While the conventional DFT method offers a convenient approach to reconstruct MRI data, it is considerably less sensitive to the phenomena we observe in perfusion MRI (i.e., blood flow) than those we observe in structural MRI. This deficiency calls for more sensitive reconstruction techniques such as those provided by Bayesian image analysis
6, in particular the K-Bayes method proposed here.
Reconstruction procedures using Bayesian image analysis are already the norm in many areas of imaging (e.g., computer vision, motion tracking) and are starting to be applied increasingly in medical imaging
7,
8 and MRI
9‐
12. The art of Bayesian modeling is to develop prior models that characterize key aspects of available prior information to be combined with the information in the data. In fact, standard reconstruction methods based on likelihood theory (such as DFT) can (loosely) be thought of as a special case of a Bayesian model where there is no informative prior information.
The Bayesian formulation is particularly useful for combining information from multiple sources. There is a recent body of literature on combining information from different medical imaging modalities. Chen et al.
13 provide a general method for fusing images from different modalities using an edge detection prior for identifying structural boundaries. The detection of edges between neighboring pixels turns off the a priori modeled smoothness between them (using Markov random fields, MRFs). This approach has spawned considerable research in applications to human brain imaging, in particular for emission computed tomography
8,
14,
15. Other methods have been developed specifically for improving MRI resolution using Bayesian image analysis. For example, Hurn et al.
10 propose a simultaneous Bayesian reconstruction and tissue classification method that reconstructs multiple MRI modalities to a common high resolution. The tissue classification information is used to impose smoothness on the reconstruction through the use of MRFs. The approach is not specifically designed for the reconstruction of low-resolution
k-space data (such as that from perfusion MRI) and is not a “true” reconstruction approach in the sense that it is only applied to data after DFT. It is therefore subject to artifacts associated with DFT when applied to perfusion MRI. Miller et al.
16 describe a likelihood-based method for reconstructing structural MRI from raw
k-space data. However, the procedure was not designed to improve the reconstruction of low-resolution data and does not incorporate prior information. The method is extended in Schaewe and Miller
17 to incorporate a MRF prior model that encourages smooth image reconstructions. However, this prior is limited by not varying smoothness levels according to whether neighboring voxels are of the same tissue type or not. Furthermore, images are only reconstructed to the same resolution that would be obtained by DFT. Denney and Reeves
18 have developed a Bayesian approach for magnetic resonance spectroscopy imaging reconstruction (that could also be applied to other MRI modalities) that models the data in
k-space and utilizes an edge-preserving prior. They use spectrally integrated estimates of the metabolites at each
k-space point as raw data. However, the edge-preserving prior they adopt does not utilize tissue class information. A number of papers
19‐
22 develop non-Bayesian approaches to improving the resolution of MRI data with tissue classification information. The basic idea is to utilize basis functions that are motivated by tissue classifications in order to smoothly represent signals within homogeneous tissue areas. Because it needs to be kept sparse (either by limiting the size of the basis set or by penalizing selection from a larger basis set), the basis function representation restricts the potential set of reconstructions.
To the best of the authors’ knowledge, the K-Bayes approach presented here represents the first attempt to combine a k-space signal modeling approach with a tissue classification prior within the Bayesian paradigm. The Bayesian approach provides a natural framework within which to balance the perfusion and anatomical information sources by quantifying them with probability distributions and combining them using Bayes’ theorem.