In MRI, volume coverage is often achieved with stacks of 2D acquired slices. For sufficient signal-to-noise ratio, 2D slice thickness is typically several times greater than in-plane resolution, i.e., voxels are non-isotropic. The apparent image intensity for each pixel is the signal integral over the slice profile along the slice selection direction, which is a complex function of MR tissue and acquisition parameters.^1,2 Exact knowledge of the actual slice profile is required in various MR applications.^3,4 For a single stack of parallel slices, conventional blind deconvolution can be utilized to recover the slice profile, where both original signals and slice profile are unknown, leading to a highly ill-posed problem to solve. We propose a convolutional forward model (CFM) to recover the slice profile by leveraging additional orthogonal stacks, which reduces complexity and leads to more accurate slice profile estimates.The proposed CFM method is illustrated in Fig. 1. Suppose an isotropic 3D physical grid $$$V(x,y,z)$$$ is the target to be imaged. A stack $$$S_X$$$ (or $$$S_Z$$$) of thick slices, acquired along direction X (or Z), is modulated by a convolution process $$$h_X$$$ (or $$$h_Z$$$) with the slice profile along X (or Z). If a subsequent convolution process is applied along the counterpart direction, e.g., apply $$$h_Z$$$ on $$$S_X$$$, then due to the commutativity property of the convolution process, we have $$$V*h_X*h_Z=V*h_Z*h_X$$$, i.e., $$S_X*h_Z=S_Z*h_X$$ where * denotes the convolution process. This equation holds at the intersections between $$$S_X$$$ and $$$S_Z$$$. Note that $$$h_X$$$ and $$$h_Z$$$ are identical 1D convolution kernels ($$$h$$$), applied along different directions X and Z. Slice profile estimation is then formulated as searching for $$$h$$$ that maximize the objective function of the similarity between $$$S_X*h_Z$$$ and $$$S_Z*h_X$$$, i.e., $$argmax(h)Φ(S_X*h_Z, S_Z*h_X)$$ where Φ is a similarity metric, such as correlation coefficient, evaluated at the intersections between $$$S_X$$$ and $$$S_Z$$$, where $$$S_X$$$ and $$$S_Z$$$ are the acquired slices (observations) along respective directions. The unknown original signal V is removed from this formulation. In practice, to handle noise and inconsistencies between the two orthogonal stacks, regularization is introduced in the optimization process to enforce smoothness of the estimated slice profile. In addition, similarity evaluation can be constrained to the regions containing rich and consistent texture.Five datasets of a variety of objects and pulse sequences were collected as shown in Fig. 2. Example slices are presented in Fig. 3. The correlation coefficient was used as the similarity metric, due to its robustness in the presence of intensity differences between stacks. An interior-point algorithm was used to solve the optimization formulation. The same algorithmic parameter configuration was applied across all experiments. The slice profile search window was set to four times the nominal slice thickness. The slice profile was initialized with equal weight across the entire search window. The head data was motion corrected using rigid registration, before CFM was applied for slice profile estimation. The resulting slice profiles are provided in Fig. 4.

Our modeling and formulation take into account the slice thickness in both orthogonal stacks, integrating two complementary convolution processes, leading to an objective function that accurately follows the physical MR imaging process. The CFM could be extended to scenarios with different contrasts provided a suitable similarity metric can be found, a promising candidate being, mutual information used in cross-modality registration. While conventional blind deconvolution requires densely oversampled stacks (stacks with large slice overlap), the presented method is able to handle large and varying slice intervals.

In cases where in-plane resolutions of the two stacks are different, e.g., in the knee dataset, the slice profile was resampled to ensure equal voxel geometries across stacks.

The slice profiles estimated for both bSSFP measurements are consistent with Bloch simulations of the bSSFP steady-state signal near resonance for varying ratios of T1/T2 shown in Fig. 5: the slice profile for the liquid filled phantom data corresponds in the simulation to the profile for a small T1/T2 ratio while the profile for the raw meat data corresponds to a profile for a large T1/T2 ratio.

We proposed a fully automated modeling algorithm for actual slice profile estimation from orthogonal stacks. Our modeling approach is data-driven and closely follows the MR forward imaging process. The proposed approach could be extended to localized analysis when spatially varying slice profiles are present. Applications that benefit from accurate actual slice profile estimates are being pursued.