# Propagation of dissection in a residually-stressed artery model

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## Abstract

This paper studies dissection propagation subject to internal pressure in a residually-stressed two-layer arterial model. The artery is assumed to be infinitely long, and the resultant plane strain problem is solved using the extended finite element method. The arterial layers are modelled using the anisotropic hyperelastic Holzapfel–Gasser–Ogden model, and the tissue damage due to tear propagation is described using a linear cohesive traction–separation law. Residual stress in the arterial wall is determined by an opening angle \(\alpha \) in a stress-free configuration. An initial tear is introduced within the artery which is subject to internal pressure. Quasi-static solutions are computed to determine the critical value of the pressure, at which the dissection starts to propagate. Our model shows that the dissection tends to propagate radially outwards. Interestingly, the critical pressure is higher for both very short and very long tears. The simulations also reveal that the inner wall buckles for longer tears, which is supported by clinical CT scans. In all simulated cases, the critical pressure is found to increase with the opening angle. In other words, residual stress acts to protect the artery against tear propagation. The effect of residual stress is more prominent when a tear is of intermediate length (\(\simeq \)90\(^\circ \) arc length). There is an intricate balance between tear length, wall buckling, fibre orientation, and residual stress that determines the tear propagation.

## Keywords

Arterial dissection Residual stress HGO model Soft tissue mechanics Buckling Tear propagation Critical pressure XFEM Cohesive traction–separation law## 1 Introduction

An arterial dissection is a tear within the wall of a large artery, such as the aorta. The dissection can lead to the creation of a false lumen through which blood flows, and propagation of the tear can quickly lead to death as a result of decreased blood supply to other organs, damage to the aortic valve, and rupture of the artery. The loading conditions on the arterial wall, the geometry of the artery and of the tear, and the material properties of the arterial wall determine whether the tear propagates. A prediction of how the critical condition for tear propagation depends on these factors could help to optimize diagnosis and treatment.

In the absence of loading, many biological soft tissues are not stress-free, but subject to residual stress. At physiological loading, the residual stress in arteries reduces variation in the stress distribution across the arterial wall and decreases the peak stress (Cardamone et al. 2009; Chuong and Fung 1986). Fung (1991) was the first to show that a radial cut along artery can release much of the residual stress. Hence, using an opening angle is a theoretical approach for recovering the stress-free configuration, and the value of the opening angle is often used to quantify the residual stress. For example, Holzapfel et al. (2000) employed this method to obtain the residual stress in a two-layer model of a rabbit carotid arterial wall.

Tear propagation in the absence of residual stress has been studied previously. For example, Sommer et al. (2008) subjected samples of human aortic media to peeling tests to estimate dissection property. This was followed by a number of computational simulations (Ferrara and Pandolfi 2010; Gasser and Holzapfel 2006). Wang et al. (2014) and Tong et al. (2011) performed similar experiments on human coronary and human carotid arteries, respectively. In these experimental and computational studies, the tear propagation is stably driven by controlling the displacement when peeling apart strips. However, blood pressure drives the propagation of dissections in vivo (Braverman 2010; Rajagopal et al. 2007). There are fewer literatures on studying this pressure-driven propagation. In experiments on a porcine thoracic aorta subject to pressure, Carson and Roach (1990) measured the peak pressure to tear the media and the work per unit area of tissue required to propagate a tear and showed that these values are independent of the tear depth, while Tam et al. (1998) studied the effect of depth of the initial tear on the critical pressure for propagation and showed that the critical pressure decreases as the depth increases. Arterial dissection during balloon angioplasty of an atherosclerotic artery was modelled by Badel et al. (2014), in which the arterial wall is compressed by inflating a balloon controlled by a displacement boundary condition. Recently, we developed a computational scheme to compute the energy release rate, a variable for quantifying the risk of propagation, for pressure-driven dissection propagation using a nonlinear energy argument in a 2D model (Wang et al. 2015), and reported how the critical pressure for arterial dissection can be affected by fibre orientation, tear length, and surrounding tissues.

The paper is organized as follows. In Sect. 2, we detail the arterial model including the geometry, the constitutive model, and the cohesive law. In Sect. 3, we show how to calculate analytically the stress-free configurations for different opening angles with a specified unloaded configuration, how to incorporate residual stress using ABAQUS, and how to determine the critical pressure. The results are shown in Sect. 4, followed by discussion and conclusions in Sect. 5.

## 2 The model

### 2.1 Geometry

*residually-stressed*configuration \(\varOmega _\mathrm{r}\). The residual stress field in \(\varOmega _\mathrm{r}\) is calculated from the value of the opening angle \(\alpha \) in the

*zero-stress*configuration \(\varOmega _0\). The configurations are listed in Table 1. The data for \(\varOmega _0\) are obtained from (Holzapfel et al. 2000), and the data for \(\varOmega _\mathrm{r}\) are computed using the analytical approach to be discussed below. We introduce an idealized dissection (Fig. 2) along an arc of constant radius in \(\varOmega _\mathrm{r}\), which is connected to the lumen of the vessel by negligibly small tears, so that the dissection surfaces are subject to the same blood pressure as in the true lumen.

The geometry of the two-layer arterial wall, \(t_\mathrm{m}\) and \(t_\mathrm{a}\) are the wall thickness of the media and adventitia

Configurations | \(r_\mathrm{i}\,(\hbox {mm})\) | \(t_\mathrm{m}\,(\hbox {mm})\) | \(t_\mathrm{a}\,(\hbox {mm})\) | \(\alpha \,(^\circ )\) |
---|---|---|---|---|

\(\varOmega _0\) | 1.430 | 0.260 | 0.130 | 160 |

\(\varOmega _\mathrm{r}\) | 0.739 | 0.259 | 0.120 | 0 |

### 2.2 Constitutive law

*c*, \(k_1\), \(k_2\) are material parameters for each layer of the artery, as listed in Table 2 (Holzapfel et al. 2000), and \(I_1, I_4\), and \(I_6\) are the invariants of the right Cauchy–Green strain tensor \(\mathbf {C}\) and the fibre-structure tensor \(\mathbf {M}_n\), defined as

*z*-component in these fibre directions is ignored. Therefore, changing \(\beta \) only changes the contribution of fibres in the circumferential direction.

The material parameters for the HGO strain- energy function (Holzapfel et al. 2000) and the cohesive law, for a rabbit carotid artery

\(c\,(\hbox {kPa})\) | \(k_1\,(\hbox {kPa})\) | \(k_2\) | \(\beta \,(^\circ )\) | \(T_\mathrm{c}\,(\hbox {kPa})\) | \(G_\mathrm{c}\,(\hbox {N/m}^2)\) | \(\Delta u_\mathrm{c}\,(\hbox {mm})\) | |
---|---|---|---|---|---|---|---|

Media | 1.5 | 2.3632 | 0.8393 | 29 | 3 | 0.001 | 0.667 |

Adventitia | 0.15 | 0.5620 | 0.7112 | 62 | 0.3 | 0.0001 | 0.667 |

### 2.3 Cohesive law

We assume that the propagation of tear is governed by a linear cohesive traction–separation law (Ferrara and Pandolfi 2010) (Fig. 3), which is specified by the maximum traction \(T_\mathrm{c}\) just before damage, the separation energy \(G_\mathrm{c}\), and the maximum displacement jump \(\Delta u_{\mathrm{c}}\), as listed in Table 2. Only two are independent. After each incremental loading step, the maximum principal stress \(\sigma _{\mathrm{mp}}\) at the centroid of each element is compared to \(T_\mathrm{c}\): if \(\sigma _{\mathrm{mp}} \ge T_\mathrm{c}\), then the displacement jump \(\Delta u\) is calculated. When \(\varDelta u > \Delta u_\mathrm{c}\), the tear propagates in the direction perpendicular to the maximum tensile principal stress.

The actual value of \(T_\mathrm{c}\) is material dependent and should be determined by experiments. In the absence of such data, we assume that \(T_\mathrm{c}/c=2\) (where *c* is the value of the media) in this study. This, together with the assumption of the plane-strain problem, means that our computed results will be qualitative. However, different values of \(T_\mathrm{c}/c\) are used later to check that the trend we observe is the same.

## 3 Methodology

All simulations are based on a residually-stressed configuration \(\varOmega _\mathrm{r}\). The residual stress and \(\varOmega _\mathrm{r}\) can be obtained both analytically and numerically.

### 3.1 Analytical approach

To ensure that the difference between various simulations is only due to the residual stress, we can also determine \(\varOmega _0\) from (5), (7), and (12), given \(\alpha \) and \(\varOmega _\mathrm{r}\).

### 3.2 Numerical approach

Meshes used for the grid independence tests

Mesh | Nodes | Elements | Relative error in stress via Eq. (15) (%) | |||
---|---|---|---|---|---|---|

Media | Adventitia | Circumference | Total | |||

Coarse | 909 | 5 | 3 | 100 | 800 | 21.37 |

Intermediate | 3417 | 11 | 5 | 200 | 3200 | 5.59 |

Fine | 13,233 | 23 | 9 | 400 | 12,800 | 5.08 |

### 3.3 Finite element implementation

## 4 Results

### 4.1 The residual stress and critical pressure

The geometries (i.e. the radius \(R_i\), thicknesses of the media and adventitia, \(T_m\), \(T_a\)) of stress-free configurations \(\varOmega _0\) associated with the *same* unloaded configuration \(\varOmega _\mathrm{r}\) (where we specify \(\alpha \), thicknesses of the media and adventitia, \(t_\mathrm{m}\), \(t_\mathrm{a}\), and inner radius \(r_\mathrm{i}\) which are equal to \(T_m\), \(T_a\), and \(R_i\) when \(a = 0)\) are shown in Table 4. The residual stress components computed analytically are shown in Fig. 7 as a function of the opening angle \(\alpha \). The absolute value of \(\sigma _{{rr}}\) is greatest at the mid-radius of the media and increases with \(\alpha \). The circumferential stress \(\sigma _{\theta \theta }\) is in compression at the inner radius of the media and is in tension at the outer radius of the media and adventitia, and \(|\sigma _{\theta \theta }|\) increases with \(\alpha \). The residual stress is smaller in the adventitia.

*c*is the value of the media). For the tear of length \(\eta =90^\circ \), the change of \(p'_{\mathrm{c}}\) with the opening angle is plotted in Fig. 9. Notice that \(p'_{\mathrm{c}}\) increases with \(\alpha \) in all the cases simulated, suggesting that existence of residual stress makes artery more resistant to the tear propagation.

Geometries of \(\varOmega _0\) (thicknesses of the media and adventitia \(T_\mathrm{m}\), \(T_\mathrm{a}\), and inner radius \(R_\text {i}\)) corresponding to the specified \(\varOmega _\mathrm{r}\) and opening angle \(\alpha \)

\(\alpha \,(^\circ )\) | \(R_\mathrm{i}\,(\hbox {mm})\) | \(T_\mathrm{m}\,(\hbox {mm})\) | \(T_\mathrm{a}\,(\hbox {mm})\) |
---|---|---|---|

0 | 0.7395 | 0.2593 | 0.1197 |

40 | 0.8472 | 0.2595 | 0.1221 |

80 | 0.9858 | 0.2597 | 0.1246 |

120 | 1.1708 | 0.2599 | 0.1272 |

160 | 1.4300 | 0.2600 | 0.1300 |

200 | 1.8191 | 0.2601 | 0.1329 |

### 4.2 Inner wall buckling and tear length

Comparison of the critical pressure for different tear lengths (Fig. 9) shows that the dimensionless critical pressure \(p'_\mathrm{c}\) increases with \(\alpha \) in all the cases simulated. Notably, the longest tear length studied \((\eta =210^\circ )\) has a higher value of \(p'_\mathrm{c}\) than that of \(\eta =90^\circ \), and \(150^\circ \). Hence, the relationship between propagation and tear length is not as simple as that of a 2D strip (Wang et al. 2015). This presumably is due to the buckling of the inner wall, which is more likely to occur for a longer tear. However, for the intermediate case, \(\eta =90^\circ \), inner wall buckling occurs only for \(\alpha =0^\circ \), and not when \(\alpha \ge 40^\circ \) (Fig. 12). This suggests that there is a subtle interplay between dissection length, residual stress, and inner wall buckling.

### 4.3 The effect of fibre orientation

Here we vary the fibre orientation. We refer to the fibre orientation listed in Table 2 as the physiological or ‘true’ case. In the following simulations, we assume that the media and the adventitia have the same fibre orientation, so that \(\beta _\mathrm{m}=\beta _\mathrm{a}=\beta \). Simulations were performed with \(\beta =0^\circ ,~10^\circ ,~15^\circ ,~20^\circ ,~30^\circ ,~60^\circ \) and \(90^\circ \). In addition, a group of simulations were run without fibres \((k_1=0)\), referred to as the ‘free’ case.

The critical pressure also changes with different fibre orientation (Fig. 14). Notice that with the inflation, fibres with \(\beta =30^\circ \) also start to bear load, but fibres at greater angles \((\beta =60^\circ \) and \(90^\circ )\) still do not take on any load, and hence, the critical pressures for these cases remain the same as in the ‘free’ case. As expected, the critical pressure when \(\beta =0^\circ \) is the highest, since the residual stress is the greatest in this case.

## 5 Discussion and conclusions

The model parameters used for the human thoracic aorta in \(\varOmega _0\) (Fereidoonnezhad et al. 2016)

\(c(\hbox {kPa})\) | \(k_1\,(\hbox {kPa})\) | \(k_2\) | \(\beta \,(^\circ )\) | \(T_\mathrm{c}\) | \(G_\mathrm{c}(\hbox {N/m}^2)\) | \(\alpha \,(^\circ )\) | \(T_\mathrm{i}(\hbox {mm})\) | \(R_\text {i}(\hbox {mm})\) | |
---|---|---|---|---|---|---|---|---|---|

Media | 20 | 112 | 20.61 | 41 | 2 | 0.001 | 80 | 0.69 | 1.13 |

Adventitia | 8 | 362 | 7.089 | 50.1 | 2 | 0.0001 | 80 | 0.48 | N/A |

We now discuss the limitations of this study. Our model is a plane-strain problem and does not include the effect of axial stretch. In our previous work (Wang et al. 2015), we found that axial stretching of fibres resists the opening of the dissection and significantly decreases the energy release rate for tear propagation. Clinically, dissections may propagate axially and may re-enter the lumen. Our model also cannot predict the absolute value of the critical pressure due to the simplifications mentioned above and lack of data on cohesive parameters, e.g. the value of \(T_\mathrm{c}\). Nevertheless, this model provides a qualitative description of the variation of the critical pressure with the residual stress. This is further illustrated in Fig. 16 for \(T_\mathrm{c}/c=2\), 6, 10 for the rabbit carotid artery, as well for the aged human thoracic aorta, where the shear modulus of the media is much greater, as shown in Table 5. In the human artery model, all the parameters for the HGO model (Table 5) are estimated by fitting the experimental data of the cyclic uniaxial tensile tests of 14 human thoracic aortas (\(60\pm 12\) year, \(\hbox {mean}\pm \hbox {SD}\)) (Fereidoonnezhad et al. 2016). Our results show that changes of the critical pressure with residual stress are similar for human aorta and rabbit carotid artery.

The other limitation is that we use an isotropic cohesive traction–separation law. The tensile testing of a porcine thoracic aorta performed by MacLean et al. (1999) showed that the stiffness in the radial direction is significantly lower than in the circumferential and longitudinal directions. MacLean et al. (1999) also performed a histological analysis to show the behaviour of elastin layers and smooth muscle cells during the aortic dissection. Work is ongoing to develop a three-dimensional and anisotropic arterial dissection model that could make use of the histological data.

## Notes

### Acknowledgments

We are grateful for the support by the UK Engineering and Physical Sciences Research Council (Grant No. EP/N014642/1). LW was supported by a China Scholarship Council Studentship and the Fee Waiver Programme of the University of Glasgow. We also thank Ashwani Goel, a Technical Specialist from SIMULIA Central, for his kind help with using Abaqus.

### Compliance with ethical standards

### Ethical standard

There are no ethical issues involved in this work.

### Conflict of interests

The authors declare that they have no conflicts of interest.

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