COMPUTATION OF EIGENSTRESSES IN THREE-DIMENSIONAL PATIENT-SPECIFIC ARTERIAL WALLS

IS Computation - Biomechanics of eigenstresses and Computational in three-dimensional Modeling of Living patient-specific Tissue arterial walls XIII International Conference on Computational Plasticity. Fundamentals and Applications COMPLAS XIII E. Oñate, D.R.J. Owen, D. Peric & M. Chiumenti (Eds) COMPUTATION OF EIGENSTRESSES IN THREE-DIMENSIONAL PATIENT-SPECIFIC ARTERIAL WALLS M. VON HOEGEN AND J. SCHRÖDER Institute of Mechanics, Faculty of Engineering / Dept. Civil Engineering University of Duisburg-Essen Universitätsstr. 15, 45141 Essen, Germany e-mail: markus.von-hoegen@uni-due.de, web page: http://www.uni-due.de/mechanika/ e-mail: j.schroeder@uni-due.de, web page: http://www.uni-due.de/mechanika/ Key words: Biomechanics, Residual Stress, Vascular Wall Abstract. In this contribution we propose a numerical scheme for the incorporation of residual stresses in arterial walls. Through the introduction of suitable stress invariants a local, time-independent residual stress tensor is introduced and computed iteratively. For the update of the residual stress tensor a non-local criterion is defined. The method is tested for patient-specific, three-dimensional healthy and degenerated arterial segments. Moreover, for a sliced and unloaded artery the opening angle caused by the eigenstresses can be predicted. 1 INTRODUCTION In order to improve simulations of mechano-biological processes in blood vessels, residual stresses should be taken into account. In [4], three-dimensional simulations with and without axial residual stresses were compared, and it was concluded that they have a major impact on determining areas of plaque rupture due to the appearing principal stress amounts. Usually material parameters of soft tissues are adjusted in uniaxial tension tests. That makes it relevant to know the stress free configuration of the artery, which is not necessary if the parameters are determined from inflation tests. Several approaches to incorporate residual stresses or strains can be found in recent literature. A couple of studies make use of the opening angle experiment to construct a deformation gradient F 0 for the mapping from a stress free to an unloaded configuration, cf. [7], [12] or [1]. In these publications, the opening angle experiment is often considered to be a problem of pure bending. A cylindrical coordinate system is introduced and F 0 is defined in terms of the principal 112

stretches, i.e. radial, circumferential and axial stretch. Required experimental input values are the axial pre-stretch, the radius in the unloaded closed configuration and the radius in the opened configuration. In these reported models the axial pre-stretch was supposed to be uniform and the method is able to reproduce the opening angle experiment. The opening angle method has also been expanded to three-dimensional models in [9] where the axial pre-stretches are incorporated. Based on the experimental measurements provided in [8], it was shown that the calculated stress distributions considerably differ when three-dimensional residual deformations are postulated. The authors concluded that only two-dimensional residual stretches deduced from the opening angle do not provide adequate results. Analytical equilibrium equations for the individual material layers were derived under certain assumptions. For this method elaborate experiments of the separated material layers are necessary to obtain the required input data. In [1], F 0 was also used to describe the residual stretch field. A fixed point iteration was implemented to determine the stress free configuration from in-vivo medical imaging, i.e. from geometries which are already exposed to physiological blood pressure. A similar approach with a multiplicative decomposition F = F e F g into an elastic part F e and a diagonal growth tensor F g was recently presented in [14]. In [13] a model to predict residual strains was developed, which is based on volumetric growth. It intends to homogenize the von Mises stresses throughout the vessel wall and uses the difference of the maximum and minimum stress values as a non-local growth criterion. Likewise in [11] the authors derived an evolution driven growth scheme and considered remodeling of material parameters at the same time. Growth models usually follow the framework of open system thermodynamics, see [5]. In so far, the method presented in this contribution, based on the work of [17], is an exception, since the residual stress distributions are directly estimated from the difference of the local stresses and associated mean values which are averaged over predefined segmental volumes. 2 CONSTITUTIVE FRAMEWORK The free energy function ψ is additively split into an isotropic part ψ iso and two superposed transversely isotropic parts ψ (1) aniso and ψ(2) aniso. Further, the function ψ( C) 2 =ψ iso + ψ (a) aniso (1) is formulated in terms of the modified right Cauchy-Green tensor C = ( ) θ 2/3 J C in order to avoid locking of quasi-incompressible formulations. In here, θ denotes the element wise constant volume dilation and J is the determinant of the deformation gradient F. For the isotropic part we choose a Mooney-Rivlin formulation and the transversely isotropic parts are represented through the Holzapfel-Gasser-Ogden model, provided in [7]: ψ iso = c 1 I 1 + c 2 I 2 + c 3 I 3 δln I 3 a=1 ψ (a) aniso = k 1 2k 2 {exp [ k 2 J (a) 4 1 2 ] 1}, (2) 113

where a := 1 (a + a ) denotes the so-called Macaulay bracket. The parameter δ = 2 2c 1 +4c 2 +2c 3 is chosen in such a way that the reference configuration becomes stressfree. Both functions in Eq. (2) are polyconvex, cf. [15], [2] and [16], and therefore guarantee in combination with the coercivity condition the existence of minimizers. In the latter equation I 1 = tr C, I 2 = tr[cof C] and I 3 = det C are the first three invariants of C and J (a) 4 = C : M (a) is a mixed invariant depending on the symmetric structural tensor M (a) = A (a) A (a). Here, A (a) denotes the fiber orientation vector of unit length. The material parameters of the two major artery components media and adventitia were fitted to uniaxial tension tests from aortic arteries, see table 1. In these tests the angle 2β f between the two considered fiber directions A (1) and A (2) is unknown and was also part of the parameter fitting. For the computation of atherosclerotic arteries the plaque is assumed to be homogeneous and purely isotropic and due to lack of experimental data we choose suitable parameters which were studied in a sensitivity analysis. Table 1: Individual material parameter sets for the different constituents. ψ iso ψ (a) aniso c 1 /kpa c 2 /kpa c 3 /kpa k 1 /kpa k 2 /- β f /rad Adventitia 2.326 6.169 60.642 3.131 10 8 147.174 0.759 Media 14.638 0.149 60.810 6.851 754.014 0.913 Plaque 60.0 15.0 800.0 - - - To enforce quasi-incompressible material behavior, the constitutive framework was extended with an augmented Lagrangian approach, see [6] and [18], with a tolerance of 1% for the change in volume. 3 ALGORITHMIC SCHEME TO INCORPORATE RESIDUAL STRESSES In order to propose a basic physiological mechanism for the computation of the eigenstresses in arteries we define suitable stress invariants capturing the distinct anisotropic behavior. These stress quantities are used for a specific homogenization scheme, where we perform an averaging procedure in radial direction, i.e. over small radial cuts, of the arteries. For that purpose the Cauchy-stress tensor σ can be additively split into a ground stress part σ and a reaction stress part σ r, i.e. σ = σ + σ r. (3) The reaction stresses are formulated in terms of the fiber stress invariants T (1) and T (2) and the structural tensors m (1) and m (2) associated to two unit fiber vectors in the actual placement: σ r = T (1) m (1) + T (2) m (2). (4) 114

This additive decomposition of σ is applied in order to generate a suitable set of measures capturing the strongly anisotropic parts. Here, σ r is defined in such a way, that tr ( σ m (1) ) = 0 and tr ( σ m (2) ) = 0 (5) holds. Similar ideas, especially for the construction of convenient sets of invariants, have been proposed in [19] and [10]. On the basis of the domain decomposition and the definition of the split of the Cauchy-stresses we are able to formulate a smoothing algorithm, see Table 2. The main idea is to load the artery initially with a physiological blood pressure and to homogenize the stress distribution afterwards. The fiber stress invariants will be considered as scalar stress measures which are capable to estimate the residual stresses. By making use of the projections in Eq. (3) we are able to derive the expressions T (1) = σ : m (1) σ : m (2) ξ and T 1 ξ 2 (2) = σ : m (2) σ : m (1) ξ (6) 1 ξ 2 for the invariants, with the abbreviation ξ = m (1) : m (2). In order to define a non-local target value for the smoothing procedure, the volume averaged mean values T,mat (1) = 1 T v,mat (1) (x) dv and T,mat (2) = 1 T B v,mat,mat (2) (x) dv (7) B,mat are introduced, which are to be computed for each sector. The spatial sector volume of each sector domain B,mat is represented by the term v,mat. Hence, the local deviations of the invariants are denoted with T (1) = T (1) T,mat (1) and T (2) = T (2) T,mat (2). (8) By inserting these deviations in the formula for the reaction stresses in Eq. (3) we obtain a tensorial representation for the deviation of the Cauchy stresses σ res and a simple push-forward yields the deviation of the 2nd Piola-Kirchoff stresses S res : 2 σ res = T (a) m (a) and S res = JF 1 σ res F T. (9) a=1 For the setup of our scheme, the total domain B of the mechanical problem is decomposed into sufficiently small sector domains B,mat, according to Fig. 1. By specification of an angle α the exemplary cross-section in a) is decomposed into n seg = 16 segments. By further dividing between the material layer with the material interface Γ, the geometry is decomposed of n 2D sec =2 n seg = 32 sectors. In the three-dimensional case, like in Fig. 1b), the geometry is also segmented in longitudinal direction. In the current example we have n 3D sec = n 2D sec n L = 160 sectors with specific sector domains B,mat. The algorithmic treatment of the computation of the residual stress tensor is facilitated in so called smoothing loops (SL). In each loop the aforementioned quantities in Eq. (6), (7), (8) and (9) are computed, see step 3.1 and 3.2 in table Table 2, respectively. In every 115

segment sector } α Γ n seg = 16 a) n 2D sec = 32 b) n L =5 n 3D sec = 160 Figure 1: Segmentation of an ideal artery, composed of media (orange) and adventitia (red) into sectors in a) two-dimensional and b) three-dimensional case. loop, the residual stresses are added to the residual stress tensor S res according to Eq. (12). Note that this step may be sub-incremented with a proportionate factor λ. After the last sub-incrementation loop we have the total proportionate factor λ which should obviously be in between zero and one. In every sub-incrementation loop the updated residual stress tensor S res is subtracted from the displacement driven stresses S( C), see Eq. (13), and a new equilibrium state is iterated. During this iteration the residual stresses are treated like a constant quantity. After solving the balance of momentum the smoothing loop is executed again based on the updated displacement and stress fields. This procedure can be repeated as often as desired until the local deviations T (1) and T (2) are satisfactory minimized. 4 NUMERICAL EXAMPLES In a first example we want to apply the discussed method on the basis of a healthy patient-specific 3 cm long artery segment. The geometry is depicted in Fig. 2. The artery was reconstructed from Virtual Histology (VH) Intravascular Ultrasound (IVUS) in-vivo imaging, see [3] for details. The smoothing algorithm described in the previous section was used to compute realistic transmural stress distributions. In the first two steps, according to Table 2, the artery was divided into n 3D sec = 72 sectors and loaded with an internal blood pressure p i of 16 kpa. Subsequently the smoothing loops were executed in step three. Fig. 3 shows the distribution of the stress invariant T (1) before smoothing (0 SL) and after different amounts of smoothing loops with λ =0.5 in all cases. Note that it was not necessary to sub-increment λ in this case. Thus, in the sub-incrementation loop in Table 2, j was set to one and λ = λ =0.5. After 0 SL, that means when the full blood pressure of 16 kpa is acting and no smoothing loop has been executed yet, considerable local stress gradients are visible. Obviously, these vanish for an increasing number of smoothing loops and the stress distribution becomes more even. The same statement also holds for the second fiber stress invariant T (2), as shown in Fig. 4 for the cross-section A-A, defined in Fig. 2. It also becomes clear that the smoothing procedure is working over the whole thickness. The 116

Table 2: Algorithmic treatment to incorporate residual stresses iteratively. 1. Divide the domain B into sufficiently small sector domains B,mat and initialize S res = 0. 2. Solve balance of momentum for the internal pressure p i. 3. Smoothing loop (SL) over k: 3.1 Calculate the local stress invariants T (1) and T (2), the non-local sector specific mean values T,mat (1) and T,mat (2) in B,mat and the local deviations T (1) and T (2) : T (1) = σ : m (1) σ : m (2) ξ σ : m (2) σ : m (1) ξ 1 ξ 2 T (2) = 1 ξ 2 T,mat (1) = T (1) 1 v,mat T,mat (2) = 1 v,mat T (1) (x) dv T (2) (x) dv B,mat B,mat = T (1) T,mat (1) T (2) = T (2) T,mat (2) T (2) 3.2 Estimate the residual stress tensors: σ res = (10) 2 T (a) m (a) and S res = JF 1 σ res F T (11) a=1 3.3 Sub-incrementation loop over j: 3.3.1 Update of the residual stress tensor S res : S res = S res + λ S res, λ = j λ [0, 1] (12) 3.3.2 Solve balance of momentum with the residual stress tensor S res which are subtracted in the following form: S = S( C) S res (13) main underlying hypothesis was to smooth the radial stress gradient in circumferential direction. Therefore, in Fig. 5 the stresses in section B-B are plotted. Here, σ 11, σ 22 and σ 33 approximate the radial, circumferential and longitudinal stresses. As we can see the stress gradient of σ 22 is significantly smoothed. The same applies for the longitudinal stresses, while the radial stresses only undergo moderate changes. Due to the low stress level on the adventitia the smoothing effect is not as pronounced as on the media. The unloaded and radially sliced arteries may be loaded with the computed residual stress tensor S res. In this case no external forces are present. In Fig. 6 the already discussed healthy artery and two degenerated arteries, suffering from atherosclerosis are depicted. Note that the plaque is not considered to be residually stressed, i.e. S res = 0. The Fig. 6 shows the simulated opening angles of the separated media after two smoothing loops with λ =0.5. In the light of the shown contour plots of the von Mises stresses 117

M. von Hoegen, J. Schro der A-A x2 x2 x3 B-B x1 x1 A-A Figure 2: Geometry of a reconstructed healthy patient-specific artery and definition of cross-section A-A and section B-B. T(1) in kpa 190 160 130 0 SL 2 SL 10 SL 100 70 40 10 Figure 3: Comparison of the first fiber stress invariant distribution after different amounts of smoothing loops with λ = 0.5. T(2) in kpa 190 160 130 0 SL 2 SL 10 SL 100 70 40 10 Figure 4: Comparison of the second fiber stress invariant distribution in section A-A after different amounts of smoothing loops with λ = 0.5. we see that the opened configurations are not stress-free which is in accordance with experimental findings. Moreover, it is not surprising, that the opening angle of the right artery is significantly smaller, since the plaque is more massive and the artery becomes 118

M. von Hoegen, J. Schro der σ11 in kpa σ22 in kpa σ33 in kpa 0 SL Media Adventitia 0 SL 2 SL 2 SL 10 SL 10 SL Adventitia Adventitia 0 SL 2 SL 10 SL Media Media Figure 5: Stress distributions in section B-B after different amounts of smoothing loops with λ = 0.5. σvm Figure 6: Opening angles of the media of three different geometries and contour plots of the von Mises stresses σvm in kpa. stiffer. In general, the presented method seems to predict reasonable magnitudes of the opening angle. 5 CONCLUSIONS The incorporation of a time-independent residual stress tensor, which is based on stress invariants seems to be appropriate to describe the residual stress state. The general algorithmic treatment presented in [17] was proven to provide reasonable results for 3D patient-specific arteries. It is capable to smooth the transmural stress gradients and to predict reasonable opening angles when the residual stresses are relieved. 119

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