Fig. 2

Transit time estimation from 4D flow CMR normalized time-resolved velocity curves. a Transit time (TT) is defined as the time shift needed to maximise the overlap between two normalized aortic velocity curves x(t) and y(t) extracted from two distinct planes positioned along the aortic centreline as illustrated in Fig. 1. Three TT estimation methods are used. b Cross-correlation transit-time (TTc): a time shift (t_i), comprised between 0 and the systolic duration is applied to x(t) while maximizing its overlap with y(t) in terms of systolic upslope. Such overlap was iteratively defined by the cross-correlation Cor(t_i) between the shifted curve x(t + t_i) and y(t) systolic upslopes. TTc was set to the time shift maximizing the cross-correlation function Cor(t_i). c Wavelet transform transit-time (TTw): 4th order Gaussian wavelet transform was applied on the normalized mean velocity curves x(t) and y(t) resulting in the provided modulus and phase of the cross-spectrum. TTw is then defined as the sum of such phase weighted by its modulus (see equation), while considering only the systolic upslope. d Fourier transform transit-time (TTf): time shift is modelled by the group delay introduced by a filter, which considers x(t) as an input and y(t) as an output. Normalized mean velocity curves x(t) and y(t) are Fourier transformed into (X(f) and Y(f)) defining the filter transfer function H(f) = Y(f)/X(f). TTf is calculated as the sum of the filter group delay (GD) weighted by the harmonics of the input signal (see equation)