# 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). Functions fitted to PTACs are also often used in the application of model-derived or population-based input functions.

PET input curves typically have a steep rise, pass through a maximum, and then undergo a slower decay. Gamma variate -based surge function

(with maximum at t=1/λ) can be used in simulations (Herscovitch et al., 1983), but it is too simplistic for fitting curves from PET studies; usually other functions, or a combination of surge functions is needed.

## Functions

### Sum of exponentials

Simple sums of exponential functions can be used to fit the decreasing tail of bolus plasma or blood curves:

For extrapolating the input curve with one-exponential function extrapol can be used. However, better approach usually is to write a script which first fits 2 or 3 exponential function (with fit_exp) to the input data starting at some time after the peak is passed, then computes the fitted TAC at desired time points using fit2dat, and finally combines the original TAC and fitted tail using dftcat.

Feng et al. (1993) proposed a formulation of exponential functions (“model 1”) for fitting PET radiotracer PTACs which include both ascending and descending phase:

Time delay term τ accounts for the delayed appearance time of the radioactivity in the measured blood data. Eigenvalues (λ1-λ4) are ≤0. Extension of this function includes surge function, and is given below.

Fitting a multiexponential function is an ill-posed problem and not that easy to solve as it might first appear (Istratov & Vyvenko, 1999).

Especially in DCE-MRI studies, but also in PET field, “biexponential” may refer to the sum of two exponentials (McGrath et al., 2009), a function presented above with N=2, or, “biexponential AIF model”, which is actually a two-compartmental model used to calculate realistic BTAC from a rectangular (boxcar) function that represents the infusion of the contrast agent or radiotracer (Pellerin et al., 2007; Poulin et al., 2013; Richard et al., 2017). The biexponential AIF model can be written as

, where D is the infusion rate of the contrast agent or radiotracer, Π(t) is boxcar function with height 1 between the infusion start and end times Ta and Ta, respectively; and ⊗ is the convolution operator, with biexponential as the response function. Fourier transform of the boxcar function can be calculated using sinc function. Compartmental models for PET input data have been suggested by Bigler et al (1981), Graham (1997), and Maguire et al (2003).

### Exponential based functions

Feng et al. (1993) proposed not only the sum of exponentials function (above) for fitting PET input TACs, but also a combination of the surge function and exponentials (“model 2”):

This function can be fitted to PET TACs with program fit_feng. Note that this function can be negative in the beginning phase, which would be non-physiological, and redundant with the time delay term τ. The derivative of the function (when t>τ) is

To ensure that the derivative function is >0 at t=0, that is, f’(0)>0, this condition has to be met:

Note that lambdas have negative values, when concentration is decreasing and approaching zero.

A simplified version of these functions contains only one exponential (in addition to the surge function), and has been used to fit the descending part of input curves in order to reduce the number of blood samples in FDG studies (Phillips et al., 1995):

, where m and n were population means from a larger dataset, leaving only two parameters, a and b, to be fitted from an individual input curve.

Function for short infusion protocols was used by Oakes et al. (1999), and these functions, optionally with a few extensions, can be fitted with program fit_sinf. Tap is the appearance time of radioactivity in the blood or plasma, and Tin is the infusion time. N should usually be set to 2 or 3, but in some situations higher number of parameters is required.

While the descending part of the bolus curve can be fitted using simple sum of exponentials, the ascending phase (before peak time, Tpeak) can be fitted using a line (Eberl et al., 1997; Vriens et al., 2009):

, or, assuming that ascending phase starts at time 0 (Su et al., 1994):

### Sigmoidal functions

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.

Input TAC in DSC- or DCE-MRI is often fitted using gamma variate functions (Thompson et al., 1964). Sum of two gamma distribution functions can fit the primary bolus and the first recirculation peak, and so on (Davenport, 1983). Additional exponential function can account for the steady-state circulation phase. Gamma variate function with recirculation term has been used in [13N]NH4+ and [15O]H2O studies (Golish et al., 2001; Lüdemann et al., 2006), and in MRI (Parker et al., 2006). Long acquisition time with DCE-MRI causes significant renal clearance of contrast agent, which can be taken into account with yet another exponential term (Duan et al., 2017).

### Rational functions

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

## 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 tracer fractions (Tsujikawa et al., 2014).

## 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.

## 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 H215O 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.

Oakes ND, Kjellstedt A, Forsberg G-B, Clementz T, Camejo G, Furler SM, Kraegen EW, Ölwegård-Halvarsson M, Jenkins AB, Ljung B. Development and initial evaluation of a novel method for assessing tissue-specific plasma free fatty acid utilization in vivo using (R)-2-bromopalmitate tracer. J Lipid Res. 1999; 40: 1155-1169.

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.

TPC modelling report: Modelling input function. TPCMOD0010.

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.

Created at: 2016-08-08
Updated at: 2018-12-05
Written by: Vesa Oikonen