In NMR studies of complex systems containing many physically and chemically distinct components, one of the principal obstacles is the linearly-additive overlap of the respective signal contributions. Post-acquisition separation of the signals corresponding to the various components is possible only in some cases. The best known fortunate case is the harmonic analysis based on Fourier transform (FT) which separates spectral components with different Larmor frequencies. Since the FT kernel is orthonormal, its spectral condition number [Wilkinson J.H., The Algebraic Eigenvalue Problem, Clarendon 1965, Chapter 2] k equals 1 and, consequently, FT inversion algorithms are numericaly quite stable.
Discrimination of various sample components on the basis of other parameters (such as their longitudinal relaxation rates R1) though highly desirable, is usually much more difficult. In the case of R1, the mathematical operation is the inversion of Laplace transform (LT) whose kernel is ill-conditioned (k>>1). This makes any LT inversion algorithm extremely sensitive to signal noise and experimental errors. To be usable at all, LT inversion algorithms must be regularized by the introduction of additional constraints on the outcome [Borgia G.C., Brown R.J.S., Fantazzini P., Magn.Reson.Imaging 16, p.549, 1998; J.Magn.Reson. 132, p.65, 1998; J.Magn.Reson. 147, p.273, 2000]. Any particular set of such conditions may be well suited in certain contexts (justifying the term prior knowledge), but criticizable and excessively subjective in others.
A different approach to the component-separation problem consists in an artful preparation of the sample magnetization prior to data acquisition. The goal is to weigh the signals arising from the individual components with respect to some discriminating parameter. An example is MRI with its gradient-induced phase encoding according to the component's spatial location. The majority of the preparatory pulse sequences used in all branches of NMR actually falls into this category - what differs from case to case is the discriminating parameter.
When the parameter is R1, one can cite MRI sequences for T1-weighed images and NMR sequences which use inversion pulse(s) to null one or more magnetization components with pre-determined values of T1. In both cases, however, there are many drawbacks such as poor discriminating power, lack of flexibility and excessive signal attenuation.
PERFIDI is a magnetization-preparation pulse-sequence preamble which introduces mathematical rigor and operational flexibility into the topic of R1-relaxation filters. It uses a particularly timed series of inversion pulses tailored to obtain what might be best described as low-, high- and band-pass relaxation-rate filters (the terminology is similar to that used for electronic filters). The timing of the series is critical since, apart from the desired filter profile, it must take into account also typical inversion imperfections (without that, the sequences would be useless). Fortunately, imperfection-tolerant solutions exist and one can pre-calculate filters using 2, 3, 4, ... inversion pulses and having different log-scale profiles. The latter can be described in terms such as cutoff relaxation rate, cutoff slope rate (discrimination power), band aperture, etc. Once a filter type has been defined, its profile can be shifted up or down the relaxation-rates scale by elementary scaling of the sequence timing.
Being a preamble, PERFIDI can be combined with nearly any existing data-acquisition technique in NMR spectroscopy, MRI, fixed-field NMR relaxometry, NMR diffusometry, ex-situ NMR, etc.
In the field of NMR relaxometry of poly-dispersed systems, PERFIDI pave the way to a direct measurement of relaxation rate distributions, avoiding the need for LT inversion (though a limited and numerically stable deconvolution may still be useful). We also discuss the fact that it is both possible and advantageous to combine the two approaches.