Ensuring congruency in multiscale modeling: towards linking agent based and continuum biomechanical models of arterial adaptation

Abstract

There is a need to develop multiscale models of vascular adaptations to understand tissue-level manifestations of cellular level mechanisms. Continuum-based biomechanical models are well suited for relating blood pressures and flows to stress-mediated changes in geometry and properties, but less so for describing underlying mechanobiological processes. Discrete stochastic agent-based models are well suited for representing biological processes at a cellular level, but not for describing tissue-level mechanical changes. We present here a conceptually new approach to facilitate the coupling of continuum and agent-based models. Because of ubiquitous limitations in both the tissue- and cell-level data from which one derives constitutive relations for continuum models and rule-sets for agent-based models, we suggest that model verification should enforce congruency across scales. That is, multiscale model parameters initially determined from data sets representing different scales should be refined, when possible, to ensure that common outputs are consistent. Potential advantages of this approach are illustrated by comparing simulated aortic responses to a sustained increase in blood pressure predicted by continuum and agent-based models both before and after instituting a genetic algorithm to refine 16 objectively bounded model parameters. We show that congruency-based parameter refinement not only yielded increased consistency across scales, it also yielded predictions that are closer to in vivo observations.

FIGURE 1

Graphical display of an ABM model of a mouse abdominal aorta (a) before and (b) after a simulated hypertension (defined herein as a sustained 30% increase in mean pressure).

FIGURE 2

Illustrative results, before parameter refinement via the genetic algorithm, for a 30% increase in mean luminal pressure using initial parameters (Table 2). CMM (dashed line) and ABM (solid line) predictions of (a) pressure and (d,e,f) stresses normalized with respect to the homeostatic values, (b) luminal diameter and (c) thickness normalized with respect to original values, (g-i) intramural constituents, and (j-o) soluble molecular masses. Elastin mass, but not mass fraction, is constant in the CMM and currently not predicted by the ABM. Axial stress cannot be estimated by the ABM and was not considered within the CMM to alter turnover rates in this illustrative simulation.

FIGURE 3

Similar to Figure 2 (for 30% increase in pressure) except based on values of the parameters (Table 2) that increased congruency between ABM and CMM model predictions of collagen and smooth muscle mass; these parameters were determined by minimizing Eq. (5) using the genetic algorithm.

FIGURE 4

Similar to Figure 3 except for a 15% increase in pressure. Note, however, that the parameter values were the same as used in Figure 3, which were found to increase congruency between the ABM and CMM for the case of a 30% increase in pressure. Hence, this result shows the broader applicability of the minimization procedure.