×

Fractional kinetics under external forcing. (English) Zbl 1345.92077

Summary: Fractional tumor development is considered in the framework of one-dimensional continuous time random walks (CTRW) in the presence of chemotherapy. The chemotherapy influence on the CTRW is studied by observations of both stationary solutions due to proliferation and fractional evolution in time.

MSC:

92C50 Medical applications (general)
92C17 Cell movement (chemotaxis, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hanahan, D., Weinberg, R.A.: The hallmarks of cancer. Cell 100, 57 (2000) · doi:10.1016/S0092-8674(00)81683-9
[2] Giese, A., et al.: Dichotomy of astrocytoma migration and proliferation. Int. J. Cancer 67, 275 (1996) · doi:10.1002/(SICI)1097-0215(19960717)67:2<275::AID-IJC20>3.0.CO;2-9
[3] Giese, A., et al.: Cost of migration: invasion of malignant gliomas and implications for treatment. J. Clin. Oncol. 21, 1624 (2003) · doi:10.1200/JCO.2003.05.063
[4] Garay, T., et al.: Cell migration or cytokinesis and proliferation? Revisiting the go or grow hypothesis in cancer cells in vitro. Exp. Cell Res. (2013). doi:10.1016/j.yexcr.2013.08.018
[5] Jerby, L., et al.: Metabolic associations of reduced proliferation and oxidative stress in advanced breast cancer. Cancer Res. doi:10.1158/0008-5472.CAN-12-2215
[6] Khain, E., Sander, L.M.: Dynamics and pattern formation in invasive tumor growth. Phys. Rev. Lett. 96, 188103 (2006) · doi:10.1103/PhysRevLett.96.188103
[7] Hatzikirou, H., Basanta, D., Simon, M., Schaller, K., Deutsch, A.: “Go or Grow”: the key to the emergence of invasion in tumour progression? Math. Med. Biol. 29, 49 (2010) · Zbl 1234.92031 · doi:10.1093/imammb/dqq011
[8] A. Chauviere, A., Prziosi, L., Byrne, H.: A model of cell migration within the extracellular matrix based on a phenotypic switching mechanism. Math. Med. Biol. 27, 255 (2010) · Zbl 1196.92006 · doi:10.1093/imammb/dqp021
[9] Kolobov, A.V., Gubernov, V.V., Polezhaev, A.A.: Autowaves in the model of infiltrative tumour growth with migration-proliferation dichotomy. Math. Model. Nat. Phenom. 6, 27 (2011) · doi:10.1051/mmnp/20116703
[10] Fedotov, S., Iomin, A., Ryashko, L.: Non-Markovian models for migration-proliferation dichotomy of cancer cells: anomalous switching and spreading rate. Phys. Rev. E 84, 061131 (2011) · doi:10.1103/PhysRevE.84.061131
[11] Iomin, A.: Toy model of fractional transport of cancer cells due to self-entrapping. Phys. Rev. E 73, 061918 (2006) · doi:10.1103/PhysRevE.73.061918
[12] Montroll, E.W., Shlesinger, M.F.: The wonderful world of random walks. In: Lebowitz, J., Montroll, E.W. (eds.) Studies in Statistical Mechanics, vol. 11. Noth-Holland, Amsterdam (1984) · Zbl 0556.60027
[13] Metzler, R., Klafter, J.: The Random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[14] Stern, J.I., Raizer, J.J.: Chemotherapy in the Treatment of malignant gliomas. Expert Rev. Anticancer Ther. 6, 755-767 (2006) · doi:10.1586/14737140.6.5.755
[15] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[16] Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.: Fractional Calculus Models and Numerical Methods, Complexity, Nonlinearity and Chaos. World Scientific, Singapore (2012) · Zbl 1248.26011
[17] Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014) · Zbl 1336.34001 · doi:10.1142/9069
[18] Fedotov, S., Iomin, A.: Probabilistic approach to a proliferation and migration dichotomy in tumor cell invasion. Phys. Rev. E 77, 031911 (2008) · doi:10.1103/PhysRevE.77.031911
[19] Murray, J.D.: Mathematical Biology. Springer, Heidelberg (1993) · Zbl 0779.92001 · doi:10.1007/b98869
[20] Petrovskii, S.V., Li, B.-L.: Exactly Solvable Models of Biological Invasion. Chapman & Hall, Boca Raton (2005) · Zbl 1151.92034 · doi:10.1201/9781420034967
[21] Tracqui, P., et al.: A mathematical model of glioma growth: the effect of chemotherapy on spatio-temporal growth. Cell Prolif. 28, 17 (1995) · doi:10.1111/j.1365-2184.1995.tb00036.x
[22] Swanson, K.R., Alvord Jr, E.C., Murray, J.D.: Quantifying efficacy of chemotherapy of brain tumors (gliomas) with homogeneous and heterogeneous drug delivery. Acta Biotheor. 50, 223 (2002) · doi:10.1023/A:1022644031905
[23] Janke, E., Emde, F., Lösh, F.: Tables of Higher Functions. McGraw-Hill, New York (1960)
[24] Bagchi, B.K.: Supersymmetry in quantum and classical mechanics. Chapman & Hall/CRC, New York (2001) · Zbl 0970.81021
[25] Glauber, R.G.: Coherent and incoherent states of the radiation field. Phys. Rev. 131, 2766 (1963) · Zbl 1371.81166 · doi:10.1103/PhysRev.131.2766
[26] Louisell, W.H.: Radiation and noise in quantum electronics. McGraw-Hill, New York (1964)
[27] Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons fractals 7, 1461 (1996) · Zbl 1080.26505 · doi:10.1016/0960-0779(95)00125-5
[28] Minniti, G., et al.: Chemotherapy for glioblastoma: current treatment and future perspectives for cytotoxic and targeted agents. Anticancer Res. 29, 5171 (2009)
[29] Blinkov, S.M., Gleser, I.I.: The Human Brain in Figures and Tables: A Quantitative Handbook. Basic Books Inc., Plenum Press, New York (1968)
[30] Usher, J.R.: Some mathematical models for cancer chemotherapy. Comput. Math. Appl. 28, 73 (1994) · Zbl 0808.92018 · doi:10.1016/0898-1221(94)00179-0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.