FEM Based Study of Concentration of Proliferating Cell in Brain Tumor

ABSTRACT

In this chapter we have developed a deterministic model of growth of abnormal cell concentration in a human subject at different positions. The diffusion-reaction Equation has been applied to satisfy growth dynamics .The whole tumor region is divided into layers which, with the growth of tumor form necrotic, quiescent and region of proliferating of tumor cells. Finite element method for one dimension has been employed for solving the Equations. Here we have taken into account the cellular motility along with proliferative growth ,which is particularly required in case of some of the brain tumors, where motility of gliomas cells differ widely in gray and white matter.

In this chapter we have developed a deterministic model of growth of abnormal cell concentration in a human subject at different positions. The diffusion-reaction Equation has been applied to satisfy growth dynamics .The whole tumor region is divided into layers which, with the growth of tumor form necrotic, quiescent and region of proliferating of tumor cells. Finite element method for one dimension has been employed for solving the Equations. Here we have taken into account the cellular motility along with proliferative growth ,which is particularly required in case of some of the brain tumors, where motility of gliomas cells differ widely in gray and white matter.

Cite this paper

S. Nema and V. Saxena, "FEM Based Study of Concentration of Proliferating Cell in Brain Tumor,"*Applied Mathematics*, Vol. 3 No. 8, 2012, pp. 935-942. doi: 10.4236/am.2012.38140.

S. Nema and V. Saxena, "FEM Based Study of Concentration of Proliferating Cell in Brain Tumor,"

References

[1] L. C. H. Cruz Jr. and A. G. Sorensen, “Diffusion Tensor Magnetic Resonance Imaging of Brain Tumors,” Neurosurgery Clinics of North America, Vol. 16, No. 1, 2005, pp. 115-134.

[2] J. Bernstein, W. Goldberg and E. J. Laws, “Human Malignant Astrocytoma Xenografts Migrate in Rat Brains: A Model for Central Nervous System Cancer Research,” Journal of Neuroscience Research, Vol. 22, No. 2, 1989, pp. 134-143. doi:10.1002/jnr.490220205

[3] M. R. Chicoine and D. L. Silbergeld, “Assessment of Brain Tumor Cell Motility in Vivo and in Vitro,” Journal of Neurosurgery, Vol. 82, No. 4, 1995, pp. 615-622. doi:10.3171/jns.1995.82.4.0615

[4] C. P. Geer and S. A. Grossman, “Interstitial Fluid Flow along White Matter Tracts: A Potentially Important Mechanism for the Dissemination of Primary Brain Tumors,” Journal of Neuro-Oncology, Vol. 32, No. 3, 1997, pp. 193-201. doi:10.1023/A:1005761031077

[5] J. D. Murray, “Mathematical Biology. II Spatial Models and Biomedical Applications,” 3rd Edition, Springer, New York, 2003.

[6] J. J. Casciari, S. V. Sotirchos and R. M. Sutherland, “Variations in Tumor Cell Growth Rates and Metabolism with Oxygen Concentration, Glucose Concentration, and Exrans-Cellular,” Cell Physiology, Vol. 151, 1992, pp. 386-394

[7] K. Swanson, E. Alvord and J. Murray, “A Quantitative Model for Differential Motility of Gliomas in Grey and White Matte,” Cell Prolific, Vol. 33, 2000, pp. 317-329. doi:10.1046/j.1365-2184.2000.00177.x

[8] A. Giese, L. Kluwe, B. Laube, H. Meissner, M. Berens and M. Westphal, “Migration of Human Glioma Cells on Myelin,” Neurosurgery, Vo. 38, No. 4, 1996, pp. 755-764. doi:10.1227/00006123-199604000-00026

[9] D. L. Silbergeld and M. R. Chicoine, “Isolation and Characterization of Human Malignant Glioma Cells from Histologically Normal Brain,” Journal of Neurosurgery, Vol. 86, No. 3, 1997, pp. 525-531. doi:10.3171/jns.1997.86.3.0525

[10] D. L. Collins, A. P. Zijdenbos, V. Kollokian, J. G. Sled, N. J. Kabani, C. J. Holmes and A. C. Evans, “Design and Construction of a Realistic Digital Brain Phantom,” IEEE Transactions on Medical Imaging, Vol. 17, No. 3, 1998, pp. 463-468. doi:10.1109/42.712135

[11] G. C. Cruywagen, D. E. Woodward, P. Tracqui, G. T. Bartoo, J. D. Murray and E. C. Alvord Jr., “The Modeling of Diffusive Tumors,” Journal of Biological Systems, Vol. 3, No. 4, 1995, pp. 937-945. doi:10.1142/S0218339095000836

[12] D. E. Woodward, J. Cook, P. Tracqui, G. C. Cruy wagen, J. D. Murray and E. C. Alvord Jr., “A Mathematical Model of Glioma Growth: The Effect of Extent of Surgical Resection,” Cell Proliferation, Vol. 29, No. 6, 1996, pp. 269-288. doi:10.1111/j.1365-2184.1996.tb01580.x

[13] P. Tracqui, G. C. Cruywagen, D. E. Woodard, G. T. Bartoo, J. D. Murray and E. C. Alvord Jr., “A Mathematical Model of Glioma Growth: The Effect of Chemotherapy on Spatio-Temporal Growth,” Cell Proliferation, Vol. 28, 1995, pp. 17-31. doi:10.1111/j.1365-2184.1995.tb00036.x

[14] D. L. Silbergeld, R. C. Rostomily and E. C. Alvord Jr., “The Cause of Death in Patients with Glioblastoma Is Multifactorial: Clinical Factors and Autopsy Findings in 117 Cases of Supratentorial Glioblastoma in Adults,” Journal of Neuro-Oncology, Vol. 10, 1999, pp. 179-185.

[15] K. R. Swanson, “Mathematical “ Modeling of the Growth and Control of Tumors,” Ph.D. Thesis, University of Washington, Washington, 1999.

[16] K. Swanson, E. Alvord and J. Murray, “Quantifying Efficacy of Chemotherapy of Brain Tumors with Homogeneous and Heterogeneous Drug Delivery,” Acta Biotheoretica, Vol. 50, No. 4, 2002, pp. 223-237.

[17] P. K. Burgess, P. M. Kulesa, J. D. Murray and E. C. Alvord Jr., “The Interaction of Growth Rates and Diffusion Coefficients in a Three-Dimensional Mathematical Model of Gliomas,” Journal of Neuropathology & Experimental Neurology, Vol. 56, 1997, pp. 704-713.

[18] O. Clatz, M. Sermesant, P. Bondiau, H. Delingette, S. Warfield, G. Malandain and N. Ayache, “Realistic Simulation of The 3d Growth of Brain Tumors in MR Images Coupling Diffusion with Biomechanical Deformation,” IEEE Transactions on Medical Imaging, Vol. 24, No. 10, 2005, pp. 1334-1346.

[19] J. D. Murray, “Mathematical Biology,” 3rd Edition, Springer, New York, 2002.

[20] N. Lyubimova and J. W. Hopewell, “Experimental Evidence to Support the Hypothesis That Damage to Vascular Endothelium Plays the Pri Mary Role in the Development of Late Radiation-Induced CNS Injury,” British Journal Radiadiolgy, Vol. 77, No. 918, 2004, pp. 488-492. doi:10.1259/bjr/15169876

[21] E. C. Alvord Jr. and C. M. Shaw, “Neoplasms Affecting the Nervous System of the Elderly,” In: S. Duckett, Ed., The Pathology of the Aging Human Nervous System, Lea and Fabiger, Philadelphia, 1991, pp. 210-286.

[22] S. Gu and J. Claridge, “Applying a Patient-Specific Bio-Mathematical Model of Glioma Growth to Develop Virtual[18F]-FMISO-PETmages,” Mathematical Medicine and Biology, Vol. 29, No. 1, 2012, pp. 31-48.

[1] L. C. H. Cruz Jr. and A. G. Sorensen, “Diffusion Tensor Magnetic Resonance Imaging of Brain Tumors,” Neurosurgery Clinics of North America, Vol. 16, No. 1, 2005, pp. 115-134.

[2] J. Bernstein, W. Goldberg and E. J. Laws, “Human Malignant Astrocytoma Xenografts Migrate in Rat Brains: A Model for Central Nervous System Cancer Research,” Journal of Neuroscience Research, Vol. 22, No. 2, 1989, pp. 134-143. doi:10.1002/jnr.490220205

[3] M. R. Chicoine and D. L. Silbergeld, “Assessment of Brain Tumor Cell Motility in Vivo and in Vitro,” Journal of Neurosurgery, Vol. 82, No. 4, 1995, pp. 615-622. doi:10.3171/jns.1995.82.4.0615

[4] C. P. Geer and S. A. Grossman, “Interstitial Fluid Flow along White Matter Tracts: A Potentially Important Mechanism for the Dissemination of Primary Brain Tumors,” Journal of Neuro-Oncology, Vol. 32, No. 3, 1997, pp. 193-201. doi:10.1023/A:1005761031077

[5] J. D. Murray, “Mathematical Biology. II Spatial Models and Biomedical Applications,” 3rd Edition, Springer, New York, 2003.

[6] J. J. Casciari, S. V. Sotirchos and R. M. Sutherland, “Variations in Tumor Cell Growth Rates and Metabolism with Oxygen Concentration, Glucose Concentration, and Exrans-Cellular,” Cell Physiology, Vol. 151, 1992, pp. 386-394

[7] K. Swanson, E. Alvord and J. Murray, “A Quantitative Model for Differential Motility of Gliomas in Grey and White Matte,” Cell Prolific, Vol. 33, 2000, pp. 317-329. doi:10.1046/j.1365-2184.2000.00177.x

[8] A. Giese, L. Kluwe, B. Laube, H. Meissner, M. Berens and M. Westphal, “Migration of Human Glioma Cells on Myelin,” Neurosurgery, Vo. 38, No. 4, 1996, pp. 755-764. doi:10.1227/00006123-199604000-00026

[9] D. L. Silbergeld and M. R. Chicoine, “Isolation and Characterization of Human Malignant Glioma Cells from Histologically Normal Brain,” Journal of Neurosurgery, Vol. 86, No. 3, 1997, pp. 525-531. doi:10.3171/jns.1997.86.3.0525

[10] D. L. Collins, A. P. Zijdenbos, V. Kollokian, J. G. Sled, N. J. Kabani, C. J. Holmes and A. C. Evans, “Design and Construction of a Realistic Digital Brain Phantom,” IEEE Transactions on Medical Imaging, Vol. 17, No. 3, 1998, pp. 463-468. doi:10.1109/42.712135

[11] G. C. Cruywagen, D. E. Woodward, P. Tracqui, G. T. Bartoo, J. D. Murray and E. C. Alvord Jr., “The Modeling of Diffusive Tumors,” Journal of Biological Systems, Vol. 3, No. 4, 1995, pp. 937-945. doi:10.1142/S0218339095000836

[12] D. E. Woodward, J. Cook, P. Tracqui, G. C. Cruy wagen, J. D. Murray and E. C. Alvord Jr., “A Mathematical Model of Glioma Growth: The Effect of Extent of Surgical Resection,” Cell Proliferation, Vol. 29, No. 6, 1996, pp. 269-288. doi:10.1111/j.1365-2184.1996.tb01580.x

[13] P. Tracqui, G. C. Cruywagen, D. E. Woodard, G. T. Bartoo, J. D. Murray and E. C. Alvord Jr., “A Mathematical Model of Glioma Growth: The Effect of Chemotherapy on Spatio-Temporal Growth,” Cell Proliferation, Vol. 28, 1995, pp. 17-31. doi:10.1111/j.1365-2184.1995.tb00036.x

[14] D. L. Silbergeld, R. C. Rostomily and E. C. Alvord Jr., “The Cause of Death in Patients with Glioblastoma Is Multifactorial: Clinical Factors and Autopsy Findings in 117 Cases of Supratentorial Glioblastoma in Adults,” Journal of Neuro-Oncology, Vol. 10, 1999, pp. 179-185.

[15] K. R. Swanson, “Mathematical “ Modeling of the Growth and Control of Tumors,” Ph.D. Thesis, University of Washington, Washington, 1999.

[16] K. Swanson, E. Alvord and J. Murray, “Quantifying Efficacy of Chemotherapy of Brain Tumors with Homogeneous and Heterogeneous Drug Delivery,” Acta Biotheoretica, Vol. 50, No. 4, 2002, pp. 223-237.

[17] P. K. Burgess, P. M. Kulesa, J. D. Murray and E. C. Alvord Jr., “The Interaction of Growth Rates and Diffusion Coefficients in a Three-Dimensional Mathematical Model of Gliomas,” Journal of Neuropathology & Experimental Neurology, Vol. 56, 1997, pp. 704-713.

[18] O. Clatz, M. Sermesant, P. Bondiau, H. Delingette, S. Warfield, G. Malandain and N. Ayache, “Realistic Simulation of The 3d Growth of Brain Tumors in MR Images Coupling Diffusion with Biomechanical Deformation,” IEEE Transactions on Medical Imaging, Vol. 24, No. 10, 2005, pp. 1334-1346.

[19] J. D. Murray, “Mathematical Biology,” 3rd Edition, Springer, New York, 2002.

[20] N. Lyubimova and J. W. Hopewell, “Experimental Evidence to Support the Hypothesis That Damage to Vascular Endothelium Plays the Pri Mary Role in the Development of Late Radiation-Induced CNS Injury,” British Journal Radiadiolgy, Vol. 77, No. 918, 2004, pp. 488-492. doi:10.1259/bjr/15169876

[21] E. C. Alvord Jr. and C. M. Shaw, “Neoplasms Affecting the Nervous System of the Elderly,” In: S. Duckett, Ed., The Pathology of the Aging Human Nervous System, Lea and Fabiger, Philadelphia, 1991, pp. 210-286.

[22] S. Gu and J. Claridge, “Applying a Patient-Specific Bio-Mathematical Model of Glioma Growth to Develop Virtual[18F]-FMISO-PETmages,” Mathematical Medicine and Biology, Vol. 29, No. 1, 2012, pp. 31-48.