The scientific work of Professor William W. Muir addresses a fundamental problem in oxygen therapeutics: increasing oxygen transport capacity does not necessarily mean that oxygen reaches the tissues where it is needed. Over several decades, Muir investigated cardiovascular physiology, respiratory gas exchange, hemorrhagic shock, microvascular perfusion, hemoglobin-based oxygen carriers (HBOCs), and tissue oxygenation. His scientific involvement with Biopure, collaboration with Carl W. Rausch, and subsequent reported advisory activities in veterinary product development form an important part of this history. This article traces the development of these ideas and their experimental evaluation, leading to a proposed next step: Precision Oxygen Therapeutics based on Biological Hemoglobin Oxygen Carrier (BHOC), with oxygen delivery treated as a distinct therapeutic function rather than an attempt to replace blood.