The Power and Utility of Mathematical Models

The phone rings and you answer. On the other end of the line is a clinician from a healthcare provider for an employee who has a set of symptoms. To help with their diagnosis, the clinician asks you to tell her about the worker’s chemical exposures. The employee works with solvents, and you have a list of chemicals used in the department where the employee works. You also have air sampling data for some chemicals in some work areas but not for other chemicals in other areas. And the concentration measurements you have aren’t comprehensive. What sort of information can you give the clinician about exposure levels?

You get a report of a worker feeling sick after a chemical exposure. She had opened an unventilated storage closet and encountered the strong smell of an organic vapor. A metal container containing Stoddard solvent had developed a leak about halfway down the side of the container. The worker suffered from a strong headache but soon felt better. Even though the situation seems to be resolved, you still want to document what sort of exposure she had. How can you get a handle on that?

The purchasing department, with input from both the production department and employee representatives, comes to you with a product recommendation. The product reportedly works better and is less expensive than the one currently used, but it contains methylene chloride. You know methylene chloride triggers exposure assessment requirements. How do you respond to the product recommendation?

These are just a few examples of many scenarios where industrial hygienists have to make decisions about exposures with limited information available to support their judgment. It’s part of the job of industrial hygienists to know about the exposures occurring at their facilities. But fully understanding them all is a monumental task.

In a world of limited budget, time, and personnel, success can hinge on the ability to leverage resources. And one of the major goals in the world of industrial hygiene is to characterize the level of risk to which workers are exposed in the course of the myriad tasks they perform. Even though the list of chemical exposures at a small facility may be short, the industrial hygiene assets for such a facility are often short as well. Larger facilities tend to have more industrial hygiene resources, but they also use more chemicals and their workers perform more tasks. In their efforts to get a firm handle on all of the exposures that need to be anticipated, recognized, evaluated, and controlled, many industrial hygienists constantly feel one or more steps behind.
Most industrial hygienists rely on professional judgment in determining the acceptability of an exposure. But research shows that the accuracy of professional judgment can vary widely.

In the basic practice of industrial hygiene, evaluating the acceptability of an exposure consists first of making some assessment of its intensity, duration, and frequency for a worker or group of workers. Then the industrial hygienist compares the exposure assessment to some decision-making criteria to judge whether the exposure is acceptable. Because the list of needed exposure evaluations is long, most industrial hygienists rely on professional judgment in determining the acceptability of an exposure. But research shows that the accuracy of professional judgment can vary widely. “According to the literature, initial judgments have a mean accuracy of approximately 30 percent; in some cases, they are no more accurate than judgments based on random chance,” write Susan Arnold and Gurumurthy Ramachandran in the
January 2014
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. And although professional judgment can be improved, initial judgments of exposure levels are often lower than the concentrations that actually exist.
Air sampling has been industrial hygienists’ go-to tool for assessing exposure levels. But a few air samples, by themselves, present only a small snapshot of the overall pattern of exposure. Determining an exposure profile based on 6 to 10 air samples as suggested by the AIHA exposure assessment strategy is resource intensive. When good air sampling data is available, it provides valuable information for judging the acceptability of an exposure. But some exposures that industrial hygienists are interested in can’t be sampled. What if a process disruption occurs? What will exposures be with a new process or an alternative chemical?
Using mathematical models to estimate chemical exposures is a way to leverage existing information to arrive at better exposure profiles and make better decisions about prioritizing efforts to gather further information and implement controls. While the world of chemical concentration models for exposure assessment can be intimidating, it doesn’t need to be. Industrial hygienists are trained and experienced applied scientists. They have the capability to integrate modeling into regular practice. Hesitation about using models can be overcome by practicing the calculations of some basic models and understanding their appropriate application.
All one needs to start using basic models to help manage workers’ exposures is an elementary understanding of math and science. A practicing industrial hygienist can begin with simple models and make conservative assumptions that will likely lead to an overestimate of true exposures. If a conservative exposure calculation results in a concentration estimate that is a fraction of the level of concern, it can support a decision that the exposure risk is minimal. But if the conservatively modeled concentration approaches or exceeds the concentration of concern, additional exposure characterization can be performed—perhaps using a more sophisticated model or prioritizing the exposure for air monitoring. It’s likely, though, that many exposures can be identified as acceptable by using simple screening models.
A simple mathematical exposure model can be used in the same way as a “worst-case” air sample. A worst-case sample of an air contaminant is one taken very close to the point of release or from a location where a worker is unlikely to be stationed for eight hours. If the lab results show that the concentration is quite a bit lower than the level of concern, the hygienist may feel comfortable in judging that worker exposures in that area are acceptable. No one thinks that the measured air concentration is representative of the full exposure profile of workers in that area. That would be asking the sampling result to do more than it can. Yet that air sample provides valuable information for decision making.
Similarly, a simple model shouldn’t be asked to do more than it can. A basic air concentration model will not fully characterize exposures of a worker or group of workers. But it can provide good information to aid in the management of exposures.
The models that can be used for worst-case screening are not complex. Any industrial hygienist can learn to use them and perform the math on a pocket calculator or smartphone. Once a hygienist is comfortable with basic models, embracing more complex models that better refine exposure estimates is easier. Practicing thinking like a modeler with simple models makes the more complex models all the more accessible.

Modeling exposures has additional benefits. By documenting the information used, the assumptions made, and the modeling calculations, an industrial hygienist has established “objective data” to support an exposure assessment. The concept of objective data is useful for regulatory compliance. The use of objective data in exposure assessment is referenced in substance-specific OSHA standards such as 1,3-butadiene, methylene chloride, and formaldehyde. For example, the butadiene standard lists the use of objective data as one way of demonstrating levels of exposure (CFR 1910.1051(a)(2)(i) and elsewhere). The standard explicitly mentions models as a type of objective data, which it defines as “monitoring data, or mathematical modeling or calculations based on composition, chemical and physical properties of a material, stream or product.”
The word “model” isn’t explicit in the formaldehyde or methylene chloride standards. But the descriptions of objective data in those standards indicate fairly clearly that it is something different than air sampling. For example, the formaldehyde standard states, “Representative samples for each job classification in each work area shall be taken for each shift unless the employer can document with objective data that exposure levels for a given job classification are equivalent for different work shifts” (see 29 CFR 1910.1048(d)(1)(iv)).
Another interesting insight to the use of mathematical models can be seen in OSHA’s
to a question regarding the necessity of doing air sampling to comply with the “reasonable estimate of exposure” requirement in the respiratory protection standard (29 CFR1910.134(d)(l)(iii)). OSHA’s reply, which is posted to the agency’s website, was that while air sampling is desirable, “other means can be used to estimate workplace exposures. These methods include, but are not limited to, the use of objective data, application of mathematical approaches, and others.”
When good air sampling data is available, it provides valuable information for judging the acceptability of an exposure. But some exposures that industrial hygienists are interested in can’t be sampled.

Access to modeling resources is steadily increasing. AIHA has print resources such as
Mathematical Models for Estimating Occupational Exposure to Chemicals
to help get industrial hygienists started in modeling. There are also periodic professional development courses, introductory and advanced, that provide training in model use. The AIHA Exposure Assessment Strategies Committee has a mathematical modeling project team that brings together modelers of different levels of experience and from various industrial sectors to network and share ideas for the advancement of modeling by industrial hygienists.

Because of its connection to nuclear weapons use, the pioneering work in ambient air modeling was supported by the government. No such support is forthcoming for occupational air pollution exposure modeling. Nevertheless, the science is progressing. Research on exposure modeling in the
Journal of Occupational and Environmental Hygiene
(JOEH) and other outlets communicates new and promising work. Soon-to-be-published research in JOEH demonstrates the effectiveness of concentration modeling in providing good information for judging the acceptability of exposures.
Modeling particles, particularly the generation rate of particles, is more challenging than modeling gases. (Note that the scenarios at the beginning of this article are all gas-phase exposures.) But even for particulate, more information is becoming available.
So what about those opening scenarios? Here’s how modeling can help in each situation:
If you’re familiar with modeling, you will likely have already done some modeling work by the time the health care provider calls. You could tell her that purchasing records helped you determine how often and how much solvent is brought into the facility. With this information, you quickly calculated an estimate of the solvent’s long-term average evaporation rate. That emission rate, along with the room air supply rate given to you by the HVAC engineer, allowed you to calculate a ballpark, first-pass estimate of the long-term average concentration in the work space.
A similar approach can work for the leak of Stoddard solvent in the closet. An inspection of the leaky container can give an estimate of the maximum mass of the solvent that may have evaporated. That mass divided by the volume of the unventilated storage closet is a reasonable estimate of the air concentration.
As for the decision to use the product containing methylene chloride, with some data on the formulation of the product and information about how it will be used and the work space in which it will be used, simple modeling can define the range of likely exposures. If you decide to use the product on a test basis, air sampling to firm up the exposure assessment would certainly be prudent. But through modeling you will have documented the decision-making process, and the workers won’t be using the product with no idea about what exposures might be.
Mathematical models play an important role in a comprehensive exposure assessment strategy. When air concentration measurements can’t be performed because the exposure took place in the past, such as a transitory accidental exposure, modeling can provide air concentration information. If the exposure hasn’t yet occurred, such as in emergency planning or process changes, modeling again can provide information. And with contemporaneous exposures, modeling can help screen exposures as well as prioritize air sampling efforts and complement air sampling results.
will step further into the nuts and bolts of mathematical modeling and provide some practical examples of how to implement this powerful tool into industrial hygiene practice.
is a professor in the Department of Geology and Environmental Science at Wheaton College in Wheaton, Ill. He can be reached at
or (630) 752-7271.

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Disadvantages of being unacclimatized:
  • Readily show signs of heat stress when exposed to hot environments.
  • Difficulty replacing all of the water lost in sweat.
  • Failure to replace the water lost will slow or prevent acclimatization.
Benefits of acclimatization:
  • Increased sweating efficiency (earlier onset of sweating, greater sweat production, and reduced electrolyte loss in sweat).
  • Stabilization of the circulation.
  • Work is performed with lower core temperature and heart rate.
  • Increased skin blood flow at a given core temperature.
Acclimatization plan:
  • Gradually increase exposure time in hot environmental conditions over a period of 7 to 14 days.
  • For new workers, the schedule should be no more than 20% of the usual duration of work in the hot environment on day 1 and a no more than 20% increase on each additional day.
  • For workers who have had previous experience with the job, the acclimatization regimen should be no more than 50% of the usual duration of work in the hot environment on day 1, 60% on day 2, 80% on day 3, and 100% on day 4.
  • The time required for non–physically fit individuals to develop acclimatization is about 50% greater than for the physically fit.
Level of acclimatization:
  • Relative to the initial level of physical fitness and the total heat stress experienced by the individual.
Maintaining acclimatization:
  • Can be maintained for a few days of non-heat exposure.
  • Absence from work in the heat for a week or more results in a significant loss in the beneficial adaptations leading to an increase likelihood of acute dehydration, illness, or fatigue.
  • Can be regained in 2 to 3 days upon return to a hot job.
  • Appears to be better maintained by those who are physically fit.
  • Seasonal shifts in temperatures may result in difficulties.
  • Working in hot, humid environments provides adaptive benefits that also apply in hot, desert environments, and vice versa.
  • Air conditioning will not affect acclimatization.
Acclimatization in Workers
Other articles on exposure assessment will address models for forecasting occupational exposures (April 2017 issue), models for studying potential exposures from products in commerce (September 2017), and models for responding to a hazardous materials emergency (December 2017, a special issue on emergency response).