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Computational MRI with Physics-basedConstraints: Application to Multi-contrast

and Quantitative ImagingJonathan I. Tamir, Frank Ong, Suma Anand, Ekin Karasan, Ke Wang, Michael Lustig

Abstract—Compressed sensing takes advantage oflow-dimensional signal structure to reduce samplingrequirements far below the Nyquist rate. In magneticresonance imaging (MRI), this often takes the form ofsparsity through wavelet transform, finite differences,and low rank extensions. Though powerful, theseimage priors are phenomenological in nature and donot account for the mechanism behind the imageformation. On the other hand, MRI signal dynamicsare governed by physical laws, which can be explicitlymodeled and used as priors for reconstruction. Theseexplicit and implicit signal priors can be synergisticallycombined in an inverse problem framework to recoversharp, multi-contrast images from highly acceleratedscans. Furthermore, the physics-based constraints pro-vide a recipe for recovering quantitative, bio-physicalparameters from the data. This article introducesphysics-based modeling constraints in MRI and showshow they can be used in conjunction with compressedsensing for image reconstruction and quantitativeimaging. We describe model-based quantitative MRI,as well as its linear subspace approximation. We alsodiscuss approaches to selecting user-controllable scanparameters given knowledge of the physical model. Wepresent several MRI applications that take advantageof this framework for the purpose of multi-contrastimaging and quantitative mapping.

Index Terms—Computational MRI, compressedsensing, quantitative imaging

I. INTRODUCTION

MRI is a flexible and rich imaging modality witha broad range of applications including visualizingsoft tissue contrast, capturing motion, and trackingfunctional and behavioral dynamics. However, longscan times remain a major limitation. A typicalMRI exam consists of several scans, each severalminutes long; in comparison, a full-body computedtomography exam takes a few seconds. To lowerscan time, concessions are often made in the acqui-sition process, leading to reduced resolution, image

J.I. Tamir, S. Anand, E. Karasan, K. Wang, and M. Lustigare with the Department of Electrical Engineering and ComputerSciences, University of California, Berkeley. F. Ong is withthe Department of Electrical Engineering, Stanford University.Corresponding author: J.I. Tamir, jtamir@eecs.berkeley.edu.

blurring, and lower signal to noise ratio (SNR).Since data collection in MRI consists of Fouriersampling, these tradeoffs can be understood froma signal processing perspective: scan time and SNRincrease with the number of Fourier measurementscollected, and sampling theory dictates resolutionand field of view (FOV).

A recent trend for faster scanning is to subsamplethe data space while leveraging computational meth-ods for reconstruction. One established techniquethat has become part of routine clinical practiceis parallel imaging [1], which takes advantage ofmultiple receive coils placed around the object toacquire data in parallel. Another avenue that isespecially suited to MRI is compressed sensing(CS) [2]. In CS MRI, data are acquired belowthe Nyquist rate using an incoherent measurementscheme such as pseudo-random sampling. A non-linear reconstruction then leverages image structuresuch as sparsity and low rank to recover the imageas if they were sampled at the Nyquist rate.

Though extremely powerful, common image pri-ors in CS fundamentally leverage the natural im-age statistics, yet often neglect to leverage theconstraints of the imaging physics that create theunderlying image in the first place. Natural imagestatistics are phenomenological in nature and do notaccount for how each voxel (3D pixel) obtainedits particular value. Modeling the signal dynamicscan help elucidate additional structure in the image,as well as remove unwanted artifacts that are notrelated to under-sampling. For example, Figure 1shows a high-resolution “gold-standard” image of avolunteer’s foot obtained with a spin-echo sequenceover 12 minutes. In comparison, the middle imagesimulates a 48-second fast spin-echo (FSE) acquisi-tion [3], one of the most common sequences usedclinically. As the name suggests, FSE is consid-erably faster, but leads to tissue-dependent imageblurring due to signal decay during the acquisition.Even when combined with CS, blurring due to FSEpersists, because the sparsity does not address the

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mechanism behind the blurring. As we will showin this article, by modeling and exploiting structurein the signal dynamics as well as leveraging CS,it is possible to mitigate the blurring and obtainsharp images [4], shown by the right-most imagein Figure 1.

Gold Standard FSE Simulation T2 Shuffling

Fig. 1. Comparison of (left) a gold-standard spin-echo imageacquired in 12 minutes, (middle) an image obtained from asimulated 48-second FSE acquisition, and (right) an image from asimulated 48-second FSE acquisition reconstructed by accountingfor and exploiting the a priori signal dynamics that occur duringacquisition (T2 Shuffling [4]).

In this article, we provide an overview of com-putational MRI methods that incorporate physics-based constraints and show how they can be usedto accelerate acquisitions, eliminate artifacts, andextract quantitative bio-physical information fromthe scan. To simplify the exposition, we limit ourscope to spin density and relaxation effects, asthese are the most commonly used contrast mech-anisms in clinical MRI. In general, the conceptscan be applied to other intrinsic tissue parameters(e.g. diffusion, spectroscopy, chemical exchange,and others), so long as an appropriate physicalsignal model is included. Importantly, many of theseapproaches have proceeded beyond the “bench-top”testing phase, and are actively used in clinicalpractice. The growing interest in characterizingthe underlying bio-physical tissue properties followsthe larger trend toward personalized, quantitativemedicine. Compared to conventional imaging, quan-titative MRI can potentially better aid in identifyingabnormal tissue, evaluating patients in longitudinalstudies, discovering novel biomarkers, and more.

II. MRI PHYSICS

A. Spin Dynamics

The MRI system can be approximated by a dy-namical system based on the Bloch equations [5],introduced by Felix Bloch and illustrated conceptu-ally in Figure 2. An aggregate of spins in each voxel

creates a net magnetization that is initially alignedwith the scanner’s main magnetic field, and evolvesbased on intrinsic biophysical tissue parameters anduser control inputs consisting of radio-frequency(RF) pulses and magnetic field gradients. RF pulsesact to rotate the magnetization vector away fromthe main (longitudinal) field direction and towardthe transverse plane. The strength and duration ofthe RF pulse determine the degree of rotation, orflip angle, experienced. The transverse componentof the magnetization is sensed by nearby receivecoils through Faraday’s Law of Induction. We willdenote the transverse magnetization as Mxy and thelongitudinal magnetization as Mz .

z

x y

z

x y

Bloch Equations

Fig. 2. The MRI dynamical system, governed by the Blochequations. An aggregate of spins at each voxel position creates anet magnetization, initially pointed in the longitudinal direction,that evolves based on user pulse sequence control inputs andintrinsic tissue parameters. The acquired image is the transversecomponent of the magnetization.

Although many tissue parameters influence thesignal evolution, here we limit our scope to relax-ation parameters, which are the most common con-trast mechanism used in MRI. Relaxation is a fun-damental component of nuclear magnetic resonanceand dictates the rate that magnetization returns tothe equilibrium state, effectively “resetting” the MRIsystem. Longitudinal magnetization exponentiallyrecovers to its initial state with time constant T1,and transverse magnetization exponentially decaysto zero with time constant T2.

As Figure 3 shows, relaxation can be understoodthrough an analogy with a toilet1, where the water inthe tank represents longitudinal magnetization, wa-ter in the bowl represents transverse magnetization,and the toilet flush is an RF excitation. When thetoilet is flushed (excitation), water transfers fromthe tank to the bowl, producing a detectable signal.As the toilet bowl drains (T2 relaxation), the tankrefills with water (T1 recovery). Successive flushes

1Introduced by Al Macovski in the 2009 ISMRM LauterburLecture.

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will transfer new water from the tank to the bowl.Neglecting other system effects, the magnetizationevolution after a 90◦ RF pulse is given by

Mz(t) = 1− e−t

T1 , (1)

Mxy(t) = e−t

T2 , (2)

where Mz(t) and Mxy(t) are the longitudinal andtransverse magnetization components at time t afterthe excitation, respectively, and have initial condi-tions of Mz(0) = 0 and Mxy(0) = 1 immediatelyfollowing the RF pulse.

RF Flip Angle

90o

Fig. 3. Signal relaxation visualized through the toilet analogy.The water in the tank represents longitudinal magnetization, waterin the bowl represents transverse magnetization, and the toiletflush is an RF excitation. When the toilet is flushed (excitation),water transfers from the tank to the bowl, producing a detectablesignal. As the toilet bowl drains (T2 relaxation), the tank refillswith water (T1 recovery). Successive flushes will transfer newwater from the tank to the bowl.

In general, the transverse magnetization distribu-tion in space and time can be described by

Mxy(r, t) = ρ(r)ft(θ(r),uτ (r)|tτ=0

), (3)

where r ∈ R3 represents the 3D spatial position,ρ(r) is the intrinsic amount of magnetization atposition r (called proton-density, or PD), and ft(·)is a spatio-temporal signal evolution that dependson both the biophysical tissue parameters givenby θ(r) and the full history of user-controllablepulse sequence and scanner parameters2 given byuτ (r), 0 ≤ τ ≤ t. Although there are numeroustissue parameters and user controls that influencethe magnetization, we will focus on the case of FSEimaging, in which the magnetization is sampled atT equispaced intervals with spacing Ts. In this case,the image of interest at position r and state i is the

2Although the pulse sequence parameters may not have aspatial component, the scanner hardware may introduce a spatialdependence, e.g. due to spatial inhomogeneities.

sampled transverse magnetization and is given by

xi(r) =Mxy(r, iTs) = ρ(r)fiTs

(θ(r),uτ (r)|iTs

τ=0

).

(4)

For FSE, the magnetization is primarily sensitiveto relaxation parameters, i.e., θ = (T1, T2), and toRF refocusing flip angles, i.e. uiTs

(r) = RFi(r),where RFi represents the flip angle of the ith RFpulse. Although the RF pulse is not prescribed foreach position, in practice the flip angles smoothlyvary spatially due to RF transmit field inhomo-geneity effects that impart varying levels of RFpower across the imaging volume. We represent thevector of magnetization points at the echo times byf (θ(r),u(r)) ∈ CT , where the ith component off is equal to fiTs

(i.e. the transverse magnetizationsignal). Based on the sequence timing and RFflip angle inputs, different types of image contrastscan be created. Figure 4 shows four common FSEimage contrasts primarily due to PD, T1, and T2,and created by using different sequence parametersin independent scans.

Proton Density T1 Weighted T2 Weighted T2 FLAIR

Fig. 4. Different image contrasts based on PD, T1, and T2

produced by careful choice of RF flip angles and sequence timingfor FSE-based scans.

In particular, f (θ(r),u(r)) can be modeled bysolutions to the Bloch equations [5], which aredifferential equations that describe the magnetiza-tion evolution as a function of time3. The Blochequations allow us to calculate the magnetization(signal) evolution for individual spins in a spatialregion, given pulse sequence inputs. For example,the relaxation behavior given by (1) and (2) isthe solution to the Bloch equations when no time-varying fields are present. Many extensions havebeen introduced to model additional contrast mech-anisms, including diffusion and chemical exchange,though these are beyond our scope.

B. Simulating Spins

Simulating the magnetization evolutions of spinscan be a powerful tool for developing and evalu-ating physics constrained reconstruction methods.For example, in Section V, we will show howto use simulated signal evolutions to construct an

3In fact the Bloch equations themselves are also phenomeno-logical!

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approximate linear subspace for reconstruction. TheBloch equations are linear differential equations,and their numerical simulation can be efficientlycomputed under reasonable assumptions and ap-proximations. In particular, when the RF pulses arediscretized, spin simulation using the Bloch equa-tions reduces to successive applications of rotationfollowed by relaxation to the magnetization vectorM = (Mxy,Mz). Figure 5 shows an illustration ofthe simulation process. The overall signal evolutionis non-linear but differentiable with respect to therelaxation and system parameters.

In real imaging applications, multiple resonantfrequencies often appear within each voxel due tolocal magnetic field inhomogeneities. That is, themagnetic field experienced by spins within a voxelcan be slightly different. For a realistic simulation,the Bloch equations have to be solved for each res-onant frequency separately that appears within oneparticular voxel, and then summed together to formthe resulting signal. This can be computationallydemanding.

An alternative to the Bloch equations is the ex-tended phase graph (EPG) formalism, which sim-ulates the signal evolution on a voxel level [6].EPG uniformly discretizes the resonant frequencieswithin one voxel. The discretization allows EPGto efficiently keep track of signal evolutions acrossmultiple resonant frequencies using the Fourier se-ries. Similar to the Bloch equations, spin simula-tion using the EPG involves successive applicationsof rotation followed by relaxation to the underly-ing magnetization representation. The overall signalevolution using EPG is also non-linear but differen-tiable.

III. MRI SAMPLING

A. Spatial Encoding

The received MRI signal represents spatial fre-quencies of the transverse magnetization distributionin space, and the linear relationship is described bythe integral

s(t) =

∫r

Mxy(r, t)e−j2πk(t)>rdr + w(t), (5)

where s(t) ∈ C is the acquired signal at time t,k(t) ∈ R3 is a trajectory through the 3D frequencyspace, and w(t) is complex-valued white Gaussiannoise. The symbol (·)> denotes the transpose oper-ation and j =

√−1. Because the spatial frequency

wave-number is typically denoted as k, MRI acqui-sitions are often described as sampling in k-space[7].

Throughout this article we consider a discreteFourier approximation of (5). Though clinical MRIsystems come standard with parallel imaging receivearrays [1], we will omit their discussion for brevity,with the understanding that they can be flexiblyincorporated into the sampling model. We first con-sider the acquisition model when the magnetizationstays constant over time, i.e. Mxy(r, t) = Mxy(r)with no relaxation effects. Then, given an underlyingimage x ∈ CN , which consists of the transversemagnetization of all N voxels, the full forwardmodel is represented in matrix form as

y = PFx+w, (6)

where y ∈ CM are the acquired k-space measure-ments, F ∈ CN×N is a discrete Fourier transformoperator, P ∈ CM×N is a sampling operator thatselects the acquired k-space measurements, andw ∈CM is the noise. The encoding operator is succintlyrepresented as E = PF .

Since the scan time is directly proportional to thenumber of measurements, we are typically interestedin solving problems for the case where M < N .Compressed sensing offers an avenue for targetingthis regime by exploiting low-dimensional structurein the image representation [2]. A common inverseproblem approach to CS MRI is a regularized least-squares optimization given by

argminx

1

2‖y −Ex‖22 + λR(x), (7)

where R is a sparsity-promoting regularization, e.g.`1 norm of the wavelet coefficients, and λ > 0 is aregularization term.

B. Multi-dimensional ExtensionsAlthough initial work on CS MRI focused on

sparsity of static, anatomical images using spatialwavelet transforms and total variation [2], manyextensions have been proposed to handle additionalimaging dimensions, including joint sparsity, lowrank, and their variants [8]–[10]. In particular, thelinear forward model can be extended to repre-sent additional image states, x =

[x1 · · · xT

],

where xi ∈ CN is the image at the ith image stateand T is the number of states. The forward model,illustrated in Figure 6 for signal relaxation, includesdifferent encoding operators for each state i basedon the user-specified sampling patterns:

Ei = PiF . (8)

This concept can be used to include additional di-mensions representing signal relaxation [4], cardiacand respiratory motion [2], [11], [12], and many

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Bloch Equations Simulation

z

x y

z

x y

z

x y

z

x y

RF Pulse Relaxation

Extended Phase Graph (EPG) Simulation

z

x y

z

x y

z

x y

RF PulseIntra Voxel Off

Resonance Relaxation

Fig. 5. When the RF pulses are discretized, spin simulation using the Bloch equations reduces to successive applications of rotationfollowed by relaxation to the magnetization vector M . An alternative to the Bloch equations is the extended phase graph (EPG)formalism, which simulates the signal evolution of a distribution of spins across a voxel. EPG efficiently keeps track of signalevolutions across multiple resonant frequencies using the Fourier series.

others [10]. Selecting the T sampling patterns ina way that maintains compatibility with parallelimaging and CS is an active research area [13], [14].

Fig. 6. The forward model extended to temporal relaxation.Each image represents a sampling time i during the acquisitionwith T time points, where each time point is Fourier transformedand sampled with a different sampling operator Pi (representedby red circles).

IV. MODEL-BASED MRIThe signal evolution model developed in the

previous sections describes how the signal receivedby the MRI scanner is formed. Combining withspatial encoding, the physics based forward modelconsiders the following non-linear evolution:

xi(r) = ρ(r)fiTs (θ(r),u(r)) ,

yi = Eixi, i = 1, . . . , T.(9)

The most explicit use of the physics model in areconstruction is to directly solve for the tissueparameters from the raw k-space measurements.This is in contrast to first reconstructing a time seriesof images followed by a parameter fit. It is possibleto formulate this model-based inversion even in thecase of undersampled k-space [15], [16]. This canbe written as a non-linear, non-convex least squaresobjective in which we aim to solve for (ρ, θ):

minimizeρ,θ

1

2‖Ex− y‖22 + λR(ρ,θ)

subject to xi(r) = ρ(r)fiTs (θ(r),u(r)) ,

i = 1, . . . , T.

(10)

In the MRI literature, equation (10) is often referredto as quantitative MRI with a model-based recon-struction, as the physical model is incorporated intothe objective function.

Compared to the conventional image series recon-struction followed by a fit, solving for the parame-ters directly serves as the ultimate dimensionalityreduction: instead of computing a set of images,the aim is to recover only the intrinsic informationrepresented by ρ and θ. The main benefit of thisapproach is that the problem size is significantlyreduced. For example, for mapping PD and T2,solving a problem of the form (7) consists of TNunknown variables, while the model-based repre-sentation (10) has only 2N unknowns. In addition,sparsity-promoting regularization penalties can beapplied directly to the parameter maps. Since thedata fidelity term is differentiable, the overall prob-lem can be optimized using first-order or second-

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order methods.Additional system-related parameters such as RF

field inhomogeneity and off-resonance can also bemodeled and used as variables to be solved foras part of the reconstruction. Explicit incorpora-tion of these systematic deviations can make thereconstruction more robust to imperfections. Withthe parameters in hand, synthetic contrast-weightedimages could be potentially generated by evaluating(3) with specific scan and sequence parameters, atechnique known as Synthetic MR [15], [17].

However, even though the individual terms aredifferentiable, the main downside is that the re-construction problem is highly non-convex, therebycomplicating the optimization. The non-convexityresults in increased computation time and depen-dence on initialization. The problem requires a goodinitial guess of ρ and θ in order to converge toa reasonable estimate. Another drawback is thatmismatches between the model and the true acqui-sition can lead to error propagation in the estimatedparameter maps. For example, radiologists havedescribed the presence of flow artifacts, white-noiseartifacts, and other artifacts in synthetic FLAIR con-trast images [17], likely due to unmodeled effectssuch as flow. Partial voluming is also known to im-pair the estimation accuracy [15]. Errors in samplingtrajectories due to eddy currents and gradient delayscan also manifest as blurring and streaking artifacts.These effects can potentially be reduced by expand-ing the signal model, e.g. to incorporate scannernon-idealities. The pulse sequence can also includenavigator components to aid in the estimation.

V. RELAXING THE MULTI-CONTRAST MODEL

Instead of fully incorporating MRI physics intothe reconstruction, it is possible to use relaxedconstraints that are more amenable to optimization[18]. The natural dynamics of the MR signal areconstrained by the Bloch equations, implying theexistence of a low-dimensional manifold, which isnon-linear in general. As Figure 7 illustrates, a linearsubspace approximation to the manifold with higherdimensionality could provide a compromise betweenrepresentation simplicity and size. This may beattractive for a few reasons; namely, to maintainconvexity and computational efficiency in the op-timization, to decouple the reconstruction from thequantitative fitting, and to reduce propagation ofmodel error. In addition, many applications do notrequire quantitative parameters, and instead rely onhigh-quality contrast-weighted images. This is thecase for nearly all clinical diagnostic imaging. Un-fortunately, contrast-weighted images derived from

parameter maps are susceptible to error propagationdue to unmodeled components, e.g. from partialvoluming and flow effects, and have seen limiteduse clinically [17].

Fig. 7. The low-dimensional manifold representing the Blochequations is captured by a linear subspace ΦK of larger dimen-sion.

A. Subspace Constraint

Continuing with the FSE sequence as a guidingexample, Figure 8 shows the signal evolutions for anFSE simulation with a particular flip angle schedule.Despite differences in relaxation parameters, thesignal evolutions for different tissues follow simi-lar trends. This correlation implies low-dimensionalstructure; namely, the signal evolutions of differenttissues form a low-dimensional subspace [4], [19]–[22].

Many approaches can be taken to design thesubspace. Here we focus on simulating a set oftraining signals derived from EPG simulation. Weassume a prior distribution p(θ) is known and drawL samples from the distribution to create the trainingsignals. The prior distribution can be taken fromknown literature values, or from a conventionalmapping procedure focused on a particular anatomy.

Consider a data matrix X ∈ CT×L consistingof an ensemble of L signal evolutions sampled at Techo times. Each column in X represents the signalevolution of a spin population with a particular θ ∈p(θ). Let Φ ∈ CT×T be an orthonormal temporalbasis, i.e.

X = ΦΦHX. (11)

The goal is to design Φ =[ϕ1 · · · ϕT

]and a K−dimensional subspace, ΦK =span{ϕ1, . . . ,ϕK}, such that∥∥X −ΦKΦH

KX∥∥ < ε, (12)

where ε is a modeling error tolerance. The choiceof norm in (12) will affect the chosen subspace and

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Time (ms) Time (ms)

Signal (a.u.)

Signal (a.u.)

Singular value index

Ensemble of signal evolutions Principal component vectorsSingular valuesParameter D

istribution

Fig. 8. Forming a subspace based on signal dynamics, shown for FSE. (left) An ensemble of signal evolutions is drawn from aprior distribution. (middle) Due to correlation in signal dynamics, the data matrix is low-rank and (right) is well-approximated withPCA.

can be used to capture average, worst-case, and othererror metrics.

When the Frobenius norm is used, the solutioncorresponds to the truncated singular value decom-position [4], [19], [20], i.e. principal componentanalysis (PCA), where ΦK consists of the left sin-gular vectors of X corresponding to the K largestsingular values. The principal component images,corresponding to the principal component vectors,are given by

α = ΦHKx. (13)

Based on (12), we also have x ≈ ΦKα.

B. Reconstruction

The subspace relationship can be incorporatedinto the reconstruction as additional prior knowledgein the form of regularization [22]:

minx

1

2‖y −Ex‖22 +

µ

2

∥∥x−ΦKΦHKx∥∥22+ λR (x) ,

(14)

where µ > 0 controls the degree of subspacemodeling. Alternatively, the subspace can be usedas an explicit constraint, i.e.,

minimizex

1

2‖y −Ex‖22 + λR (x)

subject to x = ΦKα.(15)

Although the analysis form described by (14) is amore faithful representation, the reconstruction stillrequires solving for TN parameters in x, in additionto introducing a hyper-parameter µ. In contrast,the synthesis form used in (15) introduces explicitmodel error but is significantly more computation-ally efficient.

When using the hard constraint, we can now solvefor the subspace coefficient images directly [4],

minα

1

2‖y −EΦKα‖22 + λR (α) , (16)

followed by back-projection: x = ΦKα. This isa significant dimensionality reduction! Instead ofsolving for TN variables, we only need to solvefor KN variables. In addition, when the normalequations are used in an iterative optimization (asis the case for many first-order iterative algorithms),we can take advantage of the commutativity of thesubspace operator and the Fourier transform, as theformer only operates on the parametric dimensionand the latter only operates on the spatial coordi-nates [4]:

PFΦK = PΦKF (17)

=⇒ ΦHKF

HPFΦK = FHΦHKPΦKF (18)

= FHΨKF , (19)

where ΨK ∈ CKM×KM is a block-wise diagonaloperator with a K × K symmetric block for eachspatial frequency point. In other words, the opti-mization does not have to perform any computationin the ambient space, and the complexity grows withK, independent of T .

Compared to model-based quantitative mapping,the subspace-constrained forward model is convexand easier to solve. In addition, the subspace isless sensitive to model error, as it does not strictlyimpose a specific physical model. As an exam-ple, a voxel with partial voluming will contain alinear combination of signal evolutions, x(r) =a1x

(1)(r) + a2x(2)(r), comprising different tissue

parameters θ(1) and θ(2), which is inconsistent with(3). In contrast, if x(1) and x(2) are separately repre-sented by the subspace, then so is their combination.

The main drawback to the subspace formulationis that the problem is not reduced to its intrinsicdimension governed by ρ and θ. However, in prac-tice a subspace size of K < 5 is practical forapplications even when the ambient dimension is inthe hundreds [4], [23]. A second drawback is thatthe subspace can represent points off the manifold,that are not physically meaningful. Thus, data incon-

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sistencies can manifest as inaccurate images afterback-projections.

C. Quantitative MappingA straightforward approach to quantitative map-

ping from reconstructed multi-contrast images is toperform a voxel-wise non-linear least squares fitbased on the signal model. Given the reconstructedimage x at voxel r, this amounts to solving

argminρ(r),θ(r)

1

2‖x(r)− ρ(r)f(θ(r),u(r))‖22 . (20)

As with Section IV, it is also possible to solvefor RF field inhomogeneity by additionally solvingfor the non-negative scalar η that multiplies theflip angles: u → ηu. This formulation also cov-ers dictionary-based methods which have grown inrecent popularity with the advent of MR Finger-printing [24]. Rather than solving a continuous non-linear least squares problem, dictionary-based fittingutilizes a grid search across the parameter spaceand is equivalent to matched filtering. Optimization-based methods can be used as well.

When the reconstruction strictly enforces a sub-space, there is a known model error between thereconstruction x and the signal evolution f , andwe do not expect them to match even in the ab-sence of noise and under-sampling artifacts. To im-prove parameter estimation when using a subspace-constrained reconstruction, we can solve for theparameters directly in the subspace:

argminρ,θ

1

2

∥∥α(r)−ΦHKρf(θ,u(r))

∥∥22. (21)

Since f is differentiable with respect to θ andu, first-order and second-order solvers can be used.Figure 9 shows an example of solving (21) followinga T2 Shuffling reconstruction4 [4], where K = 3 andθ = (|ρ|,∠ρ, T2). The non-linear least squares wassolved using the Trust Region Reflective algorithmincluded in the Python Scipy package, and theJacobians were calculated using the adjoint statesmethod [25].

When solving under-determined inverse prob-lems, it is important to recognize the bias introducedthrough modeling assumptions and regularization. Inparticular, the optimization problem (15) introducestwo forms of bias: model error due to the sub-space constraint, and error due to the regularization.The subspace constraint leads to a straightforwardtradeoff between model error and noise amplifica-tion: noise standard deviation increases with

√K,

4Example code available on https://eecs.berkeley.edu/∼mlustig/Software.html

PhaseMagnitude

PD

Phase

Subspace Images Reconstructed Maps

Fig. 9. Example showing voxel-wise parameter fitting fol-lowing a T2 Shuffling reconstruction using non-linear leastsquares directly in the subspace. The principal component images(magnitude and phase) were first reconstructed with the convexformulation (16), and the complex values were fit to the physicalmodel using the EPG formalism.

where K is the subspace size [4]. A small subspacewill also reduce sensitivity in the parameter space,and manifests as reduced contrast and parameter-dependent noise amplification, as all voxels arepushed to the same curve fit [20]. Bias due toregularization can also be an issue in reconstructionand parameter fitting, but its impact depends onthe specific regularization used. With non-linearregularization such as `1 commonly used in CS,the tradeoff can be harder to quantify. Blocking andsmoothing artifacts are common when using waveletregularization and total variation due to loss of high-frequency content [2]. Low-rank regularization andcan lead to washed-out contrast due to loss of spec-tral information, and its multi-scale variants can leadto blocking artifacts due to their translation-variantstructure [9]. In general, under-regularization cancause residual incoherent under-sampling artifactsand noise amplification, while over-regularizationcan lead to blurring [4], [11], [12], [15], [16], [18],[20]–[23].

VI. CHOOSING SCAN PARAMETERS

In addition to incorporating MRI physics in thereconstruction, using knowledge of the physics tooptimize the MRI pulse sequence could potentiallyimprove the quality of the acquired raw data. De-signing an MRI pulse sequence consists of two maincomponents: (i) designing the pulse waveform andtiming to guide the signal evolution, and (ii) design-ing the spatial encoding to appropriately sample k-

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space. Initial work focused on designing sequenceswith simple goals in mind such as maximizingcontrast and limiting blurring. A recent trend incomputational MRI is to use optimal control to mo-tivate scan parameter selection [25], and incoherentsampling properties to motivate k-space sampling[2]. Optimal experiment design has also been ap-plied to spatial encoding under a synthesis sparsitymodel [14]. Importantly, both spatial encoding andsequence parameters should be designed with aspecific reconstruction in mind.

A. Optimizing Sequence Parameters

Many approaches to optimizing sequence parame-ter selection involve the use of a Cramer-Rao lowerbound (CRLB), which imposes a lower bound onthe variance of an unbiased estimator. To simplifythe optimization, the spatial encoding is decoupledfrom the sequence evolution, and a 1D experimentis used for optimization:

y = f (θ,u) +w. (22)

Then, the Fisher information matrix, I(θ;u), whichmeasures the sensitivity of θ captured by y givensequence parameters u, can be used to lower boundthe variance of an unbiased estimator. The sequenceparameters can then be optimized to minimize thislower bound on the variance of the biophysicaltissue parameter estimates. This approach has beenused in [26] to develop a general framework tooptimize sequence parameters from combinations ofpulse sequences for precise estimation of T1 and T2parameters jointly. It has also been used to deter-mine sequence parameters for MR Fingerprinting toobtain maximal SNR while not violating physics-based MRI constraints [27].

Another optimal control design based approachuses the EPG formalism to develop a model for thesignal evolution in terms of various sequence andtissue parameters [25]. An optimization problem isthen formulated according to the goal, e.g. maxi-mizing signal intensity or minimizing RF power,and solved using the adjoint states method. Thisapproach can flexibly incorporate other objectivefunctions, e.g. to minimize the CRLB, similar tomethods discussed above. Figure 10 shows an ex-ample of optimizing the RF flip angles in an FSEexperiment given RF power constraints [28]. Theobjective function maximized the component in theFisher Information matrix corresponding to the vari-ance with respect to T2 estimation. Constant flipangles that achieve the same RF power limit areshown in comparison.

Data-driven methods for sequence parameter op-timization are also emerging. With these methods,a certain set of sequence parameters are chosen,the resulting time-evolution of the signal for theseparameters is either simulated or determined experi-mentally, and finally an image is reconstructed. Thesequence parameters are then updated according tothe reconstruction error. Compared to the optimalexperiment based approaches, these methods havethe advantage of incorporating spatial encoding andreconstruction into the evaluation.

When choosing sequence parameters, it is impor-tant to understand the impact of system imperfec-tions on the optimized sequence. These imperfec-tions can often be incorporated into and mitigatedin the optimization framework, though they maygreatly increase the computational complexity. Forexample, to reduce sensitivity to RF transmit fieldinhomogeneity, one may optimize the average (ormin-max) CRLB across spins experiencing a dis-tribution of RF transmit field inhomogeneities bysimulating each spin independently. This quicklyincreases in complexity as additional componentsare considered.

B. Optimizing Sampling Pattern

Conventional methods to determine CS samplingtrajectories are based on exploiting the incoher-ence conditions for CS by the use of a transformpoint spread function (TPSF), which determinesthe leakage of one transform coefficient to anothertransform coefficient due to subsampling [2]. Asampling pattern can be determined by a Monte-Carlo design procedure where a variable densitypattern is obtained by randomly drawing indicesusing a probability density function, the TPSF of theobtained trajectory is calculated, and the procedureis repeated, choosing the pattern with the lowestpeak interference. The reconstruction is performedaccording to (7) with a sparsity-promoting regular-izer on the transform coefficients. Thus, incoherencein the transform domain is a good metric for sam-pling pattern quality.

Several data-driven approaches have been devel-oped in order to improve on the current incoherence-based methods. These data-driven approaches use alearning-based framework to find the best samplingpattern for a set of training signals. In particular,methods that optimize a sampling pattern for aspecific reconstruction rule and anatomy [14] as wellas methods that jointly optimize a sampling patternand a reconstruction strategy have been developed[29]. In this way, physics-based constraints in the

10

Fig. 10. Comparison of a constant flip angle schedule vs. optimized flip angles for an FSE experiment given a maximum RF powerconstraint (left). The CRLB was optimized with respect to T1 = 1000 ms and T2 = 100 ms, and the received signal was higherfor the optimized flip angles (middle). Even though the optimization targeted a single relaxation value, the bound was lower acrossa uniform range of T2 values (right).

reconstruction are implicitly propagated through tothe sampling pattern optimization.

Although various control design based and data-driven strategies have been developed to optimizeone of scan parameters, sampling trajectories andreconstructions, the ultimate goal of developingmethods that optimize parameter, trajectory and re-construction in conjunction remain an area for futurework.

VII. SUMMARY

Since the advent of MRI, physics-based knowl-edge has been used to recover and understandthe signal dynamics and tissue parameters. Manyearly works have leveraged physical constraints tomitigate image artifacts due to systematic errorsand derive quantitative maps. However, long scantime and the lack of sophisticated reconstructionalgorithms have prevented the clinical adoption ofthese techniques.

In the last decade, compressed sensing andother computational imaging approaches have trans-formed the landscape of what is possible with MRI.Scan time can be appreciably reduced by leverag-ing natural image statistics in the reconstruction.Many advanced reconstruction algorithms have beendeveloped for this express purpose. Using thesenumerical tools and combining sparsity-based mod-eling, it is now feasible to run physics-constrainedcomputational MRI methods in the clinic with rea-sonable scan and reconstruction times [30]. Byincorporating the physical dynamics due to bothtissue-specific and scanner-specific parameters, ac-quisitions and reconstructions can be designed intandem to work across a broad patient populationin a robust manner.

The methods introduced in this article provide aframework for modeling the MR dynamics, mod-ifying the acquisition to account for the signalevolutions, and incorporating them into the recon-struction. Although we limited our focus to onlymodeling dynamics due to relaxation effects and RF

pulses, there are many other contrast mechanismsand scanner controls that can be accounted for.On the tissue characterization side, these includediffusion, water-fat imaging, susceptibility, spec-troscopy, pharmacokinetics, magnetization transfer,chemical exchange, and blood flow. On the imagingsystem side, these include field inhomogeneity, eddycurrent effects, gradient delays, temperature, andmore. Each of these components can be modeledjointly, but will greatly increase the dimensionalityof the problem. Model-based quantitative imagingreduces the inverse problem to recovering the intrin-sic parameters, but careful modeling and simulationmust be used to avoid artifacts due to model error.In contrast, subspace constraints and other low-dimensional representations can be used to flexi-bly capture the dynamics without making strongassumptions.

Many exciting modeling techniques are emergingin the signal processing community, and are greatcandidates at improving physical modeling in MRI.On the other hand, purely data-driven approacheshave grown in popularity, in the hopes of learningthe signal characteristics from real data. With thegrowing trend of applying deep learning to inverseproblems, there is great promise to incorporatingadditional physics-based constraints directly in thelearning in order to restrict the feasible solutionspace, reduce the dependence on large trainingdata sets, and model effects not described by thesimplified physics [31]. Like compressed sensing,the rapid empirical progress in deep learning-basedimaging should be suitably counter-balanced withtheoretical guarantees to guide its use in clinicalsettings. Combining physical modeling with data-driven learning is an active area of research in thesignal processing community in general, and in theMRI field in particular.

ACKNOWLEDGEMENTS

This work was supported by NIH grantR01EB009690, Sloan Research Fellowship, Bakar

11

Fellowship, and research support from GE Health-care. The authors thank Volkert Roeloffs, ZhiyongZhang, and Karthik Gopalan for valuable com-ments.

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