Experiments with retrospective gating of embryonic quail heart data were very encouraging. We found that both visual (-) and quantitative assessment () were important for evaluating our gating algorithm. A few remarks about our gating algorithm follow.
We investigated the effect of applying spatial decimation (along the x and y directions of the B-scan frame) to reduce computation time. For example, to spatially decimate by 4, we pick up every 4th pixel in both x and y. We often found that cropping the B-scan images to include only the region corresponding to the heart also helps in reducing computation time. We compared cardiac period estimates at all slice positions obtained from original and spatially decimated data (with various decimation factors, e.g., 2, 4, 8, and 16). We observed that up to a factor of about 4, the cardiac period estimates agreed numerically with those obtained from native resolution data. This suggested a lower limit on the sampling rate equal to ¼ of the native sampling rate used for out-of-order data acquisition. We then produced gated reconstructions from original resolution and each of the decimated data sets and applied the validation algorithm described in Section 2.3 in each case. The plots obtained from validation of data with different decimation factors () helped us establish a lower limit on the spatial decimation of input data to ensure accurate gated reconstructions. A spatial decimation factor > 4 clearly resulted in large departures from the ideal slope (i.e., slope of reference scan) indicating incorrect gated reconstructions (, bottom left and right). The limit on spatial decimation was therefore deduced to be equal to 4. This analysis revealed how quickly we could process the data (through the use of a decimation factor) without causing errors in resulting reconstructions.
Fig. 7 Validation of image-based retrospective gating using input data with different integer decimation factors. In each case, the B-scan frame was decimated by an integer number n in both x and y directions. Following decimation, the gating algorithm (Section (more ...)
Next, we determined the effect of temporal decimation on cardiac period estimates at all slices. We have found that using a temporal decimation factor of 2 (i.e., skipping every other frame) leads to negligible errors in estimated period (the average error in the estimates over the 80 slices was found to be 0.005 seconds, a negligible number compared to the ~0.45 s cardiac cycle time). However, a temporal decimation factor of 4 and above caused larger errors (>0.025 seconds). With slower systems these larger errors can be reduced by recording over more heartbeats at each slice location. We have typically recorded 1.5 heartbeats at each slice in our standard imaging setup. However, there could be small variations across successive heartbeats even for the same embryo. In a separate imaging experiment with a slightly older quail heart, we recorded just over 2 heartbeats at each slice (120 slices/volume) and quantitatively evaluated the effect of multiple heartbeats on gating accuracy. First, we estimated cardiac cycle time using the entire data (~2 heartbeats). We then dropped the last ½ heartbeat from each slice position, thereby using ~1.5 heartbeats/slice to estimate cardiac cycle time. The mean cardiac cycle time across 120 slices was 0.3056 s in the former case, while it was 0.3079 s for the latter case. The difference between the two estimates was 2.3 ms, or, just over half the duration of a frame, a clearly acceptable number. We concluded that our gating algorithm was equally accurate with 2 heartbeats/slice as it was with 1.5 heartbeats/slice. We have further determined that having a shorter temporal sequence at each slice (i.e., < 1.5 heartbeats) sometimes caused bimodal distributions in the correlation-based objective function during relative shift estimation. This in turn leads to false maxima and erroneous relative shifts as verified visually. In the limiting case where we have less than a single heartbeat of data at a slice, the relative shift computation algorithm fails for lack of enough image evidence to determine an accurate relative shift value.
We investigated the errors obtained in absolute shift estimation when we increased the spacing between B-scan frames (or slices) used for computing image correlations. We observed (through visual examination of gated reconstructions) that relative shift estimates obtained by correlating immediately adjacent slices (i.e., native sampling rate in volume direction) were accurate. However, when a slice was correlated with another one spaced two sampling steps away (i.e., ½ of native sampling rate), there were large numerical differences in absolute shift estimates, and the resulting error was confirmed through a visual evaluation of the resulting reconstructions. We concluded that correlating slices separated by a distance more than the spot size of the imaging system (10μm) resulted in erroneous gated reconstructions.
We discuss the possible sources of error in our validation. The small discrepancies in slope value for the validation technique can be attributed to a couple of issues. First, the real-time data were sampled spatially at a lower rate as compared to the out-of-order data, which would cause the small correlation errors leading to errors in slope. Second, the reference and the out-of-order data sets employed different scanning geometries and small positioning errors due to minor galvanometer inaccuracies would lead to small correlation errors. However, we found that in spite of these possible sources of error, the frame time offset between the starting and ending slices corresponding to the heart region in the gated data very closely matched that of the reference scan, thereby validating our image-based retrospective gating scheme. We computed errors between actual points on the validation plot and those on a straight line with the ideal slope (i.e., slope of reference scan) for the slices corresponding to the heart region. The following errors are actually worst-case estimates because of the mechanical limitations of the scanning described above. The maximum error was found to be 3.83 frames or ~18.7ms ms, and the standard deviation of the error for slices in the heart region was found to be 0.9669 frames or 4.7ms (time interval between volumes in gated data = 4.9 ms, see Section 2.2). Systolic dynamics in the heart occur in a time interval of the order of 50 ms; our gating algorithm has an error standard deviation which is an order of magnitude smaller and therefore has sufficient accuracy for visualizing systolic dynamics.
We comment on computational complexity of our algorithm. As compared to our previous (prospective) LDV gating study, the additional benefits gained from retrospective gating far outweigh the longer post-acquisition computation time needed for reconstruction (). As mentioned before, a spatial decimation factor of 4 caused negligible errors in reconstruction. We present results of computation time estimates () for some important steps of the gating algorithm on our test data set (80 slices, 120 images of 512 × 500 pixels per slice) with spatial decimation set equal to 4. Without any code optimization, MATLAB® code for cardiac cycle time estimation, data interpolation, and shift estimation took 70 seconds altogether on a Dell Precision 690 workstation with a 3.0 GHZ dual-core Intel Xeon processor and 32 GB of system RAM. Data reassembly, wrap-around, interpolation, and rearrangement on full resolution data to produce the final gated data took about 45 minutes. We are currently performing some code optimizations for reducing the running times of each of the steps above.
Computation times for different steps of the image-based retrospective gating algorithm.
In an earlier paper, we have determined using heart wall velocity estimates obtained from real-time volume scans that there could be large tissue displacement errors, as high as 205 μm [3
]. With image-based retrospective gating, the reconstructed data set has a maximum displacement error of only 8.5 μm, which would now allow us to see differences in wall velocity at the inflow and outflow of the heart tube (). We had earlier observed with our lower temporal resolution scans [2
] that the contraction appeared to be the same at the inflow and outflow. The increased temporal resolution afforded by our gating algorithm will facilitate precise measurements of many cardiac parameters (wall velocity, contraction propagation, etc.) in the developing heart tube and will allow us to develop more accurate models of cardiac dynamics at these early stages.