American Institute of Mathematical Sciences

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

Directional entropy based model for diffusivity-driven tumor growth

Marcelo E. de Oliveira 1, and  Luiz M. G. Neto 2,

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.
Keywords: fractal dimension., Glioblastomas, diffusivity tensor, Shannon entropy.
Mathematics Subject Classification: Primary: 94A17, 62P10, 68U10; Secondary: 65M06, 37M1.
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]

Physica A: Statistical Mechanics and its Applications, 387 (2008), 839-850. doi: 10.1016/j.physa.2007.10.036.  Google Scholar

[2]

Pattern Recognition, 36 (2003), 2945-2954. doi: 10.1016/S0031-3203(03)00176-6.  Google Scholar

[3]

Physica A: Statistical Mechanics and its Applications, 365 (2006), 473-480. doi: 10.1016/j.physa.2005.09.063.  Google Scholar

[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.   Google Scholar

[5]

Physica A: Statistical Mechanics and its Applications, 371 (2006), 76-79. doi: 10.1016/j.physa.2006.04.082.  Google Scholar

[6]

NeuroImage, 36 (2007), 543-549. doi: 10.1016/j.neuroimage.2007.03.057.  Google Scholar

[7]

Journal of the Neurological Sciences, 282 (2009), 67-71. doi: 10.1016/j.jns.2008.12.023.  Google Scholar

[8]

Methods San Diego Calif, 24 (2001), 309-321. Google Scholar

[9]

Neurosurgery, 39 (1996), 235-250; discussion 250-252. doi: 10.1097/00006123-199608000-00001.  Google Scholar

[10]

Medical Image Computing and Computer-Assisted Intervention, 4791 (2007), 642-650. doi: 10.1007/978-3-540-75757-3_78.  Google Scholar

[11]

Journal of Mathematical Biology, 56 (2008), 793-825. doi: 10.1007/s00285-007-0139-x.  Google Scholar

[12]

Physica A: Statistical Mechanics and its Applications, 392 (2013), 6616-6623. doi: 10.1016/j.physa.2013.08.010.  Google Scholar

[13]

Journal of theoretical biology, 203 (2000), 367-382. doi: 10.1006/jtbi.2000.2000.  Google Scholar

[14]

NeuroImage, 53 (2010), 471-479. doi: 10.1016/j.neuroimage.2010.06.050.  Google Scholar

[15]

Communications on Pure and Applied Mathematics, 9 (1956), 747-766. doi: 10.1002/cpa.3160090407.  Google Scholar

[16]

W. H. Freeman, 1982.  Google Scholar

[17]

Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.  Google Scholar

[18]

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.  Google Scholar

[19]

Journal of neuropathology and experimental neurology, 64 (2005), 479-489. Google Scholar

[20]

Physica A: Statistical Mechanics and its Applications, 388 (2009), 1303-1314. doi: 10.1016/j.physa.2008.11.038.  Google Scholar

[21]

University of Illinois Press, 1949.  Google Scholar

[22]

AJNR. American journal of neuroradiology, 23 (2002), 520-7. Google Scholar

[23]

Studies In Health Technology And Informatics, 79 (2000), 255-274. Google Scholar

[24]

Journal of the Neurological Sciences, 225 (2004), 33-37. doi: 10.1016/j.jns.2004.06.016.  Google Scholar

[25]

Cell Proliferation, 29 (1996), 269-288. doi: 10.1111/j.1365-2184.1996.tb01580.x.  Google Scholar

show all references

References:
[1]

Physica A: Statistical Mechanics and its Applications, 387 (2008), 839-850. doi: 10.1016/j.physa.2007.10.036.  Google Scholar

[2]

Pattern Recognition, 36 (2003), 2945-2954. doi: 10.1016/S0031-3203(03)00176-6.  Google Scholar

[3]

Physica A: Statistical Mechanics and its Applications, 365 (2006), 473-480. doi: 10.1016/j.physa.2005.09.063.  Google Scholar

[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.   Google Scholar

[5]

Physica A: Statistical Mechanics and its Applications, 371 (2006), 76-79. doi: 10.1016/j.physa.2006.04.082.  Google Scholar

[6]

NeuroImage, 36 (2007), 543-549. doi: 10.1016/j.neuroimage.2007.03.057.  Google Scholar

[7]

Journal of the Neurological Sciences, 282 (2009), 67-71. doi: 10.1016/j.jns.2008.12.023.  Google Scholar

[8]

Methods San Diego Calif, 24 (2001), 309-321. Google Scholar

[9]

Neurosurgery, 39 (1996), 235-250; discussion 250-252. doi: 10.1097/00006123-199608000-00001.  Google Scholar

[10]

Medical Image Computing and Computer-Assisted Intervention, 4791 (2007), 642-650. doi: 10.1007/978-3-540-75757-3_78.  Google Scholar

[11]

Journal of Mathematical Biology, 56 (2008), 793-825. doi: 10.1007/s00285-007-0139-x.  Google Scholar

[12]

Physica A: Statistical Mechanics and its Applications, 392 (2013), 6616-6623. doi: 10.1016/j.physa.2013.08.010.  Google Scholar

[13]

Journal of theoretical biology, 203 (2000), 367-382. doi: 10.1006/jtbi.2000.2000.  Google Scholar

[14]

NeuroImage, 53 (2010), 471-479. doi: 10.1016/j.neuroimage.2010.06.050.  Google Scholar

[15]

Communications on Pure and Applied Mathematics, 9 (1956), 747-766. doi: 10.1002/cpa.3160090407.  Google Scholar

[16]

W. H. Freeman, 1982.  Google Scholar

[17]

Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.  Google Scholar

[18]

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.  Google Scholar

[19]

Journal of neuropathology and experimental neurology, 64 (2005), 479-489. Google Scholar

[20]

Physica A: Statistical Mechanics and its Applications, 388 (2009), 1303-1314. doi: 10.1016/j.physa.2008.11.038.  Google Scholar

[21]

University of Illinois Press, 1949.  Google Scholar

[22]

AJNR. American journal of neuroradiology, 23 (2002), 520-7. Google Scholar

[23]

Studies In Health Technology And Informatics, 79 (2000), 255-274. Google Scholar

[24]

Journal of the Neurological Sciences, 225 (2004), 33-37. doi: 10.1016/j.jns.2004.06.016.  Google Scholar

[25]

Cell Proliferation, 29 (1996), 269-288. doi: 10.1111/j.1365-2184.1996.tb01580.x.  Google Scholar

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