Three-Dimensional Spatiotemporal Modeling of Colon Cancer Organoids Reveals that Multimodal Control of Stem Cell Self-Renewal is a Critical Determinant of Size and Shape in Early Stages of Tumor Growth

Special Issue : Mathematical Oncology

Abstract

We develop a three-dimensional multispecies mathematical model to simulate the growth of colon cancer organoids containing stem, progenitor and terminally differentiated cells, as a model of early (prevascular) tumor growth. Stem cells (SCs) secrete short-range self-renewal promoters (e.g., Wnt) and their long-range inhibitors (e.g., Dkk) and proliferate slowly. Committed progenitor (CP) cells proliferate more rapidly and differentiate to produce post-mitotic terminally differentiated cells that release differentiation promoters, forming negative feedback loops on SC and CP self-renewal. We demonstrate that SCs play a central role in normal and cancer colon organoids. Spatial patterning of the SC self-renewal promoter gives rise to SC clusters, which mimic stem cell niches, around the organoid surface, and drive the development of invasive fingers. We also study the effects of externally applied signaling factors. Applying bone morphogenic proteins, which inhibit SC and CP self-renewal, reduces invasiveness and organoid size. Applying hepatocyte growth factor, which enhances SC self-renewal, produces larger sizes and enhances finger development at low concentrations but suppresses fingers at high concentrations. These results are consistent with recent experiments on colon organoids. Because many cancers are hierarchically organized and are subject to feedback regulation similar to that in normal tissues, our results suggest that in cancer, control of cancer stem cell self-renewal should influence the size and shape in similar ways, thereby opening the door to novel therapies.

Keywords

Mathematical modeling Cancer stem cells Brain tumors Cancer therapies Feedback regulation 

Supplementary material

11538_2017_294_MOESM1_ESM.docx (8.9 mb)
Supplementary material 1 (docx 9097 KB)

References

  1. Abdullah LN, Chow EKH (2013) Mechanisms of chemoresistance in cancer stem cells. Clin Transl Med 2(1):3. doi:10.1186/2001-1326-2-3 CrossRefGoogle Scholar
  2. Aoki K, Taketo MM (2007) Adenomatous polyposis coli (APC): a multi-functional tumor suppressor gene. J Cell Sci 120(19):3327–3335. doi:10.1242/jcs.03485 CrossRefGoogle Scholar
  3. Barker N (2014) Adult intestinal stem cells: critical drivers of epithelial homeostasis and regeneration. Nat Rev Mol Cell Biol 15(1):19–33. doi:10.1038/nrm3721 CrossRefGoogle Scholar
  4. Barker N, Ridgway RA, van Es JH, van de Wetering M, Begthel H, van den Born M, Danenberg E, Clarke AR, Sansom OJ, Clevers H (2009) Crypt stem cells as the cells-of-origin of intestinal cancer. Nature 457(7229):608–611. doi:10.1038/nature07602 CrossRefGoogle Scholar
  5. Bearer EL, Lowengrub J, Frieboes HB, Chuang YL, Jin F, Wise SM, Ferrari M, Agus DB, Cristini V (2009) Multiparameter computational modeling of tumor invasion. Cancer Res 69(10):4493–4501. doi:10.1158/0008-5472.CAN-08-3834 CrossRefGoogle Scholar
  6. Beck B, Blanpain C (2013) Unravelling cancer stem cell potential. Nat Rev Cancer 13(10):727–738. doi:10.1038/nrc3597 CrossRefGoogle Scholar
  7. Biteau B, Hochmuth CE, Jasper H (2011) Maintaining tissue homeostasis: dynamic control of somatic stem cell activity. doi:10.1016/j.stem.2011.10.004
  8. Brinkmann V, Foroutan H, Sachs M, Weidner KM, Birchmeier W (1995) Hepatocyte growth factor/scatter factor induces a variety of tissue-specific morphogenic programs in epithelial cells. J Cell Biol 131(6 Pt 1):1573–1586. doi:10.1083/jcb.131.6.1573 CrossRefGoogle Scholar
  9. Buske P, Przybilla J, Loeffler M, Sachs N, Sato T, Clevers H, Galle J (2012) On the biomechanics of stem cell niche formation in the gut—modelling growing organoids. FEBS J 279:3475–3487. doi:10.1111/j.1742-4658.2012.08646.x 84865978356CrossRefGoogle Scholar
  10. Byun T, Karimi M, Marsh JL, Milovanovic T, Lin F, Holcombe RF (2005) Expression of secreted Wnt antagonists in gastrointestinal tissues: potential role in stem cell homeostasis. J Clin Pathol 58(5):515–519. doi:10.1136/jcp.2004.018598 CrossRefGoogle Scholar
  11. Cao Y, Liang C, Naveed H, Li Y, Chen M, Nie Q (2012) Modeling spatial population dynamics of stem cell lineage in tissue growth. In: Conference proceedings : annual international conference of the IEEE engineering in medicine and biology society IEEE engineering in medicine and biology society conference, pp 5502–5505. doi:10.1109/EMBC.2012.6347240
  12. Clevers H, Loh KM, Nusse R (2014) Stem cell signaling. An integral program for tissue renewal and regeneration: Wnt signaling and stem cell control. Science (New York, NY) 346(6205):1248,012. doi:10.1126/science.1248012 CrossRefGoogle Scholar
  13. Cristini V, Lowengrub J (2010) Multiscale modeling of cancer: an integrated experimental and mathematical modeling approach. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  14. Cristini V, Lowengrub J, Nie Q (2003) Nonlinear simulation of tumor growth. J Math Biol 46(3):191–224. doi:10.1007/s00285-002-0174-6 MathSciNetCrossRefMATHGoogle Scholar
  15. Cruz FD, Matushansky I (2012) Solid tumor differentiation therapy—is it possible? Oncotarget 3(5):559–567. doi:10.18632/oncotarget.512 CrossRefGoogle Scholar
  16. Dale L, Wardle FC (1999) A gradient of BMP activity specifies dorsal-ventral fates in early Xenopus embryos. Semin Cell Dev Biol 10(3):319–326. doi:10.1006/scdb.1999.0308 CrossRefGoogle Scholar
  17. Fatehullah A, Tan SH, Barker N (2016) Organoids as an in vitro model of human development and disease. Nat Cell Biol 18(3):246–254. doi:10.1038/ncb3312 CrossRefGoogle Scholar
  18. Fletcher AG, Murray PJ, Maini PK (2015) Multiscale modelling of intestinal crypt organization and carcinogenesis. Math Models Methods Appl Sci 25(13):2563–2585. doi:10.1142/S0218202515400187 MathSciNetCrossRefMATHGoogle Scholar
  19. Frieboes HB, Zheng X, Sun CH, Tromberg B, Gatenby R, Cristini V (2006) An integrated computational/experimental model of tumor invasion. Cancer Res 66(3):1597–1604. doi:10.1158/0008-5472.CAN-05-3166 CrossRefGoogle Scholar
  20. Gao CF, Vande Woude GF (2005) HGF/SF-Met signaling in tumor progression. Cell Res 15(1):49–51. doi:10.1038/sj.cr.7290264 CrossRefGoogle Scholar
  21. Gao X, McDonald JT, Hlatky L, Enderling H (2013) Acute and fractionated irradiation differentially modulate glioma stem cell division kinetics. Cancer Res 73(5):1481–1490. doi:10.1158/0008-5472.CAN-12-3429 CrossRefGoogle Scholar
  22. González-Sancho JM, Aguilera O, García JM, Pendás-Franco N, Peña C, Cal S, de Herreros AG, Bonilla F, Muñoz A (2005) The Wnt antagonist DICKKOPF-1 gene is a downstream target of beta-catenin/TCF and is downregulated in human colon cancer. Oncogene 24(6):1098–103CrossRefGoogle Scholar
  23. Gout S, Huot J (2008) Role of cancer microenvironment in metastasis: focus on colon cancer. doi:10.1007/s12307-008-0007-2
  24. Grabinger T, Luks L, Kostadinova F, Zimberlin C, Medema JP, Leist M, Brunner T (2014) Ex vivo culture of intestinal crypt organoids as a model system for assessing cell death induction in intestinal epithelial cells and enteropathy. Cell Death Dis 5(5):e1228. doi:10.1038/cddis.2014.183 CrossRefGoogle Scholar
  25. Graeber TG, Osmanian C, Jacks T, Housman DE, Koch CJ, Lowe SW, Giaccia AJ (1996) Hypoxia-mediated selection of cells with diminished apoptotic potential in solid tumours. Nature 379(6560):88–91. doi:10.1038/379088a0 CrossRefGoogle Scholar
  26. Ca Gregory, Singh H, Perry AS, Prockop DJ (2003) The Wnt signaling inhibitor dickkopf-1 is required for reentry into the cell cycle of human adult stem cells from bone marrow. J Biol Chem 278(30):28067–28078. doi:10.1074/jbc.M300373200 CrossRefGoogle Scholar
  27. Hartung N, Mollard S, Barbolosi D, Benabdallah A, Chapuisat G, Henry G, Giacometti S, Iliadis A, Ciccolini J, Faivre C, Hubert F (2014) Mathematical modeling of tumor growth and metastatic spreading: validation in tumor-bearing mice. Cancer Res 74(22):6397–6407. doi:10.1158/0008-5472.CAN-14-0721 CrossRefGoogle Scholar
  28. He XC, Zhang J, Tong WG, Tawfik O, Ross J, Scoville DH, Tian Q, Zeng X, He X, Wiedemann LM, Mishina Y, Li L (2004) BMP signaling inhibits intestinal stem cell self-renewal through suppression of Wnt-beta-catenin signaling. Nat Genet 36(10):1117–1121. doi:10.1038/ng1430 CrossRefGoogle Scholar
  29. Heddleston JM, Li Z, McLendon RE, Hjelmeland AB, Rich JN (2009) The hypoxic microenvironment maintains glioblastoma stem cells and promotes reprogramming towards a cancer stem cell phenotype. doi:10.4161/cc.8.20.9701, NIHMS150003
  30. Hirsch D, Barker N, Mcneil N, Hu Y, Camps J, Mckinnon K, Clevers H, Ried T, Gaiser T (2014) LGR5 positivity defines stem-like cells in colorectal cancer. Carcinogenesis 35(4):849–858. doi:10.1093/carcin/bgt377 CrossRefGoogle Scholar
  31. Huch M, Koo BK (2015) Modeling mouse and human development using organoid cultures. Development (Cambridge, England) 142(18):3113–3125. doi:10.1242/dev.118570 CrossRefGoogle Scholar
  32. Humphries A, Wright NA (2008) Colonic crypt organization and tumorigenesis. Nat Rev Cancer 8(6):415–424. doi:10.1038/nrc2392 CrossRefGoogle Scholar
  33. Ikari T, Hiraki A, Seki K, Sugiura T, Matsumoto K, Shirasuna K (2003) Involvement of hepatocyte growth factor in branching morphogenesis of murine salivary gland. Dev Dyn 228(2):173–184. doi:10.1002/dvdy.10377 CrossRefGoogle Scholar
  34. Jiang GS, Shu CW (1996) Efficient implementation of weighted ENO schemes. J Comput Phys 126(126):202–228. doi:10.1006/jcph.1996.0130 MathSciNetCrossRefMATHGoogle Scholar
  35. Jones CM, Smith JC (1998) Establishment of a BMP-4 morphogen gradient by long-range inhibition. Dev Biol 194(1):12–17. doi:10.1006/dbio.1997.8752 CrossRefGoogle Scholar
  36. Joo KM, Jin J, Kim E, Kim KH, Kim Y, Kang BG, Kang YJ, Lathia JD, Cheong KH, Song PH, Kim H, Seol HJ, Kong DS, Lee JI, Rich JN, Lee J, Nam DH (2012) MET signaling regulates glioblastoma stem cells. Cancer Res 72(15):3828–3838. doi:10.1158/0008-5472.CAN-11-3760 CrossRefGoogle Scholar
  37. Klaus A, Birchmeier W (2008) Wnt signalling and its impact on development and cancer. Nat Rev Cancer 8(5):387–398. doi:10.1038/nrc2389 CrossRefGoogle Scholar
  38. Kosinski C, Li VSW, Chan ASY, Zhang J, Ho C, Tsui WY, Chan TL, Mifflin RC, Powell DW, Yuen ST, Leung SY, Chen X (2007) Gene expression patterns of human colon tops and basal crypts and BMP antagonists as intestinal stem cell niche factors. Proc Natl Acad Sci USA 104(39):15418–15423. doi:10.1073/pnas.0707210104 CrossRefGoogle Scholar
  39. Krausova M, Korinek V (2014) Wnt signaling in adult intestinal stem cells and cancer. doi:10.1016/j.cellsig.2013.11.032
  40. Kunche S, Yan H, Calof AL, Lowengrub J, Lander AD (2016) Feedback, lineages and self-organizing morphogenesis. PLoS Comput Biol 12(3):e1004814. doi:10.1371/journal.pcbi.1004814 CrossRefGoogle Scholar
  41. Lander AD, Gokoffski KK, Wan FYM, Nie Q, Calof AL (2009) Cell lineages and the logic of proliferative control. PLoS Biol 7(1). doi:10.1371/journal.pbio.1000015
  42. Lee N, Smolarz AJ, Olson S, David O, Reiser J, Kutner R, Daw NC, Prockop DJ, Horwitz EM, Gregory CA (2007) A potential role for Dkk-1 in the pathogenesis of osteosarcoma predicts novel diagnostic and treatment strategies. Br J Cancer 97(11):1552–1559. doi:10.1038/sj.bjc.6604069 CrossRefGoogle Scholar
  43. Leszczyniecka M, Roberts T, Dent P, Grant S, Fisher PB (2001) Differentiation therapy of human cancer: basic science and clinical applications. Pharmacol Ther 90:105–156. doi:10.1016/S0163-7258(01)00132-2 CrossRefGoogle Scholar
  44. Li Y, Li A, Glas M, Lal B, Ying M, Sang Y, Xia S, Trageser D, Guerrero-Cazares H, Eberhart CG, Quinones-Hinojosa A, Scheffler B, Laterra J (2011) c-Met signaling induces a reprogramming network and supports the glioblastoma stem-like phenotype. Proc Natl Acad Sci USA 108(24):9951–9956CrossRefGoogle Scholar
  45. Lim YC, Kang HJ, Moon JH (2014) C-Met pathway promotes self-renewal and tumorigenecity of head and neck squamous cell carcinoma stem-like cell. Oral Oncol 50(7):633–639. doi:10.1016/j.oraloncology.2014.04.004 CrossRefGoogle Scholar
  46. Lowengrub JS, Frieboes HB, Jin F, Chuang YL, Li X, Macklin P, Wise SM, Cristini V (2010) Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity 23(1):R1–R9. doi:10.1088/0951-7715/23/1/r01 NIHMS150003MathSciNetCrossRefMATHGoogle Scholar
  47. Luqmani YA (2005) Mechanisms of drug resistance in cancer chemotherapy. Medical Princ Pract 14(Suppl 1):35–48. doi:10.1159/000086183 Google Scholar
  48. Meulmeester E, Ten Dijke P (2011) The dynamic roles of TGF-beta in cancer. J Pathol 223(2):205–218. doi:10.1002/path.2785 CrossRefGoogle Scholar
  49. Mirams GR, Fletcher AG, Maini PK, Byrne HM (2012) A theoretical investigation of the effect of proliferation and adhesion on monoclonal conversion in the colonic crypt. J Theor Biol 312:143–156. doi:10.1016/j.jtbi.2012.08.002 MathSciNetCrossRefMATHGoogle Scholar
  50. Pham K, Frieboes HB, Cristini V, Lowengrub J (2011) Predictions of tumour morphological stability and evaluation against experimental observations. J R Soc Interface 8(54):16–29. doi:10.1098/rsif.2010.0194 CrossRefGoogle Scholar
  51. Pin AL, Houle F, Huot J (2011) Recent advances in colorectal cancer research: the microenvironment impact. Cancer Microenviron 4(2):127–131. doi:10.1007/s12307-011-0070-y CrossRefGoogle Scholar
  52. Pinto D, Gregorieff A, Begthel H, Clevers H (2003) Canonical Wnt signals are essential for homeostasis of the intestinal epithelium. Genes Dev 17(14):1709–1713. doi:10.1101/gad.267103 CrossRefGoogle Scholar
  53. Pitt-Francis J, Bernabeu MO, Cooper J, Garny A, Momtahan L, Osborne J, Pathmanathan P, Rodriguez B, Whiteley JP, Gavaghan DJ (2008) Chaste: using agile programming techniques to develop computational biology software. Philos Trans Ser A Math Phys Eng Sci 366(1878):3111–3136. doi:10.1098/rsta.2008.0096 CrossRefGoogle Scholar
  54. Prasetyanti PR, Zimberlin CD, Bots M, Vermeulen L, Melo FDSE, Medema JP (2013) Regulation of stem cell self-renewal and differentiation by Wnt and Notch are conserved throughout the adenoma–carcinoma sequence in the colon. Mol Cancer 12(1):126. doi:10.1186/1476-4598-12-126 CrossRefGoogle Scholar
  55. Reynolds A, Wharton N, Parris A, Mitchell E, Sobolewski A, Kam C, Bigwood L, El Hadi A, Münsterberg A, Lewis M, Speakman C, Stebbings W, Wharton R, Sargen K, Tighe R, Jamieson C, Hernon J, Kapur S, Oue N, Yasui W, Williams MR (2014) Canonical Wnt signals combined with suppressed TGF\(\beta \)/BMP pathways promote renewal of the native human colonic epithelium. Gut 63(4):610–621. doi:10.1136/gutjnl-2012-304067 CrossRefGoogle Scholar
  56. Sato T, Clevers H (2013) Growing self-organizing mini-guts from a single intestinal stem cell: mechanism and applications. Science (New York, NY) 340(6137):1190–1194, doi:10.1126/science.1234852, arXiv:1011.1669v3
  57. Sato T, Vries RG, Snippert HJ, van de Wetering M, Barker N, Stange DE, van Es JH, Abo A, Kujala P, Peters PJ, Clevers H (2009) Single Lgr5 stem cells build crypt-villus structures in vitro without a mesenchymal niche. Nature 459(7244):262–265. doi:10.1038/nature07935 CrossRefGoogle Scholar
  58. Sato T, van Es JH, Snippert HJ, Stange DE, Vries RG, van den Born M, Barker N, Shroyer NF, van de Wetering M, Clevers H (2011a) Paneth cells constitute the niche for Lgr5 stem cells in intestinal crypts. Nature 469(7330):415–418. doi:10.1038/nature09637 CrossRefGoogle Scholar
  59. Sato T, Stange DE, Ferrante M, Vries RGJ, Van Es JH, Van Den Brink S, Van Houdt WJ, Pronk A, Van Gorp J, Siersema PD, Clevers H (2011) Long-term expansion of epithelial organoids from human colon, adenoma, adenocarcinoma, and Barrett’s epithelium. Gastroenterology 141(5):1762–1772. doi:10.1053/j.gastro.2011.07.050 CrossRefGoogle Scholar
  60. Schepers A, Clevers H (2012) Wnt signaling, stem cells, and cancer of the gastrointestinal tract. Cold Spring Harbor Perspect Biol 4(4): doi:10.1101/cshperspect.a007989
  61. Schuijers J, Clevers H (2012) Adult mammalian stem cells: the role of Wnt, Lgr5 and R-spondins. The EMBO J 31(13):3031–3032. doi:10.1038/emboj.2012.177 CrossRefGoogle Scholar
  62. Sciumè G, Shelton S, Gray WG, Miller CT, Hussain F, Ferrari M, Decuzzi P, Schrefler BA (2013) A multiphase model for three-dimensional tumor growth. New J Phys 15: doi:10.1088/1367-2630/15/1/015005
  63. Sell S (2004) Stem cell origin of cancer and differentiation therapy. doi:10.1016/j.critrevonc.2004.04.007
  64. Shamir ER, Ewald AJ (2014) Three-dimensional organotypic culture: experimental models of mammalian biology and disease. Nat Rev Mol Cell Biol 15(10):647–664. doi:10.1038/nrm3873 NIHMS150003CrossRefGoogle Scholar
  65. Smallbone K, Corfe BM (2014) A mathematical model of the colon crypt capturing compositional dynamic interactions between cell types. Int J Exp Pathol 95(1):1–7. doi:10.1111/iep.12062 CrossRefGoogle Scholar
  66. Sottoriva A, Verhoeff JJC, Borovski T, McWeeney SK, Naumov L, Medema JP, Sloot PMA, Vermeulen L (2010) Cancer stem cell tumor model reveals invasive morphology and increased phenotypical heterogeneity. Cancer Res 70(1):46–56. doi:10.1158/0008-5472.CAN-09-3663 CrossRefGoogle Scholar
  67. Stockhausen MT, Kristoffersen K, Stobbe L, Poulsen HS (2014) Differentiation of glioblastoma multiforme stem-like cells leads to downregulation of EGFR and EGFRvIII and decreased tumorigenic and stem-like cell potential. Cancer Biol Ther 15(2):216–224. doi:10.4161/cbt.26736 CrossRefGoogle Scholar
  68. Straussman R, Morikawa T, Shee K, Barzily-Rokni M, Qian ZR, Du J, Davis A, Mongare MM, Gould J, Frederick DT, Za Cooper, Chapman PB, Solit DB, Ribas A, Lo RS, Flaherty KT, Ogino S, Ja Wargo, Golub TR (2012) Tumour micro-environment elicits innate resistance to RAF inhibitors through HGF secretion. Nature 487(7408):500–504. doi:10.1038/nature11183 PMCID: PMC3711467CrossRefGoogle Scholar
  69. Tzamali E, Grekas G, Marias K, Sakkalis V (2014) Exploring the competition between proliferative and invasive cancer phenotypes in a continuous spatial model. PLoS ONE 9(8): doi:10.1371/journal.pone.0103191
  70. Tzedakis G, Tzamali E, Marias K, Sakkalis V (2015) The importance of neighborhood scheme selection in agent-based tumor growth modeling. Cancer Inf 14:67–81. doi:10.4137/CIN.S19343 Google Scholar
  71. Van De Wetering M, Francies HE, Francis JM, Bounova G, Iorio F, Pronk A, Van Houdt W, Van Gorp J, Taylor-Weiner A, Kester L, McLaren-Douglas A, Blokker J, Jaksani S, Bartfeld S, Volckman R, Van Sluis P, Li VSW, Seepo S, Sekhar Pedamallu C, Cibulskis K, Carter SL, McKenna A, Lawrence MS, Lichtenstein L, Stewart C, Koster J, Versteeg R, Van Oudenaarden A, Saez-Rodriguez J, Vries RGJ, Getz G, Wessels L, Stratton MR, McDermott U, Meyerson M, Garnett MJ, Clevers H (2015) Prospective derivation of a living organoid biobank of colorectal cancer patients. Cell 161(4):933–945. doi:10.1016/j.cell.2015.03.053 CrossRefGoogle Scholar
  72. Van Leeuwen IMM, Mirams GR, Walter A, Fletcher A, Murray P, Osborne J, Varma S, Young SJ, Cooper J, Doyle B, Pitt-Francis J, Momtahan L, Pathmanathan P, Whiteley JP, Chapman SJ, Gavaghan DJ, Jensen OE, King JR, Maini PK, Waters SL, Byrne HM (2009) An integrative computational model for intestinal tissue renewal. Cell Prolif 42(5):617–636. doi:10.1111/j.1365-2184.2009.00627.x 69549138171CrossRefGoogle Scholar
  73. Vermeulen L, De Sousa E, Melo F, van der Heijden M, Cameron K, de Jong JH, Borovski T, Tuynman JB, Todaro M, Merz C, Rodermond H, Sprick MR, Kemper K, Richel DJ, Stassi G, Medema JP (2010) Wnt activity defines colon cancer stem cells and is regulated by the microenvironment. Nat Cell Biol 12(5):468–476. doi:10.1038/ncb2048 CrossRefGoogle Scholar
  74. Wasan HS, Park HS, Liu KC, Mandir NK, Winnett A, Sasieni P, Bodmer WF, Goodlad RA, Wright NA (1998) APC in the regulation of intestinal crypt fission. J Pathol 185(3):246–255. doi:10.1002/(SICI)1096-9896(199807)185:3<246::AID-PATH90>3.0.CO;2-8 CrossRefGoogle Scholar
  75. Whissell G, Montagni E, Martinelli P, Hernando-Momblona X, Sevillano M, Jung P, Cortina C, Calon A, Abuli A, Castells A, Castellvi-Bel S, Nacht AS, Sancho E, Stephan-Otto Attolini C, Vicent GP, Real FX, Batlle E (2014) The transcription factor GATA6 enables self-renewal of colon adenoma stem cells by repressing BMP gene expression. Nat Cell Biol 16(7):695–707. doi:10.1038/ncb2992 CrossRefGoogle Scholar
  76. Wise S, Kim J, Lowengrub J (2007) Solving the regularized, strongly anisotropic Cahn–Hilliard equation by an adaptive nonlinear multigrid method. J Comput Phys 226(1):414–446. doi:10.1016/j.jcp.2007.04.020 MathSciNetCrossRefMATHGoogle Scholar
  77. Wise SM, Lowengrub J, Frieboes HB, Cristini V (2008) Three-dimensional multispecies nonlinear tumor growth-I. Model and numerical method. J Theor Biol 253(3):524–543. doi:10.1016/j.jtbi.2008.03.027 MathSciNetCrossRefGoogle Scholar
  78. Wise SM, Lowengrub J, Cristini V (2011) An adaptive multigrid algorithm for simulating solid tumor growth using mixture models. Math Comput Model 53(1–2):1–20. doi:10.1016/j.mcm.2010.07.007 MathSciNetCrossRefMATHGoogle Scholar
  79. Wong AS, Leung PC, Auersperg N (2000) Hepatocyte growth factor promotes in vitro scattering and morphogenesis of human cervical carcinoma cells. Gynecol Oncol 78(2):158–165. doi:10.1006/gyno.2000.5877 CrossRefGoogle Scholar
  80. Wong VWY, Stange DE, Page ME, Buczacki S, Wabik A, Itami S, van de Wetering M, Poulsom R, Na Wright, Trotter MWB, Watt FM, Winton DJ, Clevers H, Jensen KB (2012) Lrig1 controls intestinal stem-cell homeostasis by negative regulation of ErbB signalling. Nat Cell Biol 14(4):401–408. doi:10.1038/ncb2464 CrossRefGoogle Scholar
  81. Youssefpour H, Li X, Lander AD, Lowengrub J (2012) Multispecies model of cell lineages and feedback control in solid tumors. J Theor Biol 304:39–59. doi:10.1016/j.jtbi.2012.02.030 CrossRefGoogle Scholar
  82. Zhang L, Lander AD, Nie Q (2012) A reaction diffusion mechanism influences cell lineage progression as a basis for formation, regeneration, and stability of intestinal crypts. BMC Syst Biol 6(1):93. doi:10.1186/1752-0509-6-93 CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2017

Authors and Affiliations

  • Huaming Yan
    • 1
  • Anna Konstorum
    • 2
  • John S. Lowengrub
    • 3
  1. 1.Department of MathematicsUniversity of California, IrvineIrvineUSA
  2. 2.Center for Quantitative MedicineUniversity of Connecticut Health CenterFarmingtonUSA
  3. 3.Department of Mathematics, Department of Biomedical Engineering, Center for Complex Biological Systems, and Chao Comprehensive Cancer CenterUniversity of California, IrvineIrvineUSA

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