Aliaksandra Shymanskaya^{1}, Wieland A. Worthoff^{1}, and N. Jon Shah^{1,2}

^{23}Na-MRI can be used for non-invasive investigation of
metabolic disease, based on discrimination between sodium signals arising from
different tissue compartments due to development of multiple quantum coherences
mainly in intracellular space. Strongly modulated pulses in NMR and MRI can be
created using optimal control design to generate the most efficient transfer between the initial
and target states of the nuclear spin ensemble, defined by the density matrix
formalism. The Krotov algorithm and its implementation
by Maximov of the optimal control design were used to modify the first hard RF
pulse in the SISTINA sequence.

Simulations:
The SISTINA
sequence^{5} is an improved modification of the TQF, which allows
simultaneous creation and measurement of single-quantum coherences (SQCs) and
triple-quantum coherences (TQCs). SISTINA consists of three RF pulses with
preparatory time τ and a single ultra-short echo time UTE
read-out between the first and the second pulse for measuring SQCs, and
evolution time δ between the second and the
third pulse, followed by Cartesian read-outs for measuring TQCs. The evolution of
the ensemble of nuclear spin-3/2 during sequence performance is fully described
by density matrices and is calculated with MATLAB^{6} for any given
time.
Usually the
transfer between the states in a quantum system is provided through hard RF
pulses. However, strongly modulated pulses (SMP) can be used instead. SMP induce
the transfer between states with higher efficiency, which is given by the
quality function, calculated as an overlap between the desired state and the
achieved state, with energy deposition being the penalty factor. The search of
the optimal solution of the problem is performed using the Krotov algorithm,
with the control parameter being pulse amplitude. In this research, the method^{2,3}
is adapted to sodium imaging with the SISTINA sequence. The dynamics of the
system is given by the Liouville von Neumann equation, with the Hamilton
operator defined as:

$$\hat{H}(t)=\hat{H}_{0}+\sum_{k=x,y}\omega_{1k}(t)\hat{I}_{k}$$

It describes the behaviour of the ensemble during the RF pulse with duration T, with $$$\omega_{1k}(t)$$$ - characteristic time dependent amplitude of the RF pulse. Here the influence of the static quadrupole Hamiltonian is investigated. For the simulations the asymmetry parameter in the unordered biological tissue is $$$\eta=0$$$, so that the governing Hamiltonian during the pulse is given by:

$$\hat{H}_{0}=\frac{\omega_{Q}}6\cdot[2\hat{I}_{z}^{2}-\hat{I}_{x}^{2}-\hat{I}_{y}^{2}]$$

The starting point for the optimization is a
rectangular pulse with amplitude $$$\omega_{1x}=3925$$$ Hz, $$$\omega_{Q}=60$$$
Hz^{2,3}.

Experimental
Conditions and Measurement Parameters:
Phantom experiments
were performed on a home-built, small-bore 9.4T scanner using a Siemens console
with a home-built, single-tuned Na coil, using a 64cmx64cmx32cm field-of-view with
0.1cm isotropic resolution. A 12-step phase cycle was implemented with 2
averages and a repetition time of TR = 150ms yielding a measurement time of
approx. 24min. A preparatory time is τ = 7000µs, the evolution time is δ = 40µs. A UTE readout was performed at TE_{UTE }= 0.7ms using the DISCOBALL^{7}
scheme. A Cartesian readout was performed at 5ms after the third RF pulse. The
phantom consisted of two cylindrical tubes with diameters of 30mm and 11mm, whereas the
small tube was filled with NaCl water solution, the larger tube was filled with
agarose gel mixed with NaCl of the same concentration.

1. Ivchenko N, et al. Multiplex phase cycling. J.Magn. Reson. 2003; 160: 52–58.

2. Maximov I, et al. Optimal control design of NMR and dynamic nuclear polarization experiments using monotonically convergent algorithms. J. Chem. Phys. 2008; 128.18: 184505.

3. Maximov I, et al. A smoothing monotonic convergent optimal control algorithm for nuclear magnetic resonance pulse sequence design. J. Chem. Phys. 2010; 132.8: 084107.

4. Krotov V, Global methods in optimal control theory. CRC Press, 1995; Vol. 195.

5. Fiege D.P, et al. Simultaneous Single-Quantum and Triple-Quantum- Filtered MRI of 23Na (SISTINA). Magn Reson Med 2013; 69:1691–1696.

6. MATLAB and Statistics Toolbox Release 2015b, The MathWorks, Inc., Natick, Massachusetts, United States.

7. Stirnberg R, et al. A new and versatile gradient encoding scheme for DTI: a direct comparison with the Jones scheme. Proc Int Soc Mag Reson Med 2009; 17:3574.

An
example of a first SMP generated for the first phase cycling step with time
dependence of two amplitude components $$$\omega_{1x}$$$ (blue) and $$$\omega_{1y}$$$
(red).

The example of a time evolution of $$$\omega_{1x}$$$ component of the first SMP generated for the
first phase cycling step.

The
example of a time evolution of $$$\omega_{1y}$$$ component
of the first SMP generated for the first phase cycling step.

SISTINA
sequence with a first SMP. Image a) was acquired with the UTE read-out and is
weighted due to sodium concentration. A
triple quantum weighted image b) and a single quantum weighted image c) are
acquired after the third RF pulse with the Cartesian read-out; all three images show the expected contrast. The phantom
description is in the text.