Timing and heterogeneity of mutations associated with drug resistance in metastatic cancers

Abstract

Targeted therapies provide an exciting new approach to combat human cancer. The immediate effect is a dramatic reduction in disease burden, but in most cases, the tumor returns as a consequence of resistance. Various mechanisms for the evolution of resistance have been implicated, including mutation of target genes and activation of other drivers. There is increasing evidence that the reason for failure of many targeted treatments is a small preexisting subpopulation of resistant cells; however, little is known about the genetic composition of this resistant subpopulation. Using the novel approach of ordering the resistant subclones according to their time of appearance, here we describe the full spectrum of resistance mutations present in a metastatic lesion. We calculate the expected and median number of cells in each resistant subclone. Surprisingly, the ratio of the medians of successive resistant clones is independent of any parameter in our model; for example, the median of the second clone divided by the median of the first is √2-1. We find that most radiographically detectable lesions harbor at least 10 resistant subclones. Our predictions are in agreement with clinical data on the relative sizes of resistant subclones obtained from liquid biopsies of colorectal cancer patients treated with epidermal growth factor receptor (EGFR) blockade. Our theory quantifies the genetic heterogeneity of resistance that exists before treatment and provides information to design treatment strategies that aim to control resistance.

Keywords: cancer; drug resistance; heterogeneity; mathematical biology.

Fig. 1.

Evolution of resistance in a metastatic lesion. (A) As the lesion (green) grows from one cell to detectable size, new resistant subclones appear. Some of them are lost to stochastic drift (yellow and pink), while others survive (purple, red and orange triangle). Instead of looking at the time of appearance of new clones, our approach takes into account the total size of the lesion when the resistance mutation first occurred. (B) Agreement between computer simulations and formula (1) for the cumulative distribution function for the number of cells in the first four resistant clones. The first subclone contains 10 or fewer cells with probability 0.06, between 10 and 100 cells with probability 0.34, between 100 and 1000 cells with probability 0.47 and more than 1000 cells with probability 0.13. The second subclone contains more than 100 cells with probability 0.36. Parameters $b=0.25$, $d=0.181$, $M={10}^9$, $u=42\cdot {10}^{-9}$.

Fig. 2.

Resistant subclones in metastatic lesions. Different realizations of the same stochastic process are shown in each panel. (A) Six lesions of size ${10}^8$ and (B) six lesions of size ${10}^9$ cells. The first ten resistant clones are shown, which survived until time of detection. They are ordered according to their time of appearance. Parameter values for all simulations: $b=0.25$, $d=0.181$, $u=42\cdot {10}^{-9}$.