• Previous Article
    Mathematical model of the atrioventricular nodal double response tachycardia and double-fire pathology
  • MBE Home
  • This Issue
  • Next Article
    Classification of Alzheimer's disease using unsupervised diffusion component analysis
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:
[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

show all references

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

[1]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[2]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[3]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[4]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[5]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[6]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[7]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[8]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[9]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[10]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[11]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[12]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[13]

Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 16: 331-348. doi: 10.3934/jmd.2020012

[14]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[15]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

[16]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[17]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[18]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[19]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273

[20]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]