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I would say that organs do not represent the "personhood" of an individual. The only possible effect that occurs to me that the transplanting of an organ from the body of a Christian to the body of a non-Christian would have with respect to the eternal destiny of either person would be a testimony to the unselfishness and love for others that possibly motivated the donor's action. The organs donated would not "return" to the donor in the event of the donor being raptured. Just as God will be fully able to reconstitute every human who has ever lived in the resurrection at the close of the age, He will also be fully able to deal with the physical condition of those who are raptured -- whether that would involve giving them completely new, glorified bodies; or replacing donated organs; or some other course of action that we might not even be capable of imagining. In any event, I would emphatically say that this is not a question that should be a source of worry or concern to Christians, nor should it deter them from helping another person through organ donation if the need/possibility arises, whether the organ recipient is a Christian or not.
According to Jesus Christ we are created in the image of God. Therefore we are a soul, or spirit if you will, with a body here on earth. When we die or are caught up into the air with Jesus, we will be changed in an instant. We will then be in the same form as Jesus and our Heavenly Father, which is spirit. We will not have need of a body. We will be in their presence in the image in which He created us.
Is anything impossible for our Lord God to do? He made all that you can see that man did not make yet even then he made the matter that man uses to create things. He made all that you can not see with your mortal eyes. Whether your body is dust or parts spread all over a vast ocean, it will be resurrected. If you are redeemed, you can not be forsaken or left behind, even from the deepest cave or mine in the earth. Even if God chose not to rapture you at the time of his church being taken up into heaven, it would be to fulfill a heavenly calling during the tribulation if you are truly redeemed. If you miss the rapture, even then it is not too late for salvation, but you will suffer during the tribulation in your mortal body; however, your salvation is assured and your eternal life is fixed and sure if you're redeemed by the blood of the lamb of God. Jesus Christ, the only begotten son of God, has paid and paved the way to heaven. He can and will restore all things unto himself, even planet earth after the tribulation. No worries! Our God can come the storm and sea with a word. Why would you think that he can not do the impossible or do above what you could think or ask if it be his will and if faith is present. Impossible is not even in our God's vocabulary. We alone limit God with only two things our free will and a lack of faith. Selah!
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