Rotation relaxation splitting for optimizing parallel RF excitation pulses with T₁- and T₂-relaxations in MRI

Exact solutions of the Bloch equations with T₁- and T₂-relaxation terms for piecewise constant magnetic fields are numerically challenging. We therefore investigate an approximation for the achieved magnetization in which rotations and relaxations are split into separate operations. We develop an estimate for its accuracy and explicit first and second order derivatives with respect to the complex excitation radio frequency voltages. In practice, the deviation between an exact solution of the Bloch equations and this rotation relaxation splitting approximation seems negligible. Its computation times are similar to exact solutions without relaxation terms. We apply the developed theory to numerically optimize radio frequency excitation waveforms with T₁- and T₂-relaxations in several examples.