Inverse finite element identification of murine aortic material properties: in vivo and ex vivo comparisons

Abstract

Inverse finite element (FE) models can non-invasively estimate aortic mechanical properties from in vivo imaging. However, few studies have compared model predictions with direct mechanical characterization in the same samples. To address this, we used a mouse model of thoracic aneurysm to develop (from in vivo ultrasound imaging) and validate (from ex vivo biomechanical testing) an inverse FE approach to estimate material properties of the ascending thoracic aorta. The proposed inverse FE model was constructed based on a combination of image-based tracking of tissue deformation in 4D ultrasound images (4DUS; volumetric images over time) and non-invasive hemodynamic measures (pulsed wave Doppler velocity and tail-cuff blood pressure). Following an optimization scheme to estimate the biaxial pre-stretches that best represent the in vivo radii obtained from 4DUS images, material properties were identified, and aortic stiffness was calculated for each mouse included in the study ($n=8$ total). Inverse FE predictions were compared with paired ex vivo material characterization results for each ascending aortic sample. Multiple assumptions related to boundary conditions and unloaded tissue geometry were required to constrain the inverse identification procedure; sensitivity analysis was performed for each simplifying assumption and uncertainty in the estimated axial pre-stretch was identified as a primary contributor to the observed discrepancies between in vivo and ex vivo material property estimates. Findings from two material models (neo-Hookean and four-fiber family) were compared and all data has been provided as a benchmark for future inverse FE studies.

Introduction

Thoracic aortic aneurysm is characterized by progressive aortic dilatation that affects 17,000 patients in the United States annually and carries a 90% mortality rate in the event of rupture (Melo et al. 2022). Aortic rupture is a catastrophic biomechanical consequence of local extracellular matrix remodeling that renders the aortic wall susceptible to failure. Such degenerative aortic remodeling is a complex process mediated by an interconnected network of biochemical signals, cellular responses, and genetic factors that results in altered tissue structure and function. In the current study, we will focus on the mechanical impact of such aortic remodeling, since parameters derived from biomechanics, hemodynamics, and aortic geometry have been shown to be a better predictor of rupture risk than aortic diameter alone (Bazzi et al. 2022; Wang et al. 2023). When studying remodeling of thoracic aortic aneurysm, the estimation of aortic material properties requires a large amount of patient-specific in vivo data. However, due to limited availability of subject-specific, non-invasive, longitudinal data (e.g. biomechanics, hemodynamics, and geometry), many studies use generalized population-based remodeling parameters (e.g. Gacek et al. 2023; Gheysen et al. 2024; Irons et al. 2023), which has impeded the development of subject-specific predictive modeling. However, given the large variations in vascular remodeling associated with thoracic aortic aneurysm (Spittell 1983), subject-specific data is potentially more valuable for informing state-of-the-art growth and remodeling approaches. Motivated by the need for subject-specific modeling of aneurysm progression, it is therefore essential to improve non-invasive methods for estimating aortic wall material properties in vivo.

Various studies have attempted to estimate in vivo mechanics of healthy aorta and aortic aneurysms in humans. For example, by assuming a thin-walled cylinder, the stiffness of the aorta at a specific cross-section can be estimated from changes in diameter throughout the cardiac cycle via the law of Laplace (Claridge et al. 2009; Ng et al. 2024). Measurement of the pulse wave velocity via either ultrasound or phase contrast cine-MRI (Morgant et al. 2021) can also be used to estimate aortic stiffness (Gosling and Budge 2003). Finally, an important developing method for estimation of in vivo vascular mechanics involves tracking deformation of the aorta throughout the cardiac cycle using medical images to provide surrogate mechanical testing data (Krishnan et al. 2015; Wittek et al. 2016; Liu et al. 2019). Various material models describing the behavior of the aortic wall can then be implemented in an inverse finite element (FE) framework to estimate the stiffness of the aorta required to produce the same level of deformation observed in the images. Regardless of the approach used to estimate in vivo properties, the implementation of ex vivo (biaxial) mechanical testing and analysis remains the gold-standard for stiffness estimation of aortic tissue (Ferruzzi et al. 2013; Emmott et al. 2016; Lane et al. 2024).

A detailed review of non-invasive inverse FE analyses containing paired comparisons with ex vivo testing reported a large spread between in vivo and ex vivo stiffness estimates of the aorta; while of the same magnitude, the mean in vivo stiffness of healthy aortic tissue is generally reported to be less than that estimated by ex vivo methods (Wang et al. 2022). Unfortunately, in most prior inverse FE studies the in vivo and ex vivo data were not acquired from the same sample, hence population-based average comparisons are often required. As a result, it is difficult to separate the effects of population variation from measurement and model prediction errors. Thus, direct comparisons between in vivo and ex vivo responses from the same sample are needed.

In the current study we developed an approach to compare in vivo and ex vivo material property estimations using a mouse model of ascending thoracic aorta dilatation caused by chronic infusion of angiotensin II (AngII). We were specifically focused on: (1) developing a comprehensive inverse FE approach to estimate in vivo aortic properties from ultrasound imaging and (2) comparing the in vivo aortic mechanics obtained from inverse FE analysis with ex vivo mechanical testing estimates from the same aortic tissue sample, while (3) providing access to the raw data (DOI:10.21227/ey9t-vm54), so future vascular inverse methods can be benchmarked against this study. Furthermore, we characterized the sensitivity of the material properties obtained by inverse FE to the assumptions of aortic geometry (e.g. axial pre-stretch and wall thickness) and external support parameters. Our approach shows a reasonable agreement between in vivo and ex vivo material property estimation in the same tissues. Importantly, this work suggests multiple assumptions related to limiting the parameter search space and estimating the unloaded aortic geometry are required to constrain the inverse FE to converge to a reasonable set of material parameters.

Methods

A flowchart showing each component of the current study is shown in Figure 1. For each animal included in the study, we obtained a non-invasive ultrasound imaging dataset of the ascending thoracic aorta that was used to inform an inverse FE analysis designed to identify in vivo material properties. After euthanasia, the same sample was excised and subjected to ex vivo material characterization via biaxial extension-inflation testing. Mechanical properties obtained by in vivo and ex vivo approaches were then compared for each sample.

Figure 1.

Flow chart showing each aspect of the current study including: (1) paired in vivo and ex vivo data from the AngII infusion mouse model, (2) a reduced order flow model, and (3) inverse finite element analysis to identify mechanical properties from ultrasound image data.

Mouse model

Using a common model of thoracic aortic aneurysm development (Rateri et al. 2014), we examined structural changes in the aortic wall of male and female wild-type mice infused with a high dose of angiotensin II (AngII) for 28 days. Continuous infusion of AngII increases the central arterial blood pressure and increases vascular inflammation (Bersi et al. 2017). In response, the aorta experiences focal areas of degeneration while also thickening and dilating, thus increasing propensity to develop an aneurysm and rupture (Ferruzzi et al. 2013). This model presents an opportunity to generate the desired data set consisting of paired in vivo and ex vivo analysis in the same tissue sample.

All experiments were conducted in accordance with the Institutional Animal Care and Use Committee (IACUC) at Purdue University. A total of thirteen C57BL/6J wild-type mice (4 female, 9 male; aged 10–16 weeks; The Jackson Laboratory, Bar-Harbor, ME) were maintained in a 12h light/dark cycle with regular chow (Envigo, Indianapolis, IN) and drinking water available ad libitum. To induce ascending aortic remodeling, mini-osmotic pumps (ALZET Model 2004; DURECT Corporation, Cupertino, CA) were prepared with either normal saline (0.9% sodium chloride) or AngII dissolved in normal saline (1000 ng/kg/min) for the control and experimental groups, respectively. Pumps were primed for immediate functionality by incubating in normal saline at 37°C for 48 hrs prior to implantation.

For subcutaneous mini-osmotic pump implantation, mice were anesthetized with isoflurane (2–3%) using a low flow vaporizer (SomnoSuite, Kent Scientific, Torrington, CT). Following confirmation of anesthesia, depilatory cream was applied to the mid-scapular region to minimize intra-procedural interference and improve post-operative wound evaluation. After removing all hair from the surgical site, an approximately 2 cm incision was made in the skin and subcutaneous fat overlying the cranial aspect of the right scapula and the mini-osmotic pump was inserted in an approximately 4 cm by 3 cm subcutaneous pouch that was created through blunt dissection. Subsequently, the skin was closed with interrupted stitches using 6–0 Prolene^® suture (Ethicon, Rariton, NJ). Supplemental heat was provided during post-anesthesia care and Buprenorphine (0.05 mg/kg; intraperitoneal) was used for postoperative analgesia.

After recovery from pump implantation, the control ($n=4$; male only) and AngII infused ($n=9;5$ male, 4 female) mice were monitored weekly by in vivo ultrasound imaging and euthanized at 28 days post-implantation via isoflurane overdose and secondary thoracotomy to harvest tissue for ex vivo mechanical testing and analysis. Mice with either an incomplete in vivo dataset ($n=1$, died prematurely from AngII infusion; $n=2$, ultrasound cardiac/respiratory motion artifacts) or an incomplete ex vivo dataset ($n=2$, technical errors in mechanical testing) were removed from the analysis. Thus, the final group sizes were $n=3$ control (male only) and $n=5$ AngII infused (3 male, 2 female) mice. Several a priori assumptions used in this study were based on ex vivo information obtained from the two mice with ultrasound motion artifacts where the inverse FE could not be performed. To emulate the application of our analysis to routine monitoring of living subjects, we did not include any information that was invasively obtained from the mice in our inverse FE modeling approach.

In vivo data collection

There were three separate in vivo datasets collected for each mouse over the course of AngII infusion including 4D ultrasound scans (geometry and deformation), pulsed wave Doppler imaging (blood flow velocity), and tail cuff plethysmography (blood pressure). All in vivo data was collected at baseline just prior to mini-osmotic pump implantation as well as at 1, 3, 7, 14, 21, and 28 days after surgery (see Supplementary Material and Supplementary Figures 10–12).

4D Ultrasound scans

4D ultrasound (4DUS) images of the thoracic aorta (3D volumetric images over time) were acquired with a Vevo 3100 high frequency small animal ultrasound system (25–55 MHz, FUIJIFILM VisualSonics, Toronto, Canada) equipped with a MX550D linear array transducer. Mice were anesthetized and maintained under isoflurane (2–3%) in a supine position on a heated stage with their limbs affixed to conductive pads to monitor respiratory rate, heart rate, and electrocardiogram (ECG). Depilatory cream was applied to the ventral thorax and abdomen for optimal visualization. The 4D imaging mode on the Vevo 3100 ultrasound system from FUJIFILM VisualSonics uses a prospective cardiac- and respiratory-gating approach where high temporal resolution 2D images (via ECG-gated Kilohertz Visualization) are collected serially starting at the middle of the descending thoracic aorta and continuing up past the aortic arch. Each 4D dataset took roughly 5–7 min to acquire; given the approximate mouse heart rate of 500 bpm during acquisition, this equates to collecting data from several thousand heart beats. These 2D images were compiled to form B-mode images with an approximately 0.03 × 0.03 × 0.1 mm spatial resolution and 40 frames per cardiac cycle.

Pulsed wave Doppler ultrasound

Flow velocity in the ascending thoracic aorta, descending thoracic aorta, and the three primary branches of the aortic arch were obtained using the pulsed wave Doppler (PWD) imaging capabilities of the Vevo 3100 small animal ultrasound system. For each measurement, the Doppler imaging probe was placed at the vessel centerline with the angle oriented along the primary direction of blood flow. Volumetric flow rate through the aorta was calculated by multiplying the measured PWD by the temporally averaged area of the inlet/outlets from the tracked and segmented aorta. In the event of poor/unreliable flow velocity measurements in any of the outlet branches ($n=2$ samples, one missing the left common carotid and one missing the ascending aortic PWD), the measured waveform data was removed, and the missing flow rate was reconstructed from the other branches using conservation of mass. PWD velocity measurements were used to calibrate a resistor-capacitor-resistor (RCR) boundary condition at each outlet using a 0D reduced order model representation of the aorta (Bazzi et al. 2022; Wiputra et al. 2023). Such calibration was necessary to match specific flow rate splits across all model outlets (e.g. three aortic arch branches and descending aorta) for each mouse.

Tail cuff pressures

Blood pressure was collected from conscious mice that were comfortably restrained in a nose cone animal holder via a tail cuff-based monitoring system (CODA 2 Channel Standard, Kent Scientific, Torrington CT). Once restrained, mice were placed on a heated stage and covered with a warming blanket to minimize vasoconstriction secondary to temperature and stress. To further minimize confounding effects on blood pressure, animals were acclimated to the restraints and tail cuffs prior to data collection.

Ex vivo mechanical testing

Following euthanasia at 28 days after pump implantation, samples were prepared for dissection and ex vivo testing. Briefly, a sternotomy was performed, and the superior vena cava was severed prior to left ventricular apical injection of cold 1 M phosphate buffered saline (PBS) to perfuse the vasculature. The thoracic aorta and its branches were then gently dissected away from surrounding perivascular tissue under microscopy to preserve branch lengths for ligation. Tissues were stored in cold 1 M PBS and shipped overnight to Washington University in St. Louis for biaxial extension-inflation testing and analysis; all samples were stored at 4°C and underwent ex vivo testing within 72 h of harvesting. Note that the mechanical properties of the tissue are not expected to be significantly altered due to this short-term cold storage (Adham et al. 1996; Stemper et al. 2007; Amin et al. 2011; O’Leary et al. 2014).

After cleaning and ligating branches for pressurization, biaxial extension-inflation testing was performed following established protocols (Ferruzzi et al. 2013; Bersi et al. 2016; Kim et al. 2022). Briefly, the ascending aorta was cannulated through the aortic root (proximal) and brachiocephalic artery (distal) and placed in a custom computer-controlled mechanical testing system. Following cyclic pressurization at a reasonable axial stretch (preconditioning), the preferred in vivo axial stretch ${\lambda }_z^{iv}$ was estimated as the stretch at which a constant axial force – pressure relationship was observed (Weizsäcker et al. 1983). Ex vivo mechanical testing consisted of 3 cycles of pressurization from 10 – 140mmHg at fixed axial stretches of 0.95, 1.0, and 1.05 times the estimated in vivo axial stretch ${\lambda }_z^{iv}$ and 4 cycles of axial extension from 0 – 55mN of axial force at fixed pressures of 10, 60, 100, and 140mmHg. Together, these protocols produce the characteristic pressure-diameter and axial force-stretch behaviors of passive arterial tissue; active contractile response of vascular smooth muscle was not assessed in the current study (Ferruzzi et al. 2013). This passive mechanical data was then fitted to a four-fiber family constitutive model (Baek et al. 2007), given as:

$$\begin{gathered} W_{4fib}\left(\mathbf{C},{\mathbf{m}}^{\left(a\right)}\right)=\frac{c}{2}\left(I_1-3\right)+{\sum }_{a=1}^4\frac{c_1^{\left(a\right)}}{4c_2^{\left(a\right)}} \\ \left\{exp\left[c_2^{\left(a\right)}{\left(I{V}_c\left({\mathbf{m}}^{\left(a\right)}\right)-1\right)}^2\right]-1\right\} \end{gathered}$$ (1)

$${I_1=\mathbf{C}}_{\mathrm{i}\mathrm{i}}={\lambda }_{\theta }^2+{\lambda }_z^2+\frac{1}{{\lambda }_z^2{\lambda }_{\theta }^2}$$ (2)

$$I{V}_c\left({\mathbf{m}}^{\left(\mathrm{a}\right)}\right)={\mathbf{m}}_{\mathbf{i}}{\mathbf{F}}_{\mathbf{i}\mathbf{k}}{\mathbf{F}}_{\mathbf{k}\mathbf{j}}{\mathbf{m}}_{\mathbf{j}}$$ (3)

Where $W_{4fib}$ is the strain energy density defined per unit volume, $c$ describes the neo-Hookean matrix stiffness, ${\mathrm{c}}_1^{\left(a\right)}$ and ${\mathrm{c}}_2^{\left(a\right)}$ are the stiffness and exponential properties of a given fiber family $\left(a\right)$, $\mathbf{C}={\mathbf{F}}^T\mathbf{F}$ is the right Cauchy-Green tensor, and $\mathbf{F}=diag\left({\lambda }_z,{\lambda }_{\theta },{\lambda }_r\right)$ is the deformation gradient tensor. By enforcing the assumption of tissue incompressibility (i.e. $\mathrm{d}\mathrm{e}\mathrm{t}\left(\mathbf{F}\right)=1$), the radial stretch could be calculated as ${\lambda }_r=\frac{1}{{\lambda }_z{\lambda }_{\theta }}$. The unit vector ${\mathbf{m}}^{\left(a\right)}=\left[\mathrm{c}\mathrm{o}\mathrm{s}\left({\alpha }_o^{\left(a\right)}\right)\right.,\left.\mathrm{s}\mathrm{i}\mathrm{n}\left({\alpha }_o^{\left(a\right)}\right),0\right]$ was defined to be oriented along a fiber in the material coordinate axes locally aligned with either the axial (1), circumferential (2), or diagonal (3,4) directions along the aorta. The fiber angle ${\alpha }_o$ was defined as ${\alpha }_0^{\left(1\right)}=0^{\circ }$ in the axial direction, ${\alpha }_0^{\left(2\right)}=90°$ in the circumferential direction, and $0^{\circ }<{\alpha }_0^{\left(3\right)}<90°$ for diagonal fibers. The two diagonal fiber families were assumed to be identical and symmetric (i.e. $c_1^{\left(3\right)}=c_1^{\left(4\right)}$, $c_2^{\left(3\right)}=c_2^{\left(4\right)}$, ${\alpha }_0^{\left(3\right)}=-{\alpha }_0^{\left(4\right)}$) (Holzapfel et al. 2000).

Under this constitutive model formulation and set of assumptions, there were a total of 8 unknown material parameters: $c$, $c_1^{(\mathrm{1,2},3)}$, ${\mathrm{c}}_2^{(\mathrm{1,2},3)}$ and ${\alpha }_0^{\left(3\right)}$. Fitting was performed using a Levenberg-Marquardt non-linear least squares optimization algorithm with the following objective function:

$$obj=\sum _{i=1}^N\left[{\left(\frac{P^{th}-P^{exp}}{{\bar{P}}^{exp}}\right)}_i^2+{\left(\frac{f_z^{th}-f_z^{exp}}{{\bar{f}}_z^{exp}}\right)}_i^2\right]$$ (4)

In order to minimize the difference between theoretically-predicted ($th$) and experimentally-measured ($exp$) quasistatic configurations. Here, $P$ is the pressure and $f_z=f_T+\pi r^{2}P$ is the axial force on the vessel, where $f_T$ is the force measured by the axial force transducer and $r$ is the current inner radius. The overbar in each denominator denotes the average value over all measurements in the mechanical testing protocol $i$; a total of $N=7$ protocols were included in the optimization (e.g. 3 pressure-diameter and 4 axial force-stretch). Theoretically predicted pressures and axial forces were calculated by solving the set of average in-plane biaxial Cauchy stress equations in a thin-walled cylinder for $P$ and $f_z$

$${\sigma }_{\theta \theta }=\frac{Pr}{h},{\sigma }_{zz}=\frac{f_z}{\pi h(2r+h)}$$ (5)

Where $h$ is the current wall thickness and all other variables are defined above. Again, we enforced the assumption of volume-preserving deformations (incompressibility) and the predicted values of ${\sigma }_{\theta \theta }$ and ${\sigma }_{zz}$ were obtained from the current set of material parameters by calculating an appropriate first derivative of the strain energy function $W_{4fib}$ with respect to the strain tensor $\mathbf{C}$.

Reduced order flow model

The pressure in the aorta was represented by a voltage node that branched out to the three main branches (e.g. brachiocephalic, left common carotid, and left subclavian) and descending aorta. Flow rate (or current) through this node was prescribed by the ascending aorta flow rate obtained from PWD. Each branch was modeled by a reduced order (lumped parameter) Resistor-Capacitor-Resistor (RCR) circuit, calibrated to the flow rate splits obtained from PWD and tail-cuff blood pressure readings. Differential evolution minimization (Storn and Price 1997) of the following cost function was performed:

$$\begin{gathered} {\mathrm{c}\mathrm{o}\mathrm{s}t}_{RCR}=\Vert \left[\frac{{\bar{Q}}_{BA}}{{\bar{Q}}_{BA,PWD}}-1,\frac{{\bar{Q}}_{LCCA}}{{\bar{Q}}_{LCCA,PWD}}-1,\frac{{\bar{Q}}_{LSA}}{{\bar{Q}}_{LSA,PWD}}\right. \\ -1,\frac{{\bar{Q}}_{Desc}}{{\bar{Q}}_{Desc,PWD}}-1,\frac{P_{\text{min}}}{P_{\text{min},\text{cuff}}}-1,\frac{P_{\text{max}}}{P_{\text{max},\text{cuff}}} \\ \left.-1,\frac{P_{MAP}}{P_{MAP,\text{cuff}}}-1\right]{\Vert }_2 \end{gathered}$$ (6)

Where $\bar{Q}$ is the time averaged percent of flow split through the brachiocephalic (BA), left carotid artery (LCCA), left subclavian (LSA), and descending aorta (Desc) obtained from the circuit model. $P_{\mathit{min}}$, $P_{\mathit{max}}$, and $P_{\mathit{MAP}}$ are minimum, maximum, and mean arterial pressure obtained from the circufit model respectively. These values are normalized by the measured flow rate (e.g. ${\bar{Q}}_{PWD}$) and blood pressure (e.g. $P_{\mathit{cuff}}$). When properly calibrated, the reduced-order model returned the pressure profile over time which was applied uniformly to the inner wall of the aorta during the inverse FE analysis. Here the reduced-order model ignored the axial forces from wall shear stress, which are orders of magnitude lower than the wall stress due to the luminal pressure (Caro et al. 2012).

Inverse finite element analysis

First, the ascending aorta, aortic arch, and proximal descending thoracic aorta were segmented from the 4DUS images starting at the first time point of the cine image, and deformation was tracked over a cardiac cycle using a validated B-spline Fourier-based image registration as previously described (Wiputra et al. 2020). An iterative finite element (FE) updating approach (Ereiz et al. 2022) was implemented to solve the inverse problem of identifying material parameters for the ascending aortic wall based on the deformation observed in the ultrasound image data. Different sets of material parameters were tested until the vessel radii obtained from the finite element analysis at physiologic pressures matched the radii measured from volumetric ultrasound images. The inverse FE approach was implemented according to the following steps (Figure 2):

Figure 2.

Summary of inverse finite element approach. Estimation process involved an iterative scheme to update material parameters to match vessel radii measured by ultrasound and predicted from inverse finite element modeling. Detailed description of steps: 2,3, and 5 can be found in sections: Unloaded Configuration Estimate (Pre-stretch), Finite Element Model, and Parameter Estimation, respectively.

1. Guess an initial set of material parameter values.

2. Estimate the unloaded configuration in both the circumferential and axial direction to calculate biaxial pre-stretch.

3. Deform the geometry based on the known luminal pressure using forward FE modeling.

4. Evaluate differences in radii between model prediction and experimental measurement.

5. Repeat steps 1–4 with a new set of material parameters updated using a Levenberg-Marquardt optimization scheme until the FE model-predicted radii match those measured from ultrasound images.

When performing the inverse FE analysis in this study, we split the eight mice with complete in vivo and ex vivo paired datasets (3 controls, 5 AngII) randomly into four for fitting and four for validation. Since there are only 3 controls, we ensured that at least one of the control mice was in the validation set. The method was optimized on the four mice selected for fitting before it was applied to the four mice in the (blinded) validation set.

Unloaded configuration estimate (pre-stretch)

The unloaded geometry for each sample cannot be known a priori since the in vivo geometry observed via ultrasound imaging is always stretched circumferentially from the blood pressure and axially from surrounding anatomic tissue and cardiac contraction (Cardamone et al. 2009). Specifying the unloaded geometry is required to construct FE models, but first requires estimating the pre-stretch of the ascending aorta in the axial ${(\lambda }_{z,\mathit{pre}})$ and circumferential ${(\lambda }_{\theta ,\mathit{pre}}$) directions. It is then possible to obtain the unloaded wall thickness from loaded wall thickness obtained by 4DUS by assuming tissue incompressibility $({\lambda }_{r,\mathit{pre}}=\frac{1}{{\lambda }_{z,\mathit{pre}}{\lambda }_{\theta ,\mathit{pre}}})$. However, due to the uncertainty in the unloaded thickness $\left(h_0\right)$ measurement (see Supplementary Material), we determined $h_0$ from the average of ex vivo measurements obtained from the two excluded datasets.

The axial pre-stretch ${\lambda }_{z,\mathit{pre}}$ was experimentally determined to be the axial stretch at which the axial force-stretch curves from multiple pressures intersect (i.e. crossover point, see Supplementary Figure 7, 8). Equivalently, this crossover point can be represented as the axial stretch that results in a plateau of the axial force-pressure curve since the intersection of axial force-stretch curves marks the axial stretch where axial force was constant across pressures (Weizsäcker et al. 1983).

We model this experimental approach in silico (as step 2 of Figure 2) using a two-step pre-straining procedure:

1. Given a set of material parameters, assume ${\lambda }_{z,\mathit{pre}}={\lambda }_{\theta ,\mathit{pre}}$ and determine the value of ${\lambda }_{z,\mathit{pre}}$ that produces a constant axial force-pressure response.

2. Fix the value of ${\lambda }_{z,\mathit{pre}}$ and refine the current estimate for ${\lambda }_{\theta ,\mathit{pre}}$ by reducing the radius of the aorta until the loaded geometry matches the measured diastolic geometry.

The biaxial pre-stretches ${\lambda }_{\theta ,\mathit{pre}}$ and ${\lambda }_{z,\mathit{pre}}$ are coupled; for every value of ${\lambda }_{z,\mathit{pre}}$ there exists a ${\lambda }_{\theta ,\mathit{pre}}$ that sufficiently balances the wall stresses induced by blood pressure (Lu et al. 2007; Gee et al. 2010), and vice versa. Therefore, when determining ${\lambda }_{\theta ,\mathit{pre}}$ or ${\lambda }_{z,\mathit{pre}}$, one of the two pre-stretches must be fixed. Isotropic pre-stretch is a reasonable starting point for the ascending aorta, based on previous reports (Bersi et al. 2017) and our ex vivo measurements of ${\lambda }_{\theta ,\mathit{pre}}$ and ${\lambda }_{z,\mathit{pre}}$ (Table 1).

Table 1.

Comparison of axial (${\lambda }_{z,\mathit{pre}}$) and circumferential (${\lambda }_{\theta ,\mathit{pre}}$) pre-stretches obtained from mechanical testing (ex vivo), inverse finite element modeling (in silico), and ultrasound measurements.

	${\lambda }_{z,\mathit{pre}}$			${\lambda }_{\theta ,\mathit{pre}}$
Case	ex vivo	in silico	$I_{\mathit{US}}^{\mathit{dia}}/I_{\mathit{ex}}$	ex vivo	in silico	$D_{\mathit{US}}^{\mathit{dia}}/D_{\mathit{ex}}$
1	1.96	1.90	1.54	1.54	1.71	2.04
2	1.55	1.74	1.24	1.66	1.56	2.25
3	1.97	1.76	1.13	1.50	1.48	1.75
4	1.76	1.94	1.62	1.54	1.62	1.71
5	1.97	1.79	1.71	1.68	1.53	2.28
6	1.41	1.85	1.57	1.55	1.60	2.27
7	2.09	1.78	1.81	1.48	1.54	1.75
8	1.71	1.70	1.39	1.48	1.50	1.84
Avg	1.80	1.81	1.50	1.55	1.57	1.99

$D_{\mathit{US}}^{\mathit{dia}}$ is the diastolic diameter of the aortic root and $I_{\mathit{US}}^{\mathit{dia}}$ is the diastolic root-to-first-branch length measured from ultrasound image; ${\mathit{D}}_{\mathit{ex}}$ is the unloaded outer diameter and ${\mathit{I}}_{\mathit{ex}}$ is the unloaded axial length determined from ex vivo measurement.

In step 1, the axial pre-stretch was estimated by identifying a constant axial force-pressure response under the assumption of an idealized cylindrical geometry using the following equation:

$$\begin{gathered} {\lambda }_{z,\mathit{pre}}=\underset{{\lambda }_{z,\mathit{pre*}}}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}\left(\sum _{t=\mathit{dia}}^{\mathit{sys}}\mid \mathrm{l}\mathrm{o}\mathrm{g}\left(f_T\left(\mathit{dia},{\lambda }_{z,\mathit{pre}*}\right)\right)\right. \\ \left.-\mathrm{l}\mathrm{o}\mathrm{g}\left(f_T\left(t,{\lambda }_{z,\mathit{pre}*}\right)\right)\mid \right) \end{gathered}$$ (7)

Where $f_T(t)$ and $f_T(dia)$ are simulated axial force transducer readings during a stretch-inflation test, at time $t$ and diastole, respectively. ${\lambda }_{z,\mathit{pre}*}$ is the trial axial pre-stretch. To determine ${\lambda }_{z,\mathit{pre}}$, predicted biomechanical properties were compared between four evenly spaced time points across the cardiac cycle including end-diastole and peak systole. The axial force on the vessel $\left(f_T\right)$ is estimated by the force-balance equation in an idealized cylinder (see axial stress in Eq. (5)):

$$f_T(t)={\sigma }_{zz}(\mathrm{t})\pi h(t)(2r(t)+h(t))-\mathrm{P}(\mathrm{t})\pi r(t{)}^2$$ (8)

where $P$ is the pressure, $r$ is the inner radius and $h$ is the wall thickness obtained from the incompressibility constraint (e.g. $h(t)=\frac{h_0}{{\lambda }_z(t){\lambda }_{\theta }(t)}$) at a given time point in the cardiac cycle. To compute the force $f_{T,}$ the axial stress $\left({\sigma }_{zz}\right)$ was calculated from current set of material parameters, such that:

$${\sigma }_{zz}(t)={\lambda }_z(t)\frac{\mathrm{\Delta }W(t)}{\mathrm{\Delta }{\lambda }_z(t)}$$ (9)

where $W$ is the strain energy density function for the material model, which was defined to be either neo-Hookean ($W_{\mathit{neo}}$ as in Eq. (11)) or a four-fiber family model ($W_{\mathit{4fib}}$ as in Eq. (1)).

The axial (${\lambda }_z$) and circumferential (${\lambda }_{\theta }$) stretches could then be decomposed into the pre-stretch and the stretch observed in the ultrasound images in each direction:

$${\lambda }_z(t)={\lambda }_{z,\mathit{dia}}(t){\lambda }_{z,\mathit{pre}},{\lambda }_{\theta }(t)={\lambda }_{\theta ,\mathit{dia}}(t){\lambda }_{\theta ,\mathit{pre}}$$ (10)

The ultrasound stretches (${\lambda }_{z,\mathit{dia}},{\lambda }_{\theta ,\mathit{dia}}$) were measured as deformation relative to the diastolic geometry and were obtained from 4DUS tracking of changes in the aortic length and inner radius, respectively. Axial length measurements were defined as the distance between the aortic root and the brachiocephalic branch and inner radius was calculated from the aortic luminal area measured proximal to the brachiocephalic artery under the assumption of a circular cross-section. A minimum of three independent 4DUS length and radius measurements were collected at diastole for each animal. By defining the stretches in this way, the biaxial pre-stretches ${(\lambda }_{z,\mathit{pre}},{\lambda }_{\theta ,\mathit{pre}})$ could then be calculated between the unloaded and diastolic geometries.

We restricted the axial pre-stretch to be within a reasonable physiologic range $\left(1.4<{\lambda }_{z,\mathit{pre}}<2.5\right)$ since local minima of Eq. (7) could lead to inappropriate estimates of the axial pre-stretch. For example, the axial force-pressure curve near ${\lambda }_{z,\mathit{pre}}=1$ appears plateau-like because of small $f_z$ values, but such small axial pre-stretch is not consistent with experimental observations (Bellini et al. 2017). For each sample, the obtained ${\lambda }_{z,\mathit{pre}}$ was compared against the ex vivo measured crossover point and the ultrasound measured distance between the aortic root and the brachiocephalic branch (Table 1).

Following identification of ${\lambda }_{z,\mathit{pre}}$, the circumferential pre-stretch was determined by the pullback method (Gee et al. 2009). Namely, the in vivo aortic geometry was shrunk axially by $1/{\lambda }_{z,\mathit{pre}}$ to obtain the unloaded length and then axially pre-stretched to regain its loaded length. Once the model aorta had been axially pre-stretched, ${\lambda }_{\theta ,\mathit{pre}}$ was adjusted by gradually shrinking the aorta's unloaded radius until its radius at diastolic pressure matched the in vivo ultrasound image. As the geometry was deformed, the deformation was smoothed and regularized to ensure uniform element volumes and a well-behaved mesh. For each sample, the obtained ${\lambda }_{\theta ,\mathit{pre}}$ was compared against the ex vivo-measured and ultrasound-measured changes in diameter (Table 1). Finally, after pre-stretching was complete, the aortic geometry was pressurized to perform FE analysis.

Finite element model

The ascending and proximal descending thoracic aorta geometry were meshed with 32.1±0.8K tetrahedral elements. The mesh density, time step, and number of cardiac cycles used for the FE model were selected based on a convergence study (Supplementary Figure 1).

Boundary conditions for the finite element simulation (Figure 2, step 3), were assigned based on: 1) pressure from the reduced-order model of the aorta (prescribed on the inner wall), 2) displacement based on the B-spline Fourier image registration (prescribed at the branch ends), and 3) spring-like Robin boundary conditions to describe stiffness of the surrounding tissue (prescribed on the outer wall). When prescribing Robin boundary conditions, the spring constant $k_{Rb}$ is not a known quantity since the stiffness of the surrounding tissue was not tested. Here, it was assumed that $k_{Rb}={10}^{-4}N/m\approx {10}^{4}dyne/c{m}^3$ (Shidhore et al. 2022) at the mesh density used.

In addition to the geometry and boundary conditions, two constitutive material models were considered: 1) the four-fiber family model as in ex vivo mechanical testing (see Eq. (1), and (2)) a quasi-incompressible neo-Hookean model $W_{\mathit{neo}}$ defined as:

$$\begin{gathered} W_{\mathit{neo}}(\mathit{C})=\frac{\mu }{2}\left(I_1-3\right)-\mu \mathrm{l}\mathrm{n}J+\frac{B}{2}(\mathrm{l}\mathrm{n}J{)}^2 \\ \mu =\frac{E}{2(1+v)},B=\frac{vE}{(1+v)(1-2v)} \end{gathered}$$ (11)

Where $I_1=tr(\mathit{C})$ is the first invariant of right Cauchy-Green deformation tensor $\mathit{C}$, $J$ is the determinant of the deformation gradient tensor $\mathit{F}$, $B$ and $\mu$ are the first and second Lamé's parameter, $v$ is the Poisson ratio, and $E$ is the Young's modulus. For all simulations, a value of $v=0.45$, was chosen to be close to incompressible ($v=0.5$) yet forgiving enough for the model to converge. Importantly, when determining ${\lambda }_{z,\mathit{pre}}$ with the neo-Hookean model, the axial force-stretch crossover behavior was not observed (Supplementary Figure 8) since the stress-stretch behavior was nearly independent of ${\lambda }_z$ (Supplementary Figure 4). Therefore, a fixed value of ${\lambda }_{z,\mathit{pre}}=1.6$ (average value from two excluded mice) was used for the neo-Hookean model. All finite element analysis was performed in FEBio (Maas et al. 2012).

Parameter estimation

The vector of optimal material parameters $\overrightarrow{\mathit{c}}$ for a given strain energy function, was obtained by minimizing the L2-norm of the temporally averaged radius differences over the cardiac cycle such that the cost function $err$ was defined as:

$$\begin{array}{c}\overrightarrow{\mathit{c}}=argmin(err)\\ err=\Vert \frac{mean\left(r_{US}(t,n)-r_{FE}(t,n,\overrightarrow{\mathit{c}}),t\right)}{max\left(\left|r_{US}\left(t_{dia},n\right)-r_{US}\left(t_{sys},n\right)\right|,n\right)}{\Vert }_2\end{array}$$ (12)

Where $r_{US}$ and $r_{FE}$ are the radius obtained from ultrasound images and inverse FE simulation, respectively, at mesh node $n$ and time $t$ of the cardiac cycle. The cost function was normalized to the maximum radial displacement over all nodes, and only the inner wall radius was included in the objective function due to uncertainty in the wall thickness measured from 4DUS (see Supplementary Table 1).

For the neo-Hookean model (see Eq. (11)), there are two material parameters, $E$ and $v$. However, since $v$ was fixed at 0.45 for all cases, this resulted in $\overrightarrow{\mathit{c}}=E$. For the four-fiber model (see Eq. (1)) with symmetric diagonal fibers, the full parameter vector was defined as $\overrightarrow{\mathit{c}}=\left[c,c_1^{(1)},c_2^{(1)},c_1^{(2)},c_2^{(2)},c_1^{(3)},c_2^{(3)},{\alpha }_0^{(3)}\right]$. To restrict the parameter search space, we specified fixed values of ${\alpha }_0^{(3)}={50.6}^{\circ }$ and $c_2^{(i)}=\left[5.2\times {10}^{-13},7.7\times {10}^{-12},3.1\times \right.\left.{10}^{-1}\right]$, which were obtained from fitting the four-fiber family model to averaged ex vivo stress-stretch curves from the two excluded mice. Thus, the final vector of unknown four-fiber family model material parameters was $[c,c_1^{(1)},c_1^{(2)},c_1^{(3)}]$.

Optimization was performed in the log-domain, with random sampling (11 samples) at 0.5 to 2 times the initial guess of ${\overrightarrow{\mathit{c}}}_{\mathit{init}}=[0.33,3.78,1.08,1.18]kPa$, which was also identified from the two excluded mice. The gradient for each iteration was then calculated by forming simplices of $\overrightarrow{\mathit{c}}$ from past iterations. The Levenberg-Marquardt algorithm was used for optimization (Fletcher 1971), with a step size of 1, and a damping factor initialized at 20. The damping factor was either halved when the previous iteration yielded lower error or doubled otherwise. A momentum term was also imposed (weighted at 30%) to maintain the descent direction and reduce the influence of local minima (Qian 1999).

Material stiffness

Following identification of best-fit material parameters for each material model, material stiffness was used as an objective measure to compare model predictions. Material stiffness was calculated as the circumferential tangent modulus $\left(E_{\theta \theta }\right)$ at a specific loading state defined by a pair of axial and circumferential stretches $\left({\lambda }_z,{\lambda }_{\theta }\right)$. The tangent modulus was defined as:

$${\mathrm{E}}_{\theta \theta }\left({\lambda }_z,{\lambda }_{\theta }\right)={\left.\frac{d{\sigma }_{\theta \theta }}{d{\lambda }_{\theta }}\right|}_{\left({\lambda }_z,{\lambda }_{\theta }\right)}$$ (13)

where ${\sigma }_{\theta \theta }$ is the model-predicted circumferential stress.

Sensitivity analysis

We performed sensitivity analysis on several quantities assumed in our inverse FE approach. The neo-Hookean parameter was chosen for most of the analysis because it is more straightforward to interpret than the non-linear interactions between parameters in the four-fiber family model. We varied the assumed quantities within some percentage about the chosen value and measured relative changes in the estimated neo-Hookean stiffness. Sensitivity studies were performed on aortic wall thickness, axial pre-stretch, surrounding tissue spring constant, and choice of ${\overrightarrow{\mathit{c}}}_{\mathit{init}}$ (see Figure 5, Figure 7). We also performed a sensitivity analysis on the fixed material parameters that were used to constrain the inverse FE identification for the four-fiber family model. Namely, we calculated changes in predicted circumferential stress following +10% perturbations to the diagonal fiber exponential parameter $c_2^{(\mathrm{3,4})}$ and the fiber angle ${\alpha }_0^{(3)}$ (see Figure 6).

Figure 5.

Sensitivity analysis of the neo-Hookean parameter ($E$) identified by inverse FE modeling. Sensitivity was assessed in response to changing (A) the axial pre-stretch ${\lambda }_{z,\mathit{pre}}$, (B) the unloaded thickness of the aortic wall, and (C) the spring constant for the external tissue Robin boundary condition (plotted in log10 scale). Values selected for all inverse FE modeling simulations are circled in red. S is the sensitivity of the studied parameter taken as the slope of the tangent to the curve (%change in E divided by %change in test parameter).

Figure 7.

Representative sensitivity of the inverse FE modeling results to the choice of initial ${\overrightarrow{c}}_{\mathit{init}}$ value. Simulated circumferential stress-stretch curves from multiple runs of the inverse FE method with either the neo-Hookean (Neo) or four-fiber family (4Fib) material models. Error bars denote standard deviations across multiple rounds of optimization; the neo-Hookean model also has small error bars. Color bars depict the axial stretch of each circumferential stress-stretch curve, with lighter curves denoting higher axial stretch values.

Figure 6.

Sensitivity analysis of the proposed inverse FE method to the fiber angle $({\alpha }_0^{(3)})$ and diagonal fiber exponential parameter $(c_2^{(\mathrm{3,4})})$. (a) Predicted equibiaxial circumferential stress-stretch curves following $a+10\mathrm{\%}$ perturbation to each material parameter. (b) The corresponding percent change in stress from the original model at each value of stretch. Note that the % change at the stretch of 1 was forced to zero since both the numerator and denominator are zero in this case.

Statistical analysis

When comparing model predictions between the neo-Hookean and four-fiber family models, a two-tailed paired Student’s t-test was used. For comparisons with more than two groups, a 1-way ANOVA with post hoc Tukey tests was used. Normality and lognormality of all datasets to be statistically compared were evaluated using Shapiro-Wilk testing. Pearson’s correlation was used to determine the relationship between variables. Correlations were graded as weak (0.0–0.5), moderate (0.5–0.7), strong (0.7–0.9), or very strong (0.9–1.0). For all statistical comparisons, a value of $p<0.05$ was considered significant.

Results

Non-invasive measurement of in vivo hemodynamics and aortic diameter

We measured changes in hemodynamics and aortic geometry (diameter and wall thickness) in control and AngII-treated mice over 28 days of infusion using 4DUS and tail-cuff blood pressures. The extensive result is available in Supplementary Figure 10, 11, 12. Briefly, after 28 days of AngII infusion, we observed significant increase in both diameter and wall thickness that is weakly correlated to significant increase in its mean arterial pressure.

In vivo diastolic pre-stretch estimates

The in silico ${\lambda }_{z,\mathit{pre}}$ obtained from the plateau in the axial force-pressure plot was similar to the crossover point in the axial force-stretch plot at different pressures (Supplementary Figure 7) (Weizsäcker et al. 1983). Here, in silico estimates of ${\lambda }_{z,\mathit{pre}}$ and ${\lambda }_{\theta ,\mathit{pre}}$ (Table 1) showed no significant difference from ex vivo estimates of ${\lambda }_{z,\mathit{pre}}$ and ${\lambda }_{\theta ,\mathit{pre}}$ ($p>0.9$ for both; Table 2). Despite similar values on average, the ex vivo and in silico biaxial pre-stretch estimates were not correlated ($r=0.007$ for ${\lambda }_{z,\mathit{pre}}$ and $r=0.08{\lambda }_{\theta ,\mathit{pre}}$; Table 3). Both ex vivo and in silico measures overestimated ${\lambda }_{z,\mathit{pre}}$ and underestimated ${\lambda }_{\theta ,\mathit{pre}}$ relative to ultrasound-based stretch measurements (4DUS measurement normalized by unloaded ex vivo measurement; see Table 1). In all cases, the ex vivo and in silico measures of ${\lambda }_{z,\mathit{pre}}$ and ${\lambda }_{\theta ,\mathit{pre}}$ were significantly different (larger in axial, smaller in circumferential) from ultrasound-measured values ($p<0.02$ for axial and $p<0.0001$ for circumferential; Table 2). Both ex vivo and in silico values were more correlated, albeit weakly, with ultrasound measurements than with one another ($r<0.2$ for both), however only ex vivo estimates of ${\lambda }_{\theta ,\mathit{pre}}$ were strongly correlated $(r=0.79)$ with ultrasound-measurements (Table 3).

Table 2.

p-Values from 1-way ANOVA with post hoc Tukey test for axial (${\lambda }_{z,\mathit{pre}}$) and circumferential (${\lambda }_{\theta ,\mathit{pre}}$) pre-stretches measured ex vivo, in silico, and from 4DUS.

	${\lambda }_{z,\mathit{pre}}$			${\lambda }_{\theta ,\mathit{pre}}$
p-values	ex vivo	in silico	$I_{\mathit{US}}^{\mathit{dia}}/I_{\mathit{ex}}$	ex vivo	in silico	$D_{\mathit{US}}^{\mathit{dia}}/D_{\mathit{ex}}$
ex vivo	N/A	0.999	0.016	N/A	0.98	6e-5
in silico	0.999	N/A	0.0014	0.98	N/A	9e-5

$D_{\mathit{US}}^{\mathit{dia}}$ is the diastolic diameter of the aortic root and $I_{\mathit{US}}^{\mathit{dia}}$ is the diastolic root-to-first-branch length measured from ultrasound image; ${\mathit{D}}_{\mathit{ex}}$ is the unloaded outer diameter and ${\mathit{I}}_{\mathit{ex}}$ is the unloaded axial length determined from ex vivo measurement.

Table 3.

Correlation coefficient (r) between axial (${\lambda }_{z,\mathit{pre}}$) and circumferential (${\lambda }_{\theta ,\mathit{pre}}$) pre-stretches measured ex vivo, in silico, and from 4DUS.

	${\lambda }_{z,\mathit{pre}}$			${\lambda }_{\theta ,\mathit{pre}}$
Correlation (r)	ex vivo	in silico	$I_{\mathit{US}}^{\mathit{dia}}/I_{\mathit{ex}}$	ex vivo	in silico	$D_{\mathit{US}}^{\mathit{dia}}/D_{\mathit{ex}}$
ex vivo	1	0.007	0.29	1	0.08	0.79
in silico	0.007	1	0.43	0.08	1	0.20

$D_{\mathit{US}}^{\mathit{dia}}$ is the diastolic diameter of the aortic root and $I_{\mathit{US}}^{\mathit{dia}}$ is the diastolic root-to-first-branch length measured from ultrasound image; ${\mathit{D}}_{\mathit{ex}}$ is the unloaded outer diameter and ${\mathit{I}}_{\mathit{ex}}$ is the unloaded axial length determined from ex vivo measurement.

Neo-Hookean vs four-fiber family model

Following implementation of the inverse FE approach, the error between predicted and ultrasound-measured radii over the sample geometry was calculated (Figure 3(A)). Note that the aortic valve annulus, the aortic arch branches, and the descending aorta have zero radius error as their displacement is prescribed from 4DUS tracking. The average radius error across all studied cases using the neo-Hookean model was 19.1 ± 6.8% which was a higher than the 13.1±4.8% found when using the four-fiber family model (Figure 3(B)). Despite an overall reduction in error for the four-fiber family model, a significant difference between the radius error predicted by the two material models was not observed ($p>0.05$).

Figure 3.

Comparison of radius values between ultrasound and inverse modeling. (A) Representative distribution of predicted radius error over the ascending thoracic aorta (case 4) using either a four-fiber family $\left({\mathrm{W}}_{\mathrm{4fib}}\right)$ or neo-Hookean $\left({\mathrm{W}}_{\mathrm{neo}}\right)$ material model. (B) Surface-averaged radius error between 4DUS and inverse FE model prediction using either four-fiber family or neo-Hookean material model for all studied cases; red horizontal line indicates median values. Radius error was averaged across the diastole-to-systole radius change (see Eq. (12)).

For ease of comparison across different cases since ${\sigma }_{\theta \theta }$ is a function of both ${\lambda }_z$ and ${\lambda }_{\theta }$ (Figure 4(A)), equibiaxial stress-stretch curves for all samples are in Supplementary Figure 2. Generally, the four-fiber family model behavior was more correlated with measured equibiaxial stress-stretch responses than the neo-Hookean model across a larger range of stretch values (Figure 4(B)). However, at low stretch $\left({\lambda }_z={\lambda }_{\theta }=1.25\right)$, the neo-Hookean model better approximated the ex vivo circumferential tangent modulus $\left(E_{\theta \theta }\right)$ than the four-fiber family model (Figure 4(C)). As stretch was increased, however, the neo-Hookean model-predicted $E_{\theta \theta }$ remained constant and did not capture the exponential increase in stiffness at large stretch (Baek et al. 2007). In contrast, $E_{\theta \theta }$ was increased with the four-fiber family model, albeit to a lesser extent than the ex vivo measurements. Regardless of the material model, both inverse FE approaches underestimated the simulated equibiaxial stress-stretch behavior from ex vivo mechanical testing, similar to reports from other studies (Wang et al. 2022). For each material model, simulated non-equibiaxial stress-stretch curves (Supplementary Figure 3, 4) showed similar trends as equibiaxial loading (as in Figure 4). Using the identified material parameters for each material model, pressure-diameter curves were also simulated and compared against ex vivo fits to mechanical testing data (Supplementary Figure 5, 6). Note that the four-fiber family model showed similar concave-up pressure-diameter behavior as the ex vivo data, while the neo-Hookean model predicted an inconsistent concave-down response to pressurization.

Figure 4.

Comparison of simulated equibiaxial stress-stretch behavior from fitting of ex vivo data and inverse FE models. (A) Representative simulated equibiaxial circumferential stress-stretch curve using material parameters obtained from ex vivo mechanical testing (open circles), four-fiber model inverse FE (dashed line) and neo-Hookean inverse FE modeling (red line). Additional cases shown in Supplementary Figure 2. (B) Square of Pearson’s correlation coefficient $\left(r^2\right)$ between equibiaxial stress-stretch curves obtained from ex vivo testing and inverse FE estimation using either neo-Hookean (Neo) or four-fiber family (4Fib) material models. Note the high $r^2$ values at low stretch values (< 1.5), relative to high stretch (< 2.0). (C) Circumferential tangent modulus $\left({\mathrm{E}}_{\theta \theta }\right)$ calculated from simulated equibiaxial stress-stretch curves at different levels of equibiaxial stretch ranging from 1.25 (left) to 1.50 (middle) to 1.75 (right). ^* p < 0.05, ^** p < 0.01 from a two-tailed paired Student’s t-test (B) or 1 way ANOVA with posthoc Tukey’s test (C). Red line denotes the median.

Sensitivity analysis

The neo-Hookean parameter ($E$) was found to be most sensitive to the choice of axial pre-stretch (Figure 5(A)). It was also sensitive, albeit to a lesser extent, to the choice of unloaded wall thickness (Figure 5(B)). Combined, these two sensitivity analyses suggest the inverse FE modeling results depend upon the sample’s unloaded geometry, which is difficult, if not impossible, to obtain from in vivo images alone. Stiffness estimates of the aorta were largely insensitive to the stiffness of surrounding tissues (Figure 5(C)), as shown by minimal variations in $E$ across several orders of magnitude of $k_{Rb}$ (on the order of 0.2% reduction). For large $k_{Rb}$ values, stiffness of the aorta was irrelevant since the intraluminal pressure was resisted by the surrounding tissue and not the aorta. The small effect of $k_{Rb}$ is consistent with a previous study (Shidhore et al. 2022), where the aortic diameter changed by less than 10% with varying $k_{Rb}$. Consistent with our findings in this study, a similar inverse FE method was also found be more sensitive to vessel thickness and less sensitive to $k_{Rb}$ (Latorre Molins et al. 2025).

Since the exponent for the diagonal fibers ${\mathrm{c}}_2^{(\mathrm{3,4})}$ and the fiber angle ${\alpha }_0^{(3)}$ were fixed to constrain the inverse FE optimization, we performed a sensitivity analysis comparing the predicted circumferential stress after $a+10\mathrm{\%}$ perturbation of each material parameter (Figure 6(A)). We also calculated the corresponding percent change in circumferential stress compared to baseline stress predictions for Case 4 (Figure 6(B)). We found that the sensitivity of the inverse method to the fiber angle ${\alpha }_0^{(3)}$ was small and decreased (%|change| ≈ 7%) at larger stretches. Meanwhile, sensitivity to ${\mathrm{c}}_2^{(\mathrm{3,4})}$ was more pronounced with larger changes in stress at higher stretches (%∣change∣ ≈ 31.8% at ${\lambda }_{\mathrm{z}},{\lambda }_{\theta }=2$). Together, this suggests the inverse FE method was largely insensitive to the diagonal fiber orientation but was most sensitive to the diagonal fiber exponent.

The sensitivity of the inverse FE method was tested for both neo-Hookean and four-fiber family material models using a representative sample (case 4) and running multiple inverse FE identifications (20 times per model) with randomized initial guesses between 0.5-and-2.0 times the initial parameter values. Representative circumferential stress-stretch curves show results averaged across multiple inverse parameter identifications (Figure 7). After 20 independent parameter identifications, the mean radius fitting error (see err in Eq. (12)) for the neo-Hookean and four-fiber family models appear to be consistent (21±0.00056% and 13.9±0.173% respectively). This suggests our constrained inverse FE approach converged to a consistent minimum with each randomized initial set of parameter values. However, the variation in this minima differs between the two methods: the standard deviation of the identified neo-Hookean material property ($E$) was 0.053 kPa (0.08% of its mean), whereas the standard deviation in the identified parameters of the four-fiber family model (calculated as the square root of the first eigenvalue of the material parameter covariance matrix) was 8.29 kPa (157% of its mean). Thus, despite a consistent minimum radius error, the four-fiber family model exhibits greater variability in the minima (e.g. identified parameters) than the neo-Hookean model.

Discussion

In this work, we present a method for inversely determining material properties of the mouse ascending thoracic aorta based on 4DUS imaging. While the method is, in principle, applicable to larger animals and humans – including fetal subjects not generally suited for other imaging modalities – our focus herein was on model development and evaluation of the method’s performance based on comparisons with paired ex vivo mechanical testing. As such, there were three important outcomes from this study. First, we generated a complete set of high-resolution 4DUS images and ex vivo mechanical testing data on thoracic aortas from the same mice (see Supplementary Data, DOI:10.21227/ey9t-vm54). This data set was essential for our study and can serve as a benchmark for others developing ultrasound-based in vivo material characterization schemes. Second, we found that the proposed inverse method must be constrained to give consistent and accurate results, and that the choice of the constraint imposed (e.g. unloaded geometry and exponential material parameters) can affect the outcome considerably (see Supplementary Figure 9). Third, we found that estimation of the biaxial in vivo pre-stretch is both important for material property identification and difficult to ascertain from 4DUS images alone, thus making it a key metric that needs to be considered in future studies.

Paired comparisons between in vivo and ex vivo material property estimations in the same tissues is an important aspect of the current study. Although our study treated ex vivo mechanical testing data as the ground truth for evaluating predicted in vivo measurements, we recognize that in vivo and ex vivo mechanical behaviors might be different from one another. Rather than being truly identical, it is possible that challenges inherent to the ex vivo measurement technique may influence the ground truth material property estimation. For example, the dissection and handling required for mechanical testing may alter ex vivo tissue properties, the ex vivo testing was performed in the absence of vascular tone, and the chemical environment was not identical across the two conditions (in vivo vs. ex vivo). Nevertheless, the ex vivo mechanical testing, by allowing for precise control over multiple different testing regimens, provides a much more reliable measure of tissue properties and is thus well suited to serve as the ground truth for any validation of predicted in vivo material property estimates. While in vivo and ex vivo material property estimations from the same tissue have been performed previously, such studies are uncommon (Wang et al. 2022). For example, several studies have been performed on tissue samples acquired from patients undergoing aortic repair surgeries. Liu et al. (2019) performed inverse FE analysis on two human ascending aortic aneurysms and compared the parameter values obtained from minimizing the nodal displacement errors in the aortic geometry between diastole and systole against ex vivo planar biaxial testing. The computed material parameters were different between the in vivo and ex vivo analyses but exhibited similar stresses at small stretches ($\lambda =1.0-1.2$). In another study, Cosentino et al. (2019) compared in vivo and ex vivo responses in 9 patient samples and found a good fit at similar stretch values ($\lambda =1.0-1.2$). Finally, Trabelsi et al. (2016) performed inverse FE analysis on 5 patient samples using a regression analysis of material parameters against the systolic, mid-systolic, and diastolic volumes of the aortic wall. Again, stress-stretch curves comparing ex vivo (experimental) and in vivo (inverse FE) results matched well over small stretches $(\lambda =1.0-1.5)$ but deviated at large stretches $(\lambda >1.5)$. In these previous studies, the range of stretches for comparison was small; however, after accounting for biaxial tissue pre-stretch, the in vivo stretches were likely much larger. Moreover, in these studies, pre-stretches used in the inverse FE were not compared against paired ex vivo measurements, which we have shown is an important parameter in the in vivo estimation procedure. Direct comparisons of our results with findings from other studies are difficult due to differences in material models, vessels/animal models, expected pre-stretches, and image quality.

Similar to previous reports, predicted stresses at low equibiaxial stretch (${\lambda }_{\theta }={\lambda }_z=1.0-1.3$) were in agreement with fitted ex vivo data, with the neo-Hookean model fitting better than the four-fiber family model in our inverse FE approach (Figure 4(C) and Supplementary Figure 2). However, inclusion of pre-stretch in the model pushes the in vivo state to the exponential portion of the stress-stretch curve, thus favoring the four-fiber family model behavior (Figure 4(C)). Under certain circumstances, the neo-Hookean model has distinct advantages over the four-fiber family model given the need for fewer model parameters and simplifying assumptions during the fitting process. Thus, if material behavior at small deformations is of interest, the neo-Hookean model is sufficient and preferred; otherwise, the four-fiber family model better describes in vivo aortic behavior under large strains. Other fiber-based constitutive models with an exponential stored energy function (Gasser et al. 2006) would likely also perform well but were not tested in this study. While a non-linear mechanical response is not a novel concept in the broader context of aortic wall mechanics, we emphasize here its particular importance when inferring material properties from in vivo imaging.

In this study, we observed a consistent trend of underestimation in the aortic wall stiffness identified from the inverse FE method as compared to that obtained from ex vivo mechanical testing; this general underestimation is commonly found in literature (Wang et al. 2022). While we are unable to fully explain the discrepancy between in vivo and ex vivo stiffness estimates here, we can offer some potential contributing factors. Fundamentally, the material property estimation between in vivo and ex vivo measures are made under different assumptions and boundary conditions. Given the lack of surrounding perivascular tissue support, intra-thoracic external pressures (Ferruzzi et al. 2018), and active smooth muscle tone (Ferruzzi et al. 2013), as well as the assumptions of quasistatic loading, no spatial variations in properties, and approximation of the ascending aorta as a straight tube despite being short and curved, it is understandable that ex vivo mechanical testing will not exactly replicate in vivo loading conditions. However, the ex vivo testing method follows a common and repeatable protocol (Macrae et al. 2016; Lane et al. 2024) that provides an objective description of the tissue we chose to treat as our gold standard measurement.

All inverse FE analysis was performed with assumed boundary conditions, unloaded configuration, fixed material parameters, and uniform axial pre-stretching for each mouse. For several of these assumptions, we performed sensitivity analyses to understand their impact on material property estimation. However, we found that differences between in vivo and ex vivo stiffness estimates were driven by uncertainty in the estimated in vivo axial stretch. Namely, material property estimation was largely independent of the magnitude of the Robin boundary condition applied at the outer wall which served to model external tissue support from perivascular tissue and intrathoracic pressures (Figure 5(C) (Shidhore et al. 2022)). While not explicitly modeled herein, we expect active responses from vascular smooth muscle contractile tone to increase the apparent in vivo stiffness of the aorta which would not explain the stiffness underestimation by our models. Similar to our observations of strong model dependence on aortic wall pre-stretch, others report changes in distensibility from in vivo to in situ to ex vivo conditions that are driven by changes in vessel geometry (e.g. outer diameter, thickness, and axial pre-stretch) (Zanchi et al. 1998; Wang et al. 2017).

The biggest challenge (and therefore the biggest limitation) for the inverse FE approach implemented here stems from the assumptions required to constrain the inverse parameter identification. In general, the information provided from 4DUS imaging is comprehensive, albeit limited. It could be argued that in vivo deformation measurements provide large amounts of data for material parameter estimation. Unfortunately, we found the ascending aortic deformations measured over the cardiac cycle to be relatively small in magnitude and spatially similar to one another. For example, the average diastolic-to-systolic circumferential stretch (${\lambda }_{\theta ,\mathit{dia}}$) was about 1.06 and axial stretch $\left({\lambda }_{z,\mathit{dia}}\right)$ was about 1.03 across all samples (see Supplementary Table 4). Together, these factors result in strongly correlated in vivo deformation data and, as a result, the information that can be used for material property estimation is much smaller than the size of the overall data set. While there is spatial heterogeneity in aortic stiffness within the ascending aorta (Bersi et al. 2019; Farzaneh et al. 2019), this variation was not able to be identified with the inverse FE method given the resolution of the 4DUS images. Initial investigations with the inverse FE model included domain partitioning to identify regional material properties with the neo-Hookean model. However, significantly different regional material parameters were not identified, so uniform material properties were used across the entire geometry to reduce the number of fitted parameters. Of note, prior reports suggest that regional stiffness measurements within the ascending aorta are largely comparable to stiffness measurements in the entire arch (Farzaneh et al. 2019), suggesting our inverse approach and ex vivo characterization of the ascending aorta offers a sufficient representation of the segmented geometry’s material properties.

Furthermore, unlike biaxial extension-inflation testing, where various states of ${\lambda }_z$ at a single pressure can be tested, there is very little axial stretch in vivo between diastole and systole. Considering that the ascending aorta experiences the largest amount of dynamic in vivo axial stretch over the cardiac cycle (Ferruzzi et al. 2018), the lack of marked axial stretch in other regions of the central vasculature presents a challenge for extending our current inverse FE approach (Stålhand and Klarbring 2005; Gade et al. 2019). With little-to-no change in the in vivo axial stretch ${\lambda }_{z,\mathit{dia}}$, we found that our inverse FE approach was forced to extrapolate stress-stretch behaviors at different axial stretches which ultimately resulted in poor correlations with ex vivo estimates (see Table 3).

It should be noted that both the ex vivo and in vivo estimates of stretch are susceptible to error due to measurement noise and assumptions. For the circumferential stretch, accurate in vivo aortic diameter measurements can be challenging due to 4DUS image noise in addition to sensitivity to viewing angle, imaging mode, and high inter-observer variability (Trachet et al. 2015; AARDVARK Collaborators 2016). Ex vivo circumferential stretch measurements are also subject to error in vessel diameter tracking methods as well as changes in the external boundary conditions experienced by the tissue following dissection and removal of the perivascular environment within the body. A common method for estimating axial stretch ex vivo involves identification of a constant axial force-pressure relationship (Humphrey et al. 2009), which can vary from vessel-to-vessel and is a challenge to translate to in vivo applications (Ferruzzi et al. 2018). In both cases, image-based measurements of in vivo stretches suffer from the inability to accurately estimate an unloaded (traction-free) configuration in a manner that is similar to the ex vivo stretch estimations. Thus, the combination of these errors likely contribute to significant differences between ex vivo and in vivo estimations of aortic pre-stretch (see Table 1).

In general inverse FE problems are ill-posed and may have multiple valid solutions (Beghini et al. 2023). For example, herein both the unloaded configuration and the tissue material properties were unknown. Thus, it is possible to recover a similar level of deformation with a stiffer aorta and larger unloaded configuration (or lower pre-stretches). One common method to deal with an ill-posed problem is to reduce the number of degrees of freedom (Beghini et al. 2023). In the current study, we used prior withheld observations to specify values for the exponential parameters ($c_2^{(i)}$) and fiber angle (${\alpha }_0^{(3)}$) of the four-fiber family model which were within the range of ex vivo fitted parameters for the entire sample population (Supplementary Table 2). A sensitivity analysis was performed on two of the four fixed parameters – the exponentials for the circumferential and axial fibers were nearly zero, which made the fibers behave linearly thus diminishing the need for sensitivity analyses. Of the other two fixed parameters, the diagonal fiber exponent $c_2^{(\mathrm{3,4})}$ showed greater sensitivity with larger changes in predicted stress with increasing stretch, whereas the fiber angle ${\alpha }_0^{(3)}$ had little impact on stress predictions. Fixing these parameters constrained the length of the toe region of the stress-stretch curve, which also imposed constraints on ${\lambda }_{z,\mathit{pre}}$. Without these constraints, we found the obtained properties to be much stiffer than the ex vivo measured properties with estimated ${\lambda }_{z,\mathit{pre}}$ approaching a value of 1.0 (Supplementary Figure 9). The consistency of the obtained result was also evaluated by comparing results from multiple runs with randomized initial conditions (Figure 7) and ensuring that a consistent minimum was found.

For the neo-Hookean model, axial pre-stretch ${\lambda }_{z,\mathit{pre}}$ was assumed to be constant since predicted stresses were nearly independent of axial stretch (Supplementary Figure 4). Conversely, identification of a reasonable ${\lambda }_{z,\mathit{pre}}$ value is particularly important when using the four-fiber family model since circumferential stress-stretch behavior is highly dependent on ${\lambda }_z$ (Supplementary Figure 3). The importance of imposing fitting constraints and simplifying assumptions is further corroborated by our sensitivity analysis. We observed that inverse FE identification with the neo-Hookean model is most sensitive to errors in ${\lambda }_{z,\mathit{pre}}$ and unloaded thickness (Figure 5), suggesting a strong dependence on the sample's unloaded geometry. While unloaded geometries could be measured for each sample in our study, this will be a challenge for future implementations since unloaded geometry cannot be reliably measured without destructive methods (i.e. dissection) that are not an option when the goal is in vivo monitoring and material property assessment.

Several approaches to estimate in vivo aortic pre-stretches have been proposed. Many studies simply use fixed values (Wittek et al. 2013; Cai et al. 2014; Wittek et al. 2016; Wang et al. 2021), while others have parameterized the axial stretch based on aortic size, length, and body mass (Horny et al. 2011). Some have included ${\lambda }_{z,\mathit{pre}}$ as an unknown in the optimization scheme (Stålhand et al. 2004), whereas others have lumped ${\lambda }_{z,\mathit{pre}}$ and ${\lambda }_{\theta ,\mathit{pre}}$ together to iteratively estimate a pre-strained deformation tensor (Hsu and Bazilevs 2011; Vavourakis et al. 2016). Similar to previous studies (Wittek et al. 2013; Cai et al. 2014; Wittek et al. 2016; Wang et al. 2021), our approach assumed a uniform axial pre-stretch for each ascending aortic geometry, despite the fact that local axial pre-stretches may vary spatially from the inner to outer aortic curvature (Alford and Taber 2008; Laubrie et al. 2022). There have also been attempts to circumvent estimation of the unloaded geometry altogether by redefining the reference configuration as an in vivo configuration (Bersi et al. 2016; 2019). By introducing a framework considering deposition stretches of each constituent in a constrained mixture of aortic wall material (i.e. elastin, collagen, smooth muscle), others have shown that properly calibrated models are capable of reconstructing unloaded properties from loaded (deformed) observations (Bellini et al. 2014). This microstructural approach, however, introduces more unknown parameters related to aortic wall composition (i.e. deposition stretches, mass fractions, and distributed directionality of each constituent). That said, the fiber stiffness parameters were directionally weighted by volume fractions from imaging data as priors to reduce the overall number of unknown parameters for these models (Bellini et al. 2014). Here, we determined ${\lambda }_{z,\mathit{pre}}$ from the crossover point of simulated axial force-stretch curves and demonstrated this approach can reasonably estimate axial pre-stretch (see Table 1). To our knowledge, no other study has used the axial force-stretch crossover point in an inverse FE method despite its widespread use to estimate preferred axial stretches in ex vivo mechanical testing applications (Weizsäcker et al. 1983; Caulk et al. 2019; Tarraf et al. 2023).

Despite the challenge of determining unloaded geometry and pre-stretch from in vivo imaging, it may only be prohibitive when attempting to compare sample-to-sample variation within an experimental group. It is possible that under certain conditions, such as repeated monitoring in humans, the change in properties over time is more important than any specific property at one point in time. In other words, time-dependent changes within a single sample are likely more relevant for the prediction of aortic failure than comparisons between different samples. If so, using a standard set of appropriate priors may be sufficient for constraining the inverse FE to allow for identification of meaningful longitudinal changes, even if there is systematic error in the estimated properties at each timepoint. Some conditions, such as thoracic aortic aneurysm, may produce substantial biomechanical changes over time that render any estimation errors insignificant. Furthermore, additional non-invasive information – such as identification of calcified plaque (Yang et al. 2019) or pulse wave velocity measurements (Milan et al. 2019) – could complement the ultrasound imaging and help to further constrain the inverse problem by defining a reasonable search space for material parameters and biaxial pre-stretches. Indeed, combining multiple paired datasets across mice could be used to construct prior distributions for Bayesian statistical analysis and predictive modeling approaches that could be leveraged to refine the inverse FE approach.

The influence of many unknowns in any in vivo inverse FE estimation procedure (i.e. geometry, material parameters, hemodynamics, etc.) highlights the importance of imposing constraints on the system. Otherwise, due to the non-linear constitutive response, there are many different sets of material parameters that could return the same observed deformation. This problem of under-determinacy is aggravated by the need for large amounts of invasive ex vivo testing data to inform priors in the inverse FE method (e.g. constraints on material parameters and ${\lambda }_{z,\mathit{pre}}$). Therefore, there is a need to validate non-invasive estimations of tissue material properties against ex vivo testing and experimental methods, when possible. Importantly, these priors are likely tissue specific and might not be generalizable to different animal models, disease, and ages.

To summarize, we have shown that our inverse FE approach can estimate in vivo mechanical properties of the aorta when combined with 4DUS. While the accuracy of material property identification is strongly dependent on the unloaded geometry and imposed parameter constraints, we show that the neo-Hookean constitutive model outperforms fiber-based models when estimating material properties at low stretches and the four-fiber family model is better-suited for high stretch values (large deformation). Although identification of appropriate inverse FE modeling constraints for each sample can be a challenge, this work represents a significant step toward patient-specific modeling of aneurysm growth and remodeling. We also have provided the raw data and numerical analysis codes used to produce results from this study (DOI:10.21227/ey9t-vm54) in order to provide a benchmark for future studies focused on inverse FE model development.

Supplementary Material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/10255842.2025.2537332.