Reconstructing vascular homeostasis by growth-based prestretch and optimal fiber deposition

Abstract

Computational modeling of cardiovascular biomechanics should generally start from a homeostatic state. This is particularly relevant for image-based modeling, where the reference configuration is the loaded in vivo state obtained from imaging. This state includes residual stress of the vascular constituents, as well as anisotropy from the spatially varying orientation of collagen and smooth muscle fibers. Estimation of the residual stress and fiber orientation fields is a formidable challenge in realistic applications. To help address this challenge, we herein develop a growth based Algorithm to recover a residual stress distribution in vascular domains such that the stress state in the loaded configuration is equal to a prescribed homeostatic stress distribution at physiologic pressure. A stress-driven fiber deposition process is included in the framework, which defines the distribution of the fiber alignments in the vascular homeostatic state based on a minimization procedure. Numerical simulations are conducted to test this two-stage homeostasis generation algorithm in both idealized and non-idealized geometries, yielding results that agree favorably with prior numerical and experimental data.

Introduction

Mathematical modeling of the mechanical behavior of vascular tissue requires consideration of many important factors. While the mechanical characterization and constituent models of arterial tissue have been well developed, a challenging factor is the consideration of residual stress in the material in vivo. This residual stress determines the overall stress state of the material in the loaded in vivo configuration, as well as vascular homeostasis, which represents the preferred mechanical configuration in which vascular constituents operate (Fung, 1991). Thus, while proper constituent models are necessary to capture the detailed mechanical behavior of vascular tissue, the inclusion of residual stress is also an essential factor for predicting biomechanical behavior in vivo and vascular remodeling (Holzapfel and Ogden, 2010).

Another important factor in modeling arterial tissue is the consideration of fiber orientation. Arteries are generally composed of three layers (intima, media, and adventitia) and the major mechanical constituents are elastin, collagen, and smooth muscle fibers (Humphrey, 2008). Elastin is effective under small strain, while collagen bears the majority of loading when deformation is large (Roach and Burton, 1957), and smooth muscle fibers provide additional and vasoactive support (Fung, 1993). To model these components, different constitutive relations have been proposed, including Fung-type models (Chuong and Fung, 1983) and formulations proposed by Holzapfel et al. (2000). Notably, the arterial wall is often considered to be a two-layer fiber-reinforced material. A non-collagenous ground matrix, including elastin, is typically described as an isotropic neo-Hookean material (Dorrington and McCrum, 1977). The collagen and smooth muscle impart anisotropic behavior and are often described by a Fung-type response (Holzapfel et al., 2000; Bellini et al., 2014). To characterize the deformation of the various constituents, a constrained mixture model (Humphrey and Rajagopal, 2002; Cyron et al., 2016) is often employed, particularly in vascular growth and remodeling studies (Figueroa et al., 2009a; Zeinali-Davarani et al., 2011; Wu and Shadden, 2015). Fiber orientations have traditionally been prescribed from statistical analyses of histological data (Holzapfel, 2006) and rule-based approaches (Augustin et al., 2014), with more recent works (Gasser et al., 2006; Holzapfel and Ogden, 2017) considering the dispersion of fiber orientation. Nonetheless, prescription of fiber orientation is nontrivial in realistic branched vascular models.

The in vivo stress state of vascular tissue relies critically on residual stress and fiber orientation. Namely, the homeostatic state is hypothesized to be defined by relatively uniform stress distribution in the transmural direction of each layer in the vessel wall, and that non-uniform residual stress helps to maintain this condition (Fung, 1991). In idealized cylindrical vascular geometries, the residual stress field is often specified based on an opening angle (Alastrué et al., 2007), however, this approach is often inadequate in realistic vascular geometries. Pierce et al. (2015) prescribed residual stress by specifying the deformation gradient to map the stress-free configuration to the actual mixture configuration of the vessel wall. Bellini et al. (2014) incorporated a prestretch ratio into the total deformation to model the effect of residual stresses and showed that the stress distribution tends to be more uniform when residual stress is included. Holzapfel et al. investigated the 3D behavior of the residual stress experimentally (Holzapfel et al., 2007) and developed a theoretical framework to incorporate residual stress in different vascular layers (Holzapfel and Ogden, 2010). Previous works have often required parameter information that can be difficult to obtain or is of challenging applicability when realistic geometries are considered. Likewise, the residual stress distribution is rarely considered when the vascular homeostatic state needs to be reconstructed in non-idealized geometries, see, e.g., (Maes et al., 2019; Mousavi et al., 2017). To the best of our knowledge, this is the first time that a residual stress field is reconstructed in general vascular geometries with anomalies such as bifurcations.

We propose a two-stage approach to the reconstruction of the vascular homeostatic state. The first stage aims to generate an appropriate residual stress level for a given configuration of the vessel, and the second stage aims to generate optimal fiber alignments based on the stress field. Inspired by kinematic G&R models (Rodriguez et al., 1994), (Taber, 1998), and (Taber and Humphrey, 2001), we introduce a prestretch tensor, obtained through an iterative process to match a prescribed homeostatic stress distribution at physiologic pressure. The deposition angle of the collagen fibers is then defined by the solution of an optimization problem, which enables the vessel to sustain the stress with minimal amount of biomass, while stochasticity is incorporated into this process to account for fiber dispersion. We propose supervised learning to predict the prestretch in vascular bifurcations or other geometric anomalies, where theory-based methods struggle due to topological singularities. Based on this overall framework, the numerical stability of the generation of the residual stress field and fiber directions is improved for realistic vascular geometries. Beyond addressing the vascular homeostatic state reconstruction problem, the method proposed here provides a starting point for general biomechanical analyses, such as vascular tissue mechanics or fluid-structure interaction simulations, where residual stress and proper fiber alignment set the stage for the mechanical response of the tissue.