Variance-reduced simulation of multiscale tumor growth modeling. (English) Zbl 1362.92032
Summary: We are interested in the mean-field evolution of a growing tumor as it emerges from a stochastic agent-based multiscale model. To this end, we introduce a hybrid PDE/Monte Carlo variance reduction technique. The variance reduction on the cell densities is achieved by combining a simulation of the stochastic agent-based model on the microscopic scale with a deterministic solution of a simplified (coarse) PDE on the macroscopic scale as a control variable. We show that this technique is able to significantly reduce the variance with only the (limited) additional computational cost associated with the deterministic solution of the coarse PDE. We illustrate the performance with numerical experiments in different biological scenarios.
MSC:
| 92C50 | Medical applications (general) |
| 62P10 | Applications of statistics to biology and medical sciences; meta analysis |
| 92-08 | Computational methods for problems pertaining to biology |
Software:
EigenReferences:
| [1] | B. D. Aguda, {\it Modeling the cell division cycle}, in Tutorials in Mathematical Biosciences III, Springer, New York, 2006, pp. 1-22. |
| [2] | J. A. Aguirre-Ghiso, {\it Models, mechanisms and clinical evidence for cancer dormancy}, Nat. Rev. Cancer, 7 (2007), pp. 834-846. |
| [3] | L. Albergante, J. Timmis, L. Beattie, and P. M. Kaye, {\it A petri net model of granulomatous inflammation: Implications for il-10 mediated control of Leishmania donovani infection}, PLoS Comput Biol., 9 (2013), e1003334. |
| [4] | D. Alemani, F. Pappalardo, M. Pennisi, S. Motta, and V. Brusic, {\it Combining cellular automata and lattice Boltzmann method to model multiscale avascular tumor growth coupled with nutrient diffusion and immune competition}, J. Immunol. Methods, 376 (2012), pp. 55-68. |
| [5] | A. R. A. Anderson, A. M. Weaver, P. T. Cummings, and V. Quaranta, {\it Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment.}, Cell, 127 (2006), pp. 905-915, . |
| [6] | M. Bentele, I. Lavrik, M. Ulrich, S. Stösser, D. Heermann, H. Kalthoff, P. Krammer, and R. Eils, {\it Mathematical modeling reveals threshold mechanism in cd95-induced apoptosis}, J. Cell Biol., 166 (2004), pp. 839-851. |
| [7] | K. Bentley, H. Gerhardt, and P. a. Bates, {\it Agent-based simulation of notch-mediated tip cell selection in angiogenic sprout initialisation}, J. Theoret. Biol., 250 (2008), pp. 25-36, . |
| [8] | A. H. Berger, A. G. Knudson, and P. P. Pandolfi, {\it A continuum model for tumour suppression.}, Nature, 476 (2011), pp. 163-169, . |
| [9] | R. E. Caflisch, {\it Monte Carlo and quasi-Monte Carlo methods}, Acta Numer., 7 (1998), pp. 1-49. · Zbl 0949.65003 |
| [10] | Y. Cai, S. Xu, J. Wu, and Q. Long, {\it Coupled modelling of tumour angiogenesis, tumour growth and blood perfusion}, J. Theoret. Biol., 279 (2011), pp. 90-101. · Zbl 1397.92317 |
| [11] | P. Carmeliet, {\it Angiogenesis in life, disease and medicine}, Nature, 438 (2005), pp. 932-936, . |
| [12] | P. Carmeliet and R. K. Jain, {\it Angiogenesis in cancer and other diseases}, Nature, 407 (2000), pp. 249-257. |
| [13] | A. Chakrabarti, S. Verbridge, A. D. Stroock, C. Fischbach, and J. D. Varner, {\it Multiscale models of breast cancer progression}, Ann. Biomed. Eng., 40 (2012), pp. 2488-2500. |
| [14] | M. A. Chaplain, S. R. McDougall, and A. Anderson, {\it Mathematical modeling of tumor-induced angiogenesis}, Annu. Rev. Biomed. Eng., 8 (2006), pp. 233-257. |
| [15] | A. Cicalese, G. Bonizzi, C. E. Pasi, et al., {\it The tumor suppressor p53 regulates polarity of self-renewing divisions in mammary stem cells}, Cell, 138 (2009), pp. 1083-1095, . |
| [16] | I. N. Colaluca, D. Tosoni, P. Nuciforo, et al., {\it NUMB controls p53 tumour suppressor activity}, Nature, 451 (2008), pp. 76-80, . |
| [17] | G. D’Antonio, P. Macklin, and L. Preziosi, {\it An agent-based model for elasto-plastic mechanical interactions between cells, basement membrane and extracellular matrix}, Math. Biosci. Eng., 10 (2012), pp. 75-101, . · Zbl 1259.65001 |
| [18] | M. A. M. de Aguiar, E. Rauch, and Y. Bar-Yam, {\it Mean-field approximation to a spatial host-pathogen model}, Phys. Rev. E, 67 (2003), 047102. · Zbl 1058.92043 |
| [19] | T. S. Deisboeck, Z. Wang, P. Macklin, and V. Cristini, {\it Multiscale cancer modeling}, Annu. Rev. Biomed. Eng., 13 (2011), pp. 127-155, . |
| [20] | G. Dimarco and L. Pareschi, {\it Hybrid multiscale methods II. Kinetic equations}, Multiscale Model. Simul., 6 (2008), pp. 1169-1197. · Zbl 1185.65198 |
| [21] | R. Erban and H. G. Othmer, {\it From individual to collective behavior in bacterial chemotaxis}, SIAM J. Appl. Math., 65 (2004), pp. 361-391, . · Zbl 1073.35116 |
| [22] | G. P. Figueredo, T. V. Joshi, J. M. Osborne, H. M. Byrne, and M. R. Owen, {\it On-lattice agent-based simulation of populations of cells within the open-source chaste framework}, Interface Focus, 3 (2013), 20120081. |
| [23] | A. G. Fletcher, C. J. W. Breward, and S. J. Chapman, {\it Mathematical modeling of monoclonal conversion in the colonic crypt}, J. Theoret. Biol., 300 (2012), pp. 118-133, . · Zbl 1397.92161 |
| [24] | J. Galle, M. Loeffler, and D. Drasdo, {\it Modeling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro}, Biophys. J., 88 (2005), pp. 62-75. |
| [25] | T. Garay, E. Juhász, E. Molnár, et al., {\it Cell migration or cytokinesis and proliferation?—Revisiting the “go or grow” hypothesis in cancer cells in vitro}, Exp. Cell Res., 319 (2013), pp. 3094-3103, . |
| [26] | R. A. Gatenby and R. J. Gillies, {\it Why do cancers have high aerobic glycolysis?}, Nature Rev. Cancer, 4 (2004), pp. 891-899. |
| [27] | H. Gerhardt, {\it VEGF and endothelial guidance in angiogenic sprouting}, Organogenesis, 4 (2008), pp. 241-246, . |
| [28] | G. Guennebaud and B. Jacob, {\it Eigen v}3, (2010). |
| [29] | D. Hanahan and R. A. Weinberg, {\it The Hallmarks of Cancer}, Cell, 100 (2000), pp. 57-70, . |
| [30] | D. Hanahan and R. A. Weinberg, {\it Hallmarks of cancer: The next generation.}, Cell, 144 (2011), pp. 646-674, . |
| [31] | H. Hatzikirou, D. Basanta, M. Simon, K. Schaller, and A. Deutsch, {\it “Go or grow”: The key to the emergence of invasion in tumour progression?}, Math. Med. Biol., 29 (2012), pp. 49-65, . · Zbl 1234.92031 |
| [32] | M. Hellström, L.-K. Phng, J. J. Hofmann, et al., {\it Dll4 signalling through Notch1 regulates formation of tip cells during angiogenesis}, Nature, 445 (2007), pp. 776-780, . |
| [33] | T. Hillen and K. J. Painter, {\it A user’s guide to PDE models for chemotaxis}, J. Math. Biol., 58 (2009), pp. 183-217, . · Zbl 1161.92003 |
| [34] | M. E. Hubbard and H. M. Byrne, {\it Multiphase modelling of vascular tumour growth in two spatial dimensions}, J. Theoret. Biol., 316 (2013), pp. 70-89, . · Zbl 1397.92333 |
| [35] | L. Jakobsson, C. A. Franco, K. Bentley, et al., {\it Endothelial cells dynamically compete for the tip cell position during angiogenic sprouting}, Nature Cell Biol., 12 (2010), pp. 943-53, . |
| [36] | Y. Kam, K. A. Rejniak, and A. R. Anderson, {\it Cellular modeling of cancer invasion: Integration of in silico and in vitro approaches}, J. Cellular Physiol., 227 (2012), pp. 431-438. |
| [37] | M. Kavousanakis, P. Liu, A. Boudouvis, J. Lowengrub, and I. G. Kavrekidis, {\it Efficient coarse simulation of a growing avascular tumor}, Phys. Rev. E, 85 (2012), pp. 1-11, . |
| [38] | C. Lemieux, {\it Monte Carlo and Quasi-Monte Carlo Sampling}, Springer, New York, 2009, . · Zbl 1269.65001 |
| [39] | P. Macklin, M. E. Edgerton, A. M. Thompson, and V. Cristini, {\it Patient-calibrated agent-based modelling of ductal carcinoma in situ (DCIS): from microscopic measurements to macroscopic predictions of clinical progression}, J. Theoret. Biol., 301 (2012), pp. 122-140, . · Zbl 1397.92346 |
| [40] | N. V. Mantzaris, S. Webb, and H. G. Othmer, {\it Mathematical modeling of tumor-induced angiogenesis}, J. Math. Biol., 49 (2004), pp. 111-187. · Zbl 1109.92020 |
| [41] | S. J. Morrison and J. Kimble, {\it Asymmetric and symmetric stem-cell divisions in development and cancer}, Nature, 441 (2006), pp. 1068-1074, . |
| [42] | M. M. Olsen and H. T. Siegelmann, {\it Multiscale agent-based model of tumor angiogenesis}, Procedia Comput. Sci., 18 (2013), pp. 1016-1025, . |
| [43] | M. R. Owen, T. Alarcón, P. K. Maini, and H. M. Byrne, {\it Angiogenesis and vascular remodelling in normal and cancerous tissues}, J. Math. Biol., 58 (2009), pp. 689-721, . · Zbl 1311.92034 |
| [44] | M. R. Owen, I. J. Stamper, M. Muthana, et al., {\it Mathematical modeling predicts synergistic antitumor effects of combining a macrophage-based, hypoxia-targeted gene therapy with chemotherapy}, Cancer Res., 71 (2011), pp. 2826-2837, . |
| [45] | H. Perfahl, H. M. Byrne, T. Chen, V. Estrella, T. Alarcón, A. Lapin, R. A. Gatenby, R. J. Gillies, M. C. Lloyd, P. K. Maini, et al., {\it Multiscale modelling of vascular tumour growth in 3d: the roles of domain size and boundary conditions}, PlOS ONE, 6 (2011), e14790. |
| [46] | J. Poleszczuk and H. Enderling, {\it A high-performance cellular automaton model of tumor growth with dynamically growing domains}, Appl. Math., 5 (2014), 144. |
| [47] | G. A. Radtke and N. G. Hadjiconstantinou, {\it Variance-reduced particle simulation of the Boltzmann transport equation in the relaxation-time approximation}, Phys. Rev. E, 79 (2009), 56711, . |
| [48] | K. A. Rejniak and A. R. Anderson, {\it Hybrid models of tumor growth}, Wiley Interdiscip. Rev. Syst. Biol. Med., 3 (2011), pp. 115-125. |
| [49] | H. Rieger and M. Welter, {\it Integrative models of vascular remodeling during tumor growth}, Wiley Interdiscip. Rev. Syst. Biol. Med., 7 (2015), pp. 113-129. |
| [50] | T. Roose, S. Chapman, and P. Maini, {\it Mathematical models of avascular tumor growth}, SIAM Rev., 49 (2007), pp. 179-208, . · Zbl 1117.93011 |
| [51] | M. Rousset and G. Samaey, {\it Simulating individual-based models of bacterial chemotaxis with asymptotic variance reduction}, Math. Models Methods Appl. Sci., 23 (2013), pp. 2155-2191, . · Zbl 1291.35417 |
| [52] | K. Schleich and I. N. Lavrik, {\it Mathematical modeling of apoptosis}, Cell Commun. Signaling, 11 (2013), pp. 1-7, . |
| [53] | M. Scianna and L. Preziosi, {\it Multiscale developments of the cellular potts model}, Multiscale Model. Simul., 10 (2012), pp. 342-382. · Zbl 1402.92159 |
| [54] | J. G. Scott, A. G. Fletcher, C. L. Timlin, D. Basanta, A. R. Anderson, and P. K. Maini, {\it Towards patient-specific biology-driven heterogeneous radiation planning: Using a computational model of tumor growth to identify novel radiation sensitivity signatures}, in Proceedings of the 9th European Conference on Mathematical and Theoretical Biology, 2014. |
| [55] | J. A. Sherratt and M. A. Chaplain, {\it A new mathematical model for avascular tumour growth}, J. Math. Biol., 43 (2001), pp. 291-312. · Zbl 0990.92021 |
| [56] | S. L. Spencer and P. K. Sorger, {\it Measuring and modeling apoptosis in single cells}, Cell, 144 (2011), pp. 926-939. |
| [57] | F. Spill, P. Guerrero, T. Alarcon, P. K. Maini, and H. M. Byrne, {\it Mesoscopic and continuum modelling of angiogenesis}, J. Math. Biol., (2014), pp. 1-48, . · Zbl 1330.92026 |
| [58] | A. M. Stein, T. Demuth, D. Mobley, M. Berens, and L. M. Sander, {\it A mathematical model of glioblastoma tumor spheroid invasion in a three-dimensional in vitro experiment}, Biophys. J., 92 (2007), pp. 356-365. |
| [59] | S. H. Strogatz, {\it Exploring complex networks}, Nature, 410 (2001), pp. 268-276. · Zbl 1370.90052 |
| [60] | K. R. Swanson, {\it Quantifying glioma cell growth and invasion in vitro}, Math. Comput. Model., 47 (2008), pp. 638-648, . · Zbl 1148.92023 |
| [61] | A. Szabó and R. M. Merks, {\it Cellular potts modeling of tumor growth, tumor invasion, and tumor evolution}, Frontiers Oncology, 3 (2013), 87. |
| [62] | J. J. Tyson and B. Novak, {\it Regulation of the eukaryotic cell cycle: Molecular antagonism, hysteresis, and irreversible transitions}, J. Theoret. Biol., 210 (2001), pp. 249-63, . |
| [63] | M. G. Vander Heiden, L. C. Cantley, and C. B. Thompson, {\it Understanding the Warburg effect: The metabolic requirements of cell proliferation}, Science, 324 (2009), pp. 1029-1033. |
| [64] | J. Walpole, J. A. Papin, and S. M. Peirce, {\it Multiscale computational models of complex biological systems}, Annu. Rev. Biomed. Eng., 15 (2013), 137. |
| [65] | K. P. Wilkie and P. Hahnfeldt, {\it Mathematical models of immune-induced cancer dormancy and the emergence of immune evasion}, Interface Focus, 3 (2013), 20130010. |
| [66] | S. M. Wise, J. S. Lowengrub, H. B. Frieboes, and V. Cristini, {\it Three-dimensional multispecies nonlinear tumor growth–I: Model and numerical method}, J. Theoret. Biol., 253 (2008), pp. 524-43, . · Zbl 1398.92135 |
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