4.1. Idealized simulation
As discussed in Sec. 3.1.1, a simplified imaging scenario involving 1-to-1 mapping between image voxels and projection data was first considered. This provided an effective initial scheme for preliminary study of the properties of the proposed AB-EM algorithm in its ability to separate slope (DV) and intercept (B) contributions in (2
). In particular, the figures below depict simulations for the putamen (Btrue
=−40.0 and DVtrue
=1.4) where we imposed a wide of range of lower bounds in the AB-EM reconstruction algorithm (10
). Defining the fractional parameter α such that the lower bound a
was set as a
, depcits convergence properties for a range of α values. It is seen that for α values below 1 (i.e. the negative lower bound a
being closer to zero than Btrue
), the algorithm converges (plateaus) to incorrect values. When α is close to one, the convergence was seen to be very slow towards the correct values. By analogy, this is similar to the slow convergence rates observed (Erlandsson et al., 2000
; Verhaeghe and Reader, 2010
) in standard AB-EM applications for a cold region if the lower bound is set to 0 or only slightly negative. By contrast, there are alternative α values (~6) where convergence is considerably enhanced. Setting α to too high was also seen to slow down the convergence rates. This is attributed to the fact that while the AB-EM algorithm can be shown to converge to the true solution in the noise free case (Byrne, 1998
), in the presence of noise, it is in reality an approximation to the EM algorithm, wherein it is assumed that it is the data adjusted
by projections of the bounds that are Poisson distributed (i.e. y
in (6) or
in (9–10)), and therefore using a lower bound of substantially large magnitude can exhibit adverse convergence properties.
To better comprehend the results in , depicts plots of B and DV values for the individual 100 noise realizations (after 1000 iterative updates). Given Btrue
=−40.0 and α values of 0.1 and 0.5 (i.e. lower bounds a
of −4.0 and −20.0, respectively), it is seen that the B estimates have converged towards the lower bounds, as they are not able to further approach Btrue
. This bias in the B estimates in turns translates itself into biases in the DV estimates in order to compensate for the observed mismatch for B values. Furthermore, we see that as the lower bound further decreases, the bias is reduced but noise levels increase, as also observed in previous studies on 3D reconstruction (Erlandsson et al., 2000
; Verhaeghe and Reader, 2010
Figure 3 Estimated B (left) and DV (right) values for 100 noise realizations for the putamen. True values for B and DV were −40.0 and 1.4, respectively. The results are shown after 1000 iterations for the putamen. Clear bias is observed for α values (more ...)
depicts noise vs. bias plots for the wide range of α values. It is seen, consistent with above figure, that for α values of 0.1 and 0.5, the noise levels are considerably low, but this is at the expense of bias levels stagnating, and not further improving, after some iterations. For α values increasing from 1 onwards it is seen that noise vs. bias performance is further degraded. This is because even though convergence is improved with increasing α values (), noise levels are also further amplified, and subsequently, noise vs. bias performance is actually seen to degrade (ultimately, the criteria to use for optimization is task-based, and can vary, as discussed in Sec. 5.2).
Furthermore, we note that both bias and noise levels in are substantially lower, percentage-wise, for the slope parameter compared to the intercept parameter (as seen by comparing the scale of the axes): this is a favourable property because it is the slope parameter that is of direct interest in parametric imaging.
Finally, we note that the abovementioned studies depict that for optimum performance one must use a general AB-EM formulation wherein the lower bounds are spatially variant and in tune with the values of the intercept parameter at each given position (this is further discussed in Sec. 4.2.3).
4.2. Tomographic simulation
As discussed in Sec. 3.1.2, we also performed extensive PET simulation studies. It was found (see Sec. 2.1.4) that two factors contributed significantly to 4D algorithm performance: initialization (using standard EM followed by modelling at a particular iteration) as well as how the lower bound a was set. We elaborate upon the optimization in Sec. 4.2.3. Below we first discuss comparison of standard indirect versus proposed 4D direct methods when applied to the plasma input (Sec. 4.2.1) and reference tissue (Sec. 4.2.2) models.
4.2.1. Results for the plasma input model
shows typical reconstructed images for indirect and proposed direct parametric imaging methods, with increasing EM iterations from left to right. It is seen that while the standard EM approach results in noisy images, this trend is more controlled and qualitatively improved for the proposed 4D method.
Estimated parametric DV images. (From left to right): Increasing EM iterations of 1, 2, 3, 5 and 10 (21 subsets). No post-filtering was applied to the images shown.
To provide quantitative analysis, depicts plots of overall NSD vs. overall bias (as defined in Sec. 3.1.3) for the various parametric DV images shown in . In addition, plots NSDROI vs. BiasROI for 11 individual regions of the brain (cerebellum, caudate, putamen, cingulate Cx (cortex), occipital Cx, orbitofrontal Cx, parietal Cx, frontal Cx, temporal Cx, thalamus and brain stem). It is clearly seen that the proposed direct 4D EM reconstruction results in substantial quantitative accuracy improvements.
Figure 6 Plots of overall NSD (noise) vs. overall bias comparing parametric DV images obtained from standard indirect method (EM reconstruction followed by modeling) as well as the proposed direct 4D technique. The plasma input model was used. Points on each curve (more ...)
Figure 7 Plots of NSDROI (noise) vs. BiasROI for DV images obtained using standard EM and the proposed direct 4D EM technique (plasma input model). Points on each curve correspond to the images in each row of . The actual DV values for the 11 regions (cerebellum (more ...)
As also discussed in Sec. 3.1.3, we also performed analysis of COV (as an alternative measure of noise) vs. bias. The trade-off curves shown in for the estimated DV parametric images demonstrate substantial quantitative improvements for the proposed method, as also demonstrated in (trade-off curves for individual regions, not shown, also showed similar improvements). The scale of COV values in are smaller than NSD values in , because COV analysis calculates standard deviation of ROIs averaged values across multiple noise realization which are much less sensitive to noise than NSD calculation which performs noise analysis for individual voxels, followed by averaging across the ROIs (see discussion following Eq. (26)
Plots of overall COV (alternative measure of noise) vs. bias comparing parametric DV images obtained from standard indirect method as well as the proposed direct 4D technique. The plasma input model was used.
We also wish to note that in actual patient test-retest studies, issues of variability in radiotracer uptake as well as ROI delineation between patient scans further degrade reproducibility in parametric estimates (not simulated here). Moreoever, we note that ROI definitions in the present simulations, based on the known true images, extended across the entirety of each region. By using smaller ROI definitions (avoiding the edges), estimated COV values were seen to further increase, though similar overall performances were observed.
4.2.2. Results for the reference tissue model
The abovementioned quantitative analysis was also performed on parametric DVR images obtained using the reference tissue model (Sec. 2.1.2), with the cerebellum used as reference. For DVR images obtained using indirect and proposed direct methods, shows NSDROI vs. BiasROI as generated using increasing iterations, demonstrating quantitative improvements for the direct 4D method (similar to the results for the plasma input model in and ). Furthermore, as in , COV vs. Bias plots were also seen to depict quantitative improvements in noise vs. bias performance (not shown). It is worth noting that in conventional imaging, variability of DVR (and therefore BPND=DVR−1) estimates is too high for the relatively low uptake BPND regions of cortical grey and thalamus, and are typically not reliable. The proposed approach is thus seen to increase feasibility of imaging low BPND images.
Regional noise (NSD) vs. bias (RB) curves for DVR images obtained using standard EM vs. proposed direct 4D EM technique (reference tissue model with cerebellum used as reference). Actual DVR values are mentioned in the caption for .
4.2.3. Parameter optimization for the direct 4D method
As discussed in Sec. 2.1.4, image initialization as well as definition of the lower bound vector a noticeably impacted algorithm performance. We utilized an approach wherein initial estimates of the slope DVest and intercept Best parameters were obtained by conventional parametric imaging, as applied to dynamic images obtained after a certain number of iterations: this number is referred to as the initialized update number (IUN); for instance IUN=21 indicates that the initial estimates were obtained using kinetic analysis of dynamic images reconstructed after 21 updates (which happens to be a full iteration because each iteration consists of 21 subsets). Furthermore, as mentioned in Sec. 2.1.4, to set the lower bound a, we defined α such that a= α min(Best,0)).
depicts quantitative performance for the plasma input model for a range of (left
) α and (right
) IUN values. In the plots at (left
), similar to , it was seen that optimum noise vs. bias performance was obtained using α values close to 1 (though, setting α exactly equal to 1 was avoided to prevent stagnation of updates in Eq. (9)
: this is because for negative Best
values, if α=1 then Bold
) and thus B
values do not subsequently update; by analogy, this is equivalent to initializing image estimates in the standard EM algorithm to zero, which would then fail to change in subsequent updates).
Regional NSD (noise) vs. bias curves for DV images obtained using standard EM and the proposed direct 4D EM technique for varying (left) lower bounds (fixing IUN=21) and (right) IUN values (fixing α=1.1).
In (right), IUN optimization by comparing the various 4D EM trade-off curves was less straightforward, and IUN=21 was ultimately selected as optimum; nonetheless one sees that all initializations shown outperformed the conventional scheme. At the same time, we observed that setting IUN to smaller values resulted in sub-optimal trade-off performance curves (not shown).
4.3. Application to HRRT patient study
We then applied the proposed reference tissue model to a 11C- raclopride patient study on the HRRT scanner (elaborated in Sec. 3.2). To provide a visual/qualitative comparison, plots images of increasing iterations for the conventional indirect as well as proposed 4D direct parametric imaging techniques, depicting relatively reduced noise levels as a function of iteration.
Figure 11 Parametric DVR images for a raclopride HRRT study with increasing iterations of 2, 3, 4, 7, 15 from left to right. (Top) Standard 3D reconstruction followed by modeling; (bottom) Proposed 4D reconstruction (reference tissue model). The images include (more ...)
To additionally compare the images quantitatively (see Sec. 3.2), depicts NoiseROI vs. DVRROI plots generated by increasing iterations, for thirteen regions of: cerebellum as well as both left (L) and right (R) anterior putamen, posterior putamen, anterior caudate nucleus, posterior caudate nucleus, thalamus and ventral striatum. Across these ROIs, the proposed 4D method is seen to generally outperform the conventional method in the sense that for a given DVR value, improved noise levels are observed.
Noise vs. DVR curves (generated by increasing iterations as seen in ), comparing performance of conventional indirect vs. direct 4D estimated DVR images.
Our future work consists of application to a large pool of patients scanned on the HRRT scanner: we will explore whether the proposed direct 4D AB-EM reconstruction will produce parameter estimates with enhanced quantitative accuracy, including enhanced separability between patients vs. healthy control subjects, as well as improved consistency within healthy controls.