Calculating Breast Cancer Survival Rates

Date: 06/30/2003 at 21:30:52
From: Glenn Tisman, M.D.
Subject: Finding roots

How do I find the nth year survival:
Survival of a group of breast cancer patients is 100 at year 0, 100*X 
at year 1, 100*X*X at year 2, 100*X*X*X at year three and so on to 
year 10.

For instance for a n = 10 year survival of 40% the value of X = 0.912

Where X is percentage survival, assuming survival decreases constantly 
on a yearly basis. I would like to know how to solve for X for any 
10-year survival from 1 to 100% survival at 10 years.
n is the year from 1->10 years

Date: 06/30/2003 at 22:45:43
From: Doctor Peterson
Subject: Re: Finding roots

Hi, Glenn.

Your formula says that survival after n years (as a fraction, which 
you multiply by 100% to get a percentage) is

  s = x^n

(I'm using "x^n" to mean the nth power of x.)

This is a typical exponential decay, apparently. To solve for x, you 
take the nth root, as you suggest. A scientific calculator can do 
this in the form of the 1/nth power:

  x = s^(1/n)

Taking your example, with s = 0.40 (40%) and n = 10, x is

  x = (0.40)^(1/10) = 0.91244

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/01/2003 at 11:32:37
From: Glenn Tisman, M.D.
Subject: Re: Finding roots

Thank you, Dr. Peterson. I have applied the exponential decay results 
to actual survival data at 5, 10 and 15 years and have come up with 
an interestig observation that really does make sense. That is that 
the predicted "decay" of patients between year 0 and year 5 closely 
matches the observed survival. However, as time increases to 10 and 
15 years the predicted decay is significantly more (fewer patients 
alive) than that actually observed. This observation suggests that 
the rate of cancer deaths slows over time. If one were to have the 
actual observed 5, 10 and 15 year survivals how would one fit the 
data (changing rate of decay) to an equation?

Your help is appreciated and will be acknowledged in a Palm program I 
am trying to develop to help physicians better treat breast cancer 
patients.

Thank you,
Glenn Tisman, M.D.

Date: 07/01/2003 at 22:52:24
From: Doctor Peterson
Subject: Re: Finding roots

Hi, Dr. Tisman.

I have to say that my initial reaction to your original question was 
that a simple exponential decay seemed far too simple to fit real 
life situations like this, though it might make a reasonable first 
guess to check against observation. It sounds like you're doing just 
the right thing by looking for deviations from this prediction.

I'm not an expert in statistics, so there may be other Math Doctors 
better able to deal with this; and really it's more a matter of 
science than math in the first place. (Math can say what a particular 
model predicts, but the model and its interpretation come from 
science.) But let's see what we can tell from your observations at a 
simple level.

Your model assumes that there is a constant "rate of decay" (death 
rate) for all patients who got a certain treatment for a certain 
disease. Essentially, it supposes that treated people all have some 
residual disease that will kill a certain number of them per year; or 
alternatively, that the cancer comes back in any individual with a 
constant probability each year, never changing.

I don't know what factors there might be that would make that 
assumption false, but it seems likely to me that it would be false. 
It may be, for example, that half the patients are truly cured 
(completely), while the other half just have the disease set back by, 
say, five years in its normal course. This may be because there are 
really two different diseases that are identical except in their 
response to this particular treatment. Then the real death rate would 
be 0 for half the original patients and some kind of probability 
distribution (not constant, but looking nearly so in the short term) 
for the rest. Then in the first few years the patients would be dying 
at some rate that is nearly a constant percentage of those who are 
left (the exponential decay you are seeing), but then as the 
proportion of surviving patients consists increasingly of those who 
are completely cured, the death rate would shift down to the normal 
rate for healthy people.

That's just one possible model, and doesn't have any particular 
significance; but it does illustrate how your results might occur. 
The hard part is to decide what adjusted model best fits the data. 
That might not only give you better predictive power, but also a 
better understanding of the disease itself and its treatment!

Again, I don't think I can tell you what equation to try for an 
improved model. The proper procedure is to use an understanding of 
the underlying phenomena to choose a reasonable type of model (such 
as a combination of two distributions representing response of two 
type of patients, in my suggestion above), and then to fit the data 
to that type of equation in order to see how well it fits and then to 
use the result for prediction. If you can come up with information 
about what type of behavior to expect, then it might be worth while 
to send us (as a new question, so statistics experts can look at it) 
that information and a sample of the data, in order to get ideas for 
the next step.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/02/2003 at 11:48:25
From: Glenn Tisman, M.D.
Subject: Thank you (Finding roots)

Dr. Peterson:

I believe your analysis is correct. Your insight into how cancer kills 
is what most physicians are taught to be true. I have raw clinical 
data of the 5, 10, and 15-year survival for a large group of breast 
cancer patients. I will look at the exponential model for each 
interval and go from there. You see the problem is not fitting a 
simple curve since there are many curves depending on patient 
prognostic factors at time 0. Then there is the data of efficacy of 
therapy which has been exressed as percent decrease in death rate per 
year.

I appreciate your input. Thank you.
Glenn Tisman, M.D.