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On the optimal control of cancer radiotherapy for non-homogeneous cell populations

Part of: Applications Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

L. G. Hanin ,
S. T. Rachev  and
A. Yu. Yakovlev
L. G. Hanin*
Affiliation:
Technion-Israel Institute of Technology
S. T. Rachev*
Affiliation:
University of California at Santa Barbara
A. Yu. Yakovlev*
Affiliation:
St Petersburg Technical University
*
∗Postal address: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel.
∗∗Postal address: Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106, USA.
∗∗∗Postal address: Department of Applied Mathematics, St Petersburg Technical University, Polytechnicheskaya, St Petersburg 195251, Russia.
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Abstract

Optimization problems in cancer radiation therapy are considered, with the efficiency functional defined as the difference between expected survival probabilities for normal and neoplastic tissues. Precise upper bounds of the efficiency functional over natural classes of cellular response functions are found. The ‘Lipschitz' upper bound gives rise to a new family of probability metrics. In the framework of the ‘m hit-one target' model of irradiated cell survival the problem of optimal fractionation of the given total dose into n fractions is treated. For m = 1, n arbitrary, and n = 1, 2, m arbitrary, complete solution is obtained. In other cases an approximation procedure is constructed. Stability of extremal values and upper bounds of the efficiency functional with respect to perturbation of radiosensitivity distributions for normal and tumor tissues is demonstrated.

Keywords

TUMOR IRRADIATIONEFFICIENCY FUNCTIONALPRECISE UPPER BOUND OVER A SET OF FUNCTIONSPROBABILITY METRICSOPTIMAL FRACTIONATIONSTABILITY PROPERTIES

MSC classification

Primary: 60E05: Distributions
Secondary: 62P10: Applications to biology and medical sciences
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 1 , March 1993 , pp. 1 - 23
Copyright
Copyright © Applied Probability Trust 1993 

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References

[1] Barendsen, G. W. (1980) Variations in radiation responses among experimental tumors. In Radiation Biology in Cancer Research, pp. 333–343, Raven Press, New York.Google Scholar
[2] Eisen, M. (1979) Mathematical Models in Cell Biology and Cancer Chemotherapy. Lecture Notes in Biomathematics 30, Springer-Verlag, Berlin.Google Scholar
[3] Hanin, L. G., Rachev, S. T., Goot, R. E. and Yakovlev, A. Yu. (1989) Precise upper bounds for the functionals describing tumor treatment efficiency. In Lecture Notes in Mathematics 1421, pp. 50–67, Springer-Verlag, Berlin.Google Scholar
[4] Ivanov, V. K. (1986) Mathematical Modelling and Optimization of Cancer Radiotherapy. Energoatomizdat, Moscow. (in Russian).Google Scholar
[5] Kantorovich, L. V. and Akilov, G. P. (1982) Functional Analysis, 2nd edn. Pergamon, New York.Google Scholar
[6] Knolle, H. (1988) Cell Kinetic Modelling and the Chemotherapy of Cancer. Lecture Notes in Biomathematics 75, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[7] Rachev, S. T. and Yakovlev, A. Yu. (1988) Theoretical bounds for the tumor treatment efficacy. Syst. Anal. Model. Simul. 5, 37–42.Google Scholar
[8] Stoyan, D. (1977) Qualitatave Eigenschaften und Abschätzungen stochastischer Modell. Akademie-Verlag, Berlin.Google Scholar
[9] Swan, G. W. (1981) Optimization of Human Cancer Radiotherapy. Lecture Notes in Biomathematics 42, Springer-Verlag, Berlin.Google Scholar
[10] Swan, G. W. (1984) Applications of Optimal Control Theory in Biomedicine. Monographs and Textbooks in Pure and Applied Mathematics 81, Marcel Dekker, New York.Google Scholar
[11] Swan, G. W. (1990) Role of optimal control theory in cancer chemotherapy. Math. Biosci. 101, 237–284.Google Scholar
[12] Turner, M. M. (1975) Some classes of hit-target models. Math. Biosci. 23, 219–235.CrossRefGoogle Scholar
[13] Yakovlev, A. Yu and Zorin, A. V. (1988) Computer Simulation in Cell Radiobiology. Lecture Notes in Biomathematics 74, Springer-Verlag.Google Scholar