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Toward Predictive Multiscale Modeling of Vascular Tumor Growth

Computational and Experimental Oncology for Tumor Prediction

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Abstract

New directions in medical and biomedical sciences have gradually emerged over recent years that will change the way diseases are diagnosed and treated and are leading to the redirection of medicine toward patient-specific treatments. We refer to these new approaches for studying biomedical systems as predictive medicine, a new version of medical science that involves the use of advanced computer models of biomedical phenomena, high-performance computing, new experimental methods for model data calibration, modern imaging technologies, cutting-edge numerical algorithms for treating large stochastic systems, modern methods for model selection, calibration, validation, verification, and uncertainty quantification, and new approaches for drug design and delivery, all based on predictive models. The methodologies are designed to study events at multiple scales, from genetic data, to sub-cellular signaling mechanisms, to cell interactions, to tissue physics and chemistry, to organs in living human subjects. The present document surveys work on the development and implementation of predictive models of vascular tumor growth, covering aspects of what might be called modeling-and-experimentally based computational oncology. The work described is that of a multi-institutional team, centered at ICES with strong participation by members at M. D. Anderson Cancer Center and University of Texas at San Antonio. This exposition covers topics on signaling models, cell and cell-interaction models, tissue models based on multi-species mixture theories, models of angiogenesis, and beginning work of drug effects. A number of new parallel computer codes for implementing finite-element methods, multi-level Markov Chain Monte Carlo sampling methods, data classification methods, stochastic PDE solvers, statistical inverse algorithms for model calibration and validation, models of events at different spatial and temporal scales is presented. Importantly, new methods for model selection in the presence of uncertainties fundamental to predictive medical science, are described which are based on the notion of Bayesian model plausibilities. Also, as part of this general approach, new codes for determining the sensitivity of model outputs to variations in model parameters are described that provide a basis for assessing the importance of model parameters and controlling and reducing the number of relevant model parameters. Model specific data is to be accessible through careful and model-specific platforms in the Tumor Engineering Laboratory. We describe parallel computer platforms on which large-scale calculations are run as well as specific time-marching algorithms needed to treat stiff systems encountered in some phase-field mixture models. We also cover new non-invasive imaging and data classification methods that provide in vivo data for model validation. The study concludes with a brief discussion of future work and open challenges.

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References

  1. Abraham FF, Broughton JQ, Bernstein N, Kaxiras E (1998) Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture. Europhys Lett (EPL) 44(6):783

    Article  Google Scholar 

  2. Adams BM, Bauman LE, Bohnhoff WJ, Dalbey KR, Ebeida MS, Eddy JP, Eldred MS, Hough PD, Hu KT, Jakeman JD, Swiler LP, Vigil DM (2009) Dakota, a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis: version 5.4 user’s manual. Technical report, Sandia technical report SAND2010-2183. Sandia National Laboratories, Livermore, CA

  3. Albano G, Giorno V (2006) A stochastic model in tumor growth. J Theor Biol 242(2):329–336

    Article  MathSciNet  MATH  Google Scholar 

  4. Albano G, Giorno V, Román-Román P, Torres-Ruiz F (2012) Inference on a stochastic two-compartment model in tumor growth. Comput Stat Data Anal 56:1723–1736

    Article  MathSciNet  MATH  Google Scholar 

  5. Aldridge BB, Burke JM, Lauffenburger DA, Sorger PK (2006) Physicochemical modelling of cell signalling pathways. Nat Cell Biol 8(11):1195

    Article  Google Scholar 

  6. Almeida RC, Oden JT (2010) Solution verification, goal-oriented adaptive methods for stochastic advectiondiffusion problems. Comput Methods Appl Mech Eng 199(3740):2472–2486

    Article  MathSciNet  MATH  Google Scholar 

  7. Anderson ARA, Chaplain MAJ (1995) Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull Math Biol 60:857–899

    Article  MATH  Google Scholar 

  8. Antoine E, Vlachos P, Rylander MN (2014) Review of collagen I hydrogels for bioengineered tissue microenvironments: characterization of mechanics, structure, and transport. Tissue Eng Part B 20(6):683–696

    Article  Google Scholar 

  9. Antoine E, Vlachos P, Rylander MN (2015) Tunable collagen I hydrogels for engineered physiological tissue micro-environments. PloS (epub ahead of print) 10(3):1–18

  10. Ariffin AB, Forde PF, Jahangeer S, Soden DM, Hinchion J (2014) Releasing pressure in tumors: what do we know so far and where do we go from here? A review. Cancer Res 74(10):2655–2662

    Article  Google Scholar 

  11. Arroyo AG, Iruela-Arispe ML (2010) Extracellular matrix, inflammation, and the angiogenic response. Cardiovasc Res 86:226235

    Article  Google Scholar 

  12. Avants BB, Epstein CL, Grossman M, Gee JC (2008) Symmetric diffeomorphic image registration with cross-correlation: evaluating automated labeling of elderly and neurodegenerative brain. Med Image Anal 12(1):26–41

    Article  Google Scholar 

  13. Balay S, Abhyankar S, Adams MF, Brown J, Brune P, Buschelman K, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, McInnes LC, Rupp K, Smith BF, Zhang H (2014) PETSc web page. http://www.mcs.anl.gov/petsc

  14. Bao A, Phillips WT, Goins B, McGuff HS, Zheng X, Woolley FR, Natarajan M, Santoyo C, Miller FR, Otto RA (2006) Setup and characterization of a human head and neck squamous cell carcinoma xenograft model in nude rats. Otolaryngol Head Neck Surg 135(6):853–857

    Article  Google Scholar 

  15. Beck JL, Yuen KV (2004) Model selection using response measurements: Bayesian probabilistic approach. J Eng Mech 130(2):192–203

    Article  Google Scholar 

  16. Beg MF, Miller MI, Trouvé A, Younes L (2005) Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int J Comput Vis 61(2):139–157

    Article  Google Scholar 

  17. Bellomo N, Li NK, Maini PK (2008) On the foundations of cancer modelling: selected topics, speculations, and perspectives. Math Models Methods Appl Sci 18(4):593–646

    Article  MathSciNet  MATH  Google Scholar 

  18. Berg JM, Tymoczko JL, Stryer L (2006) Biochemistry, 6th edn. W. H. Freeman, San Francisco

    Google Scholar 

  19. Berger JO (1985) Statistical decision theory and bayesian analysis. Springer, Berlin

    Book  MATH  Google Scholar 

  20. Buchanan CF, Rylander MN (2015) Microfluidic culture models to study the hydrodynamics of tumor progression and therapeutic response. Biotechnol Bioeng 110(B):2063–2072

    Google Scholar 

  21. Buchanan CF, Verbridge SS, Vlachos PP, Rylander MN (2014) Flow shear stress regulates endothelial barrier function and expression of angiogenic factors in a 3D microfluidic tumor vascular model. Cell Adhesion Migr 8(5):517–524

    Article  Google Scholar 

  22. Buchanan CF, Voigt E, Szot CS, Freeman JW, Vlachos PP, Rylander MN (2013) Three-dimensional microfluidic collagen hydrogels for investigating flow-mediated tumor-endothelial signaling and vascular organization. Tissue Eng Part C Methods 20(1):64–75

    Article  Google Scholar 

  23. Byrne H, Drasdo D (2009) Individual-based and continuum models of growing cell populations: a comparison. J Math Biol 58(4–5):657–687

    Article  MathSciNet  MATH  Google Scholar 

  24. Cacuci DG (2007) Sensitivity and uncertainty analysis: theory, vol 1. CRC Press, Boca Raton

    MATH  Google Scholar 

  25. Calvetti D, Somersalo E (2007) Introduction to Bayesian scientific computing: ten lectures on subjective computing. Springer, Berlin

    MATH  Google Scholar 

  26. Cao Y, Jiang Y, Li B, Feng X (2012) Biomechanical modeling of surface wrinkling of soft tissues with growth-dependent mechanical properties. Acta Mech Solida Sin 25(5):483–492

    Article  Google Scholar 

  27. Chang CH, Horton J, Schoenfeld D, Salazer O, Perez-Tamayo R, Kramer S, Weinstein A, Nelson JS, Tsukada Y (1983) Comparison of postoperative radiotherapy and combined postoperative radiotherapy and chemotherapy in the multidisciplinary management of malignant gliomas. Cancer 52:997–1007

    Article  Google Scholar 

  28. Chaplain MAJ (1996) Avascular growth, angiogenesis and vascular growth in solid tumours: the mathematical modelling of the stages of tumour development. Math Comput Model 23(6):47–87

    Article  MATH  Google Scholar 

  29. Cheng G, Tse J, Jain RK, Munn LL (2009) Micro-environmental mechanical stress controls tumor spheroid size and morphology by suppressing proliferation and inducing apoptosis in cancer cells. PloS One 4(2):e4632

    Article  Google Scholar 

  30. Chevalier MW, El-Samad H (2014) A master equation and moment approach for biochemical systems with creation-time-dependent bimolecular rate functions. J Chem Phys 141:214108

    Article  Google Scholar 

  31. Chib S, Greenberg E (1995) Understanding the Metropolis–Hastings algorithm. Am Stat 49(4):327–335

    Google Scholar 

  32. Christensen GE, Rabbitt RD, Miller MI (1996) Deformable templates using large deformation kinematics. IEEE Trans Image Process 5(10):1435–1447

    Article  Google Scholar 

  33. Cimmelli V, Sellitto A, Triani V (2010) A generalized Coleman–Noll procedure for the exploitation of the entropy principle. In: Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, vol 466. The Royal Society, London, p 911–925

  34. Coleman BD, Noll W (1963) The thermodynamics of elastic materials with heat conduction and viscosity. Arch Ration Mech Anal 13(1):167–178

    Article  MathSciNet  MATH  Google Scholar 

  35. Cox RT (1946) Probability, frequency and reasonable expectation. Am J Phys 14(1):1–13

    Article  MathSciNet  MATH  Google Scholar 

  36. Criminisi A, Shotton J (2013) Decision forests for computer vision and medical image analysis. Springer, Berlin

    Book  Google Scholar 

  37. Cristini V, Li X, Lowengrub JS, Wise SM (2009) Nonlinear simulation of solid tumor growth using a mixture model: invasion and branching. J Math Biol 58:723–763

    Article  MathSciNet  MATH  Google Scholar 

  38. Cristini V, Lowengrub J (2010) Multiscale modeling of cancer: an integrated experimental and mathematical modeling approach. Cambridge University Press, Cambridge

    Book  Google Scholar 

  39. Cukier RI, Fortuin CM, Shuler KE, Petschek AG, Schaibly JH (1973) Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. I. Theory. J Chem Phys 59(8):3873–3878

    Article  Google Scholar 

  40. Curtin WA, Miller RE (2003) Atomistic/continuum coupling in computational materials science. Model Simul Mater Sci Eng 11(3):R33

    Article  Google Scholar 

  41. D’Antonio G, Macklin P, Preziosi L (2013) An agent-based model for elasto-plastic mechanical interactions between cells, basement membrane and extracellular matrix. Math Biosc Eng 10(1):75–101

    Article  MathSciNet  MATH  Google Scholar 

  42. Deakin NE, Chaplain MAJ (2013) Mathematical modeling of cancer invasion: the role of membrane-bound matrix metalloproteinases. In: Rejniak KA, Enderling H (eds) Computational models in oncology: from tumor initiation to progression to treatment. Frontiers Media SA, Switzerland

  43. Deisboeck TS, Stamatakos GS (2010) Multiscale cancer modeling. In: Britton NF, Lin X, Safer HM, Scheneider MV, Singh M, Tramontano A (eds) Chapman & Hall/CRC mathematical and computational biology series. Taylor & Francis Group, London

  44. Demicheli R, Foroni R, Ingrosso A, Pratesi G, Soranzo C, Tortoreto M (1989) An exponential-Gompertzian description of lovo cell tumor growth from in vivo and in vitro data. Cancer Res 49:6543–6546

    Google Scholar 

  45. Dupuis P, Grenander U, Miller MI (1998) Variational problems on flows of diffeomorphisms for image matching. Q Appl Math 56(3):587

    MathSciNet  MATH  Google Scholar 

  46. Elliot CM (1989) The Cahn–Hilliard model for the kinetics of phase separation. In: Rodrigues JF (ed) Mathematical models for phase change problems. Birkhauser, Switzerland

  47. Elliott CM, Songmu Z (1986) On the Cahn–Hilliard. Arch Ration Mech Anal 96(4):339–357

    Article  MathSciNet  MATH  Google Scholar 

  48. Eyre DJ (1998) Unconditionally gradient stable time marching the Cahn–Hilliard equation. In: Bullard JW, Chen LQ (eds) Computational and mathematical models of microstructural evolution, MRS proceedings, vol 529. Cambridge University Press, Cambridge, pp 39–46

  49. Feng Y, Boukhris SJ, Ranjan R, Valencia RA (2015) Biological systems: multiscale modeling based on mixture-theory. In: De S, Hwang W, Kuhl E (eds) Multiscale modeling in biomechanics and mechanobiology. Springer, Berlin

    Google Scholar 

  50. Fife PC (2000) Models for phase separation and their mathematics. Electron J Differ Equ 48:1–26

    MathSciNet  MATH  Google Scholar 

  51. Frieboes HB, Jin F, Chuang YL, Wise S, Lowengrub J, Cristini V (2011) Three-dimensional multispecies nonlinear tumor growth—II: tumor invasion and angiogenesis. J Theor Biol 264(4):1254–1278

    Article  MathSciNet  Google Scholar 

  52. Frieboes HB, Lowengrubb JS, Wise S, Zheng X, Macklin P, Bearer E, Cristini V (2007) Computer simulation of glioma growth and morphology. Neuroimage 37(Suppl. 1):S59–S70

    Article  Google Scholar 

  53. Ganapathy-Kanniappan S, Geschwind JFH (2013) Tumor glycolysis as a target for cancer therapy: progress and prospects. Mol Cancer 12:152

    Article  Google Scholar 

  54. Ghanem RG, Spanos PD (2003) Stochastic finite elements: a spectral approach, revised edn. Dover, New York

    Google Scholar 

  55. Gonalves E, Bucher J, Ryll A, Niklas J, Mauch K, Klamt S, Rochad M, Saez-Rodriguez J (2013) Bridging the layers: towards integration of signal transduction, regulation and metabolism into mathematical models. Mol Biosyst 9:1576

    Article  Google Scholar 

  56. Hanahan D, Weinberg R (2011) Hallmarks of cancer: the next generation. Cell 144(5):646–674

    Article  Google Scholar 

  57. Hanahan D, Weinberg RA (2000) The hallmarks of cancer. Cell 100:57–70

    Article  Google Scholar 

  58. Hawkins-Daarud A, van der Zee KG, Tinsley Oden J (2012) Numerical simulation of a thermodynamically consistent four-species tumor growth model. Int J Numer Methods in Biomed Eng 28(1):3–24

    Article  MathSciNet  MATH  Google Scholar 

  59. Hawkins-Daarud AJ (2011) Toward a predictive model of tumor growth. Ph.D. thesis, The University of Texas at Austin

  60. Heida M, Málek J, Rajagopal KR (2012) On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework. Z Angew Math Phys 63:145–169

    Article  MathSciNet  MATH  Google Scholar 

  61. Hesketh R (2013) Introduction to cancer biology, 1st edn. Cambridge University Press, Cambridge

    Google Scholar 

  62. Hyun AA, Macklin P (2013) Improved patient-specific calibration for agent-based cancer modeling. J Theor Biol 317:422–424

    Article  MathSciNet  Google Scholar 

  63. Jackson TL (ed) (2012) Modeling tumor vasculature—molecular, cellular, and tissue level aspects and implications. Springer, Berlin

    Google Scholar 

  64. Jain RK (2013) Normalizing tumor microenvironment to treat cancer: bench to bedside to biomarkers. J Clin Oncol 31(17):2205–2218

    Article  Google Scholar 

  65. Jaynes ET (2003) Probability theory: the logic of science. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  66. Szekely T Jr, Burrage K (2014) Stochastic simulation in systems biology. Comput Struct Biotechnol J 12(2021):14–25

    Article  Google Scholar 

  67. Kaipio J, Somersalo E (2005) Statistical and computational inverse problems. Springer, Berlin

    MATH  Google Scholar 

  68. Kansal A, Torquato S, Harsh GR IV, Chiocca E, Deisboeck T (2000) Cellular automaton of idealized brain tumor growth dynamics. Biosystems 55(13):119–127

    Article  Google Scholar 

  69. Kansal AR, Torquato S, Harsh GR IV, Chiocca EA, Deisboeck TS (2000) Simulated brain tumor growth dynamics using a three-dimensional cellular automaton. J Theor Biol 203(4):367–382

    Article  Google Scholar 

  70. Kholodenko BN, Hancock JF, Kolch W (2006) Physicochemical modelling of cell signalling pathways. Nat Cell Biol 8(11):1195

    Article  Google Scholar 

  71. Kihara T, Ito J, Miyake J (2013) Measurement of biomolecular diffusion in extracellular matrix condensed by fibroblasts using fluorescence correlation spectroscopy. PloS One 8(11):e82,382

    Article  Google Scholar 

  72. Kirk BS, Peterson JW, Stogner RH, Carey GF (2006) libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations. Eng Comput 22(3–4):237–254

    Article  Google Scholar 

  73. Knowles MA, Selby PJ (2005) Introduction to the cellular and molecular biology of cancer, 4th edn. Oxford University Press, Oxford

    Google Scholar 

  74. Le Maître OP, Knio OM (2010) Spectral methods for uncertainty quantification. Springer, Berlin

    Book  MATH  Google Scholar 

  75. Lima EABF, Almeida RC (2011) A comparative study of some fem mixed formulations for the 1D Cahn–Hilliard equation. In: CILAMCE XXXII

  76. Lima EABF, Almeida RC, Oden JT (2015) Analysis and numerical solution of stochastic phase-field models of tumor growth. Numer Methods Part Differ Equ 31(2):552–574

    Article  MathSciNet  MATH  Google Scholar 

  77. Lima EABF, Oden JT, Almeida RC (2014) A hybrid ten-species phase-field model of tumor growth. Math Models Methods Appl Sci 24(13):2569–2599

    Article  MathSciNet  MATH  Google Scholar 

  78. Liotta LA, Saidel GM, Kleinerman J (1976) Stochastic model of mestastases formation. Biometrics 32:535–550

    Article  MathSciNet  MATH  Google Scholar 

  79. Liu F, Bayarri MJ, Berger JO, Paulo R, Sacks J (2008) A Bayesian analysis of the thermal challenge problem. Comput Methods Appl Mech Eng 197:2457–2466

    Article  MATH  Google Scholar 

  80. Liu WK, Karpov EG, Park HS (2006) Nano mechanics and materials: theory, multiscale methods and applications. Wiley, New York

    Book  Google Scholar 

  81. Lo CF (2007) Stochastic Gompertz model of tumour cell growth. J Theor Biol 248:317–321

    Article  MathSciNet  Google Scholar 

  82. Lowengrub JS, Frieboes HB, Jin F, Chuang Y, Li X, Macklin P, Wise S, Cristini V (2010) Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity 23(1):R1

    Article  MathSciNet  MATH  Google Scholar 

  83. Macklin P, Edgerton ME, Thompson AM, Cristini V (2012) 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

    Article  MathSciNet  Google Scholar 

  84. Macklin P, Mumenthaler S, Lowengrub J (2013) Modeling multiscale necrotic and calcified tissue biomechanics in cancer patients: application to ductal carcinoma in situ (DCIS). In: Gefen A (ed) Multiscale computer modeling in biomechanics and biomedical engineering, chap 13. Springer, Berlin

    Google Scholar 

  85. Mallet DG, De Pillis LG (2006) A cellular automata model of tumor–immune system interactions. J Theor Biol 239:334–350

    Article  MathSciNet  Google Scholar 

  86. Mantzaris N, Webb S, Othmer HG (2004) Mathematical modeling of tumor-induced angiogenesis. J Math Biol 49:111–187

    Article  MathSciNet  MATH  Google Scholar 

  87. Martin EA (ed) (2010) A dictionary of science, 6th edn. Oxford University Press, Oxford

    Google Scholar 

  88. Menze BH, Jakab A, Bauer S, Kalpathy-Cramer J, Farahani K et al (2014) The multimodal brain tumor image segmentation benchmark (BRATS). IEEE Trans Med Imaging 1–32

  89. Milde F, Bergdorf M, Koumoutsakos P (2008) A hybrid model for three-dimensional simulations of sprouting angiogenesis. Biophys J 95:3146–3160

    Article  Google Scholar 

  90. Murphy KP (2012) Machine learning: a probabilistic perspective. MIT Press, Cambridge

    MATH  Google Scholar 

  91. Najafi A, Bidkhori G, Bozorgmehr JH, Koch I, Masoudi-Nejad A (2014) Genome scale modeling in systems biology: algorithms and resources. Curr Genomics 15:130–159

    Article  Google Scholar 

  92. National Cancer Institute (2012) Dictionary of cancer terms. http://www.cancer.gov/dictionary

  93. Naumov L, Hoekstra A, Sloot P (2011) Cellular automata models of tumour natural shrinkage. Phys A Stat Mech Appl 390(12):2283–2290

    Article  Google Scholar 

  94. Naumov L, Hoekstra A, Sloot P (2012) The influence of mitoses rate on growth dynamics of a cellular automata model of tumour growth. Proc Comput Sci 1:971–978

    Article  Google Scholar 

  95. Niemisto A, Dunmire V, Yli-Harja O, Zhang W, Shmulevich I (2005) Analysis of angiogenesis using in vitro experiments and stochastic growth models. Phys Rev E 72: 062902-1–062902-4

  96. Norton K, Wininger M, Bhanot G, Ganesan S, Barnard N, Shinbrot T (2010) A 2D mechanistic model of breast ductal carcinoma in situ (DCIS) morphology and progression. J Theor Biol 263(4):393–406

    Article  Google Scholar 

  97. Oden J, Strouboulis T, Devloo P (1986) Adaptive finite element methods for the analysis of inviscid compressible flow: part I. Fast refinement/unrefinement and moving mesh methods for unstructured meshes. Comput Methods Appl Mech Eng 59(3):327–362

    Article  MathSciNet  MATH  Google Scholar 

  98. Oden JT, Hawkins A, Prudhomme S (2010) General diffuse-interface theories and an approach to predictive tumor growth modeling. Math Models Methods Appl Sci 20(3):477–517

    Article  MathSciNet  MATH  Google Scholar 

  99. Oden JT, Prudencio EE, Hawkins-Daarud A (2013) Selection and assessment of phenomenological models of tumor growth. Math Models Methods Appl Sci 23(07):1309–1338

    Article  MathSciNet  MATH  Google Scholar 

  100. Piotrowska MJ, Angus SD (2009) A quantitative cellular automaton model of in vitro multicellular spheroid tumour growth. J Theor Biol 258(2):165–178

    Article  Google Scholar 

  101. Preziosi L (2003) Cancer modelling and simulation, 1st edn. CRC Press, Boca Raton

    Book  MATH  Google Scholar 

  102. Prudencio E, Cheung SH (2012) Parallel adaptive multilevel sampling algorithms for the bayesian analysis of mathematical models. Int J Uncertain Quantif 2(3):215–237

    Article  MathSciNet  MATH  Google Scholar 

  103. Prudencio EE, Bauman PT, Faghihi D, Ravi-Chandar K, Oden JT (2014) A computational framework for dynamic data-driven material damage control, based on bayesian inference and model selection. Int J Numer Methods Eng 102:379–403

  104. Prudencio EE, Cheung SH, Oliver T, Schulz K (2010) The parallel C++ statistical library ‘QUESO’: quantification of uncertainty for estimation, simulation and optimization (in preparation). Springer, New York

  105. Pyrz M, Baish J (2013) Effect of tumor heterogeneity on interstitial pressure and fluid flow. In: ASME 2013 summer bioengineering conference. American Society of Mechanical Engineers, Sunriver, Oregon, p V01AT07A004

  106. Quaranta V, Weaver AM, Cummings PT, Anderson ARA (2005) Mathematical modeling of cancer: the future of prognosis and treatment. Clin Chim Acta 357:173–179

    Article  Google Scholar 

  107. Ramis-Conde I, Chaplain MA, Anderson A (2008) Mathematical modelling of cancer cell invasion of tissue. Math Comput Model 47(56):533–545 (Towards a mathematical description of cancer: analytical, numerical and modelling aspects)

    Article  MathSciNet  MATH  Google Scholar 

  108. Ricken T, Schwarz A, Bluhm J (2007) A triphasic model of transversely isotropic biological tissue with applications to stress and biologically induced growth. Compu Mater Sci 39(1):124–136

    Article  Google Scholar 

  109. Rocha HL, Lima EABF, Almeida RC (2015) An agent based model of the avascular tumor growth. Congresso Latino Americano de Biomatemática—SOLABIMA (in Portuguese)

  110. Roniotis A, Marias K, Sakkalis V, Tsibidis GD, Zervakis M (2009) A complete mathematical study of a 3D model of heterogeneous and anisotropic glioma evolution. In: Annual international conference of the IEEE Engineering in Medicine and Biology Society (EMBC 2009). IEEE, pp 2807–2810

  111. Roose T, Netti PA, Munn LL, Boucher Y, Jain RK (2003) Solid stress generated by spheroid growth estimated using a linear poroelasticity model. Microvasc Res 66(3):204–212

    Article  Google Scholar 

  112. Rudd RE, Broughton JQ (1998) Coarse-grained molecular dynamics and the atomic limit of finite elements. Phys Rev B 58(10):R5893–R5896

    Article  Google Scholar 

  113. Rybinski M (2008) Analysis of mathematical models of signalling pathways. Master thesis, Uniwersytet Warszawski

  114. Saltelli A, Chan K, Scott E (2009) Sensitivity analysis no. 2008 in Wiley paperback series. Wiley

  115. Schnell S, Grima R, Maini P (2007) Multiscale modeling in biology new insights into cancer illustrate how mathematical tools are enhancing the understanding of life from the smallest scale to the grandest. Am Sci 95(2):134–142

    Article  Google Scholar 

  116. Seyfried TN, Flores R, Poff AM, DAgostino DP, Mukherjee P (2015) Metabolic therapy: a new paradigm for managing malignant brain cancer. Cancer Lett 356(2, Part A):289–300

    Article  Google Scholar 

  117. Shilkrot L, Miller R, Curtin W (2002) Coupled atomistic and discrete dislocation plasticity. Phys Rev Lett 89(2):025,501

    Article  Google Scholar 

  118. Shilkrot L, Miller RE, Curtin WA (2004) Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics. J Mech Phys Solids 52(4):755–787

    Article  MathSciNet  MATH  Google Scholar 

  119. Shirazi AS (2011) Hierarchical self-organized learning in agent-based modeling of the MAPK signaling pathway. In: IEEE congress on evolutionary computation (CEC). IEEE, New Orleans, pp 2245–2251

  120. Shrestha SMB, Joldes G, Wittek A, Miller K (2014) Modeling three-dimensional avascular tumor growth using lattice gas cellular automata. In: Computational biomechanics for medicine. Springer, Berlin, pp 15–26

  121. Sobol’ IM (1990) Sensitivity estimates for nonlinear mathematical models. Matem Model 2:112–118

    MathSciNet  MATH  Google Scholar 

  122. Sobol’ IM (1993) Sensitivity analysis for non-linear mathematical models. Math Model Comput Exp 1:407–414

    MathSciNet  MATH  Google Scholar 

  123. Stokes CL, Lauffenburger DA (1991) Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. J Theor Biol 152:377–403

    Article  Google Scholar 

  124. Stylianopoulos T, Martin JD, Chauhan VP, Jain SR, Diop-Frimpong B, Bardeesy N, Smith BL, Ferrone CR, Hornicek FJ, Boucher Y et al (2012) Causes, consequences, and remedies for growth-induced solid stress in murine and human tumors. Proc Natl Acad Sci 109(38):15101–15108

    Article  Google Scholar 

  125. Stylianopoulos T, Martin JD, Snuderl M, Mpekris F, Jain SR, Jain RK (2013) Coevolution of solid stress and interstitial fluid pressure in tumors during progression: implications for vascular collapse. Cancer Res 73(13):3833–3841

    Article  Google Scholar 

  126. Sun S, Wheeler MF, Obeyesekere M, Patrick CW Jr (2005) A deterministic model of growth factor-induced angiogenesis. Bull Math Biol 67:313–337

    Article  MathSciNet  MATH  Google Scholar 

  127. Sun S, Wheeler MF, Obeyesekere M, Patrick CW Jr (2005) Nonlinear behaviors of cappilary formation in a deterministic angiogenesis model. Nonlinear Anal 63:e2237–e2246

    Article  MATH  Google Scholar 

  128. Sunyk R, Steinmann P (2003) On higher gradients in continuum-atomistic modelling. Int J Solids Struct 40(24):6877–6896

    Article  MATH  Google Scholar 

  129. Swartz MA, Lund AW (2012) Lymphatic and interstitial flow in the tumour microenvironment: linking mechanobiology with immunity. Nat Rev Cancer 12(3):210–219

    Article  Google Scholar 

  130. Szot CS, Buchanan CF, Freeman JW, Rylander MN (2011) Collagen 1 hydrogels as a platform for in vitro solid tumor development. Biomaterials 32(32):7905–7912

    Article  Google Scholar 

  131. Szot CS, Buchanan CF, Freeman JW, Rylander MN (2013) In vitro angiogenesis induced by tumor-endothelial cell co-culture in bilayered, collagen I hydrogel bioengineered tumors. Tissue Eng Part C 19(11):864–874

    Article  Google Scholar 

  132. TACC (Texas Advanced Computing Center) (2008–2015). http://www.tacc.utexas.edu/

  133. Tadmor EB, Ortiz M, Phillips R (1996) Quasicontinuum analysis of defects in solids. Philos Mag A 73(6):1529–1563

    Article  Google Scholar 

  134. Tan SM, Fox C, Nicholls G (2005) Lecture notes on inverse problems. Physics 707:1–184

  135. Tan WY, Chen CW (1998) Stochastic modeling of carcinogenesis: some new insights. Math Comput Model 28:49–71

    Article  MathSciNet  MATH  Google Scholar 

  136. Tang L, van de Ven AL, Guo D, Andasari V, Cristini V, Li KC, Zhou X (2014) Computational modeling of 3D tumor growth and angiogenesis for chemotherapy evaluation. PloS One 9(1):e83,962

    Article  Google Scholar 

  137. Tarantola A (2005) Inverse problem theory and methods for model parameter estimation. SIAM

  138. Thomas R (1973) Boolean formalization of genetic control circuits. J Theor Biol 42:563–585

    Article  Google Scholar 

  139. Tierra G, Guillén-González F (2015) Numerical methods for solving the Cahn–Hlliard equation and its applicability to related energy-based models. Arch Comput Methods Eng 22(2):269–289

    Article  MathSciNet  Google Scholar 

  140. Travasso RDM, Poire EC, Castro M, Rodriguez-Manzaneque JC, Hernandez-Machado A (2011) Tumor angiogenesis and vascular patterning: a mathematical model. PLoS One 6(5):1–10

  141. Tustison N, Wintermark M, Durst C, Avants B (2013) Ants andarboles. Multimodal Brain Tumor Segm 47:47–50

  142. Voutouri C, Mpekris F, Papageorgis P, Odysseos AD, Stylianopoulos T (2014) Role of constitutive behavior and tumor–host mechanical interactions in the state of stress and growth of solid tumors. PloS One 9(8):e104717

    Article  Google Scholar 

  143. Van der Giessen E, Needleman A (1995) Discrete dislocation plasticity: a simple planar model. Model Simul Mater Sci Eng 3(5):689

    Article  Google Scholar 

  144. Wallace DC (2005) A mitochondrial paradigm of metabolic and degenerative diseases, aging, and cancer: a dawn for evolutionary medicine. Annu Rev Genet 39:359–407

    Article  Google Scholar 

  145. Wang R, Saadatpour A, Albert R (2012) Boolean modeling in systems biology: an overview of methodology and applications. Phys Biol 9:055001

    Article  Google Scholar 

  146. Wang Z, Birc CM, Zhang L, Sagotsky J, Deisboeck TS (2009) Cross-scale, cross-pathway evaluation using an agent-based non-small cell lung cancer model. Bioinformatics 25(18):23892396

    Article  Google Scholar 

  147. Wang Z, Birch CM, Deisboeck TS (2008) Cross-scale sensitivity analysis of a non-small cell lung cancer model: linking molecular signaling properties to cellular behavior. Biosystems 92(3):249–258

    Article  Google Scholar 

  148. Wang Z, Zhang L, Sagotsky J, Deisboeck TS (2007) Simulating non-small cell lung cancer with a multiscale agent-based model. Theor Biol Med Model 4:50

    Article  Google Scholar 

  149. Winkler R (2003) An introduction to bayesian inference and decision. Probabilistic Publishing, Sugar Land, Texas

  150. Wirtz D, Konstantopoulos K, Searson PC (2011) The physics of cancer: the role of physical interactions and mechanical forces in metastasis. Nat Rev Cancer 11(7):512–522

    Article  Google Scholar 

  151. Wise SM, Lowengrub JS, Cristini V (2011) An adaptive multigrid algorithm for simulating solid tumor growth using mixture models. Math Comput Model 53:1–20

    Article  MathSciNet  MATH  Google Scholar 

  152. Wise SM, Lowengrub JS, Frieboes H, Cristini V (2008) Three-dimensional multispecies nonlinear tumor growth—I: model and numerical method. J Theor Biol 253:524–543

    Article  MathSciNet  Google Scholar 

  153. Wu M, Frieboes HB, McDougall SR, Chaplain MA, Cristini V, Lowengrub J (2013) The effect of interstitial pressure on tumor growth: coupling with the blood and lymphatic vascular systems. J Theor Biol 320:131–151

    Article  MathSciNet  Google Scholar 

  154. Xiao S, Belytschko T (2004) A bridging domain method for coupling continua with molecular dynamics. Comput Methods Appl Mech Eng 193(17):1645–1669

    Article  MathSciNet  MATH  Google Scholar 

  155. Xiu D (2010) Numerical methods for stochastic computations: a spectral method approach. Princeton University Press, Princeton

    MATH  Google Scholar 

  156. Zhang S, Khare R, Lu Q, Belytschko T (2007) A bridging domain and strain computation method for coupled atomistic–continuum modelling of solids. Int J Numer Methods Eng 70(8):913–933

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

Oden and Lima acknowledge support of their work by the National Science Foundation under Grant DMS 1115865. Almeida acknowledges support of the National Counsel of Technological and Scientific Development (CNPq) 248622/2013-7 and support of ICES as a Visiting Scholar. Feng, Rahman and Zhou acknowledge support of their work by the National Science Foundation under Grant HRD 0932339. Feng acknowledges support of the National Institute of Health Grant K25 CA942367. Rylander acknowledges the support of the National Science Foundation Early CAREER Award CBET 0955072 and National Institute of Health Grant IR21CA158454-01A1. Fuentes acknowledges support of the ODonnell Foundation and National Institute of Health DP2OD007044-01S1.

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Authors and Affiliations

  1. Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, 201 East 24th St, POB 4.102, Austin, TX, 78712, USA

    J. Tinsley Oden, Ernesto A. B. F. Lima, Regina C. Almeida, Marissa Nichole Rylander, Danial Faghihi & J. Cliff Zhou

  2. National Laboratory for Scientific Computing (LNCC), Avenida Getúlio Vargas 333, Petrópolis, Brazil

    Regina C. Almeida

  3. Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX, USA

    Marissa Nichole Rylander

  4. Center for Simulation, Visualization and Real-Time Prediction (SiViRT), One UTSA Circle, AET 2.332, The University of Texas at San Antonio, San Antonio, TX, 78249, USA

    Yusheng Feng & Mohammad M. Rahman

  5. Department of Imaging Physics, The University of Texas MD Anderson Cancer Center, 1515 Holcombe Blvd., Unit 1902, Houston, TX, 77030, USA

    David Fuentes

  6. Virginia Tech - Wake Forest School of Biomedical Engineering and Sciences, Virginia Tech, 340 Kelly Hall, Blacksburg, VA, 24061, USA

    Matthew DeWitt

  7. Department of Biomedical Engineering, The University of Texas at Austin, 107 W. Dean Keeton, BME Building, Austin, TX, 78712, USA

    Marissa Nichole Rylander & Manasa Gadde

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  1. J. Tinsley Oden
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  2. Ernesto A. B. F. Lima
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Oden, J.T., Lima, E.A.B.F., Almeida, R.C. et al. Toward Predictive Multiscale Modeling of Vascular Tumor Growth. Arch Computat Methods Eng 23, 735–779 (2016). https://doi.org/10.1007/s11831-015-9156-x

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