Capillary operators |
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Authors: | J. M. Bateman T. R. Harris |
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Affiliation: | 1. Department of Medicine, Vanderbilt University Medical School, Biomedical Engineering Program, School of Enginering, Nashville, TN, USA
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Abstract: | The goal of this work is an examination of capillary exchange models as mathematical operators. The concentration function relations for the Krogh cylinder of a single capillary, basic to many organ models, are studied via the theory of operators on the Lebesgue normed spacesL p[0,∞], (1<-p<-∞). A discussion is included of theL p -normsvis-à-vis the coefficient of variation currently used in finding capillary parameters and evaluating parameter searches. The capillary model determines two operators on the space of locally integrable functions: O K (relating extravascular concentration to intravascular) and K a, k (relating intravascular concentration to input), wherek is the ratio of permeabilitysurface area (PS) to extravascular volume, and α is the ratio of PS to flow. These operators are shown to induce contractive (‖O K ‖ p <-1, ‖K a, k ‖ p <-1), isotone, linear operators onL p . The uniform convergence relation $$K_{a,k} = mathop {lim _{(p)} }limits_{N to infty } left( {sumlimits_{n = 0}^N {P_n (a)O_k^n } } right)$$ (as operators onL p) is derived, whereP n (a) is the Poisson probabilitye ?a a n /n!. For the important special cases ofp=∞, 1, 2 the norms are found (‖Ok‖=‖Ka,k‖p=1). Consideration is also given to the norms and operators when the functions involved are limited to a finite interval of time. |
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