Reconstruction of the residual stresses in a hyperelastic body using ultrasound techniques

International Journal of Engineering Science 70 (2013) 46–72

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International Journal of Engineering Science

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Reconstruction of the residual stresses in a hyperelastic bodyusing ultrasound techniques

0020-7225/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijengsci.2013.05.001

⇑ Corresponding author. Address: Department of Mathematics, Temple University, Philadelphia, PA 19122, USA. Tel.: +1 2104fax: +1 2152046433.

E-mail addresses: [email protected] (S. Joshi), [email protected] (J.R. Walton).

Sunnie Joshi ⇑, Jay R. WaltonDepartment of Mathematics, Texas A&M University, College Station, TX 77843, USA

a r t i c l e i n f o

Article history:Received 18 December 2012Received in revised form 24 April 2013Accepted 3 May 2013Available online 3 June 2013

Keywords:AtherosclerosisArterial wallResidual stressSturm–Liouville problemInverse spectral problemUltrasound interrogation

a b s t r a c t

This paper focuses on a novel approach for characterizing the residual stress field in softtissue using ultrasound interrogation. A nonlinear inverse spectral technique is developedthat makes fundamental use of the finite strain nonlinear response of the material to aquasi-static loading. The soft tissue is modeled as a nonlinear, prestressed and residuallystressed, isotropic, slightly compressible elastic body with a rectangular geometry. Aboundary value problem is formulated for the residually stressed and prestressed soft tis-sue, the boundary of which is subjected to a quasi-static pressure, and then an idealizedmodel for the ultrasound interrogation is constructed by superimposing small amplitudetime harmonic infinitesimal vibrations on static finite deformation via an asymptotic con-struction. The model is studied, through a semi-inverse approach, for a specific class ofdeformations that leads to a system of second order differential equations with homoge-neous boundary conditions of Sturm–Liouville type. By making use of the classical theoryof inverse Sturm–Liouville problems, and root finding and optimization techniques, severalinverse spectral algorithms are developed to approximate the residual stress distribution inthe body, given the first few eigenfrequencies of several induced static pressures.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The results reported in this paper arose as part of an extended project to understand the mechanobiology of atheroscle-rosis. Atherosclerosis is a chronic inflammatory process in which the arterial wall thickens as a result of the build up of cho-lesterol and other fatty materials in the interior surface of the wall. A soft fatty plaque is formed initially followed by thedevelopment of a calcified fibrous layer. Rupture of the plaque predisposes the flowing blood to the highly thrombogenicconstituents of the plaque, thereby leading to thrombus formation and possible embolization (Safar & Frohlich, 2007). Thisobstruction in normal blood flow eventually leads to a heart attack or a stroke.

A vast literature has been dedicated toward understanding the initiation and progression of atherosclerosis (Richardson,2006). It is now well-established that the mechanical response of the arterial wall plays a major role in understanding themechanobiology and the growth of the disease (Humphrey, 2002). For example, mechanical factors such as inner-wall shearstress and within-the-wall tensile stresses have major influence on growth and remodeling of the arterial wall, and the ini-tiation and growth of atherosclerotic lesions. Attaining a deeper understanding of the mechanobiology of arteries requiresthe ability to more accurately estimate the material properties and the stresses the arterial wall experiences in vivo. Whilethere has been progress in modeling the development of prestresses and residual stresses in arteries as the natural outcome

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S. Joshi, J.R. Walton / International Journal of Engineering Science 70 (2013) 46–72 47

of growth and remodeling processes (Cardamone, Valentin, Eberth, & Humphrey, 2002; Guillou & Ogden, 2006), devisingmodels and nondestructive experiments through which pre-stresses and residual stresses can be reliably estimated undereither in vivo or in vitro conditions remains a very challenging task.

Soft tissues, in general, exhibit a very complex mechanical behavior. Arterial wall, in particular, is anisotropic, heteroge-neous, layered, prestressed and residually stressed, exhibits an active and passive response and a highly nonlinear stress–strain behavior. Developing a model to understand the mechanical properties that takes into account all of these significant,but complicating factors is very challenging.

This paper serves the purpose of a preliminary study towards developing a mathematical model that takes into accountmost, if not all, of the significant, but complicated properties of the material to characterize the residual stress distribution inthe arterial wall. It is mainly concerned with developing a nonlinear inverse spectral technique to estimate the prestressesand the residual stresses by making a novel use of Intravascular Ultrasound (IVUS) techniques for imaging arteries. We adoptthe modeling framework for the direct problem developed in Gorb and Walton (2010) to formulate an inverse parameteridentification problem. The direct problem is constructed in two stages. First a quasi-static boundary value problem is for-mulated modeling the large deformation of the residually stressed soft tissue, the boundary of which is subjected to a time-varying pressure. Then an idealized model for the IVUS interrogation is constructed to generate small amplitude timeharmonic vibrations superimposed on the static large deformations. This is done by adding a small amplitude, time harmonicnormal traction to the pressure induced, quasi-static traction on one side of the boundary and constructing an asymptoticexpansion of the solution in powers of the amplitude of the superimposed ultrasound pressure wave. This leads to a se-quence of boundary value problems to various orders of the amplitude. The zero-order boundary value problem governsthe static finite deformation of the arterial due to applied pressure, and the first-order problem gives rises to a system ofsecond order homogeneous differential equations of Sturm–Liouville type. We then use a semi-inverse approach to studythe model for a specific class of deformations.

Most inverse spectral techniques for parameter identification use a single pressure to formulate a boundary value prob-lem and use a large number of eigenfrequencies corresponding to the pressure as data to recover the parameters. It is impor-tant to note that unlike such traditional methods, our approach is based on formulating a sequence of boundary valueproblems for different quasi-static pressures, and using only the first few eigenfrequencies of each boundary value problemas data to reconstruct the residual stresses. Making use of boundary value problems that correspond to different pressuresmakes our approach nonlinear. Such a nonlinear approach not only makes the algorithm more robust and accurate since thelower eigenfrequencies are the easiest to detect and measure accurately, it also allows us to over estimate the system anduse non-linear least squares techniques to solve the problem.

Pre-stress in a body is a stress that arises due to external forces, whereas residual stress exists in the absence of externalloads (Humphrey, 2002). A wide range of materials including polymers, composites and soft tissues possess residual stress. Itis of interest to note that a nonzero residual stress cannot be uniform (Hoger, 1986). It is widely known that residual stressexists in soft tissues through various growth and remodeling processes that occur in the living organism, and it affects theoverall stress distribution in the arterial wall under external loads. As an important special case, predicting the distributionof residual stress is a fundamental step in understanding the response and behavior of healthy and diseased blood vessels.

Modeling residual stress in arteries has been widely studied in the past few decades (Hoger, 1986; Chuong & Fung, 1986;Fung, 1991; Hoger, 1993; Johnson & Hoger, 1995; Matsumoto, Furukawa, & Nagayama, 2006). That residual stress exists insoft tissues was demonstrated and studied by Fung (1983) and Vaishnav and Vossoughi (1983). Their results were based onexperiments that involved cutting a thin segment of artery radially. The segment springs open, which shows that the un-loaded arterial segment is stressed (see Fig. 1). It has been hypothesized that this happens because the radial distributionof the circumferential stress in the aortic media is uniform in vivo, and as a result, the segment at no load has compressiveresidual stress near the inner wall and tensile near the outer, causing the opening-up of the ring-like segment (Fung, 1990). Itwas also proposed that a single radial cut is sufficient to achieve a zero-stress state, an assumption that is made in most of

Fig. 1. Existence of residual stress in arteries.

Fig. 2. IVUS catheter.

48 S. Joshi, J.R. Walton / International Journal of Engineering Science 70 (2013) 46–72

the models developed so far (Bustamante & Holzapfel, 2010; Li & Hayashi, 1996; Stalhand & Klarbring, 2006). However, it iswell known that this is not true due to the fact that the arterial wall is heterogeneous, and it takes an infinite number of cutsto achieve a natural stress free configuration. Vossoughi, Hedjazi, and Borris (1991) made an important observation in thisregard when they showed that if one cuts (circumferentially) the radially cut arterial segment into inner and outer rings,then the opening angles associated with each of the radially cut ring will be different, thus suggesting that one radial cutis not sufficient to relieve all of the residual stress (Humphrey, 2002). Physically, this means that the residual stress doesnot arise through simple deformation from a natural (stress-free) configuration.

Most experimental and theoretical attempts that have been made to characterize residual stresses in soft tissues over thepast decade use the Fung opening-angle method (Bustamante & Holzapfel, 2010; Li & Hayashi, 1996; Stalhand & Klarbring,2006). In particular, various studies are devoted to identifying residual strains instead of residual stresses (Vaishnav & Voss-oughi, 1983; Li & Hayashi, 1996). The model proposed herein is tailored towards using clinical in vivo IVUS data to estimatethe residual stress distribution using a parameter identification process. This method avoids dissecting arterial segments andthe models that we adopt from Gorb and Walton (2010) to incorporate residual stress only requires the existence of a virtualnatural configuration without having the need to specify it.

Intravascular Ultrasound is a tomographic medical technology that produces cross sectional images of the inner arterialwall. It is an invasive procedure that detects atherosclerosis plaques and blockages that cause narrowing of the vessels andobstruction of blood flow. As shown in Fig. 2, IVUS is equipped with a catheter containing a transducer that emits high fre-quency sound waves with frequency in the range of 15–40 MHz. When the catheter is inserted into the artery and the trans-ducer emits the ultrasound waves, an echo is generated which is captured and returned to an external computerized devicethat converts the information to real time images of the cross section of the arterial wall.

Unlike conventional models that allow characterization of the arterial plaques using IVUS imaging data (Nair et al., 2002;Seabra, Ciompi, Radeva, & Sanches, 2012; Zhang, McKay, & Sonka, 1998), the proposed method uses spectral characteristicsof IVUS to understand the material behavior of the arterial wall. The ultrasound wave produces a time harmonic traction onthe inner surface of the arterial wall causing the wall to vibrate at nanoscale (Gou, Joshi, & Walton, 2012). One can adjust thefrequency of the ultrasound wave to generate resonance of the arterial wall. This frequency, which gives an estimation of thenatural eigenfrequencies of the wall tissue, is utilized to develop an inverse spectral method to reconstruct the residual stres-ses in the arterial wall. We note that due to the flexibility of the method, it can be generalized to study the response of anynonlinear elastic body that is subjected to a sequence of static loads producing finite strains upon which ultrasound wavesare superimposed. Although the long term goal is to apply the technique to arteries using IVUS data, in this paper, we focusour attention on an elastic body with rectangular geometry envisioning experiments being done on either tissues samples orphantom materials mimicking soft tissue with rectangular geometry. This generalization simplifies the analysis making theinverse problem of reconstructing residual stresses easier to solve.

The rest of the paper is divided into several parts. In Section 2, a hyperelastic constitutive relation is developed for the softtissue mechanical response to large deformation. In Sections 3 and 4, two algorithms are developed, one solving the inverseproblem as a system of nonlinear equations and the other using nonlinear least squares techniques.

2. Mechanical background and model derivation

This section derives a model for the mechanical response of the nonlinear hyperelastic soft tissue that is subjected to astatic prestress. In order to model the propagation of high frequency IVUS waves on the nonlinear elastic body that


undergoes finite strain, we adopt the theory of time harmonic small amplitude vibrations superimposed upon a quasi staticlarge deformation by constructing an asymptotic expansion of the dynamic, total displacement of the vibrated, prestressedmaterial.

2.1. Basic continuum mechanics and notation

Assume that a body occupies a connected open region Bt embedded in a three dimensional Euclidean space at time t. Aparticular configuration B is called the reference configuration. A point X 2 B is called a material point. A motion of B is asmooth mapping v that assigns to each material point X and time t a point x 2 Bt such that

x ¼ vðX; tÞ:

x is referred to as the spatial point occupied by X at time t, and Bt is called the current configuration at time t. v(X, t) for fixed t,is called the deformation at time t. We denote by F the deformation gradient

FðXÞ :¼ rvðX; tÞ

and assume that J(X, t):¼detF > 0. Also the displacement of X is denoted by

uðX; tÞ ¼ vðX; tÞ � X:

A body is said to be elastic if there exists a symmetric tensor-valued function bT ðFÞ, called the material response function, suchthat the Cauchy stress tensor T is given by

T ¼ bT ðFÞ:
The first Piola Kirchhoff stress can then be defined as S ¼ bSðFÞ, where bS ¼ JbT ðFÞF�T : ð2:1Þ
s :¼ bT ðIÞ corresponds to the residual stress in B, where I is the Identity tensor. If bT ðIÞ ¼ 0, then B is called the natural con-figuration or the stress-free configuration. It is important to note that since the reference configuration B is unloaded, theresidual stress s satisfies the following equilibrium condition:

Divs ¼ 0 in Bsn ¼ 0 on @BI [ @BO

�ð2:2Þ

where

s ¼X3

i;j¼1

sijðXÞei � ej; s ¼ sT ð2:3Þ

and n is the unit normal to the corresponding boundary.An elastic body is said to be hyperelastic if there exists a scalar function W ¼ cW ðFÞ called the strain energy function per

unit reference volume, such that

S ¼ bSðFÞ ¼ @FcW ðFÞ: ð2:4Þ

2.2. Constitutive model incorporating residual stress

In order to model the constitutive response of the residually stressed elastic body, one must first develop a model for theresponse of the material to the deformation v from the reference (unloaded but residually stressed) configuration to the cur-rent configuration i.e. find a suitable form for the strain energy function cW ðFÞ, and then address the issue of how to modelthe residual stress s in the unloaded configuration. A specific form for the strain energy function is chosen based on the mate-rial properties such as isotropy, incompressibility, heterogeneity etc. Soft tissues exhibit a highly complex behavior and anonlinear elastic response to external loads. In particular, an arterial wall is layered and highly anisotropic. However, as dis-cussed above, we assume in this idealized proof-of-concept study, that the material is isotropic, even though many soft tis-sues with significant residual stress distributions are highly anisotropic. Because of its high water content, soft tissue is oftenmodeled in the literature as an incompressible elastic body. However, we adopt here the more general assumption of com-pressibility. Specifically, we study the compressible neo-Hookean constitutive law to model the material behavior which isgiven by

cW NðFÞ :¼ l uðJÞ þ 12ðjFj2 � 3Þ

� �ð2:5Þ

where u(�) is a scalar-valued function given by


uðrÞ ¼ r�2b � 12b

; b ¼ m1� 2m

: ð2:6Þ

land m are the shear modulus and Poisson’s ratio of the material respectively. Then, we have,
bSNðFÞ ¼ lðwðJÞF�T þ FÞ; where wðJÞ ¼ u0ðJÞJ: ð2:7Þ
Here, cW NðFÞ and bSNðFÞ are the strain energy function and the First Piola–Kirchhoff stress corresponding to zero residualstresses, and are called the natural strain energy function and the natural response function respectively.

In order to incorporate residual stress in the model, we use an additive form for the strain energy function as described inGorb and Walton (2010). Specifically, the strain energy function is written as

cW ðC; sÞ ¼ cW NðCÞ þ12

C � s; ð2:8Þ

where C = FTF, the right Cauchy-Green strain tensor. The corresponding Piola Kirchhoff stress is given by
bSðF; sÞ ¼ @FcW ðC; sÞ ¼ bSNðFÞ þ Fs: ð2:9Þ
It should be noted that this particular choice of cW ðC; sÞ is frame indifferent corresponding to the unloaded stress-freeconfiguration.

2.3. Problem description

We consider a nonlinear, isotropic, slightly compressible elastic body of rectangular geometry. As mentioned above, thereason for this assumption is that the mathematical equations and algorithms that are developed for this particular geom-etry can be validated through experiments envisioned on real soft tissues or phantom materials. Soft tissues is widely ac-cepted to be a nonlinear viscoelastic material. However, for frequencies as high as that used for IVUS, one can ignore thetime-dependence viscoelastic effects and simply consider the material response to be elastic (Gorb & Walton, 2010).

Consider a rectangular body in the reference configuration B (unloaded and residually stressed). The boundary @B con-sists of six different faces of the body. Let @BI and @BO denote two of the six faces of the boundary that are opposite to eachother, and let the corresponding thickness be of unit length. The equations of motion expressing the local form of balance oflinear momentum in the referential frame (Gurtin, 1981) are given by

DivS þ qb ¼ q€u in B ð2:10Þ

where Div denotes the divergence operator in B, q is the mass density in B, b is the body force, and the €u represents thesecond derivative of u with respect to time.

A constant pressure is applied on one side of the boundary in the deformed configuration. The corresponding boundarycondition is given by

Sej ¼ �pJF�>ej on @BI ð2:11Þ

where ej denotes the outward unit normal to the boundary in the reference configuration.Different boundary conditions may be applied to @BO depending on whether one is interested in in vivo or in vitro behav-

ior. One could impose

Sej ¼ �ku on @BO ð2:12Þ

where k 2 [0,1). Note that k ? 0 and k ?1 correspond to a traction-free and fixed grip boundary conditions respectively. Itshould be noted that for such cases, shear tractions must be admitted on the rest of the boundary (@B n ð@BI [ @BOÞ) in orderfor the body to be in equilibrium. Alternatively, one could also apply an equal pressure on @BO:

Sej ¼ �pJF�>ej; on @BO ð2:13Þ

along with

Sej ¼ 0; on @B n ð@BI [ @BOÞ: ð2:14Þ

This particular case will be explored in the model problem that we develop in Section 2.5.

2.4. Small vibrations superimposed upon a finite deformation

We derive the governing equations for small amplitude time harmonic waves linearized in the neighborhood of a static,finite deformation from the BVP (2.11) by subjecting the rectangular domain to a time harmonic pressure given by

pt ¼ p0ð1þ �eixtÞ

and constructing an asymptotic expansion for the solution of the form


uðX; tÞ ¼ u0ðXÞ þ �u1ðXÞeixt þ oð�Þ; ð2:15ÞFðX; tÞ ¼ I þrðu0ðXÞ þ �u1ðXÞeixt þ oð�ÞÞ ¼ F0ðXÞ þ �F1ðXÞeixt þ oð�Þ; ð2:16ÞSðX; tÞ ¼ S0ðXÞ þ �S1ðXÞeixt þ oð�Þ; ð2:17Þ

where x and � are the frequency and the maximum amplitude of the pressure wave generated by IVUS respectively. Alsonote that �� 1.

Asymptotic expansion of the form 2.15, 2.16, 2.17 of the boundary value problem (2.10) with boundary conditions 2.11,2.13, 2.14 generates a sequence of quasi static boundary value problems to all orders in �. To zero-order in �,u0 satisfies thefollowing boundary value problem

DivS0 ¼ 0; in B;S0ej ¼ �p0J0F�T

0 ej; on @BI [ @BO;

S0ej ¼ 0; on @B n ð@BI [ @BOÞ

8><>: ð2:18Þ

From (2.9), we get

DivS0 ¼ DivbSNðF0Þ þ DivF0s: ð2:19Þ

To calculate the second term in (2.19), we appeal to the definition of the divergence of a second order tensor A as the uniquevector that satisfies

k � Div ðAÞ ¼ Div ðAT kÞ ð2:20Þ

for all constant vectors k. It follows from (2.2),

k � Div ðF0sÞ ¼ Div sðFT0kÞ

� �¼ ðFT

0kÞ � Div ðsÞ þ s � r FT0k

� �¼ s � r FT

0

� �½k� ¼ k � r FT

0

� �Ts:

Hence,

Div ðF0sÞ ¼ r FT0

� �Ts: ð2:21Þ

The BVP (2.18) governs the static finite deformation of the elastic body due to the static load. Linearizing the first Piola–Kir-chhoff stress about the zero-order deformation gives us
bSðF; sÞ ¼ bSðF0; sÞ þ �EðF0; sÞ½ru1�eixt þ oð�Þ ð2:22Þ
where EðF0; sÞ½�� is the fourth-order elasticity tensor at F0 given by

EðF0; sÞ½H� :¼ DFbSðF0; sÞ½H�:

To first-order in �, u1 satisfies

DivS1 ¼ �qx2u1 in B ð2:23Þ

where

S1 ¼ EðF0; sÞ½ru1� ¼ lJ�2b0 2b F�T

0 � ru1

h iF�T

0 þ F�T0 ruT

1F�T0

� �þ lru1 þru1s: ð2:24Þ

2.5. Model problem

Let B be a rectangular domain in the unloaded residually stressed configuration as described in Section 2.3 andX ¼ ðX1;X2;X3Þ 2 B. In an effort to use the semi-inverse approach for the algorithm development and simulations, wenow consider a model problem to determine the residual stress when the material is subjected to a specific class of defor-mations of the form

vðX; tÞ ¼ vðX1; tÞe1 þ kðX1; tÞX2e2 þ gðX1; tÞX3e3; ð2:25Þ

where {e1,e2,e3} is the natural Euclidean orthonormal basis, vðX1; tÞ is the deformation in e1, and k(X1, t) and g(X1, t) are thestretches in e2 and e3 respectively with k,g > 0.

2.5.1. Static large deformation boundary value problemWe construct asymptotic expansions of F and S as suggested in (2.16) and (2.17), and of vðX1; tÞ; kðX1; tÞ and g(X1, t) in a

similar way:

vðX1; tÞ ¼ v0ðX1Þ þ �v1ðX1Þeixt þ oð�Þ ð2:26Þ


kðX1; tÞ ¼ k0ðX1Þ þ �k1ðX1Þeixt þ oð�Þ ð2:27ÞlðX1; tÞ ¼ l0ðX1Þ þ �l1ðX1Þeixt þ oð�Þ ð2:28Þ

It follows from 2.16, 2.25 and 2.26, 2.27, 2.28 that the deformation gradient F0 is given by

F0ðXÞ ¼ v00e1 � e1 þ k00X2e2 � e1 þ k0e2 � e2 þ g00X3e3 � e1 þ g0e3 � e3 ð2:29Þ

and

J0 ¼ k0g0v00: ð2:30Þ

where 0 denotes differentiation with respect to X1. We deduce the following from (2.29):

F�T0 ¼

1v00

e1 � e1 �k00X2

v00k0e1 � e2 �

g00X3

v00g0e1 � e3 þ

1k0

e2 � e2 þ1g0

e3 � e3 ð2:31Þ

DivF0 ¼ v000e1 þ k000X2e2 þ g000X3e3 ð2:32Þ

DivF�T0 ¼

1v00

� �0� k00

k0v00� g00

g0v00

� �e1 ð2:33Þ

rwðJ0Þ ¼ w0ðJ0ÞrX det F0 ¼ w0ðJ0Þ J0F�T0 � F

00

� �e1 ¼ 2bJ�2b

0v000v00þ k00

k0þ g00

g0

� �e1: ð2:34Þ

Hence, from (2.7), (2.29) and (2.31),

bSNðF0Þ ¼l v00 �J�2b

0

v00

!e1 � e1 þ J�2b

0k00X2

v00k0e1 � e2 þ J�2b

0g00X3

v00g0e1 � e3 þ k00X2e2 � e1 þ k0 �

J�2b0

k0

!e2 � e2

þ g00X3e3 � e1 þ g0 �J�2b

0

g0

!e3 � e3

!ð2:35Þ

and

DivbSNðF0Þ ¼ l �J�2b0

1v00

� �0� k00

k0v00� g00

g0v00

� �þ 2bJ�2b

0

v00v000v00þ k00

k0þ g00

g0

� �þ v000

!e1 þ lk000X2e2 þ lg000X3e3 ð2:36Þ

from 2.7 and 2.32, 2.33, 2.34. Using (2.3), (2.21) and (2.29), one gets,

Div ðF0sÞ ¼ s11v000e1 þ s11X2k000 þ 2s12k

00

� �e2 þ s11X3g000 þ 2s13g00

� �e3: ð2:37Þ

Appealing to (2.18), (2.30), (2.31) and (2.35), the boundary conditions on ð@BI [ @BOÞ reduces to

l �J�2b0

v00þ v00

!e1 þ lk00X2e2 þ lg00X3e3 ¼ �p0k0g0e1: ð2:38Þ

Additionally, the boundary conditions on @B n ð@BI [ @BOÞ reduces to

bSNðF0Þe2 ¼ l J�2b0

k00X2

v00k0e1 þ k0 �

J�2b0

k0

!e2

!¼ 0 ð2:39Þ

when j = 2 and

bSNðF0Þe3 ¼ l J�2b0

g00X3

v00g0e1 þ g0 �

J�2b0

g0

!e3

!¼ 0 ð2:40Þ

when j = 3. From 2.18, (2.36)–(2.38), v0 satisfies the following static boundary value problem:

lJ�2b0 ð1þ 2bÞ v000 þ v00

k00k0þ g00

g0

� �� þ ðlþ s11Þv000ðv00Þ

2 ¼ 0 on B ð2:41Þ

�J�2b0

v00þ v00 ¼ �

p0

lk0g0 on @BI [ @BO ð2:42Þ

The stretches in the e2 and e3 direction, k0 and g0, satisfy the following BVPs:

ðlþ s11ðXÞÞX2k000 þ 2s12ðXÞk00 ¼ 0 on B

k00 ¼ 0 on @BI [ @BO

(ð2:44Þ


ðlþ s11ðXÞÞX3g000 þ 2s13ðXÞg00 ¼ 0 on Bg00 ¼ 0 for @BI [ @BO

�ð2:45Þ

Note that the two ordinary differential equations shown above are linear, first-order equations in k00 and g00 respectively.Hence, from the boundary conditions on @BI [ @BO, we get

k00 ¼ 0; g00 ¼ 0 on B: ð2:46Þ

(2.41) then reduces to

v000 ¼ 0 or; ð2:47Þlð2bþ 1ÞJ�2b

0

v00� �2 þ lþ s11 ¼ 0: ð2:48Þ

The latter is equivalent to

v00� ��2ðbþ1Þ ¼ � k0g0ð Þ2b

1þ 2b1þ s11

l

� �:

Assuming that the magnitude of the residual stresses in arteries are small and an order of magnitude less than the magnitudeof the shear modulus (Humphrey, 2002; Gorb & Walton, 2010; Chuong & Fung, 1986), js11/lj < 1, and since b > 0, the latterequation suggests that the derivative of the deformation is a complex function, which is not possible. Thus, v0 satisfies (2.47),which means v00 is a constant and from (2.42), it satisfies

�J�2b0

v00þ v00 ¼ �

p0

lk0g0 on B: ð2:49Þ

It should be noted, from (2.39) and (2.40), that the constants k0 and g0 are given by

k20 ¼ g2

0 ¼ J�2b0 : ð2:50Þ

(2.49) and (2.50) are coupled and can be solved for v00; k0 and g0 simultaneously.The above analysis shows that the static deformation for the model problem (2.25) is homogeneous. Note that the resid-

ual stress does not appear in any explicit form in the boundary value problems derived above for the static deformation. Thisdoes not mean that the stress distribution is independent of the residual stress, but that the experiments performed withinsuch class of deformations do not indicate the presence or absence of residual stress.

2.5.2. The linearized vibrations eigenvalue problemNow, we would like to study the linearized eigenvalue problem within the class of displacements of the form

u1ðXÞ ¼ v1ðX1Þe1 þ k1ðX1ÞX2e2 þ g1ðX1ÞX3e3: ð2:51Þ

The following formulae are used to formulate the eigenvalue problem:

ru1 ¼ v01e1 � e1 þ k01X2e2 � e1 þ k1ðX1Þe2 � e2

þ g01X3e3 � e1 þ g1ðX1Þe3 � e3 ð2:52ÞDivru1 ¼ v001e1 þ k001X2e2 þ g001X3e3 ð2:53Þ

F�T0 � ru1

� �¼ tr

v01v00

e1 � e1 þX2k

01

k0e2 � e1 þ

k1ðX1Þk0ðX1Þ

e2 � e2 þX3g01g0

e3 � e1 þg1ðX1Þg0ðX1Þ

e3 � e3

� �¼ v01

v00þ k1ðX1Þ

k0ðX1Þþ g1ðX1Þ

g0ðX1Þð2:54Þ

r F�T0 � ru1

� �¼ v01

v00þ k1ðX1Þ

k0ðX1Þþ g1ðX1Þ

g0ðX1Þ

� �0e1 ð2:55Þ

Div F�T0 ruT

1F�T0

� �¼ v01

ðv00Þ2

!0þ k01

k0v00þ g01

g0v00

!e1 ð2:56Þ

Div ððru1ÞsÞ ¼ s11v001e1 þ s11X2k001 þ 2s12k

01

� �e2 þ s11X3g001 þ 2s13g01

� �e3 ð2:57Þ

Div J�2b0 F�T

0 � ru1

h iF�T

0

� �¼ J�2b

0

v00v001v00

� �þ k01

k0

� �þ g01

g0

� �� e1 ð2:58Þ

Div J�2b0 ðF�T

0 ru1ð ÞT F�T0 Þ

� �¼ J�2b

0v001ðv00Þ

2 þk01

k0v00þ g01

g0v00

!e1: ð2:59Þ

Substituting 2.52, (2.53)–(2.59), into (2.23) and (2.24) gives the boundary value problem governing the superimposed smallamplitude vibrations for pure radial resonant frequencies (with k1 = g1 = 0):

1 Whcardiac


lð2bþ 1ÞJ�2b0

ðv00Þ2 þ lþ s11

!v001 ¼ �qx2v1 on B ð2:60Þ

with homogeneous boundary conditions

v01 ¼ 0 on @BI [ @BO ð2:61Þ

We make the following remarks regarding the modeling technique:

Remark 1. The only non-zero component of the residual stress appearing in the linearized eigenvalue problem within theclass of deformations (2.25) is s11. This means that the experiments performed within such class of deformations for arectangular body only allow one to recover the stress s11. The above method can be generalized to recover the other twonormal residual stresses s22 and s33 by utilizing the eigenfrequencies obtained from a similar experiment with therectangular body rotated about the corresponding direction.

Remark 2. In order to reconstruct the shear residual stresses such as s12,s13 and s23, one needs to conduct experiments thatinvolve shearing the rectangular body as opposed to just extension and compression.

The second order eigenvalue problem given by (2.60) and (2.61) can be written as a Sturm–Liouville eigenvalue problemgiven by

� pðxÞy0ðxÞð Þ0 þ qðxÞyðxÞ ¼ kwðxÞyðxÞ;1 < a < x < b <1 ð2:62Þ

with boundary conditions

a0y0ðaÞ � a1yðaÞ ¼ 0 ð2:63Þb0y0ðbÞ þ b1yðbÞ ¼ 0; ð2:64Þ

where [a,b] = [0,1],y(x) = v1(X1),X1 2 [0,1], k = x2 and

pðX1Þ ¼ 1; qðX1Þ ¼ 0; wðX1Þ ¼1

lð1þ2bÞJ�2b0

v00ð Þ

2 þ lþ s11

0BB@1CCA; ð2:65Þ

a0 ¼ b0 ¼ 1; a1 ¼ 0 and b1 ¼k

l J�2b0 ð1þ2bÞðv00Þ

2 þ 1� � : ð2:66Þ

The problem of recovering the coefficients p(x),q(x) and/or w(x), given the the eigenvalues fkng1n¼0, is called the InverseSturm–Liouville Problem. Our interest lies in recovering s11 from the spectral data of a system of boundary value problemsof the form 2.62, 2.63, 2.64 that correspond to different pressures p applied in an experiment.1 It is important to note thatthe dependence of the problem on p is through the zero order stretch v0 = v0(X1,p). A number of constructive algorithms havebeen developed for the recovery of the coefficients of the Sturm–Liouville problem with finite spectral data (Chadan, Colton,Paivarinta, & Rundell, 1997; Fabiano, Knobel, & Lowe, 1995; Rundell & Sacks, 1992; Paine, 1984). We emphasize that such algo-rithms that reconstruct the coefficients using a number of eigenfrequencies and other spectral data correspond to using spectraldata from a single applied static pressure, and hence, cannot be implemented for our problem of reconstructing residual stressusing data from different static pressures. The rest of the paper is devoted towards constructing several numerical algorithmsfor solving the inverse Sturm–Liouville problem to recover the residual stress component s11.

2.6. Cubic spline approximation

It can be shown from (2.60) that s11 only varies in the X1 direction i.e. s11 = s11(X1). From here on, for notational conve-nience, we simply replace X1 by x and s11 by s. To that end, we assume that s(x) is continuously differentiable and can beaccurately approximated using cubic splines. More specifically, the algorithm estimates s at the finite degrees of freedomwhich are the nodal points of the spline interpolation, and a series of cubic polynomials are fitted between each of the datapoints.

We partition the domain [0,1] into N � 1 uniform sub-intervals. The nodes are denoted as xi, 1 6 i 6 N and the value of s atxi are denoted by si, 1 6 i 6 N. Define a piecewise function of the form

en applying this inverse spectral method to arteries using IVUS data, the different pressures arise from taking IVUS signals at different times during thecycle.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

Fig. 3. Cubic spline interpolation of five nodes.


ðPðxÞ ¼

p1ðxÞ if x1 6 x < x2;

p2ðxÞ if x2 6 x < x3;

..

.

pN�1ðxÞ if xN�1 6 x 6 xN ;

8>>>><>>>>:
where pi is a third degree polynomial defined by
piðxÞ ¼ aiðx� xiÞ3 þ biðx� xiÞ2 þ ciðx� xiÞ þ di; 1 6 i 6 N � 1:

The N data points are then interpolated using this piecewise function, which means

sðxÞ ¼ PðxÞ

and

PðxiÞ ¼ si; 1 6 i 6 N:

In addition, we use the natural cubic spline, which should satisfy the following conditions:

� P(x) is continuous on the interval [x1,xN], which means pi(xi) = pi�1(xi), 2 6 i 6 N � 1;� P0(x) is continuous on the interval [x1,xN], which means p0iðxiÞ ¼ p0i�1ðxiÞ, 2 6 i 6 N � 1;� P00(x) is continuous on the interval [x1,xN], which means p00i ðxiÞ ¼ p00i�1ðxiÞ; 2 6 i 6 N � 1;� P00(x1) = P00(xN) = 0.

Fig. 3 shows a cubic spline interpolation of 5 data points (0,0.5), (0.25,0.27), (0.5,0.35), (0.75,0.1875) and (1,0.2). Our fo-cus lies in finding s = (s1,s2, . . . ,sN), and the cubic spline interpolation of these N nodes gives us an approximation to theresidual stress s.

2.7. Eigenspectrum solver

In order to develop an algorithm for the Inverse Spectral problem, we need to solve the forward problem of finding theeigenvalues and the eigenfunctions of the (2.62), (2.63), (2.64). We describe a method that uses finite element bases to dis-cretize the problem to a matrix eigenvalue problem, and then utilizes the Krylov–Schur methods in a package called SLEPc tofind the eigenpairs ðki;WiÞMi¼1, where ki and Wi are the eigenvalues and the eigenvectors respectively (Bangerth, Hartmann, &Kanschat, 2007).

Let fxjgN0

j¼0 be a uniform partition of the interval [0,1]. Let f/igNi¼0 be a piecewise smooth finite element basis for W such

that /i(xj) = dij (Ern & Guermond, 2004). We can write WðxÞ ¼PN

j¼0wj/jðxÞ where w = (w1,w2, . . . ,wN) is a vector of expansioncoefficients. Multiplying (2.62) with a test function /i and integrating by parts, we get
XN
j¼0

�pðxÞ/i/0jwjj

1x¼0 þ

Z 1

0pðxÞ/0i/

0jwj þ

Z 1

0qðxÞ/i/jwj

� �¼ kXN

j¼0

Z 1

0wðxÞ/i/jwj; i ¼ 1; . . . ;N: ð2:67Þ

In matrix and vector notation, this equation can be rewritten as

Aw ¼ kMw ð2:68Þ

where A is the stiffness matrix given by


½A�ij ¼Z 1

0�pðxÞ/i/

0jj

1x¼0 þ pðxÞ/0i/

0j þ qðxÞ/i/j

� �
and B is the mass matrix given by
½B�ij ¼Z 1

0wðxÞ/i/j:

The matrix eigenvalue problem (2.68) is solved for ðki;WiÞMi¼1 using the Krylov–Schur solver which is implemented in theSLEPc library.

3. Secant method for the inverse spectral problem

The eigenvalue problem (2.60) and (2.61) depend on v0, and hence, requires one to solve the static problem which issolved using the secant method for solving a non-linear equation. The inverse spectral problem is solved using the general-ized secant method for a system of nonlinear equations and is described below.

3.1. Generalized secant method for a system of equations

Suppose we have a function f : R! R. The one dimensional secant method is used to find the roots of a non-linear equa-tion f(y) = 0. Given two initial guess y0 and y1, the recurrence relation is given by (Cheney & Kincaid, 2002)

ykþ1 ¼ yk �yk�1 � yk

f ðyk�1Þ � f ðykÞf ðykÞ; k ¼ 0;1;2; . . .

The one-dimensional secant method can be generalized to find a solution of a system of linear or non-linear equations(Wolfe, 1959). Suppose we have a system of n equations with n unknowns:

f ðxÞ ¼ 0; ð3:1Þ

where f = (f1, f2, . . . , fn) and x = (x1,x2, . . . ,xn). Just as in the single variable secant method, we require n + 1 trial solutions orinitial guesses for the method to work. Let us denote the n + 1 trial solutions by

X1 ¼ xð1Þ1 ; xð1Þ2 ; . . . ; xð1Þn

� �;

X2 ¼ xð2Þ1 ; x2Þ2 ; . . . ; xð2Þn

� �;

..

.

Xnþ1 ¼ xðnþ1Þ1 ; xðnþ1Þ

2 ; . . . xðnþ1Þn

� �:

ð3:2Þ

There are several generalizations of the secant method to solve n equations. We describe the one introduced in Wolfe (1959).The first step is to find a set of numbers g1, . . . ,gn+1 such that
Xnþ1
j¼1

gj ¼ 1 ð3:3Þ

and
Xnþ1
j¼1

gjfi xðjÞ1 ; xðjÞ2 ; . . . ; xðjÞn

� �¼ 0 for i ¼ 1;2; . . . ;n: ð3:4Þ

Construct a new vector

bX ¼Xnþ1

j¼1

gjXj: ð3:5Þ

Compute the n + 1 values

bj ¼Xn

i¼1

jfiðXjÞja for j ¼ 1;2; . . . ;nþ 1; ð3:6Þ

where a > 0 is a chosen parameter. Find m 2 [1,n + 1] such that

bm :¼ max16j6nþ1

bj: ð3:7Þ

Finally, update the set of trial solutions by replacing Xm with bX . The iteration is repeated until we have


kbXnþ1 � bXnk < e1;

where e1 is the tolerance number, e1� 1.Note that when n = 1, the above method reduces to the one dimensional secant method. One can show that the method

converges superlinearly with order a = 1.618 if the solutions to (3.3) and (3.4) remain bounded at each iteration, and then + 1 trial solutions are sufficiently close to the solution (Wolfe, 1959).

3.2. Secant method for the inverse spectral problem

We develop an algorithm for the inverse Sturm–Liouville problem, in particular, the problem of recovering the residualstress s, by making use of the algorithms described above. Our goal is to recover the N unknowns s1,s2, . . . ,sN by deriving Nequations from N different pressures p1,p2, . . . ,pN. We postulate obtaining a selection of eigenfrequencies from experimental

measurements for each pj, and denote them by ~kðjÞn

n oN0

n¼1. The multi dimensional secant method requires an initial guess of

N + 1 trial solutions sðkÞ ¼ sðkÞ1 ; sðkÞ2 ; . . . ; sðkÞN

� �; k ¼ 1; . . . ;N þ 1.

For each pj and an initial guess s = (s1,s2, . . . ,sN), we find the solution v0(s,x) from (2.49) and (2.50) using the secantmethod. Then substitute v00ðs; xÞ, s(x) and the pressure pj into (2.60) and (2.61), and use the Eigenspectrum solver described

in Section 2.7 to find the sequence of eigenvalues which depend on s and pj and denote them as kðjÞn ðsÞn oN0

n¼0, where N0 is the

total number of eigenvalues computed for each pressure. Then we define N equations of N unknowns as

djðsÞ ¼XN0n¼0

jkðjÞn ðsÞ � ~kðjÞn jC; ð3:8Þ

where C is a parameter specific to the problem. We then find s1,s2, . . . ,sN such that dj(s) � 0 by using the secant methoddescribed in Section 3.1 to solve the resulting equations.

It is important to note that in practice, only the first few lower eigenfrequencies can be measured accurately from exper-iments. However, these lower eigenfrequencies can be accurately measured for a large number of pressures. Hence, we onlyrequire the first few eigenfrequencies corresponding to different pressures to get a good reconstruction for s. In fact, in ournumerical examples, we substitute (3.8) with

djðsÞ ¼ jkðjÞ0 ðsÞ � ~kðjÞ0 jC ð3:9Þ

and use only the smallest eigenvalue for each pressure pj. The detailed algorithm is given as follows:

Algorithm 1.

1. Provide N + 1 trial solutions s(1),s(2), . . . ,s(N+1) for s, where each s(k) is a N dimensional vector given bysðkÞ ¼ sðkÞ1 ; sðkÞ2 ; . . . ; sðkÞN

� �; k ¼ 1; . . . ;N þ 1. Note that the initial guesses are required to be close enough to the true solu-

tion in order for the method to converge.2. For each s(k), find the cubic spline interpolation function.3. For each s(k) (the cubic spline interpolation from Step 2) and each pressure pj, where 1 6 j 6 N, find the numerical solu-

tion vðjÞ0 ðx; sðkÞÞ from (2.49) using the secant method.4. Substitute vðjÞ0 ðx; sðkÞÞ, s

(k) and pj into (2.60) and (2.61) and find the first eigenvalues for the SLP 2.62, (2.63)–(2.66) foreach j and k and denote it as kðjÞ0 ðskÞ.

5. Substitute kðjÞ0 ðsðkÞÞ into (3.9) to get dj(s(k)) for each j and k.6. Form a (N + 1) (N + 1) square matrix D given by

D ¼

d1ðsð1ÞÞ d1ðsð2ÞÞ � � � d1ðsðNþ1ÞÞd2ðsð1ÞÞ d2ðsð2ÞÞ � � � d2ðsðNþ1ÞÞ

..

. ... ..

. ...

dNðsð1ÞÞ dNðsð2ÞÞ � � � dNðsðNþ1ÞÞ1 1 � � � 1

266666664

377777775: ð3:10Þ

7. Find the vector g = (g1,g2, . . . ,g(N+1)) by solving the matrix–vector equation DgT = (0,0, . . . , ,0, 1)T, where the superscriptT stands for transpose.

8. Find an updated s using (3.5).9. Compute bi for 1 6 i 6 N + 1 by taking fi(Xj) = di(s(j)) in (3.6), where di(s(j)) is given by (3.9).

10. Find the value m such that (3.7) is satisfied and replace s(m) by s and form an updated N + 1 solutions s(1),s(2),. . . ,s(m�1), s, s(m+1), . . . ,s(N+1).

11. Repeat Steps 2 – 11 until jsnew � soldj < e1, where e1 is a given tolerance and snew and sold are the s from the current andthe previous iteration respectively.

0

0.2

0.4

0.6

0.8

1

1.2

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x (mm)

0

0.2

0.4

0.6

0.8

1

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x (mm)

Fig. 4. Reconstruction of Example 1 with three nodes (left figure) and five nodes (right figure).

Table 1Numerical Example 1 with three nodes.

Pressure (mmHg) 5 93 130Frequency (104Hz) 4.462 6.9143 7.7945

Initial guess 1 (kPa) 1.1 0.8 0.001Initial guess 2 (kPa) 1.3 0.97 0.01Initial guess 3 (kPa) 1.02 0.73 0.01Initial guess 4 (kPa) 0.99 0.71 0.05

True value (kPa) 1.0 0.75 0.0Returned value (kPa) 1.0005 0.7193 0.0423Iteration number 23Absolute L2 error 5.22e�02ksnew � soldkL2 8.26e�06


3.3. Numerical examples

In this section, we present various numerical examples corresponding to the algorithm developed in Section 3.2. The den-sity q is chosen to be 1gm/cm3. The thickness of the rectangular slab is taken to be 0.6 mm and @BI corresponds tox = xi = 1.39 and @BO corresponds to x = xo = 1.99.2 The values of other parameters used in the numerical simulations are chosento be: l = 27 kPa and m = 0.4. The parameter C in the algorithm is chosen to be 2. In each of the tables, the first row representsthe pressure used for that example, and the second row represents the corresponding first eigenfrequency for the pressure. Eachtable contains a particular set of initial guesses for s at the nodes x1,x2, . . . ,xN for the secant method. Note that the number ofinitial guesses depend on the number of pressures used. The true value and the returned value of s at each node is given alongwith the absolute error computed in the L2 norm and the value ksnew � soldkL2 for the last iteration. A corresponding plot of thetrue s and the iterated s for each table is presented. Note that in the figures, the solid line represents the true s and the dottedline represents the iterated s.

Example 1. The function we want to approximate is s ¼ 1� x�xi

xo�xi

� �2. We use the cubic spline interpolation of the function at

3 nodes and 5 nodes and the first eigenvalue {k0} as given data. The reconstructions are given in Fig. 4 and the iteration inputand output data are given in Tables 1,2.

Example 2. The function we want to approximate is s ¼ x�xi

xo�xi 1� x�xi

xo�xi

� �. We use the cubic spline interpolation with 3, 5 and 7

nodes. The function reconstructions are given in Figs. 5–7 and the iteration input and output data are given in Tables 3–5respectively. For 3 and 7 nodes problems, the first eigenvalue {k0} was assumed as data, and for the 5 nodes case, we comparethe results for given data {k0} and {k0,k1} for each pressure.

2 The parameters are chosen from Chuong and Fung (1986) and correspond to the measurements of rabbit carotids. The results of the simulations are onlyillustrative.

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x (mm)

Fig. 5. Reconstruction of Example 2 with three nodes.

Table 2Numerical Example 1 with five nodes.

Pressure (mmHg) 45 93 120 130 150Frequency (104Hz)(k0) 5.6967 6.9143 7.5612 7.7945 8.2512

Initial guess 1 (kPa) 1.01 0.8 0.67 0.5 0.001Initial guess 2 (kPa) 0.9 0.85 0.82 0.6 0.001Initial guess 3 (kPa) 0.8 0.75 0.625 0.35 0.002Initial guess 4 (kPa) 1.1 0.9 0.7 0.45 0.01Initial guess 5 (kPa) 1.2 1.01 0.9 0.62 0.003Initial guess 6 (kPa) 0.75 0.65 0.6 0.54 0.001

True value (kPa) 1.0 0.9375 0.75 0.4375 0.0Returned value (kPa) 0.9994 0.9528 0.7994 0.4242 0.0098Iteration number 40Absolute L2 error 5.42e�02ksnew � soldkL2 5.21e�06

0

0.05

0.1

0.15

0.2

0.25

0.3

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x (mm)

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x (mm)

Fig. 6. Reconstruction of Example 2 with five nodes using k0 (left figure) and k0 and k1 (right figure).


Example 3. The function we want to approximate is s ¼ sin 2p x�xi

xo�xi

� �. Cubic spline interpolations with 5 and 7 nodes were

used and {k0} was assumed to be given. Reconstructions are given in Fig. 8 and the iteration input and output data are givenin Tables 6 and 7.

The above numerical examples suggest that the algorithm is only locally convergent and the initial guesses need to beclose to the original function. The number of nodes N that gives the most accurate results is specific to the problem and de-pends on the shape of the function that is being approximated. For instance, for Example 1, N = 3 gives a better

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x (mm)

Fig. 7. Reconstruction of Example 2 with seven nodes.

Table 3Numerical Example 2 with three nodes.

Pressure (mmHg) 10 93 130Frequency (104Hz) 3.999 6.671 7.5996


True value (kPa) 0.0 0.25 0.0Returned value (kPa) �0.02 0.2597 0.01Iteration number 25Absolute L2 error 2.8e�02ksnew � soldkL2 4.86e�06


Pressure (mmHg) 45 93 120 130 150Frequency (104Hz)(k0) 5.357 6.671 7.3549 7.5996 8.0762Frequency (105Hz)(k1) 1.0709 1.3339 1.4707 1.5197 1.615

Initial guess 1 (kPa) 0.001 0.18 0.2 0.18 0.001Initial guess 2 (kPa) 0.01 0.21 0.32 0.21 0.01Initial guess 3 (kPa) 0.002 0.2 0.39 0.2 0.002Initial guess 4 (kPa) 0.01 0.18 0.3 0.18 0.01Initial guess 5 (kPa) 0.001 0.15 0.3 0.15 0.001Initial guess 6 (kPa) 0.955 0.92 0.875 0.71 0.49True value (kPa) 0.0 0.1875 0.25 0.1875 0.0

Returned value (kPa)(with k0) 0.0009 0.1876 0.2497 0.1876 0.0009Iteration number 27Absolute L2 error 1.3e�03ksnew � soldkL2 9.18e�07

Returned value (kPa) (with k0 and k1) �0.0006 0.1876 0.2494 0.1876 �0.0006Iteration number 44Absolute L2 error 1.1e�03ksnew � soldkL2 9.51e�06


approximation than N = 5 whereas for Example 3, N = 3 is not enough to predict the shape of the function and hence, thealgorithm does not converge. On the other hand, the absolute error for Examples 2 and 3 with N = 5 is lower than N = 7.In addition, observe that the number of iterations increase as N increases for all the examples. This is because choosing morenodes imply solving a larger system of equations that require more initial guesses which may deviate away from the truesolution. The above observations suggest that one needs to have an estimation of the shape of the function to select a rea-sonable N. Although this information might not be available beforehand, one can test for several values of N and choose thesmallest one that allows convergence of the algorithm with acceptable accuracy.

Table 5Numerical Example 2 with seven nodes.

Pressure (mmHg) 10 45 93 100 120 130 150Frequency (104Hz) 3.999 5.357 6.671 6.8516 7.3549 7.5996 8.0762

Initial guess 1 (kPa) 0.01 0.15 0.28 0.4 0.18 0.14 0.001Initial guess 2 (kPa) �0.01 0.17 0.25 0.4 0.24 0.12 �0.0001Initial guess 3 (kPa) �0.01 0.12 0.15 0.27 0.24 0.1 � 0.002Initial guess 4 (kPa) 0.01 0.2 0.3 0.4 0.3 0.19 0.01Initial guess 5 (kPa) 0.0003 0.1 0.25 0.35 0.15 0.17 0.0003Initial guess 6 (kPa) 0.001 0.14 0.22 0.28 0.2 0.2 0.1Initial guess 7 (kPa) 0.1 0.16 0.2 0.275 0.24 0.11 0.001Initial guess 8 (kPa) 0.001 0.175 0.29 0.31 0.26 0.15 0.001

True value (kPa) 0.0 0.1389 0.2222 0.25 0.2222 0.1389 0.0Returned value (kPa) �0.0004 0.1159 0.2502 0.2477 0.1980 0.1593 0.0039Iteration number 76Absolute L2 error 4.83e�02ksnew � soldkL2 9.59e�05

-1.5

-1

-0.5

0

0.5

1

1.5

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x (mm)

-1.5

-1

-0.5

0

0.5

1

1.5

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x (mm)

Fig. 8. Reconstruction of Example 3 with five nodes (left figure) and seven nodes (right figure).


Pressure (mmHg) 45 93 120 130 150Frequency (104Hz)(k0) 5.3685 6.6819 7.3655 7.61 8.0865

Initial guess 1 (kPa) 0.1 1.04 0.002 �0.98 0.001Initial guess 2 (kPa) �0.0001 0.9 �0.03 �1.02 �0.0001Initial guess 3 (kPa) �0.002 0.85 �0.02 �1.04 � 0.002Initial guess 4 (kPa) 0.01 1.04 0.003 �0.8 0.01Initial guess 5 (kPa) 0.0003 1.1 0.04 �0.9 0.0003Initial guess 6 (kPa) 0.001 1.15 0.001 �0.9 0.001

True value (kPa) 0.0 1.0 0.0 �1.0 0.0Returned value (kPa) �0.0028 1.0401 0.0089 �1.0352 0.005Iteration number 49Absolute L2 error 5.44e�02ksnew � soldkL2 1.64e�07


Table 4 of Example 2 also implies that considering the first 2 eigenfrequencies as data does not give a better approxima-tion than considering only the first eigenfrequency as data. In fact, it takes fewer iterations for the latter to converge, sug-gesting that the lowest eigenfrequency for each of the blood pressure is enough to accurately approximate the function. Thisfeature also makes the algorithm stable since the lowest eigenfrequencies are the easiest to measure with minimum error.

As we see from the above tables, the eigenfrequencies (x) of the problem that we consider are in the order of 104. Theactual eigenvalues k that we compute using SLEPc are in the order of x2, that is, 108, and they increase with increase inthe pressure. In order to avoid using such large numbers that compromises the accuracy of the algorithm, we scale the

Table 7Numerical Example 3 with seven nodes.

Pressure (mmHg) 10 45 93 100 120 130 150Frequency (104Hz) 3.999 5.3657 6.6794 6.8601 7.3631 7.6077 8.0842

Initial guess 1 (kPa) 0.1 0.9 0.98 0.002 �0.98 �0.8 0.001Initial guess 2 (kPa) �0.0001 0.5 0.6 �0.03 �0.7 �0.82 �0.0001Initial guess 3 (kPa) �0.002 0.85 0.95 �0.02 �1.04 �0.9 � 0.002Initial guess 4 (kPa) 0.01 0.5 0.89 0.003 �0.8 �1.1 0.01Initial guess 5 (kPa) 0.0003 1.1 0.92 0.04 �0.9 �0.7 0.0003Initial guess 6 (kPa) 0.001 1.15 0.7 0.001 �0.9 �0.9 0.001Initial guess 7 (kPa) 0.1 1.1 1.3 0.001 �0.9 �1.1 0.001Initial guess 8 (kPa) 0.001 1.15 0.7 0.001 �1.1 �0.9 0.001

True value (kPa) 0.0 0.866 0.866 0.0 �0.866 �0.866 0.0Returned value (kPa) �0.01 1.0044 0.7147 �0.0404 � 0.8508 �1.0114 �0.0038Iteration number 93Absolute L2 error 2.76e�01ksnew � soldkL2 8.53e�05

Table 8Numerical Example 4 with 0.01% noise.

Pressure (mmHg) 45 93 130Frequency (104Hz)(k0) 5.357 6.671 7.5996Frequency (105Hz)(k1) 1.0709 1.3339 1.5197


True value (kPa) 0.0 0.25 0.0Returned value (kPa) �0.0456 0.2385 �0.0645Iteration number 22Absolute L2 error 7.98e�02

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

Fig. 9. Reconstruction of Example 4 with 0.01% noise.


eigenvalues of the problem. We multiply both sides of (2.62) by a parameter � where �� 1, and compute the scaled eigen-value k � instead of the actual eigenvalue k. This makes the algorithm more efficient and the reconstructions more accurate.For the numerical examples above, � was chosen to be 10�6.

In addition to being accurate and efficient, an algorithm also needs to be robust in order to be reliable. The next exampleis to test the robustness of the algorithm.

Example 4. We consider the same function as in Example 2. Cubic spline interpolation of the function at 3 nodes were usedand the eigenvalues {k0,k1} with 0.01% noise were assumed to be given. The input and output details are given in Table 8 andthe reconstruction is given in Fig. 9. This example gives an idea of how accurately one needs to measure the ultrasound


frequencies in order for this method to work. We remark that although the algorithm did not converge when consideringerrors of more that 0.01%, the fact that we only use the first two eigenvalues for each pressure makes it robust.

The secant method for the inverse spectral problem is not superlinearly convergent like the generalized secant method assuggested in Wolfe (1959). This is because errors from several factors of the algorithm accumulate such as the error due toestimating the eigenvalues and the error due to approximating the zero order stretch v0 using the secant method.

4. Optimization algorithm

In the previous section, we developed a method for estimating the residual stresses by solving a system of nonlinearequations using the generalized secant method. The advantage of such methods over Newton-like methods is that the Jaco-bian matrix f0(y) need not be computed. An alternative approach is to formulate the inverse problem of approximating theresidual stress s as a minimization problem and use least squares techniques. In what follows, we describe a few numericalmethods for solving least squares problems and develop an algorithm using such methods to solve the inverse spectralproblem.

4.1. Nonlinear least squares problem

Let r1,r2, . . . ,rM be M functions of N variables b = (b1,b2, . . . ,bN). Define an objective functional F given by (Kelley, 1999)

FðbÞ ¼ 12

XM

i¼1

jriðbÞj2: ð4:1Þ

The vector R = (r1,r2, . . . ,rM) is called the residual. (4.1) can be rewritten as

FðbÞ ¼ 12

RðbÞT RðbÞ ð4:2Þ

where T represents the transpose of a vector. Nonlinear least squares problem involve finding a b 2 RN that minimizes F(b).M is the number of observations and N is the number of unknowns and we require M P N. We define the M N Jacobianmatrix R0 of R by

½R0ðbÞ�ij ¼@ri

@bj; 1 6 i 6 M; 1 6 j 6 N:

The gradient of FðbÞ 2 RN can then be written as

rF ¼ R0ðbÞT RðbÞ: ð4:3Þ

A necessary condition for b⁄ to be a minimizer is

R0ðbÞT RðbÞ ¼ 0: ð4:4Þ

Ifr2F is positive definite, then b⁄ is the unique solution to (4.4). If b is close to the minimum, (4.4) can be solved using New-ton’s method. This is equivalent to solving

r2FðbÞs ¼ �rFðbÞ; ð4:5Þbkþ1 ¼ bk þ s: ð4:6Þ

In order to solve the above equation, one needs to compute the N N Hessian matrix which is given by

r2FðbÞ ¼ R0ðbÞT R0ðbÞ þXM

i¼1

riðbÞTr2riðbÞ: ð4:7Þ

The second derivative can be rewritten as
XM
i¼1

riðbÞTr2riðbÞ ¼ R00ðbÞT RðbÞ;

where R00(b)TR(b) is a third order tensor such that for any p 2 RN , R00(b)Tp is a N N matrix

ðR00ÞT p ¼XN

i¼1

pir2ri:

Note that the computation of (4.7) requires the computation of the M Hessiansr2ri(b). In practice, this is too costly and oftennot possible to compute analytically. One way to avoid having to compute (4.7) is by using the Gauss–Newton Iteration whichsimply discards the second order term in r2F(b) and approximates the Hessian by


r2FðbÞ � R0ðbÞT R0ðbÞ: ð4:8Þ

Assumption 1. If b⁄ is a minimizer of F(b),

1. R is Lipschitz continuously differentiable near b⁄.2. R0(b⁄)TR0(b⁄) has maximal rank.

If Assumption 1 holds, the Gauss–Newton Iteration step is given by

s ¼ � R0ðbkÞT R0ðbkÞ

� ��1rF ¼ � R0ðbkÞ

T R0ðbkÞ� ��1

R0ðbkÞT RðbkÞ: ð4:9Þ

Observe that the second order term in (4.7) vanishes for zero residual problems where F(b⁄) is zero, and can be negligible forsmall residual problems where F(b⁄) is small, or for problems with very small R00(b). Hence, the Gauss Newton iteration con-verges well for zero residual problems or small residual problems with good initial data.

4.2. Line search algorithms

Let d 2 RN . d is a descent direction for F at b if

dFðbþ tdÞdt

t¼0¼ rFðbÞT d < 0:

A line search algorithm searches for decrease in the objective functional F in the descent direction d by controlling the steplength of the iteration (Kelley, 1999; Bjorck, 1996; Moré & Thuente, 1994). The step length is controlled such that there issufficient decrease in F. The general sufficient decrease condition is

Fðbk þ gkdÞ � FðbkÞ < hgkrFðbkÞT d < 0: ð4:10Þ

Here h 2 (0,1) is an algorithm parameter. The goal is to find a parameter gk in each iteration such that (4.10) is satisfied. Theiterate is then updated as

bkþ1 ¼ bk þ gkd: ð4:11Þ

One way to find the step length gk is using the Armijo rule (Armijo, 1966). For fixed k, let gk = nj, where n 2 (0,1) and j 2 Zþ.Fix n and find the smallest j such that (4.10) is satisfied. This strategy of repeatedly finding sufficient decrease in F and updat-ing the step size if (4.10) fails is also called backtracking.

Recall that the Gauss Newton direction is given by

dk ¼ � R0ðbkÞT R0ðbkÞ

� ��1R0ðbkÞ

T RðbkÞ: ð4:12Þ

If Assumption 1 holds, then (R0(bk)TR0(bk)) is positive definite, and hence,

dTkrFðbkÞ ¼ � R0ðbkÞ

T RðbkÞ� �T

R0ðbkÞT R0ðbkÞ

� ��1R0ðbkÞ

T RðbkÞ < 0 ð4:13Þ

and the Gauss Newton direction is a descent direction. The Armijo rule can also be combined with the Gauss Newton meth-od, and the method is called the damped Gauss Newton method.

Lemma 1. LetrF(b) be Lipschitz continuous with Lipschitz constant L. Let bk be the damped Gauss Newton iteration with the steps

sk ¼ bkþ1 � bk ¼ gkdk ¼ gkH�1k rFðbkÞ; ð4:14Þ

and Hk = R0(bk)TR0(bk) at each iteration k. Let j(Hk) denote the condition number Hk such that

jðHkÞ ¼kmax

kmin6 j;

where kmin and kmax are the smallest and the largest eigenvalues of Hk respectively. Then

gk P g ¼ 2nkminð1� hÞLj

ð4:15Þ

and the maximum number of stepsize reductions is

j ¼log 2kminð1�hÞ

Lj

� �logðmÞ : ð4:16Þ


The above lemma implies that the the sufficient decrease condition (4.10) can be satisfied in finitely many steps for all kand the step lengths are bounded away from zero. It also holds for other line search methods such as the Levenberg–Marquardt algorithm that is described below (Kelley, 1999).

Theorem 3.2.4 (Kelley, 1999) says that if the condition numbers of Hk and the norm of the iterates remain bounded, thenthere will be a limit point. However, uniqueness of the limit point is not guaranteed. In order for the limit point to exist, thematrices Hk need not only be non-singular, but also must be well-conditioned and uniformly bounded, which are strongassumptions. One way to guarantee these conditions are satisfied is by regularizing the matrices Hk. The Levenberg–Marqu-ardt method (Marquardt, 1963) adds a regularization parameter ck > 0 to R0(bk)TR0(bk) to obtain

Hk ¼ ckI þ R0ðbkÞT R0ðbkÞ; ð4:17Þ

where I is the N N Identity matrix and ck is called the Levenberg–Marquardt parameter. The iteration is then given by

bkþ1 ¼ bk þ dk; ð4:18Þdk ¼ �ðHkÞ�1R0ðbkÞ

T RðbkÞ: ð4:19Þ

Note that the matrix ckI + R0(bk)TR0(bk) is symmetric and positive definite. The parameter c is chosen such that the ck-

kI + R0(bk)TR0(bk) is well conditioned. The Levenberg–Marquardt method can be combined with the Armijo rule, that is, findgk such that (4.10) is satisfied where dk is given by (4.19), and update the iteration using (4.14). This method is called theLevenberg–Marquardt–Armijo algorithm.

The Levenberg–Marquardt method can be thought of as the combination of steepest descent method and the Gauss New-ton method. Both of these methods have complimentary properties and the Levenberg–Marquardt method is designed totake advantage of both the methods. When ck is large, the first term in Hk dominates and the method behaves like the steep-est descent method, and as ck decreases, it behaves like the Gauss Newton method. Since the steepest descent method exhib-its good global convergence and slower local convergence, ck is chosen to be large when bk is further away from theminimum. ck is gradually decreased as the iterate gets closer to the minimum so that the iteration behaves like the GaussNewton method. The above algorithm has the disadvantage that if the value of ck is large, the calculated Hessian Hk isnot used at all. Marquardt proposed to replace the Identity matrix in (4.17) with the diagonal of the Hessian resulting in

Hk ¼ ck diag ½R0ðbkÞT R0ðbkÞ� þ R0ðbkÞ

T R0ðbkÞ ð4:20Þ

There are several approaches to find the parameter ck (Kelley, 1999; Marquardt, 1963). The first parameter c0 is found usinga trial and error method and the rest of the ck’s are found using

ck ¼ minf1; kRðbkÞkdg;

where d 2 (0,2). This is an appropriate choice because clearly, the residual kR(bk)k decreases with the iteration number by theway we constructed the algorithm.

4.3. Least squares algorithm for the inverse spectral problem

In this section, we develop a least squares algorithm for the inverse spectral problem of estimating the residual stress sfrom the spectral data of a system of boundary value problems that correspond to different applied pressures p.

Suppose we are given the spectral data for m pressures p1,p2, . . . ,pm and for each pi, the first n eigenfrequencies are given.Consider the functional

DðsÞ ¼ 12

Xm

i¼1

Xn

j¼1

jkjpiðsÞ � ~kj

pij2 ¼ 1

2

Xmnk¼1

jkkðsÞ � ~kkj2; ð4:21Þ

where ~kjpi

is the jth eigenfrequency corresponding to the pressure pi obtained from IVUS, and kjpiðsÞ

n oare the eigenvalues

achieved from the model. We assume that s is continuously differentiable and approximate it using cubic spline interpola-tion. Just as in the algorithm developed in Section 3, the optimization algorithm estimates s = s1,s2, . . . ,sN at the nodal pointsx1,x2, . . . ,xN and the function is approximated using cubic splines. It is evident that F cannot completely vanish since theeigenfrequencies obtained from experiments contain noise and the eigenfrequencies that are achieved from the algorithmcontain numerical and round-off errors, and hence, they cannot be equal. The best one can do is to find a s that minimizesD(s). It is also important to note that the least squares problem (4.21) is non-linear. Moreover, it is different from the usualnon-linear least squares problems due to the fact that the quantities kj

piðsÞ

n oare not defined explicitly but only implicitly as

a solution to the eigenvalue problem (2.62)–(2.66). As a result, it does not have an analytical expression and its evaluationrelies on numerical methods.

The advantage of least squares method over methods for non-linear equations is that one can overestimate the number ofeigenvalues and solve the overdetermined problem to achieve more accuracy. To that end, we assume that the total numberof eigenfrequencies M = m ⁄ n is greater than or equal to the number of unknowns N. Define R 2 RM

RðsÞ ¼ ðkkðsÞ � ~kkÞ: ð4:22Þ


Then (4.21) can be rewritten as

DðsÞ ¼ 12

RðsÞT RðsÞ: ð4:23Þ

The eigenvalues fkðsÞ ¼ knðwÞ : n 2 N0g are continuously differentiable with respect to s by the differentiability of the eigen-values with respect to the coefficients p, q and w (Kong & Zettl, 1996, Theorem 4.2). Hence, R(s) is continuously differentiablewith respect to s. The gradient of DðsÞ 2 RN can then be written as

rDðsÞ ¼ R0ðsÞT RðsÞ ð4:24Þ

and the Hessian is approximated as

r2DðsÞ � R0ðsÞT R0ðsÞ; ð4:25Þ

where R0ðsÞ 2 RMN is the Jacobian matrix

½R0ðsÞ�ij ¼@kiðsÞ@sj

;1 6 i 6 M; 1 6 j 6 N: ð4:26Þ

Provided that R(s) has maximal column rank, we can adopt the gradient based algorithms such as damped Gauss–NewtonMethod and Levenberg–Marquardt–Armijo method described above to solve the minimization problem. Note that R0(s) andhence rD(s) do not have an analytical expression and can only be solved numerically. Due to this, the rank condition needsto be checked separately for each problem. R0(s) is approximated using the directional derivative. Let u be a unit vector in RN .Then the approximated directional derivative along u is given by

ruRðsÞ ¼ Rðsþ �uÞ � RðsÞ�

ð4:27Þ

where � is a small number. Then R0(s) is the unique M N matrix such that

ruRðsÞ ¼ R0ðsÞu; 8u 2 RN : ð4:28Þ

In order to solve the least squares problem, an initial guess for s, sð0Þ ¼ s01; s0

2; . . . ; s0N is required for the iteration to execute.

The corresponding cubic spline interpolation s(0)(x) of s(0) is calculated and used to find the solution v0(s,x) for each pi. Then,v0(s, x), s(x) and pj are substituted into (2.60) and (2.61), and the Eigenspectrum solver is used to find a sequence of eigen-values kj

piðsÞ

n on

j¼0for each pi. We then use the damped Gauss–Newton Method or the Levenberg–Marquardt–Armijo method

to find a s that minimizes D.The detailed algorithm is given as follows:

Algorithm 2.

1. Let the iteration number be k. Set k = 0. Provide an initial guess sðkÞ ¼ sðkÞ1 ; sðkÞ2 ; . . . ; sðkÞN

� �for s, and find the cubic spline

interpolation function for s(k). Note that the initial guess is required to be close enough to the true solution.2. For each pressure pi, where 1 6 i 6 N, find the numerical solution vðiÞ0 ðx; sðkÞÞ from (2.49).3. Substitute vðiÞ0 ðx; sðkÞÞ, s

(k) and pi into (2.60) and (2.61) to find the first n eigenvalues for the SLP 2.62, (2.63)–(2.66) foreach i and denote it as kðjÞpi

ðsðkÞÞ; j ¼ 1; . . . ;n.4. Substitute kðjÞpi

ðsðkÞÞ into (4.22) to get R(s(k)). Using this and (4.27) and (4.28), find R0(s(k)).5. Find the gradient rD(s(k)) and the Hessian r2D(s(k)) using (4.24) and (4.25) respectively.6. Depending on the algorithm that is used, find d = dk using (4.12) or 4.17, (4.18)–(4.20).7. Find the residual D(s(k)) and the gradient norm krD(s(k))k.8. If the Levenberg–Marquardt–Armijo method is used, then choose a d 2 (0,2), and let ck = min{1,kRkd}.9. Fix a parameter n 2 (0,1), say 0.5 and find the step length gk such that

DðsðkÞ þ gkdkÞ � DðsðkÞÞ < hgkrDðsðkÞÞT dk < 0: ð4:29Þ

10. Update s using s(k+1) = s(k) + gkdk.11. Fix tolerances tr and ta. Repeat Steps 1 – 10 until

krDðsðkÞÞk 6 trkrFðsð0ÞÞk þ ta: ð4:30Þ

4.4. Convergence

Theorem 1. Let sk:¼s(k) be the Levenberg–Marquardt–Armijo iterates of the least squares problem with the iteration step given by(4.20). Assume that R0(sk) is nonsingular, kR0 (sk)k is uniformly bounded and


ck ¼ minð1; kRðskÞkÞ: ð4:31Þ

Then if D(sk) is bounded from below,

limk!1rDðskÞ ¼ 0 ð4:32Þ

and hence, any limit point s⁄ such that

s ¼ limk!1

sk ð4:33Þ

is a stationary point. Furthermore, the stationary point is a minimum.

Proof. Note that R0ðsÞ ¼ @k@s is continuous. Also, since R0(s) is nonsingular, j(R0(sk)TR0(sk)) is bounded. Furthermore, for the

particular choice of ck given in (4.31), j(Hk) is also bounded, that is

j ck diag ½R0ðskÞT R0ðskÞ� þ R0ðskÞT R0ðskÞ� �

6 j:

Then Lemma 1 holds and the parameter gk appearing in (4.29) satisfies (4.15). Now since kR0(sk)k is uniformly bounded,

kR0ðskÞk 6 eC1; forall k:

Then,

kHkk ¼ kck diag ½R0ðskÞT R0ðskÞ� þ R0ðskÞT R0ðskÞk 6 eC 2 þ eC 21 6

eC3:

Here, we use the fact that kR0(sk)k = kR0(sk)Tk for the matrix norm induced by the Euclidean norm. Now, D(sk) is a decreasingsequence by construction, and if it is bounded from below, D(sk) converges and

limk!1

DðskÞ � Dðskþ1Þ ¼ 0: ð4:34Þ

Now by (4.10) and (4.20),

Dðskþ1Þ � DðskÞ < �hgkrDðskÞT H�1k rDðskÞ 6 �hgeC�1

3 krDðskÞk26 0:

Hence, by (4.34),

krDðskÞk26

eC 3ðDðskÞ � Dðskþ1ÞÞhg

! 0:

The fact that any limit point s⁄ is a minimum is clear from the fact that Hk is symmetric and positive-definite.

Remark 3. We remark that although Theorem 1 states global convergence of the scheme, the functional (4.21) that we areinterested in minimizing has multiple local minima. Due to this, the results of the iteration depend on the choice of the initialguess, and thus, we require the initial guess be close enough to the true solution. To demonstrate that the problem has multi-ple local minima, we consider the following two examples:

Suppose s = (s1,s2) 2 [0,1] [0,1] and the original function is ~s ¼ 0:45;0:67ð Þ. In the first example, we consider the first

eigenvalue k0pi

n ofor four pressures 45 mmHg, 93 mmHg, 100 mmHg and 120 mmHg. The ~k0

pi

n o4

i¼1are assumed to be given

and D(s) is plotted as a function of s1 and s2 (see Fig. 10(a)).In the second example, we consider the first three eigenvalues k0

pi; k1

pi; k2

pi

n ofor eight pressures 30 mmHg, 45 mmHg,

54 mmHg, 93 mmHg, 100 mmHg, 120 mmHg, 130 mmHg and 150 mmHg. The k0pi; k1

pi; k2

pi

n o8

i¼1are assumed to be given

and D(s) is plotted as a function of s1 and s2 (see Fig. 10(b)).In Fig. 10, we see that the multiple local minima of the objective functional D(s) are very close to one another. This feature

makes the inverse problem very ill-posed, and finding the true minima becomes very challenging. In order to get an accurateenough solution, the initial guess for the gradient based methods need to be close to the true solution. Also, note that theminima are further apart for a larger number of given eigenfrequencies. Thus, one can increase the accuracy of reconstruc-tion of the residual stress by considering a higher number of eigenfrequencies from various pressures.

4.5. Numerical examples

In this section, we present numerical simulations of reconstructing several residual stresses s using the algorithms de-scribed above and compare the results for different parameters. The material parameters are chosen as in Section 3.3. TheLevenberg–Marquardt parameter ck is chosen as in (4.31), n = 0.5 and h = 1.0 10�3. The true value strue and the returned va-lue sfinal of s at each node is given along with the L2 absolute error kstrue � sfinalkL2 and the L2 norm ofrD(s). A corresponding

Fig. 10. Plot of D(s) showing multiple local minima.

Table 9Numerical Example 5 given {k0} of 4 and 10 blood pressures.

True value (kPa) 0.0 0.222 0.222 0.0Initial guess (kPa) �0.02 0.12 0.1 �0.03

No. of pressures 4Output (kPa) �0.004 0.203 0.232 0.01Iteration number 40Absolute L2 error 2.5e�02krDkL2 7.43e�05

No. of pressures 10Output (kPa) �0.003 0.2112 0.2311 0.006Iteration number 29Absolute L2 error 1.0e�02krDkL2 7.76e�04

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x(mm)

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x (mm)

Fig. 11. Reconstruction of Example 5.


plot of the true s and the iterated s for each table is presented. Note that in the figures, the solid line represents the true s andthe dotted line represents the iterated s.

Example 5. The function we want to approximate is s ¼ 1� x�xi

xo�xi

� �2and we use the cubic spline interpolation of the

function at 4 nodes. The initial guess is taken to be (�0.03,0.1,0.12,�0.02). Table 9 shows the input and output for theiterations given the first eigenvalue {k0} of 4 pressures, {45,93,100,120} mmHg, and 10 pressures,

Table 10Numerical Example 5 given {k0,k1} of 10 blood pressures.

No. of pressures 10

True value (kPa) 0.0 0.222 0.222 0.0Initial guess (kPa) 1 2.0 2.0 1.0Output (kPa) �0.001 0.224 0.224 �0.001

Iteration number 103Absolute L2 error 1.88e�06krDkL2 5.0e�03

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

x (mm)

Fig. 12. Reconstruction of Example 5 for initial guess {1.0,2.0, 2.0,1.0}.

Table 11Numerical Example 5 given {k0} and {k0,k1} of 10 blood pressures.

No. of pressures 10Tolerance tr = 1.0e�05 ta = 1.0e�06True value (kPa) 0.0 0.222 0.222 0.0Initial guess (kPa) 5.0 5.0 5.0 5.0

Eigenfrequencies {k0}Output (kPa) 0.217 �0.005 �0.005 0.217Iteration number 41Absolute L2 error 2.4e�01krDkL2 8.5e�04

Eigenfrequencies {k0,k1}Output (kPa) �0.001 0.223 0.223 �0.001Iteration number 94Absolute L2 error 1.8e�03krDkL2 4.4e�04


{30,45,54,75,93,100,105,120,130,150} mmHg, and the reconstructions are shown in Fig. 11. Note that reconstructing thefunction with {k0} of 4 pressures is equivalent to solving a system of nonlinear equations, and it already gives a sufficientlyaccurate estimation for this particular model problem given that the initial guess is close to the solution. Although thereconstruction is only slightly better with 10 pressures than with 4, the number of iterations required for the algorithm toconverge is lower in this case.

For the next two examples, consider 10 pressures, {30,45,54,75,93,100,105,120,130,150} mmHg. In Table 10, we showthe results for Example 5 with an initial guess far away from the true function, and plot the reconstruction in Fig. 12. Twoeigenvalues {k0, k1} for each pressures is taken as data. This example shows that a high number of iterations is a trade off forhigher accuracy. It is important to note that despite the existence of multiple local minima, the algorithm converges to thetrue solution for this particular choice of the initial guess. Next we take an initial guess further away from the solution andtwo cases, first with {k0} given for each pressure, and second with {k0,k1} given for each pressure. See Table 11 and Fig. 13 foroutput and plots. This shows that the first eigenvalues {k0} are not sufficient to get a good estimate of the function if the

-0.05

0

0.05

0.1

0.15

0.2

0.25

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x (mm)

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x (mm)

Fig. 13. Reconstruction of Example 5 and Table 11.

Table 12Numerical Example 6 with 6 blood pressures.

No. of pressures 6Tolerance tr = 1.0e�05 ta = 1.0e�06True value (kPa) 0.0 0.866 0.866 0.0 0.866 0.866 0.0Initial guess (kPa) �0.01 2.0 2.0 �0.3 �2.0 �2.0 0.02

Hk updated with (4.20)Output (kPa) �0.04 1.923 1.976 �0.02 �1.99 �1.98 0.097

Hk updated with (4.17)Output (kPa) �0.08 0.832 0.894 �0.0118 �0.906 �0.858 0.146Iteration number 46Absolute L2 error 1.7e�1krDkL2 7.23e�5

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x (mm)

-1.5

-1

-0.5

0

0.5

1

1.5

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2x (mm)

(a) Hk Updated with (4.20) (b) Hk Updated with (4.17)

Fig. 14. Reconstruction of Example 6 and Table 12.


initial guess is far away from the solution, and may result in convergence to a different local minimizer. The first eigenvaluesof the final solutions {0.217,�0.005,�0.005,0.217} and {0.217,�0.005,�0.005,0.217} were computed and found to be equal,which is why the line search was unable to locate the right direction of descent.

Table 13Given {k0} of 10 blood pressures with 0.001% noise.

True value (kPa) 0.0 0.222 0.222 0.0Initial guess (kPa) �0.02 0.12 0.1 �0.03

No. of pressures 10Output (kPa) 0.088 0.237 0.216 0.078Iteration number 33Absolute L2 error 1.1e�01krDkL2 4.39e�05

0

0.05

0.1

0.15

0.2

0.25

0.3

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

Fig. 15. Reconstruction of Example 5 with 0.01% noise.


Example 6. In this example, the original function is s ¼ sin 2p x�xi

xo�xi

� �� and we use the cubic spline interpolation of the

function at 7 nodes. The function is estimated given the first three eigenfrequencies of six pressures {45,93,100,120,130}mmHg and the initial guess is taken to be (�0.01,2.0,2.0,�0.3,�2.0,�2.0,0.02). We employ two different methods forupdating the matrix Hk in the Levenberg–Marquardt algorithm, namely, using (4.17) and (4.20) with all other parametersfixed and compare the results. We observe that although the initial guess is far away from the true solution, the objective

function Dðsð0ÞÞ ¼ 12

Pmnl¼1 jk

lðsð0ÞÞ � ~klj2 and the Jacobian norm kR0(s(0))k are very small. As a result, the value of

ck diag[R0(s(k))TR0(s(k))] at the k-th iteration, with ck given by (4.31), is very small, and hence, (4.20) acts like a Gauss

Newton iteration.This makes the algorithm very slow, and in cases where it converges, it requires a lot of iterations. On the other hand, if

(4.17) is used to update the Hk, the step lengths at each iteration are large enough for the algorithm to converge to the truesolution efficiently. Our numerical experiment shown in Table 12 depict this. We halt the iteration for (4.20) in 42 steps tocompare it with the other case which converges in 42 iterations. The plots are shown in Fig. 14.

The following example is to test the robustness of the optimization algorithm. We consider Example 5 with 0.01% noise inthe eigenvalues. The input and output details are given in Table 13 and the reconstruction is given in Fig. 15. Just as thesecant method described in Section 3, this algorithm converges for data with noise up to 0.01%.

5. Conclusions

An inverse spectral method is developed to estimate the residual stresses and pre-stresses in a nonlinear rectangular elas-tic body by making a novel use of ultrasound technology. The proposed method makes fundamental use of the nonlinearmaterial behavior via spectra of the linearized vibrations problems corresponding to different quasi-static pressures. Itshould be emphasized that this approach utilizes the first few eigenfrequencies of a large number of pressures instead ofa large number of eigenfrequencies for a single pressure. This feature not only exploits the nonlinearity of the problem,but also makes the method more robust since the lower eigenfrequencies are the easiest to measure accurately via exper-iments. Several algorithms are developed to account for the ill-posedness of the problem of recovering the residual stressesvia such methods (Gorb & Walton, 2010). Numerical simulations are presented to show the viability and robustness of themethods. However, 0.01% is the level of accuracy that is unrealistic for measuring eigenfrequencies using current technology,hence, there is a need to develop other optimization algorithms to increase robustness to noise.


Acknowledgment

This work is based in part on support provided by Award No. KUS-C1–016-04 from King Abdullah University of Scienceand Technology (KAUST).

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Reconstruction of the residual stresses in a hyperelastic body using ultrasound techniques

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