Modeling a Tumor Growth under Immunological Activity

Yıl 2014, Cilt: 3 Sayı: 1, 45 - 55, 07.07.2014

Şenol Kartal

https://doi.org/10.17100/nevbiltek.210915

Öz

In this paper, a mathematical model which consists of system of differential equations with piecewise constant argument is constructed to describe tumor-immune system interaction. The system is based on the tumor growth model constructed by Kuznetsov et all. A solution of the system with piecewise constant arguments leads to a system of difference equations. Using Schur-Cohn criterion and a Lyapunov function, sufficient conditions are obtained for the local and global asymptotic stability of a positive equilibrium point of the system of difference equations. Neimark-Sacker bifurcations analysis shows that stable limit cycle occurs at the bifurcation point, thus resulting oscillations for tumor and immune system.

Anahtar Kelimeler

Tumor –immune system interaction, stability, system of difference equation, bifurcation.

Bağışıklık Sistemi Etkisi Altında Tümör Büyümesinin Modellenmesi

Öz

Bu çalışmada, tümör-bağışıklık sistemi etkileşimini tanımlamak için tam değer fonksiyonlu diferansiyel denklem sisteminden oluşan bir matematiksel model kurulmuştur. Sistem, Kuznetsov ve arkadaşlarının tümör büyümesi için önermiş olduğu matematiksel modele dayanmaktadır. Oluşturulan tam değer fonksiyonlu diferansiyel denklem sisteminin çözümünden fark denklem sistemi elde edilmiştir. Schur-Cohn kriteri ve Lyapunov fonksiyonun kullanılmasıyla fark denklem sisteminin pozitif denge noktasının yerel ve global kararlı olmasını sağlayan yeterli koşullar belirlenmiştir. Neimark-Sacker çatallanma analizi, çatallanma noktasında kararlı limit döngüsünün oluştuğu ve bunun sonucunda tümör ve bağışıklık sisteminin salınıma gittiğini göstermektedir.

Anahtar Kelimeler

Tümör –bağışıklık sistemi etkileşimi, kararlılık, fark denklem sistemi, çatallanma.

Kaynakça

• Pillis L. G. De., Gu W., Radunskaya A. E., “ Mixed immunotherapy and chemotherapy of tumors: modeling applications and biological interpretations” J. Theor. Biol., 238 (4), 841-862, 200 Ghaffari A., Naserifar, N., “ Optimal therapeutic protocols in cancer immunotherapy” Comput. Biol. Med., 40, 261-270, 2010.
• Gatenby R. A., “Models of tumor-host interaction as competing populations: implications for tumor biology and treatment” J. Theor. Biol., 176, 447-455, 1995.
• Costa O. S., Molina L. M., Perez D. R., Antoranz J. C., Reyes M. C., “Behavior of tumors under nonstationary therapy” Physica D, 178, 242-253, 2003.
• Kuznetsov, V. A., Makalkin I. A., Taylor, M. A., Perelson, A. S, “Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis” Bull. Math. Biol., 56, 295-321, 1994.
• Galach, M., “Dynamics of the tumor-immune system competition-the effect of time delay” Int. J. Appl. Math. Comput. Sci., 13, 395-406, 2003.
• Yafia, R., “Hopf bifurcation analysis and numerical simulations in an ODE model of the immune system with positive immune response” Nonlinear Anal. Real., 8, 1359-1369, 2007. Kirschner D., Panetta J.C., “Modeling immunotherapy of the tumor-immune interaction” J. Math. Biol., 37, 235-252, 1998.
• Pillis L. G. De, Radunskaya A., “A mathematical tumor model with immune resistance and drug therapy: an optimal control approach” J. Theor. Med,. 3, 79-100, 2001.
• Sarkar R. R., Banerjee S., “Cancer self remission and tumor stability- a stochastic approach” Math. Biosci., 196, 65-81, 2005.
• Merola A., Cosentino C., Amato F., “An insight into tumor dormancy equilibrium via the analysis of its domain of attraction” Bio. Sig. Proc. Control., 3, 212-219, 2008.
• Onofrio A. D., “Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy” Math. Comput. Model., 47, 614-637, 2008.
• Baker C. T. H., Bocharov G.A., Paul C. A. H., “Mathematical modeling of the interleukin-2 Tcell system: a comparative study of approaches based on ordinary and delay differential equations” J. Theor. Med., 2, 117-128, 1997. Banerjee S., “Immunotherapy with interleukin-2: a study based on mathematical modeling” Int. J. Appl. Math. Comput. Sci., 18, 389-398, 2008. Villasana M., Radunskaya A., “A delay differential equation model for tumor growth” J. Mat. Biol., 47, 270-294, 2003.
• Banerjee S., Sarkar R. R., “Delay-induced model for tumor-immune interaction and control of malignant tumor growth” Biosystems, 91, 268-288, 2008.
• Belloma N., Bellouquid A., Delitala M., “Mathematical topics on the modelling complex multicellular systems and tumor immune cells competition” Math. Mod. Meth. Appl. S., 11, 1683-1733, 2004.
• Patanarapeelert K., Frank T. D., Tang I. M., “From a cellular automaton model of tumorimmune interactions to its macroscopic dynamical equation: a drift-diffusion data analysis” Math. Comput. Model., 53, 122-130, 2011.
• Delitala M., “Critical analysis and perspectives on kinetic (cellular) theory of immune competition” Math. Comput. Model., 35, 63-75, 2002.
• Firmani B., Guerri L., Preziosi L., “Tumor/immune system competition with medically induced activation/disactivation” Math. Models Meth. Appl. Sci., 9, 491-512, 1999.
• Bellouquid A., Angelis E. De, “From kinetic models of multicellular growing systems to macroscopic biological tissue models” Nonlinear Anal. Real., 12, 1111–1122, 2011. May R. M., “Biological populations obeying difference equations: stable points, stable cycles and chaos” J. Theoret. Biol., 51, 511–524, 1975.
• Gopalsamy K., Liu P., “Persistence and global stability in a population model” J. Math. Anal. Appl., 224, 59-80, 1998.
• Muroya Y., Kato Y., “On Gopalsamy and Liu’s conjecture for global stability in a population model” J. Comput. Appl. Math., 181, 70-82, 2005.
• Uesugi K., Muroya Y., Ishiwata E., “On the global atrractivity for a logistic equation with piesewise constant arguments” J. Math. Anal. Appl., 294, 560-580, 2004.
• Muroya Y., “Persistence, contractivityand global stability in logistic equations with piesewise constant delays” J. Math. Anal. Appl., 270, 602-635, 2002.
• Muroya Y., “New contractivity condition in a population model with piecewise constant arguments” J. Math. Anal. Appl., 346, 65-81, 2008.
• Liu P., Gopalsamy K., “Global stability and chaos in a population model with piecewise constant arguments” Appl. Math.Comput., 101, 63-68, 1999.
• Ozturk I., Bozkurt F., Gurcan F., “Stability analysis of a mathematical model in a microcosm with piecewise constant arguments” Math. Biosci., 240, 85-91, 2012.
• Gurcan F., Bozkurt F., “Global stability in a population model with piecewise constant arguments” J. Math. Anal. Appl., 360, 334-342, 2009.
• Ozturk I., Bozkurt F., “Stability analysis of a population model with piecewise constant arguments” Nonlinear Anal. Real., 12, 1532-1545, 2011. Bozkurt F., “Modeling a tumor growth with piecewise constant arguments” Discrete Dyn. Nat. Soc., Article ID 841764, 2013. Li X., Mou C., Niu W., Wang D., “Stability analysis for discrete biological models using algebraic methods” Math. Comput. Sci, 5 (2011) 247-262.