## 3 The Fully Automated Cardiac Free-Running Framework
### Reconstruction  
All imaging data are processed using a previously reported fully automated motion extraction and reconstruction framework in MATLAB (Mathworks, Natick, MA, USA). First, the SI spokes of each phyllotaxis interleave are collected to obtain an SG signal (Figure 4A). By using principal component analysis (PCA) of this SG signal, the main respiratory and cardiac frequencies are derived, while respiratory and cardiac signature curves are extracted (Figure 4B). Using this cardiac motion signal, the imaging data are retrospectively sorted into a variable number of cardiac phases depending on the subject's heart rate (standard cardiac bin width of 50 ms). Simultaneously, the same imaging data are also subdivided over four equally populated respiratory motion states using the extracted respiratory motion signal. As a result, all acquired radial spokes are assigned to their respective cardiac phase and respiratory motion state, independently from their position in the phyllotaxis interleave, to maximize the reconstructed temporal resolution (Figure 4C). Following the SG signal extraction and data sorting, a CS algorithm that exploits sparsity along both the cardiac and respiratory dimensions is used to reconstruct 5D (x-y-z-cardiac-respiratory) motion-resolved images [11; 24]. An alternating direction method of multipliers algorithm with a total of 10 iterations is used to solve the CS optimization problem. The regularization weights along both the cardiac and respiratory dimensions can be chosen manually; however, they are often conservatively set to 0.001 to avoid motion compression artefacts while minimizing residual aliasing for optimal image quality. The entire reconstruction of the acquired free-running data, as mentioned above, is performed using a single mouse click without further user interaction. Figure 3: A schematic overview of enabling self-gating (SG) in 3D radial MRI by inserting a radial spoke in the superior-inferior direction at the start of each interleaf. Due to the continuous acquisition, an SG signal is obtained periodically as defined by the number of spokes per interleave (i.e., segments) and the repetition time (TR). With a TR of 3.5 ms and 20 segments per interleave (as shown), an SG signal is acquired every 70 ms and can be used to extract cardiac and respiratory motion information for total SG. Figure 4: Panel (**A**) By extracting all superior-inferior (SI) spokes from each consecutive interleave, a self-gating (SG) signal is obtained. Panel (**B**) Using principal component analysis and filtering of this SG signal, the main respiratory and cardiac frequencies can be derived, and the respiratory and cardiac signature curves can be extracted. Panel (**C**) Using the respiratory and cardiac motion signals, all acquired radial spokes can be retrospectively binned into a variable number of cardiac phases and respiratory motion states. Using a compressed-sensing reconstruction, a 3D volume is obtained for each bin, ultimately leading to a 5D cardiac and respiratory motion-resolved dataset. After reconstruction, a 5D image dataset is obtained that can be exploited in various ways. First, one can select a specific respiratory motion state (e.g., end-expiration) and loop through the cardiac phases to obtain a 3D cine image in the cardiac dimension (x-y-z-cardiac, Figure 5A). Secondly, one can select a specific cardiac phase (e.g., end-diastole) and loop through the respiratory motion states to obtain 3D cine images in the respiratory dimension (x-y-z-respiratory, Figure 5B). Finally, one can select both a specific cardiac phase and respiratory motion state to obtain a static 3D volume (x-y-z, Figure 5C). Supplemental Video S1 shows an animated overview of the abovementioned three viewing methods. For both the 3D static and 3D cine images, the isotropic spatial resolution allows for excellent MPR opportunities.