Skip to main content
SpringerLink
Log in

Mathematical models in oncology: a bird's-eye view

  • Editorial
  • Published:
Zeitschrift für Krebsforschung und Klinische Onkologie Aims and scope Submit manuscript

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

References

  • Arley,N.: Theoretical analysis of carcinogenesis. Proc. 4th Berkeley Symp. Math. Statist. Probability, Vol. IV, pp. 1–8. Berkely: Univ. California Press 1961

    Google Scholar 

  • Arley,N., Eker,R.: The dose-response relationship in radiation carcinogenesis. Nature 189, 151–152 (1961)

    Google Scholar 

  • Arley,N., Eker,R.: Mechanisms of carcinogenesis. Advances Biol. Med. Phys. 8, 375–436 (1962)

    Google Scholar 

  • Arley,N., Iversen,S.: On the mechanism of experimental carcinogenesis. III. Further development of hit theory of carcinogenesis. Acta Pathol. Microbiol. Scand. 30, 21–53 (1951)

    Google Scholar 

  • Arley,N., Iversen,S.: On the mechanism of experimental carcinogenesis. VI. Hit theoretical interpretations of some experiments of Berenblum and Shubik. Acta Pathol. Microbiol. Scand. 31, 164–171 (1952)

    Google Scholar 

  • Arley,N., Iversen,S.: Application of the quantum hit theory to virus-provoked tumours. Nature 169, 410–411 (1952)

    Google Scholar 

  • Arley,N., Iversen,S.: On the mechanism of experimental carcinogenesis. IX. Application of the hit theory to tumours produced by ultraviolet radiation. Acta Pathol. Microbiol. Scand. 33, 133–150 (1953)

    Google Scholar 

  • Armitage,P., Doll,R.: The age distribution of cancer and a multistage theory of carcinogenesis. Brit. J. Cancer 8, 1–12 (1954)

    Google Scholar 

  • Armitage,P., Doll,R.: A two-stage theory of carcinogenesis in relation to the age distribution of human cancer. Brit. J. Cancer 11, 161–169 (1957)

    Google Scholar 

  • Armitage,P., Doll,R.: Stochastic models for carcinogenesis. Proc. 4th Berkeley Symp. Math. Statist. Probability, Vol. IV, pp. 19–38. Berkeley: Univ. California Press 1961

    Google Scholar 

  • Aroesty,J., Lincoln,T., Shapiro,N., Boccia,G.: Tumor growth and chemotherapy: Mathematical models, computer simulations, and experimental foundations. Math Biosci. 17, 243–300 (1973)

    Google Scholar 

  • Ashley,D.J.B.: The two “hit” and multiple “hit” theories of carcinogenesis. Brit. J. Cancer 23, 313–328 (1969)

    Google Scholar 

  • Aspden,P., Jackson,R.P.P.: Transient solution of a system of random arrivals served in series with always sufficient servers. Res. Rep. OR Service, Dept. of Health and Social Security, London 1975

    Google Scholar 

  • Bahrami,K., Kim,K.: Optimal control of multiplicative control systems arising from cancer therapy. IEEE Trans. Autom. Control AC-20, 537–542 (1975)

    Google Scholar 

  • Barron,B.A., Richart,R.M.: A statistical model of the natural history of cervical carcinoma. I, II. J. Natl. Cancer Inst. 41, 1343–1353 (1968); 45, 1025–1030 (1970)

    Google Scholar 

  • Bartosziński,R.: A stochastic model of age dependent inheritance of cancer proneness (Abstr.). Bull. Intern. Statist. Inst. 42, Book 1, 158–159 (1967)

    Google Scholar 

  • Bartosziński,R., D'Abrera,H.: A model of age-dependent inheritance. Austral. J. Statist. 11, 14–28 (1969)

    Google Scholar 

  • Bell,G.I.: Models of carcinogenesis as an escape from mitotic inhibitors. Science 192, 569–572 (1976)

    Google Scholar 

  • Berenbaum,M.C: Dose-response curves for agents that impair cell reproductive integrity. I, II. Brit. J. Cancer 23, 426–433 (1969)

    Google Scholar 

  • Berkson,R., Gage,R.P.: Survival curve for cancer patients following treatment. J. Amer. Statist. Assoc. 47, 501–515 (1952)

    Google Scholar 

  • Bischoff,K.B.: Pharmacokinetics and cancer chemotherapy. J. Pharmacokin. Biopharm. 1, 465–480 (1973)

    Google Scholar 

  • Bischoff,K.B.: Some fundamental considerations of the applications of pharmacokinetics to cancer chemotherapy. Cancer Chemother. Rep. 59,777–793 (1975)

    Google Scholar 

  • Blum,H.F.: On the mechanism of cancer induction by ultraviolet radiation. I–IV. J. Natl. Cancer Inst. 11, 463–495 (1950); 23, 319–335, 337–342, 343–350 (1959)

    Google Scholar 

  • Blum,H.F.: Comparable models for carcinogenesis by ultraviolet light and by chemical agents. Proc. 4th Berkeley Symp. Math. Statist. Probability, Vol. IV, pp. 101–121. Berkeley: Univ. California Press 1961

    Google Scholar 

  • Blum,H.F.: Uncertainty of growth of cell populations in cancer. J. Theor. Biol. 46, 143–166 (1974)

    Google Scholar 

  • Blumenson,L.E.: Random walk and the spread of cancer. J. Theor. Biol. 27, 273–290 (1970)

    Google Scholar 

  • Blumenson,L.E.: A comprehensive modeling procedure for the human granulopoietic system: Detailed description and application to cancer chemotherapy. Math. Biosci. 26, 217–239 (1975)

    Google Scholar 

  • Blumenson,L.E.: When is screening effective in reducing the death rate? Math. Biosci. 30, 273–303 (1976)

    Google Scholar 

  • Blumenson,L.E.: Detection of disease with periodic screening: Transient analysis and application to mammography examination. Math. Biosci. 33, 73–106 (1977a)

    Google Scholar 

  • Blumenson,L. E.: Compromise screening strategies for chronic disease. Math. Biosci. 34, 79–94 (1977 b)

    Google Scholar 

  • Blumenson,L.E., Bross,I.D.J.: A mathematical analysis of the growth and spread of breast cancer. Biometrics 25, 95–109 (1969)

    Google Scholar 

  • Blumenson,L.E., Bross,I.D.J.: Use of mathematical model to bridge the clinic-laboratory gap: Local spread of endometrial cancer. J. Theor. Biol. 38, 397–411 (1973)

    Google Scholar 

  • Blumenson,L.E., Bross,I.D.J.: A possible mechanism for enhancement of increased production of tumor angiogenic factor. Growth 40, 205–209 (1976)

    Google Scholar 

  • Brodsky,A., Thompson,D.J.: A stochastic model of carcinogenesis involving two stages of excitation (Abstr.). Biometrics 23, 599–600 (1967)

    Google Scholar 

  • Burch,P.R.J.: Radiation carcinogenesis: A new hypothesis. Nature 185, 135–142 (1960)

    Google Scholar 

  • Burton,A.C: Rate of growth of solid tumors as a problem of diffusion. Growth 30, 157–176 (1966)

    Google Scholar 

  • Bühler,W.J.: Ein stochastisches Modell der Krebsentstehung (Abstr.). Biometrics 23, 870 (1967 a)

    Google Scholar 

  • Bühler,W.J.: Single cell against multicell hypotheses of tumor formation. Proc. 5th Berkeley Symp. Math. Statist. Probability, Vol. IV, pp. 635–637. Berkeley: Univ. California Press 1967 b

    Google Scholar 

  • Bühler,W.J.: Quasi-competition of two birth and death processes. Biometrische Z. 9, 76–83 (1967 c)

    Google Scholar 

  • Bühler,W.J.: Mathematische Aspekte der Krebsforschung. Naturwiss. 55, 121–125 (1968)

    Google Scholar 

  • Bühler,W. J.: Genesis, growth and therapy of tumors: Mathematical models. Meth. Inform. Med. 9, 53–57 (1970)

    Google Scholar 

  • Bühler,W.J.: Age dependence or parity dependence of inheritance. Res. Rep., Dept. of Mathematics, Univ. of Mainz 1975

  • Bühler, W.J.: Quasi-competition: A new aspect. Biometrical J. (in press) (1977)

  • Clifford, P., Sudbury,A.: Some mathematical aspects of two-dimensional cancer growth. Unpubl. paper, Tobacco Res. Council Symp., London 1971

  • Chow,I.A.: A restricted immigration and death process and its application to the distribution of polymorphonuclear granulocyte density in acute leukemia. Math. Biosci. 13, 253–264 (1972)

    Google Scholar 

  • Chow,I.A.: A stochastic approach to survival problem with concomitant variables and application to acute leukemia patients. Math. Biosci. 21, 103–118 (1974)

    Google Scholar 

  • Chuang,S.N.: Mathematic models for cancer chemotherapy: Pharmacokinetic and cell kinetic considerations. Cancer Chemother. Rep. 59, 827–842 (1975)

    Google Scholar 

  • Chuang,S.N., Lloyd,H.H.: Analysis and identification of stochastic compartment models in pharmacokinetics: Implications for cancer chemotherapy. Math. Biosci. 22, 57–74 (1974)

    Google Scholar 

  • Chuang,S.N., Lloyd,H.H.: Mathematical analysis of cancer chemotherapy. Bull. Math. Biol. 37, 147–160 (1975)

    Google Scholar 

  • Deakin,A.S.: Model for the growth of a solid in vitro tumor. Growth 39, 159–165 (1975)

    Google Scholar 

  • DeLisi,C.: Immunological reactions to malignancies: Some additional sources of complexity. In: Environmental Health (ed. A. Whittemore) pp. 149–171. Philadelphia: SIAM 1977

    Google Scholar 

  • DeLisi,C., Rescigno,A.: Immune surveillance and neoplasia. I. A minimal mathematical model. Bull. Math. Biol. 39, 201–221 (1977)

    Google Scholar 

  • De Waard,H.H.: Coincidence of mutations as a possible cause of malignancy. Intern. J. Radiol. Biol. 8, 381–387 (1965)

    Google Scholar 

  • Downham,D.Y., Green,D.H.: Inference for a two-dimensional stochastic growth model. Biometrika 63, 551–554 (1976)

    Google Scholar 

  • Downham,D.Y., Morgan,R.K.B.: A stochastic model for a two-dimensional growth on a square lattice. Bull. Intern. Statist. Inst. Vol. 45, Book 1, pp. 324–331, 1973a

  • Downham,D.Y., Morgan,R.K.B.: Growth of abnormal cells. Nature 242, 528–530 (1973b)

    Google Scholar 

  • Duchatelier,M., Israel,L.: Growth fraction, resistance, schedule-doubling time relationship, sequential versus simultaneous combination, as evaluated by a mathematical model of response to chemotherapy. Europ. J. Cancer 7, 545–549 (1971)

    Google Scholar 

  • Dubin,N.: A Stochastic Model for Immunological Feedback in Carcinogenesis. Lecture Notes in Biomath. Vol. 9. Berlin-Heidelberg-New York: Springer 1976

    Google Scholar 

  • Eisen,M., Schiller,J.: Stability analysis of normal and neoplastic growth. Bull. Math. Biol. 39, 597–605 (1977)

    Google Scholar 

  • Fellner, W.H.: Carcinogenesis in the presence of tissue regeneration. Ph.D.Thesis, Univ. of California 1974

  • Fischer,J.J.: Mathematical simulation of radiation therapy of solid tumors. I, II. Acta Radiol. 10, 73–85; 267–278 (1971)

    Google Scholar 

  • Fisher,J.C.: Multiple-mutation theory of Carcinogenesis. Nature 181, 651–652 (1958)

    Google Scholar 

  • Fisher,J.C., Hollomon,J.H.: A hypothesis for the origin of cancer foci. Cancer 4, 916–918 (1951)

    Google Scholar 

  • Fix,E., Neyman,J.: A simple stochastic model of recovery, relapse, death and loss of patients. Human Biol. 23, 205–241 (1951)

    Google Scholar 

  • Gaudin,D.: Some thoughts on a possible relationship between known gene dosage effects and neoplastic transformation. J. Theor. Biol. 41, 191–200 (1973)

    Google Scholar 

  • Green,D.H., Downham, D.Y.: An extension of the two-dimensional stochastic model of Williams and Bjerknes. Unpubl. paper, 9th European Meeting of Statisticians, Grenoble 1976

  • Greenspan,H. P.: Models for the growth of a solid tumor by diffusion. Studies Appl. Math. 51, 317–340 (1972)

    Google Scholar 

  • Greenspan,H. P.: On the growth and stability of cell culture and solid tumors. J. Theor. Biol. 56, 229–242 (1976)

    Google Scholar 

  • Gurel,O.: Dynamics of cancerous cells. Cancer 23,497–505 (1969)

    Google Scholar 

  • Gurel,O.: Qualitative study of unstable behaviour of cancerous cells. Cancer 24, 945–947 (1969)

    Google Scholar 

  • Gurel,O.: Unstable dynamic field of an individual cancerous cell. Physiol. Chem. Phys. 2, 570–580 (1970)

    Google Scholar 

  • Gurel,O.: Biomolecular topology and cancer. Physiol. Chem. Phys. 3, 371–388 (1971)

    Google Scholar 

  • Gurel,O.: Bifurcation models of mitosis. Physiol. Chem. Phys. 4, 139–142 (1972)

    Google Scholar 

  • Hahn,G.M.: Cellular kinetics, cell cycles and cell killing. Biophysik 4, 1–14 (1967)

    Google Scholar 

  • Hethcote,H.W.: Mutational models of Carcinogenesis. In: Environmental Health (ed. A. Whittemore) pp. 172–182. Philadelphia: SIAM 1977

    Google Scholar 

  • Himmelstein,K.J., Bischoff,K.B.: Mathematical representation of cancer chemotherapy effects. J. Pharmacokin. Biopharm. 1, 51–68 (1973)

    Google Scholar 

  • Horn,M., Grimm, H.: An application of birth and death processes in cancer chemotherapy. 37th Session Intern. Statist. Inst., Preprints Contributed Papers, pp. 222–224, 1969

  • Iversen,S.: Human cancer and age. Brit. J. Cancer 8, 575–584 (1954)

    Google Scholar 

  • Iversen,S., Arley,N.: On the mechanism of experimental carcinogenesis. Acta Pathol. Microbiol. Scand. 27, 773–803 (1950)

    Google Scholar 

  • Iversen,S., Arley,N.: On the mechanism of experimental Carcinogenesis. V. Application of the hit theory to virus-induced tumours. Acta Pathol. Microbiol. Scand. 31, 27–45 (1952)

    Google Scholar 

  • Iyer,K.S.S., Saksena,V.N.: A stochastic model for the growth of cells in cancer. Biometrics 26, 401–410 (1970)

    Google Scholar 

  • Jansson,B., Révész,L.: Analysis of the growth of tumor cell populations. Math. Biosci. 19, 131–154 (1974)

    Google Scholar 

  • Jansson,B., Révész,L.: Cell cycle analysis of mixed tumor cell populations. Math. Biosci. 24, 107–128 (1975)

    Google Scholar 

  • Jansson,B., Révész,L.: A deductive approach to the analysis of the growth of ascites tumor cell populatins. Methods in Cancer Res. 13, 227–290 (1976)

    Google Scholar 

  • Jansson,B., Révész,L.: Cell ecology: Deductive and dynamic models for proliferation, differentiation and competition of tumor cell populations. J. Theor. Biol. 68, 43–51 (1977)

    Google Scholar 

  • Jusko,W.J.: A pharmacodynamic model for cell-cycle-specific chemotherapeutic agents. J. Pharmacokin. Biopharm. 1,175–200 (1973)

    Google Scholar 

  • Keller,J.B., Whittemore,A.: A theory of transformed cell growth with application to initiation-promotion data. In: Environmental Health (ed. A.Whittemore) pp. 183–191. Philadelphia: SIAM 1977

    Google Scholar 

  • Kendall,D.G.: Birth-and-death processes and the theory of carcinogenesis. Biometrika 47, 13–21 (1960)

    Google Scholar 

  • Kinzler,G.: Ein stochastisches Modell zur Theorie der Karzinogenesis nach Niels Arley. Master Thesis, Univ. of Heidelberg 1966

  • Klonecki,W.: A method for derivation of probabilities in a stochastic model of population growth for carcinogenesis. Colloq. Math. 13, 273–288 (1965)

    Google Scholar 

  • Klonecki,W.: Identifiability questions for chance mechanisms underlying stochastic models for carcinogenesis. Math. Biosci. 7 365–377 (1970)

    Google Scholar 

  • Klonecki,W.: A new model of the mechanism of induction of pulmonary tumors in mice. Zastos. Mat. 15, 163–185 (1976)

    Google Scholar 

  • Klonecki, W.: An input-output model of induction of pulmonary adenomas in mice and an experiment to test it. Unpubl. paper, 10th European Meeting of Statisticians, Leuven 1977

  • Kreyberg,H.J.A.: Empirical relationship of lung cancer incidence to cigarette smoking and a stochastic model for the mode of action of carcinogens. Biometrics 21, 839–857 (1965)

    Google Scholar 

  • Lincoln,T.L., Weiss,G.H.: A statistical evaluation of recurrent medical examinations. Operations Res. 12, 187–205 (1964)

    Google Scholar 

  • Lincoln,T.L., Wells,R.E.: Predicting progress, recognizing breakthroughs and evaluating performance in the treatment of leukemia. Math. Biosci. 16, 227–245 (1973)

    Google Scholar 

  • Liotta,L.A., Saidel,G. M., Kleinerman,J.: Stochastic model of metastases formation. Biometrics 32, 535–550 (1976)

    Google Scholar 

  • Liotta,L.A., Saidel,G.M., Kleinerman,J.: Diffusion model of tumor vascularization and growth. Bull. Math. Biol. 39, 117–128 (1977)

    Google Scholar 

  • Mollison,D.: Percolation processes and tumour growth (Abstr.). Advances Appl. Probability 6, 233–235 (1974)

    Google Scholar 

  • Neyman,J.: A two-step mutation theory of carcinogenesis. Mimeographed lecture, Natl. Inst. of Health, Bethesda 1958

    Google Scholar 

  • Neyman,J.: A two-step mutation theory of carcinogenesis. Bull. Intern. Statist. Inst. 38, 123–135 (1961)

    Google Scholar 

  • Neyman,J., Scott,E.L.: Statistical aspect of the problem of carcinogenesis. Proc. 5th Berkeley Symp. Math. Statist. Probability, Vol. IV, pp. 745–776. Berkeley: Univ. California Press 1967

    Google Scholar 

  • Niskanen,E.E., Arley,N.: A stochastic model of carcinogenesis compared with mouse skin tumour enhancement after gastric instillation of 9,10-dimethyl-1,2-benzanthracene with subsequent local continuous exposure to Tween 40. Nature 199, 83–84 (1963)

    Google Scholar 

  • Nordling,C. O.: A new theory on the cancer-inducing mechanism. Brit. J. Cancer 7, 68–72 (1953)

    Google Scholar 

  • Priore,R.L.: Using a mathematical model in the evaluation of human tumor response to chemotherapy, J. Natl. Cancer Inst. 37, 635–647 (1966)

    Google Scholar 

  • Rescigno,A., DeLisi,C: Immune surveillance and neoplasia. II. A two-stage mathematical model. Bull. Math. Biol. 39, 487–497 (1977)

    Google Scholar 

  • Richardson,D.: Random growth in a tessellation. Proc. Cambridge Phil. Soc. 74, 515–528 (1973)

    Google Scholar 

  • Rittgen,W., Tautu,P.: Branching models for the cell cycle. Lecture Notes in Biomath. 11, 109–126 (1976)

    Google Scholar 

  • Rosen,G.: Mathematical model for the frequency of carcinogenesis at low doses. Math. Biosci. 18, 133–135 (1973)

    Google Scholar 

  • Rosen,G.: Model for carcinogenesis at low doses. Addendum. Math. Biosci. 34, 167–169 (1977)

    Google Scholar 

  • Rosen,G.: Physio-chemical mathematical theory for transitions in cellular metabolism: an aspect of carcinogenesis. Bull. Math. Biol. 39, 521–532 (1977)

    Google Scholar 

  • Rosen,P.: Viral carcinogenesis. J. Theor. Biol. 64, 215–220 (1977)

    Google Scholar 

  • Rossi,H.H., Kellerer,A.M.: Radiation carcinogenesis at low doses. Science 175, 200–202 (1972)

    Google Scholar 

  • Rubinow,S.I.: Human leukemia: kinetic aspects and speculations concerning its cellular origins. In: Environmental Health (ed. A. Whittemore) pp. 135–148. Philadelphia: SIAM 1977

    Google Scholar 

  • Rubinow,S.I., Lebowitz,J.L.: A mathematical model of the acute myeloblastic leukemic state in man. Biophys. J. 16, 897–910 (1976)

    Google Scholar 

  • Rubinow,S.I., Lebowitz,J.L.: A mathematical model of the chemotherapeutic treatment of acute myeloblastic leukemia. Biophys. J. 16, 1257–1271 (1976)

    Google Scholar 

  • Rubinow,S.I., Lebowitz,J.L., Sapse,A.: Parametrization of in vivo leukemic cell populations. Biophys. J. 11, 175–188 (1971)

    Google Scholar 

  • Saslaw,W.C.: A possible quantitative mechanism for carcinogenesis by ultra-violet radiation. Nature 207, 592–593 (1965)

    Google Scholar 

  • Schürger,K., Tautu,P.: Markov configuration processes on a lattice. Rev. Roumaine Math. Pures Appl. 21, 233–244 (1976a)

    Google Scholar 

  • Schürger,K., Tautu,P.: A Markovian configuration model for carcinogenesis. Lecture Notes in Biomath. 11, 92–108 (1976b)

    Google Scholar 

  • Schürger,K., Tautu,P.: A spatial stochastic model for carcinogenesis: A Markov configuration process. Unpubl. paper, 10th European Meeting of Statisticians, Leuven 1977

  • Schwartz,H., Wolff,G.: Mathematische Betrachtungen zum Wachstum von Geschwülsten. Acta biol. med. germ. 13, 378–379 (1964)

    Google Scholar 

  • Sel'kov,E.E.: Two alternative auto-oscillatory steady states in the metabolism of the thiols. Two alternative types of cell multiplication: normal an malignant. Biophysica 15, 1104–1112 (1970)

    Google Scholar 

  • Simon,R.: Application of optimization methods to the hematological support of patients with disseminated malignancies. Math. Biosci. 25, 125–138 (1975)

    Google Scholar 

  • Steward,P.G., Hahn,G.M.: The application of age response functions to the optimization of treatment schedules. Cell Tissue Kinet. 4, 279–291 (1971)

    Google Scholar 

  • Stocks,P.: A study of the age curve for cancer of the stomach in connection with a theory of the cancer producing mechanism. Brit. J. Cancer 7, 407–417 (1953)

    Google Scholar 

  • Suit,H.D., Batten,G. W.: Implications of cell proliferation kinetics for radiotherapy. In: The Proliferation and Spread of Neoplastic Cells, pp. 423–439. Baltimore: Williams & Wilkins 1968

    Google Scholar 

  • Summers,W.C.: Dynamics of tumor growth: A mathematical model. Growth 30, 333–338 (1966)

    Google Scholar 

  • Swan,G.W.: A mathematical model for the density of malignant cells in the spread of cancer in the uterus. Math. Biosci. 25, 319–329 (1975)

    Google Scholar 

  • Swan,G.W.: Solution of linear one-dimensional diffusion equations. Bull.Math. Biol. 38, 1–13 (1976)

    Google Scholar 

  • Swan,G.W.: Reduction of model for the spread of cancer in the uterus ; arbitrary specific growth rate. Bull. Math. Biol. 38, 205–207 (1976)

    Google Scholar 

  • Swan,G.W., Vincent,T.L.: Optimal control analysis in the chemotherapy of IgG multiple myeloma. Bull. Math. Biol. 39, 317–337 (1977)

    Google Scholar 

  • Swartz,J., Spear,R.C.: A dynamic model for studying the relationship between dose and exposure in carcinogenesis. Math. Biosci. 26, 19–39 (1975)

    Google Scholar 

  • Takahashi,M., Inouye,K.: Characteristics of cube root growth of transplanted tumours. J. Theor. Biol. 14, 275–283 (1967)

    Google Scholar 

  • Tautu,P.: Random systems of locally interacting cells (Abstr.). Advances Appl. Probability 6, 237 (1974)

    Google Scholar 

  • Tautu,P.: On stochastic models of growing configurations (Abstr.). Biometrics 31, 1011 (1975a)

    Google Scholar 

  • Tautu,P.: Some examples of probability models in cancer epidemiology. Bull. Intern. Statist. Inst. Vol. 46, Book 2, 144–158,(1975b)

    Google Scholar 

  • Tautu,P.: Carcinogenesis theory revisited: Reliability models avenue. Proc. 9th Intern. Biometric Conf., Vol. 2, pp. 110–124. Raleigh: The Biometrie Society 1976

    Google Scholar 

  • Tautu,P.: Blackening a d-dimensional lattice. Rev. Roumaine Math. Pures Appl. (to be published) (1977)

  • Tautu,P., Wagner,G.: Mathematical models in oncology: State of the art. Proc. 1st World Conf. on Mathematics at the Service of Man, Barcelona 1977 (to be published)

  • Tsanev,R., Sendov,B.: An epigenetic mechanism for carcinogenesis. Z. Krebsforsch. 76, 299–319 (1971)

    Google Scholar 

  • Tucker,H.G.: A stochastic model for a two-stage theory of carcinogenesis. Proc. 4th Berkeley Symp. Math. Statist. Probability, Vol. IV, pp. 387–403. Berkeley: Univ. California Press 1961

    Google Scholar 

  • Valleron,A.J.: Mathematic model of cell synchrony and drug scheduling. Cancer Treatm. Rep. 60, 1899–1911 (1976)

    Google Scholar 

  • Wagner,G., Bühler,W.J.: Über Modelle zur Carcinogenese. In: Aktuelle Probleme aus dem Gebiet der Cancerologie (eds. H.Lettré and G.Wagner) pp. 106–117. Berlin: Springer 1968

    Google Scholar 

  • Wagner,G., Tautu,P.: Inference from data: Development of mathematical models. Proc. 11th Intern. Cancer Congress, Vol. 3, pp. 11–17. Amsterdam: Excerpta Medica 1975

    Google Scholar 

  • Watson,G.S.: Age incidence curves for cancer. Prov. Natl. Acad. Sci. USA 74, 1341–1342 (1977)

    Google Scholar 

  • Waugh,W.A.O'N.: Age-dependence in a stochastic model of carcinogenesis. Proc. 4th Berkeley Symp. Math. Statist. Probability, Vol. IV, pp. 405–413. Berkeley: Univ. California Press 1961

    Google Scholar 

  • Weiss.G.H., Zelen,M.: A stochastic model for the interpretation of clinical trials. Proc. Natl. Acad. Sci. USA 50, 988–994 (1963)

    Google Scholar 

  • Weiss,G.H., Zelen,M.: A semi-Markov model for clinical trials. J. Appl. Probability 2, 269–285 (1965)

    Google Scholar 

  • Wette,R., Katz,I.N., Rodin,E.Y.: Stochastic processes for solid tumor kinetics. I, II. Math. Biosci. 19, 231–255; 21, 311–338 (1974)

    Google Scholar 

  • Wheldon,T.E.: Mitotic autoregulation of normal and abnormal cells: Alternative mechanisms for the derangement of growth control. J. Theor. Biol. 53, 421–433 (1975)

    Google Scholar 

  • Whittemore,A.: Epidemiological implications of the multistage theory of carcinogenesis. In: Environmental Health (ed. A. Whittemore) pp. 72–87. Philadelphia: SIAM 1977

    Google Scholar 

  • Whittemore,A., Keller,J.B.: Quantitative theories of carcinogenesis. SIAM Rev. (in press) (1977)

  • Williams,T.: Unpubl. working paper, Tobacco Res. Council Symp., London dy1971

  • Williams,T.: Modelli matematici della carcinogenesi. Appl. bio-mediche Calc. elettronico 7, 27–36 (1972)

    Google Scholar 

  • Williams,T.: Evidence for super-critical tumour growth (Abstr.). Advances Appl. Probability 6, 237–238 (1974)

    Google Scholar 

  • Williams,T., Bjerknes,R.: Hyperplasia: the spread of abnormal cells through a plane lattice (Abstr.). Advances Appl. Probability 3, 210–211 (1971)

    Google Scholar 

  • Williams,T., Bjerknes,R.: Stochastic model for abnormal clone spread through epitelial basal layer. Nature 236, 19–21(1972)

    Google Scholar 

  • Woo,K.B., Brenkus,L.B., Wüg,K.M.: Analysis of the effects of antitumor drugs on cell cycle kinetics. Cancer Chemother. Rep. 59, 847–860 (1975)

    Google Scholar 

  • Wright,J.K., Peto,R.: An elementary theory leading to non-linear dose-risk relationships for radiation carcinogenesis. Brit. J. Cancer 23, 547–553 (1969)

    Google Scholar 

  • Zelen,M.: Leukemia models:Mice and men. In: The Proliferation and Spread of Neoplastic Cells, pp. 463–477. Baltimore: Williams & Wilkins 1968

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Deutsches Krebsforschungszentrum, Institut für Dokumentation, Information und Statistik, Im Neuenheimer Feld 280, D-6900, Heidelberg 1, Federal Republic of Germany

    P. Tautu

Authors
  1. P. Tautu
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

“Cancer Research and Clinical Oncology” publishes in loose succession “Editorials” and “Guest Editorials” on actual and/or controversial problems in experimental and clinical oncology. These contributions represent exclusively the personal opinion of the author. The Editors

Die „Zeitschrift für Krebsforschung und Klinische Onkologie“ bringt in zwangloser Folge „Editorials“ zu aktuellen und/oder kontroversen Problemen der experimentellen und klinischen Onkologie. Diese Beiträge geben ausschließlich die persönliche Meinung des Autors wieder. Die Herausgeber

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tautu, P. Mathematical models in oncology: a bird's-eye view. Z. Krebsforsch. 91, 223–235 (1978). https://doi.org/10.1007/BF00312285

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00312285

Search

Navigation