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Computational MRI with Physics-Based Constraints: Application to Multi-Contrast and Quantitative Imaging Jonathan I

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Computational MRI with Physics-Based Constraints: Application to Multi-Contrast and Quantitative Imaging Jonathan I 1 Computational MRI with Physics-based Constraints: Application to Multi-contrast and Quantitative Imaging Jonathan I. Tamir, Frank Ong, Suma Anand, Ekin Karasan, Ke Wang, Michael Lustig Abstract—Compressed sensing takes advantage of blurring, and lower signal to noise ratio (SNR). low-dimensional signal structure to reduce sampling Since data collection in MRI consists of Fourier requirements far below the Nyquist rate. In magnetic sampling, these tradeoffs can be understood from resonance imaging (MRI), this often takes the form of sparsity through wavelet transform, finite differences, a signal processing perspective: scan time and SNR and low rank extensions. Though powerful, these increase with the number of Fourier measurements image priors are phenomenological in nature and do collected, and sampling theory dictates resolution not account for the mechanism behind the image and field of view (FOV). formation. On the other hand, MRI signal dynamics A recent trend for faster scanning is to subsample are governed by physical laws, which can be explicitly modeled and used as priors for reconstruction. These the data space while leveraging computational meth- explicit and implicit signal priors can be synergistically ods for reconstruction. One established technique combined in an inverse problem framework to recover that has become part of routine clinical practice sharp, multi-contrast images from highly accelerated is parallel imaging [1], which takes advantage of scans. Furthermore, the physics-based constraints pro- multiple receive coils placed around the object to vide a recipe for recovering quantitative, bio-physical parameters from the data. This article introduces acquire data in parallel. Another avenue that is physics-based modeling constraints in MRI and shows especially suited to MRI is compressed sensing how they can be used in conjunction with compressed (CS) [2]. In CS MRI, data are acquired below sensing for image reconstruction and quantitative the Nyquist rate using an incoherent measurement imaging. We describe model-based quantitative MRI, scheme such as pseudo-random sampling. A non- as well as its linear subspace approximation. We also discuss approaches to selecting user-controllable scan linear reconstruction then leverages image structure parameters given knowledge of the physical model. We such as sparsity and low rank to recover the image present several MRI applications that take advantage as if they were sampled at the Nyquist rate. of this framework for the purpose of multi-contrast Though extremely powerful, common image pri- imaging and quantitative mapping. ors in CS fundamentally leverage the natural im- Index Terms—Computational MRI, compressed sensing, quantitative imaging age statistics, yet often neglect to leverage the constraints of the imaging physics that create the underlying image in the first place. Natural image I. INTRODUCTION statistics are phenomenological in nature and do not MRI is a flexible and rich imaging modality with account for how each voxel (3D pixel) obtained a broad range of applications including visualizing its particular value. Modeling the signal dynamics soft tissue contrast, capturing motion, and tracking can help elucidate additional structure in the image, functional and behavioral dynamics. However, long as well as remove unwanted artifacts that are not scan times remain a major limitation. A typical related to under-sampling. For example, Figure 1 MRI exam consists of several scans, each several shows a high-resolution “gold-standard” image of a minutes long; in comparison, a full-body computed volunteer’s foot obtained with a spin-echo sequence tomography exam takes a few seconds. To lower over 12 minutes. In comparison, the middle image scan time, concessions are often made in the acqui- simulates a 48-second fast spin-echo (FSE) acquisi- sition process, leading to reduced resolution, image tion [3], one of the most common sequences used clinically. As the name suggests, FSE is consid- J.I. Tamir, S. Anand, E. Karasan, K. Wang, and M. Lustig erably faster, but leads to tissue-dependent image are with the Department of Electrical Engineering and Computer blurring due to signal decay during the acquisition. Sciences, University of California, Berkeley. F. Ong is with the Department of Electrical Engineering, Stanford University. Even when combined with CS, blurring due to FSE Corresponding author: J.I. Tamir, [email protected]. persists, because the sparsity does not address the 2 mechanism behind the blurring. As we will show creates a net magnetization that is initially aligned in this article, by modeling and exploiting structure with the scanner’s main magnetic field, and evolves in the signal dynamics as well as leveraging CS, based on intrinsic biophysical tissue parameters and it is possible to mitigate the blurring and obtain user control inputs consisting of radio-frequency sharp images [4], shown by the right-most image (RF) pulses and magnetic field gradients. RF pulses in Figure 1. act to rotate the magnetization vector away from the main (longitudinal) field direction and toward T Shuffling Gold Standard FSE Simulation 2 the transverse plane. The strength and duration of the RF pulse determine the degree of rotation, or flip angle, experienced. The transverse component of the magnetization is sensed by nearby receive coils through Faraday’s Law of Induction. We will denote the transverse magnetization as Mxy and the longitudinal magnetization as Mz. Fig. 1. Comparison of (left) a gold-standard spin-echo image acquired in 12 minutes, (middle) an image obtained from a simulated 48-second FSE acquisition, and (right) an image from a z z simulated 48-second FSE acquisition reconstructed by accounting for and exploiting the a priori signal dynamics that occur during Bloch Equations acquisition (T Shuffling [4]). 2 x y x y In this article, we provide an overview of com- putational MRI methods that incorporate physics- Fig. 2. The MRI dynamical system, governed by the Bloch based constraints and show how they can be used equations. An aggregate of spins at each voxel position creates a net magnetization, initially pointed in the longitudinal direction, to accelerate acquisitions, eliminate artifacts, and that evolves based on user pulse sequence control inputs and extract quantitative bio-physical information from intrinsic tissue parameters. The acquired image is the transverse the scan. To simplify the exposition, we limit our component of the magnetization. scope to spin density and relaxation effects, as these are the most commonly used contrast mech- Although many tissue parameters influence the anisms in clinical MRI. In general, the concepts signal evolution, here we limit our scope to relax- can be applied to other intrinsic tissue parameters ation parameters, which are the most common con- (e.g. diffusion, spectroscopy, chemical exchange, trast mechanism used in MRI. Relaxation is a fun- and others), so long as an appropriate physical damental component of nuclear magnetic resonance signal model is included. Importantly, many of these and dictates the rate that magnetization returns to approaches have proceeded beyond the “bench-top” the equilibrium state, effectively “resetting” the MRI testing phase, and are actively used in clinical system. Longitudinal magnetization exponentially practice. The growing interest in characterizing recovers to its initial state with time constant T1, the underlying bio-physical tissue properties follows and transverse magnetization exponentially decays the larger trend toward personalized, quantitative to zero with time constant T2. medicine. Compared to conventional imaging, quan- As Figure 3 shows, relaxation can be understood titative MRI can potentially better aid in identifying through an analogy with a toilet1, where the water in abnormal tissue, evaluating patients in longitudinal the tank represents longitudinal magnetization, wa- studies, discovering novel biomarkers, and more. ter in the bowl represents transverse magnetization, and the toilet flush is an RF excitation. When the II. MRI PHYSICS toilet is flushed (excitation), water transfers from A. Spin Dynamics the tank to the bowl, producing a detectable signal. As the toilet bowl drains (T relaxation), the tank The MRI system can be approximated by a dy- 2 refills with water (T recovery). Successive flushes namical system based on the Bloch equations [5], 1 introduced by Felix Bloch and illustrated conceptu- 1Introduced by Al Macovski in the 2009 ISMRM Lauterbur ally in Figure 2. An aggregate of spins in each voxel Lecture. 3 will transfer new water from the tank to the bowl. sampled transverse magnetization and is given by Neglecting other system effects, the magnetization iT ◦ x (r) = M (r; iT ) = ρ(r)f θ(r); u (r)j s : evolution after a 90 RF pulse is given by i xy s iTs τ τ=0 − t (4) Mz(t) = 1 − e T1 ; (1) − t For FSE, the magnetization is primarily sensitive M (t) = e T2 ; (2) xy to relaxation parameters, i.e., θ = (T1;T2), and to u (r) = RF (r) where M (t) and M (t) are the longitudinal and RF refocusing flip angles, i.e. iTs i , z xy th transverse magnetization components at time t after where RFi represents the flip angle of the i RF the excitation, respectively, and have initial condi- pulse. Although the RF pulse is not prescribed for each position, in practice the flip angles smoothly tions of Mz(0) = 0 and Mxy(0) = 1 immediately following the RF pulse. vary spatially due to RF transmit field inhomo- geneity effects that impart varying levels of RF power across the imaging volume. We represent the 90o vector of magnetization points at the echo times by T th RF Flip f (θ(r); u(r)) 2 C , where the i component of Angle f is equal to fiTs (i.e. the transverse magnetization signal). Based on the sequence timing and RF flip angle inputs, different types of image contrasts can be created. Figure 4 shows four common FSE image contrasts primarily due to PD, T1, and T2, and created by using different sequence parameters in independent scans.
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