# Quasi-Static Elastography and Ultrasound Plane-Wave Imaging: The Effect of Beam-Forming Strategies on the Accuracy of Displacement Estimations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{app}are the depth of the reconstruction point and aperture width, respectively. Furthermore, apodization can be applied to weigh the signals (e.g., by Hamm or Hann function) to reconstruct each point and so expected angle sensitivity and back-scatter signal intensities can be incorporated. Apodization is often applied to increase contrast in B-mode images.

## 2. Materials and Methods

_{z}and u

_{x}) after rotation can be described as:

_{0},z

_{0}) are the rotation angle and center coordinates, respectively. The gradient of u

_{x}and u

_{z}in the lateral (x) and the axial (z) directions can respectively be described as:

_{xx}and s

_{zz}in (Equations (4) and (5)) approximate zero, independently of the rotation angle and center. As these gradients (strains) yield 0 and the exact rotational angle and center were unknown in this experiment, we adopted the root-mean squared error (RMSE) of the gradients as a measure of the accuracy of the displacement estimates. The gradients of the displacements were calculated using a one-dimensional three-point least-squares strain estimator [7]. Gradients were only calculated and evaluated within a field-of-view (FoV) measuring –10 to 10 mm laterally and 17.5 to 52.5 mm axially for all transducers. However, the FoV measurement of the MS250 transducer was –8 to 8 mm laterally, since the FoV was limited by the transducer footprint. The top 17.5-mm axial depth was measured through water and was therefore neglected for all transducers. All acquisition and processing steps and intermediate results are summarized in Figure 5.

## 3. Results

_{zz}) were approximately 0% (see Figure 6f–i). For the 12L4VF transducer the results revealed some artifacts in the left top corner (Figure 6b,g). These artifacts were visible for all beam-forming strategies and line densities. The MS250 transducer showed more outliers at lower depths (30–52.5 mm) which were probably caused by attenuation of the ultrasound signal at relatively large depths for this frequency.

_{xx}). Similar artefacts were also visible for the 12L4VF and MS250 transducers since the axial and lateral displacements were estimated using two-dimensional cross-correlations, which implies these estimations were coupled. Compared to the axial displacement and strain fields (Figure 6a–j), the lateral fields (Figure 6k–t) were noisier for all transducers, beam-forming strategies and line densities. In the case of two lines per pitch and Lu’s-fk beam-forming as shown in Figure 6, s

_{zz}varied between ±0.5%, whereas s

_{xx}varied between ±1%.

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Introduction:**Element data obtained by plane-wave acquisitions were beam-formed to reconstruct radio frequency (RF) data for displacement estimation. In this study, three beam-forming strategies were used: delay-and-sum (DaS), Lu’s-fk, and Stolt’s-fk. In DaS beam-forming, dynamic focusing in receive was applied with and without apodization (Hamm function) in which the F-number had to be set (Equation (1)). Furthermore, angular weighting was applied in Lu’s-fk and Stolt’s-fk beam-forming in which the migrated k-space spectrum was multiplied with a template designed such that wave directions closer aligned to the beam-steering direction were multiplied with higher weights compared to waves deviating from the steering angle. In angular weighting, the angular range had to be set: wave directions within this range were weighted by Hann function and outside were weighted by zero. Summarized, the optimal F-number in DaS with and without apodization and the range in angular weighting were investigated.

**Method and Materials:**Plane-wave element data were obtained by three transducers (L7-4, 12L4VF, L12-5) in a multi-purpose phantom (model 539; ATS Laboratories, Norfolk, VA, USA) containing circular lesions and needles to evaluate contrast and resolution, respectively. Element data were beam-formed by above strategies in which the F-number (DaS) or angular range (angular weighting) were varied between 0 and 2.0, and ±10 and ±30. The mean contrast-to-noise ratio (CNR) of the lesions and mean axial and lateral resolution (full width at half maximum) of the needles were calculated to a depth of 50 mm (Figure A1) using software provided by the Plane-wave Imaging Challenge in Medical Ultrasound (PICMUS) of the IEEE International Ultrasonics Symposium 2016 [21].

**Results and Discussion:**In DaS, an F-number of 0.875 seemed to be most optimal for all transducers: contrast decreased while lateral resolution remained constant at values below 0.875 and contrast remained constant above 0.875 while lateral resolution increased. The axial resolution seemed to be independent of the F-number. These results were found in DaS with and without apodization. As can be seen in Table A1, apodization increased contrast, which came at the cost of increased lateral resolution.

**Figure A1.**Cross-section of the multi-purpose phantom and the needles and lesions used for analysis indicated by the orange rectangular boxes.

**Table A1.**Axial and lateral resolution, and contrast-to-noise ratio (CNR) by DaS (F-number of 0.875).

Transducer | Apodization | Ax. Res. | Lat. Res | CNR |
---|---|---|---|---|

L7-4 | No apod | 421 µm | 490 µm | 7.8 dB |

- | Hamm | 422 µm | 651 µm | 9.7 dB |

12L4VF | No apod | 422 µm | 426 µm | 5.4 dB |

- | Hamm | 419 µm | 577 µm | 7.4 dB |

L12-5 | No apod | 406 µm | 426 µm | 5.3 dB |

- | Hamm | 405 µm | 550 µm | 7.2 dB |

**Table A2.**Axial and lateral resolution, and CNR by angular weighted Lu’s-fk and Stolt’s-fk (range ±20°).

Transducer | Method | Ax. Res. | Lat. Res | CNR |
---|---|---|---|---|

L7-4 | Lu’s-fk | 423 µm | 601 µm | 9.5 dB |

- | Stolt’s-fk | 423 µm | 610 µm | 9.5 dB |

12L4VF | Lu’s-fk | 406 µm | 530 µm | 6.9 dB |

- | Stolt’s-fk | 408 µm | 616 µm | 6.7 dB |

L12-5 | Lu’s-fk | 400 µm | 510 µm | 7.4 dB |

- | Stolt’s-fk | 401 µm | 556 µm | 7.4 dB |

**Table A3.**Axial and lateral resolution, and CNR by Lu’s-fk and Stolt’s-fk without angular weighting.

Transducer | Method | Ax. Res. | Lat. Res | CNR |
---|---|---|---|---|

L7-4 | Lu’s-fk | 420 µm | 493 µm | 8.1 dB |

Stolt’s-fk | 416 µm | 507 µm | 7.7 dB | |

12L4VF | Lu’s-fk | 405 µm | 423 µm | 5.4 dB |

Stolt’s-fk | 412 µm | 584 µm | 4.7 dB | |

L12-5 | Lu’s-fk | 397 µm | 420 µm | 6.3 dB |

Stolt’s-fk | 402 µm | 532 µm | 5.6 dB |

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**Figure 1.**Illustrations of: (

**a**) a focused acquisition series in which the aperture (active elements in light gray) is shifted to acquire data line-by-line by transmitting a focused ultrasound beam and receiving the reflected signal for each aperture position; (

**b**) a plane-wave acquisition, where an unfocused ultrasound is transmitted and reflected signals are received by the full transducer aperture.

**Figure 2.**Schematic overview of delay-and-sum (DaS) beam-forming: for each reconstruction point, the time-of-flight (ToF) can be calculated by the transmit (τ

_{T}) and receive time towards each element (τ

_{R}). These ToF values are used to delay and sum the element data; if required the data can be weighted (w) by an apodization function. θ

_{inc}is the incidence angle of the signal.

**Figure 3.**Overview of beam-forming in the fk-space: acquired element data is transformed to the fk-space using 2-D Fourier transform, and Lu’s-fk or Stolt’s-fk migration is applied to convert the data into the k-space. If required, this spectrum can be multiplied by a template (green overlay) to filter the data. Finally, the reconstructed data can be obtained by the 2-D inverse Fourier transform.

**Figure 4.**Experimental setup: transducer (Table 1) was connected to a Verasonics (V1 or Vantage) research ultrasound machine; element data were acquired prior to and after rotation (θ

_{Rot}) of a container in which a gelatin phantom was positioned (light grey) and water was poured on top of the phantom to ensure ultrasonic coupling.

**Figure 5.**Summary of the processing flow described in the Material and Methods section. This work flow was repeated for every transducer (Table 1). Rectangles represent processing steps (acquisitions or calculations) and parallelograms represent (intermediate) results.

**Figure 6.**Overview of displacement and strain fields after rotation using Lu’s-fk and two lines per pitch as a beam-forming strategy and for line density, respectively: (

**a**–

**c**) axial displacements; (

**f**–

**n**) axial strains (s

_{zz}); (

**k**–

**n**) lateral displacements; (

**p**–

**s**) lateral strains (s

_{xx}); (

**e**,

**o**) color bar in millimeters related to the figures in the same row; (

**j**,

**t**) color bar in percentages related to the figures in the same row. The axis of the displacement and gradient fields represents the position below the transducer in millimeters.

**Figure 7.**The root-mean-squared error (RMSE) of the axial strains (s

_{zz}) for all beam-forming strategies and line densities for the (

**a**) L7-4, (

**b**) 12L4VF, (

**c**) L12-5, and (

**d**) MS250 transducer; (

**e**) is the legend used in (

**a**–

**d**).

**Figure 8.**The root-mean-squared error (RMSE) of the lateral strains (s

_{xx}) for all beam-forming strategies and line densities for the (

**a**) L7-4, (

**b**) 12L4VF, (

**c**) L12-5, and (

**d**) MS250 transducer; (

**e**) is the legend used in (

**a**–

**d**).

**Figure 9.**The root-mean-squared error (RMSE) without the top left artifact of the (

**a**) axial strain (s

_{zz}) and (

**b**) lateral strain (s

_{xx}) of the 12L4VF transducer; s

_{zz}and s

_{xx}values in the top left corner (axial and lateral position smaller than 28 and 0 cm respectively) were neglected in these RMSE calculations.

**Figure 10.**Lateral displacement and strain fields using Lu’s-fk for the L12-5 transducer: (

**a**,

**c**) displacement field using one (

**a**) or two (

**c**) lines per pitch. Please refer to Figure 6o for the color bar; (

**b**,

**d**) strain field using one (

**b**) or two (

**d**) lines per pitch. Please refer to Figure 6j and Figure 6t for the corresponding color bars.

Transducer | Bandwidth | f_{c} | Pitch | Manufacturer |
---|---|---|---|---|

L7-4 | 4–7 MHz | 5.0 MHz | 298 µm | ATL ^{1} |

12L4VF | 4–12 MHz | 8.2 MHz | 266 µm | Siemens ^{2} |

L12-5 | 5–12 MHz | 9.0 MHz | 198 µm | ATL ^{1} |

MS250 | 13–24 MHz | 21 MHz | 88 µm | VisualSonics ^{3} |

^{1}ATL, Bothell, WA, USA;

^{2}Siemens Healthineers, Issaquah, WA, USA;

^{3}FUJIFILM VisualSonics Inc., Toronto, ON, Canada.

Transducer | Step | Template Window ^{1} | Search Window ^{1} | Filter Size ^{2} | DispGrid ^{1} |
---|---|---|---|---|---|

L7-4 | 1 | 1.22 × 1.49 | 4.82 × 5.07 | 5 × 5 | 0.298 × 0.298 |

- | 2 | 0.62 × 0.90 | 0.89 × 1.49 | 3 × 3 | 0.298 × 0.298 |

12L4VF | 1 | 0.68 × 1.33 | 2.68 × 4.52 | 5 × 5 | 0.266 × 0.266 |

- | 2 | 0.35 ×0.80 | 0.49 × 1.33 | 3 × 3 | 0.266 × 0.266 |

L12-5 | 1 | 0.68 × 0.99 | 2.68 × 3.37 | 5 × 5 | 0.198 × 0.198 |

- | 2 | 0.35 × 0.60 | 0.49 × 0.99 | 3 × 3 | 0.198 × 0.198 |

MS250 | 1 | 0.29 × 0.44 | 1.15 × 1.50 | 5 × 5 | 0.088 × 0.088 |

- | 2 | 0.15 × 0.27 | 0.21 × 0.44 | 3 ×3 | 0.088 × 0.088 |

^{1}Axial × lateral window size or DispGrid resolution in mm;

^{2}# samples and # lines in DispGrid which is independent of # lines per pitch.

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**MDPI and ACS Style**

Hendriks, G.A.G.M.; Chen, C.; Hansen, H.H.G.; De Korte, C.L. Quasi-Static Elastography and Ultrasound Plane-Wave Imaging: The Effect of Beam-Forming Strategies on the Accuracy of Displacement Estimations. *Appl. Sci.* **2018**, *8*, 319.
https://doi.org/10.3390/app8030319

**AMA Style**

Hendriks GAGM, Chen C, Hansen HHG, De Korte CL. Quasi-Static Elastography and Ultrasound Plane-Wave Imaging: The Effect of Beam-Forming Strategies on the Accuracy of Displacement Estimations. *Applied Sciences*. 2018; 8(3):319.
https://doi.org/10.3390/app8030319

**Chicago/Turabian Style**

Hendriks, Gijs A.G.M., Chuan Chen, Hendrik H.G. Hansen, and Chris L. De Korte. 2018. "Quasi-Static Elastography and Ultrasound Plane-Wave Imaging: The Effect of Beam-Forming Strategies on the Accuracy of Displacement Estimations" *Applied Sciences* 8, no. 3: 319.
https://doi.org/10.3390/app8030319