James C Korte^{1}, Bahman Tahayori^{2}, Peter M Farrell^{1}, Stephen M Moore^{3}, and Leigh A Johnston^{1}

It is known that steady-state ring-locked trajectories are formed on an elliptical manifold under a constant amplitude and constant frequency excitation envelope. Here we demonstrate that the excitation envelope can be expressed in terms of the spin-system parameters and a target steady-state trajectory, providing control of the magnetisation on the steady-state ellipsoid when spin-system parameters, such as the relaxation constants, are known. Conversely, we exploit this relationship between excitation parameters and unknown relaxation constants to develop a volume relaxometry technique, which can potentially be extended to relaxation mapping due to the elliptical nature of balanced SSFP.

Two experiments were conducted on a 4.7T
Bruker Biospec scanner with an AVANCE III console. An iterative measurement
protocol^{13,14}
(Fig. 1) was used to incrementally measure the transverse and longitudinal
steady-state magnetisation. In both experiments, field maps and relaxation
constants were measured using standard sequences (details below).

**Reference
relaxometry:**

Rapid acquisition with relaxation enhancement with variable repetition time (RARE-VTR) scans and multi-slice multi-echo (MSME) scans were used to measure $$$T_1$$$ and $$$T_2$$$ maps respectively. The measured relaxation maps were spatially averaged to produce a single $$$T_1$$$ and $$$T_2$$$ constant per phantom.

**Field
Maps:**

The distribution of off-resonances, $$$\rho\left(\delta_{B_\text{0}}\right)$$$,
was measured using the field-mapping
sequence, MAPSHIM (Bruker Biospin). The
distribution of excitation field strength, $$$\rho\left(B_\text{1}^\text{mod}\right)$$$, was measured with a $$$B_1$$$ mapping
sequence^{15}.
Distributions were extracted via a histogram of non-background voxels.

**Verification
of excitation envelope:**

The first experiment applied the excitation envelope, $$$\gamma\,B_\text{1}^\text{e}\left(t\right)$$$, to a spherical phantom of water to demonstrate control of the steady-state magnetisation. Ring-locked trajectories were measured over a range of elevations $$$a$$$ from -0.95 to 1.0 with a 0.05 increment, using a constant excitation frequency $$$\omega$$$=50Hz, measured relaxation constants $$$T_1$$$=3.20s and $$$T_2$$$=2.02s and off-resonance $$$\Delta$$$=0Hz.

**Estimation
of relaxation constants:**

In a second experiment we measured the transverse steady-state magnetisation, $$$\boldsymbol{M}_{xy}^\text{obs}$$$, of a spherical phantom of Gadolinium doped water ($$$T_1$$$=360.23ms, $$$T_2$$$=199.50ms) over a grid of excitation parameters, amplitude $$$|\omega_1^e|$$$ from 100Hz to 1800Hz with a 100Hz increment and frequency $$$\omega$$$ from 25Hz to 400Hz with a 25Hz increment. Relaxation constants were estimated using the objective function, $$\underset{\bar{T_1},\,\bar{T_2}\in[0,\infty)}{\operatorname{minimise}}\,\LARGE{\parallel}\normalsize\left|\boldsymbol{M}^{\text{model}}_{xy}\left(|\omega_1^e|,\omega,\bar{T_1},\bar{T_2}\right)\right|-\left|\boldsymbol{M}_{xy}^\text{obs}\left(|\omega_1^e|,\omega\right)\right|\,\LARGE\parallel_{\normalsize2}$$ where $$\boldsymbol{M}^{\text{model}}_{xy}\left(|\omega_1^e|,\omega,\bar{T_1},\bar{T_2},t\right)=\int\int\,M_{xy}\left(|\omega_1^e|,\omega,\bar{T_1},\bar{T_2},\bar{\Delta}=\delta_{B_\text{0}},B_\text{1}^\text{mod},t\right)\rho\left(B_\text{1}^\text{mod}\right)\rho\left(\delta_{B_\text{0}}\right)\,dB_\text{1}^\text{mod}\,d\delta_{B_\text{0}},$$ and was solved via numerical integration and non-linear least squares optimisation with a non-negativity constraint.

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