Generalized optimization of processes of drug transport in tumors. (English. Russian original) Zbl 1455.35268
Cybern. Syst. Anal. 56, No. 5, 758-765 (2020); translation from Kibern. Sist. Anal. 2020, No. 5, 86-94 (2020).
Summary: Optimization and controllability problems for systems described by partial differential equations, where coefficients and the right-hand sides belong to different functional spaces, are considered. In particular, pharmacokinetic problems lead to such models. A model described by a general differential equation with zero initial and boundary conditions is analyzed. Coefficients are assumed positive in this area, concentrated sources are modeled by the Dirac delta function. The search of feasible control that minimizes the quality functional is performed. Based on the space of measurable and square-integrable functions, adjunction for functions smooth in the research area according to the norm, and conjugate problems are constructed. Negative spaces are introduced for the conjugate problem, and generalized solution to the problem is investigated.
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 92C45 | Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) |
| 93B05 | Controllability |
| 93C20 | Control/observation systems governed by partial differential equations |
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
References:
| [1] | L. T. Baxter and R. K. Jain, “Transport of fluid and macromolecules in tumors. I. Role of interstitial pressure and convection,” Microvasc. Res., Vol. 37, Iss. 1, 77-104 (1989). |
| [2] | L. T. Baxter and R. K. Jain, “Transport of fluid and macromolecules in tumors. II. Role of heterogeneous perfusion and lymphatics,” Microvasc. Res., Vol. 40, Iss. 2, 246-263 (1990). |
| [3] | L. T. Baxter and R. K. Jain, “Transport of fluid and macromolecules in tumors. III. Role of binding and metabolism,” Microvasc. Res., Vol. 41, Iss. 1, P. 5-23 (1991). |
| [4] | J. Lankelma, R. F. Luque, H. Dekker, W. Schinkel, and H. M. Pinedo, “A mathematical model of drug transport in human breast cancer,” Microvasc. Res., Vol. 59, Iss. 1, 149-161 (2000). |
| [5] | J. P. Ward and J. R. King, “Mathematical modelling of drug transport in tumor multicell spheroids and monolayer cultures,” Math. Biosci., Vol. 181, Iss. 2, 177-207 (2003). · Zbl 1014.92021 |
| [6] | Tzafriri, AR; Lerner, EI; Flashner-Barak, M.; Hinchcliffe, M.; Ratner, E.; Parnas, H., Mathematical modeling and optimization of drug delivery from intratumorally injected microspheres, Clin. Cancer Res., 11, 826-834 (2005) |
| [7] | S. P. Chakrabarty and F. B. Hanson, “Optimal control of drug delivery to brain tumors for a distributed parameters model,” in: Proc. the 2005, American Control Conference (Portland, Oregon, USA 8-10 June, 2005), Vol. 2, IEEE (2005), pp. 973-978. 10.1109/ACC.2005.1470086. |
| [8] | Klyushin, DA; Lyashko, NI; Onopchuk, YN, Mathematical modeling and optimization of intratumor drug transport, Cybern. Syst. Analysis, 43, 6, 886-892 (2007) · Zbl 1190.49005 · doi:10.1007/s10559-007-0113-z |
| [9] | Karabin, L.; Klyushin, D., Two-phase Stefan problem for optimal control of targeted drug delivery to malignant tumors, J. of Coupled Systems and Multiscale Dynamics, 2, 2, 45-51 (2014) · doi:10.1166/jcsmd.2014.1044 |
| [10] | Lyashko, SI; Klyushin, DA; Onotskyi, VV; Lyashko, NI, Optimal control of drug delivery from microneedle systems, Cybern. Syst. Analysis, 54, 3, 357-365 (2018) · Zbl 1401.49058 · doi:10.1007/s10559-018-0037-9 |
| [11] | S. I. Lyashko, D. A. Klyushin, D. A. Nomirovsky, and V. V. Semenov, “Identification of age-structured contamination sources in ground water,” in: R. Boucekkline, N. Hritonenko, and Y. Yatsenko (eds.), Optimal Control of Age-Structured Populations in Economy, Demography, and the Environment, Routledge, London-New York (2013), pp. 277-292. |
| [12] | S. I. Lyashko and A. A. Man’kovskii, “Controllability of impulse parabolic systems,” Autom. Remote Control, Vol. 52, No. 9, 1233-1238 (1991). · Zbl 0749.93049 |
| [13] | Lyashko, SI; Man’kovskii, AA, Simultaneous optimization of impulse and intensities in control problems for parabolic equations, Cybern. Syst. Analysis, 19, 5, 687-690 (1983) · doi:10.1007/BF01068766 |
| [14] | Nakonechnyi, AG; Lyashko, SI, Minimax estimation theory for solutions of abstract parabolic equations, Cybern. Syst. Analysis, 31, 4, 626-630 (1995) · Zbl 0855.49016 · doi:10.1007/BF02366418 |
| [15] | Lyashko, SI; Semenov, VV, Controllability of linear distributed systems in classes of generalized actions, Cybern. Syst. Analysis, 37, 1, 13-32 (2001) · Zbl 1005.93009 · doi:10.1023/A:1016607831284 |
| [16] | Lyashko, SI; Nomirovskii, DA; Sergienko, TI, Trajectory and final controllability in hyperbolic and pseudohyperbolic systems with generalized actions, Cybern. Syst. Analysis, 37, 5, 756-763 (2001) · Zbl 1037.93047 · doi:10.1023/A:1013871026026 |
| [17] | Lyashko, SI, Numerical solution of pseudoparabolic equations, Cybern. Syst. Analysis, 31, 5, 718-722 (1995) · Zbl 0859.65107 · doi:10.1007/BF02366321 |
| [18] | Lyashko, SI, The approximate solution of a pseudoparabolic equation, USSR Computational Mathematics and Mathematical Physics, 31, 12, 107-111 (1991) · Zbl 0785.65097 |
| [19] | S. I. Lyashko and S. E. Red’ko, “Approximate solution of a problem of the dynamics of a viscous stratified fluid,” USSR Computational Mathematics and Mathematical Physics, Vol. 27, No. 3, 49-56 (1987). · Zbl 0659.76122 |
| [20] | L. A. Vlasenko, A. G. Rutkas, V. V. Semenets, and A. A. Chikrii, “On the optimal impulse control in descriptor systems,” J. Autom. Inform. Sci., Vol. 51, Iss. 5, 1-15 (2019). |
| [21] | Petryk, MR; Khimich, A.; Petryk, MM; Fraissard, J., Experimental and computer simulation studies of dehydration on microporous adsorbent of natural gas used as motor fuel, Fuel, 239, 1324-1330 (2019) · doi:10.1016/j.fuel.2018.10.134 |
| [22] | Gladkii, AV, Optimization of wave processes in inhomogeneous media, Cybern. Syst. Analysis, 39, 5, 728-736 (2003) · Zbl 1097.49500 · doi:10.1023/B:CASA.0000012093.24562.50 |
| [23] | V. M. Bulavatsky and V. V. Skopetsky, “Mathematical modeling of dynamics of some distributed time-space consolidation processes,” J. Autom. Inform. Sci., Vol. 41, Iss. 9, 14-25 (2009). |
| [24] | Miller, GE, Fundamentals of biomedical transport processes, Series: Synthesis Lectures on Biomedical Engineering, 5, 1, 1-75 (2010) |
| [25] | Yosida K., Functional Analysis [Russian translation], Mir, Moscow (1967). · Zbl 0152.32102 |
| [26] | Lyashko, SI, Differentiability of regularized performance criterion for pulse-point control of pseudo-parabolic systems, Cybern. Syst. Analysis, 24, 3, 349-352 (1988) · Zbl 0694.93043 · doi:10.1007/BF01132088 |
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