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8vo (21.5 cm), 243 pp. Publisher's laminated wrappers. Mathematical Studies: Monograph Series, vol. 7. Edited by Michael Zarichnyi. A research monograph in mathematical biophysics by Vasyl Gafiychuk (Institute of Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv) and Ihor Lubashevsky (General Physics Institute of the Russian Academy of Sciences / Moscow State University), presenting a mathematical theory of bioheat transfer in living tissue regarded as a hierarchically organized, active, heterogeneous medium. The model treats biological tissue as consisting of a continuous cellular component penetrated by a highly branched arterial and venous vascular network, deriving macroscopic governing equations from mesoscopic descriptions of transport within individual vessels. Exploiting the correspondence between diffusion processes and random walks, the authors develop a systematic averaging procedure over all levels of the vascular hierarchy, yielding continuum equations that incorporate the self-regulating behaviour of living tissue. A central theme is the emergence of ideal self-regulation in large hierarchical systems, whereby global regulation arises through purely local interactions between individual elements, without any component possessing complete knowledge of the system as a whole--a mechanism the authors suggest may apply equally to ecological and economic systems. The volume is divided into six parts comprising seventeen chapters. Part I, The Basis of the Bioheat Transfer Theory, introduces transport in heterogeneous media, physiological aspects of heat transfer, hierarchical vascular models, and governing equations, concluding with a random-walk formulation of heat transport through vascular networks. Part II, Transport Phenomena Caused by Blood Flow through Hierarchical Vascular Networks, develops generalized bioheat equations describing the influence of vascular architecture and blood flow. Part III examines heat transfer under highly nonuniform blood-flow distributions and distinguishes between averaged and actual perfusion rates. Part IV, Theory of Heat Transfer in Living Tissue with Temperature Self-Regulation, investigates local thermal regulation, ideal self-regulation, and applications to tumour tissue, including mathematical models relevant to hyperthermia and cryosurgical treatment. Part V studies spatial and temporal temperature fluctuations and small-scale thermal heterogeneity, concluding with a discussion of the medical and biophysical implications of the theory. Part VI consists of an extensive appendix exploring cooperative mechanisms of self-regulation in large hierarchical systems, including applications to ecological trophic networks and market structures under perfect competition. The volume concludes with a bibliography and index. A significant interdisciplinary contribution at the intersection of applied mathematics, mathematical biology, and biophysics, offering a rigorous theoretical framework for bioheat transfer and the dynamics of complex hierarchical systems.
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