Optimal strategy in chemotherapy for Malthusian model of cancer growth

• Autorzy: Maroński R.

Identyfikatory

DOI

10.5277/ABB-00505-2015-02

• Języki publikacji: EN

Abstrakty

EN

Purpose: The problem of optimal strategy in cancer chemotherapy is reconsidered. Two incompatible goals should be completed: the number of cancer cells in the patient’s body should be reduced and the toxic effect of the therapy should be minimized. Such problem may be formulated in optimal control. The control function is the amount of the drug administered in the time unit. Methods: The Malthusian model of cell population growth is employed where the rate of increase of the number of cancer cells is proportional to the number of cells in population and an intrinsic rate that usually is assumed to be constant. The performance index is the amount of the drug cumulated in the patient’s body and it is minimized. A non-standard method of optimal control is used – method of Miele. Results: The optimal solutions are obtained for three cases: constant intrinsic rate, monotonically increasing/decreasing intrinsic rate and for periodic intrinsic rate. The optimal control is ununique for the first case – the result is irrespective of the strategy. Such result has been known earlier. The optimal control is unique for other cases and it is of bang-bang type. Conclusions: The ununique solution for constant intrinsic rate is surprising, therefore a mechanical analogy is given. The optimal strategy is in accordance with clinical experience for decreasing intrinsic rate. The optimal control is a periodic function of time for the intrinsic rate of sin/cos type – the drug should be administered, as its value is relatively high.

Słowa kluczowe

EN

optimal chemotherapy; Malthusian growth; periodic control

PL

chemioterapia; model maltuzjański; kontrole okresowe

• Wydawca: Oficyna Wydawnicza Politechniki Wrocławskiej
• Czasopismo: Acta of Bioengineering and Biomechanics
• Rocznik: 2017
• Tom: Vol. 19, no. 1
• Strony: 63--68
• Opis fizyczny: Bibliogr. 17 poz., wykr.

Twórcy

autor

Maroński R.

• Warsaw University of Technology, Institute of Aeronautics and Applied Mechanics, Warsaw, Poland

Bibliografia

• [1] CHASNOV J.R., Mathematical biology, The Hong Kong University, 2009.
• [2] ENGELHART M., LEBIEDZ D., SAGER S., Optimal control for selected cancer ODE models: A view on the potential of optimal schedules and choice of objective function, Math. Biosci., 2011, 229, 123–134.
• [3] KIMMEL M., ŚWIERNIAK A., O pewnym zadaniu sterowania optymalnego związanym z optymalną chemioterapią białaczek (On an optimal control problem connected to leukemia chemotherapy), Zeszyty Naukowe Politechniki Śląskiej, Automatyka, 1983, 65, 120–130, (in Polish).
• [4] LEDZEWICZ U., SCHÄTTLER H., Analysis of a cell-cycle specific model for cancer chemotherapy, J. Biol. Syst., 2002, 10, 3, 183–206.
• [5] LEDZEWICZ U., SCHÄTTLER H., Optimal bang-bang controls for a two-compartment model in cancer chemotherapy, J. Optimiz. Theory Appl., 2002, 114, 3, 609–637.
• [6] MAROŃSKI R., On optimal running downhill on skis, J. Biomech., 1990, 23, 435–439.
• [7] MAROŃSKI R., On optimal velocity during cycling, J. Biomech., 1994, 27, 205–213.
• [8] MAROŃSKI R., Minimum-time running and swimming – an optimal control approach, J. Biomech., 1996, 29, 245–249.
• [9] MAROŃSKI R., Optimal strategy in chemotherapy for Gompertzian model of cancer growth, Acta Bioeng. Biomech., 2008, 10, 81–84.
• [10] MIELE A., Extremization of linear integrals by Green’s theorem, [in:] G. Leitmann (ed.), Optimization Techniques with Application to Aerospace Systems, Academic Press, New York, 1962, 69–98.
• [11] PRAMOD J., Wind Energy Engineering, McGraw-Hill, New York, 2011, 198.
• [12] SMITH J.M., Models in Ecology, Cambridge University Press, Cambridge 1974.
• [13] SPIEGEL M.R., Theoretical Mechanics, McGraw-Hill, New York, 1967.
• [14] SWAN G.W., Role of optimal control problems related to optimal chemotherapy, Math. Biosciences, 1990, 101, 237–284.
• [15] ŚWIERNIAK A., DUDA Z., Some control problems related to optimal chemotherapy – singular solutions, Applied Mathematics and Computer Sciences, 1992, 2, 293–302.
• [16] ŚWIERNIAK A., POLAŃSKI A., DUDA Z., „Strange” phenomena in simulation of optimal control problems arising in cancer chemotherapy, Proc. of the 8-th Prague Symposium “Computer Simulation in Biology, Ecology and Medicine”, Nov. 9–11, 1992, 58–62.
• [17] ŚWIERNIAK A., DUDA Z., POLAŃSKI A., Symulacyjne badanie wybranych modeli sterowania cyklem komórkowym (Simulation investigation of selected models of cell cycle control), Prace VIII Symp. “Symulacja Procesów Dynamicznych SPD-8”, Polana Chochołowska, 13–17 czerwca 1994, (in Polish).

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).