# Spectral analysis (SA) in PET

In comparison to compartmental models, SA has only few presumptions. Like compartmental models, SA describes the kinetics of the radiopharmaceutical 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), in discriminating brain grey and white matter uptake on voxel-level analysis (Heurling et al., 2015), and for analysis of heterogeneous tissue data (Veronese et al., 2018).

## 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 radiotracer concentration in arterial plasma, *C _{P}(t)*, used as
the input function. Tissue curve,

*C*, can be computed (simulated) 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 non-physiological, 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 as such 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 radiopharmaceutical 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:

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images from dynamic PET studies. *In:* Quantification of brain function. Tracer kinetics and
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Cunningham VJ, Jones T. Spectral analysis of dynamic PET studies.
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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.
doi: 10.1109/TNS.2005.862982.

Hudson HM, Walsh C. Density deconvolution using spectral mixture models.
*In:* Proceedings of the Second World Congress of the IASC, Pasadena, CA,
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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.
doi: 10.1007/s002800050804.

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. doi:
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Rizzo G, Veronese M, Zanotti-Fregonara P, Bertoldo A. Voxelwise quantification of
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Schmidt K. Which linear compartment systems can be analyzed by spectral analysis of PET output
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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.
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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.
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Turkheimer F, Sokoloff L, Bertoldo A, Lucignani G, Reivich M, Jaggi JL, Schmidt K.
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10.1097/00004647-199811000-00007.

Turkheimer FE, Hinz R, Gunn RN, Aston JAD, Gunn SR, Cunningham VJ. Rank-shaping regularization of
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Veronese M, Rizzo G, Bertoldo A, Turkheimer FE. Spectral analysis of dynamic PET studies:
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Tags: Modeling, Spectral analysis, SA, Compartmental model, Exponential function

Updated at: 2019-11-22

Created at: 2014-05-07

Written by: Vesa Oikonen