Using measurements of exposure from multiple tasks to reliably estimate full-shift exposures requires some decision making. For example, let’s say we have three tasks with exposure to crystalline silica. The three tasks have extensive, high quality survey data, and each task shows significant variability in short-term exposure limits (STELs) and in their impact on the estimated time-weighted average (TWA).

How do we most appropriately use those task survey data? Should we use the 95th upper percentile of the dataset for each task to calculate the TWA? Or should we use the median, arithmetic mean, or geometric mean of each?
This may not be a common dilemma in industrial hygiene, but many of us have needed to aggregate several sets of data to make a decision. If our decision involves only one input variable, the procedure is usually simple. But what happens when we have to put together multiple pieces to reach a decision, and each piece has uncertain or variable data? Which part of the data do you use?
Maybe you’re uncertain what the correct value should be. Do you use best- and worst-case calculations? How do you quantify the effect of your input choices on the results? How do you communicate the uncertainty and its ramifications? VARIABILITY AND UNCERTAINTY Usually, none of the data or information we work with is 100 percent certain. We usually have variable data, uncertain data, or both. Even a relatively reliable and certain input parameter, such as estimates of adult male or female body weight, will still show variability at different ends of the human range.
A great tool to use when we have variable or uncertain data is Monte Carlo simulation (MCS). MCS encompasses a versatile range of tools that are useful in IH calculations, especially when we work with variable or uncertain data. “Uncertainty” in the MCS context generally refers to something we know little about. The range of human susceptibility to a new chemical, for example, is truly uncertain. 
MCS can be helpful when we use task-based data distributions to estimate total job exposures, or when we use mathematical models with known input parameter data distributions. With MCS, we can estimate the distribution of exposures for an individual or group. Perhaps the most common use of MCS so far in human health risk assessment has been to estimate the distribution of environmental exposures, such as children’s exposure to contaminated soil following a site soil remediation project.
The MCS technique IHs use most often involves assembling the statistical distributions of the various input values for our calculations. A calculation could involve a mathematical model, such as the one used to estimate the concentration of an air contaminant in a room. This equation at steady state conditions is rendered as
MCS and the Well-Mixed Room Model The following extended example illustrates the use of MCS with a common IH model, known as the well-mixed room (WMR).
Let’s say a new batch process is at the early design stage, and we are asked to assess the exposure potential in the operation and to help define the level of engineering control needed. Our task analysis suggests a possible worst-case exposure event might occur during quality-control sample collection, where workers check the level of intermediate as the reaction progresses.
The sampling is proposed to occur in a small instrument panel room (10 m3 by volume) using an open sample method to capture 5 liters of vapor in a glass sampling “bomb” from the vessel headspace. The sample container purge and collection of a sample of the vessel headspace vapor takes a few minutes. The ventilation in the panel room is proposed to be between 0.5 and 2 cubic meters per minute while occupied. The properties of the process chemical, the expected headspace concentration range, and the flow rate to the sampling container suggest an emission rate to the control panel room will vary between 500 and 1500 mg/minute. The chemical has a ceiling TLV. We’ll use the generation rate (G) as a uniform distribution, 500 to 1500 mg/min. The room ventilation (Q) will also have a uniform distribution, 0.5 to 2 m3/min.
What concentration is reached at 5 minutes, when the sample collection is ended (at the worst-case “few minutes”)? Because steady state is not achieved in this example, we will use the WMR equation in IH Mod, a flexible Excel spreadsheet that implements many basic modeling algorithms. IH MOD was developed by the AIHA Exposure Assessment Strategies Committee (EASC) and is available at the EASC Web page.
Using MCS software and IH Mod with the WMR model, we find a mean of approximately 370, 5th percentile of 200, and 95th percentile of 560 mg/m3. If the ceiling OEL is 1000 mg/m3, we may decide the exposure is acceptable, but if the ceiling value is 100 mg/m3, some design changes are needed.
Dec​isions w​ith Uncertain Data​
Practical Examples of Monte Carlo Simulations

where C is the airborne concentration of the contaminant, G is the generation rate of the contaminant, and Q is the ventilation rate in the room. Both G and Q are likely to have a range of possible values, since generation rate and ventilation rate typically vary.

To solve this equation with MCS, the software randomly selects a value from the range of possible values for both G and Q, solves the equation, and repeats the process again and again, usually thousands or tens of thousands of times. The selection of values for G and Q is weighted according to probability rules, and the results are expressed as a probability distribution.
THE MCS PROCESS The process for using MCS in industrial hygiene can be described in five steps: 1. Define the model or calculation input parameters. In the simple room air contamination model example above, values include the contaminant generation rate, the room ventilation rate, and the room volume. 2. Choose the values and distributions for the input parameters. There are several commonly used distributions. A uniform distribution can take any value between a minimum and a maximum, and is used when there is no information about likelihood in the range. A triangular distribution has a minimum value, a most likely value, and a maximum value. A normal distribution is defined as a mean and standard distribution. The log-normal distribution is well-known in IH data analyses. Other distribution forms may be appropriate in certain circumstances.
Choosing the correct distribution is very important. Estimated distributions based on limited data or with many non-detect results can give inappropriate results. Use of completely speculative data may give misleading results since the MCS calculations can appear to have a high degree of certainty. 3. Set up the calculations using the MCS software. This usually involves marking the input parameters and the output calculations. If some variables are known to be correlated, most of the software packages provide techniques to handle that.
4. Run the simulation. You’ll need to choose the number of iterations—that is, the number of times the software chooses random values for the inputs, then runs and stores the calculations. The number of iterations you run depends on the variability of the inputs and needs to be sufficient to yield stable results between successive runs of the model calculations. 5. View the results. This is in some ways the easiest stage, since most software packages present the data nicely in tabular and graphical forms, with summary statistics. SENSITIVITY AND UNCERTAINTY ANALYSES Current MCS software tools allow for sensitivity and uncertainty analyses. A sensitivity analysis shows how variability in each model input affects the variability of the results, and can identify which of the inputs has the greatest influence on the “spread” of the results. That information can help identify where improved data might most reduce the variability of the results. In contrast to the sensitivity analysis, a true uncertainty analysis is a more qualitative discussion about the state of knowledge of key model inputs. (EPA has a more advanced discussion on approaches to sensitivity and uncertainty analysis using 1- and 2-dimensional MCS.) MCS TOOLS Two recent developments in AIHA committee work illustrate the usefulness of MCS. These are a stochastic dermal risk assessment tool, and a prototype of MCS to estimate air concentrations using the well-mixed box model. Dermal Exposure Assessment Tool MCS The Dermal Exposure Assessment Tool MCS was designed to help determine the relative risk of dermal exposure based on a series of input questions regarding:
  • dermal contact area
  • dermal concentration (loading on the skin)
  • dermal contact frequency
  • dermal retention time (how long on skin)
  • dermal penetration potential
In the MCS analysis, the values of each of these inputs may be given probabilities. The resulting estimate gives weighted probabilities for each of the hazard zones. Well-Mixed Room Estimation of Air Concentration, MCS For a number of years, the AIHA Risk Assessment Committee has promoted the use of MCS in mathematical modeling to predict exposure to chemicals. Recently, AIHA member Daniel Drolet decided to show us how this would work in Microsoft Excel, without extra software to implement the calculations. A calculated example appears in Figure 1. (Click Figure 1 below for a larger view.) Each of the input variables, emission rate, ventilation rate, and room volume can be defined as distributions (uniform, triangular, normal, or log-normal).
Software for Monte Carlo Simulation In its early days, Monte Carlo Simulations (MCS) required sophisticated computer hardware and programming support. Currently, typical personal computers and off-the-shelf software are fully adequate for most needs. The software runs from freeware to commercial and from basic to sophisticated.
​For many industrial hygiene and risk assessment uses, the lower-end software, including freeware, can provide good results and insights into the analytical power of MCS. For occasional users, the costs of the commercial packages may outweigh their advantages in speed and sophistication.
Numerous commercial software packages work with Excel to implement MCS techniques, including Crystal Ball, @Risk, ModelRisk, and others. Simular and Simulacion are two freeware programs to consider for industrial hygiene purposes. Of these, Simular has some advantages in calculation speed and features, but installation can be difficult. A more thorough discussion can be found on Wikipedia.
For more information about MCS in IH practice, the AIHA Exposure Assessment Strategies and Risk Assessment Committees sponsor an introduction to MCS professional development course at AIHce. The AIHA book Mathematical Models for Estimating Occupational Exposure to Chemicals has a chapter on uncertainty, which includes a segment on MCS. THOMAS W. ARMSTRONG, CIH, PhD, is principal at TWA8HR Occupational Hygiene Consulting, LLC. He can be reached at Acknowledgements: the AIHA tools mentioned in this article are the result of collaborations from many members of AIHA volunteer groups, but especially Jennifer Sahmel, Daniel Drolet, Susan Arnold, and Mike Jayjock.