Spectral Analysis

The Spectral model allows performing a regular Spectral Analysis (SA) given a TAC and an input curve according to Cunningham and Jones [1].

Operational Model Curve

The operational equation of SA is given by

that is, tissue uptake is modeled as a sum of N possible tissue responses. Due to the constraint of first order tracer kinetics, the coefficients a_i and the decay constants b_i must be non-negative. In practice, a discrete set of the decay constants b_i is selected which covers the physiologically reasonable range, typically logarithmically spaced in the range [10^-5,1]sec^-1. The corresponding tissue responses

are the Basis Functions of spectral analysis. When fitting the operational equation above to a tissue TAC, the only unknowns are the coefficients a_i, because only a pre-defined set of discrete b_i values is considered. Therefore, the problem is that of a non-negative linear least squares estimation (NNLS) with the constraint of non-negative coefficients.

An advantage of SA is the fact that no particular compartment structure is imposed. Rather, its result can be used to estimate how many kinetic tissue compartments can be resolved by PET. To this end, the results are plotted as a spectrum with the selected decay constants b_i along the x-axis (as the "frequencies") and the estimated coefficients a_i along the y-axis (as the "amplitudes"). Because of the large range, log(b_i) is used in spectrum plotting rather than b_i. The number of peaks in this spectrum corresponds to the number of distinct compartments. A peak appearing to the far left (low frequency, slow component) indicates irreversible trapping. A peak to the far right (high frequency, fast component) corresponds to kinetics indistinguishable from the input curve, thus to vascular contributions. Intermediate peaks represent compartments which exchange reversibly with plasma or with other tissue compartments.

Parameter Fitting

In the Spectral model #Exponentials determines the number of discrete b_i decay constants which are logarithmically spaced in the range between Beta-min and the Beta-max. There is no well-defined minimum value as in the case of decay-corrected TACs where it is equal to the decay constant of the isotope.

Estimates of the tissue uptake constant K1 and the total distribution volume Vt can be calculated based on the peaks found in the spectrum. Since a blood-related peak distorts these estimates, it can be excluded from these calculations by the Exclude HF check box.

The SA results can be visualized in 3 different ways with a corresponding radio button setting:

1. Display TAC shows the TAC together with the model curve calculated from the detected peaks.
2. Display Spectrum shows the peaks in the spectrum.
3. Display unit IRF shows the Impulse Response Function which depends on the Exclude HF setting.

Reference

Cunningham VJ, Jones T: Spectral analysis of dynamic PET studies. J Cereb Blood Flow Metab 1993, 13(1):15-23.