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Friday, November 25, 2016

The Rule of 72 and PSA Doubling Time

A common approximation used in finance is that an investment which grows at x% per year will double in roughly 72/x years.  It is known as the rule of 72. What is of interest here is that it not only applies to finance but also PSA doubling time.

First consider the case of an investment.  Suppose an investment grows at 12% per year.  Then according to the aforementioned rule it will double in about
   72/12 = 6 years  (rule of 72 approximation)
This is close to the exact value (rounded to one decimal place) of
   1/log2(1+12/100) = 1/log2(1.12) = 6.1 years   (exact)
This can be double checked by entering that formula into the Google search bar.

(Here log2(x) represents the logarithm to the base 2 of x.  In the case that x is a power of 2 the value of log2(x) is the number of 2's to multiply together to get the argument.  For example, log2(8) = 3 because multiplying three twos gives 8, i.e. 2 * 2 * 2 = 8. If x is not a power of 2 then it will be between the log2 values of the nearby powers of 2. For example, log2(5)=2.321928 is between log2(4) = 2 and log2(8) = 3.)

We can use this rule to approximate PSA doubling time (PSADT).  Suppose the PSA is 0.100 at the beginning of the year and 0.112 at the end of the year.  Thus, it is increasing at .112 / .100 - 1 = .12 = 12% per year.   Using the rule of 72 this implies that the PSA will double in
   PSADT = 72/12 = 6 years   (rule of 72 approximation)
which, as before, is close to the exact value of
   PSADT = 1/log2(1+12/100) = 1/log2(1.12) = 6.1 years   (exact)
Although we used years above we can use any time period.  For example, suppose the PSA were growing at 2% per quarter.  Then it will double in
   PSADT = 72/2 = 36 quarters = 9 years   (rule of 72)
This is close to the exact number of
   PSADT = 1/log2(1+2/100) = 1/log2(1.02) = 35 quarters = 8.75 years   (exact)
There are some significant caveats.

1. The use of PSADT assumes constant exponential growth of the cancer cells.  That is, the percentage increase from period to period does not systematically change.  Such constant exponential growth would be the case if a plot of log2(PSA) vs. time were roughly linear.  If the entire plot were not roughly linear but sections of the plot were then each such section may have a different PSADT value.  For example, before and after a treatment intervention one might see different PSADT values.

2. The discussion above uses the Rule of 72 to approximate doubling time using only two PSA values but that is normally regarded as insufficient.  Typically PSADT should be calculated based on at least 3 PSA values to help eliminate the natural variation in PSA values.  Thus the above is only a first approximation before performing a more reliable calculation.

There is a more comprehensive discussion of PSADT calculations in this series of 4 blog posts:

Also, Wikipedia discusses the accuracy of the Rule of 72 on this page: [Wikipedia Rule of 72]

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