Agent-Based Modeling of Ductal Carcinoma In Situ: Application to Patient-Specific Breast Cancer Modeling

  • Paul Macklin
  • Jahun Kim
  • Giovanna Tomaiuolo
  • Mary E. Edgerton
  • Vittorio Cristini
Part of the Applied Bioinformatics and Biostatistics in Cancer Research book series (ABB)


Ductal carcinoma in situ (DCIS) of the breast is the most common precursor to invasive carcinoma (IC), the second-leading cause of death in women in USA. There has been great progress in modeling DCIS at both the cellular scale (e.g., using cellular automata and agent-based models) and the population scale (e.g., using partial differential equations or systems of ordinary differential equations), but these past efforts have been difficult to calibrate with patient-specific molecular and cellular measurements. We develop a biophysically justified, agent-based cellular model of DCIS that is well-suited to patient-specific calibration. The model is modular in nature and can thus be readily extended to incorporate more advanced biology. We give an example of recently developed, patient-specific calibration of the model and conduct parameter studies that generate testable biological hypotheses.


Invasive Breast Cancer Invasive Ductal Carcinoma Cellular Automaton Apoptotic Index Necrotic Core 
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.



This work was partially funded by a generous grant from the Cullen Trust for Health Care (VC) and by the National Science Foundation (VC). We thank Yao-Li Chuang and Sandeep Sanga at the University of Texas Health Science Center-Houston for insightful discussions on tumor growth and protein signaling modeling.


  1. Adalsteinsson D, Sethian JA (1999) The fast construction of extension velocities in level set methods. J Comput Phys 148(1):2–22. doi:10.1006/jcph.1998.6090CrossRefGoogle Scholar
  2. Adamovich TL, Simmons RM (2003) Ductal carcinoma in situ with microinvasion. Am J Surg 186(2):112–116. doi:10.1016/S0002-9610(03)00166-1PubMedCrossRefGoogle Scholar
  3. Ai L, Kim WJ, Kim TY, Fields CR, Massoll NA, Robertson KD, Brown KD (2006) Epigenetic silencing of the tumor suppressor cystatin m occurs during breast cancer progression. Cancer Res 66:7899–7909PubMedCrossRefGoogle Scholar
  4. American Cancer Society (2007) American cancer society breast cancer facts and figures 2007–2008. American Cancer Society, Inc., Atlanta. BCFF-Final.pdf
  5. Anderson E (2004) Cellular homeostasis and the breast. Maturitas 48(S1):13–17. doi:10.1016/j. maturitas.2004.02.010Google Scholar
  6. Anderson ARA, Weaver AM, Cummings PT, Quaranta V (2006) Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment. Cell 127(5):905–915. doi:10.1016/j.cell.2006.09.042PubMedCrossRefGoogle Scholar
  7. Armstrong PB (1971) Light and electron microscope studies of cell sorting in combinations of chick embryo neural retina and retinal pigment epithelium. Wilhelm Roux’ Arch 168:125–141CrossRefGoogle Scholar
  8. Bankhead III A, Magnuson NS, Heckendorn RB (2007) Cellular automaton simulation examining progenitor heirarchy structure effects on mammary ductal carcinoma in situ. J Theor Biol 246(3):491–498. doi:10.1016/j.jtbi.2007.01.011PubMedCrossRefGoogle Scholar
  9. Barros LF, Hermosilla T, Castro J (2001) Necrotic volume increase and the early physiology of necrosis. Comp Biochem Physiol A Mol Integr Physiol 130:401–409PubMedCrossRefGoogle Scholar
  10. Baxter FO, Neoh K, Tevendale MC (2007) The beginning of the end: Death signaling in early involution. J Mamm Gland Biol Neoplasia 12(1):3–13. doi:10.1007/s10911-007-9033-9CrossRefGoogle Scholar
  11. Bienz M, Clevers H (2000) Linking colorectal cancer to Wnt signaling. Cell 103:311–320PubMedCrossRefGoogle Scholar
  12. Butler LM, Khan S, Rainger GE, Nash GB (2008) Effects of endothelial basement membrane on neutrophil adhesion and migration. Cell Immunol 251:56–61. doi:10.1016/j.cellimm. 2008.04.004PubMedCrossRefGoogle Scholar
  13. Byers SM, Sommers CL, Hoxter B, Mercurio AM, Tozeren A (1995) Role of e-cadherin in the response of tumor cell aggregates to lymphatic, venous, and arterial flow: measurement of cell–cell adhesion strength. J Cell Sci 108(5):2053–2064PubMedGoogle Scholar
  14. Byrne H, Drasdo D (2009) Individual-based and continuum models of growing cell populations: a comparison. J Math Biol 58(4–5):657–687. doi:10.1007/s00285-008-0212-0PubMedCrossRefGoogle Scholar
  15. Byrne HM, Alarcon T, Owen MR, Webb SD, Maini PK (2006) Modelling cancer dynamics: a review. Philos Trans R Soc A 364:1563–1578. doi:10.1098/rsta.2006.1786CrossRefGoogle Scholar
  16. Cabioglu N, Hunt KK, Sahin AA, Kuerer HM, Babiera GV, Singletary SE, Whitman GJ, Ross MI, Ames FC, Feig BW, Buchholz TA, Meric-Bernstam F (2007) Role for intraoperative margin assessment in patients undergoing breast-conserving surgery. Ann Surg Oncol 14(4):1458–1471PubMedCrossRefGoogle Scholar
  17. Cheng L, Al-Kaisi NK, Gordon NH, Liu AY, Gebrail F, Shenk RR (1997) Relationship between the size and margin status of ductal carcinoma in situ of the breast and residual disease. J Natl Cancer Inst 89(18):1356–1360PubMedCrossRefGoogle Scholar
  18. Chuang YL, Edgerton ME, Macklin P, Wise S, Lowengrub JS, Cristini V (in preparation) Clinical predictions of bulk DCIS properties based on a duct-scale mixture modelGoogle Scholar
  19. Ciatto S, Bianchi S, Vezzosi V (1994) Mammographic appearance of calcifications as a predictor of intraductal carcinoma histologic subtype. Eur Radiol 4(1):23–26. doi:10.1007/BF00177382CrossRefGoogle Scholar
  20. Collins LC, Tamimi RM, Baer HJ, Connolly JL, Colditz GA, Schnitt SJ (2005) Outcome of patients with ductal carcinoma in situ untreated after diagnostic biopsy: results from the Nurses’ Health Study. Cancer 103(9):1778–1784PubMedCrossRefGoogle Scholar
  21. Cotran RS, Kumar V, Robbins SL (1994) Robbins Pathologic Basis of Disease, 5th edn. W.B. Saunders, Philadelphia, PAGoogle Scholar
  22. Cristini V, Lowengrub JS (in press) Multiscale Modeling of Cancer. Cambridge University Press, New York, NYGoogle Scholar
  23. Cristini V, Lowengrub JS, Nie Q (2003) Nonlinear simulation of tumor growth. J Math Biol 46:191–224. doi:10.1007/s00285-002-0174-6PubMedCrossRefGoogle Scholar
  24. Danes CG, Wyszomierski SL, Lu J, Neal CL, Yang W, Yu D (2008) 14-3-3ζ down-regulates p53 in mammary epithelial cells and confers luminal filling. Cancer Res 68:1760–1767. doi:10.1158/0008-5472.CAN-07-3177PubMedCrossRefGoogle Scholar
  25. Dillon MF, McDermott EW, O’Doherty A, Quinn CM, Hill AD, O’Higgins N (2007) Factors affecting successful breast conservation for ductal carcinoma in situ. Ann Surg Oncol 14(5):1618–1628PubMedCrossRefGoogle Scholar
  26. Dillon R, Owen M, Painter K (2008) A single-cell based model of multicellular growth using the immersed boundary method. In: Cheong K, Li Z, Lin P (eds) Moving interface problems and applications in fluid dynamics. AMS Contemporary Mathematics. American Mathematical Society, Providence, RIGoogle Scholar
  27. Duan WR, Garner DS, Williams SD, Funckes-Shippy CL, Spath IS, Blomme EAG (2003) Comparison of immunohistochemistry for activated caspase-3 and cleaved cytokeratin 18 with the TUNEL method for quantification of apoptosis in histological sections of PC-3 subcutaneous xenografts. J Pathol 199(2):221–228PubMedCrossRefGoogle Scholar
  28. Edgerton M, Chuang YL, Kim J, Tomaiuolo G, Macklin P, Sanga S, Yang W, Broom A, Do KA, Cristini V (2008) Using mathematical models to understand the time dependence of the growth of ductal carcinoma in situ. In: 31st Annual San Antonio breast cancer symposium supplement to volume 68(24), Abstract 1165Google Scholar
  29. Edgerton ME, Macklin P, Chuang YL, Tomaiuolo G, Kim J, Sanga S, Broom A, Yang W, Do KA, Cristini V (in review) A Multiscale Mathematical Models for Improved Pre-Operative Estimates of Ductal Carcinoma in Situ Size, Canc. Res.Google Scholar
  30. Edgerton ME, Mannes K, Dudek S, Jensen R, Page D (in preparation-b) Pagetoid spread of ductal carcinoma in situGoogle Scholar
  31. Franks SJ, Byrne HM, King JR, Underwood JCE, Lewis CE (2003a) Modelling the early growth of ductal carcinoma in situ of the breast. J Math Biol 47:424–452. doi:10.1007/s00285- 003-0214-xPubMedCrossRefGoogle Scholar
  32. Franks SJ, Byrne HM, Mudhar H, Underwood JCE, Lewis CE (2003b) Modelling the growth of comedo ductal carcinoma in situ. Math Med Biol 20:277–308PubMedCrossRefGoogle Scholar
  33. Franks SJ, Byrne HM, Underwood JCE, Lewis CE (2005) Biological inferences from a mathematical model of comedo ductal carcinoma in situ of the breast. J Theor Biol 232(4):523–543. doi:10.1016/j.jtbi.2004.08.032PubMedCrossRefGoogle Scholar
  34. 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):1597–1604PubMedCrossRefGoogle Scholar
  35. Frieboes HB, Lowengrub JS, Wise S, Zheng X, Macklin P, Cristini V (2007) Computer simulations of glioma growth and morphology. NeuroImage 37(S1):S59–S70. doi:10.1016/ j.neuroimage.2007.03.008Google Scholar
  36. Frieboes HB, Edgerton ME, Fruehauf JP, Rose FRAJ, Worrall LK, Gatenby RA, Ferrari M, Cristini V (2009) Prediction of drug response in breast cancer using integrative experimental/computational modeling. Cancer Res 69(10):4484–4492PubMedCrossRefGoogle Scholar
  37. Gadeau AP, Chaulet H, Daret D, Kockx M, Daniel-Lamaziè JM, Desgranges C (2001) Time course of osteopontin, osteocalcin, and osteonectin accumulation and calcification after acute vessel wall injury. J Histochem Cytochem 49:79–86PubMedGoogle Scholar
  38. Gerdes J, Lemke H, Baisch H, Wacker HH, Schwab U, Stein H (1984) Cell cycle analysis of a cell proliferation-associated human nuclear antigen defined by the monoclonal antibody Ki-67. J Immunol 133(4):1710–1715PubMedGoogle Scholar
  39. Going JJ, Mohun TJ (2006) Human breast duct anatomy, the sick lobe hypothesis and intraductal approaches to breast cancer. Breast Cancer Res Treat 97(3):285–291. s10549-005-9122-7Google Scholar
  40. Greenspan HP (1976) On the growth and stability of cell cultures and solid tumors J Theor Biol 56(1):229–242. doi:10.1016/S0022-5193(76)80054-9Google Scholar
  41. Guck J, Schinkinger S, Lincoln B, Wottawah F, Ebert S, Romeyke M, Lenz D, Erickson HM, Ananthakrishnan R, Mitchell D, Käs J, Ulvick S, Bilby C (2005) Optical deformability as an inherent cell marker for testing malignant transformation and metastatic competence. Biophys J 88(5):3689–3698. doi:10.1529/biophysj.104.045476PubMedCrossRefGoogle Scholar
  42. Hanahan D, Weinberg RA (2000) The hallmarks of cancer. Cell 100(1):57–70. doi:10.1016/ S0092-8674(00)81683-9PubMedCrossRefGoogle Scholar
  43. Hansen RK, Bissell MJ (2000) Tissue architecture and breast cancer: the role of extracellular matrix and steroid hormones. Endocr Relat Cancer 7(2):95–113. doi:10.1677/erc.0.0070095PubMedCrossRefGoogle Scholar
  44. Harms BD, Bassi GM, Horwitz AR, Lauffenburger DA (2005) Directional persistence of EGF-induced cell migration is associated with stabilization of lamellipodial protrusions. Biophys J 88(2):1479–1488PubMedCrossRefGoogle Scholar
  45. Hino Si, Tanji C, Nakayama KI, Kikuchi A (2005) Phosphorylation of β-catenin by cyclic AMP-dependent protein kinase stabilizes β-catenin through inhibition of its ubiquitination. Mol Cell Biol 25(20):9063–9072. doi:10.1128/MCB.25.20.9063-9072.2005CrossRefGoogle Scholar
  46. Hu Z, Yuri K, Ozawa H, Lu H, Kawata M (1997) The in vivo time course for elimination of adrenalectomy-induced apoptotic profiles from the granule cell layer of the rat hippocampus. J Neurosci 17(11):3981–3989PubMedGoogle Scholar
  47. Ilic D, Almeida EA, Schlaepfer DD, Dazin P, Aizawa S, Damsky CH (1998) Extracellular matrix survival signals transduced by focal adhesion kinase suppress p53-mediated apoptosis. J Cell Biol 143:547–560PubMedCrossRefGoogle Scholar
  48. Jemal A, Siegel R, Ward E, Murray T, Xu J, Thun MJ (2007) Cancer statistics, 2007. CA Cancer J Clin 57(1):43–66PubMedCrossRefGoogle Scholar
  49. Jha MK, Avlontitis VS, Griffith CDM, Lennard TWJ, Wilson RG, McLean LM, Dawes PD, Shrinmankar J (2001) Aggressive local treatment for screen-detected DCIS results in very low rates of recurrence. Eur J Surg Oncol 27(5):454–458. doi:10.1053/ejso.2001.1163PubMedCrossRefGoogle Scholar
  50. Jian B, Narula N, Li QY, Mohler III ER, Levy RJ (2003) Progression of aortic valve stenosis: TGF-β1 is present in calcified aortic valve cusps and promotes aortic valve interstitial cell calcification via apoptosis. Ann Thoracic Surg 75(2):457–465. doi:10.1016/S0003-4975(02)04312-6CrossRefGoogle Scholar
  51. Kerlikowske K, Molinaro A, Cha I, Ljung BM, Ernster VL, Stewart K, Chew K, Moore 2nd DH, Waldman F (2003) Characteristics associated with recurrence among women with ductal carcinoma in situ treated by lumpectomy. J Natl Cancer Inst 95(22):1692–1702PubMedGoogle Scholar
  52. Kerr JF, Winterford CM, Harmon BV (1994) Apoptosis. Its significance in cancer and cancer therapy. Cancer 15(8):2013–2026Google Scholar
  53. Khan S, Rogers M, Khurana K, Meguid M, Numann P (1998) Estrogen receptor expression in benign breast epithelium and breast cancer risk. J Natl Cancer Inst 90:37–42PubMedCrossRefGoogle Scholar
  54. Khan S, Sachdeva A, Naim S, Meguid M, Marx W, Simon H, et al (1999) The normal breast epithelium of women with breast cancer displays an aberrant response to estradiol. Cancer Epidemiol Biomarkers Prev 8:867–872PubMedGoogle Scholar
  55. Kopans DB, Rafferty E, Georgian-Smith D, Yeh E, D’Alessandro H, Hughes K, Halpern E (2003) A simple model of breast carcinoma growth may provide explanations for observations of apparently complex phenomena. Cancer 97(12):2951–2959PubMedCrossRefGoogle Scholar
  56. Krysko DV, Berghe TV, D’Herde K, Vandenabeele P (2008) Apoptosis and necrosis: Detection, discrimination and phagocytosis. Methods 44:205–221. doi:10.1016/j.ymeth.2007.12.001PubMedCrossRefGoogle Scholar
  57. Lampejo OT, Barnes DM, Smith P, Millis RR (1994) Evaluation of infiltrating ductal carcinomas with a DCIS component: correlation of the histologic type of the in situ component with grade of the infiltrating component. Semin Diagn Pathol 11(3):215–222PubMedGoogle Scholar
  58. Lee JS, Basalyga DM, Simionescu A, Isenburg JC, Sinionescu DT, Vyavahare NR (2006a) Elastin calcification in the rate subdermal model is accompanied by up-regulation of degradative and osteogenic cellular responses. Am J Pathol 168:490–498. doi:10.2353/ajpath.2006.050338PubMedCrossRefGoogle Scholar
  59. Lee S, Mohsin SK, Mao S, Hilsenbeck SG, Medina D, Allred DC (2006b) Hormones, receptors, and growth in hyperplastic enlarged lobular units: early potential precursors of breast cancer. Breast Cancer Res 8(1):R6PubMedCrossRefGoogle Scholar
  60. Lustig B, Jerchow B, Sachs M, Wiler S, Pietsch T, Karsten U, van de Wetering M, Clevers H, Schlag PM, Birchmeier W, Behrens J (2002) Negative feedback loop of Wnt signaling through upregulation of conductin/axin2 in colorectal and liver tumors. Mol Cell Biol 22:1184–1193PubMedCrossRefGoogle Scholar
  61. Macklin P, Lowengrub JS (2005) Evolving interfaces via gradients of geometry-dependent interior Poisson problems: application to tumor growth 203(1):191–220. doi:10.1016/ Scholar
  62. Macklin P, Lowengrub JS (2006) An improved geometry-aware curvature discretization for level set methods: application to tumor growth J Comput Phys 215(2):392–401. doi:10.1016/ Scholar
  63. Macklin P, Lowengrub JS (2007) Nonlinear simulation of the effect of microenvironment on tumor growth. J Theor Biol 245(4):677–704. doi:10.1016/j.jtbi.2006.12.004PubMedCrossRefGoogle Scholar
  64. Macklin P, Lowengrub JS (2008) A new ghost cell/level set method for moving boundary problems: application to tumor growth. J Sci Comput 35(2–3):266–299. doi:10.1007/s10915-008-9190-zCrossRefGoogle Scholar
  65. Macklin P, McDougall SR, Anderson ARA, Chaplain MAJ, Lowengrub J (2009) Multiscale modelling and nonlinear simulation of vascular tumour growth. J Math Biol 58(4–5):765–798. doi:10.1007/s00285-008-0216-9PubMedCrossRefGoogle Scholar
  66. Macklin P, Kim J, Tomaiuolo G, Edgerton ME, Cristini V (in preparation) An agent-based cell model, with application to patient-specific ductal carcinoma in situ modelingGoogle Scholar
  67. Macknight ADC, DiBona DR, Leaf A, Mortimer MC (1971) Measurement of the composition of epithelial cells from the toad urinary bladder. J Membr Biol 6(2):108–126. doi:10.1007/ BF01873458CrossRefGoogle Scholar
  68. Malladi R, Sethian JA, Vemuri BC (1995) Shape modeling with front propagation: a level set approach. IEEE Trans Pattern Anal Mach Intell 17(2):158–175. doi:10.1109/34.368173CrossRefGoogle Scholar
  69. Malladi R, Sethian JA, Vemuri BC (1996) A fast level set based algorithm for topology-independent shape modeling. J Math Imaging Vis 6(2–3):269–289. doi:10.1007/BF00119843CrossRefGoogle Scholar
  70. Mannes KD, Edgerton ME, Simpson JF, Jenson RA, Page DL (2002) Pagetoid spread in ductal carcinoma in situ: characterization and computer simulation. In: United States and Canadian Academy of Pathology (USCAP) annual meeting 2002, Chicago, ILGoogle Scholar
  71. Moffat DF, Going JJ (1996) Three dimensional anatomy of complete duct systems in human breast: pathological and developmental implications. J Clin Pathol 49:48–52. doi:10.1136/jcp.49.1.48PubMedCrossRefGoogle Scholar
  72. Ohtake T, Kimijima I, Fukushima T, Yasuda M, Sekikawa K, Takenoshita S, Abe R (2001) Computer-assisted complete three-dimensional reconstruction of the mammary ductal/lobular systems. Cancer 91:2263–2272PubMedCrossRefGoogle Scholar
  73. Øksendal B (2007) Stochastic differential equations: an introduction with applications, 6th edn. Springer, New YorkGoogle Scholar
  74. Osher S, Fedkiw R (2001) Level set methods: an overview and some recent results 169(2): 463–502. doi:10.1006/jcph.2000.6636Google Scholar
  75. Osher S, Fedkiw R (2002) Level set methods and dynamic implicit surfaces. Springer, New YorkGoogle Scholar
  76. Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations 79(1):12–49. doi:10.1016/0021-9991(88)90002-2Google Scholar
  77. Owen MR, Byrne HM, Lewis CE (2004) Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites. J Theor Biol 226(4):377–391PubMedCrossRefGoogle Scholar
  78. Page DL, Dupont WD, Rogers LW, Landenberger M (1982) Intraductal carcinoma of the breast: follow-up after biopsy only. Cancer 49(4):751–758PubMedCrossRefGoogle Scholar
  79. Panorchan P, Thompson MS, Davis KJ, Tseng Y, Konstantopoulos K, Wirtz D (2006) Single-molecule analysis of cadherin-mediated cell-cell adhesion. J Cell Sci 119:66–74. doi:10.1242/jcs.02719PubMedCrossRefGoogle Scholar
  80. Patani N, Cutuli B, Mokbel K (2008) Current management of DCIS: a review. Breast Cancer Res Treat 111(1):1–10PubMedCrossRefGoogle Scholar
  81. Rejniak KA, Dillon RH (2007) A single cell-based model of the ductal tumour architecture. Comput Math Methods Med 8(1):51–69CrossRefGoogle Scholar
  82. Sanders ME, Schuyler PA, Dupont WD, Page DL (2005) The natural history of low-grade ductal carcinoma in situ of the breast in women treated by biopsy only revealed over 30 years of long-term follow-up. Cancer 103(12):2481–2484PubMedCrossRefGoogle Scholar
  83. Scarlett JL, Sheard PW, Hughes G, Ledgerwood EC, Ku HH, Murphy MP (2000) Changes in mitochondrial membrane potential during staurosporine-induced apoptosis in jurkat cells. IFEBS Lett 475:267–272CrossRefGoogle Scholar
  84. Seidensticker MJ, Behrens J (2000) Biochemical interactions in the wnt pathway. Biochim Biophys Acta 1495:168–182PubMedCrossRefGoogle Scholar
  85. Sethian JA (1999) Level set methods and fast marching methods. Cambridge University Press, New YorkGoogle Scholar
  86. Sethian JA, Smereka P (2003) Level set methods for fluid interfaces. Ann Rev Fluid Mech 35(1):341–372. doi:10.1146/annurev.fluid.35.101101.161105CrossRefGoogle Scholar
  87. Shiryaev AN (1995) Probability, 2nd edn. Springer, New YorkGoogle Scholar
  88. Shuryak I, Sachs RK, Hlatky L, Little MP, Hahnfeldt P, Brenner D (2006) Radiation-induced leukemia at doses relevant to radiation therapy: modeling mechanisms and estimating risks. J Natl Cancer Inst 98(24):1794–1806. doi:10.1093/jnci/djj497PubMedCrossRefGoogle Scholar
  89. Silver SA, Tavassoli FA (1998) Ductal carcinoma in situ with microinvasion. Breast J 4(5):344–348CrossRefGoogle Scholar
  90. Silverstein MJ (1997a) Predicting residual disease and local recurrence in patients with ductal carcinoma in situ. J Natl Cancer Inst 89(18):1330–1331PubMedCrossRefGoogle Scholar
  91. Silverstein MJ (1997b) Recent advances: diagnosis and treatment of early breast cancer. Br Med J 314(7096):1736ffGoogle Scholar
  92. Silverstein MJ (2000) Ductal carcinoma in situ of the breast. Annu Rev Med 51:17–32. doi:10.1146/ Scholar
  93. Simpson PT, Reis-Filho JS, Gale T, Lakhani SR (2005) Molecular evolution of breast cancer. J Pathol 205(2):248–254. doi:10.1002/path.1691PubMedCrossRefGoogle Scholar
  94. Sontag L, Axelrod DE (2005) J Theor Biol 232(2):179–189. doi:10.1016/j.jtbi.2004.08.002PubMedCrossRefGoogle Scholar
  95. Stomper PC, Margolin FR (1994) Ductal carcinoma in situ: the mammographer’s perspective. Am J Roentgenol 162:585–591Google Scholar
  96. Sussman M, Smereka P, Osher S (1994) A level set approach for computing solutions to incompressible two-phase flow J Comput Phys 114(1):146–159. doi:10.1006/jcph.1994.1155Google Scholar
  97. Tannis PJ, Nieweg OE, Valdés Olmos RA, Kroon BBR (2001) Anatomy and physiology of lymphatic drainage of the breast from the perspective of sentinel node biopsy. J Am Coll Surg 192(3):399–409. doi:10.1016/S1072-7515(00)00776-6CrossRefGoogle Scholar
  98. Thorne BC, Bailey AM, Peirce SM (2007) Combining experiments with multi-cell agent-based modeling to study biological tissue patterning. Breif Bioinform 8(4):245–257. doi:10.1093/bib/bbm024CrossRefGoogle Scholar
  99. Wang R, Jinming L, Lyte K, Yashpal NK, Fellows F, Goodyer CG (2005) Role for β1 integrin and its associated α3, α5, and α6 subunits in development of the human fetal pancreas. Diabetes 54:2080–2089PubMedCrossRefGoogle Scholar
  100. Ward JP, King JR (1997) Mathematical modelling of avascular-tumour growth. IMA J Math Appl Med Biol 14(1):39–69PubMedCrossRefGoogle Scholar
  101. Ward JP, King JR (1999) Mathematical modelling of avascular tumour growth II: Modelling growth saturation. IMA J Math Appl Med Biol 16:171–211PubMedCrossRefGoogle Scholar
  102. Wei C, Larsen M, Hoffman MP, Yamada KM (2007) Self-organization and branching morphogenesis of primary salivary epithelial cells. Tissue Eng 13(4):721–735. doi:10.1089/ten.2006.0123PubMedCrossRefGoogle Scholar
  103. Wellings SR, Jensen HM, Marcum RG (1975) An atlas of subgross pathology of the human breast with special reference to possible precancerous lesions. J Natl Cancer Inst 55(2):231–273PubMedGoogle Scholar
  104. Xu Y, Gilbert R (2009) Some inverse problems raised from a mathematical model of ductal carcinoma in situ. Math Comput Model 49(3-4):814–828. doi:10.1016/j.mcm.2008.02.014CrossRefGoogle Scholar
  105. Zaman MH, Kamm RD, Matsudaira P, Lauffenburger DA (2005) Computational model for cell migration in three-dimensional matrices. Biophys J 89(2):1389–1397PubMedCrossRefGoogle Scholar
  106. Zhang L, Athale CA, Deisboeck TS (2007) Development of a three-dimensional multiscale agent-based tumor model: simulating gene-protein interaction profiles, cell phenotypes and multicellular patterns in brain cancer. J Theor Biol 244(1):96–107PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Paul Macklin
    • 1
  • Jahun Kim
  • Giovanna Tomaiuolo
  • Mary E. Edgerton
  • Vittorio Cristini
  1. 1.School of Health Information SciencesUniversity of Texas Health Science CenterHoustonUSA

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