Wideband MR elastography for viscoelasticity model identification

Abstract

The growing clinical use of MR Elastography (MRE) requires the development of new quantitative standards for measuring tissue stiffness. Here, we examine a soft tissue mimicking phantom material (Ecoflex) over a wide frequency range (200 Hz to 7.75 kHz). The recorded data are fit to a cohort of viscoelastic models of varying complexity (integer and fractional order). This was accomplished using multiple sample sizes by employing geometric focusing of the shear wave front to compensate for the changes in wavelength and attenuation over this broad range of frequencies. The simple axisymmetric geometry and shear wave front of this experiment allows us to calculate the frequency-dependent complex-valued shear modulus of the material. The data were fit to several common models of linear viscoelasticity, including those with fractional derivative operators, and we identified the best possible matches over both a limited frequency band (often used in clinical studies) and over the entire frequency span considered. In addition to demonstrating the superior capability of the fractional order viscoelastic models, this study highlights the advantages of measuring the complex-valued shear modulus over as wide a range of frequencies as possible.

Introduction

Magnetic Resonance Elastography (MRE) is widely studied for detection, classification and monitoring of disease and injury of numerous anatomical regions, including the breast, liver and skeletal muscle (1) (2) (3). MRE has also been investigated as a means of noninvasively tracking the development of engineered tissues (4) (5). Phase contrast images are obtained using motion encoding gradients (MEG) synchronized with externally-driven harmonic shear wave motion in the tissue, organ or material specimen. Changes in tissue structure and composition can alter elastic and viscous properties that in turn affect the induced mechanical shear wave motion measured by MRE. Changes in these properties are not evident in conventional T1, T2 or diffusion-weighted MR imaging. MRE can be used to develop a “viscoelastic map” of the tissue based on the measured shear wave pattern (6), (7), (8), (9).

When combining measurements taken at multiple frequencies using MRE to obtain more detailed information about the viscoelastic structure of the material, the assumed viscoelastic model type, such as Voigt or Maxwell, will affect tissue property estimates based on the imaging data. Currently, there is no agreed upon strategy or protocol to ascertain what is the most appropriate viscoelastic model type for a given specimen. Model types with the least number of parameters needing identification that still accurately capture measurable behavior are preferred. Such models can be useful from a clinical perspective as they may enable one to differentiate with greater sensitivity and specificity viscoelastic changes associated with the onset, progression and resolution of disease in a tissue or organ (10) (11).

The measurement of viscoelastic properties of a material via MRE is inherently limited in spatial resolution and spectral range given limitations of the shear wave source and measurement system. Due to increasing mechanical wave attenuation in soft tissues with increasing frequency, the mechanical actuation frequency always has an upper bound. But, as the actuation frequency must be decreased, the observed wavelength increases, which in turn diminishes spatial resolution. Some of the highest frequency MRE experiments that the authors are aware of were conducted on tissue phantoms and cartilage samples in a clinical MRI scanner using a custom built gradient coil (12) (13). Although the system had the capability of scanning from 1 kHz to 10 kHz, the bandwidth of useful frequencies was bounded by the sample dimensions to the range of 5 kHz to 9 kHz in (12) and 3 kHz to 7 kHz in (13). Besides these high frequency MRE experiments, there is another MRE study (14) which visualized compression waves, rather than shear waves, on a phantom driven by an ultrasound probe at a frequency of 515 kHz. The ultrasound frequency range is beyond the range of any commercial gradient coils by at least two orders of magnitude. Therefore, a custom-built, oil cooled gradient coil was used. Another wide band MRE study was conducted on the murine brain (15) over a 1.2 kHz bandwidth from 600 to 1,800 Hz. In this study the viscoelastic behavior of the brain was characterized by a frequency-dependent power-law relation. In a further wideband study of tissue mechanics on several animal tissues and human tissue, a common fractional model of viscoelasticity, the simple spring-pot, was investigated over a 700 Hz bandwidth from 100 Hz to 800 Hz (16). Fractional order viscoelastic models also have been used to describe human brain and liver (17) and breast tumors (18). In these MRE studies it was shown that fractional order models can make more accurate estimations than integer order models. It was also observed in (17) that the model parameters better differentiated benign and malignant tumor tissues as compared to integer order models.

In the present article, we present a study on the viscoelastic properties of a silicone-based soft tissue mimicking material known as Ecoflex (ECOFLEX-0010, Smooth-On, USA). MRE measurements were conducted over the wide range of frequencies from 200 Hz to 7.75 kHz, in a combination of 9.4 and 11.7 Tesla MRI systems. This large acoustic frequency range spans both current human and small animal MRE studies, as well as where they may go in the future to improve elastographic resolution. Common linear viscoelastic models – Maxwell, Voigt, fractional springpot, standard linear solid (SLS), generalized Maxwell and fractional Voigt – were compared in terms of their predictive ability to match experimental measurements of viscoelasticity over the 7.55 kHz frequency span.

Theory

Viscoelastic Continuum: Governing Equations

For an isotropic, homogenous, viscoelastic compressible medium one can use the following formulation of the equation of motion for small perturbations about an operating point (19)

$$(\lambda +\mu )\nabla \nabla ·\mathbf{u}+\mu {\nabla }^2\mathbf{u}=\rho \frac{{\partial }^2\mathbf{u}}{\partial t^2}$$ [1]

Here, u is the displacement vector, ρ is the density of the medium, ∂/∂t denotes a derivative with respect to time, ∇² is the spatial Laplacian operator dependent upon the chosen coordinate system, and λ and μ are the Lame constants of the medium, denoting volume viscoelasticity and shear viscoelasticity, respectively. Neglecting viscous losses, λ and μ are constants. If linear viscosity is considered, λ and μ may have rate-dependence with the nature of the dependency prescribed by the assumed viscoelastic model. For soft biological tissues or tissue-mimicking phantoms, λ is as many as 6 orders of magnitude larger than μ, which results in compression wave behavior being primarily affected by λ; in any case shear wave behavior is primarily affected by μ, not λ.

Shear Wave Propagation in a Cylindrical Cavity

Referring to Fig. 1a, considering axisymmetric vertical motion of the test tube undergoing steady state harmonic motion u_z (r = a, t) = u_zaexp (jωt), and assuming a welded contact of the test tube wall with the media inside it, for an isotropic viscoelastic medium far enough away from the free surface at the top of the medium and from the bottom of the test tube, we have (20):

$$u_z(r,t,k_{\beta })=u_{za}\frac{J_{\text{0}}(k_{\beta }r)}{J_{\text{0}}(k_{\beta }a)}{e}^{j\omega t},k_{\beta }=\omega \sqrt{\frac{\rho }{{\mu }_R+j{\mu }_I}},$$ [2a-b]

where $j=\sqrt{-1},J_0(z)$ denotes a zeroth order Bessel function of the first kind, ρ is the medium density, and μ_R and μ₁ are the real and imaginary parts of the shear modulus.

Figure 1.

(a) Mechanical vibratory motion directions on Ecoflex cylindrical sample. Dotted green arrows indicate vertical motion that creates axisymmetric (cylindrical) wave propagation. Dashed red arrows indicate the direction of surface excitation to create planar wave propagation. (b) The real and imaginary part of the planar wave motion along a linear profile from surface to bottom. (c) The real and imaginary part of the radially moving wave motion along a linear profile across the diameter of the cylindrical Ecoflex sample.

For values of ρ = 1,000 kg/m³, μ_R = 60 kPa and μ₁ = 15 kPa, Figs. 1b–c compare the analytical solutions of two types of waves. In the first case, the entire test tube is axially vibrated at 2.5 kHz with an amplitude of 1 micron as indicated by the green arrows (Fig. 1c). In the second case, shear wave motion is generated in a 25 mm inner diameter test tube from a surface source at the top of the test tube oscillating at 2.5 kHz with an amplitude of 1 micron parallel and in contact with the surface (Fig. 1b: consider the test tube on its side with the shear wave actuator as indicated by the red arrow). Clearly, the geometrically focused shear waves result in a more uniform amplitude of shear waves throughout the entire region within the test tube as compared to application of shear waves from the medium surface, which attenuate in amplitude exponentially losing an order of magnitude every 10 mm for the values used in this simulation.

Mechanical Models for Viscoelastic Materials

In this study both integer order and fractional order viscoelastic models were investigated for their ability to match experimentally obtained complex shear modulus values for a given frequency bandwidth. The following six models were used: Voigt, Maxwell, SLS and a Generalized Maxwell model of five parameters, which all include integer-order time derivatives, as well as a spring-pot and a fractional order Voigt model, which are two of the simplest models possessing fractional order time derivatives.

Although adding more components to integer order viscoelastic models is expected to increase their ability to mimic experimental measurements, this improved ability is at the expense of added model complexity and potential non-uniqueness. An alternative is to use a fractional order viscoelastic model element, a spring-pot. This single element manifests ratedependent shear deformation and is comprised of two parameters μ_α and α (17).

$$\mu ={\mu }_0+{\mu }_{\alpha }\frac{{\partial }^{\alpha }}{\partial t^{\alpha }},0<\alpha <1$$ [3]

Eq.[3] is referred to as a fractional order Voigt model, where the second term on the right hand side of the equation represents the spring-pot element. The mathematics above extends the Rouse model (21) of polymer relaxation to a continuous distribution of relaxation times, which falls off at higher frequencies as a fractional order power law. It is commonly used to describe the elastic properties of rubbery and glassy polymers (22), such as Ecoflex – a type of polysiloxane. A fractal arrangement of integer order spring (elastic) and dashpot (viscous) elements asymptotically converges into the spring-pot element (23). These fractal organizations resemble stress-strain interactions in complex multiscale cellular and extracellular structures in biological tissues.

Moreover, the fractional derivative operator does not present a mathematical difficulty when it is applied to well-conditioned functions. For harmonic functions, the Weyl definition of the fractional derivative yields Eq.[4], which is linear in nature (23).

$$\frac{{\partial }^{\alpha }(e^{j\omega t})}{\partial t^{\alpha }}={(j\omega )}^{\alpha }{e}^{j\omega t}$$ [4]

Hence Eq. [3] can be represented in Laplace (s) and frequency (jω) domains, where ω is the circular frequency, as

$$\mu ={\mu }_0+{\mu }_{\alpha }{(s)}^{\alpha },\mu ={\mu }_0+{\mu }_{\alpha }{(j\omega )}^{\alpha }$$ [5a-b]

The storage and loss moduli are defined as in Table 1 for the Maxwell, SLS, generalized Maxwell, Voigt, spring-pot and fractional Voigt viscoelastic models. In Table 1, μ₀ denotes the static stiffness, η_1,2 denotes the viscous damping coefficient multiplied with the first order time derivative of the displacement (thus α is equal to 1), and μ_1,2 denotes the dynamic stiffness, which is only effective when the loading has a non-zero time derivative. In Fig. 2, schematic diagrams of viscoelastic models in Table 1 are given.

Table 1.

Complex-Valued Shear Modulus Expressions for Viscoelastic Models

Viscoelastic Model	Real part of shear modulus μR	Imaginary part of shear modulus μI
Maxwell	$\frac{{\omega }^2{\eta }^2\mu }{{\mu }^2+{\omega }^2{\eta }^2}$	$\frac{\omega \eta {\mu }^2}{{\mu }^2+{\omega }^2{\eta }^2}$
SLS	$\frac{{\mu }_0{\mu }_1^2+{\omega }^2{{\eta }_1}^2({\mu }_0+{\mu }_1)}{{\mu }_1^2+{\omega }^2{{\eta }_1}^2}$	$\frac{\omega {\eta }_1{\mu }_1^2}{{\mu }_1^2+{\omega }^2{{\eta }_1}^2}$
Generalized Maxwell	${\mu }_0+\frac{{\omega }^2{\mu }_1{\eta }_1^2}{{\mu }_1^2+{\omega }^2{\eta }_1^2}+\frac{{\omega }^2{\mu }_2{\eta }_2^2}{{\mu }_2^2+{\omega }^2{\eta }_2^2}+\cdots$	$j\omega (\frac{{\mu }_1^2{\eta }_1}{{\mu }_1^2+{\omega }^2{\eta }_1^2}+\frac{{\mu }_2^2{\eta }_2}{{\mu }_2^2+{\omega }^2{\eta }_2^2}+\cdots )$
Voigt	μ0	ωη
Spring-Pot	${\mu }_{\alpha }{\omega }^{\alpha }cos(\frac{\pi }{2}\alpha )$	${\mu }_{\alpha }{\omega }^{\alpha }sin(\frac{\pi }{2}\alpha )$
Fractional Voigt	${\mu }_0+{\mu }_{\alpha }{\omega }^{\alpha }cos(\frac{\pi }{2}\alpha )$	${\mu }_{\alpha }{\omega }^{\alpha }sin(\frac{\pi }{2}\alpha )$

Figure 2.

Schematic representation of a Maxwell, Standard Linear Solid, Generalized Maxwell, Voigt, fractional order spring-pot, and fractional order Voigt model.

MRE

The principle upon which MRE (6) is based can briefly be described as follows. When an oscillating magnetic field is introduced at the same frequency as mechanical harmonic motion in the sample, a certain amount of phase accumulation will occur in the complex image (7), (24) of any conventional magnetic resonance imaging method. This phase accumulation depends on the duration of the motion encoding gradients, strength of the gradient field, angle between directions of mechanical motion and cycling gradients, gyromagnetic ratio, period, amplitude of mechanical wave motion and the phase difference of mechanical wave and cycling gradient field. Phase accumulation, θ, for a given point can be written in following general form,

$$\theta =\gamma {\int }_0^{\mathrm{\Delta }t}\mathbf{G}(t)·\mathbf{r}(t)dt,$$ [6]

where G(t) is the time-dependent magnetic field vector and r(t) is the time-dependent displacement vector. This phase accumulation corresponds to a linearly scaled projection of the snapshot of mechanical wave propagation. Acquisition of these wave propagation images at different time offsets of mechanical motion enables us to calculate analytical representation of real valued wave propagation images.

Methods

Electro-Mechanical Setup

MRI Scanner

Three different sample sizes were used to cover “low”, “mid” and “high” frequency ranges with overlap: {200 Hz – 1.5 kHz, 1 kHz – 3.5 kHz, and 3 kHz to 7.75 kHz}. High frequency experiments were conducted on a Bruker 11.74 Tesla 56-mm vertical bore magnet with 19 mm gradient coils (maximum strength 3000 mT/m) and a 10 mm RF coil. Low and mid frequency range experiments took place in a 9.4 Tesla Bruker BioSpec 330 mm horizontal bore scanner with 116 mm self-shielded gradient coil inserts (maximum strength 660 mT/m). The low frequency range sample with a 62 mm diameter was scanned with a Bruker (Billerica, MA) 72 mm birdcage volume coil. The mid frequency range sample with 25 mm diameter was scanned with a Bruker volume quad coil (BioSpin MRI GmbH Quad coil, OD/ID=59/35mm, length=38mm). The high frequency range sample with 8.8 mm diameter was scanned with a 10 mm Bruker volume saddle coil.

The lower frequency limit of each sample is tied to the ratio of shear wavelength to sample diameter; in order to obtain an interpretable shear wave propagation pattern there is an upper bound on this ratio. The upper frequency limit of each sample is due to increased attenuation with frequency, which decreases the SNR of the wave image. Additionally, the limited electromagnetic switching rate of the gradient coils due to the required high magnetic field to create sufficient phase accumulation imposes an overall frequency limit for this study.

The rise time of the gradient coils in the 9.4 T magnet is 130 μs; this limits the upper frequency to be theoretically less than 3.8 kHz considering a bipolar trapezoidal function is generated in one of the gradient coils, which is explained in the “MRE Method” section, with the minimum of 2 μs shoulder duration. In other words, the period of the trapezoidal function has to be longer than twice the sum of the rise time and shoulder length. Due to practical problems, such as eddy currents and magnetic inertia, it was not possible to do proper imaging at this frequency; in other words images are distorted due to distortions in the fast switching gradient fields. But, below 3.5 kHz image quality is good enough to preserve the actual geometry of the object. For the 11.74 T system theoretically the upper frequency limit is 8.3 kHz with a 60 μs rise time of the gradient. But, due to the same reasons explained above the practical limit is about 7.75 kHz.

Mechanical Actuation

Electrical Setup

Sample containers are driven by piezoceramic stack actuators; see Fig. 3. For the 10 mm setup the piezoceramic stack actuator is from Thor Labs. Inc. (6.5×6.5×18 mm) and provides 11.6 μm displacement at 100 volts. For the 25 mm and 62 mm samples the piezoceramic stack actuator is from Physik Instrumente (PI) GmbH & Co (7×7×36 mm) and provides 30 μm displacement at 100V. One end of the piezo stack was attached to the sample container via a plastic rod to avoid interference with the RF coil; the other end of piezo stack was attached to a counter mass (concrete brick for the “low” and “mid” frequency studies and a fiberglass rod for the “high” frequency study) to provide an inertial ground.

Figure 3.

Vibratory actuation setup. The element in the middle is the piezoceramic stack actuator, which expands with the positive voltage applied to its terminals. The brick on the top provides an inertial ground to one side of the piezoceramic actuator. The cylinder is the sample container centered in the middle of the RF coil. To avoid any electromagnetic interference between the charged piezoceramic stack and RF coil, the sample container is separated from the piezo stack with an insulating hard plastic (Delrin) rod.

A trigger signal generated by the MRI system goes into the “Trigger Input” of the function generator (33220A Function/Arbitrary Waveform Generator, 20 MHz, Agilent Technologies Test and Measurement, Englewood, CO) to send a certain number of sinusoidal waveforms to the power amplifier (P3500S Power Amplifier, Yamaha Corporation of America, Buena Park, CA). Since negative voltage across the terminals of the piezo stack is harmful to its mechanical and electrical integrity, a DC bias is added to the output of the power amplifier via series connection of a constant DC supply (E3634A 200W Power Supply, Agilent Technologies Test and Measurement, Englewood, CO) to the negative terminal of power amplifier. Therefore, the positive terminal of the piezo goes to the positive terminal of the power amplifier and the negative terminal of the piezo stack goes to negative terminal of the DC power supply.

Sample Preparation

Ecoflex

A polysiloxane (i.e., silicone) (ECOFLEX-0010, Smooth-On, Inc., Easton, Pennsylvania, USA) was selected as the polymer matrix due to its tissue mimicking properties (25), durability and reasonable stability over time. It is less vulnerable to change over time as compared to water-based agar gel or gelatin counterparts that are widely used as phantoms in MRE studies (26). Also, Ecoflex provides a strong and homogenous cohesion to the container walls used as the mechanical actuation source. Ecoflex is formed by mixing parts 1A and 1B in 1:1 by weight or volume and cure at room temperature with negligible shrinkage. Two methods were utilized to minimize air bubble trapping during the mixing process prior to sample curing. In the first approach the Ecoflex mixture was poured with a very thin strip into the container inside a vacuum chamber (5305-1212, Thermo Scientific-Nalgene, Rochester, NY). In the second method, the Ecoflex was poured onto larger flat plates in a vacuum chamber to speed up the escape of air bubbles from the mixture; then, the air bubble-free Ecoflex was poured into the test container slowly before it sets.

Sample Container, Acoustic Excitation Boundary

Ecoflex was cured in cylindrical tubes where both ends were open to minimize the compression wave effect that would noticeably originate from closed ends for the mid and low frequency range samples. For the high frequency range a regular glass NMR test tube was used since the effects of compression waves were not observable.

The benefits of noninvasiveness and geometric focusing of the rapidly attenuating shear waves to extend the useful shear wave frequency range, and thus achieve very short shear wavelengths that will help in providing more localized estimates of material properties were shown previously in (27). By vibrating the entire test tube or cylindrical container axially, we effectively are using the entire inner container wall as an axisymmetric shear wave source that creates waves travelling radially inward. While these waves attenuate as they travel away from the wall due to viscous effects, this attenuation is countered by the geometric focusing that occurs as they travel towards the central axis of the test tube. Three different materials were used for the sample containers: PVC tubes (low frequency), Delrin (mid frequency) and Borosilicate Glass NMR tubes (high frequency). These materials were selected for their high stiffness and consequently high resonant frequencies relative to the corresponding mechanical driving frequencies used in MRE; i.e. it is desirable that these sample containers act effectively as rigid bodies undergoing oscillating axial motion.

MRE Method

Due to mechanical structure of experiments in this study (axisymmetric vibration) and orientation of images (axial), trapezoidal motion encoding gradients (MEGs) were only placed in the slice selection direction (z-direction). It was observed that the motion in the x and y directions were quite negligible as expected due to design of the containers and actuation setup.

For studies in the 11.74 T system a gradient echo based MRE pulse sequence was utilized with the following imaging parameters: TR/TE 250/4 ms, flip angle=30°, FOV=10 mm (for 8.8 mm diameter sample), slice thickness=1 mm, acquisition matrix=128×128, eight time offsets and MEG strength = 1200 mT/m. Due to low SNR levels in the 9.4 T system a spin echo based MRE pulse sequence is preferred with the following imaging parameters: TR/TE 500/8.1 ms, FOV=65 mm (for 62 mm diameter sample) and 32 mm (for 25 mm diameter sample), slice thickness=1 mm, acquisition matrix=128×128, eight time offsets and MEG strength = 396 mT/m.

Before starting MEG pulses, the system sends a trigger to a function generator (33220A Function/Arbitrary Waveform Generator, 20 MHz, Agilent Technologies Test and Measurement, Englewood, CO) to initiate mechanical motion. Depending on the estimated speed of wave propagation, which varies with frequency, the MEG starts after some time delay to allow for the wave motion to reach steady state. The number of motion encoding gradient cycles was kept as high as possible while keeping the echo time short enough for a high SNR FID signal. Besides having too long of an echo time, a very high number of motion encoding gradients might end up with phase wrapping in the wave image depending on the excitation frequency. Although efficient algorithms are available to do 2-D phase unwrapping (28), too much mechanical motion causes intra-voxel phase dispersion (29), which leads to a decrease in SNR. Therefore, the optimal number of motion encoding gradients was identified through iteration for each frequency prior to starting the MRE experiment.

Estimation of Viscoelastic Properties from Wave Images

Shear Modulus Estimation

In order to estimate the complex-valued shear modulus of Ecoflex, eight different analytical linear profiles were taken out of each experimental complex valued MRE wave image as shown in Fig. 4a,f,k for frequencies 700 Hz, 1500 Hz and 5000 Hz, respectively. In Fig. 4c–d, h–i, m–n the real and imaginary parts of one of the profiles are shown with circular markers.

Figure 4.

First column is 62 mm diameter sample at 700 Hz, middle column is 25 mm diameter sample at 1500 Hz and the last column is 8.8 mm diameter sample at 5000 Hz. (a, f, k) Radially propagating wave images obtained from phase part of a complex MR image across an axial slice. Dashed lines indicate the linear profile directions taken for shear modulus estimation. (b, g, l) Sagittal views of radially propagating wave images. Dashed lines indicate the location of where axial images were taken. (c, h, m) Real part of a profile (circular marks) taken from the wave images indicated in (a, f, k) along with real part of estimated analytical solution (solid lines). (d, i, n) Imaginary part of a profile (circular marks) taken from the wave image indicated in (a, f, k) along with imaginary part of estimated analytical solution (solid lines). (e, j, o) Shear stiffness map obtained by LFE estimation algorithm in base 10 log scale of Pascal.

Increasing the number of profiles used beyond four resulted in no significant change; to be safe eight line profiles were used. Wave propagation does not differ across different axial slices; this is observed in the sagittal wave propagation images in Fig. 4b,g,l. Each profile has been fitted to the closed form solution of the cylindrical shear wave equation in order to estimate the real and imaginary part of the shear modulus. (See Appendix for details.) Fitted curves are shown with solid lines in Fig. 4c–d, h–i, m–n. In order to provide a reference to method presented in this study, shear stiffness maps generated by the widely used Local Frequency Estimation algorithm (30) are given in the Fig. 4e,j,o. The shear stiffness estimated with LFE at frequencies 700 Hz, 1500 Hz and 5 kHz were 44.9 kPa, 58.5 kPa and 80.9 kPa respectively, while the corresponding absolute values of shear modulus estimated by curve fitting were 45.9 kPa, 60.5 kPa and 83.4 kPa.

Viscoelastic Model Parameter Estimations

In order to estimate the parameters of a given viscoelastic model, a cost function was minimized using the same computational approach described in “Shear Modulus Estimation” section. The cost function was defined as the sum of squares of error for each frequency experiment conducted. For a particular frequency, error is the difference between the complex shear modulus of the viscoelastic model and the median of complex shear modulus estimates as described in the previous section. Error values normalized with respect to the median of the estimated complex shear modulus are given in Fig. 6d and Fig. 7d for the full frequency bandwidth and limited frequency bandwidth cases, respectively.

Figure 6.

Shear moduli estimates from 200 Hz to 7750 Hz for different viscoelastic models whose parameters are estimated via minimization of mean square error between experimental data and the predicted model. (a) Real part of shear modulus plotted versus frequency, (b) imaginary part of shear modulus plotted versus frequency, (c) real part of shear modulus plotted versus imaginary part of shear modulus, (d) normalized root mean square error between estimated shear modulus and shear modulus of fitted viscoelastic models. In part (c) the shear modulus of mechanical models begin from 50 Hz in order to demonstrate their convergence into static shear modulus values.

Figure 7.

Shear modulus estimates from 200 Hz to 7750 Hz for different viscoelastic models whose parameters are estimated via minimization of mean square error between a limited portion of experimental data (200 Hz to 900 Hz) and predicted model. (a) Real part of shear modulus plotted versus frequency, (b) imaginary part of shear modulus plotted versus frequency, (c) real part of shear modulus plotted versus imaginary part of shear modulus, (d) normalized root mean square error between estimated shear modulus and shear modulus of fitted viscoelastic model. In part (c) shear modulus of mechanical models begin from 50 Hz in order to demonstrate their convergence into static shear modulus values.

Since wavelengths become longer than the radius of the sample below 200 Hz, it wasn’t possible to acquire reliable wave images below 200 Hz. Despite that, viscoelastic models were plotted beginning from 50 Hz in order to show where they converge as frequency goes to zero (whether they converge to the experimentally measured static elasticity value of the material).

Results

Shear Modulus Estimations

Across the whole frequency range there are some frequencies that overlap between experiments. In order to explicitly show the complex-valued shear modulus estimation for each experiment, they were individually plotted on the same figure with different color codes; see Fig. 5. Although each experiment uses a different batch of Ecoflex that were not identical due to their different container sizes and human error in mixing parts A and B, μ_R and μ_I values still follow a smooth pattern across the frequency range; there are no significantly observable discontinuities at the transitions between samples.

Figure 5.

Box plot of real and imaginary parts of shear modulus of Ecoflex over frequency. Dotted rectangles indicate frequency overlap of different samples. “Low”, “Mid” and “High” frequency studies indicated with solid boxes were obtained with 62mm, 25mm and 8.8mm Ecoflex samples, respectively. The red line in the boxes marks the median value estimated from all linear profiles. The edges of the box are the 25th and 75th percentiles for a particular frequency. The whiskers extend to the most extreme data points, not considering outliers, which are plotted as markers (+).

Viscoelastic Model Parameter Estimations

Several common viscoelastic models were used to fit the experimental measurements of μ_R and μ₁ as a function of frequency. Two-parameter models are that of Maxwell, Voigt and the spring pot. Three-parameter models are the Fractional Voigt and standard linear solid (SLS). A five-parameter model is the Generalized Maxwell model. However, the static value of μ was measured by an indentation test to be 13.3 kPa (31). This value is used in the Voigt, Fractional Voigt, SLS and Generalized Maxwell models, thus reducing the number of parameters available for optimizing a fit to the MRE data to 1, 2, 2 and 4, respectively (Table 2).

Table 2.

Estimates of viscoelastic model parameters by utilizing complex-valued shear modulus values of the full frequency range (200 Hz to 7.75 kHz). Values given in parenthesis are viscoelastic model parameters estimated using complex-valued shear modulus of a limited frequency range (200 Hz to 900 Hz). In the last column, error is defined as mean of normalized difference between complex shear moduli calculated from the viscoelastic models and complex shear moduli derived from experimental MRE data. Dimensions: μ_1,2 (kPa-s); μ_α (Pa-s^α); α (unitless);η_1,2 (Pa); Error(Percentage).

Model	1st Parameter	2nd Parameter	3rd Parameter	4th Parameter	Error
Maxwell	μ=72.7 (44.4)	η = 13.4(41)			34.7% (37.1%)
SLS	μ1=68.5 (35.3)	η1=6.7 (18.1)			28.0% (32.6%)
Generalized Maxwell	μ1=78.7 (31.6)	μ2=36.9 (17)	η1=1.4 (5.6)	η2=21.3 (72.3)	8.9% (21.1%)
Voigt	η =1.1 (3.8)				70.9% (94.6%)
Springpot	μα=4,571 (6,474)	α=0.28 (0.23)			5.5% (9.8%)
Fractional Voigt	μα=1,956 (2,038)	α=0.34 (0.33)			3.8% (6.2%)

Although it is well known that two-parameter models like Maxwell and Voigt are not successful over a wide frequency range (31), they have been included in the analysis for the sake of completeness. The SLS model is a subset of the Generalized Maxwell model; hence, success of increasing the parameter number can be observed by comparing these two models. But, results show that even the two-parameter fractional order springpot model outperforms the generalized Maxwell model. The Fractional Voigt model puts in parallel a springpot with a static value for μ (whereas the springpot alone cannot sustain a static load). While both model types perform well, the Fractional Voigt model is superior in the sense of lower estimation error of complex shear modulus over a wide frequency range (See Fig. 6a,b).

The μ_R versus μ_I plot (Fig. 6c) contrasts the dynamics of each viscoelastic model and the experimental values. It is clear that the Fractional Voigt model captures the μ_R vs. μ_I curve best throughout the frequency range, followed by the springpot model. As frequency goes to zero the springpot model converges to the origin while the Fractional Voigt model and experimental values asymptotically converge to a μ₀ value, which serves as additional verification of indentation test for the μ₀ measurement.

Performance of Mechanical Models Based on Limited Bandwidth

In this part of the study we have investigated the accuracy of a viscoelastic model in the case of limited frequency range experimental data. Parameters of the mechanical models mentioned previously are estimated using frequencies from 200 Hz to 900 Hz and they have been plot over the full frequency range of the experimental data; see Fig. 7. This frequency range does not include any information taken from mid and high frequency setups. 200 Hz to 900 Hz only covers from the beginning of low frequency setup up to the beginning of mid frequency setup. The purpose here is to observe how capable the viscoelastic models are for estimating the complex shear modulus in a frequency range where they have no a priori information.

Estimated viscoelastic model parameter values are presented in Table 2 along with the error defined as mean value of differences between experimental shear moduli and shear moduli calculated from the viscoelastic model normalized by corresponding experimental shear modulus. Error for a given model is calculated for the selected frequency range. In other words, for the limited frequency bandwidth study, error does not include any information above 900 Hz. Therefore, error values given in Table 2 should be compared across the viscoelastic models but not between the full bandwidth case and limited bandwidth case.

Maxwell and Voigt models show their significant limitations. But, all other mechanical models can follow the experimental data quite well within the 200–900 Hz range. But, at higher frequencies they diverge significantly from the experimental data except for the fractional order models; see Fig. 7. Thus, we emphasize here that use of the springpot or fractional Voigt, optimized by fitting experimental data in the 200–900 Hz range, still was able to accurately predict behavior up to 7.75 kHz.

Discussion

Staging diseases and quantifying tissue malignancy with MRE (as well as with ultrasound-based elastography (32)) is an emerging technique. In addition to clinical applications, numerous in-vitro studies of MRE are also underway (33). Since post processing of elastography data is needed to extract numerical values for tissue stiffness, the choice of the viscoelastic model type may be critical for the determination of new elastic biomarkers that can, for example, distinguish between benign and malignant conditions. Viscoelastic models each exhibit a different frequency behavior and most current MRE is conducted at one or a few distinct frequencies because in many biomedical imaging situations it may not be possible to conduct a wideband MRE study like the one presented here over 7.55 kHz. For example, the target area is too large, such as when MRE is performed clinically on the human liver; higher frequency shear waves will decay with little penetration into the liver. On the other hand, if the target is too small, such as a murine brain, lower frequency shear waves could not be sustained. Hence, there is uncertainty in selecting the best model for a given clinical situation and the potential for discord between elastic biomarkers selected by different models (elastic, viscous, Maxwell, Voigt, SLS, etc.).

In this study a novel MRE technique was developed and applied to establish the viscoelastic properties of a tissue-like silicone material, Ecoflex, over a wide frequency band (200 Hz to 7750 Hz). The benefits of using viscoelastic measurements over a wide frequency range when choosing a viscoelastic mechanical model were demonstrated. These benefits can also be obtained using fractional order models when only limited bandwidth of viscoelastic property measurements are available, which is typically the case in clinical applications.

Ecoflex was chosen as an ideal phantom material standard for MRE studies because of its soft tissue like viscoelastic properties (31), its abiding mechanical properties over time (34), and its consistency between prepared samples. Ecoflex was cured into cylindrical shapes since the axisymmetric shear wave propagation (Fig. 4a) is governed by simple Bessel functions (19) in a closed from expression (Fig. 4c–d). Although geometric focusing compensates for attenuation of shear waves, as excitation frequency increases this phenomena fades. Therefore, three different size Ecoflex samples were used for the low, mid and high frequency ranges. To rule out sample size dependency, which could have affected the results, the frequency ranges were overlapped to ensure that both the slope and the magnitude of real and imaginary values of shear modulus were consistent. The small sample was investigated in a high field NMR system that has a faster gradient coil switching capability for higher frequency motion encoding, while the other two larger samples were studied in a horizontal small animal MRI scanner with a larger bore. In spite of the three different Ecoflex samples and two different MR scanners used, transitions between the frequency ranges were smooth and the variance of each measurement was low (Fig. 5).

By using the experimentally-derived complex-valued shear modulus over a 7.55 kHz frequency band, the most appropriate (in the sense of mean square error between experimental data and the predicted model) viscoelastic model type out of six common viscoelastic models shown in Fig. 2 was identified. A wide frequency range was needed to clearly identify the superior performing model types (Fig. 7).

In clinical applications, the larger dimension of internal organs and the slower switching speed of the gradient coils in human MR scanners limit the frequency band over which MRE can be performed. In this situation, identification of a viscoelastic model based on more limited frequency information is necessary. In this sense, the present study offers some insight into what may be the more appropriate model choice.

In summary, this study provides a novel approach for choosing a viscoelastic mechanical model via comparing the shear modulus of estimated mechanical models with experimentally obtained complex shear moduli of Ecoflex for both a wide frequency range and a limited frequency range. Complex-valued shear moduli of Ecoflex were estimated by fitting the exact theoretical solution of the cylindrical wave into experimental data for a wide frequency range, from 200 Hz to 7.75 kHz by utilizing two different MRI systems and three different phantom sample sizes. It is observed that the fractional order viscoelastic models provide a better fit than integer order models and this conclusion is more pronounced for the limited frequency range study, which has more relevance to clinical applications.

Appendix

Shear Modulus Estimation

A linear profile, f(r_n) taken out of an experimental complex valued MRE wave image was fit into an analytical solution of the cylindrical wave equation (Eq.[2a–b]). For additional uncertainties such as the phase of the wave propagation (θ), complex valued bias caused by axial compression waves in the cylinder (β), magnitude scale (s), and spatial position offset of the center of linear profile (δ) in experimental data, four more parameters were added to the estimated cylindrical wave equation denoted by f̂ as follows,

$$\widehat{f}(r,\mu ,\theta ,\beta ,s,\delta )=s\ast u_z(r+\delta ,t,k_{\beta }(\mu ))\ast e^{-i\theta }+\beta ,$$ [7]

where k_β(μ) is defined in Eq.[2b]. An error function, defined as the sum of squares of absolute difference between experimental data and the closed form solution of the estimated cylindrical wave equation, is as follows,

$$\mathit{Err}(\mu ,\theta ,\beta ,s,\delta )=\frac{{\sum }_{n=-N}^N{\mid f(r_n)-\widehat{f}(r_n,\mu ,\theta ,\beta ,s,\delta )\mid }^2}{2N+1},r_n=\frac{n}{N}a$$ [8]

where, 2N+1 is the total number of points in the linear profile, a is the inner radius of the cylinder.

In order to reach a minimum value for the error function, a total of seven parameters needed to be optimized, considering shear modulus and bias were complex valued parameters having both real and imaginary parts treated as two independent parameters. Because of the high number of parameters and low SNR of experimental data, as a consequence of significant attenuation at high frequencies, a global search algorithm was performed to avoid any highly probable local minima. The MultiStart function in the “Global Optimization” toolbox provided by MATLAB^® (Math-Works, Inc., Natick, MA, USA) was preferred over the GlobalSearch function provided in the same toolbox due to its better performance in circumventing local minima. The number of starting points of the global optimization procedure was empirically chosen to be 512 since an improvement was observed from 200 points to 400 points but there was no change in estimations between 400 points and 512 points. Upper and lower bounds for all the parameters are provided according to the particular size of Ecoflex sample and frequency. Stopping conditions for the optimization routine are extensively explained in MATLAB^® documentation. They shall be adjusted by trial and error since they depend on the amplitude of the experimental data and the size of geometry which can both be manipulated in any step of the programming without affecting the net results in shear modulus estimation.