The early church uniformly used leaven bread. This was because the liturgy was viewed as the victory celebration of Christ over all of His enemies, culminating in the faithful eating and drinking the new manna in the new kingdom. It is the partaking of the promise now, a foretaste of the end of the final covenant. In this regard leaven is necessary. The unleavened bread of the first Passover, which is the type or shadow of Christ who is our Passover (as St Paul teaches). The unleavened bread is the bread of bitterness, of haste. It is the not the bread of victory and joy.
Christ is also our thank offering, and our praise offering, our sin offering, everything. He is every sacrifice made in the old covenant, or really they foretell Him. To limit it to the paschal sacrifice hides this.
In the liturgy, the traditional understanding is that Christians offer themselves as living sacrifices to God, and in this offering they participate in the once for all sacrifice of Christ. They are the body of Christ and they complete what is lacking in Christ's suffering by becoming the body. And His body is sacrificed, so they as His body are sacrificed. This is the "logike latreia," the worship which is proper to rational beings, the worship of humans. We sacrifice ourselves, and we are the leaven of the world, so the bread is us as we are Him.
Later - much later - in the west and even after that somewhat in the east a view came about that was less of a common act and more of a dramatic or symbological interpretation. In this the service is a kind of re-enactment of the passion, death, and resurrection of Christ to call the laity to remembrance. In the medieval times it was even thought in some circles in the West that the laity didn't even need to take communion - that they would observe and the priests would partake in their behalf. Of course the remembrance is part of it in the earlier view as well, but it is not the main goal. In this environment the act of the Eucharist became a re-enactment if the last supper, and therefore the bread was thought to be best shown as unleavened. And under this the sacrifice became in some circles to be thought as a re-sacrifice of Christ, and not an ongoing union with His once for all sacrifice with daily sacrifice of the faithful.
All of this was part of the theological debates surrounding the use of unleavened bread.
