In vivo strain assessment of the abdominal aortic aneurysm

Alessandro Satriano a, Simone Rivolo b, Giampaolo Martufi c, Ender A. Finol d,Elena S. Di Martino e,n

a Graduate Program in Biomedical Engineering, University of Calgary, 2500 University Dr. NW, Calgary, AB, Canada T2N 1N4b Department of Biomedical Engineering, Division of Imaging Sciences and Biomedical Engineering, King's College London, King's Health Partners, St. Thomas'Hospital, London SE1 7EH, United Kingdomc Department of Civil Engineering, University of Calgary, 2500 University Dr. NW, Calgary, AB, Canada T2N 1N4d Department of Biomedical Engineering, The University of Texas at San Antonio, 1 UTSA Cir, San Antonio, TX 78249, USAe Department of Civil Engineering and Centre for Bioengineering Research and Education, University of Calgary, 2500 University Dr. NW, Calgary, AB, CanadaT2N 1N4

a r t i c l e i n f o

Article history:Accepted 11 November 2014

Keywords:Abdominal aortic aneurysmImagingDeformation maps

a b s t r a c t

The only criteria currently used to inform surgical decision for abdominal aortic aneurysms aremaximum diameter (45:5 cm) and rate of growth, even though several studies have identified theneed for more specific indicators of risk. Patient-specific biomechanical variables likely to affect rupturerisk would be a valuable addition to the science of understanding rupture risk and prove to be a lifesaving benefit for patients. Local deformability of the aorta is related to the local mechanical propertiesof the wall and may provide indication on the state of weakening of the wall tissue. We propose a 3Dimage-based approach to compute aortic wall strain maps in vivo. The method is applicable to a varietyof imaging modalities that provide sequential images at different phases in the cardiac cycle. We appliedthe method to a series of abdominal aneurysms imaged using cine-MRI obtaining strain maps atdifferent phases in the cardiac cycle. These maps could be used to evaluate the distensibility of ananeurysm at baseline and at different follow-up times and provide an additional index to clinicians tofacilitate decisions on the best course of action for a specific patient.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In clinical practice, surgery is advised to patients presenting withabdominal aortic aneurysms (AAAs) exceeding 5.5 cm in maximumdiameter (Brown et al., 1999). However, there is evidence thataneurysms smaller than this threshold can and do rupture (Scott,2002; Lederle et al., 2002; Powell and Brady, 2004). Conversely, somelarger aneurysms do not rupture during the lifespan of the patient. Ananeurysm ruptures because the local strength of the wall is insufficientto sustain the instantaneous load due to blood pressure. Local wallweakening has been reported and can be the result of an imbalancebetween production of new constituents and degradation of extra-cellular matrix (Hellenthal et al., 2009) in response to a stress-mediated mechanism that attempts to maintain homeostasis, or theoptimal wall thickness and composition (Choke et al., 2005). Therefore,a general indication based on diameter measurements alone may notbe sufficient to discriminate whether AAAs are close to rupture.Instead, individually tailored rupture indices are needed that take into

account the local properties of the wall, their evolution in time as wellas the regional stress level.

Wilson et al. (2003) measured the distensibility of the aorta in apopulation of AAAs not considered for repair due to small aneurysmsize, comorbidities or patient unwillingness. Distensibility wasmeasured at different follow-ups as the increment in diameter inresponse to the pulse pressure (diastolic minus systolic pressure).Aneurysms that ruptured during follow-up were characterized byhigher diastolic pressure and larger diameter, and their distensi-bility at baseline was higher; in addition, changes in distensibilitywere correlated with the risk of rupture. Di Martino et al. (2006)performed tensile tests on samples of aortic wall excised fromelectively repaired and ruptured AAAs, and showed that maximumtissue stiffness was inversely correlated with wall strength, suggest-ing lower stiffness as a possible predictor of AAA rupture. Inaddition, the study showed that arterial wall stiffness is nothomogeneous in the presence of an aneurysm, supporting thehypothesis that the wall remodels locally.

These results by Wilson et al. (2003) and Di Martino et al.(2006) suggest that a new predictor of AAA rupture could combinemaximum diameter and growth rate with measures of local tissuedistensibility in vivo.

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journal homepage: www.elsevier.com/locate/jbiomechwww.JBiomech.com

Journal of Biomechanics

http://dx.doi.org/10.1016/j.jbiomech.2014.11.0160021-9290/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ1 403 220 4442; fax: þ1 403 282 7026.E-mail addresses: edimarti@me.com, edimarti@ucalgary.ca (E.S. Di Martino).

Journal of Biomechanics 48 (2015) 354–360

Algorithms to measure wall strains for biological structureshave been previously proposed. Namely, the speckle trackingtechnique takes advantage of the ability of echography to detectthe velocity field in the scanned volume, and has been used toassess ventricular function (Helle-Valle et al., 2005; Edvardsenet al., 2006; Kim et al., 2007), as well as arterial strains (Estensenet al., 2013; Yang et al., 2011). Limitations related to this approachare the exclusiveness of the imaging technique from which datacan be obtained, and the strong influence of the orientation of thehand-held scanning probe on the results. Veress et al. proposedhyperelastic warping to quantify kinematics using local imageintensity changes to compute forces to be applied to the recon-structed geometry to register the images at different timestepsand compute displacements (Veress et al., 2005a,b; Phatak et al.,2009). This approach is advantageous when the elastic behavior ofthe vessel wall is the only contribution to the deformation.Conversely, in an AAA, the presence of intraluminal thrombus(ILT) and surrounding structures creates a complex non-linearsystem. Furthermore, the hyperelastic warping method requiresthe definition of a constitutive model, which may influence theoutcome. MRI tagging has been adopted to measure strains onthick tissues. However, the method has not been validated ontissues thinner than the cardiac ventricular chambers and thetechnique provides strains only on cross-sections of the structure.

Drawing inspiration from motion analysis methods used incomputer vision, we propose a 3-D image-based approach tomeasure deformations in vivo that is applicable to a variety ofbiological structures and across multiple tomographic imagingtechniques. Optical flow has been used for several years tocompute motion fields from sequential images. The basic tenetof the method is that the brightness intensity of a moving objectremains constant between two images separated by a smallinterval of time. This is used to approximate the velocity at eachpixel and to compute a displacement field. The method was firstintroduced for 2D images by Horn and Schunck (1981). Theyproposed a global method to compute displacements that mini-mize the differences between the image at time 0 and the image attime t. The algorithm uses regularity constraints to yield smoothvelocity fields and make the problem tractable. Several variationsof the original method have been proposed, including methodsthat work on individual sub-regions of the image (Lucas et al.,1981) and methods that use multi-scale strategies to improvetracking for larger displacement fields (Brox et al., 2009).

As a first step, we extended the original Horn and Schunkmethod to 3D image fields and tested its validity in the case of AAAwall strain. We computed deformation directly from the opticalflow of time-wise consecutive images, without performing a finiteelement simulation so that no constitutive model for the wall orILT is required. Instead, the deformation comes directly from thewall motion and is due to the combined effect of the pulsatileblood pressure, the properties and local thickness of the wall andILT, and the surrounding structures.

We validated our approach comparing the first principal Green–Lagrange strains obtained from a finite element simulation of a AAAgeometry under cyclic pressure and the corresponding strainsobtained from synthetic DICOM stacks generated from the meshescorresponding to 20 phases of the simulation.

Finally, as a demonstrative application, we applied the techni-que to nine cases of AAA dynamically imaged over the cardiaccycle through cine-MRI. We obtained maps of local deformationundergone by each AAA throughout the cardiac cycle. These mapsmay be used to evaluate the distensibility of an aneurysm atbaseline when the aneurysm is diagnosed and at different follow-up times and could be used, in combination with clinically adoptedindices, to inform the clinician on the best course of action for aspecific aneurysm.

2. Materials and methods

Cine-MRI stacking: Cine-MRI scans of nine human AAA patients were obtainedfrom the Division of Vascular Surgery at the Allegheny General Hospital, Pittsburgh,PA, following a protocol approved by the ethical commission. All images wereobtained on a General Electric 1.5 T imager (Signa Excite, GE Medical Systems).Repetition time/echo time/flip angle of 3.4 ms/1.5 ms/451 were adopted, with a 75%k-space acquisition, 256�256 matrix, 6-mm slice thickness, no gap.

The images consist of 20-phase ECG gated DICOM series taken at n levels acrossthe AAA, each describing the in-plane evolution of a cross-section of the imagedaorta at 20 time points throughout the cardiac cycle. For the images under study, nvaries from patient to patient, ranging from 24 to 31. The spatial resolution achieved,with 1.4 mm in the plane and 6.0 mm between planes, is highly anisotropic.

Initial aortic mesh: In order to identify and segment the AAA geometry across allslices for the same phase, we adopted an active-contour based segmentationapproach, taking advantage of the Interactive Toolkit (ITK) pipeline (Yushkevichet al., 2006; Gobbi et al., 2008). At first, we selected a region of interest (ROI)containing the AAA within the DICOM stack corresponding to Phase 0, in order toreduce the size of the 3-dimensional data set, and we resampled the ROI to a 2 mmisotropic resolution, by means of cubic interpolation. The initial active contour wasdefined by means of a spherical balloon inside the AAA that could expand in avolume identified by a lower and an upper grayscale threshold. The balloon contourevolution was dictated by weighting parameters according to the approach byCaselles et al. (1993).

The isosurface comprising the segmented regionwas exported as an stl triangularmesh. This initial segmentation was smoothed using a Taubin filter (Taubin, 1995) toobtain the desired smoothing and minimize the shrinkage of the structure. Finally,the mesh was simplified using a Quadric Edge Collapse Decimation (Garland andHeckbert, 1997) to reduce the number of shells to approximately 3000.

In vivo velocity field detection: The displacement of each node of the initial meshfrom phase n to phase nþ1 was computed approximating nodal velocities using a3D extension of the optical flow method (Horn and Schunck, 1981; Fleet and Weiss,2006). This method follows the displacement of an object between images taken atsubsequent time steps by detecting the grayscale feature corresponding to theobject and computing its velocity. The basic assumption of the method is that thebrightness intensity of an object remains unaltered across two successive images ifthe displacement is relatively small. The second assumption is that there is asmooth change of the intensity values between two consecutive phases.

The algorithm, described in detail in Appendix A, computes the velocity fieldfor each voxel in the DICOM stack using a minimization function. The velocitiesobtained between stack n and nþ1 were added to the nth nodal position of theaortic surface mesh and multiplied by dt¼1 in order to obtain the ðnþ1Þth nodalposition, describing the geometrical configuration of the aorta at timestep nþ1.

Computation of deformation: Given the deformation gradient F at point X in thereference configuration, the push-forward of a material vector V attached at X is thespatial vector v attached at point x (the current placement of X) in the currentconfiguration) and defined as

v¼ FV: ð1Þ

After defining point X as node 1 of each triangular shell element, we defined avector A1 leading from node 1 to 2, a vector A2 leading from node 1 to 3, and avector A3 defined by the normal to the element in its reference configuration. Thecorresponding vectors ai in the current configuration are obtained as

FiI ½Ak�I ¼ ½ak�i ð2Þ

with k¼ 1;2;3, ½Ak�I the Ith component of Ak , ½ak�i the ith component of ak and FiIthe iIth component of F. While A3 was imposed to have magnitude 1, a3 wasimposed to have a magnitude such to ensure incompressibility of the wall tissue. Asall three material Ak and spatial ak vectors are known, a system of equations can besolved to find all components of F.

From F, the right Cauchy–Green deformation tensor C is computed as C¼ FTF,where FT is the transposed matrix of F. The Green–Lagrange strain E can thereforebe computed as E¼ ðC�IÞ=2, with principal strains computed as its eigenvalues.

Validation: To validate our approach, we used a Computed Tomography scan ofa patient's aneurysm obtained following a protocol approved by the ethics board atthe University of Calgary. A mesh of the abdominal aneurysm wall was obtainedusing VASCOPS (VASCOPS Gmbh, Graz, Austria). A finite element simulation wasperformed on the aneurysm model using Abaqus (vs. 6.13, Dassault Systemes,Waltham, MA). The simulation consisted of a loading cycle to 80 mmHg andsymmetric unloading cycle. The upper and lower ends of the aneurysm wereconstrained in all three directions. Non-linear material properties were used,following the strain energy function proposed by Raghavan and Vorp (2000) ofthe form W

W ¼ aðI1C�3ÞþbðI1C�3Þ2 ð3Þ

where a and b are the material parameters, I1C is the first invariant of the rightCauchy–Green tensor. The parameters used, i.e. a¼113.4 kPa and b¼9.2 kPa, wereobtained from non-linear regression using uniaxial tensile tests on a polymericmaterial that exhibited a stiffness similar to that of a human aorta. The maximum

A. Satriano et al. / Journal of Biomechanics 48 (2015) 354–360 355

pressure of 80 mmHg was chosen to obtain a wide range of deformation, from nodeformation to values close to the deformation in a healthy aorta. The first principalGreen–Lagrange strains computed at the different phases with respect to phase1 were recorded and used as ground truth in the subsequent comparison. The meshat different phases from the simulation was exported in stl format, imported inSimpleware's þCAD Module (Simpleware Ltd., Exeter, United Kingdom) and usedto generate synthetic DICOM image stacks at different resolutions: 0.5 mm, 1.0 mm,1.5 mm isotropic resolutions, and 1.5 mm in-plane, 6.00 mm longitudinal resolu-tion, in analogy to the resolution available from the MRI scans. Fig. 1 shows onesection across a synthetic DICOM stack. The strain maps obtained from thesynthetic DICOM images were compared with the corresponding ground truthusing a Spearman correlation coefficient ρ. To assess the effect of noise, differentlevels of Rician noise were added to the 1.5 mm synthetic isotropic-resolutionDICOM stack corresponding to signal-to-noise ratio (SNR) levels ranging fromaround 40 to 14 dB. The effect of noise was assessed using a Spearman correlationcoefficient.

3. Results

Validation: Fig. 2 plots cross-sections of the synthetic aneurysmshowing the nodes of the mesh being tracked by the algorithm atdifferent phases (using 1.5 resolution and SNR of 42.3 dB). Table 1reports Spearman's ρ correlation coefficients calculated betweenground truth and the results obtained from the synthetic DICOMsfor each of the resolutions studied.

Table 2 reports the results obtained at 1.5-mm isotropic resolu-tion without noise and with different SNR levels.

Fig. 3 shows maps of the computed vs. ground truth displace-ments and the computed vs. ground truth strain for Phase 8 for theDICOM stack with 1.5 mm/pixel resolution. For the same resolution(1.5 mm/pixel), Fig. 4 shows medians and spread of the first prin-cipal Green–Lagrange strain for the ground truth (upper panel) andthe synthetic DICOM stacks for all the 20 phases studied.

Patient-specific AAAs: Fig. 5 plots the first principal Green–Lagrange strain distribution computed at the systolic peak ofpressure for each aneurysm analyzed. In addition to regionscorresponding to healthier walls (neck, iliac arteries), highervalues of strain were found at the junction between the neckand the bulge of the aneurysm. Median strains were computed foreach patients and are shown in Fig. 6 for each instant in thecardiac cycle. The box plots in Fig. 7 compare the median and thespread for each aneurysm.

The average maximum principal Green–Lagrange strain com-puted was 0.07, with a standard deviation of 0.02. Table 3 reports

Fig. 1. Exemplary cross-section of a synthetic DICOM data set at 0.5 mm isotropicresolution.

Fig. 2. Transversal sections of the synthetic DICOM stack for 1.5 resolution and SNRof 42.3 dB showing the nodes of the mesh after application of the deformationalgorithm: ED1 corresponds to one slice during the first phase, ES corresponds tothe mid phase, and ED2 is the last phase of the simulation.

Table 1Spearman's ρ correlation coefficients of the first principal Green–Lagrange straincomputed from synthetic DICOM stacks to the corresponding ground truthmeasures obtained from the simulation.

Resolution Correlation to ground truth p value

0.5 mm isotropic 0.81 po0:011.0 mm isotropic 0.82 po0:011.5 mm isotropic 0.84 po0:01

1.5 mm orthotropic 0.76 po0:01

Table 2Spearman's ρ correlation coefficients of the first principal Green–Lagrange straincomputed from 1.5-mm isotropic resolution synthetic images at different SNRlevels versus the results obtained with the noise-free DICOM stack at the sameresolution.

SNR (dB) Correlation to noise-free p value

42.30 0.95 po0:0130.18 0.93 po0:0123.12 0.90 po0:0118.16 0.90 po0:0114.26 0.79 po0:01

A. Satriano et al. / Journal of Biomechanics 48 (2015) 354–360356

maximum principal strain, maximum diameter of the aneurysm,age and gender for each patient studied.

4. Discussion

We developed a software technique that allows the user toperform the following steps: (1) reconstruct and mesh a 3D model

of the aneurysm at phase 0; (2) compute nodal displacements ateach subsequent phase using an optical flow technique; (3) createthree-dimensional maps of local in vivo strains for the aneurysmwall under study.

From the validation step, we concluded that the algorithm com-putes strains on a AAA geometry with strong correlation betweenmeasured and theoretical (ground truth) values. The displacementswere computed with high accuracy (errors lower than 2% of the

Fig. 3. Upper panel: absolute nodal displacements obtained from the simulation (a) and from the deformation mapping algorithm (b). Lower panel: first principal Green–Lagrange strain obtained from the simulation (c) and from the deformation mapping algorithm (d).

Fig. 4. Boxplots of the first principal Green–Lagrange strain from the simulation (upper panel) and the results obtained from our algorithm using the correspondingsynthetic DICOM stack (lower panel).

A. Satriano et al. / Journal of Biomechanics 48 (2015) 354–360 357

ground truth values). The strains computed from the algorithm appearto underestimate ground truth at the phases corresponding to higherdeformation, however the strain patterns were captured correctly forall phases at all the resolutions considered. In assessing these results itmust be considered that the validation was obtained on artificialimages that presented very little variability in brightness (see Fig. 1).Real tomographic images have a much greater variability due toheterogeneous tissue properties, providing more contrasted featuresthat can be tracked by the algorithm. We are currently assessing other

validation approaches that could allow the use of real images, such asthe use of in vivo sonomicrometry markers.

Comparing the results from noise-free stacks with those obtainedapplying varying levels of Rician noise (Gudbjartsson and Patz, 1995),we conclude that the technique is robust to noise for a wide range ofSNR levels.

A strong correlation was obtained between measured and truefirst principal Green–Lagrange strains at all the resolutions con-sidered. However, a decrease in Spearman's ρ correlation coeffi-cient can be observed in the case of anisotropic voxel resolution,due to the lower spatial resolution in the z-direction affecting thetracking of longitudinal motion.

We applied the technique to nine patients undergoing surveil-lance for AAA at the Allegheny General Hospital in Pittsburgh, PA.Strains are highly heterogeneous within each aneurysm as well asacross patients. The average maximum first principal Green–Lagrange strain is consistent with results by Imura et al. (1986)who measured 2D strain from ultrasound. The unique feature of ouralgorithm is the ability to depict the spatial distribution of defor-mation in vivo, which depends on the presence of surroundingstructures, on the complex fluid dynamics inside the lumen and onthe local wall mechanical properties (Raghavan et al., 2006). One of

Fig. 5. Colormaps of the first principal Green–Lagrange strain measured on the AAA wall for each patient at peak systolic pressure.

Fig. 6. Median values of the first principal Green–Lagrange strain obtainedthroughout the cardiac cycle for each of the studied aneurysms.

A. Satriano et al. / Journal of Biomechanics 48 (2015) 354–360358

the advantages is that the displacement of the wall behind thick orthin layers of thrombus can be obtained without requiring thedirect simulation of ILT, thus limiting assumptions on ILT propertiesor its mechanical role (Thubrikar et al., 2003). For all the AAAsinvestigated we found higher values of strains at the junctionbetween the neck and the bulge of the aneurysm (Fig. 5) wherehealthier vascular wall tissue transitions to aneurysm tissue.

Local weakening of the wall has been recognized to play animportant role in the assessment of rupture risk for AAAs (VandeGeest et al., 2006; Auer and Gasser, 2010). Deformability as well asits changes through time are correlated to the degree of weaken-ing of the wall (Wilson et al., 1998; Di Martino et al., 2006). Ourmethod allows for a local in vivo assessment of deformation whereone phase of the cardiac cycle is used as a reference. Consequently,a map of maximum strain values could be computed at baseline,and then at each of the patient's follow-ups, making it possible totrack global and local changes in distensibility.

There are several limitations to be acknowledged in thispreliminary work. The Horn–Schunk (Horn and Schunck, 1981)optical flow approach is a global gradient-based approach. In otherwords, to solve an underdetermined linear system, the methodtakes advantage of a global constraint (the spatial coherenceparameter) that forces the partial derivatives of neighboringmotion vectors to be small (Barron et al., 1994; Senst et al.,2012). We deemed this acceptable due to the relatively smalldeformation of the AAA wall. Furthermore, we chose not to adopta local gradient approach, such as the one by Lucas–Kanade (Lucaset al., 1981) that may elicit sharp discontinuities in disagreementwith our hypothesis of smooth continuum motion. An interestingcategory of optical flow methods uses a multi-scale pyramidal

approach that improves the results in the case of large deforma-tions (Brox et al., 2009). Given the good temporal resolutionachieved with cine-MR and the limited motion of the aortic wall,small displacements occur between any nth to ðnþ1Þth phase. Inaccordance, we chose a Horn–Schunk non-pyramidal approach.We assumed that objects (or wall features) maintain constantbrightness across subsequent images. This is consistent with thesequence used for the image acquisition: contrast and brightnessare mainly determined by tissue properties that do not changethrough the cardiac cycle (Bieri and Scheffler, 2013). The method isresilient to the presence of noise when synthetic Rician noise(Gudbjartsson and Patz, 1995) is introduced. It is entirely possiblethat multi-scale methods may prove better when the method isapplied to high-resolution CT images. The cine-MRI data had lowerresolution in the z-direction with respect to the in-plane resolu-tion. This certainly reduced the accuracy of the strain estimations,as demonstrated by the validation experiments; greater spatialresolution would offer improvements in the spatial results. It isimportant to remark that our approach aims at describing thespatio-temporal maps of deformation with respect to the end-diastolic configuration of the aorta, which we considered as areference. The current approach does not provide a measure ofdeformation of the arterial wall with respect to its unloaded, zero-pressure configuration, as this is beyond the scope of this study.Finally, the density of the initial mesh was chosen in relation withthe voxel size and to keep memory allocation tractable. The choiceof mesh size and mesh smoothing may impact the results andwarrants further investigation.

In conclusion, we presented a robust and repeatable method toobtain the maximum principal strains undergone by the aneur-ysmal wall throughout the cardiac cycle. There are challenges to beaddressed before this method can be proposed as a valid addi-tional instrument to discriminate aneurysms that do not needimmediate surgical intervention from life threatening cases. Inparticular, further validation efforts using in vivo images need tobe undertaken. It is primarily important to evaluate possible errorsin the calculation of strain and how they compare with themagnitude of the changes occurring between follow-ups. More-over, the implications of areas at elevated wall deformation withrespect to the risk of rupture must be elucidated. Once thesechallenges are overcome, the analysis of wall deformation couldprovide a new prospective on the mechanical behavior of patient-specific AAAs.

Conflict of interest statement

There is no conflict of interest related to this paper.

Acknowledgments

The authors gratefully acknowledge Dr. John V. Tyberg, at theUniversity of Calgary, for the useful discussion regarding interpretationof the strain curves, Dr. RandyMoore (Peter Lougheed Hospital) for theDICOM images used to create the synthetic data sets, and Dr. SalvatoreFederico for thoroughly reviewing the continuum mechanics section.This work was supported by NSERC (National Science and EngineeringResearch Council, DG 31682), Alberta Innovates – iCORE (InformaticsCircle of Research Excellence) and NSERC CREATE (CollaborativeResearch and Training Experience).

Appendix A. Supplementary data

Supplementary data associated with this paper can be found inthe online version at http://dx.doi.org/10.1016/j.jbiomech.2014.11.016.

Table 3Maximum first principal Green–Lagrange strain at peak systolic pressure, max-imum diameter (cm), age and gender for each patient studied.

Pat. no. First princ. G–L strain Max. dia. (cm) Age Gender

1 0.049 6.2 65 M2 0.055 4.2 82 M3 0.036 4.5 63 F4 0.091 4.1 62 M5 0.034 5.1 66 F6 0.032 4.8 61 M7 0.073 4.2 66 F8 0.088 2.4 71 F9 0.055 5.0 67 M

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

1 2 3 4 5 6 7 8 9

Patient no.

Peak

−Sys

tolic

1st P

rinci

pal G

reen

−Lag

rang

e St

rain

Fig. 7. Boxplots of the first principal Green–Lagrange strain for all AAAs. Diversityin medians and spreads gives a quantitative representation of the heterogeneity instrain across patients.

A. Satriano et al. / Journal of Biomechanics 48 (2015) 354–360 359

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