This paper proposes a methodology for the event stream processing of synchronous (physiological) and asynchronous (clinical) health data streams. The purpose is to illustrate the feasibility of Artemis, our extension of IBM's InfoSphere Streams, to appropriately deliver notifications from an initial clinical hypothesis within the critical care environment. We demonstrate that an positive alert can be delivered that is indicative of an onset of instability in critically ill newborns. Artemis, is also tested for its potential to allow clinicians the ability to interact directly with the rule-based system to prove certain hypothesis. We begin this methodology with a model of the clinical case study, and then transform that model into Stream's SPADE code. Subsequently, it is compiled and executed within the Streams environment to deliver notifications in real-time of the newborns health state.