@misc{oai:repo.qst.go.jp:00069796,
 author = {Hontani, Hidekata and Naganawa, Mika and Sakaguchi, Kazuya and Sakata, Muneyuki and Ishiwata, Kiichi and Kimura, Yuichi and et.al and 本谷 秀堅 and 長縄 美香 and 坂口 和也 and 坂田 宗之 and 石渡 喜一 and 木村 裕一},
 month = {Jul},
 note = {Objective: The Logan Graphical Analysis (LGA) is used for imaging a distribution volume VT. For LGA, we compute a set of {(x(t), y(t))} from the measured time-activity curves in tissue (tTAC) and plasma (pTAC) to find a best-fitting line y(t) = alpha x(t) + beta. (Eq.1).
Here, x(t) and y(t) are defined as a ratio of an integrated pTAC over tTAC and an integrated tTAC over tTAC, respectively. As known[1] , linear regression (LR) underestimates VT and its unbiased estimator is expected. 
Renormalization Method (RM) [2] enables an unbiased maximum likelihood estimation under the existence of inhomogeneous noises both in x and y by successive evaluation of bias. In this study, the applicability of RM to LGA was investigated.
\nMethods: Let Xt = (x(t),y(t),1)T and U=(u1,u2,u3)T. Then, we can rewrite (Eq.1) as XtTU = 0, where ||U|| = 1 and VT = -u1/u2. Let Ct denote the covariance matrix of the noise of Xt. The maximum likelihood estimates of ui minimize JMLE(U) = SIGMA tWt(U) (XtTU)2, where Wt(U)=1/(UTCtU). Though, the perturbation theorem tells us that the estimates become biased.
RM removes the bias by iteratively minimizing JREN(U) instead of JMLE : JREN(U) = SIGMA tWt(U) {(XtTU)2 - UTCtU}, where the last term compensates the bias. In RM, the covariance matrix Ct should be given, and it is unknown in advance. Thus, a set of voxel-based noisy TACs were simulated using physiologically plausible kinetic parameters, and the mean of Ct was calculated from the set of simulated TACs.
We applied RM and LM to synthesized tTACs and to real one of [11C]SA4503-PET. For generating the synthesized data, we simulated a set of voxel-based tTACs using a measured pTAC and the rate constant of [11C]SA4503 [3].
\nResults: [pic_01] The simulation results are summarized in Fig. (A). RM plotted in red was almost identical (y=0.99x+0.23, r2=1.00), and LR plotted in blue showed the underestimation especially in large VT (y=0.70x+6.14, r2=0.94). The estimation of deviation was larger than that of LM. However, RM successfully suppressed the bias.
The figures (B) and (C) show the results of imaging of VT obtained from the real data by RM and by LR, respectively. For the estimation, t* was set to be 15min post-injection. The computational time for RM was 10 min for 60 thousands voxels. RM gave brighter images than LR, and improved their contrast.
\nConclusions: For computing unbiased estimates, we introduced RM. We estimated the average of each Ct based on simulations. Simulation results showed that RM suppresses the bias and has the potential to realize bias-free parametric imaging of VT.
\nReferences:
[1] Slifstein et al., J Nucl Med, 41, 12, 2000.
[2] Kanatani, IEEE PAMI, 16, 3, 320-326, 1994.
[3] Sakata et al., NeuroImage, 35, 1-8, 2007., Brain'09 & BrainPET'09},
 title = {Unbiased Logan Graphical Analysis Using the Renormalization Method},
 year = {2009}
}