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Genetics, Vol. 172, 2557-2566, April 2006, Copyright © 2006
doi:10.1534/genetics.105.049791
Evolution of Resistance During Clonal Expansion
Yoh Iwasa*,1,
Martin A. Nowak
and
Franziska Michor,
* Department of Biology, Faculty of Sciences, Kyushu University, Fukuoka 812-8581, Japan, Program for Evolutionary Dynamics, Department of Organismic and Evolutionary Biology, Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138 and Harvard Society of Fellows, Cambridge, Massachusetts 02138
1 Corresponding author: Department of Biology, Faculty of Sciences, Kyushu University, Fukuoka 812-8581, Japan.
E-mail: yiwasscb{at}mbox.nc.kyushu-u.ac.jp
Acquired drug resistance is a major limitation for cancer therapy. Often, one genetic alteration suffices to confer resistance to an otherwise successful therapy. However, little is known about the dynamics of the emergence of resistant tumor cells. In this article, we consider an exponentially growing population starting from one cancer cell that is sensitive to therapy. Sensitive cancer cells can mutate into resistant ones, which have relative fitness 
prior to therapy. In the special case of no cell death, our model converges to the one investigated by Luria and Delbrück. We calculate the probability of resistance and the mean number of resistant cells once the cancer has reached detection size M. The probability of resistance is an increasing function of the detection size M times the mutation rate u. If Mu << 1, then the expected number of resistant cells in cancers with resistance is independent of the mutation rate u and increases with M in proportion to 
for advantageous mutants with relative fitness 
, to 
for neutral mutants ( = 1), but converges to an upper limit for deleterious mutants (
). Further, the probability of resistance and the average number of resistant cells increase with the number of cell divisions in the history of the tumor. Hence a tumor subject to high rates of apoptosis will show a higher incidence of resistance than expected on its detection size only.
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