Hume was born in Ahmednagar, India, where his father and grandfather had worked as teachers and missionaries. He received his B.A. from Yale (1897) and his M.D. from Johns Hopkins Medical School (1901). Married to Lotta Carswell, he was at work on plague prevention projects in India (1903-1905) when the newly formed Yale-in-China Mission invited him to launch a university medical school in China. This project became his most important lifework.
From humble beginnings in 1906 in a Changsha, Hunan, inn that served as the first hospital, Hume gathered Chinese medical co-workers, raised funds, negotiated agreements, and laid the groundwork for the Yale-China hospital which opened in 1917. He served as senior physician, dean of Hunan Medical College, professor of medicine, and liaison with Chinese medical boards and professional journals. His aim was to develop educational and medical work "under the strongest Christian influence and under the highest intellectual and scientific standards of teaching and research." Despite mission opposition, and facing intense nationalistic pressures during the 1920s, Hume advocated giving greater authority to the Chinese.
In 1926 he offered his resignation to the Yale-China board in a dispute over policy. He later lectured on Chinese medical history, published books on Eastern and Western approaches to medicine, and served as medical consultant in various church-related situations.