Nonlinear Shear Modulus Estimation with Bi-axial Motion Registered Local Strain

Abstract

Nonlinear elasticity imaging provides additional information about tissue behavior that is potentially diagnostic and avoids errors inherent in applying a linear elastic model to tissue under large strains. Nonlinear elasticity imaging is challenging to perform due to the large deformations required to obtain sufficient tissue strain to elicit nonlinear behavior. This work uses a method of axial and lateral displacement tracking to estimate local axial strain with simultaneous measurement of shear modulus at multiple compression levels. By following the change in apparent shear modulus and the stress deduced from the strain maps, we are able to accurately quantify nonlinear shear modulus. We have validated our technique with a mechanical nonlinear shear modulus measurement system. Our results demonstrate that 2D tracking provides more consistent nonlinear shear modulus estimates than those obtained by 1D (axial) tracking alone, especially where lateral motion is significant. The elastographic contrast to noise ratio in heterogeneous phantoms was 12.5%−60% higher using our method compared to 1D tracking. Our method is less susceptible to mechanical variations, with deviations in mean elastic values of 2–4% versus 5–37% for 1D tracking.

I. INTRODUCTION

NONLINEAR mechanical properties of tissues such as shear nonlinearity [1], viscoelastic nonlinearity [2], geometric nonlinearity [3] and others [4] are important in numerous biological functions as well as clinical diagnosis. In most ultrasound elasticity imaging approaches simplified, linear elastic tissue models are assumed. While measurement of nonlinear elastic properties provides additional information [5]–[6] about tissue, such measurements are difficult. Nonlinear elastic properties become evident under large deformations [7], the accurate description of which requires more sophisticated mathematical models. Material tracking methods are required to register material properties obtained in the deformed state with the original, undeformed structure.

Recently researchers have approached various aspects of elastography [8] to monitor nonlinear properties of tissues. Hall et al [9] first demonstrated that differences in the large-strain behavior of tissues is potentially diagnostic. His work quantified in-vivo nonlinear strain and tissue mechanical properties on application to cancer diagnosis. However, estimation of nonlinear elastic properties from quasi-static strain imaging sequences involves solving a challenging inverse problem of nonlinear modulus inversion. Dynamic elastography methods [10]–[15] have an advantage in providing quantitative estimates of exogeneously generated shear wave speed in tissue, and for an incompressible linear elastic material of density ρ, shear modulus μ and shear wave speed V_s, are related by $V_S^2=\frac{\mu }{\rho }$. However, the displacements induced by dynamic methods like ARFI SWEI [10] are small(10 s of μm), limiting its ability to measure tissue non-linear properties. Quasi-static methods [16]–[18], in contrast, can easily induce large (~ 10%) strains using manual pressure on the ultrasound transducer. An appropriate combination of these methods has the potential to quantify tissue non-linear properties. Simultaneous quasi-static and ARFI-SWEI elastography can be used to measure local changes in shear wave speed associated with large tissue strain.

Hughes et al [19] first used a strain energy model at large tissue deformation to relate shear wave velocity to material properties. Gennisson et al [1] demonstrated that the application of a stress to an incompressible linear elastic homogeneous medium results in a change in the apparent shear modulus(μ) that is linear on stress(σ), and can be expressed¹ as $\mu =\rho \cdot V_S^2={\mu }_0-\sigma \cdot \frac{A}{12{\mu }_0}$, where A is the Landau coefficient or coefficient of shear nonlinearity, μ₀ is the shear modulus of the material. Latorre-Ossa[20] applied this equation with strain elastography and supersonic imaging to show that spatial map of nonlinear shear modulus could be recovered. By incrementing the applied strain and estimating the local stress at each compression step from the cumulative sum of combined local Young′s modulus and local strain estimates, the strain dependence of modulus, and thus A could be obtained. Bernal et al [21] showed the utility of nonlinear shear modulus to detect breast lesions from healthy tissue. However these techniques used axial motion tracking to estimate the axial strain and ignored the lateral deformation component. In regular strain imaging the applied strain is nearly uniaxial with minimal motion, hence lateral deformation is not a problem. However, for nonlinear shear modulus imaging, large compression is needed compared to regular strain imaging, and lateral deformation of tissue becomes significant. To obtain local shear nonlinear elastic properties of tissue, the change in the shear wave speed with local strain (or the slope of shear wave speed squared times the apparent stress) needs to be measured, which requires accurate registration of deformation of the local tissue. A misestimated strain calculation of local tissues due to improper registration directly affects the slope or the estimated nonlinear shear modulus.

In this work, 2D cross-correlation [22]–[23] was used to estimate the axial and lateral deformation vectors between pre and post compression RF data. The estimated two-dimensional deformation vectors at each compression step were then registered with respect to the initial undeformed state of the material. From the registered two-dimensional deformation vectors, the axial strain was estimated. The corresponding shear wave speed map was obtained by single track location shear wave elasticity imaging(STL-SWEI) [13] at each compression step. The slope of the regression line fit to the shear wave speed squared as a function of stress derived from strain data measurements yields an estimate of the nonlinear shear modulus.

We have carried out a mechanical nonlinear shear modulus measurement experiment to validate our ultrasound based-measurement system. The performance of nonlinear shear modulus estimation with our novel motion registered 2D tracking scheme is compared to 1D deformation tracking method and global strain based method. Quantitative experimental results obtained from homogeneous agar-gelatin and Polyvinyl Alcohol(PVA) phantoms using 2D deformation tracking demonstrate that nonlinear shear modulus estimation is more robust and consistent compared to that estimated with 1D tracking. Single cylindrical inclusion phantoms also demonstrate the improvement in contrast-to-noise ratio of nonlinear shear modulus obtained with 2D vs 1D tracking. In addition, we have also evaluated the effect of mechanical conditions that impact the measurements of nonlinear shear modulus such as positional variation of transducer with respect to an inclusion, initial loading of the material, depth of the inclusion. Finally, we have examined several heterogeneous material combinations to evaluate the contrast improvement afforded by nonlinear elasticity measurements compared to linear strain imaging and linear shear modulus images.

II. Acoustoelasticity theoretical background

Acoustoelasticity theory has been developed by Toupin [24], Norris [25], Murnaghan [27], and adopted by several others [26]–[28] to describe how the speed of an elastic wave changes in a uni-axially stressed lossless solid. In this approach the equation of motion of an elastic soft-tissue solid is described in terms of the strain energy. For a quasi-incompressible soft tissue material, the strain energy [29] e is

$$e=\mu I_2+\frac{A}{3}{I}_3+D{I}_2^2$$ (1)

where μ is the zero-stress shear modulus, A and D are the third and fourth order non-linear shear modulus, and I₂ and I₃ are the second and third invariants of the Lagrangian strain tensor:

$$E=\left[\begin{array}{lll}{ϵ}_{11}\hfill & {ϵ}_{12}\hfill & {ϵ}_{13}\hfill \\ {ϵ}_{21}\hfill & {ϵ}_{22}\hfill & {ϵ}_{23}\hfill \\ {ϵ}_{31}\hfill & {ϵ}_{32}\hfill & {ϵ}_{33}\hfill \end{array}\right]$$ (2)

In terms of E the invariants are I₂ = 1/2(tr(E)² − tr(E²)) and I₃=1/6(trE)³−1/2trEtrE² + 1/3trE³. For a linearly polarized plane shear wave of low amplitude, with polarization direction parallel to the axis of compression in an isotropic, homogeneous material, the fourth and higher order terms may be neglected. As shown by Gennisson et al. [1], the wave equation in terms of μ and A may thus be expressed as

$$\rho \frac{{\partial }^2{u}_z^D}{\partial t^2}=\frac{{\partial }^2{u}_x^D}{\partial z^2}\left[\mu +2\mu \left(\frac{\partial u_x^S}{\partial x}+2\frac{\partial u_z^S}{\partial z}\right)+\frac{A}{2}\left(\frac{\partial u_x^S}{\partial x}+\frac{\partial u_z^S}{\partial z}\right)\right]$$ (3)

where and $u_{x,z}^D$ are $u_{x,z}^S$ shear displacement induced by radiation force and displacement due to static uniaxial stress respectively. By applying Hookes law and assuming the strain due to the shear wave is small compared to the static compression, equation (3) implies a wave speed as given by Gennisson et al. [1]. The value of A can thus be estimated from the slope of shear wave speed squared as a function of uniaxial stress. However, while shear wave speed and strain can be measured ultrasonically, stress cannot be determined directly. Therefore, we instead estimate the stress at the i^th compression step from the cumulative sum of incremental local strain times the shear modulus(obtained by ultrasound measurements) at each compression step as given by:

$${\sigma }_i=\sum _{j=1}^{i}3{\mu }_j\mathrm{\Delta }{ϵ}_j$$ (4)

Δϵ_jis the differential axial strain estimated at each compression step following registration of the tissue motion from estimated axial and lateral displacements. The apparent shear modulus at the i^th compression step is thus

$${\mu }_i={\mu }_0-\left[\sum _{j=1}^{i}3{\mu }_j\mathrm{\Delta }{ϵ}_j\right]\frac{A}{12{\mu }_0}$$ (5)

and A is the estimate of the third order shear modulus which is obtained by fitting the measured apparent shear modulus values to the stress estimated at each compression steps.

For mechanical testing, we measure the stress directly corresponding to different strain levels as shown in Fig. 2(b). The tangent of this curve at zero strain level gives shear modulus at undeformed state. The apparent shear modulus is obtained from tangent of this curve at given strain level. The nonlinear shear modulus A is thus estimated from fitting the estimated apparent shear modulus(μ_i) at each strain levels(ϵ_i) to the stress obtained directly from mechanical testing.

Fig. 2.

(a)Schematic illustrating the mechanical stress-strain experimental setup for a 4.9kPa homogeneous PVA phantom. (b)the method of apparent shear modulus calculation.

III. MATERIALS AND METHODS

A. Data acquisition

1. Description of phantoms: Homogeneous phantoms of 200-bloom gelatin material and hydrolyzed polyvinyl alcohol(PVA) were fabricated. The phantoms were rectangular in shape, with length of 6 cm, width of 4 cm, and height 4.5 cm. Simultaneously, cylindrical samples of 2 cm diameter and 3.5 cm height were made for mechanical testing with the same mixture used to make rectangular samples. Four cylindrical inclusion phantom types were constructed with inclusion of diameter 0.65 cm. The composition of the materials used are briefly summarized in Table. I. Aqueous solution of 200 bloom type A gelatin(Custom Collagen, Addison, IL) had 2% cornstarch to increase scattering. The soft and stiff gelatin inclusion phantoms were prepared with reverse concentrations and at same temperature condition. The hydrogel phantoms were fabricated with PVA having high levels of hydrolysis( molecular weight 44.053 g per mol, J.T.Baker™). The required concentration of PVA powder, with de-ionized (DE) water was heated to 100–120 °C. The PVA solution was stirred until the consistency was found to be homogeneous and left to cool for 20 min in an air sealed flask. The resulting solution was subjected to one freeze-thaw cycle of 12 hours freezing and 12 hours thawing to ensure a fully cross-linked hydrogel substrate. For the gel inclusion in PVA medium, first the gel inclusion was made, then PVA solution was prepared and the freeze-thaw cycle adjusted to match the elasticity of PVA with that of the gel. The process was started with lower concentration of PVA(5% PVA), so the initial elasticity of PVA is lower than the gel inclusion(8% 200 Bloom gel). With gradual freeze-thaw(2 cycles of 24 hour freeze-thaw), the elasticity of PVA increased to match the linear shear modulus of the gel(both close to 6.5kPa). Since the gel was frozen within PVA, the gel elasticity might slightly increase from the initial value, however our objective was to match the elasticity of the two. This reduces the contrast of the inclusion from background in the initial shear modulus map of gel-PVA phantom. The process was reversed for making a PVA inclusion in gel medium, where first the PVA inclusion was made, allowed to warm up at room temperature, and then embedded in the gel medium.

2. Experimental Setup:Strain and SWEI Imaging: All experiments were carried out using a conventional ultrasound probe (ATL L7–4 linear array) driven by a Verasonics Vantage 64LE ultrasound system(Verasonics Inc., Kirkland, WA, USA) shown in Fig. 1. The phantoms were placed over a natural rubber pad and a compression plate (8cm × 4.5cm) was attached to the transducer, itself mounted on a 5-axis position controller. The transducer was moved axially in 0.3 mm steps to apply quasistatic stress to the phantom with the compressor plate. Data corresponding to 30 compression steps(20% global strain) were collected. The time taken to perform such an experiment is 15–20 minutes with each compression step taking a 30 seconds. At the start of the experiment, a slight amount of compression was applied to make sure that the probe is fully coupled with the phantom surface.

Table I.

Table represents different phantom types (percentage volume).

Types	Concentration
Homogeneous Gel 1	8% Gel
Homogeneous Gel 2	13% Gel
Homogeneous PVA 1	6% PVA
Homogeneous PVA 2	8% PVA

Inclusion Phantoms	Background Medium	Inclusion
Stiff Gel-Gel	8% Gel	13% Gel
Soft Gel-Gel	13% Gel	8% Gel
PVA-Gel	5% PVA	8% Gel
Gel-PVA	10% Gel	7% PVA

Fig. 1.

Ultrasound experimental setup showing acousto-elasticity experiment in gelatin-cryogel inclusion phantom. L7–4 transducer with added compressor plate showing direction of compression and wave propagation in phantom.

Raw channel data were acquired at a ultrasound frequency of 5 MHz, 60% bandwidth. At each compression level, we acquired data from successive plane wave transmissions at thirteen different transmission angles between −7° and 7°, each separated by 1.07° and used in plane wave compounding [30] to improve estimation of lateral motion. The raw channel data was stored to disk for offline processing. Delay-and-sum (DAS) beamforming implemented on a graphics processing unit (GPU) for accelerated processing was applied to this channel data. During the beamforming, a dynamic receive F# of 1.5 was applied. Beamforming was performed on a uniform grid with 240 axial points and 128 lateral points covering 36mm and 18.5 mm, respectively.

B. Strain Mapping with 2D motion registration

The axial (dz) and lateral (dx) displacements between the pre- and postcompressed RF echo frames were estimated using a 2D cross-correlation-based similarity search algorithm. A 2.5 mm× 2.5 mm kernel was applied to track motion between the pre and post-compressed echo frames with an overlap of 80% in both axial and lateral directions. This kernel size is computed by taking 15 channel lines of RF data laterally and 7–10 wavelengths of each channel line of RF data axially. 2D spline interpolation was used to calculate the subpixel displacements [22]. The axial and lateral displacements computed at each compression step were median filtered using a 0.2mm × 0.2mm kernel. The displacement vectors were then registered with respect to the initial un-deformed state in a process summarized in Fig. 2. Mathematically, if (x_i+1, z_i+1) represents the position of a particular point in B-mode image in i + 1^th deformation, then it is expressed in terms of the un-deformed state as:

$$\left(x_{i+1},z_{i+1}\right)=\left(x_1+\sum _{k=1}^{i}d{x}_k\left(x_k,z_k\right),z_1+\sum _{k=1}^{i}d{z}_k\left(x_k,z_k\right)\right)$$ (6)

where (x₁ , y₁) is the same material point in pre-stressed state and dx_i, dz_i are the lateral and axial deformation, respectively, at i^th compression level with respect to its previous compression state. The registered axial deformation of i + 1^th step with respect to pre-stressed state, represented by dispz_i+1, is thus given as:

$$disp{z}_{i+1}=z_{i+1}-z_1$$ (7)

The normal axial strain (ϵ) was quantified using the first derivative least square strain estimator[22],[23]:

$${ϵ}_{i+1}=\frac{d\left(disp{z}_{i+1}\right)}{dz}$$ (8)

We have also compared the estimates of material nonlinearity obtained with our 2D method with those estimated by global-strain-based and 1D registered strain method. We used an adaptive strain estimator for 1D strain estimation [31]. Briefly, axial windows from pre-compression and post-compression RF data were selected for local strain estimation. The post-compression signal was modeled as a scaled (compressed) version of the pre-compression signal, where the scaling factor was ς=1-ϵ_L (ϵ_L-local strain). The local stretching factor was iteratively estimated by stretching the post-compression signal and cross-correlating with the pre-compression signal. A binary iterative search on the maximum correlation coefficient was performed to rapidly find the local stretching factor and thereby, the local strain. The global strain is calculated by following the change in thickness of the phantom at each compression level. The displacement of the top surface of the phantom, divided by the initial thickness of phantom gives the global strain [32].

C. SWEI Processing

Shear wave speed images were generated by single-track location shear wave elasticity method(STL-SWEI) [13], [33]. The STL-SWEI sequence consisted of pair of push beams with a common tracking line as illustrated in Fig. 4. In our STL-SWEI implementation, the ensemble of tracking and push beams was translated over the entire field of view (FOV) to produce the shear wave image. The distance (ΔP) between push beams was kept constant at 3.5mm and ΔT between push(left push beam) and track pair was 7.5mm. All STL-SWEI sequences contained 30 pairs of push beams to cover an FOV of 21 mm. The particle displacement (due to the shear wave) versus time at every depth in the region of interest was calculated using the 2D autocorrelation method of Loupas et al [34]. A tracking pulse repetition frequency of 7kHz was used. The shear wave arrival time difference was estimated from cross-correlation of displacement vs time profile associated with each push pulse. The distance between the push beams divided by the difference in shear wave arrival times gives the shear wave speed. The linear shear modulus (μ) is calculated by using:

$$\mu =\rho \cdot V_S^2$$ (9)

where ρ is medium density, and V_S is the shear wave speed. Table II provides the key push and tracking beam parameters used in all studies. Because soft tissues can be assumed to be quasi-incompressible materials with Poisson’s ratio 0.5, the local stiffness defined by Young’s modulus can be approximated by E ≈ 3μ. Then by using (equation 5), a 2–D non-linear shear modulus map (Fig. 3) is obtained by fitting the shear wave speed squared as a function of stress. From equation(5), as apparent shear modulus increases with strain (strain stiffening), A is negative. However while plotting the elastograms, negative of values of A are considered for convenience.

Fig. 4.

Schematic [15] illustrating the general principles of STL-SWEI. Gray and black arrows: push and tracking beams, respectively.Black dots: lateral locations where SWS is determined. Shear waves from a pair of push beams at ΔP distance apart are tracked at a common location L1. The ratio of P to arrival time difference quantifies local SWS estimates. The push pair along with the common tracking line at ΔT distance apart is translated laterally to get the entire FOV.

Table II.

Push tracking beam parameters used during STL-SWEI.

Parameters	Push Beam	Track beam
Frequency (MHz)	5	5
Pulse Duration(μsec)	200	0.2
F-number	2.5	2.5
Transmit Voltage(V)	39.8	39.8
Focus(mm)	34	34

Fig. 3.

Flow diagram showing protocol of Ultrasound based nonlinear shear modulus measurement of tissues with successive 2D deformation registration. Here three compression steps are shown and the process follows for the next compression steps. The red and green star indicate two registered speckle patterns in the inclusion and background, respectively.

D. Mechanical measurement of nonlinear shear modulus

We have performed unconfined compression measurement of the nonlinear shear modulus of the homogeneous phantoms to validate our ultrasound based measurement system. Fig. 2 shows the experimental setup of the mechanical stress-strain measurement system. A 5N load cell was used to measure applied force as the cylindrical homogeneous phantoms were deformed to obtain the stress-strain curve. The tangent of this curve at zero strain gives us the shear modulus at undeformed state. The apparent shear modulus is obtained by taking the tangent of this curve at a given strain level. With the knowledge of the stress and apparent shear modulus at each strain level and the stress free shear modulus, nonlinear shear modulus is calculated using curve fitting according to equation(4 and 5).

E. Quantitative Metric

We assessed the linear and nonlinear elastograms qualitatively by visually inspecting the images, and quantitatively using signal to noise ratio (SNR), the contrast ratio (CR) and elastographic contrast-to-noise ratio (CNR) metrics defined as follows:

$$SNR=\frac{\overline{\mu }}{S}$$ (10)

$$CR=\frac{{\mu }_{St}^{-}}{{\mu }_{So}^{-}}$$ (11)

$$CNR=\frac{|{\mu }_{St}^{-}-{\mu }_{So}^{-}|}{\surd (S_{St}^2+S_{So}^2)}$$ (12)

where $\overline{\mu }$ and S are the mean and standard deviation of shear modulus, respectively, in homogenous phantom, ${\mu }_{St}^{-}$ and S_st represent mean and standard deviation in stiffer region of inclusion phantom, ${\mu }_{So}^{-}$ and S_So the same in the softer region.

IV. Results

A. Effect of kernel size on Nonlinear shear Modulus(NLSM) Estimates

Prior to generating the nonlinear shear modulus maps, the effect of kernel size on tracking between pre- and post-compressed RF data was evaluated on a 5.3kPa gel phantom. Kernel size was varied from 1mm × 1mm to 2.5mm × 2.5mm. The cross-correlation coefficient of 2D tracking increased from 0.943 to maximum of 0.982 and the corresponding NLSM (Nonlinear Shear Modulus) maps also improved from Fig. 5(a) to Fig. 5(b). The kernel size was kept fixed at 2.5mm × 2.5mm for the remainder of the studies with an overlap of 80% both axially and laterally [23], [22]. This kernel size covers 15 channel lines of RF data laterally and minimum 7–10 wavelengths data axially. The signal to noise ratio of the elastograms improved with higher kernel size as shown in Fig. 5(d). The mean estimates of the nonlinear shear modulus was found to be more consistent spatially with larger kernel as illustrated in Fig. 5(c).

Fig. 5.

(a,b)Effect of Cross-correlation coefficient of 2D tracking displacement estimation on nonlinear shear modulus estimation. The dotted region has fewer artifacts when cross correlation of 2D tracking is higher, resulting in higher SNR. (c) Representative variations of nonlinear shear modulus spatially with different cross correlation coefficients and (d) SNR of elastograms as a function of cross-correlation estimates(5.3kPa homogeneous gel phantom). The error bars represent the variability due to repeated measurements taken for the cross-correlation estimation and the corresponding NLSM.

B. Homogeneous Phantoms.

Fig. 6 shows the linear and nonlinear shear modulus maps obtained in homogeneous gelatin phantoms by both 1D deformation tracking and our 2D tracking scheme. Qualitatively, NLSM maps estimated by 2D tracking were generally less noisy and contained fewer artifacts than those produced with 1D tracking [Fig. 6(c) and (f)].

Fig. 6.

Results obtained for homogeneous gel phantoms. (a,d) linear shear modulus maps, (b,e) and (c,f) estimated nonlinear shear modulus maps by 1D tracking and 2D tracking, respectively, for two phantoms of stiffnes 5.3 kPa and 10 kPa. (g,i) shows representative lateral displacement maps and (h,j) bar plots of the mean nonlinear shear modulus obtained at 8 rectangular sections by 2D(marked blue) and 1D(marked red) tracking.

Quantitative analysis from 8 rectangular sections(chosen with center close to push beam focal region) in the modulus map shows that nonlinear shear modulus by 2D tracking exhibits better consistency throughout the lateral sections compared to 1D tracking. The nonlinear shear modulus values towards the sides of the elastogram show higher deviations from the center for 1D tracking as illustrated in bar plots of Fig. 6(h) and (j). This is because 1D tracking does not accurately track the tissue motion due to large lateral deformations at the two sides compared to the center of the elastogram. Fig. 7 shows the nonlinear shear modulus estimates by both 2D and 1D tracking for PVA material. It can be observed from Fig. 7(h) and (j) that deviation between 1D tracking and 2D tracking is reduced in PVA compared to gelatin material.

Fig. 7.

Results obtained for homogeneous PVA phantoms. (a,d) linear shear modulus maps, (b,e) and (c,f) estimated nonlinear shear modulus maps by 1D tracking and 2D tracking, respectively, for two phantoms of stiffnes 4.9 kPa and 7.4 kPa. (g,i) presents lateral displacement maps and (h,j)bar plots of nonlinear shear modulus variation spatially. Rectangular regions of interest close to STL push focus has been chosen for evaluations.

C. Comparison of nonlinear shear modulus estimates obtained with mechanical and ultrasound methods.

Fig. 8 illustrates the percentage errors in estimates of nonlinear shear modulus as a function of strain. The errors in NLSM were obtained by calculating the difference of NLSM estimate by tracking(A_T) and final mechanical estimate(A_M) of A given by $\frac{A_T-A_M}{A_M}\times 100\%$. It can be seen that at lower strain the error in estimated NLSM, by both 1D and 2D tracking, is large, demonsMtrating higher deviation from mechanically obtained NLSM. However, at higher strain, these two methods behave differently with 2D registration giving closer estimates to mechanically observed values compared to 1D registration as seen from smaller errors in estimated NLSM. This confirms that nonlinear shear modulus estimated with 2D motion registration is a reliable estimate at higher deformation. Although it is difficult to quantify nonlinear shear modulus locally in the tissue with mechanical measurement setup, the behavior of the estimated nonlinear shear modulus with our 2D tracking scheme resembles closely with that of the mechanical system.

Fig. 8.

(a,b) Plots of error estimates(%) of NLSM for 5kPa gel and 4.9kPa PVA respectively as a function of strain. The errors are obtained by difference of NLSM A between tracking(A_T) and final mechanical estimates(A_M) given by $(\frac{A_T-A_M}{A_M}\times 100\%)$. Note this graph demonstrates how the error in estimates of A will decrease with higher strain data. At lower strain, the error is higher as it represents the bias in NLSM estimation resulting from lack of sufficient data points for fitting and applied strain not being large enough to get an estimate of A. The errorbars in the graph are computed by taking repeated 20 measurements of the mechanical data(stress-strain curve) and 20 measurements of ultrasound data(shear wave speed) at each compression step.

D. Heterogeneous Phantoms.

Fig. 9 shows the nonlinear shear modulus maps obtained in heterogeneous phantoms, using the three strain based methods described in Section II-B. The three rows of Fig. 9 represent elastograms obtained at three different strain levels. The inclusion structure in the first row is distorted for all the three methods with the rightmost NLSM image, estimated by 2D tracking, showing minimum distortion among them. As strain levels increase, inclusion structure gets properly restored in Fig. 9(i) by 2D tracking compared to the other two methods(global strain and 1D tracking) as shown in Fig. 9(g) and (h) respectively.

Fig. 9.

(I)Linear shear modulus map of a stiff gelatin inclusion phantom, (II)Representative nonlinear shear modulus maps obtained by three tracking methods for, 3%, 7% and 11% strain respectively. Inclusion structure in nonlinear elastogram was better preserved with 2D tracking.

Quantitative results(contrast ratio and CNR) were presented in Fig. 10 for the four types of heterogeneous phantoms detailed in Table. I. From the graphs, it is evident that NLSM imaging significantly improves contrast of the inclusion compared to linear shear modulus imaging. The contrast ratio values of nonlinear shear modulus by 2D tracking were estimated to be 70%–75% and 90%–115% higher than that of linear shear modulus in gelatin inclusion and PVA-gelatin inclusion phantoms respectively. On comparison with other two nonlinear shear modulus estimation techniques, our method presents 15%–60% increment in CR values. The CNR ranges between 20 and 75 for 2D tracking and 13 and 40 for 1D tracking as illustrated in bar-plots of Fig. 10. It can be seen that CNR for gelatin inclusion phantoms is higher compared to mixture of Gel-PVA phantoms.

Fig. 10.

(a–d) Plots of contrast ratio and (e–h)CNR obtained against overall strain in 4 different types of inclusion phantoms. Strain(%) levels at the starting of the experiment are ignored owing to the fact that curve fitting becomes accurate after 2–3 strain levels.

E. Effect of Lateral offset of inclusion.

Fig. 11 shows strain and nonlinear shear modulus maps obtained using both 1D and 2D tracking with the inclusion either at the center(first row) or offset 3.5 mm right of center(second row) with respect to the transducer plane. The NLSM image estimated by 1D tracking has distortions and contains more artifacts when the inclusion is offset from the center. This result is related to the fact that the strain map obtained with 1D tracking does not track the lateral motion of the tissue, which is significantly higher with the lateral offset of the inclusion. Fig. 12 plots the standard deviation of estimates of nonlinear shear modulus(bottom) of the inclusion portion, estimated by the three methods(described in section II-B), and CNR(top) as a function of lateral offset of the inclusion. It can be seen that 2D tracking presents stable nonlinear shear modulus value with lateral offset of inclusion, while for 1D tracking and global strain based method there is higher standard deviation of mean A with lateral offset. For CNR, good consistency was obtained in the 2D tracking method with lateral offset of inclusion, while for the other two methods CNR decreased from the center value. In addition, the graph on top of Fig. 12 shows that with higher deformation, CNR of nonlinear shear modulus increases. Therefore, qualitatively and quantitatively, nonlinear shear modulus estimated with 2D tracking is a more reliable measure.

Fig. 11.

(i) Linear shear modulus maps with inclusion at center and lateral offset, in a stiff inclusion phantom. (ii) Effect of lateral offset of inclusion on strain maps(a,b and e,f) and nonlinear shear modulus estimated by 1D(c,g) and 2D(a,b) tracking. Two rows represent inclusion at center and at 3.55mm right of center. 2D tracking preserves the inclusion structure better with lateral offset.

Fig. 12.

Effect of lateral offset of inclusion on CNR(top figure) and standard deviation of NLSM A(bottom figure). Representative standard B-mode images at the center.

F. Impact of stiff and soft gelatin inclusion phantoms of reverse stiffness on quantitative measurement.

In this section we studied the variations in estimated nonlinear shear modulus on stiff and soft gelatin inclusion phantoms of reverse stiffness. Table III presents the nonlinear shear modulus estimated with 1D and 2D tracking for 3 stiff inclusion and 3 soft inclusion phantom of nearly same linear shear modulus. The estimated nonlinear shear modulus of the stiff inclusion is in good agreement with the modulus of the background of the soft phantom with deviations between −2.2 kPa and + 2 kPa. For 1D tracking, the deviation range increased from −7.9 kPa to +10.2 kPa. These results suggest the importance of 2D tracking while estimating the nonlinear shear modulus. With different soft and stiff inclusion phantoms of same reverse stiffness, the lateral motion is different which when accurately tracked with 2D tracking, gives quantitatively reliable measures of nonlinear shear modulus.

Table III.

Table represents NLSM measurements estimated by 1D and 2D tracking for stiff and soft inclusion phantoms.

Inclusion Phantoms		Linear Modulus(kPa)	NLSM(kPa) (2D Tracking)	NLSM(kPa) (1D Tracking)
Stiff 1 (Inc at 20mm)	Inc.	10.2+0.6	118.2+5.5	102.1+7.2
	Med.	5.5+0.4	39.2+2.4	48.2+3.9
Stiff 2 (Inc at 30mm)	Inc.	10.9±0.8	119.1±5.5	112.5±8.9
	Med.	5.5±0.6	38.8±2.2	44.1±4.1
Stiff 3 (Inc at 40mm)	Inc.	11.3±0.6	120.4±6.5	119.7±7.1
	Med.	5.9±0.6	39.7±2.6	40.2±3.2
Soft 1 (Inc at 20mm)	Inc.	6.1±0.7	44.6±4.1	47.2±5.2
	Med.	10.7±1.2	119.3±9.4	101.6±10.4
Soft 2 (Inc at 30mm)	Inc.	6.3±0.7	44.8±4.5	57.2±7.1
	Med.	11.3±1.1	122.4±10.3	112.3±10.9
Soft 3 (Inc at 40mm)	Inc.	5.9±1.1	41.4±4.2	56.4±6.2
	Med.	10.8±l.l	120.4±9.6	116.5±11.3

G. Impact of initial loading

To assess how initial compression affects the quantification of nonlinear shear modulus, we conducted studies on the same soft inclusion phantom with given higher initial compressions. Fig. 13 shows plots of estimates of nonlinear shear modulus with pre compressions of 0.6mm, 1.2mm, 1.8mm and 2.4mm. With higher initial compression, the nonlinear shear modulus estimates initiates from higher values and gradually approaches the same stable nonlinear modulus value of 120 kPa. Higher initial loading has more profound effect on the background than on the inclusion, as evident from higher initial deviations of estimates of nonlinear modulus of the surrounding medium compared to the inclusion in Fig. 13.

Fig. 13.

Impact of pre-loading on nonlinear shear modulus estimation in a soft inclusion phantom. So by our 2D tracking, the final estimates of NLSM has good agreement even if the initial loading is varied.

H. Characterization of Material Property

This section demonstrates the importance of determination of nonlinear shear modulus in characterizing heterogenous material properties. Fig. 14 shows maps of strain, linear and nonlinear shear modulus in a gelatin inclusion embedded in PVA block at three different strain levels. The linear shear modulus images in the second column of Fig. 14 demonstrate that gelatin and PVA material have relatively closer linear shear modulus values, thus increasing the difficulty in differentiating the inclusion. The non-linear shear modulus map exhibits greater contrast of inclusion from background, apparent from the second row of Fig. 14. Thus contrast measurement of NLSM is a better way to differentiate the gelatin inclusion from PVA background when they have same linear shear modulus. Another important feature that this section captures is the difference in material non-linear property of gelatin and PVA. From the plots of estimates of nonlinear shear modulus of gelatin and PVA in Fig. 14(j), PVA material shows more nonlinearity compared to gelatin.

Fig. 14.

Representative results for gelatin inclusion embedded in PVA medium. (a,d,g) strain maps, (b,e,h) linear shear modulus maps and (c,f,i) non-linear shear map at 1.5%, 6% and 9% overall strain respectively.(j) Plots of estimates of nonlinear shear modulus obtained with fitting at gradual strain levels. Note the NLSM A does not vary with strain, at lower strain the NLSM is biased resulting from lack of data points and sufficient strain data for fitting, the best NLSM being the final stable value(dotted line).

V. Discussion

Our experiments demonstrate that non-linear shear modulus possesses distinct advantages over linear shear modulus and strain imaging in its ability to differentiate materials with similar linear elasticity. 2D motion registration improves the quantitative measurement of nonlinear shear modulus because of its ability to accurately track tissue deformation at higher strain. The error estimates between tracking and mechanical methods shown in Fig. 8 demonstrates that at higher strain 2D registration of tissue motion is a better quantitative measurement of nonlinear shear modulus. At lower strain, estimated measurement of NLsM mechanically is close to zero while ultrasound based measurement gives higher values for the same resulting in higher difference of NLSM between two methods. This result is related to the observation of an almost same apparent shear modulus as the undeformed shear modulus, very close to the origin of stress-strain curve of mechanical measurement. Hence, from equation(5), nonlinear shear modulus value estimated is near zero at lower strain. In ultrasound based system, we measure shear wave speeds locally and with small increase in strain obtained, small variation in shear wave speed gives significant change in shear modulus. We then indirectly compute the local stress(equation 4) which is different from the global stress that we obtain during mechanical experiments. Although equation 4 involves linear shear modulus, the difference in contrast between inclusion and background in Fig. 9(i), 11(h) and 14(i) is attributed solely to nonlinear shear modulus. This could be ensured from our experiments on gel-PVA inclusion phantom where the linear shear modulus was matched between gel and PVA before taking measurments. Further from Table III, where we made soft and stiff inclusions of reverse linear shear modulus, it was observed that the NLSM was also reversed. In mechanical measurements, we have the stress-strain data points, so we indirectly obtain the apparent shear modulus at different strain values. Further, it was not possible to give exactly same initial compression with the transducer as with the mechanical load which could be the reason for significant deviation in the initial nonlinear shear modulus measurements. Nevertheless, the gradual behavior of the estimates of nonlinear shear modulus obtained mechanically justifies with that of ultrasound-based measurement and at higher strain the difference gets reduced. Another observation From Fig. 8 is that, the measurements were repeatable as evident from the small percentage errorbars over multiple repeatation of the experiment. Percentage errorbars of difference of estimated NLSM A by both the tracking methods from mechanical values were same at lower strain, whereas at higher strain, errorbars were reduced with 2D tracking compared to 1D tracking, resulting in more reliable NLSM estimates and better repeatability of our method. As seen from the figure, the measurements obtained with 1D tracking have maximum percentage errorbars of ±9 and ±11.8 for gel and PVA respectively and the values get reduced to ±4.5 and ±5.7 for 2D tracking. Mechanical conditions like variation in probe position, impact of loading and distance of the inclusion with respect to the applied stress does not affect our measurements by 2D tracking. This makes the technique more flexible to measure complex tissue properties.

One limitation of this study is the difficulty to achieve ideal uniaxial compression in experiments(as seen in Fig. 6(g) and (i)) due to misalignment between the compressor plate and the phantom surface. For example, the compressor plate might not be perfectly parallel to phantom surface plane or the compressor plate might not be precisely centered with respect to the phantom. The computation time of our NLSM estimate by 2D tracking is 55–60 seconds compared to 35–40 seconds for 1D tracking. During our experiment, there might be small temperature variation due to compression. However, the phantom was incompressible and free to expand on the sides and the room temperature was maintained. Thus the scale of the temperature variation is small and we expect it not to cause any variance in our result.

In our experimental studies, shear wave speed was observed to increase with increasing strains, thereby resulting in a positive estimate of nonlinear coefficient A. This is because both gel and PVA are strain-hardening materials. However, in some of the NLSM images of inclusion phantoms, as shown in Fig. 11, there are negative measurements of nonlinear modulus values. This suggests that some portion of the material undergoes strain-softening, that is, shear modulus decreases with strain. Further studies are needed as to why in some portion of the inclusion phantoms, we are observing decrease in shear modulus with strain.

Another significant observation is the time dependent effect of shear modulus measurements with strain at each compression step. The pattern or behavior of apparent shear modulus with strain seemed to be affected by the time allowance of shear wave speed measurements after compression, particularly in PVA phantoms, which could be due to its viscoelastic nonlinearity.

The study presented here is a first step toward quantification of nonlinear shear modulus with accurate registration of tissue motion. Also single track location SWEI imaging has its own advantage in providing better quantitative shear modulus maps. A challenge for this technique is reliance on the operator to apply the deformation. In our laboratory, a robot assisted ultrasound screening system [35] has been developed to maintain the deformation and equally compress in between the deformation steps. This, will allow the operator to have real time strain, linear shear modulus and NLSM map with live compliance feedback to operate the robot for compression.

For PVA phantoms, strains of 10–15 percent are needed to get the stable NLSM estimates. Further experiments are needed to verify how this nonlinear shear modulus behaves with different stiffness of PVA. Nevertheless, the results obtained here identifies the behavior of the nonlinear shear modulus as a particular material dependent property.

VI. Conclusion

A novel 2D motion registered nonlinear shear modulus estimation technique has been implemented to measure the shear nonlinearity of soft tissues. Imaging of NLSM enables detection of inclusions that were indistinguishable in conventional linear shear modulus and strain images. The NLSM images formed by 2D motion tracking have high SNR, contrast ratio and CNR compared to those obtained by axial tracking or global strain based methods, demonstrating the importance of tracking of both axial and lateral tissue motion when applying large strains for quantifying nonlinear shear modulus. Results on combination of gel and PVA inclusion phantoms demonstrate that the behavior of nonlinear modulus can be used to characterize different material properties. We have validated our results with nonlinear shear modulus measurements obtained by mechanically unconfined compression, with 2D tracking showing better agreement at higher strain.