# Iterative 2D Tissue Motion Tracking in Ultrafast Ultrasound Imaging
## 1 Introduction  
Ultrafast ultrasound imaging has been used as the basis for the development of a number of methods intended for diagnosing and exploring different phenomena in vivo, e.g., shear wave elastography [1; 2; 3; 4], acoustic radiation force impulse imaging [5], vector flow imaging [6; 7; 8], and a method for skeletal muscle contraction [9], functional ultrasound imaging of the brain [10], and cardiac motion [11; 12]. The arterial walls have been investigated by estimating the radial strain in the common carotid artery [13] and the radial pulse wave velocity [14; 15]. In cardiovascular research, the radial movement of the arterial wall, i.e., the diameter change, has been the subject of extensive research, forming the basis for estimation of arterial wall stiffness [16]. Increased stiffness of the large central arteries has been shown to be an independent risk factor for cardiovascular mortality [17]. In contrast to the radial movement, the longitudinal movement of the arterial wall has gained less attention. We have, however, shown that in both large predominantly elastic arteries and in large muscular arteries there is a distinct bi-directional displacement of the arterial wall during the cardiac cycle [18]. The intima-media of these arteries exhibits a longitudinal displacement that is larger than that of the adventitial region [18] and thus, there is shear strain and shear stress within the arterial wall [18; 19; 20; 21]. We have recently reported that longitudinal movement and intramural shear strain undergo profound changes in response to the important circulatory hormones adrenalin and noradrenalin [22], indicating that the longitudinal movements and resulting intramural shear strain can constitute an important but overlooked mechanism in the cardiovascular system. Studies have indicated that the maximal amplitude of the longitudinal displacement of the common carotid artery is reduced in subjects with cardiovascular risk factors [23], and suspected and manifest atherosclerotic disease [24; 25]. However, the physiology behind the observed longitudinal vessel wall movement pattern is largely unknown. It is our belief that the use of ultrafast ultrasound imaging in combination with 2D tissue motion estimation can increase our understanding of this phenomenon and make it possible to estimate the propagation velocity of the longitudinal movement. However, to explore the longitudinal movement of the arterial wall by using ultrasound, the artery is scanned in the longitudinal direction and the longitudinal movement of the arterial wall occurs in the lateral direction of the ultrasound image. It is problematic to estimate lateral tissue motion in ultrafast ultrasound imaging in vivo as the tissue moves only a very short distance between consecutive frames due to the high frame rate. Thus, the motion to be estimated will be very small compared to the expected uncertainty in the motion estimates caused by the limited signal-to-noise ratio. The estimation uncertainty is larger in the lateral direction [26] as ultrasound frames normally have lower spatial resolution in the lateral direction. Consequently, Lagrangian tracking in the lateral direction and in every frame is very likely to give a large accumulated error even with an unbiased motion estimator when using ultrafast ultrasound imaging. The motion estimations can be improved by averaging motion estimations over multiple frames, but this will decrease the effective frame rate and will function as a low-pass filter on the motion estimations in the time domain. This can potentially hide vital information in the motion estimations. In this paper, we propose to estimate 2D motions with a Lagrangian viewpoint in ultrafast ultrasound cine loops using an iterative motion estimation tracking scheme in which the initial length between the used frames is larger than one.