## 1 Introduction  
The recent advances in magnetic resonance imaging (MRI) programs have led to multiple implementation challenges. The computational demands of MRI scales with advancements in imaging techniques, especially with the transition to real-time imaging [1] and model-based reconstructions [2]. Modern MRI systems face challenges including the need for higher spatial and temporal resolution, the reduction in acquisition time, and the integration of advanced techniques such as dynamic imaging [3], compressed sensing [4],and parallel imaging [5]. For instance, the motion-compensated MRI reconstruction problem relies on accurate modeling of rigid and non-rigid body motion, which implies an exact modeling of the deformation and a local tracking of the displacement fields within each voxel [6]. Hence, thousands of variables are required. Thus, it is necessary to assess the data-level parallelism in MRI reconstruction methods and employ high-performance computing (HPC) platforms to reduce the reconstruction time to an acceptable level for clinical applicability. Graphics processing units (GPUs) are suited for iterative, high-throughput operations, critical to MRI workflows, such as Fourier transforms, image interpolation, and gridding [7; 8; 9]. Unlike traditional central processing units (CPUs), GPUs are designed for data-level parallelism, enabling the processing of thousands of threads concurrently. Moreover, GPUs facilitate memory-intensive processes such as multi-frame parallel reconstruction, leveraging their superior memory bandwidth and multi-core architecture [10]. In parallel with fast GPU-driven reconstruction, a new trend is emerging toward multi-sensory pipelines that fuse imaging with real-time safety monitoring. Smart electronic-device platforms now enable continuous streaming of tissue temperature and local specific absorption rate (SAR) data, which can be co-processed on the same GPU hardware to guide adaptive control of power deposition and reconstruction parameters [11]. Integrating such wearable sensor feedback underscores the broader move toward unified, GPU-centric biomedical systems where imaging, physiology, and patient safety are handled in a single low-latency loop. Compute unified device architecture (CUDA) [12] has gained particular attention in MRI GPU-transition for its user-friendly programming model, its accessibility, and its high-level libraries. In this work, we extend our research reported in [13] by introducing the CUDA implementation of the generalized reconstruction by inversion of coupled systems (GRICS) method's reconstruction kernel [14], an iterative reconstruction framework that allows for the obtention of 3D high-resolution images, by alternatively optimizing the reconstructed image and the parameters of the motion model. A key contribution of our study lies in the design of the CUDA implementation of the reconstruction kernel, particularly, the optimal design of sparse matrix-vector/element-wise matrix multiplication to efficiently use warp-level parallelism, and the on-the-fly computation of the motion operator. Rather than repeatedly transferring precomputed elements from the CPU to the GPU, the required non-zero entries of the operator are dynamically generated within the target kernels based on the underlying motion model. Despite a higher per-iteration computational cost, this design significantly reduces data-transfer overhead and memory usage, while ensuring that all problem variables remain in GPU memory throughout the iterative process. Moreover, advanced parallel-reduction strategies using atomic operations further guarantee safe and optimal accumulation of partial results in the shared memory. Furthermore, a performance analysis is used to compare the efficiency of three implementations: the CPU-C++, the Matlab-GPU [13], and the CUDA implementations. We evaluated the eventual discrepancies between the CPU and CUDA images with the normalized root mean square error (NRMSE).