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Ultrasonic imaging of 3D displacement vectors using a simulated 2D array and beamsteering
Ultrasonics 53 (2013) 615–621
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Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Ultrasonic imaging of 3D displacement vectors using a simulated 2D array and beamsteering R. James Housden ⇑, Andrew H. Gee, Graham M. Treece, Richard W. Prager Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
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Article history: Received 30 March 2012 Received in revised form 16 October 2012 Accepted 23 October 2012 Available online 1 November 2012 Keywords: Elastography Strain imaging 3D displacement estimation Beamsteering 2D array
a b s t r a c t Most quasi-static ultrasound elastography methods image only the axial strain, derived from displacements measured in the direction of ultrasound propagation. In other directions, the beam lacks high resolution phase information and displacement estimation is therefore less precise. However, these estimates can be improved by steering the ultrasound beam through multiple angles and combining displacements measured along the different beam directions. Previously, beamsteering has only considered the 2D case to improve the lateral displacement estimates. In this paper, we extend this to 3D using a simulated 2D array to steer both laterally and elevationally in order to estimate the full 3D displacement vector over a volume. The method is tested on simulated and phantom data using a simulated 6–10 MHz array, and the precision of displacement estimation is measured with and without beamsteering. In simulations, we found a statistically significant improvement in the precision of lateral and elevational displacement estimates: lateral precision 35.69 lm unsteered, 3.70 lm steered; elevational precision 38.67 lm unsteered, 3.64 lm steered. Similar results were found in the phantom data: lateral precision 26.51 lm unsteered, 5.78 lm steered; elevational precision 28.92 lm unsteered, 11.87 lm steered. We conclude that volumetric 3D beamsteering improves the precision of lateral and elevational displacement estimates. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Ultrasonic elastography is a technique for visualising mechanical properties of tissues. The subject of this paper is quasi-static elastography, which was originally developed in [1]. This involves estimating the relative displacement from one image to the next while continually varying the compression in the tissue. It is usually only the axial displacements that are measured with high precision, because the RF data contains high resolution phase information in this direction. Axial strain is then calculated from the gradient of the axial displacement estimates. Variations in axial strain indicate relative tissue stiffness under the assumption that soft regions of the tissue will deform more than stiff regions. Several potential clinical uses of axial strain imaging have been identified, for example classification of breast lesions [2,3] and visualisation of prostate lesions [4]. Lateral (and in the case of 3D imaging, elevational) displacements are usually of secondary interest because the standard
⇑ Corresponding author. Present address: Division of Imaging Sciences, The Rayne Institute, King’s College London, St. Thomas’ Hospital, London SE1 7EH, UK. E-mail addresses: [email protected], [email protected] (R. James Housden), [email protected] (A.H. Gee), [email protected] (G.M. Treece), [email protected] (R.W. Prager). 0041-624X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2012.10.010
methods used for axial tracking produce lower precision displacement estimates laterally. However, it was shown in [5] that the precision of axial displacements can be improved by lateral tracking. Therefore, lateral tracking is potentially relevant in axial strain imaging applications. In addition, the full 3D displacement vector is of interest for measuring other mechanical properties of tissue, e.g. Poisson’s ratio for poroelastography [5,6], total shear strain for imaging tumour bonding [7] and for estimating the elastic modulus [8]. Multiple angle beamsteering is one possible method for obtaining high-precision lateral and elevational displacements. In 2D imaging, tracking axial and lateral displacements only, several studies have shown that beamsteering improves the precision of displacements perpendicular to the beam [9–11]. This involves acquiring multiple images over a region, each with a different steering angle, so that the tissue is imaged from several different directions. For each angle, displacements are measured by comparing with a corresponding steered image at a different compression. The measured displacement is the component of the actual displacement along the beam direction. The components from each angle are then combined to estimate the axial and lateral components of the displacement field. This is illustrated in Fig. 1a. Throughout this paper, the terms axial and lateral refer to the fixed vertical and horizontal directions in which we are trying to
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θ1
θ1
θ3
dm,2
d
θ3
dm,1 dm,3
(a)
(b)
Fig. 1. Beamsteering principle. (a) 2D beamsteering, with three laterally steered images at angles h1, h2 and h3. Here h1 and h3 are equal angles steering to the left and right and h2 = 0°. The actual displacement vector of a point in the image, d, is measured as components dm, 1, dm, 2 and dm, 3 along the beam direction of each image. These measurements can be used to estimate both the axial and lateral components of d. (b) In 3D beamsteering, this principle is extended to five intersecting volumes, two steered laterally and two elevationally, which allow the 3D displacement vector to be estimated.
estimate displacements. The actual directions of measurements taken along and across the steered beam are referred to as the beam direction and off-beam direction. Although there has been significant research into beamsteering, its advantages have only been shown in 2D with improved precision of lateral displacements. The same principle can be extended to estimate the full 3D displacement vector, including elevational displacements, over a volume using a 2D array probe. This would involve acquiring multiple volumes of data at different steering angles, with some volumes steered laterally and some elevationally (Fig. 1b). In [11], beamsteering equations were derived for the 3D case, but not used on 3D data. In this paper, we present a method to measure 3D displacements with multiple beamsteered volumes, using similar equations to those in [11]. We also present novel experiments comparing the accuracy and precision of 3D beamsteered displacements to those obtained using a single unsteered volume. Results are shown from a simulation of a 2D array probe and a phantom experiment imitating a 2D array using a mechanically controlled 1D array probe. As evidenced below and further highlighted in the Discussion and Conclusions text of Sections 3 and 4, the method proposed offers improvement in the precision of displacement estimation when using beamsteering. 2. Methods 2.1. Displacement estimation Displacement estimation is by one of two methods. The unsteered method tracks displacements between a single pair of volumes with beams perpendicular to the transducer. The steered method uses multiple pairs of volumes, each at a different steering angle. For each steered or unsteered volume pair, displacements along the beam direction are estimated using the Weighted Phase Separation method [12] and two-pass tracking strategy described in [13]. Each displacement estimate is accompanied by a precision estimate calculated from a weighted variance of phase differences between the pre- and post-compression RF data [12]. For the unsteered method, the displacement measurements along the beam directly give the axial displacements. Lateral and elevational displacements are calculated during axial tracking, following the method in [13]. After each axial displacement estimate, the matching correlation is calculated at offsets of ±1 vector off the beam in the lateral direction. Subvector lateral displacements are obtained by a quadratic fit to the correlation value at each lateral offset according to
dl ¼
q1 q1 2ðq1 2q0 þ q1 Þ
where q1, q0 and q1 are correlations of the RF envelope data at 1, 0 and +1 vector offset respectively from the initial lateral alignment. dl is the subvector location of the maximum correlation, giving the lateral displacement relative to the initial alignment. This is then repeated for the elevational displacements. The quadratic fit used here is an approximation to the theoretical Gaussian peak in the correlation values. This lateral and elevational tracking also improves the axial estimates, because the search range on the second tracking pass is initialised with the offsets from the first pass. Full details of the unsteered tracking method are given in our previous publications [12,13]. In the steered method, the volume pair at each angle is initially processed separately using the same phase-based tracking method in the beam direction. Off-beam displacement tracking is not used, because the non-orthogonal nature of the search directions in steered data means that searching in the off-beam direction also affects the estimate in the beam direction. The skewed grid of displacements and associated precision estimates are then linearly resampled to a regular Cartesian grid aligned with the unsteered volume. Beam-direction displacements from the individual angles are then combined in a least squares solution for axial, lateral and elevational displacements, similar to the 3D solution in [11]. The measured displacements are estimates of the projections of the actual displacement onto the beam directions, so the set of equations to solve are dm, i = dui. dm, i is a measured displacement, d = [da, dl, de]T is a vector of the axial, lateral and elevational displacements that we are estimating, and ui is a unit vector in the beam direction. In our implementation, each volume is steered in either the lateral or elevational direction, but not both at once. ui is therefore [coshi, sinhi, 0]T for volumes steered by hi laterally, [cos/i, 0, sin/i]T for volumes steered by /i elevationally and [1, 0, 0]T for an unsteered volume. The equations are solved according to the standard weighted linear least squares approach, with each measurement weighted by its associated precision estimate. It should be noted that the weighted least squares approach assumes that the errors in the displacement measurements, dm, i, are independent. In practice there will be some dependence because the images at different angles will be correlated. Having obtained displacements from either method, we then calculate strain in any one direction by simply taking the difference of the two estimates either side of each estimation location. This is followed by a Gaussian smoothing filter in each of the three directions and weighted by the precision of each estimate. The Gaussian has a standard deviation of one axial estimate spacing.
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2.2. Data acquisition and processing The two displacement estimation methods were compared using both simulated data and a phantom scan. The physical 2D array for the phantom scan was imitated using a mechanically controlled 1D array probe, as detailed below. 2.2.1. Simulated data For the simulation, we first determined a displacement field between a compressed and an uncompressed state using finite element modelling (Abaqus 6.7, Simulia, Rhode Island, USA). The model was a 3D cylinder of radius 7 cm and height 3 cm. It comprised a single linear elastic material, of Young’s modulus 10 kPa and Poisson’s ratio 0.495. The probe face was modelled as a 35 mm 35 mm square rigid compressor on the top surface of the model. In the compressed state, the compressor indented the top surface of the model by 0.3 mm (1% compression). Frictionless slip conditions were defined at the base of the model and in the contact region between the compressor and the top surface. These frictionless surfaces represent the sliding contact provided by ultrasound gel and the disconnectedness of internal tissues, e.g. muscle layers. All other surfaces were unconstrained. RF ultrasound data were simulated using Field II [14,15]. The probe was modelled as a 192 192 element fully-populated array with element spacing 0.245 mm. The transducer centre frequency was 8.5 MHz. Each A-line was sampled at 50 MHz over 1700 samples, giving a depth of 26.2 mm. Focusing for each line was achieved using a set of 32 32 active elements, with a transmit focus depth of 13 mm and dynamic receive focusing. Multiple volumes of 128 128 A-lines and different steering angles were simulated. The angles used were ±12° and ±6° laterally and elevationally, as well as an unsteered volume, giving nine angles altogether. 12° is around the maximum angle at which steering is possible without producing significant grating lobe artefacts. These nine volumes were simulated in both the compressed and uncompressed state, with the scatterer positions adjusted by the displacement in the finite element model for the compressed state. The simulated RF data then had Gaussian white noise added, reducing the SNR of the RF signal to 20 dB. Displacements for each volume were measured using approximately equal aspect ratio 3D matching windows of 8 RF cycles along the beam and 5 vectors in the off-beam directions. Window spacing was 0.6 mm, so that each window overlapped the next by 40% of its length. Over the volume, this resulted in a grid of 44 windows axially and 64 windows laterally and elevationally. These parameters are established defaults of the displacement estimation algorithm for unsteered data [13], providing an acceptable tradeoff between resolution and computation time. 2.2.2. Phantom data For the phantom scan, we used a ULA-OP (Università degli Studi di Firenze, Italy) scanner with a LA523 (Esaote, Genoa, Italy) 6–10 MHz linear array probe. Volumes were acquired using the 1D array by mounting it on a system of motorised linear slides (T-LSR300B, Zaber Technologies Inc., Vancouver, BC, Canada) and stepping the probe elevationally through 192 steps of 0.245 mm. At each elevational step, the probe was steered laterally through five angles: 0°, ±6° and ±12°. This created five volumes of 192 192 A-lines. The probe scanned through a 1.5 mm thick polyethylene compression plate of approximately 8 9 cm. The compression plate was stationary during the probe’s elevational sweep to represent the stationary face of a 2D array probe. To acquire the elevationally steered volumes using a 1D array, the scanning target and compression plate were rotated by 90° and the process repeated. Finally, a similar set of volumes was acquired with the compression plate and probe moved downwards by
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0.5 mm to compress the phantom. The complete acquisition produced ten volumes at each compression, two of which were unsteered. For displacement estimation, only the first of these two was used, so that there were nine volumes available for the steered method and one volume for the unsteered method. This produced the same set of lateral and elevational steering angles as in the simulation, so that the same beamsteering technique of Section 2.1 could be used. The rotation was achieved with the compressor and target standing on a turntable capable of rotating about a vertical axis with a precision of 0°10 . The primary source of misalignment between volumes before and after rotation is the estimate of the location of the rotation axis. Before the main acquisition, this was calibrated by scanning a point target on the turntable at several different rotational positions. The resulting set of points defined a circle, with the centre giving the location of the rotation axis. We have found that this approach is capable of aligning the volumes to within about one lateral vector spacing. This is better than the resolution of the strain images (two vector lateral and elevational window spacing) and is sufficient for our experiments measuring slowly varying displacement fields. The scanning target was an agar cylinder with a height of approximately 3.5 cm and radius 5 cm. The applied displacement of 0.5 mm resulted in an overall compression of approximately 1.4% of the phantom height. Scattering was provided by a uniform distribution of aluminium oxide powder. Contact with the compression plate and the surface the phantom was resting on was made via a combination of water and ultrasound gel. This lowfriction contact allowed the phantom to expand laterally and elevationally more easily as it was compressed. The probe specification in the phantom experiment is similar to the parameters used for simulation, except that it is a 1D array probe with a fixed elevational focus. Also, each 2D image comprised 192 A-lines, focused using 64 active elements. Displacement estimation window sizes were the same as for the simulated data, with a grid of 42 windows axially and 96 windows laterally and elevationally. The effect of these differences is that the simulated images are narrower and differently focussed. Also, the scanning target has different dimensions in the simulation and phantom experiment, which affects the details of the displacement field.1 However, these are not significant differences in the context of this paper, which is concerned with comparing different displacement estimation methods on the same data set. Therefore, corresponding estimates are compared. Also, since the methods are generally applicable and not specific to one probe or displacement field, it does not matter that the simulation and phantom data are not exactly the same.
3. Results and discussion Fig. 2 shows axial, lateral and elevational displacement images on various slices through the volume for the simulated data, using both the steered and unsteered displacement tracking methods of Section 2 and compared to the actual displacements from the finite element model. Equivalent images for the phantom data are shown in Fig. 3. Although there is no ground truth displacement data for the phantom data, it is known that the overall compression was approximately 1.4% and it is reasonable to assume that the phantom material was incompressible. The expected lateral and eleva1 These differences arise due to pragmatic considerations in the experimental design. The number of A-lines and focusing elements were reduced for the simulation to speed up the slow Field II computations. The scanning target for the simulations was generated as part of a previous experiment and so was fixed at 7 cm radius and 3 cm height, whereas the agar cylinder dimensions for the phantom experiment were fixed by the 5 cm radius of the mould in which it was made.
elevational
lateral
elevational
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elevational
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+0.30
0
+0.12
0
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+0.12
0
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Fig. 2. Displacement images for simulated data. (a) 3D vector fields of displacements on three orthogonal planes through the volume centre. For clarity, vectors are shown on the displacement estimation grid subsampled by 6 in each direction. (b–d) Axial, lateral and elevational displacement images for 2D slices through the volume. Values are in mm. Each image pixel represents one displacement estimation window from the 44 64 64 grid. The outline in the steered lateral image shows the extent of the 3D ROI used to calculate the values in Table 1.
tional strains are therefore 0.7%, producing 0.34 mm expansion over the width of the image, or ±0.17 mm at the edges if there is zero displacement at the centre. To measure quantitatively the precision of the displacement estimation, we compared the displacements from the simulated data to the known displacement field of the finite element model. Table 1 shows mean and standard deviation error values for each of the three directions using both the steered and unsteered methods. To ensure independent error estimates, values were calculated on a subset of the displacement estimation windows evenly spaced over the volume and separated by five window spacings in all directions. Also, values were taken from within a 3D region of
interest (ROI). The ROI includes only the estimates that are covered by all nine volumes, and excludes a margin of 1 mm at the edges of this overlap region. The region is outlined in one image of Fig. 2. The total number of estimates used for the results in Table 1 was 770. For the phantom data, we assume that the actual displacement field is smoothly varying within the phantom and high frequency variations in the measured displacement fields are due to measurement errors. We calculate the standard deviation of small grids (3 3 3) of displacement estimation windows, which we expect to have similar displacements. Low values therefore suggest smoother, more precise displacement fields. We take multiple
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elevational
elevational
lateral
lateral
+0.35
+0.1
+0.15
0
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+0.2
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Fig. 3. Displacement images for phantom data. (a) 3D vector fields of displacements on three orthogonal planes through the volume centre. For clarity, vectors are shown on the displacement estimation grid subsampled by 9 axially and 10 laterally and elevationally. (b–d) Axial, lateral and elevational displacement images for 2D slices through the volume. Values are in mm. Each image pixel represents one displacement estimation window from the 42 96 96 grid. The outline in the steered lateral image shows the extent of the 3D ROI used to calculate the values in Table 2.
Table 1 Displacement estimation error in simulated data. Values are in lm, and show mean ± standard deviation. Errors are relative to ground truth displacements from the finite element model. Statistical significance is for the difference in means and difference in standard deviations. The standard deviation is of most interest, since this is what affects the precision of strain calculated from the gradient of the displacements. Unsteered
Axial Lateral Elevational
0.28 ± 1.27 0.69 ± 35.69 1.20 ± 38.67
Steered
0.91 ± 1.01 0.63 ± 3.70 0.51 ± 3.64
Significance Mean
Std. dev.
p < 0.01 p > 0.05 p > 0.05
p < 0.01 p < 0.01 p < 0.01
measurements of this kind throughout a region of interest and use their average as an indication of displacement quality. Results are shown in Table 2. The region of interest is defined in the same way as for the simulated data and is outlined in one image of Fig. 3. We again use a subset of windows within the ROI, giving a total of 2091 estimates. The results of the simulated and phantom data lead to the same conclusion. As with 2D beamsteering, there is little advantage to using beamsteering for axial displacements, since the unsteered volume is sufficient to track displacements in this direction (axial displacement standard deviation is 1.27 lm unsteered compared
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Table 2 Displacement variation in phantom data. The values are averages of standard deviations of displacements, given in lm. The standard deviation is calculated on 3 3 3 grids of displacement estimation windows and averaged over several of these grids. Since actual displacements are likely to be slowly varying and continuous, lower values suggest smoother and more precise displacements. Significance is for the difference in the mean value of this standard deviation.
Axial Lateral Elevational
Unsteered
Steered
Significance
3.05 26.51 28.92
3.08 5.78 11.87
p > 0.05 p < 0.01 p < 0.01
to 1.01 lm steered in simulation and 3.05 lm unsteered compared to 3.08 lm steered in phantom data). Although the steering approach averages nine displacement values, they are not independent estimates and so do not offer any practically significant improvement in the result. In comparison, for the lateral and elevational displacements, a smoother, less noisy displacement field is produced by measurements in the steered beam direction compared to off beam lateral and elevational measurements. This is confirmed by the numerical values in Tables 1 and 2: lateral displacement standard deviation is 35.69 lm unsteered compared to 3.70 lm steered in the simulation and 26.51 lm unsteered compared to 5.78 lm steered in the phantom data; elevational standard deviation is 38.67 lm unsteered compared to 3.64 lm steered in the simulation and 28.92 lm unsteered compared to 11.87 lm steered in the phantom data. For the phantom data, it appears that the phantom has expanded more elevationally than laterally and the elevational displacements are noisier as a result. This is most likely caused by imperfect slip conditions in the experimental setup. The improved precision provided by beamsteering is particularly important for applications that require lateral and elevational strain, because even small imprecisions in the displacements become more significant when taking the gradient to measure strain.
Fig. 4 shows axial, lateral and elevational strain images for the phantom data, generated using the methods in Section 2. Table 3 shows the average values of the elastographic signal to noise ratio (SNRe) calculated over the same ROI as above. SNRe is defined as mean divided by standard deviation for a subset of 3 3 3 grids of values in the ROI. Even with beamsteering, the lateral and elevational strain is clearly noisier than the axial strain (mean SNRe is 1.41 laterally and 1.33 elevationally, compared to 9.75 axially). However, the beamsteered approach, with some post-processing from the Gaussian filters, has produced smoother, less noisy lateral and elevational strain images (mean SNRe is 1.41 steered compared to 0.41 unsteered laterally and 1.33 steered compared to 0.46 unsteered elevationally). These may prove to be useful for volumetric imaging of other mechanical properties of tissue such as Poisson’s ratio or total shear strain. For example, it was noted in [6] that the precision of Poisson’s ratio images is limited by the relatively low precision of lateral strain compared to axial strain. Improving the precision of lateral strain is therefore expected to improve the precision of Poisson’s ratio images. In this paper, we have demonstrated the principle of 3D beamsteering from a 2D array using simulated and phantom data. A particular limitation of our experimental setup is the inability to steer along more than one axis at once. An interesting future investigation would therefore be to experiment with different combinations of beams steered in any direction. Also, while our simulation represents a fully-populated 2D array with a large number of elements and ideal transmit and receive focusing, the phantom scan’s experimental setup is limited by having used a 1D array with a fixed focus in the probe’s elevational direction. Genuine 2D arrays typically have fewer elements. A realistic 2D array with more typical focusing would be required to implement the beamsteering method in practice and verify the findings of our experiments. The method necessarily reduces the frame rate by a factor of nine, which is known to have a detrimental effect on image quality [16]. In addition, the method relies on each steering angle
+0.02
0
+0.015
0
-0.015 +0.015
0
-0.015
Fig. 4. Strain images for phantom data. The figure shows axial, lateral and elevational strain images on an axial–lateral slice through the volume. The image pixels correspond to the 42 96 grid of displacement estimation windows on the central axial–lateral slice.
R. James Housden et al. / Ultrasonics 53 (2013) 615–621 Table 3 Mean SNRe of strain. The values are the averages of multiple SNRe measures over 3 3 3 grids of strain estimates within the ROI. Higher values indicate less noisy strain data. Significance is for the difference in the mean value.
Axial Lateral Elevational
Unsteered
Steered
Significance
8.63 0.41 0.46
9.75 1.41 1.33
p < 0.01 p < 0.01 p < 0.01
measuring displacements over the same compression. A practical implementation would therefore require a 2D array with a fast volume acquisition rate (perhaps using the methods in [17]) and fast displacement estimation, to overcome these limitations. Under these conditions, it would be possible to acquire volumetric data suitable for estimating the full 3D displacement vector in more practical scenarios, including in vivo scanning. 4. Conclusions We have presented a method for measuring full 3D displacement vector fields using multiple angle beamsteering from a 2D array probe. The superior displacement measurements along the beam direction give improved precision in the lateral and elevational directions. This translates to more precise strain data in these directions. Although calculating other mechanical properties is beyond the scope of this paper, a more precise measure of 3D displacement or strain is expected to improve such measures. Acknowledgement This work was supported by EPSRC Grant No. EP/E030882/1. References [1] J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, X. Li, Elastography: a quantitative method for imaging the elasticity of biological tissues, Ultrasonic Imaging 13 (2) (1991) 111–134. [2] E.S. Burnside, T.J. Hall, A.M. Sommer, G.K. Hesley, G.A. Sisney, W.E. Svensson, J.P. Fine, J. Jiang, N.J. Hangiandreou, Differentiating benign from malignant solid breast masses with US strain imaging, Radiology 245 (2) (2007) 401–410.
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Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Ultrasonic imaging of 3D displacement vectors using a simulated 2D array and beamsteering R. James Housden ⇑, Andrew H. Gee, Graham M. Treece, Richard W. Prager Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
a r t i c l e
i n f o
Article history: Received 30 March 2012 Received in revised form 16 October 2012 Accepted 23 October 2012 Available online 1 November 2012 Keywords: Elastography Strain imaging 3D displacement estimation Beamsteering 2D array
a b s t r a c t Most quasi-static ultrasound elastography methods image only the axial strain, derived from displacements measured in the direction of ultrasound propagation. In other directions, the beam lacks high resolution phase information and displacement estimation is therefore less precise. However, these estimates can be improved by steering the ultrasound beam through multiple angles and combining displacements measured along the different beam directions. Previously, beamsteering has only considered the 2D case to improve the lateral displacement estimates. In this paper, we extend this to 3D using a simulated 2D array to steer both laterally and elevationally in order to estimate the full 3D displacement vector over a volume. The method is tested on simulated and phantom data using a simulated 6–10 MHz array, and the precision of displacement estimation is measured with and without beamsteering. In simulations, we found a statistically significant improvement in the precision of lateral and elevational displacement estimates: lateral precision 35.69 lm unsteered, 3.70 lm steered; elevational precision 38.67 lm unsteered, 3.64 lm steered. Similar results were found in the phantom data: lateral precision 26.51 lm unsteered, 5.78 lm steered; elevational precision 28.92 lm unsteered, 11.87 lm steered. We conclude that volumetric 3D beamsteering improves the precision of lateral and elevational displacement estimates. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Ultrasonic elastography is a technique for visualising mechanical properties of tissues. The subject of this paper is quasi-static elastography, which was originally developed in [1]. This involves estimating the relative displacement from one image to the next while continually varying the compression in the tissue. It is usually only the axial displacements that are measured with high precision, because the RF data contains high resolution phase information in this direction. Axial strain is then calculated from the gradient of the axial displacement estimates. Variations in axial strain indicate relative tissue stiffness under the assumption that soft regions of the tissue will deform more than stiff regions. Several potential clinical uses of axial strain imaging have been identified, for example classification of breast lesions [2,3] and visualisation of prostate lesions [4]. Lateral (and in the case of 3D imaging, elevational) displacements are usually of secondary interest because the standard
⇑ Corresponding author. Present address: Division of Imaging Sciences, The Rayne Institute, King’s College London, St. Thomas’ Hospital, London SE1 7EH, UK. E-mail addresses: [email protected], [email protected] (R. James Housden), [email protected] (A.H. Gee), [email protected] (G.M. Treece), [email protected] (R.W. Prager). 0041-624X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2012.10.010
methods used for axial tracking produce lower precision displacement estimates laterally. However, it was shown in [5] that the precision of axial displacements can be improved by lateral tracking. Therefore, lateral tracking is potentially relevant in axial strain imaging applications. In addition, the full 3D displacement vector is of interest for measuring other mechanical properties of tissue, e.g. Poisson’s ratio for poroelastography [5,6], total shear strain for imaging tumour bonding [7] and for estimating the elastic modulus [8]. Multiple angle beamsteering is one possible method for obtaining high-precision lateral and elevational displacements. In 2D imaging, tracking axial and lateral displacements only, several studies have shown that beamsteering improves the precision of displacements perpendicular to the beam [9–11]. This involves acquiring multiple images over a region, each with a different steering angle, so that the tissue is imaged from several different directions. For each angle, displacements are measured by comparing with a corresponding steered image at a different compression. The measured displacement is the component of the actual displacement along the beam direction. The components from each angle are then combined to estimate the axial and lateral components of the displacement field. This is illustrated in Fig. 1a. Throughout this paper, the terms axial and lateral refer to the fixed vertical and horizontal directions in which we are trying to
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θ1
θ1
θ3
dm,2
d
θ3
dm,1 dm,3
(a)
(b)
Fig. 1. Beamsteering principle. (a) 2D beamsteering, with three laterally steered images at angles h1, h2 and h3. Here h1 and h3 are equal angles steering to the left and right and h2 = 0°. The actual displacement vector of a point in the image, d, is measured as components dm, 1, dm, 2 and dm, 3 along the beam direction of each image. These measurements can be used to estimate both the axial and lateral components of d. (b) In 3D beamsteering, this principle is extended to five intersecting volumes, two steered laterally and two elevationally, which allow the 3D displacement vector to be estimated.
estimate displacements. The actual directions of measurements taken along and across the steered beam are referred to as the beam direction and off-beam direction. Although there has been significant research into beamsteering, its advantages have only been shown in 2D with improved precision of lateral displacements. The same principle can be extended to estimate the full 3D displacement vector, including elevational displacements, over a volume using a 2D array probe. This would involve acquiring multiple volumes of data at different steering angles, with some volumes steered laterally and some elevationally (Fig. 1b). In [11], beamsteering equations were derived for the 3D case, but not used on 3D data. In this paper, we present a method to measure 3D displacements with multiple beamsteered volumes, using similar equations to those in [11]. We also present novel experiments comparing the accuracy and precision of 3D beamsteered displacements to those obtained using a single unsteered volume. Results are shown from a simulation of a 2D array probe and a phantom experiment imitating a 2D array using a mechanically controlled 1D array probe. As evidenced below and further highlighted in the Discussion and Conclusions text of Sections 3 and 4, the method proposed offers improvement in the precision of displacement estimation when using beamsteering. 2. Methods 2.1. Displacement estimation Displacement estimation is by one of two methods. The unsteered method tracks displacements between a single pair of volumes with beams perpendicular to the transducer. The steered method uses multiple pairs of volumes, each at a different steering angle. For each steered or unsteered volume pair, displacements along the beam direction are estimated using the Weighted Phase Separation method [12] and two-pass tracking strategy described in [13]. Each displacement estimate is accompanied by a precision estimate calculated from a weighted variance of phase differences between the pre- and post-compression RF data [12]. For the unsteered method, the displacement measurements along the beam directly give the axial displacements. Lateral and elevational displacements are calculated during axial tracking, following the method in [13]. After each axial displacement estimate, the matching correlation is calculated at offsets of ±1 vector off the beam in the lateral direction. Subvector lateral displacements are obtained by a quadratic fit to the correlation value at each lateral offset according to
dl ¼
q1 q1 2ðq1 2q0 þ q1 Þ
where q1, q0 and q1 are correlations of the RF envelope data at 1, 0 and +1 vector offset respectively from the initial lateral alignment. dl is the subvector location of the maximum correlation, giving the lateral displacement relative to the initial alignment. This is then repeated for the elevational displacements. The quadratic fit used here is an approximation to the theoretical Gaussian peak in the correlation values. This lateral and elevational tracking also improves the axial estimates, because the search range on the second tracking pass is initialised with the offsets from the first pass. Full details of the unsteered tracking method are given in our previous publications [12,13]. In the steered method, the volume pair at each angle is initially processed separately using the same phase-based tracking method in the beam direction. Off-beam displacement tracking is not used, because the non-orthogonal nature of the search directions in steered data means that searching in the off-beam direction also affects the estimate in the beam direction. The skewed grid of displacements and associated precision estimates are then linearly resampled to a regular Cartesian grid aligned with the unsteered volume. Beam-direction displacements from the individual angles are then combined in a least squares solution for axial, lateral and elevational displacements, similar to the 3D solution in [11]. The measured displacements are estimates of the projections of the actual displacement onto the beam directions, so the set of equations to solve are dm, i = dui. dm, i is a measured displacement, d = [da, dl, de]T is a vector of the axial, lateral and elevational displacements that we are estimating, and ui is a unit vector in the beam direction. In our implementation, each volume is steered in either the lateral or elevational direction, but not both at once. ui is therefore [coshi, sinhi, 0]T for volumes steered by hi laterally, [cos/i, 0, sin/i]T for volumes steered by /i elevationally and [1, 0, 0]T for an unsteered volume. The equations are solved according to the standard weighted linear least squares approach, with each measurement weighted by its associated precision estimate. It should be noted that the weighted least squares approach assumes that the errors in the displacement measurements, dm, i, are independent. In practice there will be some dependence because the images at different angles will be correlated. Having obtained displacements from either method, we then calculate strain in any one direction by simply taking the difference of the two estimates either side of each estimation location. This is followed by a Gaussian smoothing filter in each of the three directions and weighted by the precision of each estimate. The Gaussian has a standard deviation of one axial estimate spacing.
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2.2. Data acquisition and processing The two displacement estimation methods were compared using both simulated data and a phantom scan. The physical 2D array for the phantom scan was imitated using a mechanically controlled 1D array probe, as detailed below. 2.2.1. Simulated data For the simulation, we first determined a displacement field between a compressed and an uncompressed state using finite element modelling (Abaqus 6.7, Simulia, Rhode Island, USA). The model was a 3D cylinder of radius 7 cm and height 3 cm. It comprised a single linear elastic material, of Young’s modulus 10 kPa and Poisson’s ratio 0.495. The probe face was modelled as a 35 mm 35 mm square rigid compressor on the top surface of the model. In the compressed state, the compressor indented the top surface of the model by 0.3 mm (1% compression). Frictionless slip conditions were defined at the base of the model and in the contact region between the compressor and the top surface. These frictionless surfaces represent the sliding contact provided by ultrasound gel and the disconnectedness of internal tissues, e.g. muscle layers. All other surfaces were unconstrained. RF ultrasound data were simulated using Field II [14,15]. The probe was modelled as a 192 192 element fully-populated array with element spacing 0.245 mm. The transducer centre frequency was 8.5 MHz. Each A-line was sampled at 50 MHz over 1700 samples, giving a depth of 26.2 mm. Focusing for each line was achieved using a set of 32 32 active elements, with a transmit focus depth of 13 mm and dynamic receive focusing. Multiple volumes of 128 128 A-lines and different steering angles were simulated. The angles used were ±12° and ±6° laterally and elevationally, as well as an unsteered volume, giving nine angles altogether. 12° is around the maximum angle at which steering is possible without producing significant grating lobe artefacts. These nine volumes were simulated in both the compressed and uncompressed state, with the scatterer positions adjusted by the displacement in the finite element model for the compressed state. The simulated RF data then had Gaussian white noise added, reducing the SNR of the RF signal to 20 dB. Displacements for each volume were measured using approximately equal aspect ratio 3D matching windows of 8 RF cycles along the beam and 5 vectors in the off-beam directions. Window spacing was 0.6 mm, so that each window overlapped the next by 40% of its length. Over the volume, this resulted in a grid of 44 windows axially and 64 windows laterally and elevationally. These parameters are established defaults of the displacement estimation algorithm for unsteered data [13], providing an acceptable tradeoff between resolution and computation time. 2.2.2. Phantom data For the phantom scan, we used a ULA-OP (Università degli Studi di Firenze, Italy) scanner with a LA523 (Esaote, Genoa, Italy) 6–10 MHz linear array probe. Volumes were acquired using the 1D array by mounting it on a system of motorised linear slides (T-LSR300B, Zaber Technologies Inc., Vancouver, BC, Canada) and stepping the probe elevationally through 192 steps of 0.245 mm. At each elevational step, the probe was steered laterally through five angles: 0°, ±6° and ±12°. This created five volumes of 192 192 A-lines. The probe scanned through a 1.5 mm thick polyethylene compression plate of approximately 8 9 cm. The compression plate was stationary during the probe’s elevational sweep to represent the stationary face of a 2D array probe. To acquire the elevationally steered volumes using a 1D array, the scanning target and compression plate were rotated by 90° and the process repeated. Finally, a similar set of volumes was acquired with the compression plate and probe moved downwards by
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0.5 mm to compress the phantom. The complete acquisition produced ten volumes at each compression, two of which were unsteered. For displacement estimation, only the first of these two was used, so that there were nine volumes available for the steered method and one volume for the unsteered method. This produced the same set of lateral and elevational steering angles as in the simulation, so that the same beamsteering technique of Section 2.1 could be used. The rotation was achieved with the compressor and target standing on a turntable capable of rotating about a vertical axis with a precision of 0°10 . The primary source of misalignment between volumes before and after rotation is the estimate of the location of the rotation axis. Before the main acquisition, this was calibrated by scanning a point target on the turntable at several different rotational positions. The resulting set of points defined a circle, with the centre giving the location of the rotation axis. We have found that this approach is capable of aligning the volumes to within about one lateral vector spacing. This is better than the resolution of the strain images (two vector lateral and elevational window spacing) and is sufficient for our experiments measuring slowly varying displacement fields. The scanning target was an agar cylinder with a height of approximately 3.5 cm and radius 5 cm. The applied displacement of 0.5 mm resulted in an overall compression of approximately 1.4% of the phantom height. Scattering was provided by a uniform distribution of aluminium oxide powder. Contact with the compression plate and the surface the phantom was resting on was made via a combination of water and ultrasound gel. This lowfriction contact allowed the phantom to expand laterally and elevationally more easily as it was compressed. The probe specification in the phantom experiment is similar to the parameters used for simulation, except that it is a 1D array probe with a fixed elevational focus. Also, each 2D image comprised 192 A-lines, focused using 64 active elements. Displacement estimation window sizes were the same as for the simulated data, with a grid of 42 windows axially and 96 windows laterally and elevationally. The effect of these differences is that the simulated images are narrower and differently focussed. Also, the scanning target has different dimensions in the simulation and phantom experiment, which affects the details of the displacement field.1 However, these are not significant differences in the context of this paper, which is concerned with comparing different displacement estimation methods on the same data set. Therefore, corresponding estimates are compared. Also, since the methods are generally applicable and not specific to one probe or displacement field, it does not matter that the simulation and phantom data are not exactly the same.
3. Results and discussion Fig. 2 shows axial, lateral and elevational displacement images on various slices through the volume for the simulated data, using both the steered and unsteered displacement tracking methods of Section 2 and compared to the actual displacements from the finite element model. Equivalent images for the phantom data are shown in Fig. 3. Although there is no ground truth displacement data for the phantom data, it is known that the overall compression was approximately 1.4% and it is reasonable to assume that the phantom material was incompressible. The expected lateral and eleva1 These differences arise due to pragmatic considerations in the experimental design. The number of A-lines and focusing elements were reduced for the simulation to speed up the slow Field II computations. The scanning target for the simulations was generated as part of a previous experiment and so was fixed at 7 cm radius and 3 cm height, whereas the agar cylinder dimensions for the phantom experiment were fixed by the 5 cm radius of the mould in which it was made.
elevational
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+0.30
0
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0
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Fig. 2. Displacement images for simulated data. (a) 3D vector fields of displacements on three orthogonal planes through the volume centre. For clarity, vectors are shown on the displacement estimation grid subsampled by 6 in each direction. (b–d) Axial, lateral and elevational displacement images for 2D slices through the volume. Values are in mm. Each image pixel represents one displacement estimation window from the 44 64 64 grid. The outline in the steered lateral image shows the extent of the 3D ROI used to calculate the values in Table 1.
tional strains are therefore 0.7%, producing 0.34 mm expansion over the width of the image, or ±0.17 mm at the edges if there is zero displacement at the centre. To measure quantitatively the precision of the displacement estimation, we compared the displacements from the simulated data to the known displacement field of the finite element model. Table 1 shows mean and standard deviation error values for each of the three directions using both the steered and unsteered methods. To ensure independent error estimates, values were calculated on a subset of the displacement estimation windows evenly spaced over the volume and separated by five window spacings in all directions. Also, values were taken from within a 3D region of
interest (ROI). The ROI includes only the estimates that are covered by all nine volumes, and excludes a margin of 1 mm at the edges of this overlap region. The region is outlined in one image of Fig. 2. The total number of estimates used for the results in Table 1 was 770. For the phantom data, we assume that the actual displacement field is smoothly varying within the phantom and high frequency variations in the measured displacement fields are due to measurement errors. We calculate the standard deviation of small grids (3 3 3) of displacement estimation windows, which we expect to have similar displacements. Low values therefore suggest smoother, more precise displacement fields. We take multiple
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elevational
elevational
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+0.35
+0.1
+0.15
0
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Fig. 3. Displacement images for phantom data. (a) 3D vector fields of displacements on three orthogonal planes through the volume centre. For clarity, vectors are shown on the displacement estimation grid subsampled by 9 axially and 10 laterally and elevationally. (b–d) Axial, lateral and elevational displacement images for 2D slices through the volume. Values are in mm. Each image pixel represents one displacement estimation window from the 42 96 96 grid. The outline in the steered lateral image shows the extent of the 3D ROI used to calculate the values in Table 2.
Table 1 Displacement estimation error in simulated data. Values are in lm, and show mean ± standard deviation. Errors are relative to ground truth displacements from the finite element model. Statistical significance is for the difference in means and difference in standard deviations. The standard deviation is of most interest, since this is what affects the precision of strain calculated from the gradient of the displacements. Unsteered
Axial Lateral Elevational
0.28 ± 1.27 0.69 ± 35.69 1.20 ± 38.67
Steered
0.91 ± 1.01 0.63 ± 3.70 0.51 ± 3.64
Significance Mean
Std. dev.
p < 0.01 p > 0.05 p > 0.05
p < 0.01 p < 0.01 p < 0.01
measurements of this kind throughout a region of interest and use their average as an indication of displacement quality. Results are shown in Table 2. The region of interest is defined in the same way as for the simulated data and is outlined in one image of Fig. 3. We again use a subset of windows within the ROI, giving a total of 2091 estimates. The results of the simulated and phantom data lead to the same conclusion. As with 2D beamsteering, there is little advantage to using beamsteering for axial displacements, since the unsteered volume is sufficient to track displacements in this direction (axial displacement standard deviation is 1.27 lm unsteered compared
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Table 2 Displacement variation in phantom data. The values are averages of standard deviations of displacements, given in lm. The standard deviation is calculated on 3 3 3 grids of displacement estimation windows and averaged over several of these grids. Since actual displacements are likely to be slowly varying and continuous, lower values suggest smoother and more precise displacements. Significance is for the difference in the mean value of this standard deviation.
Axial Lateral Elevational
Unsteered
Steered
Significance
3.05 26.51 28.92
3.08 5.78 11.87
p > 0.05 p < 0.01 p < 0.01
to 1.01 lm steered in simulation and 3.05 lm unsteered compared to 3.08 lm steered in phantom data). Although the steering approach averages nine displacement values, they are not independent estimates and so do not offer any practically significant improvement in the result. In comparison, for the lateral and elevational displacements, a smoother, less noisy displacement field is produced by measurements in the steered beam direction compared to off beam lateral and elevational measurements. This is confirmed by the numerical values in Tables 1 and 2: lateral displacement standard deviation is 35.69 lm unsteered compared to 3.70 lm steered in the simulation and 26.51 lm unsteered compared to 5.78 lm steered in the phantom data; elevational standard deviation is 38.67 lm unsteered compared to 3.64 lm steered in the simulation and 28.92 lm unsteered compared to 11.87 lm steered in the phantom data. For the phantom data, it appears that the phantom has expanded more elevationally than laterally and the elevational displacements are noisier as a result. This is most likely caused by imperfect slip conditions in the experimental setup. The improved precision provided by beamsteering is particularly important for applications that require lateral and elevational strain, because even small imprecisions in the displacements become more significant when taking the gradient to measure strain.
Fig. 4 shows axial, lateral and elevational strain images for the phantom data, generated using the methods in Section 2. Table 3 shows the average values of the elastographic signal to noise ratio (SNRe) calculated over the same ROI as above. SNRe is defined as mean divided by standard deviation for a subset of 3 3 3 grids of values in the ROI. Even with beamsteering, the lateral and elevational strain is clearly noisier than the axial strain (mean SNRe is 1.41 laterally and 1.33 elevationally, compared to 9.75 axially). However, the beamsteered approach, with some post-processing from the Gaussian filters, has produced smoother, less noisy lateral and elevational strain images (mean SNRe is 1.41 steered compared to 0.41 unsteered laterally and 1.33 steered compared to 0.46 unsteered elevationally). These may prove to be useful for volumetric imaging of other mechanical properties of tissue such as Poisson’s ratio or total shear strain. For example, it was noted in [6] that the precision of Poisson’s ratio images is limited by the relatively low precision of lateral strain compared to axial strain. Improving the precision of lateral strain is therefore expected to improve the precision of Poisson’s ratio images. In this paper, we have demonstrated the principle of 3D beamsteering from a 2D array using simulated and phantom data. A particular limitation of our experimental setup is the inability to steer along more than one axis at once. An interesting future investigation would therefore be to experiment with different combinations of beams steered in any direction. Also, while our simulation represents a fully-populated 2D array with a large number of elements and ideal transmit and receive focusing, the phantom scan’s experimental setup is limited by having used a 1D array with a fixed focus in the probe’s elevational direction. Genuine 2D arrays typically have fewer elements. A realistic 2D array with more typical focusing would be required to implement the beamsteering method in practice and verify the findings of our experiments. The method necessarily reduces the frame rate by a factor of nine, which is known to have a detrimental effect on image quality [16]. In addition, the method relies on each steering angle
+0.02
0
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0
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0
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Fig. 4. Strain images for phantom data. The figure shows axial, lateral and elevational strain images on an axial–lateral slice through the volume. The image pixels correspond to the 42 96 grid of displacement estimation windows on the central axial–lateral slice.
R. James Housden et al. / Ultrasonics 53 (2013) 615–621 Table 3 Mean SNRe of strain. The values are the averages of multiple SNRe measures over 3 3 3 grids of strain estimates within the ROI. Higher values indicate less noisy strain data. Significance is for the difference in the mean value.
Axial Lateral Elevational
Unsteered
Steered
Significance
8.63 0.41 0.46
9.75 1.41 1.33
p < 0.01 p < 0.01 p < 0.01
measuring displacements over the same compression. A practical implementation would therefore require a 2D array with a fast volume acquisition rate (perhaps using the methods in [17]) and fast displacement estimation, to overcome these limitations. Under these conditions, it would be possible to acquire volumetric data suitable for estimating the full 3D displacement vector in more practical scenarios, including in vivo scanning. 4. Conclusions We have presented a method for measuring full 3D displacement vector fields using multiple angle beamsteering from a 2D array probe. The superior displacement measurements along the beam direction give improved precision in the lateral and elevational directions. This translates to more precise strain data in these directions. Although calculating other mechanical properties is beyond the scope of this paper, a more precise measure of 3D displacement or strain is expected to improve such measures. Acknowledgement This work was supported by EPSRC Grant No. EP/E030882/1. References [1] J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, X. Li, Elastography: a quantitative method for imaging the elasticity of biological tissues, Ultrasonic Imaging 13 (2) (1991) 111–134. [2] E.S. Burnside, T.J. Hall, A.M. Sommer, G.K. Hesley, G.A. Sisney, W.E. Svensson, J.P. Fine, J. Jiang, N.J. Hangiandreou, Differentiating benign from malignant solid breast masses with US strain imaging, Radiology 245 (2) (2007) 401–410.
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