Half-life (A-level only)

What you'll learn

• The difference between physical, biological, and effective half-lives.
• Why medical tracers disappear from the body faster than their physical decay rate suggests.
• How to use the equation $\frac{1}{T_E}=\frac{1}{T_B}+\frac{1}{T_P}$ to calculate clearance times.

When a patient is injected with a radioactive tracer for a medical scan (like a PET or gamma camera scan), the hospital needs to know exactly how long the patient will remain radioactive. We want the tracer to stick around long enough to get a clear image, but vanish quickly afterwards so the patient doesn't receive an unnecessary dose of ionising radiation.

The trick is that the tracer disappears from the body via two completely independent processes happening at the same time.

The Three Half-lives

First, the tracer is radioactive, so the unstable nuclei are constantly decaying into stable ones. Second, the tracer is a chemical substance, meaning the patient's body is actively trying to filter it out and excrete it (usually through urine, faeces, or sweat).

We define a specific half-life for each of these processes, and a final one for the overall effect.

The Effective Half-life Equation

Because both processes are removing active tracer from the body at the same time, the overall rate of removal is simply the sum of the two individual rates.

In nuclear physics, the rate of decay is determined by the decay constant, $\lambda$. So, the effective decay constant is the sum of the biological and physical decay constants:

$${\lambda }_E={\lambda }_B+{\lambda }_P$$

You already know from your earlier radioactivity studies that the decay constant is related to half-life by the equation $\lambda =\frac{\ln (2)}{T}$. If we substitute this into our rate equation, we get:

$$\frac{\ln (2)}{T_E}=\frac{\ln (2)}{T_B}+\frac{\ln (2)}{T_P}$$

Dividing the entire equation by $\ln (2)$ leaves us with the formula you need to use in the exam:

$$\frac{1}{T_E}=\frac{1}{T_B}+\frac{1}{T_P}$$

Working with the Formula

Let's look at how AQA tests this equation. The most common question simply asks you to find the effective half-life given the other two.

Working Backwards

Sometimes, the hospital can measure the effective half-life directly by putting a radiation detector over the patient and monitoring how fast the activity drops. If they already know the physical half-life of the isotope, they can calculate how fast the patient's body is processing the drug.

Extreme Cases

Sometimes, the biological half-life is vastly different from the physical half-life. Understanding what happens to the formula in these extreme cases is a great way to deepen your understanding:

• When $T_B$ is effectively infinite: If a tracer gets permanently locked into the body's structure (like Strontium-90 mimicking calcium and binding completely to bone), the body essentially never excretes it. Here, $T_B\approx \infty$. Because $\frac{1}{\infty }=0$, the equation becomes $\frac{1}{T_E}=0+\frac{1}{T_P}$. In this case, the effective half-life is entirely determined by the physical half-life: $T_E=T_P$.
• When $T_P$ is incredibly long: If we use an isotope with a physical half-life of thousands of years, $\frac{1}{T_P}$ approaches zero. The tracer hardly decays at all while in the body, so it only disappears because the patient urinates or exhales it out. In this case, $T_E\approx T_B$.

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