Turku PET Centre Modelling report TPCMOD0001 Appendix C

# Model equations for compartmental model with dual plasma input

## Compartments for the first tracer in parallel

This document describes the mathematical equations needed to simulate the PET time-radioactivity
concentration curves in compartmental model which has two plasma inputs (parent radiotracer and
radioactive metabolite): two parallel tissue compartments *in parallel* attached to
the first tissue compartment for the parent tracer, and one tissue compartment for the metabolite,
with possible loss from the second parallel 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 (1993) 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. (C.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. (C.9), and the compartmental concentration can be solved as in Eq. (C.10);
*T-Δt* is the time of previous sample:

To simplify the equations, we use the following substitutions:

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

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

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

Equations for C_{1} can be solved by substitutions of Eqs. (C.12) and (C.14) into
Eq. (C.5), and further substitutions of Eq. (C.9) or (C.10):

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

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

, or after substitutions of Eqs. (C.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. (C.1-3):

_{AM}=0, and, further:

Substitution of these into Eq. (C.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. C.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. (C.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.
doi: 10.1038/jcbfm.1993.7.

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

Updated at: 2019-04-04

Created at: 2013-08-20

Written by: Vesa Oikonen