Skip to main content
Log in

Some mathematical aspects of chemotherapy: I. One-organ models

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

Abstract

A model of the processes occuring in the exchange of a drug between capillary plasma, extracellular space and intracellular space is developed. This leads to an interesting set of differential differences equations, one of which is an integrodifferential equation, another a partial differential equation. Under certain conditions, these may be simplified to a set of ordinary differential equations. The application of Laplace transform techniques to the solution of these equations is discussed.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

Literature

  • Ansoff, H. and J. Krumhansl. 1948. “A General Stability Criterion for Linear Oscillating Systems with Constant Time Lag”.Q. Appl. Math.,6, 337–41.

    MATH  MathSciNet  Google Scholar 

  • bellman, R. and J. M. Danskin. 1954.A Survey of the Mathematical Theory of Time-lag, Retarded Control and Hereditary Processes. The RAND Corporation, Report R-256.

  • Chinard, F. P., G. J. Vosburgh, and T. Enns. 1955. “Transcapillary Exchange of Water and of Other Substances in Certain Organs of the Dog”.Am. J. Physiol.,183, 221–34.

    Google Scholar 

  • Collatz, L. 1947. “Über Stabilität von Regelungen mit Nachlaufzeit”.Z. Angew. Math. Mech.,25–27, 60–63.

    MathSciNet  Google Scholar 

  • Conn, H. L. and J. S. Robertson. 1955. “Kinetics of Potassium Transfer in the Left Ventricle of the Intact Dog”.Am. J. Physiol.,181, 319–324.

    Google Scholar 

  • Dobson, E. L. and G. F. Warner. 1957. “Measurement of Regional Sodium Turnover Rates and their Application to the Estimation of Regional Blood Flow”.Am. J. Physiol.,189, 269–76.

    Google Scholar 

  • Freis, E. D., T. F. Higgins, and H. J. Morowitz. 1953. “Transcapillary Exchange Rate of Deuterium Oxide and Thiocyanate in Forearm of Man”.J. Appl. Physiol.,5, 526–32.

    Google Scholar 

  • Hayes, N. D. 1960. “Roots of the Transcendental Equation Associated with a Certain Difference-Differential Equations”.J. London Math. Soc.,25, 226–32.

    MathSciNet  Google Scholar 

  • Heinz, E. 1954. “Kinetic Studies on the ‘Influx’ of Glycine-1-014 into the Ehrlich Mouse Ascites Carcinoma Cells”.J. Biol. Chem.,211, 781–90.

    Google Scholar 

  • Jacquez, J. A. 1957. “Active Transport of O-Diazoacetyl-L-Serine and 6-Diazo-5-Oxo-L-Norleucine in Ehrlich Ascites Carcinoma”.Cancer Res.,17, 890–96.

    Google Scholar 

  • Jacquez, J. A., R. Bellman, and R. Kalaba. 1959.The Distribution of a Drug in the Body. The RAND Corporation, Paper P-1560.

  • Jones, H. B. 1950. “Respiratory System: Nitrogen Elimination”. inMedical Physics, ed. O. Glasser,2, 855–71. Chicago: Yearbook Pub.

    Google Scholar 

  • Lax, P. D. 1948. “The Quotient of Exponential Polynomials”.Duke Math. J.,15, 967–70.

    Article  MATH  MathSciNet  Google Scholar 

  • Morales, M. F. and R. E. Smith. 1948. “On the Theory of Blood Tissue Exchange of Inert Gases, VI”.Bull. Math. Biophysics,10, 191–200.

    Article  Google Scholar 

  • Pappenheimer, J. R., E. M. Renkin, and L. M. Borrero. 1951. “Filtration, Diffusion and Molecular Sieving Through Peripheral Capillary Membranes”.Am. J. Physiol.,167, 13–46.

    Google Scholar 

  • Pontrjagin, L. 1942. “On Nuliakh Nekotorykh Elementarnykh Transtsendentrykh Funktsii”.Bull. Acad. Sci. URSS, Ser. Math., (Izvestiya Akad. Nauk. SSSR),6, 115–34.

    MATH  MathSciNet  Google Scholar 

  • — 1955. “On Zeros of Some Transcendental Functions”.Amer. Math. Soc. Translations, Series 2,1, 95–110.

    Google Scholar 

  • 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. Biophysics,15, 387–94.

    Article  Google Scholar 

  • Schmidt, G. W. 1952. “A Mathematical Theory of Capillary Exchange as a Function of Tissue Structure”.Bull. Math. Biophysics,14, 229–63.

    Article  Google Scholar 

  • Schmidt, G. W. 1953. “The Time Course of Capillary Echange”.Bull. Math. Biophysics,15, 477–88.

    Article  Google Scholar 

  • Smith, R. E. and M. F. Morales, 1944. “On the Theory of Blood Tissue Exchanges, I”.Bull. Math. Biophysics,6, 125–31.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bellman, R., Jacquez, J.A. & Kalaba, R. Some mathematical aspects of chemotherapy: I. One-organ models. Bulletin of Mathematical Biophysics 22, 181–198 (1960). https://doi.org/10.1007/BF02478005

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02478005

Keywords

Navigation