Turku PET Centre Modelling report TPCMOD0001 Appendix B

# Model equations for compartmental model with dual plasma input

## Compartments for the first tracer in series

This document describes the mathematical equations needed to simulate the PET
time-radioactivity concentration curves in compartmental model which has two
plasma inputs (parent tracer and metabolite): three tissue compartments
*in series* for the parent tracer, and one tissue compartment for the
metabolite, with possible loss from the last tissue compartment (C_{3})
directly to plasma.
If the compartments for the two inputs were independent, then the
model could be simulated separately for each input and resulting tissue curves
could be simply summed. In the presented equations the compartments are not
independent, because transformation of parent tracer to metabolite from
the first tissue compartment is included in the model.

The approach of Kuwabara et al. [1] is used for partial solutions of
the differential equations.
The compartment for parent tracer concentration in arterial plasma (or blood,
depending on the specific tracer), and the tissue compartments are represented
by C_{AP} and C_{1}, C_{2}, and C_{3},
respectively.
The compartment for the second tracer (radioactive metabolite) concentration
in arterial plasma, and its tissue compartment are represented
by C_{AM} and C_{4}, respectively.

The differential equations describing the concentration changes in the tissue compartments are expressed as:

, where *k _{m}* represents the rate constant for metabolism
of parent tracer in C

_{1}, adding to concentration of metabolite in C

_{4}. Rate constant

*k*represents the efflux of tracer from C

_{7}_{3}directly to plasma.

The integrated forms of Eqs. (B.1-4) are:

By applying linear interpolation [1] and assuming that initial concentrations
are zero, the integral of radioactivity concentration in compartment
*N* can be presented as in Eq. (B.9),
and the compartmental concentration can be solved as in Eq. (B.10);
*T-Δt* is the time of previous sample:

To simplify the equations, we use the following substitutions:

Substitution of Eq. (B.10) into Eq. (B.7), with rearrangements, gives the
concentration C_{3}

and substitution of Eq (B.11) into Eq. (B.7.) gives the integral of
C_{3}:

Concentration and integral in C_{2} can be solved by substitutions
of Eq. (B.12) and either Eq. (B.9) or (B.10), respectively, into Eq. (B.6.):

And equations for C_{1} can be solved similarly:

The concentration in the compartment for the second tracer, C_{4},
can be solved by substitution of Eqs. (B.9) into Eq. (B.8.):

The sum of radioactivity concentrations in tissue (C_{T}) is:

, or after substitutions of Eqs. (B.5-8),

### Volume of distribution

If the model is used to analyze PET data, a useful macro parameter for
reporting is the (equilibrium) volume of distribution of the parent tracer
(V_{TP}). It equals the sum of distribution volumes in individual
tissue compartments for the parent tracer:

In steady state, *dC _{1}(t)/dt*=0,

*dC*=0, and

_{2}(t)/dt*dC*=0. Considering only the parent tracer input (

_{3}(t)/dt*C*), the concentration ratios in steady state can be solved from Eqs. (B.1-3):

_{AM}=0, and, further:

Substitution of these into Eq. (B.20) and rearrangement gives

Note that if *k _{m}>0* this is

*not*the same as the distribution volume for parent tracer estimated using Logan plot, because Logan plot V

_{T}would include also V

_{4}, even when

*C*.

_{AM}=0Although not useful in PET data analysis, in software testing
the distribution volumes from Logan plot can be used
if, either *k _{m}=0* and C

_{AM}=0 (using Eq. B.23), or

*k*and C

_{m}>0_{AM}=C

_{AP}(we will mark both with C

_{A}). In the latter case, the concentration ratio C

_{4}/C

_{A}in steady state (

*dC*=0) can be solved from Eq. (B.4):

_{4}(t)/dt, and then

## References

Kuwabara H, Cumming P, Reith J, Léger G, Diksic M, Evans AC, Gjedde A.
Human striatal L-DOPA decarboxylase activity estimated in vivo using 6-
[^{18}F]fluoro-DOPA and positron emission tomography: error analysis and
application to normal subjects. *J Cereb Blood Flow Metab.* 1993;
13:43-56.

Tags: Compartmental model, Dual-input, Loss-rate

Updated at: 2013-08-28

Created at: 2013-08-15

Written by: Vesa Oikonen