The biomedical imaging problems are mostly less mathematical, but rather engineering oriented, still quite interesting and intriguing to think about. The scales are often easy for too large or too small, where people are neglecting the most important cases are in between.
Photoacoustic imaging has been popular in the past two decades; its nature is literally an inverse source problem, but the media are often coupled and cannot use a simple model to describe the ultrasound wave propagation. For many media, the properties are easier to quantify and tend to be homogeneous/isotropic. For some other media, like bones, the properties are rather more difficult to quantify with an elastic model only, probably due to their anisotropy and porous structures. Reconstruction under such assumptions is challenging.