Continuous monitoring of patient's physiological signs has the potential to augment traditional medical practice, particularly in developing countries that have a shortage of healthcare professionals. However, continuously streamed data presents additional security, storage and retrieval challenges and further inhibits initiatives to integrate data to form electronic health record systems. Blockchain technologies enable data to be stored securely and inexpensively without recourse to a trusted authority. Blockchain technologies also promise to provide architectures for electronic health records that do not require huge government expenditure that challenge developing nations. However, Blockchain deployment, particularly with streamed data challenges existing Blockchain algorithms that take too long to place data in a block, and have no mechanism to determine whether every data point in every stream should be stored in such a secure way. This article presents an architecture that involves a Patient Agent, coordinating the insertion of continuous data streams into Blockchains to form an electronic health record.