Graphical Analysis

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A graphical method of analysis applicable to ligands that bind reversibly to receptors or enzymes requiring the simultaneous measurement of plasma and tissue radioactivities for multiple times after the injection of a radiolabeled tracer is presented. It is shown that there is a time t after which a plot of integral of t0ROI(t')dt'/ROI(t) versus integral of t0Cp(t')dt'/ROI(t) (where ROI and Cp are functions of time describing the variation of tissue radioactivity and plasma radioactivity, respectively) is linear with a slope that corresponds to the steady-state space of the ligand plus the plasma volume,.Vp. For a two-compartment model, the slope is given by lambda + Vp, where lambda is the partition coefficient and the intercept is -1/[kappa 2(1 + Vp/lambda)]. For a three-compartment model, the slope is lambda(1 + Bmax/Kd) + Vp and the intercept is -[1 + Bmax/Kd)/k2 + [koff(1 + Kd/Bmax)]-1) [1 + Vp/lambda(1 + Bmax/Kd)]-1 (where Bmax represents the concentration of ligand binding sites and Kd the equilibrium dissociation constant of the ligand-binding site complex, koff (k4) the ligand-binding site dissociation constant, and k2 is the transfer constant from tissue to plasma). This graphical method provides the ratio Bmax/Kd from the slope for comparison with in vitro measures of the same parameter. It also provides an easy, rapid method for comparison of the reproducibility of repeated measures in a single subject, for longitudinal or drug intervention protocols, or for comparing experimental results between subjects. Although the linearity of this plot holds when ROI/Cp is constant, it can be shown that, for many systems, linearity is effectively reached some time before this. This analysis has been applied to data from [N-methyl-11C]-(-)-cocaine ([11C]cocaine) studies in normal human volunteers and the results are compared to the standard nonlinear least-squares analysis. The calculated value of Bmax/Kd for the high-affinity binding site for cocaine is 0.62 +/- 0.20, in agreement with literature values.

Help students form critical connections between abstract scientific ideas and the real world. With the Graphical Analysis app, students can visualize and interact with experiment data collected via nearly any Vernier sensor.

Create bar graphs, histograms, and FFTs. Plus, generate user-defined curve fits, understand the reliability of fitted parameters, use error bars to describe measurement uncertainty, strikethrough unwanted data, plot categorical items, and more for advanced data analysis.

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 orvolume of distribution.In some instances a reference regioncurve 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 and2003).

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.

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.

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

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.

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 estimatingand correcting for the highly variable tissue fat and lung air volume fractions which otherwisecould cause large bias for example in liver andlung PET studies.

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

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

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 Ki can be calculated by rearranging the operational equation for the Patlak plot:

, where CROI(T) and Cp(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 withthe equation for FUR, which is an approximation ofKi:

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.

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 Ki can be used to calculate the metabolic rate of the native substrate. For example, in [F-18]FDG the Ki 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 methods for reversible uptake can provide estimates of equilibrium volumes of distribution(VT). If a reference region is available, then binding potential (BPND) can be calculated from the ratio of distribution volumes for the region of interest and reference region:

, 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 VT and BPND estimates of the traditional Logan plot formulation.

When reference region is available, then VT 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):

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).

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 VT.This method can be used for analysis of radiowaterPET 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 VND and VT, and BPND as VT/VND - 1 (Ito et al., 2010 and2017). The x axis intercepts of the two fitted lines represent VND and VT.

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