Knowledge-based interpolation of curves: application to femoropopliteal arterial centerline restoration

Abstract

We present a novel algorithm, Partial Vector Space Projection (PVSP), for estimation of missing data given a database of similar datasets, and demonstrate its use in restoring the centerlines through simulated occlusions of femoropopliteal arteries, derived from CT angiography data. The algorithm performs Principal Component Analysis (PCA) on a database of centerlines to obtain a set of orthonormal basis functions defined in a scaled and oriented frame of reference, and assumes that any curve not in the database can be represented as a linear combination of these basis functions. Using a database of centerlines derived from 30 normal femoropopliteal arteries, we evaluated the algorithm, and compared it to a correlation-based linear Minimum Mean Squared Error (MMSE) method, by deleting portions of a centerline for several occlusion lengths (OL: 10 mm, 25 mm, 50 mm, 75 mm, 100 mm, 125 mm, 150 mm, 175 mm and 200 mm). For each simulated occlusion, we projected the partially known dataset on the set of basis functions derived from the remaining 29 curves to restore the missing segment. We calculated the maximum point-wise distance (Maximum Departure or MD) between the actual and estimated centerline as the error metric. Mean (standard deviation) of MD increased from 0.18 (0.14) to 4.35 (2.23) as OL increased. The results were fairly accurate even for large occlusion lengths and are clinically useful. The results were consistently better than those using the MMSE method. Multivariate regression analysis found that OL and the root-mean-square error in the 2 cm proximal and distal to the occlusion accounted for most of the error.

Fig. 1

Example of a lower extremity CT angiogram in a patient with peripheral arterial disease and bilateral femoro-popliteal artery occlusions. Maximum Intensity Projection (MIP) image (left panel) demonstrates an occlusion of the right superficial femoral artery (arrowheads) and on the left, an occlusion of the popliteal artery (arrowheads). Note the small vessels providing collateral flow in lieu of the occluded arteries. Magnified view of Curved Planar Reformation (CPR) (right panel) with virtual gauging marks (scale is in cm) through the right femoro-popliteal artery has been obtained after automated centerline extraction through the unobstructed portions of the artery, and with manual slice-by-slice centerline identification of the occluded segment. The CPR image provides a longitudinal cross section which displays the clinically important morphology of the occlusion (arrowheads) such as length, course, luminal thrombus and vessel wall calcifications. Note Images are displayed as if the viewer is facing the patient: the patient’s left leg is on the right side of figure.

Fig. 2

The original frame of reference is at the bottom of the figure and the new tilted frame of reference for the femoropopliteal artery is between points 1 and 2. On left is the Anterior-posterior (AP) view and on right is the lateral view. Point 1 is the common-femoral artery bifurcation, Point 2 is the popliteal artery bifurcation.

Fig. 3

AP view (in modified coordinate system) of 30 femoropopliteal artery centerlines

Fig. 4

Lateral view (in modified coordinate system) of 30 femoropopliteal artery centerlines

Fig. 5

Cumulative sum of eigenvalues of the correlation matrix of the 30 patients database.

Fig. 6

AP view (in modified coordinate system) of the first and the second principal component of the femoropopliteal artery

Fig. 7

Lateral view (in modified coordinate system) of the first and the second principal component of the femoropopliteal artery

Fig. 8

Box plots of MD for simulated occlusions of various lengths in the femoropopliteal artery. The box-plot whiskers represent values 1.5 times the inter-quartile range beyond the first or third quartile.

Fig. 9

Log-MD vs Log-NE Error. The line is a simple linear regression fit (r=0.367, p<0.001).

Fig. 10

Comparison of the two algorithms. Log-MD for the MMSE method vs. that for the PVSP method. The log-MD for MMSE is always above the line of identity, indicating the results are always better for the PVSP method.