Bulletin of Mathematical Biology

, Volume 49, Issue 6, pp 719–727 | Cite as

Approximate solution for capillary exchange models with and without axial mixing

  • H. D. Landahl


Explicit expressions are obtained which give quite accurate values for the effluent blood concentration for the model with axial mixing in tissue for the cases in which the initial capillary concentration is either zero or one. These simple expressions are useful since the formal solutions of Gosselin and Gosselin (1987,Bull. math. Biol 49, 329–349) are cumbersome to use. Reasonably accurate expressions are also provided for the model without axial mixing since the formal solution for the case in which the initial capillary concentration is zero (Gosselin and Gosselin, 1987) and for the case in which the initial concentration equals that of the tissue, the solution for which is provided here, require numerical integration.


Transit Time Axial Diffusion Effluent Concentration Input Concentration Distribution Volume Ratio 
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  1. Bassingthwaighte, J. B. and C. A. Goresky. 1984. “Modeling in the Analysis of Solute and Water Exchange in the Microvasculature”. InHandbook of Physiology, Sec. 2, Vol. IV, part 1, E. M. Renkin and C. C. Michel (Eds), Chap. 13, pp. 549–626. Washington, DC: Am. Physiol. Soc.Google Scholar
  2. Fletcher, J. E. 1978. “Mathematical Modeling of the Microcirculation”.Math. Biosci. 38, 159–202.MATHMathSciNetCrossRefGoogle Scholar
  3. Gosselin, R. P. and R. E. Gosselin. 1987. “The Kinetics of Solute Clearance by a Single Tissue Capillary of Limited Permeability: Explicit Solutions of Two Models and their Bounds”.Bull. math. Biol. 49, 329–349.MATHMathSciNetGoogle Scholar
  4. Hennessey, T. R. 1974. “The Interaction of Diffusion and Perfusion in Homogeneous Tissue”.Bull. math. Biol. 36, 505–526.Google Scholar
  5. Johnson, J. A. and T. A. Wilson. 1966. “A Model for Capillary Exchange”.Am. J. Physiol. 210, 1299–1303.Google Scholar
  6. Sangren, W. C. and C. W. Sheppard. 1953. “A Mathematical Derivation of the Exchange of a Labeled Substance between a Liquid Flowing in a Vessel and an External Compartment”.Bull. math. Biophys. 15, 387–394.CrossRefGoogle Scholar
  7. Schmidt, G. W. 1952. “A Mathematical Theory of Capillary Exchange”.Bull. math. Biophys. 14, 229–263.Google Scholar

Copyright information

© Society for Mathematical Biology 1987

Authors and Affiliations

  • H. D. Landahl
    • 1
  1. 1.Department of Biochemistry and BiophysicsUniversity of CaliforniaSan FranciscoUSA

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