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A simple mathematical model and alternative paradigm for certain chemotherapeutic regimens. (English) Zbl 0831.92017
Summary: A simplified two-compartment model for cell-specific chemotherapy is analysed by reformulating the governing system of differential equations as a Schrödinger equation in time. With the choice of an exponentially decaying function representing the effects of chemotherapy on cycling tumor cells, the potential function \(V(t)\) is a Morse-type potential, well known in the quantum mechanical literature; and the solutions are obtainable in terms of confluent hypergeometric functions (or the related Whittaker functions). Because the chemotherapy is administered periodically, the potential \(V(t)\) is periodic also, and use is made of existing theory (Floquet theory) as applied to scattering by periodic potentials in the quantum theory of solids. Corresponding to the existence of “forbidden energy bands” in that context, it appears that there are “forbidden” or inappropriate chemotherapeutic regimens also, in the sense that for some combinations of period, dosage, and cell parameters, no real solutions exist for the system of equations describing the time evolution of cancer cells in each compartment.
A similar, but less complex phenomenon may occur for simpler mathematical representations of the regimen. The purpose of this paper is to identify the existence of this phenomenon, at least insofar as this model is concerned, and to examine the implications for clinical activities. This new paradigm, if structurally stable (in the sense of the phenomenon occurring in more realistic models of chemotherapy), may be of considerable significance in identifying those regimens which are appropriate for effective chemotherapy, by providing a rational basis for such decisions, rather than by “trial and error”.

MSC:
92C50 Medical applications (general)
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
34A05 Explicit solutions, first integrals of ordinary differential equations
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[1] Skipper, H.E., On mathematical modeling of critical variables in cancer treatment, Bull. math. biol., 48, 253-278, (1986)
[2] Panetta, J.C.; Adam, J.A., A mathematical model of cycle-specific chemotherapy, Mathl. comput. modelling, (1995), (to appear) · Zbl 0829.92011
[3] Flügge, S., ()
[4] Hochstadt, H., The functions of mathematical physics, (1986), Dover Publications New York · Zbl 0217.39501
[5] Vezzetti, D.J.; Cahay, M.M., Transmission resonances in finite repeated structures, J. phys. D: appl. phys., 19, L53-L55, (1986)
[6] Park, D., Introduction to quantum theory, (1964), McGraw-Hill New York
[7] Seitz, D., Modern theory of solids, (1940), McGraw-Hill New York · Zbl 0025.13403
[8] Griffiths, D.J.; Taussig, N.F., Scattering from a locally periodic potential, Am. J. phys., 60, 883-888, (1992)
[9] Birkhead, B.G.; Rankine, E.M.; Gallivan, S.; Dones, L.; Rubens, R.D., A mathematical model of the development of drug resistance to cancer chemotherapy, Eur. J. cancer clin oncol., 23, 1421-1427, (1987)
[10] Adam, J.A., Non-radial stellar oscillations: A perspective from potential scattering (I). theoretical foundations, Astrophys. sp. sci., 220, 179-233, (1994) · Zbl 0812.34075
[11] Marx, J.L., Cell growth control takes balance, Science, 239, 975-976, (1988)
[12] Bajzer, Z.; Vuk-Pavlovic, S., Quantitative aspects of autocrine regulation in tumors, Crit. rev. in oncogenesis, 2, 53-73, (1990)
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