Imaging anisotropic and viscous properties of breast tissue by magnetic resonance-elastography



MR-elastography is a new technique for assessing the viscoelastic properties of tissue. One current focus of elastography is the provision of new physical parameters for improving the specificity in breast cancer diagnosis. This analysis describes a technique to extend the reconstruction to anisotropic elastic properties in terms of a so-called transversely isotropic model. Viscosity is treated as being isotropic. The particular model chosen for the anisotropy is appealing because it is capable of describing elastic shear anisotropy of parallel fibers. The dependence of the reconstruction on the particular choice of Poisson's ratio is eliminated by extracting the compressional displacement contribution using the Helmholtz–Hodge decomposition. Results are presented for simulations, a polyvinyl alcohol breast phantom, excised beef muscle, and measurements in two patients with breast lesions (invasive ductal carcinoma and fibroadenoma). The results show enhanced anisotropic and viscous properties inside the lesions and an indication for preferred fiber orientation. Magn Reson Med 53:372–387, 2005. © 2005 Wiley-Liss, Inc.

MR-elastography is a new noninvasive imaging modality that aims to measure the distribution of tissue elasticity (1–4). Pathologic changes are often accompanied by changes in the stiffness of tissue (5). Therefore, palpation represents the first modality commonly used by physicians to screen for suspicious lesions. The bare fact of a palpable finding is an indication for further mandatory diagnostic investigations in the domain of breast diagnostics. In addition, tissue stiffness represents an important parameter for surgeons during interventions. Currently, none of the available noninvasive imaging technologies, such as MR, X-ray, and ultrasound, can reliably distinguish malignant breast cancer from benign breast disease. As a consequence, there are many false-positive diagnoses, which lead to unnecessary biopsies and interventions. There is hope that MR-elastography (MRE) will provide an additional objective physical parameter that might increase the specificity for the determination of malignancy (6). However, clinical experience states that benign lesions sometimes also obtain increased values of elasticity. On the other hand, lobular carcinomas are quite often very soft. Moreover, benign fibrotic breast parenchyma can become quite hard (7). Thereby, it is reasonable to assume that the bare stiffness of a lesion will not provide a clear distinction between malignant lesions and benign diseases (8). It is thus desirable to search for additional physical parameters accessible by MRE, which might help to improve specificity.

Most approaches to elastography assume isotropy for the elasticity and thus treat it as a scalar (2, 9). In general, the viscoelastic parameters of biologic tissue show anisotropic properties (10–12), i.e., the local value of elasticity is different in the different spatial directions. The angiogenesis of malignant tumors leads to a vast growth with an irregular architecture. However, strong heterogeneity can lead to isotropy at a certain spatial scale. The observation of potential anisotropy will therefore depend on the scale at which “stiffness” is probed. In this analysis it is assumed that tissue can be described by a transversely isotropic model for the elasticity, which characterizes the elastic properties of bundles of parallel fibers. Viscosity is considered isotropic because current signal-to-noise values prevent any more sophisticated model. Certainly this represents a simplification of the true architecture and should be regarded as a first step toward a more complete description of tissue's elasticity. Results are presented for simulations, a breast phantom, muscular beef tissue, and two patients (one invasive ductal carcinoma and one fibroadenoma).


A propagating wave in an anisotropic and locally homogeneous viscoelastic medium is described by the following partial differential equation (PDE),

equation image(1)

with ui denoting the ith component of the displacement vector u(x,t), ρ the density of the medium, λiklm the rank four elasticity tensor, and ηiklm the rank four viscosity tensor (Einstein convention is assumed for identical indices). The indices i, k, l, m run over all three spatial dimensions. Spatial derivatives of the elasticity and viscosity tensors vanish due to the assumption of local homogeneity. The elasticity tensor consists in general of 21 independent quantities (the same applies to the viscosity tensor). The assessment of all these unknown parameters is beyond the current number of equations provided by the MRE experiment. It is thus necessary to limit the analysis to a simplified model. Symmetry of the assumed viscoelastic model reduces this vast number of unknowns. For instance, there are only five unknown elasticity coefficients in the case of hexagonal anisotropy (denoted the transversely isotropic model) (13). It is not required to consider attenuation in static elastography because this technique works at the limit of zero frequency (14). In dynamic elastography, however, attenuation must be taken into account and it is included into Eq. [1] using the so-called Voigt model (10). Other research groups use the so-called Maxwell model, which is more applicable to fluids than to solids (9, 15). The measurement of anisotropic viscosity is challenging and requires very high signal-to-noise (SNR) values. Therefore, isotropy is assumed for the viscosity that reduces the number of unknown viscous material parameters to 2. Furthermore, the attenuation of the longitudinal wave at low frequencies is negligible (16). Consequently, the shear viscosity (ζ) remains the sole unknown viscous parameter.

The transversely isotropic model is appealing for biologic applications because it is capable of describing the elastic properties of a bundle of fibers aligned in one direction (Fig. 1). The according five material parameters split into two groups: one group with two shear moduli (μ, μ) describes the shear wave propagation parallel (∥) and perpendicular (⟂) to the direction of the fiber. The other group (λ, λ, λM) describes the propagation of the longitudinal wave in the different directions. The typical order of magnitude of the shear moduli is in soft tissue in the kilopascal range while the λ's are in the gigapascal range. The reason for this enormous difference in magnitude is caused by the almost incompressible nature of tissue, i.e., Poisson's ratio (σ) is close to 0.5. This issue and its consequences will be discussed later in more detail.

Figure 1.

A bundle of fibers aligned such that their symmetry axis is parallel to the z-axis of the coordinate system. A rotation around the z-axis does not modify the elastic properties of this material.

The relation between the strain tensor and stress tensor components for the transversely isotropic model is given in the so-called matrix notation (13) by

equation image(2)

with uik = 1/2(∂math imageui + ∂math imageuk) the strain tensor and σik the stress tensor. Note that the matrix notation is just a convenient way to write the stress–strain relation and indices of the matrix notation refer in a more complex way to the normal coordinate system. The corresponding equation of motion for the displacement vector u (now expressed in the normal coordinate system) is obtained via the relation

equation image(3)

with ζ the shear viscosity. As mentioned previously, the viscosity of the longitudinal wave (ξ) can safely be neglected at low frequencies (Hertz to kilohertz). Moreover, ∇u has small magnitude due to the value of Poisson's ratio. Thus, ξ, ζ, and ∇u are quantities of small magnitude and it is legitimate to neglect the third part on the right-hand side of Eq. [3]. Inserting Eq. [2] into Eq. [3] leads to the following equation of motion,

equation image(4)

with the notation τ = μ − μ (note that one retrieves the isotropic wave equation for τ = 0) and the assumption that all elastic moduli related to the longitudinal wave are equal, i.e., λ = λ = λ = λM. This assumption is motivated by the following fact: the almost incompressible nature of tissue leads to a much higher compressional wave speed compared to the shear wave speed, i.e., cLcT with λ ≫ μ. It is observed in tissue that there are two transversely polarized shear waves and one longitudinally polarized compressional wave (resulting from the eigenvalues and eigenvectors of the Christoffel tensor (13)). Compressional anisotropy would lead to only one shear wave and two quasi-compressional waves, both exhibiting much higher speeds than the shear wave. This is not observed in anisotropic tissue (17), which motivates the assumption of isotropy at low frequencies for the compressional wave and potential anisotropy for the shear wave. The compressional wave probes at low frequencies mainly the fluid nature of tissue, while the shear wave probes the solid nature of tissue. These considerations do not invalidate the observation of local heterogeneity, i.e., the local variation of the speed of sound and thus variations in Poisson's ratio (18).

The arguments which allowed to neglect the compressional term of the viscosity (Eq. [3]) do not apply to the compressional term of the elasticity (second part in Eq. [4] on the right-hand side). Although ∇u is of small magnitude, one must pay attention to λ, which is of large magnitude. Moreover, neglecting this term would lead to an equation that is not capable to account for mode-conversion although it is clearly observed in tissue. The relationship between the largeness of λ and the smallness of the ∇(∇u) term can be understood via the so-called Helmholtz–Hodge decomposition for smooth data. It states that every vector field can be written as the sum of a divergence-free part (uT), a curl-free part (uL), and a so-called harmonic part (uH, which is both divergence-free and curl-free) (19–21). The harmonic part can be neglected in these considerations because any wave field in almost incompressible media can be written as the sum of plane waves, each obeying the previously mentioned property of two transversely polarized shear components (divergence-free) and one longitudinally polarized compressional component (curl-free). It can be shown that this decomposition transforms the basic wave equation, Eq. [4] (in case of isotropy and neglecting viscosity), into two separate Helmholtz equations, where the transverse wave uT and the longitudinal wave uL travel at their respective velocities (21), i.e.,

equation image(5)

with E Young's modulus. This knowledge leads to the following result for the ∇(∇u) term:

equation image(6)

One obtains for Poisson's ratio values of about σ ≈ 0.4999999 assuming a speed of sound in soft tissue of cL = 1540 m/sec and E ≈ �� (1 kPa). Thus, the ∇(∇u) term is of small magnitude. The small magnitude is minutely compensated in Eq. [4] by the second Lamé coefficient,

equation image(7)

Consequently, the compressional term of Eq. [4] exhibits the same weight as the first term, which is related to the shear wave. In conclusion, it is found that the ∇(∇u) term can only be neglected if there is no longitudinal wave component in the data set (i.e., uL = 0). This could be achieved via a careful design of the experiment, for instance, the application of a primarily shear excitation. It may be legitimate to neglect the compressional term under these circumstances ensuring negligible contributions generated via mode conversion (22). However, our analysis uses a longitudinal mode of mechanical excitation for an efficient illumination of the entire object with waves (note that it is the shear wave getting attenuated).

Equation [4] is separately valid for the components uT and uL only in case of isotropy (i.e., τ = 0). In that case all differential operators commute with the divergence operator and the curl operator. The value of τ is of the order of the shear modulus. Thus, it is a good approximation to neglect the spatial derivatives of the third term in Eq. [4], which originate from the longitudinal wave component due to its long wavelength. This results in the following PDE for the transverse component (assuming steady state),

equation image(8)

with ūT the displacement vector shifted by t = π/2ω in time (this selects the imaginary-part of uT). The knowledge of uT is required to locally solve Eq. [8]. With ∇uT = 0 it follows that there exists a vector v potential with uT = ∇xv. Thus, given the measured displacement field u one has to solve the equation

equation image(9)

to obtain the desired transverse component. This can be done using standard multigrid solvers (23). Mind that it is numerically not feasible to solve for the longitudinal component, i.e., ∇u = ∇2ω with uL = ∇ω. The small magnitude of the ∇u term leads to meaningless results for the scalar potential field ω, because finite SNR values on the measurement of u prevent a correct evaluation of the ∇u term.

Equation [8] is only valid once the fiber axis (symmetry axis) of the material coincides with the z-axis of the coordinate system (as drawn in Fig. 1). The effect of tilting the symmetry axis can be interpreted as decreasing the order of the material. The orientation of the fiber can be determined uniquely via two Euler angles, i.e., one first rotation about the x-axis (angle ϕ, see Fig. 1) and a second rotation around the new x-axis (angle θ). A third rotation around the fiber-axis does not influence the structure of the elasticity matrix (Eq. [2]) due to the symmetry of the hexagonal system. Thus, the rotation matrix Λik depends only on ϕ and θ. Rotation of the elasticity matrix can be performed via the so-called Bond transformation operator (24). We propose to mathematically rotate the displacement field at each location instead of rotating the elasticity matrix and hence changing the structure of Eq. [8]. Thus, derivatives that were previously calculated via a local fit of a hyperparaboloid to all 3 × 3 × 3 − 1 neighboring points turn into sums of derivatives weighted by the according entries of the rotation matrices. This leaves the structure of the PDE invariant and omits the appearance of entries within the elasticity matrix, which originate from combinations with the second Lamé coefficient λ. Thus, the generalized PDE forms as follows:

equation image(10)
equation image(11)
equation image(12)

Equation [10] can be evaluated at two different time points during one oscillatory cycle, i.e., t = 0 and t = π/2ω selecting real and imaginary parts of the complex valued displacement vector. This provides altogether six equations that are used to determine μ, τ, ζ, ϕ, and θ.


Hardware and Pulse Sequence

A Philips Gyroscan ACS-NT (Philips Medical System, Best, The Netherlands) is used operating at 1.5 T and equipped with the PowerTrak 6000 gradient system providing 23 mT/m within 0.2 msec. Mechanical oscillations are produced by a homebuilt transducer, which is integrated into a plastic bridge. This enables in vivo measurements with the patient in prone position. The transducer consists of a small coil, which is driven by a programmable pulse generator. The magnetic moment of the small coil couples to the main field of the magnet such that the resulting torque leads to longitudinal oscillations of a piston, which is pressed against one breast from the side (Fig. 2a). Typical excitation frequency for in vivo measurements is 65 Hz. The pulse generator is triggered via the standard trigger output channel of the spectrometer.

Figure 2.

(a) Schematic of the experimental setup. The double arrow shows the direction of the sinusoidal oscillation as generated by the piston of the transducer. (b) MR sequence for the MRE experiment. A common spin-echo sequence is extended by sinusoidal flow-encoding gradients (FEG) located to both sides of the π-pulse. The timing of the sequence is tailored to be phase-locked to the mechanical stimulation (W).

The different components of the mechanical wave are measured using a modified spin-echo pulse sequence (Fig. 2b). A sinusoidal flow-encoding gradient (FEG) is placed prior and after the π-pulse with its shape equal to the pulse shape of the mechanical excitation. The gradient channel to which the FEG is added determines which component of u is measured. Thereby, the phase of the final MR image is directly proportional to the value of the corresponding wave component at a given phase of the oscillatory cycle (2). Possible wraps within the MR phase images obstruct this straightforward relationship. The unwrapping method used in this analysis is based on the mathematical problem of minimum cost flow analysis (25).

The repetition time TR is chosen such that the sequence is phase-locked to the mechanical excitation frequency ν, i.e., TR = N · T, with T = 1000/ν [msec] the basic interval and N an integer number. The same holds for the time separation between the beginning of the first FEG and the end of the last FEG, i.e., TF = M · T/2 with M an integer number. An additional time delay TD allows the phase to be shifted between the onset of the mechanical excitation and the beginning of the imaging sequence. Typically, imaging parameters are TE/TR = 45/500 msec, FOV = 128 mm, resolution 642 pixels, and slice thickness about 2.0 mm. Seven adjacent slices are acquired, which leads to a total acquisition time of about 12.8 min.

Data Acquisition and Reconstruction

Data acquisition is performed in the stationary time regime of wave propagation, i.e., all transient effects have vanished. It remains a pure sinusoidal oscillation of each voxel with

equation image(13)

This time regime is reached about 1–2 sec after the onset of the mechanical vibration. The reconstruction of an elastogram is based on time series of images of the mechanical wave for all three spatial directions. A variation of the trigger delay TD (see Fig. 2b) allows to obtain values of ui at different phases of one oscillatory cycle T. It is possible to observe the amplitude ui(x,t) oscillating over one full period T for each pixel, if one acquires, for instance, eight phase images (26, 27). This is sketched in Fig. 3. The oscillation of each pixel is composed of all frequencies that are present in the mechanical excitation. Thus, a Fourier transformation of the oscillation allows separation of the different frequencies and the values at the mechanical excitation frequency are exactly the desired values ��i and φi. As such, MRE is capable of assessing the solution vector u of Eq. [4]. To reduce the effect of noise, the measured displacement data are Gaussian filtered before the FFT with a stealth of 3 × 3 × 3 pixels around the central point and an isotropic width equal to the size of the pixel.

Figure 3.

(a) Each MRE experiment samples a stack of adjacent slices. The spatial direction i of the motion-encoding gradient determines which spatial component of the displacement vector is measured. Eight MRE experiments with varying trigger delay TD are performed consecutively (see Fig. 2b). (b) This results in eight MR-phase images for each slice, which represent the oscillation of the ith component of the mechanical wave during one oscillatory cycle. (c) The MR-phase Φ is analyzed for each pixel via a Fourier transformation. The corresponding values for phase and amplitude (��, φ) at the mechanical excitation frequency ν represent the desired quantities for the evaluation of the elongation vector ui(x,t) (d). Thus, three experiments are necessary to measure the entire solution vector u of Eq. [1].

Helmholtz–Hodge Decomposition

The reconstruction of the local viscoelastic parameters via Eq. [10] requires knowledge about the divergence-free component uT of the displacement vector u. This is obtained by solving Eq. [9] for the unknown vector potential v. The vector potential v is calculated by minimizing the squared error function,

equation image(14)

using Gauss–Seidel relaxation with stepsize control and red–black ordering, as well as multigrid methods for an efficient relaxation of low-frequency components (23, 28). This is crucial, because errors in the Hodge decomposition lead to wrongly reconstructed viscoelastic parameters. The potential v is set to zero outside the object. The application of the curl operator to Eq. [4] is another technique for the removal of compressional contributions. This leads, however, to third-order spatial derivatives. In addition, the differential operator accounting for anisotropy does not commute with the curl operator.

Retrieval of Viscoelastic Parameters

The minimization of Eq. [10] is not straightforward due to the nonlinearities imposed by the angles of rotation. However, the equation is linear with respect to the viscoelastic parameters μ, τ, and ζ. Therefore, it can be analytically solved for a given set of angles of rotation. Equation [10] must hold also for t = π/2ω, which necessitates evaluating the pseudo-inverse (via multiplication of the transpose system matrix). This is equivalent to minimizing the squared error function:

equation image(15)

Given a certain set of angles of rotation and the three equations

equation image(16)

it is feasible to minimize Σ analytically for μ, τ, and ζ. The global minimum of Σ(ϕ,θ) is found via exhaustive search within the intervals ϕ ∈ [0, 2π) and θ ∈ [0, π/2] with typical step sizes of 0.1 rad for both angles. It is possible to find mathematical solutions of Eq. [15] with μ < 0, ζ < 0, and τ < 0. The first two regions can be excluded as nonphysical (21). The third region (τ < 0) characterizes a transversely isotropic material that is softer in the direction of the fiber than perpendicular to it. Parallel fibered soft tissue consists mainly of collagen and elastin fibers embedded in a polysaccharide matrix. These fibers typically exhibit transversely isotropic properties with μ ≥ μ, i.e., τ ≥ 0 (10). The global minimum of Eq. [15] is found given these additional conditions, i.e., μ > 0, ζ > 0, and τ > 0. The reconstructed distributions of μT, τ, and ζ are Gaussian filtered to reduce the effect of noise (stealth 3 × 3 × 3 pixels and width equal to 1 pixel).

Mode of Excitation

The mechanical mode of excitation in this analysis is longitudinal, which enforces the presence of uL in the measured displacement data. The reason for this choice is based on the requirement to obtain sufficient penetration of the mechanical wave throughout the breast. It is difficult to achieve sufficient amplitude of the mechanical wave deep inside the breast using an excitation in shear mode, because it is the shear wave getting mainly attenuated. The longitudinal mode of excitation has the advantage that the energy is efficiently propagating throughout the breast with almost no attenuation. Mode conversion at interfaces leads to the generation of shear waves. This method enables the presence of shear waves deep inside the breast. Additionally, shear waves will propagate in all directions due to reverberations and randomness of the distribution of sources for mode conversion (29, 30). Thus, each point inside the object will experience waves coming from all directions with different amplitudes and phases. This is an important feature, because otherwise one might face difficulties in assessing the transverse anisotropy. For instance, a plane wave propagating in z-direction (Fig. 1) is not capable of measuring the anisotropy because the velocities of the three waves are determined by (13)

equation image(17)

with Cik the entries of the elasticity tensor in matrix notation (Eq. [2]). Both shear waves travel at the same speed (i.e., they are degenerated) and C66 = μ, which carries the information about the anisotropy, does not contribute to the properties of these particular waves. On the contrary, a plane wave traveling perpendicular to the direction of the fibers allows the exact reconstruction of the anisotropy. Two shear waves with different speeds and one longitudinal wave are observed in that case:

equation image(18)

Thus, only the circumstance that waves are propagating in all directions will allow one to reconstruct the anisotropy in steady state. The local bias on the reconstructed values of the viscoelastic parameters depends mainly on the SNR of those components of the wave that travel perpendicular to the current fiber direction.

Analytic Simulations

Finite element or finite difference techniques for solving the forward problem of Eq. [4] reach their limits in terms of accuracy once Poisson's ratio approaches reasonable values for tissue (σ ≈ 0.4999999). Therefore, many analyses are done with values around σ ≈ 0.48 or even lower (31). This type of regularization leads to simulated displacement fields, which do not reflect the true nature of the longitudinal wave component, because cL/cT is much too low for materials such as soft tissue. Thus, one faces a problem, because it has been demonstrated under Theory that the contribution of the longitudinal wave cannot get neglected due to the compensatory effect between λ (Eq. [7]) and the ∇(∇u) term (Eq. [6]). The Helmholtz–Hodge decomposition allows one to overcome this problem, because it enables reconstruction solely on the transverse displacement field uT. Thereby, this technique omits the presence of λ (Eq. [8]).

The proposed reconstruction technique in this analysis is local. Thus, boundary effects do not influence the feasibility of the method. Therefore, it is sufficient to demonstrate the achievable accuracy of the method within a homogeneous object. Realistic conditions of an area subject to incoming waves from all directions can be simulated by superimposing the plane wave solution of Eq. [4] for different directions n̂. This solution is provided by the eigenvalues (square of the speeds) and eigenvectors (polarization vectors) of the Christoffel tensor (13), which is defined as

equation image(19)

with λiklm the elasticity tensor, which relates to the matrix Cik in Eq. [2]. Realistic values for Poisson's ratio can now be used without the hazard of problems with the convergence usually faced in numerical methods. The generation of simulated data is done as follows (Fig. 4): sources of plane-wave emitters are randomly located at infinity on a sphere around a hypothetical imaging volume. Each emitter sends a plane wave with random amplitude and random direction n̂ (pointing at least toward the inside of the sphere) with velocities and polarizations provided by the eigenvalues and eigenvectors of the Christoffel tensor. All waves are superimposed inside the imaging volume, which yields the desired displacement field. Each plane wave is additionally attenuated in an exponential way. A tilt of the hexagonal anisotropic material is performed by rotating the elasticity tensor, i.e., λ´oprs = ΛoiΛpkΛrlΛsmλiklm. This modifies the Christoffel tensor and changes the polarization vectors as well as the wave speeds. Simulations were performed with the following sets of material parameters.

Figure 4.

Sketch of the geometrical setup for the generation of simulated displacement data. Sources of plane wave emitters are homogeneously placed on the surface of a sphere. Each source emits the plane wave solution of the Christoffel tensor corresponding to the propagation direction n̂. The superposition of all individual waves provides the solution within the volume. The initial amplitudes of the sources as well as the directions n̂ are randomly chosen to avoid spherical symmetry of the superimposed wave solution. The directions n̂ are always pointing toward the inside of the sphere. (The grayscale image underneath the graphics is for purposes of illustration.)

The excitation frequency is identical to the one used for in vivo experiments, i.e., ν = 65 Hz. A voxel size of 1 mm3 is chosen with a field of view of 64 mm. Seven adjacent imaging planes are simulated and 10% random noise on the displacement is added to the amplitudes of the displacements to be comparable with in vivo measurements.

In Vivo Measurements

Formal consent of the patients has been obtained before the MRE measurement. The patients underwent surgery afterward. The pathology of the lesions was determined later by histology.


Hodge Decomposition and Error Function Minimization

Figure 5a) depicts a grayscale image of the total displacement field ux(t = t0) inside the breast of a patient. The mechanical transducer is attached from the left side and pushes in the left–right direction, i.e., in x-direction. Shear waves with wavelength of the order of 1 cm are well visible. The amplitude strongly increases toward the left side of the breast. This becomes more apparent in Fig. 5c, where the values of ux are plotted versus a horizontal line vertically located in the middle of Fig. 5a. The modulation, which originates from the propagation of shear waves, is additionally modulated by the presence of the longitudinal wave. This causes the strong rise of ux toward the source of the mechanical excitation. Figure 5b shows the result of the Hodge decomposition, i.e., the extracted transverse displacement component umath image (t = t0). Now, the shallowly varying spatial DC component of the longitudinal wave has vanished and only shear waves are apparent. This is well visible in Fig. 5c, where the displacements belonging to the transverse component of the motion field oscillate around zero.

Figure 5.

(a) Measured displacement field ux (x corresponds here to the left–right direction) within a breast. The transducer is located on the left side of the FOV. It is pushing in x-direction, which causes the strong rise of the amplitude toward the location of the transducer. Shear waves are visible underneath the coarse modulation originating from the longitudinal wave. This becomes better visible in (c) where the value of ux is plotted versus a horizontal line vertically placed in the middle of the FOV. The rise of ux cannot be explained by utilization of a Helmholtz equation. (b) Transverse displacement field as obtained after the Hodge decomposition. The DC component has vanished and pure shear waves are visible. The values of umath image oscillate around zero (see c). The bright and dark spots on the border of the object (see b) are artifacts from the masking process to exclude regions that do not belong to the object.

This example shows that a large fraction of the displacement ux can be associated with the propagation of the longitudinal wave under certain circumstances. The corresponding images of the other two spatial components (not shown) reveal that the presence of the longitudinal component is reduced in those components compared to ux (but still present). This is an expected result caused by the direction of the mechanical excitation.

Reconstruction is done locally by minimizing Eq. [15] as a function of all five unknown parameters. The curvatures ∂2umath image/∂xkxl are obtained from local fits of a hyperparaboloid to the transverse displacement field uT. Figure 6 shows the value of Σmin at one spatial location within the ϕ–θ plane in case of an anisotropic simulation. Two minima are visible: one approximately located at ϕ = 2, θ = 0.5, and another approximately located at ϕ = 4.5, θ = 1.0. The second minimum is associated with a negative value for the anisotropy τ and can therefore be excluded. The minimum search is repeated within the ϕ–θ plane for each point of the imaging volume. Divisions of 40 for ϕ ∈ [0, 2π) and 20 for θ ∈ [0, π/2] provide a reasonable resolution for the angles of rotation at total reconstruction times of about 20 min for an entire data set.

Figure 6.

Minimal value of the error function Σ (Eq. [15]) versus the angles of rotation ϕ and θ for one spatial point in case of an anisotropic simulation. The true values for the angles are ϕtrue = 2.1 rad and θtrue = 0.5 rad. A minimum is easily visible at the correct location. However, there is another minimum at ϕ = 4.5 rad and θ = 1 rad. This minimum corresponds to negative values for the anisotropy and can be excluded.


Figure 7 shows the reconstructed distributions of the viscoelastic parameters for both scenarios, i.e., isotropic and anisotropic with 10% random noise added to the displacements. In case of isotropy one observes in Fig. 7b the presence of a false anisotropy with τ = 0.1 ± 0.04 kPa, which is caused by the introduction of the noise. This effect leads to a bias in the estimation of the shear modulus with μ = 0.96 ± 0.03 kPa. The reconstruction of the viscosity is not affected by this interplay between shear modulus and anisotropy (ζ = 1.0 ± 0.04 Pa · sec). An isotropic material should lead to distributions for the angles of rotation ϕ and θ, which characterize a random distribution for the direction of the (here nonexisting) fiber. Accordingly, the distribution of ϕ is flat (Fig. 7d) and the distribution for θ exhibits a shape that describes a homogeneous distribution of points on the surface of a sphere (Fig. 7e). Reweighting by 1/sin(θ) leads to a rather flat distribution, as presented in Fig. 7f.

Figure 7.

Results of the proposed reconstruction algorithm as obtained for an isotropic and an anisotropic simulation. The distributions of the reconstructed viscoelastic parameters are shown in case of 10% noise added to the simulated displacement components. (a) shear modulus μ, (b) anisotropy τ, (c) viscosity ζ, (d) first Euler angle ϕ, and (e) second Euler angle θ. (f) Distribution of the reconstructed angle θ for the isotropic case, but reweighted by 1/sin(θ). The distribution becomes rather flat, which is an indication for a random distribution of directions. The y-axis represents for all histograms the number of entries.

The case of anisotropy shows a distribution for τ with τ = 0.37 ± 0.16 kPa, which differs significantly from that obtained for the isotropic simulation. The underestimation (τtrue = 0.5 kPa) leads to an overestimation for the shear modulus with μ = 1.1 ± 0.1 kPa. The reconstruction of the viscosity is again not biased due this under/overestimation with ζ = 1.0 ± 0.1 Pa · sec. The distributions of the angles of rotation exhibit peaks at the right values, i.e., ϕ = 2.1 ± 1.1 rad and θ = 0.5 ± 0.3 rad.

The dependency of the reconstructed viscoelastic parameters on the true second Euler angle θtrue is presented in Fig. 8 (all other parameters as well the distribution of sources of plane waves are identical to the previous anisotropic case). Open circles show the results as obtained for infinite SNR. The true viscoelastic values are correctly reconstructed, which demonstrates the validity of the proposed technique. Maximum biases of about 10% are observed in case of finite SNR (open triangles), which corresponds to the level of noise on the displacement. It becomes obvious that an overestimation of the shear modulus μ leads to an underestimation of the anisotropy τ. The viscosity is independent from the elasticity. Therefore, no dependency on θ is observed for ζ. The reconstruction of ϕ becomes ill-posed, if the fiber-axis (i.e., the symmetry axis) points toward the z-direction (i.e., θ = 0). Thus, the largest bias on the estimation of ϕ is recorded in the region of small values for θ. The reconstructed values for θ follow very well the true values. Deviations are observed at the end points of the reconstruction interval, i.e., at θ = 0 and θ = π/2. This is caused by the projection of the continuous distribution onto the interval θ ∈ [0, π/2]. Potential anisotropy of the viscosity has also been considered (solid circle in Fig. 8 at θtrue = 0.55). The value of the shear viscosity has been increased by 10% along the fiber direction (ζ = 1.1 Pa · sec, ζ = 1.0 Pa · sec). As expected, the elastic parameters remain unaffected and a 10% bias is observed for the reconstructed value of ζ.

Figure 8.

Mean and variance of reconstructed viscoelastic parameters for an anisotropic simulation without noise (open circles) and with 10% noise (open triangles) as a function of true angle θ. The true values of the individual parameters are indicated as solid lines. Reconstruction provides the correct values in case of infinite SNR. Very small errors are recorded due to the finite sampling of the wave, which leads to minor imperfections (indicated by horizontal lines at the end of the error bars). Missing lines at the end of the error bars indicate that the error is smaller than the size of the marker. Typical errors of 10% are observed in case of finite SNR. The interplay between shear modulus and anisotropy is anti-correlated, i.e., an overestimation of μ leads to an underestimation of τ. The deviation of θrecon from the 45° line at the end-points of the interval is caused by the projection onto the interval [0 − π/2]. The solid points at θtrue = 0.55 represent the results as obtained for a simulation adding 10% anisotropy to the viscosity.

Polyvinyl Alcohol Breast Phantom

A breast phantom was designed containing two small square inclusions of about 6 mm edge length (Fig. 9a). The phantom as well as the inclusions consists of polyvinyl alcohol (PVA) cryogel, which is formed by freezing and thawing an aqueous PVA solution (32). The inclusions are harder as the background material because they underwent two cycles of freezing and thawing instead of only one. A mechanical excitation frequency of ν = 406 Hz is used to provide good spatial resolution and the MR voxel size is chosen to be isotropic (1.6 mm). The measurement of the wavelength from the phase image (not shown) yields about λ ≈ 1.6 cm, which provides a coarse estimate for the shear modulus via the well-known relation μ = ρ(νλ)2 ≈ 42 kPa. The image of the reconstructed shear modulus is presented in Fig. 9b. The inclusions are quite visible and their positions correlate very well with the MR-magnitude image. The average shear modulus of the background material is μ = 42 ± 5 kPa. The shear moduli of the inclusions are μupper = 60 ± 3 kPa and μlower = 57 ± 3 kPa, which agrees, within error, with the value from literature (μ ≈ 63 ± 10 kPa (32)) for a 10% PVA solution after two cycles of freezing and thawing.

Figure 9.

Results obtained for a breast phantom made from polyvinyl alcohol. (a) Two inclusions (which are harder than the background material) are visible in the MR magnitude image as rectangular bright areas (red squares). (b) The reconstructed shear modulus μ shows a rather homogeneously reconstructed background with the two inclusions well visible as regions of increased stiffness. (c) Low values for the anisotropy τ are reconstructed. (d) Image of the reconstructed viscosity ζ. On average, very low values are observed. However, regions of increased viscosity are visible with ζ ≈ 2 Pa · sec. (e,f) Image and distribution of the rotation angle θ. The distribution is broad and does not indicate a preferred direction.

The image of the anisotropy (Fig. 9c) shows very low values with τ = 7 ± 3 kPa. This results in an overall ratio of μ = (μ + τ)/μ = 1.18 ± 0.1, i.e., very close to unity. This is an expected result because PVA should exhibit isotropic properties. The amplitude of the mechanical wave exhibits very little attenuation within the phantom (not shown). Accordingly, low values for the viscosity are recorded (Fig. 9d) with = 0.9 ± 0.7 Pa · sec. For example, Djabourov et al. found 0.21 Pa · sec in gelatin at 1.5 Hz (33). There are, however, areas of enhanced viscosity visible that have no correlation to any structure within the phantom. This demonstrates the limits on the resolution that can be achieved for the viscosity currently. The distribution of the second Euler angle θ is broad and does not show a preferred direction (Fig. 9e and f). Its shape is similar to that obtained for the isotropic simulation, hence pronouncing the isotropic properties of PVA. The distribution of the first Euler angle ϕ is flat (not shown).

Ex Vivo Muscular Beef Tissue

Figure 10 shows the results obtained from ex vivo beef muscle. The direction of mechanical excitation is from right to left at a frequency of 120 Hz. The T2-weighted MR-magnitude image (Fig. 10a) reveals highly structured tissue interrupted by cords of fatty tissue. Fiber orientation is mainly from left to right (i.e., ϕ = 0) and the tilt angle of the fibers relative to the slice orientation is about 45°. The z-component of the transverse displacement vector is presented in Fig. 10b. The pattern of varying wavelengths shows close correlation to the anatomic structures/compartments visible in the T2-weighted image. The image of the shear modulus reveals a rather homogeneous distribution with μ = 7.5 ± 2.7 kPa (Fig. 10c). Areas of reduced stiffness correlate mainly with the cords of fatty tissue in Fig. 10a. Elevated values for the anisotropy are recorded with τ = 3.4 ± 1.8 kPa, which is expected for a structured material such as muscle. This leads to an overall ratio of μ = 1.5 ± 0.2 (Fig. 10d and e). This is significantly larger than the value obtained for the isotropic PVA material. Regions of strongly enhanced anisotropy correlate roughly with stiffer regions. Areas with reduced values of the ratio μ are mainly visible along the cords of fatty tissue and at the edges of the specimen. The images of the angles of rotation demonstrate correlation to the tissue compartments. This is especially visible for the first Euler angle ϕ in the upper part of the specimen, where the left–right direction of the fibers is properly reconstructed with values for ϕ close to zero. The second Euler angle resembles, on average, the 45° tilt of the fibers with θ = 0.8 ± 0.4.. However, locale deviations are visible and occur mainly along the cords of fatty tissue. Noise in the distributions of the angles is most prominent in regions of low values for the anisotropy. This is an expected effect, because the angles of rotation can obtain any arbitrary value at zero anisotropy. Strongly absorbing properties are observed when inspecting the total amplitude of the displacement vector (not shown). This leads to increased values for the viscosity (Fig. 10h) with ζ = 3.2 ± 3 Pa · sec (note values of ζ = 0.9 ± 0.7 Pa · sec are obtained for the breast phantom).

Figure 10.

Ex vivo results from muscular tissue from the beef. (a) The T2-weighted MR magnitude image of the sample shows fiber directions mainly from left to right. (b) Extracted transverse displacement component in through-slice direction. Images of the reconstructed shear modulus μ (c), anisotropy τ (d), ratio μ (e), angles of rotation ϕ (f) and θ (g), and viscosity ζ (h).

Invasive Ductal Carcinoma

Results for the case of an invasive carcinoma are presented in Fig. 11. The tumor is located close to the mammilla and its location is indicated by the red rectangle in Fig. 11a. Figure 11b shows the reconstructed image of the shear modulus. The tumor is easily visible as a region of markedly enhanced elasticity with μtumor = 5.8 ± 1.2 kPa. Both its anisotropy and its viscosity are increased with τtumor = 6.8 ± 1.3 kPa and ζtumor = 3.0 ± 0.8 Pa · sec. This results in an average ratio of μ = 2.2 ± 0.26 (Fig. 11f). The image of the rotation angle θ (Fig. 11e) reveals a preferred direction within the area of the tumor close to θ ≈ π/2. The first Euler angle exhibits two peaks, one at ϕ ≈ 0 and the other at ϕ ≈π (not shown).

Figure 11.

In vivo resultsY from a patient with breast cancer. (a) MR-magnitude image of the breast: the tumor is located inside the red rectangle. Corresponding images of the reconstructed shear modulus (b), anisotropy (c), viscosity (d), angle of rotation θ (e), and ratio μ (f). The tumor is easily visible as a hard, anisotropic, and viscous region. The distribution of the angle of rotation exhibits a preferred value inside the tumor. The reconstructed pixels at the top edge of the FOV originate from an imperfect masking of the noise.

The background material consists of fatty tissue interrupted by remaining parenchyma, Cooper's ligaments, and scar tissue (the patient underwent surgery already). The average shear modulus is μbrd = 1.1 ± 0.5 kPa and the average anisotropy is τbrd = 0.9 ± 0.5 kPa. Certain regions exhibit very low values of anisotropy. This is easily visible in (Fig. 11f) where the ratio μ approaches unity. Other regions show fractions as high as in the tumor. However, the tumor exhibits the largest anisotropy in absolute magnitude. The viscosity of the background material is lower when compared to the results obtain in the beef muscle ((ζbrd = 1.2 ± 0.8 Pa · sec). The distribution of the rotation angle θ is similar to that obtained in case of an isotropic simulation. Reweighting by 1/sin(θ) leads to a flat distribution (not shown).


Figure 12 shows the results obtained from a patient with a fibroadenoma. The location of the lesion is indicated by the red rectangle in Fig. 12a. The lesion is very easily visible in the image of the reconstructed shear modulus (Fig. 12b) with μlesion = 2.6 ± 0.7 kPa, which is markedly lower than in the previous case of the carcinoma. The round shape of the lesion is well reproduced. The fibroadenoma exhibits also enhanced anisotropic properties as shown in Fig. 12c with τlesion = 2.4 ± 0.7 kPa, which yields a ratio μ = 2.0 ± 0.5. Enhanced viscous properties are recorded in the center of the lesion with ζlesion = 2.4 ± 1.9 Pa · sec. The anisotropic properties of the fibroadenoma are supported by the fact that the angle θ obtains a preferred direction within the lesion with θlesion = 0.9 ± 0.2 rad (note the small variance!). The first Euler angle ϕ exhibits within the lesion two prominent peaks (not shown): one at ϕ ≈ τ and the other at ϕ ≈ 5.

Figure 12.

In vivo results from a patient with a fibroadenoma. (a) The lesion is easily visible in the MR magnitude image (red rectangle) and the corresponding image of the shear modulus (b) shows the tumor well delimited from the surrounding fatty tissue. The MR magnitude image indicates that the lesion is traversed by septae, which probably lead to the enhance values of the anisotropy (c). The core of the lesion exhibits increased viscous properties (d) and the angle of rotation θ indicates a preferred direction of the fibers inside the lesion (e). (f) The ratio μ shows that parts of the lesion exhibit strong anisotropic properties.

The background material consists again of fatty tissue, parenchyma, and Cooper's ligaments. The average shear modulus is lower than in the previous case with μbrd = 0.36 ± 0.15 kPa. Probably this is caused by the circumstance that the breast of this patient has been less compressed than in the previous case (compare Figs. 11a and 12a). The strong compression in the left–right direction from both sides is easily visible in Fig. 11a. Nonlinear properties of tissue lead to an increase of the shear modulus under precompression (10). Very low values for the anisotropy are recorded in certain regions of the background tissue as visible in Fig. 12f where the ratio μ approaches unity. There are, however, also regions of enhanced anisotropy with an overall average τbrd = 0.26 ± 0.2 kPa. A band of enhanced anisotropy is visible toward the chest wall (red polygon). It agrees roughly with an anatomic structure (Fig. 12a). Low values are measured for the viscosity with ζbrd = 0.7 ± 0.55 Pa · sec. The distribution of θ is again similar to that obtained for isotropic material (not shown).


Efficient wave penetration is crucial in dynamic steady-state MRE. Unfortunately, it is the shear wave that gets strongly attenuated and all image contrast relies on the shear wave. The technique to excite with longitudinal waves is a way to overcome this problem, because attenuation of compressional waves at low frequencies is negligible. Mode conversion at interfaces leads to shear wave generation deep inside the object. Unfortunately, this advantage leads to a pronounced presence of the longitudinal component within the entire displacement vector. The almost incompressible nature of tissue prevents a correct handling of the compressional component in the partial differential equation given the finite spatial sampling of the MRE experiment and the finite SNR on the measurement (as shown under Theory). One way to overcome this obstacle is to perform the so-called Helmholtz–Hodge decomposition to extract the transverse wave component. This removes the presence of the second Lamé coefficient λ and allows one to invert the equation under the presence of noise. Plain shear excitation reduces the relative contribution of the compressional wave and reconstruction without the decomposition using the Helmholtz equation might be feasible. However, the demonstration of negligible contributions from compressional components in real data is difficult. Thus, in general the compressional component must be accounted for.

The results as obtained from simulations demonstrate the validity of the proposed technique for reconstructing anisotropic elastic parameters (Fig. 8). No biases are observed in case of infinite SNR. Small residual errors originate from the finite spatial sampling of the wave and, for instance, the impossibility of reconstructing the first Euler angle ϕ in cases where the symmetry axis coincides with the z-axis. The reconstruction of the viscoelastic parameters is feasible even with 10% noise on the displacement. Biases up to 10% are recorded for the shear modulus and the anisotropy. The results also demonstrate that reconstruction is feasible although the longitudinal component is entirely neglected. Thus, the approximation leading to Eq. [8] is legitimate. Moreover, it has been shown under Theory that the reconstruction of the anisotropy becomes impossible for certain directions of wave propagation. The steady state, however, excites all modes and thus reconstruction becomes feasible (Fig. 8). Further analyses are required to explore the limits of the ability for reconstructing anisotropic properties. For instance, a local spatial Fourier transformation of the displacement field allows performance of virtual plane-wave illumination of the region of interest. This can be used to estimate the required fraction of waves propagating perpendicular to the fiber direction in order to obtain valid results for the anisotropic parameters.

The images of the breast phantom demonstrate the feasibility of the algorithm to reconstruct small stiff objects (Fig. 9). Both hard inclusions are easily visible and the values of the shear moduli agree, within errors, with the expected ones. PVA material should exhibit isotropic properties on the millimeter scale as confirmed by very low values for the anisotropy τ. The finite SNR as well as imperfections due to the MRE data acquisition leads to locally enhanced values for the anisotropy. This is an indication for the limitations of the current technique and the achievable resolution for the anisotropy. The reconstructed values of the viscosity ζ confirm the expectation of low attenuating properties. It is obvious from Fig. 9d that the reconstruction of the viscosity is very sensitive to minute imperfections of the data acquisition because the results from simulations suggest the ability to reconstruct ζ with very good precision. For instance, imperfections can occur due to motion artifacts arising from the mechanical vibration during the read-out phase of the MR signal.

The analysis of strongly anisotropic beef muscle shows that the method is capable of distinguishing between isotropic and anisotropic material. The ratio μ is significantly enhanced when compared to the PVA phantom and the different compartments of the specimen get resembled in the images of the angles of rotation. As expected, muscle exhibits anisotropic properties with τ > 0 (34) and enhanced attenuating properties when compared to PVA. The reconstructed values for the viscosity ζ are on average about three times higher!

The in vivo examples of a carcinoma (Fig. 11) and a fibroadenoma (Fig. 12) show the lesions as stiff, anisotropic, and viscous objects that separate very well from the background tissue. The lesions exhibit the largest values of shear modulus and anisotropy within the individual breast and the rotation angle θ demonstrates that there is a preferred direction of the fibers inside the tumor. However, there are also other regions within the breast that exhibit increased anisotropic and viscous properties. This is an expected result caused by the heterogeneous and irregular architecture of the breast. The image of the ratio μ shows regions within the fatty breast tissue with values close to 1 (i.e., isotropy). Other regions exhibit fractions as large as inside the tumor. As such, the tumor is not differentiable within these images. The shear moduli of the background tissue as well as the viscosity values agree, within errors, with previous measurements (35). In that analysis, a speed cT = 0.68 ± 0.16 m/sec and an attenuation of 1.6 ± 0.2 Np/λ was measured at 50 Hz in breast tissue. This translates for the 1D case into the central values μ = 0.65 kPa and ζ = 0.38 Pa · sec.

The ability to reconstruct anisotropic properties depends on the SNR of the displacement measurement. The error on the total amplitude can be determined via a fit of a sinus function with known frequency to the temporal variation of the individual spatial components. Using the breast phantom as a reference, the regions of lowest SNR = AtotAtot are measured with values around 7–8. Similar values are obtained in the carcinoma and the fibroadenoma.

This analysis demonstrates the feasibility of imaging anisotropic elastic and isotropic viscous properties of tissue using steady-state MRE. Further analyses are required to investigate the resolution on the reconstructed parameters. Improvements can be achieved by enhancing the SNR on the measured displacement data. For example, echo-planar read-out techniques together with increased number of signal averages could provide better results at a fixed data acquisition time. Variations in the distributions of the viscosity ζ, which lack anatomic correlation, are most likely caused by imperfections in the data acquisition. Sources of errors are, for instance, motion artifacts during the MR read-out and upper harmonics generated inside the tissue due to nonlinear properties. The finite sampling duration of the motion sensitized gradients acts as a bandpass with limitations and thus allows other frequencies to enter the measured displacement distribution. A potential solution is optimized shapes for the motion encoding gradients.

Currently, clinical validation is ongoing to evaluate the diagnostic potential of the viscoelastic parameters for improving the specificity in breast cancer detection.

Table  . 
Typeμ (kPa)τ (kPa)ζ (Pa ∗ sec)ϕ (rad)θ (rad)σ