A method is described for repairing tissue of a selected organ from among heart, brain, liver, pancreas, kidney, and glands in a patient's body. In the method, stem cells that have the capability to repair tissue of the selected organ are intraluminally applied through a designated natural body vessel or duct leading to a predetermined target site of the tissue of the selected organ to be repaired. The stem cells are delivered into the respective vessel or duct through a catheter having a proximal portion of relatively larger central lumen diameter and outer diameter, and a distal portion of relatively smaller central lumen diameter and outer diameter, the two portions being integral with one another so that stem cells delivered into the central lumen of the proximal portion will flow through and exit the central lumen of the distal portion, the lumen diameter and length of each of the two portions being selected to minimize the pressure drop across the catheter during flow of the stem cells.