## An

To obtain the number of decaying atoms for a measurable period of time, the differential equation must be integrated. Therefore, the basic equation for radioactive decay is expressed as follows in terms of atoms:

Nt = No e-At where Nt (number of atoms at time t) and N0 (number of atoms at time 0). Activity (A) of a radionuclide is also expressed as the number of atoms disintegrating per unit of time (A = AN). With the substitution of A/A into the equation above for N and simplifying the result,

Ao e-At where At is the activity at time t, A0 is the original activity, and A is the decay constant.

A radionuclide's decay is routinely expressed as its physical half-life, the time in which one half of the atoms decay. Mathematically, this is expressed as ln 2

where t1/2 is the physical half-life and A is the decay constant. Substitution into the previous equation for the decay constant, A, provides

where At is the activity at time t, A0 is the original activity, t is the elapsed time interval, and tl/2 is the physical half-life of the radionuclide of interest. This equation is frequently utilized directly or indirectly in daily operations of a nuclear pharmacy or nuclear medicine department.

The activity (A) of radionuclides can be expressed in the following three ways: (a) in disintegrations per unit time (usually disintegrations per second [dps] or disintegrations per minute [dpm]), (b) in curies (Ci), and (c) in becquerels (Bq; 1 Bq = 1 dps). A curie is defined as the quantity of any radionuclide that decays at a rate of 3.7 X 1010 dps, a number closely approximating the historically calculated number of dps in 1 g of pure radium. More recently, the International System of Units (SI) has adopted the becquerel as the official unit of radioactivity. The curie is still traditionally utilized in clinical practice, and the relevant conversion factor to remember is the following:

1 millicurie (mCi) = 37 megabequerels (MBq)

An example of a routine radioactive decay calculation follows:

A sample of sodium iodide (123I) is calibrated by the manufacturer at 200 ixCi on May 14 at 12 pm C.S.T. What is the activity remaining on May 15 at 3 pm E.S.T.? The f1/2 of 123I is 13.2 hours. (Note: Calculations of elapsed time must also indicate variations in time zones incurred during transport).

(t1/2 = 13.2 hours) 4 = (200 MCi) (0.5)<26/132) 4 = (200 /xCi) (0.255) 4 = 51.1 /xCi

Radionuclides utilized in nuclear medicine are artificially produced. This is accomplished when small charged or uncharged particles bombard atomic nuclei and initiate a process of nuclear change. The artificial production of a radionuclide requires preparation of target nuclei (parent), irradiation of the target, and chemical separation of the daughter radionuclide produced from the nuclear reaction. The daughter radionuclide is converted to the desired ra-diopharmaceutical form. Quality assurance tests of the physical, chemical, and pharmaceutical qualities (i.e., sterility and apyrogenicity) of the final product are per formed. The systems used for practical production of ra-dionuclides are a nuclear reactor, cyclotron, or radioisotope generator.

The shorthand nuclear physics notation of artificial nuclear transformation reaction is as follows:

112Cd(p,2n)111In where Cd-112 is the stable target nuclei, a proton (p) is the bombarding particle, two neutrons (2n) are emitted from the nucleus during the transformation, and In-111 is the daughter radionuclide produced.

The introduction of radionuclide generators into nuclear medicine arose from the need to administer larger doses of a short half-life radionuclide to obtain higher quality images. The general principle of the radionuclide generator is that a long-lived parent is bound to some adsorbent material in a chromatographic ion exchange column and the daughter is eluted from the column with some solvent or gas. When considering potential radioactive parent and daughter pairs, two scenarios can be envisioned depending on the magnitude of difference in half-lives between the long-lived parent and the short-lived daughter. If the parent nuclide has a sufficiently longer half-life than the daughter nuclide (10-fold to 100fold), a state of transient equilibrium is ultimately reached when the daughter nuclide is being produced from the parent at the same rate as the daughter decays. In the situation where the parent half-life is much longer than that of the daughter (> 1,000-fold), a state of secular equilibrium is reached when the daughter nuclide is being produced from the parent at the same rate as the parent decays

There are many possible parent-daughter generator systems for clinical use, but there is only one in routine use in nuclear medicine, the molybdenum-99/technetium-99m system. All of the molybdenum-99 at the present time is obtained as a fission product of uranium-235 from nuclear reactors.

235U(n, fission)236U ^ 99Mo + other radionuclides

Inorganic radiochemistry techniques separate molybde-num-99 from the other radionuclides. Molybdenum-99 (t1/2 = 66 hours) decays by negatron emission to tech-netium-99m (t1/2 = 6 hours), which decays by IT to tech-netium-99 by emitting a y-ray (140 keV). Anionic molyb-date (99MoO4~2) is loaded on a column of alumina (Fig. 11.1). The molybdate ions adsorb firmly to the alumina, and the generator column is autoclaved to sterilize the system. Then the rest of the generator is assembled under aseptic conditions into its final form in a lead-shielded container. Each generator is eluted with sterile normal saline (0.9% sodium chloride). The alumina column is an inorganic ion exchange column where chloride ions (CP) exchange for pertechnetate ions (99mTcO4~) but not molybdate ions (MoO4~2). The column eluate contains sodium pertechnetate as well as sodium chloride. Elution efficiency of contemporary molybdenum-99/technetium-99m generators is approximately 80% to 90% per elution.

The method for calculating how much daughter ion present on the column at any given time is complex because it must consider the decay rates of parent and daughter nuclides, the abundance daughter nuclide produced from the parent, as well as any daughter nuclide initially present or remaining on the column after previous elution

Eluate

99mTcO4

Air Filter Air

Air Filter Air

Eluate

99mTcO4 Alumina absorbs free 99Mo to keep breakthrough to a minimum

KC'aStiC Outlet-Case Needle to Collection Vial (eluate)

Cross Section of a Generator

Alumina absorbs free 99Mo to keep breakthrough to a minimum

KC'aStiC Outlet-Case Needle to Collection Vial (eluate) Detail of Column

Eluant Inlet Needle Rubber Stopper

Disperses saline to Glass w°o1 \ obtain Silica Ge' I maximum

Band °f 99mTc yield

Alumina

(Aluminum Oxide) Glass Filter (retains Al Oxide particulates) Rubber Stopper

### Detail of Column

Figure 11.1 • Cross-sectional diagram of a molybdenum-99/technetium-99m generator. (Reprinted with permission from Bushberg, J. T., et al.: The Essential Physics of Medical Imaging, 2nd ed. Philadelphia: Lippincott Williams and Wilkins, 2002.)

(Fig. 11.2). In the case of Mo-99 (t1/2 = 66 hours), only 86% of the atoms decay to Tc-99m (t1/2 = 6 hours). Typical elution efficiencies of commercially available generators are approximately 80% to 90%. The equation for the theoretical activity of the daughter nuclide present at any time (t) after a previous elution is as follows:

Ad = (Ap)Ad[(e-V - e-Adt)/(Ad - Ap)] + A°de-Adt where A0p is the activity of the parent at the time of the previous elution, A0d is the activity of the daughter at the time of the previous elution, Ap and Ad are their respective decay constants, and t is the time since the last elution of the generator. The generator system can be eluted several times each day to obtain collective radioactivity because Tc-99m is constantly produced by Mo-99 decay. The time interval since the last elution will determine the maximal amount of Tc-99m available for elution. Approximately 40% of maximal Tc-99m is available after 6 hours, whereas maximal Tc-99m activity is achieved in 23 hours.1 