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Mathematical Medicine and Biology Advance Access originally published online on July 14, 2008
Mathematical Medicine and Biology 2008 25(3):267-283; doi:10.1093/imammb/dqn014
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Inducing catastrophe in malignant growth

Robert A. Gatenby

Department of Radiology, Arizona Health Sciences Center, Tucson, AZ 85726, USA

B. Roy Frieden{dagger}

College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA

{dagger} Email: roy.frieden{at}optics.arizona.edu

Received on February 29, 2008. Revised on May 14, 2008. Accepted on June 2, 2008.

Mathematical catastrophe theory is used to describe cancer growth during any time-dependent program a(t) of therapeutic activity. The program may be actively imposed, e.g. as chemotherapy, or occur passively as an immune response. With constant therapy a(t), the theory predicts that cancer mass p(t) grows in time t as a cosine-modulated power law, with power = 1.618···, the Fibonacci constant. The cosine modulation predicts the familiar relapses and remissions of cancer growth. These fairly well agree with clinical data on breast cancer recurrences following mastectomy. Two such studies of 3183 Italian women consistently show an immune system's average activity level of about a = 2.8596 for the women. Fortunately, an optimum time-varying therapy program a(t) is found that effects a gradual approach to full remission over time, i.e. to a chronic disease. Both activity a(t) and cancer mass p(t) monotonically decrease with time, the activity a(t) as 1/(ln t) and mass remission as t94{ – 0.382}. These predicted growth effects have a biological basis in the known presence of multiple alleles during cancer growth.

Keywords: optimal chemotherapy; including catastrophe in malignancy; catastrophe theory; cancer as chronic disease


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