To describe a tagged MR image, we designate each position within the image by its 2D image coordinate vector y = [y1 y2]T and the image brightness at each position by I(y, t). The image is obliquely oriented in 3D space and each image coordinate y can be related to its 3D position x by the function x(y) = y1 h1 + y2h2 + x0 where h1 and h2 are two 3D, orthogonal, unit vectors describing the image orientation and

x0 is the image origin. Defining H =[h1, h2], we may alternatively write x0) = Hy + xo-

The material point imaged at y is then given by p(x(y), t), which we write concisely as p(y, t). In this, we implicitly assume the images are obtained from an infinitesimally thin image plane. In reality, images are integrated across a thin slab of tissue, but the planar assumption is generally a reasonable approximation.

Basic tagged MR image acquisition is accomplished by applying a tagging sequence at end-diastole (t = 0), followed by an imaging sequence at some later time t. End-diastole is signaled by the QRS complex of the ECG, a time when the left ventricle is full of blood and the heart is relatively slow-moving. The resulting image is given by

where I0(y, t) is the image that would have been obtained without tagging and f (y, t) is the faded and deformed tag pattern. Within the myocardium, f (y, t) is given to good approximation by f (y, t) = mfo(p(y, t)) + (i - m), (6)

where fi(t) represents tag fading and is a function, mono-tonically decreasing from 1 to 0. The function fi(t) is primarily determined by longitudinal relaxation of the magnetization and the choice of imaging sequence.

The stability of the tag pattern within the myocardium leads to a 2D apparent motion of the tag pattern within the image plane, despite the fact that the true motion is 3D. The concept of apparent motion proves very useful for describing tagged image processing. To define apparent motion mathematically, we first define a 2D reference position given by q(p)=HT(p - x0), (7)

which is the projection of any material point from its reference position onto the image plane. This allows us to define a 2D correspondence between any image position y and a 2D reference position q by the mapping q(p(y, t)), which we will abbreviate q(y, t). Within the myocardium, the 2D reference map is invertible so that the forward map y(q, t) also exists. The result is a complete set of 2D motion equations — the apparent motion.

The map y(q, t) qualifies as an apparent motion of the tag pattern because tag patterns are usually applied such that all material pointsp mapping to the same q(p) also share the same initial tag pattern value f0(p). Therefore, for a fixed q, the forward map y(q, t) traces the temporal path of a particular point in the tag pattern. The required condition for this to hold is that all gradient pulses in the tagging sequence are oriented parallel to the image plane, i.e., g is a linear combination of h1 and h2. In the remainder of the chapter, we assume this to be the case except where indicated.

To obtain motion measurements throughout the contraction phase of the LV, tagged images are acquired at regular time intervals from tag application at end-diastole to full contraction at end-systole. Compiled together, each image sequence makes a movie depicting LV contraction and apparent motion within the particular image plane. Such movies are obtained for many spatial positions and image plane orientations, such as those in Fig. 3, to depict motion throughout the LV. Three (or more) tag pattern orientations, determined by g, may be used in combinations of 1 or 2 per image plane. In total, hundreds of images may be obtained to assess the motion of a single heart.

Acquisition of this number of images requires many considerations for such factors as acquisition speed and image contrast (see [8,18] for details). In particular, present technology requires multiple heartbeats in multiple breathhold periods to obtain all of the images. During acquisition, LV motion is assumed to be perfectly repeating, at least from end-diastole to end-systole. Under this assumption, the imaging equations governing MR tagging can assume that the entire imaging process takes place in one heartbeat.

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