Issue 3 (197), article 4


Cybernetics and Computer Engineering, 2019, 3(197), pp.

Bondarenko M.A., PhD (Phys and Math),
Assistant Professor, the Department of Medical and Biological
Physics and Medical Informatics

Knigavko V.G., DSc (Biology), Professor,
Head of the Department of Medical and Biological
Physics and Medical Informatics

Zaytseva O.V., DSc (Biology), Professor,
the Department of Medical and Biological Physics and Medical Informatics

Rukin A.S., PhD (Phys and Math),
Senior Lecturer of the Department of Medical and Biological Physics and Medical Informatics

Kharkiv National Medical University
4, Nauky av., Kharkiv, 61022, Ukraine


Introduction. In radiotherapy, the degree of oxygenation of tumors is of vital importance. Tumors with greater oxygenation are much more responsive to radiation therapy than tumors with significant hypoxia: well-oxygenated tumors react 2.5…3 times better. Mathematical modeling of DNA damage of irradiated cells at different degrees of their oxygenation is of current interest.

The purpose of the article is to develop a mathematical model of DNA damage in irradiated cells at different degrees of their oxygenation; to study the dependence of the number of radiation damages of DNA per unit volume of the irradiated medium on the radiation dose and the concentration of oxygen in the medium; to estimate the cell cycle duration depending on the oxygen concentration.

Results. A mathematical model of oxygen effect in cells in the case of irradiation
by X-rays or gamma-radiation is proposed. On the basis of this model, the dependence of the number of radiation DNA damages in the unit volume of the irradiated medium on the radiation dose and the concentration of oxygen in the medium is obtained. Triple damage to DNA molecules is determined by primary radiation damage and attacks of two radicals of oxygen on the DNA molecule.

The effect of potentially lethal lesions (PLL) on survival of cells under irradiation conditions is studied. The phenomenon of increasing the survival of tumor cells in their irradiation under hypoxia conditions is also due to the phenomenon of potentially lethal lesions. The optimal indicator of the severity of the PLL effect is the cell cycle duration. Thus, the task of modeling PLL was reduced to creation of a mathematical model that allows estimating the value of that indicator depending on the oxygen concentration.

Conclusions. The mathematical model created in the article allows estimating the number of radiation DNA damages in the unit volume of the irradiated medium on the radiation dose and the concentration of oxygen in the medium. The dependence of the cell cycle duration on the oxygen concentration was obtained.

Keywords: radiobiology, mathematical modeling, oxygen effect, oxygen enhancement ratio, DNA damage.

Download full text!


  1. Alper E. Introduction to Liquid–Liquid Extraction with Chemical Reaction. Proceedings of NATO ASI on “Mass transfer with chemical reaction in multiphase systems, Turkey. 1981. Vol. 72/73, pp. 577–611.
  2. Van der Schans G.P., van Loon A.A., Groenendijk R.H., Baan R.A. Detection of DNA Damage in Cells Exposed to Ionizing Radiation by Use of Anti–single–stranded DNA Monoclonal Antibody. Int. J. of Radiat. Biology. 1989. Vol. 55(5), pp. 747–760.
  3. Lücke-Huhle C., Braun A., Hagen U. Oxygen effect in gamma–irradiated DNA Z. Naturforsch B. 1970. Vol. 25(11), pp. 1264–1268.
  4. M. Zeman. Biologic Basis of Radiation Oncology. Clinical Radiation Oncology (Third Edition). 2012, pp. 3–42.
  5. Ewing D. The oxygen fixation hypothesis: a reevaluation. Am. J. Clin. Oncol. 1998. Vol. 21, pp. 355–361.
  6. Hall E., Giaccia A. Radiobiology For The Radiologist. 6th ed. Philadelphia: Lippincott William and Wilkins, 2006.
  7. Gray L., Conger A., Ebert M., Hornsey S., Scott O. The concentration of oxygen dissolved in tissues at the time of irradiation as a factor in radiotherapy. Br. J. Radiol.1953. Vol. 26, pp. 638–648.
  8. Evans S.M., Koch C.J. Prognostic significance of tumor oxygenation in humans. Cancer Lett. 2003. Vol. 195, pp. 1–6.
  9. Vaupel P, Mayer A. Hypoxia in cancer: significance and impact on clinical outcome. Cancer Metastasis Rev. 2007. Vol. 26, pp. 225–239.
  10. Wilson W.R., Hay M.P. Targeting hypoxia in cancer therapy. Nat. Rev. Cancer. 2011. Vol. 11, pp. 393–410.
  11. Knigavko V.G., Bondarenko M.A., Zaytseva O.V. The Generalized Mutation Theory of Oncogenesis. Journal of Clinical and Diagnostic Research. 2018. Vol. 12(11), pp. XE01–XE04.
  12. Bentzen S., Gregoire V. Molecular–imaging–based dose painting – a novel paradigm for radiation therapy prescription. Semin. Radiat. Oncol. 2011. Vol. 21, pp. 101–110.
  13. Howard-Flanders P, Alper T. The sensitivity of microorganisms to irradiation under controlled gas conditions. Radiat. Res. 1957. Vol. 7, pp. 518–540.
  14. Koch C.J., Stobbe C.C., Bump E.A. The effect on the Km for radiosensitization at 0 °C of thiol depletion by diethylmaleate pretreatment: quantitative differences found using the radiation sensitizing agent misonidazole or oxygen. Radiat. Res. 1984. Vol. 98, pp. 141–153.
  15. Whillans A.D.W., Hunt J.W., Whillans D.W. A Rapid–mixing comparison of the mechanisms of radiosensitization by oxygen and misonidazole in CHO cells. Radiat. Res. 1982. Vol. 90, pp. 126–141.
  16. Ling C. C., Michaels H. B., Gerweck L. E., Epp E. R., Peterson E. C. Oxygen sensitization of mammalian cells under different irradiation conditions. Radiat. Res. 1981. Vol. 86, pp. 325–340.
  17. Wouters B.G., Brown J.M. Cells at intermediate oxygen levels can be more important than the ‘hypoxic fraction’ in determining tumor response to fractionated radiotherapy. Radiat. Res. 1997. Vol. 147, pp. 541–550.
  18. Chapman J.D., Dugle D.L., Reuvers A.P., Meeker B.E., Borsa J. Letter: studies on the radiosensitizing effect of oxygen in Chinese hamster cells. Int. J. Radiat. Biol. Relat. Stud. Phys. Chem. Med. 1974. Vol. 26, pp. 383–389.
  19. Howard-Flanders P., Moore D. The time interval after pulsed irradiation within which injury to bacteria can be modified by dissolved oxygen: I. A search for an effect of oxygen 0.02 s after pulsed irradiation. Radiat. Res. 1958. Vol. 9, pp. 422–437.
  20. Bertout J., Patel S., Simon M. The impact of O2availability on human cancer. Nat. Rev. Cancer. 2008. Vol. 8, pp. 967–975.
  21. Michael B.D., O’Neill P.A. Sting in the tail of electron tracks. Science. 2000. Vol. 287, pp. 1603–1604.
  22. Yarmonenko S.P., Vainson A.A., Magdon E. Oxygen effect and radiation therapy of tumors. Moscow: Medicine, 1980. (in Russian).
  23. Voloshina E.A., Mescherikova V.V. Oxygen effect and adaptation reactions of cells. Radiobiology. 1979. Vol. 19, no. 2, pp. 283–285 (in Russian).
  24. Knigavko V.G., Bondarenko M.A., Buts V.G. Diffusion of oxygen in a malignant tumor during the early stage of its development (spheroid stage). Biophysical Bulletin. 2000. Vol. 2(7), pp. 55–59 (in Russian).
  25. Bondarenko M.A., Knigavko V.G., Gordienko V.G., Protsenko E.V., Knigavko A.V. Modeling of oxygen diffusion and consumption processes in malignant tumor strands. Biophysical Bulletin. 2001. Vol. 1(8), pp. 81–85 (in Russian).
  26. Knigavko V.G., Bondarenko M.A., Ponomarenko N.S., Radzishevska E.B. Mathematical simulation of oxygen diffusion and consumption in a flat malignant tumor. Ukrainian Journal of Radiology. 2008. Vol. 16, no. 1, pp. 61–65 (in Ukrainian).
  27. Knigavko V.G., Bondarenko M.A. Mathematical modeling of oxygen diffusion and consumption in a malignant tumor. Biophysics. 2005. V. 30, no. 3, pp. 544–549 (in Russian).
  28. Bondarenko M., Knigavko V., Zaytseva O. Approach to evaluate the risk of cancer for different number of tumor suppressor genes in the individual. East European Journal of Physics. 2018. Vol. 5, no. 2, pp. 23–26.
  29. Grimes D.R., Kelly C., Bloch K., Partridge M.A method for estimating the oxygen consumption rate in multicellular tumour spheroids. J. R. Soc. Interface. 2014. V. 11.
  30. Tannock I. Oxygen diffusion and the distribution of cellular radiosensitivity in tumours. Br. J. Radiol. 1972. Vol. 45, pp. 515–524.
  31. Grimes D. R., Fletcher A. G., Partridge M. Oxygen consumption dynamics in steady–state tumour models. R. Soc. Open Sci. 2014. Vol. 1.

Received 29.03.2019