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Capillary operators—II

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Abstract

This work continues with an examination of capillary exchange models as operators, namely the operatorsO k andK αk 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 holdsK αk +K βk =K α+β,k ; the operatorK αk has an inverse;K -1 αk , (as an operator on LebesgueL p space or on the locally integrable functions); partial derivatives are given forK αk [f](t) andO k [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 ‖K αk f-K βl f‖ p and ‖O k f-O l f‖ p for allL p . Consequent limitations on relative errors are shown. The implications for operators on a finite time interval are deduced.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Loyola University, 70018, New Orleans, LA, U.S.A.

    J. M. Bateman

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  1. J. M. Bateman
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This work supported in part by PHS Grant Nos HL-19153 (SCOR and Pulmonary Vascular Disease) and HL-19370 at Vanderbilt University Medical School.

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Bateman, J.M. Capillary operators—II. Bltn Mathcal Biology 47, 651–668 (1985). https://doi.org/10.1007/BF02460131

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  • DOI: https://doi.org/10.1007/BF02460131

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