In this paper we present a new problem arising in the context of non-emergency transportation of patients. We consider a hospital (the depot) and a set of patients with a medical appointment. Patients require either to go from home to hospital (inbound request) or from hospital to home (outbound request). The problem can be addressed as a Pickup and Delivery Problem, but the fact that all transportation requests are connected to the depot also allows tackling it as a special Multi-Trip Vehicle Routing Problem. We adopt the second standpoint and call it the Multi-Trip Vehicle Routing Problem with Mixed Pickup and Delivery, and Release and Due dates. We propose a specialized branch-and-price algorithm and demonstrate computationally that our approach outperforms a classical branch-and-price algorithm based on the Pickup and Delivery Problem modeling. We also show how our algorithm can be adapted to the solution of the Vehicle Routing Problem with Simultaneous Pickup and Delivery and Time Windows, and obtain new optimal solutions on benchmark instances.