A Prelude to the Fractional Calculus Applied to Tumor Dynamic

Rubens de Figueiredo Camargo, Arianne Vellasco Gomes, Najla Varalta


In order to refine the solution given by the classical logistic equation and extend its range of applications in the study of tumor dynamics, we propose and solve a generalization of this equation, using the so-called Fractional Calculus, i. e., we replace the ordinary derivative of order one in the usual equation by a non-integer derivative of order $ 0 < \alpha \leq 1$, and recover the classical solution as a particular case. Finally, we analyze the applicability of this model to describe the growth of cancer tumors.

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R. F. Camargo, E. C. Oliveira e Ary O. Chiacchio, Teorema de Adição para as Funções de Mittag-Leffler, TEMA, Vol 10, 1, 1-8, (2009).

R. F. Camargo, Cálculo Fracionário e Aplicações, Tese de Doutorado, IMECC, UNICAMP, (2009).

R. F. Camargo, Ary O. Chiacchio and E. Capelas de Oliveira, Differentiation to Fractional Orders and the

Fractional Telegraph Equation, J. Math. Phys., 49, 033505,(2008).

R. F. Camargo, Ary O. Chiacchio and E. Capelas de Oliveira, On anomalous diffusion and the fractional generalized Langevin equation for a harmonic oscillator,

J. Math. Phys., 50, 123518,(2009).

A.M.A El-Sayed,A.E.M. El-Mesiry, H.A.A El-Saka, One the fractional-order logistic equation, Applied Mathematics Letters, 20, 817-823 (2007).

G. H. Erjaee, M. Shahbazi and A. Erjaee, Dynamical analysis of mathematical model presented by fractional differential equations, describing the interaction between leukemic cancer cells, T cells and drug treatment with a drug optimal control, Open Access Scientific Reports, 1, 1-8 (2012).

U. Forys and A. Marciniak-Czochra, Logistic equations in tumour growth modelling, International Journal of Applied Mathematics and Computer Science, 13, 317-325 (2003).

R. A. Gatenby and T. L. Vincent, Application of quantitative models from population biology and evolutionary game theory to tumor therapeutic strategies, American Association for Cancer Research, 2, 919-927 (2013).

P. Gerlee, The model muddle: in search of tumour growth laws, Cancer Research, 73, 2407-2411 (2013).

R. Gorenflo and F.Mainardi, Fractional

Calculus: Integral and Differential Equations of Fractional

Order, CISM Lectures Notes, 223-276, (2000).

R. S. Kerbel, Tumor angiogenesis: past, present and the near future, Carcinogenesis, 21, 505-515 (2000).

A. Lotka, Meeting on the problem o forecasting city populations with special reference to New York city, Journal of the American Statistical Association, 20, (1925).

S. Michelson, A. S. Glicksman and J. T. Leith, Growth in solid heterogeneous human colon adenocarcinomas: comparison of simple logistical models, Cell Prolif, 20, 343-355 (1987).

G. M. Mittag-Leffler, G. M, Une generalisation de l. integrale de Laplace-Abel, Comptes Rendus de l.Academie des Sciences Serie II, vol. 137, pp. 537-539, (1903).

G. M. Mittag-Leffler, Sur la nouvelle fonction $E_alpha(x)$; Comptes Rendus de l.

Academie des Sciences, vol. 137, pp. 554-558, (1903).

G. M. Mittag-Leffler, Sur la representation analytiqie d.une fonction monogene (cinquieme note), Acta Mathematica, vol. 29, no. 1, pp. 101-181, (1905).

E. C. Oliveira, Funções Especiais com Aplicações, Editora Livrariada Física, São Paulo, (2005).

C. Phipps, Combination of Chemotherapy and Antiangiogenic Therapies: A Mathematical Modelling Approach, Thesis the Master Degree, University of Waterloo, Canada,(2009).

I. Podlubny, Fractional Differential Equation - An Introduction to Fractional Derivates, Fractional Differential Equations, to Methods os their Solution and some of their Applications, Academic Press, San Diego , 198, (1999).

M. Retsky, Letter to the editor ,J Theor Biol, 229, p.289, (2004).

D.S. Rodrigues, Modelagem matemática em câncer: dinâmica angiogência e quimioterapia anti-neoplásica, Dissertação de Mestrado,UNESP, IBB,(2011).

D.S. Rodrigues, P. F. A. Mancera, Suani T. R. Pinho Modelagem Matemática em Câncer e quimioterapia: uma introdução, Notas em Matemática Aplicada, SBMAC, São Carlos - SP, Brasil, Volume 58, e-ISSN 2236-5915, (2011).

J. S. Spratt, J. S. Meyer and J. A. Spratt, Rates of growth of human neoplasms: part II, J Surg Oncol, 61, 68-73 (1996).

V. G. Vaidya and F. J. Alexandro-Jr, Evaluation of some mathematical models for tumor growth, Int J Bio-med Comp, 13, 19-35 (1982).

P. F. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Correspondance mathématique et physique, 10, 113-121,(1838).

A. Wiman, Über den fundamentalsatz in der theorie der funktionen $E_alpha(z)$; Acta Math., Vol. 29, p.p. 191-201, (1905).

DOI: https://doi.org/10.5540/tema.2014.015.02.0211

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