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## Wednesday, March 14, 2007

### PSA Doubling Time (PSADT) - Part 3. DIY Formulas with Google

[updated April 20, 2008]

This is Part 3 of a 4 part series on PSA Doubling Time (PSADT):

The purpose of this post is to explore the formulas for PSADT using google's calculator.

Calculating with Google

Entering a formula into Google's search bar will cause it to calculate it and return the result. Of particular note is that Google accepts either lg(x) or log2(x) for log2(x) (see GoogleGuide) which simplifies calculations. (log2 is the number of doublings to reach a given number. For example, log2(8) is 3 because 2 x 2 x 2 = 8, i.e. 3 doublings.)

The formulas for PSADT and PSAV using google's notation are shown here (and examples of what you actually enter into google follow further down):

`PSADT = (t2 - t1) / (lg(PSA2) - lg(PSA1))Average PSAV = (PSA2 - PSA1) / (t2 - t1)Instantaneous PSAV = ln(2) * PSA / PSADT`

If PSADT is constant then PSAV will be increasing over time and the average PSAV in the second formula is the average of those various PSAV's and is normally what is calculated in a clinical setting. The instantaneous PSAV allows one to calculate the PSAV at each point in time assuming the PSADT was constant during that stretch of time.

We discuss a number of formulas to calculate doubling times that you can immediately use with google.

• Formula for two PSA values. With only two PSA values the doubling time can be obtained by entering the following into the google search bar using for our example a PSA rise from 4 to 5 over 16 months where we calculated PSADT in months and PSAV is rise per year:
`16 / (lg(5) - lg(4))`

which Google returns as 49.7.

Since we used months as the time between the two readings the result is also in months. That is the PSADT is 49.7 months.

(The Memorial Sloan Kettering calculator at http://www.nomograms.org gives 49.50 if we use start and end times of 2009-01-01 and 2010-05-01. The small difference between our calculation above and the MSK calculator is due to the fact that the MSK calculator translates the 485 days between these two dates into 465 / (365.25 / 12) = 15.93429 months rather than the 16 months we used. If we use the same input, i.e. 15.93429 months rather than 16 months, then we get the same value to two decimal places as the MSK calculator: 15.93429 / (lg(5) - lg(4)) = 49.50 .)

Now that we have calculated the PSADT let us also calculate the PSA velocity (PSAV). We calculate the average PSAV as (where we multiply by the annual rise by 12 to get rise per year):
`12*(5-4)/16`

which gives a rise of 0.75 per year. The instantaneous PSAV at the beginning and end would be:
`12 * ln(2) * 4 / 49.712 * ln(2) * 5 / 49.7`

which give PSAV rises of 0.67 and 0.84 rise per year respectively. The average PSAV of 0.75 always lies between the two instantaneous values.

The PSADT formula given above can be written even more compactly as:
`16 / lg(5/4)`

however, we will mostly use the first form since it extends to an approximation for multiple PSA values.

• Approximation Formula for More than 2 PSA Values. The first of the above two forms can also be used to get an approximation to the actual doubling time in the case of more than two values and that approximation is often surprisingly good. Divide the PSA values into two groups with approximately equal number of PSA values: an earlier group and a later group. Calculate your average age over the PSA values in each group and use the difference between those ages in place of the time, 16. Then take the average log2(PSA) in each group and use those two averages in place of lg(5) and lg(4). One simplification in the case that your two groups have an equal number of PSA values is that one can replace the averages with sums. For example, suppose at ages 60, 61, 62 and 63 PSA values are 4, 5, 6 and 7. Then enter this into the google search bar:
` (60 + 61 - 62 - 63)/(lg(4) + lg(5) - lg(6) - lg(7)) `

which gives about 3.74 years. This is very close to the actual value of 3.72 years which can be obtained from one of the methods in Part 2 in this series. Its best not to rely on these approximations but rather always double check the results of approximations against calculators that perform the full calculation.

• Rough Approximation for two PSA values. Since most people will have access to google or other scientific calculator with logs this next method is probably not of great interest but for those wishing a quick approximation on a 4 function calculator we provide this logarithm-free approximation. This approximation to the formula is not as accuate but is very simple. The approximation is (using the prior example):
` .7 * 16 / (5/4 - 1)`

Here .7 is fixed and always appears and, as before, 16 is the time between PSA tests and 4 and 5 are the successive PSA values. This equals 44.8 which in this csae is within 10% of the true value of 49.7. Its best not to rely on these approximations but rather always double check the results of approximations against calculators that perform the full calculation.

• Better Approximation for two PSA Values. A better approximation which also involves no logarithms (so it can be calculated on a 4 function calculator) but is often very close to the value calculated by the conventional formula for PSADT is 0.347 * (t2 - t1) * (PSA2 + PSA1) / (PSA2 - PSA1) where .347 is the value of log(2)/2. For our example we have:
`0.347 * 16 * (5 + 4) / (5 - 4)`

which gives 50 and is within 1% of the 49.7 value using the formula involving logarithms. Its best not to rely on these approximations but rather always double check the results of approximations against calculators that perform the full calculation.