Section16:Radioactive Calculations

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Microplate scintillation counters, used for reading SPA and filtration assays, detect flashes of light (photons) that occur when a released radioactive particle interacts with and excites a fluor molecule. Not all of the radioactive particles emitted will be detected as photons by the counting instrument. The output from the scintillation counter is the number of photons detected per unit time, typically expressed in counts per minute (CPM). The ratio between CPM detected by the instrument and actual disintegrations per minute (DPM) of the isotope is termed efficiency. The efficiency of counting depends on the geometry of the detector, scintillation properties of the fluor and the energy of the particular isotope. Determination of DPM is important for making conversions to calculate molar concentrations of radioligands and if a comparison between different instruments is required. For data that will be normalized (e.g. % Inhibition) CPM can be used as a direct read out from the instrument.

The efficiency for each isotope counting condition should be independently determined for an instrument. Steps to determine an average instrument efficiency are shown in the example below for an SPA assay using a [³H]-labeled radioligand and YSi SPA beads:

Example Determination of Efficiency

[³H]-labeled SPA Beads can be prepared by incubating [³H]-labeled biotin with YSi streptavidin beads and washing them (using centrifugation) to remove any unbound radioactivity. Alternatively, a reaction associated with an assay (e.g. WGA beads, membranes, radioligand) can be used.

Remove a 300 µl aliquot of [³H]-SPA beads to a 1.5 ml polypropylene tube.
Centrifuge the tube for 5 seconds in a microfuge to pellet the [³H]-SPA beads.
Remove the supernatant. Dispose of properly, treating as potential radioactive waste.
Add 300 µl of PBS and mix beads. Repeat centrifugation and remove supernatant.
Resuspend in a final volume of 300 µl PBS.
For a Microbeta, pipette 25 µl of beads into three different wells of a microplate. Add 175 µl of PBS. Allow the beads to settle overnight.
Count the microplate and determine the average CPM for the three replicates (example 52,800 CPM).
Add 25 µl of beads to three different scintillation vials containing scintillation cocktail. Count the vials on a liquid scintillation counter which is capable of returning results in DPM and determine the average for the three replicates (example 140,582 DPM).
Determine efficiency using the following equation:

For [¹²⁵I], a gamma counter with a known efficiency can be used for the determination of the total DPM.

Some typical instrument efficiencies for common isotope configurations on a Trilux Microbeta are shown in the table below:

Isotope	Scintillation Mode	Efficiency
H-3	Filtration	0.32
H-3	SPA (PVT)	0.23
H-3	SPA (YSi)	0.34
I-125	Filtration	0.45
I-125	SPA (PVT)	0.38
I-125	SPA (YSi)	0.56

These are approximate efficiencies for comparison. Actual efficiencies for your instrument should be determined independently. In addition, some counting conditions require special “window settings” that can impact the apparent efficiency. Alterations or repairs to an instrument (e.g. adjustment of PMT’s) may also require determination of an updated efficiency value.

Conversion from CPM to DPM

DPM are calculated from the equation shown below where efficiency is expressed as a decimal percent. Determination of instrument efficiency (Eff) is described above.

Example: 1000 CPM detected in an assay using PVT SPA beads and 3H

The instrument efficiency was determined to be 22%

Specific Activity (SA)

The amount of radioactivity per unit mole for a radioligand is referred to as the specific activity (often abbreviated as SA) and is typically given in units of Ci/mmol by the manufacturer. Since raw data from assays using radioactivity are in CPM or DPM, conversion of the specific activity from Ci/mmol to CPM/fmol or DPM/fmol is usually more convenient for further data analysis.

Equation to convert Ci/mmol to DPM/fmol: DPM/fmol = [Specific activity ~~(Ci/mmol)~~ x [2.22 x 1012 DPM/Ci] x [~~mmol~~/1012 fmol] = SA x 2.22

Example: SA = 2000 Ci/mmol DPM/fmol = SA x 2.22 = 2000 x 2.22 = 4440 DPM/fmol

'Equation to Convert Ci/mmol to CPM/fmol: CPM/fmol = [SA (Ci/mmol) x [2.22 x 1012 DPM/Ci] x [mmol/1012 fmol] x Efficiency (CPM/DPM) = SA x 2.22 x Eff

Example: Instrument efficiency = 40%, SA = 2000 Ci/mmol CPM/fmol = SA x 2.22 x Eff = 2000 x 2.22. x 0.4 = 1776 CPM/fmol

Nominal Concentration of a Radioligand

The theoretical or nominal concentration of a radioligand stock solution can be calculated from the stated radioactive concentration (RAC, in Ci/ml) and the specific activity (SA, in Ci/mmol) using the equation shown below:

[Radioligand] = RAC/SA

Example: Radioactive concentration (RAC): 50 µCi/ml Specific Activity (SA): 2000 Ci/mmol Conversion factor: 1 Ci = 106 µCi

This is the nominal concentration of the stock on the reference date. To estimate the concentration on any other day, see the Radioactive Decay section to determine the fraction remaining and the resulting concentration. See also the Dilution of Stock section to prepare a dilution of a stock radioligand.

Actual Concentration of a Radioligand

When performing radioligand binding assays, a rough estimate for the concentration of radioligand used in the assay can be computed using the information supplied with the material. This is called the theoretical or nominal concentration (shown above). In order to calculate the actual concentration of the radioligand used in an assay more accurately, count an aliquot of the stock mix and obtain the CPM or DPM for that aliquot, then use the equation below.

Equation to convert CPM to pM:

Example: Counted a 50 µl aliquot of a stock mix which yielded 50,000 CPM; SA = 1776 CPM/fmol (see above for calculation)

If values are in DPM, use specific activity (SA) expressed in DPM/fmol. Use appropriate unit conversions to determine the concentration in nM, M , etc.

It is best practice to use the actual concentration of radioligand determined for each assay in calculations such as Ki, rather than the theoretical or nominal concentration.

Radioactive Decay

Radioactive decay is a random event and follows an exponential decay trend. You can calculate the fraction remaining in a radioactive sample if you know the date (reference date) when the specific activity or radioactive concentration was known using the following equation:

where t1/2 is the half-life of the isotope (time it takes for half the isotope to decay) and time is the number of days before or after the known reference date. The term (-0.693/t1/2) is also referred to as the decay rate constant, Kdecay.

Example: I-125 radioligand with a known specific activity on 10/1/07 Half-life for I-125 = 60 days Fraction remaining on 10/20/07 (20 days):

The fraction remaining following radioactive decay can also be determined from tables. Note that for the activity on a day prior to the stated reference date, the fraction remaining will be greater than 1.

An assumption typically made is that radioactive decay results in unlabeled decay product(s) which no longer bind to the target or receptor of interest. This implies that the specific activity remains constant over time and that the concentration of ligand changes with time. This assumption may not be valid with all radioligands used.

Half-Life

A table of half-lives (time for half the isotope to decay) for common isotopes is shown below along with typical values for specific activity of a single-labeled molecule. Always consult the manufacturer's information for the exact specific activity of a radioligand.

Isotope	Half-life	Specific Activity
H-3	12.43 years	85.0 Ci/mmol
I-125	60 days	2000 Ci/mmol
P-32	14.3 days	9128 Ci/mmol
S-35	87.4 days	1493 Ci/mmol
C-14	5730 years	0.064 Ci/mmol

Note that half lives (even for the same isotope) can vary from one manufacturer to another. In addition, if software is used for tracking of decay of isotope inventories, make sure that the half life value used is consistent throughout.

Dilution of Stock

To calculate the amount of a radioligand stock solution required to prepare a specific volume of a dilution, the parameters listed below will be needed. The values listed for each parameter are for use in the example calculations.

Radioactive Concentration (RAC):	50 µCi/ml
Specific Activity (SA):	2000 Ci/mmol
Half-life for isotope:	60 days (I-125)
Reference date:	10/1/07
Date of preparation:	10/20/07
Volume of final diluted mix:	50 ml
Desired concentration of final diluted mix:	0.1 nM

Determine nominal stock concentration - described above in Stock Concentration section:
Determine stock concentration on day of use - described in Radioactive Decay section above: Date of use - Reference Date = 20 days

Therefore, stock concentration on day of use = 0.794 x 25 nM = 19.85 nM
Determine amount of stock required: C₁V₁=C₂V₂ solving for V₁, yields V₁ = C₂V₂/C₁ = (50 ml x 0.1 nM)/19.85 nM = 0.252 ml

This is the theoretical or nominal concentration. To determine actual concentration, count an aliquot of the diluted mix and calculate as shown in the Actual Concentration of Radioligand section above.