MR elastography of breast lesions: Understanding the solid/liquid duality can improve the specificity of contrast-enhanced MR mammography

Authors


Abstract

The purpose of this analysis is to explore the potential diagnostic gain provided by the viscoelastic shear properties of breast lesions for the improvement of the specificity of contrast enhanced dynamic MR mammography (MRM). The assessment of viscoelastic properties is done via dynamic MR elastography (MRE) and it is demonstrated that the complex shear modulus of in vivo breast tissue follows within the frequency range of clinical MRE a power law behavior. Taking benefit of this frequency behavior, data are interpreted in the framework of the exact model for wave propagation satisfying the causality principle. This allows to obtain the exponent of the frequency power law from the complex shear modulus at one single frequency which is validated experimentally. Thereby, scan time is drastically reduced. It is observed that malignant tumors obtain larger exponents of the power law than benign tumors indicating a more liquid-like behavior. The combination of the Breast Imaging Reporting and Data System (BIRADS) categorization obtained via MRM with viscoelastic information leads to a substantial rise in specificity. Analysis of 39 malignant and 29 benign lesions shows a significant diagnostic gain with an increase of about 20% in specificity at 100% sensitivity. Magn Reson Med, 2007. © 2007 Wiley-Liss, Inc.

The Breast Imaging Reporting and Data System (BIRADS) classification (1) represents the currently accepted system for diagnosis of MR mammography (MRM) to differentiate the likelihood of malignancy in breast lesions. They are categorized in benign (BIRADS 2), probably benign (BIRADS 3), suspicious (BIRADS 4), and probably malignant lesions (BIRADS 5). Information from different MRI sources are used for diagnosis: structural information about the shape of the lesion from high-resolution contrast-enhanced T1-weighted images and functional information from dynamic image series. Here, one follows the temporal change of the MR signal intensity within the lesion after intravenous application of a contrast medium (CM) bolus (typically gadopentetate dimeglumine). A recently developed score for classification of breast lesions seen on breast MRI can be translated into BIRADS categories and allows objective lesion assessment (2, 3). Though BIRADS categories are significantly correlated with histology, prediction of malignancy remains variable (4).

Dynamic MRM has demonstrated such an enormous sensitivity that a nonenhancing invasive malignant lesion initially merited a case report (5). Unfortunately, the technique lacks specificity (6), which limits its potential application as a modality for breast cancer screening. As such, MRM is an additional examination to X-ray mammography and breast ultrasound and adjunct information from both imaging methods are necessary to prevent an unacceptable number of false-positive cases. Breast cancer often shows a desmoplastic stroma reaction in terms of a reactive proliferation of connective tissue so that a dense layer of fibroblasts accumulates around malignant breast epithelial cells (7). This leads to a hardening of the breast tissue that can be diagnosed by palpation. The diagnostic importance of palpation within the domain of breast cancer diagnosis is undisputed in terms of breast self-examination as well as clinical breast examination (8, 9). However, this method lacks precision and objectivity because it relies on individual perception and skill. Moreover, sensitivity is low, especially in case of small tumors (10).

MR elastography (MRE) represents a novel imaging modality that is capable of overcoming this limitation of manual palpation (11–16). Based on classical MRI techniques, it allows imaging of the propagation of low-frequency acoustic waves within tissue. Since the local properties of the acoustic wave are tightly linked to the underlying viscoelastic properties of the medium, it is therefore possible to reconstruct locally the complex shear modulus G* of tissue (15). Thereby, palpation has turned into the assessment of an objective absolute physical quantity, whose diagnostic value can be quantified.

Analysis of excised liver tissue (17) as well as of individual cells (18) have demonstrated that G* follows, as a function of frequency, a power law in biological tissue. Given this experimental observation and using the causality principle (19) it is possible to establish a link between the real and imaginary part of the complex shear modulus at each single frequency and the exponent of the frequency power law. Herein lies the strength and simplicity of the current analysis, which omits the introduction of any complex rheological model, but rather straightforwardly transforms the measured data into a power law description.

THEORY

Dynamic MRE, which typically operates monochromatically, yields in the linear regime the complex shear modulus G* = Gd + iGl at one single frequency, which links the applied stress to the resulting strain. G* is deduced from the displacement measurement using the following equation

equation image(1)

with equation image the complex-valued curl of the displacement field equation image, ρ the density of the material, and ω the circular frequency. As such, the calculation of G* is not biased by the application of a particular tissue model. G* changes with frequency due to dispersion and it is thus necessary to relate its measured value at one frequency back to the natural physical parameters that cause the observed frequency dependence (20). Classical standard models are the so-called Voigt model, the Maxwell model, or the springpot model. The Voigt model (Fig. 1a) predicts a constant dynamic modulus Gd as a function of frequency and a linearly rising loss modulus Gl, which is in obvious contradiction with a power law behavior. In contrast, the Maxwell model (Fig. 1b) allows dispersion for both moduli but not according to a power law with equal exponent for Gd and Gl. Thus, both classical models fail to describe the observed dispersion typically seen in biological tissue within the accessible bandwidth. A rheological model that is finally capable of describing power law behavior for the complex shear modulus is the so-called springpot model (21, 22). This model has its mechanical interpretation in terms of an hierarchical organization of springs and dashpots (Fig. 1c) and can continuously interpolate between a pure solid material (γ = 0) and a pure liquid (γ = 1) material with respect to the power law exponent γ for G*∼ωγ. However, it has the limitations that first the power exponent γ is restricted to the interval [0,1], and second that the ratio Gl/Gd is linked to the power exponent via equation image.

Figure 1.

Mechanical representations of several classical rheological models in terms of springs and dashpots for the Voigt (a), Maxwell (b), and springpot model (c).

Here, we are proposing to follow a pure physically-motivated approach. Let us assume that the complex shear modulus follows a power law behavior within the considered frequency range. Consequently, the attenuation α and the propagation β also follow a power law given the general expression

equation image(2)

which relates the wave-vector k to the complex shear modulus G*. The causality principle (Kramers-Kronig relations) (19) enforces immediately a specific functional dependency (given by the Hilbert transform, H) of the propagation when assuming a power law for the attenuation (23, 24), i.e.,

equation image(3)

i.e., they form a Hilbert transform pair with y the power law exponent and α00 the scale for the attenuation and propagation, respectively. Here, high/low frequency limits for the propagation/attenuation are neglected since the frequency range accessible by clinical MRE is limited to the transient region. This yields directly the following relations:

equation image(4)
equation image(5)

From Eq. [ 5] it is obvious that the exponent y of the power law for α(ω) and β(ω) (see Eq. [3]) becomes the phase angle of the complex shear modulus in units of π. Note that the only assumption so far has been the power law behavior for G*. Thus, no rheological model based on arrangements of springs and dashpots has been used. The frequency dependence of the dynamic modulus in case of y = 0 resembles the one of a pure classical solid material without loss. The limit y→0.5 represents a material for which Gd → 0 and Gl → ω1; i.e., it behaves like a classical dashpot representing a liquid material.

MATERIALS AND METHODS

Patients

After approval by the local ethics committee, female patients who came into our facility for histopathologic clarification of a suspicious breast lesion seen on mammography or ultrasound were asked to participate in the study. All patients gave their informed consent before undergoing MRE of the breast. Mean age of patients with a malignant lesion was 56 ± 9 years (39 cases) and for patients with a benign lesion 46 ± 15 years (29 cases), which is according to the American Cancer Society the age interval with the highest incidence for in situ and invasive cases. Out of the 39 malignant lesions, 32 were clinically palpable (i.e., 82%), and out of the 29 benign lesions, 20 were clinically palpable (i.e., 69%). The breast lesions of our study population were rather large because of several technical considerations. First, very good lesion detection on contrast-enhanced MR images is required to perform targeted MRE afterward. Thus, inclusion criteria for the patients was a lesion size of at least 1 cm in largest diameter to assure MRI visibility of suspicious palpable, ultrasound-, or mammographically-detected lesions. Second, only those patients were included in whom breast biopsy was scheduled for later correlation of viscoelastic lesion characteristics with histology. Both facts lead to larger lesion sizes than in a typical patient population with unclear MR-detected breast lesions.

Experimental Setup and Hardware

A conventional 1.5-Tesla MR Scanner (Gyroscan Intera, Philips, The Netherlands) was used with extended software and hardware options to facilitate MRE. Both routine contrast-enhanced dynamic MRM and MRE were performed in prone position. The patients' breasts were positioned in a multiarray coil in which mechanical transducers were integrated bilaterally (Fig. 2a and b; manufactured by Philips Research Hamburg, Germany). Both measurements (MRM and MRE) were performed without repositioning of the patient, which enables a precise geometrical comparison.

Figure 2.

a: Schematic view of the patient setup: the patient is in prone position with both breasts gently compressed in the craniocaudal direction via transducer plates (red). The plates vibrate in the feet-head direction (green arrows). Four MR receiver coils (blue) are integrated into the setup to ensure high SNR. b: The actual system. Contrast-enhanced T1-weighted gradient-echo image of the right breast in transverse orientation showing an invasive ductal carcinoma (c, L1, red oval) and corresponding maximum signal enhancement after bolus injection overlaid in color (d). e: The performance of the BIRADS categorization with respect to sensitivity vs. specificity.

MRM

Initially, patients underwent standard transverse imaging to identify the lesion of interest. T2-weighted turbo-inversion-recovery sequences and T1-weighted sequences were utilized. The dynamic scan consisted of T1-weighted gradient-echo images (eight dynamics, TR = 8.5 ms, TE = 4.2 ms, effective slice thickness = 1.5 mm, field of view [FOV] = 380 mm, and flip angle = 20°) obtained before and after intravenous CM injection (3 ml/s flow) of gadopentetate dimeglumine (Magnevist®; Schering AG, Berlin, Germany; 0.16 mmol/kg. body weight), followed by 30 ml NaCl (3 ml/s flow). Each series was acquired in 75 s so that the entire dynamic scan took about 10 min. Subtraction images of each contrast-enhanced dynamic series (N = 7) were made. Average lesion diameter as measured by MRM was 28.5 ± 19.6 mm for malignant lesions and 26.3 ± 21.4 mm for benign lesions.

MRE

MRE was performed as a targeted measurement after MRM with a mechanical excitation frequency of ν = 85 Hz, a field of view = 128 mm, image resolution = 642, slices = 7, and an isotropic voxel size = 2 mm. A standard spin-echo MR sequence (TE/TR = 47 ms/412 ms) was extended by two sinusoidal motion encoding gradients (amplitude = 21 mT/m) and phase-locked to the mechanical excitation (16). Total acquisition time of the MRE-protocol was 10.5 min. The true location of each lesion was found on the images by means of morphological information and analysis of the contrast-enhanced images in the dynamic sequence. Those images were matched to the images of reconstructed viscoelastic properties.

Data Analysis

The local CM enhancement properties of the lesions were analyzed using a commercially available workstation (View Forum; Philips Medical Systems, The Netherlands). We performed dynamic analysis of the second contrast subtraction series with a manually set region of interest (ROI) that included the most intense pixels of each lesion (at least 5 pixels). The initial phase during the first 3 min after CM application and the late phase (postinitial) from the initial signal intensity peak to the last series were evaluated separately. The initial signal increase was categorized to more than 100%, between 50% and 100%, and less than 50% related to the nonenhanced images. Signal increase of more than 10% in the postinitial phase was named “continuous” and a signal decrease of more than 10% from the initial peak to the last series was categorized as “washout.” All other curves were plateau-shaped (see Table 1). The different categories of lesion shape and margins were assessed by two radiologists (K.S., T.X.) by means of consensus reading. The corresponding ROI were matched geometrically onto the reconstructed viscoelastic parameter images. The average values of viscoelastic parameters within this ROI were then used to characterize the mechanical properties of each individual lesion. Statistical analyses were performed using MedCalc for Windows, version 8.2.1.0 (MedCalc Software, Mariakerke, Belgium).

Table 1. Scoring System from Fischer et al. (2), Which Classifies MRI-Detected Breast Lesions*
 Points
  • *

    More suspicious lesions have higher scores. The score can be translated into corresponding BIRADS categories.

Morphology 
 Shape 
  Round/oval/lobulated0
  Irregular (linear, branching, stellate)1
 Margin 
  Well-defined0
  Ill-defined1
 Enhancement pattern 
  Homogeneous0
  Heterogeneous1
  Rim enhancement2
Dynamics 
 Initial enhancement 
  <50%0
 Within 3 min after contrast medium 
  50-100%1
  >100%2
 Signal intensity time course 
  Continous enhancement0
  Plateau1
  Washout2
Total breast MR imaging score0–8

Histopathology

All suspicious lesions were histologically confirmed, either by minimally invasive ultrasound-guided needle biopsy or by excision after ultrasound or mammographically-guided preoperative lesion localization. The group of 39 malignant lesions consisted of the following pathologies: invasive ductal carcinoma (22), ductal carcinoma in situ (3), invasive lobular carcinoma (10), and carcinoma (4). The group of 29 benign lesions consisted of the following pathologies: fibrocystic changes (8), sclerosing adenosis (2), scar (1), ductal hyperplasia (2), phyllodes lesion (1), fibroadenoma (14), and lipoma (1).

RESULTS

Efficacy of Contrast-Enhanced MRM

We investigated the sensitivity and specificity of routine contrast-enhanced MRM. Typically, malignant lesions were moderately visible on contrast-enhanced T1-weighted images (Fig. 2c), but clearly visible when the relative MR signal change after bolus injection of CM was overlaid in form of a colored map of enhancement or analyzed in subtraction images (Fig. 2d). As demonstrated in several studies (25–30), improved diagnostic power can be achieved when analyzing the complete time course of signal change after bolus injection of CM within a ROI over several minutes. Here, malignant lesions tend to have an initial steep signal rise followed by a so-called washout phase. In contrast, benign lesions typically show a low to moderate initial signal increase followed by a plateau or further continuous signal increase. The entire time-course of signal enhancement, together with morphological lesion information about shape and margin appearance, is condensed into a MR score (Table 1), which can be translated into BIRADS categories (2, 3) (Table 2). The receiver operating characteristics (31) (ROC) curve (Fig. 2e) shows the obtained diagnostic accuracy using the 68 patients from this study with an area under the ROC curve (AUC) = 0.88 ± 0.04 for BIRADS (throughout this work, errors on areas under ROC curves indicate the standard error). The ROC curve confirms the well-established fact that MRM provides low specificity (∼40%) at high sensitivity (∼100%).

Table 2. Translation of Breast MR Imaging Score (Table 1) Into Corresponding BIRADS Categories (3)
Breast MR imaging scoreBIRADS category
0–22
33
4–54
6–85

Efficacy of MRE Alone

In a second step, we investigated the sensitivity and specificity of MRE for lesion characterization in which contrast-enhanced MRM was used to determine the location of the suspicious lesion. Unbiased information regarding shear wave properties were obtained by applying the curl-operator to the measured displacement field (15). Omitting the curl-operator leaves one with the complete wave-equation describing both wave components, i.e., the shear and the compressional part. In case of tissue, which is quasi-incompressible, it is not possible to invert such an equation given typical SNR values or finite sampling (16). The curl-operator acts like a projection operator that selects only the shear component of the total wave field. Hereafter, fatty tissue typically showed short wavelength, regions of parenchyma showed increased wavelength, and tumors showed significantly increased wavelength (Fig. 3a–c). Solving the shear wave equation locally provides the complex shear modulus G* = Gd + iGl. The accuracy of the proposed technique for reconstructing G* is demonstrated by means of a calibrated phantom (32) (Fig. 3d–f). The expected values for the dynamic modulus Gd and the loss modulus Gl are all correctly recuperated within errors except for the artificial fibroadenoma (Table 3).

Figure 3.

a: The T2-weighted MR magnitude image of a patient with a ductal invasive carcinoma (red circle). b: The corresponding image of the real part of qx (Eq. [ 1]) gives an impression of the complexity of the shear wave pattern. c: The local wavelength λ = 2π/k (Eq. [2]) exhibits low values in fatty tissue, enhanced values within the parenchyma, and strongly elevated values at the location of the lesion (white arrows). d: “Anatomy” of a tissue mimicking breast phantom as measured by a T2-weighted sequence. C = cancer, F = fibroadenoma, G = glandular tissue, and S = subcutaneous fat. e,f: The resulting maps of Gd and Gl as obtained from an MRE experiment performed at 200 Hz. All structures are well resolved and the actual values agree very well with the expected ones except for the fibroadenoma where Gd is underestimated (Table 3).

Table 3. Comparison Between Values of G* Obtained Via MRE and a Classical Rheometer for the Tissue Mimicking Breast Phantom
RegionMRE (kPa)Rheometer (Ref.32) (kPa)
GdGlGdGl
Subcutaneous fat (S)6.6 ± 0.10.30 ± 0.15.1–6.7 ± 0.20.47–1.9 ± 0.1
Glandular (G)12.0 ± 0.60.73 ± 0.2110.8–14.2 ± 0.70.9–1.5 ± 0.2
Cancer (C)17.2 ± 1.73.8 ± 1.119.7–24.2 ± 2.31.3–2.4 ± 0.4
Fibroadenoma (F)23.0 ± 1.23.6 ± 1.230.6–33.2 ± 0.81.4–3.2 ± 0.2

A multifrequency experiment with a healthy volunteer was conducted to study the dispersion properties of normal breast tissue. Total data acquisition was approximately 60 min, which is clinically prohibitive. A rise of Gd and Gl within the parenchyma according to a power law is observed within the frequency range of 65–100 Hz (GdGl ∼ ωγ, γ = 1.67 ± 0.24; Fig. 4a). The average exponent obtained from the single frequency data (equation image = 1.74 ± 0.07; see Eq. [ 5]) agrees very well with the previous value calculated from a fit to the multifrequency data. Thereby, the calculation of the exponent γ (or equally, y = 1−γ/2) is feasible with single frequency data, which can be carried out within the clinical time constraint (about 10 min). Application of the causality principle introduces altogether two unknown parameters in the frequency range of clinical MRE: the exponent of the power law y and the scale α0 of the attenuation.

Figure 4.

a:Gd (red), Gl (green), |G*| (blue), and the exponent y (black) as a function of frequency for the parenchyma of a healthy volunteer. The data follow a power law :ωy (according lines) and the ratio between real and imaginary part; i.e., equation image remains constant. The reconstruction yields for the previous in vivo case (Fig. 2d) the complex shear modulus G* = Gd + iGl (b and c, correspondingly), which is transformed into the original variables describing a power law behavior for propagation and attenuation; i.e., y and α0 (d and e; see Eq. [ 3] and [4]). The lesion appears more liquid like (since y is elevated) and stiffer (since α0 is decreased).

An example for a ductal invasive carcinoma (Fig. 4b–e) illustrates the presence of strong alterations for Gd and Gl at the correct location of the lesion (same case as Figs. 2c–d and 3a–c). The lesion exhibits enhanced values for y and reduced values for α0 when transformed into the “language” of the power law framework (Fig. 4d and e). Note that the extend and delineation of the lesion in the image of α0 follows its true anatomy very well (compare to Fig. 2d), which is not equally the case for the other variables. The correlation of Gl with Gd using all acquired datasets (Fig. 5a) shows that malignant lesions separate significantly (P < 0.0001; Mann-Whitney test for independent samples) from benign lesions for both moduli. Interestingly, the distinction between benign and malignant lesions is significantly more pronounced (P = 0.005, power = 80%) for Gl than for Gd (AUC(Gl) = 0.91 ± 0.04; AUC(Gd) = 0.79 ± 0.04). Malignant lesions tend to populate the (high y)–(low α0) region, which is opposite to the behavior of benign lesions (Fig. 5b). Two malignant lesions, which demonstrated very low values of y≈0.13 and large values of α0≈200, had the smallest diameters among all lesions (11 mm). While Gl and Gd are rather uncorrelated (r = 0.79), y and α0 are highly linearly correlated when plotted on a log-log scale (r = –0.94). This correlation can further be used to characterize the viscoelastic state by only one variable; i.e., the distance δ on the correlation line relative to an arbitrary point on that line. The discriminating power of this variable (AUC(δ) = 0.91 ± 0.04) is rather similar to the performance of the BIRADS categorization and of Gl (Fig. 5c). The obvious correlation between y and α0 is not present between y and the scale of the propagation β0 (Fig. 5d; Eq. [ 3]).

Figure 5.

The correlation of Gl vs. Gd (a) and y vs. α0 (b) show that both classes (benign and malignant lesions) occupy different regions in the viscoelastic parameter space. The green star in (b) indicates the location for the data from Fig. 4a. c: The diagnostic performance of δ (= distance on correlation line of y vs. α0, red line) in terms of AUC is similar to the one of the BIRADS categorization (black line). d: The strong correlation between y and α0 does not hold for y vs. β0.

Efficacy of Contrast-Enhanced MRM Together With MRE

We tested in a final step the performance of combining the BIRADS categorization with the MRE variable δ. The distribution of δ (Fig. 6a) is rescaled and shifted to the interval [−1,1] and added linearly to the BIRADS categorization. This allows MRE to change the BIRADS categorization by ±1 units at most (Fig. 6b). It is important to mention that the combination of the established BIRADS categorization with MRE represents a new classification since MRE is not capable of modifying BIRADS 5 (most likely malignant) into BIRADS 6 (histopathologically-proven carcinoma). The discriminating power of this variable is significantly higher (P = 0.008) than the performance of the BIRADS categorization alone (Fig. 6c) with AUC(BIRADS + δ) = 0.96 ± 0.02. The specificity rises at 100% sensitivity from about 40% to about 60%. This 20% increase in specificity persists at 95% sensitivity. A specificity of 100% is already obtained at 67% sensitivity. A comparison between the AUC values for BIRADS and BIRADS + δ shows that the required sample size is 29 normal cases and 29 abnormal cases, assuming a Type I error of 0.05 and a Type II error of 0.20. Thus, the statistical power of the analysis is 80%. A similar combination of the BIRADS categorization with Gd (equally shifted and rescaled to the interval [−1,1]) does not provide such an increase in diagnostic gain with only AUC(BIRADS + Gd) = 0.92 ± 0.04 and a significance level of P = 0.34, which is well above the required minimum value of P = 0.05. On the contrary, the combination of BIRADS with the imaginary part Gl yields a significantly enhanced performance with AUC(BIRADS + Gl) = 0.95 ± 0.04. However, the significance level (P = 0.03) does not reach such a low value as the combination with the variable δ (P = 0.008).

Figure 6.

a: The individual distributions of δ for benign (blue) and malignant (red) lesions. The histograms follow within errors normal distributions (green and black dots/lines accordingly). b: The correlation between BIRADS and the combination BIRADS + δ shows that MRE is capable of improving the separation by shifting benign cases to lower categories and malignant cases to higher categories. c: The resulting diagnostic performance of BIRADS + δ is significantly superior to BIRADS, especially for the region of high sensitivity. d: The data of Fig. 4a in terms of velocity (red dots). The black line represents the theoretical prediction obtained from a fit using the Voigt model.

DISCUSSION

Our results show that the viscoelastic properties of the lesions measured by MRE have the potential of improving the specificity of contrast-enhanced MRM for the studied patient population. Given the protocol of this study it is important to realize that MRE is used as an adjunct to MRM, which alone has a sensitivity close to 100%. We ask the question: “Can the viscoelastic information of an enhancing lesion be used to further characterize its nature; i.e., whether it is a benign or a malignant lesion?”

MRE is capable of providing locally the complex shear modulus G* of the material, which relates stress and strain in the linear regime. The interpretation of this complex-valued “number” in terms of elastic and viscous component depends upon the underlying rheological model. Different models (Fig. 1) provide different interpretations. The correct model for a particular material can only be found by inspecting its dispersion properties; i.e., the functional dependency of G* with frequency. It is important to realize that the real and imaginary part of the complex shear modulus are not independent quantities but are linked to each other via the causality principle. Their dependency is established by the Kramers-Kronig relations. The experimental data (Fig. 4a) show that G* is rising within the considered frequency interval according to a power law with equal powers for Gd and Gl. Consequently, the Voigt model does not apply. Equally, the Maxwell model can be ruled out since it does not predict equal power exponents. Although appealing, the springpot model fails to correctly describe the relationship between the power law exponent and the ratio Gl/Gd, as seen in the data. Previous MRE analysis of the dispersion properties of ex vivo porcine specimen (33) only considered the velocity of the shear wave and demonstrated good agreement between data and a theoretical prediction using the Voigt model. Figure 6d shows the data from Fig. 4a transformed into velocity ν via:

equation image(6)
equation image(7)

A fit using the Voigt model, i.e., assuming G* = μ + iωη, does show reasonable well agreement with the data with μ = 4.2 kPa and η = 33Pa · s. However, when considering Gd and Gl individually (Fig. 4a), it is obvious that the fit does not describe the data at all since Gd is nonconstant and Gl does not rise with a power equal to one with frequency! Consequently, when trying to identify the underlying rheological model it is mandatory to consider both the real and imaginary part of the complex shear modulus.

The model proposed in the present analysis is not based on any mechanical interpretation in terms of arrangements of springs and dashpots. The only assumption is the functional dependency of α with frequency (Eq. [ 3]) as motivated by the data and the application of the causality principle. An obvious limitation of the multifrequency study presented in this analysis is the rather limited bandwidth. Studies of G* on individual cells cover over four orders of magnitude in frequency (18), making the assumption for a power law behavior more legitimate. The validity of this assumption is strongly supported in our case by the following reasoning: causality enforces the ratio Gl/Gd at each single frequency to be related to the exponent of the power law if G* follows a power law (see. Eqs. [4] and [5]). The experimental in vivo data demonstrate this behavior exactly. Furthermore, the local Kramers-Kronig relations (34, 35)

equation image(8)

predict that Gl rises with the same power exponent as Gd if Gd rises according to a power law, which is also verified by the data.

The results obtained for G* from the breast tissue mimicking phantom (Fig. 3d–f) validate the accuracy of the current reconstruction technique. The calculated values agree well within errors with the expected ones, which already have a nonnegligible uncertainty due to aging and storage conditions of the phantom. The enhanced values found for Gl at the interface between glandular tissue and subcutaneous fat (Fig. 3f) are still within the uncertainties (32) and might indicate heterogeneities originating from the manufacturing process. The dynamic modulus Gd of the inclusion, which mimics a fibroadenoma (Fig. 3c; F), is underestimated due to the limited resolution of the measurement imposed by an excitation frequency of 200 Hz. It represents a general inherent limitation of the elastography technique and can only be overcome by increasing the SNR or the excitation frequency. The latter is often limited by technical considerations of the mechanical source and enhanced absorption of the shear wave at higher frequencies. A multifrequency study between 150–200 Hz showed that G* is rising within the phantom material very shallowly with frequency (G*∼ω0.1, data not shown). Consequently, the reference values from Ref.32, can be compared to our results that are obtained at higher frequency.

The in vivo results (Fig. 5a) indicate that malignant lesions obtain on average larger values for Gd, which is in agreement with previous analysis (36–39). However, the information provided by Gd alone does not yield a significant diagnostic gain when compared to BIRADS (P = 0.34 only!). The mechanical driving device developed for this application provides strongly enhanced wave illumination of the entire breast due to a bilateral mechanical excitation. Thereby, as demonstrated in the breast phantom, reliable values for Gl can be calculated due to an improved signal to noise ratio. This allows now to explore the diagnostic value of Gl for lesion classification. It is obvious from Fig. 5a that the separation between benign and malignant lesions is significantly better for Gl, which is reflected by the diagnostic gain obtained when combining it with the BIRADS categorization (AUC(BIRADS + Gl) = 0.95 ± 0.04 with a significance level of P = 0.03). The interpretation of G* in terms of y and α0 (Fig. 5b) now provides an understanding of the underlying physics: malignant lesions are characterized by an enhanced liquid-like behavior and a decreased scale of attenuation. This link is most reasonable because the shear attenuating properties of a liquid are typically very small (40). At a reference frequency of ω = 1Hz one finds the shear wave speed ν(ω = 1Hz) = 1/β0 in malignant lesions significantly enhanced when compared to benign lesions (Fig. 5d). This distinction becomes obstructed at an operating frequency of 85 Hz leading to a poor separation between both groups when considering only Gd. The observed tendency of malignant lesions toward a mechanical liquid-like behavior is supported by their enhanced vascularization when compared to benign lesions (6). The origin for the clear correlation between y and α0 (Fig. 5b) is so far not fully understood and is a topic of current research. The combination of y and α0 into one variable δ (Fig. 6a) represents one possible way to connect in a simple manner viscoelastic information and BIRADS categorization. Other combinations are certainly possible and might yield even further diagnostic gain. Here, we observe that δ provides the most significant improvement (P = 0.008) when compared to BIRADS alone (Fig. 6b and c). This shows that it is necessary to use all information provided by G* to maximize the AUC value and not only its real or imaginary part.

Values for y close to 0.5 resemble a loosely connected network (41). Whether this interpretation applies to the actual tissue architecture of each individual lesion requires further investigations. It is however a reasonable interpretation since malignancy is typically correlated with macroscopic alterations, such as for instance stroma reaction induced by the tumor, remodeling of the normal extracellular matrix, and changes in cell density (42). Angiogenesis, tumor invasion, and metastasis all require a degradation of the extracellular matrix (43), which suggests the alteration of associated mechanical properties. Interpreted in terms of network connectivity, the clear difference between y-values obtained on a macroscopic scale for benign and malignant lesions (Fig. 5b) seems to follow such structural changes occurring on the microscopic scale.

In summary, we have established initial evidence that viscoelastic information of breast lesions have the potential to increase the specificity of MRM if interpreted with a relevant physical model. Further studies with larger populations will have to be carried out to evaluate the influence of patient collective and tumor size.

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