Hybrid modeling frameworks of tumor development and treatment

Abstract

Tumors are complex multicellular heterogeneous systems comprised of components that interact with and modify one another. Tumor development depends on multiple factors: intrinsic, such as genetic mutations, altered signaling pathways, or variable receptor expression; and extrinsic, such as differences in nutrient supply, crosstalk with stromal or immune cells, or variable composition of the surrounding extracellular matrix. Tumors are also characterized by high cellular heterogeneity and dynamically changing tumor microenvironments. The complexity increases when this multiscale, multicomponent system is perturbed by anticancer treatments. Modeling such complex systems and predicting how tumors will respond to therapies require mathematical models that can handle various types of information and combine diverse theoretical methods on multiple temporal and spatial scales, that is, hybrid models. In this update, we discuss the progress that has been achieved during the last 10 years in the area of the hybrid modeling of tumors. The classical definition of hybrid models refers to the coupling of discrete descriptions of cells with continuous descriptions of microenvironmental factors. To reflect on the direction that the modeling field has taken, we propose extending the definition of hybrid models to include of coupling two or more different mathematical frameworks. Thus, in addition to discussing recent advances in discrete/continuous modeling, we also discuss how these two mathematical descriptions can be coupled with theoretical frameworks of optimal control, optimization, fluid dynamics, game theory, and machine learning. All these methods will be illustrated with applications to tumor development and various anticancer treatments. This article is characterized under: Analytical and Computational Methods > Computational Methods Translational, Genomic, and Systems Medicine > Therapeutic Methods Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.

Keywords: mathematical modeling; mathematical oncology.

Figure 1

Schematic representation of the components of the classical hybrid models. The discrete components (on‐lattice and off‐lattice) show reciprocal relation between the number of cells handled by each modeling technique and the level of included cellular details. In each class, the model complexity rises from cells represented by single points to fully deformable bodies. The continuous components describe either time‐dependent intracellular molecular kinetics (ODEs) or time‐ and space‐dependent extracellular molecular dynamics (PDEs)

Figure 2

Schematics of hybrid modeling frameworks of tumors. Three classes of models are used for cancer problems: data‐driven, physics‐based, and optimization. Combination of models from any two classes is considered a hybrid modeling framework. The solid arrows represent interactions between different frameworks that are currently modeled. The dotted arrow represents a promising future direction. The classical hybrid models belong to the class of physics‐based models

Figure 3

Summary of the discussed hybrid modeling frameworks. Letter codes denote a section in which the given models were cited: AT, adaptive; CBT, combination; CT, chemotherapy; HT, hormone; IT, immunotherapy; NT, nanotherapy; RT, radiation; TT, targeted therapy and TD, tumor development

Figure 4

Snapshots from simulations of various hybrid frameworks of tumor development and treatment. (a) Fluid‐based model of individual deformable cells (Reprinted with permission from Rejniak (2012). Copyright 2012 Hindawi Publishing Corporation); (b) Potts model combined with PDEs (Reprinted with permission from Szabo and Merks (2017). Copyright 2017 Public Library of Science); (c) image‐based vascular network model with flow and diffusion (Reprinted with permission from Boujelben et al. (2016). Copyright 2016 Royal Society); (d) particle‐spring model for radiation (Reprinted with permission from Kempf, Bleicher, and Meyer‐Hermann (2015). Copyright 2015 Public Library of Science); (e) CA model combined with PDEs (Reprinted with permission from Scott, Fletcher, Anderson, and Maini (2016). Copyright 2016 Public Library of Science); (f) discrete vasculature model combined with PDEs (Reprinted with permission from Chamseddine, Frieboes and Kokkolaras (2018). Copyright 2018 Springer Nature); (g) agent‐based model combined with PDEs (Reprinted with permission from Bloch and Harel (2016). Copyright 2016 Springer Nature); (h) particle‐spring model combined with PDEs (Reprinted with permission from Ghaffarizadeh, Heiland, Friedman, Mumenthaler, and Macklin (2018). Copyright 2018 Public Library of Science); (i) CA model combined with PDEs (Reprinted with permission from Gong et al. (2017). Copyright 2017 Royal Society)