Four-dimensional (4D = 3D space + time) flow magnetic resonance imaging (MRI) is a dynamic phase-contrast (PC) imaging technique, which can provide spatio-temporal resolved quantification of in-vivo blood-flow velocity across the cardiac cycle (Markl et al., 2012). Previous studies have indicated that changes in blood-flow velocity patterns hold significant diagnostic value for a variety of cardiovascular diseases, such as valve insufficiency, aortic stenosis (Garcia et al., 2019), pulmonary arterial hypertension (Cerne et al., 2022), aortic dissection (Pirola et al., 2019), etc. Moreover, close connections with the brain vessel hemodynamics have also been discovered in various cerebrovascular diseases (e.g., intracranial aneurysms (Gottwald et al., 2020) and venous pulsatile tinnitus (Li et al., 2018)). Consequently, 4D-flow MRI has garnered increasing interest for in-vivo hemodynamics analysis in recent years (Zhuang et al., 2021, Marlevi et al., 2021, Wåhlin et al., 2022, Demirkiran et al., 2022, Roberts et al., 2023, Rivera-Rivera et al., 2024).

However, spatio-temporal resolved velocity encoding leads to large data sizes for 4D-flow MRI, which usually requires 20 min or more for a fully sampled acquisition (Bissell et al., 2023). In practice, 4D-flow MRI examination needs to make trade-offs between scan time, resolution and noise. Various accelerated imaging techniques have been developed for fast and robust 4D-flow MRI scan, including compressed-sensing acceleration schemes (Ma et al., 2019, Neuhaus et al., 2019, Pathrose et al., 2021), real-time acquisition strategies (Sun et al., 2017a, Sun et al., 2023), and even respiratory motion-resolved 5D imaging setups (Ma et al., 2020, Ma et al., 2021, Walheim et al., 2019). For these advanced imaging techniques, k-space undersampling is necessary to reduce the scan time. Undersampling in k-space causes aliasing artifacts and affects the accuracy of velocity quantification. Therefore, reconstruction algorithms are needed for accelerated 4D-flow MRI to recover images from undersampled measurements.

Early-stage reconstruction algorithms for flow-encoded MRI were based on parallel imaging and compressed sensing (CS) theory (Kim et al., 2012, Kwak et al., 2012). Later, low-rank modeling was introduced into PC-MRI reconstruction (Knobloch et al., 2013, Santelli et al., 2014, Sun et al., 2016a). Joint low-rank and sparse methods are also proposed to enhance the reconstruction performance (Sun et al., 2017b, Valvano et al., 2017). However, these methods are based on hand-crafted constraints, which usually have unsatisfactory performance under high undersampling rates.

In recent years, deep learning methods have shown great potential for MRI reconstruction (Muckley et al., 2021, Lyu et al., 2024). Currently, the deep unrolling method represents the leading network architecture for MRI reconstruction (Sun et al., 2016b, Aggarwal et al., 2018, Hammernik et al., 2018). Deep unrolling networks for flow-encoded MRI have also been proposed, which outperform CS algorithms under high acceleration factors (Vishnevskiy et al., 2020, Oscanoa et al., 2023). However, supervised-learning-based methods have two issues for 4D-flow MRI reconstruction. First, due to the long acquisition time of 4D-flow MRI, obtaining sufficient training data is expensive and difficult. Second, since morphology and velocity distribution vary in different vascular beds, as shown in Fig. 1, supervised methods usually have problems in their generalization performance.

To address the problems of data scarcity and model generalization ability, unsupervised methods have gained increasing research focus. Ulyanov et al. (2018) discovered that an untrained convolutional neural network (CNN) can serve as a regularizer for image restoration, which is known as the “Deep Image Prior” (DIP). DIP inspires several unsupervised methods for medical image reconstruction (Gong et al., 2018, Baguer et al., 2020, Darestani and Heckel, 2021, Yoo et al., 2021, Zou et al., 2021), which typically adopt a generative model, utilizing a CNN to transform the latent variable to image domain. However, most DIP-based models in MRI are used to reconstruct amplitude images, but rarely consider the accuracy of image phase. Besides, existing research mainly focuses on 2D image reconstruction problems with lightweight generators. For 4D-flow MRI reconstruction, a much larger model is needed to express the 4D spatio-temporal information, which poses challenges to the expression ability of the generator network.

In this work, we propose a novel unsupervised reconstruction algorithm for 4D-flow MRI based on the DIP framework. This algorithm consists of three major components. First, we designed a partially-independent network architecture for 4D-flow MRI generation. This network partly separates the generative process in the spatio-temporal domain, which can significantly reduce the model size and improve the generator efficiency. Second, we incorporate the complex-difference (CD) sparsity (Kwak et al., 2012) into the reconstruction objective to improve the accuracy of image phase recovery. Third, we introduce a joint generative and sparse optimization goal to integrate the deep neural network and compressed-sensing framework, and propose a “pretraining + ADMM finetuning” optimization algorithm to obtain the optimal solution. Experiments were conducted on two in-house acquired 4D-flow MRI datasets: one from the aorta, and the other from the brain vessels, each with 12 subjects, compared with several CS methods and supervised methods. Our contributions can be summarized as follows:

We propose a partially-independent network for the generative modeling of 4D-flow MRI images. This network architecture enables an efficient learning of the parameterized representation of 4D-flow MRI images, with reduced model size and better performance. This model only requires the single undersampled k-space data for reconstruction, addressing the problem of data scarcity.

We propose a joint generative and sparse optimization goal for 4D-flow MRI reconstruction by integrating the generator and CD sparsity constraint. We devise a “pretraining + ADMM finetuning” algorithm to solve this problem, which achieves significant improvements in the reconstruction performance, compared with only using the generative or sparse modeling.

Our method achieves leading performance in both the aorta and brain 4D-flow MRI datasets. Furthermore, our method exhibits superior generalization capability across different vascular beds, which means this algorithm can be readily used in accelerated hemodynamics imaging of various organs without acquiring additional data or readjusting the algorithm parameters.