Traction-separation law parameters for the description of age-related changes in the delamination strength of the human descending thoracic aorta

doi:10.21203/rs.3.rs-4131565/v1

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Traction-separation law parameters for the description of age-related changes in the delamination strength of the human descending thoracic aorta

https://doi.org/10.21203/rs.3.rs-4131565/v1

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Aortic dissection is a life-threatening disease that consists in the development of a tear in the wall of the aorta. The initial tear propagates as a discontinuity leading to separation within the aortic wall, which can result in the creation of a so-called false lumen. A fatal threat occurs if the rupture extends through the whole thickness of the aortic wall, as blood may then leak. It is generally accepted that the dissection, which can sometime extend along the entire length of the aorta, propagates via a delamination mechanism. The aim of the present paper is to provide experimentally validated parameters of a mathematical model for the description of the wall’s cohesion. A model of the peeling experiment was built in Abaqus. The delamination interface was described by a linear traction-separation law. The bulk behavior of the aorta was assumed to be nonlinearly elastic, anisotropic, and incompressible. The numerical values of the material parameters of the bulk constitutive model were adopted from the literature. Our simulations resulted in estimates of the material parameters for the traction-separation law of the human descending thoracic aorta, which were obtained by minimizing the differences between the FEM predicted delamination force and the force observed in the peeling experiment. The results show that, since delamination strength decreases with age, the traction-separation law parameters are also age-dependent. The material parameters provided by our study can be used in numerical simulations of the biomechanics of dissection propagation through the aorta especially when age-associated phenomena are studied.

Ageing

Cohesive model

Crack

Damage

Finite Elements Method

Fracture

The aorta is the largest artery in the human body and its main function is to take up all the blood ejected by the left ventricle and to conduct it to the branches leading to all parts of the body (Kassab 2007; Wang et al. 2023). Since the heart beats ceaselessly, the aorta faces challenging mechanical conditions at which it is under continual loading. Although its anatomical structure makes the aorta well prepared to perform its function, there are diseases that threaten its mechanical integrity and can lead to its failure (Thubrikar 2007). One of these diseases is aortic dissection, which consists in delamination of the tissue of the aortic wall so that a crack spreads along the axis and circumference of the aorta (Thubrikar 2007; Tong et al. 2016, Sherifova and Holzapfel 2019, 2020; Amabili et al. 2020; Wang et al. 2023).

From a mechanical point of view, dissection develops in a number of typical stages (Sherifova and Holzapfel 2019, 2020). First, the onset of the dissection that occurs when an initial tear is formed on the inner side of the aortic wall in the layer referred to as tunica intima. As blood enters the arterial wall, the tear spreads radially and a delamination process subsequently begins, typically in the middle layer of the artery referred to as the tunica media. The structure of the media is lamellar, and during the delamination one lamella separates from the other, causing the crack to spread further in the circumferential and axial directions, and new crack tips are formed. Finally, one or more of the following complications may occur: (1) blood flow in newly created (false) lumen can cause a blockage or a complete closure of an artery that branches out of the aorta leading to a decreased perfusion in the terminal tissue, or (2) the crack tip can propagate through the weakened aortic wall in a radial direction to the tunica adventitia until it reaches the outer surface of the aorta. The first case is dangerous, especially due to the possible blockage of the main branches of the coronary arteries (Prêtre and Von Segesser 1997), while the second means that blood starts to leak out of the aorta (Tanaka et al 2009).

The delamination mechanism described above suggests a key role for the radial integrity of the arterial wall. The organization of the internal structure itself has been studied by Wolinksy and Glagov (1967) and Clark and Glagov (1985), who described in detail the musculo-elastic fascicles that define the lamellar structure of the media. Radial mechanical properties that reflect interlamellar cohesion were studied in a direct tension test by MacLean et al. (1999), Sommer et al. (2008), and Tong et al. (2011). It was concluded that the strength in the radial direction is significantly lower than the strength in circumferential and axial direction (MacLean et al. 1999).

An important series of experiments focused on the cohesion of the arterial wall was carried out by M. R. Roach and her colleagues. They studied dissection propagation in experiments which were based on pumping a pressurized liquid into the blood vessel wall (van Baardwijk and Roach 1987; Carson and Roach 1990; Tiessen and Roach 1993). It was revealed that pressure gradient, rather than peak pressure, is correlated with the dissecting, which is in accordance with the results of Prokop et al. (1970). It was also found that dissections propagated more easily between the elastic lamellae than across them.

In the current biomechanical literature, there is a growing number of papers investigating delamination strength using the peeling experiment (Sommer et al. 2008; Tong et al. 2011, 2014; Pasta et al. 2012; Kozuń 2016; Kozuń et al. 2018; Myneni et al. 2020; Horný et al., 2022). The peeling experiment consists of creating a T-shaped specimen from the strip of an artery, which is incised in its thickness so as to achieve delamination by pulling on the arms of the T specimen (Fig. 1). In the experiment, the force required to advance the crack tip along the artery strip is measured and the delamination strength is then defined as this force divided by the width of the specimen. The experimental procedure resembles mode I crack opening and the fracture energy can be obtained in this way.

It has been found that delamination strength is a site-specific property (Tong et al. 2011; Myneni et al. 2020; Horný et al. 2022; Sokolis and Papadodima 2022; Ríos-Ruiz et al. 2022) which is in accordance with previous results obtained by Roach and Song (1994). It has also been found that delamination resistance is anisotropic because longitudinally oriented strips usually exhibit a higher delamination strength than circumferential strips (Sommer et al. 2008; Tong et al. 2011; Kozuń et al. 2016; Horný et al. 2022). Studies by Pasta et al. (2012), Angouras et al. (2019), Chung et al. (2020), Kozuń et al. (2018), Tong et al. (2023) have shown that pathologies like ascending aortic aneurysm or atherosclerosis are associated with a decreased delamination resistance of the aorta.

Significant attention has recently been paid to the mathematical modelling of cohesive properties and the numerical simulations that attempt to reproduce the propagation of dissection or delamination through the arterial wall (Gasser and Holzapfel 2003, 2006; Ferrara and Pandolfi 2010; Pal et al. 2014; Wang et al. 2018; Noble et al. 2017). The studies differ in the approaches used to introduce crack propagation. Some of them work with XFEM (eXtended Finite Element Method) where crack can propagate in any direction, but others instead use predefined interfaces through which cracks are spread. These interfaces can be based on cohesive zone (CZ) elements or on contact bond models. The fact that the location of crack propagation is known in advance is advantageously used in computational analyses of peeling experiments (Leng et al. 2015, 2016; Merei et al. 2017; Miao et al. 2020; Ríos-Ruiz et al. 2021).

However, compared to the studies that report peeling experiment results, there are still few studies that identify the parameters for the mathematical models of cohesion based on experimental data. In particular, studies dealing not with animal models but with the human aorta are rare. This limits the computational capability of predictions made for humans in biomechanical modelling. This is underscored by the fact that the physiological properties of the circulatory system, including the mechanical properties of our arteries, change significantly in humans with age and with the onset of various diseases. To the best of our knowledge, the current literature lacks experimentally validated material parameters for describing human aortic cohesion which would account for aging. Their estimation based on the use of the FEM model of the peeling experiment is the main objective of the present study.

The main objective of our study is to obtain experimentally validated estimates of the constitutive parameters for a model of cohesion at the delamination interface of the human descending thoracic aorta that would reflect age-related changes in aortic biomechanics. The displacements and forces measured during the peeling experiment play a role of observation in our study. The material parameters for the linear traction-separation law, expressing mathematically the delamination behavior, will be calibrated against the measured delamination force using the Finite Element Method (FEM) model of the peeling experiment. Figure 2 shows the phases that the FEM model goes through during the simulation of the peeling experiment. The model was built as 3D and the computational complexity was reduced by introducing the plane symmetry of the problem (cf. Figure 1 and Fig. 2). During loading, the FEM model undergoes a phase of purely elastic behavior (bending and stretching) that eventually transitions to the separation phase, which is imposed by kinematical loading set as displacement u at the end of the T-shaped specimen. The details of this procedure are expressed in the following paragraphs.

2.1 Elastic behavior

The bulk material of the aorta was considered to be hyperelastic, anisotropic and incompressible. The Holzapfel-Gasser-Ogden (HGO) model of the strain energy density function W proposed in Holzapfel et al. (2000) was employed to describe the nonlinear elastic response of the aortic wall. It can be expressed as follows

$$W=\frac{\mu }{2}\left( {{I_1} - 3} \right)+\frac{{{k_1}}}{{2{k_2}}}\sum\limits_{{j=4,6}} {\left( {{e^{{k_2}{{\left( {{I_j} - 1} \right)}^2}}} - 1} \right)}$$

where µ and k₁ are stress-like material parameters and k₂ is dimensionless. I₁ is the first principle invariant of the right Cauchy-Green deformation tensor C which is obtained from the deformation gradient F as C = F^TF. The Neo-Hooke term in (1) is assumed to be associated with the strain energy stored in the isotropic part of the extracellular matrix and smooth muscle cells, whereas the exponential term is assumed to account for the energy stored within the deformation of a network of collagen fibers that are accepted to be a cause of the anisotropic behavior of the aortic wall. In our specific case, the anisotropy is introduced by the existence of two preferred directions that are given by unit vectors M₁ and M₂ that lie in the Θ-Z cylindrical plane of the blood vessel. Vectors M₁ and M₂ are considered to be placed symmetrically with respect to the Θ axis by an angle ± β. The invariants I₄ and I₆ are given by following equations: I₄ = M₁·CM₁, and I₄ = M₂·CM₂. More details about this model can be found in Holzapfel et al. (2000), Gasser et al. (2006), or Holzapfel and Ogden (2010). Readers interested in more information on the construction of additional invariants can refer to Spencer (1982), Holzapfel (2000), Itskov (2007), or, for example, Truesdell and Noll (2004).

2.2 Cohesive behavior

Since our model simulates crack propagation in the peeling experiment, it means that the crack propagation location is known a priori and there is no need to introduce advanced concepts like XFEM and similar techniques (Belytschko and Black, 1999; Mohammadi, 2008; Belytschko et al., 2009; Gasser and Holzapfel, 2003; Wells et al., 2002; 2015; Wang et al., 2018). An approach referred to as cohesive contact was used to simulate aortic wall cohesion (Abaqus, 2019). In this approach, the contact area represents the interface at which the cohesion of the artery wall fails (Fig. 2). In Abaqus, a traction-separation (T-S) law can be assigned to the contact area, including the conditions of damage initiation and damage evolution.

The T-S law represents the constitutive equation for cohesive behavior and brings together traction (stress) and separation (displacement) vectors at the cohesive interface. In our study, linear T-S was employed (Foresell and Gasser 2011; Merei et al. 2017; Ríos-Ruiz et al. 2021). Its elastic part is expressed in (2). Here T represents the nominal traction vector, which in 3D has one normal (n) and two shear (s,t) components. The separation vector is denoted δ and K represents stiffness of the bond. In our particular case, K matrix without off-diagonal terms expresses a so-called uncoupled approach in which normal and shear separation behaviors do not affect each other.

$${\varvec{T}}=\left( {\begin{array}{*{20}{c}} {{T_n}} \\ {{T_s}} \\ {{T_t}} \end{array}} \right)=\left( {\begin{array}{*{20}{c}} {{K_{nn}}}&0&0 \\ 0&{{K_{ss}}}&0 \\ 0&0&{{K_{nn}}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{\delta _n}} \\ {{\delta _s}} \\ {{\delta _t}} \end{array}} \right)=\mathbf{K}\varvec{\delta}$$

Let’s restrict our attention to the case where the separation occurs as a result of a force interaction expressed by the normal component of the stress vector. Figure 3b shows that the value of T first increases linearly up to the value of T_c. T_c is the critical value (material parameter) at which damage initiation occurs. Using Eq. (2), it can also be expressed in terms of the separation δ as

T _c = K_nnδ_c. Initiation of the damage means that T-S law starts to decrease and follows the line segment connecting [δ_c,T_c] and [δ_f,0] (Fig. 3b). Thus, when the separation displacement reaches the value of δ_f (material parameter), permanent separation occurs and the value of the stress carried by the bond falls to zero.

$$\hbox{max} \left\{ {\frac{{{T_n}}}{{{T_{cn}}}},\frac{{{T_s}}}{{{T_{cs}}}},\frac{{{T_t}}}{{{T_{ct}}}}} \right\}=1$$

Regarding the initiation of the damage, the initiation condition employed in our study can generally be written in the form of Eq. (3). T_cn, T_cs, and T_ct here represent the material characteristics of cohesion for the normal and shear components of the stress vector. Damage D is a monotonically increasing scalar-valued function satisfying 0 ≤ D ≤ 1, where 0 corresponds to an undamaged (elastic) state, and 1 is reached when the bond fails. The damage evolves when loading continues beyond δ_c (δ > δ_c, see Fig. 3) and the decreasing phase of T-S governs mechanical response. The components of the damaged stress vector are given by equations (4).

$${T_n}=\left( {1 - D} \right){T^{\prime}_n}{\text{ for }}{T^{\prime}_n}>0{\text{ and }}{T_n}={T^{\prime}_n}{\text{ otherwise, }}{T_s}=\left( {1 - D} \right){T^{\prime}_s},{\text{ }}{T_t}=\left( {1 - D} \right){T^{\prime}_t}$$

Here T'_k (k = n,s,t) are components of the stress vector predicted by the elastic part of the T-S law and D is expressed with the help of (5) where δ_max is maximum δ attained within a loading history and in general it is given by (6) where $\left\langle x \right\rangle$ are so called Macaulay brackets ($\left\langle x \right\rangle$= 0 for x < 0 and $\left\langle x \right\rangle$= x for x ≥ 0).

$$D=\frac{{{\delta _f}\left( {{\delta _{max}} - {\delta _c}} \right)}}{{{\delta _{max}}\left( {{\delta _f} - {\delta _c}} \right)}}$$

$$\delta =\sqrt {{{\left\langle {{\delta _n}} \right\rangle }^2}+\delta _{s}^{2}+\delta _{t}^{2}}$$

The size of the area of the triangle determined in Fig. 3b by the vertices [0,0], [δ_c,T_c], [δ_f,0] can also be understood as a material characteristic that one can use to express the cohesive properties of the material. The size of this area is denoted G_c and represents the areal density of fracture energy, i.e., the areal density of the energy dissipated when the crack tip extends further in an infinitesimal step. G_c can be set as a parameter of the cohesive contact in Abaqus.

2.3 Finite element model

The software Abaqus (version 2019, Dassault Systèmes) was used to create the numerical model and to perform the simulations (Abaqus/Standard, inertial effects not considered). The model was built as 3D and plane symmetry was used to reduce the complexity of the problem. Figures 2 and 3a show the geometry and the position of the delamination interface created with the help of the contact area where T-S law was assigned. The overall dimensions of the model were 10 mm x 0.5 mm (length x thickness of the strip) and the width was set to be equal to the width of one element. Let us add that we use the term thickness here in the same way as when talking about the thickness of a blood vessel (the in-plane dimensions in Fig. 3 are length and thickness, and width is an out-of-plane dimension). The final results, which will be presented below, correspond to the width of 0.0125 mm. The model was discretized with 371 657 hybrid linear tetrahedrons C3D4H. A loading was applied by means of a kinematic boundary condition. A displacement u(t) perpendicular to the length of the delaminated strip was prescribed at the end of the strip (see Fig. 2). During the delamination phase, the displacement perpendicular to the delamination interface corresponds to the length by which the delamination interface is shortened (Fig. 2). In order to ensure the stability of the geometry during loading, the width of the model was prescribed to be constant within the loading (Gasser and Holzapfel, 2006; and Ferrara and Pandolfi, 2010).

2.4 Calibration of T-S law parameters

The aim of our study was to estimate the material parameters of the T-S law, which will be validated against the measured delamination forces. As described above, the FEM simulation of the peeling experiment was used as a regression analysis tool. The specific delamination force values to which the material parameters were fitted were adopted from our previous study Horný et al. (2022). In Horný et al. (2022), a total of 661 delamination experiments were performed with human aorta samples obtained from 46 cadavers. Here, we will limit ourselves to the results obtained for the descending thoracic aorta, as it represents one of the most dangerous sites for dissection propagation. Horný et al. (2022) conducted a total of 246 peeling experiments with samples obtained from the descending part of the thoracic aorta. The main conclusions yielded in their work are (1) cohesive properties are location-specific, (2) they are relatively independent of the extension rate (in the range 0.1–50 mms^− 1), (3) the delamination strength depends on the direction in which the crack tip propagates, and (4) the delamination strength depends on age. Table 3 in Horný et al. (2022) presents the parameters of the linear regression equation for the relationship between the delamination strength (F/w [Nmm^− 1], the delamination force per unit of the width) and age (Age [year]), F/w = a·Age + b. The parameters a and b have values as follows:

a = -0.00031 N/(year·mm) b = 0.051 N/(mm) for the strip of the aorta oriented in the axial direction of the blood vessel, and a = -0.00019 N/(year·mm) b = 0.036 N/(mm) for the strip that is oriented in a circumferential direction. In the present study, forces predicted by the above described equations were used to calibrate the parameters of the T-S law.

It remains to specify the age for which the calibration simulations were carried out. This was decided by the constitutive description adopted for the bulk material. The material parameters presented in Jadidi et al. (2020), who reported the results of mechanical tests performed with human thoracic descending aorta samples obtained from 78 donors, were used. Table 1 shows both the ages for which our simulations were performed and the specific numerical values of the parameters for W expressed by (1).

The elastic properties of the aorta, as well as its delamination strength, are anisotropic. This property is shared by model (1) and its numerical characterization in Jadidi et al. (2020), as well as by the equation for age-related changes in the delamination strength adopted from Horný et al. (2022). In the version of Abaqus used (Abaqus 2019), however, it was not possible to define a cohesive contact with an anisotropic T-S law. For this reason, simulations were split into three cases. The first one represents the delamination of an axially oriented strip (the crack face advances axially) calibrated against the delamination force obtained from an identically arranged experiment. The second case represents the same but for the circumferential direction (the circumferential strip and the delamination strength measured for it). The third case represents the calibration against the delamination force averaged from the axial and circumferential experiments. In this averaged case, the T-S law parameters were tuned so that the difference between the average of the force predicted by the simulation done with axial and circumferential strip and the average calculated form force measured in the axial and circumferential peeling test is minimal.

Table 1

– Age-related HGO model parameters of the bulk description of the descending thoracic aorta from (Jadidi et al. 2020).
Age [year]	µ [kPa]	${k}_{1}$ [kPa]	${k}_{1}$ [-]	$\beta$ [°]
15.3	41.69	1.20	2.56	33.82
24.4	41.82	1.78	2.61	35.54
36.0	40.32	0.16	6.58	61.34
46.5	44.94	0.24	9.33	64.20
54.9	49.91	0.22	16.17	65.35
66.5	51.13	0.34	17.83	63.63
73.2	51.68	0.51	27.99	60.76

Our previous experiments, against which the T-S law parameters were tuned, do not include observations of shear behavior during delamination. For this reason, we are restricting our study to the 1D case of the T-S law for the normal direction. This means that the critical value of T_c will denote the normal component of the nominal stress vector in all the following cases, and therefore we will drop the use of the subscript n in the following, i.e., T_cn = T_c. The same applies for the separation vector δ, where

δ = δ_n. As explained above, the linear T-S law can be described by the parameters K, δ_c, δ_f, T_c. However, this overdetermines the T-S law and the parameters are not independent. After pilot simulations aimed at finding conditions under which the FEM problem would converge well, it was decided to choose the value of K fixed at K = 0.5 MPa/mm for all cases. The same applies to the characterization of the separation. Abaqus operates with the quantity δ_f - δ_c, which was held constant at δ_f - δ_c = 0.1 mm. This left T_c as the only free parameter. Our choice is related to the fact that the quantity whose error is minimized is the delamination force, which was measured in the experiment. Therefore, calibration against T_c as a force quantity was chosen. The value of G_c is then calculated from the known K, δ_f - δ_c and T_c.

The results of the T-S law parameter calibration for the axially and circumferentially oriented strips are shown in Fig. 4. For the axially oriented strips, shades of blue are used, whereas for the circumferential strips, shades of red are used. The specific numerical values of T_c are given in Table 2. The monotonically decreasing height of the T_c vertices in the T-S law triangles in Fig. 4a indicates the dependence of T_c on age. The agreement between the experimentally measured delamination strength and the delamination force (per unit width) calculated by the FEM model using the estimated T-S law parameters (Table 2) is demonstrated in Fig. 4b. The FEM predicted values (isolated points) lie on the regression lines obtained in the regression analysis of the experiments. The agreement is very good. The panels of Fig. 4c and 4d show how the value of the force reaction varies with the phase of the experiment (elastic vs. delamination).

Table 2

–Age-dependent sets of T_c for axially (T_c^ax) and circumferentially (T_c^circ) oriented strip. Note that the full T-S law description includes K = 0.5 MPa/mm, and δ_f - δ_c = 0.1 mm.
Age [year]	15.3	24.4	36.0	46.5	54.9	66.5	73.2
T_c^ax [MPa]	0.230	0.218	0.200	0.186	0.175	0.163	0.155
T_c^circ [MPa]	0.177	0.171	0.165	0.155	0.148	0.139	0.134

Table 3

– Age-dependent set of T_c parameters for averaged description of the delamination strength. Note that the full T-S law description includes K = 0.5 MPa/mm, and δ_f - δ_c = 0.1 mm.
Age [year]	15.3	24.4	36.0	46.5	54.9	66.5	73.2
T_c [MPa]	0.201	0.193	0.182	0.172	0.163	0.152	0.145

The results for the averaged model are shown in Fig. 5 and the T_c values for this case are shown in Table 3. The vertices [δ_c,T_c] in Fig. 5a again show that the critical value of the normal component of the stress vector decreases, which is a consequence of age-related changes in the delamination strength. This is then evident in Fig. 5b, which documents the comparison between the regression lines for delamination strength obtained from the experimental data (blue and red line) and the predicted delamination strength for the average of the FEM model for the axial strip and of the FEM model for the circumferential strip (average indicated with gold line, individual predictions blue and red isolated points). The averaged T-S law parameters predict delamination strength in the middle between axial and circumferential behavior.

For the sake of completeness, the values of the fracture energy G_c are also presented. The magnitude of G_c always corresponds to the area of the triangle expressing the T-S law. Table 4 summarizes all the investigated cases and also documents the correlation of the fracture energy with age, which is obviously a consequence of the correlation between age and delamination strength found in the experiments in Horný et al. (2022).

Table 2

– Age-dependent sets of energy release rates G_c and linear regressions with respect to age (Fig. 6b). R² is a coefficient of determination.
Age [year]	G_c [mN/mm]	G_c^ax [mN/mm]	G_c^circ [mN/mm]
15.3	50.5	64.4	40.2
24.4	46.9	58.4	37.8
36.0	42.2	50	35.5
46.5	38.2	43.9	31.8
54.9	34.7	39.4	29.3
66.5	30.7	34.7	26.3
73.2	28.3	31.8	24.7
	− 0.077·Age + 11.220	− 0.113·Age + 14.332	− 0.055·Age + 8.919
R²	0.999	0.998	0.996

Our study provides experimentally validated estimates of the T-S law parameters for the human thoracic descending aorta. A FEM model for the peeling experiment was built in Abaqus (version 2019). Predictions of delamination force were compared with delamination forces measured in our previous experiments (Horny et al., 2022). The T-S law parameter estimates were set to achieve a convergence of the FEM problem and at the same time, to ensure that the difference between the FEM-predicted delamination strength (delamination force per strip width) and the delamination strength observed in the experiments was minimal. The linear T-S law used in our study can be generally described by K, δ_c, T_c, δ_f, and G_c, of which only three are independent of each other. In our study, two of the three parameters, K and δ_f - δ_c, were fixed so that the problem would converge (K = 0.5 MPa/mm,

δ _f - δ_c = 0.1 mm). The value of T_c was then calibrated to the delamination strengths given by the regression of the experiments. The determined T_c values were in the interval [0.134,0.230] MPa, with an average T_c equal to 0.173 MPa.

The reason for calibration against T_c rather than deformation quantities such as δ_c and δ_f, and stiffness K is that in peeling experiments the force required for delamination is the primarily measured quantity. The delamination force is also well interpretable with respect to an onset of the crack propagation. In contrast, the cohesive bond stiffness K or the deformation quantities δ_c and δ_f do not have direct analogues in peeling experiments. In Fracture Mechanics, it is common to use the fracture energy G_c to characterize the loss of cohesion during crack propagation (Irwin and Wells 1965; Zehnder 2012). However, this is again not a primary quantity in the peeling experiment, as it is calculated from the observed delamination strength and delaminated length (Tong et al. 2011).

The last decade has seen an increasing number of studies dealing with the experimental determination of the delamination strength of the human aorta and other arteries (Sommer et al. 2008; Tong et al. 2011, 2014; Pasta et al. 2012; Kozuń 2016; Kozuń et al. 2018; Myneni et al. 2020; Horný et al. 2022). These studies have revealed the association between delamination strength and the position of the artery in the circulatory system (Tong et al. 2011; Myneni et al. 2020; Horný et al. 2022; Sokolis and Papadodima 2022; Ríos-Ruiz et al. 2022), between delamination strength and the direction of the crack tip propagation (Sommer et al. 2008; Tong et al. 2011; Kozuń et al. 2016; Horný et al. 2022), between delamination strength and various types of disease (with particular reference, of course, to aortic dissection and dissecting aneurysms; Pasta et al. 2012, Angouras et al. 2019, Chung et al. 2020, Kozuń et al. 2018, Tong et al. 2023), and finally, and most importantly for our present study, between delamination strength and age (Horný et al., 2022).

The situation is somewhat different for papers that provide experimentally validated material parameters for mathematical models of aortic wall cohesion. These works are less frequent and authors also use various concepts to introduce a mathematical description of the cohesion and its failure (Gasser and Holzapfel 2006; Ferrara and Pandolfi 2010; Merei et al. 2017; Leng et al. 2018; Wang et al. 2018; Miao et al. 2020; FitzGibbon and McGarry 2021). This limits the possibility of comparing our results. However, Ríos-Ruiz et al. (2022), for example, adopted the same T-S law including the model for the damage evolution. In their FEM calibration of the peeling tests, they arrived at the following estimates of the T-S law parameters: K = 10 MPa/mm, δ_c = 0.019 mm, T_c = 0.185 MPa, δ_f = 0.086 mm, and G_c = 8 mN/mm for an aortic strip oriented in the axial direction, and K = 8 MPa/mm, δ_c = 0.02 mm, T_c = 0.160 MPa, δ_f = 0.063 mm, and G_c = 5 mN/mm for the strip where the crack tip was propagated in the circumferential direction. We would certainly not wish to suggest that the numerical specification of the T-S law parameters should be the same as in our case. On the contrary, there are many sources for the differences, including the different species (human vs. porcine tissue), the location of the delamination interface within the wall thickness (the present study deals with a homogenous wall whereas Ríos-Ruiz et al. (2022) investigated interfaces between the aortic anatomic layers), and the way the constitutive model for the bulk behavior of the tissue was found. Even so, our identified values of the T-S law parameters appear to be approximately within the order of magnitude of the parameters found in Ríos-Ruiz et al. (2022). Regarding T_c as a quantity crucial for a mathematical description of cohesive properties, there are also other studies that found values of T_c to be in the order of tenths of MPa (Ferrara and Pandolfi 2010; Leng et al. 2015, 2016, 2018; or, for example, Merei et al. 2017).

In contrast to Ríos-Ruiz et al. (2022), our study is population-based. The data used to describe the elastic behavior of the aorta were adopted from Jadidi et al. (2020) and represent population representatives. These were combined with the results of our previous study (Horný et al. 2022) dealing with the delamination strength and were adopted in the form of an age-dependent regression model, i.e., again population-averaged values. Although population averages may be distant from one particular observation, especially when considering individual health conditions, if patient-specific simulation is not the goal, they represent the most appropriate way to see what is happening in the population. Thus, for expressing the age dependence of the T-S law parameters, which was our main goal here, population averaging seems instead to be a meaningful approach.

The reliability of the results presented by this study has certain limits due to the assumptions made during the modelling process. It has already been mentioned that the results are based on population averaging. On the other hand, the trend under investigation, i.e. the effect of aging on cohesive properties, is a population characteristic and would not be detected by the observation of an individual. Another limitation that needs to be mentioned is the assumption of a homogeneous wall. The delamination strength observations on which our model is based did not distinguish variable strength along the wall thickness; the study by Horný et al. (2022) was not designed to do so. However, the results of Ríos-Rouiz et al. (2021, 2022) show that the cohesive properties depend not only on the position along the aorta, but also on the position of the crack within the wall thickness, and even on the position along the circumference (Angouras et al., 2019; Sokolis and Papadodima, 2022). This may cause differences in the future observed values of the T-S law parameters. On the other hand, it is the study of Ríos-Ruiz et al. (2022) that shows that some parameters may still agree well (T_c, δ_c).

An obvious limitation to the use of our results is that cohesive contact, as used, does not account for the anisotropy of the delamination strength. This is a limitation that arises from the chosen tool for the FEM model. Abaqus does not allow for an anisotropic cohesive contact. Therefore, the resulting parameters are divided into three sets. The set for axial and circumferential strips can be used in the modeling if the direction of crack propagation is known a priori. If the crack propagation direction is not known, the results from the averaged model can still be used. However, their use will introduce some error due to the neglecting of anisotropy.

Since our study relies on the FEM model, some attention should also be paid to the effect of mesh size. To this end, several simulations were performed in order to verify that the results are independent of the discretization. The results of the mesh size sensitivity analysis are shown in Fig. 6. FEM meshes with edge lengths of 0.1, 0.05, 0.025, and 0.00125 mm were considered (Fig. 6c). The difference in the predicted delamination strength for the meshes with element edge lengths of 0.1 mm and 0.05 mm was greater than 10% (on average, it was 8.3% for the axially oriented strip and 16.6% for the circumferential strip, Fig. 6a, b). This deviation decreased significantly when a mesh with an edge length of 0.025 mm was used (the difference between 0.05 mm and 0.025 mm was 1.75% for the axial strip and 2.1% for the circumferential strip). As mentioned in Methods, the resulting calibrated parameter values were obtained from simulations with a mesh size of 0.0125 mm, for which the average difference with respect to the 0.025 mm mesh was less than 1% for both orientations of the delaminated strips and so the results were considered to be independent of the mesh size.

Our study was aimed at finding estimates of the T-S law parameters that would be validated against delamination experiments of human aortic strips. Since the biomechanics of human arteries is significantly dependent on the age of the individual, the same can be expected not only for the material parameters of bulk constitutive models, but also for parameters expressing tissue cohesive properties. Using FEM model calibrated to the delamination strengths expressed by the regression of 246 delamination experiments with strips of the human thoracic descending aorta, the T-S law parameters for seven values of age were found. The T-S law parameters were shown to correlate with age similarly to the experimentally observed delamination strengths, which significantly decrease with age.

Conflict of interest

There are no competing interests.

Author Contribution

Z.P. carried out the FEM simulations and prepared the figures, L.H. designed the study, analyzed the results and wrote the text, P.T. built the computational model.

Acknowledgement

This study has been supported by the Czech Science Foundation in the project GA20-11186 entitled “Mechanics of Arterial Delamination and Crack Propagation”.

There are no competing interests.

This study has been supported by the Czech Science Foundation in the project GA20-11186 entitled “Mechanics of Arterial Delamination and Crack Propagation”.

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Traction-separation law parameters for the description of age-related changes in the delamination strength of the human descending thoracic aorta

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Abstract

Figures

1. Introduction

2. Methods

2.1 Elastic behavior

2.2 Cohesive behavior

2.3 Finite element model

2.4 Calibration of T-S law parameters

3. Results of the simulations: Estimated T-S law parameters

4. Discussion

5. Conclusion

Declarations

Conflict of interest

Author Contribution

Acknowledgement

References

Additional Declarations

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