Taming the Fine Particulate–Mortality Curve

Abstract

Estimating the population-level mortality burden attributable to exposure to outdoor fine particulate matter $\left(P{M}_{2.5}\right)$ requires characterizing both the magnitude and shape of the relative risk function that mathematically models how exposure affects response. The relationship can be derived using cohort studies where the association between $P{M}_{2.5}$ exposure and mortality is directly observed. As policy issues of interest can involve exposures that exceed by several-fold those observed in cohort studies, how the association is extrapolated to these high concentrations is a major source of uncertainty. To address this issue, we suggest the extrapolation criterium that the estimated proportion of deaths attributed to exposure above the observed cohort exposure range is no more than that below the range. This criterion implies that the relative risk function must be bounded from above with marginal changes in risk, rapidly declining with increasing concentration. We illustrated the approach with the use of meta-data from 44 cohorts conducted in North America, Western Europe, Asia, and China, examining the association between long-term exposure to outdoor $P{M}_{2.5}$ and non-accidental mortality.

1. Introduction

Fine particulate matter $\left(P{M}_{2.5}\right)$ is a leading contributor to the global disease burden examined by the Global Burden of Disease 2023 $\left(GBD2023\right)$ program [1]. Estimating this burden requires, in part, characterization of both the magnitude and shape of the association between exposure and a health response. For the case of mortality, this association is determined using meta-data (slope estimate from a survival model using cohort data, its standard error and its cohort-specific exposure distribution) [1,2,3]. Cohort studies of outdoor $P{M}_{2.5}$ observed exposures well below $100 \mathsf{\mu }\mathrm{g}/{\mathrm{m}}^3$, while policy analyses involving exposure scenarios can be more than 10-fold that of the figure [4].

The very first attempt at constructing a relative risk model for the association between $P{M}_{2.5}$ and mortality over the global exposure range was part of the very first $GBD$ program in 2004 [5]. At the time, only two cohort studies existed, both conducted in the United States [6,7], with the highest observed exposure concentration of $30 \mathsf{\mu }\mathrm{g}/{\mathrm{m}}^3$ [6]. A linear-in-concentration relative risk model was assumed up to $30 \mathsf{\mu }\mathrm{g}/{\mathrm{m}}^3$ with no increase in risk above this level. This extreme form of bounding was used based on concerns that this linear-in-concentration model would yield unacceptably high risks at the concentrations observed globally [5].

The next attempt was the $Integrated Exposure-Response \left(IER\right)$, where risks associated with much higher particulate exposure sources (second-hand smoke, household pollution, and active smoking) of $P{M}_{2.5}$ were integrated with outdoor sources by transforming total inhaled dose of $P{M}_{2.5}$ to the equivalent ambient concentration [8]. Several assumptions, however, had to be made, including assuming that the toxicity of $P{M}_{2.5}$ is the same for each source at the same inhaled dose, and that toxicity is not a function of the timing of exposure (i.e., smoking a cigarette versus living with a smoker) [8]. An algebraic function was used to model relative risk predictions. Since the equivalent $P{M}_{2.5}$ exposure from active smoking yielded concentrations much larger than any possible outdoor concentration $(>\mathrm{10,000} \mathsf{\mu }\mathrm{g}/{\mathrm{m}}^3)$, predicting the risk at even very high outdoor levels was a matter of interpolation. $GBD$ dropped active smoking and second-hand smoking from the $IER$, leaving household pollution as the highest $P{M}_{2.5}$ exposure source [1]. They also replaced the algebraic function with a monotonically increasing concave smoothing spline [1]. They extrapolated the risk beyond the observed exposure data with a linear function [1]. Although this method offered a mathematical framework for risk extrapolation and enabled simultaneous assessment of the mortality burden from multiple sources [1], its reliability relied on strong assumptions that have not been empirically tested or verified. In addition, since the exposure distributions of the cohorts in each of the sources can overlap, a common fitted model will not necessarily provide an adequate fit to any single source, as it is highly influenced by the differences in the magnitude in risk between sources [3]. This is a limitation of the $IER$ approach if one is interested in modeling exposure specifically from outdoor sources.

The third attempt, the $Global Exposure Mortality Model \left(GEMM\right)$, focused on the case where only risk assessment from outdoor sources of $P{M}_{2.5}$ was of interest [2], and as such, only information from cohort studies of outdoor-sourced $P{M}_{2.5}$ was used. The product of a generalized logarithmic function and a sigmodal function was used to characterize the shape of the concentration–response relationship. The algebraic form of the function formed the basis for predictions beyond the cohort exposure range.

The fourth approach, the $Fusion$ model [3], used a parametric spline with two knots composed as a fusion of three algebraic forms of the derivative of the logarithm of the relative risk model. The first form was a constant up to a concentration, the first knot, and then a function that declines with concentration up to the upper limits of the observed cohort exposure distribution, the second knot. A simplistic algebraic form was suggested as a means to extrapolate risk beyond the observable exposure range, where the derivative declines as the inverse of concentration [3]. This assumption on the derivative implies that the relative risk increases as the logarithm of concentration beyond the observed cohort exposure range. Although this assumption was conservative in the sense that the derivative of the model is always greater over the cohort exposure range than a logarithm model predicts beyond that range, relative risk predictions are unbounded, and at concentrations much higher than observed, the relative risk estimate could be extreme, so much so that they are reasonably questioned [9].

In this paper, we propose a specific criterion to guide extrapolation of the relative risk beyond concentrations observed in health studies of outdoor $P{M}_{2.5}$. We suggest that the range in the population attributable fraction (proportion of total deaths due to $P{M}_{2.5}$ ) [5] based on concentrations above the cohort exposure range should not be greater than that observed over the cohort exposure range itself. In addition, we require a relative risk estimate at the end of the cohort exposure range to be insensitive to the shape and magnitude of the relative risk predictions above this range. Our approach does not require additional assumptions on the extrapolated risk except that it is bounded from above. We illustrated our model using meta-data from 44 cohort mortality studies of outdoor $P{M}_{2.5}$ conducted globally.

2. Materials and Methods

2.1. Previous Relative Risk Models

Consider a relative risk function, ${\mathcal{R}}_g\left(z\right)$, for concentration $z$, with the general form

$${\mathcal{R}}_g\left(z\right)=g\left(\gamma {\int }_0^{z}D\left(x\right)\partial x\right)$$

where the link function $g$ is continuous, differentiable, and monotonically increasing and $\gamma D\left(z\right)$ represents the derivative of $g^{-1}\left({\mathcal{R}}_g\left(z\right)\right)$ with respect to concentration $z$ with $\gamma >0$ being the derivative at the origin, implying $D\left(0\right)=1$. We further assumed that $D\left(z\right)$ is greater than 0 and bounded above by 1; is continuous; and is both integrable and differentiable.

Two forms of $g$ have been previously considered: $g\left(x\right)=e^x$ and $g\left(x\right)=1+x$. If $D\left(z\right)=1,\forall z$ and $g\left(x\right)=e^x$, then $ln{\mathcal{R}}_{e^x}\left(z\right)=\gamma z$ is the $Log-Linear$ relative risk function [5], and if $D\left(z\right)=1/\left(1+z\right)$, then $ln{\mathcal{R}}_{e^x}\left(z\right)=\gamma \ln \left(1+z\right)$ is the $Log-Log$ relative risk function [5], with ${\mathcal{R}}_{e^x}\left(\infty \right)=\infty$ for both models. A truncated function of the form $D\left(z\right)=1$ if $z<C$ and zero otherwise, with $g\left(x\right)=1+x$ being suggested as a means of limiting the magnitude of the relative risk at high global concentrations [5] with ${\mathcal{R}}_{1+x}\left(\infty \right)=1+\gamma C<\infty$. The algebraic version of the $IER$ [8] sets $g\left(x\right)=1+x$, and ${\int }_0^{z}D\left(x\right)\partial x=1-e^{-\nu z^{\delta }}$ with ${\mathcal{R}}_{1+x}\left(\infty \right)=1+\gamma <\infty$. The power functional form ${\int }_0^{z}D\left(x\right)\partial x=z^{\psi }$ and $g\left(x\right)=1+x$ has also been proposed [10] with ${\mathcal{R}}_{1+x}\left(\infty \right)=\infty$.

When $g\left(x\right)=e^x$, a number of forms have been considered. For example, a smoothing spline [1] has been proposed for ${\int }_0^{z}D\left(x\right)\partial x$, with a linear-in-concentration functional form used to extrapolate the risk beyond observed exposures, with ${\mathcal{R}}_{e^x}\left(\infty \right)=\infty$. The $Fusion$ model assumes that $D\left(z\right)$ is a monotonically non-increasing regression spline and a logarithm-in-concentration risk model for the concentration above those observed in cohort studies, again with ${\mathcal{R}}_{e^x}\left(\infty \right)=\infty$. The $GEMM$ assumes that ${\int }_0^{z}D\left(x\right)\partial x=ln\left(z/\alpha +1\right)/\left(1+e^{-\left(z-\mu \right)/\tau }\right)$. This functional form is approximated by a logarithmic function for exposures above those observed in the cohorts, and thus, ${\mathcal{R}}_{e^x}\left(\infty \right)=\infty$.

2.2. Criterion for Risk Extrapolation

We suggest examining the population attributable fraction, $PAF\left(l,u\right)=1-1/{\mathcal{R}}_g\left(l,u\right)$, for concentrations $l<u$ as a means to define a criterion for extrapolating the risk beyond the cohort exposure range, where

$${\mathcal{R}}_g\left(l,u\right)=g\left(\gamma {\int }_0^{u}D\left(x\right)\partial x\right)/g\left(\gamma {\int }_0^{l}D\left(x\right)\partial x\right)$$

Our extrapolation criterion is based on the following assumptions:

• We have no direct (observable) information on the magnitude or shape of the relative risk function over the extrapolated exposure range;
• We should not be generating more change in the $PAF$ when extrapolating than we observe;
• The magnitude of the change in the $PAF$ over the observed exposure range is the sole determinant of the magnitude of the $PAF$ over the extrapolated exposure range;
• The magnitude and shape of the relative risk function over the extrapolated exposure range should not influence the magnitude and shape of the relative risk function over the observed exposure range.

The first assumption is the main motivating factor for our approach: that there does exist any relevant information on the risk beyond that observed from cohort studies of outdoor air pollution. The $IER$, on the other hand, assumes that the risk observed from other particulate sources, such as second-hand smoke, household pollution, and active smoking, are relevant and can provide direct information to determine the magnitude and shape of the relative risk function for concentrations above those observed in cohort studies of outdoor air pollution. However, the assumption that the toxicity of these sources is entirely due to the total inhaled dose [8] has not ever been tested or verified.

Our second assumption suggests that we should be cautious in extrapolating the risk for concentrations where no relevant information is available, and that we should not be “artificially” generating more risk than we observe. The third assumption is that we can only use information on risk that we observed to form a basis for extrapolating risk. And finally, the fourth assumption is that we do not want to alter our understanding of how exposure affects the risk over the observed exposure range in order to derive a model for extrapolation of risk.

We can mathematically model assumption (2) by assuming that the $PAF$ over the concentration range $\left[0,\theta \right]$, where $\theta$ represents a high percentile of the cohort exposure distribution (see Appendix A), should be equal to the $PAF$ over the extrapolated range $\left(\theta ,\infty \right)$, or $PAF\left(0,\theta \right)=PAF\left(\theta ,\infty \right)$. That is, we do not want to create more change in the $PAF$ when extrapolating than we observed over the cohort exposure range. This equivalence implies

$${\mathcal{R}}_g\left(0,\theta \right)={\mathcal{R}}_g\left(\theta ,\infty \right)={\displaystyle \frac{{\mathcal{R}}_g\left(0,\infty \right)}{{\mathcal{R}}_g\left(0,\theta \right)}} or {\mathcal{R}}_g\left(0,\infty \right)={\left({\mathcal{R}}_g\left(0,\theta \right)\right)}^2$$

meeting our third assumption that the magnitude of the relative risk over the cohort exposure range determines the magnitude of the relative risk over the extrapolated exposure range.

This also implies that the risk must be bounded from above: ${\mathcal{R}}_g\left(0,\infty \right)<\infty$. [To simplify the notation, we denote ${\mathcal{R}}_g\left(0,\theta \right)$ as ${\mathcal{R}}_g\left(\theta \right)$ when the lower concentration is zero.] We therefore equated the magnitude of the relative risk over the extrapolated concentrations to equal the magnitude of the relative risk over the observed cohort concentrations, without constraining the magnitude at $\theta$.

To do this, we required $D\left(z\right)$ to decline with $z$ at high concentrations at a fast enough rate such that ${\mathcal{R}}_g\left(\infty \right)<\infty$. For our model to meet this criterion, $D\left(z\right)$ must, at high concentrations—say $z>\theta$—decline at a rate faster than $1/z$, since ${\int }_{\theta }^{\infty }{z}^{-1}\partial z=\log \left({\displaystyle \frac{\infty }{\theta }}\right)=\infty$. We effectively discounted marginal changes in the relative risk when the concentrations were above $\theta$, with the discounting increasing as the concentrations get farther away from $\theta$.

We can characterize the magnitude and shape of $D\left(z\right)$ over the cohort exposure range using evidence of the association between concentrations of outdoor $P{M}_{2.5}$ and mortality observed from cohort studies. These studies are now conducted worldwide, including North America, Europe, India, China, Asia, and Africa [1,3].

All these studies use survival analysis models relating the logarithm of the instantaneous probability of death to a subject’s exposure, controlling for other known mortality risk factors such as smoking, diet, body mass index, education, and income. The exposure–response relationship is almost always assumed to be linear in concentration.

Let ${\beta }_i$ for the $i^{th}$ of $I$ cohorts represent the slope of that association and let ${\widehat{\beta }}_i$ represent the parameter estimate of ${\beta }_i$, with ${\widehat{v}}_i$ representing its standard error estimate. We interpret ${\widehat{\beta }}_i$ as the average derivative of the logarithm of the relative risk over the $i^{th}$ cohort exposure distribution when $g\left(x\right)=e^x$. If $g\left(x\right)=1+x$, we replace ${\widehat{\beta }}_i$ with the average derivative of the relative risk, denoted by ${\widehat{\delta }}_i$ (see Appendix A). Let $\left(z_i^{\left(L\right)},z_i^{\left(U\right)}\right)$ represent the lower and upper limit of that distribution.

Each ${\widehat{\beta }}_i$ (or ${\widehat{\delta }}_i$ ) gives us some insight on what $D\left(z\right)$ looks like over its respective exposure range. For example, if $D\left(z\right)$ is nearly constant, then there should be no discernible relationship with concentration. If $D\left(z\right)$ is decreasing/increasing, then the ${\widehat{\beta }}_i$ (or ${\widehat{\delta }}_i$ ) should decrease/increase with concentration. We can mathematically characterize that relationship using meta-regression [11].

3. Results

We illustrated our method with an analysis of the association between non-accidental mortality and $P{M}_{2.5}$ using meta-data on 28 cohorts that were previously examined [3], with the addition of 16 cohorts published since 2021 (one in Canada [12]; one in China [13]; one in England [14]; seven in Europe [14]; and six in Asia [15]), totaling 44 cohorts.

We defined two exposure ranges based on observed outdoor-monitored $P{M}_{2.5}$ concentrations [16] and specific high-exposure situations [4]. For the purposes of this example, we define the global outdoor exposure range as $0\text{–}120 \mathsf{\mu }\mathrm{g}/{\mathrm{m}}^3$ based on the highest global annual concentration in cities in 2024 (Byrnihat, India, at 128$\mathsf{\mu }\mathrm{g}/{\mathrm{m}}^3$ and Delhi, India, at 108$\mathsf{\mu }\mathrm{g}/{\mathrm{m}}^3$ ) [16]. We also define an extended exposure range as $0\text{–}1000 \mathsf{\mu }\mathrm{g}/{\mathrm{m}}^3$ in order to more clearly illustrate the differences in model predictions over high exposures that have been observed [4].

The meta-data are displayed in Figure 1, when the logarithm of the hazard ratio is used ${(\widehat{\beta }}_i)$ (Panel (A)) or when the hazard ratio is used directly ${(\widehat{\delta }}_i)$ (Panel (B)) as red dots, along with the fifth and 95th percentiles of the cohort-specific exposure distributions (red horizontal lines). A great deal of scatter was observed for the meta-data, for either $\widehat{\beta }$ or ${\widehat{\delta }}_i$, and there was no apparent pattern with concentration, suggesting that a constant over concentration maybe an appropriate derivative model.

If $D\left(z\right)=1,\forall z$ and $g\left(x\right)=e^x$, then ${\mathcal{R}}_{e^x}\left(z\right)=e^{\gamma z}$, defining the $Log-Linear$ model. Our estimate of $\gamma =0.0092$ yields a relative risk of 3.0 at $120 \mathsf{\mu }\mathrm{g}/{\mathrm{m}}^3$ and 9897 at $1000 \mathsf{\mu }\mathrm{g}/{\mathrm{m}}^3$. Estimates as large as 9897 appear biologically implausible.

We thus require a model that is flexible within the cohort exposure range, yet yields not such extreme values as the $Log-Linear$ model above the cohort exposure range. One such model is the $GEMM$ [2], as it was specifically designed to model a variety of shapes (near linear, sub-linear, supra-linear, and sigmodal) within the cohort exposure range, yet approximate a logarithmic model for concentrations above the range. This is accomplished by limiting the $GEMM$ parameters $\left(\alpha ,\mu ,\tau \right)$ to lie within the cohort exposure range [2].

We fit the $GEMM$ to our meta-data and displayed the relative risk predictions (blue line) and uncertainty interval (grey shaded area) over the global (Figure 2, Panel (A)) and extended (Figure 2, Panel (B)) exposure ranges. In addition, the cohort meta-distribution is displayed in Panel (A) (green line) along with our estimate of $\theta$ (see Appendix A) along with the $Log-Linear$ model predictions (black line). The $GEMM$ and $Log-Linear$ predictions are similar when concentrations are less than $\widehat{\theta }=51 \mathsf{\mu }\mathrm{g}/{\mathrm{m}}^3$. There is some departure between the model predictions between $51 \mathsf{\mu }\mathrm{g}/{\mathrm{m}}^3$ and 120$\mathsf{\mu }\mathrm{g}/{\mathrm{m}}^3$ (Figure 2, Panel (A)), with the $GEMM$ displaying more curvature than the $Log-Linear$ model. Thus, the $GEMM$ is performing as it was designed. Concerns are raised, however, when the $GEMM$ predictions are extrapolated well beyond the global concentration range (Panel (B)) with $GEMM\left(1000 \mathsf{\mu }\mathrm{g}/{\mathrm{m}}^3\right)=6.8$ and an upper uncertainty limit of 22. The $GEMM$ is thus limited by the use of a single algebraic function over both the cohort and extended exposure ranges.

Burnett and collogues [3] approached this problem by specifying two separate functions; one over the cohort exposure range (see Appendix A) and another, a logarithmic function, used to extrapolate the risk beyond the cohort exposure range, denoted as the $\mathrm{F}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$ model.

All three—the $\mathrm{G}\mathrm{E}\mathrm{M}\mathrm{M}$ (blue line: Figure 3), $\mathrm{L}\mathrm{o}\mathrm{g}-\mathrm{L}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}$ (black line: Figure 3), and $\mathrm{F}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$ (brown line: Figure 3) relative risk predictions—were similar over the cohort exposure range (Panel (A)), but differed over the extended exposure range (Panel (B)), with the $\mathrm{L}\mathrm{o}\mathrm{g}-\mathrm{L}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}$ predictions being greater than the $\mathrm{G}\mathrm{E}\mathrm{M}\mathrm{M}$ predictions, which were in turn greater than the $\mathrm{F}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$ model predictions. We noted that the $\mathrm{G}\mathrm{E}\mathrm{M}\mathrm{M}$ uncertainty interval was clearly wider than the $\mathrm{F}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$ model’s uncertainty interval (Panel (B)). However, the $\mathrm{F}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$ model prediction at $1000 \mathsf{\mu }\mathrm{g}/{\mathrm{m}}^3$, 4.7, was still very large, implying that 79% of all deaths are attributable to ${PM}_{2.5}$ exposure. In addition, $\mathrm{F}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\left(\mathrm{\infty }\right)=\mathrm{\infty }$ is an unbounded function.

We next considered applying our extrapolation criterion with the implication that ${\mathcal{R}}_g\left(\infty \right)<\infty$. The $Fusion$ model [3] specified $D\left(z\right)=D\left(\theta \right)\left({\displaystyle \frac{\theta }{z}}\right), z\ge \theta$, with the derivative declining at a rate of the inverse of the concentration above the cohort exposure range. We propose an alternative derivative model of the form $D\left(z\right)=D\left(\theta \right){\left({\displaystyle \frac{\theta }{z}}\right)}^{\alpha }, z\ge \theta$ and $\alpha >1$ with the property $D\left(\theta \right){\int }_{\theta }^{\infty }D\left(x\right)\partial x=D\left(\theta \right){\int }_{\theta }^{\infty }{\left({\displaystyle \frac{\theta }{x}}\right)}^{\alpha }\partial x={\displaystyle \frac{\theta D\left(\theta \right)}{\alpha -1}}<\infty$ being bounded from above for all concentrations. We propose using the same algebraic form of the derivative for concentrations within the cohort exposure range as the $Fusion$ model. The restriction that the derivative of $D\left(z\right)$ be continuous at $z=\theta$ in order for a smooth transition between different algebraic forms below and above $\theta$ implies that $D\left(z\right)$, when $z<\theta$, must be a function of the decay parameter $\alpha$ (see Appendix A). We term this new model the $Bounded Fusion$ model with a comparison to the $Fusion$ model presented in Figure 4. Methods for selecting the value of $\alpha$ and characterizing the magnitude and shape of $D\left(z\right)$ are provided in Appendix A.

The $\mathrm{F}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$ and Bounded Fusion relative risk predictions were similar over the cohort exposure range (Figure 4, Panel (A)), with the $\mathrm{B}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d} \mathrm{F}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$ being slightly less than the $\mathrm{F}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$ model predictions between the cohort and global concentration ranges (Panel (A)). However, the $\mathrm{B}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d} \mathrm{F}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$ model predictions were clearly less than those of the $\mathrm{F}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$ model over the extended concentration range, with narrower uncertainty intervals (Panel (B)). The property of similar risk predictions over the cohort exposure range between the $\mathrm{F}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$ and $Bounded \mathrm{F}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$ models indicates that our method meets our fourth assumption that the extrapolation model does not influence risk predictions within the cohort exposure range.

We next compared the $Fusion$ and $Bounded Fusion$ model predictions between the two specifications of the link function $\mathrm{g}$ in Figure 5. The predictions were larger when the $\mathrm{g}\left(\mathrm{x}\right)=e^x$ form was used compared to the $\mathrm{g}\left(\mathrm{x}\right)=1+\mathrm{x}$ form due to taking exponentials. The difference between the two link specifications increased with the relative risk estimate at $\mathrm{z}=\widehat{\mathsf{\theta }}$, with a larger difference observed for the $\mathrm{F}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$ model compared to the $\mathrm{B}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d} \mathrm{F}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}$ model specification.

Setting an Upper Bound on Risk

Suppose one assumes a biological bound in risk $\mathcal{R}\left(\infty \right)=\mathcal{B}>\mathcal{R}\left(\widehat{\theta }\right)$. The closer the value of $\mathcal{B}$ is to $\mathcal{R}\left(\widehat{\theta }\right)$, the larger the value of $\alpha$, resulting in more curvature in the model predictions for higher concentrations (see Appendix A). If $\mathcal{B}$ is only slightly greater than $\mathcal{R}\left(\widehat{\theta }\right)$, $\alpha$ can be very large. In the extreme case, $\mathcal{B}=\mathcal{R}\left(\widehat{\theta }\right)$ and $\alpha =\infty$. If $\mathcal{B}$ is much larger than $\mathcal{R}\left(\widehat{\theta }\right)$, $\alpha$ would be only slightly greater than unity and the extrapolated relative risk would be approximately logarithmic. In this case, the $Fusion$ and $Bounded Fusion$ relative risk predictions would be similar.

It is not clear, however, how to select such a bound in practice. Relative risks due to active smoking were used to bound the risk in the original $IER$ [8]. Using the active smoking risk as a guide, we noted that the relative risk of a current smoker to a never smoker for all causes of death was 2.78 [17]. Our upper bounds of $\mathcal{R}\left(\infty \right)=2.58$  $(g\left(x\right)=e^x)$ and $\mathcal{R}\left(\infty \right)=2.41$  $(g\left(x\right)=1+x)$ are less than the active-smoking-related relative risk. The resulting relative risk estimates for concentrations above $\widehat{\theta }$ would be larger than those displayed in Figure 4 if this smoking-based upper bound was used.

4. Discussion

Exposure to fine particulate matter is a leading contributor to mortality and morbidity globally [1]. As such, there is great interest in characterizing the association between exposure and health over the global exposure range [1,2,3]. Although there have been several cohort studies examining this association, only a few have been conducted in situations where the subject’s exposure is similar to the highest global outdoor concentrations. There are cases of interest, however, where people are exposed to much higher levels than in any of the cohorts [4]. Even if new cohorts are established in regions with high exposures, they are not likely to completely cover all cases of policy interest. Relative risk models are thus required to not only cover the global range of outdoor exposure to the general population, but also to much more extreme exposure scenarios that are observed today and for potential cases of changing future atmospheric conditions leading to even higher levels [18]. Methods are thus required to extrapolate the risk potentially well beyond the risks observed in epidemiological studies, yet meet the requirements of biological plausibility when the extrapolated risks may be potentially so large that they are viewed with skepticism [9].

We proposed a method of risk extrapolation for which the $PAF$ over the extrapolated concentration range [$\widehat{\theta }, \infty )$ is equal to that over the observed exposure range $[0,\widehat{\theta }]$. This approach does not require strong additional assumptions like those needed for the $IER$ [8]. We do not specifically bound the magnitude of the relative risk in absolute terms, but in relative terms with $\mathcal{R}\left(\widehat{\theta },\infty \right)=\mathcal{R}\left(\widehat{\theta }\right)$. The larger the value of $\mathcal{R}\left(\widehat{\theta }\right)$, the larger the magnitude of the bound $\mathcal{R}\left(\infty \right)$.

We suggest that the link function $g\left(x\right)=1+x$ be used when predictions of risk at very high concentrations are of interest, as it yields lower relative risk predictions and narrower uncertainty intervals compared to using the $g\left(x\right)=e^x$ link function. The difference between the two forms increases as risk predictions increase. We suggest that it is unnecessary to inflate these predictions at high concentrations, given that we have no data to provide guidance at concentrations well beyond the cohort data exposure concentrations.

The cohort studies typically use a multi-year average of estimates of ambient concentration at a subject’s home address [1,2,3]. However, cases of interest may involve situations where a subject is exposed to potentially very high concentrations for short periods of time, such as riding a motorcycle in Deli, India [4]. In order to use our relative risk function for burden analyses, one would have to convert their estimated time-integrated exposure to a multi-year average equivalent.

We also suggest an alternative approach to bounding relative risk predictions by fixing the value of an upper bound that is not related to the prediction of risk over the cohort exposure range. This approach allows for the policy analyst to decide on what is an acceptable upper bound on risk.

The motivation of developing our model is also its major limitation, in that we have no observed evidence of risk at these extremely high concentrations. Ideally, one should conduct studies in these cases. In the absence of such evidence, ancillary information can provide some guidance as to the reliability of our model predictions. These include information on risk from other particulate sources at very high concentrations or a comparison of our high concentration model predictions with those of the 88 mortality risk factors reported by the $GBD$ [1].

5. Conclusions

We provided a mathematical means to estimate the relative risk over the exposure range where relevant information exists, and to predict the risk at concentrations well above that range, where no information exits. The major limitation of our method, and its motivation, is the lack of direct information at very high concentrations of interest. There is a need to conduct studies at these very high concentrations to assess the reliability of our approach.