Nuclear physicists, environmental activists, pharmacists and many others like to talk about ‘half-life’. It comes across as a mathematical mystery. Perhaps they are trying to tell you when life is half over, or maybe suggesting that you only live to half your potential? Neither of those in fact! And I cannot help but bore you with a non-math discourse—-
Mathematical description of population rise and fall started back in the 1700’s when people began to worry about the truth of the proposition that the entire world population could have come from Adam and Eve, or from Noah and his family. The ideas are very simple at the core.
You know from looking at a herd of animals or a clutch of rabbits that they multiply faster and faster as time goes along. More and more arrive until everything is converted to rabbits (or to humans). So there was once an outlook called the Malthusian Gloom (1798) which said that humans would out-populate the world because their numbers will grow without limit. We understand also that if there are no rabbits at the start, there won’t be any at the end. And it turns out that the population is forever proportional to the initial population – two gets to four; four gets to eight and so on.
The rate of multiplication is always proportional to time, while the population itself escalates (exponentially) with time. Mathematically, the important parameter is the time rate of multiplication – the average time a unit takes to replicate itself. We call this the time constant (T). The rate of multiplication is equal to the initial population multiplied by time (t), divided by the time constant T. For rabbits, T is short, they multiply rapidly (seven times a year) and their population explodes. For humans T is longer, we can’t multiply as fast as rabbits and the population grows more slowly. For elephants, everything is slower again. The basic scientific feature is the time constant T for rate of population change. Since, however, we generally understand total populations better, convention is to convert T to a related number called the half life. This measures the time taken for a twofold change in the total population. We could use anything, but that is the convention.
Happily, the Malthusian Gloom wasn’t quite right. A while later, Verhulst(1838) realized after listening to some advice from his father-in-law that the multiplication rate must slow down as resources are used up – so the world would trend to a nice stable population equally balanced to the resources available. The rapidly increasing population would stabilize at some upper limit. If we are dealing with a mutant population in our bodies, for example, the risk they present in ‘logistic growth’ is defined by three factors. The first is the existing level of presence. The second is the time constant or half life for multiplication and growth. The third is the limit set by resources available. So if we are to talk sensibly about controlling a potential mutant, we cannot focus on just one or two parameters or otherwise there is perennial risk of its escape from our control. We cannot ever be sure that the initial population is zero – whatever treatment we have engaged. So we need to eliminate the resource it requires for growth (if possible) and drive its rate of multiplication down to a level that can never present threat (if possible). Since nothing is fully stable over living times, a zero multiplication rate at one time might not be sufficient downstream. So we actually have to look toward conditions that can impose an intrinsically negative multiplication rate. In crude terms, that is what the natural safeguard mechanism of the body set out to achieve. By extrapolation, one of the major adverse effects of aging is weakening of the safeguards – it is not necessarily some sudden capitulation into dysfunctional biological error. So there are many age-onset diseases which originate from cell cycle errors that have gone on virtually all our lives.
Moving along to descriptions of the decay of a population, we find exactly the same math operating in reverse. Similarly, there is a decay time constant or half life and a lower stable population. Even if the end result would be a zero final population or concentration, you would never reach it exactly. Let’s make the analogy to your memories of the stress associated with making an annual tax return. They are not pleasant memories, so you expel them quickly…but they are less intrusive as the anguish subsides and you just let them drift away. But they are never gone truly. Ed B. wrote; “Daniel is absolutely correct but I would add one caveat. In theory one NEVER eliminates all of the substance… Theoretically it never reaches 0 although this has very little real significance in the real world.” Mmmmm – Maybe true, but is that always so? It depends on the details, of course. The tax agent lurks and watches, pounces and takes advantage you don’t really want to cede! The lasting memories are dangerous if you have been hiding something. They can rebound even if they are very small indeed. All they need is opportunity!
If, then, we look at a malignant population rather than a drug concentration; it is important to understand that the remnant population does not go exactly to zero. It is always there with a non-vanishing population.
Therefore it always has potential to multiply, grow and emerge again as a dominant factor in your conscious reality. Models in mathematical and computational biology do not predict freedom from disease just because a mutant population is balanced between birth and death, or is driven into decline – any more than your physician would. The same things matter: does a population of cells exist, are they capable of replication, what resource are they able to access and can they achieve a positive effective multiplication rate? This is not a one-time question, unfortunately.
Fluctuations in the balance must be contained. Whatever treatments we engage the results are not secure, so it seems to me, unless the safeguard mechanisms are able to preclude multiplication securely and reliably. Small beginners, like the buds in the cherry tree, will otherwise turn into lush fruit with the fullness of time.