  As a future trend of healthcare, personalized medicine tailors medical
treatments to individual patients. It requires to identify a subset of patients
with the best response to treatment. The subset can be defined by a biomarker
(e.g. expression of a gene) and its cutoff value. Topics on subset
identification have received massive attention. There are over 2 million hits
by keyword searches on Google Scholar. However, how to properly incorporate the
identified subsets/biomarkers to design clinical trials is not trivial and
rarely discussed in the literature, which leads to a gap between research
results and real-world drug development.
  To fill in this gap, we formulate the problem of clinical trial design into
an optimization problem involving high-dimensional integration, and propose a
novel computational solution based on Monte-Carlo and smoothing methods. Our
method utilizes the modern techniques of General-Purpose computing on Graphics
Processing Units for large-scale parallel computing. Compared to the standard
method in three-dimensional problems, our approach is more accurate and 133
times faster. This advantage increases when dimensionality increases. Our
method is scalable to higher-dimensional problems since the precision bound is
a finite number not affected by dimensionality.
  Our software will be available on GitHub and CRAN, which can be applied to
guide the design of clinical trials to incorporate the biomarker better.
Although our research is motivated by the design of clinical trials, the method
can be used widely to solve other optimization problems involving
high-dimensional integration.
