Echo-planar imaging (EPI) suffers from geometric distortion due to magnetic field inhomogeneity. Among the distortion correction methods, the most commonly used method is the field map based approach, which provides the one-to-one correspondence relationship between points in the original space and the distortion image.¹ However, the relationship between the original and the distorted space is not one-to-one but many-to-one.² To correct the distortion, the information on the contribution to signal from adjacent points within each voxel is needed. The point spread function (PSF) approach can provide the information of the signal contribution of a voxel for the neighboring voxels because this approach can obtain the PSF imagefor each voxel.^3,4 However, this approach needs long scan time.⁵ In this study, our purpose was to develop a novel image-based method for estimating the PSF images in the distorted EPI image using T1 weighted image (T1WI).

Our basic idea for estimating the PSFs in EPI is to reproduce the distorted EPI image based on MR imaging physics. Our method synthesized the distorted image to match the measured EPI image through the generation process of EPI image according to a single-shot EPI k-space trajectory and field inhomogeneity. The field inhomogeneity was estimated by the image-based method⁶ using segmented T1WI. To synthesize the distorted EPI image, an MR signal was simulated on a voxel-by-voxel basis based on the tissue in segmented T1WI. The MR signal in the presence of field inhomogeneity ΔB is given by $$S(k_{m},k_{n},\Delta B)=A\sum_{i=0}^{M\times N-1}K_i (k_{m},k_{n})\cdot D_i (k_{m},k_{n},\Delta B_i) \hspace{95pt} [1],$$where K_i and D_i are given by $$K_i (k_{m},k_{n}) = \rho_i \cdot \exp \left\{-\frac{TE+m \Delta t_x + n \Delta t_y}{ T_{2i}} \right\} \cdot \exp \left\{ -j2 \pi (k_m x_i + k_n y_i)\right \}\hspace{3pt} [2]$$ $$D_i (k_{m},k_{n}, \Delta B_i) = \exp \left\{-j 2 \pi \left( \frac{k_m}{ G_x} + \frac{k_n}{G_y} \right) \Delta B_i\right\}\hspace{110pt} [3].$$K_i represents the contrast and position information of the tissue in voxel i, and corresponds to the k-space data from a single voxel without field inhomogeneity. D_i represents the field inhomogeneity in voxel i causing the distortion, and corresponds to the Fourier Transform of the PSF.³ The synthesized EPI image was reconstructed by taking the 2-D inverse Fourier Transform of S. The field inhomogeneity was estimated by minimizing the least-square cost function using the synthesized EPI image and the measured EPI image. By taking the inverse Fourier Transform of D_i calculated from the estimated field inhomogeneity map, we obtained the PSF image for each voxel. The spin echo (SE) EPI and T1WI data of a healthy volunteer were acquired using a 1.5-tesla clinical scanner (Magnetom, Symphony, Siemens) with an 8-channel phased-array coil. The SE EPI data was obtained by a single-shot EPI pulse sequence (FOV: 230 mm, TR=8600ms, TE=119ms, 128×128 in-plane resolution, 3 mm thickness). Three dimensional T1WI covering the same area in EPI was obtained by MPRAGE sequence (FOV: 230 mm, TR=2090ms, TE=3.93ms, TI=1100ms, FA=15°, 256×256 in-plane resolution, 1 mm thickness).

Figure 1(a, b and c) shows the measured SE EPI image from a healthy volunteer, the synthesized EPI image by our proposed method, and the absolute difference image between them, respectively. The estimated field map by the image-based method⁶ is shown in Fig. 1 (d). The image shown in Fig.2a was reconstructed by the sum of the signals from i=0 to the middle of the total voxels in Eq. [1].　It can be seen that a signal from a voxel affects the neighboring voxels. Figure 2b shows the examples of the PSFs in the voxels of the synthesized EPI image shown in Fig.2a.

The synthesized EPI image by our proposed method (Fig. 1b) was very similar to the measured EPI image (Fig. 1a). Figure 1 (c) demonstrates that our method can produce a reasonable synthesized EPI image using the estimated field inhomogeneity map (Fig. 1d). As shown in Fig. 2, our method can obtain the PSF image for each voxel. Each PSF image provides the information regarding the intensity distribution and the displacement of the voxel center due to the field inhomogeneity. Based on the PSF images obtained by our method, it could be possible to correct the geometrical distortion in EPI image using an appropriate deconvolution procedure.

We have developed an image-based estimation method for the PSF images in the distorted EPI image using the segmented T1WI and the field inhomogeneity map estimated by the image-based method. Our results demonstrate that the PSF image for each voxel in distorted EPI image can be estimated by our method using segmented T1WI instead of additional acquisitions for PSF measurement.