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 (C3) 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.  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 CAP and C1, C2, and C3, respectively. The compartment for the second tracer (radioactive metabolite) concentration in arterial plasma, and its tissue compartment are represented by CAM and C4, respectively.
The differential equations describing the concentration changes in the tissue compartments are expressed as:
, where km represents the rate constant for metabolism of parent tracer in C1, adding to concentration of metabolite in C4. Rate constant k7 represents the efflux of tracer from C3 directly to plasma.
The integrated forms of Eqs. (B.1-4) are:
By applying linear interpolation  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 C3
and substitution of Eq (B.11) into Eq. (B.7.) gives the integral of C3:
Concentration and integral in C2 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 C1 can be solved similarly:
The concentration in the compartment for the second tracer, C4, can be solved by substitution of Eqs. (B.9) into Eq. (B.8.):
The sum of radioactivity concentrations in tissue (CT) 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 (VTP). It equals the sum of distribution volumes in individual tissue compartments for the parent tracer:
In steady state, dC1(t)/dt=0, dC2(t)/dt=0, and dC3(t)/dt=0. Considering only the parent tracer input (CAM=0), the concentration ratios in steady state can be solved from Eqs. (B.1-3):
, and, further:
Substitution of these into Eq. (B.20) and rearrangement gives
Note that if km>0 this is not the same as the distribution volume for parent tracer estimated using Logan plot, because Logan plot VT would include also V4, even when CAM=0.
Although not useful in PET data analysis, in software testing the distribution volumes from Logan plot can be used if, either km=0 and CAM=0 (using Eq. B.23), or km>0 and CAM=CAP (we will mark both with CA). In the latter case, the concentration ratio C4/CA in steady state (dC4(t)/dt=0) can be solved from Eq. (B.4):
, and then
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- [18F]fluoro-DOPA and positron emission tomography: error analysis and application to normal subjects. J Cereb Blood Flow Metab. 1993; 13:43-56.
Created at: 2013-08-15
Updated at: 2013-08-28
Written by: Vesa Oikonen