You have a good concept, but if it also included a transportation service,
taxi, bus, directions, maps, to the doctors,, email reminders, placement into outlook, etc..  it would help solve two problems
a0 making sure the patenit arrives at the doctor at the appointed time - to reduce loss of revenue.
and making the patient feel special, cared bout in their health needs.
If such was incorprated you could use a slogan like the pizza delivery>

we help you order the dioctor anfd take you to see him.

patient delivery service.


like a piza delivery serivice

Just place image below into any reasonable pizza delivery add- subsibuite image for the pizza.