Modeling Multiscale Necrotic and Calcified Tissue Biomechanics in Cancer Patients: Application to Ductal Carcinoma In Situ (DCIS)

  • Paul Macklin
  • Shannon Mumenthaler
  • John Lowengrub
Part of the Studies in Mechanobiology, Tissue Engineering and Biomaterials book series (SMTEB, volume 14)


Tissue necrosis and calcification significantly affect cancer progression and clinical treatment decisions. Necrosis and calcification are inherently multiscale processes, operating at molecular to tissue scales with time scales ranging from hours to months. This chapter details key insights we have gained through mechanistic continuum and discrete multiscale models, including the first modeling of necrotic cell swelling, lysis, and calcification. Among our key findings: necrotic volume loss contributes to steady tumor sizes but can destabilize tumor morphology; steady necrotic fractions can emerge even during unstable growth; necrotic volume loss is responsible for linear ductal carcinoma in situ (DCIS) growth; fast necrotic cell swelling creates mechanical tears at the perinecrotic boundary; multiscale interactions give rise to an age-structured, stratified necrotic core; and mechanistic, patient-calibrated DCIS modeling allows us to assess our working biological assumptions and better interpret pathology and mammography. We finish by outlining our integrative computational oncology approach to developing computational tools that we hope will one day assist clinicians and patients in their treatment decisions.


Necrotic Core Tumor Spheroid Necrotic Debris Patient Pathology Mechanical Stress Relief 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



PM and SM thank the National Institutes of Health for the Physical Sciences Oncology Center grant 5U54CA143907 for Multi-scale Complex Systems Transdisciplinary Analysis of Response to Therapy–MC-START. PM thanks the USC James H. Zumberge Research and Innovation Fund (2012 Large Interdisciplinary Award) for support through the new Consortium for Integrative Computational Oncology (CICO), and the USC Undergraduates Research Associate Program (URAP) for student support. JL gratefully acknowledges partial support from the National Institutes of Health, National Cancer Institute, for funding through grants 1RC2CA148493-01, P50GM76516 for a Center of Excellence in Systems Biology at the University of California, Irvine, and P30CA062203 for the Chao Comprehensive Cancer Center at the University of California, Irvine. JL also acknowledges support from the National Science Foundation, Division of Mathematical Sciences.

PM thanks David Agus (USC Center for Applied Molecular Medicine); Andrew Evans, Jordan Lee, Colin Purdie, and Alastair Thompson (U. of Dundee/NHS Tayside); and Paul Newton (USC Department of Aerospace and Mechanical Engineering) for enlightening discussions. PM thanks Andrew Evans for Fig. 13. The authors thank Ying Chen (U. California at Irvine) for Fig. 14.


  1. 1.
    Altman, D.A., Atkinson, D.S., Brat, D.J.: Glioblastoma multiforme. Radiographics 27(3), 883–888 (2007)CrossRefGoogle Scholar
  2. 2.
    Anderson, A.R.A: A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. Math. Med. Biol. 22(2), 163–186 (2005)zbMATHCrossRefGoogle Scholar
  3. 3.
    Ayre, K.J., Hulbert, A.J: Dietary fatty acid profile influences the composition of skeletal muscle phospholipids in rats. J. Nutr. 126(3), 653–662 (1996)Google Scholar
  4. 4.
    Bacsó, Z., Everson, R.B., Eliason, J.F: The DNA of annexin V-binding apoptotic cells is highly fragmented. Cancer Res. 60(16), 4623 (2000)Google Scholar
  5. 5.
    Bai, Z.: A periodic age-structured epidemic model with a wide class of incidence rates. J. Math. Anal. Appl. 393(2), 367–376 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Barros, L.F., Hermosilla, T., Castro, J.: Necrotic volume increase and the early physiology of necrosis. Comp. Biochem. Physiol. A. Mol. Integr. Physiol. 130(3), 401–409 (2001)CrossRefGoogle Scholar
  7. 7.
    Barros, L.F., Kanaseki, T., Sabriov, R., Morishima, S., Castro, J., Bittner, C.X., Maeno, E., Ando-Akatsuka, Y., Okada, Y.: Apoptotic and necrotic blebs in epithelial cells display similar neck diameters but different kinase dependency. Cell Death Diff. 10(6), 687–697 (2003)CrossRefGoogle Scholar
  8. 8.
    Basanta, D., Strand, D.W., Lukner, R.B., Franco, O.E., Cliffel, D.E., Ayala, G.E., Hayward, S.W., Anderson, A.R.A: The role oftransforming growth factor-\(\beta\)-mediated tumor-stroma interactions in prostate cancer progression: An integrative approach. Cancer Res. 69(17), 7111–7120 (2009)Google Scholar
  9. 9.
    Basu, S., Binder, R.J., Suto, R., Anderson, K.M., Srivastava, P.K: Necrotic but not apoptotic cell death releases heat shock proteins, which deliver a partial maturation signal to dendritic cells and activate the NF-\(\kappa\)B pathway. Int. Immunol. 12(11), 1539–1546 (2000)Google Scholar
  10. 10.
    Blagosklonny, M.V., Robey, R., Bates, S., Fojo, T.: Pretreatment with DNA-damaging agents permits selective killing of checkpoint-deficient cells by microtubule-active drugs. J. Clin. Investig. 105(4), 533–539 (2000)CrossRefGoogle Scholar
  11. 11.
    Buerger, H., Mommers, E.C., Littmann, R., Diallo, R., Brinkschmidt, C., Poremba, C., Dockhorn-Dworniczak, B., van Diest, P.J., Böcker, W.B: Correlation of morphologic and cytogenetic parameters of genetic instability with chromosomal alterations in in situ carcinomas of the breast. Am. J. Clin. Pathol. 114, 854–859 (2000)CrossRefGoogle Scholar
  12. 12.
    Byrne, H., Chaplain, M.A.J: Necrosis and apoptosis: Distinct cell loss mechanisms in a mathematical model of avascular tumour growth. J. Theor. Med. 1(3), 223–235 (1998)zbMATHCrossRefGoogle Scholar
  13. 13.
    Byrne, H., Preziosi, L.: Modelling solid tumour growth using the theory of mixtures. Math. Med. Biol. 20(4), 341–366 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Cantoni, O., Guidarelli, A., Palomba, L., Fiorani, M.: U937 cell necrosis mediated by peroxynitrite is not caused by depletion of ATP and is prevented by arachidonate via an ATP-dependent mechanism. Mol. Pharm. 67(5), 1399–1405 (2005)CrossRefGoogle Scholar
  15. 15.
    Carlson, K.L., Helvie, M.A., Roubidoux, M.A., Kleer, C.G., Oberman, H.A., Wilson, T.E., Pollack, E.W., Rochester, A.B: Relationship between mammographic screening intervals and size and histology of ductal carcinoma in situ. Am. J. Roentenol. 172(2), 313–317 (1999)CrossRefGoogle Scholar
  16. 16.
    Chen, Y.: Modeling tumor growth in complex, dynamic geometries. Ph.D. dissertation, University of California, Irvine Department of Mathematics (2012)Google Scholar
  17. 17.
    Chen, Y., Lowengrub, J.S.: Tumor growth in complex, evolving geometries: A diffuse domain approach (2012, in preparation)Google Scholar
  18. 18.
    Cristini, V., Lowengrub, J.S., Nie, Q.: Nonlinear simulation of tumor growth. J. Math. Biol. 46(3), 191–224 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    D’Antonio, G., Macklin, P.: A multiscale hybrid discrete-continuum model of matrix metalloproteinase transport and basement membrane-extracellular matrix degradation (2013, in preparation)Google Scholar
  20. 20.
    D’Antonio, G., Macklin, P., Preziosi, L.: An agent-based model for elasto-plastic mechanical interactions between cells, basement membrane and extracellular matrix. Math. Biosci. Eng. (2012)
  21. 21.
    Drasdo, D., Höhme, S.: A single-scale-based model of tumor growth in vitro: monolayers and spheroids. Phys. Biol. 2(3), 133–147 (2005)CrossRefGoogle Scholar
  22. 22.
    Drasdo, D., Kree, R., McCaskill, J.S: Monte-carlo approach to tissue cell populations. Phys. Rev. E 52(6), 6635–6657 (1995)CrossRefGoogle Scholar
  23. 23.
    Edgerton, M.E., Chuang, Y.L., Macklin, P., Yang, W., Bearer, E.L., Cristini, V.: A novel, patient-specific mathematical pathology approach for assessment of surgical volume: Application to ductal carcinoma in situ of the breast. Anal. Cell. Pathol. 34(5), 247–263 (2011)Google Scholar
  24. 24.
    Eichbaum, C., Meyer, A.S., Wang, N., Bischofs, E., Steinborn, A., Bruckner, T., Brodt, P., Sohn, C., Eichbaum, M.H.R: Breast cancer cell-derived cytokines, macrophages and cell adhesion: Implications for metastasis. Anticancer Res. 31(10), 3219–27 (2011)Google Scholar
  25. 25.
    Elmore, S.: Apoptosis: A review of programmed cell death. Toxicol. Pathol. 35(4), 495–516 (2007)CrossRefGoogle Scholar
  26. 26.
    Erez, N., Truitt, M., Olson, P., Hanahan, D.: Cancer-associated fibroblasts are activated in incipient neoplasia to orchestrate tumor-promoting inflammation in an NF-\(\kappa\)B-dependent manner. Cancer Cell 17(2), 135–147 (2010)Google Scholar
  27. 27.
    Evans, A.: The diagnosis and management of pre-invasive breast disease: Radiological diagnosis. Breast Cancer Res. 5(5), 250–253 (2003)CrossRefGoogle Scholar
  28. 28.
    Evans, A., Clements, K., Maxwell, A., Bishop, H., Handby, A., Lawrence, G., Pinder, S.E: Lesion size is a major determinant of the mammographic features of ductal carcinoma in situ: Findings from the Sloane project. Radiology 53(3), 181–4 (2010)Google Scholar
  29. 29.
    Evans, A., Pinder, S., Wilson, R., Sibbering, M., Poller, D., Elston, C., Ellis, I.: Ductal carcinoma in situ of the breast: Correlation between mammographic and pathologic findings. Am. J. Roentgen. 162(6), 1307–1311 (1994)CrossRefGoogle Scholar
  30. 30.
    Fall, C.P., Bennett, J.P: Characterization and time course of MPP+-induced apoptosis in human SH-SY5Y neuroblastoma cells. J. Neurosci. Res. 55(5), 620–628 (1999)CrossRefGoogle Scholar
  31. 31.
    Festjens, N., Berghe, T.V., Vandenabeele, P.: Necrosis, a well-orchestrated form of cell demise: Signalling cascades, important mediators and concomitant immune response. Biochim. Biophys. Acta Bioenergetics 1757(9–10), 1371–1387 (2006)CrossRefGoogle Scholar
  32. 32.
    Fesus, L.: Transglutaminase-catalyzed protein cross-linking in the molecular program of apoptosis and its relationship to neuronal processes. Cell. Mol. Neurobiol. 18, 683–694 (1998)CrossRefGoogle Scholar
  33. 33.
    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(3), 1597–1604 (2006)CrossRefGoogle Scholar
  34. 34.
    Gadeau, A.P., Chaulet, H., Daret, D., Kockx, M., Daniel-Lamaziè, J.M., Desgranges, C.: Time course of osteopontin, osteocalcin, and osteonectin accumulation and calcification after acute vessel wall injury. J. Histochem. Cytochem. 49(1), 79–86 (2001)CrossRefGoogle Scholar
  35. 35.
    Galle, J., Loeffler, M., Drasdo, D.: Modeling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro. Biophys. J. 88(1), 62–75 (2005)CrossRefGoogle Scholar
  36. 36.
    Garland, J.M., Halestrap, A.: Energy metabolism during apoptosis. J. Biol. Chem. 272(8), 4680–4688 (1997)CrossRefGoogle Scholar
  37. 37.
    Grivennikov, S.I., Greten, F.R., Karin, M.: Immunity, inflammation, and cancer. Cell 140(6), 883–899 (2010)CrossRefGoogle Scholar
  38. 38.
    Hengartner, M.O: The biochemistry of apoptosis. Nature 407(6805), 770–776 (2000)CrossRefGoogle Scholar
  39. 39.
    Hofvind, S., Iversen, B.F., Eriksen, L., Styr, B.M.S., Kjellevold, K., Kurz, K.D: Mammographic morphology and distribution of calcifications in ductal carcinoma in situ diagnosed in organized screening. Acta Radiologica 52(5), 481–487 (2011)CrossRefGoogle Scholar
  40. 40.
    Huether, S., McCance, K.: Understanding Pathology, 5 edn. chap. 3. Mosby, St. Louis, MO USA (2011)Google Scholar
  41. 41.
    Iannelli, M., Ripoll, J.: Two-sex age structured dynamics in a fixed sex-ratio population. Nonlinear Anal. Real World Appl. 13(6), 2562–2577 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Jian, B., Narula, N., Li, Q.Y., Mohler, E.R. III, Levy, R.J: Progression of aortic valve stenosis: TGF-\(\beta\)1 is present in calcified aortic valve cusps and promotes aortic valve interstitial cell calcification via apoptosis. Ann. Thorac. Surg. 75(2), 457–465 (2003)Google Scholar
  43. 43.
    Jin F., Chuang Y.L., Cristini V., Lowengrub J.S.: Hybrid continuum-discrte tumor models. In: Cristini, V., Lowengrub, J.S. (eds.) Multiscale Modeling of Cancer, chap. 8, pp. 153–182. Cambridge University Press, Cambridge, UK (2010)Google Scholar
  44. 44.
    Kerr, J.F.R., Winterford, C.M., Harmon, B.V: Apoptosis. its significance in cancer and cancer therapy. Cancer 73(8), 2013–2026 (1994)CrossRefGoogle Scholar
  45. 45.
    Krause, R.F., Beamer, K.C: Lipid content and phospholipid metabolism of subcellular fractions from testes of control and retinol-deficient rats. J. Nutr. 104(5), 629–637 (1974)Google Scholar
  46. 46.
    Krysko, D.V., Berghe, T.V., D’Herde, K., Vandenabeele, P.: Apoptosis and necrosis: Detection, discrimination and phagocytosis. Methods 44(3), 205–221 (2008)CrossRefGoogle Scholar
  47. 47.
    Kumar, V., Abbas, A.K., Aster, J.C., Fausto, N.: Pathologic Basis of Disease, 8 edn. chap. 1. Saunders Elsevier, Philadelphia, PA USA (2009)Google Scholar
  48. 48.
    Lee, J.S., Basalyga, D.M., Simionescu, A., Isenburg, J.C., Sinionescu, D.T., Vyavahare, N.R: Elastin calcification in the rate subdermal model is accompanied by up-regulation of degradative and osteogenic cellular responses. Am. J. Pathol. 168(2), 490–498 (2006)CrossRefGoogle Scholar
  49. 49.
    Levin, S.A., Goodyear, C.P: Analysis of an age-structured fishery model. J. Math. Biol. 9, 245–274 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Lowengrub, J.S., Frieboes, H.B., Jin, F., Chuang, Y.L., Li, X., Macklin, P., Wise, S.M., Cristini, V.: Nonlinear modeling of cancer: Bridging the gap between cells and tumors. Nonlinearity 23(1), R1–R91 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Macklin, P.: Numerical simulation of tumor growth and chemotherapy. M.S. thesis, University of Minnesota School of Mathematics (2003)Google Scholar
  52. 52.
    Macklin, P.: Biological background. In: Cristini, V., Lowengrub, J.S. Multiscale Modeling of Cancer, chap. 2, pp. 8–24. Cambridge University Press, Cambridge, UK (2010)Google Scholar
  53. 53.
    Macklin, P.: Basic ductal carcinoma in situ (DCIS) pathobiology for modelers. (online tutorial). (2012)
  54. 54.
    Macklin, P., Edgerton, M.E., Cristini, V.: Agent-based cell modeling: application to breast cancer. In: Cristini, V., Lowengrub, J.S. Multiscale Modeling of Cancer, chap. 10, pp. 216–244. Cambridge University Press, Cambridge, UK (2010)Google Scholar
  55. 55.
    Macklin, P., Edgerton, M.E., Lowengrub, J., Cristini, V.: Discrete cell modeling. In: Cristini, V., Lowengrub, J.S. Multiscale Modeling of Cancer, chap. 6, pp. 92–126. Cambridge University Press, Cambridge, UK (2010)Google Scholar
  56. 56.
    Macklin, P., Edgerton, M.E., Thompson, A.M., Cristini, V.: Patient-calibrated agent-based modelling of ductal carcinoma in situ (DCIS): From microscopic measurements to macroscopic predictions of clinical progression. J. Theor. Biol. 301, 122–140 (2012)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Macklin, P., Kim, J., Tomaiuolo, G., Edgerton, M.E., Cristini, V.: Agent-based modeling of ductal carcinoma in situ: Application to patient-specific breast cancer modeling. In: Pham T. (ed.) Computational Biology: Issues and Applications in Oncology, chap. 4, pp. 77–112. Springer, NewYork (2009)Google Scholar
  58. 58.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Macklin, P., Lowengrub, J.S: An improved geometry-aware curvature discretization for level set methods: application to tumor growth. J. Comput. Phys. 215(2), 392–401 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Macklin, P., Lowengrub, J.S: Nonlinear simulation of the effect of microenvironment on tumor growth. J. Theor. Biol. 245(4), 677–704 (2007)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Macklin, P., Lowengrub, J.S: A new ghost cell/level set method for moving boundary problems: Application to tumor growth. J. Sci. Comp. 35(2–3), 266–299 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Macklin, P., McDougall, S., Anderson, A.R.A., Chaplain, M.A.J., Cristini, V., Lowengrub, J.: Multiscale modeling and nonlinear simulation of vascular tumour growth. J. Math. Biol. 58(4–5), 765–798 (2009)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Macklin, P., Mumenthaler, S., Jordan, L.B., Purdie, C.A., Evans, A.J., Thompson, A.M.: Integration of pathology, radiology, and in vitro data in patient-calibrated cancer simulations: Recent advances and future outlook for ductal carcinoma in situ (DCIS). To be presented at: Annual Meeting of the Society for Mathematical Biology (SMB). (2012)
  64. 64.
    Macklin, P. et al.: Agent-based cell modeling with improved subcellular fluid and solid transport: comparison with ductal carcinoma in situ pathology (2013, in preparation)Google Scholar
  65. 65.
    Majno, G., Joris, I.: Apoptosis, oncosis, and necrosis. an overview of cell death. Am. J. Pathol. 146(1), 3–15 (1995)Google Scholar
  66. 66.
    Majno, G., Joris, I.: Cells, Tissues, and Disease: Principles of General Pathology, 2nd edn. Oxford University Press, New York (2004)Google Scholar
  67. 67.
    McCarthy, J.V., Cotter, T.G: Cell shrinkage and apoptosis: a role for potassium and sodium ion efflux. Cell Death Diff. 4(8), 756–770 (1997)CrossRefGoogle Scholar
  68. 68.
    Mumenthaler, S., Macklin, P., et al.: Time-course volume and water fraction measurements in cycling and apoptotic breast cancer cell lines (2013, in preparation)Google Scholar
  69. 69.
    Muttarak, M., Kongmebhol, P., Sukhamwang, N.: Breast calcifications: which are malignant? Singapore Med. J. 50(9), 907–913 (2009)Google Scholar
  70. 70.
    Noch, E., Khalili, K.: Molecular mechanisms of necrosis in glioblastoma: The role of glutamate excitotoxicity. Cancer Biol. Ther. 8(19), 1791–1797 (2009)Google Scholar
  71. 71.
    Norton, K.A., Wininger, M., Bhanot, G., Ganesan, S., Barnard, N., Shinbrot, T.: A 2D mechanistic model of breast ductal carcinoma in situ (DCIS) morphology and progression. J. Theor. Biol. 263(4), 393–406 (2010)CrossRefGoogle Scholar
  72. 72.
    Ottesen, G.L., Christensen, I.J., Larsen, J.K., Larsen, J., Baldetorp, B., Linden, T., Hansen, B., Andersen, J.: Carcinoma in situ of the breast: correlation of histopathology to immunohistochemical markers and DNA ploidy. Breast Cancer Res. Treat. 60(3), 219–226 (2000)CrossRefGoogle Scholar
  73. 73.
    Owen, M.R., Byrne, H.M., Lewis, C.E: Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites. J. Theor. Biol. 226(4), 377–391 (2004)MathSciNetCrossRefGoogle Scholar
  74. 74.
    Pearson O.H., Manni A., Arafah B.M.: Antiestrogen treatment of breast cancer: An overview. Cancer Res. 42(8 Suppl.):3424s–3428s (1982)Google Scholar
  75. 75.
    Ramis-Conde, I., Drasdo, D., Anderson, A.R.A., Chaplain, M.A.J: Modeling the influence of the e-cadherin-beta-catenin pathway in cancer cell invasion: A multiscale approach. Biophys. J. 95(1), 155–165 (2008)CrossRefGoogle Scholar
  76. 76.
    Richards, C.H., Mohammed, Z., Qayyum, T., Horgan, P.G., McMillan, D.C: The prognostic value of histological tumor necrosis in solid organ malignant disease: a systematic review. Future Oncol. 7(10), 1223–1235 (2011)CrossRefGoogle Scholar
  77. 77.
    Richards, C.H., Roxburgh, C.S.D., Anderson, J.H., McKee, R.F., Foulis, A.K., Horgan, P.G., McMillan, D.C: Prognostic value of tumour necrosis and host inflammatory responses in colorectal cancer. Brit. J. Surg. 99(2), 287–294 (2012)CrossRefGoogle Scholar
  78. 78.
    de Roos, M.A.J., Pijnappel, R.M., Post, W.J., de Vries, J., Baas, P.C., Groote, L.D: Correlation between imaging and pathology in ductal carcinoma in situ of the breast. World J. Surg. Onco. 2(1), 4 (2004)CrossRefGoogle Scholar
  79. 79.
    Rüegg, C.: Leukocytes, inflammation, and angiogenesis in cancer: fatal attractions. J. Leukoc. Biol. 80(4), 682–684 (2006)CrossRefGoogle Scholar
  80. 80.
    Scarlett, J.L., Sheard, P.W., Hughes, G., Ledgerwood, E.C., Ku, H.K., Murphy, M.P: Changes in mitochondrial membrane potential during staurosporine-induced apoptosis in Jurkat cells. FEBS Lett. 475(3), 267–272 (2000)CrossRefGoogle Scholar
  81. 81.
    Seymour, H.R., Cooke, J., Given-Wilson, R.M: The significance of spontaneous resolution of breast calcification. Brit. J. Radiol. 72(853), 3–8 (1999)Google Scholar
  82. 82.
    Sickles, E.A: Mammographic features of 300 consecutive nonpalpable breast cancers. Am. J. Roentgen. 146(4), 661–663 (1986)CrossRefGoogle Scholar
  83. 83.
    Silva, A.S., Gatenby, R.A., Gillies, R.J., Yunes, J.A: A quantitative theoretical model for the development of malignancy in ductal carcinoma in situ. J. Theor. Biol. 262(4), 601–613 (2010)CrossRefGoogle Scholar
  84. 84.
    Silverstein, M.J., Lagios, M.D., Craig, P.H., Waisrnan, J.R., Lewinsky, B.S., Colburn, W.J., Poller, D.N: A prognostic index for ductal carcinoma in situ of the breast. Cancer 77(11), 2267–74 (1996)CrossRefGoogle Scholar
  85. 85.
    Stomper, P.C., Connolly, J.L., Meyer, J.E., Harris, J.R: Clinically occult ductal carcinoma in situ detected with mammography: analysis of 100 cases with radiologic-pathologic correlation. Radiology 172(1), 235–241 (1989)Google Scholar
  86. 86.
    Thomson J.Z., Evans, A.J., Pinder, S.E., Burrell, H.C., Wilson, A.R.M., Ellis, I.O.: Growth pattern of ductal carcinoma in situ (DCIS): a retrospective analysis based on mammographic findings. Br. J. Cancer 85(2):225–227 (2001)Google Scholar
  87. 87.
    Trump, B.E., Berezesky, I.K., Chang, S.H., Phelps, P.C: The pathways of cell death: Oncosis, apoptosis, and necrosis. Toxicol. Pathol. 25(1), 82–88 (1997)CrossRefGoogle Scholar
  88. 88.
    Walsh, G.M: Mechanisms of human eosinophil survival and apoptosis. Clin. Exp. Allergy 27(5), 482–487 (1997)CrossRefGoogle Scholar
  89. 89.
    Ward, J.P., King, J.R: Mathematical modelling of avascular tumour growth. IMA J. Math. Appl. Med. Biol. 14(1), 36–69 (1997)CrossRefGoogle Scholar
  90. 90.
    Ward, J.P., King, J.R: Mathematical modelling of avascular-tumour growth II: modelling growth saturation. Math. Med. Biol. 16(2), 171–211 (1999)zbMATHCrossRefGoogle Scholar
  91. 91.
    Wise, S.M., Lowengrub, J.S., Frieboes, H.B., Cristini, V.: Three-dimensional multispecies nonlinear tumor growth–I. model and numerical method. J. Theor. Biol. 253(3), 524–543 (2008)MathSciNetCrossRefGoogle Scholar
  92. 92.
    Yagata H., Harigaya K., Suzuki M., Nagashima T., Hashimoto H., Ishii G., Miyazaki M., Nakajima N., Mikata, A (2003) Comedonecrosis is an unfavorable marker in node-negative invasive breast carcinoma. Pathol. Int. 53, 8):501–506Google Scholar
  93. 93.
    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(2), 211–259 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Paul Macklin
    • 1
  • Shannon Mumenthaler
    • 1
  • John Lowengrub
    • 2
  1. 1.Keck School of Medicine, Center for Applied Molecular MedicineUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Departments of Mathematics, Chemical Engineering and Materials Science, and Biomedical EngineeringUniversity of CaliforniaIrvineUSA

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