# Answer to Question #11935 Submitted to "Ask the Experts"

*Category: Instrumentation and Measurements — Surveys and Measurements (SM)*

The following question was answered by an expert in the appropriate field:

I'm working at a cyclotron center which produces fluorine-18 (^{18}F) fluorodeoxyglucose ([^{18}F]FDG). I'm wondering how to convert counts per second (cps) to becquerels per meter^{3} (Bq m^{-3}). A stack monitor installed in our facility measures air discharge information in cps only, and I would like to convert it to Bq m^{-3}.

I know count results in cps, efficiency of detector (gas detector), velocity of air, measuring chamber dimensions, etc. Please let me how to calculate Bq m^{-3} using this information.

I shall attempt to give you sufficient information for you to be able to obtain the required results, although I must make specific assumptions to accomplish this. Please pay attention to these to ensure that they are appropriate. If not, alternative considerations may apply.

You state that the detector is a gas detector; being a presumably continuous stack monitor, the implication is that a representative fraction of the air being discharged through the stack is drawn continuously through the gas sampler, which presumably consists of a chamber that houses the detector and has a significant free volume. Let us assume that the free volume of the gas detection system is V in centimeters^{3} (cm^{3}), the volume flow rate through the detector system is F in cm^{3} per second (cm^{3} s^{-1}), and the activity concentration of ^{18}F in the sampled air is C (Bq cm^{-3}). You state that you know the efficiency of the gas detector; usually such efficiencies are given in terms of count rate per unit activity concentration for a particular radionuclide in the chamber volume. Apparently, this is not so in your case since, if it were, your question would be pretty much answered in that you could simply divide the observed count rate by the efficiency to obtain the apparent concentration of the radionuclide in the air at an observed time. I must assume then that the efficiency you refer to is the efficiency in dimensions of count rate per unit activity of the radionuclide of concern in the chamber volume (cps per Bq, which is the same as counts per disintegration). The radioactive species of interest is, of course, ^{18}F, used to replace one of the hydroxyl groups in deoxy-D-glucose to yield [^{18}F]FDG, the compound so commonly used in various positron emission tomography (PET) scans.

The radionuclide ^{18}F has a radioactive half-life of about 110 minutes (min). For purposes of estimation, we shall assume that air drawn through the detector chamber mixes uniformly and instantaneously with air in the chamber and that the residence time of air (and any airborne contaminants) in the free detector volume is short compared to this half-life so that radioactive decay need not be considered as a loss mechanism for ^{18}F in the free volume. This is easy enough to check: if you divide the air volume flow rate by the detector free volume, and the resulting value, which represents the air flow removal rate constant, is very large compared to the ^{18}F decay constant (ln2/110 min = 6.30 × 10^{-3} min^{-1} = 1.05 × 10^{-4} s^{-1}), then radioactive decay may be neglected; this assumption will generally prevail. If the airborne concentration of ^{18}F remains constant during an observation interval, the expected activity, A, present in the detector chamber at any given elapsed time, T, assuming a count rate of R (cps) and an efficiency of E (counts per disintegration) is given as

^{-FT/V}

When the value of FT/V is large (>>1), the value of A approaches CV, and the concentration

*C = R/EV.* (Equation 2)

If the value of FT/V is very small (<<1), then the right-hand side of the above equation reduces to CFT, and we obtain

*C = R/EFT.* (Equation 3)

For intermediate cases between these extremes, Equation 1 would apply.

As a brief example, consider a chamber volume of 1.42 × 10^{4} cm^{3} and a flow rate of 4.72 × 10^{2} cm^{3} s^{-1} for which F/V = 3.32 × 10^{-2} s^{-1} (large compared to the decay constant of 1.05 × 10^{-4} s^{-1}). Let us assume an arbitrary detection efficiency of 0.02 cps Bq^{-1}; if the system had been running for 10 min during a release of ^{18}F, and a steady net count rate of 100 cps was observed, the implied activity concentration from Equation 2 would be C = 100 cps/[(0.02 cps Bq^{-1})(1.42 × 10^{4} cm^{3})] = 0.35 Bq cm^{-3}.

It is important to keep in mind that the above considerations are based on some important assumptions, including the following:

- The
^{18}F activity in the detector chamber at any time is distributed uniformly throughout the chamber volume. - The efficiency you refer to is the detector response to the radionuclide of interest (
^{18}F) in counts per disintegration occurring in the chamber volume. - The
^{18}F activity is being released at a constant rate during the observation period.

If any of the above assumptions are not appropriate, the above development will not be accurate. If short-duration puffs of activity are released into the stack airstream, the above descriptions do not necessarily apply. In such cases a short, transient-type response may occur. In some such instances you may be able to estimate concentrations in the detector chamber volume, and if a time duration of the release can be estimated, then an effective concentration over an appropriate interval may be estimated. Depending on your license conditions and the specific regulations you are following, you may be allowed to average releases over extended periods that include intervals when no activity is being released.

I do not know what kind of readout system you have associated with the stack monitor, but if you are using a chart recorder or a continuously recording digital acquisition system you may be able to ascertain suitable time data to make appropriate estimations.

George Chabot, PhD, CHP