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The best way is through an MRI with or without contrast -- preferably without. A myelogram can diagnose it, but myelograms can also cause or worsen it. Direct observation during surgery can also diagnose it. There are no other diagnostic tests.
Be aware that many medical professionals will deny or overlook its existence, especially if they may have caused it. Many just don't know what it is and will call it failed back surgery syndrome, sciatica, spinal stenosis, etc. Make sure your diagnostician is well-versed in Arachnoiditis, its signs and symptoms.