In terms of the arthrogram - depends on what makes you nervous. As others have said, typically you are numbed first - then the dye is injected via a 'live x-ray' called a flouroscope. If you don't like needles, just turn away, don't look at the screen, etc. Typically more pinch, pressure sensation than pain. I would say vicodan would be too much for the type of discomfort. Tylenol might be better, but pain is very subjective.
If the MRI's make you nervous (i.e. the need for valium) make sure you take the valium, vicodan will not calm you down. If you do take any drug, I would recommend someone take you home. Yes people drive on this stuff all the time, but if you don't take it regularly then you don't fully know how your body and reflexes will respond. Also you are going to be stressed regardless and it is sometimes nice to have a 3rd party there to hear instructions, sympathize, get ice cream, etc.
Regarding after care - this CAN actually be a bit uncomfortable. When doing an injection of this type they are 'overfilling' a closed space (your joint). There are many 'pain' receptors in the joint that are basically pressure sensors, this can cause pain in some people. I will say, I have had an arthrogram on my shoulder and felt a decent achy pain afterwards that was helped by very gentle motion - for the hip think gentle walking, water walking or just letting the leg dangle off the edge of a step.
There are bursa sacs everywhere - they are an amazing little creation - until they are irritated!
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