Elastography: general principles and clincial applications.

Elastography visualizes differences in the biomechanical properties of normal and diseased tissues (Sarvazyan et al. 1995; Krouskop et al. 1998; Samani et al. 2007; Parker et al. 2011). Elastography was developed in the late 1980s to early 1990s to improve ultrasonic imaging (Lerner and Parker 1987; Lerner et al. 1988; Ophir et al. 1991; O’Donnell et al. 1994), but the success of ultrasonic elastography has inspired investigators to develop analogues based on magnetic resonance imaging (Muthupillai et al. 1995; Bishop et al. 2000; Sinkus et al. 2000; Weaver et al. 2001), and optical coherence tomography (Khalil et al. 2005; Kirkpatrick et al. 2006; Ko et al. 2006). In this chapter, we will focus on ultrasonic techniques with a brief reference to approaches based on magnetic resonance imaging. The general principle of elastography can be summarized as follows: (1) perturb the tissue using a quasi-static, harmonic, or transient mechanical source; (2) measure the resulting mechanical response (displacement, strain or amplitude and phase of vibration); and (3) infer the biomechanical properties of the underlying tissue by applying either a simplified or continuum mechanical model to the measured mechanical response (Manduca et al. 1998; Ophir et al. 2000; Bamber et al. 2002; Greenleaf et al. 2003; Parker et al. 2011). In this chapter, we will describe (a) the general principles of quasi-static, harmonic, and transient elastography (see Fig. 1)—the most popular approaches to elastography and (b) the physics of elastography—the underlying equations of motion that governs the motion in each approach. We also provide examples of clinical applications of each approach. Figure 1 Schematic representation of current approaches to elastographic imaging: quasi-static elastography (left), harmonic elastography (middle), and transient elastography (right). 2. The physics of elastography Like conventional medical imaging modalities, forward and the inverse problems are encountered in elastography. The former problem is concerned with predicting the mechanical response of a material with known biomechanical properties and external boundary conditions. Understanding this problem and devising accurate theoretical models to solve it has been an effective strategy in developing and optimizing the performance of ultrasound displacement estimation methods. The latter problem is concerned with estimating the biomechanical properties non-invasively using the forward model and knowledge of the mechanical response, and external boundary conditions. A comprehensive review of methods that have been developed to solve the inverse problem is given in (Doyley 2012), therefore, in this section we will focus only on the forward problem. The forward elastography problem can be described by the following system of partial differential equations (PDEs) given in compact form (Timoshenko and Goodier 1970; Fung 1981): ∇·[σij]=βi, (1) where σij is the three-dimensional stress tensor (i.e., a vector of vectors), βi is the deforming force, and ∇ is the del operator. Using the assumption that soft tissues exhibit linear elastic behavior, then the strain tensor (e) maybe related to the stress tensor (σ) as follows (Landau et al. 1986): σij=Cijklekl, (2) where the tensor C is a rank-four tensor consisting of 21 independent elastic constants (Fung 1981; Ophir et al. 1999; Greenleaf et al. 2003). However, under the assumption that soft tissues exhibit isotropic mechanical behavior then only two independent constants, λ and μ (lambda and shear modulus), are required. The relationship between stress and strain for linear isotropic elastic materials is given by: σij=2μeij+λδijΘ, (3) where Θ = ∇ · u = e11 +e22 +e33 is the compressibility relation, δ is the Kronecker delta, and the components of the strain tensor are defined as: eij=12(∂ui∂j+∂uj∂xi) (4) Lame constants (i.e., λ and μ) are related to Young’s modulus (E) and Poisson’s ratio (v), as follows (Timoshenko and Goodier 1970; Fung 1981): μ=E2(1+v),λ=vE(1+v)(1-2v) (5) The stress tensor is eliminated from the equilibrium equations (i.e., equation (2)) using equation (3). The strain components are then expressed in terms of displacements using equation (4). The resulting equations (i.e., the Navier-Stokes equations) are given by: ∇·μ∇u+∇(λ+μ)∇·u=ρ∂2u∂t2, (6) where ρ the is density of the material, u is the displacement vector, and t is time. For quasi-static deformations, equation (6) reduces to: ∇·μ∇u+∇(λ+μ)∇·u=0 (7) For harmonic deformations, the time-independent (steady-state) equations in the frequency domain give (Sinkus et al. 2000; Van Houten et al. 2001): ∇·μ∇u+∇(λ+μ)∇·u=ρω2u, (8) where ω is the angular frequency of the sinusoidal excitation. For transient deformations, the wave equation is derived by differentiating equation (6) with respect to x, y, z, which gives the following result (Timoshenko and Goodier 1970): ∇2Δ=1c12∂2Δ∂t2 (9) where ∇ · u = Δ, and the velocity of the propagating compressional wave, c1, is given by: c1=λ+2μρ (10) The wave equation for the propagating shear wave is given by: ∇2ζ=1c22∂2ζ∂t2 (11) where ζ = ∇ · u/2 is the rotational vector, and the shear-wave velocity, c2, is given by: c2=μρ (12) Analytical methods have been used to solve the governing equations for quasi-static, harmonic, and transient elastographic imaging methods (Love 1929; Sumi et al. 1995; Kallel et al. 1996; Bilgen and Insana 1998) for simple geometries and boundary conditions. However, numerical methods — namely, the finite-element method — are used to solved the governing equations for all three approaches to elastography on irregular domains and for heterogeneous elasticity distributions (Parker et al. 1990; Ponnekanti et al. 1994; Konofagou et al. 1996; Hall et al. 1997; Samani et al. 2001; Van Houten et al. 2001; Miga 2003; McLaughlin and Renzi 2006; Brigham et al. 2007).