Scientists have already grown an artificial ear on a mouse. However, we will develop the protocols to grow complex organs like a kidney, in a laboratory chamber, totally without the mouse.
One of the benefits is that the starter tissue could come from the person needing the organ. Thus once the organ was grown, there would be no chance of rejection (except when the person dates).
Judging by the spam that I get, I'm willing to bet that the first artificial organ grown for transplant, will be the male sex organ. Some men with esteem issues will pay anything for a large one.