Clinical studies often use data dictionaries with controlled sets of terms to facilitate data collection, limited interoperability and sharing at a local site. Multi-center retrospective clinical studies require that these data dictionaries, originating from individual participating centers, be harmonized in preparation for the integration of the corresponding clinical research data. Domain ontologies are often used to facilitate multi-center data integration by modeling terms from data dictionaries in a logic-based language, but interoperability among domain ontologies (using automated techniques) is an unresolved issue. Although many upper-level reference ontologies have been proposed to address this challenge, our experience in integrating multi-center sleep medicine data highlights the need for an upper level ontology that models a common set of terms at multiple-levels of abstraction, which is not covered by the existing upper-level ontologies. We introduce a methodology underpinned by a Minimal Domain of Discourse (MiDas) algorithm to automatically extract a minimal common domain of discourse (upper-domain ontology) from an existing domain ontology. Using the Multi-Modality, Multi-Resource Environment for Physiological and Clinical Research (Physio-MIMI) multi-center project in sleep medicine as a use case, we demonstrate the use of MiDas in extracting a minimal domain of discourse for sleep medicine, from Physio-MIMI's Sleep Domain Ontology (SDO). We then extend the resulting domain of discourse with terms from the data dictionary of the Sleep Heart and Health Study (SHHS) to validate MiDas. To illustrate the wider applicability of MiDas, we automatically extract the respective domains of discourse from 6 sample domain ontologies from the National Center for Biomedical Ontologies (NCBO) and the OBO Foundry.