The Forward Map and Strain

Once the 3D displacement field u(x, t) is reconstructed, it must be related to the useful motion functions defined in Section 2. From the definition of u(x, t) (12), we see that the reference map p(x, t) is easily obtained. The reference map is not, however, convenient for tracking the motion of material points over time or computing Lagrangian strains. For those purposes, the forward map x( p, t) is needed. To determine the forward map from u(x, t), we use the fact that displacement of a material point from its deformed position x to its reference position p may be turned around to provide the forward displacement from p to x. After using any of the motion models to reconstruct a set of displacements, the forward map is estimated by fitting the same model to the turned-around displacements [4,7,40].

Then Lagrangian strain is computed by differentiating the reconstructed forward map to obtain F, from which any of the strain measures discussed in Section 2 may be computed. Continuous strain functions may be found or the value may be averaged over an element to provide regionalized strain measurements. For example, Fig. 11 shows the time progression of strain for 72 distinct regions within the LV.

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