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.