# An incremental deformation model of arterial dissection

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## Abstract

We develop a mathematical model for a small axisymmetric tear in a residually stressed and axially pre-stretched cylindrical tube. The residual stress is modelled by an opening angle when the load-free tube is sliced along a generator. This has application to the study of an aortic dissection, in which a tear develops in the wall of the artery. The artery is idealised as a single-layer thick-walled axisymmetric hyperelastic tube with collagen fibres using a Holzapfel–Gasser–Ogden strain-energy function, and the tear is treated as an incremental deformation of this tube. The lumen of the cylinder and the interior of the dissection are subject to the same constant (blood) pressure. The equilibrium equations for the incremental deformation are derived from the strain energy function. We develop numerical methods to study the opening of the tear for a range of material parameters and boundary conditions. We find that decreasing the fibre angle, decreasing the axial pre-stretch and increasing the opening angle all tend to widen the dissection, as does an incremental increase in lumen and dissection pressure.

## Keywords

Arterial dissection Aortic dissection Incremental deformation Axisymmetric tear Residual stress Axial pre-stretch Holzapfel–Gasser–Ogden strain-energy## Mathematics Subject Classification

74R99 74B15 74E30 74G15 74L15 92C50## 1 Introduction

Aortic dissections resulting in rupture have an 80% mortality rate, and 50% of patients die before they even reach the hospital. Aortic dissection is divided into acute and chronic types (Khan and Nair 2002), depending on the duration of symptoms. The aortic dissection is acute when the diagnosis is made within a fortnight after the initial onset of symptoms, and chronic thereafter. About one third of patients with aortic dissection fall into the chronic category. The most common site of initiation of aortic dissection is the ascending aorta (50%) followed by the aortic regions in the vicinity of the ligamentum arteriosum.

Only a few analytical and computational models of the arterial dissection have been introduced. Geometrically nonlinear and consistently linearised, embedded strong discontinuity models for 3D problems with an application to the dissection analysis of soft biological tissues have been described by Gasser and Holzapfel (2003). They focus on the solid mechanical and structure aspects and the geometry of the artery, and, to capture the displacement discontinuity during arterial dissection, they employ the Heaviside function and consider an enriched displacement field. This is used to investigate the propagation of a dissection in a rectangular strip from a human aortic media. Recently, Wang et al. (2015) studied the conditions for tear propagation and arrest in a 2D strip using a finite-element approach. The propagation of dissection in an infinitely-long residually-stressed cylindrical tube subject to plane strain (without axial pre-stretch), was investigated numerically by Wang et al. (2017); they found regimes in which buckling of the inner wall of the dissection occurs, as is observed clinically. In addition, Wang et al. (2018) showed that deeper dissections are more likely to propagate and that propagation occurred preferentially along mutual axes with the greatest stiffness.

In this work, we treat the dissection as an axisymmetric tear in the wall of a nonlinear hyperelastic cylindrical tube model of a large artery, whose mechanical properties are described by the strain-energy function of the arterial wall, subject to residual stress and axial stretch as *in vivo*. We ignore the small connecting tear between the lumen and the main dissection, and instead maintain both at the same constant pressure. The dissection is linearised as the incremental deformation, and the incremental nominal stress is deduced from the strain energy function of the material. The equilibrium equations, boundary conditions and jump conditions for the static dissection are derived and solved numerically. We study the model’s dependence for different parameters in the strain-energy function, and determine their effects on the opening of the tear. In our analysis, we make use of studies of the axisymmetric tear problem in an isotropic linear elastic solid by Demir et al. (1992) and Korsunsky (1995), who use a jump condition to describe the displacement discontinuity for the crack. The singular stress-displacement field resulting from the introduction of a Somigliana ring dislocation is solved by Demir et al. (1992). The Burgers vector of this dislocation has two components, one being normal to the plane of the circular ring dislocation (Volterra type) and the other being in the radial direction of the ring dislocation everywhere (Somigliana type). The analytical solution, in terms of complete elliptic integrals of the first, second and third kinds, is obtained using the Love stress function and Fourier transformation. In Korsunsky (1995), the fundamental eigenstrain solutions are derived for axisymmetric crack problems.

The strain-energy function used to describe the media of a large artery has been given by Holzapfel et al. (2000). The media is modelled as a composite, reinforced by two families of collagen fibres which are arranged in symmetrical helices. The media and adventitia respond with similar mechanical characteristics and therefore the same form of strain-energy function (but a different set of material parameters) is used for each layer. In a healthy young arterial segment (with no pathological intimal changes), the thin innermost layer of the artery is not of mechanical interest. The structure of the media gives it high strength, resilience and the ability to resist loads in both the longitudinal and circumferential directions. From the mechanical perspective (Holzapfel et al. 2000), the media is the most significant layer in a healthy artery. Dissections usually happen in the media or between the media and adventitia; we focus on a dissection in the media, and model the wall as a single layer. The artery is taken to be incompressible since it does not change volume within the physiological range of deformation.

The paper is organised as follows. We introduce the solid mechanics theory in Sect. 2, including the strain-energy function of the media of a large artery given by Holzapfel et al. (2000), and then the concepts of residual stress and axial pre-stretch in the artery, followed by the ideas of arterial dissection and incremental moduli. Next the dissection is linearised as an incremental deformation, whose traction and displacement on the tear faces and vessel boundaries are expressed as the integrals of Green’s functions weighted by the displacement discontinuity along the tear. The Cauchy stress, nominal stress, and incremental nominal stress are deduced from the strain-energy function. The Green’s functions are found numerically by Fourier transform, and the displacements along the tear faces and along the inner and outer boundaries of vessel are calculated. The numerical methods are described in Sect. 4. In Sect. 5, we present our results, in terms of the changes in the width and shape of the dissection with changes in parameter values. An increase in blood pressure is modelled as an incremental pressure difference in Sect. 6, and the consequent change in the dissection is described. Conclusions are drawn in the final section.

## 2 Background

### 2.1 Strain-energy function

*c*is associated with the non-collagenous matrix of the material, and describes the isotropic part of the overall response of the tissue. The parameters \(k_1\), which is an elastic modulus, and \(k_2\), which is dimensionless, are associated with the anisotropic contribution of collagen to the overall response. The material parameters are independent of the geometry, opening angle and fibre angle, which are also different for the two layers. In Eq. (1),

### 2.2 Arterial dissection and incremental moduli

*W*is given, the nominal stress tensor \({\mathbf {S}}\) (which is the transpose of the first Piola–Kirchoff stress) and incremental nominal stress tensor \(\delta \mathbf {S}_0\) for an incompressible material are

*q*(the hydrostatic pressure) to enforce the constraint of incompressibility. Also in (5), \({\mathbf {B}}={\mathbf {A}}^{-T}\), \(\delta {\mathbf {A}}_0\) is the incremental deformation gradient in this configuration, \({\mathcal {A}}^1_{\alpha j\beta l}\) are the elastic moduli, and \({\mathcal {A}}_{0ijkl}^1\) are the incremental moduli. Since the tissue is taken to be incompressible, \(\mathrm {tr}(\delta {\mathbf {A}}_0) = 0\).

*In vivo*, the residual stress and axial pre-stretch influence the Cauchy stress and play an important role in maintaining an almost constant radial stress throughout the arterial wall. For an incompressible material, in which the deformation is given by (4), if we specify \(\lambda \) and the geometry (e.g. by specifying \(R_\text {in}\)) then there is only one equilibrium equation (the radial component) to be satisfied, derived from the equilibrium equation

*P*. It is equal to \(P_\text {ext}\) at the outer boundary and \(P_\text {in}\) at the inner boundary. \({\dot{P}}\) is the incremental pressure on the boundary that specifies the change in

*P*. Henceforth, for simplicity, we use \(\dot{\varvec{S}}_0\) to represent \(\delta {\varvec{S}}_0\) and \(\dot{q}\) to refer to \(\delta q\).

*u*and

*w*respectively and so the incremental deformation gradient is

## 3 Static tears for an axisymmetric incompressible artery

We take the stress-free artery with opening angle \(\alpha \) as the reference configuration, and the closed artery with residual stress as the current configuration. The dissection of the artery is idealised as the incremental elastic deformation on the configuration with residual stress. In the current configuration, we use \(r_c\) to represent the location of the dissection. We solve a fundamental Green’s problem in which there is a point displacement discontinuity located along the tear face, c.f. Demir et al. (1992) and Korsunsky (1995). A tear is then the convolution of this fundamental solution with a density of displacement discontinuity along the tear.

*u*, and

*w*. The normal and tangential traction components \((T_{r},T_{z})\) along the dissection, are

*U*(

*s*),

*W*(

*s*)) is the Green’s displacement discontinuity and

*s*is the arc length along the axial tear. We calculate \({{\dot{S}}}^u_{0rr}\), \({{\dot{S}}}^u_{0rz}\), \(u^u\) and \(w^u\) and \({{\dot{S}}}^w_{0rr}\), \({{\dot{S}}}^w_{0rz}\), \(u^w\) and \(w^w\) separately as follows (Li 2013). Firstly, from the strain-energy function \(\Psi \), we obtain the nominal stress \(\varvec{S}\) from Eq. (5) and the Cauchy stress, which is a function of the nominal stress. Next the incremental nominal stress \({\dot{\varvec{S}}}_0\) is calculated from Eq. (5). The Cauchy stress \(\varvec{\sigma }\) and the incremental nominal stress \({\dot{\varvec{S}}}_0\) are functions of displacements \(u^u(r,z)\), \(w^u(r,z)\) or \(u^w(r,z)\), \(w^w(r,z)\). Writing the equilibrium equations \(\mathrm {div}\,\varvec{\sigma }=0\) and \(\mathrm {div}{\dot{\varvec{S}}}_0=0\) in components, we obtain partial differentiation equations with variables \(u^u(r,z)\), \(w^u(r,z)\) or \(u^w(r,z)\), \(w^w(r,z)\). We Fourier transform these PDEs into ODEs depending on the wave number

*k*, with transformed variables \({\hat{u}}^u(r,k)\), \({\hat{w}}^u(r,k)\) or \({\hat{u}}^w(r,k)\), \({\hat{w}}^w(r,k)\). The ODEs are solved subject to boundary conditions and jump conditions using a collocation method. Finally, the inverse Fourier transforms are taken to obtain the solution of the original PDEs. Then \({\dot{\varvec{S}}}_0\) is given as a function of the displacement components \(u^u(r,z)\), \(w^u(r,z)\) or \(u^w(r,z)\), \(w^w(r,z)\), in terms of the incremental stress components \({\dot{S}}^u_{0rz}, {\dot{S}}^u_{0rr}\) or \({\dot{S}}^w_{0rz}, {\dot{S}}^w_{0rr}\).

*U*(

*s*),

*W*(

*s*)) is solved from Eqs. (18) and (19), and then the displacement (

*u*,

*w*) in Eqs. (20) and (21) can obtained. This method is used to calculate the displacements for the upper and lower dissection faces, and for the inner and outer boundaries.

### 3.1 Fourier transformations

*u*’) and tangential directions (a jump in ‘

*w*’). The relevant Fourier sine and cosine transforms in the

*z*direction and their inverses are defined as

### 3.2 Jump in *w*

*z*, we express the displacements and the stresses asand define

*r*,

*k*, material parameters and deformation parameters. (See Li 2013 and Supplementary Material for further details.) The jump conditions after Fourier transformation are

*r*,

*k*, material parameters and deformation parameters (Li 2013).

### 3.3 Jump in *u*

*r*,

*k*, the material parameters and deformation parameters (see Li 2013 and Supplementary Material). The Fourier-transformed jump conditions yield

*r*,

*k*, material parameters and deformation parameters.

## 4 Numerical solution

The quantities \({{\hat{U}}}^{w}\), \({\hat{W}}^{w}\), \(\hat{{\dot{S}}}_{0rr}^{w}\), \(\hat{{\dot{S}}}_{0rz}^{w}\), \(\hat{{\dot{S}}}_{0rr}^{u}\), \(\hat{{\dot{S}}}_{0rz}^{u}\), \({\hat{U}}^{u}\) and \({\hat{W}}^{u}\) describe the response to a point discontinuity in the radial (superscript *u*) and axial (superscript *w*) directions. These solutions can be used to formulate integral equations that relate the loading on the tear surfaces to the opening of the tear. In this section we describe a numerical method for the approximation of the integral equations that connect opening and loading. In order to achieve this, we must approximate the solutions to Eqs. (24)–(32). These equations, and their appropriate boundary conditions, depend parametrically on the wavenumber *k*; the case \(k=0\) is a special case in which some of the coefficients in Eqs. (24)–(32) are zero. The procedure described below still applies to the case \(k=0\) but with modified governing equations.

As illustrated in Fig. 4, we identify two regions; region 1 is \([r_\text {in} , r_c]\) and region 2 is \([r_c, r_\text {out}]\). In region 1 we define \(r=r_1={r_\text {in}}+R({r_c-r_\text {in}})\), and in region 2 \(r=r_2={r_\text {out}}+R({r_c-r_\text {out}})\). The range of *R* is [0, 1]. This allows us to transform all equations to be defined on the single domain \(R\in [0,1]\). The boundaries (the internal boundary for region 1 and the external boundary for region 2) are both located at \(R=0\), while \(R=1\) represents the tear faces in *both* regions (though different faces depending on whether we are in region 1 or 2). The equations and boundary conditions (24)–(32) are written in terms of the new variable *R* and solved on the domain [0, 1] using the MatLab routine bvp4c. For a discrete set of wavenumbers *k*, we obtain approximations to \({\hat{U}}^{w}(r,k), {\hat{W}}^{w}(r,k), {\hat{U}}^{u}(r,k), {\hat{W}}^{u}(r,k)\) and \(\hat{{\dot{S}}}_{0rr}^{w}(r,k)\), \(\hat{{\dot{S}}}_{0rz}^{w}(r,k)\), \(\hat{{\dot{S}}}_{0rr}^{u}(r,k)\), \(\hat{{\dot{S}}}_{0rz}^{u}(r,k)\).

*j*refers to an integral over the interval \({\left( z_j-\Delta ,z_j+\Delta \right) }\). The other terms are treated similarly; see Li (2013) and the Supplementary Material. Writing \({{\dot{S}}}_{0rr}^w\) in terms of its transform and changing the order of integration gives

*k*, and chosen so that \({\tilde{{\dot{S}}}_{0rr}^w}\rightarrow 0\) as \(k\rightarrow \infty \) to ensure convergence of the quadrature used to evaluate the integrals involving \(\tilde{\dot{S}}_{0rr}^w\).

### 4.1 Conditions at the tear face

### 4.2 Convergence of the Numerical Solution

Convergence of numerical scheme with maximum wave number (\(k_\text {max}\)) and number of intervals (\(N_k\)) in \([0, k_\text {max}]\)

\(k_\text {max}\) | \(N_k\) | Maximum dissection width |
---|---|---|

10 | 300 | 0.233178 |

10 | 500 | 0.233256 |

20 | 1000 | 0.233259 |

30 | 500 | 0.232907 |

30 | 1000 | 0.233203 |

30 | 1500 | 0.233259 |

Convergence of numerical scheme with \(N_k\) for fixed maximum wave number \(k_\text {max} = 10\)

\(N_k\) | Maximum dissection width |
---|---|

500 | 0.233256 |

600 | 0.233270 |

700 | 0.233278 |

800 | 0.233284 |

900 | 0.233287 |

1000 | 0.233290 |

1200 | 0.233293 |

1500 | 0.233296 |

### 4.3 Validation of the numerical scheme

## 5 Results

The following results are for dissection profiles with geometric parameters \(r_\text {in}=4\,\mathrm {mm}\), \(r_c=5\,\mathrm {mm}\), \(r_\text {out}=6\,\mathrm {mm}\) and \(R_i=3.9\,\mathrm {mm}\), material parameters \(k_1=2.3632\,\mathrm {kPa}\), \(k_2=0.8393\), and \(c=3\,\mathrm {kPa}\), and outer boundary pressure \(P_\text {ext}=0\,\mathrm {kPa}\). This choice of geometric parameter values ensures that there are no wrinkles on the inner boundary when the artery is deformed from the stress-free reference configuration \(\Omega _0\) to the current \(\Omega \) with residual stress and axial stretch (see Fig. 3), and are typical of a large artery. The material parameters are those used in Figure 14 of Holzapfel et al. (2000) for a rabbit carotid artery. In Figs. 6, 7 and 8, we give examples of the effects of varying the axial pre-stretch \(\lambda \), the opening angle \(\alpha \) that specifies the residual stress, and the fibre angle \(\beta \).

The fibre angle \(\beta = 30^{\circ }\) in Fig. 8, as in Fig. 7, and we show dissection profiles for three values of the axial pre-stretch \(\lambda \). Increasing \(\lambda \) is shown to lead to a narrowing of the tear as the increasing tension in the collagen fibres resists the deformation. Here the opening angle is \(\alpha = 30^{\circ }\) so that, by comparing the shape of the dissection for \(\lambda = 1.1\) with that in Fig. 7 for which \(\alpha = 45^{\circ }\), we see that a smaller opening angle (i.e. a reduced residual stress) leads to a narrower dissection. This result has been confirmed for other parameter values by Li (2013), in which it was also shown that decreasing \(r_{\text {in}}\) increases the width of the dissection.

## 6 Incremental blood pressure changes

*u*,

*w*) become

*u*,

*w*) and \({{\dot{S}}}_{0rr}^u, {{\dot{S}}}_{0rz}^u, {{\dot{S}}}_{0rr}^w, {{\dot{S}}}_{0rz}^w\), and \(u^u\), \(w^u, u^w, w^w\) are defined in (23) and (28). We must additionally obtain \({{\dot{S}}}_{0rr}^{P}\) and \(u^{P}\), which are the radial components of displacement and stress at the tear face as a result of applying the incremental pressure.

*q*is the hydrostatic pressure,

*A*(

*r*) and

*B*(

*r*) are known functions of

*r*and

*k*(related to the moduli of the material) and of the material and deformation parameters (see Li 2013 and the Supplementary Material). As in Sect. 4, we solve these equations numerically by splitting the domain \([r_\text {in},r_\text {out}]\) into the two regions \([r_\text {in},r_\text {c}]\) and \([r_\text {c},r_\text {out}]\) and transforming both onto the interval \(R=[0,1]\) so that \(R=0\) corresponds to the inner/outer boundary of the tube and \(R=1\) corresponds to the tear face (approached from either the inner or outer section).

*a*(

*r*),

*b*(

*r*) are known functions of

*r*,

*k*, material parameters, and deformation parameters (see Li 2013 and the Supplementary Material). The jump conditions are

*r*(Li 2013). Hence, once the traction \((T_r,\,T_z)\) is given, we are able to obtain the displacement \((u,\,w)\) for the upper and lower dissection faces, and the inner and outer boundaries. The traction on the tear is

## 7 Conclusions

We have derived and solved a mathematical model for the incremental deformation of a prescribed axisymmetric tear in an idealized large artery, which is a single-layer, thick-walled nonlinear incompressible axisymmetric hyperelastic tube with residual stress and two families of collagen fibres, all described by an HGO strain-energy function (Holzapfel et al. 2000).

From the results, we conclude that for the HGO material, subject to residual stress and axial pre-stretch, a dissection is widened by increasing the blood pressure \({\dot{P}}\) within the lumen and the dissection, decreasing the radius of the lumen \(r_{\text {in}}\), decreasing the fibre angle \(\beta \), by decreasing the axial pre-stretch \(\lambda \), and increasing the opening angle \(\alpha \). The dependence on \(\alpha \) implies that higher values of residual stress promote wider dissections which is unexpected and counters the effect of increasing the axial pre-stretch. We also note that displacements at the inner surface of the wall are greater than at the outer surface.

Immediate extensions of this work are to consider the full two-layer model of (Holzapfel et al. 2000), and to include a multi-cut model in which the residual stress is described by two or more cuts of the unloaded tube (Omens et al. 2003). An important limitation is that the model does not address the propagation of a dissection. To do this, a macroscopic description of the tearing, such as a cohesive zone model, is required, and the direction of tear propagation will be constrained by the layering in the wall of the artery.

We conclude by noting that this work gives insights into the key mechanisms that govern the shape of a tear in the arterial wall and can be used to inform and validate future fully-numerical models in realistic geometries.

## Notes

### Funding

Beibei Li and Lei Wang were supported by stipends from the Chinese Scholarship Council and fee waivers from the University of Glasgow. Part of this work was supported by UK Engineering and Physical Sciences Research Council (Grant No. EP/N014642/1) on which Nicholas Hill and Xiaoyu Luo are co-investigators.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## Supplementary material

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