Download age (years) cancer deaths/year

Biological physics & disease: A simple model for cancer mortality
1
Introduction
So far, this course has been mostly on understanding how living organisms work. Living organisms are amazing
things, and as we one of them, it is natural to wonder how our bodies work. However, much of the research
in biology is biomedical, and is less aimed at understanding basic biology than understanding disease, and
how best to treat it. In this part of the course, we will look a simple model of one of the two big killers in the
Western world: Cancer. The other big killer is heart diseases and other circulation diseases, such as strokes.
Most biomedical research is at least partly motivated by diseases, and these diseases are why it is relatively
generously (compared to most of physics) funded. This funding means that many jobs for scientists are in
biomedical research into disease.
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Mortality data for cancer
cancer deaths/year
Data for total cancer mortality (i.e., deaths) in the UK, is plotted in Fig. 1. The basic observation is that
overall, cancer is very rare in children and young adults, but unfortunately the mortality rate rapidly increases
with age. Also, the rate is somewhat higher in men than in women.
0.040
male
0.035
female
0.030
0.025
0.020
0.015
0.010
0.005
0.0000 10 20 30 40 50 60 70 80 90
age (years)
Figure 1:
Cancer mortality in the UK. The plot is from data from Cancer Research UK,
http://www.cancerresearchuk.org/health-professional/cancer-statistics/mortality/age,
Accessed October 2016. Data is from the Office for National Statistics. The two curves show male and female
mortality (death) rates per year, as a function of age.
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Figure 2: Lung cancer mortality in: people who smoke and continue to smoke; people who smoke but then
stop at the ages indicated; and non-smokers — people who never smoke. Plot by J Peto, British Cancer
Journal 104 (2011), p 389. Data from Halpern et al. (1993). Note that the rate in non-smokers is very
low, much less than 1 in 1000 even at 70 years of age (i.e., hC (t = 70 years) << 10−3 ), while the rate of
those who continue to smoke reaches hc (t = 70 years) ≈ 7 × 10−3 . The intermediate lines are for those who
smoke until they reach the specified range of years, 40–49, etc. Note that almost all smokers start smoking
at approximately the same time, as teenagers.
Brief Introduction to Cancer Our bodies are made of many trillions of cells. Cell growth and division,
i.e., one cell growing and then dividing to produce two daughter cells, is essential. It is required to make
our bodies of trillions of cells from one cell, the fertilised egg. The fertilised egg cell divides into two
daughter cells, which then divide again to form four cells, etc. Cell growth and division is also required
in adults as the cells in many of our tissues constantly turn over, i.e., cells die and are replaced. For
example, in our colons (bowels) the environment is tough on cells, so the cells facing inwards only live for
around 5 days before dying and so need to be constantly replaced.
So cell division is essential to our bodies developing and functioning. Normally it is very tightly
controlled, by complex mechanisms involving many genes. Cells only divide to produce more cells when
needed, e.g., to replace dead cells in the colon. However, if the complex mechanisms that regulate growth
and division are damaged, a set of cells may undergo uncontrolled cell growth. If this growth causes the
cells to invade and damage tissues, this is called cancer. Cancer is currently responsible for about 30% of
deaths in the UK. It and circulatory diseases (heart attacks, strokes etc) are the two big killers.
2.1
Smoking and lung cancer: Why if you smoke, you have already harmed yourself and
so should quit today to avoid further damage
Data for the lung-cancer mortality of smokers, non-smokers and those who smoke for a period, then quit, are
all shown in Fig. 2. Smoking greatly increases the probability of getting lung cancer, note that the death rate
for non-smokers (bottom curve) is very low, much lower than that of smokers who never quit (top curve).
But note the fact that if you smoke for say 35 years and give up, then after you have given up, you are
still much more likely to get lung cancer than if you never smoked. Almost all smokers start in their teens so
the 50–54 curve is for those who smoked for around 35 years.
Smoking permanently and irreversibly damages cells and so increases the cancer risk permanently. But
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this increased risk is smaller the sooner the smoker stops, so it is worthwhile stopping smoking as early as
possible.
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Cancer typically develops as a result of several mutations, not just a
single mutation
There is a large amount of evidence pointing to the fact that many cancers are the result of not one but
several mutations in the genes that control cell growth and division, i.e., that cancer is a multistep not a
single step process. A key piece of evidence is the fact that most cancers are very rare in young people but
the rate increases rapidly in old age. I have plotted cancer mortality statistics in the UK in Fig. 1.
3.1
Multistep model of cancer mortality
To understand Figs. 1 and 2, we need a model. It will inevitably be over simplified, but should help us to
understand the data. First we need to define a mortality rate as a function of time, hC (t), which is what is
plotted in Figs. 1 and 2. The mortality rate hC (t) is defined as the rate at which people who have survived
to age t, then die from a particular cancer at age t.
3.1.1
Model definition
Our model is that death from cancer is due to a multistep process, with K random steps (step = gene
mutation), i.e., if at time t, K or more random steps have occurred so, then so has death by cancer. Each
step is assumed independent and to occur at the same constant rate r. So after a time t, the average number
of steps that have occurred is hni = rt.
As we have assumed that each step occurs at the same rate and is independent, the probability that
exactly n steps have occurred at a time t, is just given by the Poisson distribution, with the mean number of
steps that have occurred at time t being rt, i.e.,
p(n, hni) =
hni
(rt)n
exp(−hni) =
exp(−rt)
n!
n!
Then the probability of having died of cancer, PC , in this model, is simply the probability that K or more
steps have occurred
∞
∞
X
X
(rt)n
PC (t) =
p(n, hni) =
exp(−rt)
n!
n=K
n=K
This is just the same sum we studied in the context of night vision.
To keep the maths simple, we assume that cancer mortality for a specific cancer (we will consider colon
cancer below) is relatively small, i.e., PC (t) 1, because the mean number of mutations is small, i.e., rt 1.
If we write out the terms in the sum explicitly
PC (t) =
(rt)K
(rt)K+1
exp(−rt) +
exp(−rt) + · · ·
K!
(K + 1)!
Because rt 1, then exp(−rt) ' 1. Also (rt)K+1 (rt)K , so the first term is much larger than subsequent
terms. This means we can keep only the first term, and PC (t) simplifies to
PC (t) '
3
(rt)K
K!
As PC (t) is the probability of death by age t of a cancer, the mortality rate hC (t) is just its time derivative
hC (t) =
dPC (t)
1
'
rK tK−1
dt
(K − 1)!
rt << 1
i.e., a power law, with exponent K − 1 for a K-step process.
3.1.2
Comparison of model with cancer data
Now, if the process was one step, K = 1, and
hC (t) ' r
K=1
i.e., a constant. For a one-step model of cancer mortality, the death rate is approximately a constant. As
we can see in Figs. 1, 2 and 3, the rate is absolutely not a constant, it starts off very low then in old age,
increases rapidly. Typically, cancer is very rare in children and young adults, is more common in the middle
age, and becomes rapidly more likely in old age. So, the one step model is not consistent with what we know
of the most common cancers.
In Fig. 3 we show data for the mortality from colon cancer (cancer of the bowel) plus a fit of a power law
to that data. The best fit is obtained with a power-law exponent ν = 5.0, which corresponds to a K = 6 step
process. Note that the fit is quite good. Our simple model, with 6 steps, reproduces the data on colon-cancer
quite well. Both the data and the fit give very low mortality rates in young ages before rapidly increasing
with age.
3.1.3
Conclusion
Clearly our model is grossly simplified but it gives a function that fits the data for colon cancer reasonably
well. A multistep model is also consistent with ex-smokers having higher rates of lung cancer than those who
have never smoked. While they smoked, some mutations occured, so meaning that fewer additional mutations
are needed for lung cancer to develop. There is also independent genetic evidence that most cancers do have
multiple mutations not one mutation.
0.0020
fit ν =5.0
colon cancer data (males, NY)
hC (t)
0.0015
0.0010
0.0005
0.00000
20
40
60
age t (years)
80
100
Figure 3: Data for mortality from colon cancer of males (green circles) plus a power law fit (red curve). The
data is for New York state (years 2009 to 2013), and the power law fit has a best-fit exponent ν = 5.0. Within
our simple model this implies that K = 6, i.e., that 6 mutations need to occur for colon cancer to occur.
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