4 ¼ lnx þ s2  m 2

(A36) pffiffiffiffiffiffiffi Let FCP ¼ lnð0:01ms 2p Þ þ s4 , where FCP is the critical point factor that is always positive. Then, 2s3 F1 ð0:01mÞ 



PðX  ED2nd m Þ could be expressed as follows:

¼F (A37)

pffiffiffiffiffiffiffi  lnx ¼ m  s2 ± FCP

(A38)

pffiffiffiffiffiffiffi  x ¼ exp m  s2 ± FCP

(A39)





¼ FLN ED1st;2nd ¼F P X  ED1st;2nd m m

2 s2

2 lnx þ s2  m ¼ FCP





¼F

 ! m ln ED1st;2nd m

s

 

 pffiffiffiffiffiffiffi ! m ln exp m  s2 ± FCP

s ! pffiffiffiffiffiffiffi ± FCP  s2

s

ffi  rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   pffiffiffiffiffiffiffi ¼ F ± 2sF1 ð0:01mÞ  2 ln 0:01ms 2p þ s2  s (A45)

therefore,

pffiffiffiffiffiffiffi   ED1st m  s2  FCP < exp m  s2 m ¼ exp

(A40)

pffiffiffiffiffiffiffi   m  s2 þ FCP > exp m  s2 ED2nd m ¼ exp

(A41)

Appendix E. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.chemosphere.2020.125934.

and

pffiffiffiffiffiffiffi  G’ ðxÞ < 0 c 0 < x < exp m  s2  FCP

(A42)

pffiffiffiffiffiffiffi   G’ ðxÞ > 0 c exp m  s2  FCP < x < exp pffiffiffiffiffiffiffi   m  s2 þ FCP

(A43)

therefore, GðxÞ ¼ FLN ðxÞ  Fb ðxÞ acquires the global minimum value at the first critical point, i.e., (ED1st m ;0:01m), and the local maximum value at the second critical point, i.e., (ED2nd m ;0:01m), over the interval x2 ð0; ED2nd m Þ. q At ðED1st m ; 0:01m), the global minimum value is negative, i.e., 1st 1st GðED1st m Þ < 0or FLN ðEDm Þ < Fb ðEDm Þ,according to the proof for Theorem 2. At (0, 0), lim GðxÞ ¼ lim FLN ðxÞ  lim Fb ðxÞ ¼ 0, i.e., x/0

x/0

x/0

lim FLN ðxÞ ¼ lim Fb ðxÞ. At ðED2nd m ; 0:01m) where the two functions

x/0

2nd therefore, the probabilities PðX  ED1st m Þ and PðX  EDm Þ are independent of m, i.e., lnED50 .

x/0

2nd intersect, FLN ðED2nd m Þ ¼ Fb ðEDm Þ. FLN ðxÞ Therefore, for any x2ð0;ED2nd m Þ, it makes Fb ðxÞ > FLN ðxÞ, i.e., F ðxÞ < b

2nd 2 1, and ED2nd m > mode, i.e., EDm > expðm  s Þ. The second critical

point ðED2nd m ; 0:01mÞ is also the tangent point where the tangent line Fb ðxÞjust touches the lognormal CDF FLN ðxÞ. Then, at the tangent point,



 F ’LN ED2nd m 4 F1 ð0:01mÞ  ¼ ¼1 0:01ms F ’b ED2nd m ;m

(A44)

Corollary. A straight line function Fb ðxÞ that passes through the origin point of the coordinates intersects with the real-valued continuous lognormal ðm; s2 Þ CDF FLN ðxÞ. At the first

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