Table 2.

Primary mathematical approaches and computational models for simulating biological systems.

Method/Modela	Definition	Application/Simulation
Agent-based model (ABM)b	Model contains autonomous decision-making entities termed agents. Each agent makes behavioral decisions individually based on pre-defined probabilistic rules considering agent internal state, surrounding agents, and the environment.	(1) Simulate many interactions at the individual level to uncover emergent behavior at the whole-population level; (2) Stochastic gene expression [94]; (3) Tissue formation and morphogenesis [95]; (4) Mammary stem cell subpopulation dynamics [96]; (5) Inflammation pathways [97,98]; (6) Immune system dynamics [99,100]; (7) Tumor models [101]; (8) Cell migration [102,103]; (9) Chemotaxis [104,105]
Boolean network	A discrete set of boolean variables which can be presented by a graph of linked nodes	(1) Gene regulatory networks [[106], [107], [108]]; (2) Interaction between pathogens and different cytokines [109]
Cellular automatonc	A discrete spatio-temporally extended dynamical system that includes identically programmed automaton cells and interacting units with a finite number of discrete states.	(1) 3D multicellular tissue growth [110]; (2) Cell behaviors and activities (e.g., neurons and fibroblasts); (3) Cardiac model [111]; (4) Tissue growth [112]
Cellular Potts model (CPM)	A lattice-based model where a cell can be described using a cluster of points, allowing capture of cell shape changes. The movement of the points/cluster around the lattice is governed by calculation of force equations and energy of the system therefore considering cell state and the grid environment.	(1) Simulate individual and collective cell behavior [[113], [114], [115], [116]]; (2) Tissue morphogenesis [113,117]; (3) Cancer development [118]; (4) Chemotaxis [114]; (5) Vasculogenesis and angiogenesis [119]
Differential equation modeld	Model containing differential equations that relates functions and their derivatives, to describe the dynamic aspects of biosystems. Inclusion of partial-differential equations can include some spatially dependent characteristics such as diffusion.	(1) Cancer diseases biology [120,121]; (2) Immunology and immunotherapy [[122], [123], [124], [125], [126], [127], [128]]; (3) Virus infection [129]; (4) Pharmacodynamics [130]; (5) Organ disease [131]; (6) Vascular network [132]
Hybrid multiscale model	Integrative model combining several kinds of computational models.	(1) Vascular network [[133], [134], [135]]; (2) Cancer and therapy [113,[136], [137], [138], [139]]; (3) Disease model [140]; (4) Immune system (cell and organ) [141]
Lattice modele	Model established on a 2D/3D lattice (or grid), as opposed to the continuum of spatial or spatio-temporal coordinates (off-lattice model).	(1) Predict protein structure [142]; (2) Tissue differentiation [143]; (3) Cell migration [102,116]; (4) Tumor growth [144]
Petri net formalism	Directed bipartite graph with two types of elements- places and transitions, to describe discrete-event dynamical systems.	(1) System biology [[145], [146], [147]]; (2) Gene network [148]

^aThe open source toolkits of computational approaches can be found in Ref. [149].

^bAgent-based model is also called individual-based model in some fields [150].

^cTo some extent, cellular automaton model is an agent-based model with finite grids and limited degrees of freedom.

^dOrdinary differential equation and partial differential equation models are the main differential model types used in the biological study.

^eA large number of agent-based models and Cellular-Potts models are established on lattice models.