Replace "Not working" part with this:

INNER JOIN (

SELECT PATIENT_ID, MIN(Date_Written) MIN_START, MAX(EXPIRES_ON) MAX_START

FROM [dbo].[PT_CASE]

GROUP BY PATIENT_ID ) AS Ptc ON [Pat].[PATIENT_ID] = [Ptc].[PATIENT_ID]

and simply add MIN_START, MAX_START to your SELECT.