Marvin M. Doyley, PhD∗m.doyley@rochester.edu and Kevin J. Parker, PhD,     Department of Electrical and Computer Engineering, University of Rochester, Hopeman Engineering Building 343, Box 270126, Rochester, NY 14627, USA

∗Corresponding author.

Elastography visualizes differences in the biomechanical properties of normal and diseased tissues. Elastography was developed in the late 1980s to early 1990s to improve ultrasonic imaging, but the success of ultrasonic elastography has inspired investigators to develop analogs based on MRI and optical coherence tomography. This article focuses on ultrasonic techniques with a brief reference to approaches based on MRI.

Keywords

Elastography

Ultrasonic imaging

Ultrasonic elastography

MRI

Key points

Introduction

Elastography visualizes differences in the biomechanical properties of normal and diseased tissues.1–4 Elastography was developed in the late 1980s to early 1990s to improve ultrasonic imaging,5–7 but the success of ultrasonic elastography has inspired investigators to develop analogs based on MRI8–11 and optical coherence tomography.12–14 This article focuses on ultrasonic techniques with a brief reference to approaches based on MRI.

The general principles of elastography can be summarized as follows: (1) perturb the tissue using a quasistatic, 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.2,15–18 This article describes (1) the general principles of quasistatic, harmonic, and transient elastography (Fig. 1)—the most popular approaches to elastography—and (2) the physics of elastography—the underlying equations of motion that govern the motion in each approach. Examples of clinical applications of each approach are provided.

The physics of elastography

Like conventional medical imaging modalities, forward and the inverse problems are encountered in elastography. The former problems are concerned with predicting the mechanical response of a material with known biomechanical properties and external boundary conditions. Understanding these problems and devising accurate theoretical models to solve them have been an effective strategy in developing and optimizing the performance of ultrasound displacement estimation methods. The latter problems are concerned with estimating biomechanical properties noninvasively using the forward model and knowledge of the mechanical response and external boundary conditions. A comprehensive review of methods developed to solve inverse problems is given in the article by Doyley19; therefore, this section focuses only on the forward problem.

The forward elastography problem can be described by a system of partial differential equations (PDEs) given in compact form20,21:

(1)

where σij is the 3-D stress tensor (ie, 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 (ε) maybe related to the stress tensor (σ) as follows22:

(2)

where the tensor (C) is a rank-four tensor consisting of 21 independent elastic constants.16,20,23 Under the assumption that soft tissues exhibit isotropic mechanical behavior, however, then only 2 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:

(3)

where Θ = ∇ ⋅ u = ε11 + ε22 + ε33 is the compressibility relation, δ is the Kronecker delta, and the components of the strain tensor are defined as:

(4)

Lamé constants (ie, λ and μ) are related to Young modulus (E) and Poisson ratio (v), as follows20,21:

(5)

The stress tensor is eliminated from the equilibrium equations (ie, Equation 2) using Equation 3. The strain components are then expressed in terms of displacements using Equation 4. The resulting equations (ie, the Navier-Stokes equations) are given by:

(6)

where ρ the is density of the material, u is the displacement vector, and t is time. For quasistatic deformations, Equation 6 reduces to:

(7)

For harmonic deformations, the time-independent (steady-state) equations in the frequency domain give10,24:

(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, and z, which gives21:

(9)

where ∇ ⋅ u = Δ, and the velocity of the propagating compressional wave, c1, is given by:

(10)

The wave equation for the propagating shear wave is given by:

(11)

where ζ = ∇ ⋅ u/2 is the rotational vector, and the shear wave velocity, c2, is given by:

(12)

Analytical methods have been used to solve the governing equations for quasistatic, harmonic, and transient elastographic imaging methods25–28 for simple geometries and boundary conditions. Numeric methods—namely, the finite-element method—are used, however, to solve the governing equations for all 3 approaches to elastography on irregular domains and for heterogeneous elasticity distributions.24,29–36

Approaches to elastography