Number of exponential terms describing the solution of anN- compartmental mammillary model: Vanishing exponentials

  • D. P. Vaughan
  • M. J. Dennis
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Abstract

The solution for a linear mammillary model is always described by a summation of m+1 negative exponential terms with constant coefficients m+1⩽N, where N is the number of compartments in the model, m is equal to the number of distinct values for the peripheral Ej values. Use is made of matrix notation and the theorems of Browne concerning the eigenvalues of a matrix. The consequences of vanishing exponentials are derived, and in particular the apparent volume of distribution frequently calculated from experimental data is shown not to be unique.

Key words

mammillary models compartments eigenvalues volume of distribution 
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References

  1. 1.
    L. Z. Benet. General treatment of linear mammillary models with elimination from any compartment as used in pharmacokinetics.J. Pharm. Sci. 61:536–541 (1972).PubMedCrossRefGoogle Scholar
  2. 2.
    D. P. Vaughan and A. Trainor. Derivation of general equations for linear mammillary models when the drug is administered by different routes.J. Pharmacokin. Biopharm. 3:203–218 (1975).CrossRefGoogle Scholar
  3. 3.
    J. G. Wagner. Linear pharmacokinetic models and vanishing exponential terms: Implications in pharmacokinetics.J. Pharmacokin. Biopharm. 4:395–427 (1976).CrossRefGoogle Scholar
  4. 4.
    E. T. Browne. On the separation property of the roots of the secular equation.Am. J. Math. 52:843–850 (1930).CrossRefGoogle Scholar
  5. 5.
    C. W. Sheppard and A. S. Householder. The mathematical basis of the interpretation of tracer experiments in closed steady state systems.J. Appl. Phys. 22:510–520 (1951).CrossRefGoogle Scholar
  6. 6.
    J. Z. Hearon. On the roots of a certain type of matrix.Bull. Math. Biophys. 23:217–221 (1961).CrossRefGoogle Scholar
  7. 7.
    F. R. Gantmacher.The Theory of Matrices, Vol. 1, Chap. 5, Chealsea Pub. Co., New York, 1959, p. 116.Google Scholar
  8. 8.
    J. Z. Hearon. The kinetics of linear systems with special reference to periodic reactions.Bull. Math. Biophys. 15:121–141 (1953).CrossRefGoogle Scholar
  9. 9.
    Theorems of S. Karlin and O. Taussky. Cited by R. Bellman inIntroduction to Matrix Analysis, McGraw-Hill, New York, 1960, pp. 159–182.Google Scholar
  10. 10.
    S. Gerschgorin. Ueber die Abgrenzung der Eigenwerte einer Matrix.Izv. Akad. Nauk SSSR Ser. Fiz.-Mat. 6:749–754 (1931).Google Scholar
  11. 11.
    M. Gibaldi and S. Feldman. Pharmacokinetic basis for the influence of route of administration on the area under the plasma concentration-time curve.J. Pharm. Sci. 58:1477–1488 (1969).PubMedCrossRefGoogle Scholar
  12. 12.
    D. P. Vaughan. Model independent derivation of general equations for the “first-pass” effect and extrahepatic drug elimination.Eur. J. Clin. Pharmacol. 11:57–64 (1977).PubMedCrossRefGoogle Scholar
  13. 13.
    R. L. Nation, B. Learoyd, J. Barber and E. J. Triggs. The pharmacokinetics of chlormethiazole following intravenous administration in the aged.Eur. J. Clin. Pharmacol. 10:407–415 (1976).PubMedCrossRefGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1979

Authors and Affiliations

  • D. P. Vaughan
    • 1
  • M. J. Dennis
    • 1
  1. 1.Sunderland School of PharmacySunderland PolytechnicSunderlandUK

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