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

Mathematically modeling the biological properties of gliomas: A review

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.
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:
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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.

[2]

E. C. Alvord Jr, Simple model of recurrent gliomas,, Journal of Neurosurgery, 75 (1991), 337.

[3]

M. Aubert, M. Badoual, S. Fereol, C. Christov and B. Grammaticos, A cellular automaton model for the migration of glioma cells,, Physical Biology, 3 (2006). doi: 10.1088/1478-3975/3/2/001.

[4]

E. L. Bearer, J. S. Lowengrub, H. B. Frieboes, Y.-L. Chuang, F. Jin, S. M. Wise, M. Ferrari, D. B. Agus and V. Cristini, Multiparameter computational modeling of tumor invasion,, Cancer Research, 69 (2009), 4493. doi: 10.1158/0008-5472.CAN-08-3834.

[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.

[6]

R. Chignola, A. Schenetti, G. Andrighetto, E. Chiesa, R. Foroni, S. Sartoris, G. Tridente and D. Liberati, Forecasting the growth of multicell tumour spheroids: Implications for the dynamic growth of solid tumours,, Cell Proliferation, 33 (2000), 219. doi: 10.1046/j.1365-2184.2000.00174.x.

[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.

[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.

[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.

[10]

S. Ferreira Jr, M. Martins and M. Vilela, Reaction-diffusion model for the growth of avascular tumor,, Physical Review E, 65 (2002). doi: 10.1103/PhysRevE.65.021907.

[11]

J. Folkman and M. Hochberg, Self-regulation of growth in three dimensions,, The Journal of Experimental Medicine, 138 (1973), 745. doi: 10.1084/jem.138.4.745.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

J. Godlewski, M. O. Nowicki, A. Bronisz, G. Nuovo, J. Palatini, M. De Lay, J. Van Brocklyn, M. C. Ostrowski, E. A. Chiocca and S. E. Lawler, Microrna-451 regulates lkb1/ampk signaling and allows adaptation to metabolic stress in glioma cells,, Molecular Cell, 37 (2010), 620. doi: 10.1016/j.molcel.2010.02.018.

[19]

A. Hagemann, K. Rohr, H. S. Stiehl, U. Spetzger and J. M. Gilsbach, Biomechanical modeling of the human head for physically based, nonrigid image registration,, IEEE Transactions on Medical Imaging, 18 (1999), 875. doi: 10.1109/42.811267.

[20]

H. Hatzikirou, D. Basanta, M. Simon, K. Schaller and A. Deutsch, Go or grow': The key to the emergence of invasion in tumour progression?,, Mathematical Medicine and Biology, 29 (2012), 49. doi: 10.1093/imammb/dqq011.

[21]

H. Hatzikirou, A. Deutsch, C. Schaller, M. Simon and K. Swanson, Mathematical modelling of glioblastoma tumour development: A review,, {Mathematical Models and Methods in Applied Sciences}, 15 (2005), 1779. doi: 10.1142/S0218202505000960.

[22]

C. Hogea, G. Biros, F. Abraham and C. Davatzikos, A robust framework for soft tissue simulations with application to modeling brain tumor mass effect in 3D MR images,, Physics in Medicine and Biology, 52 (2007). doi: 10.1088/0031-9155/52/23/008.

[23]

C. Hogea, C. Davatzikos and G. Biros, An image-driven parameter estimation problem for a reaction-diffusion glioma growth model with mass effects,, Journal of Mathematical Biology, 56 (2008), 793. doi: 10.1007/s00285-007-0139-x.

[24]

J. Holash, S. Wiegand and G. Yancopoulos, New model of tumor angiogenesis: Dynamic balance between vessel regression and growth mediated by angiopoietins and VEGF,, Oncogene, 18 (1999), 5356. doi: 10.1038/sj.onc.1203035.

[25]

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