Synopsis

This work aims at improving the calculation of T1 maps at high magnetic fields from a set of VFA SPGR and IR-SPGR images for direct correction of B1 inhomogeneities using DESPOT1-HIFI. In the present work we (i) call attention to an erroneous assumption regarding the IR-SPGR signal intensity equation in the original DESPOT1-HIFI method and derive an alternative equation, and (ii) propose a reduced-dimensions least-squares fitting method, which turns the complex multi-parameter fitting into a simple one-dimensional search.

PURPOSE

THEORY

The IR-SPGR acquisition involves the application of a 180° pulse followed by a delay of and a train of low-angle SPGR pulses. While the MPRAGE ² model is too complex to be used for multi-parametric nonlinear least-squares fitting, a simpler equation can be derived with the assumptions of reference ¹:

The longitudinal magnetization $$$M_z(Tr^{-})$$$ at the end of each cycle immediately prior to the next inversion pulse is related to the longitudinal magnetization $$$M_z(0^{+})$$$ at the beginning of the cycle immediately after the inversion pulse:

$$M_z(Tr^{-})=M_z(0^{+})e^{-Tr/T1}+M_0(1-e^{-Tr/T1})$$ (1)

where $$$M_0$$$ is the equilibrium magnetization and $$$Tr$$$ is the time between successive inversion pulses.

Assuming an adiabatic inversion:

$$M_z(0^{+})=-M_z(0^{-})$$ (2)

Additionally, at steady state:

$$M_z(0^{-})=M_z(Tr^{-}) $$ (3)

Combining equations (1), (2), and (3):

$$M_z(0^{+})=-\frac{1-e^{-Tr/T1}}{1+e^{-Tr/T1}}M_0$$ (4)

based on which, the longitudinal magnetization at time $$$TI$$$ at each inversion cycle is derived:

$$M_z(TI^{-})=-M_0\frac{1-e^{-Tr/T1}}{1+e^{-Tr/T1}}e^{-TI/T1}+M_0(1-e^{-TI/T1})$$ (5)

With some algebraic simplifications an approximate IR-SPGR equation is derived:

$$S_{IR-SPGR}^{(approx)}=\rho(1-\frac{2e^{-TI/T1}}{1+e^{-Tr/T1}})\sin(\kappa\alpha_p)$$ (6)

where $$$\rho$$$ is a factor proportional to $$$M_0$$$ and includes the $$$T_2^*$$$ and receive effects. $$$\alpha_p$$$ is the prescribed flip angle, and models spatial variations of the transmitted flip angle from the prescribed.

DESPOT1-HIFI derives a unique solution for $$$\kappa$$$, $$$T_1$$$, and $$$\rho$$$ by least squares minimization of the combined SPGR and IR-SPGR data to the corresponding signal equations:

$$f(\rho,T_1,\kappa)=\sum_{i=1}^{N_{IR-SPGR}}[\rho(1-\frac{2e^{-TI/T1}}{1+e^{-Tr/T1}})\sin(\kappa\alpha_{p,i})-S_{IR-SPGR,i}]^2+\sum_{j=1}^{N_{SPGR}}[\frac{\rho(1-e^{-TR/T_1})\sin(\kappa\alpha_{p,j})}{1-e^{-TR/T_1}\cos(\kappa\alpha_{p,j})}-S_{SPGR,j}]^2$$ (7)

where $$$TR$$$ is the SPGR repetition time.

Unfortunately, however, the IR-SPGR equation used in the original DESPOT1-HIFI method is that of a general IR-SE sequence, which is not applicable to IR-SPGR acquisitions, resulting in significant errors in the calculated T1 maps.

METHODS

Reference T1 data were acquired using a 2D IR-FSE sequence on a 3T GE scanner with TE/TR=11.24ms/5000ms, TI={100,400,800,1500,3000}ms. 3D SPGR/IR-SPGR data were acquired using the same scanner with TE/TR=3.71ms/8.36ms. $$$TI_{IR-SPGR}$$$=450ms, $$$\alpha_{IR-SPGR}$$$=5˚, $$$\alpha_{SPGR}$$$=4˚ and 18˚.

A custom-made agarose gel phantom comprising 9 nickel chloride doped agarose tubes and two healthy volunteers were imaged. Informed consent was obtained from the subjects. Reference T1 maps were calculated by a 3-parameter fit to the IR-FSE data. DESPOT1-HIFI T1 values were calculated by the least-squares fitting described in (7). As the range of possible $$$\kappa$$$ values is known, a global grid-search on $$$\kappa$$$ can be used, ensuring a global minimum is found. Since for each value of $$$\kappa$$$, $$$T_1$$$ and $$$\rho$$$ can be efficiently estimated using the conventional linear DESPOT1 ³, the three-parameter NLS fitting is reduced to a search over one dimension, providing a fast and efficient solution. The reduced-dimensions grid-search fitting algorithm was compared to the conventional three-parameter fitting. The fitting procedure was repeated using the original IR-SPGR equation used in ¹.

RESULTS

DISCUSSION

CONCLUSION

Acknowledgements

References

1. Deoni, S. C. L. High-resolution T1 mapping of the brain at 3T with driven equilibrium single pulse observation of T1 with high-speed incorporation of RF field inhomogeneities (DESPOT1-HIFI). J. Magn. Reson. Imaging JMRI 26, 1106–1111 (2007).

2. Mugler, J. P. & Brookeman, J. R. Three-dimensional magnetization-prepared rapid gradient-echo imaging (3D MP RAGE). Magn. Reson. Med. 15, 152–157 (1990).

3. Deoni, S. C. L., Rutt, B. K. & Peters, T. M. Rapid combined T1 and T2 mapping using gradient recalled acquisition in the steady state. Magn. Reson. Med. 49, 515–526 (2003).

4. Cheng, H.-L. M. & Wright, G. A. Rapid high-resolution T(1) mapping by variable flip angles: accurate and precise measurements in the presence of radiofrequency field inhomogeneity. Magn. Reson. Med. 55, 566–574 (2006).