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Multiple Time Grahical Analysis (MTGA)

In multiple-time graphical analysis the tracer 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 tracer or volume of distribution. In some instances a reference region curve can be used 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 tracers (Logan 2000; Logan 2003).

MTGA for irreversible uptake (Gjedde-Patlak plot)

Principles of Gjedde-Patlak analysis

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

It is assumpted that all the reversible compartments must be in equilibrium with plasma, i.e. the ratio of the concentrations of tracer in plasma and in reversible tissue compartments must remain stable. In these circumstances only the accumulation of tracer 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 descends slow enoughfor tissue compartments to follow.

Figure 1. There can be any number of reversible compartments, where the tracer can come and go. After some time, tracer concentations in these compartments start to follow the tracer 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 Gjedde-Patlak plot becomes horizontal, with slope of zero. In this case MTGA for reversible tracers (Logan plot) should be applied.

Making Gjedde-Patlak plot

When the equilibrium is achieved, the Gjedde-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 tracer in relation to the amount of tracer that has been available in plasma.

The y axis of plot contains apparent distribution volumes, that is the ratio of concentrations of tracer in tissue and in plasma. On x axis is normalized plasma integral, that is the ratio of the integral of plasma concentration and the plasma concentration.

Figure 2. Gjedde-Patlak plot becomes linear after the tracer 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.

Gjedde-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=K₁*k₃/(k₂+k₃) with plasma input, but K_i^ref=k₂*k₃/(k₂+k₃) with reference input, assuming that K₁/k₂ is similar in all regions.

Metabolic rate

When the PET tracer 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, K_i can be used to calculate the metabolic rate of the native substrate. For example, in [F-18]FDG PET study the K_i can be multiplied by 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:

$BP_{ND}=V_T//V_T^{REF}-1$ .

Operational equation for the Logan plot is

${int_0^T C_{ROI}(t)dt}/{C_{ROI}(T)}=V_T*{int_0^T C_p(t)dt}/{C_{ROI}(T)} + Int$

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

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

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

${int_0^T C_{ROI}(t)dt}/{C_p(T)}=V_T*{int_0^T C_p(t)dt}/{C_p(T)} + Int_b$

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) re-introduce 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 can be calculated directly without blood sampling by using reference region in place of the arterial plasma integral:

${int_0^T C_{ROI}(t)dt}/{C_{ROI}(T)}=({V_T}/{V_T^{REF}})*{int_0^T C_{REF}(t)dt + ( C_{REF}(T)//bar(k_2^{,REF}) ) }/{C_{ROI}(T)} + Int^,$

The population average of apparent k₂ ( k₂ of one-tissue compartment model or k₂/(1+k₅/k₆) ) must be determined from studies with plasma sampling. Fortunately, in many cases the term containing the population average of apparent k₂ can be omitted (Logan 2003).

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

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 may be impossible to fit regression line to the plots. Calculation of Fractional Uptake Rate (FUR) instead of Gjedde-Patlak plot should then be considered for irreversible tracer studies.

Summary:

Gjedde-Patlak plot for irreversible uptake, Logan plot for reversible
Linearity of plots must be checked
Plasma or reference region input can be used, depending on the tracer
Outcome from Gjedde-Patlak plot is net influx constant K_i which may be used further to calculate metabolic rate, or K_i^ref with reference region input
Outcome from Logan plot is distribution volume V_T, or distribution volume ratio DVR = V_T/V_T^REF with reference region input
Easy and fast to calculate pixel-by-pixel from dynamic PET images to produce K_i or V_T images

References:

Anteror-Dorsey JA, Markham J, Moerlein SM, Videen TO, Perlmutter JS. Validation of the reference tissue model for estimation of dopaminergic D₂-like receptor binding with [¹⁸F](N-methyl)benperidol in humans. Nucl Med Biol. 2008; 35: 335-341.

Gjedde A. Calculation of cerebral glucose phosphorylation from brain uptake of glucose analogs in vivo: a re-examination. Brain Res. 1982; 257: 237-274.

Logan J, Fowler JS, Volkow ND, Wolf AP, Dewey SL, Schlyer DJ, MacGregor RR, Hitzemann R, Bendriem B, Gatley SJ, Christman DR. Graphical analysis of reversible radioligand binding from time-activity measurements applied to [N-¹¹C-methyl]-(-)-cocaine PET studies in human subjects. J Cereb Blood Flow Metab. 1990; 10: 740-747.

Logan J, Fowler JS, Volkow ND, Wang GJ, Ding YS, Alexoff DL. Distribution volume ratios without blood sampling from graphical analysis of PET data. J Cereb Blood Flow Metab. 1996; 16: 834-840.

Logan J. Graphical analysis of PET data applied to reversible and irreversible tracers. Nucl Med Biol. 2000; 27: 661-670.

Logan J. A review of graphical methods for tracer studies and strategies to reduce bias. Nucl Med Biol. 2003; 30: 833–844.

Patlak CS, Blasberg RG, Fenstermacher JD. Graphical evaluation of blood-to-brain transfer constants from multiple-time uptake data. J Cereb Blood Flow Metab. 1983; 3: 1-7.

Patlak CS, Blasberg RG. Graphical evaluation of blood-to-brain transfer constants from multiple-time uptake data. Generalizations. J Cereb Blood FlowMetab. 1985; 5: 584-590.

Tantawy MN, Jones CK, Baldwin RM, Ansari MS, Conn PJ, Kessler RM, Peterson TE. [¹⁸F]Fallypride dopamine D2 receptor studies using delayed microPET scans and a modified Logan plot. Nucl. Med. Biol. 2009; 36: 931-940.

Yokoi T, Iida H, Itoh H, Kanno I. A new graphic plot analysis for cerebral blood flow and partition coefficient with iodine-123-iodoamphetamine and dynamic SPECT validation studies using oxygen-15-water and PET. J Nucl Med. 1993; 34(3): 498-505.

Zhou Y, Ye W, Brašić JR, Crabb AH, Hilton J, Wong DF. A consistent and efficient graphical analysis method to improve the quantification of reversible tracer binding in radioligand receptor dynamic PET studies. NeuroImage 2009 (in press).

Multiple Time Grahical Analysis (MTGA)

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References: