Abstract
Exposure to ENMs may have associated health risks, but accurate measurement of these risks is difficult due to overwhelming methodological limitations and epistemic uncertainties. This is especially the case for ENM physiochemical and toxicity measurements. A common example of controlling such risks in workplace environments where these materials are produced and used is control banding. It offers a useful framework to categorize health risk but is presently limited by existing quantitative data that is susceptible to ambiguity. With an aim to addressing these issues, this chapter develops a Bayesian regression or QSAR (Quantitative Structure Activity Relationship) model that relates hazard levels (dependent) to physical and chemical attributes (independent) but crucially takes full account of uncertainty in both the dependent and independent data sets. The developed model is applied to recover the marginal probability density distribution of a varied set of physical attribute measurements of cerium oxide nanoparticles that were supplied from a common batch. Each of the measurements in the set was carried out by one of several disparate institutions. It is in the author’s opinion that this model is successful because in principle it is able to exploit and objectively incorporate seemingly conflicting data points to produce meaningful regression fits. This is something that is not possible using conventional regression techniques that typically rely on subjective judgments to resolve such conflicts prior to analysis. The danger of the conventional approach is that potentially useful information, usually interpreted as ‘statistical outliers’, may be disregarded as a result of experimenter bias.
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Notes
- 1.
Messenger cytokines are invoked as part of the immune response to recruit antibodies from surrounding tissues to the pathogen’s location for it to be removed or destroyed. This in turn promotes the production of more cytokines that repeat and reinforce the process in a positive feedback fashion. This dynamic only occurs at the organism level and could never be predicted on the basis of in vitro observations alone.
- 2.
Reductionism attempts to understand the behaviour and properties of a system in terms of its irreducible subsystems considered in isolation from one another. The individual subsystem descriptions are then reassembled to offer a complete understanding of the parent system.
- 3.
Hypothetical scenario in which the flapping of a butterfly’s wings results in the formation of a hurricane at another point on the earth’s surface through rapid amplification of the initially small disturbance by non-linear atmospheric dynamics.
- 4.
Noosphere denotes the sphere of human intellect and its effects, through conscious intention, on the physical environment.
- 5.
The direct application of this principle to statistical mechanics leads to Maxwell–Boltzmann statistics for molecular energy distributions in high-temperature gases. In low-temperature environments, it yields the Fermi–Dirac statistics for fermions and Bose–Einstein statistics for bosons.
- 6.
Numerically, the entropy H[P(x 1, x 2, … , x n )] of a general discrete multivariate distribution, P(x 1, x 2 … x n ), is given by: \( H\left[P\right]={\displaystyle \sum}_XP\left({x}_1,{x}_2\dots {x}_n\right) \log \left(P\left({x}_1,{x}_2 \dots {x}_n\right)\right) \) in which the summation is taken over all allowable assignments of the vector \( X=\left({x}_1,{x}_2 \dots {x}_n\right) \). Informally, if P is a one of an infinite number of candidate PDFs capable of describing the distribution of a given set of observations, then H[P] is the log of the probability that P is the actual distribution underlying the observation set. Formally, on the basis of existing constraint data, for example in the form of prior moment information or marginal distributions, the most unbiased distribution that could be inferred would be the one with greatest entropy. By contrast, a biased choice of an otherwise consistent distribution would be one informed by considerations beyond the domain of available information (i.e. irrational) and with the entropy of such a choice being sub-maximal.
- 7.
A ENM’s zeta potential is one of several common physicochemical attributes that are used to characterize ENMs. Among others are size, core chemistry, crystalline structure and aspect ratio.
- 8.
The measurement of biomarkers provides a means to indirectly observe cellular activity in vitro. Unusual or elevated levels normally indicate abnormal cellular behaviour and can be used to infer the potentially toxic effects of a foreign material such as a nanoparticle. There are many varieties with probably the most cited in the literature being reactive oxygen species (ROS), cytotoxicity, cell viability, cytokine numbers and genotoxic effects. Reactive oxygen species (free radicals) result from chemical reactions between cellular components and a foreign substance. They result from normal cellular functions such as metabolism and can have elevated levels when a cell attempts to metabolize a substance that cannot be metabolized such as inorganic non-soluble foreign bodies, for example metallic nanomaterials. Cytokines are messenger molecules that support the immune system. The presence of a pathogen invokes their dispatch by the immune system to signal neighbouring white blood cells in surrounding tissues to come to the infected cell’s aid to remove or destroy the offending pathogen (white blood cells are the immune system’s vacuum cleaners). Genotoxicity effects measure changes in a cell’s DNA structure due to the presence of a pathogen.
- 9.
A polydisperse ENM is characterized as one having by a diverse range of values over a particular attribute or set of attributes.
- 10.
In general, this assumption is not necessary. It has been introduced for reasons of simplicity and ease of illustration. A linear combination of an arbitrary set of basis functions could equally have been used for nonlinear fits.
- 11.
A normal distribution tends to a delta function for vanishing variance. That is \( N\left(\overline{x},{\sigma}^2\right)\to \delta \left(\overline{x}-x\right) \) as \( \sigma \to 0 \).
- 12.
A delta function is defined as \( \delta (x)=0\kern0.5em \mathrm{when}x\ne 0\kern0.5em \mathrm{and}\kern0.5em \delta (0)\approx \infty \kern0.5em \mathrm{such}\kern0.5em \mathrm{that}\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}\delta (x)dx=1 \). It can be shown that this leads to a delta function having the following property: \( \underset{z=0}{\overset{\infty }{{\displaystyle \int }}}f(z)\delta \left(x-z\right)dz=f(x) \)for an arbitrary f. Thus \( \underset{z=0}{\overset{\infty }{{\displaystyle \int }}}\delta \left(\overline{r}(z)-r\right)\delta \left({\overline{z}}_k-z\right)dz=\delta \left(\overline{r}\left({\overline{z}}_k\right)-r\right) \) when \( f(z)=\delta \left(\overline{r}(z)-r\right) \).
- 13.
Let h i,j denote a measure of hazard defined by in vitro methods in which the cell lines used in the experiments are enumerated with the index i and the observed biomarker with the index j. The entries in the matrix are assumed to be normalized deviations from unperturbed levels of the same biomarkers for each cell line that together form a control experiment. The deviations h i,j are normalized with respect to their corresponding unperturbed levels in the control experiment. This means the entries in the matrix are dimensionless quantities and should all be equal to zero when the presence of a ENM does not elicit a response in any of the cell lines. A benign material can therefore be described by a hazard tensor in which all the entries are equal to zero. h i,j is referred to as the hazard tensor, H, given by
$$ \mathbf{H}=\left(\begin{array}{cc}\begin{array}{cc}{h}_{1,1}& {h}_{1,2}\\ {}{h}_{2,1}& {h}_{2,2}\end{array}& \begin{array}{cc}\dots & \dots \\ {}\dots & \dots \end{array}\\ {}\begin{array}{cc}\dots & \dots \\ {}\dots & \dots \end{array}& \begin{array}{cc}\dots & \dots \\ {}\dots & \dots \end{array}\end{array}\right) $$
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McAlea, E.M., Murphy, F., Mullins, M. (2016). A Bayesian Regression Methodology for Correlating Noisy Hazard and Structural Alert Parameters of Nanomaterials. In: Murphy, F., McAlea, E., Mullins, M. (eds) Managing Risk in Nanotechnology. Innovation, Technology, and Knowledge Management. Springer, Cham. https://doi.org/10.1007/978-3-319-32392-3_11
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