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Half-life is a crucial concept in nuclear physics, especially when dealing with radioactive materials. It is defined as the duration taken for half of the atoms in a radioactive sample to decay. This decay occurs naturally and at a set rate specific to each radioactive isotope. Knowing the half-life of an isotope like Oxygen-15, which has a half-life of 2.0 minutes, allows us to calculate how much of the substance remains after a certain period. When we say the half-life is 2.0 minutes, we mean that after 2 minutes, only half of the original amount of Oxygen-15 would be left due to radioactive decay.

Understanding half-life is essential when considering the delivery and use of radioactive substances in medical applications like PET scans. Healthcare providers must account for this decay to ensure that a sufficient amount of the active isotope arrives at the location of use.

Activity concentration is the measure of the decay rate of a radioactive substance within a unit volume. It's often expressed in terms like millicuries per liter (mCi/L). When dealing with radioactive materials in a hospital setting, such as Oxygen-15, the activity concentration will determine the medical use efficacy. In our exercise, we need to ensure that upon arrival at the patient's side, the Oxygen-15 has an activity concentration of 0.50 mCi/L.

This is where calculations of initial activity concentration become important. If we underestimate it, the radioactive substance won’t be effective for diagnosis or treatment. If we overestimate, it does not only waste resources, but also potentially increases exposure to unnecessary radiation.

Radioactive decay is an excellent example of exponential decay, which is a process where the quantity decreases at a rate proportional to its current value. One key feature of exponential decay is that it reduces by consistent fractions over equal time periods. Mathematically, this is often represented by the formula \(A_t = A_0 \times 0.5^{\frac{t}{t_{1/2}}}\), where \(A_t\) is the activity of the substance at time \(t\), \(A_0\) is the initial activity, and \(t_{1/2}\) the half-life of the substance.

In a practical sense, this means the amount of a substance decreases rapidly at first, then more slowly over time. For our case, this formula allows us to solve for the initial activity concentration required before transportation so that an adequate amount is available for the patient's PET scan after 3.5 minutes.

Positron Emission Tomography, or PET, is a sophisticated imaging technique used in medical diagnostics. PET scans use radioactive tracers, such as Oxygen-15, which emit positrons as they decay. These positrons quickly meet electrons in the body, leading to a matter-antimatter annihilation that produces photons. The scanner detects these photons and uses the information to create detailed images of organs and tissues.

The preciseness of a PET scan is heavily dependent on the activity concentration of the radioactive material administered to the patient. Too little activity means poor image quality, while excessive activity can be harmful. Hence, the activity must be calculated with care and precision, considering the half-life and transport time to the patient.

Nuclear physics is the field that studies the constituents and interactions of atomic nuclei. The principles of nuclear physics underlie the calculations of radioactive decay and half-life, key aspects when handling isotopes for PET scans. As in our example, knowledge of nuclear physics is vital for determining how radioactive decay will affect the preparation and delivery of medical substances.

It informs us how certain isotopes decay, which isotopes can be used for therapeutic or diagnostic purposes, and the safety measures that must be in place when using them. In a hospital, understanding these concepts ensures the safety of patients and staff, and maximises the efficacy of nuclear medicine procedures such as PET scans.