Serhat Ilbey^{1}, Michael Garwood^{2}, Michael Bock^{1}, and Ali Caglar Özen^{1,3}

^{1}Dept. of Radiology, Medical Physics, Medical Center – University of Freiburg, Freiburg, Germany, ^{2}Center for Magnetic Resonance Research and Department of Radiology, University of Minnesota, Minneapolis, MN, United States, ^{3}German Consortium for Translational Cancer Research Partner Site Freiburg, German Cancer Research Center (DKFZ), Heidelberg, Germany

MR imaging of tissue
components with extremely fast transverse relaxation times require advanced MRI
techniques such as continuous SWIFT (cSWIFT), Concurrent Excitation and
Acquisition (CEA), and full duplex MRI. Frequency modulated pulses are used in
these methods. A steady-state signal model with transverse relaxation term was
formulated and solved for an arbitrary frequency and phase modulated RF pulse.
An accurate expression for the optimal flip angle was derived. The performance
of the new formulation was verified by a home-made Bloch equation simulator for
T_{2} values between 10µs and 100ms.

To adjust the contrast and to maximize SNR in SWIFT and CEA, the steady‑state (SS) behavior of the sequence must be analyzed. For HS

FM pulses in CEA sweep through the resonance at

In this study, for comparison we used the modified Ernst equation (14) which takes T

To validate the proposed method, for a myelin (T

In Figure 3, the SS transverse magnetization M

An optimal α = 16.5° was found in the simulations for myelin, whereas the optimal calculated values were 66.2° and 19.5° for SM and EPA, respectively. With these values, the SS signal of myelin was simulated and a 2.34-fold higher SNR was found for EPA compared to SM.

For extreme T

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Figure 1 Amplitude
and frequency functions of an arbitrary HS_{1} pulse (a) and effective excitation
region (EER) and the effective pulse approximation (EPA) method (b). t_{0} and t_{1} values are the times, when α(t_{0})=0.3α_{0}
and α(t_{1})=α_{0},
where α_{0} is
the desired flip angle (Here, an example for α_{0}=30°
is shown.) From the EER of the pulse (blue area), we calculated the effective
RF energy that was used for excitation. The amplitude and duration of this
pulse region is approximated to be a hard pulse (red region). The energy of
the approximated hard pulse is equal to the energy of the EER of the HS_{1} pulse.

Figure 2 Steady-state transverse magnetization (signal intensity)
obtained with a home-made forward model Bloch equation simulator
(Semi-analytical), and with the simplified and proposed methods (SM and EPA, respectively) for a large span of
transverse relaxation constants (T_{2}s). The result of EPA is in close
agreement with the semi-analytical results. SM fails especially for T_{2}
values less than 10ms. (Tp=1 ms, TR=2 ms, T1=100 ms)

Figure 3 Steady state transverse magnetization (signal intensity)
obtained with a home-made forward model Bloch equation simulator
(Semi-analytical), and with the simplified and proposed methods (SM and EPA,
respectively) for a large span of nominal flip angles. The
vertical lines show optimal flip angles for each method. The result of EPA is in close agreement with the semi-analytical results.
Simulation Parameters: Tp=1 ms, TR=2 ms,
T1=100 ms.

Optimal
flip angles [°] for the given T_{2} values. EPA estimates the optimal
value with a good accuracy for all T_{2} values. Simulation
Parameters: Tp=1 ms, TR=2 ms, T_{1}=100
ms.