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Patient-specific parameter estimates of glioblastoma multiforme growth dynamics from a model with explicit birth and death rates

1 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
2 Department of Neurosurgery, Barrow Neurological Institute, St. Josephs Hospital and Medical Center, Phoenix, AZ 85013, USA

Special Issues: Practical Insights from Cancer Models

Glioblastoma multiforme (GBM) is an aggressive primary brain cancer with a grim prog-nosis. Its morphology is heterogeneous, but prototypically consists of an inner, largely necrotic core surrounded by an outer, contrast-enhancing rim, and often extensive tumor-associated edema beyond. This structure is usually demonstrated by magnetic resonance imaging (MRI). To help relate the three highly idealized components of GBMs (i.e., necrotic core, enhancing rim, and maximum edema ex-tent) to the underlying growth “laws,” a mathematical model of GBM growth with explicit motility, birth, and death processes is proposed. This model generates a traveling-wave solution that mimics tumor progression. We develop several novel methods to approximate key characteristics of the wave profile, which can be compared with MRI data. Several simplified forms of growth and death terms and their parameter identifiability are studied. We use several test cases of MRI data of GBM patients to yield personalized parameterizations of the model, and the biological and clinical implications are discussed.
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Keywords glioblastoma multiforme; parameter estimation; reaction-diffusion models; patient-specific models

Citation: Lifeng Han, Steffen Eikenberry, Changhan He, Lauren Johnson, Mark C. Preul, Eric J. Kostelich, Yang Kuang. Patient-specific parameter estimates of glioblastoma multiforme growth dynamics from a model with explicit birth and death rates. Mathematical Biosciences and Engineering, 2019, 16(5): 5307-5323. doi: 10.3934/mbe.2019265


  • 1. A. D. Norden and P. Y. Wen, Glioma therapy in adults, The Neurologist, 12 (2006), 279–292, URL http://www.ncbi.nlm.nih.gov/pubmed/17122724.
  • 2. W. B. Pope, J. Sayre, A. Perlina, et al., MR imaging correlates of survival in patients with high-grade gliomas, Am. J. Neuroradiol., 10 (2005), 2644–2474.
  • 3. M. R. J. Carlson, W. B. Pope, S. Horvath, et al., Relationship between survival and edema in malignant gliomas: role of vascular endothelial growth factor and neuronal pentraxin 2, Clin. Cancer Res., 13 (2007), 2592–2598.
  • 4. D. B. Hoelzinger, T. Demuth and M. E. Berens, Autocrine factors that sustain glioma invasion and paracrine biology in the brain microenvironment, J. Natl. Cancer Inst., 99 (2007), 1583–1593.
  • 5. R. Stupp, W. P. Mason, M. J. van den Bent, et al., Radiotherapy plus concomitant and adjuvant temozolomide for glioblastoma, N. Engl. J. Med., 352 (2005), 987–996, URL http://www. nejm.org/doi/abs/10.1056/NEJMoa043330.
  • 6. M. R. Gilbert, M. Wang, K. D. Aldape, et al., Dose-dense temozolomide for newly diagnosed glioblastoma: a randomized phase III clinical trial, J. Clin. Oncol., 31 (2013), 4085–4091.
  • 7. Y. Kuang, J. D. Nagy and S. E. Eikenberry, Introduction to Mathematical Oncology, 1st edition, Chapman and Hall/CRC, 2015.
  • 8. N. L. Martirosyan, E. M. Rutter, W. L. Ramey, et al., Mathematically modeling the bi- ological properties of gliomas: A review, Math. Biosci. Eng., 12 (2015), 879–905, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=11020.
  • 9. K. Harley, P. van Heijster, R. Marangell, et al., Existence of traveling wave solutions for a model of tumor invasion, SIAM J. Appl. Dyn. Syst., 13 (2014), 366–396, URL http://epubs.siam. org/doi/10.1137/130923129.
  • 10. P. Gerlee and S. Nelander, Traveling wave analysis of a mathematical model of glioblastoma growth, Math.Biosci., 276(2016), 75–81, URLhttps://www.sciencedirect.com/science/ article/abs/pii/S0025556416000602?via{%}3Dihub.
  • 11. T. L. Stepien, E. M. Rutter and Y. Kuang, Traveling waves of a go-or-grow model of glioma growth, SIAM J. Appl. Math., 78 (2018), 1778–1801, URL https://epubs.siam.org/doi/ 10.1137/17M1146257.
  • 12. R. A. Fisher, The wave of advance of advantageous genes, Annals of Eugenics, 7 (1937), 355–369.
  • 13. P. Tracqui, G. Cruywagen, D. E. Woodward, et al., A mathematical model of glioma growth: The effect of chemotherapy on spatial-temporal growth, Cell Prolif., 28 (1995), 17–31.
  • 14. D. E. Woodward, J. Cook, P. Tracqui, et al., A mathematical model of glioma growth: The effect of extent of surgical resection, Cell Prolif., 29 (1996), 269–288.
  • 15. J. D. Murray, Glioblastoma brain tumors: estimating the time from brain tumor initiation and resolution of a patient survival anomaly after similar treatment protocols, J. Biol. Dyn., 6:sup2 (2012), 118–127.
  • 16. K. R. Swanson, R. C. Rostomily and E. C. Alvord, A mathematical modellng tool for predicting survival of individual patients following resection of glioblastoma: a proof of principle, Br. J. Cancer, 98 (2008), 113–119, URL http://www.nature.com/articles/6604125.
  • 17. M. L. Neal, A. D. Trister, T. Cloke, et al., Discriminating survival outcomes in patients with glioblastoma using a simulation-based, patient-specific response metric, PLoS ONE, 8 (2013), e51951, URL https://dx.plos.org/10.1371/journal.pone.0051951.
  • 18. P. R. Jackson, J. Juliano, A. Hawkins-Daarud, et al., Patient-specific mathematical neuro-oncology: Using a simple proliferation and invasion tumor model to inform clinical prac-tice, Bull. Math. Biol., 77 (2015), 846–856, URL http://link.springer.com/10.1007/s11538-015-0067-7.
  • 19. A. van der Hoorn, J. L. Yan, T. J. Larkin, et al., Validation of a semi-automatic co-registration of mri scans in patients with brain tumors during treatment follow-up, NMR Biomed., 2 (2016), 882–889.
  • 20. C. J. Watling, D. H. Lee, D. R. Macdonald, et al., Corticosteroid-induced magn. reson. imaging changes in patients with recurrent malignant glioma, J. Clin. Oncol., 12 (1994), 1886–1889.
  • 21. H. S. Zaki, M. D. Jenkinson, D. G. Du Plessis, et al., Vanishing contrast enhancement in malignant glioma after corticosteroid treatment, Acta Neurochir. (Wien), 146 (2004), 841–845.
  • 22. S. E. Eikenberry, T. Sankar, M. C. Preul, et al., Virtual glioblastoma: Growth, migration and treatment in a three-dimensional mathematical model, Cell Prolif., 42 (2009), 511–528.
  • 23. K. R. Swanson, R. C. Rockne, J. Claridge, et al., Quantifying the role of angiogenesis in malig-nant progression of gliomas: In silico modeling integrates imaging and histology, Cancer Res., 71 (2011), 7366–7375, URL http://www.ncbi.nlm.nih.gov/pubmed/21900399http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=PMC3398690http://cancerres.aacrjournals.org/cgi/doi/10.1158/0008-5472.CAN-11-1399.
  • 24. A. Claes, A. J. Idema and P. Wesseling, Diffuse glioma growth: A guerilla war, Acta Neuropathol. (Berl.), 114 (2007), 443–458.
  • 25. J. A. Sherratt and M. A. Chaplain, A new mathematical model for avascular tumor growth, J. Math. Biol., 43 (2001), 291–312, URL http://link.springer.com/10.1007/s002850100088.
  • 26. J. Canosa, On a nonlinear diffusion equation describing population growth, IBM J. Res. Dev., 17 (1973), 307–313, URL http://ieeexplore.ieee.org/document/5391351/.
  • 27. W. Penny, K. Friston, J. Ashburner, et al., Statistical Parametric Mapping: The Analysis of Functional Brain Images, 1st edition, Academic Press, London, 2007.
  • 28. A. Fedorov, R. Beichel, J. Kalpathy-Cramer, et al., 3D Slicer as an image com-puting platform for the Quantitative Imaging Network, Magn. Reson. Imaging, 30 (2012), 1323–1341, URL https://www.sciencedirect.com/science/article/pii/ S0730725X12001816?via{%}3Dihub.
  • 29. S. Dunbar, Traveling wave solutions of diffusive lotka-volterra equations, J. Math. Biol., 17 (1983), 11–32, URL http://link.springer.com/10.1007/BF00276112.
  • 30. M. C. Eisenberg and H. V. Jain, A confidence building exercise in data and identifiability: Mod-eling cancer chemotherapy as a case study, J. Theor. Biol., 431 (2017), 63–78, URL https://www.sciencedirect.com/science/article/pii/S0022519317303454?via{%}3Dihub.
  • 31. M. Kim, J. Kotas, J. Rockhill, et al., A feasibility study of personalized prescription schemes for glioblastoma patients using a proliferation and invasion glioma model, Cancers, 9 (2017), 51, URL http://www.mdpi.com/2072-6694/9/5/51.
  • 32. A. Madzvamuse, R. Barreira and A. Gerisch, Cross-diffusion in reaction-diffusion models: Anal-ysis, numerics, and applications, in ECMI 2016: Progress in Industrial Mathematics at ECMI 2016 (eds. P. Quintela, P. Barral, D. Gmez, F. J. Pena, J. Rodrguez, P. Salgado and M. E. Vzquez-Mndez), Springer, 2017, 385–392.
  • 33. J. A. Sherratt, Wavefront propagation in a competition equation with a new motility term modeling contact inhibition between cell populations, Proc. Roy. Soc. London Series A Math. Phys. Eng. Sci., 456 (2000), 2365–2386, URL http://www.royalsocietypublishing.org/doi/10.1098/rspa.2000.0616.
  • 34. T. L. Stepien, E. M. Rutter and Y. Kuang, A data-motivated density-dependent diffusion model of in vitro glioblastoma growth, Math. Biosci. Eng., 12 (2015), 1157–1172, URL http://www. ncbi.nlm.nih.gov/pubmed/26775861.
  • 35. E. J. Kostelich, Y. Kuang, J. M. McDaniel, et al., Accurate state estimation from uncertain data and models: an application of data assimilation to mathematical models of human brain tumors, Biol. Direct, 6 (2011), 64, URL http://biologydirect.biomedcentral.com/articles/10.1186/1745-6150-6-64.
  • 36. J. McDaniel, E. Kostelich, Y. Kuang, et al., Data assimilation in brain tumor models, in Mathemat-ical Methods and Models in Biomedicine (eds. U. Ledzewicz, H. Schttler and A. F. E. Kashdan), Springer, 2013, 233–262.
  • 37. M. Kot, Elements of Mathematical Ecology, Cambridge University Press, 2001.
  • 38. N. F. Britton, Essential Mathematical Biology, Springer, 2003.


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