One method that I have had lots of success with, and others have too, is Circular Shortest Path (CSP). Here is one paper, but there are quite a few more and possibly some code floating around the web. Another technique that are perhaps more popular, and for which code might be more common, are snakes (active contours).
These techniques work by finding a connected path that minimises a certain function. In your case, it would be to find the path that goes through many edge pixels while still remaining somewhat round. The roundness constraint is what keeps the path from going off down the narrow gaps. Snakes are different from CSP in that the boundary is 'grown' out from a seed area; CSP performs a polar transformation about the centre point.
Another possibility is some kind of morphological operator. These can be used to find regions of a particular size. One would convert the image to binary, the apply a morphological opening using a structuring element of a big enough size to fit inside the area of interest, but not anywhere else.