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Appendicitis is inflammation of the appendix.
Why do you think there is no definite article in front of inflammation? Wouldn't it be better if there were one? For example, we say the handle of a knife because, even though we don't know what knife we are talking about, we do know that it has a particular handle that's part of the knife.
If you say Appendicitis is the inflammation of the appendix one might guess you talk of a definite inflammation of the appendix as in The inflammation of father's was incurable. In the sentence at hand "inflammation of the appendix" is used in a general sense and that is why it is used without the/an.
Because when it describes some 'swelling' of a human organ, it is a mass noun.
inflammation (mass noun): A localized physical condition in which part of the body becomes reddened, swollen, hot, and often painful, especially as a reaction to injury or infection.
So, Appendicitis is inflammation of the appendix.
NOTE: It seems there is no hard rule that it's mass noun always. I've also observed that this word is used as a countable noun taking indefinite article. For instance..
