# Calculation of Logan plot for regional TACs with reference tissue input

Multiple-time graphical analysis (MTGA) for reversible tracers
(Logan plot) with reference region input
is used to the distribution volume ratio (*DVR*) of the PET tracer, which is related to
binding potential (*BP _{ND}*):

*BP*.

_{ND}= DVR - 1*DVR*(and

*BP*) are unitless.

_{ND}ROI drawing, calculation of regional TACs, and calculation of Logan plot, can all be done in Carimas or PMOD.

**Figure 1.**
Logan plot is simple to calculate even in Excel. You can download Excel spreadsheets containing
simulated data and formulas for Logan plot in two situations:
input and tissue data have the same sample times
(no interpolation needed), and input and tissue data have
different sample times (interpolation required).

If there are more than just a few studies to analyze, then calculation in Excel is too slow and error-prone, and use of Carimas, PMOD, or command-line tools (below) is recommended.

## Command-line software

As an alternative to Carimas, Logan plot can also be calculated using in-house command-line tools:

### 2. Preparation of regional tissue TAC data

Save the regional TAC files in the image analysis program, in DFT format with Carimas, or in TAC format in PMOD. Calculate averages over planes and regions if needed.

Reference tissue region can be included in the datafile with other ROIs, or kept in a separate TAC file.

### 3. Computing the net uptake rate (*K*_{i}^{ref})

_{i}

^{ref}

The logan program can then be run with (at least) the following command-line parameters:

- regional tissue TAC file
- name of reference region inside the previous file, or, file name of TAC file which contains only the reference region TAC
- start time for the line fit (time where linear phase starts)
- end time for the line fit (time where linear phase ends, or end time of study)
- name for result file

With reference region input, the (population) average reference region *k _{2}*
should be used to correct for the (small) time dependence of the ROI/reference ROI -ratio
(Logan et al. 1996). This correction can be made with option

`-k2=X`

, where X is
the average *k*value determined from studies with plasma input. For some tracers, for example [

_{2}^{11}C]raclopride, this correction is not necessary (Logan et al. 1996).

#### Control the linearity of plots

MTGA methods produce correct results only if the slope is calculated from the linear phase of the plot, excluding the curvature in the beginning. You should always draw the Logan plots and control the linearity and quality of the data visually.

# See also:

- Logan plot using plasma input
- Calculation of Logan plot for images
- Reference region input compartmental models

# References:

Logan J, Fowler JS, Volkow ND, Wang G-J, Ding Y-S, Alexoff DL. Distribution volume ratios without
blood sampling from graphical analysis of PET data.
*J Cereb Blood Flow Metab.* 1996; 16: 834-840.

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

Tags: Logan plot, Excel, MTGA, Reference tissue

Updated at: 2014-06-01

Written by: Vesa Oikonen