# American Institute of Mathematical Sciences

2016, 13(1): 159-170. doi: 10.3934/mbe.2016.13.159

## The consequence of day-to-day stochastic dose deviation from the planned dose in fractionated radiation therapy

 1 National Brain Research Centre, Manesar, Gurgaon, Haryana-122051, India, India

Received  March 2015 Revised  June 2015 Published  October 2015

Radiation therapy is one of the important treatment procedures of cancer. The day-to-day delivered dose to the tissue in radiation therapy often deviates from the planned fixed dose per fraction. This day-to-day variation of radiation dose is stochastic. Here, we have developed the mathematical formulation to represent the day-to-day stochastic dose variation effect in radiation therapy. Our analysis shows that that the fixed dose delivery approximation under-estimates the biological effective dose, even if the average delivered dose per fraction is equal to the planned dose per fraction. The magnitude of the under-estimation effect relies upon the day-to-day stochastic dose variation level, the dose fraction size and the values of the radiobiological parameters of the tissue. We have further explored the application of our mathematical formulation for adaptive dose calculation. Our analysis implies that, compared to the premise of the Linear Quadratic Linear (LQL) framework, the Linear Quadratic framework based analytical formulation under-estimates the required dose per fraction necessary to produce the same biological effective dose as originally planned. Our study provides analytical formulation to calculate iso-effect in adaptive radiation therapy considering day-to-day stochastic dose deviation from planned dose and also indicates the potential utility of LQL framework in this context.
Citation: Subhadip Paul, Prasun Kumar Roy. The consequence of day-to-day stochastic dose deviation from the planned dose in fractionated radiation therapy. Mathematical Biosciences & Engineering, 2016, 13 (1) : 159-170. doi: 10.3934/mbe.2016.13.159
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##### References:
 [1] E. Budiarto, M. Keijzer, P. R. M. Storchi, A. W. Heemink, S. Breedveld and B. J. M. Heijmen, Computation of mean and variance of the radiotherapy dose for PCA-modeled random shape and position variations of the target, Phys. Med. Biol., 59 (2014), 289-310. doi: 10.1088/0031-9155/59/2/289. [2] J. F. Fowler, The linear-quadratic formula and progress in fractionated radiotherapy, Br. J. Radiol., 62 (1989), 679-694. doi: 10.1259/0007-1285-62-740-679. [3] L. Gammaitoni, P. Hänggi, P. Jung and F. Marchesoni, Stochastic Resonance: A remarkable idea that changed our perception of noise, Eur. Phys. J. B, 69 (2009), 1-3. doi: 10.1140/epjb/e2009-00163-x. [4] A. Godley, E. Ahunbay, C. Peng and X. A. Li, Accumulating daily-varied dose distributions of prostate radiation therapy with soft-tissue-based kV CT guidance, J. Appl. Clin. Med. Phys., 13 (2012), 1-3. [5] M. Guerrero and M. Carlone, Mechanistic formulation of a lineal-quadratic-linear (LQL) model: Split-dose experiments and exponentially decaying sources, Med. Phys., 37 (2010), 4173-4181. [6] W. Horsthemke and R. Lefever, Noise-Induced Transitions in Physics, Chemistry, and Biology, $2^{nd}$ edition, Springer, Berlin, 2006. [7] B. Huang, W. Wang, M. Bates and X. Zhuang, Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy, Science, 319 (2008), 810-813. doi: 10.1126/science.1153529. [8] J. L. W. V. Jensen, Sur les fonctions convexes et les inégalités entre les valeurs moyennes, Acta Math., 30 (1906), 175-193. doi: 10.1007/BF02418571. [9] M. C. Joiner and A. Kogel, Basic Clinical Radiobiology, $4^{th}$ edition, CRC Press, Boca Raton, 2009. [10] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, $1^{st}$ edition, Springer, London, 1992. doi: 10.1007/978-3-662-12616-5. [11] D. E. Lea, Actions of Radiations on Living Cells, $2^{nd}$ edition, Cambridge University Press, New York, 1962. [12] G. Murphy, W. Lawrence and R. Lenard, ACS Textbook of Clinical Oncology, $2^{nd}$ edition, The American Cancer Society, Inc., Atlanta, 1995. [13] T. Needham, A visual explanation of Jensen's inequality, Amer. Math. Monthly, 100 (1993), 768-771. doi: 10.2307/2324783. [14] J. J. Ruel and M. P. Ayres, Jensen's inequality predicts effects of environmental variation, Trends Ecol. Evol., 14 (1999), 361-366. doi: 10.1016/S0169-5347(99)01664-X. [15] F. C. Su, C. Shi, P. Mavroidis, P. R. Szegedi and N. Papanikolaou, Evaluation on lung cancer patients' adaptive planning of TomoTherapy utilising radiobiological measures and planned adaptive module, J. Radiother. Pract., 8 (2009), 185-194. doi: 10.1017/S1460396909990240. [16] E. Ullner, J. Buceta, A. Díez-Noguera and J. García-Ojalvo, Noise-induced coherence in multicellular circadian clocks, Biophys. J., 96 (2009), 3573-3581. doi: 10.1016/j.bpj.2009.02.031. [17] D. Yan, F. Vicini, J. Wong and A. Martinez, Adaptive radiation therapy, Phys. Med. Biol., 42 (1997), 123-132. doi: 10.1088/0031-9155/42/1/008. [18] E. C. Zimmermann and J. Ross, Light induced bistability in $S_2 0_6 F_2$ ⇌ $2S0_3 F$: Theory and experiment, J. Chem. Phys., 80 (1984), 720-729.
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