Non-invasive assessment of distribution volume ratios and binding potential: tissue heterogeneity and interindividually averaged time-activity curves

Abstract

Due to the stochastic nature of radioactive decay, any measurement of radioactivity concentration requires spatial averaging. In pharmacokinetic analysis of time-activity curves (TAC), such averaging over heterogeneous tissues may introduce a systematic error (heterogeneity error) but may also improve the accuracy and precision of parameter estimation. In addition to spatial averaging (inevitable due to limited scanner resolution and intended in ROI analysis), interindividual averaging may theoretically be beneficial, too. The aim of this study was to investigate the effect of such averaging on the binding potential (BP) calculated with Logan’s non-invasive graphical analysis and the “simplified reference tissue method” (SRTM) proposed by Lammertsma and Hume, on the basis of simulated and measured positron emission tomography data {[¹¹C]d-threo-methylphenidate (dMP) and [¹¹C]raclopride (RAC) PET}. dMP was not quantified with SRTM since the low k ₂ (washout rate constant from the first tissue compartment) introduced a high noise sensitivity. Even for considerably different shapes of TAC (dMP PET in parkinsonian patients and healthy controls, [¹¹C]raclopride in patients with and without haloperidol medication) and a high variance in the rate constants (e.g. simulated standard deviation of K ₁=25%), the BP obtained from average TAC was close to the mean BP (error <5%). However, unfavourably distributed parameters, especially a correlated large variance in two or more parameters, may lead to larger errors. In Monte Carlo simulations, interindividual averaging before quantification reduced the variance from the SRTM (beyond a critical signal to noise ratio) and the bias in Logan’s method. Interindividual averaging may further increase accuracy when there is an error term in the reference tissue assumption E=DV ₂−DV′ (DV ₂ = distribution volume of the first tissue compartment, DV′ = distribution volume of the reference tissue). This can be explained by the fact that the distribution volume ratio (DVR=DV/DV′) obtained from averaged TAC is an approximation for ΣDV/ΣDV′ rather than for ΣDVR/n. We conclude that Logan’s non-invasive method and SRTM are suitable for heterogeneous tissues and that discussion of group differences in PET studies generally should include qualitative and quantitative assessment of interindividually averaged TAC.

Appendices

Appendix A. Symbols

Throughout the paper, primed variables refer to the reference tissue

BP (unitless)	Binding potential, here defined as k 3 /k 4
C(t) (kBq/ml)	Decay-corrected radioactivity concentration (target region), optionally subdivided into C=C 2 +C 3 (first and second tissue compartments) Any concentration denoted C̄refers to true steady state
DV (mlblood mltissue −1)	Total distribution volume (target region), optionally subdivided into DV=DV 2 +DV 3 (first and second tissue compartments)
DVR (unitless)	Distribution volume ratio DVR=DV/DV′
IRF(t) (mlblood mltissue −1 s−1]	Impulse response function
K 1 (mlblood mltissue −1 min−1), k 2, k 3, k 4 (min−1)	Transfer rate constants within the two-tissue compartment model

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Appendix B. “Heterogeneity effect” from variable plasma curve

For different individual plasma curves C _Plasma =ΔC _Plasma + mean{C _Plasma } one obtains according to Eq. 2:

$$ aTAC = {\text{mean}}{\left\{ {IRF \otimes C_{{Plasma}} } \right\}} $$

The heterogeneity effect ΔC _Tissue (target and reference region each) according to Eq. 9can be calculated from:

$$ \Delta C_{{Tissue}} = aTAC - aTAC_{{approx}} $$

with

$$ \begin{aligned} & aTAC_{{approx}} = {\text{mean}}{\left\{ {IRF} \right\}} \otimes {\text{mean}}{\left\{ {C_{{Plasma}} } \right\}} \\ & \Rightarrow \Delta C_{{{\text{Tissue}}}} = {\text{mean}}{\left\{ {IRF \otimes C_{{Plasma}} } \right\}} - {\text{mean}}{\left\{ {IRF} \right\}} \otimes {\text{mean}}{\left\{ {C_{{Plasma}} } \right\}} = {\text{mean}}{\left\{ {\Delta IRF_{{Tissue}} \otimes \Delta C_{{Plasma}} } \right\}} \\ \end{aligned} $$

Thus ΔC _Tissue becomes small if ΔC _Plasma is not correlated with ΔIRF or if interindividual differences in the IRF or C _Plasma are small.

Appendix C. Mean{DV}/mean{DV’} versus mean{DVR}

From DV = DV ₂·(1+BP), the mean{DV} can be formulated as:

$$ \begin{aligned} {\text{mean}}{\left\{ {{\text{DV}}} \right\}} = {\text{mean}}{\left\{ {DV_{2} \cdot {\left( {1 + BP} \right)}} \right\}} & \\ \Rightarrow {\text{mean}}{\left\{ {{\text{DV}}} \right\}} = {\text{mean}}{\left\{ {DV_{2} } \right\}} + {\text{mean}}{\left\{ {DV_{2} } \right\}} \cdot {\text{mean}}{\left\{ {BP} \right\}} + {\text{mean}}{\left\{ {BP \cdot \Delta DV_{2} } \right\}} & \\ \end{aligned} $$

(12)

with:

$$ \Delta DV_{{\text{2}}} = DV_{{\text{2}}} - {\text{mean}}{\left\{ {DV_{2} } \right\}} $$

If ΔDV ₂ is not correlated with BP across the patients, the rightmost term in Eq. 12 becomes small and one can approximate:

$$ {{\text{mean}}{\left\{ {{\text{DV}}} \right\}}} \mathord{\left/ {\vphantom {{{\text{mean}}{\left\{ {{\text{DV}}} \right\}}} {{\text{mean}}{\left\{ {DV_{2} } \right\}}}}} \right. \kern-\nulldelimiterspace} {{\text{mean}}{\left\{ {DV_{2} } \right\}}} \approx 1 + {\text{mean}}{\left\{ {BP} \right\}} $$

(13)

If an error term E in the reference tissue assumption is assumed:

$$ DV = DV_{{\text{2}}} + E $$

one obtains:

$$ \begin{aligned} {\text{mean}}{\left\{ {{DV} \mathord{\left/ {\vphantom {{DV} {{\left( {DV_{{\text{2}}} + E} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {DV_{{\text{2}}} + E} \right)}}} \right\}} \ne {\text{mean}}{\left\{ {{DV} \mathord{\left/ {\vphantom {{DV} {DV_{{\text{2}}} }}} \right. \kern-\nulldelimiterspace} {DV_{{\text{2}}} }} \right\}} & \\ \Rightarrow mean{\left\{ {DVR} \right\}} \ne {\text{mean}}{\left\{ {BP + 1} \right\}} & \\ \end{aligned} $$

However, if E is distributed symmetrically (mean{E} ≈0):

$$ mean{\left\{ {D{V}'} \right\}} \approx mean{\left\{ {DV_{{\text{2}}} } \right\}} $$

and together with Eq. 13:

$$ {{\text{mean}}{\left\{ {{\text{DV}}} \right\}}} \mathord{\left/ {\vphantom {{{\text{mean}}{\left\{ {{\text{DV}}} \right\}}} {{\text{mean}}{\left\{ {D{V}'} \right\}}}}} \right. \kern-\nulldelimiterspace} {{\text{mean}}{\left\{ {D{V}'} \right\}}} \approx 1 + {\text{mean}}{\left\{ {BP} \right\}} $$