Elastography for Breast Imaging Michael I. Miga, Vanderbilt University, Department of Biomedical Engineering, Nashville, TN Marvin M. Doyley, Dartmouth College, Thayer School of Engineering, Hanover, NH Jeffrey C Bamber, Institute of Cancer Research and Royal Marsden NHS Trust, Sutton, UK John B. Weaver, Dartmouth Hitchcock Medical Center, Lebanon, NH Keith D. Paulsen, Dartmouth College, Thayer School of Engineering, Hanover, NH Jao J. Ou, Vanderbilt University, Department of Biomedical Engineering, Nashville, TN PREFACE 1 INTRODUCTION 1.1 GENERAL INTRODUCTION 1.2 INTRODUCTION TO ULTRASOUND ELASTOGRAPHY (USE) 1.3 INTRODUCTION TO MAGNETIC RESONANCE ELASTOGRAPHY (MRE) 1.4 INTRODUCTION TO MODALITY-INDEPENDENT ELASTOGRAPHY (MIE) 2 ULTRASOUND ELASTOGRAPHY 2.1 INTRODUCTION 2.2 HISTORICAL DEVELOPMENT OF ULTRASONIC ELASTOGRAPHY 2.3 CLINICAL PROTOTYPE FREEHAND ELASTOGRAPHIC IMAGING SYSTEM 2.4 CLINICAL RESULTS 2.5 MODEL-BASED ELASTOGRAPHIC IMAGING 2.6 COMPARTIVE EVALUATION OF STRAIN-BASED AND MODEL-BASED ELASTOGRAPHY
Transcript
Elastography for Breast Imaging
Michael I. Miga, Vanderbilt University, Department of Biomedical Engineering, Nashville, TN
Marvin M. Doyley, Dartmouth College, Thayer School of Engineering, Hanover, NH
Jeffrey C Bamber, Institute of Cancer Research and Royal Marsden NHS Trust, Sutton, UK
John B. Weaver, Dartmouth Hitchcock Medical Center, Lebanon, NH
Keith D. Paulsen, Dartmouth College, Thayer School of Engineering, Hanover, NH
Jao J. Ou, Vanderbilt University, Department of Biomedical Engineering, Nashville, TN
PREFACE
1 INTRODUCTION
1.1 GENERAL INTRODUCTION
1.2 INTRODUCTION TO ULTRASOUND ELASTOGRAPHY (USE)
1.3 INTRODUCTION TO MAGNETIC RESONANCE ELASTOGRAPHY (MRE)
1.4 INTRODUCTION TO MODALITY-INDEPENDENT ELASTOGRAPHY (MIE)
2 ULTRASOUND ELASTOGRAPHY
2.1 INTRODUCTION
2.2 HISTORICAL DEVELOPMENT OF ULTRASONIC ELASTOGRAPHY
1.3 INTRODUCTION TO MAGNETIC RESONANCE ELASTOGRAPHY (MRE)
Although techniques in magnetic resonance tagging were being investigated in the late
1980’s [22-24], the realization of magnetic resonance elastography (MRE) was relayed in the
September 1995 issue of Science by Muthupillai et al. [25]. In this paper, MR motion-sensitized
gradient sequences were used to phase encode the propagation of strain waves in elastic media.
One of the most important advances with the MRE framework is its ability to capture
displacement in three dimensions. While ultrasound is low-cost and less cumbersome, the
resolution of non-axial displacement is still somewhat poorer than axial measurements. The
potential for the MRE measurement to be performed in three dimensions at the same resolution
as a standard MR image series is very exciting. Subsequent to this investigation, a series of
papers were published demonstrating the potential power of MRE in the context of breast,
prostate and brain applications [26-33]. As was the case in USE, advances to the approach have
been made on many different fronts to include: new imaging sequences, new image
reconstruction techniques, and varying implementations of mechanical excitation.
1.4 INTRODUCTION TO MODALITY INDEPENDENT ELASTOGRAPHY
(MIE)
The previous two elastography methods are derived from using conventional imaging
modalities as displacement measurement devices. To generate elasticity contrast, these
measurements are further processed using assumptions regarding the mechanical behavior of the
material of interest and tissue elasticity information is presented. The method described in this
section is more akin to the original intention of medical imaging, i.e. anatomical interrogation.
This approach is derived from quantifying morphometric changes within acquired structural
images during mechanical excitation. As a result, the method is dependent on image processing
techniques but it is not inherently linked to any particular imaging modality. In some respects,
this method represents a shift from the methods presented above in that it removes the necessity
of having to accurately measure internal tissue deformations. Rather, it uses the information
shared between two images at the anatomic structural level to drive its reconstruction.
Computational techniques to non-linearly register image data via modeling methods have
considerable precedent within the medical image processing community. For example, elastic
matching has been a widely used technique to register multi-modality neuroanatomical images
since the early 1980’s [34-36]. While similar techniques using speckle tracking have been
employed to measure displacement within ultrasound images [37-39], the unique aspect to the
approach in this section is that the elastography process becomes solely a function of image
similarity and does not require the added processing step of constructing displacement fields.
The method has been called modality-independent elastography (MIE) and was first reported in
[40, 41]. Subsequent to this initial work, a series of related papers analyzing different similarity
metrics, analysis with clinical data, and phantom experiments were reported [42, 43]. Others
have also suggested that elastography reconstruction methods could be based on analyzing image
pattern [44, 45]. The transition away from a displacement driven reconstruction is perhaps one
of the most attractive characteristics of the technique, and the results in this section show its
viability as a future clinical screening method.
2 ULTRASOUND ELASTOGRAPHY
2.1 INTRODUCTION
The observation that malignant tumors are stiffer and immobile compared to surrounding
healthy breast tissue [46] have been utilized by clinicians during palpation since the earliest days
of medicine. It is important to note that the properties assessed during manual palpation are
different from bulk elastic modulus that governs the propagation of longitudinal ultrasonic
waves. The elastic properties of soft tissues are dependent on both their molecular composition
(fat, collagen, etc), and on the microscopic and macroscopic structural organization of these
components. In breast, for example, glandular tissue is firmer than fatty areas. Additionally, the
elastic modulus contrast of abnormal tissues with respect to their surrounding normal tissue can
be as much as one order of magnitude [2, 47]; however, none of the traditional medical imaging
modalities such as Ultrasound (US), Magnetic Resonance Imaging (MRI), X-ray Computed
Tomography (CT) or Positron Emission Tomography (PET) can directly measure tissue
elasticity. For example, many tumors of the breast are barely visible on standard ultrasound
examination, despite being much harder than the surrounding tissue – an expected outcome since
the interaction with soft tissue is not correlated with tissue elasticity as such. We hypothesized
that an imaging system that is capable if imaging tissue elasticity should improve both the
detection and characterization breast cancer by taking advantage of the large contrast in modulus
of elasticity between abnormal and normal breast tissue, particularly in pre-menopausal women
where the efficacy of X-ray mammography is questionable.
Although none of the established medical imaging modalities can provide a direct
measure of tissue elasticity, most can impart information about the mechanical response of soft
tissues to either an external or internal mechanical stimulation, and from this information various
mechanical parameters can be inferred. There have been substantial interests during the decade,
in developing elasticity imaging or elastography – a term that was first coined by [12] to describe
their ultrasonic elasticity-imaging approach. A central feature of this imaging technique is the
estimation of externally or internally induced internal tissue motion by employing a conventional
medical imaging modality, namely diagnostic ultrasound (US).
2.2 HISTORICAL DEVELOPMENT OF ULTRASONIC ELASTOGRAPHY
In the mid-1970s, a few investigators began to employ M-mode and static B-mode
scanners to assess the “compressibility” of breast masses by observing the response of echo
motion in response to hand-induced transducer motion [48, 49]. This technique provided
diagnostic information that was not readily available to water-bath ultrasonic breast imaging
technique. Such “relative motion assessment” was extended in the 1980s to the use of real-time
B-mode scanning, to assess so-called “dynamic features” of tissue motion in response to hand-
induced transducer motion. This approach continues to be used today but appears to be limited to
centers that have a small number of experts; however, it provides the foundation for our freehand
approach to elastographic imaging which we regard being a direct extension to the process of
evaluating the dynamic features of breast malignancy.
Current elastographic imaging techniques can be classified under two board categories
depending on the nature of the mechanical stimulation that is employed to induce motion within
the phantom or tissue under investigation (i.e. dynamic or quasi-static). Dynamic elasticity
imaging or sonoelasticity imaging as the technique is more commonly know [50] visualizes
tissue elasticity by inducing low frequency (≤ 1 kHz) shear acoustic waves within the tissue
under investigation. A stiff in-homogeneity that is surrounded by relatively soft tissue will create
a disturbance in the normal vibration pattern, which can be visualized in real-time by employing
color Doppler [50-53]. Although real-time capability of sonoelasticity makes it a very attractive
elasticity imaging technique, the images are in general very difficult to interpret owing to the
complex nature of modal patterns that are produced. A quasi-static elastography image formation
Figure 2.1. The general principle of ultrasonic elastography. Showing the reduction in the strain between two non-overlapping pre and post RF segments.
process generally consists of four-steps; first, an ultrasonic radiofrequency (RF) echo frame is
acquired from the tissue or phantom under investigation; second, a small motion is induced
within the tissue by employing either an external or internal quasi-static mechanical source;
third, a second ultrasonic RF echo frame is acquired; fourth, the spatial variation of the ensuing
internal tissue motion are estimated by performing cross-correlation analysis on the acquired RF
echo frames. Note that the displacement between consecutive pairs of pre and post-deformed RF
echo segments are estimated based on the assumption that the speed of sound is constant in soft
tissues (i.e. 1540 ms-1). Local tissue strain is subsequently computed, as illustrated in Figure 2.1
from the ratio of the separation between the post-deformed RF echo segments (i.e. (Δti -Δti-1)) to
the distance in separation in the pre-deformed RF echo segments (ΔT). The segments are
translated along the axis of RF A-line and the calculation is repeated for all depths. This
computation is performed for all A-lines pairs to produce a matrix of strain estimates, which are
displayed as a grey scale image known as a strain elastogram. The strength of this approach to
elastography resides in its simplicity and robustness, but mechanical artifacts and incomplete
contrast recovery can impede clinical utility – a consequence of interpreting strain images as
relative stiffness images (i.e. modulus elastograms) by assuming stress uniformity – a conjecture
that is applicable only for very special cases [54]. Additionally, the specialized equipment and
non-standard approach to ultrasonic examination of the breast limit its clinical usefulness, and
hence the likelihood of its rapid acceptance for widespread use. In particular, the constraining
devices that are used to reduce undesirable sources of tissue motion, makes it difficult to
examine a large proportion of the breast. Consequently, we have developed a freehand approach
to elastographic imaging [55-57].
2.3 CLINICAL PROTOTYPE FREEHAND ELASTOGRAPHIC IMAGING SYSTEM
The general idea is to employ hand-induced transducer motion as the source of
mechanical stimulation, albeit at the cost of introducing additional sources of measurement noise
and artifacts, relative to standard elastography (i.e. mechanically induced transducer motion). For
example, manual probe motion is expected to result in large transducer displacements, in plane
Figure 2.2 Acoustic footprint extender.
Figure 2.3. Schematic diagram of ICR Freehand elastographic imaging system.
and out-of-plane probe motion. To minimize these undesirable feature of probe motion and their
consequences, a transducer palpation “footprint-extender” was employed (Figure 2.2), and a fast
data acquisition was developed to allow continuous streaming of radio frequency (RF) echo data
at full frame rate (30 fps) from a commercially available ACUSON 128XP ultrasound scanner
(Mountain View CA), whilst palpating the breast with the transducer for ~ 2 s (Figure 2.3). Other
features incorporated onto the system included incremental echo tracking [56] to allow
estimation large internal tissue displacements without excessive decorrelation noise, two
dimensional RF cross-correlation tracking algorithm to reduce decorrelation due to lateral
transducer motion and lateral strains. Figure 2.4 shows representative examples of a strain
elastogram obtained from an elastically inhomogeneous phantom using mechanically induced
and hand-induced transducer motion. The signal to noise ratio (SNRe) of elastograms produced
using the hand-held transducer motion is visually lower than those produced using mechanical
induced transducer motion; however, it is apparent that quality of free hand elastograms is
sufficient to produced clinically useful results.
A theoretical framework known as the strain filter was developed by Varghese and Ophir
[54] for characterizing the performance of elastographic imaging systems. We recently extended
this concept to allow the computation of strain filters (a plot of elastographic signal to noise ratio
as a function of the applied strain) experimentally. Figure 2.5 shows an experimentally derived
strain filter obtained from elastically homogenous phantoms computed using hand-induced
transducer motion relative to those computed using mechanical deformation system. The
bandpass characteristics of the strain filter can clearly be seen in both cases (i.e. the loss of
SNRe, both at low strains, due to the dominance of electronic noise and interpolation errors, and
at high strains, due to structural decorrelation noise.
Performance Measure Freehand elastography Standard elastography
Minimum Strain 0.5 % ± 0.10 0.3 % ± 0.05
Maximum Strain 2.2 % ± 0.45 2.75 % ± 0.20
SNRe 9 16
Dynamic range 4.4 ± 1.78 9.17
Table 2.1 Comparison of elastograms produced using a hand-held transducer and mechanically induced transducer motion.
a b Figure 2.4 (a) mechanical and (b) freehand elastogram obtained from a phantom containing a 15 mm diameter inclusion that was approximately three times stiffer than the surrounding background tissue.
The performance metrics extracted from Figure 2.5 are summarized in Table 2.1, note
that the clinical prototype freehand elastographic imaging system compares favorably relative to
the standard approach to elastography.
a
b Figure 2.5 Experimentally derived strain filter produced using (A) mechanically induced and (B) hand-held transducer motion.
2.4 CLINICAL RESULTS
Using these methods, elastographic imaging on 70 breast cancer patients has been
performed, and demonstrated that freehand elastography can produce images with sufficient
spatial and contrast resolution to discriminate between normal and abnormal breast anatomy [58,
59]. For 74% of all lesions the contrast for freehand elastography was equal to or better than
conventional ultrasound, and overall lesion visibility was judged to be equal or improved for free
hand elastography (FE) in 50 % of the cases. Freehand elastography had an impact on the
diagnosis in 24 % of cases and increased the diagnostic confidence in 25%. For 9%, confidence
deteriorated, mainly in benign lesions (66%). Additionally, combining FE and diagnostic US
increased sensitivity and specificity to 90% and 46% respectively; compared with 84% and 43%
for US alone. An example of a sonogram that was obtained from a 73 year old female with a
Phylloides tumor (borderline with associated lobular carcinoma) in the upper outer quadrant of
her left breast is shown in Figure 2.6a. Phylloides tumors are relatively rare variants of
fibroadenoma with a richer stromal component and more cellularity. They grow quickly,
developing a macroscopically lobulated internal structure and may reach a large size, visibly
altering the breast profile. Sonography generally shows a solid, moderately hypoechoic nodule,
with smooth borders and good sound transmission [60]. Inhomogeneous structures may be
present because of small internal liquid areas. These appearances are non-specific and
sonography is not currently able to distinguish between benign and malignant cases, nor make a
differential diagnosis between fibroadenoma and phylloides tumor. Note that the tumor covers
most of the field of view, with the capsule of the anterior margin visible close to the top of the
image and the posterior margin visible at the bottom left. Within the tumor the appearance is
heterogeneous on a large scale, with macroscopic lobules separated by echogenic boundaries that
are probably fibrous in nature. The freehand elastogram (Figure 2.6b) confirms this appearance
but shows it much more clearly with greater contrast than the sonogram. The capsule at the top
of the image is seen to be stiffer than either the subcutaneous fat (anterior) or the tumor tissue
(inferior). The macroscopic lobules within the tumor are very clearly defined as relatively soft
regions separated by stiff septa, which is also consistent with the septa being of a fibrous nature.
Within the macroscopic lobules the stiffness appears relatively homogenous, which is consistent
with the locally homogeneous and densely cellular structure seen on the histological section in
Figure 2.6c. Finally, the small amount of tissue visible posterior to the tumor (bottom left) is
seen to be very soft relative to the other structures in the image. This information was not
available from the sonogram or any other image.
a b c Figure 2.6 (a) Sonographic, (b) elastographic, (c) histologic images of Phylloides breast tumor.
Figure 2.7 shows another example of sonogram and strain elastogram that was obtained
from a 58 year-old female with a grade 3 ductal carcinoma (with two foci of ductal carcinoma in
situ) in the upper outer quadrant of her left breast. This malignant tumor was an interesting case
because it was reported on clinical examination as non-palpable and on X-ray mammography as
displaying suspicious microcalcifications but no imagable mass. This tumor was difficult to
visualize on sonography but a suspicious ill-defined, irregular hypoechoic region may be seen in
Figure 2.7a. The freehand elastogram (Figure 2.7b) showed that this region is stiffer than the
surrounding gland and that the boundary of the region of increased stiffness corresponds very
closely to the boundary of the hypoechoic region. This tumor also appears to have a
heterogeneous internal stiffness, which may correspond to the histological appearance seen
Figure 2.7c that dense regions of tumor cells and stroma co-exist with edema and fat deposits.
Furthermore, within the general region of increased stiffness one may see two small regions of
tissue that are very stiff indeed. The histopathology report, obtained after tumor excision,
confirmed that within this carcinoma there exist two focal areas of ductal carcinoma in situ. It
was thought that these had probably been the original seed sites from which two cancers had
a b c Figure 2.7 (a) Sonographic, (b) elastographic and (c) histological images of an invasive ductal carcinoma in situ.
grown and subsequently coalesced into a larger tumor. Note that freehand elastography was the
only imaging technique employed that provided this information non-invasively.
2.5 MODEL-BASED ELASTOGRAPHIC IMAGING
Images of internal tissue strain represent the response of soft tissues to an external (or
internal) mechanical stimulus; therefore, strain alone represents an approximation measure of
tissue elasticity. Accurate quantification of tissue elasticity requires knowledge of both the axial
strain and the full three-dimensional (3D) stress state vector. At present, stress cannot be
measured in vivo; therefore, it is customary to interpret strain elastograms as modulus
elastograms based on the assumption of stress uniformity. In practice, the internal stress
distribution is seldom uniform because of stress decay and stress concentrations that appears near
modulus non-uniformity [12]. Therefore, interpreting strain elastograms as relative modulus
elastograms based on the premise of stress uniformity will generally induce mechanical artifacts
and reduce the elasticity contrast-transfer efficiency. To address this problem, reconstruction
methods have been developed for solving inverse problems – an approach to elastography that
requires an accurate formulation of the forward elasticity problem that predicts the observed
mechanical response (displacement and/or strain) based on some knowledge of the external
boundary conditions, and the intrinsic tissue mechanical parameters. Besides reducing image
artifacts, this approach to elastography (i.e. model-based elastography) should improve the
contrast-transfer efficiency [61, 62], particularly in high contrast media.
The inverse elasticity problem was solved by formulating it as a constrained parameter
optimization problem where the goal is to minimize an objective function that has the following
form Φ(μ) = Um − U{μ} 2 , where U{μ} represents the axial displacements computed from the
modulus distribution μ{ } by employing the finite element method, and Um is the ultrasonically
measured axial displacements. Minimizing this objective function with respect to modulus
variations is a nonlinear process, which is realized through an iterative solution for μ{ } based
upon an initial guess of the modulus distribution (the trial solution). The resulting matrix solution
at the (k+1) iteration has the form μ{ }k +1 = μ{ }k + J(μ k )T J(μ k ) + ρk I⎡⎣ ⎤⎦−1
• J(μ k )T Um − U μ k{ }( ), where
μ{ } is a vector of modulus updates at all coordinates in the reconstruction field of view, and
J(μ k ) is the Jacobian or sensitivity matrix. The Hessian matrix, J(E k )T J(E k )[ ], is poorly
conditioned, and may be regularized by employing either the Marquardt or Tikhonov
regularization method. Figure 2.8 shows example of modulus elastograms recovered from a
gelatin phantom that contained a single cylindrical isoechogenic inclusion. The inclusion is not
discernible in the sonogram but is highly visible in the strain and modulus elastograms,
demonstrating that strain and modulus elastography can convey new information. It is also
apparent from this figure that solving the inverse problem reduces mechanical artifacts incurred
when strain elastograms are interpreted as modulus elastograms by assuming stress uniformity.
Two implementations of this inversion scheme have been realized. One approach
computes modulus elastograms based on knowledge of known displacement boundary conditions
a b c
Figure 2.8 (a) Sonogram, (b) strain elastogram, and (c) modulus elastogram obtained from a gelatin phantom containing a single cylindrical inclusion
(DBC); whereas the other computes modulus elastograms based on knowledge of known stress
boundary conditions (SBC). The results of a comprehensive evaluation of both implementations
of the inversion scheme [63] revealed that the stress on the boundary of the tissue must be
specified to reconstruct absolute values of the modulus. Otherwise, the modulus elastogram
obtained from known DBC will have to be calibrated using either an external or internal tissue
reference of known shear or Young’s modulus. Krouskop et al. [2] have shown that among
various breast tissues, fat has a consistent and linear modulus over a wide range of applied strain
that could be employed as an internal tissue reference. Encouraging computer simulation and
phantom studies have been reported; however, this approach to elastographic imaging (i.e.,
modulus imaging) is challenging owing to the ill-posed nature of the inverse problem. Factors
such as model-to-data discrepancy, and measurement noise could compromise the quality and
accuracy of ensuing modulus elastograms. Additionally, there is no guarantee of producing
unique modulus elastograms when solving the discrete inverse elasticity problem [64]– another
characteristic trait of ill-posed problems. To minimize these potential problems, we have
imposed additional constrains (i.e., a priori information concerning the mechanical properties of
the underlying tissue structures, and the variance incurred during displacement estimation) on the
image reconstruction problem through the Bayesian framework. Figure 2.9 shows an example of
strain and modulus elastograms obtained from a gelatin phantom containing a single 20 mm
diameter inclusion with modulus contrast of 20 dB. The strain elastogram (left) was computed by
spatially differentiating the measured axial displacement using a simple gradient operator. The
tumor-like inclusion is discernible at the correct location in the strain elastogram; however,
Decorrelation noise (i.e., the dominant noise source incurred in ultrasonic elastography) is
apparent in the strain elastogram – a consequence of employing large strains (i.e., 3 %) when
performing elastographic imaging. Decorrelation noise has two effects when model-based
elastographic imaging is performed with limited a-priori information (middle); firstly, it could
generate spurious modulus estimates; secondly, it may produce erroneous modulus elastograms.
However, image reconstruction is more resilient to decorrelation noise when reconstruction
performed within the Bayesian framework – an expected outcome since imposing further
constraints on the image reconstruction process should reduce the likelihood of the
reconstruction procedure being trapped in a local minima. This observation is relevant to the
proposed research since decorrelation noise associated with in-plane and out off plane catheter
motion frequently pose problems in intravascular elastography [65] – a problem that could
reduce the clinically useful modulus elastograms computed by employing a moderately
2.6. COMPARTIVE EVALUATION OF STRAIN-BASED AND MODEL-BASED
ELASTOGRAPHY
Preliminary investigation was conducted to assess the performance of this modulus
reconstruction strategy relative to quantitative strain imaging [61]. . While the initial
investigation pertains only to quantitative strain imaging as described in [12] and the inversion
Figure 2.9 Elastograms obtained from a gelatin phantom that contained a single 20 mm diameter inclusion when elastographic imaging was performed using applied strain of 2 %. Showing, strain elastogram (left) using conventional strain imaging methodology, modulus elastogram computed using the standard least-squares estimation criterion (middle), and modulus elastogram computed within the Bayesian framework.
reconstruction approach described in [63] the analysis is applicable to all elastographic-imaging
techniques and provides insight into the relative merit of both elastographic imaging approaches.
Figure 2.10 shows a representative example of strain and modulus elastograms obtained from a
gelatin phantom that contained a single 10 mm diameter inclusion that was approximately three
times as stiff as the surrounding tissue matrix. Strain-based modulus elastograms were computed
by inverting the low resolution strain elastograms based on the assumption of stress uniformity;
whereas, model-based modulus elastograms were computed by solving the inverse problem.
Visually, the contrast-to-noise ratio (CNRe) of the modulus elastogram would appear to be
substantial higher than that of strain elastogram; however, the strain elastogram clearly possess
superior spatial resolution. The disparity in spatial resolution was foreseen because the
performance of the image reconstruction techniques, as previously discussed, was stabilized in
the presence of measurement noise by forcing the image reconstruction algorithm to converge to
Figure 2.10 Modulus elastograms computed by directly inverting strain elastograms (top panel), and solving the inverse problem (bottom panel). These elastograms were obtained from simulated phantoms containing single 10 mm diameter inclusions whose modulus contrast was progressively increased from 0.8 dB to 20 dB (going from left to right).
a smooth stable solution (i.e. the regularized solution) albeit at the cost of degrading the spatial
resolution of the resulting image. It is reasonable to assume that the observed difference in
CNRe is due solely to differences in the spatial resolution rather than any intrinsic differences in
the contrast resolution of strain and modulus imaging. Consequently, the spatial resolution of the
strain elastograms was degraded to that of the modulus elastograms by applying the spatial filter
described in [63] recursively to strain elastograms to facilitate an objective comparison of both
elastographic imaging techniques. The mean contrast-to-noise ratio (CNRe) and contrast-transfer
efficiency (CTEe) [62, 66, 67] performance metric are plotted as a function of actual modulus
contrast in Figure 2.11. Note that a fix spatial resolution, the CNRe of both elastographic imaging
approach are statistically equivalent; however, at high modulus contrast (Ec > 6 dB) the contrast-
transfer efficiency (CTEe) of modulus elastograms computed by employing model-based
inversion approach is superior, which confirms the prediction of Ophir and colleagues [54, 66]
that solving the inverse problem should substantially improve the elastographic contrast transfer
efficiency, particular in high contrast medium.
Figure 2.12 representative examples of modulus elastograms and modulus images
obtained from gelatin phantoms containing foam-reinforced cylindrical inclusion with diameters
in the range of 2-25 mm. Strain-based and model-based modulus elastograms were compared to
modulus images obtained from independent mechanical measurement with a nano-indentation®
system as described in [68]. Note that the position and extent of the inclusions were discernible
in all images (i.e. elastograms and nano-indenter® modulus images); however, small inclusions
Figure 2.11 The mean CNRe (A) and contrast transfer effeciency (CTE) (B) computed from strain-base (soild-squares) and model-base (open-circles) modulus elastograms. The error-bar represents ±1 standard deviation computed from 25 independent elastograms at each modulus contrast.
(i.e. inclusion with diameter < 5 mm) were poorly visualized in the nano-indenter modulus image
due to the low spatial resolution of the nano-indenter elastograms. In general there was good
visual correlation between the nano-indenter modulus images and modulus elastograms
computed by employing the model-based inversion technique, and simply inverting strain
elastograms.
(a)
(b)
(c)
Figure 2.12 (a) Sonograms, (b) modulus elastograms and (c) modulus images obtained from gelatin phantoms containing cylindrical inclusions with diameters of 2 mm, 10 mm and 20 mm.
3 MAGNETIC RESONANCE ELASTOGRAPHY
3.1 BACKGROUND
The primary advantage of MR elastography (MRE) over other elastography methods is that
the resolution of the displacement data is equally accurate in all three directions. Ultrasound is
less expensive and faster but the through plane resolution is poor which for ultrasound leads to
poor estimates of the displacement in that direction. Dynamic MRE methods have been more
productive than static MRE methods [69] because the data acquisition is much faster and the
mechanical properties produced are not relative values as they are with static methods where the
boundary values are not accurately known.
Several MRE approaches based on dynamic displacements have emerged which employed
different modes of displacement. In all methods, induced tissue motion is measured using phase-
contrast imaging [25], which is a powerful MR technique that is capable of sensing extremely
small tissue motions (typically on the order of 10’s of microns). The first methods induced
propagating waves in the tissue, and the shear modulus were computed directly from local
estimates of wavelength [70]. This approach is an elegant one that has produced encouraging in
vivo and in vitro results [71-73]. However, it can be difficult to generate coherent wave in
complex structures and then the accurate quantification of wavelength is limited by the presence
of longitudinal mechanical waves and reflections from internal tissue boundaries. Several
second generation reconstruction methods are being explored to remedy those limitations [74,
75]. Alternatively, the steady state approach measures the time varying displacements under
harmonic conditions [76, 77] and then reconstructs the spatial variation of shear modulus using
either a direct inversion [76] or an iterative, model-based inverse technique [78]. It is a distinct
advantage that steady state methods have no special requirements for the induced motion so any
vibration pattern will suffice as long as it is in steady state. Another advantage to steady state
methods is that the acquisitions are faster because there are no propagation delays that the first
two methods require.
Initial clinical evidence produced by the dynamic MRE methods currently used on limited
numbers of subjects show that it has promise to effectively identify malignant breast tissues. A
preliminary clinical evaluation of 15 patients with malignant tumors of the breast, 5 subjects with
benign breast lesions and 15 healthy volunteers [79] showed that MRE has the potential for
differentiating between benign and malignant tissues. McKnight et al. 2002 [72] have also
reported their clinical experience with MRE on 6 breast cancer patients and 6 healthy volunteers.
The elastograms obtained from the healthy subjects revealed moderately heterogeneous
mechanical properties with the shear modulus of fibroglandular tissue being slightly higher than
that of adipose tissue; whereas the elastograms of women with breast cancer showed focal areas
of elevated shear modulus. The mean shear modulus of breast carcinoma was observed to be
approximately four times higher than the mean shear modulus of surrounding healthy breast.
In addition, MRE has been shown to be sensitive enough to characterize normal breast tissue
in a variety of ways and in a variety of situations. Sinkus and colleagues [79] showed that the
sensitivity of MRE is sufficient to characterize breast tissue changes during the menstrual cycle
[80]. Fibroglandular tissue stiffness decreased 5 days after the onset of menses and increased
after the second week of the cycle. No significant variation in the shear modulus of adipose
tissue was observed. Kruse et al. [73] described a variety of early tissue characterization results
based on the use of local frequency estimation and an assumption of linear elastic wave
propagation. These data show frequency and temperature dependence in modulus measurements
in kidney and liver samples in vitro, as well as the possible effects of anisotropy on varying
measurements in highly ordered skeletal muscle. Additionally, an example of an in vivo breast
elasticity image was given that showed a localized area roughly two to three times stiffer than the
surrounding fibrous tissues corresponding to a biopsy-proven malignancy. Plewes et al. [81]
presented quasi-static strain images from the breast of a healthy volunteer. Elastographic signal
to noise ratio of 10 and 16 was reported for fibroglandular and fatty tissue, respectively.
However, a basic study of shear wave propagation in excised tissue by Bishop et al. [82]
indicated that wave speed exhibits relatively small changes with frequency.
3.2 MAGNETIC RESONANCE ELASTOGRAPHY METHODS
MRE systems require three elements to produce an image of the mechanical properties of
tissue: a) mechanical activation of the tissue, b) measurements of the resulting displacements
and c) reconstruction of the mechanical properties. The type of vibration must be described by
the equations used in the reconstruction model to obtain accurate mechanical properties. For
example, reflections are problematic for the dynamic methods that use local wavelength
estimations of the shear modulus but steady-state methods using the general Stokes equation
require the reflections to have reached steady state.
3.2.1 MECHANICAL ACTUATION
Our current mechanical actuation system integrated with our phased array breast coil is
shown in Figure 3.1. The system consists of a vibrating top-plate and stationary back-plate
which is fixed to the coil and MR table. The top-plate is pushed across the back-plate by a
piezoelectric actuator and returned by a hard gel spring. A dove-tail fitting on the stationary
back-plate fits tightly into a grove in the spacers that are used to establish the height and angle of
the plate required to accommodate a given breast size. The primary advantages of piezoelectric
actuation are that they produce displacements that are linear with applied voltage, the applied
Figure 3.1: Breast RF coil with vibrating plate below. The top view (far right) shows the face of the vibrating plate set up for the right breast. Side views (left) show the vibrating plate fixed on spacers to accommodate varying breast sizes.
forces generated are large enough and they can be placed in the high magnetic fields generated
by MRI in any desired orientation. The maximum mechanical actuation of the plate is currently
240 microns. The design has proven to be simple to use and generates consistent motion
successfully. Figure 3.2 shows the kind of motion maps obtained, in this case in a phantom, and
Figure 3.3 shows a typical clinical result from an MRE exam on a volunteer with normal (BI-
RADS 1) breasts.
3.2.2 ENCODING TISSUE DISPLACEMENTS
Motion is measured using motion encoding gradients (MEGs) between the RF excitation
pulse and the signal readout as shown in Figure 3.4. MEGs are applied sequentially in each of
three spatial directions and oscillate at the identical frequency as the induced mechanical motion,
itself. When a voxel of tissue is being moved sinusoidally, the phase of the MR signal is a
cosine function whose amplitude and phase uniquely determine the amplitude and phase of the
harmonic vibration.
Figure 3.2 Motion in the three directions generated by the shear plate shown in a gel phantom. The motion amplitude, given in the color bar in microns, ranges up to 300 microns.
Harmonic motion can be represented as: x = xo + A sin(ϕ) where the phase, ϕ, of the
motion and the amplitude of the motion, A, completely characterize the vibration. To estimate
the amplitude and phase of the displacement in one direction, the MR image phase must be
measured at multiple phase offsets between the motion and the MEGs, θ. The phase of the MR
image is the phase accumulated by motion during the MEG, ξm, plus the baseline phase from
spurious sources. The motion induced phase is the inner product of the MEG with the motion
which is: ξm(θ) = A cos(ϕ + θ) . The simplest way to solve for A and ϕ is to sample ξ(θ) at θ
sampled evenly over one cycle and take the Fourier transform of the phase, ξ(θ) [76]. The
amplitude of the first fundamental frequency is A and the phase of the first fundamental
frequency is ϕ. Each phase offset requires another image acquisition so the imaging times can
be longer than desired. Several methods have been suggested to reduce the imaging times.
Figure 3.3: MR magnitude (top row) and reconstructed shear modulus images (bottom row) acquired during a clinical breast exam with the RF coil and vibrating plate system in Figure 3.1.
Constraining the estimation of motion from the phase using the fact that the magnitude is
identical for all measurements reduces imaging times but increases the instability of the motion
calculation [83]. However, methods that do not cause such instabilities also exist: i) Hadarmard
encoding of the motion, and ii) elimination of the MEGs.
Acquisition w ith θ1 Acquisition w ith θ2
RF
S lice Selection
Phase Encoding
Frequency Encoding
Motion
Figure 3.4: The convensional 3D phase contrast gradient echo pulse sequence used to measure the motion of each voxel in the field-of-view. The phase between the signal driving the mechanical actuation and the MEG is labeled as θ in the figure. Separate acquisitions are made for each value of θmaking the acquisition times rather long. The MEG’s are also inverted to produce a phase cycled pair of acquisitions.
θ2 θ1
Figure 3.5 Motion measured using Hadamard encoded MEG’s (top) and convensional MEGs (bottom). The motion is very similar as expected but the acquisition time is two thirds of a convensional acquisition.
RF
Slice Selection
Phase Encoding
Frequency Encoding
Motion
Figure 3.6: Motion accumulates phase during the frequency encoding gradient just as phase is accumulated during the MEG so the frequency encoding gradient can be used to encode motion as well as position. The phase cycled pair of acquisitions using the frequency encoding gradients is shown. The TR is reduced which reduces the acquisition time by ~2.5 and the TE is reduced which improves the SNR and reduces susceptibility artifacts. However, the sensitivity to motion is also reduced so larger motions are required. There is some interaction between the position encoding and motion encoding but it is minimal for relatively small motion.
Figure 3.7: Comparison of motion measured using traditional MEG’s and the same motion measured using the frequency encoding gradient. The frequency is identical but the imaging time is roughly half. The SNR is equivalent in the two images. The sensitivity to motion is reduced when using the frequency encoding gradients but the SNR is similar.
Hadamard encoding of phase contrast measurements was suggested for 3D velocity
measurements in the 1991 [84, 85]. The first MRE paper used the method for encoding the
motion in all three directions [25], but recent papers have mostly encoded each direction
individually [75]. Hadamard encoding reduces the imaging times by a third by using baseline
measurements more efficiently. Normally a MEG is turned on in one direction at a time. For
each direction, images are acquired with the gradient on normally and then with the gradient
inverted. The phases of the two acquisitions are subtracted to remove the baseline phase caused
by a wide variety of spurious effects like eddy currents and any field inhomogeneity. Six
acquisitions are required to encode all three directions. Hadamard encoding turns on all three
MEGs simultaneously. The signs of the MEGs in all three directions are changed to acquire four
independent combinations of the four unknowns: the phase produced by motion in the three
directions and the baseline phase. The phase produced by the motion in the three directions must
then be solved. The resulting phase estimate is double that obtained by using the same gradients
acquired individually, improving the SNR of the phase angles calculated. However, gradient
heating limits the SNR improvement to ~1.2 when all three gradients are on simultaneously.
Imaging time is reduced from six to four acquisitions for each angle θ. Figure 3.5 shows
phantom data acquired using conventional MEGs compared to that acquired using Hadamard
encoded MEGs. Currently the MR manufacturer limits the number of images obtained in one
acquisition, so we can only acquire 16 slices with Hadamard encoding which is often too few for
clinical work. However, the slice number limitation will be eliminated on future systems and
Hadamard encoding should become more popular because the SNR is improved and the
acquisition time is reduced by a third.
Another alternative method of reducing the imaging times is to eliminate the MEGs
entirely; or more accurately to use the imaging gradients to record motion as well as position
[86]. The frequency encoding gradient is the most productive one to use because it is largest.
Figure 3.6 shows the pulse sequence the can be compared to Figure 3.4 which shows the
conventional one. The elimination of the MEGs reduces the TE significantly thereby increasing
the signal. It also reduces the TR which decreases the imaging time. For 100 Hz motion, the TR
and the TE have been reduced by approximately 2.5. Reduction of the TE also decreases the
sensitivity to susceptibility artifacts. The drawback is ambiguity in the encoding which produces
a convolution like error in the estimated motion. The error only occurs at discontinuities in the
motion or the magnitude of the image. Simulations show that for motion which is 2% of a pixel
dimension, the magnitude of the error is approximately 6% for full scale discontinuities in either
the magnitude or the phase or 12% for both magnitude and phase simultaneously. These errors
are very difficult to find experimentally because discontinuities in the motion are rare and
difficult to produce.
3.2.3 IMAGE RECONSTRUCTION
In this section, the reconstruction algorithm we employ will be described. We begin by
briefly summarizing our subzone approach which has served as the backbone for our
development of solutions of alternative mechanical parameters. We then describe extensions to
the model in terms of multi-parametric property estimation including the imaging of tissue
damping.
We have formulated the MRE reconstruction problem as a constrained optimization task
whose objective is to minimize the difference between a set of measured displacement fields and
those computed by a model description in which the tissue property distribution is parameterized
as a set of unknown coefficients representing the observed mechanical behavior. The typical
strategy defines a single objective function to be minimized (usually the sum of squared
differences between measured and calculated quantities) over the entire domain. In our case, we
view the total problem domain (i.e. image field-of-view) as a union of multiple subzones such
No
Start Initial Estimate Of Shear Modulus
Distribution
Divide Imageinto Subzones
Optimize Shear Modulus Distribution
on Each Subzone
End
( )
( )
( ) wxt
w
vxt
v
uxt
u
22
2
22
2
22
2
∇+Δ
+=
∇+Δ
+=
∇+Δ
+=
μ∂∂μλ
∂∂ρ
μ∂∂μλ
∂∂ρ
μ∂∂μλ
∂∂ρ
YesGlobal Error < Tolerance
Figure 3.8: Flow diagram showing the subzone reconstruction. The method is iterative starting with uniform mechanical properties. The region is subdivided into subzones that are small enough that the Stokes equations can be solved for the subzone. The set of mechanical parameters associated with the motion that most closely matches the measured motion is obtained iteratively. The process is repeated on the subsequent subzones. Then the entire process is repeated till the global error in the motion reaches an acceptable level.
that the minimization of the sum over the subzones is replaced by the sum of minimizations over
the individual subzones. The advantages of the approach are considerable. First, the nonlinear
minimization process occurs only over a single subzone at a time. The significant reduction in
the size of the inversion problem is important because the least-squares approach scales cubically
in the number of parameters to be estimated. Second, it maximizes the utilization of the
complete MR displacement data set and the concomitant tissue property resolution that can be
achieved. By dividing the problem into subzones, high resolution (e.g. MR voxel level) property
maps can be deduced that take full advantage of the high density of tissue measurements which
the MRE technique provides. Figure 3.8 shows a flow chart of the subzone inversion technique.
The 3D subzone inversion algorithm has been extensively tested and evaluated in both
simulation and phantom studies and has been found to be robust to noise in the motion data.
The power of the method resides in its stability and to its flexibility. The method
provides accurate values of the shear modulus even in the presence of significant noise [87]. For
example, lesions as small as 5 mm can be accurately characterized [88]. The flexibility of the
method is demonstrated by the wide variety of models which can be employed. Initially the
inversion algorithm used the equations of motion for a linearly elastic solid where maps of the
shear modulus and of the Lame’s constant were found. However, the inversion can be
formulated on the basis of a model which is a simple approximation to the Navier’s equations of
motion that includes the addition of a damping term in the dynamic equilibrium description:
∇ ⋅ G∇u + ∇ λ + G( )∇ ⋅ u = α∂u∂t
+ ρ∂2u∂t2 (1)
where u is the displacement vector, G and λ are the Lame's constants, α is a damping coefficient
and ρ is the tissue density. The MR measurements are generated in terms of harmonic
magnitude and phase; hence, equation (1) is recast in the frequency domain for the purposes of
image reconstruction. This is accomplished by the steady-state harmonic conversion
u x, y, z, t( ) = Re u x, y, z( )eiωt{ } where u is the spatially varying complex-valued displacement
vector phasor in which case (1) becomes
∇ ⋅ G∇u + ∇ λ + G( )∇⋅ u = iωα − pω2( )u (2)
Using (2), several options have been implemented including (i) the case where tissue density, ρ,
is assumed known and assigned a homogeneous value throughout the breast while the shear
modulus, G, Lame's constant, λ, and damping coefficient, α, are estimated spatially and (ii) the
case where tissue density, ρ, shear modulus, G, and Lame's constant, λ, are estimated spatially
while the damping coefficient, α, is set to zero.
Figure 3.9: T2 weighted FSE image of the breast with the regions of low damping coefficient outlinedin red. The regions all appear to be isolated by fibrous encapsulation.
3.2.4 SYSTEM PERFORMANCE
The system performance has been evaluated in several ways. Standard linear system
analysis has been applied: high contrast resolution was measured by the MTF and low contrast
resolution was measured by noise sensitivity methods. The MTF measures the spatial resolution
(high contrast resolution) of the MRE process and was obtained from shear modulus images of a
phantom with a cylindrical inclusion that approximated a discontinuity in the shear modulus.
There is some melting of the gels when an inclusion is added to a hot background gel so the
discontinuity is not perfect but it provides a lower bound on the MTF; i.e., the MTF may be
significantly better than that shown. The MTF’s were obtained by averaging the conditioned
ratio of the Fourier transform of the assumed discontinuity in the shear modulus image, where
the position of the discontinuity was found from the magnitude image, with the Fourier
transform of the shear modulus. The position of the discontinuity was obtained from the
magnitude image. MTF estimation is always limited by noise so we have taken a conservative
effective spatial resolution of 0.50 MTF. The MTF’s indicate the effective limiting resolution in
the shear modulus is ~1 to 3mm when a conservative detection threshold of 0.50 was used. This
can be compared to the 0.5 mm limiting resolution in the magnitude image.
The combination of noise and spatial resolution limits can be seen in the contrast detail
experiments [87]. The limiting resolution for detection and for accurate characterization of the
shear modulus are both provided in that reference. Reconstructions are not only spatially
accurate in terms of localizing the regions of increased stiffness, but also are quantitatively
correct (within 10–15%) in terms of characterizing the shear modulus. Accurate characterization
was obtained at 5mm for the lowest contrast inclusion (2:1 relative to the background).
3.3 RESULTS
3.3.1 CLINICAL IMAGES
Our clinical experience with MRE is based on normal subjects (BI-RADS 1) and a
limited number of abnormalities. Two representative examinations are shown in Figures 3.10
Figure 3.10: MRE images (magnitude left, shear modulus, right) from a breast with (BI-RAD 1) normal findings. Correlation between the stiffer properties (in red) and the glandular structure is evident
CancerCancer
Figure 3.11: An MRE examination of a subject with a malignancy in the upper outer quadrant which correlates well with the tissue that presents the stiffest mechanical properties.
and 3.11. Figure 3.10 shows the close correspondence between the fibro-glandular tissue in the
center of the breast, appearing dark in the MR magnitude images and the stiffer regions, shown
in yellow and red, in the shear modulus images. The fatty zones around the periphery are softer.
An exam with breast cancer is illustrated in Figure 3.11. The malignancy appears in the shear
modulus images as the highlighted mass. It was located in the upper outer quadrant on
mammography and this position matches well with the stiffest properties in the breast according
to the elastograms. The results of all the clinical cases imaged are shown in Figure 3.12.
3.3.2 SHEAR MODULUS COMPARISON IN REPRESENTATIVE
ABNORMALITIES
The shear modulus values of normal fatty and fibroglandular tissue found using our MRE
Figure 3.12: Shear modulus values of all of the abnormalities imaged compared to normal fat and glandular tissue. The carcinoma imaged in a screening venue (was not being treated) is much harder than normal tissues and the fibroadenoma. The neoadjuvant therapy patients are more complicated; carcinoma is always harder than normal tissue but subject 1903 was significantly softer following chemotherapy. Both patients responded to chemotherapy but 1903 was responding more fully during imaging than 1904 was.
system has been shown to agree well with the literature [89]. A plot showing the shear modulus
values for representative abnormalities is presented in Figure 3.12. There are not enough cases
to draw anything other than broad conclusions. However, all of the shear moduli of carcinomas
are stiffer than the fat and fibroglandular tissue in the same patient. The single screening
carcinoma is much stiffer than the fat and fibroglandular tissue in any of the normal subjects.
The screening carcinoma is also much stiffer than the fibroadenoma. The shear moduli of the
neo-adjuvant therapy patients is somewhat more complicated. The regions averaged were taken
from the areas of high contrast uptake and the anatomical images. Patient 1903 was imaged on
two dates, and the shear modulus of both breast cancers decreased on the second session. The
shear moduli of the carcinoma, and of the normal tissues, were much lower than in the screening
case, although still stiffer than normal tissues in the same patient. The cancer values were in the
range of stiffness present in normal subjects. This patient was well into the second
chemotherapy regimen when imaged and none of the treatments showed clinical response. The
cancer was open and ulcerative with discharge at times of examination. The left breast had a
dramatic response to an alternative therapy after imaging. The lesion in subject 1904 was very
difficult to image with mammography and was difficult to palpate because of breast size and
density. Nonetheless, the lesion was clearly seen on MRE. The cancer was very stiff in MRE
which agrees with the lack of response shown in CT and physical examination. Also, there was
substantial residual cancer in the breast on extensive lumpectomy after 4 cycles of treatment on
10/23/03.
3.4 CONCLUSIONS
Evidence that MRE will be able to contribute significantly toward earlier detection, with
the associated improvement in patient outcomes, continues to accumulate. However, as with any
new method there are more relevant clinical and biological questions about where MRE might fit
into the diagnostic process than there are answers: Can MRE reduce the number of biopsies as a
follow up study for abnormalities found on mammography? Or might it be useful in screening
high risk populations? Or might it be useful in following treatment? All are possible
applications that should be pursued, especially when the biological causes of increased stiffness
are better understood. MRE is poised to allow the biological mechanisms that influent stiffness
to be understood. The underlying biological mechanisms that increase the stiffness in cancer are
not known [92]; increased tissue pressures and increased amounts of collagen in the
extracellular matrix, as well as increased cross linking of that collagen, are all present in cancer
and all impact tissue stiffness [91, 92]. Further, the impact of increased stiffness on cancer’s
etiology is not known; there is evidence that the stiffness of the extracellular matrix influences,
or even initiates, the malignant transformation [90]. Understanding what biological mechanisms
are reflected by increased stiffness and how increased stiffness influences malignant progression
will show us where elastography can contribute clinically toward improving diagnosis and
treatment of breast cancer. MRE has a unique role to play in obtaining that understanding
because it is robust and quantitative.
3.5 ACKNOWLEDGEMENTS
Supported by P01-CA-80139, R01-NS-33900, and R01-DK-063013.
4 MODALITY-INDEPENDENT ELASTOGRAPHY
4.1 BACKGROUND
Model-based methods for non-rigid image registration may provide a potentially new
framework for characterizing breast disease. This section recasts the conventional non-rigid
image registration problem as a model-constrained constitutive property reconstruction whereby
soft tissue mechanical properties are quantified and potentially used to assess breast tissue health.
The algorithm used is a non-linear optimization that couples a biomechanical model to an image
registration framework such that the mechanical properties (e.g. Young’s modulus) of the
deforming tissue become the driving parameters for improved image registration. This technique
is multi-resolution and is used to quantitatively evaluate the elastic properties in simulation and
in a tissue-like phantom with two embedded inclusions. The results demonstrate good
localization and quantification of Young’s modulus contrast in both simulations and data.
A central goal within the field of non-rigid image registration is to provide
correspondence between spatial coordinates in one image set (often referred to as the source) to
that within a second (often referred to as the target) where the relationship between spatial
coordinates is non-linear. The alignment function may be challenged in a variety of ways. For
example, intra-modal image data of a subject may be available but non-rigid discrepancies may
be present due to physical, mechanical, or acquisition processes as in increased brain atrophy in
Alzheimer’s [93], deformations due to surgical intervention [94], or distortions by the imaging
unit itself [95]. Regardless of the source, of paramount importance within these applications, is
to automatically quantify and/or correct for these non-rigid movements using image processing
techniques.
One way of classifying non-rigid registration methods is by the underpinning numerical
technique that non-rigidly deforms the source to the target. In this classification, one possible
category would use interpolating/approximating functions such as cubic, or B-splines as the
means for the deformation process [96]. A second category would be to base deformations on a
physical model grounded within mechanics. Deformation processes described by these
techniques use the natural constitutive behaviors associated with biomechanical elastic or fluid
models to drive the non-rigid motion [97, 98].
While each category has its strengths and weaknesses, the underpinning motivation is to
transform one image set to another by any means within the computational framework possible
such that the source and target have maximum correspondence. The work presented in this
section takes a different viewpoint of the registration problem within the context of model-based
methods. Rather than viewing the model and its parameterization as an alignment with many
unconstrained degrees of freedom, this work utilizes a constrained model whereby the
constitutive parameters are the only degrees of freedom allowed to achieve a “best-fit” source-to-
target image match. By casting the problem in this manner, the parameters determined take on a
functional role that may potentially be used for tissue characterization. Although non-rigid
image registration and subsequent model analysis has been used by investigators for the
understanding of tissue mechanics (e.g. MR tagging [99], and scar assessment [100]), this
section presents a more instrumented depiction of non-rigid registration. In some sense, this
outlook represents a new class of algorithm that is focused at diagnostic probing of tissue via
model-based image registration. This methodology has been preliminarily tested within breast
and dermoscopic applications [40-43].
4.2 MODALITY INDEPENDENT ELASTOGRAPHY METHODS
4.2.1 BIOMECHANICAL MODEL
A central component to model-based inverse problems is the manner in which the
continuum is represented. While the constitutive model that best describes the deformation
mechanics of tissue is more complex, for this initial work, a linear elastic model has been
employed. The partial differential equation that expresses a state of mechanical equilibrium can
be written as:
0=σ•∇ (4.1)
where σ is the Cartesian stress tensor. The relationship between stress and strain was assumed to
be Hookean. Within the experiments presented below, two separate assumptions were made to
reduce the dimensionality of the problem from three to two dimensions. For the simulation
experiments concerned with performing reconstructions on MR and CT frontal breast image
slices, a condition of plane strain was assumed since the out-of-plane strains would be
negligible. For the experiments that involved the phantom material, a condition of plane stress
was assumed since the phantom geometry was thin and out-of-plane stresses are considered
negligible [101]. In plane stress,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
γεε
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
ν−ν
ν
ν−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
τσσ
xy
y
x
xy
y
x
/)()(
E
21000101
1 2 (4.2)
describes the constitutive relationship between the Cartesian stress tensor [σx, σy, τxy] and strain
tensor [εx, εy, γxy]. Similarly, in plane strain,
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
γεε
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
ν−ν−
ν−νν−ν
ν−ν+ν−
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
τσσ
xy
y
x
xy
y
x
)()(
)/()/(
))(()(E
122100
011011
2111 (4.3)
These assumptions allow for the simplification of Cauchy’s law from 36 stiffness constants to 2
(Young’s modulus, E, and Poisson’s ratio, ν) and a reduction in the dimensionality, respectively.
4.2.2 MODALITY INDEPENDENT ELASTOGRAPHY
The method developed to reconstruct the Young’s modulus values within tissue and
tissue-like phantoms is called Modality Independent Elastography (MIE) [40-43]. MIE begins
by acquiring a source/target image pair that differs due to an applied deformation from a
mechanical device. A finite element model is constructed from the source image and boundary
conditions are determined (in this case, by inspection of the acquired image data or by careful
device construction). After the model has been generated, two discretization processes are
performed: (1) the model domain is separated into a prescribed number of regions whereby each
has a distinct set of elastic properties that are spatially homogeneous, and (2) the target image
domain is separated into a prescribed number of zones. For the remainder of the chapter, regions
Figure 4.1. Example of K-means region formation using 16 material property regions.
will refer to the local domains associated with the mechanical properties of the model, while
zones will refer to the local domains associated with the target image. For region partitioning, a
K-means clustering approach groups element centroids into a user-prescribed number (N) of
regions such that the sum of all point-to-region centroid distances is minimized. For this work,
the implementation in the MATLAB (MathWorks, Natick, MA – www.mathworks.com)
Statistics Toolbox Version 5 was used. Figure 4.1 illustrates an example of this approach on the
rectangular domain whereby the element centroids have been clustered into 16 separate regions.
In the realization of MIE presented here, a multi-resolution strategy has been employed whereby
coarser resolutions (i.e. fewer regions) are used at the initiation of the reconstructive process and
progressively finer resolutions are employed in subsequent iterations. Previous work used only a
single property resolution [41, 43]. In more recent developments, five progressively finer
resolutions were used (16, 64, 256, 512, and 800 regions) while the number of image comparison
zones ranged between 200-400. The zones were rectangular in shape and distributed uniformly
within the deformed target image. The number of zones was based on previous work that
qualitatively studied reconstruction performance with respect to zone size [41].
Briefly stated, the reconstruction algorithm begins by assigning an initial Young’s
modulus to each of the regions at the first resolution, e.g. 16 regions (Poisson’s ratio was held
constant at 0.485). Once a material property description is prescribed, the biomechanical finite
element model is solved for the tissue displacements. These displacements are then used to
deform the source image. This model-deformed source image is then compared to the target
image over each individual zone using an image similarity method (correlation coefficient was
used in the work presented here) [102]. Other image similarity methods have been used in
previous work [41, 43]. Modulus values in the regions are updated based on maximizing the
similarity between the model-deformed source image and the target image over all the similarity
zones until a tolerance is reached or the desired number of iterations has been completed. Upon
reaching a stopping criterion, the material property description is interpolated onto the
distribution associated with the next resolution and the optimization steps are repeated. It has
been shown in [42] that use of the multi-resolution technique can result in the avoidance of local
minima and improved elastography image reconstruction.
The parameter optimization framework can be portrayed as the minimization of a least
square error objective function:
( ) ( ) ( )⎭⎬⎫
⎩⎨⎧ −=φ
2
ET ESESminErrr
(4.4)
where ( )TESr
is the similarity value achieved when comparing the target image to itself (i.e. the
maximum value for the similarity metric) and ( )EESr
is the similarity between the model-
deformed source and the target image using the current estimate of Young’s modulus. Using a
Marquardt [103] approach to equation (4.4), it can be written as:
[ ][ ]{ } [ ] ( ) ( ){ }EtTT ESESJEI]J][J[
rrr−=Δα+ (4.5)
where ]J[ is the M x N Jacobian matrix of the form ( )EES
J Er
r
∂
∂= where M is the number of zones,
and N is the number of regions ( ]I[ is the identity matrix). The details of equation (4.5) have
been reported previously [40, 41, 43]. In these types of inverse problems, the increase in
convergent parameter space afforded by Marquardt’s method is particularly important. The
regularization parameter α was determined using the methods described in [104]. Figure 4.2a is
a flowchart for the multi-resolution approach where equation (4.5) is being iteratively solved
within the ‘Stiffness Parameter Optimization’ process block. Figure 4.2b illustrates an example
Image Data
Boundary Conditions
Finite Element Model
Region & Zone Discretization
Select Resolution
Stiffness ParameterOptimization
(N Properties Found)
N regions
Final resolutionbeen achieved?
Go
to n
ext
reso
lutio
n
no
Elasticity Image
yes
Image Data
Boundary Conditions
Finite Element Model
Region & Zone Discretization
Select Resolution
Stiffness ParameterOptimization
(N Properties Found)
N regions
Final resolutionbeen achieved?
Go
to n
ext
reso
lutio
n
no
Elasticity Image
yes
Image Data
Boundary Conditions
Finite Element Model
Region & Zone Discretization
Select Resolution
Stiffness ParameterOptimization
(N Properties Found)
N regions
Final resolutionbeen achieved?
Go
to n
ext
reso
lutio
n
no
Elasticity Image
yes
a
3616 64 400 256 3616 64 400 256
bFigure 4.2. (a) Flow chart for multi-resolution MIE: a functionalized image registration method,and (b) demonstration of multi-resolution reconstruction of a single inclusion with increasing number of property regions (16, 36, 64, 256, 400 regions).
of mutli-resolution results with a single-inclusion simulation whereby the initial resolutions
provide localization while higher resolutions begin to capture more subtle shape changes. It
should be noted that in addition to regularization, spatial averaging and solution relaxation are
also employed within the elasticity imaging framework. These operations have been found to
improve the stability of the reconstruction algorithm.
4.3 BREAST AND PHANTOM EXPERIMENTS
Since this method is based on image similarity, the method is independent of any
particular imaging modality, as the name suggests. As a result, the method is solely dependent
on intensity structural contrast with a given image pair. In order to demonstrate this, three
different modalities are present in the experiments below: (1) X-ray computed tomography (CT),
(2) magnetic resonance (MR), and (3) optical.
4.3.1 BREAST EXPERIMENTS
Simulation reconstructions were performed using image slices extracted from breast
image volumes obtained from CT and MR scans (see Figure 4.3). Although these were taken
from two different patients, the images were selected to be approximately corresponding slices
~2 cm away from the chest wall in the frontal orientation of the standard anatomical position.
The simulations were set up in the same manner, using either one or two inclusions of about 1
cm in diameter embedded within the true elasticity distribution and a small compression (~8%
strain) in the cranial-caudal direction. The relative stiffness of the inclusions was designated to
Figure 4.3. Images slices of breast tissue extracted from a CT volume (left) and MR volume (right) used in simulation study of the ability of the reconstruction method to utilize disparate image data types.
be 5.7:1 for consistency with the material testing data and also because the value is fairly
representative of breast tumor properties [2]. The plane strain model approximation was used in
the breast simulation trials, progressing through resolutions of N = 24, 64, 256, and 576 regions
using M = 200 zones. The reconstruction method was then run for all four test cases (1 tumor
CT, 2 tumors CT, 1 tumor MR, 2 tumors MR).
4.3.2 PHANTOM EXPERIMENTS
In order to test our inverse problem framework with real data from yet another imaging
modality, a two-material phantom membrane model was constructed. The bulk of the phantom
was made using Smooth-On™ Evergreen 10 polyurethane (Smooth-On, 2000 Saint John Street,
Easton, PA). Two 1.5cm cylindrical inclusions were placed within the membrane phantom and
were made of a stiffer polyurethane material (Evergreen 50). The inclusion material was chosen
for its relative stiffness to that of Evergreen 10 and the similarity in color to the Evergreen 10. A
permanent marker was used to place a texture pattern on the membrane. Figure 4.4a-b shows the
skin phantom used for data collection in this series of experiments with the inclusions designated
in Figure 4.6. Membrane data was collected in a pre- and post- stretched state (compare Figure
(a) (b) (c)(a) (b) (c)Figure 4.4. Phantom membrane (a) before and (b) after mechanical stretching. (c) is a difference image.
4.4a to 4.4b with difference image Figure 4.4c). A commercial webcam (Logitech QuickCam
Pro 4000, 960 x 1280 pixel resolution) was rigidly mounted above the membrane and acquired
the image pairs.
In addition, independent material testing using a compression testing device was
performed on each of the phantom materials in separate tests. The stress-strain behavior of each
material was modestly non-linear. Figure 4.5 is an example stress-strain curve for the bulk
material used in the experiment. Young’s modulus values were determined in a piece-wise
linear fashion across the entire stress regime. For the strains observed in Figure 4.4a-b, the
Young’s modulus for the bulk and inclusion material was approximately 147 kPa and 865 kPa,
respectively, which gives an inclusion-to-bulk stiffness contrast ratio of 6:1.
With respect to model calculations, equation (4.1) was solved using the Galerkin finite
element method [105]. The computational domain involved 1255 nodes and 2367 elements
(approximate 3 mm element edge length). The mesh domain is shown below in Figure 4.6a.
Figure 4.5. Example stress-strain curve for the bulk material as measured with a material tester.
Boundary conditions for the model were determined manually from the image data and
prescribed as boundary displacements. Since Dirichlet type boundary conditions are solely used,
the elastic model is only sensitive to Young’s modulus contrast. Without applying a boundary
stress or designating one material property value within the domain, absolute properties cannot
be determined. For comparison between experimentally measured (as performed by material
testing) and reconstructed Young’s moduli, transects T1 and T2 were designated in Figure 4.6b
whereby the Young’s modulus along the transect was compared to the bulk material’s modulus.
With the optical images, two specific reconstructions were performed. The first
experiment imposed a Young’s modulus distribution on the domain shown in Figure 4.6a-b.
This description matched the inclusion sizes and locations to those of the physical phantom. The
assigned property values reflected a 6:1 Young’s modulus ratio between inclusion and bulk
material, respectively. The image shown in Figure 4.4a was deformed in simulation with the
inclusions present in the model. Beginning with a homogeneous guess at the Young’s modulus
description, the finite element domain, the source image, and the simulated model-deformed
target image, the multi-resolution MIE algorithm was initiated.
1
2
T1
T2
b
T1
T2
a
1
2
T1
T2
b
T1
T2
a Figure 4.6. (a) Finite element domain used in MIE reconstruction, and (b) image domain showing location of inclusions (dotted lines) and transects (T1, T2) used in analysis.
The second reconstruction experiment repeated the above steps but the acquired target
image shown in Figure 4.4b was used in lieu of the simulated target image. For determining the
Young’s modulus contrast ratio, it required the determination of the homogeneous value within
the reconstruction so as to generate the ratio throughout the image. To accomplish this, the
average Young’s modulus within the central area of the phantom was used as the value for the
bulk material property value.
4.3.3 RECONSTRUCTION QUALITY EVALUATION
The fidelity of the elasticity reconstruction was evaluated on its ability to detect the
presence of an inclusion based on classification of the material property distribution, and the
retrospective accuracy of localizing the lesions. The elastic properties as a whole were treated as
a Gaussian mixture of two classes and separated by a threshold established via the method
described in [106]. The likelihood of detecting a lesion in the elasticity image was found using
the contrast-to-noise ratio as defined by [87, 107]:
22
2)(2
BL
BLCNRσσμμ
+−
= (4.6)
where μ and σ2 are the sample mean and variance of a material property distribution and the
subscripts L and B denote the lesion and bulk material classes, respectively. Values of
significance for successful detection and localization were set at CNR ≥ 2.2 as noted by [87].
The average modulus contrast is found from the ratio of the means of the two material classes,
and a peak modulus contrast value is also reported by taking the ratio of two manually selected
homogeneous regions of approximately equal area known to be representative of the two
materials.
4.4 BREAST AND PHANTOM RESULTS
4.4.1 BREAST RESULTS
Figure 4.7. Reconstructions of simulation trials for the CT breast slice using a single inclusion (left) and two inclusions (right). The true inclusion boundaries are overlaid in each elasticity image.
Figure 4.8. Reconstructions of simulation trials for the MR breast slice using a single inclusion (left) and two inclusions (right). The true elasticity distributions are also shown (top row) for comparison.
Figures 4.7 and 4.8 show the final reconstruction results for the CT and MR breast slice
simulations using either one or two inclusions. In both test scenarios, the resolvability of the
stiffer material was found to be adequate according to the CNR threshold, but definitely higher in
the MR-derived elasticity images.
4.4.1 PHANTOM RESULTS
(a) (b)
(c) (d)
Contrast Ratio
(a) (b)
(c) (d)
(a) (b)
(c) (d)
Contrast Ratio Figure 4.9. Idealized Young’s modulus contrast images at resolutions of (a) 64, (b) 256, (c) 512, and (d) 800 material regions (black contour in (d) is location of inclusion).
Figure 4.9 illustrates the performance of the MIE framework on a 2-inclusion phantom
simulation. Figure 4.9a, 4.9b, 4.9c, and 4.9d represent the elasticity image progression through
resolutions of 64, 256, 512, and 800 regions, respectively (although the 16 region resolution was
used, it is not reported here). Comparing Figure 4.9a through to 4.9d, the Young’s modulus
contrast improves as well as the spatial definition of the inclusions. Figure 4.9d also has a black
contour designated which represents the true position and size of the inclusion. Figure 4.10
illustrates the Young’s modulus contrast ratio through the transects shown in Figure 4.6b for
each inclusion in Figure 4.9d.
Distance along transect (mm)
You
ng’s M
odul
us C
ontra
st
Distance along transect (mm)
Youn
g’s
Mod
ulus
Con
trast
T2
T1
Distance along transect (mm)
You
ng’s M
odul
us C
ontra
st
Distance along transect (mm)
You
ng’s M
odul
us C
ontra
st
Distance along transect (mm)
Youn
g’s
Mod
ulus
Con
trast
Distance along transect (mm)
Youn
g’s
Mod
ulus
Con
trast
T2
T1
Figure 4.10. Young’s modulus contrast along transects defined in Figure 4.6b using simulation data. The image used to construct transect was 800 region elasticity image - Figure 4.9d.
(a) (b)
(c) (d)
Contrast Ratio
(a) (b)
(c) (d)
(a) (b)
(c) (d)
Contrast Ratio Figure 4.11. Experimental Young’s modulus contrast images at resolutions of (a) 64, (b) 256, (c) 512, and (d) 800 material regions (black contour in (d) is location of inclusion).
Distance along transect (mm)
You
ng’s M
odulus
Con
trast
Distance along transect (mm)
You
ng’s M
odulus
Con
trast
T2
T1
Distance along transect (mm)
You
ng’s M
odulus
Con
trast
Distance along transect (mm)
You
ng’s M
odulus
Con
trast
Distance along transect (mm)
You
ng’s M
odulus
Con
trast
Distance along transect (mm)
You
ng’s M
odulus
Con
trast
T2
T1
Figure 4.12. Young’s modulus contrast along transects defined in Figure 4.6b using experimental data. The image used to construct transect was 800 region elasticity image -Figure 4.11d.
Figure 4.11a illustrates the performance of the MIE framework on the 2-inclusion phantom data
acquired. Figure 4.11a, 4.11b, 4.11c, and 4.11d represent the elasticity image progression
through resolutions of 64, 256, 512, and 800 regions, respectively (although the 16 region
resolution was used, it is not reported here). Comparing Figure 4.11a through 4.11d, the
Young’s modulus contrast improves as well as the spatial definition of the inclusions. In
addition, the results are similar in contrast to that of the idealized of Figure 4.9. Figure 4.11d
also has a black contour designated which represents the true position and size of the inclusion.
Figure 4.12 illustrates the Young’s modulus contrast ratio through transects shown in Figure 4.6b
for each inclusion. Figure 4.13 reports the similarity error at each resolution as calculated by
( )21
j
zones M
1i
2ij resolution CC1 ⎥
⎦
⎤⎢⎣
⎡−=ε ∑
=
for j={1,2,…,5} (4)
where CC is the correlation coefficient. Of note, this error was normalized to the maximum
Resolution (# of K-regions)
Nor
mal
ized
Sim
ilarit
y E
rror
datasimulation
Resolution (# of K-regions)
Nor
mal
ized
Sim
ilarit
y E
rror
datasimulation
Figure 4.13. Comparison of normalized error for simulation and experimental data systems. The error shown was the final error at each resolution; however, the error bars represent the standard error of the mean similarity error for each resolution.
similarity error reported in the experimental data-driven Young’s modulus reconstruction (as
opposed to the simulation). The error bars at each resolution are the standard error of the mean
similarity error for each resolution.
4.5 DISCUSSION
Although some discrepancies exist, the reconstructions performed with simulation and
experimental data across all modes are similar in fidelity. The reconstruction results in Figures
4.7 and 4.8 using the CT and MR breast images demonstrate that sufficient structural content is
present for the MIE method to achieve localization and good contrast. The membrane
experiments are also encouraging because despite nonlinear model-data mismatch, out-of-plane
distortions during stretching, and potential boundary condition mis-mapping, the elasticity
images demonstrated good localization and quantification. Furthermore, the reconstruction was
achieved with a non-pigmented lesion, thus indicating that deflections of the image pattern
(which are quite subtle in these experiments) and not the lesion image intensity itself are
responsible for the changes in the elastic modulus values within the elastography framework.
The effects of the multi-resolution strategy can also be seen more clearly in the idealized
image results when comparing Figure 4.9b to 4.9d whereby with the lower resolution, the stiff
inclusion is slightly shifted. In Figure 4.9b, the diameter of the inclusion is approximately equal
to the diameter of 2 material property regions while in Figure 4.9d, it includes 8 regions.
Undoubtedly, this increase in resolution allows for better localization and enhanced matching of
contrast.
Comparing transects among Figures 4.10 and 4.12 demonstrate that while the contrast
ratios are similar between ideal and experimental, the degree of shape conformity is qualitatively
better under the ideal circumstances. It is somewhat troubling that better transect shape
conformity and contrast matching were not achieved in the idealized reconstruction. However,
we have found, in experiments not included here, that when the amount of pattern is increased
within the inclusion, better shape integrity and contrast can be achieved. Although this is
somewhat intuitive, this suggests that developing metrics to rate the complexity and density of
image pattern in relation to algorithm success will be important and is currently under
investigation. It should also be noted that the number of regions being used even at the highest
resolution is still relatively coarse. The difference in contrast ratio through the resolutions
suggests that improved region distribution may improve contrast.
Figure 4.13 compares the error in similarity between the idealized and experimental
systems over the varying resolutions. The difference in magnitude between these curves is
undoubtedly due to inaccuracies in the model, potential lighting changes, and imperfect
boundary condition mapping (all of which cannot occur within the simulated data since these are
controlled by the user in the simulation). Another interesting observation is that the difference
between the two curves remains relatively constant. This may suggest that the degree of
reconstructed similarity is similar despite the presence of significant system noise. The standard
error bars within the figure are an indication of parameter optimization effort at each resolution.
It is interesting to note that both in simulation and data, the initial resolutions affect the
minimization process more substantially then later resolutions. It is also interesting to note that
the effort (as noted by the standard error bars) continually reduces for the simulation, which is
not the case for the data reconstructions. Based on these observations, it is evident that reduced
resolution allows for a more thorough evaluation of the function space while the increasing
resolutions allow for modest changes to refine parameters.
4.6 CONCLUSIONS
In this section, a novel multi-resolution image registration method is presented which is
capable of generating functional parameters (here, Young’s modulus) that characterize the
stiffness of soft tissue. Given the long standing history of associating tissue health with its
perceived mechanical properties via palpation, many applications exist for this work to include
breast/skin/prostate/lung cancer, vascular disease, and cardiac function evaluation.
The multi-resolution MIE architecture reported allows monitoring of reconstruction
quality and (in results not reported here) has been more robust than single-resolution versions. In
addition, the results shown here report the use of MIE on CT, MR and optical images.
MIE-based approaches to elasticity characterization represent a new class of algorithms
that may yield potentially new frameworks for disease characterization. MIE constrains the non-
rigid image registration problem to a specific deformation model. As a result, the parameter
fitting associated with maximizing source-to-target image registration becomes functionalized.
While this method is not as mature as its USE and MRE counterparts, it does offer a new avenue
for elastographic research that is more akin to the image processing and non-rigid image
registration community. With substitutions of other models, the implications for this framework
in other morphometric analyses such as Alzheimer’s disease or tumor growth are equally
intriguing.
4.7 ACKNOWLEDGEMENTS
The authors would like to thank John Boone, PhD of the University of California-Davis,
Department of Radiology and Tom Yankeelov, PhD of the Vanderbilt University Institute of
Imaging Science for their generous contributions of breast image data (CT and MR,
respectively). This work was supported in part by the Whitaker Foundation Young Investigator
Award Program, and the Congressionally Directed Medical Research Program – Breast Cancer
Research Program Predoctoral Traineeship Award.
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