Towards Mathematical Chemotherapy

  • Martin Eisen
Part of the Lecture Notes in Biomathematics book series (LNBM, volume 30)


Since the discoverer of a “wonder” drug for cancer chemotherapy is more likely to be a biological scientist than a mathematician, one may ask what contribution a mathematician can make to cancer chemotherapy. The answer to this question is that a mathematician can discover ways to use existing drugs more efficiently. Even an efficacious drug can appear worthless if it is not administered properly.


Optimal Control Problem Performance Criterion Pharmacokinetic Model Cancer Chemotherapy Mass Balance Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Aroesty, J., Lincoln, T., Shapiro, N. and Boccia, G. Tumor growth and chemotherapy: mathematical methods, computer simulations, and experimental foundations, Math. Biosciences, 17, 243–300, 1973.CrossRefMATHGoogle Scholar
  2. 2.
    Bagshawe, K.D. Tumor growth and anti-mitotic action; the role of spontaneous cell losses, Brit. J. Cancer, 22, 698–713, 1968.CrossRefGoogle Scholar
  3. 3.
    Bagshawe, K. Choriocarcinoma: The Clinical Biology of the Trophoblast and its Tumors, Edward Arnold Ltd., London, 180–220, 1969.Google Scholar
  4. 4.
    Bahrami, K. and Kim, M. Optimal Control of multiplicative control systems arising from cancer therapy, IEEE Trans. Auto. Cont., Vol. AC-20, 537–542, 1975.Google Scholar
  5. 5.
    Bekey, G. and Benken, J. Identification of biological systems: a survey, Automatica, 14, 41–47, 1978.CrossRefMATHGoogle Scholar
  6. 6.
    BBellman, R., Jacquez, A. and Kalaba, R. Some mathematical aspects of chemotherapy, Proc. Fourth Berkeley Symposium of Math. Stat, and Prob., 4, 57–66, Univ. of Cal. Press, Berkeley, 1961.Google Scholar
  7. 7.
    Bellman, R. and Dreyfys, S. Applied Dynamic Programming, Princeton University Press, Princeton, 1962.MATHGoogle Scholar
  8. 8.
    Berenbaum, M.C. Dose-response curves for agents that impair cell reproductive integrity, Brit. J. Cancer, 24, 434–445, 1969.CrossRefGoogle Scholar
  9. 9.
    Bird, R.B., Stewart, E.W. and Lightfoot, E.W. Transport Phenoma, Wiley, New York, 1960.Google Scholar
  10. 10.
    Bischoff, K.B. Some fundamental consideration of the applications of Pharmacokinetics to cancer chemotherapy. Cancer Chemotherapy Reports, 59, 777–93, 1975.Google Scholar
  11. 11.
    Bischoff, K.B. and Brown, R.G. Drug distribution in mammals, Chem. Eng. Prog. Symp. Series, 84, 64, 33–45, 1968.Google Scholar
  12. 12.
    Bischoff, K.B. and Dedrick, R.L. Generalized solution to linear, two-compartment, open model for drug distribution, J. Theor. Biol. 29, 63–83, 1970.CrossRefGoogle Scholar
  13. 13.
    Bischoff, K.B., Dedrick, R.L., Zaharko, D.S. and Slater, S. A model to represent bile transport of drugs, Proc. Ann. Conf. Eng. Med. Biol., 12, 89, 1970.Google Scholar
  14. 14.
    Bischoff, K.B. Dedrick, R.L., Zaharko, D.S. and Longstreth, J.A. Methotrexate Pharmacokinetics, J. Pharmac. Science, Vol. 60, No. 8, 1128–33, 1971.CrossRefGoogle Scholar
  15. 15.
    Bischoff, K.B., Himmelstein, K.J., Dedrick, R.L. and Zaharko, D.S. Pharmacokinetics and cell population growth models in cancer chemotherapy, in Chemical Engineering in Medicine and Biology: Advances in Chemistry, American Chemical Society, No. 118, 47–64, 1973.Google Scholar
  16. 16.
    Bloch, E.H. A quantitative study of hemodynamics in the living microvascular system, Am. J. Anat., 110, 125–145, 1962.CrossRefGoogle Scholar
  17. 17.
    Burke, P.J. and Owens, A.H. Attempted recruitment of leukemic myeloblasts to proliferative activity by sequential drug treatment, Cancer, 28, 830–836, 1971.CrossRefGoogle Scholar
  18. 18.
    Cox, E.B. Determination of possibility of cure, time to cure, and cell kill fraction in the Gompertz growth model, Duke University Medical Center, Durham, North Carolina, 1978.Google Scholar
  19. 19.
    Creasey, W., Fegley, K., Karreman, G. and Long, V. Designing optimal cancer chemotherapy regimens, in Modelling and Simulation, Vol. 9, Proceedings of the Ninth Annual Pittsburgh Conference, Instrument Society of America, Pittsburgh, 379–385, 1978.Google Scholar
  20. 20.
    Davison, E.J. Simulation of cell behavior: normal and abnormal growth, Bull. Math. Biol., 37, 427–58, 1975.MATHGoogle Scholar
  21. 21.
    Dedrick, R.L. and Bischoff, K.B. Pharmacokinetics in applications of the artificial kidney, Chem. Eng. Prog. Symp. Series No. 84, 64, 32–44, 1968.Google Scholar
  22. 22a.
    Dedrick, R.L., Forrester, D.D. and Ho, D.H.W. In vitro-in vivo correlation of drug metabolism-deamination of l-β-D-Arabinofuranosy cytosine, Biochemical Pharmacology, 21, 1–16, 1972.CrossRefGoogle Scholar
  23. 22b.
    Dedrick, R.L., Zaharko, D.S. and Lutz, R.J. Transport binding of metrotrexate in vivo, J. Pharm. Sci., 62, 882–890, 1973.CrossRefGoogle Scholar
  24. 23.
    Donaghey, C. and Drewinko, B. A computer simulation program for the study of cellular growth kinetics and its application to the analysis of human lymphoma cells in vitro, Comput. Biomed. Res., 8, 118–128, 1975.CrossRefGoogle Scholar
  25. 24.
    Donaghey, C. CELLISM II User’s Manual, Industrial Engineering Department, University of Houston, 1975.Google Scholar
  26. 25.
    Eisen, M. and Macri, N. A model for drug action at the cellular level in Modelling and Simulation, Vol. 9, Proceedings of the Ninth Annual Pittsburgh Conference, Instrument Society of America, Pittsburgh, 393–99, 1978.Google Scholar
  27. 26.
    Garfinkel, D. et al. Simulation of the Krebs Cycle, Computers and Biomed, Res, Res., 4, 1–125, 1971.CrossRefGoogle Scholar
  28. 27.
    Glass, L. Classification of biological networks by their qualitative dynamics, J. Theor. Biol., 54, 85–107, 1975.CrossRefGoogle Scholar
  29. 28.
    Griswold, D.P., Jr., Simpson-Herren, L. and Schabel, F.M., Jr. Altered sensitivity of a hamster plasmacytoma to cytosine arabinoside (NSC-63878). Cancer Chemother. Rep. 54, 338, 1970.Google Scholar
  30. 29.
    Hahn, G.M. and Steward, P.C. The application of age response functions to the optimization of treatment schedules, Cell Tissue Kinet., 4, 279–291, 1971.Google Scholar
  31. 30.
    Heinmets, P. Analysis of Norman and Abnormal Cells, Plenum Press, New York, 1966.Google Scholar
  32. 31.
    Hill, T.L. and Simmons, R.M, Free energy levels and entropy production associated with biochemical kinetic diagrams, Proc, Nat, Acad, Sci., 73, 95–99, 1976.CrossRefGoogle Scholar
  33. 32.
    Himmelstein, K.J. and Bischoff, K.B. Mathematical representations of cancer chemotherapy effects, J, of Pharmacokinetics and Biopharmaceutics, Vol. 1, No. 1, 51–68, 1973.CrossRefGoogle Scholar
  34. 33.
    Harris, E.J. Transport and Accumulation in Biological Systems, Butterworths, England, 1960.Google Scholar
  35. 34.
    Holland, J.G. Clinical studies of unmaintained remissions in acute lymphocytic leukemia, in The Proliferation and Spread of Neoplastic Cells (The University of Texas M.D. Anderson Hospital and Tumor Institute, 21st Annual Symposium on Fundamental Cancer Research, 1967) Williams and Wilkins, Baltimore, 453–62, 1969.Google Scholar
  36. 35.
    Ismail, M., Prasad, T. and Quintana, V, A methodology for modeling and simulation of biomedical systems and drug kinetics using nonlinear stochastic compartmental analysis, in Modeling and Simulation, Vol. 9, Proceedings of the Ninth Annual Pittsburgh Conference, Instrument Society of America, Pittsburgh, 373–378, 1978.Google Scholar
  37. 36.
    Jacquez, J.A., Bellman, R. and Kalaba, R, Some mathematical aspects of chemotherapy II. The distribution of a drug in the body. Bull Math, Biophys., 22, 309–322, 1960.CrossRefMathSciNetGoogle Scholar
  38. 37.
    Jansson, B. Competition within and between cell populations. In Fraction Size in Radiobiology and Radiotherapy, 51–72, editors Sugahara, T., Revesz, L. and O. Scott, Williams and Wilkins, Baltimore, 1974.Google Scholar
  39. 38.
    Jansson, B. Simulation of cell-cycle kinetics based on a multicompartmental model, Simulation, 25, 99–108, 1975.CrossRefMATHGoogle Scholar
  40. 39.
    Jusko, W.J. Pharmacodynamics of chemotherapeutic effects: dose-time-response relationships for phase-nonspecific agents, J, of Pharmaceutical Sciences, 60, 892–895, 1971.CrossRefGoogle Scholar
  41. 40.
    Jusko, W.J. Pharmacokinetic principles in pediatric pharmacology, Pediat, Clin. N. Am., 19, 81–100, 1972.Google Scholar
  42. 41.
    Jusko, W.J. A pharmacodynamic model for cell-cycle-specific chemotherapeutic agents, J. of Pharmacokinetics and Biopharmaceutics, Vol. 1, No, 3, 175–200, 1973.CrossRefGoogle Scholar
  43. 42.
    Kuzma, J.W., Valand, I. and Bateman, J. A tumor cell model for the determination of drug schedules and drug effect in tumor reduction, Bull. of Math. Biophysics, 31, 637–650, 1969.CrossRefMATHGoogle Scholar
  44. 43.
    Lewis, A.E. Principles of Hematology, Appleton Century Croft, New York, 1970.Google Scholar
  45. 44.
    Lincoln, T., Morrison, P., Aroesty, J, and Carter, G. The computer simulation of leukemia therapy: combined pharmacokinetics, intracellular enzyme kinetics, and cell kinetics of the treatment of L-1210 leukemia by ARA-C, Cancer Chemotherapy Reports, 60, 1723–1739, 1976.Google Scholar
  46. 45.
    Lincoln, Th., Aroesty, J., Meier, G. and Gross, J.F, Computer simulation in the service of chemotherapy, Biomedicine, 20, 9–16, 1974.Google Scholar
  47. 46.
    Merkle, T.C. Stuart, R.N. and Gofman, J.W, The Calculation of Treatment Schedules for Cancer Chemotherapy, UCRL-14505, Lawrence Radiation Laboratory, Livermore, California, 1965.Google Scholar
  48. 47.
    Mikulecky, D.C. A network thermodynamic two-port element to represent the coupled flow of salt and current. Improved alternative for the equivalent circuit, Biophysical J., 25, 323–340, 1979.CrossRefGoogle Scholar
  49. 48.
    Mikulecky, D.C. Huf, E.G. and Thomas, R.S. A network thermodynamic approach to compartmental analysis of Na+ transients in frog skin, Biophysical J., 25, 87–106, 1979.CrossRefGoogle Scholar
  50. 49.
    Milgram, E. and Nicolini, C.A. A preliminary report on the mathematical-numerical considerations of enzyme kinetics, Biophysics Division, Temple University Internal Report 4/75, 1–54, 1975.Google Scholar
  51. 50.
    Morrison, P.F., Lincoln, T.L. and Aroesty, J. Disposition of cytosine arabinoside (NSC-63878) and its metabolites: a pharmacokinetic simulation, Cancer Chemotherapy Reports, Part 1, Vol, 59, 861–75, 1975.Google Scholar
  52. 51.
    Oster, G., Perelson, A. and Katchalsky, A, Network thermodynamics, Nature, 234, 393–399, 1971.CrossRefGoogle Scholar
  53. 52.
    Oster, G.S., Perelson, A.S. and Katchalsky, A. Network thermodynamics: dynamic modelling of biophysical systems, Quart, Rev, Biophys., 6, 1–134, 1973.CrossRefGoogle Scholar
  54. 53.
    Petrovskii, A.M., Suchkov, V.V. and Shkhvatsabaya, I.K. Cure management of disease as a problem in modern control theory, Automatika I Telemekhanika, 34, 99–105 (English translation — Automation and Remote Control, 34, 767–771) 1973.Google Scholar
  55. 54.
    Petrovskii, A.M. Systems analysis of some medicobiological problems connected with treatment control, Automatika I Telemekhanika, 35, 54–62 (English tran translation — Automation and Remote Control, 35, 219–225) 1974.Google Scholar
  56. 55.
    Prasad, T. and Ibidapo-Obe, O. Stochastic analysis and control of physiologic systems: cancer detection and therapy, Int. J. Systems Sci., 8, 1233–42, 1977.CrossRefMATHMathSciNetGoogle Scholar
  57. 56.
    Priore, R.L. Using a mathematical model in the evaluation of human tumor response to chemotherapy, J. Natl. Cancer Inst., 37, 635–47, 1966.Google Scholar
  58. 57.
    Raughi, G.J., Liang, T. and Blum, J.J. A quantative analysis of metabolite fluxes along some of the pathways of intermediary metabolism in tetrahymena pyriformis, J. Biol. Chem., 250, 5866–76, 1975.Google Scholar
  59. 58.
    Salmon, S.E. and Durie, B.G.M. Applications of kinetics to chemotherapy for multiple myeloma, in Growth Kinetics and Biochemical Regulation of Normal and Malignant Cells, Williams and Wilkins (Drewinko, B. and Humphrey, R. M. ed.), William and Wilkins, Baltimore, 815–77, 1977.Google Scholar
  60. 59.
    Salmon, S.E. and Smith, B.A. Immunoglobulin synthesis and total body tumor cell number in IgG multiple myeloma, J. Clin, Invest., 49, 1114 – 1121, 1970.CrossRefGoogle Scholar
  61. 60.
    Sawicki, W., Rowinski, J. and Swenson, R. Change of chromatin morphology during the cell cycle detected by means of automated image analysis, J. of Cellular Physiol., 84, 423–428, 1974.CrossRefGoogle Scholar
  62. 61.
    Schmidt, G.W. A mathematical theory of capillary exchange as a function of tissue structure, Bull. Math. Biophys., 14, 229–63, 1952.CrossRefGoogle Scholar
  63. 62.
    Schackney, S.E. A computer model for tumor growth and chemotherapy and its applications to L-1210 leukemia treated with cytosine arabinoside (NSC-63878), Cancer Chemother. Rep. Part I, 54, 399–429, 1970.Google Scholar
  64. 63.
    Skipper, H.E., Schabel, F.M. and Wilcox, W.O. Experimental evaluation of potential anticancer agents, XIII, On the criteria and kinetics associated with “curability” of experimental leukemias, Cancer Chemother. Rep., 35, 111, 1964.Google Scholar
  65. 64.
    Smolen, V.F., Turrie, B.D. and Weigard, W.A. Drug input optimization: bioavailability-effected time-optimal control of multiple, simultaneous, pharmacological effects and their interrelationships, J. of Pharmaceutical Sci., 61, 1941–52, 1972.CrossRefGoogle Scholar
  66. 65.
    Stuart, R.M. and Merkle, T.C. The Calculation of Treatment Schedules for Cancer Chemotherapy, Part II, UCRL-14505, Univ. of Cal, Lawrence Laboratory, Livermore, California, 1965.Google Scholar
  67. 66.
    Sullivan, P.W. and Salmon, S.E. Kinetics of tumor growth and regression in IgG multiple myeloma, J. Clin. Invest., 51, 1697–1708, 1972.CrossRefGoogle Scholar
  68. 67.
    Swan, G.W. and Vincent, T.L. Optimal control analysis in the chemotherapy of IgG multiple myeloma, Bul. Math. Biol., 39, 317–337, 1977.MATHGoogle Scholar
  69. 68.
    Swan, G.W. Optimal control in some cancer chemotherapy problems, unpublished report.Google Scholar
  70. 69.
    Valeriote, F.A., Bruce, W.R. and Meeker, B.E. A model for the action of vinblastine in vivo, Biophys. J., 6, 145–152, 1966.CrossRefGoogle Scholar
  71. 70.
    Werkheiser, W.C. Mathematical simulation in chemotherapy, Ann. New York Acad. Sci., 186, 343–58, 1971.CrossRefGoogle Scholar
  72. 71.
    Werkheiser, W.C., Gridney, G.B., Moran, R.G. and Nichol, C.A. Mathematical simulation of the interaction of drugs that inhibit deoxyribonucleic acid biosynthesis, Mol. Pharmacol., 9, 320–29, 1973.Google Scholar
  73. 72.
    Wilson, R.L. and Gehan, E.A., A digital simulation of cell kinetics with application to L-1210 cells, Computer Programs in Biomedicine, 1, 65–73, 1970.CrossRefGoogle Scholar
  74. 73.
    Zakharova, L.M., Petrovskii, A.M. and Shtabtsov, V.I. Matrix model for selection of pharmacological treatment, Automatika I Telemkhanika, 34, 58–61 (English translation — Automation and Remote Control, 34, 1763–1764), 1973.Google Scholar
  75. 74.
    Zietz, S. Mathematical Modeling of Cellular Kinetics and Optimal Control Theory in the Service of Cancer Chemotherapy, Ph.D. Thesis, Dept. of Math., Univ. of California, Berkeley, California, 1977.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1979

Authors and Affiliations

  • Martin Eisen
    • 1
  1. 1.Department of MathematicsTemple UniversityPhiladelphiaUSA

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