Strategies for efficient numerical implementation of hybrid multi-scale agent-based models to describe biological systems

Abstract

Biologically related processes operate across multiple spatiotemporal scales. For computational modeling methodologies to mimic this biological complexity, individual scale models must be linked in ways that allow for dynamic exchange of information across scales. A powerful methodology is to combine a discrete modeling approach, agent-based models (ABMs), with continuum models to form hybrid models. Hybrid multi-scale ABMs have been used to simulate emergent responses of biological systems. Here, we review two aspects of hybrid multi-scale ABMs: linking individual scale models and efficiently solving the resulting model. We discuss the computational choices associated with aspects of linking individual scale models while simultaneously maintaining model tractability. We demonstrate implementations of existing numerical methods in the context of hybrid multi-scale ABMs. Using an example model describing Mycobacterium tuberculosis infection, we show relative computational speeds of various combinations of numerical methods. Efficient linking and solution of hybrid multi-scale ABMs is key to model portability, modularity, and their use in understanding biological phenomena at a systems level.

Keywords: Agent-Based Modeling; Hybrid Modeling; Linking Models; Multi-Scale Modeling; Numerical Implementation; Tuneable Resolution.

Figure 1. Considerations for building multi-scale models

(1) Constructing models: how to create mathematical formulations that accurately represent individual scale dynamics of a biological system, (2) Linking models: how to connect mathematical formulations of individual scale models to create multi-scale models, (3) Solving models: implementing efficient methods to solve multi-scale models, and (4) Analyzing models: Understanding and translating model predictions. Model analysis commonly iterates back to model construction in order to include new biological mechanisms of interest/relevance. In this work, we focus on how to link individual scale models and efficiently solve the resultant multi-scale model.

Figure 2. Mathematical representations of biological processes acting across different spatiotemporal scales

Individual scale models are combined with exchange of information across scales. (1) An agent-based model (ABM) represents tissue and cellular scales (e.g. cell activation states). (2) Ordinary differential equation models represent molecular scale behaviors associated with cells (e.g. receptor-ligand trafficking and intracellular signaling). (3) Partial differential equation models represent molecular scale behaviors of the environment (e.g. extracellular molecule diffusion). Together these integrated individual scale models form the basis of a hybrid multi-scale agent-based model.

Figure 3. Example of how information is exchanged across scales in a hybrid multi-scale agent-based model

Extracellular molecules in the environment (with diffusion and degradation described using partial differential equations) interact with agents through agent-associated reactions (ordinary differential equations). Based on relative levels of agent-associated species (species A, green and species B, blue) agents make different decisions: (1) if both species A and B are above specified thresholds the agent will die, (2) if only species A (green) is above the specified threshold the agent will proliferate, (3) if only species B (blue) is above the specified threshold the agent will change state, (4) if both species A and B are below specified thresholds the agent will be quiescent. Agent decision logic using thresholds is only one example of how agent-associated reactions can be linked to various dynamics. Other examples include Poisson processes based on agent-associated quantities and rate of change of agent-associated species ^,,. Figure style partially adapted from .

Figure 4. Operator splitting algorithms

The top panel represents Lie Splitting, where each operator (Θ₁, Θ₂, and Θ₃) is advanced in time one after the other. The bottom panel represents Strang splitting, where one operator (Θ₂) is advanced halfway in time, followed by the other operators being advanced all the way in time (Θ₁ and Θ₃), and then the first operator (Θ₂) is advanced another half-step in time.

Figure 5. Model layers and discretization

Implementation of multiple layers holding different types of information (extracellular molecule concentration, L) discretized into grid of spacing Δx and Δy represented by i,j coordinates. (A) The environment layer represents the extracellular space of the hybrid agent-based model and holds extracellular molecule concentrations. (B) The agent layer holds positional information of agents. Agents in the agent layer interact with the environment layer at their corresponding positions.

Figure 6. Syncing time steps across hybrid multi-scale agent-based models

Example of two different combinations of time steps for extracellular molecule diffusion and agent-associated reactions. (A) A large time step for extracellular molecule diffusion requires few sync points with the agent-associated reactions. (B) A small time step for extracellular molecule diffusion requires many sync points with the agent-associated reactions.

Figure 7. Diagram of a solution algorithm for a hybrid multi-scale agent-based

(1) Update agents (movement, states, proliferation, etc). (2) Solve a single time step (Δt_ode) for agent-associated reactions (Θ₂). Increment a counter N. (3) If the total time step (N×Δt_ode) is equal to (Δt_pde/2) then move on to extracellular molecule diffusion and degradation. If not, take another single time step for agent-associated reactions (Θ₂) and check again. (4) Solve a single time step (Δt_pde) for extracellular molecule diffusion and degradation. Increment a counter M. (5) Solve a single time step (Δt_ode) for agent-associated reactions (Θ₂). Increment a counter N. (6) If the total time step (N×Δt_ode) is equal to (Δt_pde/2) move on to the final check. If not, take another single time step for agent-associated reactions (Θ₂) and check again. (7) If the total time step (M×Δt_pde) is equal to (Δt_agent) then a full time step has been completed. Continue by updating agents as indicated in (1). If not, continue solving with step (2).