Elsevier

Medical Image Analysis

Volume 11, Issue 2, April 2007, Pages 157-168
Medical Image Analysis

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

https://doi.org/10.1016/j.media.2006.11.005Get rights and content

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.

Introduction

The problem of estimating missing data is widely encountered in medical imaging. Because human anatomy is fairly consistent from person to person, prior knowledge about typical shapes of organs might be used to estimate missing data from medical images. In contrast medium-enhanced Computed Tomography Angiography (CTA) of the lower extremities, missing data problems appear in the form of occluded blood vessels. In patients with peripheral arterial occlusive disease, complete obstruction of blood flow occurs preferentially in the region of the distal superficial femoral artery and proximal popliteal artery – or femoropopliteal artery (Zarins et al., 2005). Comprehensive visualization of peripheral arterial occlusive disease requires the computation of so-called curved planar reformations through the centerlines of perfused and occluded portions of an artery as shown in Fig. 1 (Fleischmann et al., 2006). Since contrast medium-enhanced blood cannot reach occluded portions of the vessel, the CT attenuation values of these occluded segments differ substantially from those with maintained blood flow, making them inconspicuous on a CT scan. Available automatic centerline tracking algorithms based on density or gradient information thus fail to track the clinically occluded segments. In practice, this problem is solved by inspecting the transverse CT images and manually selecting the missing points of the centerline based on the user’s knowledge of arterial anatomy.

We propose an automated knowledge-based approach to solve this problem, consisting of two main aspects, namely, characterization or description of shapes and modifying the description to match the data available from the subject. We characterize the shape of a vessel centerline by expressing it as a combination of certain basis functions. The basis functions are eigenshapes obtained by performing Principal Component Analysis (PCA) (Cootes et al., 1995, Duda et al., 2000) on a database of similar shapes. We also make use of anatomic landmarks to define oriented frames of reference for these vessels. We then express the incomplete dataset as a linear combination of the eigenshapes and use it to estimate the missing data. We correlate the error with several parameters and try to build a statistical model for predicting the error. We also compare the results with those predicted by a minimum mean squared error method.

This paper is organized as follows. In Section 2 an overview of the work done in the field of shape analysis for different applications is presented. In Section 3 the notation used in this paper is explained. Sections 4 Database of vascular centerlines, 5 Parametrization of centerline curves in oriented reference frames explain how the database of centerlines was acquired and parameterized. Section 6 explains how the database consists of similar shapes and how the linear dependency in shapes can be used to serve our purpose. Section 7 explains our algorithm and a minimum mean squared error method for comparison. In Section 8 we explain an additional step to improve the results. Sections 9 Experiments, 10 Results present the experimental framework and results in detail. In Section 11 we discuss the results and limitations of our method.

Section snippets

Related work

When the lumen of a normal arterial segment is filled with contrast medium-opacified blood, tracking can be accomplished using intensity information from the CT data by cost function minimization (Kanitsar et al., 2001), by using the connected components with minimal path technique (Cohen and Deschamps, 2001), by using flux driven algorithms (Bouix et al., 2005) and by skeletonization (Paik et al., 1998). Vessel tracking of Magnetic Resonance Angiography data has been done using these methods

Notation and mathematical formulation

Let V = f(t) represent a row vector of length p as a function of discrete parameter t = 0, 1,  , N  1. In other words f(t) is a collection of p functions. If V is unknown for t = m, m + 1,  , m + n  1, our task is to estimate f(t) for these missing indices. We are given a database of N similar functions Si = fi(t) for t = 0, 1,  , N  1 and i = 1, 2,  , M, where each Si is a variable row vector of length p. Note that each of Si and V can be completely described as an N × p matrix. For example, for p = 2:

Database:S1=S1(1,1)S1(1,2)

Database of vascular centerlines

To construct a knowledge base, we created a database of lower extremity vascular centerlines of 30 subjects (17 men and 13 woman: age 19–86 years) without peripheral arterial occlusive disease, who underwent lower-extremity CT angiography (18 for planning of free fibular flap graft harvesting, four for trauma, five for non-vascular causes of claudication due to musculoskeletal or neurologic disorders and three for miscellaneous reasons). Prior Institutional Review Board approval was obtained

Parametrization of centerline curves in oriented reference frames

Arteries of subjects of different heights may have different segment lengths and proportions. To compensate for these differences we used vascular bifurcations as anatomic landmarks to register different data sets with each other. The landmarks used to identify the femoropopliteal artery were the common femoral artery bifurcation, and the popliteal artery bifurcation. These two landmarks correspond approximately to the hip joint and the knee joint respectively and, therefore, compensate for the

Similarity and linear dependence of curves in the database

Fig. 3 shows all 30 femoropopliteal centerlines in the Anterior–Posterior (AP) view, in the oriented reference frames described earlier. The vertical axis represents the line joining the common femoral and popliteal bifurcation. Fig. 4 shows the same centerlines in the lateral view. Note that the curves have some fundamental similarities, having the same anatomic origin, although the shapes are not identical and the span of the curves in X and Y dimensions is not the same for every centerline.

Partial vector space projection (PVSP)

In this algorithm we assume that the curve to be reconstructed can be approximated as a linear combination of the curves in the database. Given the derived set of best orthonormal basis functions, the curve to be reconstructed is also a linear combination of the basis functions. Let B represent the matrix formed with kb number of basis functions as columns, where 1  kb  k. We always choose kb basis functions in the decreasing order of the corresponding eigenvalues. Let Vv be the vector form of V,

Endpoints correction

The results obtained by the PVSP method may need further correction for matching end-points of the estimated curve with the known curve, since in our application, we require that the interpolated section is contiguous with the known portions of the centerline. There are two constraints in each of the X and Y coordinates for matching the endpoints. A polynomial of degree one has two constraints and hence can be used as a correction curve. The polynomialx=f(z)=cz+dis uniquely determined by

Experiments

The basic assumption behind the PVSP algorithm is that a partially known curve can be represented as linear combination of the curves in the database. To validate this assumption, we performed PCA on the database of all 30 centerlines and observed that the curves are indeed similar to each other. Centerline paths in truly occluded arteries are not known accurately. Hence, we simulated occlusions in order to evaluate the performance of the algorithm presented. We selected each one of 30 patients

Results

We calculated the mean and standard deviation of MD for each of 9 occlusion lengths for all subjects as shown in Table 1. The distributions of the MD error for each length are shown in Fig. 8. Fig. 10 compares log-MD for the MMSE and the PVSP methods. The PVSP method consistently performs better (p < 0.001, Wilcoxen signed rank test). Table 2 shows the estimated population percentiles for the MD. For example, we estimate that 90% of 75 cm occlusions will be reconstructed with a maximum departure

Discussion and conclusion

The estimation of arterial centerlines through diseased arteries for subsequent generation of CPR images is an integral part of vascular imaging using CT Angiography (Fleischmann et al., 2006). While automated or semiautomated vessel-tracking algorithms are helpful to identify the centerlines of normal or mildly disease arteries, they fail in clinically important severely diseased and occluded portions of the arterial tree, and these segments have to be processed manually by expert users

Acknowledgements

This work was supported in part by Grant RO1 HL67194 from the National Institutes of Health (NIH). Justus E. Roos was supported by the Swiss National Science Foundation, PBBEB 106796.

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