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2016, 13(6): 1131-1142. doi: 10.3934/mbe.2016034

## Adjoint sensitivity analysis of a tumor growth model and its application to spatiotemporal radiotherapy optimization

 1 Silesian University of Technology, Institute of Automatic Control, Akademicka 16, 44-100 Gliwice 2 Silesian University of Technology, ul.Akademicka 16, 44-100, Gliwice

Received  November 2015 Revised  June 2016 Published  August 2016

We investigate a spatial model of growth of a tumor and its sensitivity to radiotherapy. It is assumed that the radiation dose may vary in time and space, like in intensity modulated radiotherapy (IMRT). The change of the final state of the tumor depends on local differences in the radiation dose and varies with the time and the place of these local changes. This leads to the concept of a tumor's spatiotemporal sensitivity to radiation, which is a function of time and space. We show how adjoint sensitivity analysis may be applied to calculate the spatiotemporal sensitivity of the finite difference scheme resulting from the partial differential equation describing the tumor growth. We demonstrate results of this approach to the tumor proliferation, invasion and response to radiotherapy (PIRT) model and we compare the accuracy and the computational effort of the method to the simple forward finite difference sensitivity analysis. Furthermore, we use the spatiotemporal sensitivity during the gradient-based optimization of the spatiotemporal radiation protocol and present results for different parameters of the model.
Citation: Krzysztof Fujarewicz, Krzysztof Łakomiec. Adjoint sensitivity analysis of a tumor growth model and its application to spatiotemporal radiotherapy optimization. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1131-1142. doi: 10.3934/mbe.2016034
##### References:
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##### References:
 [1] D. Corwin, C. Holdsworth, R. C. Rockne, A. D. Trister, M. M. Mrugala, J. K. Rockhill, R. D. Stewart, M. Phillips and K. R. Swanson, Toward Patient-Specific, Biologically Optimized Radiation Therapy Plans for the Treatment of Glioblastoma,, PLoS ONE 8, (2013). Google Scholar [2] K. Fujarewicz and A. Galuszka, Generalized backpropagation through time for continuous time neural networks and discrete time measurements Artificial Intelligence and Soft Computing - ICAISC 2004 (eds. L. Rutkowski, J. Siekmann, R. Tadeusiewicz and L. A. Zadeh),, Lecture Notes in Computer Science, 3070 (2004), 190. Google Scholar [3] K. Fujarewicz, M. Kimmel and A. Swierniak, On fitting of mathematical models of cell signaling pathways using adjoint systems,, Math. Biosci. Eng., 2 (2005), 527. doi: 10.3934/mbe.2005.2.527. Google Scholar [4] K. Fujarewicz, M. Kimmel, T. Lipniacki and A. Swierniak, Adjoint systems for models of cell signaling pathways and their application to parameter fitting,, IEEE/ACM Transacations On Computational Biology And Bioinformatics, 4 (2007), 322. Google Scholar [5] K. Fujarewicz and K. Łakomiec, Parameter estimation of systems with delays via structural sensitivity analysis,, Discrete and Continuous Dynamical Systems-series B, 19 (2014), 2521. doi: 10.3934/dcdsb.2014.19.2521. Google Scholar [6] P. Hoskin, A. Kirkwood, B. Popova, P. Smith, M. Robinson, E. Gallop-Evans, S. Coltart, T. Illidge, K. Madhavan, C. Brammer, P. Diez, A1. Jack and I. Syndikus, 4 Gy versus 24 Gy radiotherapy for patients with indolent lymphoma (FORT): a randomised phase 3 non-inferiority trial,, Lancet Oncology, 15 (2014), 457. Google Scholar [7] M. Jakubczak and K. Fujarewicz, Application of adjoint sensitivity analysis to parameter estimation of age-structured model of cell cycle,, in Information Technologies in Medicine, (). Google Scholar [8] K. Łakomiec and K. Fujarewicz, Parameter estimation of non-linear models using adjoint sensitivity analysis,, Advanced Approaches to Intelligent Information and Database Systems, (2014), 59. Google Scholar [9] K. Łakomiec, S. Kumala, R. Hancock, J. Rzeszowska-Wolny and K. Fujarewicz, Modeling the repair of DNA strand breaks caused by $\gamma$-radiation in a minichromosome,, Physical Biology, 11 (2014). Google Scholar [10] R. Rockne, E. C. Alvord Jr., J. K. Rockhill and K. R. Swanson, A mathematical model for brain tumor response to radiation therapy,, J. Math. Biol., 58 (2009), 561. doi: 10.1007/s00285-008-0219-6. Google Scholar [11] R. Rockne, J. K. Rockhill, M. Mrugala, A. M. Spence, I. Kalet, K. Hendrickson, A. Lai, T. Cloughesy, E. C. Alvord Jr and K. R. Swanson, Predicting the efficacy of radiotherapy in individual glioblastoma patients in vivo: A mathematical modeling approach,, Phys. Med. Biol., 55 (2010), 3271. Google Scholar [12] R. C. Rockne, A. D. Trister, J. Jacobs, A. J. Hawkins-Daarud, M. L. Neal, K. Hendrickson, M. M. Mrugala, J. K. Rockhill, P. Kinahan, K. A. Krohn and K. R. Swanson, Addendum to "A patient-specific computational model of hypoxia-modulated radiation resistance in glioblastoma using $^18 F-FMISO-PET$",, Journal of the Royal Society Interface, 12 (2015). Google Scholar [13] R. Rockne, J. K. Rockhill, M. Mrugala, A. M. Spence, I. Kalet, K. Hendrickson, A. Lai, T. Cloughesy, E. C. Alvord and K. R. Swanson, Reply to comment on: "Predicting the efficacy of radiotherapy in individual glioblastoma patients in vivo: A mathematical modeling approach",, Physics in medicine and biology, 61 (2016), 2968. Google Scholar [14] A. Swierniak, M. Kimmel, J. Smieja, K. Puszynski and K. Psiuk-Maksymowicz, System Engineering Approach to Planning Anticancer Therapies,, Springer, (2016). Google Scholar
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