Tagged MR image analysis has focused almost exclusively on the myocardium of the left ventricle (LV), which will be the topic of this chapter as well. Many of the techniques we will discuss are readily extendible to the right ventricle and other moving bodies. The LV is the primary heart chamber responsible for pumping blood throughout the body, with thick, muscular walls — the myocardium — delivering the required pressure. These walls form a hollow, ellipsoidal body, with the aortic and mitral valves located at one end of the ellipsoid. Within the hollow chamber are extensions of the myocardium known as papillary muscles. The LV anatomy is diagrammed in Fig. 1, along with an MR image showing these features.

The collection of all material points p within the LV myocardium comprises a 3D body in motion. Mathematically, that motion is described by the function x(p, t) that relates the reference position p to its deformed position x at time t. Traditionally, the reference position is taken to be the spatial position ofp at end-diastole (t = 0), meaningp = x(p, 0). The temporal progression of points through space x(p, t) is known as the forward map of motion and is the fundamental function required for diagnostic motion analysis. Another important function is the reference map p(x, t), which relates the spatial position x at time t to the corresponding reference position p and is often easier to measure than the forward map.

A related motion measurement, generally considered more valuable for diagnosis, is strain, which may be computed from either motion map. A detailed discussion of strain is beyond the scope of this chapter and the interested reader is directed to continuum mechanics texts. For our purposes, we will concentrate on determining the deformation gradient matrix

from which any strain function can be directly calculated. In this expression, Vp is the gradient with respect to p. In terms of the components of x = [x1 x2 x3]T and p = [p1 p2 p3]T the deformation gradient is

~ |
8x1 |
" |

8ft |
8p2 |
8ft |

8*2 |
8x2 |
8*2 |

8ft |
8p2 |
8ft |

8x3 |
8x3 |
8*3 |

.8ft |
8p2 |
8ft. |

where the dependence of F on p and t is implicit.

Examples of strains calculated from F are the left and right Cauchy-Green strain tensors given by B = FFT and C = FTF, respectively. The most common strain measurement found in

Pulmonary

Arl (try image

Plane

Pulmonary

Arl (try loft vcmritk papillaiy muscle

Bau epicardium ntyotarUwm image cntkx-ari,li um

Plane right ventricle

FIGURE 1 Heart anatomy depicted with a typical image plane superimposed (a) and imaged using standard 2-D MRI (b).

literature on cardiac motion is unit elongation in a direction ep, which is given by

IF ep

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