Evolution of resistance during clonal expansion

Abstract

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 alpha 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 M(1-1/alpha) for advantageous mutants with relative fitness alpha>1, to l nM for neutral mutants (alpha = 1), but converges to an upper limit for deleterious mutants (alpha<1). 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.

Figure 1.

The probability of resistance. We show the probability of resistance in dependence of (a) the detection size times the mutation rate, Mu, (b) the growth rate of sensitive cancer cells, r, and (c) the fitness of resistant cancer cells, α. Equations 7 and 8 are shown as lines and direct computer simulations as circles. Parameter values are , , and (a) , (line 1), and (line 2); (b) , (line 1), (line 2), and (line 3); and (c) , (line 1), (line 2), and (line 3). The results of the stochastic computer simulation are averaged over 100,000 runs.

Figure 2.

The average number of resistant cells. We show the expected average number of resistant cells in dependence of (a) the detection size, M, (b) the growth rate of sensitive cancer cells, r, and (c) the fitness of resistant cancer cells, α. Equations 11–13 are shown as lines and direct computer simulations as circles. Parameter values are and (a) , (line 1), and (line 2); (b) , , (line 1), (line 2), and (line 3); and (c) , , (line 1), and (line 2). The results of the stochastic computer simulation are averaged over 100,000 runs.

Figure 3.

The distribution of resistant cells. We show the probability distribution of resistant cells for different fitness values of resistant cells, α. Equations 15b and 16 are shown as red lines and direct computer simulations as circles. Parameter values are , , , and and (a) , (b) , (c) , (d) , and (e) . The results of the stochastic computer simulation are averaged over runs.