2.1.

Finite Element Model Simulation

Comsol Multiphysics v3.5a (COMSOL) was used to generate a 2-D plane strain model (see Appendix A), consisting of a circular inclusion (1.5 cm radius) embedded at the center of a square homogeneous background (10 cm). The simulation comprised a 2-D triangular element mesh using Comsol’s extremely fine predefined element size. Both the inclusion and background were linear, isotropic, and quasi-incompressible, with their Poisson’s ratio, $\nu$, set to 0.495 The Young’s modulus of the background was 10 kPa and varied from 1 to 100 kPa in the inclusion between successive simulations. Two versions of this model were simulated: perfect inclusion-background adhesion and perfect inclusion-background slip (that is, the inclusion boundary could slip against its surroundings with a coefficient of friction of zero).

Simulations featuring discontinuous media must be performed using contact modeling to simulate the motion of two bodies sliding past each other. Contact modeling entails mesh discontinuity across the contact boundary, in other words, elements do not exist across the boundary between contact bodies. As such, care must be taken with regards to interpretation of measurements from strain maps that are formed directly from mesh deformation data compared to those from strain images generated through spatial gradient analysis of displacement data. The latter is generally the method employed in elastography, and attention must be given to the fundamental difference in strain calculation where contact bodies are present.

The model was exposed to a uniaxial compressive force across the entire upper surface and the resulting axial displacement and strains predicted throughout the model. Global applied strain values of 0.1%, 0.2%, 0.5%, 0.8%, and 1% were simulated. Slippery boundary conditions were assigned to the upper and lower model surfaces, and rigid body motion was prevented by laterally confining a node at the center of the lower boundary and a node in the center of the inclusion. Axial strain was calculated in two ways: the “element-based” method of the FEM package and through exporting the nodal displacement values to MATLAB^® (Mathworks, Massachusetts) for conversion to strain via a least-squares estimator with a window length of 5 mm, which we call “gradient-based” strain. Although it has not previously been employed in FEM simulations of strain images in the absence of simulation of ultrasound echoes, the latter method was used because most elastography systems estimate strain in this way and, without it, the high strain halo seen in practice and mentioned above is not created.

The effect of slip-induced strain at the tumor boundary on CTE values was examined through calculating CTE using both element-based strain images directly from FEA, and gradient-based strain images. CTE is defined as the ratio (in decibels) of the mean lesion to background strain contrast to the equivalent lesion to background Young’s modulus contrast⁵⁶

The elastographically observed strain contrast, $C_{ϵ}$, is the ratio of strain observed in the background that is observed to be transmitted into the tumor. The true YM contrast, $C_{\mathrm{YM}}$, is the ground truth and must be known. The fundamental concept of CTE is to define the fidelity with which axial strain images (elastograms) reproduce the true mechanical contrast present in the material, expressed as a ratio. Although it is impossible for CTE to be measured clinically because the ground truth ($C_{\mathrm{YM}}$) cannot be known, it is an important property that determines the reliability of axial strain images under certain conditions, many of which arise clinically.

For studying the strain halo, mean strain values within an annulus centered on the inclusion boundary, of width equal to half the strain estimator length, were calculated. For studying CTE, ROIs were defined as a disc within the inclusion, avoiding the inclusion boundary, for measuring ${ϵ}_{\text{tumor}}$ and an annulus surrounding the inclusion, again avoiding the boundary, for measuring an average ${ϵ}_{\text{background}}$. The variation in these quantities with the ${ϵ}_{\text{tumor}}$ ROI radius was also studied, given the expected heterogeneity of strain distribution and the resulting effect upon calculated strain values. Throughout the remainder of this paper, “strain” refers to absolute axial normal strain.

2.2.

Tissue-Mimicking Gelatine Phantoms

In order to generate comparable data to the FEM simulations, prismatic gelatine phantoms were constructed in which the YM varied in the cross section that formed the ultrasound scan plane but was constant in the elevational direction. This provided consistency with the plane strain assumption used to predict the mechanical behavior in the FEM simulations. These phantoms comprised a cube containing a centrally placed circular rod-shaped inclusion, and the YM of the rod and cube were varied to generate phantoms of varying YM contrast. The use of a rod-shaped inclusion rather than a spherical inclusion was decided on for several reasons. First, this mimics the FEM simulation, second the conclusions drawn from results from the rod-shaped inclusion are on the whole generalizable to spherical inclusions and in addition to this there are numerous clinical situations where the ROI is more similar to a rod than to a sphere (for example, tendons). It is not thought that the difference will significantly affect the trends seen for CTE.

2.3.

Young’s Modulus Measurement

In order to establish a gold truth, measurements of the YM of the gelatine were made using gelatine cylinders manufactured from the same batch of gelatine that the phantoms were constructed from. Following the method described by Crescenti et al.,⁵⁷ cylinders of gelatine 35 mm in height and 53.5 mm in diameter were made at the concentrations of 8%, 10%, 12%, and 14%. YM was measured from force-displacement data obtained by controlled application of compressive strain along the long axis of the cylinders, and from force measurement (Instron 3342 with load cell 2519-103, Instron, Bucks, United Kingdom). 3% prestrain was used to ensure uniform contact between the compression plate and the sample. Both the compression plate and the base plate were larger than the surface of the gelatine, and they were lubricated liberally with vegetable oil to avoid measurement artifacts due to boundary friction, as described by Crescenti et al.⁵⁷ The force (measured by the load cell) and the displacement were recorded every 20 ms during a 5% strain. Thus, including prestrain, a total of 8% strain was applied. Each sample was strained six times for repeatability evaluation and between each test the sample was removed, repositioned, and relubricated. The YM values were selected to represent a similar range to those found in biological tissues.

2.4.

Phantom Construction

In order to make the phantoms, the background mixture was first poured into cuboidal moulds of side 60 mm. A 15-mm-diameter rod was suspended in the liquid so when it was removed, a well was left in the cube. Adhered inclusions were made by pouring liquid gelatine into the well at 33°C and allowing it to set at room temperature. Mobile inclusions were made by pouring liquid gelatine into the well at 28°C (the point when it was about to set). Once set, the phantom was immersed in a water bath, and a slip plane was developed between the inclusion and the background by manually releasing the inclusion with gentle force applied along its axis using a 2-mm-diameter metal rod. This did not constitute cutting the gelatine as gentle manipulation, as well as the presence of water allowed the layers to separate into inclusion and background with no damage to either layer. This was evidenced by the absence of any split in the gelatine on B-mode ultrasound imaging. Once the inclusion had been released, it was possible to see it moving relative to the background. The inclusion was released at the last possible moment in order to minimize the possibility that significant amounts of the gelatine would dissolve into the water.

A set of seven adhered and mobile inclusion phantoms with different contrast were made, detailed in Table 1. The Perspex moulds in which the phantoms were made were refrigerated in airtight containers at 4°C prior to experiments and were removed from the refrigerator at least 4 h prior to scanning to allow them to reach room temperature. They were removed from the Perspex moulds and the internal temperature was monitored to be the same as the ambient temperature of 20°C by inserting the needle thermocouple of a digital thermometer into the phantom at a point not included in the scan plane to avoid artifact from this.

Table 1

Young’s modulus contrast (CYM) in the phantoms.

2.5.

Image Acquisition

Each experiment was performed with the phantoms resting on an extremely stiff acoustic absorbent pad while immersed in a water bath at constant temperature of 20°C. Frames of radio frequency (RF) data were acquired using a Gage Compuscope 14200 (Cage Applied, Lockport, Illinois) in a personal computer running Stradwin 3.8 (Cambridge University, United Kingdom) ultrasound elastography and image acquisition software,⁵⁸ interfaced to a DIASUS (Dynamic Imaging, United Kingdom) scanner. A GE RSP6-12 mechanically swept 3-D probe (GE Healthcare, Chalfont St Giles, United Kingdom) was used in 2-D mode with the internal linear array (6 to 12 MHz) held stationary in the central position; in effect a linear array transducer with a footprint extender that covered the upper surface of the phantom. The transducer was attached to an Instron In-Spec 2200 Benchtop Portable Tester (Instron, Bucks, United Kingdom). One Newton of precompression was first applied to establish acoustic and uniform mechanical contact between the transducer and the phantom, which was comparable to the 3% prestrain during YM measurement. An image frame of RF ultrasound echo data was acquired at this position and this image was then considered as the reference frame, i.e., 0% deformation, for strain measurement in RF images acquired at the end of each of 10 subsequent compression steps of 0.5% (0.3 mm). All RF frames were sampled in the axial direction at 66.6 MHz and comprised 127 A-lines, corresponding to an image of 2.5 cm width and 6.5 cm depth. The applied force at each compression step was recorded.

2.6.

Image Analysis

Axial displacements were estimated between consecutive frames of RF data using an exhaustive search 2-D crosscorrelation echo-pattern tracking algorithm written in MATLAB^® (Mathworks, Massachusetts). Further details of the algorithm may be found in Ref. 3. The tracking parameters were search window $250\times 10$, reference window $120\times 5$, step axial 5, and step lateral 2. Axial normal strain images were then derived from the displacement data using a moving least-squares strain estimator with a window length of 50.

Each strain image frame was then read into ImageJ 1.44 (bundled with 32-bit Java 1.6.0_20) image analysis software at a resolution of $250\times 650\text{pixels}$ (10 pixels per mm). Two ROIs were defined for calculating the mean strain within the inclusion and background. A circular ROI, whose diameter was varied to investigate different regions within the inclusion, was used for the inclusion ROI. An annulus-shaped ROI of 20 pixels width (equal to approximately half the strain estimator window length), centered on the inclusion boundary, was used to measure strain across the inclusion boundary, and an annulus of width 50 pixels, starting at the outer boundary of that ROI was used to measure background strain. It was found that using ROIs from a single region of the background produced results that were not representative of the whole background and caused strain measurements to vary considerably. The approach of using circles and annuli resolved such problems for the phantom employed, which produced strain images that exhibited circular, axial, and lateral symmetry.

CTE was calculated using the method described by Kallel et al.⁵⁹ Mean strain values within the annulus centered on the inclusion boundary were used as a surrogate for mobility and were first plotted as a function of applied strain for each value of $C_{\mathrm{YM}}$ and then, after these results were used to select appropriate frames and regions for further analysis, as described below, CTE was plotted as a function of $C_{\mathrm{YM}}$. In this way, CTE results were compared for soft and hard inclusions, both adhered and mobile.

4.1.

Finite Element Model Data

In Fig. 2, axial displacements and strains from a model with inclusion modulus 20 kPa and 1% applied strain (induced by a 1-mm displacement of the top surface of the model in a downward direction) are displayed, and results from element-based and gradient-based strain estimation can be compared. The mobile inclusion demonstrates a discontinuity in displacement data along its boundary [Fig. 2(b)], which is not present for the adhered inclusion [Fig. 2(a)] where there is a smooth gradient in displacement across the boundary. The discontinuity observed in the axial displacement field causes a halo of high axial strain to surround the mobile inclusion in the gradient-based strain image [Fig. 2(f)], which is not seen in the element-based strain image [Fig. 2(d)].

Fig. 2

Left column: adhered model, right column: slippery model. (a) and (b) Axial displacement; (c) and (d) axial strain as calculated by element-based strain estimator and (e) and (f) axial strain as estimated by gradient-based strain estimator. Note differences in scale between adhered and slippery models. Scale in %.

Furthermore, there are differences in axial strain distribution between the mobile and adhered inclusion models. Inside the inclusion, for the adhered case [Fig. 2(c)], strain is homogeneously distributed, whereas in the mobile case [Fig. 2(d)] strain reaches a maximum in the inclusion center, even though the Young’s modulus is constant throughout. In the background tissue, it can be seen that the pattern of the stress concentrations also changes: it is homogeneous over more of the background in the mobile case [Fig. 2(d)] than in the adhered case [Fig. 2(c)]. In the mobile case, the strain variations, although more pronounced than in the adhered case, are located mainly in the areas close to the inclusion and decrease rapidly away from the inclusion boundary. On the circular edge of the inclusion, the locations of high and low strain regions are inverted compared to the adhered case. This inversion in the locality of the boundary is seen clearly for element-based strain [Figs. 3(c) and 3(d)], but Fig. 2(f) suggests that, depending on the resolution limit imposed by the strain estimator window size, the feature may be partially or completely obscured by the bright strain halo in the experimental elastograms. It is also interesting to note the similarity of the pattern shown in Fig. 2(f) to that seen in experimental elastograms of slippery boundaries (see Fig. 1), indicating that mechanisms such as that suggested by Konofagou for explaining the strain halo, although plausible, are not necessary, i.e., the halo may simply be a consequence of the interaction between the gradient-based strain estimator and the displacement discontinuity at the slippery boundary, seen in Fig. 2(b). Finally, for the mobile case, even far from the boundary, there is a partial inversion of the locations of high and low strain regions in the background, as compared to the adhered case. This is best seen in Fig. 2(f), as compared with Fig. 2(e); note that in the upper right quadrant of Fig. 2(e), beginning with the vertical direction as 0 deg, one can see bright (0 deg), dark (45 deg), and bright (90 deg), whereas in Fig. 2(f) the angular variation is bright (0 deg), dark (30 deg), bright (45 deg), and dark (90 deg). This pattern for the mobile inclusion is repeated in other quadrants and is also visible in the experimental elastograms of mobile inclusions (see Figs. 8, 9, and 11). However, when viewing axial strain images using the same scale, the differences in the tissue background strain distribution are less obvious, although the strain distribution differences in the inclusion are still easily visible.

Fig. 3

Mean absolute inclusion to background axial strain ratio (i.e., contrast) as a function of Young’s modulus contrast, due to a 1% applied compression, for adhered and mobile models. The terms stiff, soft, appears stiff and appears soft, apply to the inclusion and are relative to the background. Data taken from FEM “element-based” strain.

Figure 3 shows mean inclusion to background strain ratios calculated directly from FEM data, where strain in the inclusion was averaged from all mesh elements located in the inclusion, and similarly strain in the background was averaged from all elements located in the background. The applied strain was fixed at 1%. At all values of YM contrast, there is a reduction in mean strain ratio as a consequence of tumor mobility, despite the visible high strain concentration toward the center of the mobile tumor [Fig. 2(d)]. It can also be seen that, where the inclusion and background have identical Young’s moduli, a nonzero strain ratio is calculated for the mobile case. Recall that this result is generated in the absence of any strain generated in the image as a consequence of passing the gradient window of the strain estimator over the tumor boundary in the displacement data.

The CTE curves for the adhered and slippery models are compared in Fig. 4. The CTE of the adhered model is always negative, whereas in the mobile case the CTE is positive for modulus $\text{contrasts}>-0.97\mathrm{dB}$. It is shown that, for a Young’s modulus contrast of 0 dB (i.e., the inclusion is the same stiffness as the background), the slippery model has a CTE of 1.58 dB. The adhered and slippery models have the same CTE values for a modulus contrast of $-1.49\mathrm{dB}$, although it must be noted that this does not mean that the images are identical, only that the calculated CTE values are identical. The slippery inclusion has a CTE of approximately zero for Young’s modulus contrast values of $-0.97$ and 9.03 dB, although this does not mean that the strain images appear homogeneous, either in the background or within the inclusion (as is the case for the adhered model at Young’s modulus $\text{contrast}=0\mathrm{dB}$, where the strain image really is homogeneous and $\mathrm{CTE}=0\mathrm{dB}$). The point where the two CTE curves cross and where the mobile model curve crosses 0 dB CTE may change with variables such as inclusion size, position, etc., and does not mean that the two images are identical.

Fig. 4

Contrast transfer efficiency curves for adhered and slippery models. Applied strain for both models is 1%. (Element-based) strain images corresponding to points of interest, as indicated.

Figure 5 shows strain images from a 1 kPa [Figs. 5(a) and 6(d)], 10 kPa [Figs. 5(b) and 6(e)], and 100 kPa [Figs. 5(c) and 6(f)] mobile and adhered inclusions, obtained by the gradient-based method. The strain halo is present in all strain images of the mobile inclusion. The characteristic heterogeneous strain distribution within the mobile inclusion is visible in Figs. 5(d) and 6(e). Although it is present in Fig. 5(f), it is not apparent without rescaling the image because the inclusion is very stiff compared to the background. Observe also that the strain magnitude within the slip-induced halo appears approximately independent of the Young’s modulus contrast in the image.

Fig. 5

Strain images obtained using gradient-of-displacement for inclusions undergoing 1% strain: (a) background 10 kPa, inclusion 1 kPa, adhered boundary, (b) background 10 kPa, inclusion 10 kPa, adhered boundary, (c) background 10 kPa, inclusion 100 kPa, adhered boundary. (d) background 10 kPa, inclusion 1 kPa, mobile boundary, (e) background 10 kPa, inclusion 10 kPa, mobile boundary, and (f) background 10 kPa, inclusion 100 kPa, mobile boundary.

Fig. 6

Strain ratio measurements made with a constant background ROI and varying inclusion ROI radii, for (a) mobile and (b) adhered models. Strain was computed as the axial gradient of axial displacement. The applied strain was 1%.

Examining the effect of inclusion ROI size upon strain ratio measurements, Fig. 6 shows the variation in (gradient-based) strain ratio for inclusion ROI radii of 1.65 cm, which completely overlaps the tumor boundary, 1.45 cm, which encompasses the inclusion up to the start of the strain halo, and 1 cm, which is within the center of the inclusion, for mobile and adhered models. The background ROI was kept constant at the area beyond a 1.75-cm radius from the model center, which ensured that the strain halo was not included in any background strain measurement. For the mobile case, when the inclusion ROI encompasses the strain halo, the strain ratio increases, indicating an apparent softening effect, under the assumption that strain ratios are (naively) interpreted as stiffness ratios. When the inclusion ROI is 1 cm, much of the strain within the inclusion is neglected, resulting in values that differ considerably from those in Fig. 2. It is only when the inclusion ROI is set at a value that ensures that the strain halo is neglected entirely while maximizing the total area of the inclusion within it, that strain ratio values similar to that generated from FEM are obtained. In Fig. 6(b), it is seen that the adhered model strain ratio measurements are not as sensitive to inclusion ROI size, which confirms expectations from visual inspection.

Halo strain measurements with varying Young’s modulus contrast and applied strain are shown in Figs. 7(a) and 8(b). This measurement tends to zero at zero Young’s modulus contrast for the adhered case and is always less than 0.4% when the applied strain is 1%, whereas mobile inclusion halo strain is always greater than 0.8% for 1% applied strain and increases slowly with inclusion modulus. For both model types, halo strain remains approximately constant with applied strain, which is true for any modulus contrast (not shown). Finally, mobile inclusion halo strain is always greater than adhered halo strain.

Fig. 7

Variation of halo strain with (a) Young’s modulus contrast at 1% applied strain and (b) applied strain for an inclusion of modulus 20 kPa. The modulus of the background was 10 kPa.

Fig. 8

CTE variation with applied strain for phantoms where the inclusion is stiffer than the background: $C_{\mathrm{YM}}=4.27\mathrm{dB}$. CTE slip = CTE for the slippery boundary phantom. CTE adhered = CTE for the adhered boundary phantom. Mean halo strain is the mean strain measured within the annulus centered on the inclusion boundary. The nine images analyzed to produce each set of nine data points are shown as thumbnails at the bottom of the figure, for both the adhered and mobile inclusions. Strain is linear between 0% (dark) and 0.3% (light). Unscaled image is $25\mathrm{mm}\times 65\mathrm{mm}$.

5.1.

Contrast Transfer Efficiency as a Function of Young’s Modulus Contrast

In our FEM simulations of mobile inclusions with a negative $C_{\mathrm{YM}}$, the contrast reversal described above was not observed. We believe that this is due to the presence of lubricating fluid along the mobile inclusion boundary, which was the only way to create low friction in the phantom and was not simulated in FEM. Likewise, for mobile inclusions with positive $C_{\mathrm{YM}}$, the extremely high positive values of CTE at low values of applied strain are believed also to be due to fluid lubrication. As such, for the purposes of plotting CTE as a function of $C_{\mathrm{YM}}$, to make a fair comparison between simulation and experiment, we inspected the elastograms within each dynamic strain sequence for mobile inclusions with negative $C_{\mathrm{YM}}$ and selected for analyzing the frame closest to the contrast reversal point without contrast reversal occurring (i.e., at an applied strain that is equal to or just higher than that at which contrast reversal occurs). This is consistent with the frame in Fig. 9 corresponding to the disappearance of a proximal strain halo (between the transducer and the upper surface of the inclusion) without substantial disappearance of the distal strain halo (deep to the lower surface of the inclusion). This frame is identified in Fig. 9 with “*.” For positive $C_{\mathrm{YM}}$ phantoms, we selected again the frame corresponding to the loss of the proximal strain halo. In Fig. 8, this occurs at an applied strain of about 2%. The rational for these choices was to select frames where excess fluid that we hypothesized was causing the applied strain dependence of the contrast and strain halo (possibly due to its existence as a layer that is “compressible” by virtue of its ability to flow, although this is discussed further below) had been squeezed away but where sufficient lubrication remained such that the inclusions should still be mobile. The CTE values derived from analyzing only these selected frames, for all $C_{\mathrm{YM}}$ phantoms with mobile inclusions, are displayed in Fig. 10. Error bars are not displayed because points are, for the reasons described above, selected from a single experiment involving one set of phantoms. Although each experiment with each phantom had a reliably reproducible trend, the individual points varied such that a direct comparison became under representative of the trend. Also, shown are the CTE values for adhered inclusions, obtained by measuring mean strain in the inclusion and background ROIs over all frames in the sequence. Error bars in this instance were calculated as 1 standard deviation about the mean. They were constantly between 0.10 and 0.14 and are too small to display graphically. There is no point displayed for a mobile boundary phantom with $C_{\mathrm{YM}}=+2.68\mathrm{dB}$ because this phantom did not exhibit mobile features. The data from this phantom are very similar to the data from the adhered phantom and so were excluded from the mobile data series.

Fig. 10

CTE curves for adhered and mobile inclusions with varying Young’s modulus contrast. For mobile inclusions, these data were obtained by measuring CTE from the mean inclusion and mean background strain in the frames selected from dynamic strain sequences where it was judged that excess lubricating fluid had been removed but sufficient remained to ensure a slippery inclusion boundary. For the adhered inclusions, the result was independent of applied strain. For descriptions of how the error bars were calculated, see the main text.

5.2.

Stiff Inclusions

Figure 11 plots the variation in CTE with applied strain where CTE has been calculated separately for the central region under the adherence, and for the lateral regions that remain under a locally mobile boundary. As adherence occurs between the background and the upper pole of the inclusion, the central portion under the adherence appears less stiff although the areas lying either side retain their stiff appearance caused by the local effects of a partial mobile boundary. The polar adherence that results in the divergence of the lines for the adhered and slippery zones corresponds with a drop in mean halo strain reflecting the loss of the mobile boundary.

Fig. 11

Zones of slip: CTE is plotted against applied axial strain for the slip and adhered phantoms with a YMC of 4.27 dB (stiff inclusion). For the slip phantom, four ROIs were measured: All = the entire inclusion. Slip (adhered zone) = central portion under the zone of adherence. Slip (slip zone) = side regions under the local areas of slip. Slip halo strain = mean strain in the slip halo. CTE for the adhered phantom is also plotted. Strain is linear between 0% and 0.3%. Unscaled image is $25\mathrm{mm}\times 65\mathrm{mm}$.

Inclusions with a positive $C_{\mathrm{YM}}$ cause a strain concentration to appear above the inclusion as the soft background is strained against the stiff inclusion. Compression of the mobile boundary against the upper surface of the inclusion causes obliteration of the fluid layer at the upper pole of the inclusion as fluid is displaced and a point of adherence occurs between the background and the inclusion. The true strain image representation of $C_{\mathrm{YM}}$ is immediately revealed along a beam axial (vertical) line that includes this point of adherence where it behaves as an adhered inclusion but the mobile boundary regions over the lateral portions of the inclusion are not obliterated and the lateral zones of the inclusion beneath them retain their initial appearance. As the force increases, more fluid is displaced from the polar region and the zone of adherence spreads down the side of the inclusion toward the equator, accordingly the central zone of true strain representation of stiffness increases in size and the lateral zones decrease.

5.3.

Isostiff and Soft Inclusions

In order to demonstrate this phenomenon for a soft inclusion where contrast reversal occurs, Fig. 12 plots CTE that has been calculated using the signed rather than the absolute value of $C_{ϵ}$ which we have termed CTE’ thus

Fig. 12

Zones of slip. CTE’ (signed) is plotted against applied axial strain for the slip and CTE (absolute) for adhered phantoms with a YMC of $-4.27\mathrm{dB}$ (soft inclusion). For the slip phantom, signed CTE’ was calculated as Co—|Ct|. Four ROIs were measured: All = the entire inclusion. Slip (central zone) = central portion with a 20 pixel stand-off between the slip halo ($\text{image width}=250\text{pixels}$). Slip (peripheral halo) = band (10 pixels wide) immediately inside the slip halo. Slip halo strain = mean strain in the slip halo. CTE (absolute) for the adhered phantom is also plotted. Strain is linear between 0% and 0.3%. Unscaled image is $25\mathrm{mm}\times 65\mathrm{mm}$.

This results in a smooth transition from negative to positive contrast as CTE’ crosses the $C_{\mathrm{YM}}$ ($-4.27\mathrm{dB}$) in Fig. 12 rather than a dip toward it (as in Fig. 9), and more clearly illustrates the difference between the central and the peripheral ROI. CTE’ within the inclusion has been segmented into the peripheral zone (adjacent to the slip plane) and the central zone. The CTE’s for all ROIs are initially equal but the central zone ROI immediately rises above the global measurement to reflect a soft appearance. The peripheral zone rises at a slower rate and lies below the global inclusion measurement reflecting an annular “mantel” that appears stiff adjacent to the circumferential slip plane. The central zone also rises above the value for the adhered phantom demonstrating the softening effect compensating the peripheral low strain as strain is conserved within the inclusion. This explains the target appearance of mobile soft and isostiff inclusions on axial strain images: high strain in the center is related to true but amplified apparent softness, a circumference of low strain adjacent to the slip plane, and a rim or halo of high strain (the mobile boundary) and explains why mobile inclusions with no $C_{\mathrm{YM}}$ appear soft. This also provides experimental verification of the effect observed in the FEM simulations where altering the ROI radius results in differing strain values.

A possible explanation for the difference between inclusions with a positive $C_{\mathrm{YM}}$ and inclusions that have no $C_{\mathrm{YM}}$ or a negative $C_{\mathrm{YM}}$ is that it is not possible to force a stiff background against a soft inclusion or an inclusion with zero $C_{\mathrm{YM}}$, so the strain concentrations and polar adherence seen with stiff inclusions do not occur. Instead, there is uniform pressure distribution across the mobile boundary. Unlike the situation seen in stiff inclusions where polar adherence permits strain “escape” through the central subpolar region of the inclusion boundary, this does not occur with a continuous circumferential fluid layer and so the inclusion strains uniformly. As an additional effect, the mobile fluid layer causes low strain just inside the circumference of the inclusion, and because strain within the inclusion must be conserved, the center appears soft as it strains more to compensate the peripheral low strain. This is seen, albeit with elastographic noise, in the thumbnail images and the zonal analysis results of Fig. 12, and agrees with the FEM predictions in this paper.

6.1.

Relationship to Modeling

An important difference between experiment and simulation was that the simulation did not observe several of the applied strain-dependent phenomena that were observed here experimentally, viz., variation in halo strain, variation in strain contrast, and contrast reversal. In simulation, a mobile inclusion boundary is achieved simply by setting the friction coefficient to zero. Experimentally, a lubricant is required. We, therefore, hypothesize that the additional phenomena observed experimentally were due to the use of fluid lubrication to achieve a mobile inclusion boundary. Specifically, fluid could be squeezed out of the space between the inclusion and the background, escaping from the phantom at the ends of the cylindrical rod-shaped inclusion. As applied strain increased, more and more fluid may have been squeezed out until points of adherent (i.e., unlubricated) contact between background and inclusion were created, generating the observed elastogram appearances of both local reduction in strain halo contrast and local increase in inclusion strain along the axial lines where this occurred. Eventually, at sufficiently high applied strain, most of the lubricant had gone and almost complete adherence existed around the inclusion boundary, producing almost complete loss of the strain halo and an adherent-type CTE result as seen in the adherent line in Fig. 10 (derived from the mean result from the adhered phantoms in Figs. 11 and 12, for inclusions that were designed to be mobile). Indeed, when small inclusion ROIs were used to measure strain contrast for only axial lines that included points of lost halo strain (at intermediate applied strain), this too resulted in an adherent-type CTE curve (Fig. 11), which seems to confirm that localized loss of halo strain corresponds to localized adherence. The one exception to this is the presence of the proximal halo strain seen in Figs. 11 and 12 at very small applied strains. We propose that this is due to the existence of so much lubricating fluid (i.e., an excess, relative to the amount necessary simply to achieve a mobile boundary) that the applied strain in these early compression steps is largely employed (in effect, absorbed) in producing a reduction in fluid volume, resulting in less strain than would otherwise be transmitted into the inclusion and distal background. This could therefore also account for the additional negative strain contrast and the contrast reversal for the soft inclusion phantom in these early compression steps, and the “hypermobile” CTE versus YM contrast curves seen in Figs. 8 and 9.

The behavior discussed in the last paragraph above is very different, in fact virtually the opposite, to that described for the mobile boundary elastography study of Chakraborty et al.,⁶⁴ where an inclined plane phantom was used, and the applied force needed to produce slip (as judged by various elastographic criteria) was compared with the angle of the inclined plane and shown to agree with a simple model for the static coefficient of friction. However, Chakraborty et al. used very little lubrication at the inclined plane between two blocks of gelatine, which were thus designed not to slip until sufficient force was applied. In the present study, sufficient lubricant was initially present for slip to occur until the lubricant was squeezed out and adherent contact was made. It remains to be seen which model has the greatest clinical relevance and under which circumstances.

If the excess fluid hypothesis described above does indeed provide the correct explanation for the strain image behavior discussed above for the initial compression steps, then it is to be expected that the elastograms for these steps should exhibit a time-dependent behavior due to the finite time that it takes the fluid to flow and either redistribute around the boundary or escape at the ends of the phantom. In other words, the strain images should exhibit a poroelastic response^22,24 at the instant immediately after a step in applied strain, before the lubricating fluid has had a chance to flow, the incompressible fluid should transmit the strain but lubricate the boundary, producing a strain image that may be comparable to that predicted by simulating a mobile inclusion. Subsequent fluid flow will cause strain relaxations to occur, which may result in the “hypermobile” elastographic behavior mentioned above. Future work could include a high time resolution strain imaging experiment to evaluate this conjecture. Meanwhile, the force relaxation observed and mentioned in the methods section above provides circumstantial evidence in support of this suggestion.

6.2.

Clinical Significance

Physiological slip occurs in many anatomical locations in living organisms, where mobility is paramount to function. Examples include joints, tendons, and mobile internal organs such as the heart, lungs, and all abdominal viscera. In all these examples, motion is facilitated by a thin layer of lubricating fluid. Benign breast tumors, such as fibroadenomas, demonstrate slip behavior,⁵⁴ which may or may not also be in the presence of a lubricating layer.

The fluid lubricated mobile boundaries in our phantoms mimic these physiological situations, and it is important to discuss the effects that this may have on clinical imaging in situations where fluid lubrication is present. We have shown that, even without fluid lubrication, a mobile boundary can cause a soft lesion to have the appearance of a stiff lesion and to alter the axial strain contrast and heterogeneity of both soft and stiff lesions. Our results also show, however, practical ways in which misinterpretation may be avoided. As suggested in the simulation data, a strong strain band (halo) at the boundary and the characteristic strain heterogeneity, where strain inside the lesion is reduced close to the boundary and increases toward the lesion’s center, if present, would indicate a slippery boundary. Here we have also shown that if fluid lubrication is present one may also be able to take advantage of the dynamic nature of quasistatic ultrasound elastography; initial strong strain halo and high negative strain contrast (lesion appearing stiff) at small applied strains, then, as applied strain increases, a decreasing strain halo and strain contrast, with possible contrast reversal if the lesion is softer than the background, would reveal its true characteristics. It is significant finding that for any lesion where these changes occur during compression, it can be inferred that it is mobile with a well-developed fluid lubricated slippy boundary. In the case of tumor boundaries, there is likely to be no invasion into the surrounding tissues (which may aid diagnosis) and in the case of intraoperative guidance it identifies a plane where atraumatic dissection may take place.

Predicting the location of atraumatic dissection planes is important during brain tumor surgery.^41,65,66 Meningiomas, for example, are tumors that arise from the membranes surrounding the brain and grow into it. There is a physiological slip plane between the brain and its membranes (the subarachnoid space). As the tumor enlarges, it presses into the brain and although the subarachnoid space initially remains intact further growth results in loss of the subarachnoid space. Blood vessels on the surface of the brain may become surrounded by the tumor and eventually supply it through generation of new blood vessels. Eventually, direct brain invasion may occur. If the subarachnoid space is intact, a clear dissection plane exists throughout and these tumors are easily removed with little or no damage to the underlying brain. If the subarachnoid space has been breached, resulting in local areas of brain invasion, this rarely occurs over the entire brain–tumor interface. It is thus always easier to start the dissection in a region where a plane exists and work into an area where no plane exists, so location of a small area of slip is an advantage.

In summary, fluid lubricated mobile boundaries can be distinguished from adherent boundaries using the following criteria:

Evidence of these phenomena has been found in clinical scans (Uff, 2011) and this will be the subject of a future paper.

6.3.

Alternative Mobility Indices

Other authors have proposed shear strain as a means of identifying whether a tumor boundary is mobile.^17,53,54 Chakraborty⁵³ used shear strain as a measure of when slip occurred as the force progressively increased and Coutts et al.¹⁹ showed the value of shear strain in detecting the fat-muscle slip plane. We have not evaluated shear strain in the present study; however, this may provide additional information in future work.