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(BQ) Part 2 book Ultrasound imaging and therapy presents the following contents: Diagnostic ultrasound imaging (ultrasound elastography, quantitative ultrasound techniques for diagnostic imaging and monitoring, task based design and evaluation of ultrasonic imaging systems,...), therapeutic and interventional ultrasound imaging.
II Diagnostic Ultrasound Imaging 103 UltArnfdas EElnst Ultrasound Elastography Timothy J Hall, Assad A Oberai, Paul E. Barbone, and Matthew Bayer 4.1 Introduction 104 4.1.1 Brief Clinical Motivation 104 4.1.2 Theory Supporting Quasi-Static Elastography 105 4.1.3 Clinical Implementation of Strain Imaging 107 4.2 Motion Tracking and Strain Imaging 108 4.2.1 Basics 108 4.2.2 Ultrasound Image Formation 109 4.2.3 Motion Tracking Algorithms 109 4.2.4 Strain Imaging 110 4.2.5 Motion Tracking Performance and Error 111 4.2.6 Refinements 113 4.2.7 Displacement Accumulation 114 4.3 Modulus Reconstruction 115 4.3.1 Mathematical Models and Uniqueness 116 4.3.1.1 Linear Elasticity 116 4.3.1.2 Nonlinear Elasticity 117 4.3.2 Direct and Minimization-Based Solution Methods 119 4.3.2.1 Direct Method 119 4.3.2.2 Minimization Method 120 4.3.3 Recent Advances in Modulus Reconstruction 121 4.3.3.1 Quantitative Reconstruction 121 4.3.3.2 Microstructure-Based Constitutive Models 122 4.4 Clinical Applications Literature 124 4.4.1 Breast 124 4.4.2 Other Clinical Applications 125 4.4.3 Conclusions 125 Chapter References 125 Ultrasound Imaging and Therapy Edited by Aaron Fenster and James C Lacefield © 2015 CRC Press/Taylor & Francis Group, LLC ISBN: 978-1-4398-6628-3 104 Ultrasound Imaging and Therapy 4.1 Introduction Ultrasound is a commonly used imaging modality that is still under active development with great potential for future breakthroughs although it has been used for decades One such breakthrough is the recent commercialization of methods to estimate and image the (relative and absolute) elastic properties of tissues Most leading clinical ultrasound system manufacturers offer some form of elasticity imaging software on at least one of their ultrasound systems The most common elasticity imaging method is based on a surrogate of manual palpation There is a growing emphasis in medical imaging toward quantification Ultrasound imaging systems are well suited toward those goals because a great deal of information about tissues and their microstructure can be extracted from ultrasound wave propagation and motion tracking phenomena Estimating tissue viscoelasticity using ultrasound as a quantitative surrogate for palpation is one of those methods This chapter will review the methods used in quasi-static (palpation-type elastography) The primary considerations in data acquisition and analysis in commercial implementations will be discussed Moreover, methods to extend palpation-t ype elastography from images of relative deformation (mechanical strain) to quantitative images of elastic modulus and even the elastic nonlinearity of tissue will also be presented Section 4.4 will review promising clinical results obtained to date 4.1.1 Brief Clinical Motivation Manual palpation has been a common component of medical diagnosis for millennia It is well understood that physiological and pathological changes alter the stiffness of tissues Common examples are the breast self-examination (or clinical breast examination) and the digital rectal examination Although palpation is commonly used, it is known to lose sensitivity for smaller and deeper isolated abnormalities Palpation is also limited in its ability to estimate the size, depth, and relative stiffness of an inclusion or to monitor changes over time Given the long history of successful use of palpation, even with its limits, there was a strong motivation to develop a surrogate that could remove a great deal of the subjectivity, provide better spatial localization, provide spatial context of surrounding tissues, and improve estimates of tumor size and relative stiffness The spatial and temporal sampling provided by clinical ultrasound systems, as well as their temporal stability, make them very well suited to this task The first clinically viable real-time elasticity imaging system was reported in 2001 [1], and significant improvements have been made since then The typical commercial elasticity imaging system provides real-time elasticity imaging with either a side-by-side display of standard B-mode and strain images or a color overlay of elasticity image information registered on the B-mode image (or both options) Some metric of feedback to the user is also often provided so the user knows if the scanning methods are appropriate and/or if the data acquired are high quality Ultrasound Elastograph 105 4.1.2 Theory Supporting Quasi-Static Elastography A basic assumption commonly used in palpation-type elastography is that the loading applied to deform the tissue is quasi-static, meaning that motion is slow enough that inertial effects—time required and inertial mass—are irrelevant Following the presentation of Fung [2], a basic description of the underlying principles can be used to understand the assumptions used in elasticity imaging There are a variety of descriptors of the motion associated with solid mechanics, but a useful one for our purposes is the Cauchy–Almansi strain tensor: ∂u j ∂ui ∂uk ∂uk eij = + − , ∂xi ∂x j ∂xi ∂x j (4.1) ∂u j ∂ui ≈ + = εij , ∂xi ∂x j (4.2) where u(x1, x2, x3, t) is the displacement of a particle instantaneously located at x1, x2, x3, and time t, i = 1, 2, (three-dimensional [3-D] space), and repeat indices imply summation over that index (Einstein’s summation notation) The particle velocity, vi, is given by the material derivative of the displacement as follows: vi = ∂ui ∂u + v j i , ∂t ∂x j (4.3) and the particle acceleration, αi, is similarly defined as follows (replacing particle displacements with particle velocities in Equation 4.3): αi = ∂vi ∂v + α j i ∂t ∂x j (4.4) The conservation of mass is expressed by ∂ρ ∂(ρvi ) + = 0, ∂t ∂xi (4.5) where ρ is the mass density in the neighborhood of point ui, and the conservation of momentum is expressed by ρα i = ∂σ ij + Xi, ∂x j where σij are the stresses and Xi represents body forces (4.6) Chapter 106 Ultrasound Imaging and Therapy Often, the medium of interest may be accurately modeled as a linear elastic and isotropic material In that case, Hooke’s law relating stress to linearized strain is written as follows: σ ij = λε kk δij + 2µεij, (4.7) where λ and μ are the first and the second Lamé constants, respectively Equations 4.1 through 4.7 represent 22 equations with 22 unknowns (ρ, ui, vi, αi, εij, σij, i, j = 1, 2, 3) The tensors εij and σij are both symmetric, so they contain only six independent values each In many elasticity imaging contexts, displacements and particle velocities are sufficiently small in that their products may be neglected With that assumption, Equations 4.1 through 4.6 may be linearized (disregarding products and cross terms of small quantities) Dropping cross terms simplifies the Cauchy–Almansi strain tensor in Equation 4.1 into the infinitesimal strain tensor Equation 4.2 and the material derivatives in Equations 4.3 and 4.4 into simple derivatives Making these substitutions and then further substituting Equation 4.7 into Equation 4.6 yield the well-known Navier equation in solid mechanics In the indicial notation used so far, this equation is presented as follows: µ ∂ 2u j ∂ 2ui ∂ 2u ∂λ ∂u j ∂µ ∂ui ∂u j − + X i − ρ 2i = − + (λ + µ ) + ∂x j ∂x j ∂x i ∂x j ∂xi ∂x j ∂x j ∂x j ∂xi ∂t (4.8) In a homogeneous material, the right-hand side of Equation 4.8 vanishes, which gives the following equation, with the further assumption Xi = 0: µ ∂ 2ui ∂ ∂u j ∂ 2u + (λ + µ ) = ρ 2i ∂x j ∂x j ∂xi ∂x j ∂t (4.9) This equation can also be expressed in a more streamlined vector notation, in which we again drop the body forces for simplicity: µ∇ u + (λ + µ )∇(∇⋅ u ) = ρ ∂2 u ∂t (4.10) Two wave equations may be derived from the Navier equation (Equation 4.10), one for compressional waves and one for shear waves The fundamental theorem of vector calculus (i.e., Helmholtz theorem) states that any smooth nonsingular vector field can be broken into a sum of a divergence-free vector field and a curl-free vector field We call these two components us and uc, respectively, with the subscripts standing for “shear” and “compressional”; thus, the equation is presented as follows: u = us + uc (4.11) Ultrasound Elastograph 107 Equation 4.10 can then be split into two equations, one for each component of the vector field For us, the divergence vanishes, and we obtain µ∇ u s = ρ ∂2u s ∂t (4.12) For uc, we can use the identity ∇(∇ · u) = ∇2u + ∇ · (∇ · u), noting that the curls vanish, to obtain (λ + 2µ )∇ u c = ρ ∂2u c ∂t (4.13) In a heterogeneous medium, the two types of wave fields are coupled Equations 4.12 and 4.13 are wave equations The divergence-free vector field us represents a shear wave (sometimes called an S-wave), with the wave speed dependent on density and on the Lamé constant μ, also known as the shear modulus The curl-free vector field uc represents a compressional wave (sometimes called a P-wave), which is the type of wave emitted and collected by ultrasound imaging devices The compressional wave speed is determined by the density and the sum (λ + 2μ), called the P-wave modulus In soft issues, λ ≫ μ, and hence λ + 2μ ≈ λ ≈ K, the bulk modulus Quasi-static elastography is usually performed with freehand scanning, analogous to other forms of clinical ultrasound imaging Software packages to perform elasticity imaging are now implemented on clinical ultrasound imaging systems from most manufacturers These systems provide images of relative deformation (mechanical strain), which are mapped either to grayscale images (black showing effectively no strain, white showing highest strain in the field), as shown in Figure 4.1, or some other color map, where the latter option can be displayed separately or as an overlay on standard B-mode images A limiting factor for current implementations of quasi-static elastography is the dominance of 1-D array transducers used in clinical ultrasound imaging systems With these transducers, generally only 2-D radio frequency (RF) data fields are available, and from these, only 2-D displacement and strain fields can be estimated This restriction prevents tracking motion perpendicular to the image plane and limits the ability to track a particular volume element in tissue over relatively large (