From: Anders Sandberg (asa@nada.kth.se)
Date: Sat Aug 24 2002 - 13:39:30 MDT
On Sat, Aug 24, 2002 at 10:12:43AM -0700, Reason wrote:
> This seems pretty counter-intuitive. Anyone care to advance some armchair
> theories?
>
> http://www.sciencedaily.com/releases/2002/08/020823063221.htm
I think I mentioned this effect a few years ago on this list, in a study
done on rats. A low dose of radiation makes the incidence of leukemia go
down.
The explanation proposed is that radiation selectively damages cells that
are rapidly dividing, and the stress induced by the free radicals is more
than can be handled by damaged cells while healthy cells can deal with
it.
Here is a BOTE model: let H be the number of healthy cells, D the number
of slightly damaged cells and C the number of cancerous cells. A healthy
cell can spontaneously with rate P1 turn into a damaged cell, and that
cell can turn cancerous with rate P2. Healthy cells divide so that the
total number of cells remain N. Cancer cells divide at the rate R.
Healthy cells die at the rate P3, damaged cells at the rate P4 and
cancer cells at the rate P5. We end up with the following differential
equations:
H' = (N - H - D -C) -(P1+P3)*H
D' = P1*H - P4*D
C' = P2*D - P5*C + R*C
P1, P2 << 1, P3 < P4
In this model if P1 is 0, H will stabilize at N/(1+P3). If P3 is small,
there will be a "leakage" of cells to D and then to C, which will
eventually grow exponentially and overwhelm the system.
Now imagine the effect of radiation as temporarily increasing P1, P2,
P3, P4 and P5. More cells are damaged, but they die more often. It turns
out that the unhealthy stage is a kind of bottleneck that allows the body
to get rid of the cancer cells. If you have more stages (which is common
in cancer) the effect ought to be larger.
I did a matlab script (included below; is it BOTA when you start doing
Matlab runs? :-) that plots the "time of death" when C becomes larger
than H for different levels of radiation. It shows a nice (1/r+r^2) curve
of time of death, with a minimum for radiation=15. This is obviously the
wrong behavior (the curve shoudd be upside down :-) but at least shows a
nice nonlinearity and how radiation can suppress cancer. OK, I have to
fix the model a bit, but that is for another time...
N=1000;
tmax=200;
h=0.1;
dd=[];
for ra=1:40;
H=zeros(tmax,1);
D=zeros(tmax,1);
C=zeros(tmax,1);
H(1)=100;
D(1)=0;
C(1)=0;
P1=1e-4*ra;
P2=1e-4*ra;
P3=1e-4*ra;
P4=1e-2*ra;
P5=1e-2*ra;
R=1;
dtime=0;
for i=1:(tmax-1)
H(i+1) = H(i)+h*((N - H(i) - D(i) -C(i)) -(P1+P3)*H(i));
D(i+1) = D(i)+h*(P1*H(i) - P4*D(i));
C(i+1) = C(i)+h*(P2*D(i) - P5*C(i) + R*C(i));
if (C(i+1)<H(i+1)) dtime=i+1; end
end
%clf
%plot(1:tmax,D,'g');
%hold on
%plot(1:tmax,H,'b');
%plot(1:tmax,C,'r');
%pause
dd=[dd dtime];
end
plot(dd)
-- ----------------------------------------------------------------------- Anders Sandberg Towards Ascension! asa@nada.kth.se http://www.nada.kth.se/~asa/ GCS/M/S/O d++ -p+ c++++ !l u+ e++ m++ s+/+ n--- h+/* f+ g+ w++ t+ r+ !y
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