Multiple Time Graphical Analysis (MTGA)

In multiple-time graphical analysis the radiopharmaceutical concentration curves of tissue region-of-interest and arterial plasma are transformed and combined into a single curve that approaches linearity when certain conditions are reached. The data could be plotted in a graph, and line can be fitted to the linear phase. The slope of the fitted line represents the net uptake rate of the radiopharmaceutical or volume of distribution. In some instances a reference region curve can be used as input function in place of arterial plasma input.

The graphical analysis methods are independent of any particular model structure, although the slope can be interpreted in terms of a combination of model parameters for some model structure. Graphical analysis methods have been developed for reversibly and irreversibly binding radiopharmaceuticals (Logan, 2000 and 2003).

Munk (2012) proposed using vi-plot to study at which time point the quasi steady state is reached, allowing the linear fitting of Logan or Patlak plots.

MTGA for irreversible uptake (Patlak plot)

Principles of Patlak analysis

The original idea of Patlak and Blasberg was to create a model independent graphical analysis method: whatever the radiopharmaceutical is facing in the tissue, there must be at least one irreversible reaction or transport step, where the radiotracer or its labelled product cannot escape.

It is assumed that all the reversible compartments must be in equilibrium with plasma, i.e. the ratio of the concentrations of radiopharmaceutical in plasma and in reversible tissue compartments must remain stable. In these circumstances only the accumulation of radiopharmaceutical in irreversible compartments is affecting the apparent distribution volume. In practise, this can happen only after the initial sharp concentration changes when the plasma curve (input function) descends slow enough for tissue compartments to follow.

Patlak plot is model independent — **Figure 1.** There can be any number of reversible compartments, where the radiopharmaceutical can come and go. After some time, radiopharmaceutical concentrations in these compartments start to follow the radiopharmaceutical concentration changes in plasma (ratio does not change). Then, any change in the total tissue concentration (measured by PET) per plasma concentration represents the change in irreversible compartment(s).

If there is no irreversible binding in the tissue, the resulting Patlak plot becomes horizontal, with slope of zero. In this case MTGA for reversible uptake (Logan plot) could be applied instead to calculate the distribution volume.

Making Patlak plot

When the equilibrium is achieved, the Patlak plot becomes linear. The slope of the linear phase represents the net transfer rate K_i (influx constant). To make it simple, K_i represents the amount of accumulated radiopharmaceutical in relation to the amount of radiopharmaceutical that has been available in plasma.

Operational equation for the Patlak plot is

$\frac{C_{ROI}(T)}{C_{p}(T)} = K_i \times \frac{\int_0^T C_p(t)dt}{C_{p}(T)} + Int$

where linearity is achieved after the distribution volume of the reversible compartments, included in the intercept (Int) is effectively constant (Logan, 2000). The y axis of plot contains apparent distribution volumes, that is the ratio of concentrations of radiopharmaceutical in tissue and in plasma, as a function of time. On x axis is normalized plasma integral, that is the ratio of the integral of plasma concentration and the plasma concentration.

This method was actually first applied by Rutland (1979), and then by Gjedde (1982), but formal representation of the analysis method was published by Patlak & Blasberg (1983 and 1985).

Calculation of Patlak plot — **Figure 2.** Patlak plot becomes linear after the radiopharmaceutical concentrations in reversible compartments and in plasma are in equilibrium. The slope of the linear phase of plot is the net uptake (influx) rate constant *K_i*. *K_i* is in units *min^-1*, or *(mL tissue)/((mL plasma)*min)*. Note that although x axis has time units (min or sec), the values do *not* represent the PET sample times.

The y axis intercept of the Patlak plot involves tissue blood volume fraction and distribution volumes, in unknown proportions, and may be affected by input function (Laffon-Marthan et al., 2021). Therefore its use as relevant clinical parameter is limited. Yet, it may be useful for estimating and correcting for the highly variable tissue fat and lung air volume fractions which otherwise could cause large bias for example in liver and lung PET studies.

Patlak plot analysis requires that sufficiently long dynamic PET scan is performed, and that arterial plasma curve is measured starting from the radiopharmaceutical administration until the end of the PET scan. Blood sampling is not necessary, if the input function can be measured from the dynamic PET image.

Dual-time point K_i

If two static late-scans can be performed, and input function can be measured from the PET images, then it may be possible to estimate K_i (and subsequently metabolic rate) from that dual-time point (DTP) data alone (van den Hoff et al., 2013; Wu et al., 2021).

Late dynamic scan

If input function is derived from dynamic image, but scan was not started at administration time and input peak is thus missing, relative Patlak plot can be calculated, providing relative K_i image. K_i estimates are not quantitative, but comparable to true K_i values with a single scaling factor, and these parametric K_i images can be used in lesion detection and SPM (Zuo et al., 2018).

Single late-scan

If only one late-scan can be performed, but plasma data is available for the whole time, starting from the time of radiopharmaceutical administration, and a population average of the Patlak plot intercept is applicable, then K_i can be calculated by rearranging the operational equation for the Patlak plot:

$K_i = \frac{C_{ROI}(T)}{\int_0^T C_p(t)dt} - Int \times \frac{C_{p}(T)}{\int_0^T C_p(t)dt}$

, where C_ROI(T) and C_p(T) are the activity concentrations in tissue and plasma at the middle time of the PET frame. If the intercept is known to be small, but exact value for it is not available, it can be assumed to be zero; in that case, we end up with the equation for FUR, which is an approximation of K_i:

$K_i \approx \frac{C_{ROI}(T)}{\int_0^T C_p(t)dt} = FUR$

Patlak plot without plasma sampling

In brain PET studies it may be possible to have a reference region where irreversible compartments do not exist: for example cerebellum in FDOPA studies. In FDG studies this is not possible because all brain regions consume glucose. Reference region contains only reversible compartments, which also achieve an equilibrium with plasma. The reference region can be included in the model, and the plasma curve is cancelled out (Patlak and Blasberg, 1985). In practise, the only difference to the calculation using plasma input is that plasma curve is replaced with reference region curve.

The result is not the same when reference input is used instead of plasma input. In the terms of traditional three-compartmental model,

${K_i} = \frac{K_1 \times k_3}{k_2 + k_3}$

with plasma input, but

${K_i^{ref}} = \frac{k_2 \times k_3}{k_2 + k_3}$

with reference tissue input, assuming that K₁/k₂ is similar in region of interest and in reference region. If k₃≫k₂ (transport or perfusion is the limiting step), then the slope represents k₂.

Reference tissue input can be used also in analysis of dual time point data to calculate a surrogate parameter for Patlak K_i^ref (Alves et al., 2017)).

Metabolic rate

When the PET radiopharmaceutical is an analog of glucose (e.g. [F-18]FDG) or fatty acids (e.g. [F-18]FTHA) or other native substrate in the tissue, and it is metabolically trapped in tissue during the PET scan, the K_i can be used to calculate the metabolic rate of the native substrate. For example, in [F-18]FDG the K_i can be multiplied by the concentration of glucose in plasma, and divided by the appropriate lumped constant, to get an estimate of glucose uptake rate.

MTGA for reversible uptake (Logan plot)

MTGA methods for reversible uptake can provide estimates of equilibrium volumes of distribution (V_T). If a reference region is available, then binding potential (BP_ND) can be calculated from the ratio of distribution volumes for the region of interest and reference region:

$BP_{ND}=\frac{V_T}{V_T^{ref}}-1$

Operational equation for the Logan plot is

$\frac{\int_0^T C_{ROI}(t)dt}{C_{ROI}(T)}=V_T \times \frac{\int_0^T C_p(t)dt}{C_{ROI}(T)} + Int$

where linearity is achieved after the intercept (Int) is effectively constant (Logan et al., 1990, 2000 and 2003).

Tantawy et al. (2009) introduced a modified Logan plot for delayed scan protocols where the initial uptake of radiotracer is not measured.

Alternative Logan plot

An alternative form of the MTGA for reversible uptake (which looks more similar to the Patlak plot) is

$\frac{\int_0^T C_{ROI}(t)dt}{C_p(T)} = V_T \times \frac{\int_0^T C_p(t)dt}{C_p(T)} + Int_b$

, but this formulation would achieve linearity only after the true steady state condition is reached (Logan 2003), and to keep imaging session as short as possible this is therefore not commonly used. Zhou et al (2009) reintroduced this formulation (as "new plot"), because it avoids the noise-induced negative biases in the V_T and BP_ND estimates of the traditional Logan plot formulation.

Logan plot without plasma sampling

When reference region is available, then V_T ratio (distribution volume ratio, DVR) can be calculated directly without blood sampling by using reference region in place of the arterial plasma integral (Logan et al., 1996):

$\frac{\int_0^T C_{ROI}(t)dt}{C_{ROI}(T)} = \frac{V_T}{V_T^{ref}} \times \frac{\int_0^T C_{ref}(t)dt + ( C_{ref}(T) / \bar{k_2^{,}} ) }{C_{ROI}(T)} + Int^{'}$

The population average of apparent k'₂ (k₂ of one-tissue compartment model or k₂/(1+k₅/k₆) of two-tissue compartment model) must be determined from studies with plasma sampling. Fortunately, in many cases the term containing the population average of apparent k'₂ can be omitted (Logan et al., 1996; Logan, 2003). Reformulation of the equation above to

$\frac{\int_0^T C_{ROI}(t)dt}{C_{ROI}(T)} = \frac{V_T}{V_T^{ref}} \times \frac{\int_0^T C_{ref}(t)dt}{C_{ROI}(T)} + \frac{V_T}{V_T^{ref} \times \bar{k_2^{,}}} \times \frac{C_{ref}(T)}{C_{ROI}(T)} + Int^{'}$

shows that when ratio C_ref/C_ROI becomes reasonably constant the k'₂ effectively becomes part of the plot intercept.

The negative of the intercept, -Int', in the Logan plot with reference tissue input is the relative residence time (RRT), which can be used to measure the clearance of PET radiopharmaceutical from region-of-interest relative to reference region (Shoghi-Jadid et al, 2002).

PET data must be collected from the radiopharmaceutical injection time, because Logan plot method requires both tissue and input integrals starting from time 0. However, in some cases the analysis may be possible from late-scan data (Tantawy et al., 2009).

Alternative graphical analysis with reference tissue input

Based on the alternative Logan plot with plasma input, with its relatively bold assumptions, Zhou et al (2009) introduced "new plot" in form

$\frac{\int_0^T C_{ROI}(t)dt}{C_{ref}(T)} = \frac{V_T}{V_T^{ref}} \times \frac{\int_0^T C_{ref}(t)dt}{C_{ref}(T)} + Int^{''}$

which requires longer scan times but is less biased by noisy data.

For each radiopharmaceutical, the reference input methods have to be validated against plasma input methods. See for example Anteror-Dorsey et al. (2008).

Estimation of V_ND and V_T

Yokoi et al (1993) described a MTGA method for estimating V_T, which was different from the Logan plot and the alternative Logan plot. Operational equation for the "Yokoi plot" is

$\frac{C_{ROI}(T)}{\int_0^T C_p(t)dt}={Slope} \times \frac{\int_0^T C_{ROI}(t)dt}{\int_0^T C_p(t)dt} + Int$

In case of one-tissue compartmental model, the slope of the plot represents -k₂, plots y axis intercept (Int) represents K₁, and x axis intercept is the V_T. This method can be used for analysis of radiowater PET studies (Yokoi et al., 1993), but could be used for analysis of any reversible uptake data with fast kinetics.

If the plot becomes two-phasic with more steep negative slope in the beginning than in the end, then two-tissue compartmental model is needed to describe the kinetics of the radiopharmaceutical. If the two phases are clearly separable and a line can be fitted to both phases, then this two-phase graphic plot can be used to estimate both V_ND and V_T, and BP_ND as V_T/V_ND - 1 (Ito et al., 2010 and 2017). The x axis intercepts of the two fitted lines represent V_ND and V_T.

Line-fitting in MTGA

Usually line-fitting in MTGA is done using traditional regression method, where it is assumed that plot data has errors only in y values. This assumption does not hold in MTGA, where actually both plot coordinates contain variation. Patlak and logan programs for regional MTGA also use traditional regression by default. Different line fitting methods can be selected optionally.

If tissue data consists of only few PET frames or even just one late scan, then it is impossible to fit regression line to the plots. Calculation of Fractional Uptake Rate (FUR) instead of Patlak plot should then be considered for radiopharmaceuticals with irreversible uptake.

Summary:

Patlak plot for irreversible uptake, Logan plot for reversible

Linearity of plots must be verified

Plasma or reference region input can be used, depending on the radiopharmaceutical

Outcome from Patlak plot is net influx constant K_i which may be used further to calculate metabolic rate, or K_i^ref in case of reference region input

Outcome from Logan plot is distribution volume V_T, or distribution volume ratio DVR in case of reference region input

Easy and fast to calculate pixel-by-pixel from dynamic PET images to produce K_i or V_T images

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Tags: Logan plot, MTGA, Patlak plot, Ki, Volume of distribution, Yokoi plot

Updated at: 2023-06-08
Created at: 2008-04-02
Written by: Vesa Oikonen