# American Institute of Mathematical Sciences

2016, 13(2): 333-341. doi: 10.3934/mbe.2015005

## Directional entropy based model for diffusivity-driven tumor growth

 1 Robotic Systems Laboratory, Swiss Federal Institute of Technology (EPFL), Lausanne, CH-1015, Switzerland 2 Department of Mechanical Engineering, Engineering College of Sorocaba (FACENS), São Paulo, 18087-125, Brazil

Received  June 2015 Revised  October 2015 Published  November 2015

In this work, we present and investigate a multiscale model to simulate 3D growth of glioblastomas (GBMs) that incorporates features of the tumor microenvironment and derives macroscopic growth laws from microscopic tissue structure information. We propose a normalized version of the Shannon entropy as an alternative measure of the directional anisotropy for an estimation of the diffusivity tensor in cases where the latter is unknown. In our formulation, the tumor aggressiveness and morphological behavior is tissue-type dependent, i.e. alterations in white and gray matter regions (which can e.g. be induced by normal aging in healthy individuals or neurodegenerative diseases) affect both tumor growth rates and their morphology. The feasibility of this new conceptual approach is supported by previous observations that the fractal dimension, which correlates with the Shannon entropy we calculate, is a quantitative parameter that characterizes the variability of brain tissue, thus, justifying the further evaluation of this new conceptual approach.
Citation: Marcelo E. de Oliveira, Luiz M. G. Neto. Directional entropy based model for diffusivity-driven tumor growth. Mathematical Biosciences & Engineering, 2016, 13 (2) : 333-341. doi: 10.3934/mbe.2015005
##### References:
 [1] B. Brutovsky, D. Horvath and V. Lisy, Inverse geometric approach for the simulation of close-to-circular growth. The case of multicellular tumor spheroids, Physica A: Statistical Mechanics and its Applications, 387 (2008), 839-850. doi: 10.1016/j.physa.2007.10.036. [2] F. Camastra, Data dimensionality estimation methods: A survey, Pattern Recognition, 36 (2003), 2945-2954. doi: 10.1016/S0031-3203(03)00176-6. [3] P. Castorina and D. Zappalà, Tumor Gompertzian growth by cellular energetic balance, Physica A: Statistical Mechanics and its Applications, 365 (2006), 473-480. doi: 10.1016/j.physa.2005.09.063. [4] O. Clatz, M. Sermesant, P. yves Bondiau, H. Delingette, S. K. Warfield, G. Mal and N. Ayache, Realistic simulation of the 3d growth of brain tumors in mr images coupling diffusion with mass effect,, IEEE Transactions on Medical Imaging, (): 1334. [5] C. A. Condat and S. A. Menchón, Ontogenetic growth of multicellular tumor spheroids, Physica A: Statistical Mechanics and its Applications, 371 (2006), 76-79. doi: 10.1016/j.physa.2006.04.082. [6] F. J. Esteban, J. Sepulcre, N. V. De Mendizábal, J. Goñi, J. Navas, J. R. De Miras, B. Bejarano, J. C. Masdeu and P. Villoslada, Fractal dimension and white matter changes in multiple sclerosis, NeuroImage, 36 (2007), 543-549. doi: 10.1016/j.neuroimage.2007.03.057. [7] F. J. Esteban, J. Sepulcre, J. R. De Miras, J. Navas, N. V. De Mendizábal, J. Goñi, J. M. A. Quesada, B. Bejarano and P. Villoslada, Fractal dimension analysis of grey matter in multiple sclerosis, Journal of the Neurological Sciences, 282 (2009), 67-71. doi: 10.1016/j.jns.2008.12.023. [8] E. Fernández and H. F. Jelinek, Use of fractal theory in neuroscience: Methods, advantages, and potential problems, Methods San Diego Calif, 24 (2001), 309-321. [9] A. Giese and M. Westphal, Glioma invasion in the central nervous system, Neurosurgery, 39 (1996), 235-250; discussion 250-252. doi: 10.1097/00006123-199608000-00001. [10] C. Hogea, C. Davatzikos and G. Biros, Modeling glioma growth and mass effect in 3D MR images of the brain, Medical Image Computing and Computer-Assisted Intervention, 4791 (2007), 642-650. doi: 10.1007/978-3-540-75757-3_78. [11] 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), 793-825. doi: 10.1007/s00285-007-0139-x. [12] E. Izquierdo-Kulich, I. Rebelo, E. Tejera and J. M. Nieto-Villar, Phase transition in tumor growth: I avascular development, Physica A: Statistical Mechanics and its Applications, 392 (2013), 6616-6623. doi: 10.1016/j.physa.2013.08.010. [13] A. R. Kansal, S. Torquato, I. V. Harsh GR, E. A. Chiocca and T. S. Deisboeck, Simulated brain tumor growth dynamics using a three-dimensional cellular automaton, Journal of theoretical biology, 203 (2000), 367-382. doi: 10.1006/jtbi.2000.2000. [14] R. D. King, B. Brown, M. Hwang, T. Jeon and A. T. George, Fractal dimension analysis of the cortical ribbon in mild Alzheimer's disease, NeuroImage, 53 (2010), 471-479. doi: 10.1016/j.neuroimage.2010.06.050. [15] P. D. Lax, A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Communications on Pure and Applied Mathematics, 9 (1956), 747-766. doi: 10.1002/cpa.3160090407. [16] B. B. Mandelbrot, The Fractal Geometry of Nature, vol. 51, W. H. Freeman, 1982. [17] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications (Interdisciplinary Applied Mathematics) (v. 2), Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. [18] T. Neuvonen and E. Salli, Characterizing diffusion tensor imaging data with directional entropy, Conference proceedings : ... Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE Engineering in Medicine and Biology Society. Annual Conference, 6 (2005), 5798-5801. doi: 10.1109/IEMBS.2005.1615806. [19] H. Ohgaki and P. Kleihues, Population-based studies on incidence, survival rates, and genetic alterations in astrocytic and oligodendroglial gliomas, Journal of neuropathology and experimental neurology, 64 (2005), 479-489. [20] E. A. Reis, L. B. L. Santos and S. T. R. Pinho, A cellular automata model for avascular solid tumor growth under the effect of therapy, Physica A: Statistical Mechanics and its Applications, 388 (2009), 1303-1314. doi: 10.1016/j.physa.2008.11.038. [21] C. E. Shannon and W. Weaver, The Mathematical Theory of Information, vol. 97, University of Illinois Press, 1949. [22] S. Sinha, M. E. Bastin, I. R. Whittle and J. M. Wardlaw, Diffusion tensor MR imaging of high-grade cerebral gliomas, AJNR. American journal of neuroradiology, 23 (2002), 520-7. [23] G. S. Stamatakos, N. K. Uzunoglu, K. Delibasis, N. Mouravliansky, A. Marsh and M. Makropoulou, Tumor growth simulation and visualization: A review and a Web based paradigm, Studies In Health Technology And Informatics, 79 (2000), 255-274. [24] T. Takahashi, T. Murata, M. Omori, H. Kosaka, K. Takahashi, Y. Yonekura and Y. Wada, Quantitative evaluation of age-related white matter microstructural changes on MRI by multifractal analysis, Journal of the Neurological Sciences, 225 (2004), 33-37. doi: 10.1016/j.jns.2004.06.016. [25] D. E. Woodward, J. Cook, P. Tracqui, G. C. Cruywagen, J. D. Murray and E. C. Alvord, A mathematical model of glioma growth: The effect of extent of surgical resection, Cell Proliferation, 29 (1996), 269-288. doi: 10.1111/j.1365-2184.1996.tb01580.x.

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##### References:
 [1] B. Brutovsky, D. Horvath and V. Lisy, Inverse geometric approach for the simulation of close-to-circular growth. The case of multicellular tumor spheroids, Physica A: Statistical Mechanics and its Applications, 387 (2008), 839-850. doi: 10.1016/j.physa.2007.10.036. [2] F. Camastra, Data dimensionality estimation methods: A survey, Pattern Recognition, 36 (2003), 2945-2954. doi: 10.1016/S0031-3203(03)00176-6. [3] P. Castorina and D. Zappalà, Tumor Gompertzian growth by cellular energetic balance, Physica A: Statistical Mechanics and its Applications, 365 (2006), 473-480. doi: 10.1016/j.physa.2005.09.063. [4] O. Clatz, M. Sermesant, P. yves Bondiau, H. Delingette, S. K. Warfield, G. Mal and N. Ayache, Realistic simulation of the 3d growth of brain tumors in mr images coupling diffusion with mass effect,, IEEE Transactions on Medical Imaging, (): 1334. [5] C. A. Condat and S. A. Menchón, Ontogenetic growth of multicellular tumor spheroids, Physica A: Statistical Mechanics and its Applications, 371 (2006), 76-79. doi: 10.1016/j.physa.2006.04.082. [6] F. J. Esteban, J. Sepulcre, N. V. De Mendizábal, J. Goñi, J. Navas, J. R. De Miras, B. Bejarano, J. C. Masdeu and P. Villoslada, Fractal dimension and white matter changes in multiple sclerosis, NeuroImage, 36 (2007), 543-549. doi: 10.1016/j.neuroimage.2007.03.057. [7] F. J. Esteban, J. Sepulcre, J. R. De Miras, J. Navas, N. V. De Mendizábal, J. Goñi, J. M. A. Quesada, B. Bejarano and P. Villoslada, Fractal dimension analysis of grey matter in multiple sclerosis, Journal of the Neurological Sciences, 282 (2009), 67-71. doi: 10.1016/j.jns.2008.12.023. [8] E. Fernández and H. F. Jelinek, Use of fractal theory in neuroscience: Methods, advantages, and potential problems, Methods San Diego Calif, 24 (2001), 309-321. [9] A. Giese and M. Westphal, Glioma invasion in the central nervous system, Neurosurgery, 39 (1996), 235-250; discussion 250-252. doi: 10.1097/00006123-199608000-00001. [10] C. Hogea, C. Davatzikos and G. Biros, Modeling glioma growth and mass effect in 3D MR images of the brain, Medical Image Computing and Computer-Assisted Intervention, 4791 (2007), 642-650. doi: 10.1007/978-3-540-75757-3_78. [11] 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), 793-825. doi: 10.1007/s00285-007-0139-x. [12] E. Izquierdo-Kulich, I. Rebelo, E. Tejera and J. M. Nieto-Villar, Phase transition in tumor growth: I avascular development, Physica A: Statistical Mechanics and its Applications, 392 (2013), 6616-6623. doi: 10.1016/j.physa.2013.08.010. [13] A. R. Kansal, S. Torquato, I. V. Harsh GR, E. A. Chiocca and T. S. Deisboeck, Simulated brain tumor growth dynamics using a three-dimensional cellular automaton, Journal of theoretical biology, 203 (2000), 367-382. doi: 10.1006/jtbi.2000.2000. [14] R. D. King, B. Brown, M. Hwang, T. Jeon and A. T. George, Fractal dimension analysis of the cortical ribbon in mild Alzheimer's disease, NeuroImage, 53 (2010), 471-479. doi: 10.1016/j.neuroimage.2010.06.050. [15] P. D. Lax, A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Communications on Pure and Applied Mathematics, 9 (1956), 747-766. doi: 10.1002/cpa.3160090407. [16] B. B. Mandelbrot, The Fractal Geometry of Nature, vol. 51, W. H. Freeman, 1982. [17] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications (Interdisciplinary Applied Mathematics) (v. 2), Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. [18] T. Neuvonen and E. Salli, Characterizing diffusion tensor imaging data with directional entropy, Conference proceedings : ... Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE Engineering in Medicine and Biology Society. Annual Conference, 6 (2005), 5798-5801. doi: 10.1109/IEMBS.2005.1615806. [19] H. Ohgaki and P. Kleihues, Population-based studies on incidence, survival rates, and genetic alterations in astrocytic and oligodendroglial gliomas, Journal of neuropathology and experimental neurology, 64 (2005), 479-489. [20] E. A. Reis, L. B. L. Santos and S. T. R. Pinho, A cellular automata model for avascular solid tumor growth under the effect of therapy, Physica A: Statistical Mechanics and its Applications, 388 (2009), 1303-1314. doi: 10.1016/j.physa.2008.11.038. [21] C. E. Shannon and W. Weaver, The Mathematical Theory of Information, vol. 97, University of Illinois Press, 1949. [22] S. Sinha, M. E. Bastin, I. R. Whittle and J. M. Wardlaw, Diffusion tensor MR imaging of high-grade cerebral gliomas, AJNR. American journal of neuroradiology, 23 (2002), 520-7. [23] G. S. Stamatakos, N. K. Uzunoglu, K. Delibasis, N. Mouravliansky, A. Marsh and M. Makropoulou, Tumor growth simulation and visualization: A review and a Web based paradigm, Studies In Health Technology And Informatics, 79 (2000), 255-274. [24] T. Takahashi, T. Murata, M. Omori, H. Kosaka, K. Takahashi, Y. Yonekura and Y. Wada, Quantitative evaluation of age-related white matter microstructural changes on MRI by multifractal analysis, Journal of the Neurological Sciences, 225 (2004), 33-37. doi: 10.1016/j.jns.2004.06.016. [25] D. E. Woodward, J. Cook, P. Tracqui, G. C. Cruywagen, J. D. Murray and E. C. Alvord, A mathematical model of glioma growth: The effect of extent of surgical resection, Cell Proliferation, 29 (1996), 269-288. doi: 10.1111/j.1365-2184.1996.tb01580.x.
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