Quasi-Static Elastography and Ultrasound Plane-Wave Imaging: The Effect of Beam-Forming Strategies on the Accuracy of Displacement Estimations  
[PERSON]  
1 Medical UltraSound Imaging Center, Department of Radiology and Nuclear Medicine, Radboud University Medical Center, P.O. Box 9101, 6500 HB Nijmegen, The Netherlands; [EMAIL_ADDRESS] (C.C. ); [EMAIL_ADDRESS] (H.H.G.H. ); [EMAIL_ADDRESS] (C.L.d.K.) 2  
[PERSON]  
1 Medical UltraSound Imaging Center, Department of Radiology and Nuclear Medicine, Radboud University Medical Center, P.O. Box 9101, 6500 HB Nijmegen, The Netherlands; [EMAIL_ADDRESS] (C.C. ); [EMAIL_ADDRESS] (H.H.G.H. ); [EMAIL_ADDRESS] (C.L.d.K.) 2  
[PERSON]  
1 Medical UltraSound Imaging Center, Department of Radiology and Nuclear Medicine, Radboud University Medical Center, P.O. Box 9101, 6500 HB Nijmegen, The Netherlands; [EMAIL_ADDRESS] (C.C. ); [PERSON]RESS] (H.H.G.H. ); [EMAIL_ADDRESS] (C.L.d.K.) 2  
[PERSON]  
1 Medical UltraSound Imaging Center, Department of Radiology and Nuclear Medicine, Radboud University Medical Center, P.O. Box 9101, 6500 HB Nijmegen, The Netherlands; [EMAIL_ADDRESS] (C.C. ); [EMAIL_ADDRESS] (H.H.G.H. ); [EMAIL_ADDRESS] (C.L.d.K.) 2  
Footnote 1: [[https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci)]([https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci))  
Footnote 2: [[https://www.mdpi.com/journals/](https://www.mdpi.com/journals/)]([https://www.mdpi.com/journals/](https://www.mdpi.com/journals/))starting with relatively large windows analyzing the envelope signal and subsequently using finer windows of RF-signal in each subsequent iteration with the estimated displacements of the previous iteration as an offset. Finally, a least squares strain estimator can be used to reconstruct a strain map [7]. The majority of current commercial ultrasound systems collect ultrasound RF data line-by-line using focused ultrasound transmit-and-receive sequences. Back in 1979, plane wave acquisitions were proposed as alternative to line-by-line sequences [8; 9]. This type of acquisition has been getting more attention since the mid-2000s when the computational power and data transfer rates had sufficiently increased to enable the processing of the data generated by plane-wave transmissions. In plane-wave imaging, the full transducer array is used to transmit one unfocussed plane-wave and receive the reflected signals (Figure 1). In case of a 192-element array transducer, frame-rates can be increased by a factor of 192 compared to line-by-line scans. Consequently, ultrasound data can be collected with frame-rates over 10 kHz (theoretically limited by the two-way propagation of the ultrasound signal) and so fast deformations can also be quantified and visualized using quasi-static elastography. In the absence of focusing in transmit, plane-wave imaging requires software beam-forming to reconstruct RF-data from the obtained element data. In conventional ultrasound scanners, beam-forming is often applied in hardware. In this study, three software beam-forming strategies and their effect on displacement estimation were exploited: delay-and-sum (DaS) [10] in the spatial-temporal domain originating from conventional focused ultrasound, and Lu's-fk [11; 12] and Stolt's-fk [13] beam-forming methods, both operating in the Fourier domain which are specifically developed for plane-wave acquisitions. DaS is a beam-forming method based on delaying the element signal by the expected time-of-flight (ToF) and summation of the element data points (Figure 2). For a certain point to be reconstructed, the ToF is calculated as the time of the plane-wave propagating to that point and the reflected or back-scattered signal to be received by the elements. The ToF is used to identify and sum the signals originating from each point in the element data.