A free boundary problem consists of a system of partial differential equations (PDEs) which are to be solved in a domain whose boundary is unknown and has to be determined together with a solution of the PDEs.

Biomedical examples of free boundary problems include the growth of a solid tumor, the healing of a wound, abdominal aorta aneurysm, and granulomas in tuberculosis and sarcoidosis. In this talk I will focus on atherosclerosis, a disease where a plaque develops in the artery, causing potentially a heart attack or a stroke. The risk factor for the disease depends on the cholesterol levels of LDL and HDL (the "good" and "bad" cholesterols). I will present a mathematical model, by a system of PDEs and present a "risk map" which predicts whether a small plaque will grow or shrink depending on the levels of LDL and HDL in the blood. I will then describe rigorous mathematical results for a simplified model, and open problems. Finally, I will briefly mention other free boundary problems associated with biomedical processes.