The department shall assess against the patient or his or her relatives all or such part of the cost determined under section 83-365 as they are able to pay, in the judgment of the department, except that a patient who is placed in a state institution to receive appropriate special education pursuant to the Special Education Act or his or her relatives shall be assessed only for medical care and medical treatment costs as determined pursuant to rules and regulations adopted and promulgated by the department in accordance with section 83-371.