# Fitting PET input curves

Mathematical functions or compartment models can be fitted to measured plasma and blood time-activity curves (PTACs and BTACs) to make noisy curves smoother and/or to enable interpolation and extrapolation of the arterial input function (AIF) (Feng & Wang, 1993; Graham, 1997) and to reduce the number of blood samples (Feng et al., 1994). Functions fitted to PTACs are also often used in the application of model-based or population-based input functions.

PET input curves, following intravenous bolus administration
of radiotracer, typically have a steep rise, pass through a maximum, and then undergo a slower
decay. Generally, indicator dilution curves have been observed to be of the form of the *gamma
variate* function. Gamma variate -based surge function
can be used in simulations
(Herscovitch et al., 1983), but for
fitting curves from PET studies a combination of surge functions, or
a sum of surge function and its integral
function is used to account for dispersion, delay, and recirculation
effects. Infusion administration can be modelled using boxcar function convolved with surge function.

Simple sums of exponential functions can be used to fit the decreasing tail of bolus plasma or blood curves. Based on a compartmentalized model of radiopharmaceutical behaviour in the circulatory system, Feng et al. (1993a and 1993b) have proposed a four exponential function for fitting PET radiotracer PTACs which include both ascending and descending phase in the measured data, and the initial zero phase before the radiotracer appearance time. Exponential input function with a pair of repeated eigenvalues has been widely used in PET modelling and simulations; the function resembles the sum of surge function and its integral function. These functions can fit well AIF after radiotracer bolus injection, but for fitting AIF from a radiotracer infusion study the model with incorporated injection schedule works better.

The sigmoidal functions that currently are available do not usually work well with input data, but in some cases may be tried as well: fit_gvar, fit_hiad, fit_sigm. By default, fit_gvar includes a recirculation term in the fitting, providing much better fit to typical TTACs and PTACs than the plain Gamma variate function.

Rational functions (ratio of polynomials) are best suited to fit regional tissue data, but may also fit input curves in some cases: fit_ratf.

Compartmental models can be applied to fitting plasma and
blood curves, for instance for
[^{15}O]H_{2}O studies.
General AIF model for PET was presented by
Graham (1997).

Most of functions presented above contain the *appearance time* of radioactivity in the blood
curve as one of the fitted parameters. When needed, fitting of these functions to initial phases of
AIF can be used to estimate the appearance time. These functions account for the gradual decrease
of the curve slope near the peak, which is not considered in simplistic piecewise functions
(Cheong et al., 2003).

## Weighting

Input curve samples are usually either not weighted, or weighted by 1 / sampling frequency to prevent overfitting the initial part with more frequent sampling, especially if blood sampler is used followed by manual sampling. However, data could also be weighted based on count statistics, considering also the variance in unchanged radiopharmaceutical fractions (Tsujikawa et al., 2014). McGowan et al (2017) fitted function to image-derived blood TAC using weights

, where *Δt _{i}* is the duration of the

*i*th frame and

*λ*is the decay constant for the isotope.

## Using the fits

The programs mentioned above save the function parameters in specific fit file format. TAC curves can be computed from the fit parameter files using fit2dat.

## See also:

- Exponential functions as PET model input
- Gamma variate functions as PET model input
- Compartmental models for input function
- Blood sampling
- Input function
- Input for simulations
- Fitting plasma parent fractions
- Fitting compartmental models
- Fitting TTACs
- Plasma pharmacokinetics
- Area under curve (AUC)

## References

Feng D, Huang S-C, Wang X. Models for computer simulation studies of input functions for
tracer kinetic modeling with positron emission tomography.
*Int J Biomed Comput.* 1993; 32: 95-110.
doi: 10.1016/0020-7101(93)90049-C.

Feng D, Wang Z. A three-stage parameter estimation algorithm for tracer concentration
kinetic modelling with positron emission tomography.
*Proceedings, 1991 American Control Conference, vol 2* (1991): 1404-1405.
doi: 10.23919/ACC.1991.4791609.

Lüdemann L, Sreenivasa G, Michel R, Rosner C, Plotkin M, Felix R, Wust P, Amthauer H.
Corrections of arterial input function for dynamic H_{2}^{15}O PET to assess
perfusion of pelvic tumours: arterial blood sampling versus image extraction.
*Phys Med Biol.* 2006; 51: 2883-2900.
doi: 10.1088/0031-9155/51/11/014.

Motulsky HJ, Ransnas LA. Fitting curves to data using nonlinear regression:
a practical and nonmathematical review. *FASEB J.* 1987; 1: 365-374.
doi: 10.1096/fasebj.1.5.3315805.

Muzic RF Jr, Christian BT. Evaluation of objective functions for estimation of kinetic
parameters. *Med Phys.* 2006; 32(2): 342-353.
doi: 10.1118/1.2135907.

Tonietto M, Rizzo G, Veronese M, Bertoldo A. Modelling arterial input functions in positron
emission tomography dynamic studies. *In:* Engineering in Medicine and Biology Society (EMBC),
37th Annual International Conference of the IEEE. 2015, pp. 2247-2250.
doi: 10.1109/EMBC.2015.7318839.

Wagner JG. Linear pharmacokinetic equations allowing direct calculation of many needed
pharmacokinetic parameters from the coefficients and exponents of polyexponential equations which
have been fitted to the data. *J Pharmacokin Biopharm.* 1976; 4(5): 443-467.
doi: 10.1007/BF01062831.

Young P: *Everything You Wanted to Know About Data Analysis and Fitting but Were Afraid
to Ask.* Springer, 2005.
ISBN 978-3-319-19051-8.

Tags: Input function, Blood, Plasma, PTAC, Fitting, Extrapolation, Interpolation, Recirculation, Administration

Updated at: 2019-01-14

Created at: 2016-08-08

Written by: Vesa Oikonen