A Noninvasive Assessment of Flow Based on Contrast Dispersion in Computed Tomography Angiography: A Computational and Experimental Phantom Study

Abstract

Transluminal attenuation gradient (TAG), defined as the gradient of the contrast agent attenuation drop along the vessel, is an imaging biomarker that indicates stenosis in the coronary arteries. The transluminal attenuation flow encoding (TAFE) equation is a theoretical platform that quantifies blood flow in each coronary artery based on computed tomography angiography (CTA) imaging. This formulation couples TAG (i.e., contrast dispersion along the vessel) with fluid dynamics. However, this theoretical concept has never been validated experimentally. The aim of this proof-of-principle phantom study is to validate TAFE based on CTA imaging. Dynamic CTA images were acquired every 0.5 s. The average TAFE estimated flow rates were compared against four predefined pump values in a straight (20, 25, 30, 35, and 40 ml/min) and a tapered phantom (25, 35, 45, and 55 ml/min). Using the TAFE formulation with no correction, the flow rates were underestimated by 33% and 81% in the straight and tapered phantoms, respectively. The TAFE formulation was corrected for imaging artifacts focusing on partial volume averaging and radial variation of contrast enhancement. After corrections, the flow rates estimated in the straight and tapered phantoms had an excellent Pearson correlation of r = 0.99 and 0.87 (p < 0.001), respectively, with only a 0.6%±0.2 mL/min difference in estimation of the flow rate. In this proof-of-concept phantom study, we corrected the TAFE formulation and showed a good agreement with the actual pump values. Future clinical validations are needed for feasibility of TAFE in clinical use.

Introduction

Contrast-enhanced cardiac computed tomography angiography (CTA) is an established method for noninvasive anatomic assessment of coronary stenosis [1]. However, it has limited specificity for diagnosing lesions with significant hemodynamic and physiologic obstruction [2]. Novel techniques have been developed recently to address this limitation including noninvasive fractional flow reserve (FFR) delivered from CT (FFR_CT) and transluminal attenuation flow encoding equation (TAFE) [3–5] to improve the diagnostic performance of coronary CTA [6,7] using computational fluid dynamics.

Transluminal attenuation flow encoding equation is a recent contrast gradient-based method developed for quantifying the flow velocity and flowrate in a coronary vessel using transluminal attenuation gradient (TAG) as well as the time density curve acquired while bolus tracking (Fig. 1(a)), both acquired from contrast enhanced coronary CTA imaging. Other studies have also used contrast gradients based on CT and time-of-flight techniques and shown a potential for blood flow velocimetry estimation. For example, Korporaal et al. [8] introduced a new theoretical framework describing the relationship between the blood velocity, CT acquisition velocity, and contrast enhancement in CT images. Prevrhal et al. [9] derived the blood velocity from temporal changes in the sinogram. Barfett et al. [10,11] and Mahnken et al. [12] calculated the intravascular blood velocity from images that were acquired with large detectors providing a large enough time shift between the blood arriving at both ends of the detectors.

Fig. 1.

(a) Example of AIF at the descending aorta acquired in a CT acquisition of a representative patient (b) Example of AIF at “ostium” of the phantom sampled at every 2 s (c) Attenuation intensity (or contrast concentration) versus transluminal distance in a coronary artery of the same representative patient at the peak time of AIF (d) Attenuation intensity (or contrast concentration) versus cumulative volume at the peak time point of the slope figures in (c) and (d) are known to be TAG and volumetric transluminal attenuation gradient (TAG_V), respectively

Transluminal attenuation gradient (Fig. 1(b)) is defined as the linear regression coefficient between luminal attenuation and axial distance from the coronary ostium. It has been shown to have a high diagnostic performance using multidetector row CT [13–15] as well as dual source CT [16]. Furthermore, TAFE uses this information to provide a direct measurement of flowrate in coronary arteries. The flow physics underlying this method has been established and validated using computational fluid dynamics [4]. Furthermore, preliminary retrospective estimation of coronary flow rates using existing TAFE cardiac CTA images for a cohort of patients have also shown promising results [17] indicating a high degree of correlation between expected and estimated (via TAFE) flow rates. However, these studies as well as more recent clinical and animal studies [3] indicate that the absolute values of flow velocity obtained using TAFE underestimate the expected values.

Therefore, to further investigate the underestimation of TAFE for absolute flow rates, the objective of this study was to (a) design a CT compatible phantom study to assess the accuracy of TAFE based on velocity estimates in a controlled environment that incorporates CTA imaging and reconstruction-related features, (b) determine the key imaging factors absent in computational studies such as imaging artifacts (i.e., partial volume averaging, beam hardening effect) and reconstruction kernels that led to underestimation of TAFE-based flow velocity prediction, and (c) use experiments to explore corrections for this underestimation.

The previously derived TAFE formulation, together with the corrections defined in this study, will allow for a noninvasive method to accurately quantify flowrate and velocity in any parent coronary artery and its branches noninvasively. The accurate flowrate and velocity information in this study provide the cardiologists and radiologists a better understanding of the patient's physiologic condition for better diagnosis and optimum treatment options.

Materials and Methods

Theory and Derivation of Transluminal Attenuation Flow Encoding Equation.

The complete derivation of TAFE for a single vessel and branched network is described in detail previously [4]. Briefly, the analytical formulation of TAFE comes from two main equations: the Navier–Stokes for incompressible flow modeling the blood (Eq. (1)) and convection-diffusion equation to model the contrast dispersion in the arteries (Eq. (2))

$$\frac{\partial \mathit{U}}{\partial t}+\left(\mathit{U}·\mathbf{\nabla }\right)\mathbf{U}+\frac{\mathbf{\nabla }\mathrm{P}}{\mathrm{\rho }}=\nu {\mathbf{\nabla }}^2\mathit{U},\hspace{1em}\mathbf{\nabla }·\mathit{U}=0$$ (1)

$$\frac{\partial C}{\partial t}+\left(\mathit{U}·\mathbf{\nabla }\right)C=D{\nabla }^{2}C$$ (2)

Assuming unidirectional flow and minimal radial variation of contrast at each cross section in the tube/vessel and dominance of convection forces (as compared to diffusion), the mean flowrate, $Q_{\mathrm{TAFE}}$, through the vessel can be estimated using the TAFE equation (Eq. (3))

$$Q_{\mathrm{TAFE}}=\frac{\frac{\partial C}{\partial t}}{\frac{-\partial C}{\partial V_{\mathrm{cum}}\left(s\right)}}$$ (3)

where V_cum is the cumulative volume of the vessel at the axial location, s, and can be found by $V_{\mathrm{cum}}\left(s\right)=\sum A\left(s\right)\times s$ with A being the cross-sectional areas at each axial location down the vessel. The numerator, $\frac{\partial C}{\partial t}$, can be found by taking an instantaneous derivative of the time-density curve (TDC) or the arterial input function (AIF) of contrast at the ostium of the vessel in units of [HU/s] (Fig. 1(c)). $\frac{\partial C}{\partial V_{\mathrm{cum}}}\hspace{0.17em}$is the same as volumetric TAG $(\mathrm{TC}{\mathrm{G}}_V\hspace{0.17em}[\mathrm{HU}/\mathrm{c}{\mathrm{m}}^3\left]\right)$ and can be calculated by taking the derivative of contrast with respect to the cumulative volume down the vessel of interest corresponding to the same time point that the time derivative is taken (Fig. 1(d)).

Experimental Setup.

To validate the analytical formulation described in the previous section and demonstrate that it is feasible to measure blood flowrate and velocity using TAFE, phantom measurements were performed with a custom-built CT-compatible experimental phantom setup (Fig. 2(a)).

Fig. 2.

(a) Illustrative overview of the phantom experiment setup. The contrast is infused into the system by first flowing through the mixing chamber located on a magnetic stirrer to mimic the chambers of left and right ventricle and the fully mixed solution enters the phantom and finally to the waste container. (b) The experimental set used a Toshiba Aquilon One, 320 detector CT scanner, a three-dimensional (3D) printed tapered phantom, a syringe pump, mixing chamber and the magnetic stirrer.

We used a Harvard Apparatus PHD 2000 Programmable Syringe Pump with a range of 0.0001 μL/h to 1.32 × 10⁷ μL/h with an accuracy of 1% of the flowrate specified. A syringe with a 250-ml capacity was used to inject the glycerin-water (G-W) solution into the mixing chamber. The mixing chamber mimics the total average volume of the left [18] and right [19] ventricles during a cardiac phase where the contrast gets mixed with blood before entering the aortic root. A heated magnetic stirrer (kept at 37 °C) including an “X” shape magnetic propeller was used for proper mixing and temperature (Fig. 2(b)).

The phantom was then scanned by a 320-detector-row, Aquilion^® ONE, a dynamic volume CT system scanner (Canon Medical Systems Corporation, Otawara, Japan) with a gantry rotation speed of 275 milli-seconds. Dynamic imaging was performed using the following parameters: gantry rotation time = 275 ms, detector collimation = 0.5 mm × 320, tube voltage = 120 kV, tube current = 100 mA, and scan time= 99 s.

Omnipaque 240 mg (GE Healthcare, Chicago, IL) was used as the contrast agent. To match blood's viscosity at 37 °C (4 × 10⁻³ [Pa·s]), a mixture of 30/70% glycerin/water solution by volume was prepared as used in previous phantom studies. In addition, to have a similar density as the contrast agent (ρ ∼ 1200 kg/m³), for perfect mixing, sea salt was added to the G-W solution.

Since the phantom setup was a closed system (without any outlets or leaking before flow enters the phantom) and there are no branches to the phantom, based on the mass conservation law, the pump's infusion rate (true pump flow rate) is the same as the flowrate through the phantom.

In order to custom-build the phantoms with specific size and shape, we used a 3D printing machine with ABS (Acrylonitrile butadiene styrene) as this material had a desirable negative attenuation of −40 HU. Two distinct but simple phantoms were employed in this study: a straight, single vessel phantom with an internal diameter of 0.4 cm and a length of 14 cm, and a tapered phantom of the same length and same inlet diameter but with a lumen taper of 8.1 deg and an exit diameter of 0.2 cm. The diameter, length, and tapering angle of the phantoms are chosen to have similar dimensions as the major coronary arteries [20,21]. The details of the experiment steps are listed in the Supplemental Materials on the ASME Digital Collection.

Image Reconstruction.

Dynamic volume images were reconstructed every 500 ms and slice thickness of 0.5 mm, with no overlap. The convolution kernel used for the image reconstructions was FC05 with sharper edge enhancement, which is used clinically to interpret coronary CTA studies. CT scanners use a variety of filtering kernels to refine and denoise the raw image data that emerges from the scan. Therefore, this filtering process can introduce imaging artifacts, reduce effective resolution, and create errors in the TAFE-based flow velocity estimation. Our studies suggest that these filtering kernels might increase the effective pixel size by many folds, and we did a series of studies to examine this effect. The field of view employed here generates a pixel size of 0.327 mm, so we therefore have approximately 18 × 18 pixels across the lumen of the 6 mm, nontapered silicon phantom. Figure 3 shows the application of 6 different filter kernels to image reconstruction. It is clear that at a pixel resolution of 18 pixels across the diameter, the contrast concentration through the lumen is highly affected by the image reconstruction process. Based on these data, it was decided that the FC05 AIDR3D reconstruction kernel provides the best delineation of the lumen with the least noise artifact and hence we used this kernel thereafter for our phantom study reconstruction.

Fig. 3.

Effect of filter kernel on attenuation. (a)–(f) Contours of contrast concentration at a representative cross section in the phantom for six different filter kernels. (g) Contrast attenuation profile in HU for various kernels. (h)–(j) are zoomed in versions of Figure (g) to show the differences between the various reconstruction kernels used.

Image Analysis and Phantom Segmentation.

The dynamic volume image analysis was conducted using a custom written script (matlab R2015a) and consisted of two steps: (1) find the averaged contrast attenuation at each cross section of the phantom in the axial direction for each time point resulting in transluminal contrast gradient curve and (2) define the average contrast attenuation at the ostium of the phantom for each time point resulting in the time density curve (TDC).

The center point of the phantom (x_c and y_c) is determined by finding the “center of mass” based on pixel attenuation level by Eq. (4) where x_ij and y_ij are the locations of each pixel in the X- and Y-plane and C_ij is the attenuation level of contrast at that location (flowchart with steps involved shown in Fig. 4(a))

$$x_c=\frac{\sum x_{ij}{C}_{ij}}{\sum C_{ij}},\hspace{1em}{y}_c=\frac{\sum y_{ij}{C}_{ij}}{\sum C_{ij}}$$ (4)

Fig. 4.

Flowchart of custom-written algorithm for segmentation of the phantom for each cross section in a dynamic scan. (a) Original phantom cross section with the wall included (b). Background and wall level sets are assigned to have zero value and the center point is calculated shown with the star (c). Region of interest is chosen to be a 60 × 60 pixel from the center point (d). Edge of the lumen is defined using the Canny edge-detection method (e). The lumen of phantom is filled based on the edge detected (f). The filled area is corrected using active contour method (g).

With the center point determined (Fig. 4(c)) via Eq. (4), the edges of the lumen in the phantom are found after sampling the original image by a new cropped region of interest of 60 × 60 pixels via the Canny edge detection methodology [22]. Since our phantom was 3D printed, the wall is intrinsically rough with roughness enhanced by formation of small bubbles during the curing process. The use of the Canny method was therefore appropriate to not confuse imaging noise with actual model roughness. Subsequently, every pixel inside the detected edge was filled (assigned to have the value 1) by a simplified one-dimensional ray tracing method where an image is generated by tracing the path of light through pixels in an image plane [23]. Next, the lumen area was corrected using the built-in function “activecontour” in matlab with a predefined mask and 50 iterations. The active contour method used for segmentation is Chan and Vese's region-based energy model described in Ref. [24] (Fig. 4(g)).

TAFE Analysis.

TDC was extracted by estimating the mean attenuation level of the segmented cross section at the ostium of the phantom at each time point. Figure 1(c) is a representative TDC measured at the ostium after segmentation.

Correspondingly, the volumetric transluminal contrast gradient was extracted by plotting the mean concentration (attenuation level) of contrast at each cross section down the length of the phantom against the cumulative volume, $V_{\mathrm{cum}}$. The instantaneous derivative of contrast with respect to time, $\frac{\partial C}{\partial t}$, can be calculated by employing backward Euler finite difference method. However, to reduce noise in the data, $\frac{\partial C}{\partial t}\hspace{0.17em}$is determined by finding piecewise linear regression of the TDC at every 5 s. Similarly, volumetric TAG, $\mathrm{TC}{\mathrm{G}}_V$, was obtained by a linear fit (least squares) through the entire length of the phantom. As the contrast's mean concentration was fairly noisy, the outliers were removed for the data points greater than 1.5 times the standard deviation $(C_{ij}>1.5\sigma )$ (Fig. 5).

Fig. 5.

Transluminal contrast gradient curve for five consecutive time points where on the linear regression are shown in dashed lines including both original and the fit excluding the outlier data points shown with star markers

Corrections for Flow Nonuniformity.

As mentioned in the section TAFE Analysis, an assumption is made in the TAFE analysis that the radial variation of the contrast at any axial location within the phantom is minimal, i.e., the contrast has a uniform profile across any cross section. However, our observation showed that there is an apparent radial variation in the profile (Figs. 6(a) and 6(b)). There are two explanations for this observed phenomenon: the first is based on imaging artifacts, which include partial volume averaging as well as any “filtering” of the data inherent in the image reconstruction. The second mechanisms for the appearance of the radial contrast concentration variations may be related to flow physics inherent to the experiment. It is expected that flow from the mixing chamber will develop a parabolic profile as it travels to the phantom. Thus, flow in the center of the phantom will move faster than the flow near the walls, and at any given cross section, the contrast agent particles near the wall are associated with an earlier time in the AIF than the contrast agent particles in the center of the channel. Since the AIF is increasing over time, the contrast near the wall will therefore have a lower concentration than the contrast near the center. This will lower the average contrast concentration in a given cross section and modify the TAG in the phantom, and in turn, will affect the estimation of flowrate via TAFE (Fig. 6(c)).

Fig. 6.

(a) Contrast radial contour variation where the phantom wall and the air surrounding it is at 0 or negative. (b) Contrast attenuation in HU versus the radius of the phantom wall for three different cross section. The lumen radial profile is between radius values of 0 and 4. (c) Schematic of the relationship between the contrast agent dispersion in temporal and special domain. The time profile of contrast agent concentration C(t) at the vessel's ostium (phantom's inlet) is shown on the left. At time point “t,” particle (1) arrives at the peak and particle (2) has a time delay (or time lag) denoted by “t_diff.” This time delay depends on the particle's velocity(v) where t_diff = x/v. Therefore, as shown on the right side, the centerline particle (1) (faster velocity) has smaller time delay and higher concentration compared to the slower particle (2) at the walls with lower concentration.

To understand this nonuniform flow effect, we consider a simple analysis of flow and contrast through the experiment: let $C_o\left(r,t\right)=C_{\max }f\left(r\right)\frac{t}{T}$ be the concentration of contrast at the ostium of the phantom, where C_max is the maximum concentration level at the cross section, and T is the total run time. Then, assuming a general velocity profile of $u\left(r\right)=U_{\max }g\left(r\right)$, the concentration of the contrast at some location, s, downstream of phantom can be found as

$$C\left(r,t,s\right)=C_{\max }f\left(r\right)\frac{1}{T}(t-\frac{s}{u\left(r\right)})$$ (5)

With some mathematical manipulation (shown in Supplemental Materials on the ASME Digital Collection), we find that a correction factor, k, due to flow nonuniformity results in

$$k=\frac{\int g\left(r\right)dA}{\int f\left(r\right)dA}·\int \left(\frac{f\left(r\right)}{g\left(r\right)}\right)\frac{dA}{A}$$ (6)

Thus, the TAFE formulation in Eq. (3) is modified to have the form of

$$Q_{\mathrm{TAF}{\mathrm{E}}_{k-\mathrm{corr}}}=k·\frac{\frac{\partial C}{\partial t}}{\frac{-\partial C}{\partial V_{\mathrm{cum}}\left(s\right)}}$$ (7)

Since $f\left(r\right)$ is not known, computational fluid dynamics (CFD) simulations have been employed to solve for k. The details of the CFD simulations can be found in the Supplemental Materials on the ASME Digital Collection. In summary, the correction factor k is calculated as $k=\frac{Q_{\mathrm{sim}}}{Q_{\mathrm{TAFE}}}$ where $Q_{\mathrm{sim}}$ and $Q_{\mathrm{TAFE}}$ are the mean flow rate of the phantom calculated in CFD simulation and estimated by TAFE analysis, respectively. Similar to the pattern observed in the phantom experiments, Figs. S1 and S2 available in the Supplemental Materials illustrate the radial variation of contrast in the simulations of a tapered and straight phantom. It is important to note that the molecular diffusivity of contrast is not precisely known for the iodinated contrast agent (Omnipaque 240 mg). Therefore, to model the contrast flow, we examined the two upper and lower ranges of molecular diffusivity (D). This is reflected in Schmidt's number ( $\mathrm{Sc}=\frac{\nu }{D},\hspace{0.17em}\nu :$ kinematic viscosity, $\hspace{0.17em}D$: molecular diffusivity), a nondimensional number characterizing the flow's momentum and mass diffusion convection.

Here, at three different locations of s = 22, 27, and 32 (corresponding to locations at s = 2, 7, and 12 from the entrance of the phantom), the radial profiles with Sc = 1000, because of the lower molecular diffusivity, show a higher radial variation compared to that of the cases with Sc = 1 in the tapered phantom (Fig. S2 available in the Supplemental Materials).

Corrections for Imaging Artifacts.

Errors related to the finite resolution of the imaging such as partial averaging volume and filtering could particularly affect the mean attenuation across the phantom. Since separation and differentiation between these artifacts is not trivial, a general correction factor is calculated here to account for imaging artifacts. To improve upon our estimation and correct for the imaging artifacts, a premixed solution with contrast with ratio of 1–10 is prepared and inserted into the tapered phantom with the two ends affixed. With this setup, there is no flow in and out the phantom and theoretically, there should be no TAG along the vessel. However, in the tapered phantom only, there is an apparent drop in the contrast attenuation along the vessel, which can be partly explained by partial averaging volume effects. To correct for this drop, the cross-sectional average concentration, $C_{CT}$, is divided by a “reference” concentration, $C_{\mathrm{ref}}$, called $\alpha \left(A\right)$ (Eq. (8a)) and, is plotted against the area at each cross section (Fig. 7(a)). The resultant data, ${\alpha }_{\mathrm{fit}}$, are then fitted to a polynomial of the form in Eq. (8b), where A is the area and coefficients a, b, c, and d are determined to be −0.1500, 0.8578, −0.1455, and 0.0015, respectively

$$\alpha \left(A\right)=\frac{C_{\mathrm{CT}}}{C_{\mathrm{ref}}}$$ (8a)

$${\alpha }_{\mathrm{fit}}\left(A\right)=a+\frac{b}{\sqrt{A}}+\frac{c}{A}+\frac{d}{A^2}$$ (8b)

Fig. 7.

(a) Correction for imaging artifacts including partial volume averaging where averaged contrast concentration is plotted against the area at each cross section. The solid line is the fitted polynomial in Eq. (8b). Cross-sectional averaged concentration of contrast agent along the axial direction of a premixed tapered phantom experiment (b) and in a representative phantom experiment with flow (c). The dot markers at the bottom of the figure show a drop in concentration despite the stationary flow in the phantom. The dot markers at the top of the figure are the corrected using the α-correction factor determined from (a).

The reference concentration is measured with a premixed ratio of 1–10 of glycerin–water (G–W) solution to water in a test tube with a diameter of 3 cm with no known imaging artifacts. Subsequently, the concentration for each cross section is corrected employing Eq. (9), where $A_{\mathrm{CT}}\hspace{0.17em}\mathrm{and}\hspace{0.17em}{C}_{\mathrm{corr}}$ are the measured CT area and the corrected concentration, respectively. The corrected concentration results in the top of the figure (red dots) in Fig. 7(b) for the premixed tapered phantom where the slope of the corrected data is expectedly very close to zero. Similarly, a sample case with flow has been corrected for the “artificial” TAG introduced by imaging the artifact (Fig. 7(c)).

$$C_{\mathrm{corr}}\left(A\right)={\alpha }_{\mathrm{fit}}\left(A_{\mathrm{CT}}\right)\times C_{\mathrm{CT}}$$ (9)

Results

Tables 1 and 2 list the original values and the corresponding corrections for the straight and tapered phantom, respectively. The results reported here are analyzed via the simple thresholding segmentation methodology explained in Image Analysis and Phantom Segmentation section. The original results taken from phantom with no corrections compared with the true pump flow rates of Q = 20, 25, 30, 35, and 40 mL/min for the straight phantom and Q = 25, 35, 45, and 55 mL/min for the tapered phantom are shown in Fig. 8(a) by blue diamonds. For the straight phantom, the TAFE prediction shows a linear estimate with true pump velocity but the rate of rise of the estimated pump velocity is only 67% that of the true pump velocity. Consequently, the prediction becomes worse with increasing pump velocity. The estimation for the tapered phantom shows similar trends although the underprediction is exaggerated. The rate of rise of the estimated flow rate is only about 11% of the true rise, and this leads to an estimated flowrate that barely increases with the true flow rate.

Table 1.

List of flow rates estimated by TAFE and the corresponding corrections for the straight phantom

$Q_{\text{pump}}(\text{mL}/\text{min})$	$Q_{\text{TAFE}}(\text{mL}/\min )$	$Q_{{\text{TAFE}}_{k-\text{corr}}}$
20	13.95	19.52
25	17.03	23.84
30	19.65	27.51
35	24.31	34.04
40	33.37	46.71

$Q_{\mathrm{TAF}{\mathrm{E}}_{k-\mathrm{corr}}}=k·\frac{\frac{\partial C}{\partial t}}{\frac{-\partial C}{\partial V_{\mathrm{cum}}\left(s\right)}}$ is the corrected TAFE flow rate by the flow rate correction coefficient (k-factor).

Table 2.

List of flow rates estimated by TAFE and the corresponding corrections for the tapered phantom

$Q_{\text{pump}}(\text{mL}/\text{min})$	$Q_{\text{TAFE}}(\text{mL}/\min )$	$Q_{{\text{TAFE}}_{\alpha -\text{corr}}}$	$Q_{{\text{TAFE}}_{k+\alpha -\text{corr}}}$
25	4.28	17.12	23.95
35	4.15	15.86	22.24
45	4.86	21.90	30.66
55	7.83	40.08	56.11

Fig. 8.

Comparison of TAFE estimated flowrate with the true pump flow rate for straight (a) and tapered (b) phantoms. The predictions significantly improve after the corrections are applied.

We first apply the “k-correction,” i.e., the correction due to nonuniform flow, to these phantom data to both straight and tapered phantoms. Based on our analytical modeling with assuming a parabolic profile for both the flow velocity and contrast, we calculate the k-factor to have a range of 1.3–1.6. Here we used an average value of k = 1.4 for both straight and tapered phantom through all the flow rates and we use the formula in Eq. (7) to correct the TAFE estimation of the flowrate (Fig. 8(a)).

For the tapered phantom, the flow profile correction (k-factor) does lead to an improvement with a resulting slope of 0.74 and an average underprediction of 6.2 mL/min. When the imaging artifact correction (α-factor) is applied to the flow profile corrected (k-factor) TAFE prediction for the tapered phantom (Fig. 8(b)), the predicted slope became 1.04, close to unity and the mean error in the prediction 8.71 mL/min.

Discussion

Aligned with the objectives set in the Introduction, we first designed a CT compatible phantom experiment to assess the accuracy of TAFE. We validated a noninvasive analytical equation (TAFE) to estimate the flowrate in straight and tapered phantoms using the information on contrast agent dispersion in CTA imaging. However, the TAFE formulation underestimated the measured flowrate in both straight and tapered phantoms. Second, through experimental designs and computational modeling, we determined and applied a correction for a nonuniform radial contrast profile as well as imaging artifacts including partial volume averaging effect (in the tapered phantom only). Finally, we showed an excellent correlation between the measured and estimated flow rate with a Pearson correlation of r = 0.99 and 0.87 (p < 0.001) in the straight and tapered phantoms, respectively.

After the flow profile-correction was applied, the straight phantom results became very close to the dashed 45-deg angle line with the slope of 0.94 and an average underprediction of 0.2629 mL/min. This implies that the flow development effects are indeed important in the phantom experiment. We also noted that the tapered phantom does not show as strong of a linear correlation with true pump rate (R² value of 0.74) as the straight phantom (R² value of 0.98) pointing to other errors/uncertainties in the experiment. However, overall, within the limitations of the experimental setup, the current experiments and analysis demonstrate that there are systematic errors in the TAFE prediction due to inherent flow physics and imaging artifacts, and that there is a possibility of correcting for these errors.

This proof-of-concept study aligns with previous studies in the use of contrast agent information in development of tools for measuring flow rates noninvasively [8,11]. However, the pervious methods focused on the time–density curve and time of flight, whereas this study couples the time profile of contrast agent with its spatial profile (or TAG) to fully utilize the information needed to estimate the flow rate. In addition, this formulation can be used to calculate the flowrate in individual visible vessels in CTA as well as a total flow rate in major arteries.

The correction factors defined in this study are twofold: (a) flow profile correction coefficient (k-factor) can be calculated computationally a priori and will be applied to the TAFE equation for each vessel diameter at the ostium (b) the imaging artifact correction coefficient (α-factor) can also be automatically calculated and applied as a form of calibration after each image acquisition where ${\alpha }_{\mathrm{fit}}\left(A\right)$ will be determined based on the cross-sectional area along the vessel. These two correction coefficients will be then applied to the original TAFE equation and the final correct flow rate will be reported to the clinicians.

By applying these correction coefficients, a more precise assessment of physiologic flow could be performed. There are a number of clinical applications that may benefit from blood flow rate assessment. For example, CTA-based flowrate assessment can be used where flowrate calculations are desired. This might include congenital heart disease [25] or coronary arteries in patients with defibrillators where functional information about the blood flow directly from CTA in coronary arteries can be assessed and there will be no need for additional examinations. In addition, noninvasive physiological information about flow limiting lesions and classifying the physiological significance of a stenosis is also highly desirable. For example, FFR is the gold standard method to assessing flow-limiting lesions where the flow pressure information distal and proximal to the stenosis is used [7]. Therefore, flow rate measurements in vessels with stenosis may provide additional information to the pressure in determining physiological significance of a lesion.

It is important to note that the current proof-of-concept study only relies on one phantom diameter (straight and tapered) with various physiological average flow rates. However, for this methodology to be widely used and available in a clinical setting, this study will need to be repeated and the TAFE formulation will need to be corrected for (a) anatomical variation (i.e., different diameter sizes), (b) different flow rate conditions including input average flow values as well as flow pulsatility, and (c) different imaging resolution depending on the CT machine. The final flowrate value will then be corrected by a set of look-up tables in the postprocessing step when imaging is complete.

Limitations.

There are a number of limitations of the experiments to note. First, we employ steady flow rates through the phantom whereas physiological flow through the artery is pulsatile. Computational modeling has shown that TAFE might be applied to pulsatile flow with reduced accuracy, but this was not examined in the experiments due to the complexity of setting up a CT compatible pulsatile flow loop. Therefore, our experiment did not consider a Womersley flow and instead assumed that we have a parabolic flow. This assumption may have an effect on the value we derived the flow uniformity factor. For vessels such as the coronary artery with smaller Womersley number (∼2.1) as shown in previous literature [21], assuming a parabolic profile may be a sufficient assumption in this study ( $\alpha =D{\left(\frac{\omega \rho }{\mu }\right)}^{\frac{1}{2}}),\hspace{0.17em}$where $\alpha$ is the Womersley number, D is the diameter, and $\omega$ is the angular frequency). However, for higher Womersley number flows, the unsteady flow effects will have to be further studied and the TAFE formulation will need to be adjusted for it. The phantom also does not include motion artifacts that appear in coronary CT angiography imaging of patients. Since the coronary vessels are embedded in the moving myocardium, the finite temporal resolution of the scanning procedure can introduce motion artifacts that are not modeled here. In addition, when the flow-profile correction is applied, we take a universal k value of 1.4 for all flow rates and tapered phantom. However, the k-factor depends on the flow rate as well as the geometry. The TAFE formulation assumes no radial component of velocity in the vessel, whereas with the introduction of tapering to the system, the velocity will inherently have radial components and is no longer unidirectional. In addition, we acknowledge that our phantom studies are not replicates of coronary arteries in humans. Therefore, further preclinical and clinical studies need to be performed to validate this proof-of-concept method developed for blood flowrate assessment.

In terms of the phantoms, we also found that in the 3-D printing process, even with the subsequent etching of the surfaces, the inner walls of the straight and tapered phantoms are not smooth and have a high degree of roughness (Fig. S3 available in the Supplemental Materials on the ASME Digital Collection). This roughness extends into the body of the phantom, and this would introduce additional artifacts into the CT image. Finally, the phantom geometry is very simple with no curvature or branching; both of these features could also affect the image quality and TAFE prediction.

Conclusions

A CT compatible experimental phantom study has been conducted to validate the analytical TAFE formulation. Several assumptions were made in the TAFE formulation and in this study, and the TAFE formulation was modified for the radial variation of the contrast in the phantom. In addition, a general correction has been applied for the net total of imaging artifacts. The corrected estimations are in good agreement compared with the actual pump values. However, the phantom does not currently address other important features such as flow pulsatility, motion artifacts, and vessel curvature and branching. Future curved phantom studies with branches as well as in vivo studies are required to validate TAFE in more complex vessel formations and to mimic physiologic conditions.

Funding Data

• National Institute of Health's Graduate Partnership Program (Funder ID: 10.13039/100000002).

Conflict of Interest

Under a licensing agreement between HeartMetrics, Inc., and the Johns Hopkins University, RM is entitled to royalties on an invention described in this article. Dr. Mittal is a cofounder and consultant to HeartMetrics, Inc. This arrangement has been reviewed and approved by the Johns Hopkins University in accordance with its conflict of interest policies.

Supplementary Material

Supplementary Material

Supplementary Figures