Start with an apology for ignoring them.
Angela is correct - an apology is a good start (altho possible the othe rparty should be also apoligizing). I just reconnedted with a friend/co-worker I hadn;t talked to in 30 years. We connected just as tho there had been no break at all. It was worth the effort and the aploogy.
I 3rd Angela, If you are reknotting old ties I would start with an apology and then finish with something that is not demanding, expecting, or forthcoming. "Sorry I haven't been in contact, if you ever want write me feel free to send me a reply."
All you need to construct this is a simple "long time no talk, how are you?" - It Really depends on the type of relation you had. But for the most part, there is no trick of the trade. No simple way but to do it.
Here is something not to do: try not to go through anyone els in order too contact them. Especially if they have a grudge or maybe you're afraid they will take it awkwardly or creepy.
If it is important to you, and you have no other way, disregard the last statement. Life is short and that won't matter in the long run. If you really want to be in their life do what you have to.