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Background: Wall stiffness of the abdominal aorta is an important factor in the cardiovascular risk assessment. We investigated abdominal aortic wall stiffness divided in direct and cross‑ coupled stiffness components with respect to sex and age. Methods: Thirty healthy adult males (n = 15) and females were recruited and divided into three age groups: young, middle aged and elderly. Pulsatile diameter changes were determined noninvasively by an echo‑tracking system, and intra‑aortic pressure was measured simultaneously. A mechanical model was used to compute stress and stiffness in circumferential and longitudinal directions. Results: Circumferential stretch had a higher impact on longitudinal wall stress than longitudinal stretch had on circumferential wall stress. Furthermore, there were an age‑related and sex ‑independent increase in circumferential and longitudinal direct and cross‑ coupled stiffnesses and a decrease in circumferential and longitudinal stretch of the abdominal aortic wall. For the young group, females had a stiffer wall compared to males, while the male aortic wall grew stiffer with age at a higher rate, reaching a similar level to that of the females in the elderly group. Conclusion: Temporal changes in aortic stiffness suggest an age ‑related change in wall constituents that is expressed in terms of circumferential remodeling impacting longitudinal stress. These mechanisms may be active in the development of aortic tortuosity. We observed an age‑ dependent increase in circumferential and longitudinal stiffnesses as well as decrease in stretch. A possible mechanism related to the observed changes could act via chemi‑ cal alterations of wall constituents and changes in the physical distribution of fibers. Furthermore, modeling of force distribution in the wall of the human abdominal aorta may contribute to a better understanding of elastin–collagen interactions during remodeling of the aortic wall. Keywords: Abdominal aorta, Cardiovascular disease, Wall stress, Cross‑ coupled stiffness, Sex, Age, Remodeling, Tortuosity 1 Introduction The mechanical properties of the aorta are important to its physiological function. The concept of hemodynamic homeostasis enables large arteries such as the aorta to Toste Länne: Deceased. transform central pulsatile pressure and flow into con - *Correspondence: email@example.com tinuous pressure and flow in the peripheral arterioles. 1 In this transformation, central artery stiffness has been Department of Clinical Physiology in Linköping, Linköping University, 581 83 Linköping, Sweden identified as a major independent risk factor for cardio - Full list of author information is available at the end of the article vascular disease morbidity and overall mortality . © 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/. Karlsson et al. Artery Research (2022) 28:113-127 114 Arterial stiffness is attributed to, e.g., extracellular vascular obstruction. Oestrogen replacement therapy matrix (ECM) components, mainly elastin and collagen, was not prescribed to anyone of the women. The acquisi - vascular smooth muscle cell (VSMC) tone, VSMC stiff - tion of pressure and diameter is summarized below and ness and cell–ECM interactions . The cell–ECM inter - has been described elsewhere . action affects and regulates arterial mechanical function and structural integrity . Shear as well as circumfer- 3 Non‑invasive Monitoring of Diameter Changes ential and longitudinal stress is key mechanical deter- Non-invasive monitoring of pulsatile diameter change in minants of arterial wall remodeling. Aortic mechanical the distal abdominal aorta (AA) was carried out 3–4 cm properties are considered to evolve from an interdepend- proximal to the aortic bifurcation . An electronic ency of circumferential–longitudinal coupling of stress echo-tracking instrument (Diamove, Teltec AB, Lund, and stretch where stiffness may play a part [4, 5]. Sweden) was interfaced with a real-time ultrasound scan- Stress, stiffness and stretch can be measured in vitro, ner (EUB-240, Hitachi, Tokyo, Japan) and fitted with but especially stress and stiffness are difficult to measure a 3.5 Mhz linear array transducer. The instrument had in vivo. A mechanical model, usually computerized, may dual echo-tracking loops; thus, two separate echoes from be used to simulate stress, stiffness and stretch in vivo. opposite vessel walls could be tracked simultaneously. Various mechanical models describing cardiovascular The repetition frequency was 870 Hz, temporal resolu- growth and remodeling have been proposed [5, 6]. We tion was 1.2 ms, and the smallest detectable displacement have developed a model using in vivo data with non- was 7.8 µm. For static (end diastolic and systolic) aortic linear deformation and material behavior observed for diameter and for pulsatile diameter change, the coeffi - arteries to compute stress and stiffness [7 , 8]. The model cient of variation was 5% and 16%, respectively. has been validated  and produced findings relevant to age- and sex-dependent changes in vessel wall constitu- 3.1 Invasive Blood Pressure Measurements ents . Abdominal aortic (AA) blood pressure was measured In biological tissue, the stress–strain curve is nonlinear, invasively at the midpoint between the renal arteries implicating nonlinear stiffness. Our mechanical model and the aortic bifurcation with a 3-F (SPC 330A) or 4-F enables computation of wall stress as well as direct and (SPC 340) micromanometer tip catheter (Millar Instru- cross-coupled stiffnesses from the nonlinear stress– ments, Houston, TX) or with a fluid-filled catheter sys - strain curve. A commonly used variable such as Young’s tem (pressure monitoring kit DTX + with R.O.S.E, Viggo incremental elastic modulus calculated from stress– Spectramed, Oxnard, CA) depending on the availability. strain curves assumes a linear material property and is The frequency response of the Millar catheter (flat range a measure of direct stiffness. In a clinical environment, to 10 kHz) was higher than in the fluid-filled system (flat aortic pulse wave velocity (PWV) is regarded as a refer- range 35 Hz [3 dB]). However, curves from one cardiac ence parameter for central aortic stiffness  and mainly cycle from each system were superimposed on each other reflects circumferential direct stiffness. Nonlinear behav - using a Blood Systems Calibrator (Bio Tech Model 601 A, ior as well as cross-coupled stiffness is much less investi - Old Mill Street, Burlington, VT) showing similar systolic gated. Furthermore, cross-coupled stiffness links stretch blood pressures and pulsatile amplitude when compared. in one direction to stress in another direction and might The data acquisition system allowed for simultaneous reveal information related to directional interdependency monitoring of blood pressure and vessel diameter with a of stress and stretch. Therefore, the aim of this study is to maximum registration duration of 11 s. The system con - investigate direct and cross-coupled stiffness and its pos - tained a personal computer type 386 (Express, Tokyo, sible implications on aortic remodeling. Japan) and a 12-bit analog-to-digital converter (Analogue Devices, Norwood, MA) allowing a sampling frequency of 290 Hz each for both signals. Example of acquired data is found in Fig. 1. 2 Materials and Methods 3.2 The Identification of Model Parameters 2.1 Study Subjects and the Mechanical Model Thirty healthy, non-smoking male (n = 15) and female The identification of model parameters and the mechani - volunteers without any medications were included in cal model used have been described in previous publica- the study and divided in three age-groups: young (23– tions [8, 10]. Also see Appendix for further information. 30 years, n = 10), middle aged (41–54 years, n = 10) and The parameter identification method for mechanical elderly (67–72 years, n = 10). Exclusion criteria were a parameters (PIMMP) consists of two parts, a signal pro- history of cardiopulmonary disease, diabetes or regular cessing routine and a parameter identification routine medication and an ankle brachial index < 1 suggestive of Karlsson et al. Artery Research (2022) 28:113-127 115 Fig. 1 Example of acquired data. A and B illustrate measured data from the abdominal aorta and identification results from PIMMP. A Measured blood pressure (upper left panel) and inner radius (lower left panel) for a young male. Right panel shows the pressure‑radius response and the used post‑processed average signal (black). B Left panel shows wall stress vs. inner radius in the circumferential direction for a young male. Measured data with resulting fitted curve from PIMMP representing total stress is illustrated. Right panel shows circumferential and longitudinal direct as well as cross‑ coupled wall stiffness vs. inner radius for a young male Karlsson et al. Artery Research (2022) 28:113-127 116 including a nonlinear mechanical model. In the first part, These model stresses are dependent on six model measured blood pressure and diameter were processed parameters (explained below) describing the material in MATLAB. The data consisting of approximately 8–10 characteristics and the in situ pre-stress of the aortic cycles (heartbeats) were lowpass filtered with a fourth- wall [13–15]. order Butterworth filter with a cutoff frequency of 15 Hz 3. By comparing the first set of stresses from 1 to the for noise reduction and automatically adjusted for time stresses from 2, an estimate of an error (difference) delays from the measurement setup. Furthermore, pres- was obtained (Eq. 9) . If the difference between sure and radius were averaged over cycles . In part two, errors from two consecutive iterations was smaller –5 the model parameters were identified through a nonlin - than a pre-set tolerance (typically 10 ), the iteration ear curve fitting of the model response to the measured was terminated, and the parameters were considered pressure–radius loop, according to the following iterative identified. If the error exceeded the tolerance, the algorithm (Fig. 2A): parameters values were updated using standard iden- tification techniques and steps 2 and 3 were repeated. 1. The stresses in the arterial wall were computed by the Laplace’s law (Eq. 8 in Appendix) using the pressure- Six model parameters were identified, describing the radius loop together with an estimation of the cross- characteristic (material parameters: c, k , k , β) and 1 2 sectional area (A) of the aortic wall from Åstrand geometrical (geometrical parameters: R , and ) prop- et al. . erties of the aortic wall . The parameters are: 2. A second set of stresses was computed using non- linear continuum mechanics (Eqs. 11 and 12) . Fig. 2 Identification algorithm and model parameters. A and B show the identification algorithm and definitions of some model parameters. A The identification algorithm used to compute the material parameters in the abdominal aorta, see text for description. B An overview of how model parameters β, R and λ are defined. The upper cylinder represents the unloaded state with unit length; the lower cylinder represents the pressurized 0 z state Karlsson et al. Artery Research (2022) 28:113-127 117 • c (Pa)—relates to the stiffness of the isotropic constit - varies when the vessel wall is stretched in the circumfer- uents in the vascular wall, mainly elastin. ential direction. • k (Pa)—relates to the stiffness of the anisotropic constituents in the vascular wall, mainly collagen.3.3 Computed Variables • k (dimensionless)—reflects the crimpling or fold - In the parameter identification routine, the material ing, cross-linking and entanglement of collagen. parameters were computed (c, k k and β) as well as the 1, 2 • β (°)—the angle between the circumferential direc- geometry for the unloaded state (R λ ). Circumferential 0, z tion and the principal (mean) fiber direction in the stretch ( ) was approximated with = r/R where r θ θ 0 unloaded configuration (Fig. 2 B). was measured radius and R was identified through the • R (mm)—the radius in a stress and stretch free parameter identification process . Furthermore, the (ex situ) unloaded configuration (Fig. 2 B) circumferential ( σ ) and longitudinal ( σ ) stresses were θ z • (dimensionless)—longitudinal stretch between computed. Stiffness was computed as the partial deriva - the in situ configuration and the (ex situ) unloaded tive of stress with respect to stretch according to Eq. 2; configuration (Fig. 2 B). thus, there were four different stiffnesses computed: circumferential direct ( S ) and cross-coupled ( S ) a s θθ θz Values for identified parameters are found in Table 4 well as longitudinal direct ( S ) and cross-coupled ( S ) zz zθ in Appendix. stiffnesses. A single stiffness constant, e.g., Young’s modulus, can - Mean arterial pressure (MAP) was calculated as not accurately predict arterial wall stress . It must be 1/3 × (SBP − DBP) + DBP, unit Pa. To convert Pa to computed from the observed deformation using a set of mmHg, 1 mmHg = 133.32 Pa, was used. (nonlinear) equations and associated material param- Pressure, radius, circumferential and longitudinal eters [8, 17]. The model used herein is based on a stand - stretch, stress and stiffness were calculated at SBP, DBP ard Holzapfel–Gasser–Ogden (HGO) nonlinear material and MAP. Notice that the identified parameters are con - model with a neo-Hookean matrix reinforced by a two- stant during one heartbeat and are not subject to differ - family fiber structure . The stress–stretch curve in ent values at different pressures. the circumferential direction can be described by: ∂ψ iso aniso σ = = σ + σ θ θ3.4 Linearization θ θ A small amplitude perturbation analysis has been carried 1 2 2 k (I−1) 2 = 2C − + 4k (I − 1)e cos β 1 θ 2 out to determine how cross-coupled stiffness manifests ( ) θ z through stress. It was done through a linearization close (1) to a point of interest (systolic blood pressure). The line - where c, k , k > 0 guarantee energy dissipation and 1 2 arization follows standard procedures and is a first order 4πr +A 2 2 2 2 0 0 I = cos β + sin β with = . Compute d z 2 θ r 0 4πR + A Taylor expansion close to the point of interest . For circumferential stress with the result from the identifica - the present study, the linearization will be two dimen- tion routine is illustrated in Fig. 1B. sional (2D): For our purpose, incremental stiffness for a nonlin - L 0 0 0 0 σ ( , ) σ S S − θ z θ ear material such as biological tissue can be estimated θ θ θθ θz θ = + L 0 0 0 0 σ ( , ) σ S S − θ z z from the slope of the nonlinear stress–stretch curve. z z zz z zθ (3) This corresponds to the partial derivative of stress with L L respect to stretch. Hence, there will be two direct stiff - Here σ and σ are the linearized functions while σ θ z nesses and two cross-coupled stiffnesses: and σ are the nonlinear functions. S , S , S and S z θθ θz zθ zz are the first partial derivatives of σ , and σ with respect θ z ∂σ ∂σ ∂σ ∂σ θ θ z z 0 0 S = , S = , S = , S = to and . and denote the point of interest and (2) θ z θθ θz zθ zz θ z ∂ ∂ ∂ ∂ θ z θ z superscript “0” serves as a marker for a function calcu- lated at the point of interest. where S and S are circumferential direct and cross- θθ θz The linearized functions were used to quantify the coupled stiffnesses while S and S are longitudinal zz zθ influence of stretches on stress through direct and direct and cross-coupled stiffnesses. Cross-coupled cross-coupled stiffness. By changing the stretch a small stiffness carries information of how stress in one direc - amount, q (0 < q ≤ 0.03) and taking the ratio between the tion is affected when the artery is stretched in another L,jq direction. Thus, S represents how circumferential stress θz change ( σ ) and the original ( σ ) values, the effect of i i varies when the vessel wall is stretched in the longitudi- the change in stretch can be analyzed. Choosing a too nal direction and S represents how longitudinal stress zθ large value for q will end up in calculations outside the Karlsson et al. Artery Research (2022) 28:113-127 118 validity of the linearization. The higher the ratio, the 1 nS θz S × (7) θθ 0 0 more impact will stretch have on stress. Four ratios were nS nS zθ zz calculated. Here “n” denotes normalized values. The normalized L,jq σ − σ values express stiffness in terms of S ; thus, the normal- i i θθ R = (i, j = θ or z) (4) ij ized matrix element for circumferential direct stiffness will always be 1. Since S is largest, the other three stiff - θθ with nesses will always be less than one. L,jq L 0 σ − σ = S × × q (i, j = θ or z) j (5) i i ij 3.5 Statistics Arithmetic mean and standard deviation (SD) were cal- Here superscript “q” denotes stress where stretch has culated for all variables and are expressed as mean ± SD, been changed a small amount; i and j denote θ or z. The if not otherwise stated. Variable values calculated at SBP ratios express: and DBP were regarded as maximum and minimum. A two-way ANOVA test with complementing general lin- • R : circumferential (θ) stretch impact on circumfer- θθ ear models was used to compare sex and age groups as ential (θ) stress suggested by Field . Bonferroni correction was used • R : longitudinal (z) stretch impact on circumferen- θz when multiple comparisons were performed. All parame- tial (θ) stress ters were assessed for dependency of age within each sex • R : circumferential (θ) stretch impact on longitudi- zθ with a linear regression analysis with Pearson correlation nal (z) stress coefficient (R ). P < 0.05 was considered significant in the • R : longitudinal (z) stretch impact on longitudinal zz ANOVA and general linear model as well as in the lin- (z) stress ear regression analysis. Significance testing is used for a descriptive purpose. The ratios from Eq. (4) were used to assess the influ - ence of stretch on stress via direct stiffness or cross-cou - 3.6 Software pled stiffness. MATLAB (The Mathwork, Natick, MA, US) version 8.4 The impact of stiffness on stress without the effect of (R2014b) was used for computation. IBM SPSS Statistics stretch was assessed through the stiffness matrix in lin - Version 27 (IBM Corporation, Somers, NY, US) was used earized Eq. (3). Circumferential direct stiffness ( S ) wa s θθ for statistical analysis. the largest of the four stiffnesses. Normalizing the ele - ments with S will produce relative values which can be θθ compared within the matrix as well as between different 4 Results points of interest: 4.1 Aortic Blood Pressure and Vessel Diameter Baseline clinical data for the study population are found ij 0 in Table 1, while abdominal aortic (AA) diameters and nS = (i, j = θ or z) (6) ij blood pressures are shown in Table 2. In males, elderly θθ compared with young had a higher systolic blood pres- The matrix in Eq. (3) can be rewritten as: sure (SBP) value, larger AA diameter, but smaller ΔD (P < 0.05). Elderly compared with young males appeared Table 1 Characteristics of the studied population Male Female Young (n = 5) Middle (n = 5) Elderly (n = 5) Young (n = 5) Middle (n = 5) Elderly (n = 5) Age, year 24.8 ± 2.0 47.6 ± 5.6 69.6 ± 1.6 25.4 ± 2.8 49.2 ± 3.1 68.8 ± 2.0 Height, cm 177 ± 8.9 178 ± 6.7 182 ± 4.5 171 ± 8.9 170 ± 4.5 168 ± 4.5 a B C Weight, kg 71.4 ± 8.7 84.8 ± 8.9 87.2 ± 12.3 59.0 ± 9.4 67.0 ± 9.6 64.6 ± 8.0 2 a,A BMI, kg/m 22.6 ± 0.9 26.8 ± 2.3 26.3 ± 2.4 20.1 ± 2.0 23.2 ± 4.2 22.8 ± 2.4 2 B C BSA, m 1.88 ± 0.18 2.03 ± 0.1 2.08 ± 0.2 1.69 ± 0.18 1.78 ± 0.09 1.74 ± 0.1 Data are presented as mean ± SD BMI body mass index, BSA body surface area. Young: 23–30 yr, middle 41–54 yr, elderly 67–72 yr a A, B, C P < 0.05 when comparing young vs elderly for male. P < 0.05 when comparing male vs female for young, middle and elderly, respectively Karlsson et al. Artery Research (2022) 28:113-127 119 Table 2 Abdominal aortic pressure and diameter Male Female Young Middle Eldelry Young Middle Eldelry SBP, mmHg 114 ± 12 133 ± 18 135 ± 22 119 ± 14 123 ± 10 126 ± 15 DBP, mmHg 62 ± 8 71 ± 6 70 ± 12 66 ± 9 66 ± 7 64 ± 5 (a) MAP, mmHg 79 ± 9 92 ± 8 92 ± 15 84 ± 11 85 ± 7 85 ± 8 (a) PP, mmHg 52 ± 5 62 ± 17 65 ± 15 52 ± 5 57 ± 6 62 ± 12 aa,bb BB CC a Diameter SBP, mm 15.9 ± 1.2 19.4 ± 1.2 20.5 ± 2.1 15.1 ± 0.8 16.3 ± 0.7 17.2 ± 2.4 aa,bb BB CC aa Diameter DBP, mm 13.8 ± 1.4 18.2 ± 1.4 19.8 ± 2.2 13.3 ± 1.4 15.0 ± 1.2 16.6 ± 2.4 aa,bb a c Δ Diameter, mm 2.14 ± 0.32 1.14 ± 0.47 0.77 ± 0.25 1.79 ± 0.75 1.28 ± 0.45 0.67 ± 0.08 Data are presented as mean ± SD SBP systolic blood pressure, DBP diastolic blood pressure, MAP mean arterial pressure, PP pulse pressure, Diameter SBP diameter at systolic blood pressure, Diameter DBP diameter at diastolic blood pressure, Δ Diameter diameter SBP − diameter DBP a, b, c aa, bb BB, CC P < 0.05 when comparing young vs elderly, young vs middle, middle vs elderly, respectively, for male or female; P < 0.01. P < 0.01 when comparing male vs female for middle and elderly, respectively. ‘( )’ indicate P < 0.10 to have a higher mean arterial pressure (MAP) value was almost 100% higher compared to R , and when θz (P = 0.07) as well as pulse pressure (PP) value (P = 0.08). comparing R , R and R , all three were of the same θθ zθ zz In females, elderly compared with young had larger AA magnitude (Fig. 3B). diameters, but smaller ΔD than (P < 0.05). Males com- Figure 3A shows the impact of stiffness on stress, while pared to females had larger AA diameters (P < 0.05). Fig. 3B includes stretch and thus shows the combined Changes in the pulsatile diameter of the AA (ΔD) and effect of stiffness and stretch on stress. had a higher blood pressure did not differ by sex. value compared to longitudinal stretch (Table 3). The effect of this difference is not seen in Fig. 3A but is con- sidered in Fig. 3B. Comparing Fig. 3A and B expose the 4.2 Linear Model impact of stiffness and stretch on stress. First, S and S θz zθ For the linearization, the groups of elder and middle aged were of the same magnitude. Second, combining S and θz were combined to one group (n = 20) of males (n = 10) S with respective and showed that a small change zθ θ z and females. This unified group was investigated, and in will have a lesser effect on circumferential stress results are reported in Fig. 3. (second bar from left in Fig. 3B) compared to the effect a small change in will have on longitudinal stress (third bar from left in Fig. 3B). Hence, it appears that the effect 4.2.1 Comparing Normalized Linear Model Stiffness Matrix of a change in will have a greater impact on longitudi- Elements ( nS , nS , nS , nS ) at SBP, DBP and MAP θθ θz zθ zz nal stress than a change in will have on circumferential At SBP in males and females, nS had a 70–100% θθ stress. higher value compared with nS , nS and nS (males: θz zθ zz 1.00 ± 0.00 vs. 0.59 ± 0.05 vs. 0.51 ± 0.03 vs. 0.56 ± 0.10, respectively, P < 0.01, and females: 1.00 ± 0.00 vs. 4.3 Aortic Wall Stiffness and Stretch 0.57 ± 0.06 vs. 0.51 ± 0.04 vs. 0.57 ± 0.12, respectively, Stiffness and stretch are reported in Table 3. P < 0.01). In males and females, there were no differences when comparing nS , nS and nS (Fig. 3A). θz zθ zz 4.3.1 C omparing Stiffness ( S , S , S , S ) Between Age θθ θz zθ zz Groups Within a Sex at SBP, DBP and MAP (From left to right in Table 3). 4.2.2 Comparing Ratio of Stresses ( R , R , R , R ) at SBP , zz θθ θz θz At SBP, DBP and MAP, in males, elderly and middle i.e., Combined Effect of Stiffness and Stretch on Stress aged compared with young had a higher S , S , S and θθ θz zθ At SBP in males, R had a higher value compared with θθ S (P < 0.05, respectively). In females, elderly compared zz R while R had a lower value compared to R and θz θz zθ with young had a higher S , S and S (P < 0.05, respec- θθ θz zθ R , (0.053 ± 0.027 vs. 0.027 ± 0.013 vs. 0.047 ± 0.023 zz tively); there appeared to be a higher S for elderly when zz vs. 0.048 ± 0.019, respectively, P < 0.05). In females , R θθ compared to young (P < 0.1). had a higher value compared with R (0.045 ± 0.020 θz At SBP, DBP and MAP, in males, S , S , S and S θθ θz zθ zz vs. 0.023 ± 0.010, P < 0.05), while R appeared to be θz correlated positively with age (P < 0.05, respectively). lower than R and R , (0.040 ± 0.017 vs. 0.040 ± 0.015, zθ zz This holds true for females as well (P < 0.05, respectively), respectively, P < 0.1). At SBP in males and females, R θθ Karlsson et al. Artery Research (2022) 28:113-127 120 Fig. 3 Linearized stiffness matrix at SBP, elderly and middle ‑aged group. Results from linearization at SBP. Normalized linear model stiffness matrix (A) and ratio between the change and the original values (B) for males (left) and females (right) at SBP in the elderly and middle‑aged group. Panel A reveals the impact of stiffness alone on stress, while panel B include stretch and thus show the combined effect of stiffness and stretch on stress. There were no differences between sexes. nS had a higher value compared to nS , nS and nS , which, on the other hand, were of the same θθ θz zθ zz magnitude. In males, R had a lower value compared to R , R and R which, in turn, were of the same magnitude. In females, R was lower θz θθ zθ zz θz compared to R , and appeared to be lower compared to R and R which, in turn were of the same magnitude. Error bars ± 1 SD. θθ zθ zz Karlsson et al. Artery Research (2022) 28:113-127 121 Table 3 Stiffness and stretch at different blood pressures for sexes, divided into age groups Male Female 2 2 Young (n = 5) Middle (n = 5) Elderly (n = 5) Total Regr (R ) Young (n = 5) Middle (n = 5) Elderly (n = 5) Total Regr (R ) SBP aa, b (*), €€, ££ a (c) €€, ££ S (MN/m) 1.02 ± 0.48 4.12 ± 3.13 5.49 ± 2.51 3.54 ± 2.90 0.49 ↑↑ 1.38 ± 0.71 2.15 ± 1.15 4.25 ± 2.13 2.60 ± 1.84 0.38 ↑ θθ aa, b (*), €€, ££ a €€, ££ S (MN/m) 0.66 ± 0.32 2.34 ± 1.78 2.94 ± 1.31 1.98 ± 1.56 0.45 ↑↑ 0.88 ± 0.49 1.33 ± 0.72 2.24 ± 1.16 1.48 ± 0.97 0.30 ↑ θz aa, b (*), €€, ££ a €€, ££ S (MN/m) 0.53 ± 0.27 2.13 ± 1.67 2.76 ± 1.25 1.81 ± 1.49 0.47 ↑↑ 0.74 ± 0.44 1.19 ± 0.70 2.09 ± 0.70 1.34 ± 0.94 0.32 ↑ zθ aa, b €, £ (a) (€), (£) S (MN/m) 0.77 ± 0.30 2.16 ± 1.27 2.84 ± 1.11 1.92 ± 1.28 0.53 ↑↑ 1.01 ± 0.51 1.45 ± 0.76 2.10 ± 1.08 1.49 ± 0.72 0.20 ↗ zz aa, bb (c) € aa, b € λ 1.28 ± 0.06 1.19 ± 0.14 1.09 ± 0.03 1.22 ± 0.15 0.74 ↓↓ 1.33 ± 0.13 1.20 ± 0.08 1.12 ± 0.03 1.21 ± 0.12 0.56 ↓↓ § aa, (b) (c) aa, bb λ 1.050 ± 0.005 1.029 ± 0.001 1.012 ± 0.003 1.03 ± 0.02 0.66 ↓↓ 1.049 ± 0.026 1.027 ± 0.010 1.018 ± 0.006 1.03 ± 0.02 0.44 ↓↓ DBP aa, b (c) (C) a S (MN/m) 0.43 ± 0.13 1.24 ± 0.61 1.80 ± 0.72 1.16 ± 0.78 0.59 ↑↑ 0.58 ± 0.26 0.82 ± 0.39 1.24 ± 0.56 0.88 ± 0.48 0.29 ↑ θθ aa, b (c) (C) a S (MN/m) 0.22 ± 0.06 0.59 ± 0.30 0.86 ± 0.35 0.56 ± 0.37 0.57 ↑↑ 0.28 ± 0.12 0.38 ± 0.18 0.59 ± 0.26 0.41 ± 0.22 0.28 ↑ θz aa, b (c) (C) a S (MN/m) 0.20 ± 0.05 0.56 ± 0.30 0.83 ± 0.34 0.53 ± 0.36 0.58 ↑↑ 0.26 ± 0.11 0.36 ± 0.17 0.57 ± 0.27 0.39 ± 0.22 0.30 ↑ zθ aa, b (c) (C) (a) S (MN/m) 0.41 ± 0.11 1.12 ± 0.53 1.68 ± 0.59 1.07 ± 0.69 0.65 ↑↑ 0.55 ± 0.23 0.79 ± 0.40 1.11 ± 0.59 0.82 ± 0.47 0.22 ↗ zz aa, b c aa, (b) λ 1.19 ± 0.03 1.12 ± 0.09 1.05 ± 0.02 1.12 ± 0.08 0.63 ↓↓ 1.16 ± 0.05 1.10 ± 0.04 1.07 ± 0.02 1.11 ± 0.05 0.57 ↓↓ § aa, (b) (c) aa, bb λ 1.050 ± 0.005 1.029 ± 0.001 1.012 ± 0.003 1.03 ± 0.02 0.66 ↓↓ 1.049 ± 0.026 1.027 ± 0.010 1.018 ± 0.006 1.03 ± 0.02 0.44 ↓↓ MAP aa, b (c) a S (MN/m) 0.51 ± 0.17 1.40 ± 0.67 2.10 ± 1.12 1.34 ± 0.98 0.50 ↑↑ 0.67 ± 0.29 0.88 ± 0.36 1.53 ± 0.59 1.03 ± 0.55 0.38 ↑ θθ aa, b (c) a S (MN/m) 0.27 ± 0.09 0.68 ± 0.33 1.02 ± 0.56 0.66 ± 0.47 0.68 ↑↑ 0.34 ± 0.14 0.42 ± 0.16 0.74 ± 0.28 0.50 ± 0.26 0.36 ↑ θz aa, b (c) a S (MN/m) 0.24 ± 0.08 0.64 ± 0.32 0.97 ± 0.54 0.62 ± 0.46 0.47 ↑↑ 0.30 ± 0.13 0.39 ± 0.15 0.70 ± 0.28 0.46 ± 0.28 0.38 ↑ zθ aa, b (c) (C) (a) S (MN/m) 0.46 ± 0.13 0.18 ± 0.53 1.76 ± 0.68 1.13 ± 0.99 0.58 ↑↑ 0.60 ± 0.23 0.82 ± 0.38 1.19 ± 0.59 0.87 ± 0.47 0.24 ↗ zz aa, b (c) aa, b λ 1.25 ± 0.04 1.14 ± 0.11 1.07 ± 0.02 1.15 ± 0.10 0.69 ↓↓ 1.22 ± 0.07 1.14 ± 0.05 1.14 ± 0.05 1.15 ± 0.08 0.58 ↓↓ § aa, (b) (c) aa, bb λ 1.050 ± 0.005 1.029 ± 0.001 1.012 ± 0.003 1.03 ± 0.02 0.66 ↓↓ 1.049 ± 0.026 1.027 ± 0.010 1.018 ± 0.006 1.03 ± 0.02 0.44 ↓↓ Data are presented as mean ± SD SBP systolic blood pressure DBP diastolic blood pressure, MAP mean arterial pressure, S direct stiffness in circumferential direction, S cross-coupled stiffness in circumferential direction, S cross-coupled stiffness in θθ θz zθ longitudinal direction, S direct stiffness in longitudinal direction. stretch in circumferential direction, stretch in longitudinal direction. Young: 23–30 years, middle 41–54 years, elderly 67–72 years zz θ z § * €, £ is constant and therefore do not change with pressure. P < 0.05 when comparing male vs female independent of age (total), with respect to pressure. P < 0.05, when comparing pressure (SBP vs DBP, SBP vs MAP, €€, ££ respectively) with respect to male or female. indicate P < 0.01 a, b, c aa, bb C P < 0.05 when comparing young vs elderly, young vs middle, middle vs elderly, respectively, for male or female. P < 0.01. P < 0.05 when comparing male vs female for elderly. ‘()’ indicate P < 0.10 ↑↑, ↑, ↗: positive correlation; ↓↓, ↓, ↘: negative correlation; triplet from left to right: P < 0.01, P < 0.05, P < 0.10, respectively. → = no correlation Karlsson et al. Artery Research (2022) 28:113-127 122 4.3.4 C omparing Stretch Between Sexes Within an Age except for S , which appeared to correlate positively zz Group at SBP, DBP and MAP (P < 0.1). Altogether, our findings suggest that S , S , S θθ θz zθ (From left to right in Table 3). and S increased with age. zz Males compared with females showed no statistically Furthermore, elderly compared with young males as significant difference in and . well as females showed a higher value at SBP and DBP for θ z S , S , S and S (SBP: male 400%, 400%, 350%, 270%, θθ θz zθ zz respectively; female 200%, 200%, 150%, 100%, respec- 5 Discussion tively. DBP: male 300%, 300%, 300%, 300%, respectively; The main findings of this study were as follows: female 100%, 100%, 100%, 100%, respectively). u Th s, it seems that the increase in stiffness from the young to the 1. The effect of a change in circumferential stretch elderly was constant and approximately two times higher appeared to have a greater impact on longitudinal in males compared with females both at SBP and DBP, stress than a change in longitudinal stretch might i.e., for the physiological pressure range. have on circumferential stress. For elderly males and females at SBP, S compared θθ 2. Circumferential and longitudinal direct and cross- with S , S and S had a higher value (90%, 100% and θz zθ zz coupled stiffnesses increased while circumferential 95–100%, respectively). Hence, independent of sex, the and longitudinal stretches decreased with age, inde- elderly had values where S was approximately twofold θθ pendent of sex. higher than S , S and S which, in turn, were of the θz zθ zz 3. For the young group, the female abdominal aor- same magnitude. Independent of sex and pressure, it tic wall is stiffer compared to that of the male while appears that for the elderly, S had approximately the θz for the elderly group the male abdominal aortic wall same value as S . zθ seemed to be stiffer than the female wall. 4.3.2 Comparing Stiffness Between Sexes Within an Age The mechanical forces maintain a balance in the vascu - Group at SBP, DBP and MAP lar wall through remodeling. A stretch of the vessel wall (From left to right in Table 3). will cause an increased wall stress. The magnitude of the At DBP, for the elderly, males compared to females stress will be determined by the wall stiffness. Biologi - appeared to have higher S , S , S and S (P < 0.1). θθ θz zθ zz cal material such as vascular tissue will have a nonlinear Males compared to females had a lower stiffness as stress–stretch relationship, meaning that the stiffness is young but a higher stiffness as elderly (Table 3). This sug - nonlinear. Not only shear, circumferential and longitu- gests two things: first, for the young group, the female dinal stresses are fundamental mechanical properties abdominal aortic wall may be stiffer compared to that of in arterial wall remodeling, but the interdependency of the male; second: the male abdominal aortic wall grows circumferential–longitudinal coupling of stresses and stiffer at a higher rate with age. stretches is considered to be of importance as well [4, 5, 20]. Our results suggest that cross-coupled stiffness par - 4.3.3 Comparing Stretch ( , ) Between Age Groups Within θ z ticipates in the interdependency between circumferen- a Sex at SBP, DBP and MAP tial and longitudinal coupling of stresses and stretches. (From left to right in Table 3). In independent processes, we propose that an increase in At SBP, DBP and MAP in both sexes, elderly and mid- stretch in the circumferential direction affects longitudi - dle aged compared with young had a lower and θ z nal stress through cross-coupled circumferential stiffness (P < 0.05, respectively). In the model, was assumed to and the axial force with longitudinal stretch affects cir - be constant over blood pressure, . cumferential stress through longitudinal cross-coupled At SBP, DBP and MAP in both males and females, stiffness. Interdependency between circumferential and and correlated negatively with age (P < 0.05, respec- longitudinal remodeling has been shown where a longi- tively) which suggests that and decreased with age. θ z tudinally stretched vessel grew into its new length recov- From Table 3, it appears that the difference in ering a prestretched stress value, while wall thickness between young and elderly at SBP was approximately increased but not circumferential stress . Increased the same in males and females although at DBP males cell proliferation of smooth muscle and endothelial cells expressed a higher difference compared to females. as well as increased internal elastic laminae fenestrae Karlsson et al. Artery Research (2022) 28:113-127 123 size, but not density was reported . Increase in wall Our finding of an age-dependent increase in circum - thickness is a reversible process . An increase in ves- ferential and longitudinal stiffness as well as decrease in sel length is not reversible since matrix metalloproteinase stretch is in accordance with the literature, although we (MMP) activity degrades proteins and prevents longitu- cannot show a difference between sexes in stretch [14, dinal strain from returning to normal values . 24]. Furthermore, this age-dependent increase in circum- The effect of a change in circumferential stretch ferential stiffness agrees with previous reported findings appears to have a greater impact on longitudinal stress . than a change in longitudinal stretch might have on cir- For the young group, the female abdominal aortic wall cumferential stress. The linearized model showed that an may be stiffer compared to that of the male while for increase in longitudinal stress was composed by approxi- the elderly group the male abdominal aortic wall seems mately 50% of stress depending on cross-coupled stiffness to be stiffer than in the female, indicating that the male while an increase in circumferential stress was composed abdominal aortic wall may grow stiffer at a higher rate by approximately 33% of stress derived from cross-cou- with age. The gain in male abdominal aortic wall stiff - pled stiffness. This suggests that circumferential stretch ness with age is in accordance with observations of a may have a significant impact on longitudinal remodeling more accelerated ageing process in the aorta in males while longitudinal stretch probably has a minor impact compared to females . Female sex hormones seem on circumferential remodeling. to protect against aortic wall elastolysis, which in turn The blood pressure-induced alterations of circumfer - will contribute to vessel wall stiffening . The effect ential stretch may, through the proposed mechanism of postmenopausal hormone replacement therapy in of cross-coupled stiffness, contribute significantly to females seems to reduce arterial stiffness  further increased longitudinal stress and stretch, resulting in suggesting that females are better protected against the aorta growing into its new length. Considering that aortic wall stiffening than males. decreased longitudinal strain might exacerbate a length- Different mechanisms contribute to the age-related ening and create tortuosity , there are two different alterations of the vascular wall. Increased wall stress mechanisms working together in an unfavorable way has been suggested to induce synthesis of collagen in with respect to optimal aortic length. This may, at least the fiber direction and thus a maintained stretch of the in part, offer an explanation to why the aorta is prone to individual fibers [11, 29, 30]. The orientation of collagen tortuosity. fibers may also be important since it has been found to It is well accepted that a stiffer arterial wall is accompa - be more helical in the media and more longitudinal in nied by a higher blood pressure, e.g., hypertension . the adventitia [31, 32]. Additionally, increased glycation In this context, stiffness is commonly associated with of elastin and collagen as well as changed isoforms of circumferential stiffness. The role of longitudinal stiff - collagen in the aortic wall might contribute to the age- ness in hypertension is much less investigated as well as related increase in stiffness . cross-coupled stiffness. Our findings suggest that with It has been pointed out that constitutive equations increasing age, circumferential stiffness gets proportion - describing biological material such as the arterial vessel ally higher compared to longitudinal and cross-coupled wall suggest a circumferential–longitudinal coupling of stiffness while longitudinal and cross-coupled stiffnesses stresses and stretches [5, 17, 20]. Our model expresses are of the same magnitude. This indicates that the com - such an interdependency with a circumferential–longi- position of the vessel wall is changed in a stiffer direction tudinal coupling where both circumferential and lon- by way of loadbearing structures in circumferential and gitudinal stresses depend on stretch in circumferential longitudinal directions as well as the interdependency of and longitudinal directions ( σ ( , ) and σ ( , ) ) θ θ z z θ z the circumferential–longitudinal coupling. . An attempt to quantify this interdependency was Abdominal aortic aneurysms can have complicated made through a linearization of the model equations. geometries and aneurysm diameter is mainly governed To the authors knowledge, there has been no in vivo by shear stress. Aneurysms rupture when wall stress quantification of the remodeling characteristics in exceeds wall strength . Under pathological circum- one direction caused by stretching in cross-directions. stances, direct and cross-coupled stiffness may adopt Biaxial tests ex vivo provide some information regard- quite different values compared to ordinary conditions. ing the interconnection of forces. Nevertheless, as a However, our findings suggest that in an adverse situ - tool for assessing in vivo mechanisms, PIMMP may be ation, longitudinal stretch could contribute to circum- used to explore mechanical properties of the vessel wall ferential stress exceeding wall strength resulting in wall and may serve as a potential tool for risk assessment for rupture, even though the aneurysmatic diameter is small the development of cardiovascular disease. and/or the presence of a small circumferential stretch. Karlsson et al. Artery Research (2022) 28:113-127 124 Based on our results, we propose that cross-coupled different mechanical properties . An exact model stiffness may play a part in the age-related remodeling should consider this. However, such a high resolution of the aortic wall, where an increase in circumferen- in the model might introduce dependencies among the tial cross-coupled stiffness induced longitudinal stress parameters during parameter identification . Regard - constitute 50% of the change in total longitudinal stress ing the parameters from the membrane model as aver- and therefore may contribute to aortic tortuosity. Fur- ages, it might be thought of as describing the global thermore, with increasing age it seems that the inter- response of the three layers of the aortic wall. dependency of circumferential–longitudinal coupling Our model concerns the passive mechanical proper- increases possibly due to a change in composition of ties of the aorta and assumes that the outer boundary is the aortic wall which may be a contributing factor in traction free, i.e., the artery is a freestanding tube with hypertension. Additionally, under adverse conditions neglected periadventitial support. It has been shown that our findings indicate that, e.g., a longitudinal stretch the difference in circumferential stress between the free - could contribute to circumferential stress increasing standing and the tethered states is small, approximately the likelihood of aneurysm rupture. 10% . Considering the limited effect of tethering and that such mechanism would require the identification 6 Limitations of the mechanical properties of surrounding tissue, we Since there are differences between different parts of decided that a freestanding approach would be sufficient the vascular system and the abdominal aorta as well as for this study. between different segments of the aorta, with respect to histology and pulse-wave velocity, it must be emphasized 7 Conclusion that our findings should be extrapolated with caution to This paper presents quantitative estimates for circum - other segments of the aorta. Furthermore, assumptions ferential direct and cross-coupled as well as longitudi- in the model can affect the use of PIMMP. nal direct and cross-coupled stiffnesses for the human The parameter identification algorithm with the atten - abdominal aorta stratified for age and sex, based on dant mechanical model was validated against finite ele - in vivo and in situ measured radius and pressure. These ment (FE) models of an artery, a procedure which is also stiffnesses have so far only been quantified through known as “in silico” validation. The validation showed measurements ex vivo, ex situ in the laboratory. Our find - good agreement between PIMMP and the FE models . ings of an age-related circumferential remodeling act- In particular, the longitudinal response in the simulation ing through cross-coupled stiffness might have a major is sensitive to the longitudinal prestretch (λ ), and thus, impact on longitudinal stress and remodeling and possi- this parameter is a natural candidate for the validation bly aortic tortuosity. The findings suggest an age-related procedure. The PIMMP result shows a longitudinal pre - change in wall constituents. A potential explanation, stretch within the range which has been reported from although not studied here, could be related to chemical autopsy study of the human abdominal aorta . In alterations of wall constituents and changes in the physi- general, nonlinear arterial models are difficult to validate cal distribution of fibers. experimentally since each model parameter needs to be changed independently of the other parameters. There - fore, our model has not been validated against measure- ments in vivo. Appendix The strain–energy function used in this study is based The Identification of Model Parameters and the Mechanical on Holzapfel et al.  (Eq. 10). Compared to Schulze- Model Bauer and Holzapfel , it allows for a better fit to young Using measured pressure and diameter, membrane stress subjects, particularly in the low-pressure region where can be computed according to Laplace law in both cir- the mechanical behavior is primarily determined by iso- cumferential and longitudinal directions : tropic material components such as elastin and the col- 2 2 4πr + A πr P + F lp lp 0 0 lagen recruitment is small . σ = P, σ = (8) 2A A The mechanical model is based on an assumption of the cylindrical wall being a membrane, i.e., wall thickness where r is the inner radius of the artery in its physi- should be negligible when compared with the radius. ological state, A is the cross-sectional area, P is the However, wall thickness-to-radius ratio is ~ 0.1–0.2 pressure and F is the in situ axial force. Laplace law for the abdominal aorta; therefore, the validity of the in Eq. (8) is rewritten so that wall thickness is esti- assumption might be argued . Furthermore, the aor- mated through inner radius and cross-sectional area. tic wall consists of three distinct layers which all have Karlsson et al. Artery Research (2022) 28:113-127 125 Table 4 Model parameters c (kPa) k (kPa) k (–) β (°) R (mm) λ (–) 1 2 0 z Male 131.50 ± 90.00 14.18 ± 20.94 196.76 ± 264.22 42.38 ± 4.38 7.80 ± 1.78 1.03 ± 0.02 Female 101.65 ± 63.75 8.64 ± 10.00 134.95 ± 175.15 42.31 ± 3.82 6.77 ± 1.23 1.03 ± 0.02 Data are presented as mean ± SD Inner radius is measured, and cross-sectional area is model is based on a standard Holzapfel–Gasser–Ogden calculated as: males: A = 19.60 + 0.80 × age, females: (HGO) nonlinear material model with a neo-Hookean A = 20.52 + 0.56 × age; age in years and A in mm , follow- matrix reinforced by a two-family fiber structure with a ing Åstrand et al. [10, 11]. strain energy function suggested by Holzapfel et al. : The computation requires that axial force is constant k 2 k (I−1) and independent of the internal pressure while the ratio ψ = ψ + ψ = c(I − 3) + e − 1 iso aniso 1 between the longitudinal and circumferential stresses is (10) known at one internal pressure P. Assuming the stress where c, k , k > 0 to guarantee material convexity and the ratio taken to be γ = σ /σ = 0.59 at P = 13.3 kPa, 1 2 z θ −1 2 2 2 2 invariants are I = + + and the axial force can be determined explicitly following z z θ θ 4πr +A Schulze-Bauer and Holzapfel . Note that membrane 2 2 2 2 0 0 I = cos β + sin β with = . The cir - z 2 θ r 0 4πR + A stresses following Laplace law are only functions of the cumferential and longitudinal stresses can be computed applied load (pressure) and geometry (diameter) and do as: not depend on the blood vessel’s material properties. As ∂ψ a consequence, the membrane stress becomes statically iso aniso σ = = σ + σ θ θ θ θ determined . 1 2 The six model parameters are identified by comparing 2 k (I−1) 2 = 2C − + 4k (I − 1)e cos β 1 θ stresses computed according to Laplace (Eq. 8) which are ( ) θ z based on measurement, with stresses from the mechani (11) cal model (Eqs. 11 and 12). The process is based on a ∂ψ iso aniso σ = = σ + σ z z z z nonlinear least-square fitting routine using an objective function: 1 2 2 k (I−1) 2 2 = 2C − + 4k (I − 1)e sin β z z N ( ) θ z lp (12) φ(κ) = σ (κ, r , n) − σ (κ, r , n) θ 0 0 n=1 It can be noticed that computed stress consists of iso- (9) tropic and anisotropic components in both the circum- + σ (κ, r , n) − σ (κ, r , n) z 0 0 ferential and longitudinal directions [8, 10]: iso aniso iso aniso where κ = R , , c, k , k , β is the parameter vector ( ) σ = σ + σ , σ = σ + σ . (13) 0 z 1 2 θ z θ θ z z referred to as the model parameters, n is a sample, N is Isotropy and anisotropy are directional properties the total number of samples. The model parameters are linked to the constituent’s orientation. Furthermore, the solution to the minimization problem: they are independent of the material shape and vol- minφ(κ) ume. For the vascular wall, the isotropic and aniso- subject to : κ ≤ κ ≤ κ tropic components primarily reflect structures such as elastin and collagen, respectively [12, 17]. where κ and κ are the lower and upper boundaries for κ , respectively. Abbreviations Identified model parameters agree with results pub - A: Area; AA: Abdominal aorta; AAA: Abdominal aor tic aneurysm; ANOVA: lished by Åstrand et al.  (Table 4). Analysis of variance; D: Diameter; ΔD: Pulsatile diameter, ΔD = D@SBP − D@ An artery is a nonlinear material, and as such, its wall DBP; DBP: Diastolic blood pressure; S : Circumferential direct stiffness, θθ ∂σ ∂σ θ θ stress cannot be estimated from a single stiffness con - S = ; S : Circumferential cross‑ coupled stiffness, S = ; S : θθ θz θz zθ ∂ ∂ θ z ∂σ stant. Arterial wall stress must be computed from an Longitudinal cross‑ coupled stiffness, S = ; S : Longitudinal direct zθ zz ∂σ observed deformation using a set of (nonlinear) equa- stiffness, S = ; ECM: Extracellular matrix; : Circumferential stretch, zz θ tions and associated material parameters [8, 17]. Our r l = ; : Longitudinal stretch, = ; MAP: Mean arterial pressure, θ z z R L Karlsson et al. Artery Research (2022) 28:113-127 126 Consent for publication MAP = DBP + PP/3; nS : Normalized circumferential direct stiffness. θθ The authors give their consent for publication of this manuscript. Normalized element of the stiffness matrix from the linearization of the θθ nonlinear model. nS = ; nS : Normalized circumferential θθ θz θθ Author details cross‑ coupled stiffness. Normalized element of the stiffness matrix from the Department of Clinical Physiology in Linköping, Linköping University, θz linearization of the nonlinear model. nS = ; nS : Normalized 2 θθ zθ S 581 83 Linköping, Sweden. Department of Medical and Health Sci‑ θθ longitudinal cross‑ coupled stiffness. Normalized element of the stiffness 3 ences, Linköping University, 581 83 Linköping, Sweden. Solid Mechanics, zθ matrix from the linearization of the nonlinear model. nS = ; nS : Department of Management and Engineering, Linköping University, 581 θθ zz θθ 83 Linköping, Sweden. Department of Thoracic and Vascular Surgery Normalized longitudinal direct stiffness. Normalized element of the stiffness in Linköping, Linköping University, 581 83 Linköping, Sweden. Center zz matrix from the linearization of the nonlinear model, nS = ; PIMMP: θθ S for Medical Image Science and Visualization, Linköping University, 581 θθ 83 Linköping, Sweden. Parameter identification method for mechanical parameters; R : Ratio of θθ stress from a small change in baseline stretch and baseline stiffness for Received: 10 July 2022 Accepted: 9 September 2022 circumferential direct stiffness. How circumferential (θ) stretch affects L,θp L Published online: 25 November 2022 σ −σ θ θ circumferential (θ) stress. R = ; R : Ratio of stress from a small θθ θz change in baseline stretch and baseline stiffness for circumferential cross‑ coupled stiffness. How longitudinal (z) stretch affects circumferential L,zp References σ −σ θ θ (θ) stress. R = ; R : Ratio of stress from a small change in θz zθ 1. Laurent S, Cockcroft J, Van Bortel L, Boutouyrie P, Giannattasio C, Hayoz D, Pannier B, Vlachopoulos C, Wilkinson I, Struijker‑Boudier H. 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Artery Research – Springer Journals
Published: Dec 1, 2022
Keywords: Abdominal aorta; Cardiovascular disease; Wall stress; Cross-coupled stiffness; Sex; Age; Remodeling; Tortuosity
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