Figure 1.

Deformation in three dimensions. (A) Deformation in three dimensions in Cartesian coordinate system. A cube is deformed simultaneously in all three directions, in this case expanding (positive strain) along the x axis, and shrinking (negative strain) along the y and z axes. If the cube is incompressible, the three strain components are interrelated, so (1+ ${\epsilon }_x)$) × (1 + ${\epsilon }_y)$) × (1 + ${\epsilon }_z)$) = 1. (B) Deformation in three dimensions of a hollow ellipsoid. In this case a coordinate system of longitudinal, transmural and circumferential strains is more convenient. The ellipsoid shortens in the longitudinal and circumferential directions (negative strain), and expands in the transmural direction. If the object is incompressible, the three strain coordinates are interrelated in the same way: (ε_L + 1) × (ε_C + 1) × (ε_T + 1) = 1. Thus it is evident that the three strain components are coordinates of the complete three-dimensional deformation of a single object. (C) Myocardial strains explained by the ellipsoid model. There is systolic shortening of the ventricular length (longitudinal strain, ε_L) and external circumference (external circumferential shortening). The total volume reduction is shown in yellow, and the changes in external contours in black. As the wall shortens, it must thicken in order to conserve the volume, depending on the degree of myocardial compressibility. The thickening is thus mainly a function of the longitudinal shortening. As the external contour decreases, the thickening has to occur inwards. External circumferential shortening will also, to a certain degree, push the wall inwards into more limited space, thus causing some thickening. The thickening of the external layer (red) will also displace the inner layer (blue) into a region with less space, so there is more thickening of the inner layer, due to both shortening and inward displacement. Thus, there is a gradient of wall thickening (transmural strain, ε_T) from the external to the inner layers. The thickening of the two layers and total thickening (black) is shown by the length of the arrows. The external circumferential shortening is a real contraction, but is only a partial contributor to the shortening of the inner circumference. As the outer layer thickens, the midwall circumference is pushed inwards, and thus shortens more due to the wall thickening, and as there is more thickening of the inner layer the endocardial circumference shortens even more, and thus there is also a gradient of circumferential strain (ε_C). The inward movement of outer (black), midwall (red) and endocardial (blue) circumferences are indicated by the unbroken straight lines. (D) Relation to M-mode measurements. Longitudinal shortening can be measured by mitral annular plane systolic excursion (MAPSE) and longitudinal strain derived by dividing by the wall length. Transmural strain is simply relative wall thickening, which is available from transverse M-mode, while circumferential shortening equals diameter shortening, that is, shortening—endocardial, midwall and external as explained in the text.