Automatic vessel diameter quantification and vessel tracing for OCT angiography

Abstract

Optical coherence tomography angiography (OCTA) is capable of non-invasively imaging the vascular networks within circulatory tissue beds in vivo. Following improvements in OCTA image quality, it is now possible to extract vascular parameters from image data to potentially facilitate the diagnosis and treatment of human disease. In this paper, we present a method for automatic vessel diameter mapping down to the individual capillary level, through gradient-guided mimimum radial distance (MRD). Compared to traditional methods, our method demonstrated superior consistency and accuracy, as well as high tolerance to rotation of the vasculature pattern, during validation using well-characterized microfluidic flow phantoms. We also incorporated a modified A* path searching algorithm to trace vessel branches and calculate the diameter of each branch from the OCTA images. After validation in vitro, we applied these algorithms to the in vivo setting through analysis of mouse cortical vasculature. Our algorithm returned results that followed Murray’s law, until reaching the capillary level, agreeing well with known physiological data. From our tracing process, vessel tortuosity and branching angle could also be measured, establishing our approach as an automatic vasculature evaluation framework. We believe this framework would be useful for the diagnosis of vascular diseases, especially those resulting in regional early-stage morphological changes.

1. Introduction

The vasculature plays an important role in tissue development and remodeling under both normal and pathological conditions. For instance, in neurology, dysregulated angiogenesis is implicated in stroke, Alzheimer’s disease and motor neuron disease [1]; in ophthalmology, evidence of vascular remodeling provides a critical indication of disease progression [2]; in oncology, the study of irregular tumor vessels can inform drug delivery strategies [3]; and during wound healing, specific vessel parameters have been shown to correlate with wound recovery time [4]. Rigorous investigation into the properties of specific vascular beds, therefore, has the potential to further improve our understanding in disease pathogenesis, and lead to novel methods for the prevention and treatment of human disease.

Optical coherence tomography (OCT) is a promising non-contact and high resolution 3-dimensional imaging technique that utilizes backscattering light to image tissue structures in vivo [5]. Its functional extension to OCT-based angiography (OCTA) has proven to have a high clinical significance, allowing mapping of vascular networks directly from intrinsic OCT signals without requiring the use of exogenous dye [6], [7]. Since the initial applications of OCTA within the field of ophthalmology, numerous methods have been reported to measure the geometrical and morphological parameters of blood vessels from the acquired angiograms [8], [9]. Similar techniques have also been applied to small animal studies that demonstrate good correlations between vessel morphology and blood flow dynamics [10], [11]. To systematically quantify the vasculature, a common approach is to extract the key vascularity matrices, including vessel area density [9], [11], [12], vessel diameter [9], [13]–[15], vessel tortuosity [16]–[18], and vessel branching angle [15], [19]. Vessel diameter, in particular, allows for the morphologic assessment of blood transport dynamics, by characterizing changes within the vascular bed (i.e. dilation or contraction) that directly relate to differences in blood flow. As a pilot clinical study, Goldenberg et al. reported a method for the manual measurement of retinal vessel diameter from cross-sectional OCT structural images [14]. Based on the results from manual segmentation, Pilch et al. further developed a statistical shape model that utilized supervised learning for automatic vessel segmentation and diameter quantification [20]. However, only big arterioles or venules were measured from the raster-scanned cross-sectional images. In an alternate approach, diameter distributions of en face choroidal vessels were quantified through a multi-scale morphological analysis with adaptive segmentation windows [21]. This method of quantification relies on assigned window sizes, however, requiring further optimization steps to improve its accuracy. In 2015, Yousefi et al. proposed a hybrid segmentation technique by combining Hessian filtering and OCTA intensity signals through a weighted average scheme, and accordingly quantified the vessel radius through distance transformation [13]. While promising, distance transformation, by simply estimating the distance to the nearest unidirectional boundary, does not accurately predict vessel radius (in most cases, it would underestimate the radius especially for small vessels), and therefore the quantified values would likely be inconsistent even within the same vessel branch. This method of quantification is therefore limited in its application to clinical settings. Recently, the same group reported a clinically acceptable approach achieved by a series of vessel binarization, skeletonization, block-wise box counting (BWBC) and Gaussian filtering [9]. Using this method, significant difference in vessel diameter were reported when comparing control subjects with those suffering from macular telangiectasia type 2. Without consideration of vessel morphology, however, it is difficult for this method to provide consistent and robust measurement down to individual vessel branches. It is necessary to map the diameter of each specific vessel rather than the average diameter of an arbitrary block.

In this paper we present an automated framework for the quantitative characterization of blood vessel diameter, down to the level of individual capillaries, using a gradient-guided mimimum radial distance (MRD) measurement. We validated this method by imaging and quantifying parameters within well-characterized microfluidic flow phantoms. Compared to the clinically well-accepted method in [9], our approach demonstrated superior consistency and accuracy, as well as a high tolerance to rotation of the vasculature patterns. We next developed a modified A* algorithm to trace vessel branches and calculate the diameter of each branch. Our rationale for using A* tracing was based on the key observation that blood cells typically preferentially flow through the shortest vascular path between bifurcations. Following initial validation in flow phantoms, we applied our proposed algorithms to the analysis of mouse cortical vasculature in vivo. The values obtained appear to obey Murray’s law [22] until reaching the capillary level, correlating well with previously published data [23], [24]. After tracing, vessel tortuosity and vessel branching angle were automatically measured, establishing our approach as a useful vasculature evaluation framework.

2. Material Preparation and Imaging

2.1. System setup

A spectral domain OCT system was developed and utilized for imaging. The schematic setup is presented in Fig. 1. The system contained a broadband super luminescent diode (LS2000C SLD, Thorlab Inc.) with a central wavelength of 1310 nm and a spectral bandwidth of 110 nm, which provided an axial resolution of ~ 7 μm in the air. In the sample arm, a 10X objective lens was used to focus light onto the sample, corresponding to a lateral resolution of ~7 μm at full wide of half maximum. The interference signals were detected by a spectrometer that adopted a high-speed line scan camera (SU1024-LDH2, Goodrich Inc.) working at 92 kHz. With an incident light power of 3.5 mW, the system sensitivity was measured to be ~105 dB.

Figure 1.

Schematic OCT system setup for the demonstration of automatic vessel diameter quantification and automatic vessel tracing

2.2. Microfluidic fabrication and animal preparation

For validation of our algorithms, image acquisition was carried out both in vitro within flow phantoms, and in vivo within a murine model, as shown in Fig. 1. We devised microfluidic patterns to mimic 3 vascular conditions: a healthy vasculature, an ischemic vasculature with thoroughfare channels, and an ischemic vasculature with vessel dilation. Microfluidic patterns were designed using LayoutEditor software (© Juspertor, 2018) and transferred to silicon wafers via photolithographic patterning of positive photoresist followed by deep reactive-ion etching of exposed silicon. Patterns were then transferred to Polydimethylsiloxane (PDMS) using soft lithographic techniques, as previously described [25], [26]. PDMS was mixed with 0.18% TiO₂ particles prior to fabrication, in order to better mimic the optical properties of human tissue, and all PDMS devices were created with a final height of 1 mm. The microchannels within the flow phantoms were perfused with 1% intralipid solution at 1 mL/hr to mimic moving red blood cells. The intralipid solution was injected from the inlet, flowed through hierarchical vessel trees (with channels representing arterioles, capillaries, and venules), and finally emptied into the outlet reservoir. At each bifurcation point, a higher-order channel split into two equal-width lower-order channels. All tube heights were kept at 40 μm for ease of fabrication and perfusion. For in vivo experiments, an anesthetized mouse (strain: C57BL/6, age: 2 months) was surgically prepared through open-skull cranial window technique and carefully positioned in a stereotactic frame under the scanning probe, as in [27]. The animal procedures were reviewed and approved by the Institutional Animal Care and Use Committee (IACUC) of University of Washington.

2.3. OCTA imaging protocols and data pre-processing

During data collection, we utilized OCTA scanning protocols. Briefly, for the microfluidic channels, we acquired 738 A-lines along the fast scanning direction to pile into one B-frame, which covered a range of 4.8 mm. In the slow scanning direction, 800 B-frame positions were sampled with 8 repeated frames at each position, covering a range of 5.2 mm. The frame rate was set at 90 fps. For the in vivo cerebral imaging, a faster repetition rate of 180 fps was used to minimize artifacts from tissue motion. Here, we acquired 400 A-lines along the fast scanning direction, 3200 B-frames (400 B-positions × 8 frame repetitions) along the slow scanning direction, which covered a field of view of 3.5 × 3.5 mm.

After the data acquisition, complex-OCT-signal based optical microangiography (OMAG) [4] was employed to delineate the flow signal from the static tissue background. Then, the 3D angiography datasets were collapsed into 2D through layer-segmented maximum intensity projection (MIP) [28].

3. Vessel diameter quantification

Figure 2(a) shows the layout of the designed healthy vasculature: a highly symmetric pattern with five hierarchical channels (channel widths of 240 μm, 120 μm, 60 μm, 30 μm, and 15 μm) that cover a region of 3.94 × 4.93 mm. The corresponding en-face view OCTA vasculature is visualized in Fig. 2(b), in which a red line indicates a typical cross-session of the capillary network as shown in the structure and flow images in Fig. 2(c) and (d), respectively.

Figure 2.

Microfluidic channel with healthy vasculature. (a) Layout of the channel pattern consisted of channels with widths of 240 μm, 120 μm, 60 μm, 30 μm, and 15 μm. All sizes in (a) are in millimeters. (b) OCTA en-face MIP image of the channel perfused with intralipid solution. (c) and (d) Cross-sectional structure image and flow image of the 15 μm capillary network at the location indicated by the red line in (b). Scale bars: 0.5 mm

The en-face vasculature was first binarized through a combination of global thresholding, hessian filtering and adaptive thresholding [13] to result in a binary vessel map (BVM), as shown in Fig. 3(a). In the BVM, white pixels and black pixels represent the vascularized regions and avascular regions, respectively. Then, the BVM was further skeletonized by iteratively shrinking the outer boundary of the vascularized regions to a line with a single pixel width [13], as shown in the vessel skeleton map (VSM) of Fig. 3(b). The contours of the channels could also be detected from the BVM through edge detection, as in the vessel perimeter map (VPM) in Fig. 3(c). Once the vessel skeleton and vessel perimeter had been extracted, the vessel diameter could be calculated as the summation of radical distances between the skeleton line towards bilateral perimeter boundaries, as in the schematic shown as a subfigure of Fig. 3(d). The searching direction is defined by the gradient analysis of the skeleton, as shown in the horrizontal and vertical gradient maps in Fig. 3(e) and (f), respectively. Each pixel in the skeleton would generate two reversed searching directions. The positive gradients (white) represent radical search towards the right in (e) and bottom in (f), and vice versa for the negative gradients (black). Following the searching directions, we calculated the minimum distances from the skeleton to its nearest perimeter boundary within the diagnoral searching quadrants (i.e. the 2 searching quadrants partitioned by grouping the horrizontal and vertical gradients) as denoted by the diagonal hatchings in the subfigure of (f), and sumed the two distances originating from the same skeleton pixel as one diameter value, as expressed by the following equation:

$$\mathit{\text{D}}={\text{min}}_{\mathit{\text{i}} \in \mathit{\text{VPM}}, \mathit{\text{Q}}}\left(\sqrt{{{\left({\mathit{\text{x}}}_{\mathit{\text{i}}}-\mathit{\text{x}}\right)}^2+({\mathit{\text{y}}}_{\mathit{\text{i}}}-\mathit{\text{y}})}^2}\right)+{\text{min}}_{\mathit{\text{j}} \in \mathit{\text{VPM}}, \stackrel{-}{\mathit{\text{Q}}}}\left(\sqrt{{{\left({\mathit{\text{x}}}_{\mathit{\text{j}}}-\mathit{\text{x}}\right)}^2+({\mathit{\text{y}}}_{\mathit{\text{j}}}-\mathit{\text{y}})}^2}\right)$$ (1)

where D represents the vessel diameter; (x, y) represents coordinate of a typical pixel in the vessel skeleton; (x_i, y_i) and (x_j, y_j) represent coordinates of pixels in the vessel perimater map (VPM) and belonging to the diagonal searching quadrants Q and $\overline{Q}$, respectively. Then, the calculated vessel diameters were applied to the skeletons as a color-coded VSM in Fig. 3 (g). As displayed in Fig. 3 (h), by projecting each vascularized region in the BVM towards its closest vessel skeleton, we can apply the color-coded diameter values to the entire vasculature pattern, as in Fig. 3 (i). The calculated vessel diameters are displayed in parula colormaps and indicated in the colorbars in (g) and (i), respectively. Of note, here the diameter is directly measured from the vessel cross-section perpendicular to the instaneous perfusion flow, rather than the cross-section obtained from orthogonal OCT scanning, therefore providing more accurate quantifications of the physiological diameters, compared with previous approaches in [14], [20].

Figure 3.

Demonstration of MRD automatic vessel diameter quantification of the healthy vasculature pattern. (a) Binary vessel map (BVM). (b) Vessel skeleton map (VSM). (c) Vessel perimeter map (VPM) (d) Overlay between the VPM (white) and the VSM (red). A local region is zoomed to demonstrate that vessel diameter is calculated as the summation of radical distances between the skeleton line towards bilateral perimeter boundaries. (e) Horizontal gradient of the VSM. (f) Vertical gradient of the VSM. The subfigure in (f) shows that practically the diameter is calculated as minimum radical distance within the diagnoral searching quadrants. The arrows in (d), (e) and (f) indicate the radial searching directions. (g) Color-coded vessel diameter parameters on the VSM. (h) Overlay between the BVM (white) and the VSM (red). (i) Color-coded vessel diameter parameters on the BVM.

The conventional way of estimating vessel diameter (i.e. BWBC method) is designed by arbitrarily selecting a block of n × n pixels, and calculating the averaged vessel caliber inside [9] as:

$$D_b= \frac{\sum _{x_b=1}^n\sum _{y_b=1}^n\mathit{\text{BVM}}(x_b,y_b)}{\sum _{x_b=1}^n\sum _{y_b=1}^n\mathit{\text{VSM}}(x_b,y_b)}$$ (2)

where D_b represents the block-wise vessel diameter, $(x_b,y_b)$ represents the coordinate of each pixel inside the block, $\sum _{x_b=1}^n\sum _{y_b=1}^n\mathit{\text{BVM}}(x_b,y_b)$ represents the area of the vascularized regions (i.e. white pixels in (a)) within the block, and $\sum _{x_b=1}^n\sum _{y_b=1}^n\mathit{\text{VSM}}(x_b,y_b)$ represents the total lengths of the vessel segments within the block (i.e. white pixels in (b)), where the block size n is set as 20 pixels. By applying Eq.(2) to the entire vasculature pattern, a down-sampled block-wise diameter map was obtained, which was further interpolated to the original size of 800 × 738 pixels and Gaussian filtered with a 3 × 3 kernel to generate a quantitative evaluation map [9]. The locally averaged vessel diameters were visulized by integrating this map with binary masks either from BVM or VSM [9]. Indubitably, in the block-wise box counting method, the estimated vessel diameter is subject to the selected block size and the sample spacing, which requires further optimizations for various applications [11].

A series of comparisons between conventional BWBC and MRD were made based on the healthy vasculature pattern. Fig. 4(a) and (c) show the quantified vessel diameters integrated with the VSM by using BWBC and MRD, respectively. To test the robustness of two methods, the same vasculature pattern was up-sampled two-fold, and the diameters were recalculated following BWBC and MRD algorithms, respectively, as shown in the right subfigures of (a) and (c). The arrows in (a) indicate that BWBC is subject to the sample spacing, as the block size and Gaussian filter in BWBC do not take vessel morphology into consideration. MRD still provided equivalent results after up-sampling, as indicated by the arrows in (c), with the regional diameter distributions visualized in the color-coded BVM as in Fig. 4(b) and (d), respectively. The accuracy and consistency of MRD are superior to that of BWBC when quantifying single vessel branches, as shown in the vessels with the same diameter of 120 μm indicated by the red dash boxes in (b) and (d).

Figure 4.

Comparison between block-wise box counting (BWBC) and calculation of minimum radical distance (MRD) based on the healthy vasculature pattern. (a) Vessel diameters integrated with VSM by BWBC. (b) Vessel diameters integrated with BVM by BWBC. (c) Vessel diameters integrated with VSM by MRD. (d) Vessel diameters integrated with BVM by MRD. In (a) and (c), the diameters of up-sampled (× 2) vasculature patterns are zoomed as shown in the right subfigures. The arrows indicate that BWBC is subject to the sample spacing, while MRD is still robust after up-sampling. In (b) and (d), the channels with the same diameter (120 μm) are marked by red boxes that reveal the superiority of MRD in quantification accuracy and consistency.

Additionally, we analyzed the vessel diameters of two patterns mimicking ischemic vascular conditions: both with ~ 1/6 loss of capillaries compared with the healthy vasculature, but one with thoroughfare channels (60 μm) and another one with vessel dilation (doubled vessel diameter at the central zone). For the first pattern, the layout of microfluidic design and corresponding OCTA MIP image are displayed in Fig. 5 (a) and (b). The vessel diameter map integrated with VSM and that integrated with BVM by either using conventional BWBC or using MRD are shown in Fig. 5 (c) and (d), and (e) and (f), respectively. Similarly, for the second pattern, the layout of the microfluidic design, the OCTA MIP image, the vessel diameter map integrated with VSM, and that integrated with BVM by either using BWBC or MRD are displayed in Fig. 5 (g) – (l). Visually, both BWBC and MRD detect the vessel diameter changes either in the thoroughfare channels or in the dilated capillaries. However, as indicated by the red dash boxes in (d) and (j), the distribution of quantified diameters appears to be shifted for the BWBC algorithm due to the rotation of the obtained OCTA vasculature patterns (~ 3°). In contrast, MRD exhibited high tolerance to pattern rotation, as indicated in (f) and (l).

Figure 5.

Comparison between block-wise box counting (BWBC) and calculation of minimum radical distance (MRD) based on two ischemic vasculature patterns. (a) Layout of the ischemic vasculature with thoroughfare channels. (b) OCTA en-face MIP image of the pattern in (a). (c) and (d) the diameter maps of vessels in (a) integrated with VSM and integrated with BVM, as calculated by BWBC. (e) and (f) the diameter maps of vessels in (a) integrated with VSM and integrated with BVM, as calculated by MRD. (g) Layout of the ischemic vasculature with vessel dilation. (h) OCTA en-face MIP image of the pattern in (g). (i) and (j) the diameter maps of vessels in (g) integrated with VSM and integrated with BVM, as calculated by BWBC. (k) and (l) the diameter maps of vessels in (g) integrated with VSM and integrated with BVM, as calculated by MRD. During OCT imaging, the vasculatures in (a) and (g) are rotated counter-clockwise by ~3°. All sizes in (a) and (g) are in millimeters. In (d) and (j), the red squares indicate regions that are affected by the rotation of vasculature patterns, while MRD proves high tolerance to the rotation as indicated in (f) and (l).

4. Automatic vessel tracing

To trace the vessel branches and accordingly calculate the mean diameter of each branch, we first automatically detected the bifurcations, as indicated by the green dots in Figure 6 (b), by applying the following searching equation to each skeleton pixel:

$$\left\{{\sum }_{i=-1}^1{\sum }_{j=-1}^1\mathit{\text{VSM}}(x+i,\mathit{\text{ y}}+j)\ge 4\right\}\cap \mathit{\text{VSM}}(x,\mathit{\text{ y}})$$ (3)

where $\mathit{\text{VSM}}(x,\mathit{\text{ y}})$ represents a typical pixel of the vessel skeleton, and $\mathit{\text{VSM}}(x+i,\mathit{\text{ y}}+j)$ represents its neighbor pixels. The criterion ≥4 is assigned according to an observation that the bifurcation point has more neighbors (≥3) compared with the midpiece (neighbor =2) or the terminal (neighbor =1), as representatively indicated by bifurcations a, b and c compared to midpieces d, e and f in (b). For this microfluidic pattern, some corners (e.g. g and h in (b)) were mistakenly detected as bifurcations due to slight skeletonization artifacts for 90 degree corners. Fortunately, those false positive bifurcations would only add vessel segments but not affect the tracing results. Since we extract all the bifurcations, we can automatically trace between any two of them for an overview of the entire vasculature. Practically, however, in clinical scenarios physicians may only care about a small number of branches along a single vessel tree. For this reason, a bifurcation and 3 of its connected branches (yellow dot: starting; red dots: ending) within the red dash box in (c) are selected for demonstration of the tracing process.

Figure 6.

Demonstration of automatic vessel tracing based on the healthy vasculature pattern. (a) Vessel skeleton map same as Fig. 3(b). (b) The detected bifurcations (green dots) overlaid on the vessel skeleton map. (c) Selection of starting (yellow dot) and ending (red dot) of each vessel branch. (d) Frames taken from an automatic tracing movie of the region marked by a red square in (c): (1). Initialization of the starting; (2)-(4). Tracing towards 3 endings, in which red arrows indicate the tracing direction; (5). One of the downstream ending is automatically re-initialized as the starting.

As blood flow preferentially travels through the shortest (lowest-resistance) path between two bifurcations, vessel tracing was conducted using an A* path searching algorithm [29]. A* applies an informed searching on all accessible paths to find the one that incurs the smallest cost (least distance travelled in this study). Specifically, A* iteratively selects the path (pixels of the vessel skeleton) that minimizes

$$f\left(m\right)=g\left(m\right)+h\left(m\right)$$ (4)

where m is the latest selected pixel, g(m) is the distance travelled from the starting, h(m) is the heuristic that estimates the cost of the cheapest path from m to the ending. According to the design of the A* algorithm, the heuristic must never overestimate the actual distance to the ending. Here h(m) was assigned as the straight-line distance (Euclidean distance) that satisfies this criterion. In g(m), each step towards the edge-connected neighbors would be counted as 1, and that towards the diagonal-connected neighbors as $\sqrt{2}$. The searching process is visualized as movie frames in Fig. 6(d): (1). initialization of the starting; (2)-(4). tracing towards three surrounding endings, respectively; (5). automatic re-initialization of a new starting at downstream.

By integrating the tracing with BWBC- and MRD-based vessel diameter maps in Fig. 4(a) and (c), we calculated the mean diameters of ten vessel branches as marked in numbers in Fig. 7(a) and listed in Fig. 7(b). To analyze the quantification errors with respect to the true diameter values from the designed layout, we employ two evaluation parameters: the total error and the variance of error as defined by

$$\mathit{\text{Total error}}=\sum \left|D-D_t\right|$$ (5)

$$\mathit{\text{Var of error}}=\sum {(\left|D-D_t\right|-\frac{1}{N}\sum \left|D-D_t\right|) }^2$$ (6)

where D represents the quantified diameter, D_t represents the true value, N represents number of the vessel branches (N = 10). The total error suggests that BWBC is more prone to errors (~ 3 times) compared with MRD. The variance of error reveals that the stability of the diameter quantification by BWBC is much worse than that of MRD.

Figure 7.

(a) OCTA MIP image of the healthy vasculature pattern same as Fig. 2(b), in which numbers 1–10 indicate ten branches of a vessel tree. (b) The true vessel diameters of the ten branches, the calculated mean diameters by integrating automatic vessel tracing with BWBC, and the calculated mean diameters by integrating automatic vessel tracing with MRD. The quantification errors are analyzed as total errors and variances of error for BWBC and MRD, respectively.

5. Quantitative morphological evaluation of cerebral vasculature in vivo

The automatic vessel diameter quantification and automatic vessel tracing were integrated to evaluate vessel diameters in mouse cerebral cortex in vivo. Fig. 8(a) shows the microscope image of the mouse cerebral cortex through a transparent cranial window. Correspondingly, we acquired the OCT dataset and extracted the OCTA angiogram (the top 5-μm/1-pixel from the tissue surface) as in Fig. 8 (b), where the bifurcations in a superficial vein up to 7 generations are labelled in alphabetical order. First, the angiogram was binarized and skeletonized as Fig. 8 (c). Thereafter, the bifurcations were automatically detected as green dots in Fig. 8 (d). To ensure the robustness of vessel tracing within this in vivo dataset, vessel fragments with low connectivity were excluded from analysis from (c) to (d). In (d), the alphabetically labelled bifurcations are marked as starting (yellow dots), whose surrounding bifurcations are marked as ending (red dots) to trace along each vessel branch. Next, the vessel diameter information was integrated with VSM through BWBC as in Fig. 8 (e), and through MRD as in Fig. 8 (f). In BWBC, the block size n was reset as 10 pixels to match the lateral samping of 400×400 pixels.

Figure 8.

(a) The microscope image of mouse cerebral cortex through a cranial window. (b) OMAG en-face view image of the surface vasculature. The selected bifurcations up to 7 generations are labelled in alphabetical order, in which (d, e, f, g) and (d’, e’, f’, g’) correspond to bifurcations of two big branches on the same level. (c) Vessel skeleton map. (d) Bifurcations (green dots) are automatically detected from the skeleton, in which selected bifurcations in (b) are marked as yellow dots, and the surrounding red dot indicates the ending of each connected vessel branch. (e) Vessel diameters integrated with VSM by BWBC. (f) Vessel diameters integrated with VSM by MRD.

It is conceivable that analysis of vessel diameters at bifurcations is relevant to the understanding the partition of cerebral blood flow in flow-related pathological changes, such as cerebral venous thrombosis [30] and venous insufficiency [31]. The mean diameters of vessel branches that are automatically measured through MRD plus vessel tracing are listed on the OCTA angiogram in Fig. 9(a). For comparison purpose, the same branches were also evaluated through BWBC plus vessel tracing, and manually measured by one skilled technician specializing in OCTA imaging (diameter values from these two methods are not listed). Additionally, we introduced two indexes to evaluate the relations in diameters between the parent and the child branches at each bifurcation. The first index is generalized as a cubic dependence as calculated by $\mathit{\text{Cubic}}=\left({\sum }_{i=1}^N{D}_i^3\right){/D}_0^3$ (where D_i represents the diameter of each child branch, and D₀ represents the diameter of the parent branch). According to Murray’s law [32]–[34], this cubic dependence should be close to 1, such that the cost for blood transport and maintenance are minimized. The second index is described by the area ratio of vessel calibers (square dependence) as calculated by $\mathit{\text{Square}}=\left({\sum }_{i=1}^N{D}_i^2\right){/D}_0^2$. In our analyses, these two indexes are calculated for each vessel branch and for each measurement approach, as listed in Fig. 9 (b) and (c), respectively. The mean cubic dependences obtained from manual measurement (0.978±0.033) and MRD (1.026±0.116) are close to the theoretical standard of 1 that following Murray’s law. And, the mean square dependences obtained from manual measurement (1.245±0.056) and MRD (1.247±0.131) are within the range of 1.10 – 1.28 reported for cerebral vessels in the literature [19], [24], [34]. However, significant systematic errors exist in the results from BWBC (1.201±0.187 for cubic dependence and 1.401±0.157 for square dependence).

Figure 9.

(a) OCTA angiogram with labeled mean vessel diameters (cyan) quantified through MRD plus vessel tracing. (b) Cubic dependences $({\sum }_{i=1}^n{D}_n^3){/D}_0^{-3}$ measured with manual measurement, BWBC, and MRD, respectively. (c) Square dependences $\left({\sum }_{i=1}^n{D}_n^2\right){/D}_0^{-2}$ measured with manual measurement, BWBC, and MRD, respectively. In (b) and (c), each dot represents an individual bifurcation, and the horizontal bar represents the mean with standard deviation.

To enable a thorough morphological evaluation of the cerebral vasculature, vessel tortuosity and vessel branching angle were automatically measured after vessel tracing. The length of the vessel branch was measured with each step towards the edge-connected neighbors as 1, and that towards the diagonal-connected neighbors as $\sqrt{2}$. The vessel tortuosity can then be calculated as the vessel length divided by the straight-line distance between two bifurcation points. The measured tortuosity of two arterio-arterial anastomosis vessels are 1.142 and 1.129, as in Fig. 10(a) and (b) respectively, within the reasonable range in literature [16]. The vessel branching angle is obtained by selecting the first 10 pixels from each child branch, fitting them to linear approximations, and then calculated by:

$$\mathit{\text{angle}}=\text{atan}\left|\frac{k_1-k_2}{1-k_1\times k_2}\right|$$ (7)

where atan represents the calculation of arc tangent angle, k₁ and k₁ represent the fitted slopes of the instantaneous flow directions for two child branches. As this slope-based calculation does not account for angle orientation, we additionally use the length of the selected vessel segments to differentiate between acute angle or obtuse angle:

$$\mathit{\text{angle}}=\left\{\begin{array}{c}\text{atan}\left|\frac{k_1-k_2}{1-k_1\times k_2}\right|\mathit{\text{ if }}{L}_{01}^2+L_{02}^2>L_{12}^2\\ {180}^{°}-\text{atan}\left|\frac{k_1-k_2}{1-k_1\times k_2}\right|\mathit{\text{ if }}{L}_{01}^2+L_{02}^2<L_{12}^2\end{array}\right.$$ (8)

where L₀₁ and L₀₂ are the line distances of the 10^th pixels in two branches from the starting; L₁₂ is the straight-line distance between the two 10^th pixels. The automatically measured branching angles are listed in Fig. 10(c) – (f), with the data demonstrating that our algorithm performs well throughout a range of different angle orientations, including both acute and obtuse branching angles.

Figure 10.

(a) and (b) vessel tortuosity analyses of 2 arterio-arterial anastomosis vessels. Red curve: traced vessel branch. (c) - (f) Analyses of branching angles toward four different orientations. Yellow line: linear fitted instantaneous flow direction from the first 10 pixels of each branch. Yellow dot: starting. Red dot: ending.

The demonstrated methods and additional quantitative parameters of the vasculature pattern including vessel area density [9], vessel skeleton density [9] and vasculature fractal dimension [35] have been integrated into a comprehensive graphical user interface (GUI) vasculature analysis platform, which is written by PyQt5 and Python 3.6, as shown in Figure 11.

Figure 11.

GUI OCT angiography vasculature analysis platform.

5. Discussion and conclusion

We have presented an innovative approach for analyzing OCTA images through automatic vessel diameter quantification and automatic vessel tracing. The diameter was obtained by gradient-guided measurement of the mimimum radial distance between the vessel skeleton and the vessel perimeter. We preferred to calculate the MRD, instead of calculating the curvature of skeleton segments and radially searching/counting pixel numbers, for the following two reasons. (1) calculating curvature is practically more complicated and inconsistent. For instance, the calculated curvature dramatically fluctuates for the segments shown in the blue boxes in Fig. 12 (zoomed subfigure). (2) The radical search may severely overestimate the vessel diameters at the corner or branching regions. For example, in Fig. 12 (zoomed subfigure) the quantified diameter by MRD (yellow) can be overestimated as the length of vessel segment (green line), if radical searching is directly applied.

Figure 12.

The preference of MRD over curvature guided radical searching. Blue boxes: the vessel segments in which the calculation of curvature is inconsistent. Yellow line: the quantified diameter at the branching region by MRD. Green line: the overestimated diameter at the branching region by curvature guided radical searching.

In the diameter quantification, one limitation exists for vessels with 90 degree corners as indicated in Fig. 13(a), where the maximum overestimate can be ~20%. However, 1) this overestimate only occurs within a restricted small region, and even within this region the overstimate decreases rapidly following $(1-\mathit{\text{cos}}({\theta }^{\text{'}}\left)\right)/2\mathit{\text{cos}}\left({\theta }^{\text{'}}\right)$, where ${\theta }^{\text{'}}$ is the off-set angle between the radical line and the horizontal line (or its complement angle if it’s larger than 45°), as schematically shown in (a); 2) in vivo, vessels bend at angles much smaller than 90 degree, in order to maintain smooth blood flow. During vessel tracing, an A* path searching algorithm was applied to the skeletonized vessels. However, as representativly indicated by the blue circles in Fig. 13(b) – (c), the vessel skeletonization may fail at regions where large vessels are branching. Therefore, the traced skeleton pixels should be excluded from analysis if located within a given distance from the bifurcation center. In practice, this distance is designed as the maximum radius of connected vessel branches (i.e. half of the blue dashed lines in (b)). This is analogous to the analysis of conventional microscopic images, where care is generally taken to avoid measurements very close to the branching points [24].

Figure 13.

Limitation analyses of the proposed diameter quantification and vessel tracing. (a) Redisplayed figure same as Fig. 3(i), in which a 90-degree corner is zoomed for error calculation. D: quantified diameter; D_t: true diameter; $\theta$: off-set angle between the radical line and the horizontal line. (b)-(c) Redisplayed figures same as Fig. 8(b) and (c), in which the blue circle indicates a representative region removed from quantification due to poor vessel skeletonization. In (b), the blue dash line indicates the diameter of the circle which is equal to the maximum diameter of connected vessel branches.

In addition to the algorithms, the use of imported OCTA images for vessel characterization has its own limitations. Due to the restricted lateral resolution of the OCT system, the diameters of small capillaries are likely overestimated to some extent. Despite this limitation, we demonstrated that our algorithm quantifies changes in vessel diameter down to a single vessel branch, with a degree of accuracy appropriate for clinical use, as visulized in the abnormal cases in Fig. 5(f) and (l) compared with the healthy one in Fig. 3(i). Other techniqual developments including using optical coherence microscopy [36] or adaptive optics OCT [37] would provide representation of the vasculature with higher resolution, and therefore improve quantification accuracy.

For the use of our algorithm in clinical research, the computation cost/time is another important parameter to consider. Vessel diameter quantification and vessel tracing was performed in Python 3.6 on a Dell Pression T7500 with an Intel Xeon E5507 CPU (1 of 4 cores were used). The time cost of diameter quantification was ~ 3 s for the microfluidic device and ~ 5 s for the brain vasculature; the vessel tracing cost was ~1 s for all branches listed in Fig. 9(a). Further improvements can be achieved by using parallel processing with multiple CPU-cores, or alternatively adopting a GPU, which would reduce the computation time exponentially.

In conclusion, we developed an automated framework to quantitatively characterize vessel diameters down to individual capillaries. We validated our algorithm, and compared it to a clinically well-accepted method, by imaging and quantifying microfluidic flow phantoms with known vascular parameters. Compared with the currently clinically accepted method, our approach quantified vessel parameters with superior consistency, accuracy, and a high tolerance to deviations in vessel orientation. We used an A* algorithm to trace vessel branches, and accordingly calculate the mean diameter of each vessel branch. The heuristic A* algorithm provides an effective and efficient way to get positional information for individual vessels down to the capillary level. Finally, the proposed algorithms were applied to extract vessel parameters from the mouse cerebral vasculature in vivo, including vessel diameter, tortuosity, and branching angle, which agreed well with findings in the literature. Our algorithmic approach is highly adaptable, and can easily be extended to other imaging modalities such as two-photon fluorescence microscopy and photoacoustic microscopy, making it of potential interest for use in multiple clinical settings.