In many cases a partial catheter can be left where it lies and nothing much happens. However, it won't be absorbed or dissolve, and it can migrate, although if left in muscle or fatty tissue many times it will stay put. However, if it's causing discomfort, it needs to be located and removed. This really shouldn't be that difficult. Apparently this isn't the first time your doctor has had this experience, but if you're having any symptoms which may be related to the remainder of the catheter, you deserved to have the problem corrected.
I hope this is helpful. Good luck to you.
John Kenyon, EMT, CCT
Non-invasive cardiology tech, Emergency and Critical Care technician, Critical Incident Stress Mgmt. specialist