Today a wealth of knowledge and data are distributed using Semantic Web standards. Especially in the (bio)medical several sources like SNOMED, NCI, FMA, and more are distributed in the form of OWL ontologies. These can be aligned and integrated in order to create one large medical Knowledge Base. However, an important issue is that the structure of these ontologies may be profoundly different hence naively integrating them can lead to incoherences or changes in their original structure which may affect applications. In this paper we present a framework and novel approach for integrating independently developed ontologies. Starting from an initial seed ontology which may already be in use by an application, new sources are used to iteratively enrich and extend the seed one. To deal with structural incompatibilities we present a novel fine-grained approach which is based on mapping repair and alignment conservativity, formalise it and provide an exact as well as approximate but practical algorithms. Our framework has already been used to integrate a number of medical ontologies and support real-world healthcare services provided by babylon health. Finally, we also perform an experimental evaluation and compare with state-of-the-art ontology integration systems that take into account the structure and coherency of the integrated ontologies obtaining encouraging results.