There are a few things you could try. One is a disk-based backup, which, if used in conjunction with continuous data protection software, can run continuously. This address the pitfall of running a backup once every 24 hours and not having up-to-date information if, say, a patient is making a return visit to the ER. Keep in mind, though, that this will result in your backups remaining on site, so some form of tape backup may be desired as well.
Another option is asynchronous data replication, which, unlike synchronous replication, commits replicas to disk immediately after performing a write operation, rather than once all data has been replicated and the replication has been confirmed. The downside here, though, is that a server failure that occurs before data has been committed to the other replicas will wipe out all the write operations that have taken place.