Generalized Optimization of Processes of Drug Transport in Tumors
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GENERALIZED OPTIMIZATION OF PROCESSES OF DRUG TRANSPORT IN TUMORS D. A. Klyushin,1† S. I. Lyashko,1‡ N. I. Lyashko,2 O. S. Bondar,1‡ and A. A. Tymoshenko1††
UDC 517.9, 519.6
Abstract. 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. Keywords: optimization, controllability, partial differential equation. INTRODUCTION During the research of mathematical models of drag transport in cancerous tumors, there arises a need to solve optimization and controllability problems described by partial differential equations, where the coefficients and the right-hand sides belong to different functional spaces. These models, in particular, pharmacokinetic models and models of transport and distribution of chemotherapeutic agents inside a cancerous tumor have been studied in [1–6], and in many other research papers. Improving the prediction methods of drug distribution in tumors has necessitated a need for optimal control over the drug source with the aim of minimizing cancer cell density, as well as side effects. A model of optimal control over drug concentration and cancer cell density has been developed in [7]. The aim of [7] was to minimize cancer cell density and to manage side effects of drugs. This model has been expanded in [8–10] for the case of controlling point sources implanted into the tumor and controlled by the Dirac delta functions. It is expedient to study such systems in the framework of the rigged Hilbert space theory and the a priori estimates in negative norms. Optimization problems of parabolic type with concentrated sources have been examined in [11–14]. The theory underpinning these studies has been presented in [15–19]. Studies of a similar topic have been performed by A. A. Chikrii and his co-authors [20], O. M. Khimich and co-authors [21], À. V. Gladky [22] and V. M. Bulavatsky [23]. The aim of this paper is to develop a method of generalized optimization of processes of parabolic type that appear in problems of drug transfer in biological tissues [24].
1
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, †[email protected]; ‡[email protected]; [email protected]. 2V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. Translated
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