Extended Phase Graph formalism for systems with Magnetization Transfer and Chemical Exchange

An Extended Phase Graph framework for modelling systems with exchange or magnetization transfer (MT) is proposed. The framework, referred to as EPG-X, models coupled two-compartment systems by describing each compartment with separate phase graphs that exchange during evolution periods. There are two variants: EPG-X(BM) for systems governed by the Bloch-McConnell equations; and EPG-X(MT) for the pulsed MT formalism. For the MT case the "bound" protons have no transverse components so their phase graph consists only longitudinal states. EPG-X was used to model steady-state gradient echo imaging, MT effects in multislice Turbo Spin Echo imaging, multiecho CPMG for multicomponent T2 relaxometry and transient variable flip angle gradient echo imaging of the type used for MR Fingerprinting. Experimental data were also collected for the final case. Steady-state predictions from EPG-X closely match directly derived steady-state solutions which differ substantially from classic "single pool" EPG predictions. EPG-X(MT) predicts similar MT related levels of signal attenuation in white matter as have been reported elsewhere in the literature. Modelling of CPMG echo trains with EPG-X(BM) suggests that exchange processes can lead to an underestimate of the fraction of short T2 species. Modelling of transient gradient echo sequences with EPG-X(MT) suggests that measurable MT effects result from variable saturation of bound protons, particularly after inversion pulses. In conclusion, EPG-X can be used for modelling of the transient signal response of systems exhibiting chemical exchange or MT. This may be particularly beneficial for relaxometry approaches that rely on characterising transient rather than steady-state sequences.