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Mathematical Modeling in Radiation Oncology

Translating Mathematical Models into the Clinic

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Advances in Radiation Oncology

Part of the book series: Cancer Treatment and Research ((CTAR))

Abstract

The goal of precision medicine is to tailor treatments to the individual patient’s disease. In radiation oncology, this means tailoring the dose to the boundaries of the tumor, but also to the unique biology of the patient’s disease. In recent years, mathematical modeling has made inroads toward achieving these goals, through the optimization of radiation dose based on radiobiological parameters for individual patients. In this chapter, we review recent literature of mathematical models of tumor growth and response to radiation therapy (RT) and discuss the clinical utility of mathematical models, as well as provide a forward-looking perspective into how mathematical models may enhance patient outcomes through well-designed clinical trials.

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Abbreviations

BED:

Biologically equivalent dose

CT:

Computed tomography

DBCRT:

Dynamic biologically conformal radiation therapy

IMRT:

Intensity modulated radiation therapy

LQ:

Linear-quadratic

MOEA:

Multi-objective evolutionary algorithm

MRI:

Magnetic resonance imaging

OAR:

Organ at risk

PET:

Positron emission tomography

RT:

Radiation therapy

SF:

Surviving fraction

TCP:

Tumor control probability

References

  • Alomari A, Rauch PJ, Orsaria M, Minja FJ, Chiang VL, Vortmeyer AO (2014) Radiologic and histologic consequences of radiosurgery for brain tumors. J Neurooncol [Internet] 117(1):33–42, March 2014. Available from: http://www.ncbi.nlm.nih.gov/pubmed/24442402

  • Angeli S, Stylianopoulos T (2016) Biphasic modeling of brain tumor biomechanics and response to radiation treatment. J Biomech [Internet] 49(9):1524–1531. Elsevier. Available from: http://dx.doi.org/10.1016/j.jbiomech.2016.03.029

  • Badri H, Pitter K, Holland EC, Michor F, Leder K (2015) Optimization of radiation dosing schedules for proneural glioblastoma. J Math Biol [Internet]. Springer, Berlin, Heidelberg. Available from: http://link.springer.com/10.1007/s00285-015-0908-x

  • Baldock AL, Anh S, Rockne R, Neal M, Clark-Swanson K, Sterin G et al (2012) Patient-specific invasiveness metric predicts benefit of resection in human gliomas. Neuro Oncol [Internet] 14:131. Available from: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4211670/

  • Baldock AL, Rockne RC, Boone AD, Neal ML, Hawkins-Daarud A, Corwin DM et al (2013) From patient-specific mathematical neuro-oncology to precision medicine. Front Oncol [Internet] 3:62, 2013/04/09 ed. Available from: http://www.ncbi.nlm.nih.gov/pubmed/23565501

  • Belfatto A, Riboldi M, Ciardo D, Cecconi A, Lazzari R, Jereczek-Fossa B et al (2015) Adaptive mathematical model of tumor response to radiotherapy based on CBCT data. IEEE J Biomed Heal Informatics [Internet] 2194(c):1–1. Available from: http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7153523

  • Brenner DJ, Hlatky LR, Hahnfeldt PJ, Huang Y, Sachs RK (1998) The linear-quadratic model and most other common radiobiological models result in similar predictions of time-dose relationships. Radiat Res [Internet] 150(1):83–91. Available from: http://www.ncbi.nlm.nih.gov/pubmed/9650605

  • Carlson DJ, Stewart RD, Semenenko VA, Sandison GA (2008) Combined use of Monte Carlo DNA damage simulations and deterministic repair models to examine putative mechanisms of cell killing. Radiat Res [Internet] 169(4):447–459, 2008/03/28 ed. Available from: http://www.ncbi.nlm.nih.gov/pubmed/18363426

  • Chvetsov AV, Sandison GA, Schwartz JL, Rengan R (2015) Ill-posed problem and regularization in reconstruction of radiobiological parameters from serial tumor imaging data. Phys Med Biol [Internet] 60(21):8491–8503. IOP Publishing

    Google Scholar 

  • Colombo MC, Giverso C, Faggiano E, Boffano C, Acerbi F, Ciarletta P (2015) Towards the personalized treatment of glioblastoma: integrating patient-specific clinical data in a continuous mechanical model. PLoS One [Internet] 10(7):e0132887. Available from: http://dx.plos.org/10.1371/journal.pone.0132887

  • Corwin D, Holdsworth C, Rockne RC, Trister AD, Mrugala MM, Rockhill JK et al (2013) Toward patient-specific, biologically optimized radiation therapy plans for the treatment of glioblastoma. PLoS One [Internet] 8(11):e79115, Jan 2013 [cited 22 Nov 2014]. Available from: http://www.pubmedcentral.nih.gov/pubmed/3827144

  • Dale RG, Jones B (2007) British Institute of Radiology. Radiobiological modelling in radiation oncology. British Institute of Radiology, 292 p

    Google Scholar 

  • Enderling H, Chaplain MA, Hahnfeldt P (2010) Quantitative modeling of tumor dynamics and radiotherapy. Acta Biotheor [Internet] 58(4):341–353. Available from: http://www.ncbi.nlm.nih.gov/pubmed/20658170

  • Fowler JF (2006) Development of radiobiology for oncology-a personal view. Phys Med Biol [Internet] 51(13):R263–R286, 7 July 2006. Available from: http://www.ncbi.nlm.nih.gov/pubmed/16790907

  • Gallasch R, Efremova M, Charoentong P, Hackl H, Trajanoski Z (2013) Mathematical models for translational and clinical oncology. J Clin Bioinf [Internet] 3(1):23. BioMed Central Ltd, 7 Jan 2013. Available from: http://www.jclinbioinformatics.com/content/3/1/23

  • Gao X, McDonald JT, Hlatky L, Enderling H (2013) Acute and fractionated irradiation differentially modulate glioma stem cell division kinetics. Cancer Res [Internet]. Available from: http://www.ncbi.nlm.nih.gov/pubmed/23269274

  • Gong J, Dos Santos MM, Finlay C, Hillen T (2013) Are more complicated tumour control probability models better? Math Med Biol 30:1–19

    Article  PubMed  Google Scholar 

  • Hall EJ, Giaccia AJ (2011) Radiobiology for the radiologist [Internet], 7th edn. Lippincott Williams & Wilkins, Philadelphia, ix, 546 p. Available from: http://www.loc.gov/catdir/toc/ecip063/2005031128.html

  • Hathout L, Ellingson B, Cloughesy T, Pope W (2015a) Patient-specific characterization of the invasiveness and proliferation of low-grade gliomas using serial MR imaging and a mathematical model of tumor growth. Oncol Rep [Internet] 2883–2888. Available from: http://www.spandidos-publications.com/10.3892/or.2015.3926

  • Hathout L, Pope WB, Lai A, Nghiemphu PL, Cloughesy TF, Ellingson BM (2015b) Radial expansion rates and tumor growth kinetics predict malignant transformation in contrast-enhancing low-grade diffuse astrocytoma. CNS Oncol 4:247–256

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Hathout L, Patel V, Wen P (2016) A 3-dimensional DTI MRI-based model of GBM growth and response to radiation therapy. Int J Oncol [Internet] 1–7. Available from: http://www.spandidos-publications.com/10.3892/ijo.2016.3595

  • Hawkins-Daarud A, Prudhomme S, van der Zee KG, Oden JT (2013) Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth. J Math Biol 67(6–7):1457–1485

    Article  PubMed  Google Scholar 

  • Hawkins-Daarud AJ, Rockne RC, Corwin D, Anderson AR, Kinahan PE, Swanson KR (2015) In silico analysis suggests differential response to bevacizumab and radiation combination therapy in newly diagnosed glioblastoma. J R Soc Interface [Internet] 12(20150388). Available from: http://dx.doi.org/10.1098/rsif.2015.0388

  • Holdsworth C, Kim M, Liao J, Phillips MH (2010) A hierarchical evolutionary algorithm for multiobjective optimization in IMRT. Med Phys [Internet] 37(9):4986–4997. Available from: http://www.ncbi.nlm.nih.gov/pubmed/20964218

  • Holloway RP, Dale RG (2013) Theoretical implications of incorporating relative biological effectiveness into radiobiological equivalence relationships. Br J Radiol [Internet] 86(1022):20120417, 2013/02/07 ed. Available from: http://www.ncbi.nlm.nih.gov/pubmed/23385996

  • Jackson T, Komarova N, Swanson K (2014) Mathematical oncology: using mathematics to enable cancer discoveries. Am Math Mon 121(November):1–17

    Google Scholar 

  • Jeong J, Shoghi KI, Deasy JO (2013) Modelling the interplay between hypoxia and proliferation in radiotherapy tumour response. Phys Med Biol [Internet] 58(14):4897–4919, 21 July 2013. Available from: http://www.ncbi.nlm.nih.gov/pubmed/23787766

  • Jones B, Dale RG (2000) Radiobiological modeling and clinical trials. Int J Radiat Oncol Biol Phys 48(1):259–265

    Article  CAS  PubMed  Google Scholar 

  • Kim M, Ghate A, Phillips MH (2012) A stochastic control formalism for dynamic biologically conformal radiation therapy. Eur J Oper Res [Internet] 219(3):541–556. Elsevier B.V. Available from: http://dx.doi.org/10.1016/j.ejor.2011.10.039

  • Kim M, Stewart RD, Phillips MH (2015) A feasibility study: Selection of a personalized radiotherapy fractionation schedule using spatiotemporal optimization. Med Phys [Internet] 42(11):6671–6678. Available from: http://scitation.aip.org/content/aapm/journal/medphys/42/11/10.1118/1.4934369

  • Kim M, Craft DL, Orton CG (2016) Within the next five years, most radiotherapy treatment schedules will be designed using spatiotemporal optimization. Med Phys [Internet] 43(5):2009–2012. Available from: http://scitation.aip.org/content/aapm/journal/medphys/43/5/10.1118/1.4943383

  • Kirkpatrick JP, Brenner DJ, Orton CG (2009) Point/Counterpoint. The linear-quadratic model is inappropriate to model high dose per fraction effects in radiosurgery. Med Phys [Internet] 36(8):3381–3384, 2009/09/15 ed. Available from: http://www.ncbi.nlm.nih.gov/pubmed/19746770

  • Langendijk JA, Lambin P, De Ruysscher D, Widder J, Bos M, Verheij M (2013) Selection of patients for radiotherapy with protons aiming at reduction of side effects: the model-based approach. Radiother Oncol [Internet] 107(3):267–273, July 2013. Elsevier Ireland Ltd. Available from: http://www.ncbi.nlm.nih.gov/pubmed/23759662

  • Leder K, Pitter K, Laplant Q, Hambardzumyan D, Ross BD, Chan TA et al (2014) Mathematical modeling of PDGF-driven glioblastoma reveals optimized radiation dosing schedules. Cell [Internet] 156(3):603–616, 30 Jan 2014. Elsevier. Available from: http://www.ncbi.nlm.nih.gov/pubmed/24485463

  • Malinen E, Søvik A, Hristov D, Bruland ØS, Olsen DR (2006) Adapting radiotherapy to hypoxic tumours. Phys Med Biol [Internet] 51(19):4903–4921, 7 Oct 2006. Available from: http://www.ncbi.nlm.nih.gov/pubmed/16985278

  • McMahon SJ, Butterworth KT, Trainor C, McGarry CK, O’Sullivan JM, Schettino G et al (2013) A kinetic-based model of radiation-induced intercellular signalling. PLoS ONE 8(1):15–18

    Article  Google Scholar 

  • Michor F, Beal K (2015) Improving cancer treatment via mathematical modeling: surmounting the challenges is worth the effort. Cell [Internet] 163(5):1059–1063. Elsevier Inc. Available from: http://dx.doi.org/10.1016/j.cell.2015.11.002

  • Moghaddasi L, Bezak E, Harriss-Phillips W (2016) Monte-Carlo model development for evaluation of current clinical target volume definition for heterogeneous and hypoxic glioblastoma. Phys Med Biol [Internet] 61(9):3407–3426. IOP Publishing

    Google Scholar 

  • Mz H, Petitjean C, Ruan S, Vera P, Dubra B (2013) Predicting lung tumor evolution during radiotherapy from PET images using a patient specific model. IEEE 10th international symposium on biomedical imaging: from nano to macro, San Francisco, CA, pp 1404–1407

    Google Scholar 

  • Neal ML, Trister AD, Ahn S, Baldock A, Bridge CA, Guyman L et al (2013) Response classification based on a minimal model of glioblastoma growth is prognostic for clinical outcomes and distinguishes progression from pseudoprogression. Cancer Res [Internet] 73(10):2976–2986, 2013/02/13 ed., 15 May 2013. Available from: http://www.ncbi.nlm.nih.gov/pubmed/23400596

  • O’Rourke SFC, McAneney H, Hillen T (2009) Linear quadratic and tumour control probability modelling in external beam radiotherapy. J Math Biol 58(4–5):799–817

    Article  PubMed  Google Scholar 

  • Poleszczuk JT, Luddy KA, Prokopiou S, Robertson-Tessi M, Moros EG, Fishman M et al (2016) Abscopal benefits of localized radiotherapy depend on activated T-cell trafficking and distribution between metastatic lesions. Cancer Res 76(5):1009–1018

    Article  CAS  PubMed  Google Scholar 

  • Powathil G, Kohandel M, Sivaloganathan S, Oza A, Milosevic M (2007) Mathematical modeling of brain tumors: effects of radiotherapy and chemotherapy. Phys Med Biol [Internet] 52(11):3291–3306. Available from: http://www.ncbi.nlm.nih.gov/pubmed/17505103

  • Powathil GG, Adamson DJA, Chaplain MAJ (2013) Towards predicting the response of a solid tumour to chemotherapy and radiotherapy treatments: clinical insights from a computational model. PLoS Comput Biol 9(7)

    Google Scholar 

  • Powathil GG, Swat M, Chaplain MAJ (2015) Systems oncology: towards patient-specific treatment regimes informed by multiscale mathematical modelling. Semin Cancer Biol [Internet] 30C:13–20, Feb 2015. Elsevier Ltd. Available from: http://www.ncbi.nlm.nih.gov/pubmed/24607841

  • Powathil GG, Munro AJ, Chaplain MAJ, Swat M (2016) Bystander effects and their implications for clinical radiation therapy: insights from multiscale in silico experiments. J Theor Biol [Internet] 401:1–14. Elsevier. Available from: http://dx.doi.org/10.1016/j.jtbi.2016.04.010

  • Prokopiou S, Moros EG, Poleszczuk J, Caudell J, Torres-Roca JF, Latifi K et al (2015) A proliferation saturation index to predict radiation response and personalize radiotherapy fractionation. Radiat Oncol [Internet] 10(1):159. Available from: http://www.ro-journal.com/content/10/1/159

  • Raman F, Scribner E, Saut O, Wenger C, Colin T, Fathallah-Shaykh HM (2016) Computational trials: unraveling motility phenotypes, progression patterns, and treatment options for glioblastoma multiforme. PLoS One [Internet] 11(1):e0146617. Available from: http://dx.plos.org/10.1371/journal.pone.0146617

  • Ribba B, Colin T, Schnell S (2006) A multiscale mathematical model of cancer, and its use in analyzing irradiation therapies. Theor Biol Med Mod 3(7)

    Google Scholar 

  • Rockne R, Alvord EC, Rockhill JK, Swanson KR, Alvord Jr EC, Rockhill JK et al (2009) A mathematical model for brain tumor response to radiation therapy. J Math Biol [Internet] 58(4–5):561–578, April 2009. Available from: http://www.ncbi.nlm.nih.gov/pubmed/18815786

  • Rockne R, Rockhill JK, Mrugala M, Spence AM, Kalet I, Hendrickson K et al (2010) Predicting efficacy of radiotherapy in individual glioblastoma patients in vivo: a mathematical modeling approach. Phys Med Biol [Internet] 55(12):3271–3285. Available from: http://www.ncbi.nlm.nih.gov/pubmed/20484781

  • Rockne RC, Trister AD, Jacobs J, Hawkins-daarud AJ, Neal ML, Hendrickson K et al (2015) A patient-specific computational model of hypoxia-modulated radiation resistance in glioblastoma using 18F-FMISO-PET. J R Soc Interface [Internet] 12. Available from: http://classic.rsif.royalsocietypublishing.org/content/12/103/20141174.short

  • Saberian F, Ghate A, Kim M (2016) Optimal fractionation in radiotherapy with multiple normal tissues. Math Med Biol 33:211–252

    Article  PubMed  Google Scholar 

  • Scott J (2012) Phase i trialist. Lancet Oncol [Internet] 13(3):236, March 2012. Elsevier. Available from: http://www.ncbi.nlm.nih.gov/pubmed/22489289

  • Scott JG, Fletcher AG, Anderson ARA, Maini PK (2016) Spatial metrics of tumour vascular organisation predict radiation efficacy in a computational model. PLoS Comput Biol 12(1):1–24

    Article  Google Scholar 

  • Siam FM, Grinfeld M, Bahar A, Rahman HA, Ahmad H, Johar F (2016) A mechanistic model of high dose irradiation damage. Math Comput Simul [Internet]. Elsevier B.V. Available from: http://linkinghub.elsevier.com/retrieve/pii/S0378475416000562

  • Smith WP, Kim M, Holdsworth C, Liao J, Phillips MH (2016) Personalized treatment planning with a model of radiation therapy outcomes for use in multiobjective optimization of IMRT plans for prostate cancer. Radiat Oncol [Internet] 11(1):38. Available from: http://ro-journal.biomedcentral.com/articles/10.1186/s13014-016-0609-7

  • Stamatakos GS, Antipas VP, Uzunoglu NK, Dale RG (2006) A four-dimensional computer simulation model of the in vivo response to radiotherapy of glioblastoma multiforme: studies on the effect of clonogenic cell density. Brit J Radiol 79:389–400

    Article  CAS  PubMed  Google Scholar 

  • Tariq I, Humbert-Vidan L, Chen T, South CP, Ezhil V, Kirkby NF et al (2015) Mathematical modelling of tumour volume dynamics in response to stereotactic ablative radiotherapy for non-small cell lung cancer. Phys Med Biol [Internet] 60(9):3695–3713. Available from: http://iopscience.iop.org/0031-9155/60/9/3695/

  • Tariq I, Chen T, Kirkby NF, Jena R (2016) Modelling and Bayesian adaptive prediction of individual patients’ tumour volume change during radiotherapy. Phys Med Biol [Internet] 61(5):2145–2161. IOP Publishing. Available from: http://stacks.iop.org/0031-9155/61/i=5/a=2145?key=crossref.511fc11bdea8efaa5a2bdcb3669ad645

  • Tinganelli W, Durante M, Hirayama R, Krämer M, Maier A, Kraft-Weyrather W et al (2015) Kill-painting of hypoxic tumours in charged particle therapy. Sci Rep [Internet] 5:17016. Nature Publishing Group. Available from: http://www.nature.com/articles/srep17016

  • Titz B, Jeraj R (2008) An imaging-based tumour growth and treatment response model: investigating the effect of tumour oxygenation on radiation therapy response. Phys Med Biol [Internet] 53(17):4471–4488, 7 Sept 2008. Available from: http://www.pubmedcentral.nih.gov/pubmed/2819145

  • Unkelbach J, Menze B, Konukoglu E, Dittmann F, Ayache B, Shih H (2014a) Radiotherapy Planning for Glioblastoma Based on a Tumor Growth Model: improving Target Volume Delineation. Phys Med Biol 59(3):771–789

    Google Scholar 

  • Unkelbach J, Craft D, Hong T, Papp D, Ramakrishnan J, Salari E et al (2014b) Exploiting tumor shrinkage through temporal optimization of radiotherapy. Phys Med Biol [Internet] 59(12):3059–79, 21 June 2014. Available from: http://www.ncbi.nlm.nih.gov/pubmed/24839901

  • Watanabe Y, Dahlman EL, Leder KZ, Hui SK (2016) A mathematical model of tumor growth and its response to single irradiation. Theor Biol Med Model [Internet] 13(1):6. Available from: http://www.tbiomed.com/content/13/1/6

  • Yankeelov TE, Atuegwu N, Hormuth D, Weis JA, Barnes SL, Miga MI et al (2013) Clinically relevant modeling of tumor growth and treatment response. Sci Transl Med [Internet] 5(187):187ps9, 2013/05/31 ed. Available from: http://www.ncbi.nlm.nih.gov/pubmed/23720579

  • Zaider M, Hanin L (2011) Tumor control probability in radiation treatment. Med Phys [Internet] 38(2):574 [cited 14 Jan 2015]. Available from: http://scitation.aip.org/content/aapm/journal/medphys/38/2/10.1118/1.3521406

  • Zaw TM, Pope WB, Cloughesy TF, Lai A, Nghiemphu PL, Ellingson BM (2014) Short-interval estimation of proliferation rate using serial diffusion MRI predicts progression-free survival in newly diagnosed glioblastoma treated with radiochemotherapy. J Neurooncol [Internet] 116(3):601–608, Feb 2014. Available from: http://www.ncbi.nlm.nih.gov/pubmed/24395348

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Authors and Affiliations

  1. Division of Mathematical Oncology, Department of Information Sciences, Beckman Research Institute, City of Hope National Medical Center, 1500 E Duarte Rd., Duarte, CA, 91010, USA

    Russell C. Rockne

  2. Division of Biostatistics, Department of Information Sciences, Beckman Research Institute, City of Hope National Medical Center, 1500 E Duarte Rd., Duarte, CA, 91010, USA

    Paul Frankel

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Correspondence to Russell C. Rockne .

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Editors and Affiliations

  1. Department of Radiation Oncology, City of Hope, Duarte, California, USA

    Jeffrey Y.C. Wong

  2. Department of Radiation Oncology, Division of Radiation Physics, City of Hope, Duarte, California, USA

    Timothy E. Schultheiss

  3. Department of Radiation Oncology, City of Hope, Duarte, California, USA

    Eric H. Radany

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Rockne, R.C., Frankel, P. (2017). Mathematical Modeling in Radiation Oncology. In: Wong, J., Schultheiss, T., Radany, E. (eds) Advances in Radiation Oncology. Cancer Treatment and Research. Springer, Cham. https://doi.org/10.1007/978-3-319-53235-6_12

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