Statistical Modelling of Variability and Uncertainty in Risk Assessment of NanoparticlesInsight into risks of nanotechnology and the use of nanoparticles is an essential condition for the societal acceptance and safe use of nanotechnology. Risk assessment of ENPs has been hampered by lack of knowledge about ENPs, their environmental fate, toxicity, testing considerations, characterisation of nanoparticles and human and environmental exposures and routes. This lack of knowledge results in uncertainty in the risk assessment. Moreover, due to the novelty of nanotechnology, risk assessors are often confronted with small samples of data on which to perform a risk assessment. Dealing with this uncertainty and the small sample sizes are main challenges when it comes to risk assessment of ENPs. The objectives of this thesis are (i) to perform a transparent risk assessment of nanoparticles in the face of large uncertainty in such a way that it can guide future research to reduce the uncertainty and (ii) to evaluate empirical and parametric methods to estimate the risk probability in the case of small sample sizes. To address the first objective, I adapted an existing Integrated Probabilistic Risk Assessment (IPRA) method for use in nanoparticle risk assessment. In IPRA, statistical distributions and bootstrap methods are used to quantify uncertainty and variability in the risk assessment in a two-dimensional Monte Carlo algorithm. This method was applied in a human health (nanosilica in food) and an environmental (nanoTiO _{2} in water) risk context. I showed that IPRA leads to a more transparent risk assessment and can direct further environmental and toxicological research to the areas in which it is most needed.For the second objective, I addressed the problem of small sample size of the critical effect concentration (CEC) in the estimation of R = P(ExpC > CEC), where ExpC is the exposure concentration. First I assumed normality and investigated various parametric and non-parametric estimators. I found that, compared to the non-parametric estimators, the parametric estimators enable us to better estimate and bound the risk when sample sizes and/or small risks are small. Moreover, the Bayesian estimator outperformed the maximum likelihood estimators in terms of coverage and interval lengths. Second, I relaxed the normality assumption for the tails of the exposure and effect distributions. I developed a mixture model to estimate the risk, R = P(ExpC > CEC), with the assumption of a normal distribution for the bulk data and generalised Pareto distributions for the tails. A sensitivity analysis showed significant influence of the tail heaviness on the risk probability, R, especially for low risks. In conclusion, to really be able to focus the research into the risks of ENPs to the most needed areas, probabilistic methods as used and developed in this thesis need to be implemented on a larger scale. With these methods, it is possible to identify the greatest sources of uncertainty. Based on such identification, research can be focused on those areas that need it most, thereby making large leaps in reducing the uncertainty that is currently hampering risk assessment of ENPs. `Integrated Probabilistic Risk Assessment for Nanoparticles in Food` Insight into risks of nanotechnology and the use of nanoparticles is an essential condition for the societal acceptance and safe use of nanotechnology. One of the problems with which the risk assessment of nanoparticles is faced is the lack of data, resulting in uncertainty in the risk assessment. We attempt to quantify some of this uncertainty
by expanding a previous deterministic study on nanosilica (5-200nm) in food into a fully integrated probabilistic risk assessment. We use the integrated probabilistic risk
assessment method in which statistical distributions and bootstrap methods are used to quantify uncertainty and variability in the risk assessment. Due to the large amount
of uncertainty present, this probabilistic method, that separates variability from uncertainty, contributed to a better understandable risk assessment. We found that quantifying
the uncertainties did not increase the perceived risk relative to the outcome of the deterministic study. We pinpointed particular aspects of the hazard characterisation that
contributed most to the total uncertainty in the risk assessment, suggesting that further research would benefit most from obtaining more reliable data on those aspects.There is a growing need for good environmental risk assessment of engineered nanoparticles (ENPs). Environmental risk assessment of ENPs has been hampered by lack of data and knowledge about ENPs, their environmental fate and their toxicity. This leads to uncertainty in the risk assessment. To effectively deal with uncertainty in the risk assessment, probabilistic methods are advantageous. In this chapter, we develop a method to model both the variability and uncertainty in environmental risk assessment of ENPs. This method is based on the concentration ratio (CR), the ratio of the exposure concentration to the critical effect concentration, both considered to be random. In our method, variability and uncertainty are modelled separately, so as to allow the user to see which part of the total variation in the CR is due to uncertainty and which part is due to variability. We illustrate the use of our method using a simplified aquatic risk assessment of nanoTiO _{2}. Our method allows a more transparent risk assessment and can also direct further environmental and toxicological research to the areas in which it is most needed.Estimating the risk, R = P(X > Y ), in probabilistic environmental risk assessment of nanoparticles is a problem when confronted by potentially small risks and small sample sizes of the exposure concentration X and/or the effect concentration Y . This is illustrated in the motivating case study of aquatic risk assessment of nanoAg. A nonparametric estimator based on data alone is not sufficient as it is limited by sample size. In this chapter, we investigate the maximum gain possible when making strong parametric assumptions as opposed to making no parametric assumptions at all. We compare maximum likelihood and Bayesian estimators with the non-parametric estimator and study the influence of sample size and risk on the (interval) estimators via simulation. We found that the parametric estimators enable us to estimate and bound the risk for smaller sample sizes and small risks. Also, the Bayesian estimator outperforms the maximum likelihood estimators in terms of coverage and interval lengths and is, therefore, preferred in our motivating case study. Estimation of P(X > Y): Normal-GPD ModelIn estimating the risk in a risk assessment, we are interested in the tails of the exposure (X) and effect (Y ) distributions. In the case of risk assessment of nanoparticles, we are often confronted with small sample sizes. In this situation, empirical estimation of the risk fails due to the lack of data points in the tails of the exposure and/or effect distributions. Although the normal distribution is customarily the first choice when moving from empirical to parametric estimation, its tails are often found to be too thin. In this chapter, we allow for thicker tails by using the generalised Pareto distribution to estimate the tails of the exposure and effect distributions. We develop a mixture model to estimate the risk, P(X > Y ), with the assumption of a normal distribution for the bulk data and generalised Pareto distributions for the tails of X and Y . A sensitivity analysis showed significant influence of the tail thickness on the risk value, especially for low risks. We also studied the effect of small sample sizes on the estimation of the tail index and illustrate the proposed methods on a real data set. |

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