We propose a pipeline for estimating the parameters of the WK3 model of the aortic arch branch arteries when the artery-specific flow rate and pressure waveforms are not available (Figure 1). The goal is to use the artery-specific total resistance (R_tot) and cardiac period (T) to scale the population-averaged, artery-specific normalized R₁, C, and R₂ values (R_1nom, C_norm, and R_2norm) to estimate the non-normalized parameters as: R 1 = R 1 n o r m ∙ R t o t ;   R 2 = R 2 n o r m ∙ R t o t ;   C = C n o r m ∙ T R t o t (1)

To encompass the patient-to-patient variability of mean flow rates and pressures, normalized R₁, C, and R₂ parameters are used and defined as: R 1 n o r m = R 1 R t o t ;   R 2 n o r m = R 2 R t o t = 1 − R 1 n o r m ;   C n o r m = R t o t ∙ C T (2)

T_tot is calculate from mean pressure ( P ¯ ) and mean flow rate ( Q ¯ ) as described in Eq. 3. R t o t = R 1 + R 2 = P ¯ Q ¯ ; (3)

The mean arterial pressure can be estimated from the non-invasively measured systolic (SBP) and diastolic (DBP) brachial blood pressures via a commonly used equation P ¯ = 1 3 S B P − D B P + D B P (4) assuming systole is one-third of the cardiac cycle (Sesso et al., 2000). Mean flow rate can be measured by Doppler ultrasound if available or estimated from the previously published flow rate-diameter relationships based on the diameter extracted from CTA images, such as those reported by us (Tricarico et al., 2020a). CTA is routinely performed for diagnosis and treatment planning of aortic pathologies and is often used in clinical research to extract three dimensional arterial models. Cardiac period can be calculated from heart rate. Furthermore, by defining the normalized flow rate, pressure, and time as each variable divided by its mean (flow rate, pressure) and T, respectively, the differential equation governing the relation between flow rate and pressure in the WK3 model is maintained (Supplementary Material). The population-averaged, artery-specific R_1norm, C_norm, and R_2norm values can be extracted from measured flow rate and pressure data. We describe the extraction and assessment of the combinations of normalized parameters in the next sections.

We developed a prospective study to measure patient-specific waveforms of flow rate and pressure data in aortic arch branch arteries in 10 patients suffering aortic aneurysms or dissections, undergoing TEVAR. The protocol for this prospective study was approved by the University of Florida College of Medicine Institutional Review Board (Gainesville, FL, United States), and informed consent was signed by every patient. Duplex ultrasound measurements were acquired under resting conditions (with awake patients in supine position) using a Philips iU22 system. Measurements were collected on multiple locations of bilateral common carotid (proximal, middle, and distal), subclavian (distal to the vertebral artery, thyrocervical and costocervical trunks, and internal thoracic artery), and vertebral arteries at pre-operative and 1-week post-operative time in the vascular laboratory. We did not take intra-operative flow rate measurements as this would interrupt the standard TEVAR procedure. Arterial diameters and flow rates were extracted from ultrasound images. Details on the methodology of flow rate calculation have been described in our previous work (Tricarico et al., 2020a).

Intra-operative pressure measurements were performed invasively using a TruWave disposable pressure transducer (Edwards Lifesciences, Irvine, CA, United States) before and after stent graft deployment in brachiocephalic artery, left common carotid, and left subclavian arteries, all carried out after catheter flushing. All pressure waveforms were traced and smoothed in Matlab R2017b (Mathworks Inc., Natick, MA, United States) using the heart rate extracted from ultrasound images, due to patient sedation at the time of the intra-operative pressure measurements. In addition, non-invasive brachial artery pressures were collected by a cuff at the time of the ultrasound imaging and mean arterial pressure was estimated using Eq. 4.

We identified the non-normalized R₁, C, and R₂ values for each artery (common carotid, subclavian, and vertebral arteries) of the patients to obtain the average of each parameter for this patient cohort using the Simulink Design Optimization toolbox (Mathworks Inc.), where the WK3 model was built, and the governing equation of the WK3 model is embedded into the block diagram (additional details in Supplementary Figure S1). Measured patient-specific, pre-operative flow and pre-deployment pressure waveforms (referred to as training dataset in Section 2.5) were used as imposed input and expected output of the model, respectively. Since there were no intra-operative pressure measurements in the right common carotid artery and subclavian artery, pressure waveforms measured at the corresponding left side arteries were utilized for simulations on the right vasculature. In addition, subclavian artery pressures were used for vertebral artery simulations, due to invasiveness and risks of intra-deployment measurements in vertebral arteries.

Because R_tot = R₁+R₂, there are only two independent parameters in the WK3 model that need to be identified. The mean flow rate and mean pressure used to calculate R_tot, were obtained from the corresponding waveforms. The Runge-Kutta method was chosen as the solver with a fixed time step of 10^–4 s to maximize the accuracy of pressure waveform calculation iteratively. A non-linear least square method and trust-region-reflective algorithm were utilized with both parameter tolerance and function tolerance of 0.001. The sum of squared error (SSE) between measured and simulated pressure of the WK3 model was chosen as the cost function for the optimization problem. At each iteration, the cost function quantified the quality of the pressure matching, and at the end of the optimization process, the optimized R₁, C, and R₂ combination was collected. Additionally, the L²-norm of the relative error for pressure estimation in time was calculated as in Eq. 5 and collected for each R₁, C, and R₂ combination on each artery. e L 2 = ∑ k = 1 N P s i m u l t k − P exp t k 2 ∑ k = 1 N P exp t k 2 * 100 (5) where P_simul and P_exp are simulated and experimentally measured pressures, respectively; N is the number of time steps in a cardiac period. Additional details of the parameter estimation process are given in Supplementary Figure S2. After obtaining the optimized, non-normalized R₁, C, and R₂ values for each artery, the corresponding normalized R₁, C, and R₂ values were calculated as Eq. 2.

A Monte Carlo method was performed in the Simulink Design Optimization Toolbox to examine the sensitivity of predicted pressure to the parameters of the WK3 model. For purposes of comparison, all flow rate and pressure waveforms were normalized to their means in Matlab. In addition, time-normalization by the cardiac period and interpolation (every 0.01) were performed. For each type of artery, the averages of the normalized pre-operative flow rate and pre-deployment pressure waveforms were imposed respectively as input and expected output of the WK3 model. Accordingly, normalized WK3 model parameters were used as the input. 1500 sets of the two independent parameters (R_1norm and C_norm) were randomly generated under the hypothesis of a uniform distribution for the three arterial-specific Monte Carlo analyses. The simulation ran until a minimum cost function was achieved.

The normalized R₁, C, and R₂ values obtained from pre-operative flow and pre-deployment pressure data (Section 2.3, training dataset) were assessed using the post-operative flow at 1 week and post-deployment pressure data (testing dataset) for the left arteries of all patients. Specifically, the medians of the normalized R₁, C, and R₂ obtained from the training dataset were scaled to each R t o t and T to estimate the patient-specific, non-normalized R₁, C, and R₂ parameters of each artery, as detailed in Eq. 1. R_tot was calculated from the mean flow rate and pressure calculated from the measured waveforms of the testing dataset. Each estimated R₁, C, and R₂ and patient-specific (non-normalized) input flow rate were used in the Winkessel model to estimate pressure waveforms using Simulink Design Optimization toolbox. Relative errors between estimated and measured artery-specific pressure waveforms were calculated. Additionally, the estimated R₁, C, and R₂ values were tested on the Windkessel model, imposing non-normalized pressure waveforms as input and flow rate waveforms as expected output, to quantify relative errors between estimated and measured flow rate waveforms. Simulated pressure and flow rate waveforms are presented in normalized form for comparison purposes.

Both non-normalized and normalized parameter values are presented as artery-specific median and 25th-75th percentiles (interquartile range, IQR). Other data were presented as mean ± standard deviation. T-tests or Mann-Whitney rank sum tests were performed to detect statistical differences in flow rate and pressure means. Statistical analyses were performed in Sigmaplot (SYSTAT Software Inc., Chicago, IL, United States). A p-value <0.05 was considered statistically different.

The analyzed population, 10 patients with age 64 ± 3 (range, 40–82) years and body surface area 2.0 ± 0.3 (range, 1.6–2.3) m², was 50% male. Four of ten patients underwent percutaneous transluminal repair only, either with TEVAR isolated to the descending thoracic aorta or TEVAR with a fenestrated branch to the left subclavian artery. The remaining six patients underwent a hybrid TEVAR procedure, which involved left subclavian artery coverage and a left common carotid artery-left subclavian artery bypass. The average heart rate for these patients was 63 ± 10 beats per minute (range, 47–76 beats per minute). The most common comorbidities for this set of subjects are listed in Table 1. The majority of subjects suffered hypertension (80%) and were active or former tobacco users (60%).

No statistically significant difference of the mean flow rate was observed between the right and left sides. Therefore, the main results are presented on left and right arteries combined hereafter. The mean flow rates of the pre-operative dataset for common carotid artery (458 ± 139 mL/min), subclavian artery (185 ± 116 mL/min) and vertebral artery (90 ± 59 mL/min) were not statistically different from the means of the post-operative 1 week dataset (428 ± 147, 228 ± 116, and 92 ± 48 mL/min for common carotid, subclavian, and vertebral arteries, respectively). No significant differences of the mean pressures between left common carotid artery and subclavian artery of the pre-deployment (83 ± 11 vs. 85 ± 12 mmHg for common carotid artery and subclavian artery, respectively) or post-deployment (89 ± 13 vs. 88 ± 10 mmHg for common carotid and subclavian arteries, respectively) dataset were observed, neither between pre- and post-deployment mean pressures in each artery.

Figures 2A–C present the patient-specific normalized arterial flow rate waveforms for the pre-operative dataset. Notably, the subclavian artery showed systolic high peak and reversal flow at early diastole (Zhang et al., 2022), not present in the mono-phasic waveforms of common carotid artery and vertebral artery. Figures 2D, E present the normalized, pre-deployment pressure waveforms in the left common carotid artery and subclavian artery.

When the patient-specific optimized R₁, C, and R₂ parameters were obtained from the WK3 simulation (as described in Section 2.3), the relative errors on the pressure waveform matching were small and 4 (3–4) %, 4 (3–6) %, and 3 (3–5) % for common carotid, subclavian, and vertebral arteries, respectively (example of pressure waveform matching and corresponding relative error in Supplementary Figure S3). The individual estimated and average of measured pressure waveforms are shown in Figure 3. The patient-specific non-normalized and normalized parameters of the WK3 model for each type of artery are shown in Table 2.

The normalized WK3 model parameters are also shown in Figure 4 as boxplots. The fairly large ranges of normalized resistances and compliance on each artery can be related to the apparent variation of the normalized waveform profiles among patients (Figure 2).

The Monte Carlo analysis provided information on the sensitivity of the pressure waveform to the variation of the R₁, C, and R₂ parameters for each type of artery. From the contour plots describing the output quality (minimized values) with different parameter combinations, we can see that the sensitivity of the pressure output to the WK3 parameters varies with different combinations of these parameters (Figure 5). For example, when C equals 0.5, the common carotid artery pressure is more sensitive to R₁ when R₁ is small (R₁ < 0.05), and less sensitive when R₁ is larger (Figure 5A). The combination of the medians of normalized R₁ and C parameters obtained from WK3 simulations for both common carotid artery and subclavian artery falls into the area of minimized value equal to 0.1, indicating good matching between expected and simulated pressure waveform (Figures 5A, B). The combination of the medians of normalized parameters for the vertebral artery falls into the area of minimized value equal to 0.3, suggesting the existence of alternative R₁ and C combinations which could better represent the pressure outlet waveform (Figure 5C). For all three arteries, the pressure output is more sensitive to variations of R₁ than C (smaller R₁ ranges than C ranges). Around the medians of the normalized parameters, the common carotid artery pressure is not sensitive to relatively small deviations of R₁ and C. The pressure of subclavian artery is even more tolerable to deviations of C from their corresponding medians (Figure 5B). The contour plot of vertebral artery is similar to that of carotid artery (Figure 5C), although with different axes’ ranges. Around the medians of the normalized parameters, the vertebral artery pressure is more sensitive to small deviations of R₁ and C for smaller R₁ and larger C. Plots of the normalized pressure waveforms under some minimized values are shown for common carotid, subclavian, and vertebral arteries (Figures 5D–F). Note that the simulated pressure waveform may be different for the same minimized value although the overall differences are the same.

First, the individual flow rate waveform was used as the testing input, and the average flow rate waveform of the testing dataset for each type of artery is reported in Figure 6. The estimated pressure showed fairly high tolerance to patient-specific inlet flow rate waveforms using the estimated R₁, C, and R₂ parameters. Relative errors of pressure output of the testing dataset were: 13 (9–15) % for common carotid artery, 16 (10–17) % for subclavian artery, and 12 (11–16) % for vertebral artery. The estimated pressure waveforms are shown in Figure 7. Compared with the average of the measured waveforms, only two subclavian arteries presented largely out-of-range estimated pressure waveforms at the systolic peaks. An example of the measured and corresponding estimated pressure waveforms is shown in Supplementary Figure S4. These resulted from flow rate waveforms with very high peak systolic and large reversal diastolic flow. Using the pressure waveform as input, the relative errors of estimated flow rate waveform matching were higher: 34 (29–43) % for common carotid artery, 94 (74–116) % for subclavian artery, 30 (23–31) % for vertebral artery (Figure 8). One example of the measured and estimated flow rate waveforms of a subclavian artery using the measured pressure waveform as the input is shown in Supplementary Figure S5.