During embryonic development, the foramen ovale is a normal opening between the two atria. Its function is to divert the blood that comes into the right side of the heart directly into the left side, thus preventing it from entering the still immature and non-functioning lungs. After birth, the foramen ovale closes, channeling blood through the now mature lungs, where it can be loaded with oxygen. However, the foramen ovale stays open in approximately 25% of all people, creating an abnormal passage of blood between the two atria.
People who have a PFO usually don't realize it, as it typically does not cause any detectable symptoms. However, PFO has been associated with migraines with aura, especially in young people.
For instance, a small but provocative study published in 2005 showed that 62% of people who suffer from migraines with aura have a PFO. Nonetheless, subsequent studies failed to show an improvement of migraine headaches after surgical closure of the PFO, suggesting that the association between PFO and migraine is either incidental, or a lot more complicated than we can fathom at this time.
The PFO-migraine link continues to be a subject of intense investigation, and therefore, a final consensus has yet to be achieved.
When people develop a blood clot in one or more of their leg veins (a DVT), a PFO places them at risk of stroke caused by a paradoxical embolism.
Sources:
Dowson A, Mullen MJ, Peatfield R, et al. Migraine Intervention With STARFlex Technology (MIST) trial. A prospective, multicenter, double-blind, sham-controlled trial to evaluate the effectiveness of patent foramen ovale closure with STARFlex septal repair implant to resolve refractory migraine headache. Circulation 2008; 117:1397-1404.
Giorgio Dalla Volta, Massimiliano Guindani, Paola Zavarise, Simona Griffini, Alessandro Pezzini, Alessandro Padovani; Prevalence of patent foramen ovale in a large series of patients with migraine with aura, migraine without aura and cluster headache, and relationship with clinical phenotype J Headache Pain (2005) 6:328330
Meissner I, Whisnant JP, Khandheria BK, et al. Prevalence of potential risk factors for stroke assessed by transesophageal echocardiography and carotid ultrasonography: the SPARC study. Stroke prevention: assessment of risk in a community. Mayo Clin Proc. 1999;74:862869.

