A system is configured to bring about anastomosis between two lumens in a patient or between two sections of a single lumen in a patient. The anastomosis system includes a first tissue-compressing element, a second tissue-compressing element, and an energy source. The energy source can be a thermal energy source or laser energy source. Tissue is interposed between the elements. Magnetic material incorporated into the tissue-compressing elements facilitates the alignment of the elements as well as compression of the interposed tissue. The energy source can deliver energy to tissue. This delivery of energy can cause local changes to the tissue that can help maintain positional stability of the implants, can bring about immediate patency of the anastomosis and can otherwise facilitate achieving desired outcomes for the patient.