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and provides the fractional change in length in the given direction. The subscript p indicates that the direction is specified in the reference frame relative to p. Unit elongation measurements — sometimes called normal or axial strains — are often calculated for directions oriented radially outward from the center of the heart (radial strain), circumferentially within the myocardium (circumferential strain), or parallel to the long axis of the heart (longitudinal strain). Other cardiac strain measurements that have been made include shear strain, eigenvalues and eigenvectors of strain tensors, and the temporal rate of change in strain.

Finally, strain may be calculated relative to a Lagrangian or Eulerian description of motion. For Lagrangian strain, parameters are specified in the reference frame, relative to p. In (2), for example, ep is specified in the reference frame, making e(cp) a Lagrangian strain. Alternatively, Eulerian measurements are made in the current frame relative to the spatial position x. For example, the inverse of the deformation gradient may be calculated at x using

F—1 = Vxp(x, t). Then, unit elongation at that spatial position is given by sequences — to impose temporary spatial variation in the longitudinal magnetization of protons inside the body. In subsequent images, the varying magnetization manifests itself as an alternating light and dark tag pattern, such as the parallel lines apparent in Fig. 2a. The advantage of tagging is that the pattern is induced within the tissue itself. Therefore, any motion that occurs between tag application and imaging is clearly visible as a distortion of the applied pattern, as depicted in Fig. 2b. Figure 2b also shows fading of the tag pattern relative to Fig. 2a, reflecting the temporary nature of tags.

We designate the applied tag pattern as f0(p), a function that modulates the steady-state tissue magnetization of each material point p and must satisfy |f,(p)| < 1. The design of tagging sequences for creative tag patterns has received much attention in the literature (e.g. [1,14,15]), but the most commonly used sequence produces the relatively simple spatial modulation of magnetization (SPAMM) [2,16]. SPAMM tag patterns are imposed by applying a sequence of N hard RF pulses, generally with different tip angles. Between the RF pulses, a gradient pulse with integrated magnitude and direction 1 g is applied, where y is the gyromagnetic ratio. The whole sequence is then followed by a crusher to remove the effect of transverse magnetization. For a SPAMM sequence applied at t = 0, the tag pattern is (see [17])

Because ex is specified in the current frame as indicated by the subscript x, (3) is an Eulerian strain. Eulerian and Lagrangian strains are equivalent measures, as evidenced by the fact that (2) and (3) produce the same result for a particular material point if ex and Fep point in the same direction. Either may be calculated depending on which frame, the reference or the current, facilitates the calculation.

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