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:

**Display TAC**shows the TAC together with the model curve calculated from the detected peaks.**Display Spectrum**shows the peaks in the spectrum.**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.