Peto's Paradox

The larger an animal is the more cells it has and the longer lived an animal has the more time cells have to mutate so one would think that larger and longer lived animals would have more cancer; however, that is not the case.   Science World Report quotes researchers as saying that "less than 5 percent of elephants develop cancer compared to 25 percent of humans". That larger animals do not have higher rates of cancer is called Peto's paradox. [Wikipedia] [PMID: 21296451] [Full text] While it may be that larger animals do not get more cancer larger, or at least taller, humans have higher rates of cancer than shorter humans so this observation does seem to hold on the scale of an individual species. Evidently methods of cancer prevention have evolved in larger animals; however, taller humans may not have evolved over a sufficiently long period to elicit an evolutionary response.

We can get a bit more insight by reviewing the simple probability model of cancer developed by Calabrese and Shibata. [PMID: 20051132] [Full text] [Excel model] [Powerpoint slides] [Also see Box 2 of this paper] Assume that mutations must occur in k critical genes for cancer to occur. Assume each gene divides into d copies in its cell lineage and that there are M cells at risk in a particular organ such as the prostate -- all cells in an organ are not necessarily at risk so M is, in general, less than the number of cells in an organ.  Let u be the prob of a mutation in one critical gene. Then as shown by Calabrese and Shibata (also see proof at end of this post) the probability of cancer, p, is:

       p = 1 - (1 - (1 - (1 - u)^d)^k)^M

Larger animals have more cells, M, and longer lived animals undergo more cell divisions d.  The number of mutations needed for cancer, k, could vary among animals and the even by the type of cancer within a species.  Thus a number of other factors may be at play as well as size and life expectancy.

In the case of elephants two independent papers have concluded that they have multiple copies of the p53 tumor suppressor gene (which kills cancer cells) whereas humans only have a single set.  This had been previously hypothesized; however, these two 2015 papers seem to have independently determined this to be the case.

Abegglen et al
http://jama.jamanetwork.com/article.aspx?articleid=2456041

Sulak et al
http://biorxiv.org/content/early/2015/10/06/028522

An interview with Joshua Schiffman, an author of the first paper, appears here:
http://medicalxpress.com/news/2015-10-elephants-rarely-cancer-potential-mechanism.html

In a New York Times article
http://www.nytimes.com/2015/10/13/science/why-elephants-get-less-cancer.html
Dr. Patricia Muller mentions that this research does not establish the mechanism of action so further work needs to be done even before attempting to replicate this in humans.  Furthermore, the following paper: http://www.impactaging.com/papers/v2/n7/full/100178.html indicates that in
some contexts p53 accelerates aging in mice whereas in other contexts it promotes longevity so there is some question as to whether it would be feasible to apply p53 at all to humans.

Other large or long lived animals with low cancer rates might have evolved different mechanisms of combating cancer.  Discovering the various mechanisms that evolved over millions of years might lead to cancer treatments in humans.  Schiffman is quoted in the same New York Times article as speculating "that parrots, tortoises and whales may all have special longevity tactics of their
own".  http://www.nytimes.com/2015/10/13/science/why-elephants-get-less-cancer.html

Note:
The formula given above can be derived as follows:

p = 1 - Prob of no cancer in any of the M cells at risk)
       = 1 - (Prob of no cancer in one cell at risk)^M
       = 1 - (1 - Prob of cancer in one cell at risk)^M
       = 1 = (1 - Prob of one cell accumulating k mutations)^M
       = 1 - (1 - (Prob of one mutation)^k)^M
       = 1 - (1 - (Prob of mutation in 1 critical gene)^k)^M
       = 1 - (1 - (1 - Prob of no mutation in one cell lineage)^k))^M
       = 1 - (1 - (1 - (1 - Prob of no mutation in one of d divisions)^d)^k)^M
       = 1 - (1 - (1 - (1 - u)^d)^k)^M