Toward a multi-scale computational model of arterial adaptation in hypertension: verification of a multi-cell agent based model

Abstract

Agent-based models (ABMs) represent a novel approach to study and simulate complex mechano chemo-biological responses at the cellular level. Such models have been used to simulate a variety of emergent responses in the vasculature, including angiogenesis and vasculogenesis. Although not used previously to study large vessel adaptations, we submit that ABMs will prove equally useful in such studies when combined with well-established continuum models to form multi-scale models of tissue-level phenomena. In order to couple agent-based and continuum models, however, there is a need to ensure that each model faithfully represents the best data available at the relevant scale and that there is consistency between models under baseline conditions. Toward this end, we describe the development and verification of an ABM of endothelial and smooth muscle cell responses to mechanical stimuli in a large artery. A refined rule-set is proposed based on a broad literature search, a new scoring system for assigning confidence in the rules, and a parameter sensitivity study. To illustrate the utility of these new methods for rule selection, as well as the consistency achieved with continuum-level models, we simulate the behavior of a mouse aorta during homeostasis and in response to both transient and sustained increases in pressure. The simulated responses depend on the altered cellular production of seven key mitogenic, synthetic, and proteolytic biomolecules, which in turn control the turnover of intramural cells and extracellular matrix. These events are responsible for gross changes in vessel wall morphology. This new ABM is shown to be appropriately stable under homeostatic conditions, insensitive to transient elevations in blood pressure, and responsive to increased intramural wall stress in hypertension.

Keywords: agent-based modeling; constrained mixture modeling; hypertension; multi-scale modeling; vascular remodeling.

Figure 1

Screen shot of the ABM during a homeostatic run, displaying a 2-D representation of a transverse section of the model mouse abdominal aorta on the right and user controls as well as a display of progress on the left. Inner diameter of the vessel is 460 μm.

Figure 2

Parameter sensitivity analysis for the parameter δ in the rule for ET-1 production (cf. Table 3.1). This rule states that the production of ET-1 by ECs depends on wall shear stress in a sigmoid fashion. Each solid line represents the mean value based on 100 replications of the ABM. Blue indicates the response when the parameter was increased an order of magnitude, green when the parameter was decreased an order of magnitude, and red when the parameter remained at its original value. The pastel colors represent the 95% confidence intervals surrounding each result.

Figure 3

Sensitivity of ABM outputs to changes in proliferation rate of SMC in response to PDGF-AB. Note that for smaller changes in this parameter, SMC number and collagen mass reach a new equilibrium as hoop stress increases enough to support PDGF-AB and TGF-β production and therefore cell proliferation and collagen production.

Figure 4

Homeostatic Conditions. ABM results are shown in blue, CMM results in black. ABM results are an average of 100 simulations. Stochastic fluctuations in cell number lead to some changes in stress and growth factor production. Overall change in cell number is <1%. (A) Pressure, (B) SMC, (C) Collagen, (D) PDGF-AB, (E) TGF-β.

Figure 5

Transient Pressure Increases. ABM results are shown in blue, CMM results in black. ABM results are an average of 100 simulations. Transient increases in pressure of 10% for 6 h drive short-term changes in growth factor expression, but not SMC mass or collagen production. (A) Pressure, (B) SMC, (C) Collagen, (D) PDGF-AB, (E) TGF-β.

Figure 6

Agent-based model Hypertension Response. ABM results are an average of five simulations. Panel (A) ABM is subjected to a step increase in mean arterial pressure of 30%. (B, C) Response of SMC proliferation and collagen production to increased growth factor levels. (D, E) Production of the growth factors PDGF-AB and TGF-β in response to the elevated pressure and circumferential stress.