Liver stiffness measurement techniques: Basics

Soft tissue mechanics

Material stiffness, or elasticity, is a measure of the degree of resistance to elastic deformation in response to an applied force. Stiffness is measured in terms of pressure in Pascals and is calculated as the ratio between the applied stress (applied force per unit area) and resulting strain (change in length per unit length) of a material. Stress and strain are second-order tensor quantities and together completely describe the state of deformation of a material. Elasticity imaging thereby relies upon disturbing soft tissue with an external force and measuring the tissue’s displacement to calculate stiffness as a biomarker for disease. Deriving tissue stiffness in terms of measurable quantities utilized by common imaging techniques follows.

Stress can be represented as

where $\overrightarrow{a}$ is particle acceleration, $\overrightarrow{f}$ represents applied force per unit volume, ρ is material density, and σ _ij is the stress tensor. In elasticity imaging methods, the force is commonly applied using static external compression, acoustic radiation force, or an external vibration mechanism. The material density is commonly assumed to be that of water and is not estimated from the imaging data.

The strain tensor, ${\epsilon }_{\mathrm{ij}}$ , can be represented as

where $\overrightarrow{u}$ represents displacement. Displacement is commonly measured in ultrasonic imaging modalities by correlating time shifts using ultrasonic radio frequency data, phase shifts of in-phase and quadrature (IQ) data ^, at a pulse repetition frequency of several kilohertz.

However, for complex, nonhomogenous, nonlinear materials or for complex applied stress fields (e.g., those generated from focused acoustic radiation force), it can be difficult to derive stress–strain relations that model elasticity under any loading. Soft tissues are viscoelastic (strain dependent on rate of stress application), nonhomogenous, and anisotropic (stress–strain response is orientation dependent). To simplify modeling of soft tissue complexity, many assumptions are made to model linear, elastic behavior under loading used in imaging.

Assuming small strains, soft tissues can be described as linear, elastic solids.

Three moduli are used to characterize the material’s elastic properties: Young’s modulus, E , which describes resistance to deformation uniaxially; shear modulus or rigidity, G or μ , which characterizes resistance to shear; bulk modulus, $K$ , which measures a material’s resistance to compression. Deformation orthogonal to the axis of loading is described by Poisson’s ratio, $v$ . Three linear elastic constitutive equations relate these four constants,

Assuming units of tissue are homogenous and isotropic further allows a single equation to be used to relate stress and strain because isotropic materials behave the same in every direction ; therefore all coefficients of stress and strain tensors as represented in equations (1) and (2) can be represented in terms of material coefficients and their relations in equations (3), (4), and (5). The constitutive equation arises from the derivation of Hooke’s law in three dimensions and is represented as

The two material coefficients, λ where λ = K + 2 μ /3, and μ is also the shear modulus, are known as the Lamé constants.

Elastic materials can support two main modes of wave propagation. In longitudinal or compressional waves, the wave particles oscillate in the direction of wave propagation, whereas in transverse or shear waves, the particles oscillate in the direction normal to the direction of wave propagation. The other principal wave modes, surface and plate waves, are not relevant unless the structures being imaged are considered “thin” (relative to the shear wavelength) or shear waves are incident on a structural boundary (e.g., the edge of an organ). By taking the spatial derivative of equation (7) and decomposing it with respect to the Helmholtz theorem, two wave equations can separately describe the longitudinal and transverse wave propagation, represented respectively as

where $c_L$ and $c_T$ are the longitudinal and shear wave speeds, respectively. These equations can be related to material parameters instead of Lamé constants of soft tissue by

For incompressible materials, ν = 0.5 and (3) reduces to 3 G = E . In soft tissues that have ultrasonic (compressional) wave speeds ranging between 1490–1540 m/s (instead of an infinite compressional wave speed for a truly incompressible material), the factor of 3 relating E and G is >99.9% accurate.

Accurately measuring shear waves is only possible in the far-field as in the near-field coupling between longitudinal and shear waves can occur. In the case of viscoelastic materials like soft tissue, shear wave speed depends on elastic properties. The shear modulus can be represented as

where ρ is roughly constant in soft tissues at 1000 kg/m ³ . Combined with 3 G = E under the assumption of incompressible materials, both Young’s modulus and shear modulus can be represented in terms of shear velocity.

To model the elastic properties of soft tissue in which stiffness is dependent upon strain magnitude and is time-dependent, the elastic model can be coupled with a viscous component to calculate a complex shear modulus. Elasticity is often quantified by the magnitude of $G$ whereas the real and imaginary components can be decomposed to the viscoelastic storage and loss components of deformation, respectively. Viscosity leads to a frequency-dependent shear modulus and can be a source of difference between different elasticity imaging methods.