\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

A tumor growth model with autophagy: The reaction-(cross-)diffusion system and its free boundary limit

  • *Corresponding author: Zhennan Zhou

    *Corresponding author: Zhennan Zhou
Abstract Full Text(HTML) Figure(8) Related Papers Cited by
  • In this paper, we propose a tumor growth model to incorporate and investigate the spatial effects of autophagy. The cells are classified into two phases: normal cells and autophagic cells, whose dynamics are also coupled with the nutrients. First, we construct a reaction-(cross-)diffusion system describing the evolution of cell densities, where the drift is determined by the negative gradient of the joint pressure, and the reaction terms manifest the unique mechanism of autophagy. Next, in the incompressible limit, such a cell density model naturally connects to a free boundary system, describing the geometric motion of the tumor region. Analyzing the free boundary model in a special case, we show that the ratio of the two phases of cells exponentially converges to a "well-mixed" limit. Within this "well-mixed" limit, we obtain an analytical solution of the free boundary system which indicates the exponential growth of the tumor size in the presence of autophagy in contrast to the linear growth without it. Numerical simulations are also provided to illustrate the analytical properties and to explore more scenarios.

    Mathematics Subject Classification: Primary: 35Q92, 92-10; Secondary: 35R35.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Illustration of the ODE model. The two gray squares denote the densities of normal cells $ n_1 $ and autophagic cells $ n_2 $. And the green square denotes the nutrient concentration $ c $. Normal cells change into autophagic cells with a transition rate $ K_1 $. And autophagic cells change into normal cells with a transition rate $ K_2 $. Normal cells and autophagic cells both grow with a net growth rate $ G $. Dashed lines show the consumption or supply of nutrients by cells. Both normal cells and autophagic cells consume nutrients with a rate $ \psi $. The key assumption is that autophagic cells will "kill" themselves with an extra death rate $ D $ to provide nutrients with a supply rate $ a $. Nutrients are added with a rate $ \lambda c_B $ and discharged with a rate $ \lambda c $

    Figure 2.  Densities of cells at time $ t = 0,1.5,3 $ with a heterogeneous initial density fraction. Blue dash-dotted line: the total density $ n $. Red solid line: the density of autophagic cells $ n_2 $. Orange dashed line: the theoretical value for the density of autophagic cells in the "well-mixed" limit. Parameters: $ G(c) = gc,\psi(c) = c,K_1 = K_2 = 1,g = 1,a = 0.4,D = 0.3 $

    Figure 3.  Evolution of $ ||\mu(\cdot,t)-\mu^*||_{L^{2n}({\Omega(t)})} $ with respect to time under different parameter regimes. Parameters: $ \gamma = 80 $, $ G(c) = gc,\psi(c) = c $ with $ g = 1,a = 0.5 $ and $ c_B = 1 $. Here the threshold value of nutrient concentration $ c_0 = 0.5 $

    Figure 4.  Graph of nutrient concentration $ c $ and moving speed of the boundary for different $ \mu $. When $ \mu = 1 $ there is no autophagic cells, when $ \mu = 0 $ all cells are autophagic cells. In the presence of autophagy, the nutrient is more sufficient and the growth rate tends to be linear in $ R $, which leads to an exponential growth of $ R $. Parameters: $ c_B = 1,a = 0.5,g = 1,D = 0.3 $

    Figure 5.  Graph of pressure $ p $ for cell density model for $ \gamma = 80 $. Left: $ ga>D $. Right $ ga<D $, there is a necrotic core in the middle

    Figure 6.  Plots of the total density $ n $ and the pressure $ p $ for different $ \gamma $ at $ t = 1 $. Left: $ n $. Right $ p $. $ \gamma = \infty $ stands for the analytical solution of the free boundary problem. As $ \gamma $ increases, the solution of the compressible model approximates the solution of the free boundary model. Parameters: $ g = 1,a = 0.5,D = 0.3,c_B = 1,K_1 = K_2 = 1 $

    Figure 7.  Growth of tumor radius w.r.t time for different $ D $. Other parameters: $ g = 1,a = 0.5 $. Left: radius w.r.t. time. Solid line: $ D = 0.3 $ and thus $ ga>D $. Dashed line: $ D = 0.5 $ and thus $ ga = D $. Right: $ \log R $ w.r.t. time for $ D = 0.3 $ from $ t = 10 $ to $ t = 20 $

    Figure 8.  Plots of the total cell density $ n $ and the autophagic cell density $ n_2 $, and the evolution of tumor radius in the case $ K_1,K_2 $ are not constants. Parameters $ a = 0.5,D = 0.3,\gamma = 80,K_1(c) = (\frac{1-c}{c+0.1})_+,K_2(c) = \frac{2c}{c+1} $

  • [1] M. BertschR. Dal Passo and M. Mimura, A free boundary problem arising in a simplified tumour growth model of contact inhibition, Interfaces and Free Boundaries, 12 (2010), 235-250.  doi: 10.4171/IFB/233.
    [2] M. BertschD. HilhorstH. Izuhara and M. Mimura, A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, Differential Equations & Applications, 4 (2012), 137-157.  doi: 10.7153/dea-04-09.
    [3] S. Bialik, S. K. Dasari and A. Kimchi, Autophagy-dependent cell death–where, how and why a cell eats itself to death, Journal of Cell Science, 131 (2018), jcs215152. doi: 10.1242/jcs.215152.
    [4] D. BreschT. ColinE. GrenierB. Ribba and O. Saut, Computational modeling of solid tumor growth: The avascular stage, SIAM Journal on Scientific Computing, 32 (2010), 2321-2344.  doi: 10.1137/070708895.
    [5] F. BubbaB. PerthameC. Pouchol and M. Schmidtchen, Hele–shaw limit for a system of two reaction-(cross-) diffusion equations for living tissues, Archive for Rational Mechanics and Analysis, 236 (2020), 735-766.  doi: 10.1007/s00205-019-01479-1.
    [6] J. A. CarrilloS. FagioliF. Santambrogio and M. Schmidtchen, Splitting schemes and segregation in reaction cross-diffusion systems, SIAM Journal on Mathematical Analysis, 50 (2018), 5695-5718.  doi: 10.1137/17M1158379.
    [7] X. ChenS. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior, Transactions of the American Mathematical Society, 357 (2005), 4771-4804.  doi: 10.1090/S0002-9947-05-03784-0.
    [8] X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: An application to tumor growth, SIAM Journal on Mathematical Analysis, 35 (2003), 974-986.  doi: 10.1137/S0036141002418388.
    [9] H. CheongU. NairJ. Geng and D. J. Klionsky, The atg1 kinase complex is involved in the regulation of protein recruitment to initiate sequestering vesicle formation for nonspecific autophagy in saccharomyces cerevisiae, Molecular Biology of the Cell, 19 (2008), 668-681.  doi: 10.1091/mbc.e07-08-0826.
    [10] S. Cui, Asymptotic stability of the stationary solution for a hyperbolic free boundary problem modeling tumor growth, SIAM Journal on Mathematical Analysis, 40 (2008), 1692-1724.  doi: 10.1137/080717778.
    [11] S. Cui and A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, Journal of Mathematical Analysis and Applications, 255 (2001), 636-677.  doi: 10.1006/jmaa.2000.7306.
    [12] S. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth, Interfaces and Free Boundaries, 5 (2003), 159-182.  doi: 10.4171/IFB/76.
    [13] S. Cui and X. Wei, Global existence for a parabolic-hyperbolic free boundary problem modelling tumor growth, Acta Mathematicae Applicatae Sinica, 21 (2005), 597-614.  doi: 10.1007/s10255-005-0268-1.
    [14] N. David and B. Perthame, Free boundary limit of a tumor growth model with nutrient, Journal de Mathématiques Pures et Appliquées, 155 (2021), 62–82. doi: 10.1016/j.matpur.2021.01.007.
    [15] K. Degenhardt, R. Mathew, B. Beaudoin, K. Bray, D. Anderson, G. Chen, C. Mukherjee, Y. Shi, C. Gélinas, Y. Fan et al., Autophagy promotes tumor cell survival and restricts necrosis, inflammation, and tumorigenesis, Cancer Cell, 10 (2006), 51–64. doi: 10.1016/j.ccr.2006.06.001.
    [16] P. DegondS. Hecht and N. Vauchelet, Incompressible limit of a continuum model of tissue growth for two cell populations, Networks and Heterogeneous Media, 15 (2020), 57-85.  doi: 10.3934/nhm.2020003.
    [17] C. M. Elliott and V. Janovskỳ, A variational inequality approach to hele-shaw flow with a moving boundary, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 88 (1981), 93-107.  doi: 10.1017/S0308210500017315.
    [18] J. Escher and G. Simonett, Classical solutions of multidimensional hele–shaw models, SIAM Journal on Mathematical Analysis, 28 (1997), 1028-1047.  doi: 10.1137/S0036141095291919.
    [19] A. Friedman, A hierarchy of cancer models and their mathematical challenges, Discrete and Continuous Dynamical Systems Series B, 4 (2004), 147-159.  doi: 10.3934/dcdsb.2004.4.147.
    [20] A. Friedman, Mathematical analysis and challenges arising from models of tumor growth, Mathematical Models and Methods in Applied Sciences, 17 (2007), 1751-1772.  doi: 10.1142/S0218202507002467.
    [21] H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Studies in Applied Mathematics, 51 (1972), 317-340.  doi: 10.1002/sapm1972514317.
    [22] N. GuillenI. Kim and A. Mellet, A hele-shaw limit without monotonicity, Archive for Rational Mechanics and Analysis, 243 (2022), 829-868.  doi: 10.1007/s00205-021-01750-4.
    [23] F. Guillén-González and J. V. Gutiérrez-Santacreu, From a cell model with active motion to a hele-shaw-like system: A numerical approach, Numerische Mathematik, 143 (2019), 107-137.  doi: 10.1007/s00211-019-01053-7.
    [24] B. Gustafsson, On a differential equation arising in a hele shaw flow moving boundary problem, Arkiv för Matematik, 22 (1984), 251–268. doi: 10.1007/BF02384382.
    [25] B. Gustafsson and A. Vasil'ev, Conformal and Potential Analysis in Hele-Shaw Cells, Springer Science & Business Media, 2006.
    [26] P. GwiazdaB. Perthame and A. Świerczewska-Gwiazda, A two-species hyperbolic–parabolic model of tissue growth, Communications in Partial Differential Equations, 44 (2019), 1605-1618.  doi: 10.1080/03605302.2019.1650064.
    [27] H. J. HwangY. Oh and M. A. Fontelos, The vanishing surface tension limit for the hele-shaw problem, Discrete & Continuous Dynamical Systems-B, 21 (2016), 3479-3514.  doi: 10.3934/dcdsb.2016108.
    [28] H. Jin and J. Lei, A mathematical model of cell population dynamics with autophagy response to starvation, Mathematical Biosciences, 258 (2014), 1-10.  doi: 10.1016/j.mbs.2014.08.014.
    [29] C. Kang and L. Avery, To be or not to be, the level of autophagy is the question: Dual roles of autophagy in the survival response to starvation, Autophagy, 4 (2008), 82-84.  doi: 10.4161/auto.5154.
    [30] T. KawamataY. KamadaY. KabeyaT. Sekito and Y. Ohsumi, Organization of the pre-autophagosomal structure responsible for autophagosome formation, Molecular Biology of the Cell, 19 (2008), 2039-2050.  doi: 10.1091/mbc.e07-10-1048.
    [31] I. Kim and N. Požár, Porous medium equation to hele-shaw flow with general initial density, Transactions of the American Mathematical Society, 370 (2018), 873-909.  doi: 10.1090/tran/6969.
    [32] I. Kim and J. Tong, Interface dynamics in a two-phase tumor growth model, Interfaces and Free Boundaries, 23 (2021), 191-304.  doi: 10.4171/IFB/454.
    [33] I. C. Kim, Uniqueness and existence results on the hele-shaw and the stefan problems, Archive for Rational Mechanics & Analysis, 168 (2003), 299-328.  doi: 10.1007/s00205-003-0251-z.
    [34] D. J. Klionsky, Autophagy: From phenomenology to molecular understanding in less than a decade, Nature Reviews Molecular Cell Biology, 8 (2007), 931-937.  doi: 10.1038/nrm2245.
    [35] D. J. Klionsky, Autophagy participates in, well, just about everything, Cell Death & Differentiation, 27 (2020), 831-832.  doi: 10.1038/s41418-020-0511-6.
    [36] M. Komatsu, S. Waguri, T. Ueno, J. Iwata, S. Murata, I. Tanida, J. Ezaki, N. Mizushima, Y. Ohsumi, Y. Uchiyama et al., Impairment of starvation-induced and constitutive autophagy in atg7-deficient mice, Journal of Cell Biology, 169 (2005), 425–434. doi: 10.1083/jcb.200412022.
    [37] B. Levine, Autophagy and cancer, Nature, 446 (2007), 745-747.  doi: 10.1038/446745a.
    [38] X. LiS. He and B. Ma, Autophagy and autophagy-related proteins in cancer, Molecular Cancer, 19 (2020), 1-16.  doi: 10.1186/s12943-020-1138-4.
    [39] L. Lin and E. H. Baehrecke, Autophagy, cell death, and cancer, Molecular & Cellular Oncology, 2 (2015), e985913. doi: 10.4161/23723556.2014.985913.
    [40] J.-G. LiuM. TangL. Wang and Z. Zhou, An accurate front capturing scheme for tumor growth models with a free boundary limit, Journal of Computational Physics, 364 (2018), 73-94.  doi: 10.1016/j.jcp.2018.03.013.
    [41] J.-G. LiuM. TangL. Wang and Z. Zhou, Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics, Discrete & Continuous Dynamical Systems-series B, 24 (2019), 3011-3035.  doi: 10.3934/dcdsb.2018297.
    [42] J.-G. LiuM. TangL. Wang and Z. Zhou, Toward understanding the boundary propagation speeds in tumor growth models, SIAM Journal on Applied Mathematics, 81 (2021), 1052-1076.  doi: 10.1137/19M1296665.
    [43] J.-G. Liu and X. Xu, Existence and incompressible limit of a tissue growth model with autophagy, SIAM Journal on Mathematical Analysis, 53 (2021), 5215-5242.  doi: 10.1137/21M1405253.
    [44] T. LorenziA. Lorz and B. Perthame, On interfaces between cell populations with different mobilities, Kinetic and Related Models, 10 (2017), 299-311.  doi: 10.3934/krm.2017012.
    [45] S. LorinA. HamaïM. Mehrpour and P. Codogno, Autophagy regulation and its role in cancer, Seminars in Cancer Biology, 23 (2013), 361-379.  doi: 10.1016/j.semcancer.2013.06.007.
    [46] J. S. Lowengrub, H. B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumours, Nonlinearity, 23 (2010), R1–R91. doi: 10.1088/0951-7715/23/1/R01.
    [47] A. MelletB. Perthame and F. Quiros, A hele–shaw problem for tumor growth, Journal of Functional Analysis, 273 (2017), 3061-3093.  doi: 10.1016/j.jfa.2017.08.009.
    [48] N. Mizushima, Autophagy: Process and function, Genes & Development, 21 (2007), 2861-2873.  doi: 10.1101/gad.1599207.
    [49] N. Mizushima and M. Komatsu, Autophagy: Renovation of cells and tissues, Cell, 147 (2011), 728-741.  doi: 10.1016/j.cell.2011.10.026.
    [50] H. NakatogawaK. SuzukiY. Kamada and Y. Ohsumi, Dynamics and diversity in autophagy mechanisms: Lessons from yeast, Nature Reviews Molecular Cell Biology, 10 (2009), 458-467.  doi: 10.1038/nrm2708.
    [51] T. P. Neufeld, Autophagy and cell growth–the yin and yang of nutrient responses, Journal of Cell Science, 125 (2012), 2359-2368.  doi: 10.1242/jcs.103333.
    [52] Y. Ohsumi, Historical landmarks of autophagy research, Cell Research, 24 (2014), 9-23.  doi: 10.1038/cr.2013.169.
    [53] H. Pan and R. Xing, Bifurcation for a free boundary problem modeling tumor growth with ecm and mde interactions, Nonlinear Analysis: Real World Applications, 43 (2018), 362-377.  doi: 10.1016/j.nonrwa.2018.02.013.
    [54] B. Perthame, Some mathematical models of tumor growth, https://www.ljll.math.upmc.fr/perthame/cours_M2.pdf.
    [55] B. PerthameF. Quirós and J. L. Vázquez, The hele–shaw asymptotics for mechanical models of tumor growth, Archive for Rational Mechanics and Analysis, 212 (2014), 93-127.  doi: 10.1007/s00205-013-0704-y.
    [56] B. PerthameM. Tang and N. Vauchelet, Traveling wave solution of the hele–shaw model of tumor growth with nutrient, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2601-2626.  doi: 10.1142/S0218202514500316.
    [57] G. J. PettetC. P. PleaseM. J. Tindall and D. L. S. Mcelwain, The migration of cells in multicell tumor spheroids, Bulletin of Mathematical Biology, 63 (2001), 231-257.  doi: 10.1006/bulm.2000.0217.
    [58] B. C. Price and X. Xu, Global existence theorem for a model governing the motion of two cell populations, Kinetic & Related Models, 13 (2020), 1175-1191.  doi: 10.3934/krm.2020042.
    [59] J. RanftM. BasanJ. ElgetiJ. JoannyJ. Prost and F. Julicher, Fluidization of tissues by cell division and apoptosis, Proceedings of the National Academy of Sciences of the United States of America, 107 (2010), 20863-20868.  doi: 10.1073/pnas.1011086107.
    [60] M. Reissig and L. v. Wolfersdorf, A simplified proof for a moving boundary problem for hele-shaw flows in the plane, Arkiv för Matematik, 31 (1993), 101–116. doi: 10.1007/BF02559501.
    [61] T. RooseS. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Review, 49 (2007), 179-208.  doi: 10.1137/S0036144504446291.
    [62] R. C. RussellH.-X. Yuan and K.-L. Guan, Autophagy regulation by nutrient signaling, Cell Research, 24 (2014), 42-57.  doi: 10.1038/cr.2013.166.
    [63] M. Tsukada and Y. Ohsumi, Isolation and characterization of autophagy-defective mutants of saccharomyces cerevisiae, FEBS Letters, 333 (1993), 169-174.  doi: 10.1016/0014-5793(93)80398-E.
    [64] Y. P. Vinogradov and P. Kufarev, On a problem of filtration, Akad. Nauk SSSR. Prikl. Mat. Meh, 12 (1948), 181-198. 
    [65] J. P. Ward and J. R. King, Mathematical modelling of avascular-tumour growth, Mathematical Medicine and Biology-A Journal of The IMA, 14 (1997), 39-69.  doi: 10.1093/imammb/14.1.39.
    [66] J. P. Ward and J. R. King, Mathematical modelling of avascular-tumour growth ii: Modelling growth saturation, Mathematical Medicine and Biology-A Journal of The IMA, 16 (1999), 171-211.  doi: 10.1093/imammb/16.2.171.
    [67] E. White, Deconvoluting the context-dependent role for autophagy in cancer, Nature Reviews Cancer, 12 (2012), 401-410.  doi: 10.1038/nrc3262.
    [68] F. YiY. Tao and Z. Liu, Quasi-stationary stefan problem as limit case of mullins-sekerka problem, Science in China Series A: Mathematics, 40 (1997), 151-162.  doi: 10.1007/BF02874434.
    [69] L. Yu, C. K. McPhee, L. Zheng, G. A. Mardones, Y. Rong, J. Peng, N. Mi, Y. Zhao, Z. Liu, F. Wan et al., Autophagy termination and lysosome reformation regulated by mtor, Nature, 465 (2010), 942–946. doi: 10.1038/nature09076.
  • 加载中

Figures(8)

SHARE

Article Metrics

HTML views(197) PDF downloads(142) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return