Angiographic velocimetry analysis using contrast dilution gradient method with a 1000 frames per second photon-counting detector

Abstract.

Purpose

Contrast dilution gradient (CDG) analysis is a quantitative method allowing blood velocity estimation using angiographic acquisitions. Currently, CDG is restricted to peripheral vasculature due to the suboptimal temporal resolution of current imaging systems. We investigate extension of CDG methods to the flow conditions of proximal vasculature using 1000 frames per second (fps) high-speed angiographic (HSA) imaging.

Approach

We performed in-vitro HSA acquisitions using the XC-Actaeon detector and 3D-printed patient-specific phantoms. The CDG approach was used for blood velocity estimation expressed as the ratio of temporal and spatial contrast gradients. The gradients were extracted from 2D contrast intensity maps synthesized by plotting intensity profiles along the arterial centerline at each frame. In-vitro results obtained at various frame rates via temporal binning of 1000 fps data were retrospectively compared to computational fluid dynamics (CFD) velocimetry. Full-vessel velocity distributions were estimated at 1000 fps via parallel line expansion of the arterial centerline analysis.

Results

Using HSA, the CDG method displayed agreement with CFD at or above 250 fps [mean-absolute error (MAE): $2.6\pm 6.3\mathrm{cm}/\mathrm{s}$, $p=0.05$]. Relative velocity distributions correlated well with CFD at 1000 fps with universal underapproximation due to effects of pulsatile contrast injection (MAE: 4.3 cm/s).

Conclusions

Using 1000 fps HSA, CDG-based extraction of velocities across large arteries is possible. The method is sensitive to noise; however, image processing techniques and a contrast injection, which adequately fills the vessel assist algorithm accuracy. The CDG method provides high resolution quantitative information for rapidly transient flow patterns observed in arterial circulation.

1. Introduction

Accurate estimation of patient specific hemodynamic conditions has been at the forefront of vascular disease research for many decades.^1,2 The approaches may be broadly divided in four categories: direct measurements with sensors, quantitative imaging (such as Doppler ultrasound), simulations using computational fluid dynamics (CFD) and quantitative angiography. In the neurovascular space, direct measurement of the blood flow conditions with sensors or doppler is not feasible or accurate given the location within the cranial cavity and tortuous access.

With the advancement of computational capabilities, research in CFD applications to assess neurovascular disease severity and treatment optimization has advanced tremendously. The method allows unmatched detail for hemodynamic conditions estimations, including pressure gradients, wall shear stress, and velocities with high spatial and temporal resolution.³ The CFD simulations adhere to the physical principles of fluid flow by solving the Navier-Stokes equation,⁴ a set of non-linear partial differential equations based on the principles of conservation of mass and momentum and can be used to derive velocities and pressures of many varieties of moving fluids within a solution space. This concept becomes useful for neurovascular flow studies when used to solve incompressible blood flow in patient-specific geometries extracted via tomographic angiography (CTA).⁵ Using measured flow conditions, as well as approximated/assumed boundary conditions, these computational models can resolve hemodynamics in many regions of the neurovascular space. There are several limitations to the CFD simulations, including high sensitivity to boundary conditions and intensive computational resources for complex geometries found in vascular lesions treated with endovascular devices. For this reason, although CFD analysis of patient-specific models is invaluable to imaging-based velocimetry research, its lack of real-time feedback limits it from clinical practice.

Angiography-based methods to estimate hemodynamics in the brain have been proposed and used for several years.⁶ These methods include imaging biomarker analysis (such as angiographic parametric imaging) based on the time-density curves (TDCs) of the iodine,^7,8 bolus tracking,⁹ gradient-based methods,¹⁰ etc. These methods can provide great clinical impact since they rely on intra-operative angiograms, thus allowing instantaneous feedback to the interventionalist.

Bolus tracking algorithms compare peak iodinated contrast intensity between two regions and determine the elapsed time between peaks. In this way, these algorithms estimate bulk convective velocity in an attempt to quantify the flow in and surrounding neurovascular pathologies of interest. An early bolus tracking study found success modeling blood flow in human iliac arteries using this method.¹¹ Imaging biomarker analysis utilizes the same plots of contrast intensity over time; however, rather than comparing the peaks of two TDCs, biomarker analysis derives discrete values that semi-quantitatively describe hemodynamic conditions. These values include the bolus arrival time, mean transit time, time to peak contrast intensity, peak height, and area under the TDC. Although these biomarkers do not yield velocity information directly, a retrospective study between pre- and post-treatment endovascular aneurysm treatments concluded these biomarkers could be predictive of treatment success.⁷ Although these methods do provide diagnostic power, velocimetry based upon bolus tracking or TDC analysis fails to account for the pulsatile nature of blood flow, as well as the complex flow patterns that arise in the tortuous arteries of the neurovascular system.

Gradient-based velocimetry methods derive flow characteristics by tracking contrast intensity over individual pixels or a region of interest (ROI), similar to TDC determination. However, these methods calculate the rate of change of contrast intensity (the slope of the TDC) rather than singular value discrete parameters based on the curve. In this way, gradient-based methods allow determination of velocimetric information directly from the image, giving insight into the flow in and surrounding neurovascular pathologies. For example, one gradient-based method, optical flow (OF), which estimates velocity information from an analysis of the contrast gradients in an angiography sequence, has been validated using Doppler ultrasound in vivo,¹² and has been used to assess the hemodynamic environment in and around an arteriovenous malformation (AVM).¹³ This AVM study showed the robustness of the algorithm to tortuous geometries and an expected correlation between pre- and post-treatment flow environments, suggesting its potential use in treatment efficacy assessment in the neurovascular domain. The common limitation of these approaches has been the low temporal resolution of the current commercial detectors which constrains their accuracy in clinical practice for neurointerventionalist applications.

Recent advancements in photon counting detector (PCD) technologies led to development of high-speed/high-resolution detectors which may enable use of quantitative angiography in neurovascular applications. Krebs et al. have recently reported on the application of a PCD system capable of high-speed angiography (HSA) with 1000 fps within the neurovascular space, demonstrating feasibility and qualitative assessment using transient flow patterns in patient-specific 3D printed vascular phantoms.¹⁴

Following the initial qualitative evaluation of HSA, various quantitative methods, including OF and X-ray particle imaging velocimetry (XPIV) were investigated to extract velocimetry data from HSA acquisitions. These velocimetry methods were each compared to the reference standard CFD.

Although OF velocimetry broadly refers to algorithms which convert contrast intensity within a ROI into velocities, these platforms can be generally classified within two categories: polynomial-fit OF algorithms and model-based OF algorithms. Polynomial fit algorithms track the TDCs of individual pixels in the image sequence and typically fit the curves to a fourth-order polynomial function. This fitting is performed to overcome the effects of quantum mottle on velocimetry results, and a fourth-order function fitting is utilized due to the presence of at most three stationary points in the TDC.¹⁵ Once these curves are fit, blood velocity information may be extracted from the slope of the curve. Model-based algorithms make use of principal component analysis (PCA)-derived pulsatile waveforms (based upon several sample waveforms throughout the vessel) to achieve denoising of velocimetry results. These algorithms first calculate gradient-based velocimetric data and smooth high frequency responses using a low-pass filter, then adjust outputs according to the pulsatile flow function derived from PCA. Shields et al. leveraged HSA and an open-source OF algorithm to estimate non-localized convection flow in stenotic phantoms¹⁶ and time-independent localized flow in more generalized vasculature.¹⁷

Although the discussed OF algorithm has proven effective on HSA data, the pre- and post-processing involved in handling quantum mottle makes it time-intensive (on the order of several minutes). Compounding this with the large number of frames and decreased signal to noise ratio (SNR) per frame present in HSA acquisitions, OF methods require more aggressive filtering schemes, which present computational challenges to real-time feedback.

The XPIV refers to a group of algorithms in which particle tracking, rather than gradient analysis, is used to determine velocimetry data.¹⁸ The method attempts to track single particles as they pass through the vessel of interest, marking their location in each frame. Then, provided the frame rate and pixel pitch are known, velocity distributions may be obtained simply by charting the change in particle position (in cm) versus the change in time between frames (in seconds).

The advantage of this method is that it utilizes the centroids of particles to extract velocity information, making it more robust to the effects of quantum mottle than the previously discussed OF methods, and a strong candidate for HSA velocimetry studies.¹⁹ This aside, several conditions must be met in order for correct extraction of velocity distributions. First, the particles must be sufficiently iodinated to be identified above the quantum mottle present in the image sequence to be automatically detected and tracked throughout the acquisition. Second, the particles must be separable from one another, such that the algorithm consistently tracks the same particle throughout the sequence, to avoid misrepresentation of the flow within the vessel. Third, there needs to be a sufficient number of particles to cover flow through the entire vessel lumen. There is a tradeoff between the second and third conditions, requiring careful selection of the number of particles used for velocity calculations. Additionally, injection of iodinated microspheres is not currently practiced clinically, meaning the method is more difficult to integrate into the clinical workflow than the previously discussed OF methods.

One current solution to this problem, called propagation-based phase contrast imaging²⁰ uses blood as a surrogate marker, which presents as a speckle pattern. This method tracks the propagation of blood cells directly, eliminating the need for iodinated microspheres as well as the upper limit on particle density. Although these problems are mitigated, the extreme energy and spatial resolution (on the order of 0.2 microns) required to both separate and resolve individual blood cells from the surrounding blood plasma necessitates careful beam placement and a monoenergetic beam. These conditions make clinical implementation of a robust system difficult, and, as current work has focused on in vitro measurement, the method does not factor the effects of anatomical noise.

While each discussed imaging-based approach shows promise for delivering velocimetry data from HSA acquisitions, algorithm convergence typically necessitates either a loss in spatial resolution via aggressive low-pass filtering, obscuring localized flow detail, or temporal resolution, via temporal averaging, losing the advantage HSA holds over clinically standard digitally subtracted angiography. We seek a velocimetry method, which does not require a tradeoff between temporal and spatial resolution, and which operates at maximum capacity of our HSA detector. This would afford clinicians high resolution velocimetric information in and surrounding vascular pathologies of interest, defining the direct, quantitative impacts of endovascular treatments and further improving upon current qualitative analyses of contrast stagnation during endovascular intervention.²¹

2. Materials and Methods

2.1. Velocimetry Modality

Initially proposed for clinically standard 3 fps acquisitions,¹⁰ contrast dilution gradient (CDG) analysis is a method that relates change in contrast intensities in time and space to blood velocity. The fundamental equation relating changes in contrast intensity to velocity is

$$v=\frac{\frac{\partial C}{\partial t}}{\frac{\partial C}{\partial x}}(\text{diffusion negligible}),$$ (1)

where $\partial C/\partial t$ is the change in contrast in a pixel with respect to time, $\partial C/\partial x$ is the change in contrast with respect to pixel position, and $v$ is the change in position per unit time (pixels per frame). One may then achieve velocimetric data in the form of cm/s via simple dimensional analysis, given the pixel pitch, magnification and frame rate of the detector is known.

2.2. Experimental Setup

HSA imaging was performed in in-vitro simulations using CTA-derived patient-specific 3D printed carotid bifurcation phantom ($144\mathrm{mm}\times 32.5\mathrm{mm}\times 34\mathrm{mm}$, shown in Fig. 1) in a flow loop. The image sequence was acquired using the XC-Actaeon PCD (Direct Conversion Inc./Varex Inc.) at 1000 fps (the maximum temporal resolution of this detector), a $100\mu \mathrm{m}$ pixel pitch, with 70 kVp, 100 mA, and 3 s continuous exposure time, and under 480 mL/min constant flow conditions. The 3-s exposure time was selected to ensure adequate acquisition of undiluted contrast injection, since the X-ray source, Actaeon detector, and programmable syringe injector were not synchronized for this study.

Fig. 1.

Carotid bifurcation model at outlets (a), as shown in AP view (b), and cropped to the FOV of our detector (c).

Data were saved in binary format, with an unsigned integer 16-bit depth, shown in Fig. 2(a). Pre-contrast arrival frames were used to enhance visualization of contrast flow patterns, shown in Fig. 2(b).²² Once subtracted, the image sequence was automatically trimmed (by identifying and removing frames with no signal present) to account for desynchronization between the start of the detector’s acquisition and the X-ray exposure, respectively.

Fig. 2.

Temporally averaged HSA acquisition before (a) and after (b) background subtraction.

The XC-Actaeon detector includes a detector circuitry reset which results in two dropped frames every 100 ms in which there is little to no signal output from the detector. A detector reset correction algorithm automatically identifies these dropped frames via thresholding of global pixel intensities, removes the frames entirely, and performs a weighted interpolation scheme to recover the lost frames.

When using 1000 fps HSA with the XC-Actaeon detector, small temporal fluctuations in X-ray beam fluence cause intra-frame intensity variations. An image intensity correction was implemented to ensure background signal, i.e., uniform air regions around the phantom are at a constant gray level throughout the HSA sequence. The frame-by-frame offset was estimated by subtracting mean background gray value (per frame) from the global mean background gray value (of all frames). Next, a bilateral filtering scheme is used to reduce the effect of quantum mottle in the acquisition while best preserving contrast edge resolution, shown in Fig. 3.²³ We implemented a $10\times 10\mathrm{pixel}$ neighborhood bilateral filter with an allowable color space tolerance of 50 gray values and an allowable distance for color mixing of 50 pixels, and we observed a 1.61:1 CNR improvement between filtered and unfiltered image sequences.

Fig. 3.

Randomly selected background-subtracted frame (a) and corresponding bilaterally filtered frame (b).

2.3. CDG Analysis

For one-dimensional velocity estimation, we developed an algorithm for vessel centerline extraction. This necessitates an automated segmentation of the vessel lumen, accomplished via simple binary thresholding. A skeletonization operation finds the arterial centerline, and a series of rule-based algorithms, which detect separation in the centerline, identify, and separate bifurcating line profiles for individual analysis, shown in Fig. 4. The centerline should approximately capture the convectional flow in the longitudinal direction (along the long axis of the vessel) in which the majority of arterial velocity lies.²⁴

Fig. 4.

Background subtracted HSA acquisition (a) is masked to obtain a centerline (b) in which bifurcations are identified and separated accordingly (c).

With the arterial centerline properly identified, the CDG algorithm plots each centerline profile over the entire angiographic sequence to form a “contrastagram” of line profile pixel intensities versus time. Partial derivatives of this map, taken with respect to time and with respect to line profile pixel position, may then be divided, yielding velocimetric data, as shown in Eq. (1). To reduce the effects of derivative singularities caused by quantum mottle on the partial derivative maps, the partial derivatives are calculated via a $1\times 7\mathrm{pixel}$ kernel spatial or temporal sampling (depending on the partial derivative calculation). This necessitates a dropout of three frames at the beginning and end of the image sequence and a dropout of three pixels on either side of the arterial centerline profile during the calculation of the velocity map; however, the benefit to the output velocity stability is considered adequate to outweigh the loss in data. The results for a single line profile within a carotid bifurcation model under constant flow and non-constant contrast injection conditions are shown in Fig. 5.

Fig. 5.

Line profile (a) is plotted versus frame number (time axis) to form contrastagram (b) before partial derivative maps (c) are calculated with respect to time (top) and line position (bottom).

When dividing partial derivative maps, division by zero and other confounding artifacts create infinite or otherwise undefined entries to the velocity map calculation. By convention, the CDG algorithm sets these values to 0. Another confounding factor of the CDG method is the amplification of quantum mottle effects, which result in partial derivative singularities and subsequent large outliers in the final velocity map. These outliers are identified and replaced with neighboring values using a selective Z-score filtration scheme. The “selective” component of this filter refers to the omission of velocities outside biological ranges when calculating the mean velocity across the centerline of the vessel, and across the acquisition. Using this mean value, outlier velocities are identified via Z-score thresholding ($\mathrm{Z}>3$) and are replaced with a localized $3\times 3\mathrm{pixel}$ neighborhood selective mean, which exclude neighboring outlier values.

2.4. CFD Simulations

A detailed report of the CFD parameters used to generate our ground truth data is summarized in Table 1. CTA-derived patient-specific vascular phantoms studied via HSA were imported into ICEM (ANSYS Inc., Canonsburg, PA), a CFD mesh generation software. For each model, inlet and outlet boundaries were determined via the flow direction both biologically assumed, and which corresponded to the benchtop flow studies using HSA. The robust octree method was used to generate the mesh, forming prism layers at the surface while generating a high-quality tetrahedral mesh throughout the volume of the carotid bifurcation. Global smoothing was performed to improve the quality of the mesh. The final mesh model contained $\sim \mathrm{250,000}$ cells, which was satisfactorily evaluated for grid independence ($<5\%$ difference), as shown in Fig. 6. This resolution balanced mesh fidelity and computational resources while preserving physical flow details. We performed transient, laminar flow simulations by numerically solving the incompressible Navier-Stokes equations in ANSYS Fluent (ANSYS Inc., Canonsburg, Pennsylvania, United States), assuming rigid walls and a Newtonian fluid with density $1060{\mathrm{kg}/\mathrm{m}}^3$ and viscosity of 0.0035 Pa-s. A user-defined function file was used to define a parabolic velocity profile across the inlet cross-section, with a mean velocity of 25 cm/s at the vessel centerline (based on the flow rate set for the pump used in benchtop flow studies). All arterial walls were considered no-slip walls, and zero pressure gradient was assumed at each outlet for simplicity. The time step was set at 1 ms for a total duration of 1 s to provide sufficient temporal resolution to resolve all unsteady motions in the flow, ensuring the convergence criteria of $1{\mathrm{e}}^{-6}$ was achieved at each time step. 3D velocity data was exported every 50 ms and then temporally averaged over the full one second interval for analysis. This was deemed acceptable as the CFD simulation, and corresponding HSA benchtop flow study was operated under constant flow conditions. The resulting velocity data, a temporally averaged constant flow velocity map, were used as a ground truth result for this study.

Table 1.

CFD simulation parameters used to generate the ground truth velocity map.

CFD simulation parameters
Meshing	Model	Carotid bifurcation
	Mesh generation software	ICEM (ANSYS Inc.)
	MESH generation method	Robust Octree
	Mesh fidelity	$\approx \mathrm{250,000}\mathrm{cells}$
Fluid properties	Fluid density	$1060{\mathrm{kg}/\mathrm{m}}^3$
	Fluid viscosity	$0.0035{\mathrm{Pa}}^{-\mathrm{s}}$
	Mean centerline velocity	25 cm/s
	Fluid type	Newtonian
CFD simulation	CFD simulation software	ANSYS fluent
	FLOW	Laminar, transient
	Time step	1.0 ms
	Pressure-velocity coupling	SIMPLE
	Convergence criteria	$1{\mathrm{e}}^{-6}$
	Spatial discretization	Second order
	Transient formulation	First order implicit
Assumptions	Wall	Rigid, no-slip
	Velocity	Constant, parabolic
	Outlets	Zero pressure

Fig. 6.

Mesh independency test used to determine satisfactory grid independence for the CFD simulation.

2.5. Frame Rate Study

To study the effect of the frame rate on velocity calculation via CDG method, simulated angiographic sequences with frame rates of 10, 25, 50, 100, 250, and 500 fps were obtained via retrospective temporal binning of 1000 fps HSA data. This was considered an adequate simulation of each frame rate, as PCDs, such as the XC-Actaeon detector used for HSA studies, sum photon counts over a specified interval to form each frame.²⁵

Acknowledging the tradeoff between image noise and temporal resolution in photon counting systems, varying CDG filtering schema were utilized at each frame rate. At each frame rate, CDG results were obtained with and without each HSA pre-processing step, with and without increased spatiotemporal sampling discussed in Sec. 2.3, and with and without each post-processing step discussed in Sec. 2.3 (with additional testing regarding the optimal number of $\mathrm{Z}$-score filtration iterations and $\mathrm{Z}$-score threshold), totaling 112 different combinations of filters, as shown in Fig. 7.

Fig. 7.

List of processes examined during the frame rate study. Comparisons were obtained with and without each pre-processing step (green) and each post-processing step (orange).

The filtration scheme which most closely agreed with CFD results was identified automatically using a loss function which incorporated both the mean-absolute error (MAE) across the time sequence (between temporally averaged CFD and CDG results), as well as the standard deviation of the velocity map, shown in Eq. (2). Temporal averaging was performed to account for the difference in temporal resolutions between various frame rates; however, the HSA image sequence and corresponding CFD simulation were both acquired under constant flow conditions meaning temporal averaging should have little effect on comparison accuracy. This process was repeated at each frame rate, pairing frame rate to a filtration method suitable for the quantum mottle and temporal resolution present in each acquisition.

$$\mathrm{Loss}=\mathrm{mean}\left(\frac{\mathrm{abs}(\mathrm{CDG}(x)-\mathrm{CFD}(x))}{\mathrm{CFD}(x)}\right)+\frac{\mathrm{std}(\mathrm{CDG})}{\mathrm{mean}(\mathrm{CDG})},$$ (2)

where $\mathrm{CDG}(x)$ refers to the temporally averaged CDG result at pixel $x$ of the centerline, $\mathrm{CFD}(x)$ is the corresponding temporally averaged CFD result, and CDG is the temporally resolved CDG velocity map.

The CFD projection view most closely correspondent with the HSA projection view was obtained via rigid 3D image registration and subsequent depth-integration.²⁶ This process allows a 1:1 geometrical comparison between CFD velocimetry and HSA-derived velocimetry calculations. The arterial centerline-generating algorithm was applied to the 2D CFD projection to obtain constant velocity values at each pixel. The percent difference, as shown in Eq. 3, across each pixel in the CDG centerline velocity map (across the all time steps), relative to the CFD centerline velocity map, was calculated and tabulated for each frame rate

$$\text{Percent difference}=\mathrm{mean}(\frac{\mathrm{CDG}(x)-\mathrm{CFD}(x)}{\mathrm{CFD}(x)}\times 100),$$ (3)

where $\mathrm{CDG}(x)$ refers to the temporally averaged CDG result at pixel $x$ of the centerline and $\mathrm{CFD}(x)$ is the corresponding temporally averaged CFD result.

2.6. Expansion of Contrast Dilution Gradient Analysis to 2D

Once CDG analyses were conducted and validated at the arterial centerline, the process could be expanded to include the entire vessel lumen. This was accomplished by algorithmically generating line profiles parallel to the arterial centerline and constraining them within the segmented vessel geometry. First, the bounds of the vessel are assessed to determine its orientation (horizontal or vertical) in the image frame and then parallel lines radiating from the centerline are generated. This creates a series of parallel lines, as shown in Fig. 8(b), which will be used to assess the distribution of longitudinal flow velocities across the vessel lumen. The line sampling density is determined by a user-specified maximum spacing between adjacent lines (in pixels), as well as the maximum width of the vessel itself. Once the parallel lines are generated, the algorithm rasters through the rows or columns of the vessel segmentation (depending on the orientation) constrains the lines to within the vessel walls and then evenly spaces the lines at the given row or column, allowing sampling of the volume at equal increments across the vessel width [Fig. 8(c)].

Fig. 8.

Algorithmic definition of longitudinal flow lines (b), based on arterial centerline acquisition (a) and constrained to the vessel geometry (c).

Once the line profiles are properly defined, the same CDG analysis discussed in Sec. 2.3 is performed at each line profile, resulting in time-dependent longitudinal flow distributions spanning the 2D geometry of the vessel. Due to the independence of one line’s velocities from the others’, the analysis may be easily parallelized for increased efficiency. Similar to the frame rate analysis described in Sec. 2.5, velocity distribution results obtained from CDG analyses were compared to temporally averaged CFD results. This comparison was performed by co-registering the 2D vessel projections from CFD and HSA acquisitions, then using the line profile distribution from the CDG algorithm to sample temporally averaged CFD results across each line profile. This allows comparison of both velocity magnitudes and spatial distributions between temporally averaged results of each velocimetry method, including MAE [Eq. (4)] and spatial standard deviation calculations [Eq. (5)]. Once again, this temporally averaged comparison was deemed acceptable since both HSA and CFD data were acquired under constant flow conditions. To ensure temporal averaging did not obscure errors in CDG results, the temporal standard deviation (per pixel) was also calculated and reported [Eq. (6)]

$$\mathrm{MAE}=\mathrm{mean}(\mathrm{abs}(\mathrm{CDG}(x)-\mathrm{CFD}(x))),$$ (4)

$$\text{Standard deviation}=\mathrm{std}(\mathrm{CDG})\mathrm{or}\mathrm{std}(\mathrm{CFD}),$$ (5)

$$\text{Standard deviation}=\mathrm{mean}({\mathrm{std}}_t(\mathrm{CDG}(x,t))),$$ (6)

where $\mathrm{CDG}(x)$ is the temporally averaged CDG result at pixel $x$ in the velocity distribution, $\mathrm{CFD}(x)$ is the corresponding temporally averaged CFD result, CDG is the entire temporally averaged CDG velocity map, CFD is the entire temporally averaged CFD velocity map, and $\mathrm{CDG}(x,t)$ is the temporally resolved CDG velocity map.

Based on the enhanced accuracy at 1000 fps determined from the frame rate analysis discussed in Sec. 2.5, these results were only obtained at the full 1000 fps temporal resolution available using the Actaeon detector. It should be noted that the minimum displacement per frame, $100\mu \mathrm{m}$, limits the minimum measurable velocity to 10 cm/s; however, this lower limit is sufficient to capture arterial flows through large vessels.

3. Results

3.1. CDG Analysis of HSA Acquisitions

The algorithm performed well to define and handle the bifurcating geometry of the carotid artery phantom used for benchtop flow simulations, and correctly separated line profiles into two separate analyses, as shown in Fig. 4, removing errors due to inappropriate line profile definitions. It is noteworthy that the algorithm used to segment the mask, generate the arterial centerline, and separate bifurcating regions into separate lines using acquisition-specific information to generate a mask threshold and should be considered robust to other acquisition geometries and image intensity ranges as well.

The best agreement between CFD and CDG results at 1000 fps (percent error = $-1.2\%$) was obtained using the full pre-processing scheme (background subtraction, dark frame correction, bilateral filtration, and intensity equalization), $3\times 3\mathrm{pixel}$ spatiotemporal sampling during derivative calculations, and two rounds of Z-score filtration with a threshold of $\mathrm{Z}=3$. The spatial distributions are given in Fig. 9.

Fig. 9.

Temporally averaged CDG results (a), temporally averaged CFD results (b), and their difference (c).

As frame rate decreases, a general trend of a decrease in the magnitude of velocities emerged, as seen in Table 2. A substantial decrease in velocity values appears between 250 and 100 fps with decreasing performance (relative to CFD) as frame rate approaches 10 fps.

Table 2.

The mean velocity calculated with CDG at different frame rates and the percent difference compared to that calculated by CFD.

Optimized filtration results
Frame rate (fps)	Mean velocity (cm/s)	% Difference (%)
1000	25.2	$-1.2$
500	26.2	2.4
250	27.3	6.7
100	5.6	$-78.0$
50	2.9	$-88.5$
25	1.2	$-95.3$
10	0.4	$-98.5$

3.2. Expansion of CDG Analysis to 2D

At the full spatial and temporal resolution afforded for this study ($256\times 256$ 100-micron pixel pitch, 1500 frames, 1000 fps) and utilizing 8 CPU cores for parallelization, the full time-dependent velocity distribution is obtained within 33 s of HSA image sequence input, including all pre- and post-processing of CDG results. In practice, analysis times were linearly correlated with user-defined line density (sampling rate across the vessel) and inversely proportional to the number of CPU cores used during parallelization. Comparing CDG analyses to temporally averaged CFD (mean velocity: 31.3 cm/s) results, as shown in Fig. 10, we observe spatial agreement of relative velocity values, with underapproximation in CDG velocimetry (mean velocity: 27 cm/s; MAE: 4.3 cm/s).

Fig. 10.

Temporally averaged CDG velocity distribution (a), CFD results (b), and their difference matrix (c).

The standard deviation of the velocity distribution is 6.5 cm/s for the temporally averaged CDG results and 7.9 cm/s for the CFD results with ranges of 25.1 cm/s and 25.0 cm/s, respectively. The average standard deviation across the time sequence of the CDG method (per pixel) is 10.8 cm/s.

4. Discussion

Fundamentally, the CDG method performs contrast velocity calculations which have been previously reported using other methods;^12–16 however, CDG makes no assumptions regarding the puslatility of blood flow or the shape of the TDC. Although this makes the method more sensitive to quantum mottle than the other methods discussed previously, its unrivaled speed and relation to HSA imaging procedures make it a strong candidate for clinical implementation of quantitative blood flow analyses in rapidly transient, large-vessel applications.

The main challenge of this method was reducing the errors in calculating the time and space derivatives from 1000 fps HSA acquisitions due to the quantum mottle. To avoid severe errors, the study proposes bilateral filtration scheme applied exclusively in 2D image space (i.e., single frame), reducing the effects of quantum mottle in the space domain, but neglecting the effects of quantum mottle on the temporal gradient. The time domain gradient is adjusted such that the average background intensity in the image sequence remains constant via the intensity equalization; however, effects of quantum mottle on individual pixels in the time domain are not addressed directly in frame-wise HSA pre-processing.

Gradient smoothing on spatial and temporal derivative maps via $3\times 3$ neighborhood spatiotemporal sampling is utilized during partial derivative calculations; however, this smoothing is highly localized, allowing improved algorithm convergence with minimal cost to temporal and spatial resolution. Post-processing $\mathrm{Z}$-score filtration further reduces the prevalence of quantum mottle-based errors to improve algorithm precision; however, this is performed only on the output velocity map. It is assumed that derivative singularities in the partial derivative maps will be propagated into (via division of spatial and temporal velocity maps) and filtered out of the output velocity map (via $\mathrm{Z}$-score filtration).

For 1000 fps acquisitions the relative distribution of centerline velocities agrees well with CFD distributions. It is additionally noteworthy that CDG centerline velocimetry completion occurs within 8.7 s of image sequence input at 1000 fps, allowing real time feedback regarding high resolution centerline velocity distributions. This benefit is further compounded when considering optimal input into the algorithm, in which a single contrast bolus, rather than a flow-disturbed contrast injection (a non-uniform contrast profile containing multiple edges), is used, allowing more accurate and complete definition of the vessel geometry (due to greater contrast between the vessel and the background) and analysis of the contrast gradients. Provided the injection technique is kept relatively constant between acquisitions,²⁷ this tool may also be useful for comparing endovascular flow environments pre- and post-clinical treatment, giving clinicians valuable insight into the direct and timely physical effects of their work during the time of the intervention.

As shown in Table 2, a clear relationship between decreasing frame rate and increasing error appears to emerge. However, it should be noted that filtering schemes designed for use at 1000 fps were applied to the acquisitions created at each frame rate, potentially leading to over-smoothing of gradients at lower frame rates in which quantum mottle effects are less apparent. It is additionally possible that the temporal binning used to simulate lower frame rates decreased the resolution of the various contrast bolus edges in the injection, leading to misrepresentation of contrast flow and subsequent underapproximation of velocities throughout the vessel.

We demonstrate a method to calculate velocity distributions of longitudinal flow patterns by extending the CDG method used to calculate velocity along the arterial centerline. The discussed method calculates the 2D distribution of the velocity in a plane parallel to the detector, along the centerline in the projected angiogram. In our experiment, the imager was parallel with the vessel, and thus the velocities calculated via the CDG method should be representative of the flow through the phantom. Overall, there was strong agreement between 2D-CDG results, CFD simulations and the flow rates recorded during HSA acquisition of the in-vitro flow loop.

The 2D CDG underapproximation of velocities in the temporally averaged results relative to the CFD results [shown in Fig. 10(c)] is the result of the suboptimal injection. The HSA acquisition used for this study was under constant flow conditions; however, the contrast bolus was disturbed during injection, leading to several narrow contrast edges rather than a single, continuous gradient, a phenomenon captured in Fig. 5(b). In regions with very little contrast, the observed velocity values tended toward 0 cm/s, leading to an artificial deflation of the temporally averaged results. Additionally, the CFD velocities represent the velocity magnitude at each point, regardless of direction. While this should be along the longitudinal direction of flow, a direction approximated by the centerline acquisition algorithm, CDG velocity magnitudes will be deflated should the line profile not be defined perfectly in the direction of longitudinal (maximal) flow, as the contrast gradients present in non-longitudinal flow patterns are much smaller than those in the longitudinal direction.

Additional differences are primarily caused by regions in which the line profiles generated to cover the lumen of the vessel are directed perpendicular to the direction of flow. In these regions, radial velocities, not longitudinal velocities, are calculated and compared to the longitudinal velocities observed in the CFD results in Fig. 9(c), demonstrating the importance of proper directionality and geometry of the line profiles when calculating correct velocity measurements.

It is also worth noting the tradeoff in this method between obtaining results strictly based on the raw acquisition data and the handling of quantum mottle during CDG calculations. Filtering is inherently necessary to reduce the effects of quantum mottle-based derivative singularities; however, extreme filtering may also make incorrect assumptions of the velocity distributions present in the acquisition, leading to slightly erroneous estimations. The standard deviation of velocities per pixel is deemed acceptable considering both the fluctuation in velocity values in regions and times lacking proper contrast gradients, and the quantum mottle in the HSA acquisition.

It is additionally noteworthy that CDG calculations at the line density shown in Fig. 8 occur in 33 s. Compared to the CFD simulation workflow (including mesh generation, proper boundary condition definition, and flow field solution) for a large-vessel geometry under constant flow conditions, the CDG method is several orders of magnitude faster and significantly less computationally expensive. This allows the method to be used to derive velocity distributions in pseudo-real time, posing a significant advantage of the CDG method over the comparable results generated from CFD.

The results of this study warrant more rigorous, quantitative comparisons of velocity fields generated from CFD (more flow patterns and geometries), as well as other velocimetry methods implemented previously. Such comparisons, which would include accuracy to ground truth flow fields, computing times, temporal and spatial resolution of results, each of which with an associated statistical analysis, would assist understanding of both the advantages and limitations of the CDG method.

With this promising feasibility study, future work aims to remove the directional dependence of the CDG calculations to boost the robustness of the velocimetry method. This should allow resolution of velocity fields in more complex vasculature, such as the circle of Willis in the neurovascular domain. In addition, the high temporal fidelity of the method opens the door to cardiac applications in which the rapid deformity of coronary arteries and rapid flows through the aorta bar standard imaging modalities from accurate flow velocimetry. Finally, determination of wall shear stresses, a factor determining platelet and leukocyte activation leading to atherosclerosis, from CDG may be useful in determining risk of stroke or aneurysm rupture.

5. Conclusions

This study presents a velocimetry method using 1000 fps HSA and CDG analysis to resolve flow details at both the arterial centerline, and across the arterial diameter, of larger neurovasculature. This is a significant improvement over previous methods, which were constrained to resolve velocities in small and straight vessel geometries due to lack of temporal resolution. The method is shown to be functional at several temporal resolutions, provided they are sufficiently high to fully capture the contrast gradients present at the arterial centerline (over 250 fps). Overall, the temporal information afforded by the 1000 fps frame rate of the Actaeon PCD are invaluable to both the clinical observer and to the CDG method itself, providing the most accurate velocimetry results observed within this study.

The described algorithm is capable of generating endovascular flow details in real time, affording clinicians rapid feedback on the effects of endovascular treatments on bulk arterial flow patterns solely from HSA data.

Biographies

Kyle A. Williams received his BS degree in biomedical engineering from SUNY University at Buffalo in 2020 and is pursuing a PhD in biomedical engineering. Currently, he is working as a research assistant at SUNY University at Buffalo (Department of Biomedical Engineering). His research interests include imaging informatics, computed tomography reconstruction, machine learning, and computer aided diagnostics.

Biographies of the other authors are not available.

Disclosures

The authors of this manuscript have no financial interests to disclose.