Chapter 10

Magnetic Resonance Elastography

Magnetic resonance elastography (MRE) is an imaging modality capable of visualizing the stiffness of biological tissues by measuring the propagating strain waves in an object of interest (Low et al. 2010; Muthupillai et al. 1995; Papazoglou et al. 2005; Sack et al. 2002; Sinkus et al. 2000). Since MRE provides non-invasive assessment of variations in tissue elasticity, it has been used for non-invasive diagnosis of liver disease or for detecting prostate cancer. Tissue elasticity can change with disease, and the shear modulus (or modulus of rigidity) varies over a wide range, differentiating various pathological states of tissues (Venkatesh et al. 2008; Yin et al. 2007). It also has potential applications in studying skeletal muscle biomechanics (Papazoglou et al. 2005; Uffmann et al. 2004).

Tissue elasticity refers to the ability of a tissue to deform its shape when a mechanical force is applied and to regain its original shape after the force is removed; tumor tissue is less compressible than normal tissue. For centuries, palpation using the surface of the finger or palm has been used to measure tissue stiffness; it can be viewed as an elasticity measurement technique to feel the degree of tissue distortion (strain) due to pressure (stress) on the tissue. As a visual palpation, elastography using ultrasound was developed in the late 1980s (Lerner and Parker 1987; Ophir et al. 1991). This ultrasound elastography influenced the early development of MRE.

MRE provides a quantitative assessment of tissue stiffness (shear modulus) with a non-invasive method. It is based on the fact that the speed of shear wave propagation is closely related to tissue stiffness; the stiffer the tissue, the faster the speed. For a linear elastic medium, the shear modulus is proportional to the square of the shear velocity. To measure the shear velocity, MRE uses magnetic resonance imaging (MRI) techniques to detect the propagation of transverse acoustic strain waves in the object of interest (Muthupillai et al. 1995). It visualizes the time-harmonic displacement in the tissue induced by a harmonically oscillating mechanical vibration. With a fixed frequency, the shear velocity is proportional to the wavelength, which can be viewed as the peak-to-peak distance of the time-harmonic displacement. Hence, we can evaluate local values of tissue elasticity from the time-harmonic displacement. We refer to the review articles of Doyley (2012) and Mariappan et al. (2010) for overviews of MRE techniques. In this chapter, we will introduce several MRE approaches, from basic physics to potential applications.

10.1 Representation of Physical Phenomena

When a force is applied to a solid body, the body deforms in shape and volume to some extent. If the deformed body returns to its original shape once the force is removed, it is said to have experienced elastic deformation. If the deformation is irreversible, it is said to be plastic deformation. Stress is a description of the average force per unit normal area of a surface within a body. The unit of stress is the pascal (Pa = N m², newtons per square meter). Strain measures the extent of deformation in terms of a relative displacement, that is, the change of distance between two adjacent points in the deformed state with respect to that distance in the undeformed state. In this section, we will deal only with linear elastic materials, and we give a brief overview of some elementary concepts in linear elasticity regarding the relationship between stress and strain. We refer to the books of Landau and Lifshitz (1986) and Pujol (2002) for detailed explanations.

10.1.1 Overview of Hooke's Law

Hooke's law states that strain is proportional to stress. In a one-dimensional simple model of a rod of elastic material,

10.1

where E is Young's modulus. Here, we regard the rod of elastic material as a spring; Young's modulus can be viewed as the spring constant. Young's modulus E = σ/ can be used to predict compression or elongation as a result of axial stress. A compressive force causes the rod to get shorter, whereas a tensile force makes the rod longer.

A generalized Hooke's law for a three-dimensional elastic body can be derived by viewing the elastic body as a network of linear springs. For a three-dimensional inhomogeneous elastic body, the stress at any given point r within the body is defined by a 3 × 3 matrix called the Cauchy stress tensor:

The first column of the stress tensor represents the force acting on a differential area dA normal to the x axis (see Figure 10.1). Similarly, the second and third columns of are the forces acting on the differential area dA normal to the y and z axes, respectively. There are three main types of stress: compression, tension and shear stress. The diagonal components σ₁₁, σ₂₂ and σ₃₃ are normal stresses, the forces perpendicular to the area dA. The remaining off-diagonal components are shear stresses, the forces parallel to the area dA.

Strain can be obtained by comparing deformed geometry with undeformed geometry. The strain at any given point r within the body is expressed by the 3 × 3 matrix

Then, the generalized three-dimensional Hooke's law can be expressed as

10.2

where C = (C_ijk) is the fourth-order elastic tensor, which is symmetrical: C_ijk = C_ijk = C_jik. The tensor (C_ijk) is called the stiffness tensor, and it links the stress tensor and the strain tensor.

Adopting the Voigt notation (e.g. C₁₁₂₃ = C₁₄), we can rewrite (10.2) in the following matrix representation:

10.3

In the case of isotropic materials, (10.3) can be simplified as

10.4

with

where E is Young's modulus, ν is Poisson's ratio and μ is the shear modulus.

Next, we will briefly describe the derivation of (10.4) by considering a cubic material aligned with the axes {x, y, z}. We assume that three “tension tests”, labeled as ′, ′′ and ′′′, are conducted along x, y and z, respectively. Then, normal strains will be produced as follows:

The combined strains can be obtained by superposition:

This provides us with the three equations

10.5

The shear strains and stresses are connected by the shear modulus μ as

where

which leads to

10.6

The identity (10.4) follows from (10.5) and (10.6).

10.1.2 Strain Tensor in Lagrangian Coordinates

We now provide basic descriptions of the strain tensor. Let Φ_t:Ω → Ω_t be the mapping from the undeformed body Ω at time t = 0 to the deformed state Φ_t(Ω) = Ω_t at time t. Let R = (X, Y, Z) denote a position of a particle in the undeformed frame, and let r = r(R, t) = Φ_t(R) indicate the position of the same particle in the deformed frame at time t (see Figure 10.2). Hence, the motion of the specified particle at R in the reference frame is described by r(R, t) = Φ_t(R). This approach is called the Lagrangian description. One may use the Eulerian description R = R(r, t) when we are interested in a particle that occupies a given point r at a given time.

The displacement vector (the total movement of a particle with respect to the undeformed frame) is given by

With the assumption of small deformations, we can approximate

where I is the identity matrix and is the 3 × 3 matrix

To analyze the change in length elements, we compare the distance |dr|² in the deformed frame with the distance |dR|² in the reference frame:

From the assumption of small deformations, we can neglect the term to get the approximation

Introducing Cauchy's infinitesimal strain tensor

the difference |dr|² − |dR|² can be approximated by

Since |dr|² − |dR|² ≈ 2|dR|(|dr| − |dR|), the relative change in length in the dR direction is

The above identity provides some geometric meaning of the strain tensor.

10.2 Forward Problem and Model

Newton's second law (force = mass × acceleration) provides a description of motion. We will use the Eulerian description of motion in order to focus on a particle at r = r(R, t). As t varies, different particles occupy the same spatial point r. With the Eulerian description, the velocity v = ∂u(R, t)/∂t can be expressed as

Here, D/Dt is the derivative with respect to time t keeping R constant (called the material derivative), and ∂/∂t is the derivative with respect to time t keeping r constant. Similarly, the acceleration of a particle can be expressed as

Assuming that the density of the body, denoted by ρ, is locally constant, let us examine the equation of motion in a small cube Q with surface ∂Q. From the balance of linear momentum,

10.8

where n is the unit outward normal vector and f is the body force per unit volume. From the divergence theorem, (10.8) becomes

10.9

Since the cube Q is arbitrarily small, (10.8) leads to

10.10

From the assumption of small deformations, the acceleration Dv/Dt is approximated by

Assuming that the body is isotropic, the generalized Hooke's law (10.2) leads to

10.11

where μ and λ are the Lamé coefficients given by

with Poisson's ratio ν. Hence, substituting (10.11) into (10.10) yields the equation of motion in terms of the displacement u:

10.12

In the case of a locally homogeneous isotropic medium, (10.12) can be simplified to

10.13

In the case of an anisotropic medium, u = (u₁, u₂, u₃) is dictated by the following elasticity system (Landau and Lifshitz 1986):

10.14

10.3 Inverse Problem in MRE

Imaging methods in MRE can be roughly divided into three steps (Mariappan et al. 2010):

• Excitation. Apply a sinusoidal vibration with an angular frequency of oscillation , through the surface of the object. This induces tissue vibrations in an imaging region inside the human body.
• Tissue response measurement. Measure the induced tissue vibrations that are magnetically encoded by oscillating magnetic field gradients (Muthupillai et al. 1995).
• Shear modulus reconstruction. Visualize the shear modulus distribution using the measured tissue-displacement field.

The sinusoidal excitation at the angular frequency induces the internal displacement u(r, t) within the body at the same angular frequency , and its time-harmonic displacement u(x) is given by

Here, for simplicity, we use the same notation for the time-harmonic displacement u(x) as the time-dependent displacement u(r, t). Hopefully, this will not cause any confusion from the context. From now on, the notation u will be used for the time-harmonic displacement. Assume that ρf ≈ 0 in (10.13) and (10.14).

For a linear isotropic body, substituting into (10.13) leads to the governing equation of the induced internal time-harmonic displacement vector u:

10.15

Here, taking the viscosity effect into account, μ and λ can be complex-valued. The real part is the shear modulus and is the Lamé coefficient, with

is the shear viscosity accounting for attenuation within the medium and is the viscosity of the compressible wave.

Similarly, in the case of an anisotropic material, substituting into (10.14) with leads to

10.16

The corresponding inverse problem is to recover the distribution of tissue elasticity from the time-harmonic displacement u(r) inside the body.

10.4 Reconstruction Algorithms

Let Ω be the domain occupying the object to be imaged. Assume that soft tissues exhibit linear, isotropic mechanical properties. Most reconstruction methods for shear modulus imaging have used the following scalar equation:

10.17

This model has the major advantage of requiring one component of u, and it simplifies the underlying mathematical theory to the corresponding inverse problem; the inverse problem using the vector equation (10.15) can be reduced to the inverse problem using the scalar equation (10.17). We refer the interested reader to Manduca et al. (2001), McLaughlin and Renzi (2006), McLaughlin and Yoon (2004), Oliphant et al. (2001) and Sinkus et al. (2005b).

This model (10.17) uses the assumption that the longitudinal wave can be filtered out, since the longitudinal wave varies slowly compared with the shear wave (McLaughlin and Renzi 2006; Sinkus et al. 2005b). We should note that the scalar equation (10.17) for u is not accurate because is not negligible; although is very small, λ is very large. However, it seems that the scalar equation (10.17) for μ is reasonably accurate; for a given axial component of time-harmonic shear wave in (10.15), the shear modulus μ approximately satisfies (10.17).

The scalar equation (10.17) can be derived under the assumptions that and are small (Lee et al. 2010). From the elasticity equation (10.15), u can be decomposed into

10.18

where

Neglecting ϒ ≈ 0, we get the following approximation:

10.19

 

Writing

the decomposition (10.19) is expressed as

10.20

Application of divergence to both sides of (10.19) leads to

10.21

Since , the above approximation yields . Therefore, u_L satisfies both and . This means that each component of u_L satisfies the Laplace equation approximately

10.22

From the elasticity equation (10.15) with the assumptions and , we have

10.23

From (10.22), u_L is approximately harmonic in the region of interest, thereby varying very slowly in the internal region with negligible contribution to u. Hence, u_L can be treated as noise in the data (McLaughlin and Renzi 2006) and we may assume that the shear modulus μ and the displacement u satisfy the forward equation

10.24

For the above derivation of (10.24), we have used the assumption of in two places.

10.4.1 Reconstruction of μ with the Assumption of Local Homogeneity

Assuming local homogeneity on μ (or ), (10.24) can be expressed as

10.25

Then, μ can be directly recovered from

10.26

This direct inversion method is the most commonly used MRE algorithm, which requires the local homogeneity assumption on μ (Kruse et al. 2000; Manduca et al. 2001; Manduca et al. 2002; Manduca et al. 2003; Oliphant et al. 2000a; Oliphant et al. 2000b; Oliphant et al. 2001; Sinkus et al. 2005b). Figure 10.3 shows the performance of the direct algebraic inversion method (Mariappan et al. 2010).

One drawback of the direct algebraic inversion method (10.26) is that double differentiation of the measured displacement data u can cause undesirable noise effects owing to the tendency of the operation to amplify noise. To alleviate the noise amplification from measured data, this method typically needs some filtering to reduce high-frequency noise (Manduca et al. 2003). Another drawback of this method is that the modeling error from the assumption of local homogeneity produces artifacts around regions of differing elastic properties even with noiseless data (Kwon et al. 2009).

10.4.2 Reconstruction of μ without the Assumption of Local Homogeneity

Next, we explain a method for reconstructing shear modulus images without the assumption of local homogeneity (Kwon et al. 2009; Lee et al. 2010).

According to the Helmholtz–Hodge decomposition, the vector field in (10.17) can be decomposed into a curl-free component and a divergence-free component:

10.27

where the scalar potential f satisfies

10.28

and the vector field W satisfies

10.29

Taking the inner product of , the complex conjugate of , on both sides of the identity (10.27) leads to the following identity:

10.30

Assuming , μ can be decomposed into

10.31

where

10.32

Let μ_d denote the recovered shear modulus using the direct inversion formula (10.26):

The following theorem provides some characteristics of μ* and μ_d.

 

For the image reconstruction, the first quantity μ* in (10.31) can be computed explicitly by solving the problem (10.28) from knowledge of and the Neumann boundary condition. However, we cannot compute μ^** directly since the equation contains the unknown quantity . Hence, we need an iterative procedure to get μ^**. We summarize the reconstruction procedure with the iterative method.

We can test the iterative method via numerical simulations using a rectangular two-dimensional model of 10 × 10 cm² with the origin at its bottom-left corner. Figure 10.4(a) shows the image of the simulated target shear modulus. The target shear modulus included vertical and lateral thin bars with shear storage modulus with 0.2 kPa viscosity and background shear storage modulus with 0.1 kPa viscosity. To get the displacement data, we solved the wave equation in (10.25) with the boundary conditions given by u(x, y) = 1 for y = 10, 0 ≤ x ≤ 10 and otherwise. Figure 10.4(b) and (c) show the real and imaginary parts of the simulated displacement image, respectively. Figure 10.5(a) and (b) show the intensity of and , respectively. Since the wave propagates from the top to bottom, Figure 10.5(a) highlights ∂μ/∂y and Figure 10.5(b) emphasizes ∂μ/∂x.

Figure 10.6 shows the reconstructed image of the shear storage moduli μ_d, μ* and . The quantity μ* itself is a good approximation of the true μ, as shown in Figure 10.6. The reconstructed μ* was close to the target μ in the left region where and the updated μ⁵ recovered the missed information of μ* in the region where the intensity of was high in Figure 10.5(b). The recovered in Figure 10.6(a), even without added noise, shows some artifacts by neglecting the term in the left region where the intensity of was high in Figure 10.4(a). Figure 10.7 shows the results of in vivo human liver experiments (Kwon et al. 2009).

10.4.3 Anisotropic Elastic Moduli Reconstruction

Most research in MRE has made the assumption that soft tissues exhibit linear and isotropic mechanical properties. However, various biological tissues such as skeletal muscle are known to have anisotropic elastic properties (Chaudhry et al. 2008; Dresner et al. 2001; Gao et al. 1996; Gennisson et al. 2003; Heers et al. 2003; Humphrey 2003; Kruse et al. 2000; McLaughlin et al. 2007). Techniques using the scalar equation (10.17) definitely cannot be applied to such anisotropic cases.

The simplest model of the anisotropy would be a transversely isotropic model. Papazoglou et al. (2005) adopted the model with transverse isotropy in which the principal axis of symmetry x₃ = z is aligned to be parallel to the muscle fibers. In this model, the full elasticity system (10.16) is reduced to the two-dimensional elasticity system with elastic moduli (shear modulus, Young's modulus and Poisson's ratio) depending only on r: = (x, z). To be precise, let Ω be the xz plane of the elastic object to be imaged. The time-harmonic displacement wave vector u(r) = (u₁(r), u₃(r))^T, r ∈ Ω, satisfies the following simplified elasticity system:

10.37

where

ρ is the tissue density and μ_β: = 4μ₁₂E₃/E₁. Here, E₃ and E₁ are Young's modulus with respect to the x₃ and x₁ axes, respectively.

This transversely isotropic model for the elasticity has a potential application of determining the mechanical properties of human skeletal muscle, including the elastic properties of bundles of parallel fibers aligned in one direction. Papazoglou et al. (2005) and Sinkus et al. (2005a) studied this transversely isotropic model with the assumption of local homogeneity. Papazoglou et al. (2005) made a step toward a better description of two-dimensional shear wave patterns to reveal the anisotropy of the muscle fibers. Sinkus et al. (2005a) developed a direct reconstruction scheme for imaging the anisotropic property of breast tissues.

10.5 Technical Issues in MRE

MRE is a non-invasive imaging technique for recovering the mechanical properties of tissues. The image reconstruction algorithm in MRE is not perfect yet, owing to difficulties in handling the general elasticity equations. There are many challenging issues in the inverse problem of MRE. For soft tissues, is very small, whereas the Lamé parameter λ is very large. Thus, even though , the total effect of may not be negligible and may produce errors or artifacts in reconstructed images. Hence, it would be desirable to investigate a stable algorithm to reconstruct the Lamé parameters simultaneously without the incompressibility assumption.

There have been increasing demands for anisotropic models, but it is very difficult to achieve a robust reconstruction. If MRE alone is insufficient to provide a robust reconstruction of high-resolution images, one may try combining it with other techniques to get some complementary information.

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