- Proof:
- The equation for a repeated inversion recovery sequence is
- where
- TI = inversion time
- TR = repetition time
- T1 = spin-lattice relaxation time
- T2 = spin-spin relaxation time
- r = spin density
- k = proportionality constant
We start with the Bloch equations introduced in Chapter 3. The signal in a pulse sequence is proportional to the amount of longitudinal magnetization rotated in to the xy-plane. For this reason, we only need to examine the component which describes in differential form the time (t) evolution of the Mz magnetization to equilibrium Mzo with time constant T1 after perturbation by an RF pulse.
Grouping like terms and setting up the integrals between the time limits of 0 and TI, and the Mz limits of -Mzo(1-e-(TR-TI)/T1) and Mz (please pardon the confusing symbolism) we have
Integration yields
and evaluating between the limits yields the following equations.
Which after rearrangement gives the IR equation.
