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Cancers 2011, 3, 340-367; doi:10.3390/cancers3010340 OPEN ACCESS cancers ISSN 2072-6694 /journal/cancers Review Why Victory in the War on Cancer Remains Elusive: Biomedical Hypotheses and Mathematical Models Leonid Hanin Department of Mathematics, Idaho State University, 921 S. 8th Avenue, Stop 8085, Pocatello, ID 83209-8085, USA; E-Mail: hanin@ Received: 3 December 2010; in revised form: 6 January 2011 / Accepted: 11 January 2011 / Published: 17 January 2011 Abstract: We discuss philosophical, methodological, and biomedical grounds for the traditional paradigm of cancer and some of its critical flaws. We also review some potentially fruitful approaches to understanding cancer and its treatment. This includes the new paradigm of cancer that was developed over the last 15 years by Michael Retsky, Michael Baum, Romano Demicheli, Isaac Gukas, William Hrushesky and their colleagues on the basis of earlier pioneering work of Bernard Fisher and Judah Folkman. Next, we highlight the unique and pivotal role of mathematical modeling in testing biomedical hypotheses about the natural history of cancer and the effects of its treatment, elaborate on model selection criteria, and mention some methodological pitfalls. Finally, we describe a specific mathematical model of cancer progression that supports all the main postulates of the new paradigm of cancer when applied to the natural history of a particular brea