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Model-based elastography: a survey of approaches to the
inverse elasticity problem
M M Doyley
University of Rochester, Department of Electrical and Computer Engineering, Hopeman
Engineering Building 413, Box 270126, Rochester, NY 14627, USA m.doyley@rochester.edu
Abstract
Elastography is emerging as an imaging modality that can distinguish normal versus diseased
tissues via their biomechanical properties. This article reviews current approaches to elastography
in three areas — quasi-static, harmonic, and transient — and describes inversion schemes for each
elastographic imaging approach. Approaches include: first-order approximation methods; direct
and iterative inversion schemes for linear elastic; isotropic materials; and advanced reconstruction
methods for recovering parameters that characterize complex mechanical behavior. The paper’s
objective is to document efforts to develop elastography within the framework of solving an
inverse problem, so that elastography may provide reliable estimates of shear modulus and other
mechanical parameters. We discuss issues that must be addressed if model-based elastography is
to become the prevailing approach to quasi-static, harmonic, and transient elastography: (1)
developing practical techniques to transform the ill-posed problem with a well-posed one; (2)
devising better forward models to capture the transient behavior of soft tissue; and (3) developing
better test procedures to evaluate the performance of modulus elastograms.
1. Introduction
Elastography is an emerging imaging modality that exploits differences in the biomechanical
properties of normal and diseased tissues (Krouskop
et al
., 1998; Samani
et al
., 2007;
Sarvazyan
et al
., 1995; Parker
et al
., 2011). Several groups have investigated the diagnostic
value of elastography in various clinical settings; these include detecting and characterizing
atherosclerotic plaques (de Korte
et al
., 2002; de Korte
et al
., 2000; Doyley
et al
., 2001;
Brusseau
et al
., 2001; Woodrum
et al
., 2006); guiding minimally invasive therapeutic
techniques (Kallel
et al
., 1999; Righetti
et al
., 1999; Varghese
et al
., 2003); and improving
the differential diagnosis of breast and prostate cancers (Hiltawsky
et al
., 2001).
Elastography was developed in the late 1980s to early 1990s to improve the diagnostic value
of ultrasonic imaging (Lerner and Parker, 1987; Lerner
et al
., 1988; O’Donnell
et al
., 1994;
Ophir
et al
., 1991), but the success of ultrasonic elastography has inspired other
investigators to develop analogues based on other imaging modalities; these include
magnetic resonance elastography (Muthupillai
et al
., 1995; Bishop
et al
., 2000; Weaver
et
al
., 2001; Sinkus
et al
., 2000), and optical coherence tomography elastography (Khalil
et al
.,
2005; Kirkpatrick
et al
., 2006; Ko
et al
., 2006).
Although current approaches to elastography vary considerably, we can summarize the
general principles of elastography as follows: (1) perturb the tissue using a quasi-static,
harmonic, or transient mechanical source; (2) measure the internal tissue displacements
using a suitable ultrasound, magnetic resonance, or optical displacement estimation method;
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and (3) infer the mechanical properties from the measured mechanical response, using either
a simplified or continuum mechanical model. Several review articles provide a
comprehensive overview of different approaches to elastography (Bamber
et al
., 2002;
Greenleaf
et al
., 2003; Manduca
et al
., 1998; Ophir
et al
., 2000; Parker
et al
., 2011). This
article provides a brief description of current approaches. To simplify our discussion, we
classify current approaches to elastography into three groups: quasi-static, harmonic, and
transient; and review a selection of inverse reconstruction schemes (methods for
reconstructing shear modulus, viscosity, nonlinearity, and anisotropy within soft tissues) that
have been proposed for each elastographic-imaging approach. The primary goal of this
review is to document the efforts made by several groups, including that of the author, to
develop elastography within the framework of solving an inverse problem (i.e., model-based
elastography), a strategy that is beginning to transform elastography from an imaging
modality that provides only an approximate estimate of shear modulus to one that can
provide reliable estimates of shear modulus and other mechanical parameters — namely
viscosity, anisotropy, poro-elasticity, and nonlinearity.
This paper surveys elastographic approaches from the simplest to the most complex and
computationally intensive, and is organized into six sections. Section 2 examines the basic
first-order approximation methods employed in earlier work. Section 3 covers the direct (or
forward) problem in elasticity imaging. Section 4 then considers the major categories of
inversion schemes under linear elastic, isotropic models. Section 5 reviews approaches to
more complex models including viscoelastic, poroelastic, nonlinear, and anisotropic
behaviors. Finally, section 6 describes issues that still remain to be addressed.
2. First order approximation to shear modulus
Several groups have developed approaches for obtaining approximate estimates of shear
modulus, and despite their limited accuracy these techniques are fast and robust — traits that
make them clinically appealing. In this section, we review the methods that have been
proposed for obtaining approximate estimates of shear modulus in quasi-static, harmonic,
and transient elastography. Figure 1 provides a pictorial representation of all three
approaches to elastography.
2.1 Quasi-static elastography based on stress uniformity
Ophir
et al
. (1991) proposed a quasi-static method that is arguably the most established
approach to elastography. Quasi-static elastography was originally developed as an
ultrasound imaging technique (O’Donnell
et al
., 1994; Ophir
et al
., 1991; Bamber and Bush,
1995), but a magnetic resonance (MR) analogue was later described (Plewes
et al
., 2000;
Fowlkes
et al
., 1995). Quasi-static elastography measures the axial strain induced within the
tissue using either an external or internal source. A small motion is induced within the tissue
(typically on the order of 2 % of the axial dimension) with a quasi-static mechanical source;
the axial component of the internal tissue displacement is then measured, usually by
performing cross-correlation analysis on pre- and post-deformed radio-frequency (RF) echo
frames in the time or frequency domain; and strain elastograms are produced by spatially
differentiating the axial component of displacement, using either a finite difference or a least
squares strain estimator (Kallel and Ophir, 1997). In quasi-static elastography, soft tissues
are typically envisioned as a series of one-dimensional springs that are arranged in a simple
fashion. For this simple mechanical model, the measured strain (
ε
) is related to the internal
stress (
σ
) as follows (Hooke’s Law):
(1)
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where
E
is the Young’s modulus of the tissue. Currently, no method can measure the
internal stress distribution
in vivo
; consequently, in quasi-static elastography, the internal
stress distribution is typically assumed to be constant (i.e.,
σ
≈ 1). Using this assumption, an
approximate estimate of Young’s modulus is computed from the reciprocal of the measured
strain, but the disadvantage of computing modulus elastograms in this manner is that stress
concentration and target hardening artifacts (Konofagou
et al
., 1996; Ponnekanti
et al
.,
1994) may compromise the diagnostic quality of the ensuing images. In spite of this
limitation, Kallel
et al
. (1998) demonstrated that this approach can provide a good relative
estimate of the shear modulus of low-contrast focal lesions, in cases for which uniform
stress is induced within the imaging field of view. Several groups have adopted this
approach to obtain reasonable relative estimates of shear modulus for which accurate
quantification of shear modulus is not essential; these include guiding minimally invasive
therapeutic techniques, and detecting abnormalities in several organs such as the breast,
prostate, and liver tissue. Figure 2 shows an example of an approximate modulus elastogram
computed using the assumption of stress uniformity; the strain images were filtered using
the spatial filter described in (Doyley
et al
., 2005) prior to inversion.
2.2 Harmonic elastography base ed on local frequency estimation
Like its quasi-static counterpart t, harmonic elastography was first proposed as an ultrasound
imaging method (Lerner and Parker, 1987; Lerner
et al
., 1988; Parker
et al
., 1990;
Yamakoshi
et al
., 1990), but was later extended to magnetic c resonance imaging (MRI)
(Muthupillai
et al
., 1995), and is now the prevailing approach to magnettic resonance
elastography (MRE). In harmonic elastography, a low-frequency acoustic wave (typiccally <
1 kHz) is transmitted within the tisssue using a sinusoidal mechanical source. The phase and
a amplitude of the propagating waves are visuualized using either color Doppler imaging
(Parker
et al
., 1990; Lerner
et al
., 1990; Yamakoshi
et al
., 1990) or phase-contrast MR
imaging (Muthupillai
et al
., 1995; Sinkus
et al
., 2000; Weaver
et al
., 2001). Assuming that
shear waves propagate with plane wavefronts, then an approximate estimate of the local
shear modulus (
μ
) may be computed from local estimates of o the wavelength as follows:
(2)
where
c
2 is the velocity of the shear wave, and
ρ
is the density of the tissue. Manduca et al.
(2001) showed that in a homogeneous tissue, shear modulus can be estimated from local
estimates of instantaneous frequency, which they computed with a bank of wavelet filters as
described in (Knutsson
et al
., 1994). Wu
et al
. (2006) used a similar approach to compute
shear modulus images with sonoelastography (i.e., the ultrasound analogue to MRE).
Although this shear-modulus estimation approach is relatively insensitive to measurement
noise, the spatial resolution of the ensuing modulus elastograms is limited. A further
weakness of the approach is that the plane-wave approximation breaks down in complex
organs, such as the breast and brain, when waves reflected from internal tissue boundaries
interfere constructively and destructively.
2.3 Transient elastography based on arrival time estimation
A major limitation of harmonic elastography is that shear waves attenuate rapidly as they
propagate within soft tissues, which limits the depth of penetration. To overcome this
limitation, Sarvazyan
et al
. (1998) proposed a transient approach to elastography that uses
the acoustic radiation force (ARF) of an ultrasound transducer to perturb tissue locally.
Nightingale
et al
. (2003) were the first to implement this technique on a clinical ultrasound
scanner to assess the viscoelastic properties of the liver. A MR analog has been reported in
(McCracken
et al
., 2004; Souchon
et al
., 2008). Bercoff
et al
. (2004) also developed a
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transient approach to elastography that they call supersonic shear-wave imaging (SSI),
which is rapidly becoming the most established approach to ultrasonic elastography. This
elastographic imaging method uses an ultrasound scanner with an ultra-high frame rate (i.e.,
10,000 fps) to track the propagation of shear waves. As in harmonic elastography, local
estimates of shear modulus are estimated from local estimates of wavelength via equation
(2) in transient elastography. However, the reflections of shear waves at internal tissue
boundaries make it difficult to measure shear-wave velocity. Ji
et al
. (2003) proposed to
overcome this limitation by computing wave speeds directly from the arrival times.
McAleavey
et al
. (2009) proposed to measure the shear modulus distribution within soft
tissues using a technique known as spatially modulated ultrasound radiation force (SMURF)
imaging. This elastographic imaging technique uses radiation force to generate a shear wave
of known spatial frequency, and then measures the temporal frequency response of the
vibrating tissue as the wave propagates past a point.
Fatemi and Greenleaf (1998) proposed a technique known as vibroacoustography that uses
radiation force to vibrate tissues in the kHZ range, by using two overlapping ultrasound
beams with slightly different frequencies. The resulting tissue mechanical response is
dependent on the local acoustic mechanical properties of tissue that are obtained using a
hydrophone. Using this technique, Fatemi and Greenleaf demonstrated that
vibroacoustography could visualize microcalcification with high contrast resolution.
3. Solving the direct elasticity problem
To solve the inverse elasticity problem, we need an accurate model to predict the strains
and/or displacements incurred when a tissue of known biomechanical properties and
boundary conditions is perturbed using a pseudo-static, harmonic or transient mechanical
source (i.e., solving the direct problem). Solving the direct problem provides the
computational basis for solving the inverse problem: estimating the mechanical properties
(i.e., the unknowns) from the measured mechanical responses. The advantage of this
approach to elastography is that both the direct and inverse problems are formulated from
well-established physical laws, which provide equations that relate the biomechanical
properties (namely shear modulus, Poisson’s ratio, anisotropy, viscosity, non-linearity, and
poroelasticity) to the measured mechanical response.
3.1 Equations of motion
We can derive a system of partial differential equations (PDEs) from the conservation of
linear momentum to describe the direct elasticity problem. These equations are given in
compact form by (Timoshenko and Goodier, 1970; Fung, 1981):
(3)
where
σ
ij is the three-dimensional stress tensor (i.e., a vector of vectors), f
i
is the deforming
force, and ▽ is the del operator. Assuming that soft tissues exhibit linear elastic behavior,
which is valid for infinitesimal deformations, then we can relate the strain tensor (
εkl
) to the
stress tensor (
σij
) as follows (Landau
et al
., 1986):
(4)
where the tensor
C
is the Christoffel rank-four tensor consisting of 21 independent elastic
constants (Greenleaf
et al
., 2003; Ophir
et al
., 1999; Fung, 1981). Developing methods to
estimate all 21 elastic parameters is not trivial; therefore, several assumptions are used to
reduce the number of independent elastic constants. The most common assumption, and
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perhaps the simplest, is that soft tissues exhibit linear, purely elastic isotropic mechanical
behavior (i.e., that soft tissues behave like Hookian material). In this case, only two
independent constants,
λ
( the second Lamé constant) and
μ
(shear modulus), are required to
describe the mechanical behavior of soft tissues. However, this assumption is clearly not
true for some tissues, as discussed in Section 5. The constitutive relation that describes the
relationship between stress and strain for linear isotropic elastic materials is given by:
(5)
where
δij
is the Kronecker delta, Δ = △ ·
u
= ε11 +ε22 +ε33 is the compressibility relation,
and the components of the strain tensor are defined as:
(6)
where and
uj
are the displacement components in the Cartesian coordinates
xi
Lamé
constants (i.e.,
λ
and
μ
) are related to traditional engineering constants, such as Young’s
modulus (
E
) and Poisson’s ratio (
v
), as follows (Timoshenko and Goodier, 1970; Fung,
1981):
(7)
Since stress cannot be measured
in vivo
, it is typically eliminated from the equilibrium
equations (i.e., equation (3)) using equation (5), and the strain components can be expressed
in terms of displacements, using equation (6). The resulting equations of equilibrium (i.e.,
the Navier-Stokes equations) are given by:
(8)
where
ρ
is density is the density of the material, and
t
is time. For quasi-static deformations,
equation (8) reduces to:
(9)
For harmonic deformations, the displacement field is assumed to have a time harmonic form
that is given by (Sinkus
et al
., 2000; Van Houten
et al
., 2001):
(10)
where 𝕽is the real component of the time harmonic displacement. The time-independent
(steady-state) equations in the frequency domain give:
(11)
where
ω
is the angular frequency of the sinusoidal excitation. For transient deformations, the
wave equation is derived by differentiating equation (8) with respect to
x,y,z
, which gives
the following result (Timoshenko and Goodier, 1970):
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(12)
where the velocity of the propagating compressional wave,
C
1, is given by:
(13)
To derive the wave equation for the propagating shear wave, an operation of curl is
peformed on equation (8), which gives:
(14)
where
ζ
= △ × u /2 is the rotational vector, and the shear-wave velocity,
c
2, is given by:
(15)
3.2 Numerical solution of governing equations
The equations that describe the direct problem for quasi-static, harmonic and transient
elastographic imaging methods have been solved using analytical methods (Bilgen and
Insana, 1998; Kallel
et al
., 1996; Love, 1929; Sumi
et al
., 1995a) for simple geometries and
boundary conditions. However, it is more useful to solve these equations on irregular
domains for elastically heterogeneous tissue, which is difficult to perform using analytical
methods. Consequently, investigators have employed numerical methods — namely, the
finite-element method (McLaughlin and Renzi, 2006; Parker
et al
., 1990; Ponnekanti
et al
.,
1994; Van Houten
et al
., 2001; Samani
et al
., 2001; Miga, 2003; Konofagou
et al
., 1996;
Hall
et al
., 1997; Brigham
et al
., 2007), and the finite-difference method (Raghavan and
Yagle, 1994; Sinkus
et al
., 2000; O’Donnell
et al
., 1994) — to solve the governing equations
for all three approaches to elastography. However, the finite element method (FEM) is
currently the most popular approach for solving PDEs, which is not surprising because: (1) it
analyzes structures with complex geometries and boundary conditions more easily than
other numerical methods; and (2) several powerful FEM packages are commercially
available (such as ANSYS, MARC, COMSOL, Abaqcus, and NASTRAN).
A finite-element representation of the governing PDEs involves four steps. First, the
geometry of the tissue is segmented into a series of finite elements, through a process known
as mesh generation. The design of efficient, two- and three-dimensional mesh generators is
an area of active research (Geuzaine and Remacle, 2009; Pinheiro
et al
., 2008;
Triantafyllidis and Labridis, 2002). However, the main requirement for an efficient mesh
generator is that it should be capable of meshing an object that is composed of both smooth
and irregular surfaces (Lionheart, 2004). Second, a weak form of the governing PDEs is
derived, using either the variational or the weighted residual method (Reddy, 1993; Cook
et
al
., 1989). Third, a basis or shape function is substituted in the derived equation to produce a
system of linear algebraic equations, which has the following form:
(16)
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where the matrix [K] is the global stiffness or coefficient matrix, f is the global force vector,
and u is the vector of unknown displacements. The final step involves imposing the external
boundary conditions associated with equation (16) before solving the resulting equations.
3.2.1 Weak form—The weak form is a weighted-integral statement that is equivalent to
the governing partial differential equation and the natural boundary conditions of the
problem. The weak form of the governing elasticity equation for each approach to
elastographic imaging can be derived as described in (Reddy, 1993). For simplicity, we will
describe only the finite element implementation for the harmonic case (i.e., equation (8)),
since the procedure is identical for the quasi-static and transient cases. The weak form of
equation (8) with a scalar weighting function ϕ
i
(
x
,
y
,
z
) is given by:
(17)
where represents an integration over the boundary ( Γ) of the element, ϕ
i
is a scalar basis
associated Γ with the element, n̂ represents the outward-pointing normal vector, and t̂
represents the Nuemann boundary condition. If the scalar components of the displacement
vector, u, are
u
(
x
,
y
,
z
),
v
,(
x
,
y
,
z
),
w
(
x
,
y
,
z
) in the x,y, and z directions, respectively, then the
Galerkin approximation of the displacement may be derived by expanding the scalar
components in the ϕ basis to give:
(18)
where
N
is the number of nodes associated with each element in the finite element mesh and
uj
,
vj
,
wj
are the axial, lateral, and elevational displacement components at each element
node. The Galerkin weak-form finite element model is obtained by substituting equation
(18) into equation (17):
(19)
where f is the 3
N
× 1 global force vector and
K
(
μ
,
λ
,
ρ
) that is given by
(20)
where the coefficients of the global stiffness matrix are determined by the basis function and
the material properties (i.e.,
μ
and
λ
).
3.2.2 Dimensionality reduction using the plane-strain and plane-stress
approximation—Despite the technological advances in 3D ultrasound imaging, ultrasound
is predominately a two-dimensional imaging modality. Thus, most investigators in
ultrasonic imaging typically reduce the 3D elasticity problem to a two-dimensional problem,
using either a plane-strain or plane-stress approximation. To illustrate the differences
between the two approximations, let’s consider a linear elastic 1 1 solid, Ω, of uniform
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thickness,
h
, bounded by two parallel planes ( and ) and a closed 2 2
boundary, Γ. If the thickness of
h
≫Ω, then the problem can be considered a plane-strain
problem. In this case, we assume that
ε
13 =
ε
23 =
ε
33 = 0. The plane-strain approximation is
typically used when structures above and below the ultrasound scan plane have motion
confined to the elevation (z) direction (Kallel
et al
., 2001; Doyley
et al
., 2000; Skovoroda
et
al
., 1995; Barbone and Bamber, 2002). However, when
h
≪Ω, the problem is considered a
plane-stress problem. In this case, we assume that
σ
13 =
σ
23 =
σ
33 = 0). The plane-stress
approximation generally applies to thin plates; however, (Sumi
et al
., 1995b) used a plane-
stress approximation to reduce the 3D elasticity problem to 2D. In reality, the plane-strain
and plane-stress assumptions are valid only for special cases such as phantoms with
cylindrical inclusions, or when elastography is performed using the constrained imaging
method described in (Kallel and Ophir, 1997). Therefore, errors are typically incurred when
the 3D elasticity problem is modeled using either approximation.
4. Computing shear modulus by solving the inverse elasticity problem
Several groups have proposed inversion schemes for computing the mechanical properties
within soft tissues. Figure 3 summarizes the inversion schemes proposed for the three
different approaches to elastography. These inversion schemes were formulated based on the
premise that soft tissue behaves like an Hookian material (i.e., it behaves like a linear, purely
isotropic material).
4.1 Quasi-static elastographic inversion schemes
4.1.1 Direct inversion—Raghavan and Yagle (1994) proposed a direct inversion scheme
for recovering shear modulus. They derived a linear system of equations by re-arranging the
PDEs that describe the direct problem for the plane-strain condition. The PDEs that
(Raghavan and Yagle, 1994) derived are given by:
(21)
where , the unknowns are shear modulus (
μ
) and hydrostatic pressure (
p
)1, and the
coefficients are functions of the internal tissue strains that are related to the measured
displacements (see equation (6)). The weakness of this approach is that both the shear
modulus (
μ
) and the hydrostatic pressure (
p
) on the boundary must be known to solve
equation (21).
Although Rhagavan and Yagle’s computer simulation demonstrated that this inversion
scheme could produce very encouraging displacements, they observed that the performance
of the reconstructed elastograms degraded rapidly with increasing measurement noise.
Another weakness of this inversion scheme is that no imaging system to date can measure
hydrostatic pressure. To overcome this limitation, (Skovoroda
et al
., 1995) used an
analytical method to eliminate the pressure term from equation (21). Eliminating the
pressure term gives:
(22)
As in equation (21), the shear modulus must be known on the boundary of the region of
interest (ROI) to solve equation (22). Skovoroda
et al
. (Skovoroda
et al
., 1995) were the first
1Rhagavan (1994) used the form of Hooke’s law (i.e.,
σij
=
pδij
+ 2
μεij
), which includes a hydrostatic pressure term,
p
.
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to demonstrate that this information could be obtained by exploiting the stress-continuity
properties of soft tissues. They demonstrated that contours of small shear modulus variations
could easily be defined, and that the modulus along the boundary could be computed by
processing axial strain elastograms as follows:
(23)
where
μr
is the relative shear modulus, and and are the axial strains within and
outside the boundary of ROI. Equation (22) contains high-order derivatives that will amplify
measurement noise, which could compromise the quality of ensuing modulus elastograms as
demonstrated in figure 4. Despite this potential limitation,, the viability of this inversion
scheme has been demonstrated in gelatin and
ex vivo
kidney phantoms (Skovoroda
et al
.,
1999; Skovoroda and Aglyamovv, 1995; Chenevert
et al
., 1998). Skovoroda
et al
. (1999)
cast the inverse problem as an integral rather than a differential form — an approach that
was first described in (Sumi
et al
., 1995b) — to make the technique less susceptible to
measurement noise. To bolster performance further, they computed lateral displacement
using the incompressible method described in (Lubinski
et al
., 1996).
Bishop
et al
. (2000) proposed to eliminate the pressure term appearing in equation (21) by
partitioning the matrix, rather than by doing this analytically as described in (Skovoroda
et
al
., 1995). But the resulting formulation proved to be ill-conditioned. Consequently, (Bishop
et al. a
, 2000) constrained the solution by employing the Tikhonov regularization method.
Sumi
et al
. (1995b) also proposed a direct inversion scheme; however, they solved the
inverse problem for the plane-stress case, using the following PDEs:
(24)
where the unknowns are spatial derivatives of relative Young’s modulus, and the
coefficients are strains and their spatial derivatives. They computed the shear modulus at a
given point (
x
,
y
) within the tissue relative to a reference point —let’s say: (
A
,
B
) — by
employing a line integral. The feasibility of this inversion scheme has been demonstrated in
phantoms and excised tissues (Sumi, 2007; Sumi and Nakayama, 1998); nevertheless, the
plane-stress condition is typically not relevant for most clinical applications. To make the
technique more clinically relevant, (Le Floc’h
et al
., 2009) extended the concept to the
plane-strain case. The equations they derived for solving the inverse problem for the plane-
strain condition are given in compact form as follows:
(25)
Although theoretically feasible, equation (28) is difficult to solve, because cannot be
measured in practice. However, (Le Floc’h
et al
., 2009) demonstrated that the second term
could be used to highlight the boundaries of different tissue types.
4.1.2 Iterative Inversion—The inverse problem can also be viewed as a parameter-
optimization problem, where the goal is to find the shear modulus that minimizes the error
between measured displacement or strain fields, and those computed by solving the direct
problem. This inversion approach has been successfully used in several emerging medical
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imaging modalities, as discussed in (Paulsen
et al
., 2005; Yorkey
et al
., 1987), and was
applied to elastography by (Kallel and Bertrand, 1996; Doyley
et al
., 1996). The objective
function that is minimized typically has the following form:
(26)
where u is the solution to the direct problem computed from the shear modulus distribution,
μ
, by solving equation (9). Minimizing equation (29) with respect to shear modulus
variations is a nonlinear process; however, we can do this iteratively using a variety of
techniques as discussed in (Vogel, 2002). Proposed optimization techniques can be
classified into three broad categories: (1) those that require only the value of the functional
π
for different values of the parameters
μ
(i.e., non-gradient approaches); (2) those that
require the value of the functional and its derivative with respect to the material parameters
(called the gradient vector); and (3) those that require the first and second derivative of the
functional with respect to the material parameters (called the Hessian matrix). All three
optimization methods are illustrated below.
(a) Hessian-based optimization method: The Gauss-Newton method is perhaps one of the
most established optimization methods. Minimizing equation (26) using the Gauss-Newton
method produces a matrix solution at the (i+1) iteration that has the general form:
(27)
where T denotes the transpose; I denotes an Identity matrix; Δ
μi
is a vector of shear
modulus updates at all coordinates in the reconstruction field; and
J
(
μ
i
) is the Jacobian, or
sensitivity, matrix. The Hessian matrix, [
J
(
μ
i
)
T
J
(
μ
i
)], is usually ill conditioned; therefore,
to stabilize performance in the presence of measurement noise, the matrix is regularized
using one of three variational methods: the Tikhonov (Kallel and Bertrand, 1996), the
Marquardt (Doyley
et al
., 2000), or the total variational method (Richards
et al
., 2009; Jiang
et al
., 2009).
Moulton
et al
. (1995) computed the JacobIan matrix [
J
] column-wise, where each column
represents the difference between displacement computed when the direct problem was
solved, once for a given shear modulus distribution and then again when the shear modulus
of a single node, or element, was perturbed by unity. However, computing the Jacobian
matrix in this fashion is very demanding. Kallel and Bertrand (1996) proposed a more
efficient approach, which involved computing the derivative of the forward problem with
respect to shear modulus at a given node — let’s say j — as follows:
(28)
The global stiffness matrix on the left-hand side of equation (28) requires factorization, and
is the same matrix used to solve the direct problem at the previous iteration. Therefore, all
that is required to compute each is a simple matrix back-substitution.
Solving the inverse problem using displacement boundary conditions (DBC) will provide
only relative estimates of shear modulus. Doyley
et al
. (2000) demonstrated that to recover
absolute values the pressure on the boundary must be known.
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Rather than developing a custom optimization code, an increasing number of investigators
have demonstrated that clinically useful elastograms can be computed using the FMINCON
algorithm, which is part of MATLAB optimization toolbox (Jiang
et al
., 2009; Baldewsing
et al
., 2005a; Le Floc’h
et al
., 2009). The advantage of using FMINCON is that it doesn’t
require the global stiffness matrix and force vector, and thus, allows the implementation of
the inverse problem based on commercially available finite-element codes. Jiang
et al
.
(2009) developed an edge-preserving, iterative, inverse-reconstruction approach based on
FMINCON. They used this algorithm to demonstrate the feasibility of using model-based
elastography to guide and monitor radio-frequency (RF) ablation. Figure 5 shows an
example of modulus elastograms computed using this reconstruction approach.
The boundary between the ablated and normal tissue was better delinated in the modulus
elastograms than the strain elastograms, which provided a better estimate of the extent of the
thermal zone.
Miga
et al
. (2003) proposed a novel approach to elastography that they refer to as modality
independent elastography (MIEE) which is based on a combination of the finite element
method, the Gauss-Newton iterative scheme, and image similarity. In this technique the
objective function that is minimized has the following form:
(29)
where
STRUE
is similarity values computed when the target image is compared to itself,
S
(
E
)
EST
is the similarity between the target and model-deformed source image using the
current estimate of Young’s. Minimizing equation (29) with the Gauss Newton iteration
scheme gives the following property updates:
(30)
where the Jacobian matrix.
Ou
et al
. (2008) demonstrated that 3D MIE was able to successfully reconstruct modulus
elastograms using data obtained from magnetic resonance and x-ray computed systems.
(b) Gradient-based optimization method: Although the gradient can be computed using
the method described in (Kallel and Bertrand, 1996), the computational expense required to
compute the Jacobian matrix (i.e., the gradient) increases in proportion to the number of
parameters. Oberai
et al
. (2003) were the first to demonstrate that this limitation could be
circumvented by using the adjoint method to compute the gradient of the objective function.
To do this, they added a scalar, L, (i.e., the Lagrangian) to equation (26) to produce a new
objective function, which was defined as:
(31)
where w is the adjoint displacement field (i.e., a vector of Lagrangian multipliers). Note that
L
is a function of u, w and . Equations (26) and (31) are equivalent (i.e., when w
T
(
K
u –
f )=0 is satisfied for an arbitrary value of w. The change in
L
denoted by δ
L
, due to small
changes in u, w and , denoted by δ u,δ w and , is given by:
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(32)
Since equation (32) is valid for any u and w, these vectors were chosen so that the terms
multiplying δ u and δ w were zero, which gives:
(33)
With these choices, δ
L
reduces to:
(34)
Since
δπ
=
δL
this yields:
(35)
where the terms in the parentheses are equal to the gradient vector, G . Computing the
gradient vector requires only two forward solves: (1) to compute w i.e., solving equation
(33), and (2) to compute u.
(c) Gradient-free optimization methods: In principle, we can use the generalized Hooke’s
law to compute shear modulus directly from the axial strain; however, the principal stress
components cannot be measured
in vivo
, an issue that (Ponnekanti
et al
., 1995) attempted to
solve using the analytic method described in (Love, 1929). Given the limitations of
computing stress using analytical models, Doyley
et al
. (1996) and then later Samani
et al
.
(2001), used the finite-element method to compute the principal components of the stress
tensors iteratively as follows (Samani
et al
., 2001; Doyley
et al
., 1996; Ponnekanti
et al
.,
1995):
(36)
where is the measured strain, ν is Poisson’s ratio, and
σ
11,
σ
22, and
σ
33 are normal stress
tensors. More specifically, modulus is assumed to be constant at the start of the
reconstructive process. The modulus is updated by combining the measured axial strain
with the principal stress components (i.e.,
σ
11,
σ
22, and
σ
33) that were computed by solving
the direct problem with the current estimate of Young’s modulus. Samani et al. (2001) were
the first group to demonstrate that imposing geometrical constraints enhanced the
performance of modulus elastograms. More specifically, they showed that dividing the
reconstruction field of view (ROI) into segments based on anatomical features, then
computing the average Young’s modulus over each segmented region, produced more stable
elastograms.
Baldewsing
et al
. (2006) applied this technique to reconstructive intravascular ultrasound
elastography. More specificially, they segmented strain elastograms into different segments
using deformable curves, and reconstructed the shear modulus within each segment using a
combination of FMINCON and Sepran (Sepra Analysis, Technical Univeristy Deflft, The
Netherlands)(Baldewsing
et al
., 2005b) finite element packages. They demonstrated the
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feasibility of their technique using simulated and thin-cap fibroatheromas (TCFAs) as well
as to
in vivo
and
in vitro
human coronary plaques (Baldewsing
et al
., 2005b).
Le Floc’h
et al
. (2009) also constrained their inversion approach using structural information
which they obtained by segmenting radial strain obtained from coronary arteries using
equation (25). Figure 6 shows examples of modulus elastograms recovered from excised
tissues using their technique.
Stochastic methods, such as with genetic algorithms (GA) have also been proposed for
solving the inverse problem (Zhang
et al
., 2006; Khalil
et al
., 2005). Stochastic methods can
find the global minimum of the objective function without imposing any additional
constraints; therefore, this reconstruction approach strategy may prove useful.
4.2 Harmonic elastography inversion schemes
Harmonic elastography can produce absolute modulus elastograms when all components of
internal tissue motion are readily available. Consequently, this method is the most
established MR elastographic imaging approach. Like their quasi-static counterpart,
modulus elastograms are reconstructed using either direct or iterative inversion schemes.
Harmonic elastography can measure modulus directly from shear speed estimates via
equation (2). However, measuring shear speed in organs with complex geometries or when
the propagating wave is reflected internally is challenging as discussed in section 2.2.
Nevertheless using equation (8), we can properly account for complex geometries and edge
reflection given appropriate discretization of the solution domain. Consequently, a variety of
model-based inversion methods have been propose for harmonic MRE.
4.2.1 Direct inversion—Skinus
et al
. (2000) solved the inverse harmonic elastography
problem using a linear system of PDE that they derived by solving the wave propagation
model described in (Landau
et al
., 1986) in the frequency domain, and expressed Young’s
modulus as a symmetric tensor. Lorenzen
et al
. (2003) used this inversion scheme to
demonstrate that MRE can detect changes in breast tissue elasticity during the monthly
hormonal cycle. Using the medical standard, the first day of menses is counted as day 1 in
the woman’s cycle. Using MRE, (Lorenzen
et al
., 2003) showed that on day 5, the median
value of elasticity for fibroglandular adipose tissue declined significantly, but at day 14, the
same tissue’s elasticity increased noticeably.
Manduca
et al
. (2001) proposed to solve the inverse harmonic elastography problem using a
direct inversion scheme, which they referred to as algebraic inversion of differential
equation (AIDE). By assuming local homogeneity, the equations were solved separately at
each pixel using only data from a local neighborhood to estimate local derivatives as
described in (Oliphant
et al
., 2001). Very encouraging phantoms and patient results have
been obtained using this technique. However, the large difference in magnitude of shear
modulus and the second Lamé coefficient (i.e., kPa for shear modulus and GPa)prevents
simultaneous estimation of
μ
and
λ
, Manduca
et al
. (2001) assumed that the divergence of
the displacement vector in equation (8) was negligible (i.e., ∇ ·
u
≈ 0). Using this
assumption,
μ
was estimated from a single component of motion as follows:
(37)
Romano
et al
. (1998) developed a direct-inversion method by using the variational or weak-
form of equation (8), and appropriately chosen test functions to estimate both Lamé
constants. The advantage of using the weak form of equation (8) is that the derivative is
calculated not from the measured data, but rather from a smooth test function.
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Park and Maniatty (2006) also described a direct-inversion scheme for reconstructing shear
modulus from displacements measured during time-harmonic excitation. Their approach
assumed that the time-harmonic displacements were divergence-free, which reduces the
governing equation to:
(38)
If the gradient of the shear modulus is also neglected, equation (38) reduces to the
Helmholtz equation.
4.2.2 Iterative inversion method—Like its quasi-static counterpart, shear modulus can
also be computed by treating the image-reconstruction problem as a parameter-optimization
problem. However, to accosmplish this all component of the internal tissue displacement
must be known. Using computer simulation, Van Houten
et al
. (2001) demonstrated that the
displacement fields of an oscillating, 3D, isotropic, linear elastic body was not acurately
characterized using 2D plane approximation. Consequently, no ultrasonic methods have
been proposed to solve the inverse-harmonic-elastographic problem because without
symmetries, the three dimensional case cannot be approximated.
The objective function to be minimized is identical to that used in the quasi-static
elastography equation (29). However, the computational overhead required to solve the full
3D elasticity problem at the resolution MR data set with the Hessian method would make
the computations infeasible on contemporary processors. For example, if a typical MR
displacement data set were discretized to 16 × 256 × 256 image slices, the corresponding
parameter set matching the MR resolution would have over a million elements (assuming a
description based on single parameter of elasticity). Each parameter update for this large
property description would require over ≈ 1
e
18 floating-point operations to invert the
Hessian matrix, and an additional 1
e
18 operations to generate the matrix beforehand. Van
Houten
et al
. (2001) demonstrated that this issue can be circumvented by dividing the
reconstruction field-of-view into a series of overlapping subzones, and expanding equation
(29) as a sum over all the subzones as follows:
(39)
where u
z
(
μz
) represents the displacements on the zth subzone computed by solving the
direct problem from the shear modulus (
μz
), and represents the corresponding MR
measured tissue displacements. They assumed that minimizing the sum in equation (39) was
equivalent to the sum of minimization of the individual subzones:
(40)
which involves equating derivatives of displacements with respect to subzone shear modulus
to zero, and solving the resulting set of nonlinear equations with the Gauss-Newton method.
The resulting matrix solution at iteration ( i+1 ) has the form:
(41)
where represents the Jacobian matrix for a given subzone, and
αz
is the regularization
parameter on the subzone level. Van Houten
et al
. (2001) deployed the subzones in a
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random and overlapping manner from a diminishing list of seed points that corresponded to
locations in the computational domain that had n not been previously updated; however the
manner in which the subzones are deployed is not essential as demonstrated in (Doyley
et
al
., 2007). When the shear modulus at all locations is updated, a global sweep is concluded
and the process is repeated. Doyley
et al
. (2007) demonstrated that the computatiional load
of this inversion technique increased linearly with increasing subzones — a limitation that
they circumvented by reducing the size of the reconstruction field of view and the overlap
between subzones. Doyley
et al
. (2005) also demonstrated that the spatial resolution of
elastograms computed using this technique was on the order of 5 mm. Figure 7 shows a
representative example of an elastogram obtained from a health volunteer using the
subzone-inversion technique, which illustrates that the resolution of the elastograms was
sufficiently high to visualize fibroglandular tissue from the adipose tissue. The advantage of
this inversion scheme is that it is ideally suited for a parallel-computing platform, because
the sub-domain discretizations are computationally independent as demonstrated in (Doyley
et al
., 2005).
Van Houten
et al
. (2003) reported results of a pre-clinical study that they performed on five
healthy volunteers. Their results demonstrated that the elastic properties of the breast fibro-
glandular and fatty breast tissues, measured in
in vivo
with the subzone inversion technique,
were comparable to those reported in literature.
5. Advanced reconstruction methods
Soft tissues display several biomechanical properties, including viscosity, nonlinearity,
porosity, anisotropy and permeability, which may improve the diagnostic value of
elastography when visualized alone or in combination with shear modulus. Krouskop
et al
.
(1998) demonstrated that clinicians could use mechanical nonlinearity to differentiate
between benign and malignant breast tumors. They performed mechanical tests on excised
breast tissue, which revealed that benign breast tumors displayed linear mechanical
behavior, while malignant breast tumors exhibited nonlinear mechanical tendencies. There is
mounting evidence that other mechanical parameters, namely viscosity (Qiu
et al
., 2008;
Sinkus
et al
., 2005b), anisotropy (Sinkus
et al
., 2005a), and porosity can also differentiate
between benign and malignant tissues – similar claims have also been made for shear
modulus (Sinkus
et al
., 2005a). Not only can these mechanical parameters discriminate
between different tissue types, but they may provide value in other clinical areas, including
brain imaging (Hamhaber
et al
., 2010; Sack
et al
., 2009), distinguishing the mechanical
properties of active and passive muscle groups (Asbach
et al
., 2008; Hoyt
et al
., 2008;
Perrinez
et al
., 2009), characterizing blood clots (Schmitt
et al
., 2007), and diagnosing
edema (Righetti
et al
., 2007a). Several investigators are actively developing model-based
techniques to visualize different mechanical properties, either alone or in combination, using
quasi-static, harmonic, and transient elastographic imaging approaches.
In the proceeding subsections, we review several promising model-based, elastographic
imaging approaches that have been proposed for visualizing other biomechanical parameters
besides shear modulus.
5.1 Viscoelasticity
In most approaches to model-based elastography, the mechanical behavior of soft tissues is
modeled using the theory of linear elasticity (Hooke’s law), which is an appropriate model
for linear elastic materials (i.e., Hookian materials). However, it is well known that most
materials, including soft tissues, deviate from Hooke’s law in various ways. Materials that
exhibit both fluid-like and elastic (i.e., viscoelastic) mechanical behavior deviate from
Hooke’s law (Fung, 1981). For viscoelastic materials, the relationship between stress and
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strain is dependent on time. Viscoelastic materials exhibit three unique mechanical
behaviors: (1) strain increases with time when stress (externally applied load) is sustained
over a period of time, a phenomenon that is known as viscoelastic creep; (2) stress decreases
with time when strain is held constant, a phenomenon known as viscoelastic relaxation; (3)
during cyclic loading, mechanical energy is dissipated in the form of heat.
Viscoelastic materials are usually characterized by suddenly applying a uniform stress or
strain that is sustained over a period of time. Several investigators have proposed
elastographic imaging methods as a way to visualize the mechanical parameters that
characterize linear viscoelastic materials (i.e., viscosity, shear modulus, Poision’s ratio).
Such methods usually involve fitting dispersive shear-wave speed and attenuation
coefficients to a rheological model, such as the Voigt, Kelvin or Maxwell model.
5.1.1 Quasi-static methods—Sridhar
et al
. (2007a, b) developed an elastographic
imaging approach for characterizing viscoelastic materials. More specifically, they acquired
time-varying axial strain elastograms when a viscoelastic material was subjected to a
constant stress, and constructed a creep curve at each pixel from the time-varying strain
elastograms. They computed the creep compliance from the the ratio of the time-varying
strain, ε (
t
) , to the applied stress; and computed the complex compliance,
D
*(
s
) , from the
Laplace transform of the creep compliance:
(42)
where
D’
(
s
) and
D
” (
s
) represent the storage and loss compliance, respectively. Sridhar
et al
.
(2007a) used a three-parameter Kelvin-Vogit rheological model to predict the strain
response at each pixel within the time-varying strain elastogram. In the time domain, the
resulting strain response is given by:
(43)
where εo is the instantaneous strain, i.e., the strain incurred immediately after compression;
ε1 is the viscoelastic strain amplitude; and
t
1 is the retardation time, i.e., the time required
for the tissue to become fully deformed. They estimated the three model pameters by fitting
the rheological model to the measured creep response at each pixel. Using this tecnique, this
group demonstated that
t
1 is the most useful parameter for discriminating between malignant
and benign breast tumors (Qiu
et al
., 2008). More specifically,
t
1 values measured from
malignant tissue were smaller than those measured from the surrounding healthy breast
tissue; whereas, the converse was observed for benign tumors.
5.1.2 Harmonic methods—Hoyt
et al
. (2008) proposed a viscoelastic approach based on
sonoelastography imaging. They measured the visoelastic properties of gelatin and muscle
samples, by fitting a Voigt model to the dispersive shear speed obtained using the crawling-
wave method described in (Wu
et al
., 2006). These authors validated their technique by
conducting simulations, phantom studies, and human studies. The simulation studies
revealed a 2.3 % difference in the computed and true shear speeds under ideal measurement
conditions. The phantom studies revealed a 1 % error in shear modulus that had been
computed using the proposed technique, relative to that measured using a mechanical testing
system. They also observed a slight increase in shear speed with increased frequency.
Statistically significant differences were observed in the shear and loss moduli of relaxed
and volunteered contracted muscles. The authors also observed a slight increase in shear
speed with frequency. To minimize any anisotropic effects, they obtained data parallel to the
muscle fiber.
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Asbach
et al
. (2008) used their multifrequency MR elastography method to measure the
viscoelastic properties of normal liver tissue versus diseased liver tissue taken from patients
with grade 3 and 4 liver fibrosis. Like Hoyt
et al
. (2008), they measured shear-wave and
shear-attenuation dispersion; but, rather than using ultrasound, they computed shear wave
and shear attenuation from the Fourier transform of the complex MR displacement fields.
They computed the shear modulus and viscosity variations within the tissue by fitting a
Maxwell rheological model to the measured data, by solving the linear viscoelastic wave
equation in the frequency domain. They observed that fibrotic liver tissue had a higher
viscosity (14.4 ± 6.6 Pa s) and elastic modulus (
μ
1 = 2.91 ± 0.84
kPa
and
μ
2 = 4.83 ±
1.77
kPa
) than normal liver tissue. Their results revealed that although liver tissue is
dispersive, it appeared as non-dispersive between the frequency range of 25 — 50 Hz. This
research group has also measured the viscoelastic properties of brain tissues (Hamhaber
et
al
., 2010; Sack
et al
., 2009), but in that case, they characterized the viscous properties of the
brain by fitting a Voigts model to measured complex modulus. They also observed that the
viscosity didn’t agree with the predictions of the Kramers-Kronig relation (Klatt
et al
. ,
2010; Madsen
et al
., 2008; Urban and Greenleaf, 2009). Klatt
et al
. (2010) also measured the
viscoelastic properties of the liver. In this case, the spring-pot model was used to study the
dispersive behavior of the viscoelastic properties between frequency ranges of 25, 37.5, 50
and 62.5 Hz. Like Hoyt
et al
. (2008), they observed that the stiffness and viscosity of muscle
increased with voluntary contraction.
5.1.3 Transient methods—Catheline
et al
. (2004) were the first to propose a method to
visualize the viscoelastic properties of soft tissues. Using transient elastography, they
measured the spatial variation of the time-harmonic displacement field, and used a plane-
wave approximation to compute the complex wave number (
k
=
k
’ +
i
α
T
) of the
propagating waves, as follows:
(44)
where
uz
(
x
) is the measured displacement, and
FT
is the Fourier Transform. The wave speed
and attenuation coefficients of the propagating shear wave relate to the complex wave
number as follows:
(45)
where
𝕽
and represents the real and imaginary component of the complex wave number,
respectively.
Catheline
et al
. (2004) used Agar-gelatin phantoms to demonstrate that the shear wave speed
(
c
2 ) and attenuation coefficient (α
T
) computed using equation (45) were comparable to
those measured independently from phase and amplitude measurements. They computed the
shear modulus (
μ
) and viscosity (
η
) by fitting the measured speed of sound and attenuation
to Voigt’s and Maxwell’s rheological models, whose wave speed and attenuation were given
by:
(46)
and similarly, the attenuation was given by:
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(47)
where the superscripts
V
and
M
represent the Voigt and Maxwell models, respectively. The
recovered shear modulus values were independent of the rheological model employed, but
researchers observed an order of magnitude difference in viscosity values recovered with the
two models. This was not surprising, because the goodness of fit of the wave-speed
dispersion data to both rheological models were comparable, but Catheline
et al
. (2004) had
demonstrated that the Voigt model provided a better fit to the attenuation-dispersion data
than the Maxwell model, which proved that attenuation coefficents were independent of
frequency despite the strong frequency dependency that was apparent in the data. Catheline
et al
. (2004) also had conducted studies on excised bovine muscles, which revealed two
apparent wave speeds: either a fast or a slow wave was observed, depending on the
polarization. They obtained a good estimate of both wave speeds using equation (46), but
there was an overestimation of attenuation, because the model assumed the displacement
field arose solely from transverse waves.
Sinkus
et al
. (2005b) developed a direct-inversion scheme to visualize the mechanical
properties of visocelastic materials, in which a curl operation was performed on the time-
harmonic displacement field u(
x
,
y
,
z
,
t
) = u(
x
,
y
,
z
)
ej
ω
t
to remove the displacement
contribution of the compressional wave. They dervived the governing equation that
describes the motion incurred in an isotropic, viscoelastic medium by computing the curl of
the PDEs that describe the motion incurred by both transverse and compressional shear
waves. The resulting PDEs for transverse waves are given in compact form by:
(48)
Sinkus
et al
. (2005b) developed a direct-inversion scheme from equation (48), in which
μ
and
η
(viscosity) were the unknowns. They evaluated the inversion scheme using (a)
computer simulations, (b) phantom studies, and (c) patient studies. Their simulation studies
revealed that the proposed algorithm could accurately recover shear modulus and viscosity
from ideal displacement data. However, with noisy displacements, a good estimate of shear
modulus was obtained only when the shear modulus of the simulated tissue was < 8 kPa; the
inversion scheme overestimated the shear modulus values when actual stiffness of the tissue
was larger than 8 kPa. A similar effect was observed when estimating viscosity, albeit much
earlier (i.e., the algorithm provided good estimates of of viscosity when
μ
< 5 kPa).
Although the shear modulus affected the bias in the viscosity measurement, the authors
demonstrated that the converse did not occur; i.e., the viscosity did not affect the bias in
shear modulus. Despite these issues, their phantom studies revealed that inclusions were
discernible in both
μ
and
η
-elastograms, and the viscosity values agreed with previously
reported values for gelatin (0.21 Pa s). The patient studies revealed that the shear modulus
values of malignant breast tumors were noticeably higher than those of benign
fibroadenomas, but there was no significant difference observed in the viscosity of the tumor
types, a result that would appear to contradict those reported in (Qiu
et al
., 2008).
Vappou
et al
. (2009) proposed a two-step approach for quantifying the viscoelastic
properties of tissue. They measured the real component of the wave number (
k’
) from the
phase ( ϕ) of the Fourier transform of the time-varying displacement at the excitation
frequency:
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(49)
They also measured the ratio of the real to the imaginary component of the complex shear
modulus, from the phase between shear strain images and radiation force:
(50)
From
R
and
k’
, they computed the
k
” (the imaginary component of the complex wave
number) using the following relation:
(51)
Using this relation, they computed the storage (
G’
(
ω
) ) and the loss (
G”
(
ω
)) as follows:
(52)
They validated their technique by performing phantom studies, and observed good
agreement between the storage moduli that had been computed with the proposed technique,
and those that had been measured with a commercially available rheometer. However, there
was large discrepancy between the loss moduli.
Schmitt
et al
. (2010) used a similar approach to characterize the viscoelastic properties of
vascular tissues. In their approach, they computed the real and imaginary components of the
complex wave number from the Fourier transform of the real component of the complex
time-harmonic displacement (i.e., u(
x
,
y
,
t
) = u
ei
(
k
’+
j
α
T
)
o
ei
ϕ ). They computed the
attenuation coefficient by fitting a line to the natural log of the absolute value of the
complex time-harmonic displacement. Storage and loss moduli were computed using using
equation (60) as described in (Vappou
et al
., 2009), where
k”
= α . For materials with
vascular geometries, they fitted the complex modulus that was computed using an analytic
model to the measured data, using MATLAB’s nonlinear solver. They measured the
dispersion of
G’
and
G
” in aortic samples between 540-670 Hz, and conducted experiments
on rat liver where
G’
(
ω
) and
G”
(
ω
) were 119.24 ± 61.6 Pa and 96.7 ± 7.9 Pa, respectively,
at 250 Hz. The group also demonstrated that the viscoelastic properties of blood clots could
be characterized using the proposed technique. However, in these studies of complex
moduli, they estimated the storage and loss moduli by fitting a rheological model to the
measured data. More specifically, they compared the goodness of fit of five rheological
models—Maxwell, Kelvin-Voigt, Jeffery, Zener, and a third-order Maxwell—and observed
that the Zener model gave the best fit to the data.
G’
had the maximum chang at the
beginning, and stabilized after 120 minutes; whereas
G”
was constant, with a spike between
38 and 81 Hz, followed by a gradual decrease in amplitude.
5.2 Poroelasticity
Poroelastic materials also display a transient mechanical response (i.e., they display creep
and stress relaxation when a load is applied and is held constant for a while). This is a
phenomenon that occurs because the matrix of poroelastic materials is porous, and
interstitial fluid may flow through the pores when a load is applied. Multi-phasic mechanical
models are typically used to predict the temporal mechanical response of poroelastic
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materials. These mechanical models assume that poroelastic materials consist of two or
more distinct phases. For example, the biphasic mechanical model described in Armstrong
et al
. (1984) assumes that poroelastic materials consist of both a solid and a fluid phase, and
that the material’s transient behavior is governed by the interaction of the two phases, i.e.,
the mechanical properties of the solid matrix and the permeability of the matrix to fluid.
Although porous and viscoelastic materials both exhibit transient mechanical behavior, the
porous material is compressible, whereas the viscoelastic material is incompressible. More
specifically, volume is conserved when viscoelastic materials are deformed, but this is not
the case for poroelastic materials.
Poroelastic models, such as the biphasic model described in Armstrong
et al
. (1984) and the
consolidation mechanical model described in (Miga
et al
., 2000), have been used to model
the transient response of brain tissue, cartilage, and other soft tissues. It has also been
demonstrated that the mechanical behavior of pathological conditions, such as edema, may
be described with a poroelastic model, and elastography could be extended to allow
visualization of the poroelastic mechanical parameters — important information that may be
used to characterize and monitor the treatment of edema (Righetti
et al
., 2007a).
Consequently, several groups are now actively developing methods to extend both
ultrasound and MR elastography to poroelastic materials.
5.2.1 Quasi-static methods—Konofagou
et al
. (2001) introduced poroelastography.
Using the finite-element method, they studied the temporal behavior of the radial-to-axial
strain ratio within a homogenous, cylindrically shaped poroelastic material during stress
relaxation. They predicted that three mechanical responses would be seen in poroelastic
materials; specifically, that:
•(a) The radial-to-axial strain ratio elastograms would have a uniform value of 0.5
immediately after compression, which was consistent with the theoretical
predictions made earlier by Armstrong
et al
. (1984) using an analytical model.
•(b) The radial-to-axial strain ratio within the sample would tend towards an
equilibrium value that was equal to the Poisson’s ratio of the solid matrix (
νs
)
when the compression was sustained.
•(c) The time taken to reach equilibrium (the time constant) would depend on
permeability of the solid matrix and the length of the fluid path — i.e., that
materials with a high permeability would reach equilibrium more quickly than
materials with a low permeability.
Righetti
et al
. (2004) demonstrated the feasibility of poroelastography experimentally. By
studying the transient mechanical behavior of drained and undrained tofu samples (Wu,
2001), Righetti and colleagues demonstrated the following transient mechanical behaviors:
(1) the mean radial-to-axial strain ratio of undrained tofu samples was approximately 0.5
immediately after compression, which was consistent with the numerical predictions of
Konofagou
et al
. (2001); (2) the mean radial-to-axial strain ratios of drained and undrained
tofu samples were noticeably different, but no significant difference was observed in
rehydrated and undrained tofu samples; (3) the mean radial-to-axial strain ratio of undrained
tofu samples decayed from 0.5 towards an equilibrium value that was slightly higher than
Poisson’s ratio measured for the drained state; however, the rate of decay of the measured
radial-to-axial strain ratio was noticeably slower than that predicted by the analytical model
described in (Armstrong
et al
., 1984); and finally, (4) the mean radial-to-axial strain ratio of
drained tofu samples displayed some transient mechanical behavior, albeit only slight, which
suggests that besides being porous, the tofu samples were also viscoelastic.
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Berry et al. (2006a) derived analytical expressions to predict the internal strain field of
cylindrically shaped poroelastic materials that are undergoing stress relaxation. By
extending the analytical model described in (Armstrong
et al
., 1984), they derived analytical
expressions for predicting the spatio-temporal behavior of radial strain, axial strain, and
radial-to-axial strain ratio elastograms. Using the derived equations, they predicted that
immediately after compression, the radial-to-axial strain ratio would be equivalent to 0.5,
and both the radial-to-axial ratio would decay to towards an equilibrium value that is equal
to the Poisson’s ratio of the solid matrix when compression is sustained. This theoretical
prediction was consistent with that predicted by Konofagou
et al
. (2001) using the finite-
element method. They also demonstrated that there was good agreement between the mean
radial-to-axial strain ratio predicted using the derived equation and those computed using
either the analytical model described in (Armstrong
et al
., 1984), or the finite-element
method. Using the modified analytical model, Berry
et al
. (2006a) predicted two mechanical
behaviors that were not previously reported. More specifically, they predicted that (1) a
plateau would be present in the radial-to-axial strain ratio profile in the early stages of stress
relaxation, because fluid flow does not begin simultaneously throughout poroelastic
materials; and that (2) near the surface, the radial-to-axial strain ratio would overshoot the
Poisson’s ratio of the matrix analytical model to the time-dependent strain elastograms.
In a companion paper, Berry et al. (2006b) reported the results of experiments that they had
performed to corroborate both the theoretical predictions reported in Berry et al. (2006a),
and predictions made using the finite-element method concerning the effect of non-slip
boundary conditions on the strain response within a poroelastic material undergoing stress
relaxation. Using the finite-element method, they predicted that the transient strain response
of cylindrically shaped poroelastic materials with slip boundary conditions would be
noticeably different from those observed in samples with non-slip boundary conditions.
More specifically, they predicted that: (1) immediately after compression and at equilibrium,
region variations would be present in the radial, axial, and radial-to-axial strain elastograms
obtained from poroelastic materials with non-slip boundary conditions; (2) axial strain
elastograms would be time-dependent and spatially varying, because during stress
relaxation, the axial strain in the central region would transfer to regions near non-slip
boundaries; and (3) radial-to-axial strain profiles acquired from regions far away from the
nonslip boundaries would be similar to those acquired for samples with slip boundary
conditions. Berry et al. (2006b) also introduced a new elastogram known as the volumetric
strain elastogram, which was the most useful elastogram for studying the transient
mechanical behavior of poroelastic materials. They showed that, immediately after
compression, this parameter was zero (incompressible) for samples with slip and non-slip
boundary conditions. Although their experimental results departed from the theoretical
predictions, they demonstrated that useful parametric images could be obtained
experimentally using their model-based reconstruction approach. However, this approach
was not without problems; local variations were observed in the parametric images obtained
from a homogeneous sample, but the results were sufficiently encouraging to warrant further
investigation.
The main limitations of stress-relaxation studies are that temporal variation is not typically
observed in the axial direction during unconfined testing, and that the poor lateral resolution
of current ultrasound systems compromises the quality of radial-to-axial strain elastograms.
To address this issue, Righetti et al. (2007b) studied the temporal behavior of porous
materials during creep. They demonstrated that the quality of transient axial strain
elastograms was sufficiently high to differentiate between various grades of tofu samples.
They generated time-constant elastograms by fitting an exponential function to each pixel in
the time-sequence axial-strain elastogram. Using this technique, they demonstrated that they
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could detect focal poroelastic inclusions embedded in a poroelastic background with a
contrast-to-noise ratio.
5.2.2 Harmonic Methods—Poroelasticity has also been explored using steady-state
harmonic magnetic resonance elastography (Perrinez
et al
., 2010). More specifically,
Perrinez
et al
. (2010) described a novel, 3D inversion scheme for recovering the mechanical
parameters from volumetric displacement data obtained with MR in a porous media. They
formulated the inverse-elasticity problem by treating it as a parameter-optimization problem,
in which the goal was to minimize the difference between (1) measured displacements and
(2) those computed by solving the partial differential equations that describe time-harmonic
behavior of a poroelastic medium, which is composed of a porous, compressible, linear
elastic solid matrix and a viscous, incompressible fluid. The resulting system of PDEs that
describes the forward problem is given by:
(53)
The parameter
β
is defined as:
(54)
where p is the time-harmonic pore pressure;
κ
is the hydraulic conductivity; ϕ is the matrix
porosity; and
ρf
and
ρa
represent the pore-fluid density and the apparent mass density,
respectively.
Minimizing the objective function with respect to Lamé constants is a nonlinear process that
can be solved using the Gauss-Newton iterative scheme. At each iteration, updates to the
mechanical parameters are computed as follows:
(55)
where {δδ } = {δ
μ
,δ
λ
, δ
p
},
J
*
T
J
is the self-adjoint Hessian matrix, α is the regularization
parameter,
J
* is the complex Jacobian or sensitivity matrix, and u
m
and u
c
are the measured
and computed displacement fields, respectively.
Perrinez
et al
. (2010) solved equation (55) using the sub-zone inversion approach described
in (Van Houten
et al
., 2001; Van Houten
et al
., 1999), which they implemented on a
parallel-computing platform as described in (Doyley
et al
., 2004). They performed
simulation and phantom studies to assess the performance of the proposed algorithm. The
simulation study revealed that the algorithm could recover good elastograms (i.e., shear
modulus, the second Lamé constant, and pore-pressure amplitude) in the presence of 5%
additive noise, and that hydraulic conductivity influenced the performance of the
reconstruction method. More specifically, the reconstruction method overestimated shear
modulus when image reconstructions were performed using hydraulic conductivity less than
1×10−9, and considerable variability was observed in recovered images when hydraulic
conductivities were greater than the true values. Perrinez
et al
. also performed a phantom
study to assess the performance of the poroelastic reconstruction method relative to the
linear-elastic reconstruction method described in (Van Houten
et al
., 2001; Van Houten
et
al
., 1999; Doyley
et al
., 2004) . In this study, they perfoormed MR elastographic imaging on
a phantom that contained a cylindrically shaped gelatin inclusion that was embedded in a
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block of tofu. Elastograms recovered with the linear -elastic inversion scheme showed high
variations of the recovered shear modulus in both the surrounding poroelastic background
and the linear elastic inclusion; whereas, the poroelastic-inversion scheme produced
noticeably better images.
The advantage of the poroelastic inversion scheme as a reconstruction method is that it
produces images of the pore-pressure that could prove useful in studying edema or
hydrocephalus. Figure 8 shows examples of poroelastic elastograms obtained from the brain
of a healthy volunteer using the poroelastic inversion scheme described in (Perrinez
et al
.,
2010).
5.3 Nonlinearity
Soft tissues display nonlinear mechanical behavior because of geometric and/or material
nonlinearity as discussed in (Taber, 2004).
5.3.1 Geometric nonlinearity—When soft tissues deform by a small amount (an
infinitesimal deformation), their geometry in the undeformed and deformed states is similar,
and thus the deformation is characterized using engineering strain. However, when soft
tissues experience a finite deformation, their geometries are noiceably different in the
undeformed and deformed states. In such cases, errors are incurred when deformations are
significantly different; thus, engineering strain provides an accurate measure of the
deformation. To characterize finite deformation, we first have to define a reference
configuration, which is the geometry of the tissue under investigation in either the deformed
or undeformed state. The Green-Lagrangian strain tensor can be used to characterize the
deformation incurred during finite deformations, which is defined as:
(56)
The nonlinear term is neglected when the magnitude of the spatial derivative is small, to
produce the linear strain tensor as defined in equation (6). The relationship between stress
and strain is nonlinear even for a linearly elastic material when it is undergoing finite
deformations. Conseqently, Skovoroda
et al
. (1999) proposed an iterative technique to
compute shear modulus for materials undergoing finite deformations.They performed
studies using a linear elastic phantom that was undergoing finite deformation, to evaluate the
quality of the ensuring elastograms relative to those produced using equation (22). The
results of this investigation revealed that a smaller standard deviation was incurred in
elastograms computed using the nonlinear reconstruction method for large deformations,
versus elastograms computed using the linear-elastic reconstruction method (i.e., based on
equation (22)).
5.3.2 Material nonlinearity—Some materials exhibit nonlinear material properties that
are typically described using a strain energy density function. Among the strain engery
functions proposed in the literature, the most widely used for modeling tissues are (a) the
Neo-Hookean hyperelastic model, and (b) the Neo-Hookean model with anexponential term.
Oberari
et al
. (2009) used a different model, the Veronoda-Westman strain energy density
function, to describe the finite displacement of a hyperelastic solid that is undergoing finite
deformation, which is defined by:
(57)
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where the terms
I
1 and
I2
are the first and second invariants of the Cauchy-Green strain
tensor,
μ
0 is the shear modulus, andΓ denotes the nonlinearity. The authors proposed to
produce a secant modulus elastogram, by minimizing the following function:
(58)
Where ϑ is the total variation diminishing (TVD) regularization functional. For the
nonlinear case, they proposed to reconstruct a nonlinear parameter, and the shear modulus at
zero strain by minimizing the following functional:
(59)
In this case, there were two displacements: one obtained at small strain value, denoted by
and the other obtained at larger strain, denoted by ; the corresponding predicated axial
displacements were denoted by u1 and u2, respectively. The weighting factors
w
1 and
w
2
were selected to ensure that both the large deformation data and the small deformation data
contribute in roughly equal measures.
Equations (58) and (59) were minimized using the quasi-Newton methods (.i.e., the
Broyden-Fletcher-Goldfarg-Shanno method). They compared the performance of modulus
elastograms computed using the secant and nonlinear iteriative inversion techniques, using
data obtained from voluteer breast-cancer patients, one with a benign fibroadenoma tumor
and the other with an invasive ductal carcinoma(IDC). For the fibroadenoma case, the tumor
was visible in modulus elastograms that had been computed using small strain and large
strain (12 %), although the contrast of the elastograms computed at large strain (7:1) was
lower than that computed at smaller strain (10:1). The fibroadenoma tumor was not visible
in nonlinear-parameter elastograms. The inclusion in the patient with IDC was discerible in
shear modulus elastograms recovered using small and larger strains; however, the stiffness
contrast of the modulus elastograms recovered at both small and high strains were
comparable, and the IDC tumor was visible in nonliear-parameter elastograms. This result is
one of several that have demonstrated the clinical value of nonlinear elastographic imaging.
Specifically, model-based elastography can characterize the nonlinear behavior of soft
tissues and may be used to differentiate between benign and malignant tumors. Figure 9
shows examples of shear modulus and nonlinear parameter images of the breast that were
computed using the nonlinear reconstruction technique.
5.4 Anisotropy
The stiffness tensor given in equation (4) contains 21 independent coefficients; however,
neither ultrasound nor MR elastography imaging technology allows us to measure all these
parameters in a practical manner. Consequently, to solve this dilemma, we can use
simplified mechanical models whose stiffness matrices contains fewer independent
coefficients. The stiffness matrix of the simplest mechanical model currently used in
elastography — a model that is valid for an elastically isotropic material — contains two
independent coefficients: shear modulus (
μ
), and the second Lamé’s coefficient (
λ
). This
model has been adequate for most tissues with the exception of muscle, however all tissues
will exhibit anisotropic mechanical behavior when they are probed deeply enough, which
also emphasizes the need for exploring the anisotropic model.
Several anisotropic models have been proposed to describe the mechanical behavior of
polymers, in which the models depend both on the material’s crystalline morphology and the
molecular orientation. The transversely anisotropic model is perhaps the simplest anisotropic
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model; it assumes the mechanical properties are isotropic in the plane orthogonal to the
molecular orientation. The stiffness matrix of the transversely anisotropic model contains
five independent coefficients; it is ideally suited for tissues containing parallel fibers, such
as muscles.
Papazoglou
et al
. (2005) developed an analytical tool to compute the coefficients for a
transversely anisotropic medium. Several groups have observed that shear waves patterns of
isotropic material are concentric, while those of muscle and tendons (anisotropic materials)
have V-shaped patterns (Dresner
et al
., 2001; Sack
et al
., 2002; Uffmann
et al
., 2004).
Papazoglou
et al
. (2005) used an elliptical approximation of the shear waves to model the V-
shape wave patten, which they derived by assuming incompressiblity from a transversely
anisotropic elasticity model. Using this anlytical tool, they characterized the V-shaped
wavefront in terms of its straightness, slope, and interference; they used these results to
estimate the coefficient of the transverely anisotropic models. They analyzed 2-D shear
wave patterns from images of human biceps, obtained via MR elastographic imaging using
the proposed technique, and demonstrated that shear-wave speeds parallel to the muscle
fibers were approximately four times faster than those perpendicular to the fibers.
Sinkus
et al
. (2005a) developed a direct inversion scheme for reconstructing the mechanical
parameters of transversely anisotropic materials from time-harmonic displacement estimates
obtained using MRE. They removed the dependence of Poisson’s ratio on the reconstruction
procedure by applying the Helmholtz-Hodge decomposition, which states that every vector
field can be written as a sum of the divergence-free part, the curl-free part, and the harmonic
part. Using computer simulations, they demonstrated that the proposed inversion scheme
produced good modulus elastograms in the presence of 10 % additive noise, albeit the
modulus elastograms were biased. Sinkus and colleagues also conducted phantom studies to
evaluate the performance of the method. They observed that they could discern hard
inclusion in the shear modulus elastograms, but the magnitude of anisotropy elastograms
was low owing to the absence of anisotropy in the phantom. When Sinkus and colleagues
applied this technique and took images in volunteers with benign and malignant tumors,
they observed enhanced anisotropic and viscous properties within the tumors.
6. Discussion
Developing elastography within the framework of solving an inverse problem should
provide more accurate estimates of the mechanical parameters of human tissues than the
simple approaches described in section 2 of this article. However, several concerns remain
to be resolved before model-based elastography could become the prevailing approach to
quasi-static, harmonic, and transient elastography. These concerns include: (1) developing
practical techniques to transform ill-posed problems into a well-posed ones; (2) minimizing
model-data mismatch; and (3) developing better test procedures to evaluate and optimize the
performance of advanced reconstruction methods.
6.1 Transforming the reconstruction problem to a well-posed one
Solving the inverse elasticity problem may produce a non-unique solution. More
specifically, both valid and invalid modulus distributions could yield identical mechanical
responses. We can recognize invalid modulus distributions in cases when the truth is known
(as in simulation and phantom studies), but misdiagnosis could occur if invalid modulus
distributions were to masquerade as the truth. Several researchers have applied the
uniqueness theorem to elastography. Barbone and Bamber (2002) found that solving the
quasi-static inverse elasticity problem with one displacement field did not produce unique
modulus elastograms. McLaughlin and Yoon (2004) found that transient elastography could
provide unique modulus elastograms when the full 3D displacement field is available.
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Although the uniqueness theorem has not been applied to harmonic elastography, accurate
estimates of shear modulus have been recovered with steady state harmonic elastography
(Doyley
et al
., 2003). The author would like to caution the reader that though solving the
transient elastography problem yields a unique solution for simple materials (i.e., linear,
purely elastic, isotropic), there is no guarantee that this method will provide a unique
solution when it is applied to soft tissues, which exhibit complex mechanical behavior
(McLaughlin and Yoon, 2004). In the proceeding sub-sections, we describe strategies for
introducing
a
priori
information into the image reconstruction process that is applicable to
all approaches to elastography.
(a) Regularization—The ill-pose issue occurs when pertinent information about the
solution is not available; therefore, the goal of regularization is to introduce
a priori
information, such as smoothness, in the reconstruction process. The two challenges we may
encounter when using regularization are (i) selecting the most appropriate method, and (ii)
deciding the optimum value of the regularization parameter. Too little regularization
produces unusable modulus elastograms, while excessive regularization typically produces
low-resolution modulus elastograms.
Discovering the most appropriate regularization technique for elastography is still an open
research question, but the Tikhonov regularization is currently the most commonly used
method. From our experience, elastograms produced with the total variation diminishing
(TVD) regularization method usually possess better (contrast recovery and contrast-to-noise
ratio) than elastograms computed using the Tikhonov regularization method, although the
TVD regularization method typically does produces blotchy images (Richards and Doyley,
2011). The H1-seminorm regularization method could prove to be a better choice, since this
regularization method typically produces elastograms that do not contain “blotchy” artifacts,
its performance is comparable to modulus elastograms produced with the TVD method.
However, before settling on any given regularization method more detailed studies must be
conducted.
Developing objective methods to select the optimum value of the regularization parameter is
another concern. To avoid the temptation of “tweaking”, the regularization parameter should
be selected objectively using either the L-curve or the generalized cross-validation method
(Vogel, 2002). However, since the L-curve and the generalized cross valiation methods are
not appropriate for clinical applications, the author recommends that a statistical approach to
the image reconstuction problem be employed, since statistical-based reconstruction
methods provide a precise description of the regularization parameter (Van Houten
et al
.
2003).
(b) Spatial priors—In addition to regularization, other methods could transform the ill
posed inverse elasticity problem to a well-posed one. For example, Barbone and Bamber
(2002) suggested that model-based elastography be performed with multiple, independent
displacement fields. Although feasible for quasi-static breast elastography, this method
would be difficult to implement in other approaches to elastography. An alternate approach
is to incorporate structural information in the image reconstruction process (Richards
et al
.,
2010; Baldewsing
et al
., 2006; Baldewsing
et al
., 2005b; Le Floc’h
et al
., 2009; Le Floc’h
et
al
., 2010). Structural information can be obtained by segmenting images obtained from
ultrasound, MR, or other sources. In quasi-static elastography, spatial priors are typically
used to impose hard constraint on the reconstruction process through a procedure known as
parameter reduction (referred to here as “hard prior reconstruction”). More specifically, the
shear moduli of all pixels in a given region as defined by the spatial prior are lumped
together. Hard-prior reconstruction methods do not require regularization because the
reconstruction problem is well conditioned; however, the technique is prone to errors
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because the segmentation of congruent features may contain classification errors. The author
recommends that spatial prior should be used to impose soft constraints but not hard
constraints, since this should minimize the effect of segmentation errors on the
reconstruction process.
6.2 The consequence of modeling errors
Errors are guaranteed to occur in model-based elastography because of (a) measurement
noise, and (b) discrepancy between the model and reality. However, without additional
information, we cannot distinguish between errors due to measurement noise, and errors due
to a limited model. Nonetheless, if we were to consider a hypothetical case in which
modeling errors exceeded the measurement noise, an important question to ask is: “how
would modeling errors impact the resulting modulus elastograms?” The short answer to this
question would be that it depend on the severity of the error. More specifically, the
reconstruction process would produce erroneous modulus elastograms if the modeling error
were severe; however, it would provide plausible modulus elastograms that might contains
artifacts if the the modeling error were small, yet significant. Another important question to
ask would be: “what is the consequence of using the plane strain approximation to
approximate a 3D elasticity problem?” In quasi-static elastography, the plain-strain
approximation does not represent a major challenge, because the motion of the tissue can be
confined in the in-plane direction during imaging (Kallel
et al
. 1997b) — which should be
consistent with the motion predicted with the plane strain elasticity model. Unfortunately, in
transient and harmonic elastography, it is difficult to confine the propagation of shear waves
to the in-plane direction; therefore, erroneous elastograms would be produced if image
reconstruction were performed using an approximate mechanical model. Another question to
ask would be “what would happen if linear elastic reconstruction methods were used to
reconstruct the shear modulus of tissues that exhibit complex mechanical behavior (i.e.,
viscous, nonlinear, anisotropic materials)?” We have observed, when using steady state
harmonic elastography (Perreard
et al
., 2010), that when linear elastic reconstruction was
applied to frequency-dependent phantoms, the shear modulus estimates were consistently
less accurate — in somewhat unpredictable ways that resulted from a complex interplay
between multiple factors (i.e., size, shape and contrast of inclusions). The author expects
that a similar behavior to occur if linear elastic reconstruction methods were applied to
materials that exhibits strong poroelastic, anisotropic or nonlinear elastic behavior.
There is recent considerable interest in using elastography to visualize the viscoelastic
behavior of soft tissues. It is also clear that if viscoelastic elastography is to become a viable
clinical approach, then better mechanical models must be employed to accurately capture the
viscoelastic behavior of soft tissues. The author agrees that transient and harmonic
elastographic imaging is perhaps the most natural approach for viscoelastic elastography,
because the harmonic solution of the wave equation can easily be transformed into a
dispersive relation. By fitting a rheological model to wave-speed data and attenuation
dispersion data, viscosity and shear modulus can be estimated. The problem with this
approach is that although most rheological models provide good estimates of wave-speed
dispersion, there is often a large discrepancy between the measured attenuation dispersion
and computed attenuation dispersion. An alternative approach could be to develop a
reconstruction method based on a viscoelastic continuum mechanical model, since this
might provide a more accurate prediction of wave-speed and attenuation dispersion over the
small frequency range employed in elastography.
Poroelasticity is another rapidly developing imaging modality. However, there is
disagreement concerning the most appropriate models for poroelastic imaging of the brain
and cartilage. Some researchers prefer biphasic models, while others other researchers prefer
consolidation models — such as those employed in soil mechanics. Still other researchers
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suggests that neither biphasic nor consolidation models suffice; they believe that viscoelastic
models should be used. As with viscoelastic elastography, in poroelasticity, there also needs
to be consensus among the elastographic imaging community about which is the most
appropriate model.
6.3 Evaluating performance
Several research groups are developing model-based elastographic imaging systems to
characterize the different mechanical properties within soft tissues (i.e., shear modulus,
viscosity, Poisson’s ratio, anisotropy, nonlinearity). The ability to compare image quality
based upon spatial and contrast resolution must be addressed if model-based elastography is
to progress beyond simple anecdotal reports. One approach towards achieving this objective
is to adopt a method of x-ray mammography system characterization. The low contrast
performance of modulus elastograms computed with different model-based elastographic
approaches could be assessed using studies similar to those described in (Doyley
et al
.,
2003). However, to facilitate such studies, new materials must be developed to fabricate
more complex phantoms. More specifically, fabrication techniques should be developed to
allow the investigators to vary each mechanical parameter over a wide dynamic range. In
addition, efforts should be made to develop better mechanical testing devices – especially
for harmonic and transient elastography – similar to those described in (Madsen
et al
.,
2008). Alternatively, techniques such as time-temperature superposition could be used to
extend the useful range of commercial dynamic mechanical analyzers as discussed in
(Doyley
et al
., 2010).
Acknowledgments
This work is supported by NIH National Heart and Lungs research grant R01-HL 088523. The author would like to
thank Prof. Kevin Parker in the Department of Electrical & Computer Engineering at the University of Rochester
for many valuable discussions and comments on the early draft of this manuscript, and Dr. Jiang Yao in the
Department of Mechanical Engineering at the University of Rochester for her useful discussions.
Appendix
Appendix A:
Abbreviations & Symbols
αregularization parameter
αrregularization parameter applied to nonlinearity
αμregularization parameter applied to shear modulus
αTabsorption or attenuation coefficient
attenuation coefficient of the Maxwell model
attenuation coefficient of the Voigt model
αzregularization parameter applied to a zone
Cijkl the Christoffel rank-four tensor
c1velocity of the compressional wave
c2velocity of the shear wave
shear wave speed of the Maxwell model
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shear wave speed of the Voigt model
D *(s)complex compliance
D ’(s)storage compliance
D ”(s)loss compliance
Δdel operator
δij Kronecker delta
EYoung’s modulus
ε0instantaneous strain
ε1viscoelastic strain amplitude
εstrain tensor
ε0axial strain
ε (t ) time-varying strain
fforce vector
ʒimaginary component of a complex number
Ggradient vector
G ’ real component of complex shear modulus (storage modulus)
G ” imaginary component of complex shear modulus (loss modulus)
Γboundary of element
γnonlinearity
HAaggregate modulus
HAkproduct of aggregate modulus and permeability
hthickness
Iidentity matrix
I1first variant of the Cauchy-Green strain tensor
I2second variant of the Cauchy-Green strain tensor
iiteration number
JJacobian or sensitivity matrix
jcomplex number
j ” imaginary component of the wave number
J *Tcomplex Jacobian matrix
J *TJcomplex Hessian matrix
Kglobal stiffness or coefficient matrix
kcomplex wave number
k ’ real component of the wave number
κhydraulic conductivity
LLagrangian scalar
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λsecond Lamé’s coefficient
μshear modulus
μrrelative shear modulus
μzshear modulus in a zone
Nnumber of nodes
n̂outward pointing normal vector
ηviscosity
Ωarea of finite element
Ωangular frequency
phydrostatic pressure
ptime-harmonic pore pressure
Φphase of time harmonic displacement
ϕbasis function associated with element
πobjective function
πzobjective function within a zone
Kreal component of a complex number
ρdensity
ρfpore-fluid density
ρaapparent mass density
σthree dimensional stress tensor
Ttranspose
ttime
t̂Neumann boundary condition
t1retardation time
ummeasured displacement field
ucomputed displacement field
uzcalculated displacement within a zone
measured displacement within a zone
vPoisson’s ratio
ϑTVD regularization functional
vsPoisson’s ratio of the solid matrix
WVeronoda-Westman strain energy density function
wadjoint displacement field
wjdisplacement component
w1weighting factor
w2weighting factor
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ζrotational vector
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Figure 1.
Schematic representation of current approaches to elastographic imaging: quasi-static
elastography (left), harmonic elastography (middle), and transient elastography (right).
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Figure 2.
Modulus elastogram obtained from a phantom containing a single 10 mm diameter inclusion
whose modulus contrast was approximately 6.03 dB. The modulus elastogram was derived
by taking the reciprocal off the strain elastogram after spatial filtering.
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Figure 3.
Hierarchical diagram of proposed approaches to shear modulus estimation for harmonic,
transient, and quasi-static elastography, assuming linear elastic isotropic mechanical
behavior.
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Figure 4.
Modulus elastograms computed from (a) ideal axial and lateral strain estimates and (b) strain
estimates that were corrupted with 4 % additive white noise. The simulated phantom m
contained an inclusion with a Gaussian modulus distribution that had a peak contrast of 4:1.
Courtesy off Dr. P. Barbone, Boston University Department of Mechanical and Aeronautical
Engineering.
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Figure 5.
Sonogram (a), strain elastogram (b), and modulus elastogram (c) of RF ex vivo ablated
bovine liver. Courtesy of Drs. T. J. Hall H , T. Varghese, and J. Jiang (University of
Wisconsin -Madison).
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Figure 6.
caption. Sonograms, radial strain, and modulus elastgraphy obtained from homogenous and
heterogenous vessel phantoms. The modulus elastgrams were reconstructed with a
constrained inversion scheme. Courtesy of Drs. S. Le Floc’h, J. Ohayon, and G. Cloutier,
University of Montreal Department of Radiology.
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Figure 7.
Montage of MR magnitude images (A) and shear modulus elastograms (B) recovered from a
healthy volunteer using the subzone inversion scheme. Courtesy of Drs. J. B. Weaver and K.
D. Paulsen, Dartmouth College, Thayer School of Engineering.
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Figure 8.
Shear images (a) and pore-pressure images (b) showing 16 coronal slices through the brain.
Images cover most of the ventricles, which are depicted by the lower shear modulus (blue
inn image 10a). Voxel size was 3.0 mm3. Courtesy of Drs. J. B. Weaver and K. D. Paulsen,
Dartmouth College, Thayer School of Engineering.
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Figure 9.
Reconstructions of the shear modulus (Mu) and the nonlinear parameter (Gamma) for breast
tissue using
in vivo
free-hand compression data. The tissue behavior is governed by an
exponential stress-strain law, where the parameter Mu represents the shear modulus at small
strain, and the nonlinear parameter, Gamma, represents the exponential increase in stiffness
with increasing strain. (A) and (B) are the images for a Fibroadenoma (FA), and (C) and (D)
are the images for an Invasive Ductal Carcinoma (IDC). Courtesy of Dr. A. Oberai,
Rensselaer Polytechnic Institute.
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