Dual time point estimation of K_i

Net influx rate (K_i) of a PET tracer is usually estimated from a dynamic PET data using either Patlak plot or compartmental model, and it requires the measurement of arterial plasma curve starting from the tracer administration until the end of the PET scan. Blood sampling is not necessary, if the input function can be measured from the dynamic PET image. An estimate of K_i, FUR, does not require dynamic imaging, but can be calculated from a static late-scan image; however, FUR calculation still needs the input function (integral) from the whole time, starting from the tracer administration until the end of the PET scan, and late-scan image can not provide that. If the concentration in the blood can be measured during the late-scan, then tissue-to-blood ratio can be calculated, and it is superior to SUV as a surrogate parameter of K_i (van den Hoff et al., 2013).

If two static late-scans can be performed, and input function can be measured from the PET images, then K_i (and subsequently metabolic rate) can be calculated from that dual time point (DTP) data alone (van den Hoff et al., 2013), assuming that

there is an irreversible compartment in tissue in which the tracer is trapped (Patlak plot and FUR methods have this same requirement), and
input function follows mono-exponential decay between the two PET scans, and
a population-based estimate of the y axis intercept of Patlak plot is available (this would also improve the FUR method).

If the tracer concentration in plasma and tissue at time t are represented by C_P(t) and C_T(t), respectively, and the PET scan times are t₁ and t₂, then the estimate of K_i can be calculated from equation:

$K_i \approx \frac{C_{T}(t_2) - C_{T}(t_1)}{(t_2 - t_1) \sqrt{C_{P}(t_1) \times C_{P}(t_2)}} + \frac{\overline{I}_r \times ln(C_{T}(t_1) / C_{T}(t_2))}{t_2 - t_1}$

, where Ī_r is the population-based estimate of the y axis intercept of the Patlak plot.

Alternatively, if the time course of the input function is assumed to follow an inverse power law (hyperbola) for t ≳ 1-2 min p.i.,

$C_{P}(t) = A \times t^{-b}$

, then even simpler equation for an estimate of K_i can be derived (Hofheinz et al., 2016):

$K_i \approx (1-b) \frac{ \frac{C_{T}(t_2)}{C_{P}(t_2)} - \frac{C_{T}(t_1)}{C_{P}(t_1)} }{t_2 - t_1}$

, but usage of this method requires that the inter- and intra-individual variability of b can be ignored and a population average of b has been calculated.

If input function is available from the injection time, dual-time point K_i can be calculated from equation

$\frac{C_{ROI}(t_2)}{C_{p}(t_2)} - \frac{C_{ROI}(t_1)}{C_{p}(t_1)} = K_i \times \ \left( \frac{\int_0^{t_2} C_p(\tau)dt}{C_{p}(t_2)} - \frac{\int_0^{t_1} C_p(\tau)dt}{C_{p}(t_1)} \right)$

, from which the intercept of the Patlak plot has been cancelled out (Wu et al., 2021). If input function is not measured, but blood radioactivity concentration can be assessed from the two PET scans, then those values could be used to scale population-based input function to individual level (Wu et al., 2021).

DTP with reference tissue

If arterial plasma data at the times of the two PET scans is not available, not even from the images, but a reference region can be found in the images, then it can be used to calculate a surrogate parameter for K_i^ref, that in case of dynamic imaging could be calculated using Patlak plot with reference tissue input. Alves et al (2017) have applied this method to [¹⁸F]FDOPA studies, using occipital cortex as reference region:

$K_{dtp}^{ref} = \frac{ \frac{C_{T}(t_2)}{C_{R}(t_2)} - \frac{C_{T}(t_1)}{C_{R}(t_1)} }{{RPT}(t_2) - {RPT}(t_1)}$

C_R represents the radioactivity concentration in the reference tissue as a function of time. RPT (reference Patlak time) is the area-under-curve (AUC) of the reference tissue from tracer administration time (t=0) to the scan time (t₁ or t₂) divided by the concentration in the reference region at the corresponding scan time. For the method to be useful, the RPT(t₁) must be assumed to have sufficiently low inter- and intra-individual variability, so that a population average of RPT(t₁) can be calculated and used in the previous equation, and in calculation of RPT(t₂) from equation

${RPT}(t_2) = \frac{ C_{R}(t_1) \times {RPT}(t_1) }{C_{R}(t_2)} + \frac{ \int_{t_1}^{t_2} C_{R}(t)dt }{C_{R}(t_2)}$

Literature

Alves LI, Meles SK, Willemsen AT, Dierckx RA, Marques da Silva AM, Leenders KL, Koole M. Dual time point method for the quantification of irreversible tracer kinetics: A reference tissue approach applied to [¹⁸F]-FDOPA brain PET. J Cereb Blood Flow Metab. 2017; 37(9): 3124-3134. doi: 10.1177/0271678X16684137.

van den Hoff J, Hofheinz F, Oehme L, Schramm G, Langner J, Beuthien-Baumann B, Steinbach J, Kotzerke J. Dual time point based quantification of metabolic uptake rates in ¹⁸F-FDG PET. EJNMMI Res. 2013; 3: 16. doi: 10.1186/2191-219X-3-16.

Hofheinz F, van den Hoff J, Steffen IG, Lougovski A, Ego K, Amthauer H, Apostolova I. Comparative evaluation of SUV, tumor-to-blood standard uptake ratio (SUR), and dual time point measurements for assessment of the metabolic uptake rate in FDG PET. EJNMMI Res. 2016; 6(1): 53. doi: 10.1186/s13550-016-0208-5.

Logan J. Graphical analysis of PET data applied to reversible and irreversible tracers. Nucl Med Biol. 2000; 27: 661-670. doi: 10.1016/S0969-8051(00)00137-2.

Patlak CS, Blasberg RG. Graphical evaluation of blood-to-brain transfer constants from multiple-time uptake data. Generalizations. J Cereb Blood Flow Metab. 1985; 5: 584-590. doi: 10.1038/jcbfm.1985.87.

Tags: Patlak plot, Ki, Late-scan

Updated at: 2022-01-14
Created at: 2018-02-14
Written by: Vesa Oikonen

Dual time point estimation of Ki

DTP with reference tissue

See also:

Literature

Dual time point estimation of K_i