Development of customized pTx MRexcitation methods and their safe

application

Der Naturwissenschaftlichen Fakultatder Friedrich-Alexander-Universitat

Erlangen-Nurnberg

zur

Erlangung des Doktorgrades Dr. rer. nat.

vorgelegt von

Rene Gumbrecht

aus Erlangen

Als Dissertation genehmigt

von der Naturwissenschaftlichen Fakultat

der Friedrich-Alexander-Universitat Erlangen-Nurnberg

Tag der mundlichen Prufung: 25.10.2013

Vorsitzender des Promotionsorgans: Prof. Dr. Johannes BarthGutachter/in: Prof. Dr. Paul Muller

Prof. Dr. Oliver Speck

Contents

1 Introduction 5

2 Theory 72.1 Parallel transmission at high fields . . . . . . . . . . . . . . . . . . . . . . 72.2 Patient heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Experiments 153.1 Local SAR supervision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Pulse design evaluation workflow . . . . . . . . . . . . . . . . . . . . . . . 18

4 Methods 214.1 High flip-angle pTx excitation using Composite Pulses . . . . . . . . . . . 314.2 Accurate saturation pulses (pTx CHESS) . . . . . . . . . . . . . . . . . . 334.3 Hybrid pulse design: 2D Composite Pulses . . . . . . . . . . . . . . . . . . 354.4 Local SAR optimized RF pulse design . . . . . . . . . . . . . . . . . . . . 384.5 Quality controlled SAR reduction . . . . . . . . . . . . . . . . . . . . . . . 39

5 Results & Discussion 435.1 Local SAR supervision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 pTx Pulse design performance . . . . . . . . . . . . . . . . . . . . . . . . . 465.3 Flip-angle homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.4 SAR optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6 Conclusions 61

7 Appendix 637.1 RF chain calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.2 Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.3 Hardware control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3

1 Introduction

Magnetic resonance imaging (MRI) is a non-invasive imaging modality that is routinelyused for clinical diagnosis to visualize the anatomical structure and function of the humanbody in vivo. The development of novel techniques for visualizing properties of humantissue is an active area of research.

MRI is based on the effect of nuclear magnetic resonance (NMR). It only occurs whencertain nuclei with a nuclear spin are placed in a strong static magnetic field. To inducean NMR signal, it is necessary to excite the system of nuclei with nuclear spin in amagnetic field. This is realized by applying a radio-frequency (RF) pulse. The RF pulseis a circular polarized electro-magnetic wave with a circular polarized magnetic fieldcomponent B+

1 . The resulting magnetization is spatially encoded and the associatedmagnetic fields are measured. This thesis focuses on the spin excitation part of the MRIexperiment.

To increase signal-to-noise ratio (SNR), improve image contrast and introduce newcontrasts, a push towards high static magnetic flux densities of B0 = 7T or higher tookplace in the past years. B0 is later referred to as static magnetic field. These fieldstrengths render several assumptions invalid that worked reliable at lower fields. Inparticular, the increased Lamor frequency reduces the wavelength to be similar to thesize of the human head. This causes highly inhomogeneous RF field distributions thatcounteract the expected SNR gain and contrast improvements.

To compensate for inhomogeneities of the RF field, specialized RF pulses are availableon conventional single channel transmit systems. To gain new degrees of freedom, a newtechnology was introduced to spatially vary the RF field distribution by superposingRF fields from different RF antennas at different spatial positions. The transmit arrayallows to independently control the driving voltage of each transmit channel to create atemporally and spatially varying RF field.

Based on the transmit array technology, spin excitation techniques have been de-veloped to tailor the spatial excitation pattern. This is typically achieved by a linearapproximation of the Bloch equation that describes magnetization dynamics. This ap-proximation holds true for small flip-angle excitations.

In contrast to other imaging techniques based on X-Rays, there is no evidence thatMRI causes a health risk. However, the RF exposition of the patient can cause tissuedamage due to heating if the local RF power deposition is not limited properly. Dueto the increased Lamor frequency that also increases the RF power necessary to excitespins, the issue of patient heating is much more pressing at high fields. Additionally, theability of the transmit array to spatially vary the magnetic field component also affectsthe electric field component. This complicates the supervision of RF exposure to thepatient.

5

In this work, RF pulses for high flip-angle excitation are developed that compensatefor B+

1 inhomogeneities and reduce localized patient heating. For high flip-angles, non-linear effects of the Bloch equation are pronounced and exploited in combination with atransmit array to achieve the spatially tailored magnetization excitation.

After a short introduction to the basics of MRI imaging and RF pulse design forparallel transmission, the experimental setup used to validate the developed excitationtechniques is shown in chapter 3. This includes the RF supervision system needed toguarantee patient safety.

Chapter 4 introduces the the optimization techniques developed to calculate 5 novelexcitation methods for parallel transmission. A non-linear RF pulse optimization toolboxthat is able to calculate high flip-angle RF pulses is shown. The method was prototypedas part of the diploma thesis [28,36]. In this thesis, the method was improved to optimizenovel RF pulse types. For practical workflow reasons, one important design criteria wascalculation efficiency.

Excitation and refocussing pulses for up to 180 flip-angle are shown. These paralleltransmit (pTx) Composite Pulses consist of several conventional sub-pulses that create anon-trivial magnetization trajectory on the Bloch sphere to mitigate B+

1 inhomogeneity.These pulses were developed for non slice-selective and slice-selective excitation.

A novel approach extends Composite Pulses by incorporation the principles of slice-selection in two dimensions. A 2D selective excitation is achieved by exciting two ortho-gonal slabs repeatedly after each other with carefully chosen excitation phases.

In contrast to spin excitation techniques introduces so far, a new approach for signalsuppression in the presence of RF field inhomogeneities is shown. Based on conventionalchemical shift selective suppression pulses (CHESS), this technique is combined with atransmit array resulting in a novel suppression mechanism where each sub-pulse achievesgood suppression only at different spatial regions.

To minimize localized patient heating, the nonlinear optimization routine was exten-ded to optimize the electric field distribution in addition to the flip-angle distribution.Another approach to minimize RF heating is to prolong the RF pulses which reducesRF power. However, long RF pulses are prone to B0 inhomogeneities as the bandwidthof such pulses is reduced. Local frequency offsets consequently lead to signal voids inthe image due to reduced excitation flip angle. In combination with the transmit array,prolonged RF pulses are optimized to restore signal over the full field of view.

Chapter 5 contains results and discussions for all spin excitation techniques introducedabove. Additionally, the performance of the RF supervision system is discussed.

The thesis closes with a set of conclusions. The appendix contains a detailed descrip-tion of the local SAR supervision concept. Due to the high power requirements of highflip-angle RF pulses shown in this thesis, there is a need for a non-conservative supervi-sion of patient heating. Currently available commercial Transmit array systems do notprovide a local SAR supervision of any kind. Therefore, in-vivo scanning is only possibleusing highly conservative RF power limits. As part of this thesis, a local SAR super-vision concept was realized to allow the safe application of high RF power for in-vivohuman scanning.

6

2 Theory

Magnetic resonance imaging is based on the magnetic moments associated with thenuclear spins present in a static magnetic field B0. The most commonly used nucleusfor imaging is the proton of water.

The magnetic moment that is aligned to the external static magnetic field can berotated on the Bloch sphere using short RF pulses at the Lamor frequency. This cancreate an excitation of the magnetization M with a flip-angle α = cos−1(Mz). The Blochequation [5] describes magnetization dynamics depending on the external magnetic fieldvector B in a frame rotating at the Lamor frequency:

M(t) = γM(t)×B(t)− exMx(t)

T2− ey

My(t)

T2− ez

Mz(t)− 1

T1(2.1)

where γ is the gyromagnetic factor and T1 and T2 are tissue specific relaxation constants.Spins can be excited by a circular polarized high frequency magnetic field componentB+

1 (x) that is the transversal component of B. B+1 (x) is the spatial distribution of the

transmitter sensitivity of a transmit coil. B−1 is the corresponding receiver sensitivity.

Due to the limited bandwidth of applied RF pulses, a simple spatial selective excitationcan be achieved by shifting the Lamor frequency of spins not to be excited outside of theRF pulse bandwidth. This is possible using linear magnetic gradient fields. For smallflip-angles, a Fourier relation can be established between the RF pulse waveform u(t)and the spatial distribution of the excitation m(x) [53]. The field gradients g(t) are usedto travel through the Fourier space (k-space) and deposit RF energy at specific k-spacelocations:

m(x) ∝T∫

0

u(t)eik(g(t))xdt (2.2)

2.1 Parallel transmission at high fields

At high B0 field strength of 7T or above and the associated high RF frequencies of300MHz or above, the spatial distribution of the B+

1 fields created by the RF coil isinhomogeneous in human tissue [39]. This was first found at 4T by [2,6]. The reason isa wavelength that is comparable to the object size. This reduces image quality in termsof contrast and SNR. To compensate for this effect, the Fourier relation between RFpulses and spatial magnetization distribution introduced above can be used. However,this approach requires very long RF pulses.

7

To increase the degrees of freedom to control the spatial distribution of B+1 , a novel

technology was introduced that allows to control several transmit RF coil elements inde-pendently [1, 13, 21]. Currently, transmit arrays with eight channels are most common.Assuming linear independent B+

1 profiles increases the number of degrees of freedomavailable for B+

1 mitigation by a factor of eight.

RF shimming, a straight forward method to control the transmit array applies asubject-specific static amplitude and phase scaling factor to each transmit channel. Allapplied conventional RF pulses are scaled by this factor to reduce B+

1 inhomogeneity[40,63]. This technique can be combined with specialized image reconstruction methodsto further increase image quality [49].

On the other hand, specialized RF pulses can be calculated for each transmit channel.In combination with linear field gradients, almost any magnetization distribution canbe created. The transmit array allows to accelerate the resulting pulses. This tech-nique is similar to an accelerated image acquisition method called SENSE [54,55] and isconsequently called Transmit SENSE [42,60].

RF pulse design

The target of all RF pulse design approaches is to control the creation of a spatial distri-bution of magnetization as flexible as possible. The foundation of many RF pulse designtechniques is the Fourier relation between RF pulse shape and spatial magnetizationdistribution shown in figure 2.2. However, this approach requires homogeneous B+

1 andB0 distributions to work properly. Additionally, only one transmit channel is supported.

To overcome this issue an iterative approach was proposed [65] where the originalFourier relation is time-discretized to form a system of linear equations. This system ofequations can then be solved for example by using the LSQR algorithm [51] as shownin [67]. This allows to account for inhomogeneous B+

1 and B0 distributions. Later, theconcept was improved to support transmit arrays [26].

While these approaches were initially used to create complex magnetization patters bycovering large areas of k-space, further developments also focused on simpler approachesto just mitigate B+

1 inhomogeneities. Therefore, multiple conventional sub-pulses areplaced at selected k-space positions [56]. Only 2 to 10 k-space positions are necessary toachieve good results depending on the application. This approach was successfully shownusing transmit arrays with two or 8 channels at field strengths of 3T and 7T [57,59,62].Due to the sparse k-space coverage, methods were propose to include the selected k-space position to the optimization process [10, 66]. These approaches combine a linearmodel of the Bloch equation with the non-linear problem of optimizing k-space positions.Another hybrid approach using the approximated Bloch equation is local SAR optimizedpulse design [27]. Here, local SAR described by quadratic forms is added to the problem,creating a convex optimization.

All approaches discussed so far rely on a linear approximation of the Bloch equation,that limits the maximum target flip-angle. Additionally, non-linear features of the Blochequation are not modeled. These can be used to enlarge the optimization space andimprove effectivity.

8

A first step towards optimizing the full Bloch equation was introduced in [58]. There,a magnitude least square optimization is performed, optimizing only the target mag-netization amplitude and not the magnetization phase. Calculating the amplitude is anon-linear operation. The optimization is performed by repeating the linear optimiz-ation and adapting the target magnetization iteratively. A similar, but more generalapproach was shown in [25] to calculate high flip-angle RF pulses. The error betweenlinear approximation and Bloch equation is compensated by a new linear approximationin each iteration. The first full non-linear optimization method for high flip-angle pTxRF pulse optimization was shown in [64]. It is based on optimal control theory. However,computation times for this approach are rather long.

The non-linear optimization algorithm proposed in this thesis is based on a trust-regionsub-space solver for systems of non-linear equations [16, 46]. Calculating the functionvalue and Jacobi matrix of this system of equations is performed on a graphics processingunit which is a highly parallel processor. This approach to speed up calculations becomesincreasingly popular [18].

To evaluate the optimization results, a Bloch simulator [61] is used that calculates themagnetization vector after a RF pulse based on the full Bloch equation.

B+1 and B0 mapping

All RF pulse optimization techniques rely on the spatial distribution of the B+1 fields

per unit voltage of all transmit channels, called B+1 maps. Additionally, the spatial

static magnetic field distribution (B0 map) needs to be known. The B+1 maps are meas-

ured using a pre-saturation technique [20]. There, a reference GRE [37] image sref(x)is measured followed by one pre-saturated GRE image sprep(x) per transmit channel.The pre-saturation is performed using a single transmit channel, while the GRE imageacquisition uses all transmit channels in a circular polarized mode. The pre-saturationencodes the flip-angle profile of the transmit channel in the longitudinal magnetization.Assuming that the measured signal is proportional to the longitudinal magnetization,the division of the pre-saturated image by the reference image is proportional to thenormalized longitudinal magnetization:

α(x) = cos−1

(sprep(x)

sref(x)

)(2.3)

where α is the flip-angle distribution of the pre-saturation pulse. This equation showsthat the theoretical dynamic range of this techniques is 0 to 180. Due to the inversecosine, the SNR for small flip-angles and large flip-angles is reduced significantly, leavinga dynamic range of approximately 20 to 160. This technique only allows to measureB+

1 amplitudes. B+1 phases are measured by acquiring one GRE image per transmit

channel with one transmit channel active.

Because the dynamic range of the B+1 field of individual transmit channels is much

larger than this, eight linear independent combinations of all transmit channels aremeasured that reduce the overall dynamic range. The flip-angle maps of individual

9

channels are then restored in a post-processing step:

αindiv(x) = X−1αcomb(x) =

0.3 1 11 0.3 11 1 0.3

−1

αcomb(x) (2.4)

where αindiv(x) is a vector of individual flip-angle maps, αcomb is a vector of flip-anglemaps with combined channel excitation and X is a matrix describing the linear combin-ations of the transmit channels. This B+

1 map acquisition and reconstruction techniqueis called interferometric B+

1 mapping [8] [17].

To acquire a B0 map, a second echo is added to the GRE image acquisitions used tomeasure B+

1 phase and the B0 map is calculated from the image phase differences:

∆B0 = ∆φ/γ (2.5)

Composite Pulses

Composite pulses [44] are a class of RF pulses to mitigate B+1 inhomogeneities by ex-

ploiting non-linear effects of the Bloch equation. They consist of a set of rectangularsub-pulses with different flip-angle and excitation phase. The following example showsa 180 Composite Pulse using a flip-angle/excitation phase notation. E.g. 900 denotesa 90 flip-angle pulse with a 0 excitation phase.

900 18090 900 (2.6)

A variation of the target flip-angle is compensated by the pulse as follows. The firstsub-pulse moves most of the magnetization to the transverse plane, leaving a minorlongitudinal magnetization component of ±δ. The second sub-pulse is applied with anexcitation phase similar to the magnetization phase that approximately changing thesign of δ and leaving the transverse magnetization untouched. Now, magnetization thatwas flipped by more (less) than 90 in with the first sub-pulse has now a flip-angle ofless (more) than 90. The final sub-pulse is identical to the first sub-pulse. With thetransformation achieved by the second sub-pulse, the final magnetization is closer to180 compared to a conventional 180 pulse. The final longitudinal magnetization of theshown Composite Pulse is given as:

Mz =

[[cos(2α)

[1− sin(α)

]]− sin(α)

]= cos(2α)︸ ︷︷ ︸

conventional

−[sin(α)

[1 + cos(2α)

]]︸ ︷︷ ︸B+

1 homogenization

(2.7)

with α = 90±δ. The B+1 homogenization term is always larger than 0 and thus pushing

the magnetization closer to 180 (Mz = −1) as a conventional RF pulse if δ 6= 0. Theseeffects are clearly non-linear and cannot be modeled using RF pulse design techniquesbased on a linear approximation of the Bloch equation. A extension of Composite Pulses

10

to a transmit array without considering B0 effects and slice-selection was shown in [14]in a simulation-only study. In the following, all pTx pulses that consist of sub-pulseswith amplitude and phase variations between sub-pulses are called Composite Pulses.

Flexible control over the magnetic field distribution of a multi-channel transmit RFcoil as it is achieved with a transmit array also implies similar degrees of freedom onthe electric fields created by that RF coil. This complicates RF supervision for patientsafety.

2.2 Patient heating

The electric fields produced by a transmit RF coil cause heating in human tissue andtherefore may be a serious instantaneous risk for patients. Because temperature is hardto model and measure in vivo in the presence of human thermo-regulations, the specificlocal power absorption rate (local SAR) is supervised instead. Local SAR is defined asthe average power dissipated within a 10g volume of human tissue over a given timewindow. IEC regulations [41] do not allow localized temperature increase above 1C.This limit is transformed to IEC local SAR limits shown in table 2.1 that are relevantfor transmit array RF coils. A 10 second and 6 minute moving average power windowhas to be supervised. It is distinguished between a normal operation mode and a firstlevel controlled mode. The latter allows higher local SAR limits and in turn requires amore extensive monitoring of the patient.

10s interval 6 minute interval

normal 20W/kg 10W/kgfirst level controlled 40W/kg 20W/kg

Table 2.1: IEC local SAR limits

The localized RF power deposition created by conventional RF coils driven by a singleRF signal is given by:

SARlocal(xi, t) = σ(xi)u∗(t)u(t) u ∈ C1, σ ∈ R (2.8)

where u(t) is the complex RF voltage in units of [V ] and σ(xi) is the conductivity permass of a 10g volume at position xi out of nx positions i in units of [S/kg].

This allows a simple supervision procedure that only depends on the applied RFvoltage over time, because the spatial power distribution is not changed by the appliedRF voltage. Thus, to make sure local SAR limits are never exceeded, only the volumewith maximum k has to be supervised:

max(SARlocal(xi, t)

)= max

(σ(xi)

)|u(t)|2 (2.9)

The use of multiple transmit coil elements with individual excitation allows to manip-ulate the spatial local SAR distribution using adapted RF pulse shapes [68]:

11

SARlocal(xi, t) = u∗(t)S(xi)u(t) u ∈ Cn, S ∈ Cn×n (2.10)

where u(t) is a vector of complex RF voltages of all transmit channels and S(x) is theelectrical conductivity per mass matrix which is positive semi-definite.

In contrast to the conventional case, the RF phase relation between transmit channelsis now relevant for the local SAR distribution. Equation 2.9 is a special case of equation2.10 where S = σ and n = 1.

The safe supervision of RF heating based on this more complex equation is a dominantlimiting factor for high performance pTx applications.

Current transmit array systems as provided by the vendors do not provide any mech-anism to supervise local SAR. To be able to stay within regulatory limits in this case, aworst-case transmit phase combination must be assumed for the whole examination. Asshown in [15], this worst-case situation can cause a maximum local SAR that is morethan an order of magnitude higher than the local SAR created by a conventional singlechannel RF coil. With such RF power limitations, patient scanning is extremely limited.Additionally, any efforts taken to minimize the local SAR by optimizing the applied RFpulses can not be leveraged.

To overcome this issue by calculating exact local SAR values from equation 2.10,the complex RF voltage must be measured [7, 23, 24]. Additionally, the conductivitymatrices S must be obtained using electromagnetic field simulations [48]. Therefore, adetailed RF coil model together with multiple human body models [11] are simulated.Multiple body models are used to cover a variety of possible subjects. Using this data, Smatrices are calculated for each 10g volume within the exposed body parts. These 10gvolumes have to be averaged using a standardized averaging algorithm [41].This resultsin a number of S matrices in the order of one million per body model.

Evaluating equation 2.10 for all 10g volumes nx of the subject exposed to the RF isnot possible in real-time using current computing technology. However, it is necessaryevaluate local SAR at the time it is applied to guarantee patient safety. To overcome thisissue, a compression method was developed [19,43] to increase calculation efficiency andreduce the number of matrices to evaluate to nv (j = 1...nv). The algorithm calculatesa set of matrices Vj which is orders of magnitudes smaller than the number of all10g volumes, but yields peak local SAR values that are never underestimated and onlyslightly overestimated by a predefined maximum overestimation factor ε. The calculatedVOPs need to fulfill the following equation:

u(t)S(xi)uH(t) ≤ u(t)Vju

H(t) ≤(u(t)S(xi)u

H(t))

(1 + ε) ∀u(t) (2.11)

where nx nv and ε is the maximum overestimation.These virtual observation points (VOP) combine multiple clusters of 10g volumes to a

single matrix Vj . This allows to reduce the number of quadratic forms that need to bevalidated from millions to a few hundreds and thus allows peak local SAR calculationwithin micro-seconds based on equation 2.10.

As part of this thesis, an online local SAR calculation and supervision system usingmeasured complex transmit array RF pulse shapes was built. Local SAR is calculated

12

in near real-time based on pre-calculated compressed electric field models [3, 4] of thecurrent RF coil and patient. Calculations were performed using the FDTD method.u(t) is measured using as many components as possible from the existing multi-channelMRI system.

Additionally to local SAR calculations based on measured complex RF pulses, safetyconcepts typically provide a SAR prediction strategy. Only sequences that will notexceed SAR limits are allowed to be executed. To function properly, both look-aheadand online supervision must give valid results. Additionally, a look-ahead system needsto know the exact sequence timing of all executed sequences to function properly.

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3 Experiments

Experiments are performed using a Magnetom 7T (Siemens AG, Erlangen, Germany)MRI scanner. This corresponds to a Lamor frequency of 297MHz. The system has threemain components: A superconducting magnet creating the static magnetic field, a fieldgradient system that creates linear magnetic field gradients for spatial encoding and aRF system to excite spins and receive the MR signal.

The main focus of this thesis is the transmit RF subsystem. The standard transmitRF system is replaced by a transmit array (TX Array) consisting of eight RF channelsthat can be controlled completely independent. A schematic is shown in figure 3.1.

Figure 3.1: Transmit array architecture including hardware for local SAR supervision. Fourtransmit channels are shown.

The intermediate frequency RF signal and the timing is created digitally by eightindependent measurement control units (MCU). It is converted to an analog signal,mixed to MR frequency using a 295MHz local oscillator and amplified by the RF poweramplifier (RFPA) (1). These signals are transmitted to a specialized multi-channel RFcoil (2) to excite the spins. The MR signal received with the RF coil is sent to thereceivers where the signals are mixed down and are available as digital base-band signalsat the image reconstruction computer (6).

The standard TX Array system is an experimental prototype to explore the possibil-ities of parallel transmission. It contains eight independent forward RF power meters.They can be configured to limit the average RF power transmitted within given timeperiods to guarantee patient safety in terms of global SAR. These units are broadbandand cannot measure RF phase. This supervision system requires very conservative powerlimits due to the unknown phase relationship between transmit channels. To overcome

15

this issue, these components were replaced by a local- and global-SAR supervision systemdescribed in the following section.

All 7T MRI measurements were performed at the Erwin L. Hahn Institute for MagneticResonance Imaging, Essen, Germany. 3T measurements were performed on a MagnetomTIM Trio equipped with a transmit array.

3.1 Local SAR supervision

Figure 3.1 shows the hardware architecture of the proposed online local SAR supervisionsystem. To supervise local SAR based on measured RF pulse shapes, the precise RFamplitude and relative phase at a reference position (RF coil plug) needs to be known atany time. This work was presented at the Annual Scientific Meeting ISMRM 2013 [29,33]

To supervise this RF waveform transmitted by the RFPA (1), directional couplers(3) are inserted in the transmit chain close to the RF coil to provide small signal RFwaveforms proportional to the forward and reflected complex RF signal. This enablesthe calculation of local SAR because therefore, the RF transmitter phase is essential.

These signals are fed to the standard MR receivers (4) through a switch matrix (5).This allows using the same receiver circuitry for RF supervision and MR signal reception.The digitized base-band signal is available at a sample rate of 1us (±500kHz bandwidth)at the image reconstruction computer (6). There, the local SAR is calculated from pre-calculated electromagnetic field simulations of the RF transmit coil and the measuredRF waveforms.

Figure 3.2 shows the eight directional couplers that are placed at the back of themagnet (a) and the switch matrix which is part of the electronics box inside the magnetroom (b).

The image reconstruction computer is not a real-time component. Therefore, the aFPGA watchdog logic is used to supervise the image reconstruction computer and createa pseudo realtime system. The logic makes sure that a system shutdown occurs if timinglimits are violated.

This logic, the directional couplers and the switch are the only hardware componentsadded to the standard MRI system. All other needed components are reused to reduceimplementation efforts and costs.

Based on equation 2.10, the local SAR must be calculated for a 10s and 6min averageaccording to IEC regulations:

SARlocal(x, tsw, t) =∑ij

t∫t−tsw

ui(t′)u∗j (t

′) dt′

Sij(x) (3.1)

where tsw is the moving average window size, e.g. ten seconds or six minutes. t is thetime where local SAR is evaluated. The equation was reordered to separate temporalfrom spatial quantities. This allows to evaluate the compute intensive temporal integralbefore applying the result to the electrical field model data Sij(x). This data is providedto the calculation routine in a compressed form using virtual observation points (VOP).

16

Figure 3.2: Directional couplers (a) and switch matrix (b) placed at the back of the magnet

The complex RF waveforms measured are essential for evaluating equation 3.1. Thisdata contains systematic errors caused by the receiver hardware and cabling. The MRreceivers have non-predictable time delays in the order of 0ns to 200ns. Additionally,relative RF phase errors occur due to imperfect cabling. 2cm cable length differencecreate a relative phase deviation of 11. Also, a phase and amplitude calibration factorneeds to be obtained to convert the signal measured by the receivers to complex RFwaveforms at the reference plane, the coil plug. Therefore, a calibration and correctionprocedure is required. This results in a transmit channel and frequency dependentcalibration function c(ω, ...).

In summary, the following system imperfections were observed and have to be correc-ted for reliable local SAR supervision:

• Receiver NCO delays between channels τ2 (hardware specific)

• Receiver signal path delays between channels τ1 (measured after system boot)

• Receiver phase differences φrx (measured once a year)

• Receiver amplitude gain A (measured once a year)

All these errors can be written in a single complex calibration function c(ω, ωNCO)depending only on the signal frequency ω and the NCO frequency ωNCO:

17

ci(ω, ωNCO) = Aeiφeiτ1(ω−ωref)eiτ2(ωNCO−ωref)

= Aei(φ′+τ1ω+τ2ωNCO) (3.2)

where φ′ = φ− ωref(τ1 + τ2).

To allow a reliable RF supervision, the receiver electronics have to be able to measureall frequency components that can be transmitted. Because the receivers only deliverreliable results within a bandwidth of approximately 500kHz, the measurement is abortedif signal is detected outside this frequency band.

A detailed description of these system imperfections and the calibration and correc-tions methods can be found in the appendix.

3.2 Pulse design evaluation workflow

RF pulses for the transmit array are calculated based on patient and system specificB+

1 and B0 maps. Thus, the B+1 and B0 mapping procedure is an integral part of the

workflow used to validate the RF pulses.

All calculated RF pulses are validated using the following workflow. Depending onthe RF pulse properties, a quantitative or qualitative validation method is chosen.

1. After the TX Array system is successfully started, safety test procedures are per-formed to make sure all safety relevant components are working as expected.

2. An adjustment procedure is run to calibrate the receivers for local SAR supervision.

3. The RF coil is connected and a phantom or a human subject is placed in thescanner.

4. A B+1 mapping sequence with integrated B0 mapping is run to get the patient

specific input data for the pulse design algorithm. A pre-saturation based techniqueis used used [20]. A interferometric reconstruction is used [8].

5. The anatomical planning of the regions of interest is done using an integratedtask-card. This data, including B+

1 and B0 maps, are exported to the pulse designcomputer.

6. RF pulses are calculated for the specific application and saved on the scanner.

7. The target sequence is run using the pre-calculated RF pulses.

If the subject is not moved, further measurements are performed by repeating steps5 to 7. Otherwise, steps 3 to 7 are repeated. The following target sequences are run totest and validate the calculated RF pulses:

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• Pre-saturation based flip-angle mapping: This sequence is used to measure the flip-angle distribution achieved by the calculated RF pulse. This allows a quantitativevalidation of RF pulse behavior if flip-angles are in a range of 30 to 160. Outsidethis range, the mapping accuracy degrades significantly.

• Gradient recalled echo sequence: This sequence is used to get a qualitative val-idation of calculated RF pulses if flip-angles are small. Using this measurementseveral other effects than RF pulse flip-angle are visible such as receive sensitivityeffects and relaxation effects.

• Spin-echo sequence: This sequence is used to get a qualitative validation of cal-culated RF pulses if flip-angles are large. Therefore, the calculated RF pulse isused as a refocussing pulse. Using this measurement several other effects than RFpulse flip-angles are visible such as receive sensitivity effects, relaxation effects andimperfections of the excitation RF pulse.

All computations of RF pulses for parallel transmission were done on a standalonecomputer equipped with a dual-core Intel Xeon processor running at 3.4GHz and aNvidia GeForce GTX285 graphics processing unit with 240 processing cores running at1.476 GHz. This gives a peak floating point performance on the graphics processing unitof 708Gflop/s according to Nvidia specifications [47]. The pulse calculation computeris connected to the MRI scanner using ethernet for data exchange. Prior pulse applica-tion, they are checked using a Bloch equation simulator to verify the correctness of theoptimization algorithm and to quantify residual errors in the achieved magnetizationpattern.

Either a spherical phantom with a diameter of 20cm or a human head is imaged. Thephantom was filled with a gel that has a conductivity similar to the human head [22].

An eight channel transmit/receive head-coil [50] is used. The transmit elements areorganized as one ring of eight stripline elements.

19

4 Methods

All pTx pulse applications shown later are based on a generic high flip-angle non-linearpulse design algorithm introduced here. This method was first shown in the Diplomathesis. Here, after a short introduction, modifications and improvements of this methodare shown. The following sections show different applications of this pulse design toachieve flip-angle homogenization, signal suppression or SAR reduction. The modifica-tions necessary to achieve these goals are shown in the corresponding sections.

The proposed pulse design method is based on the solution of the Bloch equationwithout relaxation.

m(t) = γm(t)× b(t) (4.1)

where m(t) is the magnetization vector, b(t) is the magnetic field vector with threecomponents b(x),b(y),b(z) and γ is the gyromagnetic factor. The equation is shown in therotating frame of the radio frequency field at the Lamor frequency.

To solve the Bloch equation, its differential formulation shown above is discretized intime. Each time step is assumed to have constant magnetic field. In this case, the Blochequation can be solved analytically. The solution is a rotation of the magnetizationvector in three dimensional space. The magnetization is rotated around an axis parallelto the magnetic field vector b and an angle of α = γ∆t||b||2 with a time-step length of∆t. By concatenating these rotation operations, a coupled literal system of non-linearequations can be formulated to model an RF pulse with constant sub-pulses. This isdone for every voxel in the region of interest for spins at any frequency of interest.

The degrees of freedom of the pulse design problem are complex RF scaling factorsuct for a transmit channel c out of Nc, a given time step t out of Nt with a duration of∆tt. Every time step within the RF pulse can then be described as a rotation Rtxf =R(uc=0...Nc,t, ff , B

0x,gt,xx, Ax,c=0...Nc) with an axis and angle that depends on uct and

the system parameters spin frequency ff, static magnetic field inhomogeneity B0x, the

gradient field amplitude gt, the location of the voxel xx and the complex RF sensitivitymaps Axc for each channel. f identifies one spin frequency out of Nf frequencies and xidentifies one voxel out of Nx in the region of interest.

For a given voxel and frequency, the resulting total rotation Rtotxf is given by the

concatenation of the rotation operations of every time span.

Rtotxf =

∏t=0...Nt

Rtxf (4.2)

where Rtotxf and Rtxf are rotation quaternions.

21

The resulting magnetization Mxf is then given as a function of Rtotxf and the initial

magnetization M0x :

Mxf =(Rtotxf

)∗M0

xRtotxf (4.3)

The 3D magnetization vector is given by the imaginary part of the magnetization qua-ternion Mxf.

In addition to the system of equations describing magnetization dynamics, other equa-tions can be added directly to introduce constraints on the RF pulse to be designed. Inthe simplest case, an equation describing global forward RF power can be added. Thiscan be generalized to global or even local SAR constraints [19]. Because the used op-timization routine described in this work is able to handle non-linear equations, theseadditional equations do not change the optimization procedure at all. These additionalconstraints are a function of the RF pulse: C(uct).

The final cost function is solved using a numerical solver for non-linear coupled systemsof equations to yield an RF pulse uct for a given target magnetizationMdes

xf . Here, a trust-region sub-space method based on the Levenberg-Marquardt algorithm is chosen [9]. Thecost function for this solver can be stated as:

minuct

∑

x=0...Nx,f=0...Nf

(Mxf (Rtot

xf ,M0x )−Mdes

xf

)2+ C(uct)

2

(4.4)

This method searches iteratively for a local minimum of the function value. In eachiteration, a linear approximation of the problem is calculated for the neighborhood ofthe current RF pulse. At this point, the gradient and Newton step are obtained to spana two dimensional subspace within the solution space. There, an update for the currentRF pulse is calculated within a neighborhood of the current point in the solution space,the so-called trust-region. The updated RF pulse is used in the next iteration.

For each iteration, the following computation steps are done:

1. Calculate the magnetization created by the current RF pulse using equation 4.5.

2. Calculating the Jacobi matrix Jxfct with NxNf rows and 2NcNt columns for thecurrent RF pulse

3. Calculate the gradient v(1)ct =

∑xf JxfctMxf of the non-linear system of equations

for the current RF pulse

4. Calculate the approximate Newton step v(2)ct by using a well-known linear solver

(CGLS, LSQR). ∑xfc′t′

(Jxfct)∗Jxfc′t′v

(2)c′t′ = −

∑xf

(Jxfct)∗Mxf

5. Find a trust-region solution in a sub-space that is spanned by the newton step andgradient. This is a correction for RF that is likely to result in a magnetization

22

pattern that is closer to the desired magnetization pattern for the new RF pulse.The norm of the correction is limited by a dynamically adjusted upper bound, thetrust-region.

This process starts with a predefined RF pulse.

The parts of the optimization procedure fall into a number compute intensive partsand a number of easy to calculate parts. If possible, the compute intensive parts areimplemented on a highly parallel compute unit like a graphics processing unit. All otherparts are straight forward implementations using the C++ programming language.

The most compute intensive parts of the optimization process are the calculation ofthe magnetization of a given RF pulse (1), the Jacobi matrix calculation (2) and thematrix-matrix multiplication associated with the Newton step calculation (4).

Steps 1 to 4 shown above are described in detail in the following sections. There, themathematical formulation and an efficient implementation is shown. Step 5 is part ofthe well known trust-region method. Details can be found in the literature [9].

Solving the Bloch equation

To solve the Bloch equation for a given RF pulse consisting of constant sub-pulses,the concatenation of associated rotations needs to be calculated. This can be doneanalytically. To yield a high performance and numerical stability, quaternions are chosento formulate the rotations operations [38]. Multiplying two rotation matrices needs 45multiplications and additions, while the multiplication of two quaternions needs only 28multiplications and additions. Also, creating a rotation matrix from a magnetic fieldvector needs 22 operations, while the rotation quaternion can be created using only 4operations. Any rotation is then given by:

Rtxf (btxf ) = cos(γ||btxf ||2∆tt/2

)+

sin(γ||btxf ||2∆tt/2

)||btxf ||2

(ib

(x)txf + jb

(y)txf + kb

(z)txf

)

where i,j and k are the imaginary units of quaternions.

The magnetization Mxf resulting from the effect of an RF pulse on an initial magnet-ization M0 is given by:

Mxf =(Rtotxf

)∗M0xR

totxf (4.5)

where ()∗ denotes the conjugate quaternion. Repeating this for all voxels and frequen-cies, the resulting system of coupled non-linear equations describes the full nonlinearpTx pulse design problem.

23

M11 =(Rtot

11

)∗M0xR

tot11

M21 =(Rtot

21

)∗M0xR

tot21

...

MNxNf=(RtotNxNf

)∗M0xR

totNxNf

(4.6)

These equations are implemented on a graphics processing unit. Every compute coreof these highly parallelized units calculates the magnetization for one voxel. Becausethese units are built for single precision floating point operations, their double preci-sion capabilities are limited. However, concatenating a lot of rotations gets numericalunstable using single precision operations. The cumulated rotation is denormalized dueto these instabilities. Therefore, after concatenating a few rotations, the cumulatedrotation is renormalized:

∏t=0...

Rtxf =

∏t=0...Rtxf

|∏t=0...Rtxf |

(4.7)

This restores numerical stability. The results of this calculation is reused when calcu-lating the Jacobi matrix.

Table 4.1 shows all compute intensive steps of the process and how their calculationtime depends on the problem size, i.e. Nt,Nx,Nc and Nf . This is shown for straight-forward and optimized implementations. Straight forward means a finite difference ap-proach for the Jacobi matrix and a normal matrix matrix multiplication without anyproblem-specific optimizations.

part straight forward optimized

Bloch simulation NtNp NtNp

Jacobi calculation N2t NpNc NtNp

Matrix-matrix multiplication N2t Np N2

t Np/4

Table 4.1: Complexity of compute intensive parts of the optimization process

Jacobi matrix calculation

For the system of equations describing magnetization dynamics (eq. 4.6), the Jacobimatrix can be calculated analytically. This yields advantages in compared to a straightforward finite differences method. This sections shows a method of calculating theJacobi matrix with a calculation time that scales linearly with the number of time stepsNt compared to a quadratic dependency when using finite differences.

The full Jacobi matrix is defined by:

24

Jxfct =

[∂Mxf

∂<utc,∂Mxf

∂=utc

]where Jxfct is a matrix with NxNf rows and 2NcNt columns. Because utc is a complex

quantity, the Jacobi matrix has to be calculated for the derivative with respect to thereal (<) and imaginary (=) part of uct. The following equations are similar for bothcases. Thus, the derivative is written with respect to uct, which represents both, realand imaginary parts. Jxfct represents the corresponding part of the Jacobi matrix. Theanalytic Jacobian including equation 4.5 is:

Jxfct =∂Rtot

xf

∂utcM0x

(Rtotxf

)∗+RtotxfM

0x

(∂Rtot

xf

∂utc

)∗(4.8)

This is a complex non-linear equation which includes the derivative of a concatenationof rotations that depends on the voxel position, the resonance frequency, the time-stepand the transmit channel. However, not all parts of this equation are non-linear.

Going back from the derivative to magnetization dynamics itself, the problem can beseparated into two different physical processes for each time-step. First, the magneticfields of the different transmit coils are superimposed at a given position. This is alinear process. The magnetization then reacts on the total magnetic field accordingto the Bloch equations. This is a non-linear process where the magnetization is notaware of the individual transmit channels. That means, all processes related to paralleltransmission are linear.

For a given time-step, voxel and frequency, the Jacobi matrix shown in equation 4.8has 2Nc elements. Two for each transmit channel. It is shown that the relation betweenthese elements is purely linear. The non-linear part of these elements is independent ofthe transmit channel and thus, equation 4.8 can be written

Similar to magnetization dynamics shown above, it is shown that all non-linear com-putations for these components are identical and the difference between them is given bya linear relation. This allows to reformulate equation 4.8 using a reduced Jacobi matrixJ redxft where all non-linear calculations are done:

Jxfct ≡∂Mxf

∂<utc+ i

∂Mxf

∂=utc= J red

xftSxc (4.9)

Note that the reduced Jacobian is independent of the transmit channel. To restorethe full Jacobian, the reduced Jacobian is weighted linearly by the spatial B+

1 sensitivitymaps.

This separation is possible because the derivative with respect to the transmit channelis a scalar value. As shown in equation 4.10, this derivative is enclosed by quaternions.Only scalar values are commutative with quaternions, and thus allow to move this deriv-ative to the end of this equation. A detailed derivation of these simplifications in shownin later in this chapter.

As shown in equation 4.18, the reduced Jacobi matrix contains only one computeintensive expression:

25

Figure 4.1: Derivative exchange process used for Jacobi matrix calculation

∂Rtotxf

∂uxt= Rl

txf

∂Rtxf∂uxt

Rrtxf (4.10)

where uxt can be <u1xt or =u1

xt. Rltxf and Rr

txf are defined according to equation4.16. This equation has to be calculated for every voxel, time-step and frequency. Asstated here, every evaluation needs Nt quaternion multiplications. This is not efficientand leaves a quadratic computation complexity in Nt. However, this equation is verysimilar for two neighboring time-steps. Figure 4.1 shows this similarity for time-step4 (upper line) and 3 (lower line) using 7 time-steps total. The gray part of the figurehighlights the difference between the Jacobi elements for these two time-steps.

Splitting each line into three parts which correspond to the three factors of equation4.10, allows to calculate every time-step recursively from the previous one. Only thederivative has to be calculated from scratch in every time-step. The two other partscan be modified with the rotation and inverse rotation corresponding to the derivative.These quantities are a byproduct of the calculation of the derivative.

Mathematically, the removal of one rotation from the left set of rotations Rl can beformulated as:

Rl(t−1)xf = Rl

txfR∗(t−1)xf (4.11)

The inverse rotation of R(t−1)xf is applied to Rltxf . Using quaternions, the inverse ro-tation quaternion is the complex conjugate of this quaternion. Appending an additionalrotation in front of the right set of rotations Rr is done using the following equation:

Rr(t−1)xf = RtxfR

rtxf (4.12)

Starting from the last time-step, the procedure to calculate the reduced Jacobi matrixfor all time-steps is shown in the following enumeration:

1. Start with t := Nt and set Rr = 1 and Rl = Rtotxf

2. Calculate the rotation Rtxf and and its derivative

3. Remove Rtxf from the left set of rotations using equation 4.11

26

4. Concatenate Rltxf , the derivative of Rtxf and Rr

txf to form the final derivative (eq.4.10)

5. Multiply Rtxf to the right set of rotations using equation 4.12

6. Set t := t− 1 and go to step 2 until t = 0.

This process needs 4 quaternion multiplications for each Jacobi element and no longerNt. Thus, the complexity of the total Jacobi calculation depends linear on Nt and Nx.Additionally, as shown above, these complex operations need to be performed only twiceper time-step, instead of 2Nt times.

In total, 3 steps need to be done for each time-step and voxel to get the full Jacobimatrix. First, the recursive procedure shown above is applied. Because of the quaternionmultiplication and the calculation of the derivative, this part takes approximately anorder of magnitude longer than the other two steps. Second, the reduced Jacobi matrixis calculated using equation 4.18. Finally, the full Jacobi matrix is restored by weightingthe reduced Jacobian with the B+

1 maps according to equation 4.9.

Derivative of a single rotation

To be able to compute the equations shown above, the derivative of a single rotation Rwith respect to the total RF field b1 applied needs to be calculated:

∂R

∂<b1 ,

∂R

∂=b1

The following use the total magnetic field vector which is defined as:

b = <

∑c

utcAxc

ex + =

∑c

utcAxc

ey + (γB0

x + gtxx + f/γ)ez (4.13)

where b1xt =∑

c utcAxc which is the complex weighted sum of the transmit voltage andthe spatial B+

1 sensitivity maps.

A rotation quaternion that rotates according to the Bloch equation around an axisparallel to the total magnetic field with an angle of α = γ∆t|b|/2 is written as

R = R0 + iR1 + jR2 + kR3 = cosα+ sinα/|B|(iBx + jBy + kBz

)For simplification, this calculation is done in a new coordinate system that is rotated

around the z axis. The new first basis vector x is chosen to be parallel to the RF field.The second basis vector y is orthogonal to x. The resulting derivative is then rotatedback to the original coordinate system.

The RF field in the new coordinate system is Bx = |Bxy| and By = 0. The derivativeneeds to be calculated for this point:

27

∂R0

∂By

∣∣∣∣∣0

=∂R1

∂By

∣∣∣∣∣0

=∂R2

∂Bx

∣∣∣∣∣Bx

=∂R3

∂By

∣∣∣∣∣0

= 0 (4.14)

∂R0

∂Bx

∣∣∣∣∣Bx

= −γ∆t

2

Bx|B|

sin(α)

∂R1

∂Bx

∣∣∣∣∣Bx

=1

|B|sin(α) +

B2xγ∆t

2|B|2cos(α)− B2

x

|B|3sin(α)

∂R2

∂By

∣∣∣∣∣0

=1

|B|sin(α)

∂R3

∂Bx

∣∣∣∣∣Bx

=γ∆tBxBz|B|2

cosα− BzBx|B|3

sinα

To get the derivative with respect to Bx and By, the product rule for derivatives leadsto this equation:

∂R

∂Bx=

∂R

∂Bx

∂Bx∂Bx

+∂R

∂By

∂By∂Bx

=∂R

∂Bx

Bx|Bxy|

− ∂R

∂By

By|Bxy|

=

Bx|Bxy|

∂R0

∂Bx

∣∣∣∣∣Bx

+ i∂R1

∂Bx

∣∣∣∣∣Bx

+ k∂R3

∂Bx

∣∣∣∣∣Bx

− j By|Bxy|

∂R2

∂By

∣∣∣∣∣0

(4.15)

The derivative with respect to By can be obtained analog. To get the rotation qua-ternion in the original coordinate system, it is rotated by the phase of the RF fieldaround the z axis (which has the imaginary unit k here)

∂R

∂Bx=∂R0

∂Bx+∂R3

∂Bxk +

(Bx|Bxy|

∂R1

∂Bx+

By|Bxy|

∂R2

∂Bx

)i

+

(− Bx|Bxy|

∂R1

∂Bx+

By|Bxy|

∂R2

∂Bx

)j

=Bx|Bxy|

∂R0

∂Bx

∣∣∣∣∣Bx

− k ∂R3

∂Bx

∣∣∣∣∣Bx

+ i

∂R1

∂Bx

∣∣∣∣∣Bx

B2x

|Bxy|2+∂R2

∂By

∣∣∣∣∣0

B2y

|Bxy|2

+ j

BxBy|Bxy|2

∂R1

∂Bx

∣∣∣∣∣Bx

− ∂R2

∂By

∣∣∣∣∣0

28

Again, the derivative with respect to By can be calculated analog.

Reduced Jacobi matrix

Starting from equation 4.8, Rtotxf needs to be calculated. This is a concatenation ofrotations with only one rotation depending on the parameter btc for a given time step:

∂Rtotxf∂uct

=

∏t′=0...t−1

Rt′xf

∂R(uct, ff , B0x, Axc)

∂uct

∏t′=t+1...Nt

Rt′xf

= Rl

txf

∂R(uct, ff , B0x, Axc)

∂uctRrtxf (4.16)

where Rltxf and Rr

txf are defined as the products of the rotations on the left and righthand side of the derivative. This equation shows that most parts of the Jacobi calculationare rotation operations R that are already calculated earlier in the optimization process.Only the derivative of a single rotation needs to be calculated additionally.

The problem further simplifies by separating the complex rotation concatenations fromthe individual transmit channels.

Therefore, the derivative of a single rotation is analyzed. During one time span, themagnetization vector is affected by the magnetic field vector btxf (see eq. 4.13) whichis assumed constant in each time step.

This shows that the rotation R depends on the weighted sum of the RF scaling factorsutc with the sensitivity maps Axc. b1xt is the total RF field seen by the spin system.That way, the derivative can be split up using the product rule and thus separate thedependency of the transmit channels.

∂R(uct, ff , B0x, Axc,xx,gt)

∂uct=∂R(b1xt, ff , B

0x,xx,gt)

∂<b1xt∂<(b1xt)

∂uct

+∂R(b1xt, ff , B

0x,xx,gt)

∂=b1xt∂=(b1xt)

∂uct(4.17)

The first factor on the right hand side of both summands of equation 4.17 is a qua-ternion, but the second factor is a scalar. Putting this simplification back to equation4.8 allows to move the scalar factor to the end of this equation because a scalar, andonly a scalar, is commutative to a quaternion:

∂Rtotxf

∂uct= Rl

txf

∂Rtxf∂<b1xt

Rrtxf∂<b1xt∂uct

+Rltxf

∂Rtxf∂=b1xt

Rrtxf

∂=b1xt∂uct

The resulting equation can then be integrated in equation 4.8 where the scalar men-tioned above is again moved to the end.

29

Finally, this can be used to define the reduced Jacobi matrix J redxft that is independent

of the individual transmit channel:

Jxfct ≡∂Mxf

∂<utc+ i

∂Mxf

∂=utc= J red

xftSxc with

J redxft = Rl

txf

∂Rtxf∂<b1xt

RrtxfM0

(Rtotxf

)∗+RtotxfM0

(Rltxf

∂Rtxf∂<b1xt

Rrtxf

)∗+

i

Rltxf

∂Rtxf∂=b1xt

RrtxfM0

(Rtotxf

)∗+RtotxfM0

(Rltxf

∂Rtxf∂=b1xt

Rrtxf

)∗ (4.18)

In this formulation, J redxft is the only part of the full Jacobi matrix that contains complex

non-linear rotation operations. If equation 4.10 is calculated, M0 is along theB0 directionand the optimization target is the longitudinal magnetization, equation 4.18 can besimplified to 8 scalar multiply-add operations:

J redxft = 2

−M0xf

(∂Rtot

xf

∂<b1xt

)0

+M1xf

(∂Rtot

xf

∂<b1xt

)1

+M2xf

(∂Rtot

xf

∂<b1xt

)2

−M3xf

(∂Rtot

xf

∂<b1xt

)3

+ i2

−M0xf

(∂Rtot

xf

∂=b1xt

)0

+M1xf

(∂Rtot

xf

∂=b1xt

)1

+M2xf

(∂Rtot

xf

∂=b1xt

)2

−M3xf

(∂Rtot

xf

∂=b1xt

)3 (4.19)

The optimization algorithm introduced above is now adopted to four specific magnet-ization generation applications. The following sections demonstrate their features anddescribe the adaptions of the optimization algorithm necessary to design the correspond-ing RF pulses.

All these applications are only possible using a non-linear pulse design as it was shownhere.

30

4.1 High flip-angle pTx excitation using Composite Pulses

To make use of the non-linear high flip-angle features of the proposed pulse optimiz-ation, Composite Pulses [44] are calculated. These pulses allow for example flip-anglehomogenization based on non-linear properties of the Bloch equation. Here, the conceptof Composite Pulses is extended to the transmit array and to slice-selective pulses.

Composite Pulses consist of several conventional RF sub-pulses that are concatenated.The sub-pulse shapes are identical except for global complex scaling factors applied toeach sub-pulse on each transmit channel. Figure 4.2 shows a schematic Composite Pulseusing rectangular sub-pulses for nonselective excitation. The sub-pulse shape can bechosen arbitrarily as required by the application. For example, windowed sinc pulses ornon-linear Shinnar-Le Roux [52] pulses can be used to create slice selective CompositePulses.

t

RF channel 1

channel 2

Figure 4.2: Schematic RF amplitudes for a three sub-pulse composite pulse using rectangularsub-pulses. Two transmit channels are shown. RF phases change in every sub-pulseand channel but are not shown in this figure.

The optimization routine is used to optimize the complex scaling factors defined foreach sub-pulse and transmit channel. The design target is a homogeneous flip-angledistribution with arbitrary flip-angles up to 180. Non-selective Composite Pulses can bemodeled using a single time step per sub-pulse. Each time-step represents a rectangularsub-pulse. In this case, the rotation of the magnetization during this time step is directlygiven by the magnetic field multiplied by the sub-pulse length. The magnetic field createdby the coil for the given time step is calculated as shown in equation 4.13.

To compensate for inaccuracies of the B0 map, the optimization routine is adapted tooptimize a Composite Pulse for multiple virtual B0 maps. Therefore, the measured B0

map is duplicated and user-defined offsets are added to each one. For example, to designa Composite Pulse that is immune to B0 variations of ±30Hz, three virtual B0 mapscan be introduced: B0,1 = B0 − 30Hz, B0,2 = B0, B0,3 = B0 + 30Hz. The optimizationroutine then optimizes one Composite Pulse to achieve homogeneous excitation for allthree B0 maps and thus makes the pulse more robust to errors of the B0 map. Thenumber of equations necessary increases by a factor equal to the number of frequencies

31

to optimize. This creates additional calculation effort on the GPU implemented code.The actual non-linear solver implemented on the CPU works with Newton step andGradient data which is independent of number of equations. Equation 4.20 shows theadapted system of non-linear equations for multiple B0 maps. This is based on equation4.6. The final magnetization and the total rotation operator now depend on the B0 mapand thus the optimization frequency selected:

M11(B0,1) =(Rtot

11 (B0,1))∗M0xR

tot11 (B0,1)

...

MNxNf(B0,1) =

(RtotNxNf

(B0,1))∗M0xR

totNxNf

(B0,1)

M11(B0,2) =(Rtot

11 (B0,2))∗M0xR

tot11 (B0,2)

...

MNxNf(B0,2) =

(RtotNxNf

(B0,2))∗M0xR

totNxNf

(B0,2)

... (4.20)

When using slice-selective Composite Pulses, potential signal loss due to B0 inhomo-geneities using non-selective pulses is transformed to a slice shift (a). Additionally, theconcatenation of sub-pulses introduces B0 related signal loss (b). The signal intensityversus frequency is shown in figure 4.7 for slice-selective Composite Pulses. This beha-vior is modeled without simulating the full slice-selective RF pulse shape using manytime-steps to reduce computational complexity. Therefore, each sub-pulse is representedby two time-steps. One ultra-short rectangular pulse that has a high bandwidth wheresignal amplitude is not affected by B0 inhomogeneities. This models effect (a). Afterthat, a gap is introduced where no optimization is performed to keep the spacing betweensub-pulses correct. This makes sure that the phase shift due to B0 is modeled correctly(b). After optimization, the true sub-pulse shapes are restored. This post-processingstep does not affect the flip-angle distribution. All Composite Pulses are simulated usingthe final pulse shape.

A variation of the pulse schematic shown in figure 4.2 allows to optimize spokes basedRF pulses for flip-angle homogenization [12,56] in the high flip-angle regime. Therefore,gradient blips are introduced between the sub-pulses. All pulse types shown later basedon Composite Pulses can be extended to use gradient blips between sub-pulses exceptpTx CHESS pulses.

32

4.2 Accurate saturation pulses (pTx CHESS)

Chemical shift selective suppression (CHESS) [45] is a technique that is used for ex-ample in MR spectroscopy to suppress the water signal. This is necessary because thewater peak of the resulting spectra is orders of magnitude larger than the signal of themetabolites of interest and causes severe artifacts that make these signals invisible. Tosolve this issue, CHESS can be used to suppress the longitudinal magnetization of waterprior to the spectroscopic measurement.

CHESS pulses consist of typically three frequency selective RF sub-pulses that aredesigned to transfer all longitudinal magnetization of water in the transversal plane.After each sub-pulse, spoiler gradients are applied to dephase the created transversalmagnetization. Figure 4.3 shows the sequence diagram of a three sub-pulse CHESSpulse. Using multiple sub-pulses improves suppression and allows to null signal withdifferent T1 values.

Figure 4.3: Sequence diagram of a CHESS pulse.

This concept relies on a fairly homogeneous B+1 field across the field of view. Other-

wise, longitudinal magnetization will persist and cause artifacts in the measured spectra.Due to the strong B+

1 inhomogeneities at high field strengths, this method is not dir-ectly applicable at 7T. Therefore, this section shows an adapted technique using paralleltransmission in order to achieve water suppression in the presence of strong B+

1 homo-geneities. This work was presented at the Annual Scientific Meeting ISMRM [34].

For this adaption, the sequence diagram shown in figure 4.3 is duplicated for alltransmit channels, but allowing for different complex scaling factors on each sub-pulseand transmit channel. These scaling factors are the degrees of freedom optimized bythe pulse design algorithm shown above. Therefore, this algorithm has to be adapted tomodel spoiler gradients correctly. So far, this is not the case due to the small number ofspins per volume that are simulated which is typically less than 10 per cubic centimeter.

Therefore, the cost function 4.4 is modified to assume a transverse magnetization ofzero after each spoiling gradient. This approach models the complex spoiling mechanismin the optimization routine.

Instead of a magnetization vector, only the longitudinal magnetization is of interest.The target of the optimization is a longitudinal magnetization Mdes

xf of zero for this ap-plication. Transverse magnetization is zero after the CHESS pulse by definition becauseof the last spoiler gradient.

33

Because the frequency selective RF sub-pulses are rather long, T1 effect might be relev-

ant. This makes a recursive definition of the longitudinal magnetization m(z)xft necessary.

minuct

∑

x=0...Nx,f=0...Nf

(m

(z)xft

(Rxft,m

(z)xf,t−1

))2

+ C(uct)2

(4.21)

where m(z)xf0 = 1.

The cost function calculation and the reduced Jacobi matrix calculation are the onlyparts of the optimization routine that have to be adapted.

Without T1 effects, the final longitudinal is given as:

m(z)xft(Rxft) =

∏nt

RxftR∗(k)xft (4.22)

where nt is the number of sub-pulses. Including T1 effects results in:

m(z)xft

(Rxft,m

(z)xft−1

)= 1 +

(m

(z)xft−1RxftR

∗(k)xft − 1

)e−∆t/T1 (4.23)

where R∗(k)xft is Rxft with a conjugation of the imaginary unit k only.

To calculate the reduced Jacobi matrix as defined in equation 4.18, the longitudinalmagnetization of a three sub-pulse CHESS pulse is written as:

m(z)xf2 = 1+

1 +

(1 +

((m

(z)xf0 − 1

)e−∆t/T1

)m

(z)xf1 − 1

)e−∆t/T1

m(z)xf2 − 1

e−∆t/T1

(4.24)

where m(z)xft = RxftR

∗(k)xft .

The derivative of this equation with respect to b1t can be derived analytically:

dm(z)xf2

db10=

dm(z)xf0

db10m

(z)xf1m

(z)xf2e−3∆t/T1 (4.25)

dm(z)xf2

db11= m

(z)xf0

dm(z)xf1

db11m

(z)xf2e−2∆t/T1 (4.26)

dm(z)xf2

db12= m

(z)xf1

dm(z)xf2

db12e−∆t/T1 (4.27)

wheredm

(z)xft

db1tis calculated as shown equation 4.19 replacing Rtot

xf with Rxft. These

equations define the three columns of the NxNf × Nt reduced Jacobi matrix. Repla-cing the functions of the optimization routine to calculate the final magnetization andreduced Jacobi matrix with equations 4.24 and 4.27 are the only changes necessary.The optimization routine is then capable of calculating complex weighting facts for each

34

transmit channel and sub-pulse. As a post-processing step, the rectangular sub-pulsesare replaced with frequency selective sub-pulses. This does not affect the resultingmagnetization distribution if the flip-angle of both pulses at the desired frequency isidentical.

In contrast to conventional CHESS where the spatial flip-angle distribution of all sub-pulses is identical, pTx CHESS is based on independent flip-angle distributions betweensub-pulses.

4.3 Hybrid pulse design: 2D Composite Pulses

MRI acquisition techniques require to sample the whole field of view that was excitedin order to reconstruct images without folding artifacts. This increases scan-time if theexcited area is larger than the region of interest. Therefore 2-D selective excitationtechniques based on k-space sampling have been proposed. The technique shown here isa novel approach of 2-D selective excitation that does not depend on k-space samplingbut a combinatorial approach. This approach extends conventional slice selection to asecond dimension. That way, the slice definition (e.g. edge sharpness) can be definedprior to the Composite Pulse optimization by selecting appropriate conventional slice-selective sub-pulses. This work was presented at the Annual Meeting ISMRM 2012 [30].

Figure 4.4 shows the excitation strategy used for 2D Composite Pulses. The excitationtarget is a rectangular area that is smaller than the size of the object to measure, e.g.a spherical phantom. Using two excitation modes that excite orthogonal slabs usingconventional slab-selective excitation, the object can be divided into four logical regions:

• (A) A overlap region of both excitation modes which is chosen identical to theexcitation target

• (B) A region where none of the excitation modes creates transversal magnetization

• (C) A region of the object where only the first, but not the second excitation modecreates transversal magnetization

• (D) A region of the object where only the second, but not the first excitation modecreates transversal magnetization

This shows that the excitation target region has a unique feature. This suggests that itis possible to excite only the target region by applying a both excitation modes multipletimes in a specific ordering.

One possible sequence diagram is shown in figure 4.5 where both excitation modes areapplied in an alternating fashion. In general, the excitation modes do not have to beorthogonal. Choosing other angles than 90 allows for diamond shape excitations.

Assuming a homogeneous B+1 and B0 distribution of a volume RF coil using a single

transmit channel, a simple excitation scheme can be found for a 90 pulse (4.28). Anamplitude/phase notation is used, where E1(9045) denotes a RF pulse using the firstexcitation mode with a flip-angle of 90 and an excitation phase of 45.

35

Figure 4.4: Excitation strategy of 2D-Composite Pulses

E1(900) E2(9090) E1(90−90) E2(90180) (4.28)

The first sub-pulse creates 90 flip-angle in region (A),(C). The second sub-pulsecreates 90 flip-angle in region (D) and does not affect region (A) because excitation andmagnetization phase are identical and no longitudinal magnetization is left in region(A). Sub-pulse three flips regions (A) and (C) back to longitudinal direction. Sub-pulsefour flips region (D) back to the longitudinal direction and region (A) to 90 flip-angle.After this pulse sequence, only region (A) has transverse magnetization.

However, this simple scheme does not achieve useful results in case of inhomogeneousB+

1 and B0. Regions (C) and (D) suffer most from B0 inhomogeneities as the magnet-ization is not entirely flipped back into longitudinal direction. This is caused by thephase shift ∆φ = 2π∆B0∆t of the transverse magnetization where ∆t is two times thespacing of the sub-pulses. B1+ inhomogeneities do only affect region (A) by causinginhomogeneous excitation.

To overcome these issues, parallel transmission is used to compensate for both, B+1

and B0 inhomogeneities. For each sub-pulse one complex scaling factor for each transmitchannel is used as a degree of freedom. The intra-subpulse RF shape and gradientshapes are not optimized because they only affect the spatial location and quality of theexcitation slabs. These are predefined. For the optimization algorithm, the sub-pulseis a black box that can only be modified using one complex scaling factor per transmitchannel.

The B+1 maps that are an input for the optimization routine are used to hide the sub-

pulse internals from the algorithm. It is shown that simulating a full sub-pulse based onstandard B+

1 maps has the same result as simulating a delta-pulse using modified B+1

36

Figure 4.5: 2D Composite pulse sequence diagram

maps where the slab-profile of the sub-pulse is already integrated. This works properlyas long as the slab-selective RF pulse behaves linear as it can be seen for sinc basedpulses for flip-angles smaller than 90.

For each excitation mode used, one modified B+1 map is created. The optimization

routine is modified to support multiple B+1 maps depending on the sub-pulse. The

excitation scheme used by the optimization routine uses rectangular RF sub-pulses withthe same sub-pulse spacing as before and no gradients. Because rectangular RF pulseshave a limited bandwidth, they are chosen as short as possible with a gap between pulsesto maintain the desired RF sub-pulse spacing.

The optimization target is then chosen such that the overlap region where both excit-ation modes overlap (A) targets the desired flip-angle of e.g. 90. All other regions havea target flip-angle of 0. A deviation from the desired flip-angle in regions (B-D) causesimage artifacts due to aliasing. The same deviation in region (A) causes slight imagecontrast variations. The former is the more pronounced artifact. Therefore, a weightingfactor is introduced to penalize flip-angle deviations in regions (B-D) more.

Prior to the pulse optimization a conventional slab-selective RF pulse is created. Therethe time-bandwidth product of the slab-selective sinc pulse is defined to control thetransition regions from region (A) to (B-D). This gives the possibility to trade-off edgesharpness for peak RF power prior to the optimization.

After optimization, the true RF pulse train is calculated from the pre-defined sincsub-pulses and the optimized scaling factors. This pulse is then simulated using theBloch simulator that is part of the standard optimization routine.

37

4.4 Local SAR optimized RF pulse design

Using a transmit array, not only the spatial distribution of the B+1 field can be controlled,

but also the associated electric fields. Because the electric fields directly affect the RFenergy deposition in the subject, local SAR can be added as a second optimizationtarget in addition to the target on flip-angle distribution. This work was presented atthe Annual Scientific Meeting ISMRM 2012 [31].

The optimization target is to limit the peak local SAR. This leads to the followingadapted cost function:

minut

(||Mxf (ut)−Mdes

xf ||2 + λ||SARlocal(ut)||∞)

(4.29)

where SARlocal(ut) is the local SAR vector with one entry for each VOP dependingon the RF pulse shape ut. t is the optimization time step, e.g. one sub-pulse.

Local SAR is described as a set of quadratic forms (eq. 3.1). The degrees of freedomto control local SAR and flip-angle distribution are identical. To maximize calculationspeed, the VOP compressed SAR matrices are used. The maximum of the local SARvalues calculated for each VOP is minimized. This is described using the infinity normover all VOPs in equation 4.29.

The SAR equation is discretized using the same time steps as the rest of the costfunction. For several applications, these time steps represent more complex RF sub-pulses. It is shown that as long as the relative phase between transmit channels withinone sub-pulse is constant, this representation is possible without affecting local SAR.

Starting from equation 3.1, the local SAR of VOP k for one sub-pulse can be writtenas:

SARlocal,sub-pulse,k =∑ij

T∫0

ui(t′)u∗j (t

′) dt′

Vijk

=∑ij

T∫0

|ui(t′)|eiφi(t′)|u∗j (t′)|e−iφj(t′) dt′

Vijk

=∑ij

T∫0

|ui(t′)| |u∗j (t′)| dt′

eiθijVijk

∝∑ij

|ui||u∗j |TeiθijVijk (4.30)

where T is the sub-pulse length and φi(t) is the transmit voltage phase of transmitchannel i. eiθij = eiφi(t)e−iφj(t) is independent of t because the relative phases betweenchannels do not change over time.

38

This leads to the following equation:

T∫0

|ui(t′)| |u∗j (t′)|dt′ ∝ |ui||u∗j | (4.31)

If ui is the transmit voltage of the optimization time step of channel i, local SARoptimization is possible using these sub-pulses. This is the case for all RF sub-pulseswhere the flip-angle is proportional to the transmit voltage.

The local SAR for the whole RF pulse is given as:

SARlocal,k =∑ijl

uilu∗jlTlVijk (4.32)

dSARlocal,k

duil=∑ij

Vijkujl (4.33)

Due to the infinity norm introduced in the local SAR term, the associated Jacobimatrix is not defined. To overcome this issue, an approximation of the infinity norm isused:

minu(t)

(||Mxf (u(t))−Mdes

xf ||2 + λ||SARlocal(u(t))||m)

1 m∞ (4.34)

The parameter λ is used to trade off local SAR versus excitation quality. The para-meter m has to be chosen as large as possible to get a good approximation of the infinitynorm while maintain numerical stability.

4.5 Quality controlled SAR reduction

For ultra high field strengths of 7T and above, peak RF power and SAR are domin-ant limiting factors for high performance human imaging. One way to minimize thesequantities, is to prolong the RF pulses. However, long RF pulses are prone to B0 in-homogeneities as the bandwidth of such pulses is reduced. This causes flip-angle dropsin case of non-selective rectangular excitation pulses:

α ∝ sin(π∆B0t)

π∆B0tE ∝ t−2 (4.35)

where α is the flip-angle depending on the rectangular pulse length t and the B0 offset∆B0. E is the required pulse energy.

In case of windowed sinc shaped slice-selective RF pulses, a slice-shift occurs dependingon the B0 offset:

∆x ∝ ∆B0

BWE ∝ BW2 (4.36)

39

where ∆x is the slice shift and BW is the bandwidth of the RF pulse. The slice thicknessis assumed to be constant.

A long non-selective RF pulse will reduce SAR at the cost of flip-angle homogeneity.A low bandwidth slice-selective pulse will reduce SAR at the cost of slice profile fidelity.

In this section, the potential of prolonging large-tip-angle composite RF pulses forpower and SAR reduction is investigated. The goal is to break the trade-off between RFenergy of a pulse and its associated bandwidth to achieve both, homogeneous excita-tion/high quality slice profiles and low SAR at the same time. Therefore pTx CompositePulses for large tip angle excitation or refocusing are calculated that compensate for re-duced global bandwidth. The designed pulses should yield an excitation pattern whichis at least as homogeneous as an RF shim optimized excitation.

The key feature of the designed pulses is to adjust their resonance frequency locallyand adapt to the B0 map included in the optimization routine.

Figure 4.6 shows multiple slices of a B0 map of the human head at 7T acquiredusing [35]. Strong field distortions are visible near the sinuses and the oral cavity causedby air-tissue boundaries. The peak-to-trough difference of B0 values is approximately350Hz in this case. However, the B0 distortions are smooth and do not contain highspatial frequency components. This is a similarity to multi-channel B+

1 maps.

Figure 4.6: B0 map of several transversal slices of the human head at 7T (a) and the corres-ponding anatomical images (b)

Splitting a long nonselective RF pulse into n sub-pulses, reduces the sub-pulse lengthto t/n. The pulse design algorithm is then configured to calculate complex weightingfactors for each sub-pulse and channel to adjust the B+

1 phase to the magnetizationphase that experiences a phase shift depending on the local B0 offset. An upper boundfor this compensation if given by:

α ∝ sin(π∆B0t/n)n

π∆B0t(4.37)

40

because a pulse that is a concatenation of sub-pulses can never exceed the bandwidthof its sub-pulses. Figure 4.7 shows the flip-angle over frequency of a three sub-pulseComposite Pulse with gaussian sub-pulses. The bandwidth of the sub-pulses is approx-imately 600Hz.

0

10

20

30

40

50

60

70

-600 -400 -200 0 200 400 600

Fli

p a

ngle

[deg

rees

]

Frequency offset [Hz]

Voxel 1Voxel 2

Figure 4.7: Flip-angle over frequency of a three sub-pulse composite pulse with gaussian sub-pulses. Two voxels optimized for different B0 are shown

Splitting a long slice-selective RF pulse into n sub-pulses increases the sub-pulse band-width to n · BW. This directly reduces the B0 induced slice shift.

∆x ∝ ∆B0

nBW(4.38)

However, the magnetization de-phasing between sub-pulses creates reduced flip-anglesas it can be seen with long nonselective sub-pulses. The pulse design algorithm isconfigured to calculate complex weighting factors for each sub-pulse and channel tocompensate for B0 induced image quality reduction.

In both situations, the same mechanism is used to compensate for image qualityreduction. In a simple example, two voxels at positions x1 and x2 experience differentB0 offsets B0(x1) and B0(x2). Using a two sub-pulse RF pulse with a sub-pulse spacingof ts, a B0 induced phase shift occurs depending on the spatial location of the voxel xi:

∆φ(xi) = γB0(xi)ts (4.39)

Using a two channel transmit array, transmit phases can always be found that havethe same phase shift as the magnetization.

The following equation shows the relation between the B+1 field created by a two

channel transmit array with transmit sensitivities Sij and a transmit voltage ui:(B+

1 (x1)B+

1 (x2)

)= Sij

(u1

u2

)(4.40)

41

If the transmit sensitivity matrix is linear independent, two transmit voltage settingscan be found to create two B+

1 field distributions where the phase difference is exactlythe B0 induced phase shift ∆φ(xi).

Figure 4.7 shows shows an example where central peak of the frequency response isshifted by 35Hz for the second voxel. The gaussian envelope stays the same for bothvoxels.

For realistic scenarios, the number of voxels within the region of interest is muchlarger than the number of transmit channels. On the other hand, the B0 distribution issmooth, containing only low spatial frequencies. The optimization routine tries to findan approximate solution that minimizes the sum of squares of the error of all voxels.

42

5 Results & Discussion

5.1 Local SAR supervision

To be able to apply large amounts of RF energy to the scanned subject, a supervision sys-tem was built to accurately measure RF voltage and calculate local SAR from measuredRF voltage data. This was achieved by reusing as many components already availableon the scanner as possible. This approach was chosen because an implementation onreal-time hardware involves rebuilding the whole receiver electronics chain based on areal-time FPGA design.

The supervision was evaluated with regards to calculation speed and accuracy. Fur-thermore, SAR performance was analyzed compared to previous SAR supervision meth-ods.

To be able to run any possible sequence with local SAR supervision, the calculationtime for a given RF pulse must be shorter than the RF pulse itself. To validate thisaspect, a high duty-cycle GRE sequence was run with an excitation pulse of 1ms lengthand minimum TE and TR. The VOP dataset used for testing contains 500 VOPs andthe RF is sampled at a dwell-time of 1us. To measure the peak SAR calculation time,the maximum allowed time of unsupervised RF configured in the watchdog was reduceduntil a measurement abort was triggered by the watchdog. This experiment showed aprocessing time of less than 120us is typical.

Figure 5.1: Online calculated local SAR vs. predicted local SAR for several arbitrary TX ArrayRF pulses

To evaluate the full local SAR supervision setup, multiple sequences with randomlychosen pTx pulses were applied on the scanner. For each sequence, the expected localSAR is calculated from electric fields, conductivity distribution, density distribution and

43

instructed RF pulse shapes. Figure 5.1 compares the theoretical peak local SAR to thelocal SAR calculated by the supervision system based on a VOP compressed dataset andmeasured RF pulse shapes. The error estimation technique shown in section 7.2 that isused to make sure local SAR is never underestimated in the presence of measurementerrors was disabled. That way, the overall local SAR detection accuracy can be shown.

The error between expected and online calculated local SAR was always less than6% of the expected local SAR. This data contains accumulated errors caused by theVOP compression, the RF amplifier and the voltage measurement system. The errordistribution shown is better than what could be expected with a amplitude error of 10%and a phase error of 5 as shown in figures 7.6 and 5.1. This has two reasons: First,the measurement errors calculated from component specifications are worst case values.Second, the error distributions in figure 5.1 were calculated using single voltage settings,while in the real world experiment, multiple voltage settings are used and the resultingunder-/over-estimations are averaged.

For further validation, local SAR was calculated based on the following data:

1. Electric fields, conductivity and density using predicted pulse

2. SAR matrices using predicted pulse

3. VOPs using predicted pulse

4. SAR matrices using measured pulse

5. VOPs using measured pulse

Table 5.1 shows the differences between these results where eVOP is the overestimationof local SAR introduced by the VOP compression and emeas is the SAR error caused bymeasurement errors of the complex RF voltages, as it is estimated in figure 5.1. eVOP

was always below 1% as specified for the compression algorithm.

1 2 3 4 5

1 0 0 eVOP emeas eVOP+emeas

2 0 eVOP emeas eVOP+emeas

3 0 eVOP+emeas emeas

4 0 eVOP

5 0

Table 5.1: Local SAR calculation errors

Method (1) and (2) do show identical results because they use algorithms than canbe reformulated into each other. The errors of all other cases are below or equal to theerrors shown in figure 5.1 where results (1) are compared to (5).

Figure 5.2 shows the probability of a given local SAR overestimation factor for threesupervision strategies. α3 was calculated for a local SAR supervision strategy based

44

only on measured RF amplitudes. Due to the lack of phase information, significant localSAR overestimation occurs regularly. 50% of all measurements show a overestimationfactor more than 9, resulting in an extreme reduction of scanning performance.α1 was calculated for a straight forward local SAR calculation based on complex

measured RF voltages. This approach is a significant risk for subjects due to the localSAR underestimation occurring in approximately 50% of all cases.α2 includes worst case measurement errors calculated from component specifications

into the local SAR calculation procedure. That way, local SAR is never underestimatedand therefore, the supervision is safe. This comes at the cost of a slightly higher SARoverestimation that is below 25% for 50% of all evaluated cases.

0

1

2

3

4

5

0 1 2 3 4 5 6

Pro

bab

ilit

y

Local SAR over-estimation

α1α2α3

Figure 5.2: Online calculated local SAR vs. predicted local SAR for several arbitrary TX ArrayRF pulses

A real-time high performance local SAR calculation and supervision system withoutuser interaction after a coil/patient model is selected was presented. It was shownthat the system is capable of supervising high RF duty-cycle sequences. This is onlypossible using compressed electric field models. The error between expected and onlinecalculated local SAR was always less than 6% for a large number of real world test cases.This error includes imperfections in the transmitted RF, which are correctly modeledby the supervision system. Because the standard image reconstruction system is usedfor local SAR calculation, additional supervision tasks based on measured forward andreflected complex RF pulse shapes can be implemented easily. The proposed systemworks independently of a local SAR look-ahead which can be implemented as a separateand independent second safety shell.

This local SAR supervision is the foundation for applying complex RF pulses forparallel transmission in-vivo. Especially RF pulses that are optimized to minimize localSAR are only useful with a full local SAR supervision in place.

45

5.2 pTx Pulse design performance

To analyze the efficiency of the pulse design algorithm introduced above, the runtime ofthe optimization procedure is evaluated.

Table 5.2 shows calculation runtime for the full algorithm together with the corres-ponding magnetization quality and the number of iterations needed in each case. Themagnetization quality is measured using the quadratic deviation from the target flip-

angle:√∑

xf (Mxf −Mdesxf )2/(NxNf ).

Pulse Type Qu

alit

y

Ru

nti

me

[ms]

#of

iter

atio

ns

RF-shimming 20 67 93 sub-pulse composite 90 4 79 93 sub-pulse composite 180 7 91 13spectral-spatial composite 5 104 14

Spokes 4 87 9

Table 5.2: Pulse design algorithm runtimes

All pulses shown above can be calculated in less than one second. Pulses whichcan be modeled using a modest number of time-steps (approx. 20), can be calculated inapproximately 100 milliseconds. Adding additional equations for local SAR optimizationdoes not significantly increase calculation time because these calculations are performedon the CPU in parallel to GPU calculations.

To analyze the effectiveness of the GPU implementation to calculate the Jacobi mat-rix, the individual runtime of this algorithm is compared to a CPU version implementedin Matlab and theoretical relationships stated above. The Jacobi matrix is calculatedfor 7680 voxels and eight transmit channels. As shown in figure 5.3, the GPU imple-mentation shows a linear dependency between runtime and the number of time-stepscalculated. This is in agreement with the theoretical prediction and in contrast to thequadratic dependency that is expected when using a finite differences approach. Run-ning the same setup with 16 transmit channels shows an increase in computation timeof only 5%. This is because most of the calculations performed are independent of thenumber of transmit channels by using the reduces Jacobi matrix. Calculating the Jacobimatrix for eight transmit channels and 50 time-steps with varying number of voxels hasno effect on calculation time until the number of voxels is larger than the number ofparallel execution units of the GPU. Each voxel can be calculated completely independ-ent on a single execution unit. The GPU is able to process 2.8 instructions per clockcycle which is close to the theoretical throughput of 3 instructions per clock. The GPUversion of the algorithm is approximately a factor of 100 faster than the CPU versionbased on C++ code.

46

0

0.5

1

1.5

2

2.5

0 50 100 150 200 250 300

runti

me

[ms]

# of timesteps calculated

runtime Jacobi calculationlinear fit

Figure 5.3: Runtime of the jacobi matrix calculation using the optimized GPU code.

It has been shown that GPU acceleration is very suitable to speed up MRI pTxpulse design calculations. In conjunction with an efficient mathematical formulation ofthe optimization problem, this approach allows to calculate a variety of pTx RF pulseclasses in less than one second. A pulse calculation step of this duration can be easilyincluded in a typical workflow on the scanner.

5.3 Flip-angle homogenization

This section shows results that improve the flip-angle homogeneity by the applicationof parallel transmission. These RF pulses can be used to compensate for B+

1 fieldinhomogeneities present at high B0 field strengths. All results shown in the followingsections are obtained using the workflow defined in section 3.2. An example of subject-specific B+

1 maps that are input to the optimizer are shown in figure 5.4.

Composite pulses

Composite Pulses for 90 excitation and 180 refocussing were calculated, simulatedand compared to measurements based on flip-angle mapping. Pulse performance wascompared to RF shimming and pTx spokes pulses.

A 90 non-selective pTx Composite Pulse was calculated with the following paramet-ers:

• 3ms total length

• Peak voltage limited to RFPA peak voltage of 160V per channel

• 90 magnitude least square composite excitation design

Figure 5.5 shows the flip-angle distribution of this pulse in a spherical phantom usingBloch simulation (a) and flip-angle mapping (b). Flip-angles are plotted from 60 to90 to show residual inhomogeneities. The elevated noise level of the simulation resultis caused by noise in the B0 map. The flip-angle distributions were calculated based on

47

0

1e-08

2e-08

3e-08

4e-08

5e-08

6e-08

7e-08

8e-08

9e-08

(a) Amplitude (T/V)

-4

-3

-2

-1

0

1

2

3

4

(b) Phase (rad)

Figure 5.4: B+1 map of a spherical phantom measured using individual channels of an eight

channel transmit array.

48

60

65

70

75

80

85

90

(a) Simulation

60

65

70

75

80

85

90

(b) Measurement

Figure 5.5: Measured and simulated 90 composite excitation pulse. Flip-angle distribution isshown in degree

a transverse magnetization distribution. This limits the flip-angle range visible from 0

to 90.

Both results are in good agreement. This shows that magnetization dynamics areproperly modeled in the optimization and simulation environment and all relevant effectsare included.

The pulse has the following properties compared to a conventional circular polarizedexcitation pulse with the same total length:

pTx Composite Pulse circular polarized

relative pulse energy 3 1flip angle standard deviation [] 6 20

Table 5.3: Pulse properties for a 8 channel composite excitation pulse

For the given energy/quality trade-off, a reduction of factor 3 in flip-angle standarddeviation requires three times more RF pulse energy. Removing power constraints resultsin a flip-angle homogeneity of less than 1 with power requirements that cannot berealized on the scanner.

Figure 5.6 shows the flip-angle distributions of three slice-selective RF pulses in aspherical phantom. Figure 5.6a is a measured RF shim calculated with the proposedoptimization routine. A peak-to-trough B+

1 ratio of 3:1 can be seen. Figure 5.6b showsa measured slice-selective Composite Pulse with a target flip-angle of 90. Three sub-pulses with a sub-pulse length of 2ms were used. The sub-pulse shape is a windowedsinc. A significant improvement in flip-angle homogeneity is visible. RF pulse energy is3 times higher compared to the RF shimmed pulse with the same length. Figure 5.6cshows a simulation result of a 180 refocussing pulse. A simulation is shown because

49

it is not possible to measure a flip-angle map with 180 flip-angle using the chosen B+1

mapping technique. Three sub-pulses with a length of 3ms each were used. A similarflip-angle homogeneity is achieved compared to the 90 pulse. The total pulse energyis 2.4 times higher compared to the 90 composite pulse. In all cases, RF power wascontrolled to stay within hardware limits.

Figure 5.6: Flip-angle distribution of a conventional slice-selective RF pulse with RF shimapplied (a), a 90 optimized Composite Pulse (b) and a 180 optimized CompositePulse (c). (a) and (b) are measured results, (c) is a simulation result because flip-angle distributions of 180 pulses could not be measured. Note the different scalingnecessary to show residual magnetization inhomogeneities.

Figure 5.7 compares a 90 Composite Pulse excitation to a 90 spokes pulse excitation.Both pulse types use slice-selective sub-pulses. In terms of sequence timing, spokes arevery similar to Composite Pulses and can therefore be calculated using the proposedoptimization routine. Compared to Composite Pulses, short gradient blips are addedbetween sub-pulses. Therefore, spokes pulses are based on a k-space approach, where foreach sub-pulse, a different k-space position is selected using the gradient blips. Spokesdo not explicitly use non-linear effects of the Bloch equation, but are affected by thesefor high flip-angles. The shown pulses have the same length of 6ms total. Both achievea similar flip-angle homogeneity. However, spokes pulses need 30% less RF energy. Thisadvantage is mitigated for higher flip-angles.

Figure 5.7: Flip-angle distribution of a 90 optimized Composite Pulse (a) a 90 optimizedSpokes pulse (b). Results were measured using the B+

1 mapping sequence.

50

Figure 5.8 shows two Composite Pulses measured using five different RF center fre-quencies. This shows the effects of these pulses being played out on a system where theB0 has changed from the measurement of the B0 map to the application of the CompositePulse. The first Composite Pulse (a-e) was optimized using a single B0 map only. Thesecond Composite Pulse (f-j) was optimized using three B0 maps with B0,1 = B0−40Hz,B0,2 = B0, B0,3 = B0 + 40Hz. The target flip-angle was 90 in both cases. The RFpower needed for both pulses is the same. The results show a significant improvementwhen using multiple B0 maps over a range of frequencies from -40Hz to +40Hz. On theother hand, the quality of the 0Hz result decreases slightly when optimizing for severalB0 maps. This shows that the optimization routine creates extremely narrow-band RFpulses to achieve a high flip-angle homogeneity. Enforcing a higher bandwidth comes atthe cost of slightly decreased flip-angle homogeneity.

Figure 5.8: Measured flip-angle distribution of a Composite Pulse optimized on resonance fre-quency (0Hz) (a-e) and optimized for a frequency band of ±40Hz (f-j). Both pulsesare measured with RF center frequencies of -40Hz, -20Hz, 0Hz, 20Hz and 40Hz.

In summary, it was shown that pTx Composite Pulses are well suited for homogeneoushigh flip-angle excitation and refocussing in the presence of inhomogeneous B+

1 and B0.Simulations and measurements were in good agreement showing that all relevant effectswere modeled correctly in the pulse optimization procedure.

Compared to RF shimming, a significant improvement in flip-angle homogeneity wasachieved at the cost of higher RF energy requirements. Compared to high flip-angle pTxspokes pulses optimized with the same algorithm, a similar pulse quality is achievable.pTx spokes performed better in terms of RF energy for 90 pulses.

Including multiple frequencies to the optimization procedure allows to calculate RFpulses which are more robust to unknown B0 effects at almost no cost in pulse qualityand RF energy requirements.

All RF pulses shown were calculated in less than 100ms. This does not disturb atypical scanning workflow.

51

pTx CHESS

Conventional CHESS pulses for water suppression were adapted for parallel transmissionto compensate for B+

1 inhomogeneities.Figure 5.9 shows the longitudinal magnetization that is left after application of in-

dividual sub-pulse and the final longitudinal magnetization after all three sub-pulses.Please note the different scaling. For each voxel in the field of view, at least one sub-pulse exists that reaches a longitudinal magnetization close to zero. The different spatialflip-angle distribution are only possible due to the spatial degrees of freedom introducedby the transmit array. Applying the full pTx CHESS pulse results in a very homogeneouswater suppression over the full field of view.

The optimization routine uses experimentally collected B+1 maps to optimize pulse

amplitude and phase for all transmit channels and all sub-pulses to achieve the bestpossible water suppression. The gap between CHESS sub-pulses was 20ms and theywere designed to suppress water in a range of T1 of one to five seconds. Gaussian sub-pulses were used to achieve frequency selection to only suppress water.

Figure 5.9: Longitudinal magnetization left from individual sub-pulses (left) and final longit-udinal magnetization after application of the full pTx CHESS pulse (right).

Figure 5.10 shows the achievable suppression for pTx CHESS for different T1. Al-though the pulse is explicitly calculated only for a T1 of 1s, good suppression is alsoachieved in simulation for T1 = 3s and 5s as shown in figure 5.10b,c.

Figure 5.10: Simulated water suppression achievable for pTx CHESS over for T1=1s (a), T1=3s(b) and T1=5s (c).

Figure 5.11 shows the mean suppression over the field of view with respect to T1

for conventional CHESS and pTx CHESS pulses in the presence of B+1 inhomogeneities.

Conventional CHESS pulses were applied using a circular polarized excitation mode. pTx

52

CHESS pulses achieve a mean water suppression of better than -45dB for T1 betweenone and five seconds (blue). Their minimum suppression is always better than -30dB.In contrast, conventional CHESS have a mean water suppression of only around -25dB.

Figure 5.11: Simulated water suppression achievable for conventional CHESS (red) and pTxCHESS (blue) over T1.

pTx CHESS is a novel approach for chemical shift selective signal suppression. Incontrast to conventional CHESS, different spatial magnetization patterns are createdfor each sub-pulse. To create a good suppression within a given voxel, only one sub-pulse needs a flip-angle of 90 for that voxel. Because a homogeneous B+

1 distributioncannot be achieved using RF shimming at high fields, this feature enables high qualitysuppression. A non-linear optimization is not necessary here to describe magnetizationexcitation using RF. Here, linear RF shimming occurs in every sub-pulse. However, theuse of spoiler gradients between sub pulses requires a non-linear optimization routine.

The shown pulses achieve a substantially better water suppression compared to con-ventional CHESS. This is achieved without compromising the ability of CHESS pulsesto suppress spins over a large range of T1. The pulses shown here can be calculated inless than 100ms.

2D Composite Pulses

In contrast to the RF pulses shown above, 2D Composite Pulses allow a spatial selectionin two dimensions.

Figure 5.12 shows the spatial flip-angle distribution of a conventional excitation usinga RF shimmed windowed since pulse (a) compared to pTx optimized 2D composite pulseexcitation (b) with a target flip-angle of 90. Significant magnetization is created onlywithin the targeted rectangular region of interest. The pulse shown here is non-selectivein the third dimension not shown in this figure. In addition, a flip-angle homogenizationcompared to the conventional excitation within the region of interest is visible. Theedge sharpness of the excited region is similar the edge sharpness of the predefinedslab-selective sub-pulses.

53

Figure 5.12: Simulation results showing the flip-angle of a conventional excitation (a) and a2D Composite excitation (b)

Figure 5.13 shows two slices of a cylindrical phantom of a 3D gradient recalled echosequence with a 2D Composite pulse as excitation. Here, spatial selection is achieved indimensions 2 and 3 compared to 1 and 2 in the simulation results shown above. In thethird dimension, a 5cm slab is excited. Figure 5.13a shows the center slice where theexcitation region (A) is excited and homogenized as expected. Within region (C), goodsignal suppression can be seen. Figure 5.13b shows an off-center slice outside the 5cmslab using a different windowing. Region (D) shows residual magnetization. Region (B)is never excited.

Both pulses shown consist of four sub-pulses of 2.5 milliseconds each using windowedsinc sub-pulses with a time-bandwidth product of 4. RF power was limited to keep peakvoltages within hardware limits. To achieve the signal suppression outside the field ofview, suppressed regions are weighted 10 times higher in the cost function compared toexcited regions.

Similar 2D selective excitations can be achieved using k-space sampling techniquewhere typically more than 10 sub-pulses are concatenated. Each sub-pulse is applied ata different k-space position selected by field gradients between sub-pulses. Compared tothese conventional 2D selective pulses, a similar effect can be achieved using only 4 sub-pulses when using 2D Composite Pulses. Additionally, the edge sharpness is predefinedand not created as a part of the optimization process as it is the case with k-space based2D selective excitation.

2D Composite pulses are a concatenation 4 sub-pulses with a flip-angle of approxim-ately 90 each. Compared to a 1D selective excitation with the same bandwidth to keepB0 robustness similar, the proposed pulses require approximately 4 times more energy.

In summary, 2D composite pulses allow high quality 2D selective excitation withpre-defined transition regions and short total pulse length. This comes at the cost ofincreased power requirements and a higher sensitivity to unpredicted B0 deviations.

54

Figure 5.13: 2D Composite pulses measured using a 3D gradient recalled echo sequence on aphantom. (a,b) show two slices of the measured data, (c,d) show a schematic ofthe corresponding excitation regions as they are defined in figure 4.4. (a) showsthe center slice and (b) an off center slice using a different windowing to showresidual magnetization in region D.

5.4 SAR optimization

Local SAR optimization

Explicit local SAR optimization was added to the non-linear pulse optimization problem.The optimization targets are a homogeneous flip-angle distribution and low peak localSAR. Because these are conflictive targets, multiple solutions are calculated with differ-ent weighting of both targets. Figure 5.14 shows l-curves with the calculated solutionswhere the flip-angle root-mean-square-error is plotted versus the normalized local SAR.

As a reference, optimizations were performed using a penalty on RF power as shownin equation 4.4 where C(uct) = λ

∑ct uctu

∗ct. λ is used to configure the weighting of

both optimization targets. If λ equals zero, no penalty on RF power is applied. Thissolution corresponds to the top-left edge of all l-curves shown. Increasing λ allows tomove solutions along the l-curve. Optimization results with penalized RF power areshown as dashed lines.

To optimize local SAR, equation 4.34 is solved. Again, the optimization is repeatedfor several values of λ. The results are plotted as solid lines. Best results were obtainedusing the 8-norm with m = 8 in equation 4.34.

Figure 5.14 shows l-curves for non-selective Composite Pulses using three sub-pulses(a) and slice-selective Spokes pulses using three sub-pulses (b). Both pulse types wereoptimized using RF power (dashed line) or local SAR (solid line) constraints. For a value

55

of λ = 0, both optimization strategies show the same result (top-left edge of l-curves).For all other values of λ, the optimization routine with explicit local SAR constraintsresults in lower peak local SAR values. For Composite Pulses, the local SAR reductionis up to 20% and for Spokes, the reduction is up to 30%. Best local SAR reduction isachieved if both optimization targets are weighted equally. This is the case where theeuclidian distance of the l-curve to the origin of the coordinate system is minimal.

Figure 5.14: Normalized peak local pulse power versus flip-angle root mean square error forComposite Pulses (a) and Spokes (b). Optimization results using local SAR op-timization (solid) and global RF power optimization (dashed) are shown.

It has been shown that local SAR minimization is possible by adding additional equa-tions to the system of non-linear equations that describe local SAR. These equationsare approximated to create a differentiable system of equations. Therefore, the infinitynorm is replaced with the 8-norm.

The approach works for both, non-selective rectangular Composite RF pulses andslice-selective Spokes pulses. The latter do not include the slice-selective sub-pulses inthe optimization model. Without significant impact on pulse optimization speed, localSAR reduction up to 30% is achievable.

A parameter λ is used to trade-off flip-angle homogeneity versus local SAR. Thisapproach does not allow to specify a hard local SAR limit or a target flip-angle homo-geneity. To achieve this goal, the optimization routine has to be rerun with differentparameters λ. For many applications, this still seems feasible due to pulse calculationtimes in the order of 100ms.

It was shown that the local SAR optimization has less effect on the result if flip-anglehomogeneity is weighted much higher than local SAR optimization. Also, less efficientlocal SAR minimization is possible in cases where SAR minimization is the primary goal.In this case, there is no flexibility to cancel electric fields and thus reduce local SAR,because this would reduce flip-angle homogeneity and/or increase RF power. This inturn increases local SAR.

To make this concept feasible, VOP compression of the electric field models used to

56

calculate local SAR is essential to achieve reasonable pulse optimization times.

To take advantage of this optimization concept, a local SAR supervision system onthe scanner is required. In contrast, the SAR reduction strategy shown in the followingsection also works without local SAR supervision, because it also reduces RF power,which is not necessarily the case here.

Quality controlled SAR reduction

To reduce SAR, a non-selective rectangular excitation pulse was prolonged and split intothree sub-pulses. The target flip-angle homogeneity was kept similar to the distributionof RF shimming by adjusting λ accordingly. Figure 5.15 shows the simulated flip-angledistributions of a prolonged Composite Pulse with three sub-pulses (a) and a conven-tional rectangular RF pulse (b). Both pulses have a total length of 9ms. The CompositePulse has a flip-angle distribution that is similar to conventional RF pulse with 1mstotal length. The conventional RF pulse suffers from significant signal loss in some areasdue to B0 inhomogeneities. The Composite Pulse is able to compensate for these B0

inhomogeneities.

Figure 5.15: Conventional (b) and composite pulse (a) excitation using the same RF energy.The simulated excitation flip-angle distribution is plotted. Both pulses have atotal length of 9ms. (a) has the same excitation quality as a 1ms conventionalexcitation pulse.

Figure 5.16 shows the RF pulse energy required by Composite Pulses to achieve aflip-angle homogeneity similar to a short RF shimmed pulse for a given total RF pulselength (solid line). The dashed line of the figure is a lower bound for the required RFenergy assuming no B0 inhomogeneity to compensate for. This is a 1/x dependency. Upto a pulse length of 10ms, no additional RF energy is required to compensate for theB0 induced signal loss as shown in figure 5.15b. For RF pulse lengths of 10ms to 15ms,the RF energy reduction gained from prolonging the pulse is significantly reduced. Forpulses longer than 15ms, the Composite Pulses are no longer able to reduce total RFenergy requirements.

Figure 5.17 shows phantom results measured at a 3T transmit array. A spin echosequence was run with in-plane slice excitation and through-plane slab refocussing. Therefocussing pulse slab profile is visible. Figure 5.17a is measured using a 4.5ms con-ventional windowed sinc refocussing pulse. Due to B0 inhomogeneities, slab bendingis visible. Figure 5.17b uses a split refocussing pulse without optimization and three

57

Figure 5.16: Forward RF energy for 90 excitation pTx composite pulses (solid) versus pulselength. The dashed line is a theoretical lower bound for the forward RF energy,assuming compensation of B0 effects does not require additional RF energy.

sub-pulses. The total pulse length 9ms. A reduced slab bending is visible due to in-creased RF pulse bandwidth. Additionally, the SAR of the refocussing pulse is reducedby a factor of 2. This comes at the cost of a B0 induced signal void in the center. Fig-ure 5.17c uses the same RF pulse configuration including RF pulse optimization. Thesignal void is removed and SAR of the refocussing pulse is reduced by a factor of 1.7compared to (a). Furthermore, the flip-angle of each sub-pulse is now much lower than180. Windowed sinc pulses are based on a linear approximation of the Bloch equationand therefore exhibit distortions due to non-linear effect for high flip-angles. The lowersub-pulse flip-angles reduce these slice profile distortions and side lobes.

Figure 5.18 shows in-vivo results of the human head measured at 3T. The focus wason improving slab-selection quality while keeping SAR constant. The acquisition schemeis similar to the results shown in figure 5.17. Figure 5.18a uses a 4.5 ms conventionalrefocussing pulse with a bandwidth that is limited by RFPA peak power. Slab distortionsare visible and the excited fat slab is clearly shifted to the left. Figure 5.18b uses athree sub-pulse Composite Pulse for refocussing with a total length of 6ms. The time-bandwidth product was increased to improve edge sharpness, and due to the shorter sub-pulses, the slice-select gradient amplitude was increased. This reduces slab distortions.Additionally, the fat shift is reduced significantly. Due to a bi-polar slice-select gradientscheme, fat shift is now visible in both directions. The SAR of both refocussing pulsesis approximately the same. The longer total length of the Composite Pulse was used toincrease the time-bandwidth product.

Prolonging RF pulses in combination with a pTx Composite Pulse design showed to bea powerful strategy to reduce SAR or improve slab-selection quality in the presence of B0

inhomogeneities. This approach works for non-selective and slice-selective excitations.The advantage is that the trade-off between RF pulse energy and bandwidth is removed.A spatial adaption of the pulse to the local Lamor frequency is achieved using the

58

Figure 5.17: Spin echo images measured at a 3T transmit array with refocussing plane or-thogonal to excitation plane creating a 2D selective excitation. The refocussingpulse slab-profile is shown. (a) is measured with a conventional windowed sincrefocussing pulse. (b) is measured with a three sub-pulse Composite Pulse forrefocussing without pulse optimization. The pulse is two times longer than (a).(c) is measured similar to (b) but with pTx pulse optimization.

transmit array and Composite Pulses. Prolonging RF pulses reduces global SAR, localSAR and RFPA peak power requirements.

For slice-/slab-selective pulses, several advantages for the slab profile quality exist thatcan be traded for SAR. These include higher slice-select gradient strength for reducedslice bending and fat shift, and lower sub-pulse flip-angles that reduce distortions of theslice profile caused by non-linearities of the Bloch equation. For moderate pulse lengths(in the example shown above, less than 10ms), B0 compensation is possible withoutneeding additional RF power.

This allows flexible solutions for SAR and quality requirements. This approach isuseful especially for the excitation of large slabs where pulse bandwidth is limited byRFPA peak power and thus extremely low slice-select gradients must be used.

The SAR reduction approach works also if no local SAR supervision based on phase-sensitive RF measurements is available. This is the case because the SAR reduction isachieved by reducing RF power, which scales with local SAR.

59

Figure 5.18: Spin echo images measured at a 3T transmit array with refocussing plane ortho-gonal to excitation plane creating a 2D selective excitation. The refocussing pulseslab-profile is shown. (a) is using a conventional windowed sinc refocussing pulse.(b) uses a three sub-pulse Composite Pulse that has approximately the same pulselength compared to (a).

60

6 Conclusions

Parallel transmission technology for high field magnetic resonance imaging showed tobe of high value to control magnetization dynamics during excitation. Compared tosingle channel transmission, the increased flexibility in shaping the excitation RF fieldB+

1 allows to counteract field inhomogeneities more efficiently.

Most current approaches to calculate excitation pulses for parallel transmission arebased on a linear approximation of the Bloch equation useful for small flip-angles. Highflip-angle pulse design algorithms suffer from long computation times or the inability totake advantage of the non-linear properties of the Bloch equation.

In this work, a generic algorithm for non-linear high flip-angle pulse design was intro-duced that allows to take advantage of the non-linear properties of the Bloch equation.This also allows to improve results for the extension to low-flip angle concepts to highflip-angles. To improve scanning workflow, care has been taken to allow a fast RF pulseoptimization. The optimization framework was designed to minimize calculation ef-forts by exploiting redundancies in the equations. Furthermore, compute intensive partswere transferred to a GPU. This allows to calculate all pulses shown in this thesis inapproximately 100ms.

The proposed optimization framework was used to calculate Composite Pulses forflip-angle homogenization. This pulse class requires a non-linear optimization frame-work because it relies on non-linear properties of the Bloch equation. High flip-angleComposite Pulses were designed for flip-angles up to 180. An improvement in flip-anglestandard deviation of factor 3 to 4 was achieved at the cost of increased SAR comparedto conventional RF pulses. Also, an extension was shown that makes these pulses morerobust to unknown B0 variations of ±40Hz without a penalty on SAR. These inac-curacies of the B0 map are common for in vivo scanning due to movement or breathing.These results were compared to pTx spokes pulses that were designed with the sameoptimization routine. Both pulse types showed a similar excitation quality, but Spokesrequire 30% less RF energy for 90 pulses. This difference is mitigated for 180 pulses.

Slice-selective Composite Pulses were extended to allow 2D-selective excitation. Thenovel approach shown in this thesis is based on a combinatorial technique. Accurate2D-selective excitation was achieved with similar quality compared to k-space based 2D-selective excitation techniques. However, the proposed technique allows much shorterRF pulses and a predefinition of the transition regions. On the other hand, short 2DComposite Pulses require high RF power. This makes this technique favorable in caseswhere only a few pulses are needed in the presence of tight timing constraints.

While all other pulse types so far were designed for homogeneous excitation, thelast method demonstrated is used for signal suppression. pTx CHESS pulses achievean improvement of the suppression of approximately -20dB compared to conventional

61

CHESS pulses. This improvement is achieved by varying the flip-angle distribution ofdifferent CHESS sub-pulses in a way that every voxel is suppressed well by at least onesub-pulse. Although very exact suppression was shown in simulations, the proposedmethod is sensitive to errors in the B+

1 maps which circumvents good measurementresults so far.

Local SAR is a dominant limiting factor for high performance imaging. To overcomethis issue, either RF power needed by RF pulses must be reduced, or a local SARsupervision must be available to provide a more accurate limitation and thus reducesafety margins needed otherwise.

Using the transmit array in conjunction with Composite Pulses allowed to break thetrade-off relation between RF pulse bandwidth and RF power. This allows a reductionof RF power without compromising image quality. It was shown that RF pulses can bedesigned that have a spatially variable center frequency, allowing to adapt to local B0

field changes. This allows to reduce the RF pulse bandwidth below the peak-to-troughresonance frequency range by prolonging the pulse. This approach can reduce SARsignificantly by a factor of 2 to 3 without negative impact on image quality. Additionally,when applying this technique to slab-selective pulses, slab selection is improved and canbe traded for SAR.

Another approach to reduce local SAR is to adjust the relative transmitter RF amp-litudes and phases in a way to cancel out local electric field peaks. Therefore, local SARwas added as an additional optimization target to the proposed non-linear optimization.This approach yields local SAR reduction of up to 30% in the shown examples. Thisapproach can be used additionally to prolonging RF pulses. Compared to prolongedpulses, the SAR reduction potential is significantly smaller.

To leverage this SAR reduction, a specialized local SAR supervision for transmitarrays is essential. A conservative approximation of local SAR may overestimate localSAR by an order of magnitude, rendering the excitation techniques proposed in thisthesis impractical in vivo. To overcome this issue, an easy to use, integrated localSAR supervision concept was realized. The proposed supervision system reduces theoverestimation of local SAR for typical use cases to less than 50%. This is a reductionof the safety margin by a factor of 12 for the RF coil used in this thesis. Based onmeasured complex RF pulses using DICOs close to the coil, a local SAR supervision ispossible. However, this system is unable to detect a failure of the RF coil. To furtherimprove patient safety in cases where the RF coil gets damaged during scanning, thissystem can be extended to measure the scattering parameters of the RF coil [32] whichis a good indicator for RF coil health.

While current commercially available transmit array systems still lack an integrationlevel comparable to a clinical scanner, the work done for this thesis allowed to demon-strate novel excitation techniques that leverage the potential of a transmit array forextensive control over the flip-angle distribution and local SAR.

62

7 Appendix

To allow safe application of high power RF pulses for transmit arrays, an online localSAR supervision system was built. This system consists of a measurement setup forcomplex RF voltages and a software component to calculate and limit local SAR basedon this data.

This work was presented at the Annual Scientific Meeting ISMRM 2013 [29,33]

7.1 RF chain calibration

In order to define a calibration procedure for the measurement setup shown in figure3.1, a characterization of the RF system used for transmission and local SAR super-vision is performed. Therefore, the impulse response is measured. A RF pulse withminimal length is transmitted on each transmit channel and measured using the MRreceivers. Due to the sampling rate limitation to fs = 1MHz of the digital base-bandsignal circuitry, the shortest possible pulse si(t) is 1us long and has rectangular shape.The bandwidth on the receive side is limited to ±500kHz.

The allowed frequency range on intermediate frequency level is limited to 1250-2500kHz.The intermediate frequency signal is digitally created from the base-band signal usinga numerically controlled oscillator (NCO) and a digital mixer. The corresponding RFfrequency range of MR resonance band is 296.25 to 297.5Mhz with a local oscillator(LO) frequency of 295MHz using the upper side-band. The RF signal chain is shown infigure 7.1.

To cover the full frequency range, several impulse responses are measured using dif-ferent NCO settings. The resulting frequency response H(ω) is calculated from themeasured signal so(t) by:

H(ω) =Fso(t)Fsi(t)

= Fso(t) ωeiωtrect

2fs sin(ω/(2fs))(7.1)

where si(t) is the input signal which is rectangular RF pulse of length trect.Figure 7.2a shows the frequency response of all eight transmit channels for a NCO

frequency of 2.18MHz (297.18MHz RF). Here, the low-pass filter of the digital downconverter that transforms the signal from the intermediate frequency to the base-bandis the dominant effect seen. This is a receive-only effect. Figure 7.3 shows the frequencyresponse of a single receive channel with a external signal generator connected. Bothfigures are qualitatively similar. Within an interval ±250kHz, the signal amplitudedeviation is below 2%. Only this interval shall be used for local SAR supervision. Ifsignal is detected outside this frequency band, the scanner is stopped. This check isperformed in the frequency domain during signal correction.

63

Figure 7.1: RF signal chain for one transmit channel including all components relevant for localSAR supervision. The directional coupler (DICO) close to the RF coil is the onlycomponent added compared to a standard MRI system.

Figure7.2b shows the corresponding group delay:

τgr = −dφ(ω)

2π dω(7.2)

where φ(ω) is the phase of H(ω).Here, a transmit-receive delay of approximately 4us is observed on all channels within

a frequency interval of ±300kHz. If this delay is the same for all channels, the localSAR measurement is unaffected (see equation 7.5). Also, the non-constant group delayis not an issue if it is the same for all channels. However, the peak-to-peak receiver-receiver delay of approximately τ = 170ns will cause phase differences between channelsand affect the local SAR calculation. Over a frequency interval of ∆f = ±300kHz, amaximum phase difference of 18.4 between channels can be observed with:

∆φ = 2πτ∆f (7.3)

The relative delays between channels are shown in table 7.1. It can be observed thatthe delays are approximately quantized in steps of 25ns. This shows that additionallyto delays caused by different cable lengths, the analog-digital converters are locking onto different edges of the common 40MHz digital clock. Furthermore, this locking isindeterministic and changes after system reset.

Figure 7.4 shows the frequency response of the phase. Two measurement results areplotted, one with a receiver NCO frequency of 297.18MHz and one with a receiver NCOfrequency of 297.08MHz. On first sight, both results show good agreement. Howeverfour out of eight channels show a minor phase difference of approximately 3.6. Thiscorresponds to a delay of 100ns with a receiver NCO frequency offset of 100kHz (see

64

0

0.01

0.02

0.03

0.04

296.68 296.88 297.08 297.28 297.48 297.68 297.88 298.08

Am

pli

tude

[1]

Frequency [MHz]

TX 1TX 2TX 3TX 4TX 5TX 6TX 7TX 8

3.9

4

4.1

4.2

296.68 296.88 297.08 297.28 297.48 297.68 297.88 298.08

Gro

up d

elay

[us]

Frequency [MHz]

Figure 7.2: Frequency response and derived group delay for eight transmit channels(TX1...TX8) . Data was measured at a center frequency of 297.18MHz

equation 7.3). The receiver NCOs of channels 3,4,7 and 8 lack behind 100ns whichis one sample of the 10 mega-samples per second (MSPS) NCO signal (see table 7.1).Repeated measurements on different machines and after several weeks show the sameresults. Thus, the receiver NCO delays are assumed to be constant and are not calibratedduring system installation and after system restart.

Figure 7.4 also shows a global phase offset between transmit channels, although thetransmitted RF signal has identical phase. These offsets have two major components:

1. Different signal delays on different channels. For example, a delay of 100ns causesa phase offset of ±18 peak over a frequency range of 300kHz. Delays of this orderof magnitude are caused internally by some components and are not related tocable length differences. 100ns delay correspond to 20m cable.

2. Small signal path length differences. For example, 2cm of additional cable creates adelay of 0.1ns. This is not within measurement accuracy of the delay measurementsshown above, but will cause a phase offset of 11.

65

0

0.2

0.4

0.6

0.8

1

296.68 296.88 297.08 297.28 297.48 297.68

Am

pli

tude

[1]

Frequency [MHz]

Figure 7.3: Digital downconverter lowpass filter amplitude response. Measured using an ex-ternal signal generator. The line connecting the measurement points is drawn toimprove clarity.

# τ1 τ2

1 −163ns± 1.5ns −2ns± 1.6ns2 −101ns± 1.1ns 1ns± 1.3ns3 3ns± 1.2ns 96ns± 1.3ns4 15ns± 1.1ns 99ns± 1.2ns5 −100ns± 1.1ns −9ns± 1.3ns6 −79ns± 1.2ns −4ns± 1.2ns7 −21ns± 1.1ns 94ns± 1.2ns8 9ns± 1.1ns 108ns± 1.3ns

Table 7.1: Relative ADC delays (τ1) and receiver NCO delays (τ2).

Thus, phase offset and delays have to be measured using different approaches and areseparate correction factors in the final correction function.

In summary, the following system imperfections were observed and have to be correc-ted for reliable local SAR supervision:

• Receiver NCO delays between channels τ2 (hardware specific)

• Receiver signal path delays between channels τ1 (measured after system boot)

• Receiver phase differences φrx (measured once a year)

• Receiver amplitude gain A (measured once a year)

All other errors revealed do either not affect local SAR calculations or are negligible.These are:

66

-1440

-1080

-720

-360

0

360

296.68 296.88 297.08 297.28 297.48 297.68

Phas

e [d

egre

e]

Frequency [MHz]

TX 1TX 2TX 3TX 4TX 5TX 6TX 7TX 8

Figure 7.4: Frequency response of phase for two measurements with different receiver NCO fre-quencies: 297.18MHz, 297.08MHz. Eight transmit channels are shown (TX1...TX8)

• Signal delays identical on all channels

• Non-constant, but similar groupdelay on all channels

• Signal outside of ±250kHz band (forbidden region for RF pulses)

The measurements shown in this section all use the signal path shown in figure 7.1.Using this setup, errors from different components are mixed together. Thus, thesemeasurements are not used for system calibration. There, dedicated calibration signalloops are used as shown in figure 7.5. One signal loop is connecting a single modulatorwith all receivers (1) and a second is connecting all modulators to a single receiver(2). A third signal loop is established during system installation connecting an externaldirectional coupler (DICO) to a reference receiver (2).

The correction function Fc(ω, ...) needed to obtain exact RF voltage measurementcreates a subsample shift of the measured signal. In the time domain, this causes theGibbs phenomenon. A reformulation of the equation 3.1 allows to calculate local SARin the directly in the frequency domain. This avoids time-domain sampling artifacts.

67

Figure 7.5: Signal loops used for calibration

.

Uij =

∫ui(t)u

∗j (t) dt =

∫ui(ω)u∗j (ω) dω

=

∫dω dt1 dt2 e

iωt1u∗i (t1)e−iωt2uj(t2)

=

∫dt1 dt2 δ(t2 − t1)u∗i (t1)uj(t2)

=

∫ui(t)u

∗j (t) dt

where u(ω) = Fu(t)

Using this reformulation leads to the following final equation for local SAR calculation:

SARlocal(x) =∑ij

ωmax∫ωmin

ci(ω′)c∗i (ω

′)ui(ω′)u∗j (ω

′) dω′

Sij(x) (7.4)

It is also shown, that a phase drift φd(ω) = Fφd(t) as a function of time or frequencyapplied to the measured voltage ui(t) := eiφd(t)ui(t) does not change Uij as long as alltransmit channels experience the same drift:

.

Uij =

∫ui(t)u

∗j (t) dt =

∫eiφd(t)e−iφd(t)︸ ︷︷ ︸

=1

ui(t)u∗j (t) dt (7.5)

68

This includes for example delays when the signal is passing through cables or evennon-linear phase shifts caused by filters.

Phase and amplitude calibration

To calibration method shown here is used to calculate the receiver phase correction factorφrx and the receiver amplitude correction factor A. These factors are used to convertmeasured data at the receiver to complex RF voltages at the reference plane, the coilplug.

To measure the relative phases at the coil plug, one receive channel is used as reference.Its input is connected to one transmit pin of the coil plug via an external directionalcoupler which is terminated with 50Ω. The calibration is repeated for all transmit pins ofthe coil plug. This calibration targets passive component imperfections and is assumednot to change with time. Therefore, this calibration is repeated only once a year.

During the calibration process, a rectangular RF pulse is transmitted through thestandard transmit chain. It is measured using the integrated voltage measurement setup(DICO and MR receivers) and the reference receiver simultaneously. Due to the simul-taneous measurement, RFPA thermal drift does not affect the measurement.

From the measured data, two sets of correction factors are calculated: The first set isapplied to the data received through the integrated DICOs to yield the same phase andamplitude values as measured by the reference receiver:

∆φrx = −φint + φext (7.6)

Arx =Aext

Aint(7.7)

where φint is the phase measured using the internal DICO and φext is the phase measuredusing the external DICO. Arx is an amplitude scaling factor to convert the amplitudemeasured using the internal DICO Aint to the amplitude measured with the externalDICO Aext.

The second set is used to create a plane wave at the coil plug and is applied to theRF characteristic:

∆φtx = −φext (7.8)

The calibration values including standard deviation are shown in table 7.2.

To calibrate the signal amplitude in units of Volt, the power absorption limiter (PALI)is used as reference because it has calibrated power meters integrated. Therefore, RF istransmitted within a 10s interval and the power integral of simultaneous measurementstaken from PALI and DICOs is evaluated. Because the PALI does not measure powerat the reference plane (coil plug), but much earlier in the transmit chain, the cable losshas to be measured separately.

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# ∆φrx Arx ∆φtx

1 −22.972 ± 0.0029 1.3894± 8.1 · 10−5 94.94 ± 0.068

2 −29.855 ± 0.0034 1.4755± 10 · 10−5 44.23 ± 0.073

3 −108.529 ± 0.0057 1.4210± 9.4 · 10−5 112.73 ± 0.059

4 −34.023 ± 0.0085 1.4522± 10 · 10−5 159.37 ± 0.043

5 −103.29 ± 0.033 1.5585± 10 · 10−5 105.21 ± 0.064

6 −19.64 ± 0.034 1.4953± 18 · 10−5 −76.48 ± 0.048

7 −128.32 ± 0.027 1.5348± 10 · 10−5 123.86 ± 0.048

8 −16.45 ± 0.026 1.4331± 9.7 · 10−5 140.65 ± 0.060

Table 7.2: Amplitude and phase calibration data acquired with signal loops (2) as shown infigure 7.5

# / APALI day 0 day 7 day 14

1 1386.5 1386.7 1386.12 1351.4 1351.7 1351.63 1376.2 1376.2 1376.14 1388.6 1388.8 1388.45 1378.7 1379.4 1378.86 1371.4 1373.3 1373.17 1377.7 1376.7 1377.68 1365.5 1363.4 1365.1

Table 7.3: Amplitude calibration to convert ADC values to units of Volt

For each transmit channel, a scaling factor APALI,i can be calculated to convert thesignal measured by the external DICO to calibrated voltage values:

APALI,i =

√∫Pfwd,i(t) dt− Pcable∫

(Arx,iso,i(t))2 dt(7.9)

where Pcable is the cable loss from the PALI to the coil plug, Pfwd,i is the forwardRF power measured for transmit channel i at the PALI. so,i(t) is the signal amplitudemeasured using the internal DICO for channel i. This gives a calibration value forthe transmit amplitude at the coil plug. The cable loss can either be measured byevaluating the reflected power at PALI level, or the value can be calculated based oncable specifications.

Table 7.3 shows the measured values for APALI over three weeks. The peak-to-peakdifference of all values is 2.7%.

From this data, the amplitude correction factor Ai can be calculated for each channelto convert the signal amplitude measured using the internal DICO to units of Volt. Toimprove accuracy, the mean value of APALI over all transmit channels is applied. The

70

resulting receiver amplitude gain is:

Ai = Arx,i

∑j

APALI,j

nc(7.10)

where nc is the number of transmit channels.

Component delay calibration

The calibration procedure shown here is used to calculate the receiver delays τ1. Thisprocedure is repeated after each system startup, because the receiver delays and themaster modulator delay may change after system reset. Therefore, specific calibrationsignal loops are used to minimize the number of involved components. To measure thereceiver delay, loop (1) (fig. 7.5) is used to transmit a small signal rectangular RF pulsefrom one modulator to all receivers. Two data-sets are measured using a frequencyoffset of foffset = 200kHz. The resulting data is calibrated in the frequency domain toremove the NCO delays and then transformed to the time domain. The resulting phasedata is averaged over the RF pulse to increase SNR. Then, the transmit-receive delay iscalculated for each channel individually:

τ1,i =

6

( ∑ts0,i(t)∑

tsfoffset,i

(t)

)2πfoffset

(7.11)

where s?,i(t) is the complex signal measured a frequencies 0 and foffset.Because a common transmit-receive delay of all channels does not affect local SAR

calculation, all delays are finally applied relative to one channel:

τ1,i = τ1,i −max(τ1,i

)(7.12)

The same procedure is repeated using loop (2) to calibrate the master modulator delay.There, all modulators transmit to a single receiver one after the other. The resultingdelay τ3,i is not relevant for local SAR calculation, but is necessary for the DICO testto succeed. Furthermore, it needs to be known to achieve good image quality, becausethe relative transmit phases need to be predictable for pulse design (see chapter 2) .

7.2 Error estimation

To guarantee patient safety, measurement errors have to be accounted for. Due to thehigh SNR of the voltage measurement, statistical errors are small compared to systematicerrors caused by inaccurate system calibration. These errors do change slowly within longtime periods. Therefore, they are assumed to be constant within a six minute intervalof SAR averaging. The measured voltage including channel-dependent time-invariantmultiplicative errors e is given as:

ui(t) = eiui(t) (7.13)

71

Putting this together with equation 3.1 leads to:

SARlocal(x, tsw, t) =∑ij

t∫t−tsw

eiui(t′)u∗j (t

′)e∗i dt′

Vij(x)

=∑ij

ei

t∫t−tsw

ui(t′)u∗j (t

′) dt′

Vij(x)e∗i

=∑ij

eiUij(t, tsw)Vij(x)e∗i = eV(t, tsw,x)e∗ (7.14)

where Uij(t, tsw)Vij(x) = V(t, tsw,x). This is a quadratic form that is positive semi-definite.

Considering component specifications and regular consistency checks, a maximumphase φe and amplitude error Ae can be estimated. To calculate the worst-case localSAR including errors, equation 7.15 has to be maximized within the given error margins.

SARlocal,max(x, tsw, t) = max(eV(t,x)e∗

)s.t. (1−Ae) < |ei| < (1 +Ae)

−φe < 6 ei < φe (7.15)

An easy solution of this problem can be obtained using a linear approximation if errorsare small. This gives a direction towards higher SAR values for each component of e.Because equation 7.15 is convex, the maximum allowed error is then assumed in thisdirection. The first derivative is given by:

SAR′local,i =dSARlocal

de

∣∣∣∣e=1

= 2∑j

Ui,jVi,j (7.16)

The approximated maximum error vector is then given by:

ei =(

1 +Ae sgn(<(SAR′i)

))eiφe sgn(=(SAR′i)) (7.17)

To validate this approximation, the approximated SAR and the real SAR is calculatedfor 108 random u, a VOP data set of a eight channel head coil and error vectors witha upper error bound of Ae = 0.1 and φe = 5. Figure 7.6 shows a probability densityfunctions comparing the local SAR calculated with the proposed error approximation(green, α1) and without (red, α2). The local SAR over-estimation factor is plotted, i.e.values greater than one show a safe, but performance limiting over-estimation of localSAR and values smaller one show a SAR under-estimation which has to be avoided toguarantee patient safety.

72

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5 3

Pro

bab

ilit

y

Local SAR over-estimation

α1α2

Figure 7.6: Probability density functions for local SAR over-estimation calculated with pro-posed error approximation (α1) and without (α2).

α1 =SARlocal,max

SARlocalα2 =

SARlocal,ideal

SARlocal(7.18)

While the approximated SAR never under-estimates the local SAR, the local SARvalues calculated without accounting for errors are approximately gaussian distributedaround one. Even significant upscaling of these values does not remove under-estimationsreliably. In the shown example, in 73ppm of the evaluated cases, the calculated SAR isless than 50% of the true value.

Using the approximated local SAR, 50% of all evaluated cases show a over-estimationof less than 25%, for 90% of all cases, the over-estimation is less than 50% (see fig. 7.7).

7.3 Hardware control

To test if all system components are within specifications, two regular consistency checksare implemented. All signal paths used for local SAR supervision (see figure 3.1) aretested. All components except the cabling from the DICOs to the RF coil plug can betested by transmitting RF to the coil and checking the data measured for local SARsupervision (DICO test). To check the cabling to the coil, RF is transmitted to the opencoil plug which causes total reflection at this point with a phase jump of 180. Both,reflected and forward signal is evaluated by the receive system (Reflection test). Thistest can only be performed without RF coil connected. Therefore it is performed aftersystem boot. The DICO test is performed prior to each measurement.

DICO test

The DICO test is used to test the cabling from the DICOs to the receivers, the switchmatrix, the DICOs, the receivers (especially for changed delays) and all associated calib-

73

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

p

local SAR over-estimation

α1α2

Figure 7.7: Accumulated probability density functions for local SAR over-estimation calculatedwith proposed error approximation (α1) and without (α2).

ration parameters. Therefore, a 50V rectangular RF pulse is transmitted on all channelswith a phase difference of 0. This is repeated for three different frequencies. Becausethe receive NCO is automatically shifted to the center transmit frequency, both, NCOand receiver delays affect the signal.

The received signal is calibrated in the frequency domain using equation 3.2 and thenFourier transformed to the time domain. Signal amplitudes and phases are compared toinstructed amplitudes and phases. Because the application of the calibration data causesgibbs ringing in the time domain, only the center half of the rectangular RF pulse isevaluated.

It is evaluated if the accumulated errors of all involved components are below themaximum allowed error margins.

min(|u(t)|) ≥ (1−Ae)50V max(|u(t)|) ≤ (1 +Ae)50V (7.19)

maxijt

(6(ui(t)u

∗j (t)))

< φe (7.20)

If these equations are not fulfilled, the test fails and further scanning is not possible.

Reflection test

The reflection test is used to detect any cable issues that might occur between the direc-tional couplers and the coil plugs. This includes broken cables accidentally interchangedcables. Therefore, a defined RF pulse is transited to the open coil plug. Forward andreflected voltage is measured using the directional couplers. The derived parameterβi =

∫ufwd,i(t)/uref,i(t) dt is compared to values measured at a time when all connec-

tions are known to be within specifications. If 6(βref,i/βi

)is larger than 1, the test is

failed. This corresponds to a cable length difference of approximately 2mm.

74

Establishing realtime behaviour

In this work, the local SAR supervision algorithms shown above were implemented on thestandard non-real-time image reconstruction computer. However, real-time processing isstrictly necessary to guarantee patient safety. Implementing this functionality tends tobe difficult as add-on to existing MRI systems because, for example, components suchas MR receivers are not accessible in real-time and calculation delays on a standardcomputer occur. This latency tl between the application of RF to the patient and itssupervision must be controlled precisely. Therefore, a simple real-time logic implementedon an FPGA in combination with a conventional real-time global power supervisionunit is used to establish a local SAR calculation system similar to a true real-timeimplementation.

Figure 7.8 shows the accumulated unsupervised RF time created by measurementand calculation latencies. After one chunk of RF is measured (e.g. 4ms or one RFpulse) it is sent to the image reconstruction computer where the evaluation takes place.This evaluation needs a short, but undefined amount of time. This undefined behavioris especially caused by other tasks such as image reconstruction running on the samecomputer. The associated latencies are of the order of milliseconds.

Figure 7.8: Unsupervised local SAR due to calculation latency.

Until this evaluation is finished, no local SAR value is known for RF power that alreadyheated the patient. For this time period, a faster, but less accurate local SAR estimate isused. Therefore, the non-phase-sensitive power supervision unit, the power absorptionlimiter (PALI) with an update-rate of 25ns is used. The local SAR calculated fromequation 3.1 is split in two components: A time period with exact local SAR supervisionbased on phase-sensitive data, and a much shorter time period with approximated localSAR supervision based on RF power data only:

75

SARlocal(x, tsw, t0) = SARlocal(x, tsw − tl, t0 − tl) + SARlocal(x, tl, tl)

= SARlocal(x, tsw − tl, t0 − tl) +

t0∫t0−tl

u(t)S(x)u∗(t) dt

≤ SARlocal(x, tsw − tl, t0 − tl) + λmax50Ω

t0∫t0−tl

P (t) dt (7.21)

where λmax is the maximum eigenvalue of all VOPs and P (t) is the total current RFpower transmitted on all channels.

For the latter time period, an approximation based on the Rayleighquotient is used:

u∗uλmin ≤ u∗V(x)u ≤ u∗uλmax (7.22)

This approximation is chosen such that the local SAR may be overestimated, butnever underestimated.

This maximum processing latency tl can be chosen freely. To minimize the secondaddend in equation 7.21 and thus maximize system performance, tl should be chosen assmall as possible.

To make sure that the actual calculation delay time never exceeds the defined max-imum delay time, a simple watchdog logic is implemented on a FPGA as shown in figure7.9. This logic uses two one bit inputs and one output to disable the RFPA if necessary.A counter measures the actual processing delay. While the RFPA is transmitting RF,it sends logical true signals through the RFPA TX line. The counter counts up. If theonline local SAR calculation has checked another millisecond of RF, it sends a pulseon the RF check line which decrements the counter by one millisecond. If the counterreaches zero or the maximum allowed calculation latency, the RFPA is turned off andthe measurement is aborted. Note that this logic supervises RF time, not measurementtime.

The final value for the maximum allowed delay time was chosen as 10ms. The localSAR estimation used here can be an order of magnitude larger than the true value.Therefore, it is very important to choose the maximum allowed delay time as smallas possible. However, it is very unlikely that this estimated value is actually reached.Multiple conditions have to be fulfilled:

• The running sequence must create more SAR than predicted by the local SARlookahead

• The transmit phases have to resemble the worst case phase combination associatedwith λmax

• The RFPA must transmit at high power for a relatively long time which is unlikelygiven typical RF pulses

76

Figure 7.9: FPGA logic to implement a watchdog that supervises the calculation delay time.

For practical reasons, the two addends in equation 7.21 have independent SAR limits.The regular local SAR limit (first addend) is chosen as 20W/kg for 10s as defined fornormal mode operation. The second addend (approximated supervision) has a limit of10W/kg for 10s. This gives a total local SAR limit of 30W/kg over 10s which is abovenormal mode operation limits, but below first level controlled mode operation limits.Local SAR above normal mode limits is only reached in case of a system failure. Thisleads to a power limit of 3.5kW for a 10ms average for a eight channel local body coil.Given a peak RFPA power of 4.5kW at the coil plug, this does not limit the user fromusing full RFPA power.

77

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[56] S. Saekho, C.-y. Yip, D. C. Null, F. E. Boada, and A. V. Stenger. Fast kz three-dimensional tailored radiofrequency pulse for reduced b1 inhomogeneity. MagneticResonance in Medicine, pages 719–742, 2006.

[57] K. Setsompop. Design Algorithms for Parallel Transmission in Magnetic ResonanceImaging. PhD thesis, MIT, 2008.

[58] K. Setsompop, L. Wald, V. Alagappan, B. Gagoski, and E. Adalsteinsson. Mag-nitude least squares optimization for parallel radio frequency excitation designdemonstrated at 7 tesla with eight channels. Magnetic Resonance in Medicine,59(4):908, 2008.

[59] K. Setsompop, L. Wald, V. Alagappan, B. Gagoski, F. Hebrank, U. Fontius,F. Schmitt, and E. Adalsteinsson. Parallel rf transmission with eight channelsat 3 tesla. Magnetic Resonance in Medicine, 56(5):1163, 2006.

[60] A. Stenger, S. Saekho, Z. Zhang, S. Yu, and F. E. Boada. B1 inhomogeneityreduction with transmit SENSE. Proceedings of the 2nd International Workshop onParallel Imaging, Zurich, Switzerland, 2004.

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List of peer-reviewed conference abstracts

R. Gumbrecht, B. Gagoski, and E. Adalsteinsson. Optimized chemical shift selectivesuppression for ptx systems at 7t. Proceedings 18th Scientific Meeting ISMRM, Stock-holm, Sweden, 2010.

R. Gumbrecht, J. Lee, H.-P. Fautz, D. Diehl, and E. Adalsteinsson. Fast high- flip ptxpulse design to mitigate b1+ inhomogeneity using composite pulses at 7t. Proceedings18th Scientific Meeting ISMRM, Stockholm, Sweden, 2010.

B. Gagoski, R. Gumbrecht, M. Hamm, K. Setsompop, B. Keil, J. Lee, K. Makhoul,A. Mareyam, K. Fujimoto, T. Witzel, U. Fontius, J. Pfeuffer, E. Adalsteinsson, and L.L. Wald. Real time rf monitoring in a 7t parallel transmit system. Proceedings 18thScientific Meeting ISMRM, Stockholm, Sweden, 2010.

R. Gumbrecht, E. Adalsteinsson, P. Mueller and H.-P. Fautz. RF energy reduction byparallel transmission using large-tip-angle composite pulses. Proceedings 20th ScientificMeeting ISMRM, Montreal, Canada, 2011.

R. Gumbrecht and H.-P. Fautz. 2D composite pulses: A novel method for spatiallyselective excitation. Proceedings 20th Scientific Meeting ISMRM, Sydney, Australia,2012.

R. Gumbrecht and H.-P. Fautz. Fast non-linear ptx pulse design with integrated peaklocal rf energy minimization. Proceedings 20th Scientific Meeting ISMRM, Sydney, Aus-tralia, 2012.

R. Gumbrecht and H.-P. Fautz. Measurement of the scattering parameter matrix of apTx coil using a transmit array. Proceedings 30th Annual Scientific Meeting ESMRMB,Toulouse, France, 2013.

R. Gumbrecht, U. Fontius, H. Adolf, T. Benner, F. Schmitt, E. Adalsteinsson, L. Wald,and H.-P. Fautz. Online local SAR supervision for transmit arrays at 7T. Proceedings21th Scientific Meeting ISMRM, Salt Lake City, USA, 2013.

R. Gumbrecht, T. Benner, U. Fontius, H. Adolf, A. Bitz, and H.-P. Fautz. Safe onlinelocal sar calculation for transmit arrays using asynchronous data processing. Proceed-ings 21th Scientific Meeting ISMRM, Salt Lake City, USA, 2013.

R. Gumbrecht, P. Gross, and H.-P. Fautz. Flexible multi-echo b0 mapping with voxelbased phase unwrapping. Proceedings 30th Annual Scientific Meeting ESMRMB, Toulouse,France, 2013.

A. Bitz, R. Gumbrecht, S. Orzada, H.-P. Fautz, and M. Ladd. Evaluation of virtualobservation points for local sar monitoring of multi-channel transmit rf coils at 7 tesla.Proceedings 21th Scientific Meeting ISMRM, Salt Lake City, USA, 2013. (Awarded withthe 2nd poster price in the category MR Safety)

List of patents

R. Gumbrecht, D. Diehl, J. Nistler, M. Vester, S. Wolf and W. Renz. 2012. Magneticresonance apparatus and method for determining a pulse sequence to feed an RF radi-ating coil. U.S. Patent 8,120,359, filed June, 19, 2009, and issued February, 21, 2012.

R. Gumbrecht and D. Diehl. 2013. Method to determine parameters to control thegradient coils and radio-frequency coils of a magnetic resonance device. U.S. Patent8,354,845, filed June, 18, 2010, and issued January, 15, 2013.

R. Gumbrecht and H.-P. Fautz. 2013. pTX Breitbandpulse mit optimiertem SAR,Schichtprofil und raumlicher Anregungstreue. German Patent DE102010081509, filedAugust, 24, 2011, and issued March, 28, 2013

R. Gumbrecht and H.-P. Fautz. 2012. Method and device for determining a mag-netic resonance system activation sequence. U.S. Patent application 2012/0268130, filedMarch, 23, 2011, published October, 25, 2012.

R. Gumbrecht and H.-P. Fautz. 2012. 2D Composite Pulses for spatially selective ex-citation using parallel transmission. German Patent application 102012205664.5, filedApril, 5, 2012.

R. Gumbrecht and H.-P. Fautz. 2012. Fast high flip-angle RF pulse design with integ-rated local SAR optimization. German Patent application 102012205297.6, filed March,30, 2012.

D. Paul, H.-P. Fautz, R. Gumbrecht, and T. Bachschmidt. 2012. Fat-water separationin SPACE using band-selective excitation. German Patent application. 102012216353.0,filed September, 14, 2012.

R. Gumbrecht and H.-P. Fautz. 2012. SAR Reduktion mittels B0 spezifischer HF An-regung. German Patent application 102012220462.8, filed November, 9, 2012.

R. Gumbrecht, H.-P. Fautz, H. Adolf, J. Fontius, and T. Benner. 2013. Online SAR su-pervision with non-real-time system. German Patent application 102013205651.6, filedMarch, 28, 2013

R. Gumbrecht, H.-P. Fautz, H. Adolf, J. Pontius, T. Benner. 2013. System zur Kalibrier-ung und Funktionsuberwachung von Mehrkanal-Hochfrequenz-Sendesystemen. GermanPatent application 102013206570.1, filed April, 12, 2013

List of Figures

3.1 Transmit array architecture including hardware for local SAR supervision.Four transmit channels are shown. . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Directional couplers (a) and switch matrix (b) placed at the back of themagnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1 Derivative exchange process used for Jacobi matrix calculation . . . . . . 26

4.2 Schematic RF amplitudes for a three sub-pulse composite pulse usingrectangular sub-pulses. Two transmit channels are shown. RF phaseschange in every sub-pulse and channel but are not shown in this figure. . 31

4.3 Sequence diagram of a CHESS pulse. . . . . . . . . . . . . . . . . . . . . . 33

4.4 Excitation strategy of 2D-Composite Pulses . . . . . . . . . . . . . . . . . 36

4.5 2D Composite pulse sequence diagram . . . . . . . . . . . . . . . . . . . . 37

4.6 B0 map of several transversal slices of the human head at 7T (a) and thecorresponding anatomical images (b) . . . . . . . . . . . . . . . . . . . . . 40

4.7 Flip-angle over frequency of a three sub-pulse composite pulse with gaus-sian sub-pulses. Two voxels optimized for different B0 are shown . . . . . 41

5.1 Online calculated local SAR vs. predicted local SAR for several arbitraryTX Array RF pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Online calculated local SAR vs. predicted local SAR for several arbitraryTX Array RF pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3 Runtime of the jacobi matrix calculation using the optimized GPU code. . 47

5.4 B+1 map of a spherical phantom measured using individual channels of an

eight channel transmit array. . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.5 Measured and simulated 90 composite excitation pulse. Flip-angle dis-tribution is shown in degree . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.6 Flip-angle distribution of a conventional slice-selective RF pulse with RFshim applied (a), a 90 optimized Composite Pulse (b) and a 180 op-timized Composite Pulse (c). (a) and (b) are measured results, (c) isa simulation result because flip-angle distributions of 180 pulses couldnot be measured. Note the different scaling necessary to show residualmagnetization inhomogeneities. . . . . . . . . . . . . . . . . . . . . . . . 50

5.7 Flip-angle distribution of a 90 optimized Composite Pulse (a) a 90 op-timized Spokes pulse (b). Results were measured using the B+

1 mappingsequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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5.8 Measured flip-angle distribution of a Composite Pulse optimized on reson-ance frequency (0Hz) (a-e) and optimized for a frequency band of ±40Hz(f-j). Both pulses are measured with RF center frequencies of -40Hz,-20Hz, 0Hz, 20Hz and 40Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.9 Longitudinal magnetization left from individual sub-pulses (left) and finallongitudinal magnetization after application of the full pTx CHESS pulse(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.10 Simulated water suppression achievable for pTx CHESS over for T1=1s(a), T1=3s (b) and T1=5s (c). . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.11 Simulated water suppression achievable for conventional CHESS (red) andpTx CHESS (blue) over T1. . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.12 Simulation results showing the flip-angle of a conventional excitation (a)and a 2D Composite excitation (b) . . . . . . . . . . . . . . . . . . . . . . 54

5.13 2D Composite pulses measured using a 3D gradient recalled echo sequenceon a phantom. (a,b) show two slices of the measured data, (c,d) show aschematic of the corresponding excitation regions as they are defined infigure 4.4. (a) shows the center slice and (b) an off center slice using adifferent windowing to show residual magnetization in region D. . . . . . . 55

5.14 Normalized peak local pulse power versus flip-angle root mean squareerror for Composite Pulses (a) and Spokes (b). Optimization results usinglocal SAR optimization (solid) and global RF power optimization (dashed)are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.15 Conventional (b) and composite pulse (a) excitation using the same RFenergy. The simulated excitation flip-angle distribution is plotted. Bothpulses have a total length of 9ms. (a) has the same excitation quality asa 1ms conventional excitation pulse. . . . . . . . . . . . . . . . . . . . . . 57

5.16 Forward RF energy for 90 excitation pTx composite pulses (solid) versuspulse length. The dashed line is a theoretical lower bound for the for-ward RF energy, assuming compensation of B0 effects does not requireadditional RF energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.17 Spin echo images measured at a 3T transmit array with refocussing planeorthogonal to excitation plane creating a 2D selective excitation. Therefocussing pulse slab-profile is shown. (a) is measured with a conventionalwindowed sinc refocussing pulse. (b) is measured with a three sub-pulseComposite Pulse for refocussing without pulse optimization. The pulse istwo times longer than (a). (c) is measured similar to (b) but with pTxpulse optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.18 Spin echo images measured at a 3T transmit array with refocussing planeorthogonal to excitation plane creating a 2D selective excitation. The re-focussing pulse slab-profile is shown. (a) is using a conventional windowedsinc refocussing pulse. (b) uses a three sub-pulse Composite Pulse thathas approximately the same pulse length compared to (a). . . . . . . . . 60

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7.1 RF signal chain for one transmit channel including all components relevantfor local SAR supervision. The directional coupler (DICO) close to theRF coil is the only component added compared to a standard MRI system. 64

7.2 Frequency response and derived group delay for eight transmit channels(TX1...TX8) . Data was measured at a center frequency of 297.18MHz . . 65

7.3 Digital downconverter lowpass filter amplitude response. Measured usingan external signal generator. The line connecting the measurement pointsis drawn to improve clarity. . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7.4 Frequency response of phase for two measurements with different receiverNCO frequencies: 297.18MHz, 297.08MHz. Eight transmit channels areshown (TX1...TX8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.5 Signal loops used for calibration . . . . . . . . . . . . . . . . . . . . . . . . 687.6 Probability density functions for local SAR over-estimation calculated

with proposed error approximation (α1) and without (α2). . . . . . . . . . 737.7 Accumulated probability density functions for local SAR over-estimation

calculated with proposed error approximation (α1) and without (α2). . . . 747.8 Unsupervised local SAR due to calculation latency. . . . . . . . . . . . . . 757.9 FPGA logic to implement a watchdog that supervises the calculation delay

time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

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