Survival distribution function?

[JeanTate]

A widely used cancer survival number ("statistic") is "5-Year Relative Survival". In medical papers one often sees "median survival".

Assuming the two are of the same thing - say, Stage 4 Colorectal cancer - is there a known distribution function which connects the two? One which is, perhaps, often used (but rarely mentioned)? Or is it known that this function varies a lot, by cancer type perhaps?

I tried searching the literature, but came up empty.

 

[jimbob]

A widely used cancer survival number ("statistic") is "5-Year Relative Survival". In medical papers one often sees "median survival".

Assuming the two are of the same thing - say, Stage 4 Colorectal cancer - is there a known distribution function which connects the two? One which is, perhaps, often used (but rarely mentioned)? Or is it known that this function varies a lot, by cancer type perhaps?

I tried searching the literature, but came up empty.

Someone mentioned this ages ago (probably between 2002 and 2010) in a column I read. The columnist found (s)he had a particular cancer and looked up the survival statistics. They were surprised at the choice of average, and when they investigated found a strongly bimodal distribution, where people either died very quickly or seemed to survive for decades.

Obviously those details are not enough to even google the article, but assuming my recollection is correct, it would say that there can be different types of distributions even for a single type of cancer.

In a weibull plot, it would show up as a change in gradient as the at least two different mortality mechanisms kick in

 

[casebro]

Or, like Pancreas cancer, there are two different two of Colo-rectal cancer?

OR: Those who chose not to treat last 6 months, those who opt for chemo etc. last 4-5 years. My bro is in the latter group. Looks like he'll make it 4-5 years, on his last hope chemo- if it works. Keytruda. I suppose if it doesn't work, the 6 months is the estimate.

 

[Camillus]

You might be interested in the Kaplan-Meier estimator, which is commonly used in medicine to estimate how survival changes over time following treatment.

 

[JeanTate]

Thanks everyone.

As expected, a few clarifications are in order.

First, the function I’m interested in is not, really, a distribution function (though is related to one).

Think of a chart/plot/graph, with x and y axes. The x axis goes from zero to the right, and represents time; units years, say. The y axis is bounded, [0,1], and represents relative survival (more details later).

The curve starts at (0,1), which means that 100% of the patients/people-diagnosed-with-cancerA are alive. At time of diagnosis.

At x=5 there will be a number/percentage of people (still) alive; if 30%, say, then there’s a second point at (5,0.3).

Finally, median survival: let’s assume it means the time at which 50% of the people are still alive (so y=0.5). If it’s 12 months, say, then the third point is (1,0.5).

Now with just three points an uncountably infinite number of curves can be fitted. But because you can go from alive to dead but not vv , the curve must be monotonically decreasing.

(The curve will also be discontinuous - y values change in fixed increments - but for all but the rarest cancers, we can ignore that).

So, what shape is the curve, for real cancers? Assume a “simple” cancer, say colorectal, and Stage 4, for which complete remission is generally very rare (though perhaps not so rare for some cancer cancers?).

Highly unlikely it’s a straight line, or close to one. Somewhat like the RHS of a normal curve, or Weibell? Or more like an exponential? Or does it vary considerably between cancers?

 

[Camillus]

Are figures 1 and 2 from here the sort of thing that you mean?

This is a common presentation of survival curves for various sorts of cancer (and other conditions / interventions).

 

[Steve001]

A widely used cancer survival number ("statistic") is "5-Year Relative Survival". In medical papers one often sees "median survival".

Assuming the two are of the same thing - say, Stage 4 Colorectal cancer - is there a known distribution function which connects the two? One which is, perhaps, often used (but rarely mentioned)? Or is it known that this function varies a lot, by cancer type perhaps?

I tried searching the literature, but came up empty.

As I recall that "5 year" mark only means if one survives for that length after treatment the chances of recurrence drops and continues to do so beyond 5 years.

 

[JeanTate]

Here in the US, SEER publishes cancer statistics, in particular, "How Many People Survive 5 Years Or More after Being Diagnosed with [...]":

Relative survival statistics compare the survival of patients diagnosed with cancer with the survival of people in the general population who are the same age, race, and sex and who have not been diagnosed with cancer. Because survival statistics are based on large groups of people, they cannot be used to predict exactly what will happen to an individual patient. No two patients are entirely alike, and treatment and responses to treatment can vary greatly.

In contrast, the "median survival" numbers one reads in the literature (for example), do not "correct" for those in the general population who have not been diagnosed with cancer. This makes comparison difficult.

Another, obvious, caveat is that the SEER numbers are "backward looking" ... to be among those in the "5 years or more" group, you need to have been first diagnosed 5+ years earlier. So more recent developments - e.g. a new chemo regimen - are not reflected in those numbers.

One other thing I don't know: if you were first diagnosed with a Stage 3 cancer (say) and later it spread (become a Stage 4 one), do you appear twice (assuming you survive to "year 5" in both cases)?

 

[Giordano]

Just to supply a non-mathematical discussion behind the math:

Most survival curves have an initially decreasing line (a negative slope part) and a later plateau. Most important is that the shape of the survival curves depend on the nature of the cancer and its treatment. Depending on the cancer (and the staging) many treatments decrease the negative "slope" of the survival curve, often substantially (meaning a lot of patients survive longer than if they were not treated) and also produce a "plateau" later, representing the people who have made it through the initially risky part and are truly cured and who will then be at no greater risk of that cancer than the general population (we all are in the middle of survival curves whether we had cancer or not). For many cancers if you make it 5 years without reoccurrence you have shown that you are likely a member of the "cured" group. You have made it to the plateau. But not for all cancers/treatments. Sometimes making it for 3 years is enough to show a cure, sometimes 7, or see below.

The slope part and the plateau part can be quite distinct. For multi-myeloma for example recent revolutions in treatment have dramatically flattened (improved) the slope such that many people who would have died 2 years after diagnosis now survive 5, or 10, or 15. But there is still no cure, so virtually no plateau. You may make it 15 years longer but you are never "out of the woods" remain on a slope and retain a virtual certainty that it will eventually return. But now you are 70 instead of 57, which is not a bad trade-off. Whereas other cancers can be "cured" but only a tiny percent of the patients achieve this (e.g. only 10% make it 5 years to the plateau but those who do have no greater risk of reoccurrence than the general population).


Personally I would rather play the odds and be diagnosed with a cancer that can be treated to give most patients 15 extra years even if it cannot be cured, than diagnosed with a cancer that can be treated to give 90% of the patients only 2 extra years and 10% a chance of permanent cure. Of course one's wishes change in the latter in retrospect if one finds oneself at 5 years and therefore in the 10% population.

 

[Giordano]

Although not typically recognized by the general public: improved treatments continue to appear for many cancers at a surprising rate, such that if an older treatment buys you some "extra" time, a newer treatment may appear during that time to buy you even more time. I presume the shape of the survival curves change as they reflect this. I'll just note that I am around to post here for this very reason.

These stepwise improvements in cancer treatment can add up to a significant change in survival and cure, but are not typically noted by the general public because they are cancer-type specific and not a magic overnight CURE!

 

[Giordano]

Here in the US, SEER publishes cancer statistics, in particular, "How Many People Survive 5 Years Or More after Being Diagnosed with [...]":



In contrast, the "median survival" numbers one reads in the literature (for example), do not "correct" for those in the general population who have not been diagnosed with cancer. This makes comparison difficult.

Another, obvious, caveat is that the SEER numbers are "backward looking" ... to be among those in the "5 years or more" group, you need to have been first diagnosed 5+ years earlier. So more recent developments - e.g. a new chemo regimen - are not reflected in those numbers.

One other thing I don't know: if you were first diagnosed with a Stage 3 cancer (say) and later it spread (become a Stage 4 one), do you appear twice (assuming you survive to "year 5" in both cases)?

Not twice. The survival curves usually start with the initial diagnosis and continue as a single line for each individual (merged with multiple individuals in the overall survival curve). Progression to stage 5 at year 3 but the disease begins to disappear or the person otherwise survives to year 5 and after: that person stays on a 100% survival plateau. Progression to stage 5 at year 3 and death at year 4: that person is on a line that stayed at 100% until year 4, then plunged to 0% survival at year 4. Each individual's curve contributes its part to the overall survival curve.

There are survival curves, detectable-cancer free-curves, non-progression curves, etc. All different

 

[JeanTate]

You might be interested in the Kaplan-Meier estimator, which is commonly used in medicine to estimate how survival changes over time following treatment.

Yes, this is interesting, thank you.

 

[JeanTate]

Are figures 1 and 2 from here the sort of thing that you mean?

This is a common presentation of survival curves for various sorts of cancer (and other conditions / interventions).

Yes they are. And they are just the sorts of charts/graphs/plots/figures one reads in the literature.

What I'm curious about is whether there's anything on what functional form(s) the curves take (other than monotonically decreasing), and whether these vary from one cancer (and stage) to another. In a later post, Giordano noted that for at least some (cancers/stages) there's an initial, more rapid decline (steeper slope) followed by a gentler decline (almost a plateau).

 

[JeanTate]

Although not typically recognized by the general public: improved treatments continue to appear for many cancers at a surprising rate, such that if an older treatment buys you some "extra" time, a newer treatment may appear during that time to buy you even more time. I presume the shape of the survival curves change as they reflect this. I'll just note that I am around to post here for this very reason.

These stepwise improvements in cancer treatment can add up to a significant change in survival and cure, but are not typically noted by the general public because they are cancer-type specific and not a magic overnight CURE!

Indeed.

This is why, among other things (e.g. better detection, earlier treatment), the SEER data (for example) are almost uniformly "pessimistic" ... a person today who is at t=0 (i.e. just diagnosed) will more likely survive than one who was diagnosed five years earlier. Today the five-year survival rate may be 40% (for cancer A at stage X); in five years' time it may well be 50%.

 

[Giordano]

Indeed.

This is why, among other things (e.g. better detection, earlier treatment), the SEER data (for example) are almost uniformly "pessimistic" ... a person today who is at t=0 (i.e. just diagnosed) will more likely survive than one who was diagnosed five years earlier. Today the five-year survival rate may be 40% (for cancer A at stage X); in five years' time it may well be 50%.

For me, my hematological malignancy has become basically a chronic disease, with my neoplasia having been driven down by some of my treatments into undetectable levels for a year period or more, but then stirring back up to a detectable level, only to be driven back down by the ability of my physician to pull out a yet-more clever treatment from his tool bag. "Ah, another reoccurrence? Well, not great news but not unexpected and we have something for that! In fact this next treatment is pretty new!"

A bit disturbing that he and I are using up tools as we go along (and they in general become somewhat less effective), but nice that new tools have continued to arrive and partly make up for the ones we have depleted from his bag of tricks. I have to accept that at some point we will likely end up with no tools left that will work against my particular form of this cancer. At that point if I continue to post here it will be actual proof of life after death rather than simple participation in a political thread. But at least my disease has become a drawn-out holding action, rather than the rapid death I would have experienced 15 years ago. I am incredibly grateful to all the researchers and my oncologist for that! Maybe at some point a new treatment will appear that will represent a cure, but I can't allow myself to even begin consider that while maintaining an emotionally healthy acceptance of my situation.

 

[JeanTate]

Thanks again everyone who has posted here, with various inputs.

Having dug around a bit more, I now realize that my initial approach is far too naive. There may be a functional form to describe the survival curves, but cancer is just so complex a disease/disorder that their importance or meaning is likely ~zero (beyond "fitting function" convenience).