Capillary operators—II

This work continues with an examination of capillary exchange models as operators, namely the operatorsOk andKαk relating extravascular and intravascular concentration to input for the Krogh cylinder model of a single capillary, a model basic to many organ models. Fundamental algebraic and analytic properties are presented: the operators belong to a commutative Banach algebra; an addition theorem holdsKαk +Kβk =Kα+β,k; the operatorKαk has an inverse;Kαk-1, (as an operator on LebesgueLp space or on the locally integrable functions); partial derivatives are given forKαk[f](t) andOk[f](t) (sensitivity functions); and inequalities are established for the derivatives. Dominance relations between model curves are inferred. Error bound formulas are presented forK andO as bounds on ‖Kαkf-Kβlf‖p and ‖Okf-Olf‖p for allLp. Consequent limitations on relative errors are shown. The implications for operators on a finite time interval are deduced.