# Net influx rate (*K*_{i})

_{i}

*K _{i}* is the unidirectional uptake rate constant that incorporates both net
inward transport and trapping of the radiotracer in tissue.

Lets imagine a theoretical situation, where we keep the concentration of radioligand in plasma at a steady level for a very long time.

After some time has passed, radiotracer concentrations in all reversible tissue compartments will reach a dynamic equilibrium with each other and with plasma. Meanwhile the total tissue concentration will continue to increase because of the irreversible process.

The total tissue concentration will increase linearly, with a slope that gives the net influx
rate, *K _{i}* = slope/plasma.
If there were no irreversible process, then tissue curve would reach a steady level,
representing the total distribution volume,

*V*= tissue/plasma.

_{T}In bolus infusion PET studies, calculation of *K _{i}* is possible when
input function is measured from the administration time (0-time)
to the end time of the PET scan.

*K*can be calculated with compartmental model fitting or multiple-time graphical analysis for irreversible uptake (Patlak plot). Calculation can be done for regional tissue time-activity curves to get regional

_{i}*K*values, or for dynamic PET images to get parametric K

_{i}_{i}image for further analysis. Instead of a full dynamic PET scan, if certain assumptions are met, estimation of

*K*may be possible from two static late-scans using dual time point method.

_{i}While bolus infusion methods are normally applied in PET, steady-state methods can be used, too,
to estimate the *K _{i}*.
Steady-state methods enable monitoring of transient metabolic changes

*in vivo*. These techniques have been applied in [

^{18}F]FDG PET in small animals (Bérard et al., 2006) and in human brain activation studies, termed as fPET (Villien et al., 2014).

##
*K*_{i} and compartmental model

_{i}

*K _{i}* is not dependent on compartmental
model, although it can also be derived from the differential equations of compartmental models.

### Three-tissue compartmental model

Assuming that compartments are in series,

### Two-tissue compartmental model

Assuming that *k _{4}=0*:

### One-tissue compartmental model

Assuming that *k _{2}=0*:

Notice that *K _{i}* is partially, and non-linearly,
dependent on perfusion (a component of

*K*and

_{1}*k*). This may lead to difficulties when perfusion is low compared to the irreversible process that we try to quantitate.

_{2}## Metabolic rate

When the PET radiopharmaceutical is an analog of glucose (e.g.
[^{18}F]FDG) or
fatty acids (e.g.
[^{18}F]FTHA) or other native substrate in
the tissue, and it is metabolically trapped in tissue during the PET scan, *K _{i}*
can be used to calculate the metabolic rate of the native substrate.
For example, in [

^{18}F]FDG the

*K*can be multiplied by concentration of glucose in plasma, and divided by the appropriate lumped constant, to get an estimate of glucose uptake rate.

_{i}*LC* for [^{18}F]FDG accounts for the differences in transport and phosphorylation
rates between glucose and [^{18}F]FDG.

## See also:

- Calculation of
*FUR*image - Calculation of regional
*FUR* - Standardized Uptake Value (SUV, DUR)
- Patlak plot
- Interpretation of Patlak plot by Lars Jødal
- Dual time point estimation of
*K*_{i} - Carimas software

# Literature

Allman KC, Stevens MJ, Wieland DM, Hutchins GD, Wolfe ER Jr, Greene DA, Schwaiger M.
Noninvasive assessment of cardiac diabetic neuropathy by carbon-11 hydroxyephedrine and positron
emission tomography. *J Am Coll Cardiol.* 1993; 22: 1425-1432.
doi: 10.1016/0735-1097(93)90553-D.

Camici P, Araujo LI, Spinks T, Lammertsma AA, Kaski JC, Shea MJ, Selwyn AP, Jones T, Maseri A.
Increased uptake of ^{18}F-fluorodeoxyglucose in postischemic myocardium of patients with
exercise-induced angina. *Circulation* 1986; 74(1): 81-88.
doi: 10.1161/01.CIR.74.1.81.

Villien M, Wey H-Y, Mandeville JB, Catana C, Polimeni JR, Sander CY, Zürcher NR, Fowler JS,
Rosen BR, Hooker JM. Dynamic functional imaging of brain glucose utilization using fPET-FDG.
*NeuroImage* 2014; 100: 192-199. doi:
10.1016/j.neuroimage.2014.06.025.

Tags: Patlak plot, Ki, Macroparameter

Updated at: 2022-10-18

Created at: 2014-01-27

Written by: Vesa Oikonen