Phys. Med. Biol. 45 (2000) 1511–1520. Printed in the UK
PII: S0031-9155(00)06938-4
An effective ultrasonic strain measurement-based shear
modulus reconstruction technique for superficial
tissues—demonstration on in vitro pork ribs and in vivo
human breast tissues
Chikayoshi Sumi†, Kiyoshi Nakayama† and Mitsuhiro Kubota‡
† Department of Electrical and Electronics Engineering, Sophia University, Tokyo 102-8554,
Japan
‡ School of Medicine, Tokai University, Kanagawa 259-1193, Japan
E-mail: [email protected]
Received 16 August 1999, in final form 7 April 2000
Abstract. An effective shear modulus reconstruction technique is described which uses ultrasonic
strain measurements for diagnosis of superficial tissues, i.e. our previously developed ultrasonic
strain measurement and shear modulus reconstruction methods are combined and enhanced. The
technique realizes very low computational load, yet yields fairly high quantitativeness, high stability
and spatial resolution, and large dynamic range. The suitability of the method is demonstrated on
in vitro pork ribs and in vivo human breast tissues (fibroadenoma and scirrhous carcinoma).
1. Introduction
Over the past few years we have been working towards the development of an ultrasonic
strain measurement-based shear modulus reconstruction technique for diagnosis of living soft
tissues, e.g. breast, liver and kidney (Yagi and Nakayama 1988, Sumi and Sudou 1998). Our
efforts parallel those of several other research groups who have been developing breast strain
measurement/imaging methods (Ophir et al 1991, Garra et al 1997, Chaturvedi et al 1998).
Using a previously reported iterative rf-echo phase-matching method, we recently obtained
fairly accurate estimates of small strain tensor components (<1%) generated in vivo in soft
tissues by extracorporeally applied pressures or spontaneous heart motion (Sumi et al 1995b,
Kubota et al 1996, Sumi 1999); and in other closely related work (Sumi et al 1993, 1995a,
Sumi and Nakayama 1998) we proposed (i) a suitable configuration technique for mechanical
sources/reference regions and (ii) a stable numerical solution of an inverse problem in which
only strain data are used for reconstructing a globally relative shear modulus distribution
with respect to reference shear moduli. Here we describe how these methods are suitably
combined and enhanced for particular application to reconstructing the absolute shear modulus
distribution in superficial tissues such as those in the human breast (Sumi et al 1999, Sumi
and Kubota 1999). The suitability of the technique is demonstrated on in vitro pork ribs and
in vivo human breast tissues (fibroadenoma and scirrhous carcinoma).
0031-9155/00/061511+10$30.00
© 2000 IOP Publishing Ltd
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2. Description of the technique
Our reconstruction technique is constructed assuming that a conventional linear array type
transducer will be used. Since the subcutaneous region of interest (ROI) has a macroscopically
layered tissue structure that does not include obstacles such as bones, by placing a thin reference
material with a suitable shear modulus value between the transducer and body surface, and by
mechanically compressing it using the transducer such that the ROI predominantly deforms in
the axial direction (x) in the scanning plane, it is reasonable to assume that a 1D axial stress
distribution is generated on each ultrasonic beam line (y) in the ROI. Such a deformation
condition can be realized by mounting the transducer onto a large plate (Ophir et al 1991,
Garra et al 1997) or by placing the target tissue between plates such that the ROI is parallel
to the plates (Sumi 1999). These configurations allow the acquisition of paired rf-echo data
frames, i.e. ones under pre- and post-compression.
The following ordinary differential equation (ODE) holds for an unknown shear modulus
distribution G(x, y) on each line y in the ROI (Sumi et al 1993, 1995a), i.e.
d
d
xx
(1)
ln G = − xx .
dx
dx
Therefore, if iterative phase matching (Sumi et al 1995b, Sumi 1999) is applied to the paired
echo data frames in order to measure the strain distribution xx (x, y), as done in simple phantom
experiments, then it might be possible to quantitatively estimate the relative shear modulus
distribution in the ROI by calculating the ratios of the measured strains with respect to that
of a reference point on each line y. Although 1D phase matching will normally result in a
lower computational load than 2D matching, obtaining sufficient measurement accuracy for
practical shear modulus reconstruction dictates using the latter method. On the other hand,
since it is possible to control tissue displacement such that it is sufficiently small (<0.5 mm),
by accordingly employing a small local window size, matching iterations are substantially
reduced.
In any case, when dealing with soft tissues, even if 2D matching is performed, due to
measurement errors, singular points occur where the ratio cannot be calculated, namely where
the strain is estimated to be positive. It is for this reason that we developed a regularizationbased robust numerical reconstruction approach (Sumi and Nakayama 1998) for solving
equation (1). Briefly, by assuming that the gradient of the logarithm of the globally relative
shear modulus distribution G(x, y) in the ROI is bounded and smoothly varies throughout
the ROI, the unknown globally relative shear modulus distribution Ĝ(x, y) with respect to
reference shear moduli can be stably obtained as a raw vector g through minimization of the
following functional with respect to g :
2
2
2
T
g g g .
error(g ) = e − EDx
+ α 1 D
+ α 2 D D
(2)
gr gr gr That is, raw vectors g and gr are respectively comprised of the logarithms of the unknown
globally relative shear moduli ln Ĝ(x, y) and the given reference shear moduli in the
ROI, the matrix E of the lexicographically ordered reasonably smoothed strains xx (x, y),
the differential operator Dx of the lexicographically ordered coefficients of the first-order
approximation of derivative of ln G(x, y) along the axial direction x, the gradient operator D of
the lexicographically ordered coefficients of the approximations of the partial derivatives along
the axial and lateral directions, the raw vector e of the lexicographically ordered reasonably
smoothed (−d/dx · xx (x, y)), and the so-called regularization parameters α1 and α2 , which
are pre-specified positive parameters. The regularization parameters adjust the relative fidelity
of the smoothed measurement data and the a priori knowledge. Practically speaking, the
Ultrasonic shear modulus reconstruction
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Figure 1. Conventional B-mode image of in vitro pork rib under pre-compression (ROI size:
15.6 mm × 72.8 mm).
minimization is performed by the conjugate gradient method, and the reconstructed globally
relative shear modulus distribution is imaged using the log grey scale in which relatively soft
regions are bright and relatively stiff regions are dark.
3. Pork rib in vitro experiment
In vitro experiments were performed on the rib cage of a swine slaughtered 6 h previously. A
conventional linear array type transducer of nominal frequency 3.5 MHz, beam pitch 0.75 mm
and contact surface 11.8 cm × 1.8 cm was mounted onto the central part of a 14 cm × 6 cm
plate. As the transducer was intended for abdomen diagnosis, a 35 mm thick high-polymer
ultrasound standoff pad (Kiteco, 3M Health Care Co. Ltd, Tokyo, Japan) was placed between
the plate (transducer) and the rib cage. These configurations allowed collection of pre- and
post-compression (0.32 mm) rf-echo data that were subsequently stored after digitization with
10 bit resolution at a 30 MHz sampling rate. We analysed a rectangular ROI (15.6 mm (x)
×72.8 mm (y)) spreading from 39.3 to 54.9 mm in the axial direction. Figure 1 shows the
B-mode image under pre-compression, where a multilayered structure of fat and muscle is
clearly apparent.
The ROI strain distribution xx (x, y) was accurately measured by just one application of
phase matching to the paired echo data frames using a local window size of 0.8 mm × 6.0 mm,
with the resultant mean value of the measured strains being −0.438%. Unfortunately, six
singular points occurred in the ROI. For example, figure 2 shows the measured strain profile on
an axial line of y = 21.8 mm, where a singular point exists at 52.2 mm (strain = 2.37×10−5 ).
Subsequently, we reconstructed the globally relative shear modulus distribution by setting
nearly all the reference points in the second fatty layer of the ROI, i.e. the second hypoechoic
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Figure 2. Measured strain profile on the axial line of y = 21.8 mm, i.e. xx (x, 21.8).
layer in the B-mode image of figure 1. Specifically, by taking the original reference point as
(40.0 mm, 0.75 mm), i.e. G(40.0, 0.75) = 1.0, and by calculating the ratios of the strains on the
lateral line of x = 40.0 mm with respect to those of the original reference point, the reference
shear moduli on the line were calculated. In addition, by substituting the gradient operator D
in equation (2) for the differential operator D , minimization of equation (2) was performed at
each line y. The regularization parameters α1 and α2 were empirically set at 0 and 5.0 × 10−8
respectively. Figure 3(a) shows the reconstructed globally relative shear modulus profile on
y = 21.8 mm in comparison with that obtained for the non-regularized case in which α2 is
also set at 0, where similar to the results obtained at the five other singular points, the singular
point at x = 52.2 mm is suitably dealt with. It is therefore demonstrated that a reasonably
smooth reconstruction can be obtained over a large dynamic range of 38.6 dB. Note that the
corresponding log grey scaled image in figure 3(b) clearly indicates the multilayered structure
of fat and muscle.
4. Breast tissue in vivo experiments
For the in vivo experiments carried out to obtain shear modulus images of human breast tissue
previously diagnosed as cases of fibroadenoma or scirrhous carcinoma, we used a reference
material (agar phantom; Sumi et al 1995b) with shear modulus value of 1.4 × 106 N m−2 .
Specifically obtained was a 2D reconstruction of the absolute shear modulus distribution
(α1 = 0 and α2 = 1.0×10−6 in equation (2)). Female volunteers were supinely positioned and
both their breasts and the reference material were placed between plates to confine elevational
motion. Although a thin reference is most desirable, a thickness of 40.0 mm was necessary
to prevent the presence of multiply reflected echos in the echo from the ROI. The ultrasound
Ultrasonic shear modulus reconstruction
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(a)
(b)
Figure 3. Reconstruction of the globally relative shear modulus distribution (α1 = 0 in
equation (2)). (a) Profile on the axial line of y = 21.8 mm, i.e. G(x, 21.8)/G(40.0, 0.75), when
α2 = 5.0 × 10−8 and α2 = 0, and (b) image in log grey scale with a dynamic range of 38.6 dB
when α2 = 5.0 × 10−8 .
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(a)
(b)
Figure 4. Breast fibroadenoma tissue elasticity images of a 44.5 mm × 37.4 mm ROI spreading
from 17.8 to 62.3 mm in the axial direction. The reference material (thickness: 40.0 mm) covering
the volunteer’s skin surface had a shear modulus of 1.2 × 106 N m−2 . First trial results: (a) Bmode image under pre-compression, (b) reconstructed absolute shear modulus image in the log
grey scale ranging from 2.4 × 104 N m−2 to 4.9 × 106 N m−2 . Second trial results in mostly the
same region: (c) B-mode image under pre-compression and (d) shear modulus image ranging from
2.5 × 104 N m−2 to 5.1 × 106 N m−2 .
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(c)
(d)
Figure 4. (Continued)
nominal frequency was 7.5 MHz and the beam pitch was 0.48 mm. Two-dimensional phase
matching was performed only once. Volunteers with biopsy-diagnosed breast fibroadenoma
or scirrhous carcinoma were 43 and 59 years old respectively.
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(a)
(b)
Figure 5. Breast scirrhous carcinoma tissue elasticity images of 50.0 mm×42.2 mm ROI spreading
from 17.1 mm to 67.1 mm in the axial direction. The reference material (thickness: 40.0 mm)
covering the volunteer’s skin surface had a shear modulus of 1.2 × 106 N m−2 . (a) B-mode image
under pre-compression and (b) reconstructed absolute shear modulus image in the log grey scale
ranging from 2.7 × 104 N m−2 to 8.6 × 106 N m−2 .
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4.1. Fibroadenoma tissue
Figure 4(a) shows the fibroadenoma tissue B-mode image of a 44.5 mm × 37.4 mm ROI
spreading from 17.8 to 62.3 mm in the axial direction, where an elliptical hypoechoic region
with approximate axial and lateral lengths of 8 and 13 mm appears at the central bottom part
of the image. The mean value of the measured strains in the ROI was −0.623% (0.45 mm
compression). By using reference points inside the ROI upper boundary, i.e. x = 17.8 mm, a
stable reconstruction was obtained as shown in figure 4(b). At a dynamic range of 46.2 dB, the
absolute shear modulus distribution ranges from 2.4×104 N m−2 to 4.9×106 N m−2 with spatial
resolution of 0.8 mm × 3.8 mm. Since the target lesion was predicted to have a rather complex
shear modulus distribution, the reliability of the reconstruction was verified by obtaining an
additional reconstruction using different paired echo data collected mostly in the same region.
Figure 4(c) shows the B-mode image under pre-compression, while figure 4(d) shows the
reconstructed shear modulus distribution ranging from 2.5 × 104 N m−2 to 5.1 × 106 N m−2 .
It is quite noteworthy that these results indicate nearly the same tissue structure.
4.2. Scirrhous carcinoma tissue
Figure 5(a) shows the scirrhous carcinoma tissue B-mode image of a 50.0 mm × 42.2 mm ROI
spreading from 17.1 to 67.1 mm in the axial direction, where at the central part there exists a
very large irregular shaped hypoechoic region whose periphery part appears hyperechoic. The
axial length of the scirrhous carcinoma lesion is about 15 mm. By using the mean value of the
measured strains in the ROI (−0.605%, 0.40 mm compression) in conjunction with reference
points inside the ROI upper boundary (x = 17.1 mm), the absolute shear modulus distribution
was reconstructed as shown in figure 5(b). At a dynamic range of 50.1 dB, it ranges from
2.7 × 104 N m−2 to 8.6 × 106 N m−2 with spatial resolution of 0.8 mm × 3.8 mm; results
indicating a rather stiff lesion, as expected.
5. Concluding remarks
Although the stable shear modulus reconstruction images presented here were obtained in only
5 min using a typical workstation (Compaq XP 1000, 500 MHz), they nevertheless possess
high stability and spatial resolution over a large dynamic range. It is therefore expected that this
newly developed imaging technique has great potential to be practically used as an effective
diagnostic tool for superficial tissues. At present, a variety of in vivo shear modulus images are
being obtained to clarify any limitations for further application as a differentiation technique.
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