INSTITUTE OF PHYSICS PUBLISHING INVERSE PROBLEMS

Inverse Problems 22 (2006) 681–706 doi:10.1088/0266-5611/22/2/018

Shear wave speed recovery in transient elastographyand supersonic imaging using propagating fronts

Joyce McLaughlin and Daniel Renzi

Mathematics Department, Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York,12180, USA

E-mail: mclauj@rpi.edu and renzid@rpi.edu

Received 2 March 2005, in final form 26 February 2006Published 27 March 2006Online at stacks.iop.org/IP/22/681

AbstractTransient elastography and supersonic imaging are promising new techniquesfor characterizing the elasticity of soft tissues. Using this method, an ‘ultrafastimaging’ system (up to 10 000 frames s−1) follows in real time the propagationof a low frequency shear wave. The displacement of the propagating shearwave is measured as a function of time and space. The objective of thispaper is to develop and test algorithms whose ultimate product is images ofthe shear wave speed of tissue mimicking phantoms. The data used in thealgorithms are the front of the propagating shear wave. Here, we first developtechniques to find the arrival time surface given the displacement data from atransient elastography experiment. The arrival time surface satisfies the Eikonalequation. We then propose a family of methods, called distance methods, tosolve the inverse Eikonal equation: given the arrival times of a propagatingwave, find the wave speed. Lastly, we explain why simple inversion schemesfor the inverse Eikonal equation lead to large outliers in the wave speed andnumerically demonstrate that the new scheme presented here does not haveany large outliers. We exhibit two recoveries using these methods: one is withsynthetic data; the other is with laboratory data obtained by Mathias Fink’sgroup (the Laboratoire Ondes et Acoustique, ESPCI, Universite Paris VII).

1. Introduction and overview

The goal of elastography is to create high-resolution shear stiffness images of human tissuefor diagnostic purposes. Shear stiffness is targeted because the shear wave speed is 2–4times larger in abnormal tissue than in normal tissue. The goal is to take advantage of thislarge contrast to generate high-resolution images. The expectation is that shear stiffnessimages will identify abnormal tissue not identified by standard ultrasound techniques. Initial

0266-5611/06/020681+26$30.00 © 2006 IOP Publishing Ltd Printed in the UK 681

682 J McLaughlin and D Renzi

experiments indicate that this can be achieved, see Fink [41], for preliminary work in thisarea.

Four experiments have been proposed to detect shear stiffness variations:

• Static experiment: the tissue is compressed;• Dynamic sinusoidal excitation: a time harmonic excitation made on the boundary creates

a time harmonic wave in the tissue;• Acoustic radiation force impulse (ARFI): focused acoustic pulses are used to generate

interior sources of displacement;• Transient elastography or supersonic shear imaging: a time-dependent pulse on the

boundary, or along a line in the interior, creates a propagating wave in the tissue.

In each of these cases, the interior displacement is measured on a fine grid of points usingultrasound [3–5, 8, 12, 13, 20–23, 32–34, 37, 41–46] or magnetic resonance imaging [7, 10,15, 24–31]. The aim is to create a shear wave speed image from the displacement data. Havinginterior displacement data enables us to develop a local algorithm, that is, to obtain the wavespeed at a point from displacement data at only neighbouring points.

In the static experiment, see [37], the tissue is compressed creating a large deformation.Naturally, less change in the tissue thickness is observed in stiff regions. In [37], inversemethods, that is methods that find the material parameters of the governing partial differentialequation, are not applied to obtain shear stiffness images, but rather images are made of thedisplacement or strain itself. It is shown that large displacements or strain occur in normaltissue and smaller displacement or strain regions indicate where stiff inclusions occur. If oneapplies inverse methods to reconstruct the elastic parameters from a single experiment, becausethe boundary conditions are unknown, there is some non-uniqueness. Inverse methods areapplied to image shear stiffness in [1, 2, 6, 9, 10, 35, 36, 38] using linear elastic mathematicalmodels for the forward problem.

Images have also been created using dynamic sinusoidal excitation. In these experiments,a time harmonic excitation is made on the boundary creating a time harmonic wave and dataare recorded after the tissue has reached steady state. We review two measurement techniquesfor this problem. The first technique is sonoelastography [12, 45, 46]. Inverse methods are notapplied to obtain shear stiffness images using sonoelastography, but rather images are madeof the maximal amplitude of the harmonic displacement obtained using Doppler ultrasound.It is shown that regions of small displacement indicate stiff inclusions. Examples are shownin [12, 45, 46] when small displacement created by a dynamic sinusoidal excitation identifiesabnormal tissue which conventional ultrasound does not. The second technique is magneticresonance elastography (MRE) [7, 15, 24–27, 31, 47]. In MRE, the displacement of thetime harmonic wave is measured as a function of space and time, and inverse techniques areapplied to recover the shear modulus. Interior displacement data are obtained by taking severalmeasurements at each of a set of times equally spaced with respect to the sinusoidal excitation.Data are high quality and data acquisition time is of the order of minutes. Once more, commentneeds to be made. Boundary conditions or some a priori assumptions, for example the ‘locallyconstant’ assumption that neglects terms in the model that contain derivatives of the unknown,need to be determined to enable recovery of a unique solution.

Acoustic radiation force impulse (ARFI) imaging [32–34] was developed in part toovercome difficulties with unknown boundary conditions. Brief, high energy, focused acousticpulses are used to generate well-localized radiation force in tissue, and the resulting tissuedisplacements near the focus of the acoustic pulses are measured with ultrasound. Indirectmeasures of the stiffness such as the peak displacement are imaged. In [34], in vivo images ofablated cardiac tissue clearly demonstrate small amplitude in the ablated regions.

Shear wave speed recovery in transient elastography 683

Transient elastography or supersonic imaging is the focus of this paper. In this experiment,tissue initially at rest is excited with a broadband pulse at the boundary or along a line in theinterior. Typically, the pulse has a central frequency of 50–200 Hz. This pulse creates twodifferent types of waves that propagate into the medium. The first wave displaces tissue inthe same direction that the wave propagates. This wave is comparatively fast and is called thecompression wave. Tissue in the human body is composed mostly of water. For this reason,the compression wave travels at approximately the speed of sound in water, 1500 m s−1. Thesecond wave displaces tissue orthogonally to the direction that the wave propagates. Thiswave is very slow compared to the compression wave, approximately 3 m s−1, and is calledthe shear wave. The propagating shear wave is measured with the ultrafast ultrasound-basedimaging system developed in the laboratory of Mathias Fink, ESPCI [8]. Data acquisition isof the order of 100 ms.

We will take advantage of several properties of the transient elastography experiments todevelop algorithms for shear wave speed recovery. First of all, the propagating waves havelow amplitude: the displacements are of the order of microns. This suggests we can use alinear model. The linear model assumption is also argued as follows. Mathematical modelsdescribing wave motion are based on the stress–strain curve. This curve describes how tissuedeforms (strain) when a force (stress) is applied. Assuming a linear stress–strain relationleads to a linear mathematical model, in our case the linear equations of elasticity. In [42],the stress–strain curve for transient elastography experiments is shown to be approximatelylinear. For these reasons, in this initial work, we use a linear model.

Secondly, the compression wave speed, Cp, is several orders of magnitude greater thanthe shear wave speed, Cs , in biological tissue with very long wavelength and so it is slowlyvarying. Furthermore, for the constant coefficient case in an elastic half space, the exactsolution has been found in [30], and from this solution it is clear that the amplitude of thecompression wave is very small, O((Cs/Cp)4), when the ratio Cp/Cs is large. In addition,the experimenters measure only one component of the elastic wave, and it is measured usingultrasound which is based on reflections of high frequency compression waves. For thesereasons, we will model the compression wave contributions as noise in the displacement dataand assume that the measured component satisfies the acoustic wave equation. In a laterpaper, we will use the linear equations of elasticity as the mathematical model when eitherwe have more than one component, in which case we will model the compression wave partof the model as pressure, or when we use a geometric optics approximation (see the remarkbelow).

Lastly, the boundary or line pulse generates a shear wave front that propagates throughthe medium. In the experiment, the interior displacement of the shear wave is measured as afunction of space and time, see [5, 8, 41, 44]. Because we have interior data, we can determinethe spacetime position of this propagating shear wave front. The final goal then is to use asubset of the full data set, that is, the spacetime position of the propagating front to reconstructthe shear wave speed.

So, in this paper, we consider the following inverse problem: given interior displacementmeasurements from a transient elastography experiment, determine the propagating shearwave front; and from the knowledge of the spacetime position of this front, determine theshear wave speed. To accomplish this, we have already shown [19] that using the assumptionof a wave equation model, Lipschitz continuous arrival times of the propagating shear wavefront satisfy the Eikonal equation. This extended previously known results which requiredthat the front be C1 in order to establish that the Eikonal equation is satisfied. Our interest inestablishing this extension arises because our medium is heterogeneous with large contrasts.This means that a C1 propagating front assumption is unrealistic.

684 J McLaughlin and D Renzi

The Eikonal equation relates the arrival times to the shear wave speed. This implies thatthe shear wave speed can be determined from a hyper-subsurface of the shear displacementdata: the spacetime position of the propagating shear wave front. The primary benefits tousing the Eikonal equation is that it is both lower order and lower dimensional than the waveequation. It contains only first-order spatial derivatives. We will develop a fast shear wavespeed recovery algorithm that takes advantage of these benefits. In a later paper, we will showthat when the mathematical model is the linear equations of elasticity and under the geometricoptics approximation assumption, the front of the shear wave still satisfies this same Eikonalequation.

To accomplish our goal of creating images of shear wave speed, we develop an algorithmcomposed of two sub-algorithms:

• Finding the arrival time front from the shear wave displacements.• Using the Eikonal equation to find the shear wave speed.

The first sub-algorithm is based on the following observation. Let y be a point near the linewhere the source is applied. Consider a time trace of the data at y, starting with the time theshear wave is generated, say t = 0. Now consider a time trace of the displacement at a point xin the interior. The shear wave takes some time to reach this point, but the shape of the waveis nearly preserved as it propagates through the medium. That is, we should observe roughlythe same time trace at x as we observed at y, only at x it starts at a later time, say t = t1, whichis the arrival time of the shear wave at x. We then determine t1 by matching the pattern weobserve from the shear wave at y near the source with the pattern we observe from the shearwave at any x in the interior. We implement this idea using a cross-correlation procedure.

The second sub-algorithm uses the Eikonal equation to find the shear wave speed from thearrival time information output from the first sub-algorithm. Extracting the wave speed fromthe arrival times is an ill-posed problem for two reasons. First, determining the wave speedinvolves derivatives of the arrival times. Second, the wave speed is inversely proportional tothe gradient of the arrival times, so derivatives of the arrival times appear in the denominator.Unfortunately, the quantity in the denominator is of the order of the noise level, particularlyin the high speed region of interest, when using experimental data. This means that withthese data, the quantity in the denominator can become very small, leading to large outliers inthe recovered wave speed. We remove this problem by developing a linear relation betweenapproximations to the wave speed and distances between level sets of the arrival time surface.This leads to difference schemes where the derivatives are in the numerator. These differenceschemes could be of arbitrarily high order depending on the local smoothness of the arrivaltimes. A simple implementation of these schemes, which we refer to as distance methods,involves finding distances from each point to neighbouring contours. Here, we implement thefirst- and second-order distance methods on a test problem. We demonstrate that the numericalconvergence rates agree with the theoretical convergence rates and that the methods are robustwith respect to noise. With the distance implementation the running time is O(m3/2), wherem is the number of grid points. Finally, we test the complete algorithm on both syntheticdata computed numerically from the acoustic wave equation and experimental data from thelaboratory of Mathias Fink (Laboratoire Ondes et Acoustique, ESPCI, Universite Paris VII).

The remainder of this paper is composed as follows. Section 2 contains the mathematicalmodel and theory. The sub-algorithms are developed in sections 3 and 4. The algorithm istested on synthetic data and laboratory data in section 5. Finally, we give some concludingremarks in section 6.

Prior work of the authors to show uniqueness, to create images from single frequency datacontent and to establish the equation satisfied by the arrival time is contained in [16–19, 29].

Shear wave speed recovery in transient elastography 685

2. Reducing the linear equations of elasticity to the wave equation and uniqueidentifiability of the shear wave speed

For an isotropic model, the material parameters that describe the medium are the Lameparameters, λ and µ, and the density, ρ. The vector elastic displacement �u is governed by thefollowing hyperbolic system:

∇(λ∇ · �u) + ∇ · (µ(∇�u + ∇�uT )) = ρ�utt in � × (0, T ). (2.1)

Here, (·)T denotes the transpose of matrices and I is the identity matrix and ∇· represents thedivergence of vectors or matrices according to the context. We arrive at the wave equationmodel for any single component by neglecting the terms ∇(λ∇·�u) and ∇·(µ∇�uT ) and treatingtheir contribution as noise. One reason this is successful is that tissue is mostly water and sothe volumetric change in tissue, measured by ∇ · �u, is nearly zero. (In a later paper, we willexplain how to include these terms as well and still arrive at the same arrival time equationgiven below.)

For the reasons given above, we assume that the downward component of the elastic wavesatisfies the following wave equation:

∇ · (µ(x)∇u(x, t)) = ρ(x)utt (x, t) in � × (0, T ). (2.2)

Consistent with the experiment, we assume that the medium is initially at rest satisfying thehomogeneous initial condition

u(x, 0) = ut (x, 0) = 0 on �, (2.3)

and one of the following boundary conditions:

u(x, t) = f (x, t) on ∂� × (0, T ), (2.4)

or

µ(x)∇u(x, t) · ν(x) = g(x, t) on ∂� × (0, T ), (2.5)

where ν is the outward normal to ∂� which is assumed for the transient elastographyexperiment. Note that in the experiment now being used (2.5) is the correct representation forthe applied force.

We here review uniqueness results for the following inverse problem: given interior timeand space-dependent displacement data for the model above, recover ρ ∈ C(�) and Lipschitzcontinuous µ. This problem was studied in [29] and [19]. There they show that there is atmost one shear wave speed,

√µ/ρ, corresponding to a solution of the scalar wave equation

measured throughout � × (0, T ). If µ is either given on the boundary or determined from theboundary traction force, then at most one pair (µ, ρ) corresponds to a given solution.

These results show that the displacement data as a function of time and space in �×(0, T )

is a very rich data set. At least two coefficients, µ and ρ, can be simultaneously determinedfrom this data set. In a later paper, we will establish the same result for the supersonicimaging experiment where the non-homogeneous boundary condition will be replaced bya non-homogeneous acoustic equation. In this paper, we are only interested in recoveringthe wave speed,

√µ/ρ, and to achieve this we only use a data subset, the position of the

propagating front. To achieve our goal, we review another result given in [19]. There weassume that the propagating front is Lipschitz continuous, µ, ρ ∈ C1; we define the arrivaltime of the propagating front as

T (x) := inf{t ∈ (0, T ) : |u(x, t)| > 0}, (2.6)

and we consider the following inverse problem: given the arrival times, T (x), of thepropagating front recover the wave speed,

√µ/ρ. We summarize the results stated in [19]

here.

686 J McLaughlin and D Renzi

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Figure 1. (A) Time trace of synthetic data at the source boundary. (B) Time trace of synthetic dataat the interior.

• Under the assumption that T (x) is Lipschitz continuous then the arrival timescorresponding to a solution of the scalar wave equation satisfy the Eikonal equation,

|∇T |√

µ/ρ = 1, a.e. (2.7)

• There is at most one wave speed,√

µ/ρ, that corresponds to a given arrival time, T (x),in the region of the medium that has experienced wave propagation.

• The wave speed√

µ/ρ depends continuously on |∇T | under the assumption that |∇T |has a positive lower bound.

In this paper, we will recover the wave speed using the Eikonal equation. The primaryadvantages of using the arrival times are: (1) there is one less derivative in equation (2.7).Taking derivatives of noisy data is always an issue, as it is an ill-posed operation, and the fewerderivatives the better; (2) the Eikonal equation has one less dimension (there is no time variablein the Eikonal equation). This suggests that either simpler algorithms or faster algorithms arepossible. In this paper, we present an elementary algorithm based on the concept of the speedof the propagating front. In a later paper, we will present a second fast algorithm.

3. Techniques for finding the arrival times

Before we can utilize arrival times we must extract them from the data. So, in this sectionwe investigate techniques for finding the arrival times, T (x), given the displacement, u(x, t).We will use a pattern matching idea based on the following observation. Let y be a point onthe boundary where the source is applied and let f (t) be the time trace at y. Now let g(t) bethe time trace at a point x in the interior. The shape of time trace f (t) should be roughly thesame as g(t − t1) when t1 is arrival time of the wave at x. To illustrate, figure 1(A) showsa time trace of synthetic data at a point y on the source boundary and figure 1(B) shows atime trace of synthetic data at a point x in the interior. Looking at these two time traces, the

Shear wave speed recovery in transient elastography 687

human eye can easily see and match the two patterns. Now we discuss some pattern matchingalgorithms.

One way to do the pattern matching is to try to identify a feature of the pattern the shearwave creates in the displacement data and track this feature as the wave propagates throughthe medium. The astute reader may want to take advantage of the fact that the displacementis zero before the shear wave arrives. So a detecting feature could be that the pattern is the‘first’ nonzero displacement observed in the time trace. Looking at the definition of arrivaltimes in (2.6), an obvious approach when u is continuous would be to generate an estimate ofthe arrival time by

T (x) ≈ inf{t ∈ (0, T ) : |u(x, t)| > δ},where δ > 0 is a fixed threshold above the noise level. The problem with this approach isthat as the wave propagates through the medium, if the wave speed is not constant, then theamplitude is decreased due to wave spreading and backscattering. This means that the timeselected, even without noise, is not a consistent shift from the true arrival time. See [39] for amore detailed description. In deciding what alternative to use for finding the arrival times, wenote that the most appropriate feature to track may depend on the wave pulse. Naturally, wewould prefer to use a method that works on a wide variety of wave pulses and automaticallyselects the detecting features based on the input wave pulse. We accomplish this by using across-correlation technique. The displacement time history u(x, t) at each spatial point x iscross-correlated against the displacement time history u(xref, t) at a single reference point xref ,chosen near the source. We compute the biased cross-correlation of u(x, t) and u(xref, t) foreach x using the following formula:

C(x, δt) := 1

T

∫ T

0u(xref, t)u(x, t − δt) dt,

u(x, t) =

u(x, t) if 0 � t � T ,u(x, t − T ) if t > T ,u(x, t + T ) if t < 0.

Now we estimate the arrival times by T (x) ≈ δtmax, where

δtmax := argmaxδt∈[0,T ]C(x, δt).

So T (x) is estimated by the time delay δt that maximizes the correlation between the signalsu(xref, t) and u(x, t −δt). The idea behind this method is that as the wave pulse travels throughthe medium it is highly correlated with the wave pulse in the reference signal, while the restof the data are relatively uncorrelated. To illustrate, figures 2(A) and (B) show the referencesignal and a displacement time trace of synthetic data. Note that the displacement time tracein figure 2(B) is given at a point after the shear wave has passed through a high speed regionand significant amplitude loss has occurred. Figure 2(C) shows the correlation as a functionof −δt . The time-delayed displacement time trace that has the maximum correlation with thereference signal is shown in figure 2(D) along with the reference signal. As we can see, thecross-correlation technique uses the entire phase content of the two signals as the detectingfeature. We show arrival time surfaces for synthetic data and laboratory data obtained froma phantom experiment in section 5. Below we also point out artefacts that result in the shearwave speed image because of this approach. In a later paper, we will improve this method toreduce these artefacts.

688 J McLaughlin and D Renzi

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(D) reference signaltime trace

Figure 2. (A) Reference signal; (B) time trace of synthetic data; (C) cross-correlation function ofthe signals shown in (A) and (B); (D) the reference signal (solid line) from (A) is shown with thesynthetic data time trace (dashed line) from (B) that is time delayed by an amount that maximizesthe correlation of the two signals.

4. Two solution techniques for the inverse Eikonal equation

In this section, x ∈ Rn, n = 2, 3, and we develop two elementary solution techniques for theinverse Eikonal equation (2.7). As discussed in section 2, there are two causes of ill-posednessin this problem. The first is that the wave speed,

√µ/ρ, depends on derivatives of the arrival

times, T . The second is that if you solve for the wave speed,√

µ/ρ = 1/|∇T |, then noise inthe data drives the quantity |∇T | even closer to zero, which can cause very large outliers in theapproximate wave speed

√µ/ρ . See [39] for a simple 1D example where Tx is approximated

by a forward Euler approximation that illustrates this phenomena. Note that when we calculateour approximation to the wave speed by making a forward approximation to the derivativesin |∇T |, we call this a simplistic method. To avoid outliers, our more robust method toapproximate the wave speed uses the following elementary idea: instead of approximating|∇T |−1 we calculate the wave speed directly. We make the approximation

õ(x)

ρ(x)≈ 1

�tinf{|x − x| : x satisfies T (x) = T (x) + �t}, (4.1)

for a small fixed �t > 0. This is essentially the distance from the point x to the level curveT (x) = T (x) + �t divided by �t , and when �t → 0, since T is differentiable at a.e. x, theright-hand side of (4.1) yields the speed of the level set of T at x in the normal direction.Whenever we use this idea directly we refer to this as the first-order distance method. As onemight expect, the accuracy of (4.1) is O(�t) when |∇T | is Lipschitz continuous. To provethis, we will use the following lemma:

Shear wave speed recovery in transient elastography 689

Lemma 1. Suppose T (x) ∈ C1(�) and satisfies the Eikonal equation, |∇T |√µ(x)/ρ(x) = 1,with 0 <

√µ(x)/ρ(x) < M . Then for sufficiently small �t

|�t | = O(inf{|x − x| : x satisfies T (x) = T (x) + �t}|). (4.2)

Proof. Without loss of generality assume �t > 0. We first note that given any x ∈ �, since∇T is uniformly continuous in �, there exists a δ > 0 satisfying∣∣∣∣∣|∇T (x)| − ∇T (y) · ∇T (z)

|∇T (z)|

∣∣∣∣∣ < |∇T (x)|/2, (4.3)

for |z − x| < δ, y = s ∗ (z − x), 0 � s � 1. Second, for any fixed but arbitrary x, choose�t < max|x−x|�δ/2T (x) − T (x). Third, let

x+ = argminT (x)=T (x)+�t |x − x| (4.4)

and define dx+ = (x+ − x)/|x+ − x|. The Euler–Lagrange equations for the constrainedoptimization problem are

x − x

|x − x| = λ∇T (x), (4.5)

implying that

dx+ = �t

|�t |∇T (x+)

|∇T (x+)| = ∇T (x+)

|∇T (x+)| . (4.6)

Furthermore, |x+ − x| < δ since there exists z, |z − x| < δ, with T (z) − T (x) = �t implyingthat

�t = T (x+) − T (x) = |x+ − x|∇T (y) · dx+ >|x+ − x|

2M, (4.7)

where y = x + s(x+ − x), 0 � s � 1. �

Theorem 1. Suppose T (x) ∈ C2(�) and satisfies the Eikonal equation, |∇T |√µ(x)/ρ(x) =1, with 0 <

√µ(x)/ρ(x) < M . Then,∣∣∣∣∣

õ(x)

ρ(x)− 1

�tinf{|x − x| : x satisfies T (x) = T (x) + �t}

∣∣∣∣∣ = O(�t). (4.8)

Proof. Using the notation in the proof of lemma 1, we first reduce δ, if necessary, so thatthe absolute value of any eigenvalue of the Hessian, H = (∂T /(∂xi∂xj ))

2i,j=1, is less than

1/(4Mδ). Now Taylor expanding T (x+) − T (x) about x+ when |x − x+| < δ yields

�t

|x+ − x| = T (x+) − T (x)

|x+ − x| = ∇T (x+) · dx+ − dTx+H(y)dx+ |x+ − x|, (4.9)

where y = x + h(x+ − x), 0 � h � 1. Inverting, using lemma 1 together with|1 − 1/(1 − |z|)| � 2|z| when |z| < 1/2, we obtain∣∣∣∣ |x − x+|

�t− 1

∇T (x+) · dx+

∣∣∣∣ � 2dTx+H(y)dx+ |x+ − x|(∇T (x+) · dx+)2

. (4.10)

Hence,

|x − x+|�t

= 1

∇T (x+) · dx+

+ O(�t) =√

µ(x+)

ρ(x+)+ O(�t) =

õ(x)

ρ(x)+ O(�t). (4.11)

�

690 J McLaughlin and D Renzi

Furthermore, we can obtain an O(�t)2, second-order distance method, from the formula√µ(x)

ρ(x)≈

{1

2�t(inf

x+|x − x+| + inf

x−|x − x−|) : x±satisfies T (x±) = T (x) ± �t

}, (4.12)

which we verify as follows:

Theorem 2. Suppose T (x) ∈ C3(�) and satisfies the Eikonal equation, |∇T |√µ(x)/ρ(x) =1, with 0 <

√µ(x)/ρ(x) < M. Then,∣∣∣∣∣

õ(x)

ρ(x)−

{1

2�t(inf

x+|x − x+| + inf

x−|x − x−|) : x±satisfies T (x±) = T (x) ± �t

}∣∣∣∣∣= O(�t2). (4.13)

Proof. Using the notation in the proof of lemma 1, we first let x− = argminT (x)=T (x)−�t |x−x|.Theorem 1 implies∣∣∣∣∣√

µ(x)

ρ(x)�t − |x+ − x|

∣∣∣∣∣ = O(�t2) and

∣∣∣∣∣√

µ(x)

ρ(x)�t − |x− − x|

∣∣∣∣∣ = O(�t2).

(4.14)

Now the triangle inequality yields the estimate

|x+ − x| − |x− − x| = O(�t2). (4.15)

Second, we let dx− = (x − x−)/|x − x−| = ∇T (x−)/|∇T (x−)| and Taylor expand√µ(x+)/ρ(x+) +

√µ(x−)/ρ(x−) about x to obtain√

µ(x+)

ρ(x+)+

√µ(x−)

ρ(x−)= 2

õ(x)

ρ(x)+ ∇

õ(x)

ρ(x)· dx+ |x −x+| − ∇

õ(x)

ρ(x)· dx−|x − x−| + O(�t2)

= 2

õ(x)

ρ(x)+ ∇

õ(x)

ρ(x)· (dx+ − dx−)|x − x|

− (|x+ − x| − |x− − x|)∇√

µ(x)

ρ(x)· dx− + O(�t2)

= 2

õ(x)

ρ(x)+ O(�t2). (4.16)

Third, Taylor expansion of dTx−H(x−)dx− about x+ yields that

dTx−H(x−)dx−|x− − x| − dT

x+H(x+)dx+ |x+ − x|= (

dTx−H(x−)dx− − dT

x+H(x+)dx+

)|x+ − x| + dTx−H(x−)dx−(|x− − x| − |x+ − x|)

= O(�t2). (4.17)

Now, we let a1 = |∇T (x+)|, a2 = |∇T (x−)|, b1 = −dTx−H(x−)dx− |x− − x| + O(�t2) and

b2 = dTx+H(x+)dx+ |x+ − x| + O(�t2). Taylor expansion of |x+ − x|/(T (x+) − T (x)) + |x− −

x|/(T (x) − T (x−)), along with equations (4.16) and (4.17), yields

|x − x−| + |x − x+|�t

= |x − x−|T (x) − T (x−)

+|x − x+|

T (x+) − T (x)= 1

a1(1 + b1

a1

) +1

a2(1 + b2

a2

)

Shear wave speed recovery in transient elastography 691

Table 1. L∞ error for the simplistic method.

h L∞ error forward Euler L∞ error centred difference

0.1 0.0504 0.00180.05 0.0251 0.000 458 710.025 0.0125 0.000 114 69

= 1

a1

(1(

1 + b1a1

) − 1

)+

1

a1+

1

a2

(1(

1 + b2a2

) − 1

)+

1

a2

= 1

a1+

1

a2+

1

a1

( −b1a1(

1 + b1a1

) +b1

a1

)− b1

a21

+1

a2

( −b2a2(

1 + b2a2

) +b2

a2

)− b2

a22

= 1

a1+

1

a2− b1

a21

− b2

a22

+1

a21

b1a1

2(1 + b1

a1

) +1

a22

b2a2

2(1 + b2

a2

)= 1

a1+

1

a2− b1

a21

+b1

a22

− b1

a22

− b2

a22

+ O(�t2)

= 1

a1+

1

a2− b1

(1

a1+

1

a2

) (1

a1− 1

a2

)− 1

a22

(b1 + b2) + O(�t2)

= 2

õ(x)

ρ(x)+ O(�t2). �

4.1. Numerical accuracy and stability tests of the distance method

In this section, we numerically test the accuracy and stability of equations (4.1), (4.12) on thefollowing test problem, where (x, y) ∈ R2:√

µ(x, y)/ρ(x, y) = 2 + sin(x), (x, y) ∈ [0, 10] × [0, 10], T (0, y) = 0, (4.18)

where T satisfies the Eikonal equation. The wave speed√

µ(x, y)/ρ(x, y) is shown infigure 3(A). This problem has the exact solution

T (x, y) =∫ x

01/(2 + sin(x1)) dx1. (4.19)

The arrival times T (x, y) are shown in figure 3(B). We first test first- and second-ordersimplistic methods where ∇T is calculated using a forward Euler approximation for thefirst-order method and a centred difference approximation for the second-order method. Thesolutions with constant grid spacing h = 0.1 are shown in figures 3(C) and (D), respectively.We repeat the calculation for h = 0.05, 0.025 and tabulate the errors in table 1. As expected,the errors appear to be O(h) for the first-order method and O(h2) for the second-ordermethod.

Next, we test the first- and second-order distance methods on the same test problem. Weimplement this method in Matlab. First, we generate a grid of arrival times with constant gridspacing and then use a Matlab contour plotter to find the contours T (x, y) = m ∗ �t,m =1, 2, 3, . . . . Finally, we solve equations (4.1) and (4.12) to approximate the wave speed√

µ(x, y)/ρ(x, y). The solutions for �t = 0.1 are shown in figures 3(E) and (F), respectively.We repeat these calculations for �t = 0.5, 0.025 and tabulate the errors in table 2. Asexpected, the errors appear to be O(�t) for the first-order distance method, and O((�t)2) for

692 J McLaughlin and D Renzi

(A) (B)

(C) (D)

(E) (F)

Figure 3. (A) Exact wave speed; (B) exact arrival times; (C) wave speed recovery without noiseusing the first-order simplistic method; (D) wave speed recovery without noise using the second-order simplistic method; (E) wave speed recovery without noise using the first-order distancemethod and (F) wave speed recovery without noise using the second-order distance method.

Table 2. L∞ error for the distance methods.

�t L∞ error equation (4.1) L∞ error equation (4.12)

0.1 0.1131 0.01850.05 0.0561 0.00450.025 0.0285 0.0011

Shear wave speed recovery in transient elastography 693

Table 3. Running times for the distance methods.

�t h Distance method running time (s)

0.2 0.2 4.79600.1 0.1 38.41380.05 0.05 330.8757

the second-order distance method. In terms of accuracy when using exact arrival times, there isno advantage in determining the wave speed with a distance method over a simplistic method.However, the distance method does have much better stability properties. To demonstrate this,we add random Gaussian noise to the arrival times

Tnoisy(x) = T (x) + γ rand(x),

where rand(x) is a random number generated in Matlab from the normal distribution withmean zero and variance one, and γ is the noise level. We will consider a noisy arrival timeexample with γ = 0.02. Now we repeat the wave speed calculations with these noisy arrivaltimes. The noisy arrival times are shown in figure 4(B). The solution obtained by the first- andsecond-order simplistic methods with h = 0.1 are shown in figures 4(C) and (D), respectively.The solutions obtained using the first- and second-order distance methods with �t = 0.1 areshown in figures 4(E) and (F), respectively. The reader should take particular note of thevertical scale in figures 4(C)–(F).

In this simulation, the solutions obtained by the simplistic methods are terrible, exhibitingwild oscillations. The solutions using the distance methods do exhibit some oscillation, butthere are no large outliers in the solution. The solution using the second-order distance methodis clearly the best, and we will concentrate on these methods in the rest of this paper.

The last comment we want to make is about running times. For simplicity, we restrictour discussion to two dimensions. While the distance methods have very desirable stabilityproperties, the implementation described above is very slow. In table 3, we tabulate the timeit takes to run the second-order distance algorithm with various size grids and step sizes �t .Note that whenever we halve �t , we also halve the grid spacing h. This is because weinterpolate the grid values to find the contours, so to achieve higher accuracy by shrinking�t the grid spacing h also must shrink. As you can see from table 3, the running time ofthe above implementation appears to be O(n3), or O(m(3/2)), where n(m) is the number ofgrid points in time (space). Looking at equation (4.1), it is easy to see why the method is soslow. For every point on every contour, one has to calculate the distance to every point ona neighbouring contour to find the minimum distance. Implementation of this is a triple forloop. If we assume that the number n of points in time is approximately the number

√m of

points in one of the space dimensions, then the running time is obviously O(n3). Of course,one does not have to calculate the distance to every point on the neighbouring contour to findthe smallest one. It would be sufficient to only calculate the distance to points that are ‘closeby’. However, it is difficult to develop algorithms that do this quickly, especially in higherdimensions. In a later paper, we will improve the running time considerably, to O(m log m),by finding the distance simultaneously from sets of points to contours.

5. Testing the complete shear wave speed recovery algorithm

5.1. Numerical tests on synthetic data

Combining the ideas of sections 3 (finding arrival times from displacement data) and 4 (findingwave speed from arrival times) gives a complete algorithm to recover the shear wave speed

694 J McLaughlin and D Renzi

from displacement data. We test this algorithm using synthetic data with a range of frequencies.For these numerical simulations, we generate data by solving (2.2), (2.3) and (2.4)/(2.5) foru(x, t), using a finite-element PDE solver and a mesh generator [11, 14]. The units we areusing to display the results using synthetic data are seconds for time, centimetres for lengthand cm s−1 for speed. Our domain, � := {x = (x1, x2) ∈ R2 : 0 � x1 < 6,−6 � x2 � 6}, isa two-dimensional domain, and a mixed boundary condition is given by

∂u

∂x1(0, x2, t) = 1

2√

πct

cos(τ (t − t0)) exp

{− (t − t0)

2

2c2t

},

u(x1, 6, t) = u(x1,−6, t) = u(6, x2, t) = 0,

where we choose t0 = 0.02 and calculate for three values of (ct , τ ), which are (ct , τ ) =(0.0018, 250/π), (0.009, 125/π), (0.0045, 62.5/π). To simulate a short duration wave pulse,a cut-off function should be applied. However, the above Gaussian decays rapidly, and aftera short time, about 0.03 seconds, the source function drops below machine precision. Weassume that for the density ρ = 1, and for the stiffness parameter, µ, we use

µ(x1, x2) =(

300 + 900 exp

{− (x1 − c1)

2

w21

− (x2 − c2)2

w22

})2

,

where we choose w1 = 0.25, w2 = 0.5, c1 = 3 and c2 = 0. Note that with ρ = 1, then thewave speed is

õ.

With our choice of parameters, the background wave speed is 300 cm s−1 and the maximumwave speed is 1200 cm s−1. The simulation is stopped before the arrival time front hits any ofthe Dirichlet boundaries at tmax = 0.05. Finally, for our discretization, we use 250 time stepsand a 120 × 240 spatial grid. With all of these choices, the stiffness is changing rapidly on a6 × 12 section of the grid.

Now we test the shear wave speed recovery algorithm using the correlation method forfinding the arrival times. We first give results using the highest frequency τ/(2π) = 250. Tocalculate the arrival times, T (x), at the point x from the synthetic data, we first compute thecross-correlation between u(x, t) and a reference signal u(xref, t) with a Matlab discrete cross-correlation routine. Then, we estimate the time delay that maximizes the cross-correlationusing thresholding. We take the derivative of the correlation with respect to the time delayusing a forward Euler discretization scheme and then we pass this result to a Matlab zerofinding routine. This routine uses cubic interpolation between grid points with the thresholdestimate as an initial guess. The result of this procedure is the time delay δtmax that maximizesthe correlation between u(xref, t) and u(x, t−δt) and we use it as our arrival time T (x) = δtmax.The arrival times, using the correlation method, when the central frequency is τ/(2π) = 250,are shown in figure 5(A). Following this, we calculate the wave speed,

√µ/ρ, directly by

finding the level curves and using the second-order distance method. For illustration, the levelcurves of the arrival time surface when the central frequency is τ/2π = 250 are shown infigure 5(B). They bend when passing through the high speed region. In figure 5(C), we showthe exact wave speed we want to recover with our algorithm. We begin showing our recoveredwave speeds with the data obtained when the central frequency is again τ/(2π) = 250 andrecover the wave speed from the arrival times first using the second-order distance method.This recovery is shown in figure 5(D). The recovery of the high speed region is excellent. Boththe support of this region and the values of the wave speed are very accurate. The backgroundwave speed is also recovered very well, with the exception of one artefact. Directly in frontof the high speed region, at about (0, 2.2), there is a bright spot and then a dip. This is exactlythe region where the wave pulse we are tracking is mixing with wave pulses that have reflectedfrom the inhomogeneous region and this reflected wave influences our arrival time calculation.

Shear wave speed recovery in transient elastography 695

(A) (B)

(C) (D)

(E) (F)

02

46

810

0

2

4

6

8

100

10

20

30

40

50

60

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80

xy

wav

e sp

eed

02

46

810

0

2

4

6

8

100

5

10

15

20

25

30

35

40

wav

e sp

eed

Figure 4. (A) Exact wave speed; (B) noisy arrival times; (C) wave speed recovery with noise(γ = 0.02) using the first-order simplistic method; (D) wave speed recovery with noise (γ = 0.02)

using the second-order simplistic method; (E) wave speed recovery with noise (γ = 0.02) usingthe first-order distance method and (F) wave speed recovery with noise (γ = 0.02) using thesecond-order distance method.

We explain this further now. Usually, correlation is well known for giving good recoverieseven in the presence of very large quantities of noise. This is because the method uses thephase content of the signals more than the amplitude. However, in this case, in the subregion

696 J McLaughlin and D Renzi

2

2.5

3

3.5

4

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−1

0

1

20.006

0.007

0.008

0.009

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0.013

0.014

depth

(A)

x

time

dela

y

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4−2

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0

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1

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2

depth

x

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1200

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x

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x

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2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 42

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1

0.5

0

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1

1.5

2

depth

x

(B)

(C) (D)

(E) (F)

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

Figure 5. (A) Arrival time for 250 Hz synthetic data; (B) level curves of the arrival time surfacefor 250 Hz synthetic data; (C) exact wave speed; (D) wave speed recovery with arrival times for250 Hz synthetic data using the second-order distance method; (E) wave speed recovery witharrival times for 125 Hz synthetic data using the second-order distance method and (F) wave speedrecovery for 62.5 Hz synthetic data using the second-order distance method.

where we have the artefact, the reflected wave has, up to a sign, the exact same phase as theforward wave. For this reason, the correlation method has difficulty separating out the forwardpropagating wave from the reflected waves.

Now we continue to test the second-order distance algorithm, reducing the frequencyand repeating these calculations with frequency τ/(2π) = 125, 62.5. Figure 5(E) shows the

Shear wave speed recovery in transient elastography 697

recovery with τ/(2π) = 125. As you can see, the recovery overshoots the fast change andblurs somewhat with the lower frequency. Figure 5(F) shows the recovery with τ/(2π) = 62.5.The recovery has blurred further and overshoots more. As we remarked earlier in this paper,in a later paper we will improve the method to find the arrival times and expect more accuraterecovery of the amplitudes in the high speed region in that case. For ease of comparison,we again show the recoveries for τ/(2π) = 125, 250 in figures 6(B) and (C), where in thesefigures these two recoveries are plotted on the same scale as the recovery for τ/(2π) = 62.5.For convenience, we also replot the exact wave speed with the new scale in figure 6(A). Wenote that the artefact that appeared in the 250 Hz synthetic data (figure 5(D)) did not disappearat lower frequency. At lower frequencies, the wavelength is longer, and therefore the distancebetween peaks is greater. Because of this, the region where the wave pulse we are trackingmixes with earlier reflected pulses is farther from the high speed region. Figures 6(D)–(F)show the artefacts generated at the different frequencies τ/(2π) = 62.5, 125, 250. Note that,in each of these figures, we have cut off the colour bar at 800 cm s−1 to make the artefact easierto see. Any recovered speeds higher than 800 cm s−1 are shown as bright red. At the lowestfrequency, as shown in figure 6(B), there is an artefact about 1 cm above (depth is on thehorizontal axis) the high speed inclusion . As the frequency is increased in figures 6(D) and(F), the size of the artefact decreases and the artefact moves closer to the high speed region.

Here, we would like to make some comments on the frequency dependence of the results.While the proof that the arrival times satisfy the Eikonal equation has no frequency dependence,the methods we employ to find the arrival times do. Whenever there is noise (recall that weare considering reflections as noise), the resolution of the cross-correlation method dependson the sharpness of the peak in the correlation function. Since the correlation function flattensas the frequency drops, we should expect reduced resolution at low frequencies.

To test the stability of the second-order distance algorithm, we add Gaussian randomnoise to the τ/(2π) = 250 frequency displacement data

unoisy(x, t) = u(x, t) + γ umax rand(x, t),

where umax is the maximum displacement, γ is the noise level and rand(x, t) is a randomnumber generated in Matlab from the normal distribution with mean zero and variance one.We repeat the calculations above to get recoveries with 7% noise (γ = 0.07). Here, we usea low-pass filter on the cross-correlation function to remove small oscillations before we findthe location of the maximum value. The recovered wave speed is shown in figure 7(B). Theexact wave speed is again shown in figure 7(A). Our noisy recovery compares favourablyto the recovery in figure 5(D) using exact synthetic data with the same central frequency,250 Hz.

Now we will give some concluding remarks about the performance of our algorithmswith synthetic data. For our synthetic data examples, the wave speed used in our forwardalgorithms in the high speed region is up to four times greater than the background wave speed.Our algorithms work well for this example and are robust with respect to noise. They do tendto overshoot the fast change at low frequencies. However, there is one artefact in all of therecoveries. It corresponds to regions where reflections from earlier wave pulses are mixingwith later wave pulses.

5.2. Numerical tests on experimental data

Here, we test our algorithm on data measured from a phantom experiment in the laboratoryof Mathias Fink. The phantom is made of a 3% concentration of agar powder throughout thephantom. The stiff inclusion is a circular cylinder with 5 mm radius. The gelatin concentration

698 J McLaughlin and D Renzi

xx

x

xx

x

depth depth

depth depth

depth depth

(A) (D)

(B) (E)

(C) (F)

-

-

-

-

Figure 6. (A) Exact wave speed plotted on the same scale as the 62.5 Hz recovery; (B) wavespeed recovery for 125 Hz synthetic data using the second-order distance method and plotted onthe same scale as the 62.5 Hz recovery; (C) wave speed recovery for 250 Hz synthetic data usingthe second-order distance method and plotted on the same scale as the 62.5 Hz recovery; (D)wave speed recovery for 62.5 Hz synthetic data using the second-order distance method containingthe artefact; (E) wave speed recovery for 125 Hz synthetic data using the second-order distancemethod containing the artefact and (F) wave speed recovery for 250 Hz synthetic data using thesecond-order distance method containing the artefact.

Shear wave speed recovery in transient elastography 699

is 2% outside of the cylinder and 4% in the cylinder and the result is that the wave speed insidethe cylinder is double the wave speed everywhere else. The experiments are performed byexerting a force simultaneously with two parallel bars on the boundary of the phantom. Bothbars have the same central frequency. The experiment is performed five times, each time witha different central frequency. The five different central frequencies are 50, 60, 70, 80 and90 Hz. The axis of the cylinder is parallel to the boundary and perpendicular to the axes ofthe parallel bars. The downward displacement, that is perpendicular to the phantom surfacewhere the 2 bar force is applied, is measured in the plane equidistant between the two bars.Note that the agar and gelatin concentrations for this phantom are different from the phantomexperiment reported in [41]. We present results from the experiment performed at 50, 60 and70 Hz.

The units used here are different from the units we use for the synthetic data experiments.Here, we use the units seconds for time, millimetres for length and m s−1 for speed,when we describe our measured data recovery. In the cross section, the domain is(0, 70.492)×(−20.955, 20.955) and the inclusion for the high speed region is Bp(5) where thecentre point is p = (36.746, 0). The wave speed in the inclusion is twice the wave speed in thebackground. Before we describe the process to recover the wave speed from the experimentaldata, we first discuss some of the characteristics of the experimental displacement data. Theprimary difference between the experimental data and the synthetic data is that there are twodifferent waves propagating through the medium, a very fast compression wave and a relativelyslow shear wave. To use an acoustic equation model, we must treat the compression waveas noise in the data. In figure 8, we show the displacement along a line in depth. Note thatat about t = 0.01 a pulse appears almost simultaneously at all depths and can be seen in thefigure as a horizontal line. This pulse does not propagate at the shear wave velocity and isnot predicted by the acoustic model. This pulse is either a fast compression wave experiencedbecause of its fast wave speed almost everywhere at once or is a bulk motion due to the initialforce of the bars. The shear wave does not appear until later, so we cut off the first 0.014 sof the displacement data. After about 0.06 s the shear wave has passed through the region ofinterest and the data only contain reflections from the boundary of the container, so we willtruncate the displacement data from 0.014 < t < 0.06. For ease of illustration, we replotthese data with the scale reduced to the times of interest and the depth just below the inclusionin figure 9. You can clearly see the pulses propagating through the medium. Compared to thesynthetic data, the reflections from the high speed region are much harder to see. The circledregion in figure 9 contains an area where there is mixing between the forward and reflectedwaves. Visually, it appears that the circled pulse is more heavily influenced by reflectionsthan the other pulses. The acoustical model we are assuming only allows reflection of theshear wave, but in the experiment conversions from shear to compression wave or vice versaare also possible. All of these conversions must be considered as noise. As we remarkedabove, correlation is well known for giving good recoveries even in the presence of very largequantities of noise because the method uses the phase content of the signals more than theamplitude. However, in time the compression wave has the same phase as the shear wave soone should expect the correlation method to have some difficulty in regions containing a largeamplitude compression wave.

Now we will describe how we recover the wave speed from experimental displacementdata. The first step is to find the arrival times. In section 5, we described a correlationprocedure for use with synthetic data. Keeping in mind the characteristics of the experimentaldata discussed above, we will modify these procedures as follows.

Our method to find the arrival times in experimental data is a cross-correlation method.The first step is to truncate the time traces and only use data for t = [0.014, 0.06]. Then, at each

700 J McLaughlin and D Renzi

300

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2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 42

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(A)

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(B)

depth

x

−

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−22

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−

Figure 7. (A) Exact wave speed and (B) correlation wave speed recovery for 250 Hz syntheticdata with noise using the second-order distance method.

20 25 30 35 40 45 50 55 60 65 700

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

depth

time

Figure 8. Displacement data along a line in depth. Along the horizontal line, t = 0.01, the fastpulse appears almost simultaneously at all depths. Above the horizontal line, t = 0.06, the shearwave passes through the region of interest.

Figure 9. Displacement data along a line in depth. The vertical bars are the edges of the highspeed region. In the circle, reflections are mixing with the forward travelling pulses.

Shear wave speed recovery in transient elastography 701

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

−2

0

2

timedi

spla

cem

ent

(A)

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

−2

0

2

time

disp

lace

men

t

(B)

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1−1

0

1

δ t

corr

elat

ion

(C)

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

−2

0

2

time

disp

lace

men

t

(D)reference signaltime delayed signal

Figure 10. (A) Reference signal; (B) time trace at a point near the inclusion (note the changein vertical scale); (C) correlation function and (D) time-delayed signal of maximum correlationplotted with the reference signal.

point, x0, we compute the cross-correlation between the time trace, u(x, t), and a referencesignal, u(xref, t), using a Matlab discrete cross-correlation routine. The anomalies in the datadiscussed earlier cause errors in the cross-correlation function. To make peak identification ofthe cross-correlation function easier, we again low-pass filter the cross-correlation function,C(x, δt), and use the time delay that maximizes the smoothed cross-correlation function asthe arrival time, T (x). Once again, the arrival times are only calculated to the nearest timestep, as we see no improvement in accuracy of the wave speed recovery by interpolating to asubgrid with real experimental data.

We use the modified cross-correlation method described above to find the arrival times ofthe experimental displacement data. For illustration, in figures 10(A) and (B) we show twotime traces of the experimental data, one near the source (A) and one further in the mediumnear the inclusion (B). For the correlation method, the time trace in figure 10(A) is used as thereference signal. Figure 10(C) shows the correlation as a function of the time delay andfigure 10(D) shows the time-delayed signal with maximum correlation plotted with thereference signal. It is clear from this figure that the wave pulse has broadened significantly asit travelled into the medium. This is the effect of visco-elasticity and is not modelled by thecurrent model. The correlation method has the effect of centring the pulse of the referencesignal inside the broadened pulse of the displacement time trace.

The reconstructed arrival time surface is shown in figure 11(A). The slight stair steppingeffect is because we have truncated the arrival times to the nearest time step. There is a smallregion centred at about (37, 0) where the arrival time surface has flattened out because thespeed is larger than in the surrounding regions. The effects of the increased wave speed can

702 J McLaughlin and D Renzi

(A) (B)

Figure 11. (A) Arrival times of 50 Hz experimental data using correlation and (B) level curves ofthe arrival time surface generated with correlation.

1.5

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−15

−10

−5

0

5

10

15

20

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x

(A) (B)

(C)

x

x

depth

depth

Figure 12. A wave speed recovery for 50 Hz experimental data using the second-order distancemethod; (B) wave speed recovery for 60 Hz experimental data using the second-order distancemethod and (C) wave speed recovery for 70 Hz experimental data using the second-order distancemethod; in all figures the wave speed is given in units of m s−1.

Shear wave speed recovery in transient elastography 703

also be seen by looking at the level curves of the arrival time surface, which are shown infigure 11(B), and bend when travelling through the high speed region.

We use these data in our distance method to calculate the solution to our inverse problem.The presence of the small closed level curves in the data causes problems and is non-physical.So, prior to implementing our algorithm the occasional small closed level curves are eliminatedby keeping only the largest connected portion of any given level curve. The eliminated curvesare shown in red.

Before we show the results with experimental data, we explain a last step taken in ourinverse problem. For laboratory data, we generate an initial guess (

√µ/ρ)0 using the second-

order distance or second-order level curve method. We then add a total variation minimizationstep. That is, we find the

√µ/ρ that minimizes∫

�

|∇√

µ/ρ| dx + λ

∫�

|√

µ/ρ − (√

µ/ρ)0|2dx,

where λ is a regularization parameter. To determine the minimum value, the associatedEuler–Lagrange equation is

∇ ·( ∇√

µ/ρ

|∇√µ/ρ|

)− 2λ(

√µ/ρ − (

√µ/ρ)0) = 0.

Following [40], introducing the evolving variable s, we solve

∂

∂s

õ

ρ= ∇ ·

( ∇√µ/ρ

|∇√µ/ρ|

)− 2λ(

√µ/ρ − (

√µ/ρ)0),

õ

ρ

∣∣∣∣s=0

= (√

µ/ρ)0 (5.1)

to steady state. This method is only guaranteed to converge to a local minimum, but in practicethe initial guess is close enough to the solution that this method converges to the desired result.In our reconstructions, we use λ = 2. The final wave speed reconstruction using the second-order distance method is shown in figure 12(A). The wave speed recoveries focus very well onthe high speed region. Directly after the high speed region, there is an artefact. This artefacthas been spread out by the total variation denoising, but the centre of the low spots (darkestblue) appears to be at about (53, 0). This remains to be investigated. There is also an artefactin front of the high speed region as well. In this region, the reflected waves are mixing withthe forward waves. Note that by including the total variation minimization step, we can takea smaller time step in the level curve method. The result is an improvement in the contrastbetween the high speed and background regions.

Now we repeat all of the above calculations with the 60 Hz and 70 Hz experiment. Forthe 60 Hz data, the final wave speed reconstruction using the second-order distance method isshown in figure 12(B). Similar to the 50 Hz recoveries, the correlation recoveries focus verywell on the high speed region, but have artefacts directly above and below the high speedregion. Finally, we give the results for the 70 Hz data. The final wave speed reconstructionusing the second-order distance method is shown in figure 12(E). Similar to the 50 Hz and60 Hz recoveries, the correlation recoveries focus very well on the high speed region, but haveartefacts directly above and below the high speed region.

To summarize, our complete algorithm to determine the shear wave speed,√

µ/ρ, by thesecond-order distance using the arrival time from the measured displacement data u is (seefigure 13 for the flowchart) as follows:

1. Choose arrival time T (x) = −δtmax at each x to maximize the correlation betweenu(xref, t) and u(x, t − δt).

2. For each level set of T (x), eliminate occasional small closed level curves.3. Generate initial guess (

√µ/ρ)0 using the second-order distance method.

4. Minimize the total variation of√

µ/ρ by solving (5.1) to steady state.

704 J McLaughlin and D Renzi

Figure 13. Flowchart for our complete algorithm to determine the shear wave speed√

µ/ρ usingthe arrival time from the measured displacement data.

6. Concluding remarks

In this paper, we develop a family of stable methods, which we name distance methods,to accomplish wave speed inversion of the Eikonal equation. We test these methods usingsynthetic arrival time data and compare it with a simplistic approach. When using exact arrivaltimes, there is almost no difference between the simplistic and distance methods. When usingnoisy arrival times, the simplistic methods give a rather poor recovery. There are large outliersin the recovered wave speed whenever |∇T | is small. The distance methods give a very goodrecovery. There are some oscillations, but no large outliers are observed. We also show shearwave speed images made from synthetic displacement data and laboratory displacement datausing the second-order distance method. In a later paper [28], we will first use level sets tocreate a much faster family of inverse Eikonal solvers that we name level curve methods. Inthat paper we will compare results obtained with the distance method and the faster level curvemethod.

Acknowledgments

We have benefited from discussions with J-R Yoon, L Ji, A Maniatty, E Park, P Barbone,S Catheline, M de Hoop, M Fink, B Isakov, S Kurylev, M Lassas, C Nolan, S Osher, J Sethian,M Sini, M Tanter and D Tataru. We would like to thank members of M Fink’s laboratory forsharing the measured data. JM is partially supported by NSF Focus Research Group Grant NoDMS 0101458, ONR Grant No N000 14-96-1-0349 and NIBIB Grant No 1R21EB003000-01.DR is partially supported by NSF VIGRE Grant No DMS 9983646.

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