Practices receive payments for rendered services after weeks, and sometimes it takes months for them to get paid. It is not a deal between patients and doctors, but there is a third player involved - payer (the one who pays). The word P3 stands for Patients, Providers, and Payers, the three main players and 'Care' joins them in a meaningful way turning it into P3Care.
An idiom we often use to motivate someone to keep practicing. It is through practice we achieve perfection and perfection means we're passionate about it.