Electron. J. Diff. Eqns.,
Vol. 2007(2007), No. 171, pp. 1-24.
Seeking bang-bang solutions of mixed immuno-chemotherapy of tumors
Lisette G. de Pillis, K. Renee Fister, Weiqing Gu,
Craig Collins, Michael Daub, David Gross,
James Moore, Ben Preskill
Abstract:
It is known that a beneficial cancer treatment approach for a
single patient often involves the administration of more than one
type of therapy. The question of how best to combine multiple
cancer therapies, however, is still open. In this study, we
investigate the theoretical interaction of three treatment types
(two biological therapies and one chemotherapy) with a growing
cancer, and present an analysis of an optimal control strategy for
administering all three therapies in combination. In the
situations with controls introduced linearly, we find that there
are conditions on which the controls exist singularly. Although
bang-bang controls (on-off) reflect the drug treatment approach
that is often implemented clinically, we have demonstrated, in the
context of our mathematical model, that there can exist regions on
which this may not be the best strategy for minimizing a tumor
burden. We characterize the controls in singular regions by
taking time derivatives of the switching functions. We will
examine these representations and the conditions necessary for the
controls to be minimizing in the singular region. We begin by
assuming only one of the controls is singular on a given interval.
Then we analyze the conditions on which a pair and then all three
controls are singular.
Submitted October 3, 2007. Published December 6, 2007.
Math Subject Classifications: 37N25, 49J15, 49J30, 49N05, 62P10, 92C50.
Key Words: Cancer modelling; mixed immuno-chemo-therapy;
immunotherapy; chemotherapy; linear optimal control.
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Lisette G. de Pillis
Department of Mathematics,
Harvey Mudd College
Claremont, CA 91711, USA
email: depillis@hmc.edu |
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K. Renee Fister
Department of Mathematics and Statistics
Murray State University, Murray, KY 42071, USA
email: renee.fister@murraystate.edu |
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Weiqing Gu
Department of Mathematics,
Harvey Mudd College
Claremont, CA 91711, USA
email: gu@math.hmc.edu |
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Craig Collins
Department of Mathematics and Statistics
Murray State University, Murray, KY 42071, USA
email: craig.collins@murraystate.edu |
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Michael Daub
Department of Mathematics and Statistics
Williams College, Williamstown, MA 01267, USA
email: Michael.W.Daub@williams.edu |
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David Gross, James Moore, Ben Preskill
Department of Mathematics,
Harvey Mudd College
Claremont, CA 91711, USA
email: david_gross@hmc.edu, jmoore@hmc.edu, bpreskill@hmc.edu
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