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Journal of Biomechanics 40

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model, and identify the scope of error in stress estimation in the conventional approach within a realistic range of material

et al., 2002, 2003). The image-based geometry, however, provided solutions to uniform expansion of thick

strate its ability to accurately determine the load-free

ARTICLE IN PRESSgeometry. We further investigate the scope of error inthe conventional forward approach within a realistic

0021-9290/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jbiomech.2006.01.015

Corresponding author. Tel.: +1319 3356405; fax: +1319 3355669.E-mail address: [email protected] (J. Lu).corresponds to a pressurized state rather than the initialstress-free state. A common approach is to take the invivo geometry as the initial conguration, and proceedto nd the stress using the standard forward niteelement analysis. In other words, the pre-deformationdue to in vivo pressure is not accounted for. This iscertainly a limitation in the current analyses; the inducederror could be nontrivial if the pre-deformation issignicant, as what could happen under hypertensionconditions.In this communication, we demonstrate that the

assumption made to the initial conguration is com-

spherical or cylindrical shell that allow either thedeformed or the undeformed conguration to befound knowing the other. Recently, Govindjee andMihalic (1996, 1998) and Yamada (1995) proposed aprocedure of solving the inverse problem via thenite element method (FEM). Govindjees formulation,in particular, involves only minor changes to theelements designed for forward analysis. Here, weimplement a material model reported specically forabdominal aortic aneurysm (AAA) in Govindjeesinverse framework. We use the inverse method on arepresentative patient-specic AAA model and demon-r 2006 Elsevier Ltd. All rights reserved.

Keywords: Inverse elastostatics; Inverse nite element method; Aortic aneurysm; Patient-specic analysis

1. Introduction

A recent trend in aneurysm stress analysis is to utilizepatient-specic models with aneurysm geometry con-structed from diagnostic images (Wang et al., 2002;Raghavan et al., 2000; Thubrikar et al., 2001; Fillinger

pletely unnecessary. The initial stress-free congurationof an elastic body can be determined from agiven deformed state. The feasibility of nding theinitial geometry is a unique feature of elastostaticproblem. Inverse solutions have been reported in theliterature. For example, Green and Zerna (1954)parameter variations.Short com

Inverse elastostatic stress anastructures: Demonstration usi

Jia Lua,, Xianlian ZhouaDepartment of Mechanical and Industrial Engineering, Center for Com

bDepartment of Biomedical Engineering, Th

Accepted 1

Abstract

In stress analysis of membrane-like biological structures, the

to a deformed state, is routinely taken as the initial stress-fr

completely removed using an inverse elastostatic approach, nam

a given deformed state. We demonstrate the utility of the inv(2007) 693696

nication

is in pre-deformed biologicalabdominal aortic aneurysms

adhavan L. Raghavanb

Aided Design, The University of Iowa, Iowa City, IA 52242-1527, USA

versity of Iowa, Iowa City, IA 52242, USA

uary 2006

etry constructed from in vivo image, which often corresponds

eometry. In this paper, we show that this limitation can be

method for nding the initial geometry of an elastic body from

approach using a patient-specic abdominal aortic aneurysm

www.elsevier.com/locate/jbiomech

www.JBiomech.com

relative to c and is

where in index notation, Ic1ijkl 12 c1ik c1jl c1il c1jk ,and means the standard tensor product. The inverseprocedure and this material model have been imple-mented in an in-house version of a nonlinear FEM code(FEAP) originally developed at University of CaliforniaBerkeley (Taylor, 2003).

3. Results

The surface mesh shown in Fig. 1 was constructed fromcomputed tomography (CT) images of an abdominalaortic aneurysm. The triangular surface mesh wasextruded outward to create the 3D nite element modelwith one layer of elements in the thickness direction. Thewall thickness was assumed to be 1.9mm. We employedthe material model (2) with the population mean materialparameters a 17:39N=cm2 and b 188:08N=cm2. Thepenalty parameter was set to k 100 000N=cm2.Assuming that the CT-reconstructed geometry corre-

ARTICLE IN PRESSomechanics 40 (2007) 6936962qrqc

j4a 8btr c1 3Ic1 8bc1 c21 1range of AAA wall material behavior in the patientpopulation.

2. Methods

The inverse formulation by Govindjee et al. startsfrom the Eulerian weak form of the static equilibriumproblemZOsijZi;j dv

ZOrbiZi dv

ZqOt

tiZi da, (1)

where O is the given current conguration, r is theCauchy stress, b is the body force, t is the prescribedsurface traction and qOt is the boundary where thetraction is applied. The solution for the initial geometryis facilitated via the introduction of the inverse motionU : X Ux, which is the mathematical inverse of theforward motion u : x uX, the primary variable in aforward problem. The cornerstone in Govindjeesapproach is to reparameterize the Cauchy stress, whichis normally a function of the forward deformationgradient F, in inverse deformation gradientf:qU=qx F1. In this manner, the weak form isformulated in the inverse motion, which can be solved tond the initial geometry. In implementation, this schemeresults in a FEM formulation that involves minimumchange to the standard element. The details ofimplementation, including a mixed treatment of thepressure eld, are described in Govindjee and Mihalic(1996, 1998).We implemented a hyperelastic material model

reported specically for abdominal aortic aneurysms(Raghavan and Vorp, 2000) in the inverse nite elementframework. The model is specied by the energyfunction

W aI1 2 log J 3 bI1 32 klog J2. (2)Here, I1 tr FTF, J detF, and a;b;k are materialconstants. Comparing to the original energy function inRaghavan and Vorp (2000), a volumetric penalty term isaugmented for enforcing quasi-incompressibility con-straint. The Cauchy stress in inverse kinematics can beobtained by converting the normal forward stressfunction, and this gives

r 2jac1 I 4jbtr c1 3c1 2kj log jI, (3)where c fTf is the inverse deformation tensor, I is thesecond-order identity tensor, and j det f. The materialtangent tensor, which is needed a NewtonRaphson-based iterative solver, is dened as the derivative of r

J. Lu et al. / Journal of Bi694 2kI c r c , 4(a) (b)

Fig. 1. Views of the stress-free conguration of a pressurized

aneurysm. The predicted initial geometry is depicted in solid and is

visibly smaller than the in vivo shape by surface mesh. In the FE

model, the nodes at the top and bottom cross sections are xed in z-

direction, in addition to a set of six constraints necessary to eliminatesponds to the deformed state at 100mmHg mean aorticpressure, we predicted the load-free geometry and thestress in this state. The load-free geometry computedusing the inverse approach is shown in Fig. 1 super-imposed on the CT-reconstructed geometry. As avalidation, the in vivo pressure was subsequently appliedto the recovered load-free conguration and a forwardstress analysis was performed. This forward analysisexactly recovered the in vivo shape, and resulted in astress distribution that exactly matches those from theinverse approach.It is of practical interest in assessing the error in the

conventional approach. For this, we performed forwardstress analysis assuming the CT-reconstructed geometrythe rigid body motion.

There are some limitations in the present inversemethodology in AAA applications. First, the inversemethodology as we have described here exists on the

ARTICLE IN PRESS

(a)

24.8022.73

(b

nalys

max

(a; b) smax smax Error (%)Present Conventional smax smax

smax 100

(14.40, 188.08) 21.9 25.4 16.0

(20.40, 188.08) 21.4 24.3 13.6

(17.39, 188.08) 21.6 24.8 14.8

(17.39, 115.20) 21.1 24.7 17.1

(17.39, 261.00) 21.9 24.9 13.7

iomecas itself stress-free, as has been done in this eld so far,and estimated the stress distribution under a pressure100mmHg. Fig. 2 shows the stress distribution from theinverse approach and the conventional forward ap-proach. The stress distributions have remarkably similarpatterns but the latter is elevated uniformly. To furtherinvestigate the scope of error within a realistic range ofmaterial parameter variation, we performed repetitiveanalyses by varying the parameters a and b within thereported 95% condence interval (Raghavan and Vorp,2000). Specically, we assigned a to the upper and lowerlimits in the condence domain while holding b xed atthe population mean, and vice versa. The stress distribu-tions at 100mmHg arterial pressure were computed usingboth approaches. The maximum von Mises stresses arelisted in Table 1. The conventional approach for thisrepresentative AAA was found to result in 13.6%17.1%

Fig. 2. von Mises stress at 100mmHg arterial pressure. (a) Inverse a21.5819.8018.0216.2414.4612.6810.909.127.345.563.782.00

J. Lu et al. / Journal of Berror within the given range of parametric variation.Although the error does not uctuate signicantly overthe considered range, one can clearly see that errorincreases as the tissue becomes more compliant.

4. Discussion and conclusion

In this communication, we demonstrated that theload-free conguration of an AAA can be exactlyrecovered using the inverse elastostatic approach, andthus conclusively addressed the issue of nding thereference conguration in imaging-based patient-specicstress analysis. We found that the conventionalapproach over-predicts the stress, and the error magni-es as the material becomes more compliant. This isexpected, because a compliant wall would result ingreater differences between the load-free and in vivogeometries, which is precisely what the conventionalapproach neglects.20.6518.5816.5114.4412.3610.298.226.154.072.00

)

is; (b) Forward analysis using the in vivo conguration as reference.

Table 1

Comparison of the maximum von Mises stress at 100mmHg arterial

pressure predicted from the present approach smax and theconventional approach s units: N=cm2hanics 40 (2007) 693696 695premise that the residual stress is indeed negligible. Ifresidual stress was found to be signicant, furthermodications to the approach not described in thisreport would be needed. Second, the use of a single layerof prism elements can affect stress estimation slightly.However, because all comparisons used the same prismelements, they should remain reasonably valid.Although we have used AAA as a test case for

demonstration, the inverse approach is applicable to avariety of biological structures that are eternally loadedand could be treated as residual stress free. Membrane-like structures such as brain aneurysms, pericardium,urinary bladder, cell membranes are potential applica-tion areas for inverse elastostatics.

Acknowledgments

We would like to thank Dr. Mark F. Fillinger(Dartmouth-Hitchcock Medical Center) for providing

the AAA surface mesh. We also thank Mr. Wenyi Houand Mr. Weixue Yang for assistance in 3D meshgeneration. The rst author (J. L.) acknowledges thesupport by the National Science Foundation GrantCMS 03-48194.

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ARTICLE IN PRESSJ. Lu et al. / Journal of Biomechanics 40 (2007) 693696696

Inverse elastostatic stress analysis in pre-deformed biological structures: Demonstration using abdominal aortic aneurysmsIntroductionMethodsResultsDiscussion and conclusionAcknowledgmentsReferences