Noninvasive monitoring of pulsatile diameter change in the distal abdominal aorta (AA) was carried out 3–4 cm proximal to the aortic bifurcation [10]. An electronic echotracking instrument (Diamove, Teltec AB, Lund, Sweden) was interfaced with a realtime ultrasound scanner (EUB240, Hitachi, Tokyo, Japan) and fitted with a 3.5 Mhz linear array transducer. The instrument had dual echotracking loops; thus, two separate echoes from opposite vessel walls could be tracked simultaneously. The repetition frequency was 870 Hz, temporal resolution was 1.2 ms, and the smallest detectable displacement was 7.8 µm. For static (end diastolic and systolic) aortic diameter and for pulsatile diameter change, the coefficient of variation was 5% and 16%, respectively.
3.1 Invasive Blood Pressure Measurements
Abdominal aortic (AA) blood pressure was measured invasively at the midpoint between the renal arteries and the aortic bifurcation with a 3F (SPC 330A) or 4F (SPC 340) micromanometer tip catheter (Millar Instruments, Houston, TX) or with a fluidfilled catheter system (pressure monitoring kit DTX + with R.O.S.E, Viggo Spectramed, Oxnard, CA) depending on the availability. The frequency response of the Millar catheter (flat range to 10 kHz) was higher than in the fluidfilled system (flat range 35 Hz [3 dB]). However, curves from one cardiac cycle from each system were superimposed on each other using a Blood Systems Calibrator (Bio Tech Model 601 A, Old Mill Street, Burlington, VT) showing similar systolic blood pressures and pulsatile amplitude when compared.
The data acquisition system allowed for simultaneous monitoring of blood pressure and vessel diameter with a maximum registration duration of 11 s. The system contained a personal computer type 386 (Express, Tokyo, Japan) and a 12bit analogtodigital 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.
3.2 The Identification of Model Parameters and the Mechanical Model
The identification of model parameters and the mechanical model used have been described in previous publications [8, 10]. Also see Appendix for further information.
The parameter identification method for mechanical parameters (PIMMP) consists of two parts, a signal processing routine and a parameter identification routine including a nonlinear mechanical model. In the first part, measured blood pressure and diameter were processed in MATLAB. The data consisting of approximately 8–10 cycles (heartbeats) were lowpass filtered with a fourthorder Butterworth filter with a cutoff frequency of 15 Hz for noise reduction and automatically adjusted for time delays from the measurement setup. Furthermore, pressure and radius were averaged over cycles [8]. In part two, the model parameters were identified through a nonlinear curve fitting of the model response to the measured pressure–radius loop, according to the following iterative algorithm (Fig. 2A):

1.
The stresses in the arterial wall were computed by the Laplace’s law (Eq. 8 in Appendix) using the pressureradius loop together with an estimation of the crosssectional area (A) of the aortic wall from Åstrand et al. [11].

2.
A second set of stresses was computed using nonlinear continuum mechanics (Eqs. 11 and 12) [12]. These model stresses are dependent on six model parameters (explained below) describing the material characteristics and the in situ prestress of the aortic wall [13,14,15].

3.
By comparing the first set of stresses from 1 to the stresses from 2, an estimate of an error (difference) was obtained (Eq. 9) [8]. If the difference between errors from two consecutive iterations was smaller than a preset tolerance (typically 10^{–5}), the iteration was terminated, and the parameters were considered identified. If the error exceeded the tolerance, the parameters values were updated using standard identification techniques and steps 2 and 3 were repeated.
Six model parameters were identified, describing the characteristic (material parameters: c, k_{1}, k_{2}, β) and geometrical (geometrical parameters: R_{0}, and \({\lambda }_{z}\)) properties of the aortic wall [10]. The parameters are:

c (Pa)—relates to the stiffness of the isotropic constituents in the vascular wall, mainly elastin.

k_{1} (Pa)—relates to the stiffness of the anisotropic constituents in the vascular wall, mainly collagen.

k_{2} (dimensionless)—reflects the crimpling or folding, crosslinking and entanglement of collagen.

β (°)—the angle between the circumferential direction and the principal (mean) fiber direction in the unloaded configuration (Fig. 2B).

R_{0} (mm)—the radius in a stress and stretch free (ex situ) unloaded configuration (Fig. 2B)

\({\lambda }_{z}\) (dimensionless)—longitudinal stretch between the in situ configuration and the (ex situ) unloaded configuration (Fig. 2B).
Values for identified parameters are found in Table 4 in Appendix.
A single stiffness constant, e.g., Young’s modulus, cannot accurately predict arterial wall stress [16]. It must be computed from the observed deformation using a set of (nonlinear) equations and associated material parameters [8, 17]. The model used herein is based on a standard Holzapfel–Gasser–Ogden (HGO) nonlinear material model with a neoHookean matrix reinforced by a twofamily fiber structure [12]. The stress–stretch curve in the circumferential direction can be described by:
$${\sigma }_{\theta }={\lambda }_{\theta }\frac{\partial \psi }{\partial {\lambda }_{\theta }}={\sigma }_{\theta }^{\mathrm{iso}}+{\sigma }_{\theta }^{\mathrm{aniso}}=2C\left[{\lambda }_{\theta }^{2}\frac{1}{{\left({\lambda }_{\theta }{\lambda }_{z}\right)}^{2}}\right]+4{k}_{1}\left(I1\right){\mathrm{e}}^{{k}_{2}{\left(I1\right)}^{2}}{\lambda }_{\theta }{\mathrm{cos}}^{2}\beta$$
(1)
where c, k_{1}, k_{2} > 0 guarantee energy dissipation and \(I={\lambda }_{\theta }^{2}{\mathrm{cos}}^{2}\beta +{\lambda }_{z}^{2}{\mathrm{sin}}^{2}\beta\) with \({\lambda }_{\theta }=\frac{{R}_{0}}{{r}_{0}}\frac{4\pi {r}_{0}^{2}+A}{4\pi {R}_{0}^{2}+{\lambda }_{z}A}\). Computed circumferential stress with the result from the identification routine is illustrated in Fig. 1B.
For our purpose, incremental stiffness for a nonlinear material such as biological tissue can be estimated from the slope of the nonlinear stress–stretch curve. This corresponds to the partial derivative of stress with respect to stretch. Hence, there will be two direct stiffnesses and two crosscoupled stiffnesses:
$${S}_{\theta \theta }=\frac{\partial {\sigma }_{\theta }}{\partial {\lambda }_{\theta }}, {S}_{\theta z}=\frac{\partial {\sigma }_{\theta }}{\partial {\lambda }_{z}}, {S}_{z\theta }=\frac{\partial {\sigma }_{z}}{\partial {\lambda }_{\theta }},{S}_{zz}=\frac{\partial {\sigma }_{z}}{\partial {\lambda }_{z}}$$
(2)
where \({S}_{\theta \theta }\) and \({S}_{\theta z}\) are circumferential direct and crosscoupled stiffnesses while \({S}_{zz}\) and \({S}_{z\theta }\) are longitudinal direct and crosscoupled stiffnesses. Crosscoupled stiffness carries information of how stress in one direction is affected when the artery is stretched in another direction. Thus, \({\mathrm{S}}_{\mathrm{\theta z}}\) represents how circumferential stress varies when the vessel wall is stretched in the longitudinal direction and \({\mathrm{S}}_{\mathrm{z\theta }}\) represents how longitudinal stress varies when the vessel wall is stretched in the circumferential direction.
3.3 Computed Variables
In the parameter identification routine, the material parameters were computed (c, k_{1,} k_{2} and β) as well as the geometry for the unloaded state (R_{0,} λ_{z}). Circumferential stretch (\({\lambda }_{\theta }\)) was approximated with \({\lambda }_{\theta }=r/{R}_{0}\) where r was measured radius and R_{0} was identified through the parameter identification process [8]. Furthermore, the circumferential (\({\sigma }_{\theta }\)) and longitudinal (\({\sigma }_{z}\)) stresses were computed. Stiffness was computed as the partial derivative of stress with respect to stretch according to Eq. 2; thus, there were four different stiffnesses computed: circumferential direct (\({S}_{\theta \theta }\)) and crosscoupled (\({S}_{\theta z}\)) as well as longitudinal direct (\({S}_{zz}\)) and crosscoupled (\({S}_{z\theta }\)) stiffnesses.
Mean arterial pressure (MAP) was calculated as 1/3 × (SBP − DBP) + DBP, unit Pa. To convert Pa to mmHg, 1 mmHg = 133.32 Pa, was used.
Pressure, radius, circumferential and longitudinal stretch, stress and stiffness were calculated at SBP, DBP and MAP. Notice that the identified parameters are constant during one heartbeat and are not subject to different values at different pressures.
3.4 Linearization
A small amplitude perturbation analysis has been carried out to determine how crosscoupled stiffness manifests through stress. It was done through a linearization close to a point of interest (systolic blood pressure). The linearization follows standard procedures and is a first order Taylor expansion close to the point of interest [18]. For the present study, the linearization will be two dimensional (2D):
$$\left(\begin{array}{c}{\sigma }_{\theta }^{\mathrm{L}}\left({\lambda }_{\theta },{\lambda }_{z}\right)\\ {\sigma }_{z}^{\mathrm{L}}\left({\lambda }_{\theta },{\lambda }_{z}\right)\end{array}\right)=\left(\begin{array}{c}{\sigma }_{\theta }^{0}\\ {\sigma }_{z}^{0}\end{array}\right)+\left(\begin{array}{cc}{S}_{\theta \theta }^{0}& {S}_{\theta z}^{0}\\ {S}_{z\theta }^{0}& {S}_{zz}^{0}\end{array}\right)\left(\begin{array}{c}{\lambda }_{\theta }{\lambda }_{\theta }^{0}\\ {\lambda }_{z}{\lambda }_{z}^{0}\end{array}\right)$$
(3)
Here \({\sigma }_{\theta }^{\mathrm{L}}\) and \({\upsigma }_{z}^{\mathrm{L}}\) are the linearized functions while \({\sigma }_{\theta }\) and \({\sigma }_{z}\) are the nonlinear functions. \({S}_{\theta \theta }\), \({S}_{\theta z}\), \({S}_{z\theta }\) and \({S}_{zz}\) are the first partial derivatives of \({\sigma }_{\theta }\), and \({\sigma }_{z}\) with respect to \({\lambda }_{\theta }\) and \({\lambda }_{z}\). \({\lambda }_{\theta }^{0}\) and \({\lambda }_{z}^{0}\) denote the point of interest and superscript “0” serves as a marker for a function calculated at the point of interest.
The linearized functions were used to quantify the influence of stretches on stress through direct and crosscoupled stiffness. By changing the stretch a small amount, q (0 < q ≤ 0.03) and taking the ratio between the change (\({\sigma }_{i}^{\mathrm{L},\mathrm{j}q})\) and the original (\({\sigma }_{i}^{\mathrm{L}})\) values, the effect of the change in stretch can be analyzed. Choosing a too large value for q will end up in calculations outside the validity of the linearization. The higher the ratio, the more impact will stretch have on stress. Four ratios were calculated.
$${R}_{ij}=\frac{{\sigma }_{i}^{\mathrm{L},jq}{\sigma }_{i}^{\mathrm{L}}}{{\sigma }_{i}^{\mathrm{L}}} \quad (i, j =\theta \; \mathrm{ or } \; z)$$
(4)
with
$${\sigma }_{i}^{\mathrm{L},jq}{\sigma }_{i}^{\mathrm{L}}={S}_{ij}^{0}\times {\lambda }_{j}\times q \quad (i, j =\theta \; \mathrm{ or } \; z)$$
(5)
Here superscript “q” denotes stress where stretch has been changed a small amount; i and j denote θ or z. The ratios express:

\({R}_{\theta \theta }\): circumferential (θ) stretch impact on circumferential (θ) stress

\({R}_{\theta z}\): longitudinal (z) stretch impact on circumferential (θ) stress

\({R}_{z\theta }\): circumferential (θ) stretch impact on longitudinal (z) stress

\({R}_{zz}\): longitudinal (z) stretch impact on longitudinal (z) stress
The ratios from Eq. (4) were used to assess the influence of stretch on stress via direct stiffness or crosscoupled stiffness.
The impact of stiffness on stress without the effect of stretch was assessed through the stiffness matrix in linearized Eq. (3). Circumferential direct stiffness (\({S}_{\theta \theta }\)) was the largest of the four stiffnesses. Normalizing the elements with \({S}_{\theta \theta }\) will produce relative values which can be compared within the matrix as well as between different points of interest:
$${nS}_{ij}^{0}=\frac{{S}_{ij}^{0}}{{S}_{\theta \theta }^{0}} \quad (i, j=\theta \; \mathrm{ or } \; z)$$
(6)
The matrix in Eq. (3) can be rewritten as:
$${S}_{\theta \theta }^{0}\times \left(\begin{array}{cc}1& {nS}_{\theta z}^{0}\\ {nS}_{z\theta }^{0}& {nS}_{zz}^{0}\end{array}\right)$$
(7)
Here “n” denotes normalized values. The normalized values express stiffness in terms of \({S}_{\theta \theta }\); thus, the normalized matrix element for circumferential direct stiffness will always be 1. Since \({S}_{\theta \theta }\) is largest, the other three stiffnesses will always be less than one.
3.5 Statistics
Arithmetic mean and standard deviation (SD) were calculated for all variables and are expressed as mean ± SD, if not otherwise stated. Variable values calculated at SBP and DBP were regarded as maximum and minimum. A twoway ANOVA test with complementing general linear models was used to compare sex and age groups as suggested by Field [19]. Bonferroni correction was used when multiple comparisons were performed. All parameters were assessed for dependency of age within each sex with a linear regression analysis with Pearson correlation coefficient (R^{2}). P < 0.05 was considered significant in the ANOVA and general linear model as well as in the linear regression analysis. Significance testing is used for a descriptive purpose.
3.6 Software
MATLAB (The Mathwork, Natick, MA, US) version 8.4 (R2014b) was used for computation. IBM SPSS Statistics Version 27 (IBM Corporation, Somers, NY, US) was used for statistical analysis.