Physics-based reconstruction methods for magnetic resonance imaging

Abstract

Conventional magnetic resonance imaging (MRI) is hampered by long scan times and only qualitative image contrasts that prohibit a direct comparison between different systems. To address these limitations, model-based reconstructions explicitly model the physical laws that govern the MRI signal generation. By formulating image reconstruction as an inverse problem, quantitative maps of the underlying physical parameters can then be extracted directly from efficiently acquired k-space signals without intermediate image reconstruction-addressing both shortcomings of conventional MRI at the same time. This review will discuss basic concepts of model-based reconstructions and report on our experience in developing several model-based methods over the last decade using selected examples that are provided complete with data and code. This article is part of the theme issue 'Synergistic tomographic image reconstruction: part 1'.

Keywords: inverse problems; magnetic resonance imaging; model-based reconstruction.

Figure 1.

The forward operator F can be formally factorized into operator $\mathcal{M}$ that describes the spin physics, the multiplication with the coil sensitivities $\mathcal{C}$, the (non-uniform) Fourier transform $\mathcal{F}$, and a sampling operator $\mathcal{P}$. (Online version in colour.)

Figure 2.

(a) (Leftmost) Model-based reconstructed T₁ map and (left middle) the ROI-analysed quantitative T₁ values for the numerical phantom using the single-shot IR radial FLASH sequence. (Right middle and rightmost) Similar results for T₂ mapping using the multi-echo spin-echo sequence. (b) (Top) The reconstructed parameter maps (M_ss, M₀, R*₁)^T for the T₁ model and (bottom) (M₀, R₂)^T for the T₂ model with the corresponding T₁ / T₂ maps in the rightmost column. (Online version in colour.)

Figure 3.

Real-time liver images acquired during free-breathing using a radial multi-echo (ME) FLASH sequence. Model-based reconstruction directly and jointly estimates separated water and fat images, as well as $R_2^{\star }$ and B₀ field maps. (Online version in colour.)

Figure 4.

Comparison between (top) the model-based reconstruction and (bottom) the conventional phase-difference reconstruction. A section crossing the ascending and descending aorta was selected as the imaging slice. Displayed images are (left) anatomical magnitude image and (right) phase-contrast velocity map at systole. With direct phase-difference regularization, the model-based reconstruction largely reduces random background phase noise in the velocity map. (Online version in colour.)

Figure 5.

Demonstration of subspace-based methods for (a) single-shot inversion-recovery and (b) multi-gradient-echo signal, respectively. (Left) Simulated (top) T₁ relaxation and (bottom) $T_2^{\ast }$ relaxation and off-resonance phase modulation curves. (Centre) Plot of the first 30 principle components. (Right) The temporal subspace curves that can be linearly combined to form (top) T₁ relaxations and (bottom) multi-gradient-echo relaxations. (Online version in colour.)

Figure 6.

Comparison of linear and nonlinear model-based reconstructions on the simulated phantom. (a) Linear subspace reconstructed T₁ maps using 2, 3, 4, 5 complex coefficients and their relative difference to the reference. (b) Linear subspace reconstructed T₁ maps using four complex coefficients with changing regularization parameters. (c) Model-based reconstructed T₁ maps using different regularization strengths. The numerical phantom used here is simulated using 208 frames, one spoke per frame, and TR = 20.5 ms. All reconstructions are done with L2-regularization. The regularization strength and the normalized relative errors to the reference are shown on the top-left and bottom-left of each figure, respectively. (Online version in colour.)

Figure 7.

(a) Reconstructed four complex coefficient maps (only magnitude is shown) using the linear subspace method for a human brain study. (b) Synthesized images (at inversion time 40 ms, 400 ms, 800 ms, 4000 ms) using (top) the above four complex coefficient maps of the linear subspace method and (bottom) the three physical maps of the nonlinear model-based reconstruction, respectively. The corresponding T₁ maps are presented in the rightmost column. (Online version in colour.)

Figure 8.

Reconstructed subspace coefficients maps (top) along with its root-sum-squares composite image for a individual slice within the acquired 3D volume. Synthesized bSSFP images are computed from these coefficient maps for different virtual frequency offsets (bottom). (Online version in colour.)