# Fractional uptake rate (*FUR*)

*FUR*is a simple PET quantifier, calculated as a ratio of tissue activity

*C*at time

_{T}*T*and integral of plasma activity

*C*from time

_{P}*0*to

*T*(Camici et al., 1986; Allman et al., 1993; Ishizu et al. 1994 and 1995; Rutland et al., 2000):

Thus the unit of *FUR* is min^{-1}
(Thie, 1995), the same as for
*K _{i}*.

*FUR*can be calculated from a single late PET scan, like

*SUV*, but blood sampling from the injection to the scan time is required.

Fractional uptake rate was previously calledRetention index(R), especially in analysis of cardiac PET studies. This is discouraged, because the term Retention index has several other uses, also in nuclear medicine._{i}

*FUR* is an approximation to the Patlak plot
(MTGA for irreversible uptake) slope *K _{i}*,
to the extent that at large

*T*(late time after injection) the effective distribution volume term in Patlak analysis is not important (y axis intercept is assumed to be 0).

*FUR*and

*SUV*are proportional, related by plasma clearance rate and a dimensionless initial distribution volume (Thie, 1995).

Equation of FUR can be derived by rearrangement of
the operational equation of the Patlak plot, assuming
that the Patlak plot intercept is zero. If a population average for the Patlak plot intercept is
applicable, then that can be used to calculate
*K _{i}* from a single measured tissue activity
as if using Patlak plot with fixed y axis intercept:

## Simulations

Mathematical function that describes the population average of
FDG concentrations in arterial plasma is used to create FDG
PTAC at 3 s sample intervals. Tissue curve (TTAC) is simulated
using the PTAC as input function, and irreversible
2TCM, with
typical parameters for the brain grey matter:
*K _{1}*=0.09 mL*(min*mL)

^{-1},

*K*=0.41 mL*mL

_{1}/k_{2}^{-1}, and

*k*=0.11 min

_{3}^{-1}(Fig 2a). Blood volume is not considered in this simulation (

*V*=0). With these parameters, net influx rate can be calculated from equation for

_{B}*K*in 2TCM,

_{i}, giving value *K _{i}*=0.0300444 mL*(min*mL)

^{-1}. Patlak plot indeed gives a very precise estimate for

*K*with these simulated TACs, if line fit is started from 15 min. Instead,

_{i}*FUR*is overestimated, approaching the correct value, but not reaching it during 120 min (Fig 2b).

The bias in *FUR*, as compared to *K _{i}*, is not only dependent on
the time of

*FUR*calculation, but also on the intercept of the Patlak plot.

In case of an irreversible 2TCM, and assuming that *V _{B}*=0, the theoretical
y axis intercept of the Patlak plot can be calculated from the rate constants:

If we do the simulation again, this time with 15% lower and 15% higher *k _{3}*,
but keeping other rate constants as they were, we will get the TTACs shown in Fig 3a, and FURs
as shown in Fig 3b. Patlak plot will again provide correct

*K*s (0.0300444, 0.02688, and 0.03293 mL*(min*mL)

_{i}^{-1}, respectively). The 15% change in

*k*changes

_{3}*K*-10.5% and +9.5%, and an opposite change in intercept (+11% and -9%, respectively).

_{i}*FUR*overestimates

*K*in all situations, but the overestimation is lower when

_{i}*k*(and

_{3}*K*) is increased (Fig 3c); thus true differences in

_{i}*K*will be underestimated when

_{i}*FUR*calculation is used.

In the next simulation, *k _{2}* is set to 15% lower and 15% higher value,
but other rate constants are kept unchanged. This will provide the TTACs shown in Fig 4a, and FURs
as shown in Fig 4b. Patlak plot will again provide correct

*K*s (0.0300444, 0.03338, and 0.02731 mL*(min*mL)

_{i}^{-1}, respectively). The 15% change in

*k*changes

_{2}*K*+11.1% and -9.1%, and causes a parallel but smaller changes in the intercept (+5% and -5%, respectively).

_{i}*FUR*is overestimated in all situations, and not much dependent on the value of

*k*(Fig 4c). Even though the effect is small, the smaller

_{3}*k*(and increased

_{2}*K*) leads to lower overestimation; thus true differences in

_{i}*K*will be underestimated when

_{i}*FUR*calculation is used, whether the change in

*K*is due to changed

_{i}*k*or

_{2}*k*.

_{3}The data and scripts used in these simulations are available in gitlab.utu.fi/vesoik/simulations.

## Metabolic rate

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

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

Naturally, the *FUR*-based metabolic rates contain the same bias as *FUR* itself
when compared to metabolic rates based on *K _{i}*.

## See also:

- Calculation of
*FUR*image - Calculation of regional
*FUR* - Area-under-curve (AUC) in PET
- Standardized Uptake Value (SUV, DUR)
- Patlak plot
- Carimas software
- Simulations in GitLab

# References:

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.
10.1161/01.CIR.74.1.81.

Ishizu K, Nishizawa S, Yonekura Y, et al. Effects of hyperglycemia on FDG
uptake in human brain and glioma. *J Nucl Med* 1994; 35:1104-1109.

Ishizu K, Yonekura Y. Clarification of a fractional uptake concept – Reply. *J Nucl
Med* 1995; 36:712.

Rutland M, Que L, Hassan IM. “FUR” – one size suits all. *Eur J Nucl Med.* 2000;
27:1708-1713. doi: 10.1007/s002590000.

Thie JA. Clarification of a fractional uptake concept. *J Nucl Med* 1995; 36:
711-712.

Tags: FUR, Retention index, Ki, Simulation, AUC

Updated at: 2019-03-28

Created at: 2007-11-01

Written by: Vesa Oikonen