Physical, biological and effective half-life

This technical mathematics article is written by Dr Chris Robbins of Grallator Limited. Information and links for Chris and his company will be found at the end of this article.

Physical, biological and effective half-life

Dr Chris Robbins, Grallator

Radioactive decay is based on the assumption that it is a random Poisson process and that over a given time interval there is a fixed probability that any given nuclide will decay. To obtain the instantaneously total number from a collection that will decay we just multiply up by the population number. This gives a differential equation for the rate of change of population number due to radioactive decay as \[ \frac{dN}{dt} = -\lambda N. \tag{1} \] where:

\(\qquad \lambda \qquad \)is the decay constant
\(\qquad N \qquad \)is the population size at time \(t\)

In the above a negative sign is used to indicate that decay reduces the population number.

The solution to this equation is given by simple re-arrangement and direct integration with the boundary condition \(N(t=0) = N_0\) \[ \begin{align} \frac{dN}{dt} &= -\lambda N \Rightarrow \int_{N_0}^{N} \frac{dN}{N} = -\lambda \int_{0}^{t} dt, \\ \\ \left [\ln(N) \right ]_{N_0}^{N} &= - \lambda \left [t \right ]_{0}^{t}, \\ \\ \ln(N) - \ln(N_0) &= -\lambda t, \\ \\ \ln \left ( \frac{N}{N_0} \right ) &= -\lambda t. \tag{2} \end{align} \]

Taking the exponential of both sides gives \[ \begin{align} \frac{N}{N_0} &= e^{-\lambda t} \Rightarrow N=N_0 e^{-\lambda t}. \tag{3} \end{align} \]

i.e. the standard exponential decay equation.

The value of \(\lambda\) is determined by timing how long a reduction in the number of atoms to a measurable value takes, i.e., \(t\) and the ratio \(\frac{N}{N_0}\) are known. This can be done using (2) as \[ \lambda = - \frac{\ln \left ( \frac{N}{N_0} \right )}{t} = \frac{\ln \left ( \frac{N_0}{N} \right )}{t} \tag{4} \]

A useful measurement point is the time taken for half the initial amount of material to decay, better known as the half-life. At this time, denoted \(t_{1/2}\), \(N = \frac{1}{2} N_0 \Rightarrow \frac{N_0}{N}=2 \), so that (4) gives \( \lambda \) in terms of the half life as \[ \lambda = \frac{\ln(2)}{t_{1/2}} \tag{5} \]

The above relates to removal by physical decay. In biological systems there can be removal by excretion. As with decay there is a fixed probability that any one particle will be removed per unit time and so (1) can also be used to describe biological removal. When calculating a biological half-life the time taken for the body to remove half the initial material is considered and a biological removal rate is constructed: \[ \lambda_b = \frac{\ln(2)}{t_{1/2,b}} \tag{6} \\ \] where the b subscripts are used to differentiate from physical decay.

We can now ask "what happens when both physical decay and biological removal occur?" To consider this, we will re-visit (1) and write removal terms from both mechanisms. Note they are independent removal mechanisms so the total probability of removal per unit time is obtained by summing the two individual probabilities to give \[ \frac{dN}{dt} = -(\lambda+\lambda_b) N. \tag{7} \\ \] which can be written as \[ \frac{dN}{dt} = -\lambda_{eff} N. \tag{8} \\ \] where \(\lambda_{eff}\) is the effective removal rate given by \( \lambda_{eff} = \lambda+\lambda_b \). Rearranging (5) we can calculate the effective half-life from the effective removal rate as \[ t_{1/2,eff} = \frac{\ln(2)}{\lambda_{eff}} \tag{9} \\ \] Using \( \lambda_{eff} = \lambda+\lambda_b \) with (5) and (6) we can write the effective half life in terms of the physical and biological half life values as \[ \begin{align*} t_{1/2,eff} &= \frac{\ln(2)}{\lambda_{eff}} \\ \\ &= \frac{\ln(2)}{\lambda + \lambda_b} \\ \\ &= \frac{\ln(2)}{ \frac{\ln(2)}{t_{1/2}} + \frac{\ln(2)}{t_{1/2,b}} } = \frac{\ln(2)}{ \frac{t_{1/2,b}\ln(2)+t_{1/2}\ln(2)}{t_{1/2}t_{1/2,b}} } =\frac{t_{1/2}t_{1/2,b}}{t_{1/2}+t_{1/2,b}} \tag{10} \\ \end{align*} \] Looking at the example of Po-210 in the body with \(t_{1/2}\ = 138\) days and \(t_{1/2,b} = 40\) days. The effective half-life in the body is given by \[ \begin{align*} t_{1/2,eff} &= \frac{t_{1/2}t_{1/2,b}}{t_{1/2}+t_{1/2,b}} \\ \\ &= \frac{138 \times 40}{138+40} = \frac{5520}{178} = 31 \text{ days} \end{align*} \]

More about the half life and the mathematics behind it can be found on the Grallator website (opens in a new tab).

This article was written by a good friend, and mathematics and physics wizard, Chris Robbins. His website is well worth a visit if you need assistance in animation, software design, physical modelling or educational resources.