As people age, the main valve controlling the flow of blood out of the heart can narrow, causing heart failure, and sometimes death. In the past the only way to repair the damage was risky open-heart surgery. But an ingenious medical device now allows the heart to be repaired using a catheter that introduces a replacement valve through a main artery in the leg—another miracle of modern medicine.

In 2011, more than four years after they hit the European market, the Food and Drug Administration finally approved aortic heart valves...