The signal model is the link between data acquisition and image reconstruction. Simple models are tempting, and in MR the Fourier model has been predominantly used because of its simplicity and efficiency. However, this simple model ignores many things and is often insufficient. This lecture will give an overview of the modern techniques for model-based reconstruction that apply an extended signal model to better describe the data acquisition process.
In 1973 Paul Lauterbur published the first MR images of two small water-filled glass capillaries, which were acquired as a series of 1D projections in different directions obtained through applying static magnetic field gradients to the sample [1] . These data were then mathematically back-projected using an iterative reconstruction to form a 2D image. The image formation by this procedure was called zeugmatography, derived from the Greek word “zeugma” meaning “that which is used for joining”. Although these first images were exciting and triggered the further development of MRI, they were only 20x20 pixels and the image reconstruction was quite a tedious procedure. According to Lauterbur’s Nobel Lecture in 2003 [2], it involved manually transferring the measurements to punch cards that were fed in a computer with so limited memory that the intermediate results had to be punched out on a deck of cards and reentered later on the next step. The final result was then printed by a typewriter as a 20x20 array of numbers and the image was produced by hand drawn contours on that array.
It wasn’t until the introduction of “spin warp” imaging [3], which allowed using the Fast Fourier Transform (FFT) for image reconstruction, that MRI reconstruction became practical. For the next almost two decades, image reconstruction in MRI was almost exclusively based on the FFT and the main effort was to tune the scanner to better fit the Fourier model. The k-space formalism [4,5] and the FFT made MRI reconstruction efficient, but this simple signal model does not tell the whole story. The measured signal is a complex function of the tissue relaxation constants (T1, T2, T2*), the proton density, motion at different scales like diffusion, perfusion, cardiac or respiratory motion. The tissue composition (chemical shift), magnetization transfer, and magnetic susceptibility also play a role, as well as system characteristics like the B0 and B1 fields, gradient performance, and the coil (array) used for data acquisition. Finally, the specific sequence parameters TR, TE, TI, flip angle, etc. and the k-space trajectory will also influence the measurements.
To obtain an artifact free image or a quantitative map of a tissue or system property, this simple data model needs to be extended to include all the physics behind the image formation. This can be done by formulating the image reconstruction as an inverse problem and applying the available math to solve this problem.
A simple extension of the signal model is to include the coil sensitivity maps:
$$y_j (t) = \int c_j(r)\rho(r)e^{-2\pi ik(t)r}dr, $$
where $$$ y_j (t)$$$ denotes the measured NMR signal, $$$\rho(r)$$$ is the transversal magnetization and $$$c_j(r) $$$ are the coil sensitivity maps. This gives a general formulation of the SENSE reconstruction problem [6]. This formulation shows that data measured with a coil array are not simply the Fourier transform of the image, but the coil array provides an additional encoding to the gradient encoding. The sensitivity encoding allows image reconstruction from reduced number of gradient encoding steps, which is the principle behind parallel imaging.
B0 field inhomogeneity is always present to some extent. Even with a perfectly shimmed magnet, the tissue magnetic susceptibility locally alters the magnetic field, which disrupts the relation between resonance frequency and spatial position in the Fourier model. Ignoring this effect leads to geometric distortions in Cartesian scans and blurring for spiral trajectories. Including the resonance offset due to B0 inhomogeneity $$$\omega(r)$$$ leads to a more accurate signal model [7]:
$$ y_j (t) = \int c_j(r)\rho(r)e^{-i\omega(r)t}e^{-2\pi ik(t)r}dr $$
In addition to the static field inhomogeneity, higher order dynamic field perturbations can occur due to eddy currents, concomitant fields, thermal drifts, higher order shimming imperfections, as well as dynamic susceptibility effects caused by patient motion, breathing or cardiac pulsation. These field perturbations can be expanded in terms of spherical harmonics and used to further extend the data model [8, 9]:
$$y_j (t) = \int c_j(r)\rho(r)e^{-i\omega(r)t}e^{-2\pi i\sum_l k_l(t)b_l(r)}dr $$
Relaxation and chemical shift can also be included in the signal model, either to reduce the associated artifacts or estimate the relaxation parameters and the chemical composition of tissue (e.g. water-fat separation) [10, 11]. Including motion in the signal model allows not only the reconstruction of motion free images but also obtaining motion information from the data [12, 13]. This approach of using a generalized extended signal model can be applied to any other property that influences the measured MR signal giving a more complete and more accurate description of the acquisition process.
The image reconstruction problem using an extended signal model is usually more complex and often requires iterative method to obtain the image [14]. However, considering image reconstruction in MRI as a generalized inverse problem gives a lot of flexibility to accurately describe the data acquisition process and potentially include prior knowledge in the reconstruction. In its simplest form, the image reconstruction can be formulated as minimizing the distance between the forward model and the measurements:
$$\hat{x} = argmin\>||f(x)-y||_2^2, $$
where $$$f(x)$$$ is the signal model and $$$x$$$ is the image or parameter map of interest. Additional prior knowledge can be incorporated in the reconstruction in the form of regularization
$$\hat{x} = argmin\>||f(x)-y||_2^2 + \lambda R(x). $$
One particular type of regularization that has gained a lot of attention in the last 10 years is the $$$\ell_1$$$ norm of the image in a certain transform domain, which is known to promote sparse solutions and is used in the context of compressed sensing [15]. Compressed sensing introduces an interesting link between data acquisition and the information content in the image: the sparser the image, the less data we need to acquire. For a successful reconstruction, we also need to know an appropriate signal representation, in which the image is sparse. While generic transforms like wavelets and finite differences lead to sparse representations of most images, data adapted signal representations (dictionaries) can achieve higher sparsity. Using data adapted dictionaries is already an example, where machine learning is used in MR reconstruction.
Recently, neural networks have been also explored for image reconstruction. The application of neural networks ranges from learning the regularization parameters and data adaptive sparsity transform [16] to using the networks for solving the entire inverse problem, including the Fourier transform [17]. Also in the case of neural networks, feeding in accurate information about the acquisition system behavior is essential for a successful reconstruction.
[1] P. Lauterbur, „Image formation by induced local interactions: Examples employing Nuclear Magnetic Resonance,“ Nature, pp. 190-191, 1973.
[2] P. Lauterbur, All science is interdisciplinary – from magnetic moments to molecules to men, 2003.
[3] W. Edelstein, J. Hutchinson, G. Johnson und T. Redpath, „Spin warp NMR imaging and applications to human whole body imaging,“ Phys Med Biol, pp. 751-756, 1980.
[4] S. Ljunggren, „A simple graphical representation of Fourier-based imaging methods,“ Journal of Magnetic Resonance, pp. 338-343, 1983.
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[9] B. Wilm, C. Barmet, S. Gross, L. Kasper, J. Vannesjo, M. Haeberlin, B. Dietrich und D. Brunner, „Single-Shot Spiral Imaging Enabled by an Expanded Encoding Model: Demonstration in Diffusion MRI,“ Magn Reson Med, pp. 83-91, 2017.
[10] D. Hernando, Z.-P. Liang und P. Kellman, „Chemical Shift-Based Water/Fat Separation: A Comparison of Signal Models,“ Magn Reson Med , pp. 811-822, 2010.
[11] J. Tamir, M. Uecker, W. Chen, P. Lai, M. Alley, S. Vasanawala und M. Lustig, „T2 shuffling:Sharp, multicontrast, volumetric fast spin echo imaging,“ Magn Reson Med, pp. 180-195, 2017.
[12] F. Odille, N. Cindea, D. Mandry, C. Pasquier, P.-A. Vuissoz und J. Felblinger, „Generalized MRI reconstruction including elastic physiological motion and coil sensitivity encoding,“ Magn Reson Med, pp. 1401-1411, 2008.
[13] L. Feng, L. Axel, H. Chandarana, K. T. Block, D. K. Sodickson und R. & Otazo, „XD-GRASP: Golden-Angle Radial MRI with Reconstruction of Extra Motion-State Dimensions Using Compressed Sensing,“ Magnetic Resonance in Medicine, p. 775–788, 2016.
[14] J. Fessler, „Model-based image reconstruction for MRI,“ IEEE Signal Process Mag, pp. 81-89, 2010.
[15] M. Lustig und J. P. David Donoho, „Sparse MRI: The application of compressed sensing for rapid MR imaging,“ Magn Reson Med , pp. 1182-1195, 2007.
[16] K. Hammernik, T. Klatzer, E. Kobler, M. P. Recht, D. K. Sodickson, T. Pock und F. Knoll, „Learning a variational network for reconstruction of accelerated MRI data,“ Magn Reson Med , pp. 3055-3071, 2017.
[17] B. Zhu, J. Liu, S. Cauley, B. Rosen und M. Rosen, „Image reconstruction by domain-transform manifold learning,“ Nature, 2018.