# Multilinear models in PET data analysis

Non-linear fitting algorithms are used to estimate parameters of compartmental models (CM) from regional PET data. For pixel-by-pixel computation of parametric maps these methods tend to be too slow, and therefore much faster linear analysis methods are preferably used, including multiple-time graphical analysis (Logan and Patlak plots) and multilinear models.

The compartmental models for PET data are described in terms of a set of linear, first-order, constant-coefficient, ordinary differential equations (ODEs). In a reversible one-tissue compartment model (1TCM) one ODE is sufficient to describe the exchange between the tissue and arterial input compartments:

Assuming that initial concentration in tissue compartment is zero (*C _{1}(0)=0*),
Eq 1 can be integrated to provide the tissue concentration at time

*T*:

The radioactivity concentration in tissue, measured using PET, is contaminated by spillover from adjacent blood vessels and vascular volume inside the region-of-interest (or image voxel):

Substituting *C _{1}* with Eq 2 gives

Integration and rearrangement of Eq 3 gives

which is substituted in Eq 4, providing equation

If the radioactivity concentration in blood can be represented by the model input function,
as is the case with [^{15}O]H_{2}O, then
*C _{B}=C_{0}*, and the Eq 6 simplifies to
Eq 7. This assumption can be made also with some other radiotracers,
including [

^{18}F]FDG.

The three coefficients in Eq 7, *p _{1}=V_{0}*,

*p*, and

_{2}=K_{1}+V_{0}k_{2}*p*, can be estimated by multilinear regression

_{3}=k_{2}, where *p _{i}* denotes the coefficient of independent variable

*x*. The number of coefficients and model parameters must be the same for the linearization method to be useful.

_{i}Since the *p _{i}* coefficients are either ≥0 or ≤0, the
non-negative least squares (NNLS) method is well-suited for the
estimation. As its input, NNLS requires matrix A and vector b, filled with y and x values,
respectively. Values from time frames

*1 - n*are placed on their own rows:

NNLS algorithm returns *p _{i}* coefficient vector, from which the compartment
model parameters can be solved, in this case simply as:

The multilinear solution method has been extended to two-tissue compartment model
(2TCM) by
Blomqvist (1984) and Evans (1987), and further
for [^{18}F]FDOPA model with rate constants *K _{1}-k_{5}*
(Gjedde, 1991). Equations for three-tissue compartmental models with six rate constants are given in
TPCMOD0023 and
TPCMOD0024.
Equations for the irreversible and reversible multilinear
2TCMs are

and

The net influx rate *K _{i}* can be calculated
from the parameters of the irreversible 2TCM model, and from the coefficient vector of the
multilinear solution in Eq 8 as:

The distribution volume *V _{T}* can
be calculated from the parameters of the reversible 2TCM model, and from the coefficient vector of
the multilinear solution in Eq 9 as:

Thus, the calculation of macroparameters *K _{i}* or

*V*from the

_{T}*p*coefficients or rate constants requires division, which leads to very high variation and too noisy parametric maps. Rearranging of equations can provide the macroparameters directly with much less noise although with somewhat increased biases (Zhou et al., 2003; Kim et al., 2008). For the irreversible 2TCM (Eq 8), the solution of rearranged equation

_{i}provides the *K _{i}* estimate directly as

*p*. For the reversible 2TCM (Eq 9), the solution of rearranged equation

_{4}provides the *V _{T}* estimate without division as

*p*.

_{5}- p_{1}## Reference region input compartmental models

Simplified reference tissue model (SRTM) can be
linearized to compute *BP _{ND}* maps
(Zhou et al., 2003):

, where *R _{1}* is the ratio of

*K*s in the region of interest and in the reference region, and by assumption of reference region models, also the ratio of

_{1}*k*s:

_{2}To reduce noise in *BP _{ND}*, the linearized model can be restructured to compute

*DVR*(

*BP*) without division:

_{ND}+1Since the *k _{2}'* should not vary across brain pixels, in
SRTM2 the SRTM model is constrained by fixing

*k*to its global median (Wu & Carson, 2002; Ichise et al., 2003). In multilinear version (MRTM2), the

_{2}'*k*(

_{2}'*=k*) could first be estimated by minimizing model

_{2}/R_{1}and then, using the global median of *k _{2}'*,

*BP*map can be computed by minimizing

_{ND}The transport-limited reference tissue model
(TRTM), where the reference region is such that irreversible binding or metabolism is very rapid
(*k _{3}'>>k_{2}'*), can be linearized to produce

*k*maps (TPCMOD0002):

_{3}or to estimate *k _{3}* directly without division:

## Applications

One-tissue compartment model (1TCM) is sufficient to
describe the kinetics of many radiotracers, including radiowater.
Linearization of radiowater model was described by
van den Hoff et al (1993), and it was
applied to skeletal muscle
[^{15}O]H_{2}O PET data by
Burchert et al (1997),
and to myocardial muscle by
Lee et al (2005).
Linearized 1TCM has been used to assess BBB permeability
from [^{68}Ga]EDTA PET data
(Zhou et al., 2001)
and histamine receptor occupation with [^{11}C]dozepin
PET (Zhou et al., 2002).

Multilinear form of irreversible 2TCM
(*k _{4}=0*) and reversible 2TCM (

*k*) has been used to compute parametric images from PET studies where 1TCM is not suitable, for example [

_{4}>0^{11}C]glucose, [

^{11}C]deoxyglucose, and [

^{11}C]methionine (Blomqvist, 1984), [

^{18}F]FDG (Cai et al., 2002; Huang et al., 2007; Pouzot et al., 2013), [

^{18}F]fluoride (Kim et al., 2007; Sanchez-Crespo et al., 2017), dopamine 2 receptor radioligand [

^{11}C]FLB457 (Hagelberg et al., 2004), and μ opioid receptor radioligand [

^{11}C]MeNTI (Kim et al., 2008) PET data.

SRTM can be linearized to compute
BP_{ND} maps
(Zhou et al., 2003;
TPCMOD0002). The method has been used
in analyses of brain receptor studies, for example [^{11}C]DASB
(Kim et al., 2006;
Endres et al., 2011).

Time delay between arterial blood and tissue curves can be rapidly estimated using linearized model (van den Hoff et al., 1993).

## Bias

Noise in the data leads to bias in parameter values estimated using linear least squares methods, (Feng et al., 1993 and 1996). The results of multilinear model fitting can be used as initial values for bias-free but slower non-linear optimization methods (Feng et al., 1993) or refined with general linear least squares (GLLS) method) (Feng & Ho, 1993; Feng et al., 1995 and 1996; Negoita & Renaut, 2005; Wen et al., 2007).

## Multilinear equations with time frames

The radioactivity concentration curves (time-activity curves, TACs), measured with PET,
consist of consecutive time frames; we do not have measurements of
radioactivity concentrations at specific time points (as assumed in the equations above), but
instead we have the average concentrations during each time frame, and the
*AUCs* at the end of each time frame.
At the time zero, *AUC=0*.
Assuming that the frames are contiguous (the start time of frame *i* equals the
end time of frame *i-1*), at the end of frame *i* the *AUC* is

, where *Δt* is the length of time frame
(*Δt = t ^{e}_{i} - t^{e}_{i-1}*) and

*C*is the average concentration during the frame

_{i}*i*.

Regarding the one-tissue compartment model in Eq 2, the concentrations in
the tissue compartment *C _{1}* at the start and end of frame

*i*are:

Substitution of Eq 5 into these gives

The average concentration during frame *i* is

The average *AUC* during frame *i* is

and thus the PET time frame based multilinear equation for one-tissue compartment can be written as

, where all concentrations and *AUCs* represent the averages during the frame *i*.
This equation has the same form as Eq 7 for the non-framed data.
Equations for the other compartmental models can be similarly applied to PET data with time frames.
The *AUC* average during PET frame *i* can be calculated as

## See also:

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Tags: Compartmental model, NNLS

Updated at: 2023-07-01

Created at: 2023-06-07

Written by: Vesa Oikonen