Fitting functions to PET TTACs

Mathematical functions or compartment models can be fitted to regional tissue time-activity curves (TTACs). Fitting functions to TTACs is usually not necessary, and should be avoided as a pre-processing step before compartmental modelling. Instead, Input functions and plasma parent fractions frequently need to be fitted in order to enable interpolation and extrapolation or to reduce noise. Noise may be a problem with TTACs when assessing TTAC peak value, for example for the purpose of estimating binding potential using transient equilibrium method, and fitting an empirical function to the TTACs may help to determine the peak values without noise-induced bias and variance. Another possible use is to estimate the tracer appearance time for time delay correction, or accounting for the time delay in AUC calculation for ARG methods.

Functions

Feng et al (1993) proposed a set of functions for fitting input curves, and these functions can also be used to fit TTACs. The "model 2" is a combination of the surge function and exponentials:

$\begin{equation} f(t) = \left\{ \begin{array}{l l} 0 & \quad \text{if } t \leq \tau\\ [A_1 (t-\tau) - A_2 - A_3] e^{\lambda_1 (t-\tau)} + A_2 e^{\lambda_2 (t-\tau)} + A_3 e^{\lambda_3 (t-\tau)} & \quad \text{if } t > \tau \end{array} \right. \end{equation}$

, and can be fitted to PET TTACs with program fit_feng.

Rational functions (ratio of polynomials),

$\begin{equation} f(t) = \left\{ \begin{array}{l l} 0 & \quad \text{if } t < \tau\\ \frac{p_1 \: + \: p_2 (t-\tau) \: + \: p_3 (t-\tau)^2 \: + \: p_4 (t-\tau)^3}{q_1 \: + \: q_2 (t-\tau) \: + \: q_3 (t-\tau)^2 \: + \: q_4 (t-\tau)^3} & \quad \text{if } t \geq \tau \end{array} \right. \end{equation}$

, can well represent the regional tissue data, but are not suitable for extrapolation because of the discontinuities at the zeroes of the divider function. Parameters p₁ and q₁ are usually fixed to 0 and 1, respectively, and other parameters are ≥0. Rational functions can be fitted to TTACs using fit_ratf.

Specific tumour vascularization function (patent WO/2008/053268, entitled Method and System for Quantification of Tumoral Vascularization),

$\begin{equation} f(t) = \left\{ \begin{array}{l l} 0 & \quad \text{if } t < \tau\\ a_1 \: (\frac{A \: + \: (\frac{t-\tau}{a_2})^p}{B \: + \: (\frac{t-\tau}{a_2})^q}) & \quad \text{if } t \geq \tau \end{array} \right. \end{equation}$

has been used to fit tissue curves in [¹⁸F]FDOPA glioma studies (Zaragori et al., 2021).

Sigmoidal functions can work well with certain types of TTACs, and the following programs can be tested: fit_gvar, fit_hiad, fit_sigm.

Function based on the sum of Weibull probability density function and its integral (Weibull cumulative density function),

$\begin{equation} f(t) = \left\{ \begin{array}{l l} 0 & \quad \text{if } t \leq \tau\\ A \frac{n}{B} (\frac{t - \tau}{B})^{n-1} e^{-(\frac{t-\tau}{B})^n} + k A (1 - e^{-(\frac{t-\tau}{B})^n}) & \quad \text{if } t > \tau \end{array} \right. \end{equation}$

, can be fitted to PET TTACs with program fit_wcdf.

Simple sums of exponential functions

$\begin{equation} f(t) = \sum_{i=1}^{N} A_i e^{-\lambda_i t} \end{equation}$

have been traditionally used to fit not only input curves but also regional TTACs. If bolus injection is administered into local tissue artery, then tracer washout curve can be well fitted with decreasing exponentials. Program fit_exp can be used for these purposes. Initial parameters for unconstrained multiexponential functions can be calculated using linear methods as suggested by (Jean Jacquelin); this method can be tested with program llsqe3.

Weighting

TTAC samples can be weighted by 1 / frame length to prevent overfitting the initial part with shorter frames.

Using the fits

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

Literature

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.

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.

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: TTAC, Fitting, Interpolation

Updated at: 2023-05-09
Created at: 2018-05-17
Written by: Vesa Oikonen

Fitting functions to PET TTACs

Functions

Weighting

Using the fits

See also:

Literature