The purpose of the study is to test if the drug doxycycline is effective in slowing the progression of lung disease in LAM. Lymphangioleiomyomatosis (LAM) is a rare lung disease which affects young women. Women with LAM develop enlarged air spaces in the lungs called cysts, caused by an excess of matrix metalloproteinases (MMPs), protein-digesting enzymes. LAM is associated with kidney tumours, called angiomyolipomas, and causes recurrent lung collapse, breathlessness and death or need for lung transplant. There is no proven treatment. Doxycycline, a commonly used antibiotic can block MMP production and a small number of patients have shown some benefit from doxycycline. The investigators will perform a study to test if doxycycline can slow the fall in lung function in patients with LAM. Forty patients who consent to participate will take doxycycline or a placebo (dummy) tablet for two years in addition to their standard treatment.