The explosive growth of small voids in vulnerable cap rupture; cavitation and interfacial debonding

Abstract

While it is generally accepted that ruptures in fibrous cap atheromas cause most acute coronary deaths, and that plaque rupture occurs in the fibrous cap at the location where the tissue stress exceeds a certain critical peak circumferential stress, the exact mechanism of rupture initiation remains unclear. We recently reported the presence of multiple microcalcifications (μCalcs) <50 μm diameter embedded within the fibrous cap, μCalcs that could greatly increase cap instability by introducing up to a 5-fold increase in local tissue stress. Here, we explore the hypothesis that, aside from cap thickness, μCalc size and interparticle spacing are principal determinants of cap rupture risk. Also, we propose that cap rupture is initiated near the poles of the μCalcs due to the presence of tiny voids that explosively grow at a critical tissue stress and then propagate across the fibrous cap. We develop a theoretical model based on classic studies in polymeric materials by Gent (1980), which indicates that cavitation as opposed to interfacial debonding is the more likely mechanism for cap rupture produced by μCalcs <65 μm diameter. This analysis suggests that there is a critical μCalc size range, from 5 μm to 65 μm, in which cavitation should be prevalent. This hypothesis for cap rupture is strongly supported by our latest high resolution μCT studies in which we have observed trapped voids in the vicinity of μCalcs within fibrous caps in human coronaries.

Introduction

Most acute coronary deaths are caused by the rupture of a fibrous cap atheroma, a complex biomechanical phenomenon, not yet fully understood. The most widely accepted factor for increasing cap vulnerability is the thickness of the fibrous cap overlying the necrotic core (Burke et al., 1997). Necrotic core size, tissue composition and mechanical properties of the cap have also been recognized to play a role in plaque vulnerability (Ohayon et al., 2008). However, finite element analysis (FEA) has shown that individually these latter factors lead to only a 20% to 30% increase in local tissue stress (Maldonado et al., 2012) and are unlikely to elevate the tissue stress in the cap above the minimum rupture threshold of 300 kPa proposed by Cheng et al. (1993). Furthermore, almost half of observed ruptures occur in the center of the cap (Maehara et al., 2002), or in thick caps (Tanaka et al., 2008), where 3D FEA (Maldonado et al., 2012) and fluid-structure-interaction calculations (Rambhia et al., 2012) predict tissue stresses significantly lower than 300 kPa. These observations suggest that other unforeseen factors may play an important role.

Although coronary calcification is clinically related to poor prognosis and is used as a marker of the advancement of the disease, it has not been successfully correlated with plaque rupture (Thilo et al., 2010). Key to understanding the role of calcified tissue in plaque stability was the study by Vengrenyuk et al. (2006) in which the counterintuitive idea that microcalcifications (μCalcs) embedded in the fibrous cap proper, lying below the resolution of current in vivo imaging techniques, could greatly increase cap instability by introducing a 2-fold increase in local tissue stress.

High resolution micro-computed tomography (HR-μCT) was successfully used to provide the first evidence of the existence of such μCalcs in the human fibrous cap (Vengrenyuk et al., 2006), and later confirmed in a much larger set of samples (Maldonado et al., 2012) where 81 μCalcs <60 μm diameter were analyzed using 3D finite element analysis (FEA). This study revealed that closely spaced μCalcs could increase local tissue stresses by a factor of five. However, little is known about the exact mechanism of cap rupture.

In this study, we propose that cap rupture is initiated near the μCalcs when the fibrous cap is subject to tensile stresses that exceed a critical value, wherein tiny precursor voids (minute bubbles) greatly expand due to large tensions generated in vicinity of a μCalc (henceforth called cavitation), or where the fibrous tissue detaches from the μCalc surface initiated by a small initial separation (debonding). Cavitation in a hyperelastic solid differs from cavitation in a fluid in which bubbles will grow when subjected to pressures at or below their vapor pressure.

Based on the study by Gent (1980) for spherical beads in polymeric materials, we develope herein theoretical models to investigate this cavitation/debonding cap failure hypothesis by considering the local elastic adhesion energy released when debonding at the μCalc surface occurs or when cavitation is initiated in the tissue itself if the cap critical yield stress is exceeded. We also investigate the effect of the μCalc size and spacing between multiple μCalcs on peak cap stress using FEA and HR-μCT. Human coronary atheromas in which caps with μCalcs were previously detected using HR-μCT (Maldonado et al., 2012) are analyzed herein to obtain insight into the cap rupture initiation process.

Our idealized mathematical model consists of a typical fibroatheroma where one or more spherical μCalcs are embedded in the fibrous cap as shown in Fig. 1, Fig. 2. Blood pressure is applied in the lumen to deform the artery and create a tensile stress in the cap, which is calculated as the σ₂₂ component of the Cauchy stress tensor. μCalcs are assumed to be rigid spherical particles and the fibrous cap is assumed to have incompressible hyperelastic material properties (Holzapfel, 2000), such that the Cauchy stress tensor,

$$\mathit{\sigma }=-p\mathbf{I}+\mathbf{F}{\left(\frac{\partial \psi \left(\mathbf{F}\right)}{\partial \mathbf{F}}\right)}^T$$

In Eq. (1a), F is the deformation gradient, ψ is the strain energy, and p is an arbitrary pressure-like scalar. Simplifying (1a) for a Neo-Hookean, isotropic and homogeneous solid,

$$W=C(I_1-3)$$

$$C=\mu /2,\mu =\frac{E_t}{2\left(1+v\right)}$$

where W is the strain energy density, I₁ is the first strain invariant, μ is the shear modulus, ν is the Poisson ratio, and E_t is the Young elastic modulus of the soft tissue. Assuming ν≈0.5, one finds from (1c) that C≈E_t/6 (Ohayon et al., 2007).

Two failure modes have been previously described by Gent in layers of hyperelastic materials with rigid spherical inclusions: cavitation and debonding (Gent and Park, 1984). Based on the theoretical model initially developed for the study of reinforced polymers (Gent and Lindley, 1958, Gent and Tompkins, 1969, Gent, 1980, Cho and Gent, 1988), we studied the failure mode of a hyperelastic fibrous cap with embedded μCalcs. Since the presence of a μCalc introduces local stress concentrations in a cap under tension, elastic energy is stored in the vicinity of the μCalc. Both cavitation and debonding lead to the release of this stored energy. The preferential mode of failure and the magnitude of the stored energy are determined by the size of the μCalc, the strength of the bond between the μCalc and the tissue and their respective Young's modulus of elasticity.

Debonding mechanism: A very small initially debonded area is assumed to grow in accordance with Griffith's criterion when the stored strain energy is greater than the energy required for debonding. From Gent (1980), the minimum applied stress necessary to cause debonding, σ_d, at the interface of a μCalc at its a pole is

$${\sigma }_d^2=\frac{8\pi G_a{E}_t}{3rk\sin \theta }$$

where G_a is the bond fracture energy per unit a of bonded surface, r is the radius of the μCalc and θ is the initial debonding angle (see Fig. 1; parameter values in Table 1). When the μCalc is a sphere, k has a value of 2.

Cavitation mechanism: If we assume a fibrous cap under tension containing an extremely small spherical void, the pressure, P_m, within the void acting on the tissue takes the form (Gent and Lindley, 1958)

$$P_m=C\left(5-\frac{4}{\lambda }-\frac{1}{{\lambda }^4}\right)$$

where λ is the extension ratio of the void. As the void grows and λ»1, P_m approaches the limiting value 5/6 E_t causing catastrophic rupture of the tissue layer. The presence of a void as a free surface in the tissue, creates a surface tension γ that opposes to the growth of the cavity, and if the initial void is assumed to be very small, of radius a=r/10, this energy is not negligible (Gent and Tompkins, 1969, Gent, 1980). The local tissue stress σ_c for a small void to grow is given by

$${\sigma }_c=\frac{5}{6}{E}_t+2\frac{\gamma }{a}$$

Eqs. (2), (4) determine which mode of failure will occur. σ_c represents the minimum tissue stress required to induce cavitation. Any larger local tissue stress would trigger the unbounded growth of the void in the tissue itself, while σ_d is the minimum tissue stress required to start debonding of the tissue at the tensile pole of the μCalc starting as a small separation void of prescribed initial debonding angle at the interface. Thus, if σ_c<σ_d then cavitation in the tissue will occur before interfacial debonding; if σ_c>σ_d then the failure mode will be interfacial debonding. The limiting behavior for cavitation, P_m=5E_t/6, is approached when the void is large enough to neglect its surface energy for expansion.