Hi Eternita,
Thanks for your question, I think it is a good one. But I cannot answer this question myself, I was hoping some of our scholars here would be able to respond to it.
Of course, it is possible that this does not fit the frame of reference for Byzantine theology. I don't know which Fathers, if any, have spoken or written on the subject.
Some thoughts: suffering due directly to the beliefs and lifestyle choices of a Christian is witnessing. Even though it may not be desired or sought after if it is suffering due ones faith it would be a form of martyrdom.
Askesis (or asceticism) is a deliberate re-orienting of ones life, a striving for holiness. It typically involves denying ones desires or restricting ones intake or mobility, a sort of regimen to restrain the passions like lust and greed. This could be a very difficult, even painful, lifelong experience for most people, but would also be the result of conscious/deliberate choice.
Having a natural affliction can make these practices even more difficult. Arthritis could make a difficult lifestyle even more difficult, and torture even more debilitating.
All I can think is that perhaps offering up the pain and suffering might fit in there somewhere. Pain and suffering may change ones thinking about things, inspire repentance. Giving one a natural heart in place of a heart of stone.
A rich and powerful person raised in comfortable circumstances might get polio for example, and learn empathy. This could have world-changing implications. Suffering like this might be considered a gift but cannot be a willful decision.
Jesus, of course, when encountering the sick and incapacitated who believed He could heal them would say "your sins are forgiven" before He would say "get up and walk". Were these poor ones offering up their pains and sufferings? Jesus could see into their hearts and knew if they had true faith. Was it because they suffered that they were forgiven or was it because they believed? Were they asking for forgiveness or were they asking for relief?
It may be possible that "offering it up" is not a concept that is understood from an eastern perspective.
So forgive me, I cannot answer this question. I only respond to it because I am hoping that someone more knowledgable than I would take up the question and enlighten us both.
Michael