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A 3-D model of tumor progression based on complex automata driven by particle dynamics

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Abstract

The dynamics of a growing tumor involving mechanical remodeling of healthy tissue and vasculature is neglected in most of the existing tumor models. This is due to the lack of efficient computational framework allowing for simulation of mechanical interactions. Meanwhile, just these interactions trigger critical changes in tumor growth dynamics and are responsible for its volumetric and directional progression. We describe here a novel 3-D model of tumor growth, which combines particle dynamics with cellular automata concept. The particles represent both tissue cells and fragments of the vascular network. They interact with their closest neighbors via semi-harmonic central forces simulating mechanical resistance of the cell walls. The particle dynamics is governed by both the Newtonian laws of motion and the cellular automata rules. These rules can represent cell life-cycle and other biological interactions involving smaller spatio-temporal scales. We show that our complex automata, particle based model can reproduce realistic 3-D dynamics of the entire system consisting of the tumor, normal tissue cells, blood vessels and blood flow. It can explain phenomena such as the inward cell motion in avascular tumor, stabilization of tumor growth by the external pressure, tumor vascularization due to the process of angiogenesis, trapping of healthy cells by invading tumor, and influence of external (boundary) conditions on the direction of tumor progression. We conclude that the particle model can serve as a general framework for designing advanced multiscale models of tumor dynamics and it is very competitive to the modeling approaches presented before.

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References

  1. Bellomo N, de Angelis E, Preziosi L (2003) Multiscale modeling and mathematical problems related to tumor evolution and medical therapy. J Theor Med 5:111–136

    Article  Google Scholar 

  2. Gee MS, Procopio WN, Makonnen S, Feldman MD, Yeilding NM, Lee WF (2003) Tumor vessel development and maturation impose limits on the effectiveness of anti-vascular therapy. Am J Pathol 162:183–193

    Google Scholar 

  3. McDougall SR, Anderson ARA, Chaplain MAJ, Sherratt JA (2002) Mathematical modelling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies. Bull Math Biol 64:673–702

    Article  CAS  Google Scholar 

  4. Stéphanou A, McDougall SR, Anderson ARA, Chaplain MAJ, Sherratt JA (2005) Mathematical modelling of flow in 2D and 3D vascular networks: Applications to anti-angiogenic and chemotherapeutic drug strategies. J Math Comput Model 41:1137–1156

    Article  Google Scholar 

  5. Eberhard A, Kahlert S, Goede V, Hemmerlein B, Plate KH, Augustin HG (2000) Heterogeneity of angiogenesis and blood vessel maturation in human tumors: Implications for antiangiogenic tumor therapies. Cancer Res 60:1388–1393

    CAS  Google Scholar 

  6. Folkman J (1971) Tumor angiogenesis: Therapeutic implications. N Engl J Med 285:1182–1186

    CAS  Google Scholar 

  7. Folkman J, Hochberg M (1973) Self-regulation of growth in three dimensions. J Exp Med 138:745–753

    Article  CAS  Google Scholar 

  8. Folkman J, Hochberg M (1973) Self-regulation of growth in three dimensions. J Exp Med 138:745–753

    Article  CAS  Google Scholar 

  9. Ferrara N, Chen H, Smyth DT, Gerber HP, Nguyen TN, Peers D, Chisholm V, Hillan KJ, Schwall RH (1998) Vascular endothelial growth factor is essential for corpus luteum angiogenesis. Nat Med 4:336–340

    Article  CAS  Google Scholar 

  10. Folkman J (2003) Angiogenesis and apoptosis. Semin Cancer Biol 13:159–167

    Article  CAS  Google Scholar 

  11. Asosingh K, de Raeve H, Menu E, van Riet I, van Marck E, van Camp B, Vanderkerken K (2004) Angiogenic switch during 5T2MM murine myeloma tumorigenesis: Role of CD45 heterogeneity. Blood 103:3131–3137

    Article  CAS  Google Scholar 

  12. Chaplain MAJ (2000) Mathematical modelling of angiogenesis. J Neuro-Oncol 50:37–51

    Article  CAS  Google Scholar 

  13. Hellstrom M, Phng L-K, Hofmann JJ, Wallgard E, Coultas L, Lindblom P, Alva J, Nilsson A-K, Karlsson L, Gaiano N, Yoon K, Rossant J, Iruela-Arispe M-L, Kale´n M, Gerhardt H, Betsholtz CH (2007) Dll4 signaling through Notch1 regulates formation of tip cells during angiogenesis. Nature 445:422–425

    Article  Google Scholar 

  14. Nogueratroise I, Daly Ch, Papadopoulos NJ, Coetzee S, Boland P, Gale NW, Lin HC, Yancopoulos GD, Thurston G (2006) Blockade of Dll4 inhibits tumour growth by promoting non-productive angiogenesis. Nature 444:1032–1037

    Article  CAS  Google Scholar 

  15. Mantzaris N, Webb S, Othmer HG (2004) Mathematical Modeling of Tumor-induced Angiogenesis. J Math Biol 49:1432–1416

    Article  Google Scholar 

  16. Preziozi L (ed) (2003) Cancer modelling and simulation. Chapman & Hall/ CRC Mathematical Biology & Medicine 426

  17. Alarcon T, Byrne H, Maini PK (2005) A multiple scale model for tumor growth. Multiscale Model Simul 3:440–475

    Article  CAS  Google Scholar 

  18. Lee D-S, Rieger H, Bartha K (2006) Flow correlated percolation during vascular remodeling in growing tumors. Phys Rev Let 96:058104–1-4

    Article  Google Scholar 

  19. Rieger H, Bartha K (2006) Vascular network remodeling via vessel cooption, regression and growth in tumors. J Theor Biol 241:903–918

    Google Scholar 

  20. Carmeliet P, Jain RK (2000) Angiogenesis in cancer and other diseases. Nature 407:249–257

    Article  CAS  Google Scholar 

  21. 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  CAS  Google Scholar 

  22. Darland DC, Massingham LJ, Smith SR, Piek E, Saint-Geniez M, D’Amore PA (2003) Pericyte production of cell-associated VEGF is differentiation-dependent and is associated with endothelial survival. Dev Biol 264:275–288

    Article  CAS  Google Scholar 

  23. Nehls V, Denzer K, Drenckhahn D (1992) Pericyte involvement in capillary sprouting during angiogenesis in situ. Cell Tissue Res 270:469–474

    Article  CAS  Google Scholar 

  24. Amyot F, Small A, Gandjbakhche AH (2006) Stochastic modeling of tumor induced angiogenesis in a heterogeneous medium, the extracellular matrix. In: Proc 28th IEEE EMBS Annual International Conference New York City, USA, 30 Aug- 3 Sept 2006

  25. Milde F, Bergdorf M, Koumoutsakos PA (2008) Hybrid model of sprouting angiogenesis. Lect Notes Comp Sci 5102:167–176

    Article  Google Scholar 

  26. Godde R, Kurz H (2001) Structural and biophysical simulation of angiogenesis and vascular remodeling. Dev Dyn 220:387–401

    Article  CAS  Google Scholar 

  27. De Angelis E, Preziosi L (2000) Advection-diffusion models for solid tumour evolution in vivo and related free boundary problem. Math Mod Meth Appl Sci 10:379–407

    Google Scholar 

  28. Stein AM, Demuth T, Mobley D, Berens M, Sander LMA (2007) Mathematical model of glioblastoma tumor spheroid invasion in a three-dimensional in vitro experiment. Biophys J 92:356–365

    Article  CAS  Google Scholar 

  29. Stéphanou A, McDougall SR, Anderson ARA, Chaplain MAJ (2006) Mathematical modelling of the influence of blood rheological properties upon adaptative tumour-induced angiogenesis. Math Comput Model 44:96–123

    Article  Google Scholar 

  30. Greenspan HP (1976) On the growth and stability of cell cultures and solid tumours. J Theor Biol 56:229–242

    Article  CAS  Google Scholar 

  31. Luo S, Nie Y (2004) FEM-based simulation of tumor growth in medical image. Medical Imaging 2004:Visualization, Image Guided Procedures, and Display. In: Galloway RL (ed) Proceedings of SPIE 5367:600-608

  32. Szczerba D, Lloyd BA, Bajka M, Szekely GA (2008) Multiphysics model of Myoma growth. Lect Notes Comput Sci 5102:187–196

    Article  Google Scholar 

  33. Cavalcante FSA, Moreira AA, Costa UMS, Andrade JS Jr (2002) Self-organized percolation growth in regular and disordered lattices. Stat Mech Appl Phys A 311:313–319

    Article  Google Scholar 

  34. Bauer AL, Jackson TL, Jiang YA (2007) Cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis. Biophys J 92:3105–3121

    Article  CAS  Google Scholar 

  35. Dormann S, Deutsch A (2002) Modeling of self-organized avascular tumor growth with a hybrid cellular automaton. In Silico Biol 2:393–406

    CAS  Google Scholar 

  36. Moreira J, Deutsch A (2002) Cellular automaton models of tumor development: A critical review. Adv Complex Syst 5:247–269

    Article  Google Scholar 

  37. Topa P (2008) Dynamically reorganising vascular networks modelled using cellular automata approach. Lect Notes Comput Sci LNCS 5191:494–499

    Article  Google Scholar 

  38. Wcisło R, Dzwinel W (2008) Particle based model of tumor progression stimulated by the process of angiogenesis. Lect Notes Comput Sci ICCS 2008 LNCS 5102:177–186

    Article  Google Scholar 

  39. Hockel M, Vaupel P (2001) Tumor hypoxia: Definitions and current clinical, biologic, and molecular aspects. J Natl Cancer Inst 93:266–276

    Article  CAS  Google Scholar 

  40. Dzwinel W, Alda W, Yuen DA (1999) Cross-scale numerical simulations using discrete-particle models. Mol Simul 22:397–418

    Article  CAS  Google Scholar 

  41. Dzwinel W, Alda W, Pogoda M, Yuen DA (2000) Turbulent mixing in the microscale. Phys D 137:157–171

    Article  CAS  Google Scholar 

  42. Dzwinel W, Boryczko K, Yuen DA (2003) A discrete-particle model of blood dynamics in capillary vessels. J Colloid Int Sci 258:163–173

    Article  CAS  Google Scholar 

  43. Dzwinel W, Yuen DA, Boryczko K (2006) Bridging diverse physical scales with the discrete-particle paradigm in modeling colloidal dynamics with mesoscopic features. Chem Eng Sci 61:2169–2185

    Article  CAS  Google Scholar 

  44. Hoekstra AG, Lorenz E, Falcone LC, Chopard B (2007) Towards a complex automata framework for multi-scale modeling: Formalism and the scale separation map. Lect Notes Comput Sci 4487:1611–3349

    Google Scholar 

  45. Kadau K, Germann TC, Lomdahl PS (2004) Large-scale molecular-dynamics simulation of 19 billion particles. Int J Mod Phys C 15:193–201

    Article  CAS  Google Scholar 

  46. Hoogerbrugge PJ, Koelman JMVA (1992) Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Europhys Let 19:155–160

    Article  Google Scholar 

  47. Español P (1998) Fluid particle model. Phys Rev E 57:2930–2948

    Article  Google Scholar 

  48. Monaghan JJ (1992) Smoothed particle hydrodynamics. Annu Rev Astronomy Astrophys 30:543–574

    Article  Google Scholar 

  49. Dzwinel W, Boryczko K, Yuen DA (2006) Modeling Mesoscopic Fluids with Discrete-Particles. Methods, Algorithms and Results. In: Spasic AM, Hsu JP (eds) Finely Dispersed Particles: Micro-, Nano-, and Atto-Engineering. Taylor&Francis, CRC Press, pp 715-778

  50. Haile PM (1992) Molecular Dynamics Simulation. Wiley&Sons, New York

    Google Scholar 

  51. Vaupel P, Kallinowski F, Okunieff P (1989) Blood Flow, Oxygen and Nutrient Supply, and Metabolic Microenvironment of Human Tumors: A Review. Cancer Res 49:6449–6465

    CAS  Google Scholar 

  52. Muller M, Charypar D, Gross M (2003) Particle-Based Fluid Simulation for Interactive Applications. In: Proceedings of Eurographics/SIGGRAPH Symposium on Computer Animation. San Diego 27-31 July 2003:154-372

  53. Grote J, Suskind R, Vaupel P (1977) Oxygen diffusivity in tumor tissue (DS-Carcinosarcoma) under temperature conditions within the range of 20–40C. Pflugers Arch 372:37–42

    Article  CAS  Google Scholar 

  54. Maxwell PH, Ratcliff PJ (2002) Oxygen sensors and angiogenesis. Semin Cell Dev Biol 13:29–37

    Article  CAS  Google Scholar 

  55. Moulder JE, Rockwell S (1984) Hypoxic fractions of solid tumors: experimental techniques, methods of analysis, and a survey of existing data. Int J Radiat Oncol Biol Phys 10:695–712

    CAS  Google Scholar 

  56. Filho IPT, Leunigt M, Yuant F, Intaglietta M, Jaint RK (1994) Noninvasive measurement of microvascular and interstitial oxygen profiles in a human tumor in SCID mice. Proc Nad Acad Sci USA 91:2081–2085

    Article  Google Scholar 

  57. Gridley T (2007) Vessel guidance. Nature 445:722–723

    Article  CAS  Google Scholar 

  58. Dorie MJ, Kallman RF, Rapacchietta DF, Van Antwerp D, Huang YR (1982) Migration and internalization of cells and polystyrene microsphere in tumor cell spheroids. Exp Cell Res 141:201–209

    Article  CAS  Google Scholar 

  59. Newman MEJ (2003) The structure and function of complex networks. SIAM Review 45:167–256

    Article  Google Scholar 

  60. Boryczko K, Dzwinel W, Yuen DA (2002) Parallel implementation of the fluid particle model for simulating complex fluids in the mesoscale. Concurrency and Computation: Practice and Experience 14:1–25

    Article  Google Scholar 

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Acknowledgments

This research is financed by the Polish Ministry of Education and Science, Project No.3 T11F 010 30, internal AGH Institute of Computer Science grant and Vlab project of National Science Foundation. The exemplar movies from simulations can be downloaded from http://www.icsr.agh.edu.pl/∼wcislo/Angiogeneza/index.html.

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Authors and Affiliations

  1. Institute of Computer Science, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059, Krakow, Poland

    Rafał Wcisło & Witold Dzwinel

  2. Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, MN, 55455-0219, USA

    David A. Yuen

  3. Division of Hematology, Oncology, and Transplantation, Department of Medicine, University of Minnesota Medical School, Minneapolis, MN, 55455-0219, USA

    Arkadiusz Z. Dudek

Authors
  1. Rafał Wcisło
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  2. Witold Dzwinel
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  4. Arkadiusz Z. Dudek
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Correspondence to Witold Dzwinel.

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Wcisło, R., Dzwinel, W., Yuen, D.A. et al. A 3-D model of tumor progression based on complex automata driven by particle dynamics. J Mol Model 15, 1517–1539 (2009). https://doi.org/10.1007/s00894-009-0511-4

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  • DOI: https://doi.org/10.1007/s00894-009-0511-4

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