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Review
. Jan-Feb 2011;3(1):115-25.
doi: 10.1002/wsbm.102.

Hybrid Models of Tumor Growth

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Free PMC article
Review

Hybrid Models of Tumor Growth

Katarzyna A Rejniak et al. Wiley Interdiscip Rev Syst Biol Med. .
Free PMC article

Abstract

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.

Figures

Figure 1
Figure 1. Reciprocal relation between the numbers 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
Figure 2
Figure 2. Snapshots from simulations of various hybrid models of tumor growth
a) 3D tumor spheroid, simulated by a hybrid cellular automaton; reprinted from [8] with permission from Birkhauser-Verlag. b) Tumor invasion in prostate ducts simulated by a hybrid cellular automaton; Reprint permission requested from [16]. c) 3D tumor spheroid simulated by an agent-based on-lattice model; reprinted from [10] with permission from Springer. d) 3D tumor self-metastatic spheroids simulated by a hybrid cellular automaton; reprinted from [5] with permission from Nature Publishing Group. e) 3D model of ductal carcinoma in situ simulated by a square-grid cellular automaton; reprinted from [18] with permission from Elsevier. f) 2D tumor spheroids simulated by a hexagonal cellular automaton; reprinted from [34] with permission from BioMed Central, the Open Access Publisher. g) 3D vascularized tumor spheroid simulated by Potts model; reprinted from [46] with permission from Public Library of Science, open access article. h) 2D tumor spheroid in a heterogeneous environment composed on ECM fibers simulated by Potts model; reprinted from [42] with permission from Elsevier. i) 2D model of colorectal tumor simulated using the particle model with Voronoi triangulation; reprinted from [61] with permission from John Wiley and Sons. j) 2D tumor spheroid modeled using the cell-centered off-lattice model; reprinted from [50] with permission from Springer. k) 2D hybrid model of tumor growth simulated by particle center-based ellipsoid model; reprinted from [56] with permission from World Scientific. l) 2D multiclonal tumor growth simulated by a model of deformable fluid-based cells; reprinted from [69] with permission from Birkhauser-Verlag.
Figure 3
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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