Skip to main content
SpringerLink
Log in

Structure and kinetic behavior of linear multicompartment systems

  • Published:
The bulletin of mathematical biophysics Aims and scope Submit manuscript

Abstract

The general kinetic behavior of a multicompartment system is shown to depend upon certain general structural features, including its connectivity, whether it is open, and whether it contains cyclic pathways. Structural influences are clarified by putting the system matrix in a certain form. For systems not strongly connected, a distinction is drawn between partially and completely open systems. Necessary and sufficient conditions are given for non-singularity of the system matrix and for asymptotic stability of the system. Sufficient conditions are given for non-overshooting and monotonic transitions. A system is demonstrated whose solution may contain a prolonged series of damped oscillations; but the oscillations are very slow and small; and it seems unlikely that oscillations could be detected experimentally in any biological system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature

  • Bellman, R. 1960.Introduction to Matrix Analysis. New York: McGraw-Hill Book Company, Inc.

    MATH  Google Scholar 

  • Brauer, A. 1946. “Limits for the Characteristic Roots of a Matrix”.Duke Math. J.,13, 387–395.

    Article  MATH  MathSciNet  Google Scholar 

  • Coddington, E. A. and N. Levinson. 1955.Theory of Ordinary Differential Equations. New York: McGraw-Hill Book Company, Inc.

    MATH  Google Scholar 

  • Gantmacher, F. R. 1959.The Theory of Matrices, Vol. 2. New York: Chelsea Publishing Company.

    MATH  Google Scholar 

  • Gerschgorin, S. 1931. “Uber die Abgrenzung der Eigenwerte einer Matrix”.Isvest. Akad. Nauk SSSR,7, 749–754.

    Google Scholar 

  • Goldberg, K. 1956. “A Matrix with Real Characteristics Roots”.J. Res. Natl. Bur. Standards,56, 87.

    MATH  Google Scholar 

  • Guggenheim, E. A. 1926. “On the Determination of the Velocity Constant of a Unimolecular reaction”.Phil. Mag. (series 7),2, 538–543.

    Google Scholar 

  • Hearon, J. Z. 1953. “The Kinetics of Linear Systems with Special Reference to Periodic Reactions”.Bull. Math. Biophys.,15, 121–141.

    MathSciNet  Google Scholar 

  • — 1963. “Theorems on Linear Systems”.Ann. N. Y. Acad. Sci.,108, 36–68.

    MATH  MathSciNet  Google Scholar 

  • Lederman, W. 1950. “On the Asymptotic Probability Distribution for Certain Markoff Processes”.Proc. Cambridge Phil. Soc.,46, 581–594.

    Article  Google Scholar 

  • Muir, T. and W. H. Metzler. 1930.A Treatise on the Theory of Determinants. Menasha, Wisconsin: The Collegiate Press, George Banta Publishing Company.

    Google Scholar 

  • Parter, S. 1960. “On the Eigenvalues and Eigenvectors of a Class of Matrices”.J. Soc. Indust. Appl. Math.,8, 376–388.

    Article  MATH  MathSciNet  Google Scholar 

  • Rescigno, A. and G. Segre. 1966.Drug and Tracer Kinetics. Waltham, Massachusetts: Blaisdell Publishing Company.

    Google Scholar 

  • Taussky, O. 1949. “A Recurring Theorem on Determinants”Am. Math. Monthly,56, 672–676.

    Article  MATH  MathSciNet  Google Scholar 

  • Zierler, K. 1963. “Effect of Circulatory Beds on Tracer Experiments, or Noncompartmental Analysis”.Ann. N.Y. Acad. Sci.,108, 106–116.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pharmacology and Toxicology, Dartmouth Medical School, 03755, Hanover, New Hampshire

    C. D. Thron

Authors
  1. C. D. Thron
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was supported by U.S. Public Health Service Research Grant NS 08710 from the National Institute of Neurological Diseases and Stroke.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Thron, C.D. Structure and kinetic behavior of linear multicompartment systems. Bulletin of Mathematical Biophysics 34, 277–291 (1972). https://doi.org/10.1007/BF02476521

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02476521

Keywords

Search

Navigation