# Different models of chemotherapy taking into account drug resistance stemming from gene amplification

• Volume: 13, Issue: 3, page 297-305
• ISSN: 1641-876X

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## Abstract

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This paper presents an analysis of some class of bilinear systems that can be applied to biomedical modelling. It combines models that have been studied separately so far, taking into account both the phenomenon of gene amplification and multidrug chemotherapy in their different aspects. The mathematical description is given by an infinite dimensional state equation with a system matrix whose form allows decomposing the model into two interacting subsystems. While the first one, of a finite dimension, can have any form, the other is infinite dimensional and tridiagonal. A methodology of the analysis of such models, based on system decomposition, is presented. An optimal control problem is defined in the l^1 space. In order to derive necessary conditions for optimal control, the model description is transformed into an integro-differential form. Finally, biomedical implications of the obtained results are discussed.

## How to cite

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Śmieja, Jarosław, and Świerniak, Andrzej. "Different models of chemotherapy taking into account drug resistance stemming from gene amplification." International Journal of Applied Mathematics and Computer Science 13.3 (2003): 297-305. <http://eudml.org/doc/207644>.

@article{Śmieja2003,
abstract = {This paper presents an analysis of some class of bilinear systems that can be applied to biomedical modelling. It combines models that have been studied separately so far, taking into account both the phenomenon of gene amplification and multidrug chemotherapy in their different aspects. The mathematical description is given by an infinite dimensional state equation with a system matrix whose form allows decomposing the model into two interacting subsystems. While the first one, of a finite dimension, can have any form, the other is infinite dimensional and tridiagonal. A methodology of the analysis of such models, based on system decomposition, is presented. An optimal control problem is defined in the l^1 space. In order to derive necessary conditions for optimal control, the model description is transformed into an integro-differential form. Finally, biomedical implications of the obtained results are discussed.},
author = {Śmieja, Jarosław, Świerniak, Andrzej},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {biomedical modelling; multivariable control; infinite dimensional systems; infinite-dimensional systems},
language = {eng},
number = {3},
pages = {297-305},
title = {Different models of chemotherapy taking into account drug resistance stemming from gene amplification},
url = {http://eudml.org/doc/207644},
volume = {13},
year = {2003},
}

TY - JOUR
AU - Śmieja, Jarosław
AU - Świerniak, Andrzej
TI - Different models of chemotherapy taking into account drug resistance stemming from gene amplification
JO - International Journal of Applied Mathematics and Computer Science
PY - 2003
VL - 13
IS - 3
SP - 297
EP - 305
AB - This paper presents an analysis of some class of bilinear systems that can be applied to biomedical modelling. It combines models that have been studied separately so far, taking into account both the phenomenon of gene amplification and multidrug chemotherapy in their different aspects. The mathematical description is given by an infinite dimensional state equation with a system matrix whose form allows decomposing the model into two interacting subsystems. While the first one, of a finite dimension, can have any form, the other is infinite dimensional and tridiagonal. A methodology of the analysis of such models, based on system decomposition, is presented. An optimal control problem is defined in the l^1 space. In order to derive necessary conditions for optimal control, the model description is transformed into an integro-differential form. Finally, biomedical implications of the obtained results are discussed.
LA - eng
KW - biomedical modelling; multivariable control; infinite dimensional systems; infinite-dimensional systems
UR - http://eudml.org/doc/207644
ER -

## References

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1. Arino O., Kimmel M. and Webb G.F. (1995): Mathematical modelling of the loss of telomere sequences. — J. Theor. Biol., Vol. 177, No. 1, pp. 45–57.
2. Axelrod D.E., Baggerly K.A. and Kimmel M. (1994): Gene amplification by unequal chromatid exchange: Probabilistic modelling and analysis of drug resistance data. — J. Theor. Biol., Vol. 168, No. 2, pp. 151–159.
3. Bate R.R. (1969): The optimal control of systems with transport lag, In: Advances in Control and Dynamic Systems (C.T. Leondes, Ed.). — Academic Press, Vol. 7, pp. 165–224.
4. Brown B.W. and Thompson J.R. (1975): A rationale for synchrony strategies in chemotherapy, In: Epidemiology (D. Ludwig and K.L. Cooke, Eds.). — Philadelphia: SIAM Publ., pp. 31–48.
5. Coldman A.J. and Goldie J.H. (1986): A stochastic model for the origin and treatment of tumors containing drug-resistant cells. — Bull. Math. Biol., Vol. 48, No. 3–4, pp. 279–292. Zbl0613.92006
6. Connor M.A. (1972): Optimal control of systems represented by differential-integral equations. — IEEE Trans. Automat. Contr., Vol. AC-17, No. 1, pp. 164–166. Zbl0259.49013
7. Curtain R.F. and Zwart H.J. (1995): An Introduction to Infinite- Dimensional Linear Systems Theory. — New York: Springer. Zbl0839.93001
8. Gabasov R. and Kirilowa F.M. (1971): Qualitative theory of optimal processes. — Moscow: Nauka, (in Russian).
9. Harnevo L.E. and Agur Z. (1993): Use of mathematical models for understanding the dynamics of gene amplification. — Mutat. Res., Vol. 292, No. 1, pp. 17–24.
10. Kimmel M. and Axelrod D.E. (1990): Mathematical models of gene amplification with applications to cellular drug resistance and tumorigenicity. — Genetics, Vol. 125, No. 3, pp. 633–644.
11. Kleinrock L. (1976): Queuing Systems. Vol. 1: Theory. — New York: Wiley. Zbl0361.60082
12. Olofsson P. and Kimmel M. (1999): Stochastic models of telomere shortening. — Math. Biosci., Vol. 158, No. 1, pp. 75– 92. Zbl0943.92015
13. Polański A., Kimmel M. and Świerniak A. (1997): Qualitative analysis of the infinite-dimensional model of evolution of drug resistance, In: Advances in Mathematical Population Dynamics — Molecules, Cells and Man (O. Arino, D. Axelrod and M. Kimmel, Eds.). — Singapore: World Scientific, pp. 595–612. Zbl0929.92018
14. Pontryagin L.S., Boltyanski V.G., Gamkrelidze R.V. and Mischenko E.F. (1962): Mathematical Theory of Optimal Processes. — New York: Wiley.
15. Ramel C. (1997): Mini- and microsatellites. — Environmental Health Perspectives, Vol. 105, Suppl. 104, pp. 781–789.
16. Śmieja J., Duda Z. and Świerniak A. (1999): Optimal control for the model of drug resistance resulting from gene amplification. — Prep. 14th IFAC World Congress, Beijing, China, Vol. L, pp. 71–75.
17. Śmieja J., Świerniak A. and Duda Z. (2000): Gradient method for finding optimal scheduling in infinite dimensional models of chemotherapy. — J. Theor. Med., Vol. 3, No. 1, pp. 25–36. Zbl0986.92018
18. Swan G.W. (1990): Role of optimal control theory in cancer chemotherapy. — Math. Biosci., Vol. 101, No. 2, pp. 237– 284. Zbl0702.92007
19. Świerniak A., Śmieja J., Rzeszowska-Wolny J. and Kimmel M. (2001): Random branching walk models arising in molecular biology — control theoretic approach. — Proc. IASTED MIC Conf., Innsbruck, Austria, Vol. II, pp. 584– 589. Zbl1176.93011
20. Świerniak A., Kimmel M., Polański A. and Śmieja J. (1997a): Asymptotic properties of infinite dimensional model of drug resistance evolution. — Proc. ECC’97, Brussels, TUA- C-4, CD-ROM. Zbl0929.92018
21. Świerniak A., Polański A., Duda Z. and Kimmel M. (1997b): Phase-specific chemotherapy of cancer: Optimisation of scheduling and rationale for periodic protocols. — Biocybern. Biomed. Eng., Vol. 16, No. 1–2, pp. 13–43.
22. Świerniak A., Kimmel M. and Polański A. (1998): Infinite dimensional model of evolution of drug resistance of cancer cells. — J. Math. Syst. Estim. Contr., Vol. 8, No. 1, pp. 1–17. Zbl0897.92015
23. Świerniak A., Polański A., Kimmel M., Bobrowski A. and Śmieja J. (1999): Qualitative analysis of controlled drug resistance model — inverse Laplace and semigroup approach. — Contr. Cybern., Vol. 28, No. 1, pp. 61–74. Zbl0949.93070
24. Świerniak A., Polański A., Śmieja J., Kimmel M. and Rzeszowska-Wolny J. (2002): Control theoretic approach to random branching walk models arising in molecular biology. — Proc. ACC Conf., Anchorage, pp. 3449–3453.
25. Zadeh L.A. and Desoer C.A. (1963): Linear System Theory. The State Space Approach. — New York: McGraw-Hill. Zbl1145.93303

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