This is a working document prepared by NIOSH or its contractor for use in discussions with the ABRWH or its 

Working Groups or Subcommittees. Draft, preliminary, interim, and White Paper documents are not final NIOSH 

or ABRWH (or their technical support and review contractors) positions unless specifically marked as such. This 

document represents preliminary positions taken on technical issues prepared by NIOSH or its contractor. 

NOTICE: This report has been reviewed to identify and redact any information that is protected by the 

Privacy Act 5 USC §552a and has been cleared for distribution.

1

White Paper

Square Function Approximation to Estimating Inhalation Intakes 

Prepared by David Allen, DCAS

June 2013

Background

For General Steel Industries, NIOSH proposed estimating inhalation intakes using the 95th percentile of 

the distribution of surrogate airborne data collected while handling uranium metal. In the NIOSH 

estimate, the airborne material was assumed to cause an intake only during the time individuals were 

present and working with uranium metal. Resuspension of uranium contamination was accounted for 

separately.

During GSI work group deliberations, SC&A raised a concern that the airborne activity does not diminish 

instantly when the metal operations end and, therefore, the NIOSH approach could underestimate the 

intakes received by workers. This paper is intended to explore this issue.

Equations Describing Airborne Activity

The airborne active created from any operation can be described in terms of the production and 

removal mechanisms for the airborne material. The rate of change of the airborne activity can be 

described as the production rate minus the removal rate as described in the equation below:

where A is the airborne activity, P is the production rate and λ is the removal rate. For multiple removal 

mechanisms, λ represents the sum of the removal rates. The air concentration at any time after the 

beginning of operations can be expressed as:

It can be seen from each of these equations that, without a removal mechanism, the airborne activity 

will continue to rise as long as there is a production term. In reality, there are removal mechanisms and 

the airborne activity will reach a maximum value equal to P/ λ, when the removal rate equals the 

production rate. Also, from these equations, it can be seen that while the level of this maximum activity 

is dependent on both the production rate and the removal rate, the time necessary to reach the 

maximum activity is driven solely by the removal rate. 
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Working Groups or Subcommittees. Draft, preliminary, interim, and White Paper documents are not final NIOSH 

or ABRWH (or their technical support and review contractors) positions unless specifically marked as such. This 

document represents preliminary positions taken on technical issues prepared by NIOSH or its contractor. 

NOTICE: This report has been reviewed to identify and redact any information that is protected by the 

Privacy Act 5 USC §552a and has been cleared for distribution.

2

Single Episode

In the simple case of a single episode of airborne generating activity, the airborne concentration will 

increase during the operation (up to a maximum) and decrease after the cessation of the operation. 

Figure 1 depicts this effect with the airborne generating activity occurring between time zero and time 

t1.

Figure 1 – Buildup and Decline of Airborne Activity from a Single Episode

In Attachment A, the equations necessary to describe this curve are shown, as well as the mathematics 

associated with integrating the curves over time from time equal to zero to time equal to infinity. The 

resulting area is equal to P/λ*t1. Realizing that P/λ is the maximum airborne activity for continuous 

operations, this area is equivalent to assigning that airborne activity only for the duration of the 

operation. This demonstrates that any intakes received while the airborne activity is decreasing (after 

operations) are accounted for by overestimating the intake during the buildup portion of the curve.

Intermittent Episodes

If, after the first airborne causing episode, a second episode begins, the airborne activity in the area will 

include both sources. These would have an additive affect and cause the airborne activity during the 

second episode to increase to higher levels than experienced in the first episode. Figure 2 shows a 

graph depicting this effect.

t = 0 t = t1
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Working Groups or Subcommittees. Draft, preliminary, interim, and White Paper documents are not final NIOSH 

or ABRWH (or their technical support and review contractors) positions unless specifically marked as such. This 

document represents preliminary positions taken on technical issues prepared by NIOSH or its contractor. 

NOTICE: This report has been reviewed to identify and redact any information that is protected by the 

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Figure 2 – Buildup and Decline of Airborne Activity from a Multiple Episodes

The airborne activity from multiple events can be calculated by adding the overlapping times of multiple 

single events. Since Attachment A demonstrates that an intake from a single event can be calculated 

using the square function, it follows that the intake from multiple events can be estimated using 

multiple square functions. The total area of multiple square functions would be the airborne activity 

multiplied by the total time of operation. 

Intakes

Attachment A provides the mathematical basis for indicating the square function is a good estimate of 

the intakes. For illustrative purposes, Figure 3 below superimposes that function on the saw toothed 

function above.
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Working Groups or Subcommittees. Draft, preliminary, interim, and White Paper documents are not final NIOSH 

or ABRWH (or their technical support and review contractors) positions unless specifically marked as such. This 

document represents preliminary positions taken on technical issues prepared by NIOSH or its contractor. 

NOTICE: This report has been reviewed to identify and redact any information that is protected by the 

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Figure 3 – Airborne Activity from a Multiple Episodes with Square Function Superimposed

From the figure, it is not easily seen that the area under both curves is equal. In reality, they are not. 

They are only equal if the saw toothed curve is extended out to infinity. Until then, the square function 

is higher. It is more illustrative to show a graph of the area under the curves (Figure 4).

Figure 4 – Integrated Airborne Activity from Multiple Episodes

While Attachment A indicated the area under both curves is equal, this graph shows the square function 

is actually higher. The reason for this is the equations in Attachment A were integrated to infinity. 
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Working Groups or Subcommittees. Draft, preliminary, interim, and White Paper documents are not final NIOSH 

or ABRWH (or their technical support and review contractors) positions unless specifically marked as such. This 

document represents preliminary positions taken on technical issues prepared by NIOSH or its contractor. 

NOTICE: This report has been reviewed to identify and redact any information that is protected by the 

Privacy Act 5 USC §552a and has been cleared for distribution.

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Figures 5 and 6 below show Figures 3 and 4 extended out further in time after the end of intermittent 

operations.

Figure 5 – Figure 3 with Extended Time

Figure 6 – Figure 4 with Extended Time

From these graphs, it can be seen that the area under the two curves will eventually be equal but the 

square function reaches the final value faster.
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Working Groups or Subcommittees. Draft, preliminary, interim, and White Paper documents are not final NIOSH 

or ABRWH (or their technical support and review contractors) positions unless specifically marked as such. This 

document represents preliminary positions taken on technical issues prepared by NIOSH or its contractor. 

NOTICE: This report has been reviewed to identify and redact any information that is protected by the 

Privacy Act 5 USC §552a and has been cleared for distribution.

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Effect of Different Removal Rates

In the graphs presented so far in this report, the values of parameters were chosen to illustrate the 

dynamics of the airborne activity. This leads to the question of what would the effect be of different 

values for the removal rate. All the graphs (except Figures 5 and 6) show a 190 hour time period with a 

removal rate of 0.03 hours. Below are graphs showing various removal rates. It is clear from these 

graphs that a fast removal rate would closely resemble the square function. This makes intuitive sense 

because a fast removal rate would cause the airborne activity to reach an equilibrium value quickly after 

the start of operations and decrease to zero quickly after operations. That pattern would closely mimic 

the square function. For slower removal rates, the square function over-predicts the airborne even 

more during the operations.

Figure 7 – Integrated Airborne with λ = 1
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Working Groups or Subcommittees. Draft, preliminary, interim, and White Paper documents are not final NIOSH 

or ABRWH (or their technical support and review contractors) positions unless specifically marked as such. This 

document represents preliminary positions taken on technical issues prepared by NIOSH or its contractor. 

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Figure 8 – Integrated Airborne with λ = 0.1

Figure 9 – Integrated Airborne with λ = 0.01
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Working Groups or Subcommittees. Draft, preliminary, interim, and White Paper documents are not final NIOSH 

or ABRWH (or their technical support and review contractors) positions unless specifically marked as such. This 

document represents preliminary positions taken on technical issues prepared by NIOSH or its contractor. 

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Figure 10 – Integrated Airborne with λ = 0.001

Approximation

An alternative method of estimating intakes would be to assume a continuous intake with the 

production rate prorated to account for the fraction of time operations occurred. For example, if 

operations occurred 8 hours per day every day, the production rate would be multiplied by 1/3 (8 

hours/24 hours). An example of this effect is shown in the figure below over the extended time frame.

Figure 3 – Airborne Activity from Multiple Episodes with Approximation Superimposed
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Working Groups or Subcommittees. Draft, preliminary, interim, and White Paper documents are not final NIOSH 

or ABRWH (or their technical support and review contractors) positions unless specifically marked as such. This 

document represents preliminary positions taken on technical issues prepared by NIOSH or its contractor. 

NOTICE: This report has been reviewed to identify and redact any information that is protected by the 

Privacy Act 5 USC §552a and has been cleared for distribution.

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This approach would use the same equations in Attachment A, but in this case, “P” is replaced with “fP” 

where f is the prorating factor. From Attachment A, this would result in the integrated airborne activity 

being described by fP/λ * t1. This would be equivalent to assuming the airborne concentration (reduced 

by f) being applied only for the duration of the operations. In this case, the duration of operations 

would be continuously over the entire time frame of intermittent operations.

This analysis implies that any airborne activity applied throughout a period of intermittent operations 

must be prorated for the amount of time actual operations occurred.

Conclusion

Intakes from airborne activity can be shown mathematically to be accounted for using the square 

function. The square function simply uses the airborne activity times the length of time the airborne 

causing evolutions occur. The square function overestimates the intakes at the start of the operation 

and underestimates after the end of the operation. The two effects mathematically cancel. Exploration 

of multiple episodes indicates the same is true for multiple episodes. Exploration of various removal 

rates also shows the same results.

It is also important to note that this analysis assumes someone is always present in the area. Further 

adjustments should normally be made for period of time when an individual leaves the area.
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Working Groups or Subcommittees. Draft, preliminary, interim, and White Paper documents are not final NIOSH 

or ABRWH (or their technical support and review contractors) positions unless specifically marked as such. This 

document represents preliminary positions taken on technical issues prepared by NIOSH or its contractor. 

NOTICE: This report has been reviewed to identify and redact any information that is protected by the 

Privacy Act 5 USC §552a and has been cleared for distribution.

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Attachment A

Mathematical Comparison of Buildup and Decay to Square Function

Figure A-1 – Building up and Decline of Airborne Activity from a Single Episode

The graph above shows the buildup and decline of airborne activity from a single airborne generating

operation starting at time equal to zero and ending at time equal to t1. The buildup can be described by 

equation 2 (reiterated here).

The airborne level at t1 is then described by replacing t in the equation above with t1. At that point, 

that airborne activity begins to decline at a rate described by:

For someone continuously in the area, the intake can be described as the area under the curve which is

the integration of the two equations above over the appropriate time periods:

t = t1 t = 0
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Working Groups or Subcommittees. Draft, preliminary, interim, and White Paper documents are not final NIOSH 

or ABRWH (or their technical support and review contractors) positions unless specifically marked as such. This 

document represents preliminary positions taken on technical issues prepared by NIOSH or its contractor. 

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11

Noting that in the last term:

The equation for the area can be expanded to

This can be solved as:

Noting that:

This simplifies to:

or

Note that P/λ is the equilibrium airborne activity and t1 is the time the airborne activity was produced. 

This demonstrates that for an individual present in the area the entire time, the intake can be estimated 

by assuming the equilibrium activity was present from the start but only for the duration of the airborne 

causing evolution.