This type of design is more complicated than might first appear, because the patient profile is not something that the respondent is evaluating and choosing or not choosing, but is instead acting like a context variable. So what you are saying through this type of a design is that someone's preference for treatments changes depending on the patient profile. So instead of having a single utility for a treatment, you potentially have a modifying utility based on the patient profile.
These are typically done using our free format questions and a CBC design as you mention, but then our standalone MBC (Menu Based Conjoint) software is used in order to specify the model. It can't be analyzed inside Lighthouse Studio because the context variable would need to exist in each profile presented, and would effectively become a constant.