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Improving Blood Vessel Tortuosity Measurements Via Highly Sampled Numerical Integration of the Frenet-Serret Equations Alexander Brummer, David Hunt, Van Savage

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Improving Blood Vessel Tortuosity Measurements Via Highly Sampled Numerical Integration of the Frenet-Serret Equations Alexander Brummer, David Hunt, Van Savage 1 Improving blood vessel tortuosity measurements via highly sampled numerical integration of the Frenet-Serret equations Alexander Brummer, David Hunt, Van Savage Abstract—Measures of vascular tortuosity—how curved and vessels can be linked to local pressures and stresses that act twisted a vessel is—are associated with a variety of vascular on vessels, connecting physical mechanisms to morphological diseases. Consequently, measurements of vessel tortuosity that features [17–19]. are accurate and comparable across modality, resolution, and size are greatly needed. Yet in practice, precise and consistent measurements are problematic—mismeasurements, inability to calculate, or contradictory and inconsistent measurements occur within and across studies. Here, we present a new method As part of the measurement process, researchers have used of measuring vessel tortuosity that ensures improved accuracy. sampling rates—the number of points constituting a ves- Our method relies on numerical integration of the Frenet- sel—nearly equal to the voxel dimensions, or resolution limits, Serret equations. By reconstructing the three-dimensional vessel for their images. This practice has proven sufficient for binary coordinates from tortuosity measurements, we explain how to differentiation between diseased and healthy vessels as long identify and use a minimally-sufficient sampling rate based on vessel radius while avoiding errors associated with oversampling as there is no significant variation in modality, resolution, and and overfitting. Our work identifies a key failing in current vessel size [6, 8, 20]. However, we show that measurements practices of filtering asymptotic measurements and highlights made at voxel-level sampling rates generally underestimate inconsistencies and redundancies between existing tortuosity common tortuosity metrics and obscure existing equivalen- metrics. We demonstrate our method by applying it to manually cies between different proposed metrics, thereby reducing constructed vessel phantoms with known measures of tortuousity, and 9,000 vessels from medical image data spanning human the diagnostic and prognostic potential of these valuable cerebral, coronary, and pulmonary vascular trees, and the biomarkers. This is of interest since growing experimental carotid, abdominal, renal, and iliac arteries. evidence demonstrates a causal link between vascular en- Index Terms—Biomechanical modeling, shape analysis, vessels dothelial growth factor signaling and tumor angiogenesis [21– 23]. As fluid shear stress and pressure can be mathematically expressed in terms of curvature and torsion [19, 24], improved accuracy in these two measures of tortuosity may better inform I. INTRODUCTION underlying disease pathology. ECENT work in both clinical and mathematical mod- R eling studies has shown that measures of vessel tor- tuosity—the extent of ‘curliness’, ‘squiggliness’, or ‘wiggli- ness’—serve as biomarkers of diseases such as atherosclero- We present a method based on numerical integration that sis, hypertension, arteriovenous malformations, recovery from uses curvature and torsion measured at sub-voxel sampling stroke or stent implantation, and classification of tumors rates. From this we reconstruct vessel centerline coordinates and their response to intervention [1–13]. In the research to an accuracy related to the vessel radius, thereby allowing for literature there exists many different definitions of tortuosity, comparisons across modality, resolution, and size. To test this with researchers constructing measures designed specifically method we first examine manually constructed vessel phan- to target particular biomarkers and for a given cohort [2– toms of semi-circles (constant, non-zero curvature yet zero arXiv:1911.12316v2 [physics.med-ph] 6 Aug 2020 16]. Using two components of tortuosity—curvature and tor- torsion), helices (constant, non-zero curvature and torsion), sion—it is possible to completely reconstruct the measured and salkowski curves (constant curvature and non-constant vessel, thereby providing a systematic check for measurement torsion) [25], all of which have tortuosity metrics that can be error. Yet, we have not found a single study that exploits this expressed analytically. Following this, we examine previously capability. Furthermore, these two mathematical measures of published data of the common, external, and internal carotid arteries [26]; the abdominal aorta, associated right renal artery, This work was supported in part by the National Science Foundation grant and both left and right common iliac arteries [27]; complete 1254159 coronary arterial trees [28]; the anterior, posterior, left, and A. Brummer and V. Savage are with the Institute for Quantitative and Computational Biosciences, and the Departments of Computational Medicine, right middle cerebral vascular trees [29]; and pulmonary and Ecology and Evolutionary Biology, University of California, Los Angeles, arterial and venous trees from clinical imaging of patients Los Angeles, CA, 90095 USA e-mail: [email protected]. with and without pulmonary hypertension [30] (see Fig. 1). D. Hunt is with the Institute for Learning and Brain Sciences, University of Washington, Seattle, WA, 98195 USA. Excluding the patients with pulmonary hypertension, all data V. Savage is also with the Santa Fe Institute, Santa Fe, NM, 87501. are from healthy human patients. 2 II. BACKGROUND A. Curvature, Torsion, and the Frenet-Serret Coordinate Frame Differential geometry was developed to deal with details of curved surfaces [31, 32]. Here we provide the standard set of techniques borrowed from differential geometry to estimate the tortuosity of vessels. Spatial curves are described in terms of position and vectors, #» #» r (sj) that assume Cartesian coordinates— r (sj) = x(sj)xb + y(sj)yb + z(sj)bz with (x(sj); y(sj); z(sj)) defined relative to the origin of the medical images. The subscript j represents either the indexed voxel-space of the original images or a subsampled space. We choose sj to be the arc length, defined numerically as the sum of the euclidean distances between all neighboring points from the first to the jth point, so Pi=j−1 #» #» sj = i=1 jj r i+1 − r ijj. The first and second derivatives of points along a spa- tial curve define the osculating plane such that changes in the curves shape can be described in two ways—in-plane changes and out-of-plane changes. Curvature, κ(sj), measures the rate of in-plane changes with respect to the curve’s arc length sj. Torsion, τ(sj), measures the rate of out-of-plane changes. Together these quantities are fundamental descriptors of any continuously differentiable curve that can be used to Fig. 1. Visualizations of vessel phantom, vessels, and vessel trees studied. reconstruct the spatial xj; yj; and zj-coordinates. To calculate (a) A segment of a Salkowski curve, or spiral, noted for having constant curvature and torsion, it is essential to introduce the Frenet- curvature and non-constant torsion (see Supplementary Material I-C and Serret coordinate frame. [25]). (b) Common, external, and internal carotid arteries from Kamenskiy et al. [26]. (c) Abdominal, iliac, and renal arteries from O’Flynn et al. [27]. The Frenet-Serret (FS) frame is a moving coordinate sys- (d) Posterior (green), anterior (brown), left middle (purple), and right middle tem defined at all points along a curve. This frame exists (blue) cerebral arterial trees from Bullitt et al. [29]. (e) Coronary arterial trees wherever the second derivative is continuous. The FS-frame from Vorobstova et al. [28]. (f-g) Pulmonary arterial (red) and venous (blue) trees from patients with pulmonary hypertension from Helmberger et al. [30]. can be calculated from the position vector of the curve, #» Solid black bars represent 20 mm scales. r (sj). The unit-vectors that constitute the FS-frame are the tangent, Tb(sj), the normal, Nb (sj), and the binormal, Bb (sj), vectors (Figure 2). The tangent vector is the normalized, first- derivative of the position vector with respect to arc length, #»0 #»0 Tb(sj) = r (sj)=jj r (sj)jj, and the normal vector is the nor- malized, second-derivative of the position vector with respect #»00 #»00 to arc length, Nb (sj) = r (sj)=jj r (sj)jj. These two unit- vectors point respectively in the instantaneous directions of velocity and acceleration of an object in motion. Finally, the #» binormal vector is the cross-product of the tangent and normal Fig. 2. Origin of coordinates, position vector, r (sj ), and unit tangent, vectors, Bb (sj) = Tb(sj) × Nb (sj) that points in the direction Tb (sj ), normal, Nb (sj ), and binormal, Bb (sj ), vectors (in blue, red, and green of angular velocity for rigid body rotation. respectively) drawn on simulated 2D-data (one period of sinusoidal curve). Curvature κ(sj) and torsion τ(sj) are the rates of change of the tangent and binormal vectors, which point in the direction of the normal vector. This is expressed formulaically as, This interpretation will be beneficial when we investigate the relationship between increased sampling rates and smooth rotations of the Frenet-Serret frame (see Fig. 3). Finally, the d Tb(sj) = νjκ(sj)Nb (sj) (1) rate of change of the normal vector in terms of curvature and dsj torsion is, d d Bb (sj) = −νjτ(sj)Nb (sj) (2) N(s ) = −ν κ(s )T(s ) + ν τ(s )B(s ) (3) dsj b j j j b j j j b j dsj where ν is the “speed” of the curve at s , defined as j #» j νj = jjd r j=dsjjj. Of note, curvature and torsion can be Using Eqs. (1)-(3) and the definitions of the FS-frame, interpreted as the rates of rotation of the Frenet-Serret frame curvature and torsion can be expressed solely in terms of about the binormal and tangent unit-vector axes, respectively. derivatives of the position vector. 3 3) Sum-of-Angles Metric (SOA): This metric integrates #»0 #»00 k r (sj) × r (sj)k angular changes in orientation between (at least) four neigh- κ(sj) = (4) #»0 3 boring points along a vessel. It is widely used in lieu of k r (sj)k curvature and/or torsion [2–8, 11, 33, 35, 36].
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