In order to overcome the limitations of the linear-quadratic model and include synergistic effects of heat and radiation, a novel radiobiological model is proposed. The model is based on a chain of cell populations which are characterized by the number of radiation induced damages (hits). Cells can shift downward along the chain by collecting hits and upward by a repair process. The repair process is governed by a repair probability which depends upon state variables used for a simplistic description of the impact of heat and radiation upon repair proteins. Based on the parameters used, populations up to 4-5 hits are relevant for the calculation of the survival. The model describes intuitively the mathematical behaviour of apoptotic and nonapoptotic cell death. Linear-quadratic-linear behaviour of the logarithmic cell survival, fractionation, and (with one exception) the dose rate dependencies are described correctly. The model covers the time gap dependence of the synergistic cell killing due to combined application of heat and radiation, but further validation of the proposed approach based on experimental data is needed. However, the model offers a work bench for testing different biological concepts of damage induction, repair, and statistical approaches for calculating the variables of state. 1. Introduction In radiation oncology, mathematical models are used to describe clonogenic survival, tumour control probabilities (TCP), or normal tissue complication probabilities (NTCP). The most widely used model for cell survival is the linear-quadratic (LQ) model. The (originally empiric) model was first used by Lea and Catcheside [1] to fit radiation chromosome damage. The model is based on the observation that the logarithmic plot of the surviving cell fraction (with = number of viable cells after and number of cells before radiation) versus radiation dose can be described by a linear and a quadratic dose-dependent term (). Based on this relationship, adaption of doses for hyper- or hypofractionated radiotherapies can be calculated (e.g., application of the BED concept in clinical oncology [2]). There is also a certain need to calculate equivalent doses in the case of application of moderate hyperthermia (40–43°C) in combination with radiation (HT-RT). But the extension to combined therapies requires some knowledge of the underlying dynamic processes (radiation and heat induced formation of cellular damages, repair, etc.). Theories about DNA lesion formation or cell survival (e.g., Chadwick and Leenhouts [3]) led to mechanistic interpretations of the LQ
References
[1]
D. E. Lea and D. G. Catcheside, “The mechanism of the induction by radiation of chromosome aberrations in Tradescantia,” Journal of Genetics, vol. 44, no. 2-3, pp. 216–245, 1942.
[2]
B. Jones, R. G. Dale, C. Deehan, K. I. Hopkins, and D. A. Morgan, “The role of biologically effective dose (BED) in clinical oncology,” Clinical Oncology, vol. 13, no. 2, pp. 71–81, 2001.
[3]
K. H. Chadwick and H. P. Leenhouts, “A molecular theory of cell survival,” Physics in Medicine and Biology, vol. 18, no. 1, article 7, pp. 78–87, 1973.
[4]
M. Guerrero and X. A. Li, “Extending the linear-quadratic model for large fraction doses pertinent to stereotactic radiotherapy,” Physics in Medicine and Biology, vol. 49, no. 20, pp. 4825–4835, 2004.
[5]
R. L. Wells and J. S. Bedford, “Dose-rate effects in mammalian cells. IV. Repairable and nonrepairable damage in noncycling C3H10T1/2 cells,” Radiation Research, vol. 94, no. 1, pp. 105–134, 1983.
[6]
R. G. Dale, “The application of the linear-quadratic dose-effect equation to fractionated and protracted radiotherapy,” The British Journal of Radiology, vol. 58, no. 690, pp. 515–528, 1985.
[7]
R. Oliver, “A comparison of the effects of acute and protracted gamma-radiation on the growth of seedlings of Vicia faba,” International Journal of Radiation Biology and Related Studies in Physics, Chemistry, and Medicine, vol. 8, pp. 475–488, 1964.
[8]
P. H. Hardenbergh, P. Hahnfeldt, L. Hlatky et al., “Distinct mathematical behavior of apoptotic versus non-apoptotic tumor cell death,” International Journal of Radiation Oncology Biology Physics, vol. 43, no. 3, pp. 601–605, 1999.
[9]
A. M. Kellerer and H. H. Rossi, “A generalized formulation of dual radiation action,” Radiation Research, vol. 75, no. 3, pp. 471–488, 1978.
[10]
P. Lambin, E. P. Malaise, and M. C. Joiner, “Might intrinsic radioresistance of human tumour cells be induced by radiation?” International Journal of Radiation Biology, vol. 69, no. 3, pp. 279–290, 1996.
[11]
C. Mothersill, C. B. Seymour, and M. C. Joiner, “Relationship between radiation-induced low-dose hypersensitivity and the bystander effect,” Radiation Research, vol. 157, no. 5, pp. 526–532, 2002.
[12]
C. Streffer, “Molecular and cellular mechanisms of hyperthermia,” in Thermoradiotherapy and Thermochemotherapy, M. H. Seegenschmiedt, P. Fessenden, and C. C. Vernon, Eds., pp. 47–74, Springer, New York, NY, USA, 1995.
[13]
S. A. Sapareto, L. Hopwood, and W. Dewey, “Combined effects of x-irradiation and hyperthermia on CHO cells for various temperatures and orders of application,” Radiation Research, vol. 44, pp. 221–233, 1978.
[14]
N. A. Franken, A. L. Oei, H. P. Kok et al., “Cell survival and radiosensitisation: modulation of the linear and quadratic parameters of the LQ model (Review),” International Journal of Oncology, vol. 42, no. 5, pp. 1501–1515, 2013.
[15]
S. B. Curtis, “Lethal and potentially lethal lesions induced by radiation—a unified repair model,” Radiation Research, vol. 106, no. 2, pp. 252–270, 1986.
[16]
O. N. Vassiliev, “Formulation of the multi-hit model with a non-Poisson distribution of hits,” International Journal of Radiation Oncology, Biology, Physics, vol. 83, no. 4, pp. 1311–1316, 2011.
[17]
R. G. Bristow and R. P. Hill, “Hypoxia, DNA repair and genetic instability,” Nature Reviews Cancer, vol. 8, no. 3, pp. 180–192, 2008.
[18]
S. Scheidegger, G. Lutters, and S. Bodis, “A LQ-based kinetic model formulation for exploring dynamics of treatment response of tumours in patients,” Zeitschrift fur Medizinische Physik, vol. 21, no. 3, pp. 164–173, 2011.
[19]
S. Scheidegger, R. M. Füchslin, G. Lutters, and S. Bodis, “Dynamic modelling of the synergistic effect of hyperthermia and radiotherapy,” Radiotherapy and Oncology, vol. 103, Supplement 1, pp. S360–S361, 2012.
[20]
C. C. Ling, L. E. Gerweck, M. Zaider, and E. Yorke, “Dose-rate effects in external beam radiotherapy redux,” Radiotherapy and Oncology, vol. 95, no. 3, pp. 261–268, 2010.
[21]
G. E. Adams and D. G. Jameson, “Time effects in molecular radiation biology,” Radiation and Environmental Biophysics, vol. 17, no. 2, pp. 95–113, 1980.
[22]
J. R. Lepock, “Protein denaturation: its role in thermal killing,” in Radiation Research: A Twentieth-Century Perspective, W. C. Dewey, M. Edington, R. J. M. Fry, E. J. Hall, and G. F. Withmore, Eds., pp. 992–998, Academic Press, New York, NY, USA, 1991.
[23]
S. Scheidegger and R. M. Füchslin, “Kinetic model for dose equivalent—an efficient way to predict systems response of irradiated cells,” in Proceedings of the ASIM Workshop (ASIM '11), 2011.
[24]
A. Szasz and G. Y. Vincze, “Dose concept of oncological hyperthermia: heat-equation considering the cell destruction,” Journal of Cancer Research and Therapeutics, vol. 2, no. 4, pp. 171–181, 2006.
[25]
P. A. Corning and S. J. Kline, “Thermodynamics, information and life revisited—part I: “To Be or Entropy,” Systems Research and Behavioral Science, vol. 15, no. 4, pp. 273–295, 1998.
[26]
P. A. Corning and S. J. Kline, “Thermodynamics, information and life revisited—part II: “Thermoeconomics” and ‘Control information‘,” Systems Research and Behavioral Science, vol. 15, no. 6, pp. 453–482, 1998.
[27]
H. Wolters and A. W. T. Konings, “Membrane radiosensitivity of fatty acid supplemented fibroblasts as assayed by the loss of intracellular potassium,” International Journal of Radiation Biology, vol. 48, no. 6, pp. 963–973, 1985.
[28]
E. H. Johnson and M. J. Polisar, The Kinetic of Molecular Biology, John Wiley & Sons, New York, NY, USA, 1954.
[29]
E. K. Rofstad and T. Brustad, “Arrhenius analysis of the heat response in vivo and in vitro of human melanoma xenografts,” International Journal of Hyperthermia, vol. 2, no. 4, pp. 359–368, 1986.
[30]
M. Forlin, I. Poli, D. de March, N. Packard, G. Gazzola, and R. Serra, “Evolutionary experiments for self-assembling amphiphilic systems,” Chemometrics and Intelligent Laboratory Systems, vol. 90, no. 2, pp. 153–160, 2009.
[31]
D. Ferrari, M. Borrotti, and D. de March, “Response improvement in complex experiments by co-information compostite likelihood optimization,” Statistics and Computing, 2013.
[32]
F. Hutter, H. H. Hoos, K. Leyton-Brown, and T. Stützle, “ParamILS: an automatic algorithm configuration framework,” Journal of Artificial Intelligence Research, vol. 36, pp. 267–306, 2009.
[33]
I. Lohse, S. Lang, J. Hrbacek et al., “Effect of high dose per pulse flattening filter-free beams on cancer cell survival,” Radiotherapy and Oncology, vol. 101, no. 1, pp. 226–232, 2011.
[34]
C. C. Ling, I. J. Spiro, J. Mitchell, and R. Stickler, “The variation of OER with dose rate,” International Journal of Radiation Oncology, Biology, Physics, vol. 11, pp. 1367–1373, 1985.
[35]
H. B. Michaels, E. R. Epp, C. C. Ling, and E. C. Peterson, “Oxygen sensitization of CHO cells at ultrahigh dose rates: prelude to oxygen diffusion studies,” Radiation Research, vol. 76, no. 3, pp. 510–521, 1978.
[36]
Y. Shibamoto, S. Otsuka, H. Iwata, C. Sugie, H. Ogino, and N. Tomita, “Radiobiological evaluation of the radiation dose as used in high-precision radiotherapy: effect of prolonged deliverytime and applicability of the linear-quadratic model,” Journal of Radiation Research, vol. 53, no. 1, pp. 1–9, 2012.
[37]
S. A. Sapareto and W. C. Dewey, “Thermal dose determination in cancer therapy,” International Journal of Radiation Oncology Biology Physics, vol. 10, no. 6, pp. 787–800, 1984.
