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Turku PET Centre Modelling report TPCMOD0023 2003-11-02
Vesa Oikonen
Multilinear solution for 4-compartment model: I. Tissue
compartments in series
The compartmental models presently used in the analysis of positron emission
tomography data lead into similar systems of differential equations. Usually, the
unknown model rate constants are solved with iterative non-linear least-squares
methods. Blomqvist (1984) proposed a linear algorithm for solution of two-
compartment and irreversible three-compartment models. The same method has been
extended to reversible three-compartment model with vascular contribution to the
total radioactivity concentration (Gjedde 1990). This document extends further the
method by applying it to compartmental model with three tissue compartments in
series.
Compartmental model
The measured radioactivity concentration of the parent tracer in arterial plasma, C0(t),
is the input to tissue compartmentsC1(t),C2(t) and C3(t). The total tissue
concentration that is measured with PET, CT(t), is defined as the sum of separate
tissue compartments and the product of vascular volume fraction in tissue, V0, and
radioactivity concentration in vascular volume:
)1()()()()()( 32100 tCtCtCtCVtCT+++=
The contribution of vascular radioactivity is included in the model in a way that is
valid only if the radioactivity concentration in vascular volume can be assumed to be
equal to concentration of parent tracer in plasma; this is the case usually in [18F]FDG
studies. Another possibility is to assume that V0 represents only arterial volume
fraction, and that the radioactivity concentration in venous blood is similar or at least
proportional to the sum of radioactivity concentrations in tissue compartments. All
radioactivities are corrected for physical decay, and they are therefore proportional to
the concentration of tracer molecules.
The total distribution volume, DVT, equals the sum of distribution volumes of separate
tissue compartments:
C0C1C2C3
K1
k2
k3
k4
k5
k6
( )
642
6463531
6
5
4
3
2
1
6
5
4
3
2
1
4
3
2
1
2
1
321
11 kkk kkkkkkK
k
k
k
k
k
K
k
k
k
k
k
K
k
k
k
K
k
K
DVDVDVDVT
++
=
++
=++=++=
Differential equations
Solution to the full model
The sum of equations 2-4 is substituted into differential format (differentiation with
respect to time) of equation 1, giving equation 5.
)5()()(
)()( 1201
0
0tCktCK
dt
tdC
V
dt
tdCT−+=
The concentration of the first tissue compartment is solved from it (Eq. 6) and its first,
second and third order differential equations are represented in equations 7-9.
)9(
)(
)()(
)(
)8(
)(
)()(
)(
)7(
)(
)()(
)(
)6(
)(
)(
)(
)(
0
1
0
0
1
2
0
1
0
0
1
2
0
1
0
0
1
2
01
0
012
dtdtdtdt tddddC
dtdtdt
tdddC
K
dtdtdtdt
tddddC
V
dtdtdt tdddC
k
dtdtdt tdddC
dtdt
tddC
K
dtdtdt
tdddC
V
dtdt tddC
k
dtdt tddC
dt tdC
K
dtdt tddC
V
dt tdC
k
dt tdC
tCK
dt tdC
VtCk
T
T
T
T
−+=
−+=
−+=
−+=
Concentration of the third tissue compartment can be solved from the equation 3 (Eq.
10), and differentiated with respect to time (Eq. 11):
( )
( )
)11(
)()()(
)(
)10()()(
)(
)(
2
54
1
3
23
6
25413
2
36
dt
tdC
kk
dt
tdC
k
dtdt
tddC
dt
tdC
k
tCkktCk
dt
tdC
tCk
++−=
++−=
These can be substituted into equation 4, which gives, after rearrangement:
( )
( )
)4()()(
)(
)3()()()(
)(
)2()()()(
)(
3625
3
3625413
2
2413201
1
tCktCk
dt tdC
tCktCkktCk
dt tdC
tCktCkktCK
dt tdC
−=
++−=
++−=
( )
)12()(
)(
)(
)()( 163
1
3264
2
654
2tCkk
dt tdC
ktCkk
dt tdC
kkk
dtdt tddC +=++++
Similarly, concentration of the second tissue compartment can be solved from the
equation 2 (Eq. 13), and differentiated twice with respect to time (Eqs. 14 and 15):
( )
( )
( )
)15(
)(
)(
)()(
)14(
)()()()(
)13()()(
)(
)(
1
32
0
1
12
4
1
32
0
1
12
4
13201
1
24
dtdt tddC
kk
dtdt tddC
k
dtdtdt tdddC
dtdt tddC
k
dt tdC
kk
dt tdC
k
dtdt tddC
dt tdC
k
tCkktCk
dt tdC
tCk
++−=
++−=
++−=
Equations 13-15 can be substituted into equation 12, which gives, after rearangement:
( )
( ) ( )
[ ]
( )
)16(
)(
)()(
)(
)(
)()(
0641
0
6541
0
1
1642
1
432632542
1
65432
1
tCkkK
dt tdC
kkkK
dtdt tddC
K
tCkkk
dt tdC
kkkkkkkkk
dtdt tddC
kkkkk
dtdtdt tdddC
++++=
++++++
++++++
Equations 6-9 are substituted into equation 16. Rearrangement, and three integrations
over the time interval (0,T) and using the initial conditions
CT(0)=C1(0)=C2(0)=C3(0)=0
and
dCT(0)/dt=dC1(0)/dt=dC2(0)/dt=dC3(0)/dt=0, gives
( )
[ ]
( ) ( ) ( )( )
[ ]
( )
[ ]
( ) ( )
[ ]
( )
)17(
)(
)(
)(
)(
)(
)(
)(
)(
0
65432
00
432632542
000
642
00
006543201
00 0432632542065431
000 064206463531
∫
∫∫
∫∫∫
∫
∫∫
∫∫∫
++++
−
+++++
−
−
++++++
+
+++++++++
+
+++
=
T
T
TT
T
TTT
T
T
TT
TTT
T
dttCkkkkk
dtdutCkkkkkkkkk
dtdudvtCkkk
TCV
dttCkkkkkVK
dtdutCkkkkkkkkkVkkkkK
dtdudvtCkkkVkkkkkkK
TC
After numerical integration of the measured plasma and tissue concentrations, the
coefficients can be estimated with any least-squares method, e.g. NNLS (Lawson and
Hanson, 1974). The model parameters can then be solved from these coefficients. If
the coefficients are represented with P1, ..P7, the equations for model parameters are:
)18(
)*/(
/)//)*((
/)/)*((
/)*(
*
643275
4256
32515413274
215416273
164272
7431
40
−−−−=
=−−−−−= −−−−= −−= −=
=
kkkkPk
kkPk
kkPKPPPkkPk
kKPPPPkPk
KPPPPk
PPPK
PV
Total volume of distribution could be calculated as DVT=P1/P5-P4. However, if DVT is
the only parameter of interest, it may be preferable to get an estimate for it directly
without division. For that purpose, equation 17 can divided by k2k4k6 and rearranged
to give
( )
( ) ( ) ( )( )
( )
( ) ( )
)19(
)(
1
)(
)(
)(
)(
)(
)(
)(
642
0
642
65432
00
642
432632542
0
642
0
00
642
6543201
00 0
642
432632542065431
000 00
642
6463531
000
TC
kkk
dttC
kkk kkkkk
dtdutC
kkk kkkkkkkkk
TC
kkk V
dttC
kkk kkkkkVK
dtdutC
kkk kkkkkkkkkVkkkkK
dtdudvtCV
kkk kkkkkkK
dtdudvtC
T
T
T
TT
T
T
TT
TTT
TTT
T
−
++++
−
+++++
−
+
+++++
+
+++++++++
+
+
++
=
∫
∫∫
∫
∫∫
∫∫∫
∫∫∫
, where the first coefficient equals DVT+V0.
Solution to the model when V0=0
If the contribution of vascular concentration to the total tissue radioactivity can be
assumed negligible, then V0 can be set to zero in the equation 17, giving
( )
( )
( ) ( )
[ ]
( )
)20(
)(
)(
)(
)(
)(
)(
)(
0
65432
00
432632542
000
642
001
00 065431
000 06463531
∫
∫∫
∫∫∫
∫
∫∫
∫∫∫
++++
−
+++++
−
−
+
+++
+
++
=
T
T
TT
T
TTT
T
T
TT
TTT
T
dttCkkkkk
dtdutCkkkkkkkkk
dtdudvtCkkk
dttCK
dtdutCkkkkK
dtdudvtCkkkkkkK
TC
, where, if the coefficients are represented with P1, ..P6, the equations for solving
model parameters are:
)21(
)*/(
/)//(
/)/(
/
643265
4246
324113264
2115263
1262
31
−−−−=
=−−−−= −−−= −=
=
kkkkPk
kkPk
kkPKPkkPk
kKPPkPk
KPPk
PK
Solution to the model when k6=0
If there is irreversible uptake in the tissue, then k6 must be set to zero in the equation
17, giving
( ) ( )( )
[ ]
( )
[ ]
( )
[ ]
( )
)22(
)(
)(
)(
)(
)(
)(
)(
0
5432
00
32542
00
00543201
00 03254205431
000 0531
∫
∫∫
∫
∫∫
∫∫∫
+++
−
++
−
+++++
+
+++++
+
=
T
T
TT
T
T
TT
TTT
T
dttCkkkk
dtdutCkkkkk
TCV
dttCkkkkVK
dtdutCkkkkkVkkkK
dtdudvtCkkK
TC
, where, if the coefficients are represented with P1, ..P6, the equations for solving
model parameters are:
)23(
)*/(
/)/(
/)*(
*
53264
3115
2115263
154262
6431
40
−−−=
=−−−= −−= −=
=
kkkPk
kKPk
kKPPkPk
KPPPPk
PPPK
PV
Equation (22) coeld be rearranged to a form that was previously given by Gjedde
(1991). Because the model is irreversible, the total volume of distribution cannot be
calculated, but instead the influx rate constant, Ki, can be solved as Ki = P1/P5. In this
four-compartmental model setting
535242
531 kkkkkk
kkK
Ki++
=
If Ki is the only parameter of interest, it may be preferable to get an estimate for it
directly without division. For that purpose, equation 23 can divided by k2k4+k2k5+
k3k5 and rearranged to give
( )
( )
)24(
)(
1
)(
)(
)(
)(
)(
)(
535242
0
535242
5432
0
535242
0
00
535242
543201
00 00
535242
5431
000 0
535242
531
00
TC
kkkkkk
dttC
kkkkkk kkkk
TC
kkkkkk V
dttC
kkkkkk kkkkVK
dtdutCV
kkkkkk kkkK
dtdudvtC
kkkkkk kkK
dtdutC
T
T
T
T
TT
TTT
TT
T
++
−
++ +++
−
++
+
++ ++++
+
+
++ ++
+
++
=
∫
∫
∫∫
∫∫∫
∫∫
Solution to the model when k6=0 and V0=0
If there is irreversible uptake in the tissue and vascular volume can be ignored, then k6
and V0 can be set to zero in the equation 17, giving
( )
( )
[ ]
( )
)25(
)(
)(
)(
)(
)(
)(
0
5432
00
32542
001
00 05431
000 0531
∫
∫∫
∫
∫∫
∫∫∫
+++
−
++
−
+
++
+
=
T
T
TT
T
T
TT
TTT
T
dttCkkkk
dtdutCkkkkk
dttCK
dtdutCkkkK
dtdudvtCkkK
TC
, where, if the coefficients are represented with P1, ..P5, the equations for solving
model parameters are:
)26(
)*/(
/)/(
/
53254
3115
2114253
1252
31
−−−=
=−−−= −=
=
kkkPk
kKPk
kKPPkPk
KPPk
PK
Solution to the reversible model with only two tissue compartments
If the third tissue compartment does not exist, then k5 and k6 can be set to zero in the
equation 17, giving
( )
[ ]
( )
[ ]
( )
)27(
)(
)(
)(
)(
)(
)(
0
432
00
42
00
0043201
00 0420431
∫
∫∫
∫
∫∫
++
−
−
++++
+
++
=
T
T
TT
T
T
TT
T
dttCkkk
dtdutCkk
TCV
dttCkkkVK
dtdutCkkVkkK
TC
, which is the same equation that has been previously formed from the two-tissue
compartment model (Gjedde and Wong, 1990). If the coefficients are represented
with P1, ..P5, the equations for solving model parameters are:
)28(
/
/)*(
*
4253
244
143152
5321
30
−−=
=−−= −=
=
kkPk
kPk
KPPPPk
PPPK
PV
Total volume of distribution in this model setting is
+=+=+= 4
3
2
1
4
3
2
1
2
1
21 1k
k
k
K
k
k
k
K
k
K
DVDVDVT
It can be calculated from the coefficients as DVT = P1/P4 – P3. However, if DVT is the
only parameter of interest, it may be preferable to get an estimate for it directly
without division. For that purpose, equation 17 can divided by k2k4k6 and rearranged
to give
( )
)29(
)(
1
)(
)(
)(
)(1
)(
42
0
42
432
0
42
0
00
42
43201
00 00
4
3
2
1
00
TC
kk
dttC
kk kkk
TC
kkV
dttC
kk kkkVK
dtdutCV
k
k
k
K
dtdutC
T
T
T
T
TT
TT
T
−
++
−
+
+++
+
+
+
=
∫
∫
∫∫
∫∫
If also vascular volume is ignored (V0=0), then the following equations can be
formed:
( )
( )
)30(
)(
)(
)(
)(
)(
0
432
00
42
001
00 0431
∫
∫∫
∫
∫∫
++
−
−
+
+
=
T
T
TT
T
T
TT
T
dttCkkk
dtdutCkk
dttCK
dtdutCkkK
TC
)31(
/
/
4243
234
1142
21
−−=
=−=
=
kkPk
kPk
KPPk
PK
Total volume of distribution could be calculated directly without division using the
following equation (instead of equation 30):
)30(
)(
1
)(
)(
)(1
)(
42
0
42
432
00
42
1
00 0
4
3
2
1
00
TC
kk
dttC
kk kkk
dttC
kkK
dtdutC
k
k
k
K
dtdutC
T
T
T
T
TT
TT
T
−
++
−
+
+
=
∫
∫
∫∫
∫∫
References
1. Blomqvist G. On the construction of functional maps in positron emission
tomography. J. Cereb. Blood Flow Metab. 1984; 4: 629-632.
2. Gjedde A. Modeling the dopamine system in vivo. In: In vivo imaging of
neurotransmitter functions in brain, heart, and tumors. (Ed. Kuhl DE). ACNP
Publication 91-2, USA, 1991, 157-179.
3. Gjedde A, Wong DF. Modeling neuroreceptor binding of radioligands in vivo.
In: Quantitative Imaging: Neuroreceptors, Neurotransmitters, and Enzymes.
(Eds. Frost JJ, Wagner HN Jr). Raven Press, Ltd., New York, 1990, 51-79.
4. Lawson CL, Hanson RJ. Solving least squares problems. Prentice-Hall,
Englewood Cliffs, New Jersey, 1974.
