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Some implications of scale relativity theory in avascular stages of growth of solid tumors in the presence of an immune system response. (English) Zbl 1397.92316
Summary: We present a traveling-wave analysis of a reduced mathematical model describing the growth of a solid tumor in the presence of an immune system response in the framework of scale relativity theory. Attention is focused upon the attack of tumor cells by tumor-infiltrating cytotoxic lymphocytes (TICLs), in a small multicellular tumor, without necrosis and at some stage prior to (tumor-induced) angiogenesis. For a particular choice of parameters, the underlying system of partial differential equations is able to simulate the well-documented phenomenon of cancer dormancy and propagation of a perturbation in the tumor cell concentration by cnoidal modes, by depicting spatially heterogeneous tumor cell distributions that are characterized by a relatively small total number of tumor cells. This behavior is consistent with several immunomorphological investigations. Moreover, the alteration of certain parameters of the model is enough to induce soliton like modes and soliton packets into the system, which in turn result in tumor invasion in the form of a standard traveling wave. In the same framework of scale relativity theory, a very important feature of malignant tumors also results, that even in avascular stages they might propagate and invade healthy tissues, by means of a diffusion on a Newtonian fluid.
MSC:
92C50 Medical applications (general)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C37 Cell biology
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[1] Abbott, L.F.; Wise, M.B., Dimension of a quantum-mechanical path, Am. J. phys., 49, 37-39, (1981)
[2] Abbott, R.G.; Forrest, S.; Pienta, K.J., Simulating the hallmarks of cancer, Artif. life, 12, 617-634, (2006)
[3] Adam, J., General aspects of modeling tumor growth and the immune response, ()
[4] Aguirre Ghiso, J.A., Inhibition of FAK signaling activated by urokinase receptor induces dormancy in human carcinoma cells in vivo, Oncogene, 21, 16, 2513-2524, (2002)
[5] Alarcón, T.; Byrne, H.M.; Maini, P.K., A cellular automaton model for tumour growth in inhomogeneous environment, J. theor. biol., 225, 257-274, (2003)
[6] Alberts, B.; Johnson, A.; Lewis, J.; Raff, M.; Roberts, K.; Walter, P., Molecular biology of the cell, (2002), Garland Science New York
[7] Alsabti, A., Tumour dormant state, Tumour res., 13, 1, 1-13, (1978)
[8] Ambrosi, D.; Guana, F., Stress-modulated growth., Math. mech. solids, 12, 3, 319-342, (2007) · Zbl 1149.74040
[9] Ambrosi, D.; Mollica, F., On the mechanics of a growing tumor, Int. J. eng. sci., 40, 1297-1316, (2002) · Zbl 1211.74161
[10] Ambrosi, D.; Preziosi, L., On the closure of mass balance models for tumor growth, Math. mod. meth. appl. sci., 12, 737-754, (2002) · Zbl 1016.92016
[11] Anderson, A.R.A., A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion, IMA math. appl. med. biol., 22, 163-186, (2005) · Zbl 1073.92013
[12] Anderson, A.R.A.; Chaplain, M.A.J., Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull. math. biol., 60, 857-900, (1998) · Zbl 0923.92011
[13] Araujo, R.; McElwain, D., A history of the study of solid tumour growth: the contribution of mathematical modelling, Bull. math. biol., 66, 1039-1091, (2004) · Zbl 1334.92187
[14] Araujo, R.P.; McElwain, D.L.S., A linear-elastic model of anisotropic tumor growth, Eur. J. appl. math., 15, 365-384, (2004) · Zbl 1057.92034
[15] Araujo, R.P.; McElwain, D.L.S., A mixture theory for the genesis of residual stresses in growing tissues II: solutions to the biphasic equations for a multicell spheroid, SIAM J. appl. math., 66, 447-467, (2005) · Zbl 1130.74311
[16] Auffray, C.; Nottale, L., Scale relativity theory and integrative systems biology: 1 founding principles and scale laws, Progr. biophys mol. biol., 97, 79-114, (2008)
[17] Bellomo, N.; de Angelis, E.; Preziosi, L., Multiscale modelling and mathematical problems related to tumor evolution and medical therapy, J. theor. med., 5, 111-136, (2003) · Zbl 1107.92020
[18] Bowman, F., Introduction to elliptic functions with applications, (1953), English Universities Press Ltd London · Zbl 0052.07102
[19] Breslow, N.; Chan, C.W.; Dhom, G.; Drury, R.A.; Franks, L.M.; Gellei, B.; Lee, Y.S.; Lundberg, S.; Sparke, B.; Sternby, N.H.; Tulinius, H., Latent carcinoma of prostate at autopsy in seven areas, Int. J. cancer, 20, 5, 680-688, (1977)
[20] Buzea, C.Gh.; Bejinariu, C.; Boris, C.; Vizureanu, P.; Agop, M., Motion of free particles in fractal space – time, Int. J. nonlinear sci. numer. sim., 10, 11-12, 1399-1414, (2009)
[21] Buzea, C.Gh.; Rusu, I.; Bulancea, V.; Badarau, G.; Paun, V.P.; Agop, M., The time dependent ginzburg – landau equation in fractal space – time (II), Phys. lett. A, 374, 2757-2765, (2010) · Zbl 1248.35202
[22] Byrne, H.; Preziosi, L., Modelling solid tumour growth using the theory of mixtures, Math. med. biol., 20, 341-366, (2003) · Zbl 1046.92023
[23] Byrne, H.M.; Chaplain, M.A.J., Modelling the role of cell – cell adhesion in the growth and development of carcinomas, Math. comput. model, 24, 1-17, (1996) · Zbl 0883.92014
[24] Byrne, H.M.; Chaplain, M.A.J., Growth of necrotic tumors in the presence and absence of inhibitors, Math. biosci., 135, 187-216, (1996) · Zbl 0856.92010
[25] Byrne, H.M.; Matthews, P., Asymmetric growth of models of avascular solid tumors: exploiting symmetries, IMA J. math. appl. med. biol., 19, 1-29, (2002) · Zbl 1014.92019
[26] Byrne, H.M.; Alarcon, T.; Owen, M.R.; Webb, S.D.; Maini, P.K., Modelling aspects of cancer dynamics: a review, Philos. trans. R. soc. A, 364, 1563-1578, (2006)
[27] Célérier, M.N.; Nottale, L., Quantum-classical transition in scale relativity, J. phys. A: math. gen., 37, 931-955, (2004) · Zbl 1098.81730
[28] Chaplain, M.A.J.; Ganesh, M.; Graham, I.G., Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth, J. math. biol., 42, 387-423, (2001) · Zbl 0988.92003
[29] Chaplain, M.A.J.; Graziano, L.; Preziosi, L., Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development, Math. med. biol., 23, 192-229, (2006) · Zbl 1098.92037
[30] Conger, A.D.; Ziskin, M.C., Growth of Mammalian multicellular tumour spheroids, Cancer res., 43, 558-560, (1983)
[31] Coulie, P.G., Human tumor antigens recognized by T cells: new perspectives for anti-cancer vaccines?, Mol. med. today, 3, 261-268, (1997)
[32] Cresson, J., Quelques problèmes de physique mathématique, Memoire d’habilitation diriger des recherches, (2001), Franche-Comté University Besançon, France
[33] Cresson, J., Scale calculus and the Schrödinger equation, J. math. phys., 44, 4907-4938, (2003) · Zbl 1062.39022
[34] Cristini, V.; Lowengrub, J.S.; Nie, Q., Nonlinear simulation of tumor growth, J. math. biol., 46, 191-224, (2003) · Zbl 1023.92013
[35] Cristini, V.; Frieboes, H.B.; Gatenby, R.; Caserta, S.; Ferrari, M.; Sinek, J., Morphological instability and cancer invasion, Clin. cancer res., 11, 6772-6779, (2005)
[36] Deweger, R.A.; Wilbrink, B.; Moberts, R.M.P.; Mans, D.; Oskam, R.; den Otten, W., Immune reactivity in SL2 lymphomabearing mice compared with SL2-immunized mice, Cancer immunol. immunother., 24, 1191-1192, (1987)
[37] Einstein, A., Die grundlage der allgemeinen relativitätstheorie, Ann. phys., 49, 769-822, (1916) · JFM 46.1293.01
[38] Feynman, R.P.; Hibbs, A.R., Quantum mechanics and path integrals, (1965), MacGraw-Hill New York, NY · Zbl 0176.54902
[39] Folkman, J., Tumour angiogenesis: therapeutic implications, New engl. J. med., 285, 1182-1186, (1971)
[40] Folkman, J.; Hochberg, M., Self-regulation of growth in three dimensions, Exp. med., 138, 745-753, (1973)
[41] Forni, G.; Parmiani, G.; Guarini, A.; Foa, R., Gene transfer in tumour therapy, Ann. oncol., 5, 789-794, (1994)
[42] Freyer, J.P., Role of necrosis in regulating the growth saturation of multicell spheroids, Cancer res., 48, 2432-2439, (1988)
[43] Freyer, J.P.; Schor, P.L., Regrowth of cells from multicell tumour spheroids, Cell tissue kinet., 20, 249, (1987)
[44] Freyer, J.P.; Sutherland, R.M., Regulation of growth saturation and development of necrosis in EMT6VRO multicellular spheroids glucose and oxygen supply, Cancer res., 46, 3504-3512, (1986)
[45] Frieboes, H.B.; Zheng, X.; Sun, C.-H.; Tromberg, B.; Gatenby, R.; Cristini, V., An integrated computational/experimental model of tumor invasion, Cancer res., 66, 1597-1604, (2006)
[46] Frieboes, H.B.; Lowengrub, J.S.; Wise, S.; Zheng, X.; Macklin, P.; Bearer, E.L.; Cristini, V., Computer simulation of glioma growth and morphology, Neuroimage, 37, S59-S70, (2007)
[47] Frieboes, H.B., Wise, S.M., Lowengrub, J.S., Cristini, V. Three dimensional diffuse-interface simulation of multispecies tumor growth-II: investigation of tumor invasion. Bull. Math. Biol., in preparation. · Zbl 1406.92049
[48] Greenspan, H.P., On the growth and stability of cell cultures and solid tumors, J. theor. biol., 56, 229-242, (1976)
[49] Haji-Karim, M.; Carlsson, J., Proliferation and viability in cellular spheroids of human origin, Cancer res., 38, 1457-1464, (1978)
[50] Hogea, C.S.; Murray, B.T.; Sethian, J.A., Simulating complex tumor dynamics from avascular to vascular growth using a general level-set method, J. math. biol., 53, 86-134, (2006) · Zbl 1100.92029
[51] Holmberg, L.; Baum, M., Work on your theories!, Nat. med., 2, 8, 844-846, (1996)
[52] Inch, W.R.; McCredie, J.A.; Sutherland, R.M., Growth of modular carcinomas in rodents compared with multicell spheroids in tissue culture, Growth, 34, 271-282, (1970)
[53] Isaeva, V.V.; Puschina, E.V.; Karetin, Y.A., The quasi-fractal structure of fish brain neurons, Russ. J. mar. biol., 30, 127-134, (2004)
[54] Jackson, E.A., Perspectives in nonlinear dynamics, (1991), Cambridge University Press Cambridge, London
[55] Jones, A.F.; Byrne, H.M.; Gibson, J.S.; Dold, J.W., A mathematical model of the stress induced during avascular tumor growth, J. math. biol., 40, 473-499, (2000) · Zbl 0964.92025
[56] Jumarie, G., Fractionalization of the complex-valued Brownian motion of order n using riemann – liouville derivative. applications to mathematical finance and stochastic mechanics, Chaos solitons fractals, 5, 1285-1305, (2006) · Zbl 1099.60025
[57] Kumar, V.; Abbas, A.K.; Fausto, N., Robbins and cotran pathologic basis of disease, (2004), W.B. Saunders Co.
[58] Landau, L.; Lifchitz, E., Fluid mechanics, (1959), Mir, Pergamon Moscow, Russia, New York, NY
[59] Landry, J.; Freyer, J.P.; Sutherland, R.M., Shedding of mitotic cells from the surface of multicell spheroids during growth, J. cell. physiol., 106, 23-32, (1981)
[60] Li, X.; Cristini, V.; Nie, Q.; Lowengrub, J.S., Nonlinear three dimensional simulation of solid tumor growth, Disc. cont. dyn. syst. B, 7, 581-604, (2007) · Zbl 1124.92022
[61] Lindahl, T., Instability and decay of the primary structure of DNA, Nature, 362, 709-715, (1993)
[62] Lodish, H.; Berk, A.; Zipurski, S.L.; Matsudaira, P.; Baltimore, D.; Darnell, J., Molecular cell biology, (2000), W.H. Freeman and Company New York
[63] Loeffler, D.; Ratner, S., In vivo localization of lymphocytes labeled with low concentrations of HOECHST-33342, J. immunol. methods, 119, 95-101, (1989)
[64] Lord, E.M.; Burkhardt, G., Assessment of in situ host immunity to syngeneic tumours utilizing the multicellular spheroid model, Cell. immunol., 85, 340-350, (1984)
[65] Macklin, P., 2003. Numerical simulation of tumor growth and chemotherapy. Master’s Thesis. University of Minnesota School of Mathematics.
[66] Macklin, P.; Lowengrub, J., Nonlinear simulation of the effect of microenvironment on tumor growth, J. theor. biol., 245, 677-704, (2007)
[67] Macklin, P.; Lowengrub, J., A new ghost cell/level set method for moving boundary problems: application to tumor growth, J. sci., 35, 2-3, 266-299, (2008) · Zbl 1203.65144
[68] Macklin, P.; Lowengrub, J.S., Evolving interfaces via gradients of geometry-dependent interior Poisson problems: application to tumor growth, J. comput. phys., 203, 1, 191-220, (2005) · Zbl 1067.65111
[69] Mallett, D.G.; de Pillis, L.G., A cellular automata model of tumor immune system interactions, J. theor. biol., 239, 334-350, (2006)
[70] Mandelbrot, B., LES objets fractals, (1975), Flammarion Paris · Zbl 0900.00018
[71] Mansury, Y.; Kimura, M.; Lobo, J.; Deisboeck, T.S., Emerging patterns in tumor systems: simulating the dynamics of multicellular clusters with an agent-based spatial agglomeration model, J. theor. biol., 219, 343-370, (2002)
[72] Martins, M.L.; Ferreira, S.C.; Vilela, M.J., Multiscale models for the growth of avascular tumors, Phys. life rev., 4, 128-156, (2007)
[73] Matzavinos, A.; Chaplain, M.A.J., Travelling-wave analysis of a model of the immune response to cancer, C. R. biol., 327, 995-1008, (2004)
[74] ()
[75] McCormack, P.D.; Crane, L., Physical fluid mechanics, (1973), Academic Press London-NewYork · Zbl 0289.76001
[76] Nottale, L., Fractals and the quantum theory of spacetime, Int. J. mod. phys. A, 4, 5047-5117, (1989)
[77] Nottale, L., The theory of scale relativity, Int. J. mod. phys. A, 7, 4899-4936, (1992) · Zbl 0954.81501
[78] Nottale, L., Fractal space – time and microphysics: towards a theory of scale relativity, (1993), World Scientific Singapore · Zbl 0789.58003
[79] Nottale, L., Scale relativity-from quantum mechanics to chaotic dynamics, Chaos solitons fractals, 6, 399-410, (1995) · Zbl 0905.58053
[80] Nottale, L., Scale relativity and fractal space – time: applications to quantum physics, cosmology and chaotic systems, Chaos solitons fractals, 7, 877-938, (1996) · Zbl 1080.81525
[81] Nottale, L., Scale-relativity and quantization of extra-solar planetary systems, Astron. astrophys. lett., 315, L9-L12, (1996) · Zbl 1101.85316
[82] Nottale, L., 2004. The theory of scale relativity: non-differentiable geometry and fractal space – time. In: American Institute of Physics Conference Proceedings, vol. 718, pp. 68-95.
[83] Nottale, L., 2008. Origin of complex and quaternionic wavefunctions in quantum mechanics: the scale-relativistic view. In: Angle’s, P. (Ed.), Proceedings of the Seventh International Colloquium on Clifford Algebra, Birkhäuser Verlag, Springer, New York, NY. · Zbl 1181.81053
[84] Nottale, L.; Schneider, J., Fractals and nonstandard analysis, J. math. phys., 25, 1296-1300, (1984)
[85] Ord, G.N., Fractal space – time—a geometric analog of relativistic quantum mechanics, J. phys. A math. gen., 16, 1869-1884, (1983)
[86] Ord, G.N.; Deakin, A.S., Random walks, continuum limits, and Schrödinger’s equation, Phys. rev. A, 54, 3772-3778, (1996)
[87] Ord, G.N.; Galtieri, J.A., The Feynman propagator from a single path, Phys. rev. lett., 89, 250403, (2002)
[88] Pastor, J.; Bru, A.; Fernaud, I.; Berenguer, C.; Fernaud, M.J.; Melle, S., Fractal dimension and temporal evolution of neurons, An fis (RSEF), 10, 1-2, (1998)
[89] Pissondes, J.C., Quadratic relativistic invariant and metric form in quantum mechanics, J. phys. A: math. gen., 32, 2871-2885, (1999) · Zbl 0941.35085
[90] Quaranta, V.; Weaver, A.M.; Cummings, P.T.; Anderson, A.R.A., Mathematical modeling of cancer: the future of prognosis and treatment, Clin. chim. acta, 357, 173-179, (2005)
[91] Roose, T.; Netti, P.A.; Munn, L.L.; Boucher, Y.; Jain, R., Solid stress generated by spheroid growth estimated using a linear poroelastic model, Microvasc. res., 66, 204-212, (2003)
[92] Sanga, S.; Sinek, J.P.; Frieboes, H.B.; Fruehauf, J.P.; Cristini, V., Mathematical modeling of cancer progression and response to chemotherapy, Expert rev. anticancer ther., 6, 1361-1376, (2006)
[93] Schirrmacher, V., T-cell immunity in the induction and maintenance of a tumour dormant state, Semin. cancer biol., 11, 285-295, (2001)
[94] Sinek, J.; Frieboes, H.; Zheng, X.; Cristini, V., Two-dimensional chemotherapy simulations demonstrate fundamental transport and tumor response limitations involving nanoparticles, Biomed. microdev., 6, 197-309, (2004)
[95] Sutherland, R.M., Cell and environment interactions in tumour microregions: the multicell spheroid model, Science, 240, 177-184, (1988)
[96] Udagawa, T.; Fernandez, A.; Achilles, E.G.; Folkman, J.; D’Amato, R.J., Persistence of microscopic human cancers in mice: alterations in the angiogenic balance accompanies loss of tumor dormancy, Faseb j., 16, 11, 1361-1370, (2002)
[97] Uhr, J.W.; Marches, R., Dormancy in a model of murine B cell lymphoma, Semin. cancer biol., 11, 277-283, (2001)
[98] van der Velden, Lud F.J.; Francke, Anneke L.; Hingstman, Lammert; Willems, Dick L., Dying from cancer or other chronic diseases in The Netherlands: ten-year trends derived from death certificate data, BMC palliative care, 8, 4, (2009)
[99] Vaupel, P.W.; Frinak, S.; Bicher, H.I., Heterogenous oxygen partial pressure and ph distribution in C3H mouse mammary adenocarcinoma, Cancer res., 41, 2008-2013, (1981)
[100] Waliszewski, P.; Molski, M.; Konarski, J., On the holistic approach in cellular and cancer biology: nonlinearity, complexity and quasideterminism of dynamic cellular network, J. surg. oncol., 68, 70-80, (1998)
[101] Waliszewski, P.; Molski, M.; Konarski, J., Self-similarity, collectivity and evolution of fractal dynamics during retinoid-induced differentiation of cancer cell population, Fractals, 7, 139-149, (1999)
[102] Waliszewski, P.; Konarski, J.; Molski, M., On the modification of fractal self-space during cell differentiation or tumor progression, Fractals, 8, 195-203, (2000)
[103] Waliszewski, P.; Molski, M.; Konarski, J., On the relationship between fractal geometry of space and time in which a cellular system exists and dynamics of gene expression, Acta biochim. Pol., 48, 209-220, (2001)
[104] Wilson, K.M.; Lord, E.M., Specific (EMT6) and non-specific (WEHI-164) cytolytic activity by host cells infiltrating tumour spheroids, Br. J. cancer, 55, 141-146, (1987)
[105] Wise, S.M., Lowengrub, J.S., Frieboes, H.B., Cristini, V. Three dimensional diffuse-interface simulation of multispecies tumor growth-I: numerical method. Bull. Math. Biol., in preparation. · Zbl 1398.92135
[106] Yefenof, E., Cancer dormancy: from observation to investigation and onto clinical intervention, Semin. cancer biol., 11, 269-270, (2001)
[107] Zheng, X.; Wise, S.M.; Cristini, V., Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite element/level set method, Bull. math. biol., 67, 211-259, (2005) · Zbl 1334.92214
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