Automated In Vivo Sub-Hertz Analysis of Viscoelasticity (SAVE) for Evaluation of Breast Lesions

Abstract

We present an automated method for acquiring images and contrast parameters based on mechanical properties of breast lesions and surrounding tissue at load frequencies less than 1 Hz. The method called sub-Hertz analysis of viscoelasticity (SAVE) uses a compression device integrated with ultrasound imaging to perform in vivo ramp-and-hold uniaxial creep-like test on human breast in vivo. It models the internal deformations of tissues under constant surface stress as a linear viscoelastic response. We first discuss different aspects of our unique measurement approach and the expected variability of the viscoelastic parameters estimated based on a simplified one-dimensional reconstruction model. Finite-element numerical analysis is used to justify the advantages of using imaging contrast over viscoelasticity values. We then present the results of SAVE applied to a group of patients with breast masses undergoing biopsy.

I. Introduction

Pathologic changes in tissues can manifest as changes in their mechanical properties. Physical examination via hand palpation has been traditionally used to detect these abnormalities as a large group of malignant changes occur as stiff solid masses [1]. This is the basis for elastography techniques, which aim at estimating tissue stiffness noninvasively. In quasi-static strain elastography, an axial compression is applied and tissue deformation is visualized by analyzing the pre-compression and post-compression ultrasonic data [2]–[4]. In these maps, called “elastograms”, softer tissue shows higher levels of deformation (strain) compared to the stiffer parts. In breast tissue, methods based on strain elastography have shown promising results in increasing the diagnostic accuracy of the ultrasound-based methods [5]–[8]. While analysis of the elasticity alone can predict architectural changes in tissue to some extent, the complexity of tissue biomechanics requires a more comprehensive model to predict a wider range of pathologies. In breast tumors, particularly, finding a mechanical property that not only shows sensitivity to malignancy, but also to benign changes can be extremely beneficial as it can potentially enhance diagnosis specificity, which in turn may reduce a significant number of unnecessary, painful and costly biopsies.

Many recent studies have aimed at extending the notion of elastography to more comprehensive models that account for the biphasic nature of the tissue. These works have resulted in models based on viscoelasticity [9] as well as poroelasticity [10], [11] and poroviscoelasticity models [12]. These studies have shown that two major components play important roles in governing tissue deformation under external compression: a) the drained matrix viscoelasticity, which is mostly defined by the fiber density, orientation and cross-linking density, and b) the interstitial fluid motion that creates retardation in the deformation rate via frictional forces. However, a comprehensive model that can describe tissue biomechanics at a wide range of frequencies is still unknown. For example, using magnetic resonant elastography (MRE) the study in [13] shows that due to biphasic nature of tissue a single constitutive model cannot simultaneously explain tissue behavior at different frequency ranges. Hence, a unified approach to separate the dynamics of the two phases, i.e., a viscoelastic deformable solid response and a hydraulic fluid motion, in tissue-like materials is still an open problem [14], [15]. Due to these difficulties, alternative approaches that describe tissue deformation as purely poroelastic or purely viscoelastic solids are favored. Unlike quasi-static strain elastography, which relies only on pre and post deformation states of the tissue, these methods require continuous observation of the tissue response under an external stimulation. For example, methods based on poroelasticity imaging [11], require observation of the tissue dynamic deformation during hundreds of seconds while a constant surface compression is maintained. The methods based on viscoelastic models, on the other hand, have been shown to predict mechanical properties in smaller time scales [9] that are more favorable for in vivo applications. Tissue deformations at these time scales reside in a frequency range of less than 1 Hz. Stress-relaxation and creep-compliance tests are two standard mechanical testing methods suitable for this range of frequency. While stress-relaxation is not feasible for in vivo scenarios (due to lack of an accurate estimate of the stress distribution), creep-like tests can be performed with the aim of a force control mechanism that mimics a ramp-and-hold stress excitation while monitoring internal strain field via ultrasonic speckle tracking algorithms [9], [16]. The preliminary study in a group of patients with non-palpable lesions has shown that malignant and benign masses present different viscoelastic features [17] when employing a creep-like test with limited force feedback information [18]. However, the added noise due to manual operation, lack of an accurate estimate of the exerted force (to ensure operation in a linear range while maintaining a constant force) and operator dependency in interpretation of the imaging contrasts based on different viscoelastic parameters limits the utility and reproducibility of such method for patient studies.

In this study we introduce an automated method that utilizes a custom made proportional-integrator-differentiator (PID)-controlled compression device integrated with ultrasound imaging for accurate force application and viscoelastic modelling of the internal deformations. Our previous study showed the utility of this device in viscoelasticity characterization of tissue mimicking phantoms [19]. Here, we show through simulations that intrinsic viscoelastic parameters cannot be reliably recovered when a 1-D inversion model is used, even in uniform blocks of viscoelastic solids. However, simulation study highlights the benefits of viscoelastic imaging contrast features as more appropriate measures for lesion differentiation. We also propose an automated region of interest selection for calculation of the contrast features based on different viscoelasticity parameters, which removes subjectivity in interpretation of the estimated viscoelastic maps in terms of imaging contrast features. The results of SAVE in a group of breast patients undergoing biopsy will be presented and different viscoelasticity parameters will be analyzed for significant differences in the two groups of benign and malignant lesions.

II. Materials and Methods

A. Linear Viscoelastic Model

Uniaxial creep testing is a standard procedure for studying long-term viscoelastic properties of materials [19]. In this study, in order to analyze tissue viscoelastic properties, a standard linear solid model is adopted to analyze tissue response under uniaxial creep compression. Fig. 1(a) and (b) show a 3-elememt constitutive model and its stress-strain temporal behavior under a creep test respectively. Under this model, in response to a fast ramp-and-hold stress, tissue behaves mostly as an elastic solid during an initial fast compression phase followed by a mostly viscoelastic response.

Fig. 1.

(a) Standard linear solid model based on first order Kelvin-Voigt element (b) model response to a ramp-and-hold input stress. Shaded area represents the mostly elastic part of the strain response.

Based on this model, the stress-strain relationship can be formulated as

$$\begin{gathered} \sigma (t)=\{\begin{array}{ll}\frac{{\sigma }_0}{T_r}t\hfill & t\le T_r\hfill \\ {\sigma }_0\hfill & t>T_r\hfill \end{array} \\ \epsilon (t)=\frac{\sigma (t)}{E_0}u(t)+\left(\frac{d}{dt}\sigma (t)\right)\ast \frac{1}{E_1}\left(1-e^{-\frac{t}{T_1}}\right)u(t) \\ \approx \{\begin{array}{ll}\frac{{\sigma }_0}{E_0}\frac{t}{T_r}\hfill & t\le T_r\hfill \\ \frac{{\sigma }_0}{E_0}+\frac{{\sigma }_0}{E_1}\left(1-e^{-\frac{t}{T_1}}\right)\hfill & t>T_r\hfill \end{array} \end{gathered}$$ (1)

where σ₀ is the final value of applied stress, T_r is the stress ramp time, u(t) is the Heaviside step function, T₁ = η/E₁ is the viscoelastic retardation time and * indicates convolution operation. Given that T_r ≪ T₁, the dashpot shown in Fig. 1(a) does not allow any sudden deformations during ramp time, hence a mostly elastic response is observed due to the spring element in series. The dashpot also does not play any role in the final deformation when time approaches infinity hence the final strain reaches a plateau based on this model. A formal derivation of these approximations is given in Appendix A.

B. Finite Element Modeling

Equation (1) describes the deformation of a 1-D element. Characterization of 3-D physical materials using this model requires complete knowledge about deformations in all directions. However, ultrasonic tracking of the tissue deformations only provides strain data in one imaging plane, which is mostly accurate along the transducer transmission (axial) direction. Hence, to examine the amount of error incurred when using the 1-D model in (1) for heterogeneous 3-D materials, which is also suitable for studying breast masses, a finite element numerical simulation was performed. Two sets of simulations were considered. In one set of simulations, two uniform blocks of viscoelastic materials with different parameters were analyzed under uniaxial creep with a ramp-and-hold surface stress. The viscoelastic parameters of these materials were chosen such that their combinations present viscoelastic values and contrast features similar to those reported in the previous studies of human breast [17]. Since efficiency of the contrast transfer for purely elastic models have been widely investigated [20], [21], the same elastic properties were used for all models (uniform blocks and inclusion). Using the model shown in Fig. 1, for material 1, E₀ = 45 kPa, E₁ = 450 kPa, η = 450 kPa·s and for material 2 E₀ = 45 kPa, E₁ = 450 kPa, η = 1350 kPa·s were selected. Hence the two materials had the same elastic parameters but different viscosity which resulted in retardation times of 1 s and 3 s for material 1 and 2 respectively. In another set of simulations, a cylindrical inclusion of one material with diameter of 1 cm was inserted in the other material to mimic a breast tumor. Both material 1 and material 2 were used as inclusions embedded in the other material. The model that had the material with retardation time of 1 s embedded in the material with retardation time of 3 s is called Model 1–3 and the model that had the material with retardation time of 3 s embedded in the material with retardation time of 1 sis called Model 3-1.

In order to mimic the loading asymmetry condition which might occur in realistic in vivo situations, the cylindrical inclusion was shifted up 6mm from the center of the cube and moved 1 mm to the right side of the cubic background.

In all examples, we considered the bottom of the cube under no-slip conditions and the top surface interacting with the compression plate as a free-slip condition. The compression plate had a surface area of 6 cm × 4 cm, similar to the one used in the in vivo studies, and its center was aligned with the center of the cubic block.

In each simulation, a ramp-and-hold force was applied with a ramp time of T_r = 0.25 s and final force value of 2 N. Nodal axial strain data obtained from each simulation were fitted to the viscoelastic response in (1) in two steps. In the first step, the strain amplitude after 0.25 s was used to estimate E₀ based on (1) for t ≤ T_r. The initial 0.25 s of strain data was removed and a single exponential viscoelastic response ((1); t > T_r) was fitted to the remaining portion of the strain curve time series to obtain E₁ and T₁ using Marquette-Leveque nonlinear least-square fitting in MATLAB (Mathworks, Natick, MA).

C. Force Application and Ultrasonic Displacement/Strain Tracking

The compression force measurement and imaging setup explained in [16] was used to create axial force as well as acquiring ultrasonic radiofrequency data for in vivo studies. A schematic of the setup is shown in Fig. 2. The ultrasound probe surface was extended via rigid plastic back and front plates. While the back plate was fixed, the front plate was kept in minimal contact with the load cells and was able to move axially. The edges around the two plates were open to allow free motion of the ultrasound gel during compression. Four load cells were mounted at each corner of the extension plate (between the two plates) for simultaneous measurement and control of the desired force profile as explained in [16].

Fig. 2.

A schematic diagram of the uniaxial creep test setup for in vivo viscoelasticity imaging of breast lesions.

The entire compression setup had a portable weight and was mounted on a lockable flexible arm during patient studies (Fig. 3).

Fig. 3.

Compression device mounted on a lockable flexible arm. The setup is positioned on a breast elasticity phantom to demonstrate the utility of the device for in vivo studies.

In order to mimic step-like stress stimulation a fast ramp force is required. Such fast deformation of tissue, if not imaged at appropriate frame rates, can cause severe decorrelation of the axial data, which in turn can lead to speckle tracking failure [23]. In order to image all phases of deformation, plane wave imaging was used to acquire raw radiofrequency data at 200 frames per second using a Verasonics programmable ultrasound machine (Verasonics Inc., Kirkland, WA) and a linear array transducer L11-4v (Verasonics Inc., Kirkland, WA). A two dimensional autocorrelation technique [23] was used for displacement tracking. The fast axis windows length was 3λ (λ = 0.25 mm is the imaging wavelength at 6.25 MHz with sound speed of 1540 m/s). The slow axis window was 15 ms. The displacement data was then used to estimate the induced strain for each pixel in the imaging plane using the staggered strain estimation method in [24] with a window length of 20λ. The resulted strain field was post processed using a 9λ × 9λ median filter.

D. In Vivo Patient Studies

Data collected from 16 female patients referred to Mayo Clinic’s breast imaging were analyzed. All subjects signed an informed Institutional Review Board (IRB) consent form prior to the study. Pathology results from needle biopsy were obtained for all patients after the study. During the study patients were scanned while in a supine position. The compression/imaging setup (Fig. 3) was then applied to each patient’s breast. A sonographer with more than 28 years of experience found the lesion sonographically and positioned the compression device appropriately for each acquisition. Prior to each data acquisition, test compressions were applied to ensure minimal lateral and out-of-plane tissue motions will occur during the actual test. Depending on the location of the lesion, patients were instructed to slightly reposition in order to make best use of the chest wall as a supporting structure for uniform axial compression as shown in Fig. 2. A total of 10 seconds of data was collected from each patient at 200 frames per seconds imaging rate. During this time, patients were asked to hold their breath to minimize respiration induced strain artifacts. For each patient a ramp-and-hold force profile was applied with a final force value of 2 N and ramp speed of 8 N/s. This amount of force created a final stress value of σ₀ = 833.33 Pa, given the compression plate surface of 24 cm², which ensured linear tissue response based on previous studies [9]. The pre-compression was maintained less than about 0.3 N to avoid any effect due to preloading. The ultrasound strain tracking provided strain curves for each pixel in the imaging domain. Due to variations in the breast tissue viscoelasticity in different patients, the force ramp time was expected to be slightly different from the programmed values. Hence, instead of 0.25 s, 0.5 s of the initial strain data was omitted to separate the instantaneous elastic response. The amplitude of the instantaneous strain response was used to estimate the elasticity value for the spring in series based on the model presented in Fig. 1. The remaining viscoelastic compliance curves were fitted to the model in (1) using Marquette-Leveque nonlinear least square fitting in MATLAB in which complete two dimensional maps of different parameters were created. For each pixel in the imaging plane located at grid point (m, n), the viscoelastic parameters E₀, E₁ and T₁ were estimated as

$$\begin{gathered} {\widehat{E}}_0(m,n)=\frac{{\sigma }_0}{{\epsilon }_{m,n}(T_r)} \\ \left({\widehat{E}}_1(m,n),{\widehat{T}}_1(m,n)\right) \\ =\underset{(E_1,T_1)}{argmin}{\Vert \frac{{\sigma }_0}{E_1}\left(1-e^{-\frac{t}{T_1}}\right)-({\epsilon }_{m,n}(\mathrm{t})-{\epsilon }_{m,n}(T_r))\Vert }^2,t>T_r \end{gathered}$$ (2)

A normalized root mean square (NRMS) fitting error was calculated for each pixel located at axial-lateral grid point (m, n) in the imaging plane as

$$e\left(m,n\right)=\frac{\Vert {\epsilon }_{m,n}(t)-{\widehat{\epsilon }}_{m,n}(t)\Vert }{\Vert {\epsilon }_{m,n}(t)\Vert }$$ (3)

where ε_m,n (t) and ε̂_m,n (t) are the measured and fitted strain profiles respectively and ||.|| indicates the Euclidian norm. This measure can be regarded as a parameter for assessing goodness of the fitted viscoelastic model. The residual term in (3) incorporates both slow deviations from the model (e.g., due to a non-representative model) as well as strain variations due to the inability of the force-control instrument to remove fast fluctuations (e.g., due to natural cardiac motions). The latter appears as zero-mean high-frequency fluctuations imposed upon the strain curves; it has a minor effect on viscoelastic parameter estimates. Hence the NRMS fitting error is rescaled based on its energy contained in the frequency range expected for the slow creep response (i.e., <1 Hz).

E. Automated Region of Interest (ROI) Definition

The results of nonlinear fitting process are two dimensional maps of viscoelasticity parameters E₀, E₁ and T₁. The main assumption is that lesions present viscoelasticity features that are different from those of the surrounding non-lesion breast tissue. In practice, however, this difference may not clearly delineate the lesion boundary in the acquired viscoelasticity maps, or parameters may indicate different contrast features. In order to assess tumor margins, an automatic method was devised. This method was comprised of the following steps:

1) Initial Seeding of Lesion Boundary

Prior to any compression, each lesion boundary was defined from the first B-mode image in the image sequence. The speckle tracking displacement data was then used to deform this pre-compressed boundary according to the motion of each point in the imaging plane. The boundary deformation was stopped just before the creep response started. This deformed boundary was then used for all subsequent analysis of the estimated viscoelastic parameters based on the reduced strain data (the strain data after 0.5 s).

2) ROI Formation for In Vivo Measurement of Contrast Values

Automatic selection of the quantification areas significantly removes subjectivity from the interpretation of viscoelasticity values associated with lesion and background tissue. Toward this goal and to limit the quantification of viscoelastic parameters to lesions area and its surrounding tissues, the deformed lesion boundary explained in the previous section was used to create region of interests (ROIs). It is known that in some malignant breast lesions there is a desmoplastic reaction surrounding the lesion so that lesions appear larger in stiffness maps than that observed in B-mode images. Hence, in order to reduce ambiguity in classification of the lesion and surrounding non-lesion tissue, a quantification exclusion mask around the lesion area was established. This mask was created using a dilation operation on the lesion boundary. The dilation was performed using morphological operation to select ROI regions which adapt to the shape and location of different lesions. The dilation size can vary, however, here a dilation factor of 0.3 was empirically chosen for the in vivo data based on the results of lesion size differences reported for the strain elastograms compared to B-mode [25]. A second ROI was then formed using a dilation factor of 1.3 to create an annular area in the surrounding non-lesion tissue with a radius equal to that of tumor radius. As shown in Fig. 4, area 2 was excluded from any quantification and area 1 and 3 were used for quantification of the lesion and back-ground normal tissue viscoelastic parameters, respectively. In cases of elongated lesions, the radius along the longest axis was considered. Since the initial elastic and viscoelastic responses were analyzed separately, different quantification regions were used. For E₀ parameter, all ROIs were created based on the lesion boundary seen in the pre-compressed B-mode. For E₁ and T₁, however, ROIs were formed based on deformed lesion boundary obtained by continuous motion tracking of the initial boundary during the ramp part of the applied stress.

Fig. 4.

Automated ROI selection based on lesion geometry and morphology dilation. Area 1 represents the lesion and is formed based on lesion appearance on the B-mode image and continuous tracking of the boundary during deformations. This area is used for quantification of the lesion viscoelastic properties. Area 2 is the results of dilating area 1 by a factor of 0.3 and subtracting it. This area is excluded from any quantification due to uncertainty in the lesion size obtained from the B-mode images. Following similar procedure, area 3 is formed and used for quantification of the viscoelastic properties of the surrounding non-lesion tissue.

3) Contrast Values Versus Viscoelastic Parameters

The estimated viscoelasticity parameters based on direct inversion of the 1-D strain data (as demonstrated in the results section) present strong sensitivity to the boundary conditions and geometries. The situation may be even more challenging for in vivo scenarios due to natural heterogeneity of breast tissue and surrounding anatomies. Hence, addition of contrast measures based on viscoelastic parameters may provide a systemic way to reduce these sensitives due to inherent normalization. In this study, for viscoelastic parameter X measured within the lesion and surrounding non-lesion tissues, contrast was defined as

$$X\text{Contrast}=\frac{(X_{\text{lesion}}-X_{\text{nonlesion}})}{(X_{\text{lesion}}+X_{\text{nonlesion}})/2}$$ (4)

F. Statistical Analysis

All statistical analyses were performed using MedCalc, version 15.0 (MedCalc Software, Ostend, Belgium). Due to unrealistic fluctuations of E₀ and E₁ near boundaries (as presented in simulation and in vivo results), 25% of the quantitative values outside the median distribution were excluded before performing any statistical analysis for these parameters.

Student t-test was used to analyze significant differences in different parameters in the two groups of benign and malignant lesions.

III. Results

Fig. 5 summarizes the results of finite element modeling. Each column of images presents the spatial distribution of parameters for a different viscoelastic phantom estimated using axial strain data. The first two columns are from media with uniform properties including retardation times of 3 s and 3 s, respectively. Columns 3 (4) represents the results of the inclusion model where inclusion had a retardation time of 3 s (1 s) and background had a retardation time of 3 s (1 s).

Fig. 5.

Two dimensional maps of viscoelasticity parameters based on direct inversion of the 1-D strain data (a)–(c) uniform block with retardation time of 1s, (d)–(f) uniform block with retardation time of 3 s, (g)–(i) inclusion Model 3-1 and (j)–(l) inclusion Model 1-3.

In all cases, the SAVE measurement shows a high degree of error in estimation of the E₀ and E₁ near the top and bottom surfaces. Estimates approach the true values (E₀ = 45 kPa and E₁ = 450 kPa) only near the center of the model. Unlike elastic parameters, retardation time maps do not present noticeable sensitivity to boundary effects in the uniform block models (Fig. 5(c) and (g)) where average estimated T₁ values are very close to the input values (less than 1% error in both materials) (Fig. 5(c), (f)). In the inclusion models, however, the estimated T₁ maps present spatial variations. The estimated T₁ values from background and inclusion approach each other’s value, which in turn, results in a loss of contrast as seen in Fig. 5(i), (l). Table I summarizes all the numerical results from the simulation models. In order to minimize estimation bias due to increased node density near edges (as usually occurs in finite element meshing), the viscoelastic parameters estimated from each node’s strain data were spatially interpolated to a uniform rectangular grid. Additionally, to remove rapid fluctuations, 25% of the data residing outside the median values were excluded from averaging and calculation of the standard deviations. As it can be seen in Table I, for media with uniform properties, there is significant bias in the estimation of E₀ and E₁. However the estimated viscoelastic retardation times are within 1% of the input values for both uniform materials. In contrast, the estimated viscoelastic retardation times in the inclusion models show larger deviations. In Model 3-1, the inclusion retardation time decreases from 3 s to 1.77 s (more than 40% bias) while it increases from 1 s to 1.1 s (10% bias) in the background. Similarly, in Model 1-3, the inclusion retardation time increases from 1 s to 2.16 s (116% bias), while it decreases from 3 s to 2.91 s (3% bias) in the background. In both inclusion models, while the background elastic parameters E₀ and E₁ present significant bias (more than 44% for E₀ and more than 24% for E₁), the inclusion elastic parameters E₀ and E₁ are in strong agreement with the input values (less than 10% bias for E₀ and less than 3% bias for E₁).

TABLE I.

Viscoelastic Parameters Estimated From the Simulated Axial Strain Data in Different Models

	Model parameters	Uniform block	Model3-1	Model1-3
Material 1	E0 = 45 kPa	E0 = 108 ± 152.28 kPa	E0 = 65.05 ± 8.71 kPa	E0 = 65.39 ± 8.75 kPa
	E1 = 450 kPa	E1 = 602.61 ± 81.18 kPa	E1= 621.52 ± 86.78 kPa	E1 = 562.73 ± 78.60
	T1 = 1 s	T1 = 1.01 ± 0.02 s	T1 = 1.10 ± 0.07 s	T1 = 2.16 ± 0.07 s
Material 2	E0 = 45 kPa	E0 = 105.66 ± 142.90 kPa	E0 = 49.12 ± 1.60 kPa	E0 = 49.17 ± 1.59 kPa
	E1 = 450 kPa	E1 = 537.66 ± 70.55 kPa	E1 = 459.56 ± 14.39 kPa	E1 = 450.06 ± 14.30 kPa
	T1 = 3 s	T1 = 3.0215 ± 0.22 s	T1 = 1.77 ± 0.07 s	T1 = 2.91 ± 0.10 s

A. In Vivo Study Results

1) Patient Demographics and Histopathological Results

Participants had a mean age of 54.19 ± 14. The breast imaging reporting and data system (BI-RADS) score was 5, 4 and 3 in 5, 10 and 1 patients respectively. Pathological findings revealed 10 lesions as malignant and 6 lesions as benign.

2) Reconstruction Examples

Fig. 6 summarizes the results of SAVE processing on two representative subjects; one with a malignant invasive carcinoma and another with a benign fibroadenoma.

Fig. 6.

left panel: SAVE results from a malignant breast lesion (a) normalized applied stress and normalized strain curves from lesion area and surrounding non-lesion tissue, (b) B-mode image, (c) automated ROI created for quantification of E₀, (d) E₀ map with boundaries of quantification regions overlaid, (e) automated ROI created for quantification of E₁ and T₁, (f) E₁ map with boundaries of quantification regions overlaid, (g) T₁ map with boundaries of quantification regions overlaid, (h) map of NRMS fitting error e. right panel: SAVE results from a benign breast lesion (j) normalized applied stress and normalized strain curves from lesion area and surrounding non-lesion tissue, (i) B-mode image, (l) automated ROI created for quantification of E₀, (k) E₀ map with boundaries of quantification regions overlaid, (n) automated ROI created for quantification of E₁ and T₁, (m) E₁ map with boundaries of quantification regions overlaid, (o) T₁ map with boundaries of quantification regions overlaid, (p) map of NRMS fitting error e.

In each case, the lesion boundary is highlighted using a red dashed line in the B-mode image and dashed black line in the corresponding maps of different viscoelasticity parameters. Using the automated ROI creation method explained in the previous section, lesion area, a quantification exclusion mask and surrounding non-lesion tissue area are defined. These quantification areas are defined for both initial elastic responses (Fig. 6(c), (l)) and viscoelastic responses (Fig. 6(e), (n)). The colored areas highlighting different quantification regions are similar to those in Fig. 4 where in each viscoelasticity map the magenta and blue dashed lines highlight the boundary of region 2 and 3 respectively. Fig. 6(a), (j) show the normalized surface stress and representative normalized strain time curves from the lesion area and surrounding non-lesion tissue. As can be seen in both cases, the strain response comprises an initial fast elastic response followed by a slow viscoelastic response. The vertical dashed blue lines in the graphs indicate the time boundary that separates the mostly elastic response from mostly viscoelastic. In both cases, the stress time curves present cyclic fluctuations which can be attributed to the cardiac-induced stress not completely removed due to the limited bandwidth of the force-control feedback loop. Similar to the simulation models, the estimated E₀ and E₁ maps show significant overestimation near the skin as can be seen in Fig. 6(d), (f), (k), (m). In the case of a malignant lesion (Fig. 6 left panel), clear lesion contrast can be seen in all viscoelasticity parameters, E₀, E₁ and T₁, which is consistent with the lesion boundary obtained from the B-mode images. The elasticity maps E₀ and E₁ present a positive contrast in the lesion area indicating higher stiffness of the lesion compared to the background tissue. The viscoelasticity retardation time maps, T₁, on the other hand presents a negative contrast, indicating a faster creep response in the lesion area compared to the surrounding tissue. In the case of a benign lesion (Fig. 6 right panel), no significant contrast is observed in the maps of elasticity parameters E₀ and E₁ while a distinct positive contrast is seen in the T₁ map. This finding is in strong agreement with the appearance of the lesion in the B-mode image in terms of shape and location. Figs. 6(h), (p) show that the NRMS fitting error, e, is negligible throughout most of the reconstruction maps in both cases. Regions with low values of e indicate a close fit between the data and best-fit model function for those pixels. Using the quantification regions shown in Figs. 6(c), (l), (e), (n), for each case, the mean and standard deviation of E₀ and E₁ and T₁ for lesion and surrounding non-lesion tissue were calculated and are shown in Table II. Additionally, contrast parameters based on E₀ and E₁ and T₁ measurements were also found and presented in this table. In the malignant lesion, E₀ was 61.85 ± 10.12 kPa and 28.36 ± 5.97 kPa in the lesion and non-lesion surrounding areas respectively. These values resulted in a E₀ contrast of 0.74 which is in agreement with the conspicuous imaging contrast seen in Fig. 6(d). In the benign lesion, however, the lesion and non-lesion areas had very similar E₀ values (11.98 ± 0.78 kPa and 11.31 ± 1.92 kPa respectively) which resulted in a small contrast value of 0.06. It is important to note the even though there is visible heterogeneity in E₀ values in areas far from the lesion (Fig. 6(k)), the automated ROI definition method has properly excluded those areas from quantification of the viscoelastic parameters of lesion and surrounding non-lesion tissue.

TABLE II.

Viscoelastic Parameters Estimated From Two in vivo Patient Data Files Using SAVE

	E0 (nonlesion)(kPa)	E0 (lesion)(kPa)	E1 (nonlesion)(kPa)	E1 (lesion)(kPa)	T1 (nonlesion)(s)	T1 (lesion)(s)	E0 contrast	E1 contrast	T1 contrast
Malignant	28.36 ± 5.97	61.85 ±10.12	110.77 ± 25.86	169.59 ± 22.94	1.75 ± 0.74	1.14 ± 0.39	0.74	0.42	−0.43
Benign	11.31 ± 1.92	11.98 ± 0.78	54.14 ± 9.93	45.87 ± 3.27	1.75 ± 0.42	2.23 ± 0.39	0.06	−0.17	0.24

In the malignant lesion, E₁ was 169.59 ± 22.94 kPa and 110.77 ± 25.86 kPa in the lesion and surrounding non-lesion areas respectively with a contrast value of 0.42. In the benign case, however, the E₁ value was 45.87 ± 3.27 kPa and 54.14 ± 9.93 kPa in the lesion and surrounding non-lesion areas respectively. These values resulted in a contrast value of −0.17. In the malignant case, T₁ was 1.14 ± 0.39 s and 1.75 ± 0.74 s in the lesion and surrounding non-lesion areas. These values resulted in a T₁ contrast of −0.43. In the benign case, T₁ was 2.23 ± 0.39 s and 1.75 ± 0.42 s in the lesion and surrounding non-lesion areas respectively. These values resulted in a T₁ contrast of 0.24. While T₁ contrast presented opposite trends in the malignant and benign cases, T₁ value of surrounding non-lesion tissue was very similar (0.32s difference) in the two cases.

3) Statistical Results

Fig. 7 shows the error-bar plots of means for viscoelasticity parameters E₀ and E₁ and T₁ acquired by SAVE associated with lesion and surrounding tissue in the two groups of patients with biopsy-proven benign and malignant lesions. The mean viscoelasticity parameters E₀ and E₁ and T₁ were 17.12 ± 8.95 kPa, 51.55 ± 46.49 kPa and 1.36 ± 0.33 s respectively in the malignant group. The mean viscoelasticity parameters E₀ and E₁ and T₁ were 15.00 ± 8.39 kPa, 28.11 ± 21.41 kPa and 1.84 ± 0.80 s respectively in the benign group. The mean elasticity values E₀ and E₁ were higher in the malignant cases compared to benign but the differences were not statistically significant.

Fig. 7.

Error-bar plots of mean for different viscoelasticity parameters E₀ and E₁ and T₁ associated with lesion and surrounding tissue in benign (N = 6) and malignant (N = 10) lesions. Black bars represent the data range.

The mean T₁ value was lower in the malignant group compared to benign but it was not found to be statistically significant. In the group of benign lesions, the mean values of E₀ and E₁ and T₁ for the surrounding tissue were 11.54 ± 3.66 kPa, 24.51 ± 24.86 kPa and 1.56 ± 0.71 s respectively. These values were not found to be significantly different from those associated with the lesion. In the group of malignant lesions, the mean values of E₀ and E₁ and T₁ for the surrounding tissue were 11.54 ± 3.66 kPa, 34.33 ± 24.46 kPa and 1.43 ± 0.45 s respectively. These values were not found significantly different from those associated with the lesion. Additionally, none of the viscoelastic parameters associated with the surrounding tissue in benign cases were found to be significantly different from the corresponding values in the malignant group. Fig. 8 shows the error-bar plot of contrast values based on different viscoelasticity parameters. As it can be seen, while contrast values based on elastic parameters (E₀ and E₁) do not present a significant difference in the two groups of benign and malignant patients (P = 0.7562 and P = 0.3505 respectively), T₁ contrast presents a mostly positive contrast for benign lesions and a negative contrast for malignant cases. The mean T₁ contrast was found to be significant in the two groups of patients (P = 0.0079).

Fig. 8.

Error-bar plot of contrast values based on different viscoelasticity parameters in benign and malignant lesions.

IV. Discussion

In this paper we introduced SAVE as a novel method for imaging linear viscoelastic properties of breast lesions in vivo. Our method comprised an automated force-controlled compression device for applying accurate ramp-and-hold stress on breast tissue and analyzing the resulting strain behaviors in terms of a linear viscoelastic response. Our simulation model showed that even in ideal uniform blocks, the 1-D inversion of the elastic parameters present significant sensitivity near boundaries which results in highly unrealistic moduli of elasticity. The viscoelasticity retardation time, however, was not seen to be sensitive to boundary effects and 1-D inverse modelling on the uniform blocks resulted in accurate and spatially uniform recovery of the retardation times. When testing inclusion models, elasticity parameters presented similar sensitivity issue near boundaries. The viscoelasticity retardation time, though not affected by the boundary conditions, showed spatial variability and bias in both inclusion and background materials. This effect was also seen in our earlier results in tissue mimicking phantoms [19]. Although the estimated retardation times from the 1-D model cannot accurately represent intrinsic material parameters, when the task is benign-malignant classification, contrast features acquired by SAVE are more reliably discriminating. When applied in vivo, in order to interpret different viscoelasticity parameters in terms of mechanical parameters and their corresponding contrast values, the definition of an ROI is necessary to separate the parameters of lesion from those of background tissue. ROI selection has been a major obstacle in objective interpretation and quantification of the mechanical properties acquired by different elastography methods. This is especially important as lesion B-mode features (e.g., size) do not always coincide with the features seen in the maps acquired from viscoelastic reconstructions [25]. In this paper, we presented a novel ROI selection method based on an initial seeding of the lesion boundary in pre-compressed B-mode images and continuous tracking of this contour through different phases of deformation. This method enabled automatic definition of the quantification regions and exclusion of the regions where lesion-background margin cannot be reliably estimated from the B-mode images. In case of the malignant lesion, the estimated lesion boundary derived from the B-mode image favorably coincided with the margin of an area with elevated E₀ and E₁ values and decreased T₁. In case of a benign lesion, only the T₁ map presented a noticeable contrast with distinct margins that corroborated with the automatic ROI boundaries derived from the B-mode image. In both cases the numerical contrast values were in good agreement with the contrast features seen in different viscoelasticity reconstruction maps on and near the lesion area. Natural complexity of the human breast, both in terms of tissue composition and anatomical geometries can create heterogeneity in the estimated viscoelastic parameters. However, multiple elements played important roles in our unique method which helped capture the best diagnostic values from this test. First, utilization of an automated compression device with large contact area provided excellent conditions for mostly uniaxial and plane strain deformation of the tissue and simultaneous force control. This in turn, enabled accurate separation of initial elastic from slow viscoelastic responses from the stain curves which can directly impact the accuracy of estimated viscoelastic parameters. This feature alone provided a significant improvement over manual force application presented previously [17] with the added value of ensuring operation in linear regime (via accurate implementation of small forces). Second, the use of automatic ROIs, not only provided a systematic quantification method with minimal subjectivity, but also enabled limiting the quantification areas very close to the lesion location. While compression plate can provide excellent uniaxial strain in the areas near the center, unwanted dynamics might adversely impact the accuracy of tissue viscoelastic estimation near the edges. Hence, the center area where lesion is targeted provides the best quantification region. The automated ROI, enabled this smart selection by adjusting to the lesion location and size without manual intervention. As a result of these two important features, in the two in vivo cases, while lesion T₁ value was different (mostly due to different pathologies), surrounding non-lesion areas had very similar T₁ values. The preliminary results of SAVE in 16 patients with breast lesions indicated that malignant lesions had higher elasticity values (both E₀ and E₁) compared to the benign lesions. Additionally, benign lesions showed larger lesion viscoelastic retardation times compared to the malignant cases. However, none of these findings were statistically significant. Among other factors, the sensitivity of the 1-D inversion on lesion geometry, location and boundary condition (as also observed in the simulations) can greatly limit the ability of SAVE in resolving intrinsic viscoelasticity values under different conditions which may be experienced during in vivo studies. However, the T₁ contrast values were mostly positive in the benign cases and mostly negative in the malignant lesion. The difference in T₁ contrast values in the two groups was found to be statistically significant (P = 0.0147). Hence, contrast values may provide better discrimination powers due to inherent normalization which is included in calculation of these values (4).

V. Conclusion

In this study we presented SAVE as a novel method for characterization of breast masses based on their viscoelastic features in vivo at very low frequencies (<1 Hz). Using numerical modelling and in vivo examples, we discussed different aspects of creep-like tests for estimation of viscoelastic parameters of the breast lesions and surrounding non-lesion areas from which reliable contrast features were derived. These features, if proved in a larger pool of patients, may introduce new biomarkers for differentiation of breast masses and provide reliable tools to help decrease the number of unnecessary biopsies.

Appendix

A. Viscoelastic Response Approximations With Ramp-and-Hold Input Stress

Given the ramp-and-hold stress as

$$\sigma (t)=\{\begin{array}{ll}\frac{{\sigma }_0}{T_r}t\hfill & t\le T_r\hfill \\ {\sigma }_0\hfill & t>T_r\hfill \end{array}$$

the resulting strain can be written as

$$\epsilon (t)=\frac{\sigma (t)}{E_0}u(t)+\left(\frac{d}{dt}\sigma (t)\right)\ast \frac{1}{E_1}\left(1-e^{-\frac{t}{T_1}}\right)u(t)$$

where * indicates convolution operation. Additionally

$$\frac{d}{dt}\sigma (t)=\frac{{\sigma }_0}{T_r}(u(t)-u(t-T_r))$$

Hence

$$\left(\frac{d}{dt}\sigma (t)\right)\ast \frac{1}{E_1}\left(1-e^{-\frac{t}{T_1}}\right)u(t)=\frac{{\sigma }_0}{T_r}(u(t)-u(t-T_r))\ast \frac{1}{E_1}\left(1-e^{-\frac{t}{T_1}}\right)u(t)$$

For 0 ≤ t < T_r

$$\begin{gathered} \frac{{\sigma }_0}{T_r}(u(t)-u(t-T_r))\ast \frac{1}{E_1}\left(1-e^{-\frac{t}{T_1}}\right)u(t) \\ =\underset{0}{\overset{t}{\int }}\frac{{\sigma }_0}{T_r}\frac{1}{E_1}\left(1-e^{-\frac{\tau }{T_1}}\right)d\tau \\ =\frac{{\sigma }_0}{T_r}\frac{1}{E_1}\left[t-T_1\left(1-e^{-\frac{t}{T_1}}\right)\right]\approx \frac{{\sigma }_0}{T_r}\frac{1}{E_1}\left[t-T_1\frac{t}{T_1}\right] \\ =0 \end{gathered}$$

where in the last equation we used $1-e^{-\frac{t}{T_1}}\approx \frac{t}{T_1}$ (since t < T_r ≪ T₁).

For t > T_r

$$\begin{gathered} \frac{{\sigma }_0}{T_r}(u(t)-u(t-T_r))\ast \frac{1}{E_1}\left(1-e^{-\frac{t}{T_1}}\right) \\ =\underset{t-T_r}{\overset{t}{\int }}\frac{{\sigma }_0}{T_r}\frac{1}{E_1}\left(1-e^{-\frac{\tau }{T_1}}\right)d\tau \\ =\frac{{\sigma }_0}{T_r}\frac{1}{E_1}\left(T_1{e}^{-\frac{t}{T_1}}\left(1-e^{\frac{T_r}{T_1}}\right)+T_r\right) \\ =\frac{{\sigma }_0}{E_1}\left(1+\frac{1-e^{\frac{T_r}{T_1}}}{T_r/T_1}{e}^{-\frac{t}{T_1}}\right)\approx \frac{{\sigma }_0}{E_1}\left(1-e^{-\frac{t}{T_1}}\right) \end{gathered}$$

where in the last equation we used

$$\underset{\frac{T_r}{T_1}\to 0}{lim}\frac{1-e^{\frac{T_r}{T_1}}}{T_r/T_1}=-1$$

Hence

$$\epsilon (t)\approx \{\begin{array}{ll}\frac{{\sigma }_0}{E_0}\frac{t}{T_r}\hfill & t\le T_r\hfill \\ \frac{{\sigma }_0}{E_0}+\frac{{\sigma }_0}{E_1}\left(1-e^{-\frac{t}{T_1}}\right)\hfill & t>T_r\hfill \end{array}$$

B. Kelvin-Voigt to Maxwell Standard Linear Solid Model Conversion

In order to define material properties in Abaqus, the constituent model needs to be in the form of a Prony series based on Maxwell standard linear solid model (Fig. 9). However, since parameters of a standard linear solid model based on Kelvin-Voigt model are desired a conversion is required. These parameters can be obtained by equating the creep-compliance response of the two models. By doing so, the parameters of a Maxwell equivalent of the model shown in Fig. 1 can be found as

Fig. 9.

Standard linear solid model base on first order Maxwell element.

$$\begin{gathered} E_{2,m}=\frac{1}{(1/E_0+1/E_1)} \\ {E}_{1,m}=E_{2,m}{E}_0/E_1 \\ {\eta }_{1,m}=\frac{\eta }{E}\frac{1}{(1/E_{1,m}+1/E_{2,m})} \end{gathered}$$

In order to define a first order viscoelastic material in Abauqs, a Prony series in the form of

$$G(t)=G_0\left(1-g_1{e}^{-t/T_1^R}\right)$$

is used where G₀ is the instantaneous shear modulus, g₁ is the normalized shear relaxation moduli associated with the relaxation time $T_1^R$. Using this notation for the model shown in Fig. 9, the parameters of a first order Prony series required for Abaqus simulation can be obtained as $T_1^R=\frac{{\eta }_{1,m}}{E_{1,m}}$

$$\begin{gathered} G_0=\frac{E_{1,m}+E_{2,m}}{2(1+\nu )} \\ {g}_1=\frac{E_{1,m}}{E_{1,m}+E_{2,m}} \end{gathered}$$