Synopsis

The novel contrast mechanism, MREPT, is still difficult to use in the clinical field. This is because the reconstruction of MREPT is performed under many assumptions, which lead to limitations for solving the governing equations, and is highly noise sensitive characteristics to noise in practical use. In this work, we propose a new discrete Laplacian estimation method using Median kernel sets, which has fast and noise-robust features, and show its performance by using in-vivo experiments and EM simulation. For validation of the proposed method, three healthy volunteers were scanned on a clinical MRI system with identical imaging parameters.

Introduction

Methods

1. Generalized Median Sets for Laplacian Estimation

The one-dimensional Laplacian kernel, L, can be represented as a difference of the Mean and identity kernels:

$$ L=\left[1 -2 1\right]=3\times(\frac{1}{3}\times\left[1 1 1\right]-\left[0 1 0\right]) $$

In this equation, for noisy images, the mean can be approximated from the Median value. Then the Laplacian can be estimated from the difference of two Median kernels with higher and lower center weights. It can be simply generalized as:

$$ \frac{1}{n_{L}}\times M_{L}-\frac{1}{n_{H}}\times M_{H}=\frac{W_{H}-W_{L}}{n_{H}n_{L}}L $$

where the center weighted Median kernel, $$$M_{H}$$$ and $$$M_{L}$$$ , whose components are $$$ [1 W_{H} 1]$$$ and $$$[1 W_{L} 1]$$$ with higher and lower weighting values, $$$W_{H}$$$ , and $$$W_{L}$$$, and normalizing constant, $$$n_{H}$$$ and $$$n_{L}$$$ is sum of all components in $$$M_{H}$$$ and $$$M_{L}$$$.

2. Projection onto ROtating Median Sets (PROMS)

Since the Laplacian is very sensitive to noise, Laplacian kernel is defined by combining two sets of Median kernels of 4 directions whose difference has the form of the central finite difference approximation of second-order derivatives.Then the Laplacian stack for iterative process with 4-directions, $$$L_{i}$$$, is given by:
$$L_{i}=\frac{1}{c}(f_{i}*M_{i,L}-f_{i}*M_{i,H})$$
where the iteration step number, i=1,2,3,…,4$$$\times$$$number of cycles, scaling constant, $$$c=\frac{W_{H}-W_{L}}{n_{H}n_{L}}$$$, $$$f_{i}$$$ and $$$M_{i}$$$ represent the input and Median kernel at i-th iteration step, respectively. Finally, the estimated Laplacian can be obtained from the average of Laplacian stack.

To prevent intensity loss near the boundary during iterative Median projections, the input is updated by using the MAX operator. The input for the next iterative step can be expressed as:
$$f_{i+1}=MAX[f_{i},\hat{f_{H}}],$$
where the input is convolved with a high weighting Median kernel, $$$\hat{f_{H}}=f_{i}*M_{i,H}$$$.

From here, each values of pixels in the input is bounded in the previous iterative step and it will protect intensity loss near the boundary in iterative Median projections. It means that increase of conductivity with negative values, which is known as boundary artifacts, are blocked during the update process.The proposed processing scheme is explained in Fig. 1.

3. EPT Reconstruction

The phase-based electrical conductivity reconstruction can be simplified as ⁷:
$$\sigma\approx\frac{\triangledown^{2}\phi_{0}}{2\omega\mu_{0}}$$
where $$$\phi_{0}$$$, $$$\omega$$$, and $$$\mu_{0}$$$ denote transceiver phase, Larmor frequency in Rad/sec, and the magnetic permeability, respectively.

4. Data Acquisition

For validation of the proposed method, FDTD simulations were conducted using Sim4Life (ZMT AG, Zurich, CH)⁸.

In-vivo experiments were performed on 3.0T MRI system (Philips Achieva, The Netherlands). For reproducibility test, three healthy volunteers were scanned using multiple gradient-echo sequence and the transceiver phase was estimated using the linear fitting method.⁹

Detailed imaging parameters are shown in Fig.2.

Results

Discussion and Conclusions

Acknowledgements

References