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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Atherosclerotic plaque rupture can initiate stroke or myocardial infarction. Lipid-rich plaques with thin fibrous caps have a higher risk to rupture than fibrotic plaques. Elastic moduli differ for lipid-rich and fibrous tissue and can be reconstructed using tissue displacements estimated from intravascular ultrasound radiofrequency (RF) data acquisitions. This study investigated if modulus reconstruction is possible for noninvasive RF acquisitions of vessels in transverse imaging planes using an iterative 2D cross-correlation based displacement estimation algorithm. Furthermore, since it is known that displacements can be improved by compounding of displacements estimated at various beam steering angles, we compared the performance of the modulus reconstruction with and without compounding. For the comparison, simulated and experimental RF data were generated of various vessel-mimicking phantoms. Reconstruction errors were less than 10%, which seems adequate for distinguishing lipid-rich from fibrous tissue. Compounding outperformed single-angle reconstruction: the interquartile range of the reconstructed moduli for the various homogeneous phantom layers was approximately two times smaller. Additionally, the estimated lateral displacements were a factor of 2–3 better matched to the displacements corresponding to the reconstructed modulus distribution. Thus, noninvasive elastic modulus reconstruction is possible for transverse vessel cross sections using this cross-correlation method and is more accurate with compounding.

Rupture of atherosclerotic plaques and the successive formation of thrombus is regarded as one of the major causes of stroke and myocardial infarction [

Elastic modulus reconstructions have been performed for coronary arteries based on ultrasound strain information that was derived from raw radiofrequency (RF) data obtained intravascularly using a catheter-mounted ultrasound device [

The main goal of this study is to examine if the displacements resulting from this three-angle compounding also allow a better reconstruction of the elastic moduli than can be obtained through conventional single-angle imaging. To compare the accuracy of the reconstructions based on the two methods, simulated and experimental phantom data for various vessel geometries were generated. To our knowledge this is the first study that combines compounding methods for elasticity reconstructions in transverse cross sections of vessel shaped structures based on noninvasive ultrasound recordings obtained with a linear array transducer.

To test our reconstruction methods, vessels with three different configurations were considered. The geometries are presented in

To perform the estimation of the relative elasticity modulus, an iterative algorithm was used which requires the axial and lateral displacement fields of the tissue as input. The algorithm has been described extensively for radial displacement fields obtained from intravascular ultrasound (IVUS) data [_{x}(μ) and u_{y}(μ) ] that match the input displacement fields best (

Here
_{μ} is a smoothness term that restricts the amount of variation in the modulus field. α is a weighting factor that determines the amount the smoothness term contributes to the penalty function. α was set to 1e-9 for this entire study. _{0}_{0}

For the heterogeneous case with two layers, the constant _{0}

To create the finite mesh for the modulus reconstruction, we segmented the inner and outer vessel boundaries, populated them with regularly spaced nodes and meshed them in a regular fashion, both radially and angularly, with respect to the lumen center

To obtain the finite mesh for a certain angle, the first displacement estimate within the vessel wall with respect to the lumen center, was considered as the inner radius value for that angle. Then, for the same angle the last displacement estimate within the vessel wall was determined and defined as the outer radius value for that angle. The procedure was repeated for each angle at an angular increment of 1° resulting in a value for the inner and the outer radius for each angle. Next, a small and a large ellipse were fitted through the inner and outer radius values using least squares fitting. Because the radii of the lumen ellipse were a little underestimated and the outer vessel radii were overestimated by the fitting procedure again discontinuities were generated at the boundaries. To reduce these discontinuities as well, the lumen ellipse was slowly expanded and the wall ellipse was slowly shrunk until all values laid within the ROI. Once the ellipses were known, the wall thickness for each angle was calculated (_{wall}_{p}

As stated before, first the reconstruction algorithm was applied to axial and lateral displacement fields obtained by finite element modeling. Finite element models (FEMs) of the three vessel geometries were constructed using the Partial Differential Equation Toolbox of Matlab 2007a (MATLAB, The Mathworks, Natick, MA, USA). The displacement fields for the three different vessel configurations were calculated for an intra-luminal pressure increase of 4 mmHg under the assumption of plane strain. Over 60,000 two-dimensional linearly elastic finite elements were defined and distributed over the vessel volume. All elements were assumed to be nearly incompressible (Poisson's ratio, ν = 0.495) and isotropic. The Young's moduli of the various layers were set to the values shown in

To test the strain estimation in combination with the reconstruction method in a controlled situation, ultrasound RF data were simulated using the ultrasound simulation software package Field II [_{c}) of 8.7 MHz and 288 physical elements. The element pitch was 135 μm and the element height 6 mm. Each physical element was subdivided into 10 by 10 mathematical elements. RF data were simulated at a sampling rate of 117 MHz and down sampled to 39 MHz afterwards to match the experimental situation. The three times higher sampling rate during the simulations was chosen, because Field II uses digitally sampled versions of the excitation and impulse responses, which are continuous signals in reality. In the axial-elevational direction, a fixed focus of 2.5 cm was set and Hanning apodization was used in both transmit and receive to mimic an acoustical lens. In the axial-lateral direction, a fixed transmit focus of 2.5 mm was set. In transmit, no apodization was applied; in receive, dynamic focusing was applied with an F-number of 0.875. The maximum number of simultaneously active elements was restricted to 128. Lateral apodization with a Hamming window was applied in receive. Toward the edges of the transducer a lower, but symmetric, number of elements were active in receive and transmission. All these parameters were chosen to match the transducer used in the experiments. To generate the pre-deformation RF data, one million scatterers were randomly distributed over the cross sections of the vessels shown in

In analogy with the finite element modeling and the simulations, vessel mimicking phantoms were constructed for each vessel geometry. The phantoms were constructed from various gelatin-agar solutions with Young's moduli as shown in _{c} = 8.7 MHz, pitch = 135 μm, f_{s} = 39 MHz) was used to acquire the data. The focus was set at 2.5 mm.

Before performing displacement estimation, all RF data were low-pass filtered to remove the signal caused by grating lobes. Additionally, a correction was applied to correct for the skewness of the beam-steered data. Both the grating lobe filtering and the correction for the skewness were described in detail previously [

Displacement estimation was carried out separately for each beam steering angle. The axial component was estimated directly from the non-steered 0° acquisitions both for the three-angle method and the conventional single-angle acquisition method. The lateral component was either estimated from the non-steered 0° acquisitions (single-angle imaging) or indirectly, by projecting the axial displacement estimates from the acquisitions at the positive and negative beam steering angles (three-angle compound imaging) using the equation:
_{ax,θ} and u_{ax,-θ} are the axial displacements estimated at the positive and negative beam steering angle, respectively. u_{lat,0} is the lateral displacement component.

To determine the performance of the modulus reconstruction, the median and interquartile range (IQR) of the absolute differences between the axial and lateral displacement input and output of the reconstruction method were calculated. For example, for the axial component this absolute difference Δu_{x} was calculated following:
_{x} is the reconstructed axial displacement, and u_{x}^{m} is the estimated axial displacement for a point on the mesh with radius r, and angle θ.

Next to these absolute displacement differences, the median value and the IQR of the relative modulus estimates over the vessel wall were calculated. For the heterogeneous vessel, the modulus analysis was performed separately for both layers.

The left column of

The results illustrate that the combination of the reconstruction algorithm, the interpolation method, and the iterative 2D cross-correlation based displacement estimation techniques allow an accurate reconstruction of relative Young's moduli based on linear array ultrasound data. As can be expected, because the lateral displacements improve with compounding [

As aforementioned, most errors in the reconstructed images were observed at the lumen-vessel interface. These errors were observed even in the FEM results (

In previous modulus reconstruction studies using intravascular ultrasound data, no specific increase of error in the vicinity of the lumen interface was observed. This can probably be explained by the fact that IVUS imaging is already performed in a polar grid, thus no interpolation is required. The errors are in the order of 10% when performing compounding and larger without compounding, which seems quite large. However, also without compounding the errors are hardly noticeable when considering a contrast difference between different parts of a plaque of a factor of 10 as is illustrated for instance by

Another point of discussion is the fact that we reconstructed relative modulus images instead of absolute modulus images. The reason we did not investigate absolute moduli, is because that also requires knowledge of the frame-to-frame intraluminal hydrostatic pressure change. Although the hydrostatic pressures were known in the present study, these pressures will not be known in an

Finally a short discussion about the applicability of the soft prior condition

Accurate modulus reconstructions for transverse cross sections of vascular structures can be obtained given axial and lateral displacement estimates which are estimated based on raw radio frequency ultrasound data obtained noninvasively with a linear array transducer. Displacements obtained using a recently developed three-angle compounding method allow a more accurate reconstruction of the modulus than can be obtained from displacements estimated from conventional single-angle images. The next step will be to explore the limits of the proposed modulus reconstruction method and to determine its performance

This research is supported by the Dutch Technology Foundation STW (NKG 07589), Applied Science Division of NWO and the Technology Program of the Ministry of Economic Affairs. Furthermore, this research is supported by the NIH National Heart and Lung research grant R01-HL 088523. The authors also acknowledge Philips Medical Systems for their support.

The geometries and Young's moduli of the vessels investigated. (

(

(

Relative modulus images for the three vessels obtained from ultrasound simulations. (

Relative modulus images for the three vessels obtained from phantom experiments. (

Modulus reconstructions for the heterogeneous two-layered phantom for different soft-prior settings. On the left the regions are shown that were considered to have a different modulus. On the right the corresponding relative modulus reconstruction using soft priors.

Median and interquartile ranges (IQR) of the modulus reconstruction of the three vessels based on finite element modeling (FEM) and simulations. The median and IQR of the absolute differences between the input and output displacement fields are reported also.

(a) |
1.000 (0.997–1.003) | 1.000 (0.996–1.005) | 0.7 (0.3–1.3) | 4.8 (2.3–8.3) |

(a)^{C} |
1.000 (0.997–1.002) | 0.5 (0.2–0.8) | 1.3 (0.6–2.4) | |

(b) |
1.000 (0.995–1.006) | 1.000 (0.982–1.030) | 1.5 (0.7–2.8) | 8.0 (3.9–13.6) |

(b)^{C} |
1.000 (0.991–1.015) | 1.3 (0.6–2.3) | 3.1 (1.4–5.7) | |

(c)_{in} |
0.105 (0.105–0.105) | 0.236 (0.228–0.252) | 2.4 (1.0–4.4) |
8.3 (3.8–14.0) |

(c)_{in}^{C} |
0.128 (0.128–0.128) | 1.2 (0.5–2.0) |
3.4 (1.6–5.9) | |

(c)_{out} |
1.000 (0.996–1.006) | 1.000 (0.984–1.015) | 2.4 (1.0–4.4) |
8.3 (3.8–14.0) |

(c)_{out}^{C} |
1.000 (0.993–1.000) | 1.2 (0.5–2.0) |
3.4 (1.6–5.9) |

(a) = concentric homogeneous vessel; (b) = eccentric homogeneous vessel; (c)_{in} = inner layer heterogeneous vessel; (c)_{out} = outer layer heterogeneous vessel,

= single-angle,

= with compounding.

= No differentiation between the two layers of the heterogeneous vessel.

Median and interquartile ranges (IQR) for the modulus reconstruction of the three vessels based on experiments. The median and IQR for the absolute differences between the input and output displacement fields are reported also.

(a) |
1.000 (0.994–1.004) | 1.0 (0.5–1.9) | 3.4 (1.6–5.7) |

(a)^{C} |
1.000 (0.997–1.002) | 0.8 (0.3–1.5) | 1.0 (0.5–1.7) |

(b) |
1.000 (0.988–1.015) | 1.5 (0.4–2.6) | 4.1 (2.0–7.0) |

(b)^{C} |
1.000 (0.996–1.008) | 1.3 (0.5–2.1) | 1.6 (0.8–2.8) |

(c)_{in} |
0.200 (0.195–0.203) | 3.1 (1.5–5.3) |
3.6 (1.7–6.1) |

(c)_{in}^{C} |
0.214 (0.210–0.220) | 2.6 (1.2–5.5) |
3.4 (1.0–3.5) |

(c)_{out} |
1.000 (0.992–1.009) | 3.1 (1.5–5.3) |
8.3 (1.7–6.1) |

(c)_{out}^{C} |
1.000 (0.996–1.008) | 2.6 (1.2–5.5) |
3.4 (1.0–3.5) |

(a) = concentric homogeneous vessel; (b) = eccentric homogeneous vessel; (c)_{in} = inner layer heterogeneous vessel; (c)_{out} = outer layer heterogeneous vessel,

= single-angle,

= with compounding.

= No differentiation between the two layers of the heterogeneous vessel.