Lymphangioleiomyomatosis (LAM) is a rare cystic lung disease that appears to behave like a slowly growing cancer. Since clinical progression is very slow, new blood tests have been used to speed the time required to find safe and effective medications. A large National Institute of Health study called MILES showed that sirolimus (also known as Rapamycin) improved lung function in individuals with LAM. Since most individuals with LAM and impaired lung function are now on sirolimus, future studies may prove more difficult. Laboratory studies suggested that Imatinib mesylate (imatinib), an FDA-approved drug for leukemia, initiates LAM cell death. A pilot trial with imatinib titled Imatinib Mesylate for the treatment of Lymphangioleiomyomatosis - (LAMP-1) was funded by the Department of Defense in 2016, and documented (1) the safety of use of tyrosine kinase inhibitors in patients with LAM; (2) the safety of concurrent use of tyrosine kinase and mTOR inhibitors; and, (3) short term variability in vascular endothelial growth factor D (VEGF-D) - a LAM biomarker, as a response to therapies. Due to the short-term LAMP-1 trial, LAMP-2 will be a longer-term 6-month clinical study evaluating the safety and tolerability of imatinib in patients with LAM. Patients that participate in the trial will come in for 5 office visits and check-up phone calls every 2 weeks over the course of 6 months.