## 2 Materials and Methods
### Semi-Automated Hemodynamic Force Analysis  
The method was adapted from the work of [PERSON] et al. for HDF calculation. This study used the semi-automated 'Segment' software v2.2 R6410 (Lund, Sweden, Medviso) for image processing. This software is freely available, and all the post-processing steps were described and validated in detail by [PERSON] et al. The 'Segment' software is designed to analyze 4D-flow data, provided that the cine and 4D-flow MRI data can be loaded into it [21]. The overall force extraction method is illustrated in Figure 1. Intraventricular pressure gradients (g) were computed from 4D-flow velocity data using Navier-Stokes equations (Equation (1)) [35], and HDF was computed by integrating 'g' over the entire LV cavity (Figure 1). \[\textit{Intraventricular pressure gradients, (g)}=-\boldsymbol{\rho}\frac{\boldsymbol{\partial v}}{ \boldsymbol{\partial t}}-\boldsymbol{\rho}(\boldsymbol{v}.\boldsymbol{\
abla v })+\boldsymbol{\mu}\boldsymbol{\
abla}^{2}\boldsymbol{v} \tag{1}\]  
where \(v\) is the velocity (m/s), \(\boldsymbol{\rho}\) the density (1060 kg/m\({}^{3}\)) and \(\mu\) is blood viscosity (0.004 Ns/m\({}^{2}\)) [36]. HDFs were calculated using pressure gradients along three axes (Figure 1, pressure gradients), following a spatial reference system originating at the atri-overticular (AV) plane (Figure 1, defined force direction). The heart's orientation was defined in three perpendicular directions. Firstly, the apical-basal (B-A) direction was set perpendicular to the AV plane. Secondly, the lateral wall-septum (S-L) direction was set perpendicular to the apical-basal (B-A) direction and aligned to the LV outflow tract (LVOT). Lastly, the inferior-anterior (I-A) direction was set perpendicular to the apical-basal and lateral-septal directions. As mentioned by [PERSON] et al., the automated segmentation was only available for the LV; however, the directions were placed manually following the reference mentioned above. A standardized time axis has been created to effectively compare the heart displacement function among subjects with different heart rates. This was achieved by linearly resampling the force curves for systole and diastole separately, as previously described [21; 22; 37]. The mean amplitude of the longitudinal force along the entire cardiac cycle is usually expressed as the root mean square (RMS). The RMS HDF was computed for each direction (Equation (2)), where \(\boldsymbol{f_{n}}\) is the force in time frame \(n\), and \(N\) is the number of timeframes in a cardiac phase. The RMS HDF in each direction was computed for systole and diastole separately. \[\textit{RMS}=\sqrt{\frac{1}{N}\sum_{n=1}^{N}|\textit{f}n|^{2}} \tag{2}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Details** \\ \hline Scan Technique & Free-breathing retrospective ECG-gated technique \\ Flip angle (FA) & 15\({}^{\circ}\) \\ Velocity encoding range in all directions (VENC) & 1.5–2.0 m/s \\ Spatial resolution & 2.0-3.6 \(\times\) 2.0–3.0 \(\times\) 2.5-3.5 mm\({}^{3}\) \\ Temporal resolution & 25–35 ms \\ Number of phases & 30 \\ Bandwidth (BW) & 455–495 Hz/Pixel \\ Echo time (TE) & 2.01–2.35 ms \\ Pulse repetition time (TR) & 4.53–5.07 ms \\ \hline \hline \end{tabular} Notes: The overall scan time varied between 8 and 12 min, depending on the physiological factors, defined scan parameters, and respiratory gating efficiency. Abbreviations: 4D = four-dimensional; MRI = magnetic resonance imaging; ECG = echocardiography.  
\end{table} 
Table 1: Scan parameters for 4D-flow MRI. Figure 1: Illustrates the methodology of non-invasive hemodynamic force (HDF) extraction using ‘Segment’ software. The pressure gradient (g) was calculated using the Navier-Stokes equation from 4-dimensional flow magnetic resonance imaging (4D-flow MRI) data. The pressure gradient (g) was calculated in the x, y, and z directions. Then, the magnitude of the pressure gradient was integrated over the segmented left ventricular (LV) volume using short-axis views to calculate the left ventricular HDF.