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Consider using the Rules module (together with Rules Scheduler, a sub-module of it), as explained in the 3 steps below.

Step 1: Create a field to store the appointment date

Add a field to the "user" entity with machine name (say) field_next_appointment. Allow only 1 date (the store the "next" appointment), but don't make it required.

Step 2: Create a Rules "Component"

Here is the Rules Component to be created (in Rules export format, just import it in your own environment via copy-paste):

{ "rules_send_email_notification" : {
    "LABEL" : "Send eMail notification",
    "PLUGIN" : "action set",
    "OWNER" : "rules",
    "REQUIRES" : [ "rules" ],
    "USES VARIABLES" : { "user_to_notify" : { "label" : "User to be notified", "type" : "user" } },
    "ACTION SET" : [
      { "mail" : {
          "to" : [ "user-to-notify:mail" ],
          "subject" : "Friendly reminder about your upcoming event",
          "message" : "This is a reminder about the upcoming event that you have an upcoming event on [user-to-notify:field_next_appointment] ...",
          "from" : "[site:mail]",
          "language" : [ "" ]
        }
      }
    ]
  }
}

Obviously, the Subject and Content of the eMail may need review / tuning, while you might also want to use some other From eMail ID.

Step 3: Create a "Rule" using the Rules Component

Here is the Rules to be created (in Rules export format, just import it in your own environment via copy-paste):

{ "rules_on_new_user_event_registration" : {
    "LABEL" : "On new user event registration",
    "PLUGIN" : "reaction rule",
    "OWNER" : "rules",
    "REQUIRES" : [ "rules", "rules_scheduler" ],
    "ON" : { "user_update" : [] },
    "IF" : [
      { "entity_has_field" : { "entity" : [ "account" ], "field" : "field_next_appointment" } }
    ],
    "DO" : [
      { "schedule_delete" : {
          "component" : "rules_send_email_notification",
          "task" : "[account:name]"
        }
      },
      { "schedule" : {
          "component" : "rules_send_email_notification",
          "date" : {
            "select" : "account:field-next-appointment",
            "date_offset" : { "value" : 259200 }
          },
          "identifier" : "[account:name]",
          "param_user_to_notify" : [ "account" ]
        }
      }
    ]
  }
}

Note that this rule above refers to the Rules Component from the previous step.

The date_offset (=259200) corresponds to "7 days in advance", which may need review / tuning. Also note that schedule_delete part of it to ensure that previously "scheduled" reminders get removed when updating the data field.

If needed, you could also use the Field Permissions module to set field-level permissions to edit, view and create fields on any entity.

Note: Using Rules to send an email x days after a user registers also contains quite some details that may help you.