Calculations with radioactivity

This section explains how to perform basic computations with radioactivity. Use the Radioactivity calculator to perform the calculations on a web form.

It is not possible to detect every radioactive disintegration. The fraction of radioactive disintegrations detected by your counter is called efficiency. Determine efficiency by counting a standard sample under conditions identical to those used in your experiment.

It is relatively easy to detect gamma rays emitted from isotopes such as 125I, so efficiencies are usually over 90%, depending on the geometry of the counter. The detector doesn't entirely surround the tube, so a small fraction of gamma rays (photons) miss the detector.

With 3H, the efficiency of counting is much lower, often about 40%. When a tritium atom decays, a neutron converts to a proton and the reaction shoots off an electron and neutrino. The energy released is always the same, but it is randomly partitioned between the neutrino (not detected) and an electron (that we try to detect). When the electron has sufficient energy, it can travel far enough to encounter a fluor molecule in the scintillation fluid. This fluid amplifies the signal and gives a flash of light detected by the scintillation counter. If the electron has insufficient energy, it is not captured by the fluor and is not detected.

Since the decay of a large fraction of tritium atoms does not lead to a detectable number of photons, the efficiency of counting is much less than 100%. This low efficiency is a consequence of the physics of decay, and you can't increase the efficiency much using a better scintillation counter or a better scintillation fluid. However, poor technique can reduce the efficiency. Electrons emitted by tritium decay have very little energy, so can only be detected when they immediately encounter a fluor molecule. Radioactivity trapped in tissue chunks won't be detected. Nor will radioactivity trapped in an aqueous phase not well mixed into the scintillation fluid.

When you buy radioligands, the packaging usually states the specific radioactivity as Curies per millimole (Ci/mmol). Since you measure counts per minute (cpm), the specific radioactivity is more useful if you change the units to be in terms of cpm rather than Curie. Often the specific radioactivity is expressed as cpm/fmol (1 fmol = 10^-15 mole).

To convert from Ci/mmol to cpm/fmol, you need to know that 1 Ci equals 2.22 x 10^12 disintegrations per minute. The following equation converts Z Ci/mmol to Y cpm/fmol when the counter has an efficiency (expressed as a fraction) equal to E.

In some countries, radioligand packaging states the specific radioactivity in Gbq/mmol. A Becquerel, abbreviated bq, equals one radioactive disintegration per second. A Gbq is 10^9 disintegration per second. To convert from Gbq/mmol to cpm/fmol, use this equation:

Rather than trust your dilutions, you can accurately calculate the concentration of radioligand in a stock solution. Measure the number of counts per minute in a small volume of solution and use the equation below. C is cpm counted, V is volume of the solution you counted in ml, and Y is the specific activity of the radioligand in cpm/fmol (calculated in the previous section).

Radioactive decay is entirely random. A particular atom has no idea how old it is, and can decay at any time. The probability of decay at any particular interval is the same as the probability of decay during any other interval. Therefore decay follows an exponential decay as explained in Example model 2. Exponential decay.

If you start with N0 radioactive atoms, the number remaining at time t is:

Kdecay is the rate constant of decay expressed in units of inverse time. Each radioactive isotope has a different value of Kdecay. The half-life (t½) is the time it takes for half the isotope to decay. Half-life and decay rate constant are related by this equation:

This table below shows the half-lives and rate constants for commonly used radioisotopes. The table also shows the specific activity assuming that each molecule is labeled with one isotope. (This is often the case with 125I and 32P. Tritiated molecules often incorporate two or three tritium atoms, which increases the specific radioactivity.)

You can calculate radioactive decay from a date where you (or the manufacturer) knew the concentration and specific radioactivity using this equation.

For example, after 125I decays for 20 days, the fraction remaining equals 79.5%. Although data appear to be scanty, most scientists assume that the energy released during decay destroys the ligand so it no longer binds to receptors. Therefore the specific radioactivity does not change over time. What changes is the concentration of ligand. After 20 days, therefore, the concentration of the iodinated ligand is 79.5% of what it was originally, but the specific radioactivity remains 2190 Ci/mmol. This approach assumes that the unlabeled decay product is not able to bind to receptors and has no effect on the binding. Rather than trust this assumption, you should always try to use newly synthesized or repurified radioligand for key experiments.

The decay of a population of radioactive atoms is random, and therefore subject to a sampling error. For example, the radioactive atoms in a tube containing 1000 cpm of radioactivity won't give off exactly 1000 counts in every minute. There will be more counts in some minutes and fewer in others, with the distribution of counts following a Poisson distribution. This variability is intrinsic to radioactive decay and cannot be reduced by more careful experimental controls.

After counting a certain number of counts in your tube, you want to know what the "real" number of counts is. Obviously, there is no way to know that. But you can calculate a range of counts that is 95% certain to contain the true average value. So long as the number of counts, C, is greater than about 50 you can calculate the confidence interval using this approximate equation:

GraphPad StatMate does this calculation for you using a more exact equation that can be used for any value of C. For example, if you measure 100 radioactive counts in an interval, you can be 95% sure that the true average number of counts ranges approximately between 80 and 120 (using the equation here) or between 81.37 and 121.61 (using the exact equation programmed into StatMate).

When calculating the confidence interval, you must set C equal to the total number of counts you measured experimentally, not the number of counts per minute.

Example: You placed a radioactive sample into a scintillation counter and counted for 10 minutes. The counter tells you that there were 225 counts per minute. What is the 95% confidence interval? Since you counted for 10 minutes, the instrument must have detected 2250 radioactive disintegrations. The 95% confidence interval of this number extends from 2157 to 2343. This is the confidence interval for the number of counts in 10 minutes, so the 95% confidence interval for the average number of counts per minute extends from 216 to 234. If you had attempted to calculate the confidence interval using the number 225 (counts per minute) rather than 2250 (counts detected), you would have calculated a wider (incorrect) interval.

The Poisson distribution explains the advantages of counting your samples for a longer time. For example, the table below shows the confidence interval for 100 cpm counted for various times. When you count for longer times, the confidence interval will be narrower.

The table below shows percent error as a function of the number of counts. Percent error is defined as the width of the confidence interval divided by the number of counts.

Counts		Percent Error
50		27.72%
100		19.60%
200		13.86%
500		8.77%
1000		6.20%
2000		4.38%
5000		2.77%
10000		1.96%
25000		1.24%
50000		0.88%
100000		0.62%
C