The subsampled data in 3D Cartesian k-space can be restored by using combined parallel imaging (PI)^1,2 and compressed sensing (CS)³ image reconstruction methods. In the case of same acceleration factor (AF), different choices of undersampling pattern in k-space also have a large effect on the accuracy of image estimation. Variable density Poisson Disk Sampling (vd-PDS) pattern⁴ can provide a good compromise between local and global sampling distribution in k-space which are beneficial for PI and CS respectively. However, the minimum radius parameter is usually difficult to perfectly match the given AF in 3D Cartesian imaging and the optimization of variable density scheme is also demanded. For these purposes, a parametric variable radius sampling scheme, termed Cartesian Under-Sampling with Target Ordering Method (CUSTOM), was presented in this study and investigated in both retrospective and prospective experiments.Parametric Variable Radius Sampling: Given the current sample position (k_y^s,k_z^s), a parametric weight function w(k_y,k_z) centered at (k_y^s,k_z^s) is constructed to approximate the local support effect within a circular neighborhood defined by radius r(k_y^s,k_z^s). Once (k_y^s,k_z^s) is labeled as sampled position, the weights of all positions are updated. The k-space position with minimum weight among all unsampled positions is determined as the next sample position. Repeat this weight ordering process can sequentially generate a sampling mask matched with AF requirement as shown in figure 1. This sampling process can maintain a consistently effective interpolation condition within local k-space regions for PI reconstruction. To make the global k-space sampling density preferable to CS reconstruction, the radius r(k_y,k_z) is defined as a spatially variant function indicating variable density sampling. In this study, generalized Gaussian function was exploited for definition of w(k_y,k_z) and r(k_y,k_z), where (α_l, β_l)/(α_g, β_g) was a pair of scale and shape parameters belonging to function w(k_y,k_z)/r(k_y,k_z). α_l/α_g can be determined by β_l/β_g given the decay threshold/radius magnification. Therefore, further optimization for β_l and β_g parameters can be implemented to balance the local and global characteristics of sampling distribution. Image Reconstruction: STEP⁵ with optimized Gaussian mixture model regularization⁶ was used in this study as one typical combination of calibrationless PI⁷ and CS image reconstruction method. Normalized Root Mean Square Error (nRMSE) and mean Structural SIMilarity index (mSSIM)⁸ were exploited as two criterions for quantitative image quality measurement. Data Acquisition: To optimize parameters in CUSTOM and compare the impact of different undersampling patterns, a 3D isotropic 0.8mm T1 weighted joint intra- and extracranial dataset⁹ was fully acquired on a Philips Achieva 3.0T TX scanner (Philips Healthcare, Best, Netherland) with dedicated 36-channel neurovascular coil¹⁰. Another 8-channel T1 weighted brain dataset was obtained from author’s webpage¹¹, in order to evaluate the generalization ability of the previously optimized CUSTOM. In addition, a prospectively subsampled large coverage 3D-MERGE scan¹⁰ was performed to demonstrate the feasibility of CUSTOM in practical usage.Figure 2 showed that a combination of β_l = 0.22 and β_g = 0.33 can provide more accurate and stable reconstruction result. This optimized CUSTOM was further compared with uniform PDS¹¹, variable density random sampling³, VDRad¹² methods and figure 3 demonstrated that a better tradeoff between local and global sampling distribution characteristics can improve the quality of image reconstruction. The similar comparison was performed on another brain dataset, but the central area of 30x30 in k-space was fully acquired in order to compare the impact of different undersampling patterns in high-spatial-frequency area. In figure 4, it indicated that the peripheral undersampling pattern of CUSTOM can improve the detail image information restoration although the overall accuracy measurements are similar among different undersampling patterns. Furthermore, it showed that this parametric optimization had certain generalization ability in different applications. Figure 5 illustrated that similar vessel wall delineation can be retrieved from a prospectively AF = 4 subsampled dataset in comparison to the fully sampled reference.In this work, we presented a parametric variable radius sampling scheme called CUSTOM that can be optimized to design improved undersampling pattern for joint PI and CS image reconstruction, providing a good matching between total number of sampling points and the expected AF in 3D Cartesian imaging. Arbitrary parametric function can be used as weight and radius function in CUSTOM. Considering generalized Gaussian function as one specific implementation, experimental results demonstrated that optimized CUSTOM undersampling pattern can improve the accuracy of image estimation particularly the detail information. Also, the feasibility of CUSTOM for prospective undersampling has been validated.