The Palpable Prostate

Prostate cancer topics, links and more. Now at 200+ posts!

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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:

http://palpable-prostate.blogspot.ca/2007/03/psa-doubling-time-psadt-part-1.html
http://palpable-prostate.blogspot.ca/2007/03/psa-doubling-time-psadt-part-2.html
http://palpable-prostate.blogspot.ca/2007/03/psa-doubling-time-psadt-part-3-diy.html
http://palpable-prostate.blogspot.ca/2007/03/psa-doubling-time-psadt-part-4-online.html

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

Tuesday, November 15, 2016

2016 AUA Meeting

[Updated: December 2, 2015] >br />
Highlights of the 2016 American Urology Association (AUA) meeting have recently been published.
[review]

Key topics relating to prostate cancer included:

- Molecular imaging. PSMA is expressed in prostate cancer and radiotracers can detect it.  The highlights include discussion of such imaging as well as the use of antibodies targeting PSMA to deliver toxic drugs directly to the cancer cells.

- Active Surveillance

- new prognostic markers aimed at distinguishing between disease having significant potential for malignancy vs. more benign disease

See the link above for the summary.

Another summary of some recent results can be found here: [PMID:27895809] [free full text]

Sunday, October 2, 2016

Blog updates for September 2016

Oct 2, 2016. In Grape Seed Extract we add: Although the 62% reduction in prostate cancer among the high grape seed extract group is eye opening there is only one clinical trial on grape seed extract and prostate cancer: [clinicaltrials.gov - prostate cancer and grape seed extract]. In comparison there are 27 clinical trials on metformin and prostate cancer: [clinicaltrials.gov - prostate cancer and metformin] so the bets seem to be much more on [metformin] at this point.

Friday, September 23, 2016

Grape Seed Extract

[Updated October 2, 2016]

This 2011 report on the VITAL study of 35,239 men found no effect on the development of prostate cancer for chondroitin, coenzyme Q10, fish oil, garlic, ginkgo biloba, ginseng, glucosamine or saw palmetto; however, it did find a 41% reduction in the development of prostate cancer among those who took grape seed oil supplements and a 62% reduction in the development of prostate cancer among those who took doses higher than those found in multivitamins over a 10 year period. Low doses of grape seed extract, primarily from multivitamins, did not show any reduction in prostate cancer development.

There were some limitations to the VITAL study and the authors did not recommend that taking such supplements but if any of the above substances work to reduce the likelihood of the development of prostate cancer then it would seem that grape seed oil supplements would be more likely than the others. Also, the study did not look at patients who already have prostate cancer but of course if it works in one set of people it might work in another. [PMID: 21598177] [full free text].

This 2014 study of grape seed extract found an anti-cancer effect against prostate cancer cells in test tubes. [PMID: 24191894] which adds to the evidence in the VITAL study.

Although the 62% reduction in prostate cancer among the high grape seed extract group is eye opening there is only one clinical trial on grape seed extract and prostate cancer: clinicaltrials.gov - prostate cancer and grape seed extract. In comparison there are 27 clinical trials on metformin and prostate cancer: clinicaltrials.gov - prostate cancer and metformin so the bets seem to be much more on metformin at this point. For more information on grape seed extract (side effects, etc.) see the pages at drugs.com and Memorial Sloan Kettering. Additional discussion and references on grape seed extract can be found in a 2014 review of chemoprevention strategies [PMID: 24389535] [full free text] .

Monday, August 1, 2016

Blog updates for July 2016

July 22, 2016. In Metformin and Prostate Cancer we added statin references: Other recent work suggests benefit from combination therapy of metformin with statins [PMC3329836 full free text], [news release], p53 stabilizers [PMID: 26900800], chemotherapy [PMID: 27118574].

Wednesday, June 22, 2016

Brachytherapy problem

In a perineal saturation biopsy, a brachytherapy-like grid of rectangular squares is spaced at s = 5mm. Biopsy samples are taken. The cancer will be assumed to be a disc, i.e. circle, of diameter d. It will be detected if  the disc intersects a vertex on the grid. What is the probability of detecting the cancer? of missing the cancer? Consider a disc of diameter d around each of the 4 corners of a particular square in the grid. (See accompanying diagram.)


One quarter of each of those discs lies in the particular square so the total area of those discs within that square divided by the area of the square is the probability of detection. (This assumes a single focus of cancer d mm in diameter, that the probability distribution of its center is uniform and further assumes that the quarter discs shown in the diagram are not so large that they overlap.)

probability of detection of cancer 
= probability that the center of the cancer lies in one of the 4 quarter discs 
= (sum of 1/4 of the area of 4 discs of diameter d) / (area of one grid square) 
= 4 * 1/4 * (area of a disc of diameter d) / (area of one grid square) 
= (area of a circle of diameter d) / (area of one grid square) 
= (pi * (d^2)/4) / s^2
 
We used the fact that the area within a circle is pi*r^2 where r = d/2 is its radius. In terms of the diamter this can be written as pi * d^2/4. The above only works if the 4 quarter circles do not overlap and that is guaranteed if d is less than s. If d is greater than s then the quarter circles would overlap and adding their areas would represent double counting in the overlapping regions.

To double check we can compare the above formula to a simulation. If the cancer has a diameter of d = 2mm and each side is s = 5mm, then the probability of detecting the cancer, is 0.50265 by the formula and 0.4955 via the simulation below -- these two numbers are equal to two decimal places. We review the formula calculation and then the simulation:

1. Formula. Using the formula above and substituting in d = 2 and s = 5 we find that the probability of detection is 0.50265 (and so the probability of missing the cancer is 1 - 0.50265 = 0.49737).

(pi * (d^2)/4) / s^2 = (3.1415926 * 4^2/4) / 5^2 = 0.50265

2. Simulation. Using the R code at the end we create a simulation again assuming d = 2 and s = 5. d as well as s and n (number of iterations) can all be changed as needed by modifying their values near the top of the code. The code generates n = 10,000 points uniformly in an s by s square and then counts the fraction lying within d/2 of a corner as the probability of detection. This approach is more general since it also works even in the case where the discs at the 4 corners of the square overlap. Run it by copying the code below to the clipboard and paste it into the text input box at http://www.r-fiddle.org -- be sure to erase anything already in the r-fiddle text entry box first. Then press Run. (You may need to press Run twice if the answer does not appear the first time.) The code below gives the probability of detection and 1 minus that number is the probability of missing the cancer.

set.seed(123)
n <- 10000  # number of iterations in simulation
s <- 5 # length of side of a grid square
d <- 4  # diameter of each of the 4 discs centred at the 4 corners
r <- d/2
x <- runif(n, 0, s)
y <- runif(n, 0, s)
mean(x^2 + y^2 < r^2 | (s - x)^2 + y^2 < r^2 | x^2 + (s-y)^2 < r^2 | (s-x)^2 + (s-y)^2 < r^2)


Thanks to Don Morris for raising this problem.

Friday, June 3, 2016

Blog Updates for May 2016

June 3, 2016. In Metformin and Prostate Cancer we added: A May 2016 update by Sayyid and Fleshner [PMID: 27195314] [Full Free Text] reviews studies that show a decreased prostate cancer risk with metformin as well as beneficial effects for prostate cancer patients after treatment. Other recent work suggests benefit of combination therapy of metformin with statins, p53 stabilizers [PMID: 26900800], chemotherapy [PMID: 27118574].