There have been numerous clinical trials between 2009 and 2014, which focused on locating fetal DNA in maternal blood. The main goal for these trials was to develop an accurate, non-invasive, screening test that would permit doctors to identify chromosomal abnormalities through maternal blood samples. This was a success in mid-2014, and is just starting to be introduced around the world with companies, such as Verifi™ and Panorama™.
On the other hand, there is hardly any research being done on the cure for Patau syndrome. This is because it is a very rare disease with many—seemingly— irreversible symptoms. Also, biomedical science is not nearly advanced enough to deal with removing chromosomes from cells. At the moment, the best that could be done to help mitigate the symptoms thereof is surgery—for the severe physical abnormalities—and occupational therapy.