  Since penicillin was discovered about 90 years ago, we have become used to
using drugs to eradicate unwanted pathogenic cells. However, using drugs to
kill bacteria, viruses or cancer cells has the serious side effect of selecting
for mutant types that survive the drug attack. A key question therefore is how
one could eradicate as many cells as possible for a given acceptable risk of
drug resistance evolution. We address this general question in a model of drug
resistance evolution in spatial drug gradients, which recent experiments and
theories have suggested as key drivers of drug resistance. Importantly, our
model takes into account the influence of convection, resulting for instance
from blood flow. Using stochastic simulations, we study the fates of individual
resistance mutations and quantify the trade-off between the killing of
wild-type cells and the rise of resistance mutations: shallow gradients and
convection into the antibiotic region promote wild-type death, at the cost of
increasing the establishment probability of resistance mutations. We can
explain these observed trends by modeling the adaptation process as a branching
random walk. Our analysis reveals that the trade-off between death and
adaptation depends on the relative length scales of the spatial drug gradient
and random dispersal, and the strength of convection. Our results show that
convection can have a momentous effect on the rate of establishment of new
mutations, and may heavily impact the efficiency of antibiotic treatment.
