RADAR - The Kinetic Models

Here's the really hard part. Conceptually, it's not too hard to explain the part of the system. You understand the basics already if you understand N_S. How we get values of N_S is a little bit complicated, but not too bad. We like the system of representing kinetics of an organ (or the total body, whatever) as a sum of exponential terms involving a 's and l 's, where

Here, A(t) is the activity as a function of time (t), a _j is the activity associated with exponential term j and l _j is the biological removal constant. Does that look like Martian heiroglyphics to you? OK, consider that we have I-131 sodium iodide in the whole body. Let's assume that the kinetic model says that for sodium iodide 75% is eliminated with a biological half-time of 6 hours and 25% with a biological half-time of 1560 hours. The relationship between half-time and removal constant is T_1/2 = 0.693/l , so that gives us l 's of 0.693/6 = 0.116 hr^-1 and 0.693/1560 = 0.000444 hr^-1. So, assuming for example a 1 Bq administration, our equation for whole body retention of sodium iodide looks like:

We have 0.75 Bq being removed with the first exponential term and 0.25 Bq being removed with the second term. This would apply to any isotope of iodine. To calculate the number of disintegrations specifically for I-131 (assuming that we are integrating from zero to infinity) one needs only to calculate the following simple ratios:

where l _p is the physical decay constant for I-131 = 0.693/(8 days x 24 hrs/day) = 0.0036 hr^-1. We would normally change the units of l to sec^-1, and the units of this result would be Bq-sec, or disintegrations. You can use other unit combinations, like mCi and hr; you can always calculate the number of disintegrations from first principle considerations (1 mCi = 3.7 x 10⁷ dis/sec, 1 hr = 3600 sec, etc.).

Now if you don't want to integrate to infinity, you make life a little harder on yourself, each term in the N equation has a (1-e^{-l t}) term in it, but generally we don't have to worry about that. So, if we just set up a nice table of a 's and l 's, as was done in many of the MIRD Dose Estimate Reports, we can calculate values of N for all of our source regions S, and then just do matrix multiplication by the DCFs to get a set of doses (that's where computers come in, who wants to do all that garbage by hand?). Here's an example table, the complete table from MIRD Dose Estimate Report No. 5 for sodium iodide:

ORGAN:	a 1	l 1	a 2	l 2	a 3	l 3	a 4	l 4
Intestine	0.169	0.114	0.00115	0.0488	-0.0000962	0.00496	0.000221	0.000444
Liver	0.0159	0.114	-0.00206	0.0488	-0.00506	0.00496	0.00667	0.000444
Stomach	0.149	0.114	0.00103	0.0488	-0.0000913	0.00496	0.000204	0.000444
Thyroid	-0.255	0.114	--	--	--	--	0.255	0.000444
Total Body	0.729	0.114	--	--	--	--	0.271	0.000444

This is easy stuff - drag the a 's and l's into a spreadsheet and try it yourself. Just remember to apply the physical decay constant for I-131 and the value to convert hours to seconds. The hard part is deciding on the actual values of the a 's and l 's for every compound. We've put in the database (see the Internal Sources resource page) what we believe are the best available for many pharmaceuticals in nuclear medicine and other compounds for the occupational radionuclide situations. In this table, we give you fractions (a 's) and, instead of l 's, we give the corresponding biological half-times, as this is usually better understood by more people. The model for I-131 NaI is just slightly different than what is shown above - these kinetic models normally change over time as new information is gained about a pharmaceutical. Most of our kinetic models (but not all) come from the ICRP 53/ICRP 80 documents, and their model for NaI was a little different than the MIRD model from 1975. We intend to keep the database current and to use the latest and most accurate values for any compound. If you know of some data we haven't considered, or think one of the models is off, let us know.

<Rant Mode On>

In our table, we continue to show the value given in the ICRP reports of Ã/A₀, which the MIRD Committee has referred to as "residence time". We do not like the use of this term (because it has other very specific meanings in pharmacology, biology, and engineering, and often confuses people). We also don't think it's a good idea to show the number of disintegrations that occurred in a source organ with units of time. The number of disintegrations that occurred in an organ has nothing to do with units of time. Can I say that again? It's my web page, sure I can. The number of disintegrations that occurred in an organ has nothing to do with units of time. What is going on here? Well the number of disintegrations comes from the area under the time activity curve. So this might have units of Bq-sec or mCi-hr. We like to give the number of disintegrations per unit of administered activity, so that we can give doses per unit of administered activity. So if we have doses in mGy/MBq (administered), if one person gives 500 MBq and another gives 350 MBq, they can both use the same dose estimate to calculate how many mGy their patients received. OK, so if I have the number of disintegrations in Bq-sec and I divide by the number of Bq administered to the patient, I have Bq-sec/Bq, which gives units of seconds if you cancel the Bq's (you can do the same thing with m Ci-hr and mCi, and get units of hr, which the ICRP and the MIRD Committees have generally done). But this unit is not a measure of seconds in any way, shape, or form!!! And many users find it confusing to think of in this way. So we use in our system the number of disintegrations per unit of administered activity (dis/Bq), but we show the values of Ã/A₀ (in hr) for reference. OK, I'll quit ranting on about residence times now, I'm sure you've heard enough. But it is a peeve of mine - why? Because when things are confusing, it makes your life harder, and I want to make your life easier.