A Lagrangian cylindrical coordinate system for characterizing dynamic surface geometry of tubular anatomic structures

To develop a Lagrangian cylindrical coordinate system which can accurately quantify the surface of a tubular anatomical structure, the first step is to develop a continuous coordinate system for the longitudinal and angular dimensions. The luminal surface of a tubular structure, such as a blood vessel, can be defined by a sequence of n cross-sectional contours S_i, in 3D Cartesian space (Fig. 1a). These contours, formally denoted as \( {\left\{{S}_i\right\}}_{i=1}^n \), are defined as \( {S}_i={\left\{{b}_{ij}\right\}}_{j=1}^{m_i} \) where b_ij = (x_ij, y_ij, z_ij) defines the m_i individual points on each contour. For each contour, the centroid \( {C}_i=\left({\overline{x}}_i,\kern0.5em {\overline{y}}_i,\kern0.5em {\overline{z}}_i\ \right) \) can then be used to construct a centerline curve \( {\left\{{C}_i\right\}}_{i=1}^n \). The arc length of this centerline curve, σ, is used as the longitudinal coordinate for this coordinate system.

Next, the angular coordinate, θ, can be defined by specifying a particular circumferential reference point among the boundary points b_ij for each contour S_i, which is called γ_i (Fig. 1b). To define this reference point γ_i in a consistent, non-arbitrary way, an initial material point must be identified with a fiducial marker. For vascular structures, a bifurcation point serves as a natural fiducial marker. The lumen boundaries of the mother and daughter branches are defined as two separate sets and the most distal intersection of these two sets is selected as the material point δ and will be the landmark on the mother vessel that will serve as the natural reference point (Fig. 2). Then, the closest boundary point to this landmark is denoted the Greenwich point γ of the vessel. That is, γ ∈ {b_ij} where ∣δ − γ ∣  = min_{i, j} ∣ δ − b_ij∣. Now, suppose that γ is selected on the kth contour S_k. This point is used as the reference point for the centerline curve, such that C(0) = C_k and C(σ) is proximal to S_k if σ < 0 and distal to S_k if σ > 0. Corresponding reference points on every contour can then be assigned by letting γ_k = γ, and defining γ_k + 1 using a projection onto the S_k + 1 section as explained in more detail later. Iterating this process defines Greenwich points for each contour. The piecewise linear curve defined by these points is defined as the Greenwich curve.

Using the longitudinal σ and angular θ dimensions, the Lagrangian cylindrical surface function r(σ, θ) is created (Fig. 1b). By using linear interpolation in both dimensions, a continuous vessel coordinate system provides a Lagrangian description of every vessel boundary point. The radial function r(σ, θ) can be expanded to include a time parameter, i.e., r(σ, θ, t), in order to define changes of the vessel surface with respect to time.

This Lagrangian coordinate system can be used to quantify and monitor the surface curvature in both the circumferential (θ) and longitudinal (σ) directions. To compute the curvature at a given surface point, a circle is fit around the initial point and two of its symmetric neighboring points in the direction of interest (Fig. 3a, b). The reciprocal of the radius of this circle is defined as the magnitude of the curvature. The product and mean of the circumferential and longitudinal curvatures at a given point could be used to serve as proxies for the Gaussian curvature and mean curvature, respectively.

Additionally, this system can be used to quantify both the orientation and magnitude of the eccentricity of the structure. Classically, the eccentricity of an ellipse with major axis a and minor axis b, defined by \( \frac{x^2}{a^2}+\frac{x^2}{a^2}=1 \), has eccentricity \( \sqrt{1-\frac{b^2}{a^2}} \). This definition can be generalized to irregular contours by selecting the two orthogonal diameters through the center point C_i while minimizing the quotient \( \frac{d}{D} \) where D is the major axis and d is the axis perpendicular to D and defining the eccentricity as \( \sqrt{1-\frac{d^2}{D^2}} \). The direction of eccentricity is defined as the positive angle θ_e between the major diameter D and the Greenwich point γ_i (Fig. 3c). By tracking the direction of eccentricity as a function of σ, the static and dynamic spirality of eccentricity can be quantified.

In order to apply this method to arbitrary complex tubular structures, derivation of the Greenwich curve and prescribed window sizes to compute curvature need to be optimized. For this optimization exercise, two idealized software phantoms were utilized. The first was a simple tubular phantom with circular contours and a longitudinal bend causing variation in longitudinal curvature (Fig. 4a). The second phantom was designed with non-circular cross sections and two bends in different planes, exhibiting variable circumferential curvature, longitudinal curvature, eccentricity, and orientation of eccentricity (Fig. 4b).

For defining the points that comprise the Greenwich curve, three methods were attempted. The first method started with the initial point γ_k and projected the vector γ_k − C_k onto the S_k + 1th section along the vector C_k + 1 − C_k. Then, γ_k + 1 is selected to be the closest point in the sequence \( {\left\{{b}_{\left(k+1\right)\ j}\right\}}_{j=1}^{m_{k+1}}. \) The second method instead projected the vector γ_k − C_k along the normal of the S_kth section onto the S_k + 1th section and defined γ_k + 1 in the same way. The third method simply selected γ_k + 1 to be the closest point in \( {\left\{{b}_{\left(k+1\right)\ j}\right\}}_{j=1}^{m_{k+1}} \) from γ_k. Using any of these methods, Greenwich points on every section can be defined. The piecewise linear curve formed by these Greenwich points is then designated as the Greenwich curve. In this paper, we selected the first method, i.e., projection of the vector γ_k − C_k onto the S_k + 1th section along the vector C_k + 1 − C_k, since we would like to obtain a Greenwich curve that more robustly tracks the centerline curve.

To determine the optimal span length of the three points to calculate curvature (i.e., window size), the idealized phantoms, where the analytic curvatures were known, were used. A variety of window sizes were evaluated for both the longitudinal and circumferential directions and window sizes which adequately resolved the true curvature, yet did not produce substantial spurious oscillations, were selected.

The optimized coordinate system was applied to the two phantoms described above, as well as three human data sets to exemplify the full range of quantifications and analyses possible. For the human data, high-resolution computed tomography (CT) imaging data were acquired of a thoracic aortic endograft implanted via thoracic endovascular aortic repair (TEVAR) (Fig. 5a), an abdominal aorta and the visceral artery branches after endovascular aneurysm repair (EVAR) (Fig. 5b), and the iliofemoral veins after stent implantation (Fig. 5c). From the idealized phantoms and the human data sets, centerlines and orthogonal segmentations were created using SimVascular (Open Source Medical Software Corporation, San Diego, CA) [28]. Modeling was started by creating an initial lumen path manually along the center of vessel lumens. Then, semi-automatic 2D level set segmentation was performed orthogonally to this initial path, at every one half radius of the vessel. From every segmentation contour, mathematical centroid was extracted and connected to form the centerline. To assure the orthogonality of lumen contours, subsequent round of 2D segmentation was performed along the centerline instead of initial path, with consistent interval. The centerline was updated according to the second-round contours, which together used as the input for this application. The cylindrical coordinate system methods were applied to quantify the longitudinal and circumferential curvature, as well as the amplitude and direction of eccentricity at each cross section. Finally, for the thoracic aortic endograft, dynamic changes in geometry due to cardiac pulsation are shown by comparing the surface geometries between systole and diastole.

Figure 6 illustrates a subset of the trials run to optimize window sizes to calculate the surface curvature in the longitudinal (Fig. 6a–c) and circumferential (Fig. 6d–f) directions. For longitudinal curvature, a window size of 20 mm produced spurious noise, as indicated by the deep ridges in Fig. 6a, while a window size of 40 mm underestimated the peak longitudinal curvature of 1 cm⁻¹ on the inner curve of the second bend by 7% (Fig. 6c). A longitudinal window size of 30 mm, however, did not exhibit spurious noise and was able to calculate the peak curvature to within 0.5% (Fig. 6b). In the circumferential direction, a window size of π/8 exhibited spurious oscillations (Fig. 6d) while a window size of 3π/8 underestimated the peak circumferential curvature of 2 cm⁻¹ on the elliptical cross sections by 35% (Fig. 6f). In this case, a window size of π/4 rad (Fig. 6e) was determined to be optimal. Similar relative window sizes were utilized for analysis of idealized phantom and human data sets.

The constructed 3D model of the simple phantom with 2D cross-sectional contours, centerlines, and Greenwich curve is shown in Fig. 7a. The pointwise measurements of the longitudinal and circumferential curvature along the entire surface are shown in Fig. 7b, c. As shown in Fig. 7b, the longitudinal curvature was calculated to be 0 cm⁻¹ proximal and distal to the bend and a maximum value of approximately 0.5 and 0.25 cm⁻¹ along the inner and outer curve of the bend, respectively. In Fig. 7c, the circumferential curvature was calculated to be approximately 1.0 cm⁻¹ along the entire surface.

Likewise, the constructed model of the complex phantom is shown in Fig. 8a and the results of the analyses are shown in the remainder of Fig. 8. Surface calculations of longitudinal and circumferential curvature are shown in Fig. 8b, c, and the quantification of magnitude and orientation of eccentricity are shown in Fig. 8d, e. As shown in Fig. 8b, the straight sections exhibit 0.0 cm⁻¹ longitudinal curvature, the outer curve of the first bend exhibits 0.5 cm⁻¹ curvature, and the inner curve of the second bend exhibits 1.0 cm⁻¹ curvature. Figure 8c depicts circumferential curvatures of 1.5 cm⁻¹ along the vertices of the major of the elliptical cross sections, approximately 0.25 cm⁻¹ along the flat sections of the elliptical cross sections and 1.0 cm⁻¹ at the circular cross sections. In Fig. 8d, the magnitude of eccentricity along the longitudinal direction was calculated as 0.87 for the proximal and distal straight sections, with a steep dip at σ = 96 mm, reflecting the location of the circular cross sections. In Fig. 8e, the orientation of eccentricity was calculated as 1.6 and 0.0 rad for the proximal and distal straight sections, respectively, with a sudden transition at σ = 96 mm. As complement to these images, Table 1 shows the absolute difference between the numerically estimated curvatures and the analytic solution with respect to minimum, maximum and median.

Figure 9a–c illustrates orthogonal segmentations, centerlines, and derived Greenwich curves for each human image data set. Utilizing the coordinate system method on these examples, the amplitude and orientation of the vessel eccentricity, and pointwise longitudinal and circumferential surface curvatures were quantified. Examples of these analyses can be seen in Fig. 9d–f. Figure 9d shows the amplitude and orientation of the eccentricity of the thoracic aortic endograft (from Fig. 5a). Figure 9e shows the pointwise circumferential surface curvature of the aneurysmal abdominal aorta (from Fig. 5b). Figure 9f shows the pointwise longitudinal curvature of the iliac vein (from Fig. 5c).

To demonstrate the ability of the developed method to compare dynamic changes of surface geometry, Fig. 10 depicts a coordinate system analysis of pointwise longitudinal surface curvature for the thoracic aortic endograft (from Fig. 5a) at systolic and diastole phases of the cardiac cycle.