The Eddy Diffusion Near Field Model is Now Useable

I am going back to a “nuts and bolts” piece on inhalation exposure modeling this week.   The subject is a near field model that has been around for many years but was not very useful until recent work has promised to make it so.

The model is the Eddy Diffusivity Model.   The basic model is presented below:

It may not look like it but the math is pretty straightforward especially if you let IH MOD do the work (see previous blog MODELING MATH MADE EASY OR AT LEAST EASIER to learn about IH MOD Excel Spreadsheet Models and documentation).    Conceptually, the model is pretty simple.   If you have a small source its vapors will radiate out as a sphere if it is suspended in air,  They will radiate as a hemisphere if it’s on a flat surface, as a quarter-sphere if it is on a floor surface along a wall and as a 1/8 sphere if it’s in the corner.  The above equation is for a sphere, the 4 become a 2 for a hemisphere, a 1 for a quarter-sphere and 0.5 for a 1/8 sphere.  What is cool about it is that the concentration is a continuously decreasing gradient as you go away from the point source.   That is, as the distance from the source (r) increases then C decreases.   It does NOT need or use the well-mixed assumption of the 1 zone or 2 zone models.

Seems like it would be the ideal model for such sources but there was one major problem.   All the parameters in the model are relatively easy or straightforward to estimate or measure except D.  Indeed, the predictions of this model are highly depended on D as defined above.   D is dependent on how the air moves about randomly in the indoor environment and it has historically proven itself to be very difficult to measure.   As a result we have had to use a very wide range of estimates for D and as such the utility of this model was quite limited.

Enter some sharp researchers from Stanford University and their work on estimating D from parameters in the room that are much easier to measure; namely, ventilation rate expressed as air changer per hour (ACH) and the dimensions of the room.  They published their work in the journal of Environmental Science and Technology (ES&T) which has a very good reputation.    This part of that paper boils down to the following simple regression relationship:

D  = L²  (0.60 (ACH) + 0.25)/60    (units:  m²/min)

                L = cube root of the room volume (m)

               ACH  = mixing air changes per hour in the room volume (hr^-1)

The R²  regression fit for this sub-model is 0.70 which means that 70% of the relationship between D and the room volume and ventilation and D is explained or predicted by the model and about 30% is unexplained or random noise.  In my experience, given the other uncertainties involved, this is pretty good.  This algorithm is applicable over an ACH range of 0.1 to 2.0.    Dr. Kai-Chung Cheng was first author on this paper and it is my understanding that he is pursuing additional work to sharpen up this relationship and to add to its applicability.   Dr. Rachael Jones (University of Illinois at Chicago, School or Public Health) is also a brilliant modeler and a very active researcher in this area.  I understand that she is also planning research to deepen our quantitative understanding of these relationships.   In the mean time I have put the above algorithm to estimate D into a spreadsheet which I will happily send to whoever asks me for it at mjayjock@gmail.com.

I plan to use it whenever I use the 2 box model (see previous blog: THE MOST VERSATILE AND WELL-TESTED INHALATION MODEL) to compare the results and try and learn something about what these different models are telling us. 

The reference for the Stanford paper is:

Kai-Chung Cheng, Viviana Acevedo-Bolton, Ruo-Ting Jiang, Neil E. Klepeis, Wayne R. Ott, Oliver B. Fringer, and Lynn: Modeling Exposure Close to Air Pollution Sources in Naturally Ventilated Residences: Association of Turbulent Diffusion Coefficient with Air Change Rate M. Hildemann,  dx.doi.org/10.1021/es103080p | Environ. Sci. Technol. 2011, 45, 4016–4022

I am quite sure that Dr. Cheng will be happy to send you a pdf copy if you write to him at: KaiChung Cheng kccheng78@gmail.com

                                                                                                                                    

Dimensional Analysis is an Important Modeling Tool

Modeling is about units and keeping the units straight is critical.   Dimensional analysis assures that we are comparing “apples to apples” and that we are in the correct ball park with our answers.  Most of you probably already know this but some of you perhaps do not.

I once reported an answer that should have been in units of milligrams (mg) as micrograms (µg) and thus released a report with an error of 1000x !   That mistake provided two lessons.   First, ALWAYS have some peer review by a trusted colleague and second, take your time and do a thorough dimensional analysis of your math. 

Most exposure models represent a string of algebraic calculations.   Sometime the string can get pretty long and complicated with all the factors that go into making the model prediction on the left hand side of the equation and the answer (either final or intermediate answer) on the right side of the equal sign.  If you break it down using dimensional analysis, it becomes much easier to handle.

Let’s do an example of a relatively simple equilibrium concentration model with constant source rate and ventilation rate:   C = G/Q     Note:  we want our answer in mg/ m³

The scenario is an open drum evaporating into a room at room temperature.

For Q we are told that it is a 30 m³ room with 0.5 mixing air changes of fresh air ventilation per hour.

So  Q = (room vol)(air changes per hour) =  units of  (m³)(1/hr) or m³/hr  ans:  15 m³/hr

Simple enough, indeed I think we have all done this; however, look closely at what we really did.  We multiplied a variable with units (m³) times a variable with units  (1/hr or hr^-1) to get an entity with units m³/hr.   That is really all there is to dimensional analysis.

Let’s say we measured the evaporative loss of liquid from the drum over time as 2 grams in 400 minutes.   That is 2/400 or 0.005 grams/min; however, we are looking for units of mg/hr.     So:

G = (0.005 grams/min)(60 min/hr)(1000 mg/gram) =   300 mg/hr   (we cancel out the grams and the minutes and are left with mg/hr

As such,  we have C = to G (300 mg/hr) divided by Q (15 m³/hr).   For clarity I am just showing just the units or dimensions below:    

                 (mg/hr)/(m³/hr) and the hrs cancel out leaving mg/m³

In the above equation we are left with 20 mg/m³ as an estimated equilibrium airborne concentration in this room.  

If we knew the molecular weight of the compound we could calculate the concentration as volume parts per million (ppm v/v) in air using the molar volume (gaseous volume of 1 mole of any gas or vapor) of 24 liters (L) at 25C.   For a vapor at 1 mg/m3 with a MW of 100 g/mole we can determine the linear conversion factor:

(1g/1000mg)  (24L/mole) (1 mole/100 g) (0.001 m³/L)(1 mg/m³)= 2.4 x 10^-7 (unitless conversion factor)

Thus, for every 1 mg/m³ of a gas (with MW 100 at 25C) there are 2.4 x 10^-7 parts of gas per volume parts of atmosphere.  Multiply this part per part by one million (10⁶) and you have the parts per million conversion factor of 0.24 for the conversion between mg/m³ and ppm v/v or 1/0.24 =  4.2 to convert ppm v/v to mg/m³ for this gas at this temperature.  The dimensional analysis for this is below:

For every 1 mg/m³ there are (2.4 x 10^-7 parts/parts) (1,000,000 parts/million parts by volume) =  0.24 ppm v/v and the receprical 1/0.24 or 4.2 mg/m³ for every ppm v/v of this vapor.  For our example it would be 20 mg/m³ times 0.24 or 4.8 ppm v/v assuming its MW was 100 and it was at 25C.

If this explanation of dimensional analysis is a little fuzzy, I found one that is clearer and is only 9 minutes long on YouTube:    http://www.youtube.com/watch?v=fEUaQdaOBKo

Believe me, dimensional analysis is your friend and will help to keep you sane in doing problems associated with modeling.

Risk Assessment Uncertainty or How Safe is Safe? Part2 Exposure Limits

In the last blog I discussed the inherent uncertainty around measured or model estimated exposure.  This week it is time to talk about the uncertainty in any exposure limit.

We have all seen changes in the exposure limits we have used over time.  The changes are almost invariably downward toward lower limits.   Does this mean that the chemical became more toxic?   Of course not, it just means that the uncertainty inherent in that particular exposure limit was not very well handled.  To guard against these surprises, I believe that uncertainty should be explicitly addressed during the documentation process. 

The current definition of the risk present at the exposure limits that most of us use is that exposures controlled to these limits will protect “nearly all”.    Although the intent is clearly to protect the vast majority of folks exposed at the limit, there is currently no attempt to quantify what is meant by “nearly all”.   For a long time I have thought that the level of risk present at any exposure limit worthy of documentation should be quantified to the extent possible and, more important, the uncertainty around that estimated quantitative level of risk should also be provided.

In truth, the risk of an adverse health effect occurring is a distribution of values which is low at low exposure levels and high at high exposures.   The exposure limit is but one value on that distribution.  We (Jerry Lynch, Phil Lewis and I) wrote a paper in 2001 about how one could estimate the risk at any exposure limit and how the uncertainty might be estimated.  I would be happy to send a copy of that paper to anyone who asked me at mjayjock@gmail.com.    A more definitive scientific treatment of this subject was put forth in 2009 by in the National Academy of Sciences – Science and Decisions: Advancing Risk Assessment, also known as the “Silver Book”.   The hard copy of the book will set you back about $55 but the NAS offers it for FREE as a PDF Download!

http://www.nap.edu/catalog.php?record_id=12209      

The meat of this subject is in Chapter 5.

So in the final analysis, risk is a combination of uncertain (a distribution of) exposure and (a distribution of) hazard (or toxicological response).  Combining both distributions presents an output distribution of risk at any particular nominal or median exposure.   If the following conditions are met then the risk will be shown to be relatively low or “safe”:

·         An exposure limit that is relatively high versus the median estimated exposure.

·         The distributions for exposure and exposure limit are relatively narrow such that they do not have a lot of overlap.

Please note there will still be some finite level of predicted risk – it will never be zero.

When the exposure goes up relative to the exposure limit and/or the distributions for exposure or exposure limit are relatively wide then the predicted potential risk goes up as well. 

I believe that this is how we might start to get our arms around “How safe is safe?” 

Describing uncertainty in this or a similar manner will keep us from being surprised like we have been in the past.  It is also important to understand that much (perhaps most) of the uncertainty in the estimated hazard (exposure limit) is a result of our lack of knowledge around the actual mechanisms of toxicology.   Some modeled exposure estimates are also fraught with this uncertainty born of a lack of knowledge.   Thus, this type of analysis will also show us where we need to sharpen up our tools to narrow either the exposure limit or exposure distributions and allow much more confident estimates of risk for our clients.

Risk Assessment Uncertainty or How Safe is Safe? Part1 Exposure

In the last blog I discussed the client’s expectation that the risk assessments we do represent our professional certification of the relative safety of any scenario under consideration.   Of course, the thoughtful reader will then question:  What is safe?  

The above assumes that the risk assessment will end with a “happy face”.   That is, that the scenario is deemed in the report to be relatively safe.   The reality is that I have rarely written an assessment that was not so.   Most clients do not want a determination of significant or unacceptable risk documented.   Typically, if the client has committed to doing a risk assessment then they are committed to either refining the assessment (with additional testing and data) to the point of allowing a conclusion of safety (see previous blog) or applying risk management options that choke down the exposure and reduce the risk to acceptable (or at least not unacceptable) levels. 

Again we are at essentially the same question:  What is safe or at least not unacceptably risky?

One answer to that question is that a “safe” exposure is an exposure that does not exceed the exposure limit.   For the purpose of this blog we will assume that the exposure limit is a “bright line” that defines a safe exposure and then look at it from the exposure end of things.    The factors that make up exposure are not constant and indeed they are quite variable.  In fact, if you look at monitoring data for the same person doing the same job, the spread in values is quite large and is often described as a lognormal distribution with a geometric standard deviation (GSD) of 2 or greater.   When we have a GSD of 2, it means that the ratio of the 84^th percentile/50^th percentile of this distribution and the 50^th %tile/16^th %tile is equal to 2.     Thus, the 84^th%tile/16%tile is 4 fold.  That still leaves 32% of the exposures either less than 1/2^th or greater than 2x of the median exposure.   As practical example, a measured distribution with a median exposure of 100 and a GSD of 2 will have 16% of its values below 50 and 16% above 200.     If the exposure limit is 200 then 16% of the time the exposure limit will be exceeded by the exposure.

Considering such statistics, many in our profession consider an exposure “safe” or at least in compliance if it does not exceed the exposure limit greater than 5% of the time.   Thus a median exposure of 100 with a GSD 2 would not be considered “safe” given an exposure limit of 200.   The median measured exposure would have to be significantly lower than 100 assuming the GSD remains at 2.   

The above is an ideal case, when we have a lot of data and can accurately estimate the actual distribution of exposures. 

Consider what most often is the case.  We take a few samples and if they are below the exposure limit some of us might often declare the situation safe.    For the above example, it should be obvious that we should do some statistical analysis on the samples we take.  IH STAT was designed to do just that. This important tool for evaluating our monitored data is available at:
http://www.aiha.org/get-involved/VolunteerGroups/Pages/Exposure-Assessment-Strategies-Committee.aspx

I will cover this important tool in a future blog.   It will tell you how good your data really are at predicting exposure and risk.

If you want a very sobering experience.  Download the free app IH  DIG (by Adam Geitgey) on your Android device (available at the Play Store) and see how good you are at predicting the actual exposure potential using the above criteria of "safe" from a few measured values.   Like I said, it is a very sobering experience.

Modeling exposure has the same issue.  If you are honest about the variables you put into the models you know that they are not single values but distributions as well.   That means that the model output of estimated exposure is also a distribution of exposures which can be compared to an exposure limit.  Monte Carlo analysis is the best way to gauge the input distribution and obtain an output distribution of predicted exposures. Not surprizing, most output distribution appear to be shaped like lognormal curves.  I will go over a simple example in a future blog but the point is that there will almost always be some level of predicted exposure in these distributions that is above the exposure limit. 

So "how safe is safe?”  It turns out that it is a question to be decided by the body politic as a subjective judgment.   I personally think the 5% level of exceedance mentioned above seems reasonable to me but that is just my opinion.   The point here is that there is almost always some level of predicted exceedance based on the inherent variability of reality.

I think it is important to let the client in on this game of uncertainty analysis to show him/her that there is no such thing as absolute safety only relative safety expressed in terms of uncertainty.

Just to really complicate matters, the above is just the exposure half.   Can we really think that there is no uncertainty in the toxicity benchmark or exposure limit half as well?   More above this in next week's blog.

Balancing the Risk Assessment Client’s Needs with Yours

I used to work at the now defunct Rohm and Haas Company.  For many years I did risk assessment for the businesses.  I mentioned to a colleague once that I was having trouble figuring out who the client was on a particular project I was working on.   He seemed perplexed and ask me what I meant by the term “client”.    I told him that clients are the folks that get and use our analyses.  Doing risk assessment in a corporate setting they may not be (and often are not) the ones who we report to or those who determine our rank and salary but they are critical nonetheless.   To the extent that we do a good job for them is the extent that we remain gainfully employed.

It should be noted that in our business we have both clients and charges.   Clients are roughly defined above but our charges are the folks on the receiving end of the exposures that we estimate.   We have a professional and moral responsibility to all these folks to get it right.

Clients, because of their position, can be typically demanding.   Clearly and appropriately, they want to get an answer that satisfies their needs using the least possible expense in the process.  For your part, you essentially want to do the same; however, it is our responsibility to render these answers in a realistic manner.   We need to admit to and deal with the inevitable uncertainty borne of any analysis and put that uncertainty into context for the client.

A prime example of this balancing act for me comes to mind involving an additive that was used in motor oil.   The additive existed in new motor oil but not in oil that was used.   I was asked to do a risk assessment on the additive in this application.  Data and modeling indicated that inhalation exposure was not a factor; however, dermal exposure to fresh oil during the oil changing process could be.    I assumed the following scenario.  

•  Commercial oil changing (e.g., Jiffy Lube)
• 10 oil changes per day
•  Fingers of both hands covered with new oil during change
•  Instantaneous and complete absorption of the additive by this dermal exposure

Let me know if you are interested in some of the details of the subsequent assessment and I will let you know by email or cover it in a future blog if enough folks are interested.

Because of my own experience at changing oil (I am very sloppy) and my lack of data otherwise, I felt comfortable that this scenario would definitely and appropriately OVERESTIMATE the exposure potential of this material.  More important, given a classical precautionary approach, I did not feel personally comfortable changing any of these assumptions on my own.

The client argued that it was indeed a worst case and told me that the above assumptions should be less stringent.   At that point, I told him/her that it was in fact their business and that they definitely should have more information/insight than I relative to these assumptions and that they were free to change any of them. However, I would need them to write down their assumptions and the bases for them which would be incorporated into the risk assessment as a reference.   Faced with this possibility, they declined to take this approach and agreed with the above assumptions as a working worst case.

An alternative approach would be to commission studies of commercial oil changing facilities to determine a distribution of number of changes per day and a dosimeter (e.g., washed cotton gloves that would be extracted and analyzed afterwards) study of amount of new oil that gets on the hands of these workers per oil change.   Another approach would be to do a dermal absorption study using human or animal skin.  (Note: I will get into dermal absorption testing and modeling in a future blog)

Both of the above approaches could be quite expensive but would almost certainly significantly lower the estimated level of exposure to workers.

The bottom line here is that I had to draw line relative to where my comfort level was regarding these assumptions.   I had to use my best judgment as to my skill level to ultimately trade conservatism for data and vice versa.    The client needed me to tell him/her (i.e., to professionally certify) that their product was “safe” in its intended use.  Indeed, I needed to provide an analysis that accomplished this same end for me as well.

Ultimately, the assessment using the above assumptions did not serve the client’s needs.   Indeed, it turned out that more data was needed and obtained to make the case for safety in which both the client and I were comfortable with the results.   The client, of course, became temporarily poorer having paid for the study and data but ultimately richer in the confident knowledge of that their product was safe.   Arguably the “charges” or folks receiving the exposure in this assessment were also reasonable well-served.

This brings me to the topic of the next blog:  Risk Assessment Uncertainty or How Safe is Safe?