A doctor is a medical professional that gives care to people who are sick. "Patience" sounds like "patients". Patients are people that receive medical attention from doctors, or other healthcare professionals. To lose patience means to become easily frustrated. To lose patients means to stop seeing caring for someone that once sought medical treatment from you. This pun is a play on the word "patience" since it sounds like "patients". 
OLD DOCTORS never die they just lose their patience .