In MRI, at ultrahigh field, the use of parallel transmit radiofrequency (RF) arrays is very beneficial to better control spin excitation spatially. In that framework, the so-called “universal pulse” technique, proposed recently for head imaging at 7 tesla, gives access to “plug-and-play” nonadiabatic solutions exhibiting good robustness against intersubject variations in the resonant transmit fields. This new type of solution has been defined so far as the result of numerical pulse optimizations performed across a collection of RF field maps acquired on a small population sample (pulse design database). In this work, we investigate an alternative universal pulse design approach in the linear small tip angle regime whereby the database of RF field maps is first transformed into a second-order statistical model and which then exploits a statistical robust design formalism for the optimization of the RF and magnetic field gradient waveforms. Experimental validation with an eightfold transmit RF coil for 7 tesla brain imaging shows that this new approach brings some benefit in terms of computational efficiency. Hence, for a design database composed of 35 maps, the computation time initially of 50 min could be reduced down to 3 min. The proposed statistical approach thus enables integration of large databases, presumably necessary to ensure robust solutions. Finally, it provides means to compute flip angle statistics and, along with it, simple performance metrics for quality assurance (RF pulse performance) or guidance in the optimization of TX array architectures.

In magnetic resonance imaging at high magnetic field, the apparition of standing wave effects [_{T}-point pulses [

However, in addition to being nonuniform in space, the transmit RF field (

To better control the actual performance of UP on the entire population, we propose in this work to model the transmit RF field as a stochastic process, whereby every new subject (giving rise to a new

We thus present in this work a statistical representation of the TX field distribution

Let

The control field vector being subject dependent is viewed as a stochastic variable where each new subject gives rise to a realization of

Let

For the covariance matrix, we shall use the Ledoit–Wolf [

Given a statistical model for

As the control field distribution is typically obtained by MR acquisition, it is common in practice that

Let

Note that the flip angle created by the pulse

In this model, the effect of off-resonances is taken into account but is not modelled as a random process

In the following, we present a method to compute universal FA shimming pulses which is entirely based on a second-order statistical model for the control field. Along with this statistically driven framework, we provide a method to evaluate the performance of a universal FA shimming pulse.

Let

Denoting now by

Hence, the problem described by equation (

If, now, for a given pulse candidate

Retrospectively, the link with the database-driven objective function (simultaneous design across all subjects of the database of

Now, if we assimilate sample mean and statistical mean, we obtain

Let

In Appendix

The expression for

Furthermore, it is shown in Appendix

Hence, under the condition

The condition

It is also of interest to define the ultimate FA-CV—in the sense of equation (

In what follows, we present an application of the proposed statistical model of the intersubject variability of the control field and their implication in terms of pulse design and flip angle shimming performance assessment.

The data consisted of a collection of

The control field second-order statistics

The lower bounds for the CV of the effective control field

For that study, the spins were assumed to be at resonance _{T}-point [

A first FA shimming pulse, referred to as SRD-UP, was designed from the second-order control field statistics of the Nova TX array head coil (estimated in

In the above expression,

For comparison with SRD-UP, the solution (same parameterization and constraints as for the design of SRD-UP) of the database-driven UP objective was also computed. The database-driven UP objective was defined as follows:

This pulse is referred to as UP. Finally, a standard hard pulse where the transmit array is used in CP mode was defined. The amplitude of this pulse thus was set to return on average on the design database the target flip angle.

We then studied the spatial distribution of the mean and CV of the flip angle for UP, SRD-UP, and CP and compared it with the ultimate FA shimming performance

Using the five last subjects of the cohort (the 35 first constituting the design database), the flip angle normalized root mean squares error (FA-NRMSE) of UP, SRD-UP, and CP (_{T}-point pulse (minimization of the

All pulse optimizations were performed using the second-order active-set algorithm, available in the Optimization Toolbox of MATLAB R2019a (The Mathworks, Natick, MA), which allows solving constrained optimization problems numerically. The active-set algorithm used the analytical expressions for the Jacobian of the objective function and the RF power, RF energy, and MFG constraints (Hoyos-Idrobo et al. [

In Figure

(a) Axial view of the mean control field (in Hz/volt) of the Nova TX array (images 1 to 8 correspond to the 8 TX channels and the last image correspond to the CP excitation mode) and (b) the covariance of the control field (Hz^{2}/volt^{2}) for the Nova TX array.

Figure

Sagittal, coronal, and axial views of the lower bound for CV of the effective control field (

Cumulative histograms of the lower bound for the control field CV for the CP (dashed curves) and pTX (solid curves) drive modes. With pTX, the 80% quantile of control field CV is

Cumulative histogram of the ratio

The FA shimming performance analysis of

Median sagittal view of the mean (a) and (b) CV of the flip angle for (from top to bottom)

Mean value (a, b) and (c) CV of the flip angle for

The result of the FA-NRMSE analysis made separately on the design and test subjects is presented in Figure

FA-NRMSE performances of (a)

Denoting by _{T}-points), the design of SRD-UP is computationally more efficient than the UP design when the size of the design database is greater than

In this work, we have shown that the statistical robust and the UP design methods are theoretically nearly equivalent when the size of the database is sufficiently large to embrace the variability of the resonant TX RF field across the population of interest. Despite its apparent complexity, the statistical robust design algorithm defined by equation (

The theory presented in this work defines the performance of a FA shimming pulse as the coefficient of variation (std/mean) of the flip angle. Furthermore, in the linear STA regime, it is shown that this CV cannot be lower than a certain value

The present study was performed with the _{T}-point pulses computed in this study) can deviate from the analysis that one would obtain with a larger database to estimate the control field second-order statistics.

The theory developed in this study is based on the assumption that the statistical distribution of the control field vector

Extending this work to large flip angle pulse is one interesting prospect. However, in this case, the linear relationship between

An important difference with the pulse design method reported in [

In this work, for simplicity, the statistical robust pulse design implementation did not include specific absorption rate (SAR) constraints. However, as it was done in the past for the design of UPs, there would be no obstacle in adding explicit local SAR constraint to the statistical robust design procedure in the form of virtual observation points [

We have presented in this work a statistical robust approach to design calibration-free pTX universal pulses for brain imaging at ultrahigh field. Experimentally, this method has been shown to provide comparable FA-NRMSE results as the conventional database-driven UP design, but presents a significant advantage in terms of computational efficiency when the size of the design database exceeds 4. Furthermore, in contrast with the database-driven design of UP, the computation time of the statistical robust UP does not depend on the size of this database and consequently enables integration of a very large design database. This is an important criterion to ensure optimal UP performance. Furthermore, the theoretical lower bound for the coefficient of variation of the flip angle derived in this work,

In this section, we focus on the relaxed SRD problem (infinite norm replace by the

It follows from condition (

In Figure

Plot of

As a result, for a candidate pulse

Noting that (i) the leading term in equation (

We note that the condition given by equation (

Below we first solve the phase-constrained problem and then the phase-unconstrained problem. While the first problem can be solved analytically in general, the resolution of the second problem—the one that we focus on in this study—is conducted under the assumption that

First, we look for the solution of the minimization problem:

Using the equality constraint, we have

The Lagrangian of the above problem reads

If

From equation (

From equations (

The solution to equation (

The phase-unconstrained problem (

Now, if we suppose that

Under this hypothesis, the problem (

The Lagrangian of this optimization problem reads

The second-order optimality condition thus reads

Given now that

As a result, the solution to the phase-unconstrained problem is given (up to a uniform phase term across TX channels) by

Taking the above expression for

Given a FA shimming pulse

This new pulse is simply obtained by scaling the RF waveform

In other words,

To complete the proof, we note that the function

The datasets generated and/or analyzed during the current study and the pulse design source codes are available from the corresponding author on reasonable request.

The authors declare that they have no conflicts of interest.

This research received funding from the RETP program (Research Equipment and Technology Platforms) of the Leducq Foundation, the European Research Council under the European Union’s Seventh Framework Program (FP7/2013-2018), and ERC Grant Agreement #309674 and Proof of Concept #700812. The authors are grateful to Dr. B. Thirion for his valuable input regarding the estimation of the covariance matrix of the control field.