Hybrid models of tumor growth

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Katarzyna A. Rejniak, Alexander R. A. Anderson

Published Online: Jul 07 2010

DOI: 10.1002/wsbm.102

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Cancer is a complex, multiscale process in which genetic mutations occurring at a subcellular level manifest themselves as functional changes at the cellular and tissue scale. The multiscale nature of cancer requires mathematical modeling approaches that can handle multiple intracellular and extracellular factors acting on different time and space scales. Hybrid models provide a way to integrate both discrete and continuous variables that are used to represent individual cells and concentration or density fields, respectively. Each discrete cell can also be equipped with submodels that drive cell behavior in response to microenvironmental cues. Moreover, the individual cells can interact with one another to form and act as an integrated tissue. Hybrid models form part of a larger class of individual‐based models that can naturally connect with tumor cell biology and allow for the integration of multiple interacting variables both intrinsically and extrinsically and are therefore perfectly suited to a systems biology approach to tumor growth. WIREs Syst Biol Med 2011 3 115–125 DOI: 10.1002/wsbm.102

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Figure 1.

Reciprocal relation between the number of cells handled by the models and the level of included cellular details. In each class (on‐lattice and off‐lattice), the models complexity rises from cells represented by single points to fully deformable bodies.

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Figure 2.

Snapshots from simulations of various hybrid models of tumor growth. (a) Three‐dimensional (3D) tumor spheroid simulated by a hybrid cellular automaton (Reprinted with permission from Ref 12. Copyright 2007 Birkhauser‐Verlag). (b) Tumor invasion in prostate ducts simulated by a hybrid cellular automaton (Reprinted with permission from Ref 25. Copyright 2009 American Association of Cancer Research). (c) 3D tumor spheroid simulated by an agent‐based on‐lattice model (Reprinted with permission from Ref 14. Copyright 2009 Springer). (d) 3D tumor self‐metastatic spheroids simulated by a hybrid cellular automaton (Reprinted with permission from Ref 5. Copyright 2009 Nature Publishing Group). (e) 3D model of ductal carcinoma in situ simulated by a square‐grid cellular automaton (Reprinted with permission from Ref 29. Copyright 2007 Elsevier). (f) Two‐dimensional (2D) tumor spheroids simulated by a hexagonal cellular automaton (Reprinted with permission from Ref 34. Copyright 2008 BioMed Central, the Open Access Publisher). (g) 3D vascularized tumor spheroid simulated by Potts model (Reprinted with permission from Ref 46. Copyright 2009 Public Library of Science, open access article). (h) 2D tumor spheroid in a heterogeneous environment composed on extracellular matrix (ECM) fibers simulated by Potts model (Reprinted with permission from Ref 42. Copyright 2008 Elsevier). (i) 2D model of colorectal tumor simulated using the particle model with Voronoi triangulation (Reprinted with permission from Ref 47. Copyright 2009 John Wiley and Sons, Inc.). (j) 2D tumor spheroid modeled using the cell‐centered off‐lattice model (Reprinted with permission from Ref 48. Copyright 2009 Springer). (k) 2D hybrid model of tumor growth simulated by particle center‐based ellipsoid model (Reprinted with permission from Ref 49. Copyright 2007 World Scientific). (l) 2D multiclonal tumor growth simulated by a model of deformable fluid‐based cells (Reprinted with permission from Ref 50. Copyright 2007 Birkhauser‐Verlag).

[ Normal View 112K | Magnified View 177K ]

Figure 3.

A schematic of modeling scales and techniques. Multiple biological scales can be bridged by various types of mathematical models.

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