The number of disintegrations of a sample of nuclei (atoms) per unit of time (decay rate) is called the radioactivity or simply the activity of a sample of a radionuclide. For example, if we have a collection of 1,000 atoms of a radionuclide at a given time and 50 of these disintegrate in the next 5 seconds, the activity of the sample is 50/5 = 10 disintegrations per second (dis/s). In algebraic terms, if out of a number N_t of radioatoms at a time t, a number of radioatoms dN_t decays in a small time interval dt, the radioactivity R_t of the sample is calculated by dividing the number of radioatoms decayed (dN_t) by the interval (dt) during which the decay took place. In other words,

In this equation, the minus sign denotes that the radioatoms are decreasing in number.

A caveat: radioactivity is defined in terms of disintegrations or decays of parent nuclei and not in terms of a particular type of radiations or total number of radiations emitted. As we saw in Chapter 2, p. 17, the number of radiations emitted per decay of parent nucleus varies from radionuclide to radionuclide. Therefore, the number of radiations being emitted per unit time from a radioactive sample will be different from the actual disintegration per unit time of nuclei in the sample.

The unit of radioactivity is the becquerel (Bq) in the current system of units système international (SI) and the curie (Ci) in the older system of units (centimeter-gram-seconds). SI units have now replaced the older system of units in scientific work. However, in clinical nuclear medicine, old units are still sometimes used. We use the new units here. For important data, results, or problems, we also include old units in parentheses. The major differences between the two systems of units are given in Appendix B.

A becquerel is defined as the radioactivity of a sample that is disintegrating at a rate of 1 dis/s, whereas a curie is defined as the radioactivity of a sample that is disintegrating at a rate of 3.7 × 10¹⁰ dis/s. Because the becquerel is a small unit, larger units derived from it are used. On the other hand, the curie is a large unit; therefore, smaller units derived from it are often used. Some of these derived units for the curie and the becquerel and their interrelationships are given as follows:

Bq = 1 dis/s = 27.03 × 10^-12 Ci = 27.03 pCi

Kilobecquerel (KBq) = 10³ Bq = 27.03 nCi

Megabecquerel (MBq) = 10⁶ Bq = 27.03 µCi

Gigabecquerel (GBq) = 10⁹ Bq = 27.03 mCi

and

Curie (Ci) = 3.7 × 10¹⁰ dis/s = 37 GBq

Millicurie (mCi) = 10^-3 × Ci = 3.7 × 10⁷ dis/s = 37 MBq

Microcurie (µCi) = 10^-6 × Ci = 3.7 × 10⁴ dis/s = 37 KBq

Nanocurie (nCi) = 10^-9 × Ci = 3.7 × 10 dis/s = 37 Bq

Picocurie (pCi) = 10^-12 × Ci = 3.7 × 10^-2 dis/s.

In nuclear medicine, the amount of radioactivity (MBq or mCi) administered to a patient is called dosage and not dose (to avoid confusion with another term, radiation dose, discussed in Chapter 7).

When a sample of a radionuclide is observed for a long period of time, it is found that the radioactivity R of a sample at a given time t depends on the number of radioatoms N_t present at that time and a constant λ that is characteristic of a given radionuclide and is different for various radionuclides.

The constant λ, also known as the decay constant, is defined as the probability of decay per unit time for a single radioatom. Therefore, if there are N_t radioatoms present at time t, then the number of nuclei disintegrating per unit time R_t at time t will be simply the product of λ and N_t or

From this equation, it can be seen that for the different radionuclides (i.e., different λ), the number of radioatoms N_t that must be present to produce the same amount of radioactivity R_t varies. For example, the number of radioatoms present in a sample of ^99mTc, which has a radioactivity of 37 MBq (1 mCi), is not the same as the number of radioatoms present in a sample of ¹³¹I, which has a radioactivity of 37 MBq (1 mCi). This is true because the decay constant for ^99mTc (λ = 3.2 × 10^-5/second) is different from the decay constant for ¹³¹I (λ = 10^-6/second). The number of radioatoms of ^99mTc or of ¹³¹I that must be present in a sample to produce a radioactivity of 37 MBq can easily be determined from equation (3.2). For ^99mTc,

R_t = 37 MBq (1 mCi) = 37 × 10⁶ dis/s and λ (^99mTc) = 3.2 × 10^-5/s

Therefore, from equation (3.2),

For ¹³¹I,

R_t = 37 MBq (1 mCi) and λ (¹³¹I) = 10^-6/second

Therefore, from equation (3.2),

(i.e., for the same amount of radioactivity, there are approximately 32 times more radioatoms present in a sample of ¹³¹I than in a sample of ^99mTc).

Once we know the number of radioatoms N_t present in a 1-mCi sample of a radionuclide of mass number A, it is simple to calculate the mass of the radionuclide present in that sample. Here, we assume that the radioactive sample contains only the radionuclide under consideration and none of its isotopes.

From equation (3.2), for a 1-MBq sample,

From Avogadro’s hypothesis, A grams of this radionuclide contain 6 × 10²³ radioatoms.

or, substituting value of N_t,

What is the mass of a 1 MBq sample of ^99mTc? The mass number for this radionuclide is 99, and the decay constant is 3.2 × 10^-5/second. Therefore,

In the above calculations, the mass of a 1-MBq sample was determined. In many instances, however, one is interested in the reciprocal of the above (i.e., the amount of radioactivity per unit mass of the radionuclide of interest). This, the radioactivity per unit mass of a radionuclide, is known as the specific activity of a radioactive sample. It is generally expressed in units such as MBq/mg (mCi/mg) or disintegration/min/mg. In the above example, the specific activity of the ^99mTc sample is simply the inverse of M or equal to 1/(5.17 × 10^-12) MBq/g or equal to 1.93 × 10¹¹ MBq/g or 1.93 × 10⁸ MBq/mg.

In this example, it was assumed that all atoms of technetium were radioactive and no stable or longer-lived technetium atoms were present in the sample. Such a sample that contains only the radionuclide of interest and no other isotope or long-lived radioisotope of that radionuclide is generally called a “carrier-free” sample. Therefore, the above calculation of the specific activity applies only to a carrier-free sample. The specific activity of a sample with carrier (which contains stable or longer-lived isotopes of the radionuclide of interest) will always be less than that of a carrier-free sample and will depend on the amount of the carrier in the sample.

When a radionuclide is not present in its elemental form but is a part of a molecule, the concept of specific activity can be generalized to the mass of the molecule rather than the element. For instance, in the above sample, if the technetium radioactivity is present as pertechnetate (^99mTc), rather than as ^99mTc, the specific radioactivity of the sample can also be expressed as radioactivity per unit mass of pertechnetate. Again, if the sample is carrier free (i.e., no molecules of pertechnetate exist in the sample that does not contain ^99mTc as its part or ⁹⁹Tc molecules are absent), the specific activity of a pertechnetate sample can be calculated as before by simply taking the inverse of M and substituting the molecular weight of Tc for A in the above formula. Because the molecular weight of Tc is about 163, the specific activity of a 1-MBq sample of ^99mTc

Of course, when the sample is not carrier free, the specific activity of the sample will be lower than that of a carrier-free sample and will depend on the amount of carrier present. In nuclear medicine, knowledge of the specific activity of a sample is important for several reasons. First, the toxicity of a chemical or drug is always dependent on the amount of the chemical or drug administered to the patient. Therefore, it is important to know the amount of a particular chemical or drug in the radioactive sample. Second, the distribution of a radiochemical in a biologic system often depends on the amount of carrier present in the sample. Third, the labeling efficiency of a radionuclide to a chemical is also sometimes dependent on the specific activity of the radionuclide.

If we combine equations (3.1) and (3.2), then

This is a differential equation whose solution is given by the following expression:

This equation relates the number of radioatoms N_t remaining at time t when we initially started (at t = 0) with a number N₀ and is known as the exponential law of decay. When we plot this expression
as a linear graph, curves similar to those shown in Figure 3.1 result for different radionuclides (different λ). When we plot the same expression on semilog graph paper, as in Figure 3.2, we obtain straight lines of different slopes (steepness) for different radionuclides.

The exponential law of decay is also true for the radioactivity R_t, that is, R_t is also related to the original radioactivity R₀ by an expression similar to equation (3.3). This can be easily seen by substituting equation (3.3) on the right side of equation (3.2):

where R₀ = λ · N₀ is the radioactivity at time 0. If one plots activity R_t as a function of time, curves similar to those shown in Figures 3.1 and 3.2 result except that the y-axis will now represent the radioactivity.

How does one determine the decay constant λ? Experimentally, it is more convenient to measure another parameter T_1/2, known as the half-life. This is defined as the time interval for a given number of nuclei (or their radioactivity) to decay to one-half of the original value. For example, if initially there are 10,000 nuclei in a radionuclide and 5,000 of them decay in 5 days, then the half-life of this radionuclide is 5 days. Therefore, by definition of the half-life, when

If we substitute these values in equation (3.3), we obtain

or

Because e^-0.693 is equal to 1/2, it follows that

Equation (3.5) relates the decay constant to the half-life of a radionuclide. From this, if one knows the half-life of the radionuclide, one can calculate the decay constant or vice versa.

Half-life of ¹³¹I = 8 days. Determine the decay constant for ¹³¹I

0.691 × 10⁶ × λ = 0.693 or λ = 10^-6/second

Equations (3.1), 3.2, 3.3, 3.4 and (3.5) are the basic equations necessary for solving routine problems in nuclear medicine. In general, to solve such problems, the use of a scientific calculator is necessary. Interestingly, equation (3.3) can be written in a simpler form by substituting the value of λ from equation (3.5).