Capillary operators

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

$$K_{a,k} = \mathop {\lim _{(p)} }\limits_{N \to \infty } \left( {\sum\limits_{n = 0}^N {P_n (a)O_k^n } } \right)$$

(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 (‖O_k‖=‖K_a,k‖_p=1). Consideration is also given to the norms and operators when the functions involved are limited to a finite interval of time.