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The Role of Layer-Specific Residual Stresses in Arterial Mechanics: Analysis via a Novel Modelling Framework

The Role of Layer-Specific Residual Stresses in Arterial Mechanics: Analysis via a Novel... The existence of residual stresses in unloaded arteries has long been known. However, their effect is often neglected in experimental studies. Using a recently developed modelling framework, we aimed to investigate the role of residual stresses in the mechanical behaviour of the tri-layered wall of the pig thoracic aorta. The mechanical behav- iour of the intact wall and isolated layers of n = 3 pig thoracic aortas was investigated via uniaxial tensile testing. After modelling the layer-specific mechanical data using a hyperelastic strain energy function, the layer-specific deforma- tions in the unloaded vessel were estimated so that the mechanical response of the computationally assembled tri-layered flat wall would match that measured experimentally. Physiological tension–inflation of the cylindrical tri- layered vessel was then simulated, analysing changes in the distribution of stresses in the three layers when neglect- ing residual stresses. In the tri-layered model with residual stresses, layers exhibited comparable stresses throughout the physiological range of pressure. At 100 mmHg, intimal, medial, and adventitial circumferential load bearings were 16 ± 3%, 59 ± 4%, and 25 ± 2%, respectively. Adventitial stiffening at high pressures produced a shift in load bearing from the media to the adventitia. When neglecting residual stresses, in vivo stresses were highest at the intima and lowest at the adventitia. Consequently, the intimal and adventitial load bearings, 23 ± 2% and 18 ± 3% at 100 mmHg, were comparable at all pressures. Residual stresses play a crucial role in arterial mechanics guaranteeing a uniform distribution of stresses through the wall thickness. Neglecting these leads to incorrect interpretation of the layers’ role in arterial mechanics. Keywords: Residual stresses, Tri-layered arterial wall modelling, Intima, Media, Adventitia 1 Introduction arterial function. The intima, the innermost layer, inter - The arterial wall is characterised by a strong relationship acts directly with the blood flow and provides a small between structure and function [1, 2], so that arterial contribution to the wall’s mechanics due to its small structure varies considerably at different locations in the thickness. The media, the middle and thickest layer, arterial tree [2]. Large elastic arteries are located in prox- determines the elastic behaviour that characterises large imity of the heart. At a macroscopic level, these arteries arteries at physiological pressures [3] and endows them are organised in three concentric layers, the intima, the with their blood pressure buffering (or Windkessel) func - media and the adventitia, each playing a pivotal role in tion, which smooths the intermittent pumping action of the heart into a more continuous blood pressure ensur- ing organ perfusion also in diastole. The adventitia is the *Correspondence: a.giudici@maastrichtuniversity.nl 3 outermost layer and is typically described as a protective Department of Biomedical Engineering, CARIM School for Cardiovascular Diseases, Maastricht University, Universiteitssingel 50, sleeve that preserves the wall’s integrity at high pressures Room 3.353, 6229 ER Maastricht, The Netherlands [2]. Full list of author information is available at the end of the article © The Author(s) 2022. 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Giudici and Spronck Artery Research (2022) 28:41–54 42 To fulfil their specific function, arterial layers differ the unloaded vessel represents an equilibrium configura - significantly at the microstructural level. The intima is tion between residual stresses acting throughout the arte- constituted mainly by endothelial cells, a thin basement rial wall thickness that are compressive and tensile in the membrane, a proteoglycan rich matrix and collagen inner and outer part of the wall, respectively [26]. When fibres oriented in the axial direction [4, 5]. The media is a radial cut is performed, the geometrical constraint that organised in concentric medial lamellar units, alternating guarantees this equilibrium is lost and the artery deforms elastin-rich lamellae and inter-lamellar spaces composed to release said residual stresses, reaching a new equilib- of vascular smooth muscle cells, elastin and collagen rium in its stress-free configuration. Knowledge of the fibres which are predominantly oriented in the circum - inverse of this deformation is, therefore, crucial to deter- ferential direction [3, 4, 6]. The adventitia is mainly con - mine the layer-specific prestress in the artery unloaded stituted by bundles of diagonally oriented collagen fibres, state and constitutes the link to fully understand the com- showing larger angular dispersion than in the media. Fur- plex tri-layered mechanics of the arterial wall. ther, unlike the intima and the media, the adventitia is The circumferential component of this deformation relatively acellular [7, 8]. is commonly quantified via an opening angle (OA), e.g. Understanding the role each layer plays in arterial the angle formed by connecting the two endpoints of the function has been the objective of several investigations, arc-shaped circumferential wall sample to its mid-point using different experimental techniques such as selective [22]. Knowledge of the OA allows estimating the distri- enzymatic digestion of specific wall constituents [9, 10] bution of the  circumferential deformation throughout and imaging the wall’s microstructure at different levels the wall thickness of the unloaded artery, thus allowing of distending pressure [3, 11, 12]. The separation and to determine its pre-stressed state [23]. Further, a more mechanical testing of isolated layers is a useful technique comprehensive characterisation of the layer-specific that directly assesses how their different microstructures three-dimensional geometry of excised circumferential determine their mechanical behaviour. This technique and axial wall strips allows formulating complex mod- has been used to characterise the layer-specific mechan - els that predict the distribution of residual stresses in all ics in several arterial locations, species and in both physi- three principal directions throughout the wall thickness ological and pathological conditions [13–19]. Moreover, [20, 27]. However, the implementation of such models is constitutive modelling, i.e. fitting structurally motivated non-trivial and residual stresses are often neglected when mathematical models that describe the wall’s mechanical modelling the tri-layered arterial wall [28, 29]. behaviour as the summed contribution of the response of We recently developed a new tri-layered modelling its constituents, has allowed investigating how layer-spe- framework that allows accounting for layer-specific cific microstructural features affect the layer’s mechani - residual stresses using only intact wall and layer-specific cal properties, including anisotropy and recruitment of uniaxial test data [30]. Using the newly developed mod- collagen fibres [13–19]. elling framework, the present study aimed to illustrate While characterising the behaviour of isolated lay- the effects of residual stresses on arterial mechanics and ers provides some insight into the tri-layered arterial how neglecting these can lead to considerable misinter- wall mechanics, it is not sufficient to fully understand pretation of the specific role arterial layers play in the their role in the macroscopic behaviour of arteries. macroscopic behaviour of the arterial wall. We will simu- Indeed, while an excised unloaded artery is subjected to late three different scenarios: (1) a complete model with null external loads in all three principal directions (i.e. residual stresses, (2) a model where the unloaded vessel circumferential, radial and axial), local stresses at any is considered stress free (i.e. completely neglecting all generic point within the arterial wall’s volume are, in residual stresses), and (3) a model where a flat slab of wall general, non-zero [20, 21]. As shown in previous stud- tissue (as typically tested in planar biaxial tensile experi- ies [22–24], this particular feature of arteries likely arises ment) is used as an approximation of a cylindrical vessel. from the fact that their wall develops and remodels in its loaded state (i.e. in  vivo), attempting to preserve a uni-2 Methods form distribution of stresses throughout its thickness. 2.1 Theoretical Background When all external loads are removed, however, its inho- 2.1.1 T ri‑layered Wall Model mogeneous material properties and in  vivo deformation A complete description of the mathematical formulation field lead to the existence of residual stresses. of the tri-layered wall model can be found in our previous Residual stresses were first discovered in 1983 when study [30]. Briefly, the arterial wall is assumed to be com - two studies [21, 25] observed that, when cut radially, an prised of three adequately spaced membranes representing arterial ring typically does not retain its circular shape but the intima, media, and adventitia. The composition of the springs open and assumes an arc shape. This implies that wall from isolated layers to unloaded cylindrical vessel is Giudici and Spronck Artery Research (2022) 28:41–54 43 described via two mapping steps: first, the three layers are The total deformation each layer is subjected to in assembled into a flat rectangular slab of arterial wall tis - κ is given by tension−inflation sue. Due to the presence of residual stresses, layers change F = F F total,k 2 residual,k their shape when isolated from the wall. The layer-specific deformation gradient G (where k ∈{i, m, a} , i=intima, m k k = diag  �  , ,  �  . θ � z Z X Z =media, and a=adventitia) describes the inverse of this k k �   � θ � z Z X Z process, i.e. the deformation from the isolated layer con- (7) figuration κ in Cartesian coordinates (X ,Y,Z ) to tri- isolated layered flat wall slab κ in ( X,Y, ). For each layer k, composite 2.1.2 Lay erSp ‑ ecific Constitutive Modelling k 1 k , ,  The mechanical behaviour of the three isolated arterial G = diag , k k (1) X   Z X Z layers was modelled using the Holzapfel–Gasser–Ogden k k k k (HGO) two fibre family strain energy function (SEF) [32]. = l /L where and  = l /L are the circumferen- X X Z Z Z The passive behaviour of each layer is assumed to be tial and axial components of G , l and l are the circum- determined by the summed contribution of an isotropic ferential and axial dimensions of the composite wall, and k k matrix, typically associated with elastin, and two sym- L and L are the circumferential and axial lengths of the X Z metrically oriented families of collagen fibres that deter - isolated layer k . Note that the radial stretch is determined mine the wall’s anisotropy: from incompressibility [31]. The second mapping step describes the deformation of k k k k c ρ (I )+ 1−3ρ I −1 c 1 4,i k k 2 2 the flat tri-layered wall, κ , into an unloaded cylin- composite � = µ (I − 3) + (e − 1), i=1 k 2c drical vessel, κ in cylindrical coordinates ( , R , Z ). unloaded (8) The related deformation gradient is where µ is an isotropic stiffness-like parameter 1 ( k ∈{i, m, a} , i=intima, m=media, and a=adventitia), c F = diag  , ,  , Z (2) is a collagen stiffness-like parameter, c is a dimension- less collagen non-linearity parameter, and ρ ∈[0, ] is where the axial stretch  is assumed to be constant 3 a fibre dispersion coefficient, with ρ = 0 denoting fully throughout the wall and the circumferential component aligned and ρ = denoting fully dispersed fibres (i.e. is calculated as 3 isotropic behaviour). I and I are the first and fourth 1 4,i 2 invariant of the right Cauchy–Green tensor, respectively, 4π R 4πY internal � = + , (3) l � l X Z with i ∈ {1, 2} indicating the collagen fibre family with k k k principal orientation α = {−α , α } with respect to the 1,2 where R is the radius of the artery at the luminal internal circumferential orientation. Note that the layer-specific surface. fibre orientation α was not derived from microscopy 1,2 The layer-specific residual stresses in κ arise from unloaded data but phenomenologically estimated from the meas- the total deformation from κ to κ isolated unloaded ured mechanical behaviour. For this reason, these angles should not be regarded as exact measures of the fibre k 1 k , , F = F G = diag  Z . residual,k 1 X k k Z orientation in the arterial layers but rather as a general X Z quantification of the collagen-induced tissue anisotropic (4) properties. The deformation gradient F maps the tension–inflation The Cauchy stress tensor is defined as of the cylindrical vessel to a simulated in  vivo configura - tion, κ in cylindrical coordinates ( θ , r , z ). As tension−inflation ∂� k T σ = −pI + 2F F , (9) total,k ∂C total,k for F , the axial component of F ( ) is assumed to be con- total,k 1 2 z stant throughout the wall. F is defined as where I is the spatial second order identity tensor and p is 1 the Lagrange multiplier enforcing incompressibility [31]. F = diag  , ,  , θ z (5) 2.2 Experimental Methods with Three pig plucks (i.e. the content of the thorax: 2 2 2 r R −R heart, thoracic aorta, lungs, trachea, and liver) (age internal internal = + , (6) 2 2 R R 6–12  months) were purchased from a local abattoir (samples from school, UK). Delivered frozen, each where r is the luminal radius in κ . internal tension−inflation pluck was immediately stored at − 20 °C in a laboratory Giudici and Spronck Artery Research (2022) 28:41–54 44 2.3.1 Lay erSp ‑ ecific Constitutive Parameters freezer until the day of testing when it was left to thaw The five layer-specific constitutive parameters in at room temperature for ~ 4 h. The aorta was then care - Eq.  8 were fitted on the layer-specific mechani - fully dissected from the rest of the tissues using a scalpel. cal data, minimising the error between the individ- Two circumferentially and two axially oriented arterial ual layer’s experimental (Eq.  10) and modelled (Eq.  9) strips (approximately 4 × 25 mm ) were isolated from Cauchy stress in the loading direction for the uni- the region of the descending thoracic aorta character- axial tests in the circumferential and axial direction ised by the branching of the intercostal arteries. Their simultaneously. Since the off-axis deformation  (i.e. width and thickness were carefully measured three times j ∈{axial, circumferential} corresponding to the loading along the strip length using a high precision digital cal- directions i ∈{circumferential, axial} ) was not measured liper. The mean of the three measurements was used during the experiments, this was determined by impos- for later analysis. Each strip was then subjected to uni- ing σ = 0 and enforcing incompressibility. axial tensile testing in its long direction using a high- jj resolution uniaxial tensile device (MFS Stage with 20N 2.3.2 D etermination of the LayerSp ‑ ecific Deformation load cell, Linkam Scientific Ltd., UK) equipped with ser - in κ rated jaws to prevent slipping of the sample. The inter- unloaded As shown in Eq.  4, determining the layer-specific defor - jaw distance was initially set to 15  mm and then slowly mation gradient in κ requires estimating G and increased until the sample laid flat. This flat length was unloaded F . The axial and circumferential components of G of considered as the unloaded sample length. Strips were the three layers were fitted on the mechanical data of the uniaxially tested up to a Cauchy stress of 250  kPa after intact wall, simultaneously minimising the error between performing five loading/unloading cycles up to the same the wall experimental Cauchy stress (Eq.  10) and that Cauchy stress. After testing, each strip was separated obtained when simulating the uniaxial tensile test of the into its three layers using tweezers, first carefully peeling tri-layered composite wall in both circumferential and the adventitia from the intima-media and then isolating axial directions. The average modelled Cauchy stress of the intima from the media. As done for the intact wall, the composite wall was calculated as the layer’s dimensions were measured at three locations along the strip length and the average values used for i i m m a a wall h σ +h σ +h σ σ = . (11) later analysis. Then, each layer was tested following the wall same protocol described for the wall strips. As described in the introduction, the unloaded con- The wall’s and layer’s experimental Cauchy stress was figuration of the vessel ( κ ) implies zero average unloaded calculated as [33] wall stresses in all three principal directions. Therefore, k F and the layer-specific  were estimated iteratively by σ =  , (10) ii,exp A i wall wall wall simultaneously minimising σ , σ , and σ . �� RR ZZ where F is the force measured by the load cell, A is 2.3.3 Simulation of the Tension–Inflation the sample unloaded cross-sectional area, and  is the To assess the contribution of residual stresses to the stretch in which direction i (circumferential or axial) the wall mechanics, we simulated three tension–inflation sample k ∈ wall, i, m, a is loaded during the uniaxial scenarios: (1) full tri-layered model including layer-spe- test. cific residual stresses ( F = F G ), (2) tri-layered residual,k 1 model where residual stresses are completely neglected 2.3 Parameter Estimation ( F = I ), and (3) assuming F = I ( F = G ) The fitting routine of all the model parameters has been residual,k 1 residual,k (Table  1 and Fig.  1). Case 1 represents the closest described previously [30], and is briefly summarised approximation of the in  vivo condition; cases 2 and 3 below. Table 1 Description of the three modelling scenarios analysed in this study Scenario F Description residual,k 1 Layer-specific residual stresses in the unloaded cylindrical vessel ( κ ) arise from two subsequent F G unloaded deformations of the isolated layers ( κ ): (1) into a tri-layered flat wall ( κ ) that is (2) then bent composite isolated into κ unloaded 2 I Residual stresses in κ are completely neglected as no deformation occurs from κ to κ unloaded isolated unloaded 3 Layer-specific residual stresses in κ arise solely from the deformation of the isolated layers G unloaded ( κ ) into κ . Hence, κ is used as an approximation of κ isolated composite composite unloaded Giudici and Spronck Artery Research (2022) 28:41–54 45 Fig. 1 Schematic representation of the different configurations and linking deformation gradients analysed in this study: (1) isolated layers ( κ ), (2) composite wall ( κ ), (3) unloaded cylindrical vessel ( κ ), and (4) pressurised axially stretched cylindrical vessel isolated composite unloaded ( κ ). Note that the deformation from κ to κ , mapped by the deformation gradient F , defines the layer-specific tension−inflation isolated unloaded residual,k residual deformations. Deformation gradients were differentially defined in the three analysed scenarios s ∈{1,2,3} ( Table 1) as indicated. Note the use of different coordinate systems in the respective configurations. Superscript k ∈ {i, m, a} indicates the intimal ( i ), medial (m ), or adventitial ( a ) layer, respectively replicate two possible modelling approximations. In case Then, pseudo-physiological tension–inflation was sim - 2, κ is assumed to be a stress-free configuration ulated by axially stretching the vessel to  and increasing unloaded (i.e. all three layers are undeformed in these configura - luminal pressure from 0 to 200 mmHg. tions). In case 3, the flat arterial wall (which is the typi - Layer-specific and whole-wall stresses were determined cal configuration of uniaxial and biaxial tensile tests) is using Eqs.  9 and 11, respectively. The percentage of load assumed to be a good approximation of the cylindrical bearing for each layer was determined as vessel, thus neglecting the deformation from κ to composite k k σ h θθ κ . Load bearing % = · 100%. unloaded (14) wall wall σ h θθ Except for the definition of F , all three cases fol- residual,k lowed the same modelling process. First, the simulated The circumferential material stiffness C was calcu- θθθθ in  vivo axial stretch  was determined as the average lated according to the small-on-large formulation [35] cross-over point between simulated reduced axial force- and evaluated continuously as a function of the distend- axial stretch relationships at the distending pressure lev- ing pressure: els P = 60 , 100 , and 140 mmHg [34]. The reduced axial ∂ � force F was calculated as C = 2(σ + p) + 4 . θθθθ θθ θ 2 (15) k k k 2 F = 2πσ r h − πr P, z (12) zz k=i,m,a internal 2.4 Statistical Analysis where r is the mid-wall radius of the layer k . Given P , For each aorta, tensile tests were conducted on two axi- the corresponding r was estimated iteratively to internal ally oriented and two circumferentially oriented strips. The satisfy the Laplace equation layer-specific constitutive and tri-layered modelling was then k h conducted in pairs (i.e. circumferential strip 1 with axial strip P = (σ − σ ) . (13) k=i,m,a θθ rr k r Giudici and Spronck Artery Research (2022) 28:41–54 46 3.2 Tri‑layered Wall Modelling 1 and circumferential strip 2 with axial strip 2) and the pre- 3.2.1 Lay erSp ‑ ecific Deformation in  κ sented results of each artery are the mean of the two. unloaded The layer-specific components of the deformation gra - Results are presented as mean ± standard devia- k k dients G , F , and F = F G are presented in tion of the three tested aortas. Outcome parameters 1 residual,k 1 Table  4. In the flat wall configuration κ , the (wall stresses, material stiffness, load bearing %) were composite intima was subjected to tensile stretches in both circum- first evaluated continuously as a function of pressure. ferential and axial directions. Conversely, from κ to Then, comparison between modelling conditions (i.e. isolated κ , both media and adventitia experienced defor- with and without residual stresses) was carried out composite mations of opposite nature in the circumferential and at the reference normotensive mean arterial pres- axial directions: circumferential extension and axial com- sure (100  mmHg) and at 160  mmHg, representing the pression for the media and circumferential compression average hypertensive systolic pressure, using paired and slight axial extension for the adventitia. As expected, student’s t tests. p < 0.05 was taken as statistically the deformation from κ to κ ( F ) intro- significant. composite unloaded 1 duced compressive and tensile circumferential deforma- tions in the inner half and outer half of the wall thickness, 3 Results respectively, with negligible deformations in the axial 3.1 W all and Layer‑Specific Response to Uniaxial Testing direction. As a result, the total circumferential deforma- The wall and layer geometrical features of the three tion from κ to κ ( F ) was strongly com- pig thoracic aortas tested in this study are reported isolated unloaded residual pressive (0.93 ± 0.01) for the intima, slightly tensile (1.01 in Table  2. The mean radius and wall thickness were ± 0.01) for the media, and tensile (1.04 ± 0.00) for the 7.79 ± 0.45  mm and 1.74 ± 0.31  mm, respectively. adventitia. The isolated layer thicknesses were 0.28 ± 0.10  mm for the intima, 0.89 ± 0.24  mm for the media and 3.2.2 Simulated Tension–Inflation 0.51 ± 0.07  mm for the adventitia, corresponding to Figure  3A–C presents the mean simulated pressure– 17 ± 2%, 52 ± 3% and 31 ± 1% of the wall thickness, diameter relationships for the three scenarios consid- respectively. ered in this study (Table 1 and Fig. 1). Interestingly, while Figure 2 presents the wall and layer-specific response to pressure–diameter relationships appeared very similar uniaxial tensile testing in both circumferential and axial in the three considered scenarios, the distribution of directions. The wall was stiffer circumferentially than stresses between the three layers differed considerably: axially at stretches below 1.3 but became increasingly First, in the full tri-layered wall model with residual isotropic at higher deformations. This complex behav - stresses, σ −P and σ −P relationships of the three lay- iour reflects the heterogeneity of the layer responses. θθ zz ers appeared almost superimposed in the physiological Layer-specific model parameters of the three aortas are pressure range (Fig.  4A, D). At a physiological pressure reported in Table 3. Both intima and media showed con- of 100  mmHg, σ and σ were 0.078 ± 0.017 and 0.058 siderable anisotropy (Fig. 2), with α ranging from 36.2°– θθ zz ± 0.014 MPa, 0.107 ± 0.013 and 0.062 ± 0.010 MPa, and 38.9° (intima) and 28.8°–35.5° (media), with respect to 0.078 ± 0.012 and 0.067 ± 0.004 MPa for intima, media, the circumferential direction. Conversely, the adventi- and adventitia, respectively. The percentage of circum - tia displayed a nearly isotropic behaviour (Fig.  2), if not ferential load borne by the media (62 ± 3%) was more slightly stiffer in the axial direction ( α =43.0°–49.5°). than twice that of the adventitia (24 ± 2%) and four times Further, the non-linearity parameter c was more than an that of the intima (14 ± 3%; Fig.  5A, B). When pressure order of magnitude higher in the adventitia than in both increased to 160  mmHg, the corresponding increase intima and media, signifying a more pronounced stiffen - in σ and σ was comparable in the intima, 72 ± 15% ing (non-linearity) with increasing stretch. θθ zz and 38 ± 11%, and media, 65 ± 12% and 34 ± 7%, but Table 2 Wall and layer geometrical features of the three pig thoracic aortas tested in this study intima wall media adventitia Sample # R (mm) h (mm) h (mm) h (mm) h (mm) I 8.00 2.09 0.29 1.15 0.59 II 8.20 1.62 0.29 0.83 0.49 III 7.16 1.50 0.27 0.68 0.46 Mean ± SD 7.79 ± 0.45 1.74 ± 0.25 0.28 ± 0.01 0.89 ± 0.19 0.51 ± 0.06 SD standard deviation Giudici and Spronck Artery Research (2022) 28:41–54 47 Fig. 2 Wall (A) and individual-layer (B intima; C media; D adventitia) Cauchy stress–stretch relationships of the three pig thoracic aortas (I, II, III) included in this study. Solid and dashed lines indicate uniaxial tests in the circumferential (circ.) and axial directions, respectively. Each curve was obtained by averaging the sample behaviour obtained in the two tests executed for each loading direction Table 3 Layer-specific parameters of the Holzapfel-Gasser-Ogden strain energy function of the three aortas included in this study k k k k k Layer Sample # R µ (kPa) c (kPa) c (–) α (°) ρ (–) 1 2 Intima I 12.7 163.5 0.1 36.6 0.20 0.98 II 22.7 165.8 3.3 38.9 0.21 0.97 III 6.5 249.5 1.1 36.2 0.20 0.96 Media I 24.6 129.0 3.6 29.1 0.24 1.00 II 30.1 95.7 4.0 28.8 0.21 1.00 III 29.0 182.4 3.4 35.5 0.20 1.00 Adventitia I 20.7 22.2 66.3 49.5 0.24 0.99 II 26.5 13.7 40.9 48.8 0.17 0.99 III 25.2 49.6 29.1 43.0 0.26 0.98 Parameters are defined in Eq. 8; R , coefficient of determination of the simultaneous fitting of both circumferential and axial uniaxial tensile test data approximately twice as high in the adventitia (135 ± relationship of the adventitia; for pressures < 90  mmHg, 25% and 111 ± 37%). As a result, while the intimal load the adventitial C −P relationship was compara- θθθθ bearing remained almost unaltered (13 ± 3%), medial ble to the intimal and medial relationships, which were and adventitial load bearings dropped and grew by -6 ± nearly linear over the entire investigated pressure range 3% and + 7 ± 4%, respectively. This shift in circumfer - (Fig.  4G). Conversely, adventitial stiffening with increas - ential load bearing was caused by the biphasic C –P ing pressure was markedly higher than that of both θθθθ Giudici and Spronck Artery Research (2022) 28:41–54 48 k k Table 4 Layer-specific circumferential and axial components of the deformation gradients G , F and F = F G . 1 residual,k 1 Intima Media Adventitia i i m m a a X Z X Z X Z I 1.04 1.02 1.04 0.99 0.94 1.02 II 1.02 1.05 1.03 0.97 0.98 0.99 III 1.03 1.00 1.01 0.99 0.97 1.02 Mean ± SD 1.03 ± 0.01 1.03 ± 0.02 1.03 ± 0.01 0.98 ± 0.01 0.96 ± 0.01 1.01 ± 0.01 m m a a i i Z  Z I 0.89 1.00 0.98 1.00 1.09 1.00 II 0.92 1.00 0.98 1.00 1.07 1.00 III 0.92 1.00 0.98 1.00 1.07 1.00 Mean ± SD 0.91 ± 0.01 1.00 ± 0.00 0.98 ± 0.00 1.00 ± 0.00 1.07 ± 0.01 1.00 ± 0.00 m a a k i i m m m a a F = F G i residual,k 1 X  X Z Z X  Z Z Z Z I 0.92 1.02 1.02 0.99 1.03 1.02 II 0.93 1.06 1.01 0.98 1.04 1.00 III 0.95 1.00 1.00 0.99 1.03 1.02 Mean ± SD 0.93 ± 0.01 1.03 ± 0.02 1.01 ± 0.01 0.98 ± 0.00 1.04 ± 0.00 1.01 ± 0.01 F defines the residual stretches experienced by each layer in the unload cylindrical vessel configuration ( κ ) residual,k unloaded Fig. 3 Mean simulated pressure–internal diameter relationship for the three cases analysed in this study: (1) full tri-layered model with residual deformations (A), (2) tri-layered model neglecting residual deformations (B), and (3) tri-layered model neglecting F (C). SD standard deviation intima and media at pressures > 90 mmHg, so that C θθθθ were 60 ± 22% (p = 0.014), 39 ± 18% (p = 0.032) and 24 at 160  mmHg was 4.20 ± 1.71  MPa for the adventitia, ± 9% (p = 0.018) higher than corresponding values in approximately three and four times higher than that of the full model with residual stresses, respectively, while media (1.34 ± 0.28  MPa) and intima (0.95 ± 0.27  MPa), those of the adventitia were 32 ± 5% (p = 0.091) and 16 respectively. ± 8% (p = 0.11) lower. As a result, the intimal load bear- Second, fully neglecting the layer residual stresses ing at 100 mmHg increased to 23 ± 1% (p = 0.009), that ( F = I ), strongly affected the distribution of of the adventitia dropped to 16 ± 2% (p = 0.002), while residual,k both σ and σ among layers; σ and σ were high- θθ zz θθ zz that of the media was unaltered (Fig.  5C, D). Further, est and lowest at the intima and adventitia, respectively, the shift in load bearing in response to the 60  mmHg over most of the investigated pressure range (Fig.  4B, pressure increased was milder at – 3 ± 3% and + 4 ± 4% E). At 100  mmHg, intimal σ and σ and medial σ θθ zz zz for media and adventitia, respectively. Indeed, while Giudici and Spronck Artery Research (2022) 28:41–54 49 Fig. 4 Layer-specific circumferential ( σ ) and axial ( σ ) Cauchy stress and circumferential material stiffness ( C ) as a function of pressure θθ zz θθθθ for the three cases analysed in this study: (1) full tri-layered model with residual deformations (A, D, G), (2) tri-layered model neglecting residual deformations (B, E, H), and (3) tri-layered model neglecting F (C, F, I) still exhibiting a biphasic C −P relationship, adventi- C at 160 mmHg was comparable in the three layers: θθθθ θθθθ tial stiffening with increasing pressure was less marked 1.58 ± 0.34 MPa, 1.39 ± 0.26 MPa and 2.29 ± 1.35 MPa than that observed for the model with residual stresses. for intima, media and adventitia, respectively (Fig. 4H). Giudici and Spronck Artery Research (2022) 28:41–54 50 Fig. 5 Mean circumferential Cauchy stress–circumferential stretch relationships with load partitioning between layers and layer-specific circumferential load bearing for the three cases analysed in this study: (1) full tri-layered model with residual stresses (A and B), (2) tri-layered model neglecting residual stresses (C and D), and (3) tri-layered model neglecting F (E and F). Circumferential stretch at the inner radius was computed m a a as  =r /R . In Panels A, C and E, the intimal line was obtained using Eq. 11 with σ = 0 and σ = 0 , and the media line with σ = 0 . The internal internal adventitial line was obtained using the full version of Eq. 14. This means that, for any given  /pressure, the amplitude of each coloured area is given k h by the corresponding layer’s circumferential stress multiplied by its loaded relative thickness (i.e. σ where k indicates a generic layer) wall θθ Third, Fig.  4C, F present the layer-specific σ and increased further to 0.140 ± 0.022 MPa (+ 82 ± 21% with θθ σ as a function of pressure for the model where respect to the full model, p = 0.010, and + 14 ± 2% with zz F = G (i.e. neglecting the contribution of F ). respect to the model with no residual stresses, p = 0.041) residual,k σ and σ were lowest and highest at the adventitia and 0.106 ± 0.022 MPa (+  86 ± 14%, p = 0.028, and + 35 θθ zz and intima, respectively, throughout the entire investi- ± 16%, p = 0.13). Intimal circumferential load bearing at gated pressure range. Intimal σ and σ at 100  mmHg 100 mmHg increased to 27 ± 2% (p = 0.002 and p = 0.013 θθ zz Giudici and Spronck Artery Research (2022) 28:41–54 51 with respect to cases 1 and 2, respectively; Fig.  5E, F). stresses aim at counterbalancing this effect, guaranteeing Further, C was lower in the adventitia than in the an almost uniform distribution of stresses throughout the θθθθ intima and media at most pressures (Fig.  4I), so that the wall thickness in the physiological range of pressures [21, adventitia did not act as a protective sleeve against high 24, 36]. Indeed, when residual stresses were included in pressures. The shift in load bearing between adventitia the tri-layered arterial model, the stress-pressure rela- and media when moving from 100 to 160  mmHg was tionships of intima, media, and adventitia were almost negligible (− 1 ± 2% and + 2 ± 2%, respectively). superimposed throughout most of the pressure range investigated in this study. However, at high pressures (i.e. > 140–150  mmHg), the adventitial stress consider- 4 Discussion ably deviated from that of the intima and media due to Residual stresses play a fundamental role in arterial its rapid stiffening with increasing pressure. This finding mechanics [21, 22]. However, modelling residual stresses suggests a strong functional coupling between the layer- from complex three-dimensional geometrical features of specific microstructure, mechanical properties, and dep - layer-specific stress-free configurations is non-trivial [20, osition stretches (i.e. F ), allowing the adventitia residual,k 27] and such stresses are hence sometimes neglected in to behave similarly to intima and media at normal physi- experimental studies on arterial structure and mechanics ological pressures and act as a protective sleeve that pre- [28, 29]. Using a recently developed tri-layered modelling serves the wall integrity at high- and supra-physiological framework that relies solely on data from wall and layer- pressures [2]. specific uniaxial mechanical  testing, this study aimed In agreement with previous findings [37], neglecting to illustrate the effects of totally or partially neglecting the layer-specific residual stresses affected the role indi - residual stresses on the mechanical behaviour of the tri- vidual layers play in the macroscopic behaviour of the layered arterial wall [30]. wall; the intimal load bearing increased from 16 to 23%, In 1986, Chuong and Fung [21] were the first to esti - reflecting a 55% increase in its circumferential stress. Fur - mate the distribution of stretches throughout the wall thermore, the adventitia lost its role of protective layer at thickness of the unloaded rabbit thoracic aorta from high pressures. Recently, De Lucio et al. [28] performed a measures of its OA. They found that the circumferential finite element simulation of the inflation of a tri-layered stretch ranged from approximately -0.86 at the luminal model of the human aneurysmatic aorta using previously surface to 1.14 at the outer adventitial surface, cross- determined layer-specific HGO-SEF model parameters ing 1 (i.e. null deformation) at 37% of the wall thickness. of the human abdominal aorta with non-atherosclerotic More recently, Holzapfel and colleagues [20] developed a intimal thickening [13]. Their model did not account for thick-walled modelling framework to estimate the three- residual stresses (comparable to scenario 2 in our study) dimensional layer-specific deformation of an unloaded and estimated that the intima, constituting approximately vessel from measurements of the curvature of both cir- 25% of the wall thickness, bore approximately 71% of the cumferentially (i.e. estimating the  OA) and axially ori- circumferential load, with only marginal contributions ented strips. When applied to the human aorta, their from media (14%) and adventitia (15%). While the non- method estimated compression in both circumferential atherosclerotic intimal thickening and the pathological and axial directions for the intima, circumferential ten- geometrical features of the aneurysmatic wall likely con- sion and axial compression for the media, and tension in tributed to concentrating circumferential stresses at the both directions for the adventitia [20, 36]. In the present intima, these findings need to be considered with caution study, the circumferential stretch in κ was 0.93 for unloaded since neglecting the layer-specific residual stresses of the the intima, 1.01 for the media and 1.04 for the adventitia. aneurysmatic wall [18] likely considerably altered the dis- It is worth noting, however, that, given the thin-walled tribution of stresses across the wall thickness. modelling approach used here, these values refer to the Among experimental methods, tension–inflation mid-wall point of each layer and are assumed to be con- experiments most closely mimic the physiological multi- stant throughout the layer thickness. The intimal, medial directional loading condition the arterial wall is subjected and adventitial mid-wall points were located at approxi- to in vivo [38]. However, the required specialised equip- mately 9%, 42% and 85% of the total wall thickness from ment is not available in every biomechanics laboratory the luminal surface, respectively. Therefore, deformations and might pose some limits when simultaneously imag- at the luminal and outer adventitial surface likely exceed ing the wall microstructure in human-sized arteries [6]. average values for the intima and adventitia [21]. Therefore, planar uniaxial and biaxial tensile tests are still The inflation of a cylindrical structure induces circum - widely used to investigate arterial mechanics and micro- ferential deformations that decrease monotonically from structure ex  vivo [10–14, 17]. Their inherent methodo - the luminal surface to the outer surface of the adventi- logical limitations should, however, always be considered tia (see Eq.  6). Researchers generally agree that residual Giudici and Spronck Artery Research (2022) 28:41–54 52 when analysing their results, and the tri-layered model- intact wall (on average 5–9% of the sample thickness). ling framework herein represents an attempt, in part, This suggests that the variability in layer thickness along to address this issue. For example, planar uniaxial and the sample length mainly reflected actual variability in biaxial tensile tests are commonly used to estimate func- intact wall thickness [42] rather than being a sign of sub- tions that define the collagen recruitment with increasing optimal layer separation. stretch [39, 40]. Chow and colleagues [11] combined pla- nar biaxial mechanical testing (comparable to scenario 3 5 Conclusions in our study) and multiphoton microscopy to investigate In this study, a recently developed tri-layered modelling the rearrangement of collagen and elastin microstruc- framework was used to investigate the effect of residual tures in response to load and found that the recruitment stresses on the mechanics of the tri-layered arterial wall. of adventitial collagen showed a 20% strain delay with While isolated layers exhibited considerably different respect to that of the media. It is worth noting, however, behaviour, their pre-deformed state make their response that our model predicted a 7% circumferential elonga- strikingly similar at physiological pressures. Conversely, tion of the adventitia in the deformation from κ at high pressures, the adventitial mechanical response composite to κ ( F ). This deformation would compensate, at deviated from that of intima and media, conferring its unloaded 1 least in part, the observed delayed recruitment of adven- protective function. titial collagen. Indeed, in our model,  neglecting F con- siderably limited the contribution of the adventitia to the Abbreviations macroscopic wall behaviour (Fig. 4C, F, I). a: Adventitia; α: Collagen fibre orientation parameter; c : Collagen fibre stiffness-like parameter; c : Collagen fibre non-linearity parameter; C : Right Cauchy–Green strain tensor; C: Wall material stiffness; F : Deforma- 4.1 Limitations tion gradient mapping the deformation from κ to κ composite unloaded ; F : Deformation gradient mapping the deformation from κ to In this study, the three layers were modelled as three con- 2 unloaded κ ; G : Layer-specific deformation gradient mapping the tension−inflation centric membranes (thin-walled approach). Therefore, deformation from κ to κ ; HGO: Holzapfel–Gasser–Ogden; isolated composite all modelled parameters, including F , and biome- residual,k h: Arterial wall thickness in κ ; i: Intima; κ : Isolated tension−inflation isolated chanical variables represent mean values across the layer layer configuration in Cartesian coordinates ( X ,Y ,Z ); κ : Flat, composite tri-layered wall configuration in Cartesian coordinates ( X,Y,Z); κ : unloaded thickness. In reality, arterial layers are not as homogene- Unloaded, cylindrical, tri-layered vessel configuration in cylindrical coordinates ous and F is not constant across the layer thick- residual,k ( , R , Z); κ : Axially stretch and pressurised, cylindrical, tension−inflation ness. This leads to the existence of layer-specific opening tri-layered vessel configuration in cylindrical coordinates ( θ , r , z); l : Length of the flat, tri-layered wall in the j-direction; L : Length of the isolated layer k angles, possibly affecting the stress distribution across k k in the j-direction;  : j-direction component of G ;  : j-direction compo- the layer thickness [14, 20]. A thick-walled modelling j nent of F ;  : j-direction component of F ; m: Media; µ : Elastin stiffness- 1 j 2 approach would further refine the analysis, but model like parameter; OA: Opening angle; : Strain energy function; R : internal complexity would also increase considerably as would the Unloaded luminal radius; r : Loaded luminal radius; ρ: Collagen fibre internal dispersion parameter; SEF: Strain energy function; σ: Modelled Cauchy stress complexity of estimating the related geometrical param- tensor; σ : Modelled Cauchy stress tensor. exp eters (layers’ three-dimensional curvatures). The layer-specific model parameters were estimated by Acknowledgements We thank Ashraf W. Khir for his support and valuable advice. fitting the intact wall and layer-specific response to uni - axial testing in the circumferential and axial directions, Author Contributions simultaneously, as done previously [13, 14, 30]. While AG and BS contributed to the study conception and to the development of the tri-layered modelling framework used in the study. AG contributed to the this approach indirectly yields biaxial information, biaxial experimental data collection and analysis, and to the manuscript drafting. BS experiments could further refine our analysis, providing contributed to the critical appraisal and manuscript editing. direct information on the coupling between circumferen- Funding tial and axial responses. This work was supported by the ARTERY (Association for Research into Arterial We did not perform any imaging of the layers’ cross- Structure and Physiology) society (2019 Research Exchange Grant to AG) and section to visually verify the accuracy of the peeling pro- by the European Union’s Horizon 2020 Research and Innovation programme (Grant 793805 to BS). The sole role of the funding sources was providing cess. Nevertheless, layer-specific thicknesses found here financial contribution. are in line with those reported in other studies [13, 14, 41]. Further, the thickness of each layer was measured Availability of Data and Materials The data supporting this study are available from the corresponding author. three times along the strip length, thus allowing to eval- uate whether the peeling process was performed uni- Declarations formly. 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The Role of Layer-Specific Residual Stresses in Arterial Mechanics: Analysis via a Novel Modelling Framework

Giudici, Alessandro; Spronck, Bart
Artery Research , Volume 28 (2) – Jun 1, 2022
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10.1007/s44200-022-00013-1
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Abstract

The existence of residual stresses in unloaded arteries has long been known. However, their effect is often neglected in experimental studies. Using a recently developed modelling framework, we aimed to investigate the role of residual stresses in the mechanical behaviour of the tri-layered wall of the pig thoracic aorta. The mechanical behav- iour of the intact wall and isolated layers of n = 3 pig thoracic aortas was investigated via uniaxial tensile testing. After modelling the layer-specific mechanical data using a hyperelastic strain energy function, the layer-specific deforma- tions in the unloaded vessel were estimated so that the mechanical response of the computationally assembled tri-layered flat wall would match that measured experimentally. Physiological tension–inflation of the cylindrical tri- layered vessel was then simulated, analysing changes in the distribution of stresses in the three layers when neglect- ing residual stresses. In the tri-layered model with residual stresses, layers exhibited comparable stresses throughout the physiological range of pressure. At 100 mmHg, intimal, medial, and adventitial circumferential load bearings were 16 ± 3%, 59 ± 4%, and 25 ± 2%, respectively. Adventitial stiffening at high pressures produced a shift in load bearing from the media to the adventitia. When neglecting residual stresses, in vivo stresses were highest at the intima and lowest at the adventitia. Consequently, the intimal and adventitial load bearings, 23 ± 2% and 18 ± 3% at 100 mmHg, were comparable at all pressures. Residual stresses play a crucial role in arterial mechanics guaranteeing a uniform distribution of stresses through the wall thickness. Neglecting these leads to incorrect interpretation of the layers’ role in arterial mechanics. Keywords: Residual stresses, Tri-layered arterial wall modelling, Intima, Media, Adventitia 1 Introduction arterial function. The intima, the innermost layer, inter - The arterial wall is characterised by a strong relationship acts directly with the blood flow and provides a small between structure and function [1, 2], so that arterial contribution to the wall’s mechanics due to its small structure varies considerably at different locations in the thickness. The media, the middle and thickest layer, arterial tree [2]. Large elastic arteries are located in prox- determines the elastic behaviour that characterises large imity of the heart. At a macroscopic level, these arteries arteries at physiological pressures [3] and endows them are organised in three concentric layers, the intima, the with their blood pressure buffering (or Windkessel) func - media and the adventitia, each playing a pivotal role in tion, which smooths the intermittent pumping action of the heart into a more continuous blood pressure ensur- ing organ perfusion also in diastole. The adventitia is the *Correspondence: a.giudici@maastrichtuniversity.nl 3 outermost layer and is typically described as a protective Department of Biomedical Engineering, CARIM School for Cardiovascular Diseases, Maastricht University, Universiteitssingel 50, sleeve that preserves the wall’s integrity at high pressures Room 3.353, 6229 ER Maastricht, The Netherlands [2]. Full list of author information is available at the end of the article © The Author(s) 2022. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Giudici and Spronck Artery Research (2022) 28:41–54 42 To fulfil their specific function, arterial layers differ the unloaded vessel represents an equilibrium configura - significantly at the microstructural level. The intima is tion between residual stresses acting throughout the arte- constituted mainly by endothelial cells, a thin basement rial wall thickness that are compressive and tensile in the membrane, a proteoglycan rich matrix and collagen inner and outer part of the wall, respectively [26]. When fibres oriented in the axial direction [4, 5]. The media is a radial cut is performed, the geometrical constraint that organised in concentric medial lamellar units, alternating guarantees this equilibrium is lost and the artery deforms elastin-rich lamellae and inter-lamellar spaces composed to release said residual stresses, reaching a new equilib- of vascular smooth muscle cells, elastin and collagen rium in its stress-free configuration. Knowledge of the fibres which are predominantly oriented in the circum - inverse of this deformation is, therefore, crucial to deter- ferential direction [3, 4, 6]. The adventitia is mainly con - mine the layer-specific prestress in the artery unloaded stituted by bundles of diagonally oriented collagen fibres, state and constitutes the link to fully understand the com- showing larger angular dispersion than in the media. Fur- plex tri-layered mechanics of the arterial wall. ther, unlike the intima and the media, the adventitia is The circumferential component of this deformation relatively acellular [7, 8]. is commonly quantified via an opening angle (OA), e.g. Understanding the role each layer plays in arterial the angle formed by connecting the two endpoints of the function has been the objective of several investigations, arc-shaped circumferential wall sample to its mid-point using different experimental techniques such as selective [22]. Knowledge of the OA allows estimating the distri- enzymatic digestion of specific wall constituents [9, 10] bution of the  circumferential deformation throughout and imaging the wall’s microstructure at different levels the wall thickness of the unloaded artery, thus allowing of distending pressure [3, 11, 12]. The separation and to determine its pre-stressed state [23]. Further, a more mechanical testing of isolated layers is a useful technique comprehensive characterisation of the layer-specific that directly assesses how their different microstructures three-dimensional geometry of excised circumferential determine their mechanical behaviour. This technique and axial wall strips allows formulating complex mod- has been used to characterise the layer-specific mechan - els that predict the distribution of residual stresses in all ics in several arterial locations, species and in both physi- three principal directions throughout the wall thickness ological and pathological conditions [13–19]. Moreover, [20, 27]. However, the implementation of such models is constitutive modelling, i.e. fitting structurally motivated non-trivial and residual stresses are often neglected when mathematical models that describe the wall’s mechanical modelling the tri-layered arterial wall [28, 29]. behaviour as the summed contribution of the response of We recently developed a new tri-layered modelling its constituents, has allowed investigating how layer-spe- framework that allows accounting for layer-specific cific microstructural features affect the layer’s mechani - residual stresses using only intact wall and layer-specific cal properties, including anisotropy and recruitment of uniaxial test data [30]. Using the newly developed mod- collagen fibres [13–19]. elling framework, the present study aimed to illustrate While characterising the behaviour of isolated lay- the effects of residual stresses on arterial mechanics and ers provides some insight into the tri-layered arterial how neglecting these can lead to considerable misinter- wall mechanics, it is not sufficient to fully understand pretation of the specific role arterial layers play in the their role in the macroscopic behaviour of arteries. macroscopic behaviour of the arterial wall. We will simu- Indeed, while an excised unloaded artery is subjected to late three different scenarios: (1) a complete model with null external loads in all three principal directions (i.e. residual stresses, (2) a model where the unloaded vessel circumferential, radial and axial), local stresses at any is considered stress free (i.e. completely neglecting all generic point within the arterial wall’s volume are, in residual stresses), and (3) a model where a flat slab of wall general, non-zero [20, 21]. As shown in previous stud- tissue (as typically tested in planar biaxial tensile experi- ies [22–24], this particular feature of arteries likely arises ment) is used as an approximation of a cylindrical vessel. from the fact that their wall develops and remodels in its loaded state (i.e. in  vivo), attempting to preserve a uni-2 Methods form distribution of stresses throughout its thickness. 2.1 Theoretical Background When all external loads are removed, however, its inho- 2.1.1 T ri‑layered Wall Model mogeneous material properties and in  vivo deformation A complete description of the mathematical formulation field lead to the existence of residual stresses. of the tri-layered wall model can be found in our previous Residual stresses were first discovered in 1983 when study [30]. Briefly, the arterial wall is assumed to be com - two studies [21, 25] observed that, when cut radially, an prised of three adequately spaced membranes representing arterial ring typically does not retain its circular shape but the intima, media, and adventitia. The composition of the springs open and assumes an arc shape. This implies that wall from isolated layers to unloaded cylindrical vessel is Giudici and Spronck Artery Research (2022) 28:41–54 43 described via two mapping steps: first, the three layers are The total deformation each layer is subjected to in assembled into a flat rectangular slab of arterial wall tis - κ is given by tension−inflation sue. Due to the presence of residual stresses, layers change F = F F total,k 2 residual,k their shape when isolated from the wall. The layer-specific deformation gradient G (where k ∈{i, m, a} , i=intima, m k k = diag  �  , ,  �  . θ � z Z X Z =media, and a=adventitia) describes the inverse of this k k �   � θ � z Z X Z process, i.e. the deformation from the isolated layer con- (7) figuration κ in Cartesian coordinates (X ,Y,Z ) to tri- isolated layered flat wall slab κ in ( X,Y, ). For each layer k, composite 2.1.2 Lay erSp ‑ ecific Constitutive Modelling k 1 k , ,  The mechanical behaviour of the three isolated arterial G = diag , k k (1) X   Z X Z layers was modelled using the Holzapfel–Gasser–Ogden k k k k (HGO) two fibre family strain energy function (SEF) [32]. = l /L where and  = l /L are the circumferen- X X Z Z Z The passive behaviour of each layer is assumed to be tial and axial components of G , l and l are the circum- determined by the summed contribution of an isotropic ferential and axial dimensions of the composite wall, and k k matrix, typically associated with elastin, and two sym- L and L are the circumferential and axial lengths of the X Z metrically oriented families of collagen fibres that deter - isolated layer k . Note that the radial stretch is determined mine the wall’s anisotropy: from incompressibility [31]. The second mapping step describes the deformation of k k k k c ρ (I )+ 1−3ρ I −1 c 1 4,i k k 2 2 the flat tri-layered wall, κ , into an unloaded cylin- composite � = µ (I − 3) + (e − 1), i=1 k 2c drical vessel, κ in cylindrical coordinates ( , R , Z ). unloaded (8) The related deformation gradient is where µ is an isotropic stiffness-like parameter 1 ( k ∈{i, m, a} , i=intima, m=media, and a=adventitia), c F = diag  , ,  , Z (2) is a collagen stiffness-like parameter, c is a dimension- less collagen non-linearity parameter, and ρ ∈[0, ] is where the axial stretch  is assumed to be constant 3 a fibre dispersion coefficient, with ρ = 0 denoting fully throughout the wall and the circumferential component aligned and ρ = denoting fully dispersed fibres (i.e. is calculated as 3 isotropic behaviour). I and I are the first and fourth 1 4,i 2 invariant of the right Cauchy–Green tensor, respectively, 4π R 4πY internal � = + , (3) l � l X Z with i ∈ {1, 2} indicating the collagen fibre family with k k k principal orientation α = {−α , α } with respect to the 1,2 where R is the radius of the artery at the luminal internal circumferential orientation. Note that the layer-specific surface. fibre orientation α was not derived from microscopy 1,2 The layer-specific residual stresses in κ arise from unloaded data but phenomenologically estimated from the meas- the total deformation from κ to κ isolated unloaded ured mechanical behaviour. For this reason, these angles should not be regarded as exact measures of the fibre k 1 k , , F = F G = diag  Z . residual,k 1 X k k Z orientation in the arterial layers but rather as a general X Z quantification of the collagen-induced tissue anisotropic (4) properties. The deformation gradient F maps the tension–inflation The Cauchy stress tensor is defined as of the cylindrical vessel to a simulated in  vivo configura - tion, κ in cylindrical coordinates ( θ , r , z ). As tension−inflation ∂� k T σ = −pI + 2F F , (9) total,k ∂C total,k for F , the axial component of F ( ) is assumed to be con- total,k 1 2 z stant throughout the wall. F is defined as where I is the spatial second order identity tensor and p is 1 the Lagrange multiplier enforcing incompressibility [31]. F = diag  , ,  , θ z (5) 2.2 Experimental Methods with Three pig plucks (i.e. the content of the thorax: 2 2 2 r R −R heart, thoracic aorta, lungs, trachea, and liver) (age internal internal = + , (6) 2 2 R R 6–12  months) were purchased from a local abattoir (samples from school, UK). Delivered frozen, each where r is the luminal radius in κ . internal tension−inflation pluck was immediately stored at − 20 °C in a laboratory Giudici and Spronck Artery Research (2022) 28:41–54 44 2.3.1 Lay erSp ‑ ecific Constitutive Parameters freezer until the day of testing when it was left to thaw The five layer-specific constitutive parameters in at room temperature for ~ 4 h. The aorta was then care - Eq.  8 were fitted on the layer-specific mechani - fully dissected from the rest of the tissues using a scalpel. cal data, minimising the error between the individ- Two circumferentially and two axially oriented arterial ual layer’s experimental (Eq.  10) and modelled (Eq.  9) strips (approximately 4 × 25 mm ) were isolated from Cauchy stress in the loading direction for the uni- the region of the descending thoracic aorta character- axial tests in the circumferential and axial direction ised by the branching of the intercostal arteries. Their simultaneously. Since the off-axis deformation  (i.e. width and thickness were carefully measured three times j ∈{axial, circumferential} corresponding to the loading along the strip length using a high precision digital cal- directions i ∈{circumferential, axial} ) was not measured liper. The mean of the three measurements was used during the experiments, this was determined by impos- for later analysis. Each strip was then subjected to uni- ing σ = 0 and enforcing incompressibility. axial tensile testing in its long direction using a high- jj resolution uniaxial tensile device (MFS Stage with 20N 2.3.2 D etermination of the LayerSp ‑ ecific Deformation load cell, Linkam Scientific Ltd., UK) equipped with ser - in κ rated jaws to prevent slipping of the sample. The inter- unloaded As shown in Eq.  4, determining the layer-specific defor - jaw distance was initially set to 15  mm and then slowly mation gradient in κ requires estimating G and increased until the sample laid flat. This flat length was unloaded F . The axial and circumferential components of G of considered as the unloaded sample length. Strips were the three layers were fitted on the mechanical data of the uniaxially tested up to a Cauchy stress of 250  kPa after intact wall, simultaneously minimising the error between performing five loading/unloading cycles up to the same the wall experimental Cauchy stress (Eq.  10) and that Cauchy stress. After testing, each strip was separated obtained when simulating the uniaxial tensile test of the into its three layers using tweezers, first carefully peeling tri-layered composite wall in both circumferential and the adventitia from the intima-media and then isolating axial directions. The average modelled Cauchy stress of the intima from the media. As done for the intact wall, the composite wall was calculated as the layer’s dimensions were measured at three locations along the strip length and the average values used for i i m m a a wall h σ +h σ +h σ σ = . (11) later analysis. Then, each layer was tested following the wall same protocol described for the wall strips. As described in the introduction, the unloaded con- The wall’s and layer’s experimental Cauchy stress was figuration of the vessel ( κ ) implies zero average unloaded calculated as [33] wall stresses in all three principal directions. Therefore, k F and the layer-specific  were estimated iteratively by σ =  , (10) ii,exp A i wall wall wall simultaneously minimising σ , σ , and σ . �� RR ZZ where F is the force measured by the load cell, A is 2.3.3 Simulation of the Tension–Inflation the sample unloaded cross-sectional area, and  is the To assess the contribution of residual stresses to the stretch in which direction i (circumferential or axial) the wall mechanics, we simulated three tension–inflation sample k ∈ wall, i, m, a is loaded during the uniaxial scenarios: (1) full tri-layered model including layer-spe- test. cific residual stresses ( F = F G ), (2) tri-layered residual,k 1 model where residual stresses are completely neglected 2.3 Parameter Estimation ( F = I ), and (3) assuming F = I ( F = G ) The fitting routine of all the model parameters has been residual,k 1 residual,k (Table  1 and Fig.  1). Case 1 represents the closest described previously [30], and is briefly summarised approximation of the in  vivo condition; cases 2 and 3 below. Table 1 Description of the three modelling scenarios analysed in this study Scenario F Description residual,k 1 Layer-specific residual stresses in the unloaded cylindrical vessel ( κ ) arise from two subsequent F G unloaded deformations of the isolated layers ( κ ): (1) into a tri-layered flat wall ( κ ) that is (2) then bent composite isolated into κ unloaded 2 I Residual stresses in κ are completely neglected as no deformation occurs from κ to κ unloaded isolated unloaded 3 Layer-specific residual stresses in κ arise solely from the deformation of the isolated layers G unloaded ( κ ) into κ . Hence, κ is used as an approximation of κ isolated composite composite unloaded Giudici and Spronck Artery Research (2022) 28:41–54 45 Fig. 1 Schematic representation of the different configurations and linking deformation gradients analysed in this study: (1) isolated layers ( κ ), (2) composite wall ( κ ), (3) unloaded cylindrical vessel ( κ ), and (4) pressurised axially stretched cylindrical vessel isolated composite unloaded ( κ ). Note that the deformation from κ to κ , mapped by the deformation gradient F , defines the layer-specific tension−inflation isolated unloaded residual,k residual deformations. Deformation gradients were differentially defined in the three analysed scenarios s ∈{1,2,3} ( Table 1) as indicated. Note the use of different coordinate systems in the respective configurations. Superscript k ∈ {i, m, a} indicates the intimal ( i ), medial (m ), or adventitial ( a ) layer, respectively replicate two possible modelling approximations. In case Then, pseudo-physiological tension–inflation was sim - 2, κ is assumed to be a stress-free configuration ulated by axially stretching the vessel to  and increasing unloaded (i.e. all three layers are undeformed in these configura - luminal pressure from 0 to 200 mmHg. tions). In case 3, the flat arterial wall (which is the typi - Layer-specific and whole-wall stresses were determined cal configuration of uniaxial and biaxial tensile tests) is using Eqs.  9 and 11, respectively. The percentage of load assumed to be a good approximation of the cylindrical bearing for each layer was determined as vessel, thus neglecting the deformation from κ to composite k k σ h θθ κ . Load bearing % = · 100%. unloaded (14) wall wall σ h θθ Except for the definition of F , all three cases fol- residual,k lowed the same modelling process. First, the simulated The circumferential material stiffness C was calcu- θθθθ in  vivo axial stretch  was determined as the average lated according to the small-on-large formulation [35] cross-over point between simulated reduced axial force- and evaluated continuously as a function of the distend- axial stretch relationships at the distending pressure lev- ing pressure: els P = 60 , 100 , and 140 mmHg [34]. The reduced axial ∂ � force F was calculated as C = 2(σ + p) + 4 . θθθθ θθ θ 2 (15) k k k 2 F = 2πσ r h − πr P, z (12) zz k=i,m,a internal 2.4 Statistical Analysis where r is the mid-wall radius of the layer k . Given P , For each aorta, tensile tests were conducted on two axi- the corresponding r was estimated iteratively to internal ally oriented and two circumferentially oriented strips. The satisfy the Laplace equation layer-specific constitutive and tri-layered modelling was then k h conducted in pairs (i.e. circumferential strip 1 with axial strip P = (σ − σ ) . (13) k=i,m,a θθ rr k r Giudici and Spronck Artery Research (2022) 28:41–54 46 3.2 Tri‑layered Wall Modelling 1 and circumferential strip 2 with axial strip 2) and the pre- 3.2.1 Lay erSp ‑ ecific Deformation in  κ sented results of each artery are the mean of the two. unloaded The layer-specific components of the deformation gra - Results are presented as mean ± standard devia- k k dients G , F , and F = F G are presented in tion of the three tested aortas. Outcome parameters 1 residual,k 1 Table  4. In the flat wall configuration κ , the (wall stresses, material stiffness, load bearing %) were composite intima was subjected to tensile stretches in both circum- first evaluated continuously as a function of pressure. ferential and axial directions. Conversely, from κ to Then, comparison between modelling conditions (i.e. isolated κ , both media and adventitia experienced defor- with and without residual stresses) was carried out composite mations of opposite nature in the circumferential and at the reference normotensive mean arterial pres- axial directions: circumferential extension and axial com- sure (100  mmHg) and at 160  mmHg, representing the pression for the media and circumferential compression average hypertensive systolic pressure, using paired and slight axial extension for the adventitia. As expected, student’s t tests. p < 0.05 was taken as statistically the deformation from κ to κ ( F ) intro- significant. composite unloaded 1 duced compressive and tensile circumferential deforma- tions in the inner half and outer half of the wall thickness, 3 Results respectively, with negligible deformations in the axial 3.1 W all and Layer‑Specific Response to Uniaxial Testing direction. As a result, the total circumferential deforma- The wall and layer geometrical features of the three tion from κ to κ ( F ) was strongly com- pig thoracic aortas tested in this study are reported isolated unloaded residual pressive (0.93 ± 0.01) for the intima, slightly tensile (1.01 in Table  2. The mean radius and wall thickness were ± 0.01) for the media, and tensile (1.04 ± 0.00) for the 7.79 ± 0.45  mm and 1.74 ± 0.31  mm, respectively. adventitia. The isolated layer thicknesses were 0.28 ± 0.10  mm for the intima, 0.89 ± 0.24  mm for the media and 3.2.2 Simulated Tension–Inflation 0.51 ± 0.07  mm for the adventitia, corresponding to Figure  3A–C presents the mean simulated pressure– 17 ± 2%, 52 ± 3% and 31 ± 1% of the wall thickness, diameter relationships for the three scenarios consid- respectively. ered in this study (Table 1 and Fig. 1). Interestingly, while Figure 2 presents the wall and layer-specific response to pressure–diameter relationships appeared very similar uniaxial tensile testing in both circumferential and axial in the three considered scenarios, the distribution of directions. The wall was stiffer circumferentially than stresses between the three layers differed considerably: axially at stretches below 1.3 but became increasingly First, in the full tri-layered wall model with residual isotropic at higher deformations. This complex behav - stresses, σ −P and σ −P relationships of the three lay- iour reflects the heterogeneity of the layer responses. θθ zz ers appeared almost superimposed in the physiological Layer-specific model parameters of the three aortas are pressure range (Fig.  4A, D). At a physiological pressure reported in Table 3. Both intima and media showed con- of 100  mmHg, σ and σ were 0.078 ± 0.017 and 0.058 siderable anisotropy (Fig. 2), with α ranging from 36.2°– θθ zz ± 0.014 MPa, 0.107 ± 0.013 and 0.062 ± 0.010 MPa, and 38.9° (intima) and 28.8°–35.5° (media), with respect to 0.078 ± 0.012 and 0.067 ± 0.004 MPa for intima, media, the circumferential direction. Conversely, the adventi- and adventitia, respectively. The percentage of circum - tia displayed a nearly isotropic behaviour (Fig.  2), if not ferential load borne by the media (62 ± 3%) was more slightly stiffer in the axial direction ( α =43.0°–49.5°). than twice that of the adventitia (24 ± 2%) and four times Further, the non-linearity parameter c was more than an that of the intima (14 ± 3%; Fig.  5A, B). When pressure order of magnitude higher in the adventitia than in both increased to 160  mmHg, the corresponding increase intima and media, signifying a more pronounced stiffen - in σ and σ was comparable in the intima, 72 ± 15% ing (non-linearity) with increasing stretch. θθ zz and 38 ± 11%, and media, 65 ± 12% and 34 ± 7%, but Table 2 Wall and layer geometrical features of the three pig thoracic aortas tested in this study intima wall media adventitia Sample # R (mm) h (mm) h (mm) h (mm) h (mm) I 8.00 2.09 0.29 1.15 0.59 II 8.20 1.62 0.29 0.83 0.49 III 7.16 1.50 0.27 0.68 0.46 Mean ± SD 7.79 ± 0.45 1.74 ± 0.25 0.28 ± 0.01 0.89 ± 0.19 0.51 ± 0.06 SD standard deviation Giudici and Spronck Artery Research (2022) 28:41–54 47 Fig. 2 Wall (A) and individual-layer (B intima; C media; D adventitia) Cauchy stress–stretch relationships of the three pig thoracic aortas (I, II, III) included in this study. Solid and dashed lines indicate uniaxial tests in the circumferential (circ.) and axial directions, respectively. Each curve was obtained by averaging the sample behaviour obtained in the two tests executed for each loading direction Table 3 Layer-specific parameters of the Holzapfel-Gasser-Ogden strain energy function of the three aortas included in this study k k k k k Layer Sample # R µ (kPa) c (kPa) c (–) α (°) ρ (–) 1 2 Intima I 12.7 163.5 0.1 36.6 0.20 0.98 II 22.7 165.8 3.3 38.9 0.21 0.97 III 6.5 249.5 1.1 36.2 0.20 0.96 Media I 24.6 129.0 3.6 29.1 0.24 1.00 II 30.1 95.7 4.0 28.8 0.21 1.00 III 29.0 182.4 3.4 35.5 0.20 1.00 Adventitia I 20.7 22.2 66.3 49.5 0.24 0.99 II 26.5 13.7 40.9 48.8 0.17 0.99 III 25.2 49.6 29.1 43.0 0.26 0.98 Parameters are defined in Eq. 8; R , coefficient of determination of the simultaneous fitting of both circumferential and axial uniaxial tensile test data approximately twice as high in the adventitia (135 ± relationship of the adventitia; for pressures < 90  mmHg, 25% and 111 ± 37%). As a result, while the intimal load the adventitial C −P relationship was compara- θθθθ bearing remained almost unaltered (13 ± 3%), medial ble to the intimal and medial relationships, which were and adventitial load bearings dropped and grew by -6 ± nearly linear over the entire investigated pressure range 3% and + 7 ± 4%, respectively. This shift in circumfer - (Fig.  4G). Conversely, adventitial stiffening with increas - ential load bearing was caused by the biphasic C –P ing pressure was markedly higher than that of both θθθθ Giudici and Spronck Artery Research (2022) 28:41–54 48 k k Table 4 Layer-specific circumferential and axial components of the deformation gradients G , F and F = F G . 1 residual,k 1 Intima Media Adventitia i i m m a a X Z X Z X Z I 1.04 1.02 1.04 0.99 0.94 1.02 II 1.02 1.05 1.03 0.97 0.98 0.99 III 1.03 1.00 1.01 0.99 0.97 1.02 Mean ± SD 1.03 ± 0.01 1.03 ± 0.02 1.03 ± 0.01 0.98 ± 0.01 0.96 ± 0.01 1.01 ± 0.01 m m a a i i Z  Z I 0.89 1.00 0.98 1.00 1.09 1.00 II 0.92 1.00 0.98 1.00 1.07 1.00 III 0.92 1.00 0.98 1.00 1.07 1.00 Mean ± SD 0.91 ± 0.01 1.00 ± 0.00 0.98 ± 0.00 1.00 ± 0.00 1.07 ± 0.01 1.00 ± 0.00 m a a k i i m m m a a F = F G i residual,k 1 X  X Z Z X  Z Z Z Z I 0.92 1.02 1.02 0.99 1.03 1.02 II 0.93 1.06 1.01 0.98 1.04 1.00 III 0.95 1.00 1.00 0.99 1.03 1.02 Mean ± SD 0.93 ± 0.01 1.03 ± 0.02 1.01 ± 0.01 0.98 ± 0.00 1.04 ± 0.00 1.01 ± 0.01 F defines the residual stretches experienced by each layer in the unload cylindrical vessel configuration ( κ ) residual,k unloaded Fig. 3 Mean simulated pressure–internal diameter relationship for the three cases analysed in this study: (1) full tri-layered model with residual deformations (A), (2) tri-layered model neglecting residual deformations (B), and (3) tri-layered model neglecting F (C). SD standard deviation intima and media at pressures > 90 mmHg, so that C θθθθ were 60 ± 22% (p = 0.014), 39 ± 18% (p = 0.032) and 24 at 160  mmHg was 4.20 ± 1.71  MPa for the adventitia, ± 9% (p = 0.018) higher than corresponding values in approximately three and four times higher than that of the full model with residual stresses, respectively, while media (1.34 ± 0.28  MPa) and intima (0.95 ± 0.27  MPa), those of the adventitia were 32 ± 5% (p = 0.091) and 16 respectively. ± 8% (p = 0.11) lower. As a result, the intimal load bear- Second, fully neglecting the layer residual stresses ing at 100 mmHg increased to 23 ± 1% (p = 0.009), that ( F = I ), strongly affected the distribution of of the adventitia dropped to 16 ± 2% (p = 0.002), while residual,k both σ and σ among layers; σ and σ were high- θθ zz θθ zz that of the media was unaltered (Fig.  5C, D). Further, est and lowest at the intima and adventitia, respectively, the shift in load bearing in response to the 60  mmHg over most of the investigated pressure range (Fig.  4B, pressure increased was milder at – 3 ± 3% and + 4 ± 4% E). At 100  mmHg, intimal σ and σ and medial σ θθ zz zz for media and adventitia, respectively. Indeed, while Giudici and Spronck Artery Research (2022) 28:41–54 49 Fig. 4 Layer-specific circumferential ( σ ) and axial ( σ ) Cauchy stress and circumferential material stiffness ( C ) as a function of pressure θθ zz θθθθ for the three cases analysed in this study: (1) full tri-layered model with residual deformations (A, D, G), (2) tri-layered model neglecting residual deformations (B, E, H), and (3) tri-layered model neglecting F (C, F, I) still exhibiting a biphasic C −P relationship, adventi- C at 160 mmHg was comparable in the three layers: θθθθ θθθθ tial stiffening with increasing pressure was less marked 1.58 ± 0.34 MPa, 1.39 ± 0.26 MPa and 2.29 ± 1.35 MPa than that observed for the model with residual stresses. for intima, media and adventitia, respectively (Fig. 4H). Giudici and Spronck Artery Research (2022) 28:41–54 50 Fig. 5 Mean circumferential Cauchy stress–circumferential stretch relationships with load partitioning between layers and layer-specific circumferential load bearing for the three cases analysed in this study: (1) full tri-layered model with residual stresses (A and B), (2) tri-layered model neglecting residual stresses (C and D), and (3) tri-layered model neglecting F (E and F). Circumferential stretch at the inner radius was computed m a a as  =r /R . In Panels A, C and E, the intimal line was obtained using Eq. 11 with σ = 0 and σ = 0 , and the media line with σ = 0 . The internal internal adventitial line was obtained using the full version of Eq. 14. This means that, for any given  /pressure, the amplitude of each coloured area is given k h by the corresponding layer’s circumferential stress multiplied by its loaded relative thickness (i.e. σ where k indicates a generic layer) wall θθ Third, Fig.  4C, F present the layer-specific σ and increased further to 0.140 ± 0.022 MPa (+ 82 ± 21% with θθ σ as a function of pressure for the model where respect to the full model, p = 0.010, and + 14 ± 2% with zz F = G (i.e. neglecting the contribution of F ). respect to the model with no residual stresses, p = 0.041) residual,k σ and σ were lowest and highest at the adventitia and 0.106 ± 0.022 MPa (+  86 ± 14%, p = 0.028, and + 35 θθ zz and intima, respectively, throughout the entire investi- ± 16%, p = 0.13). Intimal circumferential load bearing at gated pressure range. Intimal σ and σ at 100  mmHg 100 mmHg increased to 27 ± 2% (p = 0.002 and p = 0.013 θθ zz Giudici and Spronck Artery Research (2022) 28:41–54 51 with respect to cases 1 and 2, respectively; Fig.  5E, F). stresses aim at counterbalancing this effect, guaranteeing Further, C was lower in the adventitia than in the an almost uniform distribution of stresses throughout the θθθθ intima and media at most pressures (Fig.  4I), so that the wall thickness in the physiological range of pressures [21, adventitia did not act as a protective sleeve against high 24, 36]. Indeed, when residual stresses were included in pressures. The shift in load bearing between adventitia the tri-layered arterial model, the stress-pressure rela- and media when moving from 100 to 160  mmHg was tionships of intima, media, and adventitia were almost negligible (− 1 ± 2% and + 2 ± 2%, respectively). superimposed throughout most of the pressure range investigated in this study. However, at high pressures (i.e. > 140–150  mmHg), the adventitial stress consider- 4 Discussion ably deviated from that of the intima and media due to Residual stresses play a fundamental role in arterial its rapid stiffening with increasing pressure. This finding mechanics [21, 22]. However, modelling residual stresses suggests a strong functional coupling between the layer- from complex three-dimensional geometrical features of specific microstructure, mechanical properties, and dep - layer-specific stress-free configurations is non-trivial [20, osition stretches (i.e. F ), allowing the adventitia residual,k 27] and such stresses are hence sometimes neglected in to behave similarly to intima and media at normal physi- experimental studies on arterial structure and mechanics ological pressures and act as a protective sleeve that pre- [28, 29]. Using a recently developed tri-layered modelling serves the wall integrity at high- and supra-physiological framework that relies solely on data from wall and layer- pressures [2]. specific uniaxial mechanical  testing, this study aimed In agreement with previous findings [37], neglecting to illustrate the effects of totally or partially neglecting the layer-specific residual stresses affected the role indi - residual stresses on the mechanical behaviour of the tri- vidual layers play in the macroscopic behaviour of the layered arterial wall [30]. wall; the intimal load bearing increased from 16 to 23%, In 1986, Chuong and Fung [21] were the first to esti - reflecting a 55% increase in its circumferential stress. Fur - mate the distribution of stretches throughout the wall thermore, the adventitia lost its role of protective layer at thickness of the unloaded rabbit thoracic aorta from high pressures. Recently, De Lucio et al. [28] performed a measures of its OA. They found that the circumferential finite element simulation of the inflation of a tri-layered stretch ranged from approximately -0.86 at the luminal model of the human aneurysmatic aorta using previously surface to 1.14 at the outer adventitial surface, cross- determined layer-specific HGO-SEF model parameters ing 1 (i.e. null deformation) at 37% of the wall thickness. of the human abdominal aorta with non-atherosclerotic More recently, Holzapfel and colleagues [20] developed a intimal thickening [13]. Their model did not account for thick-walled modelling framework to estimate the three- residual stresses (comparable to scenario 2 in our study) dimensional layer-specific deformation of an unloaded and estimated that the intima, constituting approximately vessel from measurements of the curvature of both cir- 25% of the wall thickness, bore approximately 71% of the cumferentially (i.e. estimating the  OA) and axially ori- circumferential load, with only marginal contributions ented strips. When applied to the human aorta, their from media (14%) and adventitia (15%). While the non- method estimated compression in both circumferential atherosclerotic intimal thickening and the pathological and axial directions for the intima, circumferential ten- geometrical features of the aneurysmatic wall likely con- sion and axial compression for the media, and tension in tributed to concentrating circumferential stresses at the both directions for the adventitia [20, 36]. In the present intima, these findings need to be considered with caution study, the circumferential stretch in κ was 0.93 for unloaded since neglecting the layer-specific residual stresses of the the intima, 1.01 for the media and 1.04 for the adventitia. aneurysmatic wall [18] likely considerably altered the dis- It is worth noting, however, that, given the thin-walled tribution of stresses across the wall thickness. modelling approach used here, these values refer to the Among experimental methods, tension–inflation mid-wall point of each layer and are assumed to be con- experiments most closely mimic the physiological multi- stant throughout the layer thickness. The intimal, medial directional loading condition the arterial wall is subjected and adventitial mid-wall points were located at approxi- to in vivo [38]. However, the required specialised equip- mately 9%, 42% and 85% of the total wall thickness from ment is not available in every biomechanics laboratory the luminal surface, respectively. Therefore, deformations and might pose some limits when simultaneously imag- at the luminal and outer adventitial surface likely exceed ing the wall microstructure in human-sized arteries [6]. average values for the intima and adventitia [21]. Therefore, planar uniaxial and biaxial tensile tests are still The inflation of a cylindrical structure induces circum - widely used to investigate arterial mechanics and micro- ferential deformations that decrease monotonically from structure ex  vivo [10–14, 17]. Their inherent methodo - the luminal surface to the outer surface of the adventi- logical limitations should, however, always be considered tia (see Eq.  6). Researchers generally agree that residual Giudici and Spronck Artery Research (2022) 28:41–54 52 when analysing their results, and the tri-layered model- intact wall (on average 5–9% of the sample thickness). ling framework herein represents an attempt, in part, This suggests that the variability in layer thickness along to address this issue. For example, planar uniaxial and the sample length mainly reflected actual variability in biaxial tensile tests are commonly used to estimate func- intact wall thickness [42] rather than being a sign of sub- tions that define the collagen recruitment with increasing optimal layer separation. stretch [39, 40]. Chow and colleagues [11] combined pla- nar biaxial mechanical testing (comparable to scenario 3 5 Conclusions in our study) and multiphoton microscopy to investigate In this study, a recently developed tri-layered modelling the rearrangement of collagen and elastin microstruc- framework was used to investigate the effect of residual tures in response to load and found that the recruitment stresses on the mechanics of the tri-layered arterial wall. of adventitial collagen showed a 20% strain delay with While isolated layers exhibited considerably different respect to that of the media. It is worth noting, however, behaviour, their pre-deformed state make their response that our model predicted a 7% circumferential elonga- strikingly similar at physiological pressures. Conversely, tion of the adventitia in the deformation from κ at high pressures, the adventitial mechanical response composite to κ ( F ). This deformation would compensate, at deviated from that of intima and media, conferring its unloaded 1 least in part, the observed delayed recruitment of adven- protective function. titial collagen. Indeed, in our model,  neglecting F con- siderably limited the contribution of the adventitia to the Abbreviations macroscopic wall behaviour (Fig. 4C, F, I). a: Adventitia; α: Collagen fibre orientation parameter; c : Collagen fibre stiffness-like parameter; c : Collagen fibre non-linearity parameter; C : Right Cauchy–Green strain tensor; C: Wall material stiffness; F : Deforma- 4.1 Limitations tion gradient mapping the deformation from κ to κ composite unloaded ; F : Deformation gradient mapping the deformation from κ to In this study, the three layers were modelled as three con- 2 unloaded κ ; G : Layer-specific deformation gradient mapping the tension−inflation centric membranes (thin-walled approach). Therefore, deformation from κ to κ ; HGO: Holzapfel–Gasser–Ogden; isolated composite all modelled parameters, including F , and biome- residual,k h: Arterial wall thickness in κ ; i: Intima; κ : Isolated tension−inflation isolated chanical variables represent mean values across the layer layer configuration in Cartesian coordinates ( X ,Y ,Z ); κ : Flat, composite tri-layered wall configuration in Cartesian coordinates ( X,Y,Z); κ : unloaded thickness. In reality, arterial layers are not as homogene- Unloaded, cylindrical, tri-layered vessel configuration in cylindrical coordinates ous and F is not constant across the layer thick- residual,k ( , R , Z); κ : Axially stretch and pressurised, cylindrical, tension−inflation ness. This leads to the existence of layer-specific opening tri-layered vessel configuration in cylindrical coordinates ( θ , r , z); l : Length of the flat, tri-layered wall in the j-direction; L : Length of the isolated layer k angles, possibly affecting the stress distribution across k k in the j-direction;  : j-direction component of G ;  : j-direction compo- the layer thickness [14, 20]. A thick-walled modelling j nent of F ;  : j-direction component of F ; m: Media; µ : Elastin stiffness- 1 j 2 approach would further refine the analysis, but model like parameter; OA: Opening angle; : Strain energy function; R : internal complexity would also increase considerably as would the Unloaded luminal radius; r : Loaded luminal radius; ρ: Collagen fibre internal dispersion parameter; SEF: Strain energy function; σ: Modelled Cauchy stress complexity of estimating the related geometrical param- tensor; σ : Modelled Cauchy stress tensor. exp eters (layers’ three-dimensional curvatures). The layer-specific model parameters were estimated by Acknowledgements We thank Ashraf W. Khir for his support and valuable advice. fitting the intact wall and layer-specific response to uni - axial testing in the circumferential and axial directions, Author Contributions simultaneously, as done previously [13, 14, 30]. While AG and BS contributed to the study conception and to the development of the tri-layered modelling framework used in the study. AG contributed to the this approach indirectly yields biaxial information, biaxial experimental data collection and analysis, and to the manuscript drafting. BS experiments could further refine our analysis, providing contributed to the critical appraisal and manuscript editing. direct information on the coupling between circumferen- Funding tial and axial responses. This work was supported by the ARTERY (Association for Research into Arterial We did not perform any imaging of the layers’ cross- Structure and Physiology) society (2019 Research Exchange Grant to AG) and section to visually verify the accuracy of the peeling pro- by the European Union’s Horizon 2020 Research and Innovation programme (Grant 793805 to BS). The sole role of the funding sources was providing cess. Nevertheless, layer-specific thicknesses found here financial contribution. are in line with those reported in other studies [13, 14, 41]. Further, the thickness of each layer was measured Availability of Data and Materials The data supporting this study are available from the corresponding author. three times along the strip length, thus allowing to eval- uate whether the peeling process was performed uni- Declarations formly. 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Journal

Artery Research – Springer Journals

Published: Jun 1, 2022

Keywords: Residual stresses; Tri-layered arterial wall modelling; Intima; Media; Adventitia

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