# Interpretation of the Patlak plot

The following is an attempt to make the Patlak plot understandable as more than just a mathematical description of data. First the basic math behind the Patlak plot is described, using a simple model. Then the math is commented and interpreted. Finally, generalizations to more complex models are discussed.

## Basic math in a simple model

The Patlak plot is useful in compartment models with *irreversible*
uptake of a tracer. The following is a very simple example of such a
model with just one tissue compartment (1TC model):

,where *P(t)* is the plasma input
function, and *C _{i}(t)* is the signal within the given
region-of-interest (ROI) from the tissue compartment irreversibly trapping
the tracer. The signal measured from the ROI will then be

The above simple system is described by the equation

Integration gives *C _{i}(t)*:

Inserting *C _{i}(t)* from

*Eq. 3*into

*Eq. 1*yields

As in *Eq. 1*, this is signal from the tracer irreversibly taken up
plus background. The background signal can be reduced to a constant value if we
divide by the input function:

Defining

it becomes clear that *Eq. 5* can be expressed as a straight line
– the Patlak plot:

Given the input function *P(t)* and the measured ROI signal
*ROI(t)*x and *y* can be computed for each
time point. The slope of the line is the rate of
net influx, *K _{i}*.

Further comments:

- In practice, the Patlak plot is only linear after some time, when steady-state conditions apply outside the compartment of reversible uptake. Even for the simple 1TC model discussed above, a bolus injection of tracer will need a short time to be well-mixed within the plasma.
- For the simple 1TC model, the line's intercept with the
yaxis can be interpreted as the blood volumeV(_{B}Eq. 8). In more general models, this volume will be a mixture of the blood volume and the reversible volumes but with no simple interpretation of the precise value.- Interpretations of
xandyare given in the following section.

### Interpretation of the math in the simple model

#### General comments

*Eq. 2*: The net influx happens with speed
*K _{i}*×

*P(t)*, that is, a constant times the concentration of tracer presented to the system through the input function.

*Eq. 3*: The amount of tracer taken up by time *t* is
calculated by integration of the speed. By analogy with a car: The
integral over speed corresponds to the distance driven since time
zero.

*Eq. 4*: The measured signal comes from the tracer irreversibly taken
up, plus a background signal proportional to the input function.

*Eq. 5*: Division by the input function makes the background
a constant value. The disadvantage is that the other two terms
(*x* and *y*) becomes difficult to interpret.

#### Interpretation of x

*Eq. 6*: Using the car analogy from the comment on *Eq. 3*,
the integral corresponds to distance driven, while *P(t)* corresponds to
speed at point *t*.
If the “speed” has been constant, then *x* is simply the time
since start:

*x* = distance/speed = time = *t*
(only valid for constant speed)

However, in most cases, the speed has not been constant, and *x*
is a kind of “normalized time” or more precisely:

*x* = distance / current speed = the time needed to drive the
distance using current speed

For example, if a car drives at speed 100 km/h for one hour, then at speed
80 km/h for an hour, and then at speed 50 km/h for an hour, the values of
*x* will be:

*x*(1 hour) = (100 km) / (100 km/h) = 1 hour

*x*(2 hours) = (100 km + 80 km) / (80 km/h) = 2.25 hour

*x*(3 hours) = (100 km + 80 km + 50 km) / (50 km/h) = 4.6 hour

That is, after 3 hours the car has driven 230 km, which would have taken
4.6 hours with a constant speed of 50 km/h. Since the car has *not*
driven with constant speed, this “normalized time” of 4.6
hours is *not* equal to the actual time.

Turning to the kinetic model, we can interpret

xas “normalized time” in the following sense:

xis the hypothetical time it would have taken to irreversibly accumulate this amount of tracer, if the input function had been at the current value from the beginning of the study.Put another way:

xis time normalized for the variations in plasma concentration.

For a tracer study with constant infusion, the input function would indeed
be constant, and we would obtain *x* = *t*.
In fact, a tracer study with constant
infusion will automatically be linear and therefore not need the Patlak plot.

For studies with bolus injection, the plasma concentration (input function)
will usually peak and then decrease for most of the study.
Such continuous “deceleration” will result in
“normalized time” (*x*) being increasingly higher than
real time (*t*).

#### Interpretation of y

*Eq. 7*: Like *x* is normalized time, the *y* value is
a kind for normalized measured signal. Again, the normalization, division by
plasma concentration *P(t)*, can be interpreted as a compensation
for the variations in plasma concentration over time. Assuming that signal and
input function are measured in the same units (e.g. Bq/mL), *y*
has no unit.

yis the measured signal normalized for the variations in plasma concentration, expressed as the unitless valuey= (measured signal)/(input function).

#### The Patlak plot

For tracer studies using constant infusion of tracer, the Patlak plot is
not needed: When steady-state has been achieved, net influx will have
a constant rate = *K _{i}*, and signal plotted as a function of
time will be a straight line with slope being

*K*times the value of the constant input function.

_{i}*Eq 8*: For tracers studies with non-constant plasma concentration,
e.g. bolus injection, plasma concentration will vary with time.
But normalization of both time (as *x*) and signal (as *y*)
corrects for this variation and results in a straight line with slope
*K _{i}* = net influx rate.

## More complex models

It turns out that the Patlak plot also is useful in far more complex models than the simple model discussed above.

A two-tissue compartmental (2TC) model with irreversible uptake can be used as an example:

The Patlak plot will in then become linear when the reversible compartment
is in steady-state equilibrium with the plasma, i.e. when the ratio
*C _{1}(t) / P(t)* becomes stable. When that is the case,
the reversible compartment(s) essentially behaves as an extension of the plasma
input function, and the irreversible uptake can be described as an effective
net influx rate

*K*.

_{i}In 2TC model from above, the effective net influx rate constant becomes

This can be interpreted as *K _{1}* being the primary uptake
constant, but of this only the fraction

*k*ends up in the irreversible compartment.

_{3}/(k_{2}+k_{3})For models with several reversible compartments, the Patlak plot will be
linear from the time when all reversible compartments are in steady-state
equilibrium with the plasma. The slope of the Patlak plot will again be
*K _{i}*, the effective net influx rate, although the
analytic expression for

*K*is not necessarily simple to derive.

_{i}If the Patlak plot is used for a system without irreversible compartment,
this corresponds to *K _{i}*= 0, and the
Patlak plot will become a flat line.

## See also:

- K
_{i} - Multiple-time graphical analysis
- Fractional Uptake Rate (
*FUR*) - Equations for graphical analysis of irreversible tracers (Gjedde-Patlak plot)
- Patlak plot from regional TACs with plasma input
- Patlak plot from regional TACs with reference tissue input

## References:

Gjedde A. Calculation of cerebral glucose phosphorylation from brain uptake
of glucose analogs in vivo: a re-examination.
*Brain Res.* 1982; 257: 237-274.

Logan J. Graphical analysis of PET data applied to reversible and
irreversible tracers. *Nucl Med Biol.* 2000; 27: 661-670.

Patlak CS, Blasberg RG, Fenstermacher JD. Graphical evaluation of
blood-to-brain transfer constants from multiple-time uptake data.
*J Cereb Blood Flow Metab.* 1983; 3: 1-7.

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.

Rutland MD. A single injection technique for subtraction of blood background
in 131-I-hippuran renograms. *Br J Radiol.* 1979; 52: 134-137.

Tags: MTGA, Patlak plot, Ki

Updated at: 2014-02-06

Created at: 2014-01-21

Written by: Lars Jødal