# Spectral analysis (SA) in PET

In comparison to compartmental models, SA has only few presumptions. Like compartmental models, SA describes the kinetics of the tracer using homogeneous compartments, but there is no need to know the number of compartments; SA can instead be used to estimate the number of compartments. Therefore SA can be used for selecting or validating a compartmental model. SA has also been applied in determination of time delay (Hinz and Turkheimer, 2006), and in discriminating brain gray and white matter uptake on voxel-level analysis (Heurling et al., 2015).

## Impulse response function (IRF)

IRF represents the tissue tracer concentration curve that would be measured
(with PET) after an ideal instantaneous bolus injection. In spectral analysis
IRF is assumed to be the sum of *M+1* (0, …, *M*)
exponential functions:

, where *α _{j}≥0*,

*β*, and

_{0}=0*β*.

_{j}≥0## Simulation of tissue curve

In practice, the input function to
the tissue is far from ideal bolus. Spectral analysis requires
arterial blood sampling to get the tracer
concentration in arterial plasma, *C _{P}(t)*,
used as the input function. Tissue curve,

*C*, can be computed by convolution between

_{T}(t)*h(t)*and

*C*:

_{P}(t)## Estimation of the spectrum

First a fixed list of *β _{j}* values is defined, with
a range starting from zero (

*β*). Since input function

_{0}=0*C*is measured, we can calculate a table of basis functions for each

_{P}(t)*β*,

_{j}, to replace the nonlinear part in equation (2).
Non-negative least-squares (NNLS) method (Lawson and Hanson, 1974) can then
be used to solve the *α _{j}* values, minimizing the
weighted residuals sum of squares (WRSS) between the simulated tissue curve and
measured tissue curve,

*C*:

_{PET}(t), where *N* is the number of PET time frames, and
*w _{i}* are the weights of the time frames.
Negative

*α*values would be nonphysiological, and NNLS method is therefore suitable for estimating

_{j}*α*.

_{j}The estimated *α _{j}* values can be called the
spectrum of the regional tissue TAC, and the structure of the model
(number of compartments, reversibility and irreversibility) can be derived
from the spectrum.

NNLS method is fast to compute, but SA should still not be applied to pixel-by-pixel calculations because of its sensitivity to noise.

## Dual-input

Spectral analysis is applicable to analysis of PET data when the tracer has a radioactive metabolite which is transported into tissue, and the plasma input curves of both parent tracer and radioactive metabolite are measured and incorporated in the model (thus the name dual-input or double-input) (Tomasi et al., 2012).

## See also:

## References:

Cunningham VJ, Ashburner J, Byrne H, Jones T. Use of spectral analysis to
obtain parametric images from dynamic PET studies. *In:*
Quantification of brain function. Tracer kinetics and image analysis in brain
PET. Elsevier, 1993, pp 101-111.

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

Cunningham VJ, Gunn RN, Byrne H, Matthews JC. Suppression of noise artifacts
in spectral analysis of dynamic PET data. *In:* Quantitative functional
brain imaging with positron emission tomography, p 329-334, Academic Press,
1998.

Hinz R, Turkheimer FE. Determination of tracer arrival delay with spectral
analysis. *IEEE Trans Nucl Sci.* 2006; 53(1): 212-219.

Hudson HM, Walsh C. Density deconvolution using spectral mixture models.
*In:* Proceedings of the Second World Congress of the IASC, Pasadena, CA,
pp 593-599.

Lawson, C. L., Hanson, R.J.: *Solving Least Squares Problems*,
Prentice-Hall, Englewood Cliffs, New Jersey, 1974.

Meikle SR, Matthews JC, Brock CS, Wells P, Harte RJA, Cunningham VJ,
Jones T. Pharmacokinetic assessment of novel anti-cancer drugs using spectral
analysis and positron emission tomography: a feasibility study.
*Cancer Chemother Pharmacol.* 1998; 42: 183-193.

Reutens DC, Andermann M. Constraints in spectral analysis. *In:*
Quantitative functional brain imaging with positron emission tomography.
Academic Press, 1998, pp 335-337.

Riaño Barros DA, McGinnity CJ, Rosso L, Heckemann RA, Howes OD, Brooks DJ,
Duncan JS, Turkheimer FE, Koepp MJ, Hammers A. Test-retest reproducibility of
cannabinoid-receptor type 1 availability quantified with the PET ligand
[^{11}C]MePPEP. *Neuroimage* 2014; 97: 151-162.

Rizzo G, Veronese M, Zanotti-Fregonara P, Bertoldo A.
Voxelwise quantification of [^{11}C](*R*)-rolipram PET data: a
comparison between model-based and data-driven methods.
*J Cereb Blood Flow Metab.* 2013; 33: 1032-1040.

Schmidt K. Which linear compartment systems can be analyzed by spectral
analysis of PET output data summed over all compartments.
*J Cereb Blood Flow Metab.* 1999; 19: 560-569.

Sederholm K. Using NNLS in multilinear PET problems. TPCMOD0020.pdf.

Suominen H. Yleistettyyn lokeromalliin perustuva spektraalianalyysi positroniemissiotomografia-mallintamisessa. Pro gradu, 2005.

Tomasi G, Kimberley S, Rosso L, Aboagye E, Turkheimer F. Double-input
compartmental modeling and spectral analysis for the quantification of positron
emission tomography data in oncology.
*Phys Med Biol.* 2012; 57: 1889-1906.

Turkheimer F, Moresco M, Lucignani G, Sokoloff L, Fazio F, Schmidt K.
The use of spectral analysis to determine regional cerebral glucose utilization
with positron emission tomography and [^{18}F]fluorodeoxyglucose:
theory, implementation, and optimization procedures.
*J Cereb Blood Flow Metab.* 1994; 14: 406-422.

Turkheimer F, Sokoloff L, Bertoldo A, Lucignani G, Reivich M, Jaggi JL,
Schmidt K. Estimation of component and parameter distributions in spectral
analysis. *J Cereb Blood Flow Metab.* 1998; 18: 1211-1222.

Turkheimer FE, Hinz R, Gunn RN, Aston JAD, Gunn SR, Cunningham VJ.
Rank-shaping regularization of exponential spectral analysis for application to
functional parametric mapping. *Phys Med Biol.* 2003; 48: 3819-3841.

Tags: Modeling, Spectral analysis, SA, Compartmental model

Updated at: 2017-11-18

Created at: 2014-05-07

Written by: Vesa Oikonen