American Institute of Mathematical Sciences

Advanced
2015, 12(4): 879-905. doi: 10.3934/mbe.2015.12.879

Mathematically modeling the biological properties of gliomas: A review

Nikolay L. Martirosyan 1, , Erica M. Rutter 2, , Wyatt L. Ramey 3, , Eric J. Kostelich 4, , Yang Kuang 5, and  Mark C. Preul 6,

1. 

Division of Neurosurgery, University of Arizona, Tucson, AZ 85724, United States
2. 

School of Mathematical & Statistical Sciences, Arizona State University, Tempe, AZ 85281, United States
3. 

Creighton Medical School, Phoenix Campus, St. Joseph's Hospital and Medical Center, Phoenix, AZ 85013, United States
4. 

School of Mathematical & Statistical Sciences, Arizona State University, Tempe, AZ 85287, United States
5. 

School of Mathematics and Statistical Sciences, Arizona State University, Tempe, AZ 85281
6. 

Division of Neurological Surgery, Barrow Neurological Institute, St. Joseph's Hospital and Medical Center, Phoenix, AZ 85013, United States

Received  May 2014 Revised  December 2014 Published  April 2015

Although mathematical modeling is a mainstay for industrial and many scientific studies, such approaches have found little application in neurosurgery. However, the fusion of biological studies and applied mathematics is rapidly changing this environment, especially for cancer research. This review focuses on the exciting potential for mathematical models to provide new avenues for studying the growth of gliomas to practical use. In vitro studies are often used to simulate the effects of specific model parameters that would be difficult in a larger-scale model. With regard to glioma invasive properties, metabolic and vascular attributes can be modeled to gain insight into the infiltrative mechanisms that are attributable to the tumor's aggressive behavior. Morphologically, gliomas show different characteristics that may allow their growth stage and invasive properties to be predicted, and models continue to offer insight about how these attributes are manifested visually. Recent studies have attempted to predict the efficacy of certain treatment modalities and exactly how they should be administered relative to each other. Imaging is also a crucial component in simulating clinically relevant tumors and their influence on the surrounding anatomical structures in the brain.
Keywords: tumor growth simulation, proliferation., glioma, Biomathematical modeling, forecasting, invasion.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Nikolay L. Martirosyan, Erica M. Rutter, Wyatt L. Ramey, Eric J. Kostelich, Yang Kuang, Mark C. Preul. Mathematically modeling the biological properties of gliomas: A review. Mathematical Biosciences & Engineering, 2015, 12 (4) : 879-905. doi: 10.3934/mbe.2015.12.879
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show all references

References:
[1]

T. Alarcón, H. M. Byrne and P. K. Maini, A multiple scale model for tumor growth,, Multiscale Modeling & Simulation, 3 (2005), 440.  doi: 10.1137/040603760.  Google Scholar

[2]

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[3]

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[4]

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[5]

P.-Y. Bondiau, O. Clatz, M. Sermesant, P.-Y. Marcy, H. Delingette, M. Frenay and N. Ayache, Biocomputing: Numerical simulation of glioblastoma growth using diffusion tensor imaging,, Physics in Medicine and Biology, 53 (2008).  doi: 10.1088/0031-9155/53/4/004.  Google Scholar

[6]

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[7]

V. Cristini, X. Li, J. S. Lowengrub and S. M. Wise, Nonlinear simulations of solid tumor growth using a mixture model: Invasion and branching,, Journal of Mathematical Biology, 58 (2009), 723.  doi: 10.1007/s00285-008-0215-x.  Google Scholar

[8]

T. Deisboeck, M. Berens, A. Kansal, S. Torquato, A. Stemmer-Rachamimov and E. Chiocca, Pattern of self-organization in tumour systems: Complex growth dynamics in a novel brain tumour spheroid model,, Cell Proliferation, 34 (2001), 115.  doi: 10.1046/j.1365-2184.2001.00202.x.  Google Scholar

[9]

S. E. Eikenberry, T. Sankar, M. Preul, E. Kostelich, C. Thalhauser and Y. Kuang, Virtual glioblastoma: Growth, migration and treatment in a three-dimensional mathematical model,, Cell Proliferation, 42 (2009), 511.  doi: 10.1111/j.1365-2184.2009.00613.x.  Google Scholar

[10]

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[11]

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[12]

J. Fort and R. V. Sole, Accelerated tumor invasion under non-isotropic cell dispersal in glioblastomas,, New Journal of Physics, 15 (2013).  doi: 10.1088/1367-2630/15/5/055001.  Google Scholar

[13]

H. B. Frieboes, J. S. Lowengrub, S. Wise, X. Zheng, P. Macklin, E. L. Bearer and V. Cristini, Computer simulation of glioma growth and morphology,, Neuroimage, 37 (2007).  doi: 10.1016/j.neuroimage.2007.03.008.  Google Scholar

[14]

H. B. Frieboes, X. Zheng, C.-H. Sun, B. Tromberg, R. Gatenby and V. Cristini, An integrated computational/experimental model of tumor invasion,, Cancer Research, 66 (2006), 1597.  doi: 10.1158/0008-5472.CAN-05-3166.  Google Scholar

[15]

S. Gao and X. Wei, Analysis of a mathematical model of glioma cells outside the tumor spheroid core,, Applicable Analysis, 92 (2013), 1379.  doi: 10.1080/00036811.2012.678335.  Google Scholar

[16]

R. A. Gatenby and E. T. Gawlinski, The glycolytic phenotype in carcinogenesis and tumor invasion insights through mathematical models,, Cancer Research, 63 (2003), 3847.   Google Scholar

[17]

J. L. Gevertz and S. Torquato, Modeling the effects of vasculature evolution on early brain tumor growth,, Journal of Theoretical Biology, 243 (2006), 517.  doi: 10.1016/j.jtbi.2006.07.002.  Google Scholar

[18]

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