Diffusion MRI using hyperpolarized gases is sensitive to changes in lung microstructure [1]. Theoretical models that estimate airway dimensions from the diffusion signal [2,3] require the acquisition of images with several b-values. Since the acquisition time is limited by the duration of a breath hold (~15 s), there is a compromise between achievable image resolution, number of slices and number of b-values, thus limiting the accuracy and number of parameters of the theoretical models.

Compressed sensing (CS) has been suggested for accelerating acquisition for hyperpolarized gas MRI [4]. This approach used spatial total variation (TV), exploiting sparsity only in the spatial encoding direction. However, diffusion images are more sparse in the b-direction than in the spatial domain.

In this work, we incorporate the knowledge of the diffusion signal behavior into the reconstruction to accelerate the acquisition of MR diffusion data by undersampling in both spatial and b-value dimensions. The proposed method is compared to TV by assessing its effect on the estimated parameters of a stretched exponential model, which has been used to estimate mean alveolar dimensions [5].

Image reconstruction method

Our novel compressed sensing approach incorporates a model of the signal decay into the reconstruction (SIDER) method, as prior information. It combines spatial total-variation with a penalty function that promotes sparsity across the b-direction as follows:

$$$min_{u} \alpha \parallel M_{b}S(b)\parallel _{1} +\beta\parallel \triangledown_{x,y}S(b)\parallel_{1} st.FS(b)=f $$$

where S(b) are the images acquired for different b values, F represents the Fourier transform and f denotes the measured undersampled data. M_b is an operator that encodes the relationship between signals for consecutives b values; when these are close enough, this relationship can be approximated by a mono-exponential function:

$$$ M_{b}S(b_{j})=S(b_{j})-S(b_{j-1})exp(-\overline{D}(b_{j}-b_{j-1})) $$$

where $$$\overline{D}$$$ is an estimated average value of diffusion. The problem define by these equations was solved using the Split Bregman method [6,7,8].

Data sets and undersampling patterns

Fully sampled diffusion datasets of a normal volunteer (control) and a patient with moderate COPD were available from earlier work [5]. Data consisted of five slices (10 mm thick with 10 mm gap between slices), 64x64 resolution and 5 b-values (0, 1.6, 3.2, 4.8 and 6.4 s/cm²). These datasets were retrospectively undersampled to simulate CS methods. Quasi-random undersampling patterns (Fig. 1) were created, in which randomization was performed in the phase encoding and b-directions, exploiting data redundancy in two dimensions. We analyzed the results for acceleration factors of x2, x4, x5, x7, x10 and x15.

Evaluation

To evaluate the results, maps of the distributed diffusion coefficient D and heterogeneity index α [3, 5] were estimated by fitting the reconstructed signal S(b), on a pixel-by-pixel basis, to the stretched exponential model:

$$$ S(b)=S(0)exp(-(bD)^{\alpha}) $$$

Methods were evaluated by comparing the estimated maps of D and α, their histograms and their mean and standard errors with those obtained from the fully sampled data. In addition, we verified that the errors introduced by the undersampling were smaller than the reported differences between control and patient data sets [5].

For acceleration factors up to x7, SIDER provided maps of D and α that were almost identical to the fully sampled data set (Fig. 3). On the contrary, TV led to large errors and artefacts that were more apparent for α. For an acceleration factor x15 SIDER still provided reasonable results but maps presented localized errors.

The SIDER method produced only minor changes (more noticeable above x7) in the shape of the distributions of D and α over the whole lung (Fig. 4) with respect to those obtained from the fully sampled data. However, the changes only resulted in negligible errors in the mean values of D and α for all accelerations (Fig. 5), except for α at x15. Furthermore, these errors were much smaller than the difference in mean D and α between control and patient found in this work (Fig. 5) and the one reported for the larger subject sample studied in [5]. The TV method produced larger errors in the distributions and mean values of the diffusion parameters for accelerations as low as x5, which were more evident for α. Our results suggest that using SIDER accelerations of at least x7 are achievable with negligible effect on the estimates of diffusion parameters. This would allow increasing the amount of data acquired during a breath-hold (e.g. by doubling the number of slices and b values) thus improving the accuracy of estimated lung airway dimensions. SIDER could also be extended to other hyperpolarized gas MR applications (e.g. pO2 mapping) where the signal behavior is also known.