Introduction

Recently the feasibility of sodium Magnetic Resonance Fingerprinting (²³Na-MRF) was presented as a proof of concept^1,2, enabling the simultaneous quantification of T₁,T_2s^*,T_2l^*,T₂^* and ΔB₀. Here, non-steady state conditions were generated via temporally varying flip angles (FAs) and sequence timings to generate a unique signal evolution for a given parameter set. In this work, ²³Na-MRF was extended in two ways: Firstly, the coverage was changed from 2D to 3D, allowing for simultaneous quantification of T₁,T_2s^*,T_2l^*,T₂^* and ΔB₀ in a 3D volume within approximately the same measurement time. Secondly, irreducible spherical tensor operators (ISTOs) were exploited for the simulations, which allow a more sophisticated signal simulation for spin 3/2-nuclei.

Methods

The 3D-radial implementation of the previously proposed ²³Na-MRF sequence¹ uses a variable non-selective excitation pulse followed by a variable echo time (Fig.1) and a density adapted center-out radial readout³. Subsequently, a gradient rewinder is played out and a 2π spoiler gradient is switched along the z-direction, followed by the next excitation pulse. Each time frame was acquired with 150 center-out spokes which are distributed equidistantly in k-space, whereas subsequent time frames were rotated via the 13^th tiny golden angle⁴ to achieve full sampling over the entire measurement. The dictionary simulation was based on ISTOs as proposed by Hancu et al.⁵, which are based on spectral density parameters ($$$J_0\geq J_1\geq J_2$$$), the off-resonance ΔB₀ and the residual quadrupolar moment ω_q. Relaxation towards M₀ was enabled⁶. Under the assumption that ω_q is negligible, the relaxation times can be calculated as⁷:
$$T_{1l}=\frac{1}{2J_2};T_{1s}=\frac{1}{2J_1} [1] $$
$$T_{2l}=\frac{1}{J_1+J_2};T_{2s}=\frac{1}{J_0+J_1} [2]$$
To introduce the apparent transverse relaxation T₂^* (T_2l^*,T_2s^* respectively) each parameter set (J₀,J₁,J₂,ω_q), with central off-resonance ΔB₀, was simulated for an off-resonance range of $$$\overline{ΔB_0}=[ΔB_0-100,ΔB_0-99,...,ΔB_0+100]$$$ Hz. The resulting signal evolutions were interpolated onto a 0.5Hz grid and summed along the $$$\overline{ΔB_0}$$$-direction, weighted with a Lorentzian distribution p_L. Different T₂^* values were constructed by varying the width of p_L such that $$$T_2>T_2^*>0.4T_2$$$ ($$$T_{2l}>T_{2l}^*>0.6T_{2l}$$$ respectively) in 1ms steps.
The expected signal evolution of a ²³Na inversion recovery (IR) experiment is given by:
$$S(t)=S_0(1-2(0.8e^{-t/T_{1l}}+0.2e^{-t/T_{1s}})) [3]$$
However, in literature T₁ is mostly modelled monoexponentially since the separation of T_1l and T_1s based on fitting the inversion recovery signal curve is challenging. Therefore, the first order Taylor expansion of a biexponential function with $$$a+b=1$$$ is considered:
$$ae^{\overline{a}t}+be^{\overline{b}t}=a\sum_{n=0}^\infty \frac{(\overline{a}t)^n}{n!}+b\sum_{n=0}^\infty \frac{(\overline{b}t)^n}{n!}\approx(a+b)+a\overline{a}t+b\overline{b}t=1+(a\overline{a}+b\overline{b})t\approx \sum_{n=0}^\infty \frac{((a \overline{a}+b \overline{b})t)^n}{n!}=e^{(a \overline{a}+b \overline{b})t} [4]$$
Comparison with equation [3] yields:$$$a=0.8$$$,$$$\overline{a}=\frac{-1}{T_{1l}}$$$,$$$b=0.2$$$ and $$$\overline{b}=\frac{-1}{T_{1s}}$$$, resulting in the monoexponential estimate:
$$\frac{1}{T_1}=\frac{0.8}{T_{1l}}+\frac{0.2}{T_{1s}} [5]$$
The dictionary was simulated as follows:
1. Set parameter space: biexponential: T₁=[20,21,…70]ms, T_2l=[15,16,…60], T_2s=[1.0,1.3,…14.8], ΔB₀=[-50,-48,…50]Hz;
monoexponential: T₁=[30,31,…90]ms, T₂=T_2l=T_2s=[5,6,…80]ms, ΔB₀=[-50,-48,…50]Hz
2. Calculate T_1l and T_1s using equations [1],[2] and [5]
3. Delete parameter sets that violate $$$T_{1l} \geq T_{2l} \geq T_{1s} \geq T_{2s}$$$ ($$$\widehat{=}J_0 \geq J_1 \geq J_2$$$)
4. Convert parameter sets into spectral density parameters using equations [1] and [2]
5. Simulate signal evolution for each parameter set
6. Interpolate along $$$\overline{ΔB_0}$$$-axis to a step size of 0.5Hz
7. Sum up signal evolutions weighted with variable p_L along the $$$\overline{ΔB_0}$$$-axis such that $$$T_{2l}>T_{2l}^*>0.6 T_{2l}$$$ (monoexponential: $$$T_{2}>T_{2}^*>0.4 T_{2}$$$) ; T_2l^* (T₂^*) in 1ms steps; central ΔB₀ = [-50,-48,…,50]Hz
8. Concatenate biexponential and monoexponential parts
9. Compress dictionary using singular value decomposition (SVD) up to rank 12

The reconstruction was performed via a low rank alternating direction method of multipliers approach^8,1.
Measurements (resolution: 3x3x3mm³) were performed on a 7T whole-body system (Siemens,Germany) using a ²³Na-birdcage coil (Rapid,Germany). A cylindrical phantom filled with 0.9% NaCl solution (compartment 0), containing seven vials with additional 1%-7% agar (compartments 1-7), was imaged. A reference T₁ map was determined based on an IR sequence (sequence parameters in Fig.2). Transverse relaxation parameters were determined by fitting a multi-echo GRE dataset (Fig.3). The phase difference yielded reference ΔB₀ maps (Fig.2) and B₁⁺ was measured using the phase sensitive method⁹ (FA=90°;TR=185ms;TE=0.55ms). The combined reference scan time (excluding B₁⁺) was 7h42min, whereas the ²³Na-MRF data was acquired within 1h4min.
Mean and SD of all quantified relaxation times were calculated in all vials for each slice.

Results

The resulting relaxation times and ΔB₀ of the central slice are shown in Figs.2 and 3. All mean relaxation times obtained by ²³Na-MRF are in agreement with the reference results within the SD (Tab.1), except for T₁ in compartment 7 and T₂^* in compartment 1. Fig.4 displays the mean relaxation time in each compartment in dependence on the z-position, as well as the mean B₁⁺ averaged over the whole phantom in each slice.

Discussion and Conclusion

This work demonstrates the feasibility of simultaneously quantifying T₁,T_2s^*,T_2l^*,T₂^* and ΔB₀ within approximately 1h by means of 3D ²³Na-MRF in combination with ISTOs for the dictionary simulation. Despite using a fairly simple FA and TE pattern, good agreement between the results of the reference measurements and 3D ²³Na-MRF was found. The differences in T₁ in compartment 7 can be explained as here strong a biexponential T₁ could be present¹⁰, violating equations [4] and [5]. In compartment 1, T₂^* can be described both bi- and monoexponentially almost equally well (see Fig.3), which could explain the discrepancy between reference and MRF.
Fig.4 indicates that B₁⁺ correction could improve the parameter quantification.
These experiments are promising as this technique could allow full 3D-mapping of the relaxometric parameters in less than 1h.

Acknowledgements

No acknowledgement found.