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In medical magnetic resonance imaging, parallel radio frequency excitation pulses have to respect a large number of specific absorption rate constraints. Geometrically, each of these constraints can be interpreted as a complex, centered ellipsoid. We propose to replace a collection of such constraints by the single constraint which corresponds to the associated maximum volume inscribed ellipsoid and implies all original constraints. We describe how to compute this ellipsoid via convex programming. Examples show that this reduction has very short computation times but cuts away parts of the feasible power domain.

Parallel radio frequency (RF) excitation pulses for magnetic resonance imaging (MRI) deposit energy in the irradiated tissue. In the affected areas this creates an amount of heat which depends on the strength and duration of the superimposed RF fields and the conductivity and density of the irradiated tissue. Specific absorption rate (SAR) constraints are responsible for restricting this heating to a harmless level. Current International Electrotechnical Commission [

Physics dictates that the thermal energy deposited in a specific volume by oscillating electromagnetic fields is given by a time-homogeneous quadratic functional applied to the RF excitation waveforms; compare equation (4) in [

In the case of a single RF excitation coil the SAR constraints can easily be reduced to the single constraint which belongs to the most affected specific volume. However this is not possible in the case of parallel excitation with more than one coil, where the most affected specific volume can depend on the utilized coil voltages and phases. In this case the SAR constraints can present a serious additional burden for the design of RF waveforms, especially because of their large number. This obstacle might not only increase pulse design computation time, but also impact the quality of the calculated excitation pulse.

Special algorithms have been developed for coping with parallel RF pulse design problems in the presence of SAR constraints. In [

A popular SAR reduction technique was introduced by Eichfelder and Gebhardt in [

In this work we present a novel pure and extreme reduction technique. Geometrically, the quadratic functional defining a SAR constraint can be interpreted as a complex, centered ellipsoid with a dimension corresponding to the number of RF excitation coils. We propose to replace a collection of SAR constraints by the single virtual SAR constraint, which is associated with the maximum volume inscribed ellipsoid (MVIE). This single MVIE SAR constraint implies all original SAR constraints. The maximum volume characteristic makes sure that maximum room remains for the RF pulse design after this reduction. This intuitive idea has the appealing mathematical property that the parameters of the MVIE SAR constraint can be obtained as the solution of a convex optimization problem. We take advantage of this fact to develop methods for a quick numerical calculation of the parameters of the MVIE SAR constraint. The developed “implicit active constraints selection” heuristic in the iterations of an interior point algorithm shows particularly short numerical run times.

Besides these algorithmic aspects, this work focuses on the mathematics, which underlies the MVIE SAR constraints reduction. To this end we strengthen relevant theoretical results for MVIEs from the real-valued case to the complex-valued one, as required in our application; see the Appendix. Noticeable is the result that blowing up the complex MVIE ellipsoid by a factor corresponding to the square root of its dimension yields an ellipsoid, which contains the intersection of all original ellipsoids, which, to the best of our knowledge, has not been proved for the complex case before.

We present only few tentative numerical results. The two main examples consist of collections of 448480 and 111939 SAR constraints for 2 and 8 RF excitation coils, respectively. These examples show not only the feasibility of our method, but also its speed: it only takes seconds to reduce the example SAR constraint collections to the MVIE SAR constraint on a standard personal computer.

As drawback of our method we have to mention the missing ability to control how much power the MVIE SAR constraint is cutting away from the feasible RF space of the original SAR constraints. Such a reduction can impact the achieved flip angle or lead to an increased inhomogeneity of the magnetization (compare Figure 7 in [

Maximum surface inscribed ellipse for three ellipses in light gray. Cut away difference between this ellipse and the intersection of the original ellipses in dark gray. Contact points are marked with circles and cut away corners with squares.

An overview of this work is as follows: we formally introduce SAR constraints in Section

We let

We let

We consider

The RF pulse design problem is to find a SAR-compliant RF waveform, which accurately produces the desired magnetization and also respects the physical capabilities of the scanner hardware; see [

If value and speed of an RF waveform

For a positive definite Hermitian matrix

If an RF waveform

It is well known that there exists a unique MVIE for each closed compact convex set with a nonempty interior [

For a matrix

Finding the maximum volume ellipsoid can be casted as a convex optimization problem: Since the negative of the concatenated functions

It is interesting to note that when the maximum volume inscribed ellipsoid

We unify the objective function (

The calculation of

We note that the numerical calculation of the inverses

The idea for dealing with convex positive definiteness or semidefiniteness constraints

By modifying constraints (

The converse inclusion,

Interior point algorithms [

In the first variant we replace the

The second variant is what we call an

The

Since, after the evaluation and sorting steps,

The following numerical examples show that MVIE SAR parameters can be calculated quickly and that an overall approach consisting of the calculation of the MVIE SAR constraint and its subsequent utilization in an RF pulse optimization can reduce the overall computation time without affecting the excitation quality. However, these examples are not comprehensive enough to make statements to which MRI applications these observations can be carried over.

The calculation times which we report in this section have been obtained with an Intel(R) Core(TM) i7-4800MQ CPU at 2.70 GHz. We are using the interior point solver IpOpt [

In order to assess how much RF power can be lost by reducing

Here we consider three example sets of SAR matrices:

A Siemens internal example with

An example of Nicolas Boulant with

The example from Hoyos-Idrobo et al. [

In Section

For each of these examples we calculate a virtual SAR constraint with each of the three different methods of Section

By solving the convex optimization problem (

By solving variant (

As the first method, but with the

Each of these methods delivers an approximation

Numerical calculation of maximum volume inscribed ellipsoid for the three examples and the three proposed methods.

Example characteristics | |||
---|---|---|---|

Example number | | | |

| |||

RF coils | 2 | 8 | 8 |

SAR constraints | 448480 | 111939 | 491 |

| |||

Method | |||

| |||

Number of iterations | 33 | 100 | 22 |

Overall algorithm [s] | 657.7 | 847.3 | 0.5 |

Symbolic factorization [s] | 630.8 | 607.0 | 0.1 |

Update Hessian [s] | 0.6 | 127.2 | 0.2 |

Compute direction [s] | 22.2 | 97.0 | 0.1 |

Acceptable trial point [s] | 2.4 | 0.04 | 0.0 |

Number of back steps | 15 | 76 | 22 |

| |||

Method | |||

| |||

Number of iterations | 100 | 100 | 61 |

Overall algorithm [s] | 27.9 | 129.9 | 0.9 |

Symbolic factorization [s] | 0.1 | 0.1 | 0.1 |

Update Hessian [s] | 0.7 | 115.5 | 0.5 |

Compute direction [s] | 0.1 | 0.06 | 0.1 |

Acceptable trial point [s] | 26.3 | 0.001 | 0.1 |

Number of back steps | 1920 | 412 | 117 |

| |||

Method | |||

| |||

Number of iterations | 19 | 58 | 46 |

Overall algorithm [s] | 2.9 | 4.4 | 0.6 |

Symbolic factorization [s] | 0.1 | 0.1 | 0.1 |

Compute Hessian [s] | 0.0 | 0.3 | 0.2 |

Compute direction [s] | 0.0 | 0.2 | 0.1 |

Acceptable trial point [s] | 0.0 | 3.6 | 0.1 |

Number of back steps | 15 | 127 | 137 |

Power losses for the three examples.

Example number | | | |

RF coils | 2 | 8 | 8 |

| |||

| 1.00002 | 1.11045 | 1.10287 |

| 1.1802 | 2.56831 | 2.83011 |

| 1.88863 | 5.18516 | 6.2459 |

In examples (1) and (2) with a large number of SAR matrices the solution of the problem with individual SAR constraints utilizes most of the computation time for the symbolic factorization of the Hessian matrix. This part of the computation is negligible in the other proposed algorithms. The external active constraint selection method largely reduces the number of back steps in the first example but increases this number in the other examples. As intended it spends little time for the calculation of the Hessians and achieves the smallest overall computation time.

Up to numerical inaccuracies all methods deliver the same maximum volume inscribed ellipsoid. In Table

We consider the inversion pulse design problems of [

For this optimization we follow the approach described in [

The

The

We are applying these two approaches to each of the two pulse design problems with either the full set of 491 SAR constraints or the single MVIE SAR constraint obtained in Section

Optimization of AFA and STA objective functions for the excitation and inversion examples without and with proposed MVIE SAR reduction.

| | |

| ||

Number of SAR constraints | 491 | 1 |

Average power constraints | 8 | 8 |

| ||

| ||

| ||

Number of RF constraints | 40 | 40 |

| ||

Arbitrary flip angle optimization (AFA) | ||

| ||

Computation time [s] | 2.63 | 1.24 |

Iteration count | 41 | 38 |

Root mean square error [deg] | 1.63 | 1.87 |

| ||

Small tip angle approximation (STA) | ||

| ||

Computation time [s] | 1.87 | 0.89 |

Iteration count | 38 | 41 |

Root mean square error [deg] | 1.64 | 1.90 |

| ||

| ||

| ||

Number of RF constraints | 56 | 56 |

| ||

Arbitrary flip angle optimization (AFA) | ||

| ||

Computation time [s] | 8.52 | 2.87 |

Iteration count | 84 | 55 |

Root mean square error [deg] | 15.44 | 15.44 |

| ||

Small tip angle approximation (STA) | ||

| ||

Computation time [s] | 4.84 | 1.44 |

Iteration count | 65 | 52 |

Root mean square error [deg] | 10.38 | 10.38 |

The use of the MVIE SAR reduces the computation time by a factor of about 1/2 to 1/3 in the STA and AFA optimization. This reduction is larger than the time it takes to calculate the MVIE SAR parameters. Only in the 30-degree excitation example with 5

This RF pulse design example shows that even in the case where the SAR matrices have been already been reduced to about 500 they put a burden on the pulse optimization which dominates computation time. This burden is largely reduced by utilizing the MVIE SAR constraint.

Parallel RF excitation pulses not only have to form an accurate magnetization, but also must make sure that a large number of SAR constraints are respected. In this work we suggested to reduce this additional burden by replacing a collection of SAR constraints by a single virtual MVIE SAR constraint. This MVIE SAR constraint is characterized by the properties that it implies each of the original SAR constraints and that its associated ellipsoid has the maximum volume. The

The theory of maximum volume inscribed ellipsoids has been developed for ellipsoids in Euclidian spaces

There exists a unique matrix

As in [

If the matrix

As preparation we define the mapping

Moreover, we define for a nonempty set

The MVIE matrix

According to Theorem

We let

The data sets are not publicly available.

The author declares no conflicts of interest.

_{T}