Gradient echo and 180 degree spin echo method

This pulse sequence uses a 180 degree spin-echo refocusing pulse and acquires two images using an excitation pulse α and $2\alpha$ . It assumes that there is full signal recovery (long TR), and because it refocuses T₂^*, it eliminates signal variability caused by B₀ in the resulting B₁ map Insko & Bolinger, 1993. Alternatively, a gradient echo could be used?

Assuming a refocusing pulse is used (i.e. isn’t dependent on B₁), we can develop the equation for a gradient echo and spin echo case.

M_{\alpha}=M_{0}\text{sin}\left( \alpha \right)\text{e}^{\left( -\frac{TE}{T_{2}} \right)}

(4.1)

M_{2\alpha}=M_{0}\text{sin}\left( 2\alpha \right)\text{e}^{\left( -\frac{TE}{T_{2}} \right)}

(4.2)

Thus

\frac{M_{\alpha}}{\text{sin}\left(\alpha \right)}=\frac{M_{2\alpha}}{\text{sin}\left(2\alpha \right)}

(4.3)

and

\frac{M_{2\alpha}}{M_{\alpha}}=\frac{\text{sin}\left(2\alpha \right)}{\text{sin}\left(\alpha \right)}

(4.4)

Using a well known trigonometry identity (see Appendix A for derivation),

\text{sin}\left( 2\alpha \right)=2\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)

(4.5)

We can simplify Eq. 4.5,

\frac{M_{2\alpha}}{M_{\alpha}}=\frac{2\text{sin}\left( \alpha \right)\text{cos}\left( \alpha \right)}{\text{sin}\left(\alpha \right)}

(4.6)

\frac{M_{2\alpha}}{M_{\alpha}}=2\text{cos}\left( \alpha \right)

(4.7)

And the true flip angle can be calculated from the ratio of these two magnetizations / signals / images:

\alpha=\text{arcos}\left( \frac{M_{2\alpha}}{2M_{\alpha}} \right)

(4.8)

Knowing that $\alpha = B_{1}$ alpha_nominal, B₁ is thus:

B_{1}=\frac{\text{arcos}\left( \frac{M_{2\alpha}}{2M_{\alpha}} \right)}{\alpha_{nominal}}

(4.9)

Source:Jupyter Notebook

Figure 4.2:B₁ computed from analytical GRE equations for DA sequence

This equation is also used for α-180 spin echo pulses, however it assumes no dependency on of the refocusing pulse on B₁. Figure 4.3 explores this using Bloch simulations

Source:Jupyter Notebook

Figure 4.3:B₁ computed from bloch simulations for ideal spin echo and refocusing pulse where FA = 180*B₁

Source:Jupyter Notebook

Figure 4.4:B₁ computed from bloch simulations for spin echo with refocusing pulse where FA = 180*B₁, and composite pulse 90_x-180_y-90_x where each 90 and 180 are also multiplied by B₁.

References¶

Insko, E. K., & Bolinger, L. (1993). Mapping of the radiofrequency field. J. Magn. Reson. A, 103(1), 82–85.

Double Angle Methods

Double Angle method(s)

Double Angle Methods

Appendix A