Multiscale Coupling of an Agent-Based Model of Tissue Fibrosis and a Logic-Based Model of Intracellular Signaling

Abstract

Wound healing and fibrosis following myocardial infarction (MI) is a dynamic process involving many cell types, extracellular matrix (ECM), and inflammatory cues. As both incidence and survival rates for MI increase, management of post-MI recovery and associated complications are an increasingly important focus. Complexity of the wound healing process and the need for improved therapeutics necessitate a better understanding of the biochemical cues that drive fibrosis. To study the progression of cardiac fibrosis across spatial and temporal scales, we developed a novel hybrid multiscale model that couples a logic-based differential equation (LDE) model of the fibroblast intracellular signaling network with an agent-based model (ABM) of multi-cellular tissue remodeling. The ABM computes information about cytokine and growth factor levels in the environment including TGFβ, TNFα, IL-1β, and IL-6, which are passed as inputs to the LDE model. The LDE model then computes the network signaling state of individual cardiac fibroblasts within the ABM. Based on the current network state, fibroblasts make decisions regarding cytokine secretion and deposition and degradation of collagen. Simulated fibroblasts respond dynamically to rapidly changing extracellular environments and contribute to spatial heterogeneity in model predicted fibrosis, which is governed by many parameters including cell density, cell migration speeds, and cytokine levels. Verification tests confirmed that predictions of the coupled model and network model alone were consistent in response to constant cytokine inputs and furthermore, a subset of coupled model predictions were validated with in vitro experiments with human cardiac fibroblasts. This multiscale framework for cardiac fibrosis will allow for systematic screening of the effects of molecular perturbations in fibroblast signaling on tissue-scale extracellular matrix composition and organization.

Keywords: agent-based model; cardiac fibroblast; fibrosis; multiscale modeling; myocardial infarction; network model; systems biology.

Figure 1

Components of individual ABM and network models. The ABM is comprised of agents that store information about attributes and perform methods. Value layers can be modified independently by defined parameters or by the activity of agents. Individual agents store a network state, which is updated by the fibroblast network model.

Figure 2

Coupled model process diagram. A detailed process diagram illustrates the methods and order in which they occur at each time step (1 h), and components of the ABM and network model that interact. Boxed numbers refer to the equation number which describes that process.

Figure 3

Agent-based model is initialized with cytokine gradients. (A) Four phenotypic regions are created by a combination of fibrotic and inflammatory cues. (B) TGFβ is initialized with an increasing gradient from bottom to top. (C) IL-1β, IL-6, and TNFα are initialized with an increasing gradient from left to right.

Figure 4

Verification tests confirm that coupled model and network model produce equivalent fibroblast network states. (A) In an unstimulated condition (0.25 input for all nodes), and a (B) stimulated condition (0.5 input for TGFβ, TNFα, IL-1β, and IL-6 nodes), the network state of a fibroblast using the coupled model or network model alone are comparable with a SSE of 3.865e-7 (A) and 1.168e-6 (B).

Figure 5

Coupled model predicts collagen profile over a range of physiological conditions. (A) Collagen area fraction for an unstimulated condition (0.25 input for all nodes) is compared to baseline collagen area fraction (4%) in a healthy rat. Model predictions for a stimulated condition (0.5 input for TGFβ, TNFα, IL-1β, and IL-6 nodes) are compared to results from a rat model of myocardial infarction up to 6 weeks post-MI. Error bars = SEM. (B) Collagen area fraction predictions at 6 weeks from a model simulation with gradient initial conditions and a fibroblast in each grid space (n = 100).

Figure 6

Signaling network exhibits a range of activation patterns in response to extracellular cues. Node activity level of individual network nodes for each fibroblast at steady state (6 weeks). Model simulation with gradient initial conditions and a fibroblast in each grid space (n = 100). Heat maps show network states for input receptors (A–C), intermediate network nodes (D–F), and network outputs (G–I).

Figure 7

Key parameters affect spatial gradient of collagen deposition. (A) Sensitivity coefficients calculated based on Equation (11) with individual parameter perturbations of 0.1×. State variable outputs include total collagen area fraction, global semivariance, semivariance in the x direction, and semivariance in the y direction. Collagen area fraction heat maps at 6 weeks with (B) all parameters at baseline, (C) k_gen,IL−1β parameter multiplied by 0.1, and (D) K_d,TGFβ parameter multiplied by 0.1.

Figure 8

Individual fibroblasts respond dynamically to extracellular environment. (A,C) Fibroblast migration path for a single fibroblast over a period of 50 h. Fibroblast starting location indicated by filled black triangle and end location indicated by open white triangle. (B,D) Corresponding TGFβ and IL-6 inputs for the fibroblasts tracked in (A,C), and their respective collagen and MMP mRNA activity over the time course of 50 h.

Figure 9

Fibroblast migration speed and density affect spatial heterogeneity of collagen. Collagen area fraction heat map at 6 weeks for simulations with 20 randomly migrating fibroblasts with a migration speed of (A) 1 grid/10 h, (B) 1 grid/h, and (C) 10 grids/h. (D) Average collagen area fraction at 6 weeks for each migration speed and simulations with 20, 40, 60, 80, or 100 fibroblasts. Mean reported for 5 runs. Error bars = standard deviation. (E) Semivariance calculated globally, in the x direction, and y direction for each migration speed. Mean reported for 10 runs. Error bars = standard deviation. ^*p < 0.05.

Figure 10

Coupled model predictions were compared to independent in vitro experiments using human cardiac fibroblasts treated with TGFβ1 and/or IL-1β. (A) Pro-collagen 1, αSMA, and F-actin expression from in vitro experiments with human cardiac fibroblasts were quantified by image analysis to measure the median fluorescence for all individual cells in each well (n = 3). Treatment conditions included control, TGFβ1 (20 ng/mL), IL-1β (10 ng/mL), and TGFβ1 (20 ng/mL) + IL-1β (10 ng/mL). Error bars = standard deviation. ^*p < 0.05 with reference to control condition. ^∧p < 0.05 with reference to TGFβ1 condition. (B) Coupled model predicts network expression of collagen I mRNA, αSMA, and F-actin when simulating the addition of TGFβ1 (20 ng/mL), IL-1β (10 ng/mL), and TGFβ1 (20 ng/mL) + IL-1β (10 ng/mL), compared to a simulation with all parameters at baseline. (C) Representative images of human cardiac fibroblast expression of pro-collagen 1 (green), αSMA (orange), and F-actin (purple) when treated with TGFβ1 (20 ng/mL), IL-1β (10 ng/mL), and TGFβ1 (20 ng/mL) +IL-1β (10 ng/mL), compared to control. Nuclei are stained with DAPI (blue). Scale bar = 500 microns.