Modeling Medicine Propagation in Tissue: Generalized Statement
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SYSTEMS ANALYSIS MODELING MEDICINE PROPAGATION IN TISSUE: GENERALIZED STATEMENT I. T. Selezov1 and Iu. G. Kryvonos2
UDC 531/534:57
Abstract. A new generalized model of propagation of a medicine in tissue is considered. Instead of the traditional diffusion model described by a parabolic equation, the model described by a more general hyperbolic equation is postulated, which predicts finite velocity of disturbance propagation. As a result, the medicine is delivered to the invaded tissue with a finite velocity. The first disturbance (precursor) carries information from the injection to any point in the tissue and, what is important, to the affected zone, wherefrom information arrives at the brain in the form of neurological disorder. The generalized solution of the problem is considered. A brief generalization of the Bellman problem is given concerning the injection of medicine with respect to the time of injection. Keywords: medicine, tissue, medicine propagation, diffusion, hyperbolic equation. INTRODUCTION In continuum consideration, i.e., in terms of the theory of differential equations, propagation of wave disturbances is described by hyperbolic equations with a finite propagation velocity, which agrees well with the reality (the Maxwell (1864) and Einstein (1907) theories). Kalashnikov [1] represents the mathematical interpretation of the velocity finiteness. However, the cases are possible where hyperbolic equations do not describe disturbances with a finite velocity [2, 3]. In terms of differential equations, we can conventionally distinguish three fields [4]: (i) elliptical equations, (ii) parabolic and hyperbolic–parabolic equations, and (iii) hyperbolic equations. The first field represents non-propagating disturbances, i.e., frozen world (for example, Poisson equation Ñ 2 j = - r / e 0 , which describes electric field of a point charge). The second field describes propagation of disturbances with infinite velocity (for example, diffusion equation DÑ 2 j = ¶ j / ¶ t). The third field corresponds to the real physical process of propagation of disturbances with a finite velocity (for example, acoustics equation Ñ 2 u = c0-2 ¶ 2 u / ¶ t 2 ). In this connection, the well-known parabolic models of diffusion type were generalized and transferred to hyperbolic models [5–7]. Hyperbolic diffusion model is developed in [8]. Unlike the classical diffusion equation, it takes into account the velocity of particles between collisions and hence the duration of free run of the particles. In case of heat propagation within rather short time interval, the classical parabolic equation should be replaced with a more general hyperbolic equation [9]. Computations and comparisons based on improved models with experimental data have shown that in many cases important for modern applications, diffusion equation leads to less real values of temperature at wave front [9]. The qualitative effect of strong energy concentration in peak zone is exhibited at the wave front, which corresponds to hyperbolic equation [10], while in diffusi
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