Geodesic
Optimal control for a bone metastasis with radiotherapy model using a linear objective functional
A. Camacho1 ; E. Díaz-Ocampo2 ; S. Jerez3
1 Facultad de Ciencias, Universidad Autónoma de Baja California, Mexico.
2 UNINTER, Mexico.
3 Department of Mathematics, CIMAT, Mexico.
Mathematical modelling of natural phenomena, Tome 17 (2022), article no. 32.

Voir la notice de l'article provenant de la source EDP Sciences

Radiation is known to cause genetic damage to highly proliferative cells such as cancer cells. However, the radiotherapy effects to bone cells is not completely known. In this work we present a mathematical modeling framework to test hypotheses related to the radiation-induced effects on bone metastasis. Thus, we pose an optimal control problem based on a Komarova model describing the interactions between cancer cells and bone cells at a single site of bone remodeling. The radiotherapy treatment is included in the form of a functional which minimizes the use of radiation using a penalty function. Moreover, we are interested to model the ‘on’ and the ‘off’ time states of the radiation schedules; so we propose an optimal control problem with a L 1-type objective functional. Bang-bang or singular arc solutions are the obtained optimal control solutions. We characterize both solutions types and explicitly give necessary optimality conditions for them. We present numerical simulations to analyze the different possible radiation effects on the bone and cancer cells. We also evaluate the more significant parameters to shift from a bang-bang solution to a singular arc solution and vice versa. Additionally, we study a fractionated radiotherapy model that yields an output solution that resembles intermittent radiotherapy scheduling.
DOI : 10.1051/mmnp/2022038

A. Camacho 1 ; E. Díaz-Ocampo 2 ; S. Jerez 3

1 Facultad de Ciencias, Universidad Autónoma de Baja California, Mexico.
2 UNINTER, Mexico.
3 Department of Mathematics, CIMAT, Mexico. 
@article{MMNP_2022_17_a15,
     author = {A. Camacho and E. D{\'\i}az-Ocampo and S. Jerez},
     title = {Optimal control for a bone metastasis with radiotherapy model using a linear objective functional},
     journal = {Mathematical modelling of natural phenomena},
     eid = {32},
     publisher = {mathdoc},
     volume = {17},
     year = {2022},
     doi = {10.1051/mmnp/2022038},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2022038/}
}
TY  - JOUR
AU  - A. Camacho
AU  - E. Díaz-Ocampo
AU  - S. Jerez
TI  - Optimal control for a bone metastasis with radiotherapy model using a linear objective functional
JO  - Mathematical modelling of natural phenomena
PY  - 2022
VL  - 17
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2022038/
DO  - 10.1051/mmnp/2022038
LA  - en
ID  - MMNP_2022_17_a15
ER  - 
%0 Journal Article
%A A. Camacho
%A E. Díaz-Ocampo
%A S. Jerez
%T Optimal control for a bone metastasis with radiotherapy model using a linear objective functional
%J Mathematical modelling of natural phenomena
%D 2022
%V 17
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2022038/
%R 10.1051/mmnp/2022038
%G en
%F MMNP_2022_17_a15
A. Camacho; E. Díaz-Ocampo; S. Jerez. Optimal control for a bone metastasis with radiotherapy model using a linear objective functional. Mathematical modelling of natural phenomena, Tome 17 (2022), article  no. 32. doi : 10.1051/mmnp/2022038. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/2022038/

[1] A. Alvarez-Arenas, K.E. Starkov, G.F. Calvo, J. Belmonte-Beitia Ultimate dynamics and optimal control of a multi-compartment model of tumor resistance to chemotherapy Discr. Continu. Dyn. Syst. B 2019 2017

[2] J.W. Bachman and T. Hillen , Mathematical optimization of the combination of radiation and differentiation therapies for cancer. Front. Oncol. 3 (2013). https://doi.org/10.3389/fonc.2013.00052

[3] H.E. Barker, J.T.E. Paget, A.A. Khan, K.J. Harrington The tumour microenvironment after radiotherapy: mechanisms of resistance and recurrence Nat. Rev. Cancer 2015 409 425

[4] G.C. Barnett, C.M.L. West, A.M. Dunning, R.M. Elliott, C.E. Coles, P.D.P. Pharoah, N.G. Burnet Normal tissue reactions to radiotherapy: towards tailoring treatment dose by genotype Nat. Rev. Cancer 2009 134 142

[5] G. Bedard, E. Chow The failures and challenges of bone metastases research in radiation oncology J. Bone Oncol. 2013 84 88

[6] F.J. Bonnans , D. Giorgi , V. Grelard , B. Heymann , S. Maindrault , P. Martinon , O. Tissot and J. Liu , Bocop – a collection of examples. INRIA (2017).

[7] B. Bonnard and M. Chyba , Vol. 40 of Singular trajectories and their role in control theory. Springer-Verlag (2003).

[8] D.J. Brenner The linear-quadratic model is an appropriate methodology for determining isoeffective doses at large doses per fraction Semin. Radiat. Oncol. 2008 234 239

[9] C. Bruni, F. Conte, F. Papa, C. Sinisgalli Optimal weekly scheduling in fractionated radiotherapy: effect of an upper bound on the dose fraction size J. Math. Biol. 2015 361 398

[10] A. Camacho, S. Jerez Bone metastasis treatment modeling via optimal control J. Math. Biol. 2019 497 552

[11] A. Camacho, S. Jerez Nonlinear modeling and control strategies for bone diseases based on TGFβ and Wnt factors Commun. Nonlinear Sci. Numer. Simul. 2021 105842

[12] A. Chandra, T. Lin, M.B. Tribble, J. Zhu, A.R. Altman, W.-J. Tseng, L. Qin PTH1-34 alleviates radiotherapy-induced local bone loss by improving osteoblast and osteocyte survival Bone 2014 33 40

[13] A. Chandra, T. Lin, T. Young, W. Tong, X. Ma, W.-J. Tseng, I. Kramer, M. Kneissel, M.A. Levine, Y. Zhang, K. Cengel, X.S. Liu, L. Qin Suppression of sclerostin alleviates radiation-induced bone loss by protecting bone-forming cells and their progenitors through distinct mechanisms J. Bone Mineral Res. 2017 360 372

[14] D. Chappard, B. Bouvard, M.F. Baslé, E. Legrand, M. Audran Bone metastasis: histological changes and pathophysiological mechanisms in osteolytic or osteosclerotic localizations. A review Morphologie 2011 65 75

[15] E. Chow, K. Harris, G. Fan, M. Tsao, W.M. Sze Palliative radiotherapy trials for bone metastases: a systematic review J. Clinic. Oncol. 2007 1423 1436

[16] L.G. De Pillis, W. Gu, K.R. Fister, T. Head, K. Maples, A. Murugan, K. Yoshida Chemotherapy for tumors: An analysis of the dynamics and a study of quadratic and linear optimal controls Math. Biosci. 2007 292 315

[17] R. Di Franco , S. Falivene , V. Ravo and P. Muto , Radiotherapy for the Treatment of Bone Metastases, in Interventional Neuroradiology of the Spine, edited by M. Muto . Springer, Milano (2013) pp. 221–230.

[18] H. Enderling, A.R.A. Anderson, M.A.J. Chaplain, A.J. Munro, J.S. Vaidya Mathematical modelling of radiotherapy strategies for early breast cancer J. Theor. Biol. 2006 158 171

[19] A. Ergun, K. Camphausen, L.M. Weinx Optimal scheduling of radiotherapy and angiogenic inhibitors Bull. Math. Biol. 2003 407 424

[20] W. Fleming and R. Rishel , Deterministic and Stochastic Optimal Control. Springer-Verlag, New York (1975).

[21] J.F. Fowler 21 years of biologically effective dose Br. J. Radiol. 2010 554 568

[22] A. Ghaffari, B. Bahmaie, M. Nazari A mixed radiotherapy and chemotherapy model for treatment of cancer with metastasis Math. Methods Appl. Sci. 2016 4603 4617

[23] C.E. Gross, R.M. Frank, A.R. Hsu, A. Diaz, S. Gitelis External beam radiation therapy for orthopaedic pathology J. Am. Acad. Orthopaed. Surg. 2015 243 252

[24] W.S. Hong , S.G. Wang and G.Q. Zhang , Lung cancer radiotherapy: simulation and analysis based on a multicomponent mathematical model. Comput. Math. Methods Med. 2021 (2021) Article 6640051.

[25] S. Jerez, A. Camacho Bone metastasis modeling based on the interactions between the BMU and tumor cells J. Comput. Appl. Math. 2018 866 876

[26] S. Jerez , E. Pliego , F.J. Solís and A.K. Miller , Antigen receptor therapy in bone metastasis via optimal control for different human life stages. J. Math. Biol. 83 (2021) Article 24.

[27] S.V. Komarova, R.J. Smith, S.J. Dixon, S.M. Sims, L.M. Wahl Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling Bone 2003 206 215

[28] J.Y. Krzeszinski, Y. Wan New therapeutic targets for cancer bone metastasis Trends Pharmacolog. Sci. 2015 360 373

[29] U. Ledzewicz, H. Schättler Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy Math. Biosci. 2007 320 342

[30] U. Ledzewicz, H. Maurer, H. Schättler Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy Math. Biosci. Eng. 2011 307 323

[31] U. Ledzewicz, M. Naghnaeian, H. Schättler Optimal response to chemotherapy for a mathematical model of tumor–immune dynamics J. Math. Biol. 2012 557 577

[32] U. Ledzewicz, H. Maurer, H. Schättler Optimal combined radio- and anti-angiogenic cancer therapy J. Optim. Theory Appl. 2019 321 340

[33] J.M. Lemos, D.V. Caiado, R. Coelho, S. Vinga Optimal and receding horizon control of tumor growth in myeloma bone disease Biomed. Signal Process. Control 2016 128 134

[34] S. Lenhart and J.T. Workman , Optimal control applied to biological models. CRC Press (2007).

[35] S. Lutz The role of radiation therapy in controlling painful bone metastases Curr. Pain Headache Rep. 2012 300 306

[36] M. Mcasey, L. Mou, W. Han Convergence of the forward-backward sweep method in optimal control Comput. Optim. Appl. 2012 207 226

[37] G.R. Mundy Metastasis to bone: causes, consequences and therapeutic opportunities Nat. Rev. Cancer 2002 584 593

[38] P.D. Ottewell The role of osteoblasts in bone metastasis J. Bone Oncol. 2016 124 127

[39] S. Paget The distribution of secondary growths in cancer of the breast The Lancet 1889 571 573

[40] J. Poleszczuk, R. Walker, E.G. Moros, K. Latifi, J.J. Caudell, H. Enderling Predicting patient-specific radiotherapy protocols based on mathematical model choice for proliferation saturation index Bull. Math. Biol. 2018 1195 1206

[41] L.S. Pontryagin , Mathematical theory of optimal processes. CRC Press (1987).

[42] E. Ratajczyk, U. Ledzewicz, H. Schättler Optimal control for a mathematical model of glioma treatment with oncolytic therapy and TNF-α inhibitor J. Optim. Theory Appl. 2018 456 477

[43] A.M.A.C. Rocha, M.F.P. Costa, E.M.G.P. Fernandes On a multiobjective optimal control of a tumor growth model with immune response and drug therapies Int. Trans. Oper. Res. 2018 269 294

[44] R. Rockne, E.C. Alvord, J.K. Rockhill, K.R. Swanson A mathematical model for brain tumor response to radiation therapy J. Math. Biol. 2009 561 578

[45] H. Schättler and U. Ledzewicz , Optimal control for mathematical models of cancer therapies. Springer-Verlag, New York (2015).

[46] R. Serre, S. Benzekry, L. Padovani, C. Meille, N. Andre, J. Ciccolini, D. Barbolosi Mathematical modeling of cancer immunotherapy and its synergy with radiotherapy Cancer Res. 2016 4931 4940

[47] G.W. Swan Role of optimal control theory in cancer chemotherapy Math. Biosci. 1990 237 284

[48] A. Swierniak , Control problems arising in combined antiangiogenic therapy and radiotherapy, in Proc. 5 IASTED Conf. on Biomedical Engineering (2007) 113–117.

[49] Team Commands, Inria Saclay (2017). BOCOP: an open source toolbox for optimal control. http://bocop.org

[50] M.S. Tiwana, M. Barnes, E. Yurkowski, K. Roden, R.A. Olson Incidence and treatment patterns of complicated bone metastases in a population-based radiotherapy program Radiother. Oncol. 2016 552 556

[51] J.M. Ubellacker, S.S. Mcallister The unresolved role of systemic factors in bone metastasis J. Bone Oncol. 2016 96 99

[52] L.A.M.-L. Vakaet, T. Boterberg Pain control by ionizing radiation of bone metastasis Int. J. Dev. Biol. 2004 599 606

[53] A. Wächter, L.T. Biegler On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming Math. Program. 2006 25 57

[54] P. Warman, A. Kaznatcheev, A. Araujo, C. Lynch, D. Basanta Fractionated follow-up chemotherapy delays the onset of resistance in bone metastatic prostate cancer Games 2018 19

[55] Y. Yang, L. Xing Optimization of radiotherapy dose-time fractionation with consideration of tumor specific biology Med. Phys. 2005 3666 3677

[56] J. Zhang, X. Qiu, K. Xi, W. Hu, H. Pei, J. Nie, G. Zhou Therapeutic ionizing radiation induced bone loss: a review of in vivo and in vitro findings Connective Tissue Res. 2018 509 522

Cité par Sources :