The purpose of this medical research study is to evaluate the safety and effectiveness of a new medication called imatinib mesylate in the treatment of Lymphangioleiomyomatosis (LAM). LAM is a rare disease in which abnormal cells (called LAM cells) grow out of control. Over time, LAM cells destroy healthy lung tissue and cause respiratory disease or failure.

Many patients with LAM are currently treated with a medication called sirolimus (rapamycin). Sirolimus slows the growth of LAM cells.

Imatinib mesylate (hereafter called imatinib) is approved by the Food and Drug Administration (FDA) for the treatment of some cancers that share common pathways with LAM cells. Laboratory studies suggest that imatinib could completely block the growth of LAM cells through initiation of targeted cell death.

An important purpose of this research is to determine the safety of imatinib in people with LAM. This study will also evaluate the short-term effectiveness of imatinib. Participants will be randomized to receiving imatinib (study medication) or placebo (no treatment) for the 180 day duration of participation. The study is being conducted at the Medical University of South Carolina and at Columbia University in New York (CUMC). Each site will enroll 10 participants.