I’m aware of the differences in units and measures…but is there any guesstimate, WAG, or “rule of thumb” which can convert gamma radiation counts per minute per minute (yes, I mean to say that), in a human being, to Sieverts? My girl is going to be radioactive in a few days due to a medical procedure, and the last time this happened my Geiger counter went off the scale (999 cpm) within 6 feet of her. Any idea about a WAG on converting “100 gamma cpm for 1 minute is about X Sievert”, keeping in mind I know there is no real direct conversion?
This sounds to me almost like a mass transfer problem, but with gamma radiation. Fick’s Law might get you somewhere but you would have to make a lot of assumptions.
Nope. Sieverts are a biological dose equivalent, and the unit is equivalent to joules per kg. So you first need to know the energy of the gamma photons (easily calculated from frequency or wavelength, or you can guesstimate about 1MeV = 1.6 x 10^-13 J), then you need the body mass being exposed (so you might calculate, say your exposure), then you’d need to know the relative biological effectiveness of the radiation on the type of tissue being exposed. The RBE of gamma radiation on human tissue is (very roughly) 0.1 for most internal tissues, and about 0.01 for bones and skin.
So, you can calculate J/kg per unit time from your count rate, and multiply by 0.1 to get sieverts per unit time. Your 1000 counts per minute, hand-waving style, will be about 0.1 microsieverts per year for a good-sized man – far below background.
Of course, to get an accurate number for her exposure, you’ll need to consider she’s intercepting the entire dose, not just the angular fraction you’re picking up with your meter. Got any idea what isotope they’re using, and how much?
Iodine-123. Is it possible you could help me determine if that radically changes the estimate which you’ve made above? (for which I do thank you) It says it decays with either 159 or 127 keV.
I must say, not to criticize, but 0.1 mSv for 1,000 cpm sounds awfully low, since the normal background of our house is typically 20-30 cpm. 
Don’t you mean counts per minute times minutes? Or just counts?
I meant counts per minute per minute, since cpm would be the intensity of the radiation, and the 2nd minutes would be the duration, so I figured the intensity times duration would be a somewhat analogous to a received dose.
My estimate was counts/minute x energy/count x time/body weight x RBE
= 1000 x (1.6 x 10[sup]-13[/sup]) x (60 x 24 x 365) x 0.1 x 1/100 kg = 0.084 microsieverts
The typical background exposure is 2.4 millisieverts.
The typical diagnostic dosage of I-123 emits about 10-20 million counts per second – let’s call it 15 – primarily at 159 keV = 2.54 x 10[sup]-14[/sup]. At a half-life of 13 hours, this will drop off rapidly on the first day, so let’s just take the first day’s exposure and call it constant; let’s also assume a petite patient at 50 kg.
(15 x 10[sup]6[/sup]) x (2.54 x 10[sup]-14[/sup]) x (60 x 60 x 24) x 0.1 x 1/50 kg = 66 microsieverts – nearly 1000 times what we calculated above, but still well below a year’s background exposure.
What’s making you and your counter choke on this is that your counter and your training, IIRC, are oriented toward environmental radiation, consisting of relatively small amounts of radioactive material, where the concern is years of exposure to, say, inhaled particles of long-lived alpha-emitting isotopes (alpha particles do tremendous damage to biological molecules). This is a much greater hazard than brief exposure to a small amount of I-123, yet the count rate in such an environment is much smaller. Again, it’s mostly alpha particles from minerals like uranium, radium, and radon, and these particles are 30 times more energetic and have 20 times the RBE of the gamma photons from I-123.
Whee! My degree in biophysics is good for something!
That’s about the correct weight (she weighs about 45 kg). So that’s the value for the first day’s exposure then, 66 microsieverts? That’s not too bad I guess, but damn, it sure is scary to see the Geiger counter react! Thank you very much for doing the calculations for me.
Some comments, and a different approach:
(1) I-123 converts via electron capture (EC) to an excited state of Te-123 that decays down to its ground state via a cascade of nuclear rearrangements, eventually releasing 1.242 MeV. The 0.159 MeV line is only one of the transitions; all 1.242 MeV eventually comes out as gamma rays. However, your counter will only see one blip per EC, meaning the each count is a factor of eight more potent than you would think just by multiplying the count rate by 0.159 MeV.
(2) Geiger counters have very low efficiency for detecting gamma rays (maybe 1%), and this efficiency is somewhat energy dependent. Do you have any specs on your particular counter? If not, a worst-case assumption would be that the source is 40 to 100 times stronger than the count rate suggests. Let’s just take 2%, or 50x.
(3) At 6 ft, and assuming an 80 cm[sup]2[/sup] collection area for the Geiger counter, I get a source power for 100 cpm of:
(1.242 MeV) (100 cpm) (4[symbol]p[/symbol]) (6 ft)[sup]2[/sup] / (80 cm[sup]2[/sup]) / (2%)
or
5.2x10[sup]-6[/sup] J/minute (per 100 cpm)
Gamma radiation at these energies isn’t terribly penetrating. The attenuation length at 0.159 MeV is around 4 cm, increasing to 9 cm at the highest relevant energies. So, assuming the I-123 is concentrated in the thyroid, then anything that goes generally up or down (say, half of the gamma rays) will lead to fairly local deposition, and anything that goes generally sideways (the other half) will drop maybe a third of its energy on its way out. So, a very rough estimate would be that about 15 kg of body mass sees about half of the above power. Thus, for 100 cpm in the counter:
local dose rate = (0.5) (5.2x10[sup]-6[/sup] J/minute) / (15 kg) per 100 cpm
local dose rate = 0.17 microsievert per minute per 100 cpm
If you measure your count rate at t=2 hr after the I-123 is (instantaneously) taken up, then the total dose is:
(0.17 [symbol]m[/symbol]Sv/min) e[sup](2 hr / 19.1 hr)[/sup] (19.1 hr) per 100 cpm
or simply:
local dose = 0.22 millisievert per 100 cpm
So, if you measure…
1000 cpm…
at 6 ft…
2 hours after uptake…
with a 2% efficient counter…
with a 80 cm[sup]2[/sup] acceptance area…
then our subject will receive in total a dose of 2.2 mSv.
Yipe, you think my Geiger counter has only a 2% efficiency? If that’s true, then my house has a background radiation of 20/0.02 to 30/0.02 counts per minute. That sounds way too high.
And that also means, if I interpret your post properly, that in 2 hours she receives 2.2mSv? That would be quite high over the time it takes for the combination of excreting the iodine + half life decay.
To gamma radiation, yes. Most gamma rays will pass through invisibly. It could be lower or higher than 2%, but it certainly won’t be 100%. (Assuming it’s actually a Geiger counter, that is… Scintillator counters can approach 100%.)
No, the 2.2 mSv is the total. I assumed you measure the rate 2 hours after the I-123 is injected, and then I did the time integral of the dose rate to get dose, folding in the 19.1 hr lifetime. (That’s the step with the exponential.)
Ah. So that’s about the same as the normal worldwide background radiation, according to Wikipedia.