Influence of Pulsatility and Inflow Waveforms on Tracheal Airflow Dynamics in Healthy Older Adults

Abstract

Tracheal collapsibility is a dynamic process altering local airflow dynamics. Patient-specific simulation is a powerful technique to explore the physiological and pathological characteristics of human airways. One of the key considerations in implementing airway computations is choosing the right inlet boundary conditions that can act as a surrogate model for understanding realistic airflow simulations. To this end, we numerically examine airflow patterns under the influence of different profiles, i.e., flat, parabolic, and Womersley, and compare these with a realistic inlet obtained from experiments. Simulations are performed in ten patient-specific cases with normal and rapid breathing rates during the inhalation phase of the respiration cycle. At normal breathing, velocity and vorticity contours reveal primary flow structures on the sagittal plane that impart strength to cross-plane vortices. Rapid breathing, however, encounters small recirculation zones. Quantitative flow metrics are evaluated using time-averaged wall shear stress (TAWSS) and oscillatory shear index (OSI). Overall, the flow metrics encountered in a real velocity profile are in close agreement with parabolic and Womersley profiles for normal conditions, however, the Womersley inlet alone conforms to a realistic profile under rapid breathing conditions.

Introduction

The knowledge of airflow characteristics in human airways is crucial in understanding the physiological and pathological aspects of breathing. Chronic obstructive pulmonary disease (COPD) is a chronic inflammatory lung disease that is caused by tobacco smoking, exposure to passive smoke, and environmental pollution [1]. Among the obstructive lung diseases, COPD is the third leading cause of death worldwide, causing 3.23 million deaths in 2019 as reported by World Health Organization (WHO), and is associated with long-term disability [2]. Additionally, a certain subset of patients with COPD experience expiratory central airway collapse (ECAC) [3,4] that contributes to additional airflow obstruction resulting in respiratory morbidity [5]. Understanding airflow patterns in the airways is critical in predicting the particle deposition in inhalation drug therapy and the dispersion of exhaled air, which can be used as awareness to reduce the spread of lung disease. Clinically, computed tomography (CT) and pulmonary function tests (PFTS) are commonly used to diagnose obstructive lung diseases [6,7]. CT scans provide geometrical details, and the velocity data, however, achieved from MRI is of low resolution, so to understand the airflow characteristics, there is an additional need to employ the computational fluid dynamics (CFD) approach. Airflow simulations [8–12] shed new perspectives to improve clinical diagnostics and treatment planning for patients with COPD. In a complex airway, CFD provides information on the underlying flow patterns, pressure, and distribution of wall shear stress (WSS) that helps to understand the pathophysiology of the disease.

Furthermore, advancement in computational efficiency and noninvasive technology has led researchers to explore respiratory fluid dynamics in patient-specific geometries [13]. In contrast to simulations in idealized geometries [14], a crucial aspect involves enforcing accurate boundary conditions (BCs) to acquire realistic airflow dynamics during breathing. In general, patient CT scans are available for diagnostic purposes and can be easily used to simulate tracheal airflow characteristics by imposing a generic inlet velocity profile. CFD simulations in the trachea for various flow parameters as well as inlet and outlet boundary conditions have been investigated [15,16]. However, a limited number of studies have focused on airflow patterns within the patient-specific trachea. The unsteadiness of airflow in the human trachea is characterized by flow separation and the formation of secondary structures on the cross-planes due to complex tracheal geometry [17]. Oscillatory flow imposed on human airways showed that the size of the secondary flow structures depends on the Reynolds number (Re) and the Womersley number (Wo) [15]. Consequently, the velocity profile varies due to the change in the Womersley number. This study includes the effect of the Womersley profile that accounts for flow unsteadiness [18,19], which has been explored to a limited extent. In the past, flat and parabolic velocity profiles [8,15] have been popular steady-state solution choices for exploring airflow patterns in patient-specific geometries. To assess the accurate flow patterns in airways, the focus of our present work relies on generating (a) flat, parabolic, and Womersley profiles that have similar flow rates that appear in a real velocity inlet and (b) carrying CFD simulations while imposing these boundary conditions. They resemble normal and rapid breathing rates, and we identify the velocity profile that is most suitable for exploring realistic airflow simulations.

Additionally, the time-averaged wall shear stress (TAWSS), and oscillatory shear index (OSI) in patient-specific geometries obtained from CT scans are further examined. The velocity and vorticity contours are shown for different boundary conditions, i.e., flat, parabolic, and Womersley, and comparing these against a real profile taken from an experimental study [20] that considers different breathing rates. The results are discussed at the peak inhalation, and the inspiration phase is only considered for simulations. In turn, this avoids changes resulting from tracheal movement during the exhalation process.

The airflow simulation considers two conditions including normal and rapid breathing conditions. The typical frequency (f) and flowrate ${(Q}_{\max })$ that exist in normal: f = 0.28 Hz, i.e., 17 breaths/min and 364 ml/s, while for rapid: f = 1.08 Hz, i.e., 65 breaths/min and 360 ml/s, respectively. Here, rapid condition refers to an abnormally high breathing rate. These values are embraced from a benchtop study conducted by Jalal et al. [20] and correspond to different Womersley numbers, i.e., Wo = 2.42–4.82. Following this, the present study simulates the influence of different idealized profiles on flow metrics in ten patients during the inhalation phase. Consequently, the profile closest to the real inlet is identified by establishing velocity and vorticity contours. This study explores an appropriate idealized profile that can act as a surrogate model for modeling realistic airway simulations.

Largely, this study has several distinctive features, first, it incorporates a unique patient cohort that includes ten normal nonsmoking controls from the COPDGene^® database, which provides a perspective of unique patient airflow features between different conditions of breathing. Secondly, it investigates the influence of boundary conditions on airflow in the trachea by implementing different inlet velocity profiles and proposes the closest surrogate model for different breathing conditions that can be readily used for patient-specific diagnostics, providing a new perspective to clinical approach. Lastly, it specifically focuses on the comparison between normal and rapid breathing conditions, which is not explored in the literature to our knowledge. Overall, it complements previous research and adds to the current understanding of the influence of boundary conditions on airflow in the trachea between different breathing conditions.

Materials and Methods

Patient Cohort.

COPDGene® [21] is a cohort study that enrolled 10,000 participants who were current and former smokers between 45 and 80 years of age in 21 clinical centers in the United States. The study cohort comprised a diverse group of individuals, including smokers and nonsmokers with and without COPD. Specifically, ten normal nonsmoking controls without COPD were selected from the COPDGene^® database for the present study. Further, the sample size undertaken represents the sample of convenience. The age, sex, BMI, weight, and height of these patients are given in Table 1.

Table 1.

Demographic data of the patient (n = 10)

Parameters	Mean ± SD
Age (years)	63.3 ± 7.3
Gender (female/male)	8/2
Body mass index (kg/m2)	28.1 ± 5.5
Weight (kg)	78.5 ± 17.9
Height (cm)	166.9 ± 10.6

Data are reported as mean ± standard deviation.

Preprocessing of Patient-Specific Models.

Each patient-specific trachea geometry is entirely different from the straight pipe, as shown in Fig. 1(a), which is obtained from computed tomography (CT) scans. These scans were obtained from the COPDGene database, which used a standard protocol and underwent rigorous quality control, with a resolution of 0.644 mm × 0.644 mm × 0.625 mm. Further, it may also be noted that the primary region of interest was only the tracheal portion, and thus major bifurcations originating from it were not included in the present study. Also, flow dynamics in straight pipe served as a baseline study for understanding the underlying flow patterns in patient-specific geometries. Ideally, features such as ring-shaped cartilages, curvature, and tortuosity exist within tracheal geometries, however, a straight pipe approximation makes it easier to explicitly understand the contribution of each entity toward modifying the airflow dynamics within patient-specific geometries. The CT scans were smoothed and clipped using a three-dimensional slicer2 along with the Vascular Modeling Toolkit3 to obtain trachea models. The extracted trachea had an inlet with a noncircular cross section that increases the complexity of imposing the boundary condition. To avoid this, the trachea inlet was extruded by 2 mm to obtain a circular cross section in solidworks ^® (SolidWorks Corp., Waltham, MA), as shown in Fig. 1(a).

Fig. 1.

(a) CT scan of lung airways—tracheal flow dynamics is first modeled as a straight pipe for the baseline study, then simulations are performed in ten patient-specific trachea models, (b) mesh on cross-sectional and sagittal planes using ICEM-CFD, (c) spatio-temporal velocity extracted for implementation in CFD simulations, (d) mesh independence and optimal time-step selection while evaluating TAWSS, (e) two-dimensional velocity profile plotted along the centerline of the inlet, and (f) pressure outlet boundary condition imposed for normal and rapid breathing condition

The final trachea model was imported to ICEM CFD (ansys 19.5, ANSYS Inc., PA) and the Octree mesh method was used to obtain a surface mesh. This was followed with the Delaunay method for a smooth transition of volumetric mesh. Six concentric rows of the prism layer with a smooth cell transition ratio (∼1.11) are used for the boundary wall. The prismatic layers of the wall were refined during the grid independence test to ensure that the y+ values were less than 1, and the equivalent wall-adjacent cell height is 0.01 mm. To resolve the high-velocity gradients, the generated mesh was a mixture of prism layers with unstructured tetrahedral elements accompanied by a few pyramids near the tracheal wall. For increased stability and reduced computational time, the tetrahedral mesh was processed to a polyhedral mesh in fluent ^® (ansys 19.5, ANSYS Inc., PA) and is shown in Fig. 1(b). Moreover, mesh independence tests were conducted while increasing the mesh size from 0.4 M to 5.75 M to ensure that computed results are independent of the chosen grid. A time independence test was also conducted with a step size range from 0.1 s–0.0001 s to get the temporal accuracy for capturing the detailed flow dynamics. Finally, a mesh size of 1.27 million with a time-step size of 0.01 s was selected while accounting for TAWSS (Fig. 1(d)) over the entire trachea.

Computational Fluid Dynamics Simulations and Boundary Conditions.

3-D, pulsatile CFD simulations of air flow through the trachea with air assumed as a Newtonian fluid with a constant density of 1.225  $\mathrm{kg}/{\mathrm{m}}^3$ and kinematic viscosity ( $\mathrm{\nu }$ ) of 1.524 × 10⁻⁵  ${\mathrm{m}}^2/\mathrm{s}$ , referenced at the room temperature of 25  $°\mathrm{C}$ , and standard atmospheric pressure were carried out using ansys fluent ^®. With the Reynolds number range in Table 2, being in the transitional flow regime, unsteady flow simulations were conducted using the transition shear stress transport (SST) k-ω viscous model. Although in some cases Re ≤ 2000, flow instabilities are prevalent due to the complexity of the geometry that justifies the use of the transition model [22]. A second-order discretization scheme is used for pressure and momentum, with second-order implicit schemes for transient flow. Furthermore, warped-face gradient correction enabled an increase in accuracy of the gradient calculation in the polyhedral mesh.

Table 2.

Computed average velocity, inlet diameter, Reynolds number, and Womersley number of ten-patient data for normal and rapid breathing

	Normal Breathing	Rapid Breathing
Average velocity (u)	2.40  $\pm$  0.51 m/s	2.42  $\pm$  0.17 m/s
Diameter (D)	14.34  $\pm$  1.02 mm
Reynolds number $\left(\mathrm{Re}=\frac{uD}{\nu }\right)$	2230  $\pm$  364	2206  $\pm$  359
Womersley number $\left(\mathrm{Wo}=\frac{D}{2}\sqrt{\frac{\omega }{\nu }}\hspace{0.17em}\right)$	2.42  $\pm$  0.17	4.82  $\pm$  0.34

Data are reported as mean ± standard deviation.

A residual of 1 × 10⁻³ was used as the convergence criteria with the time-step size of 1 × 10⁻² s. Though the simulations were run for the convergence criteria of 1 × 10⁻⁴, however, the values converged for 1 × 10⁻³, as well. Further, in order to avoid the time involved in 3-D unsteady simulations for 10 patient cases 1 × 10⁻³ was chosen as the convergence criterion. For each case, four different velocity profiles, i.e., flat, parabolic, Womersley, and real, were enforced at the model inlet (Fig. 1(c)). In principle, apart from cartilaginous rings, the membrane portion of the trachea is flexible and conforms to structural deformations under the influence of oscillatory flow loading induced during breathing. In turn, it becomes a fluid-structure interaction problem where the appropriate material properties are of significant interest. However, in the present study, the trachea was assumed to be rigid, and a no-slip boundary condition (BC) was imposed on the wall. The real profile (Fig. 1(c)) was taken from an experimental study conducted by Jalal et al. [20] using the Magnetic Resonance Velocimetry technique (MRV) to obtain the three-dimensional velocity flow field in benchtop tracheal models. The velocity obtained was extracted at the inlet of the trachea and interpolated spatially as well as temporally to obtain the data at each mesh element of computational study. The Womersley, flat, and parabolic profiles were implemented in fluent ^® using a user-defined function (UDF) based on real velocity data while preserving the mass flowrate. The Womersley profile used in this study was constructed using an in-house matlab code that generated the data based on the mass flowrate obtained from the MRV experiment. Moreover, parabolic or Womersley profile enforced in the spatial domain is a fully developed velocity profile. In the temporal sense, these inlet waveforms represented a portion of the oscillatory wave that mimics normal and rapid breathing conditions, respectively. It may be further noted that the velocity values are reported after a finite time, i.e., 2 breathing cycles, in order to avoid the initial transient effects. This was accomplished by establishing cycle-to-cycle convergence where intercycle variations in TAWSS value were obtained within 2%. Furthermore, the overall length of the trachea did not interfere with the imposed outflow boundary condition. Thus, it ensures that the flow patterns obtained are redundant to the initial transient effects and the outflow conditions. For normal and rapid breathing conditions these two-dimensional inlet velocity profiles are explicitly represented in Fig. 1(e). The time-varying pressure drop was implemented at the outlet using the following equation

$${\hspace{0.17em}P}_t\left(t\right)=R_{g}Q\left(t\right)+\frac{V\left(t\right)}{C_g}+P_i\left(t\right)$$ (1)

where $Q\left(t\right)$ is the flowrate, $V\left(t\right)\hspace{0.17em}=\hspace{0.17em}\int Q\left(t\right)\hspace{0.17em}\mathrm{dt}$ is the time-dependent breathing volume, $P_t$ is the pressure at the trachea, $P_i$ is the intrapleural pressure in the pleural cavity, which drives the breathing flow, and finally, $R_g$ (∼1.5 × 10⁻³  $\mathrm{cm}{\mathrm{H}}_2\mathrm{O}\hspace{0.17em}\mathrm{s}\hspace{0.17em}{\mathrm{ml}}^{-1}$ ), $C_g$ (∼1.81 × 10⁻²  $\mathrm{ml}\hspace{0.17em}\mathrm{cm}{\hspace{0.17em}\mathrm{H}}_2{\mathrm{O}}^{-1}$ ) are the resistance and compliance of the lower airways, respectively, taken from healthy adult data [23]. BCs were prescribed with an intermittency of 0.05, a turbulence intensity of 1%, and a turbulence viscosity of 10, respectively [24].

Data Analysis

Wall Shear Stress (WSS).

Wall shear stress is defined as the tangentially acting force per unit surface area by the air flowing on the walls of the tube in the opposite direction. Wall shear stress $\left({\mathbf{\tau }}_w\right)$ along the trachea is

$${\mathbf{\tau }}_w=\mu \hspace{0.17em}{\frac{\partial \mathbf{u}}{\partial y}|}_{y=0}$$ (2)

where μ is the air viscosity, $\mathbf{u}$ is velocity, and y is the distance perpendicular to and away from the wall.

Time-Averaged Wall Shear Stress (TAWSS).

Time-averaged wall shear stress accounts for the total effect of WSS on pulsatile flow averaged during the inhalation phase of the breathing cycle. The TAWSS is

$$\mathrm{TAWSS}=\frac{1}{T}{\int }_0^T\left|{\mathbf{\tau }}_w\right|\mathrm{d}t$$ (3)

It acts as a diagnostic indicator to study the effects of airflow on the tracheal wall. The wall shear stress corresponds to compressive or stretching mechanical forces experienced by the trachea, which directly weakens it. The TAWSS was normalized by inlet velocity and diameter for each inlet velocity profile. Normalized TAWSS is defined as

$$\mathrm{Normalized}\hspace{0.17em}\mathrm{TAWSS}\hspace{0.17em}\left({\tau }^{\mathrm{*}}\right)\hspace{0.17em}=\hspace{0.17em}\frac{\mathrm{TAWSS}}{\mu \frac{u_i}{D_i}}$$ (4)

where $\mu \hspace{0.17em}$ is the dynamic viscosity, $\hspace{0.17em}{u}_i$ is the inlet average velocity, and $D_i$ is the inlet diameter.

Oscillatory Shear Index (OSI).

It is a nondimensional metric that describes the cyclic departure of the WSS vector over a breathing cycle and exhibits values in context to its alignment with the TAWSS vector. The OSI [25] is calculated by

$$\mathrm{OSI}=\frac{1}{2}\hspace{0.17em}\left(1-\frac{\left|{\int }_0^T|{\mathbf{\tau }}_w|\mathrm{d}t\right|}{{\int }_0^{\mathrm{T}}\left|{\mathbf{\tau }}_w\right|\mathrm{d}t}\right)$$ (5)

The OSI varies within 0–0.5, where 0 represents the unidirectional flow with no cyclic variation of the WSS vector, and 0.5 signifies complete oscillatory flow with disturbed flow behavior. The flow near the wall can be simple or disturbed according to the normal or diseased condition that changes the instantaneous alignment of the WSS vector with the TAWSS. It is introduced here as it relates closely to the remodeling of the airway wall and the deposition of aerosol particles.

Statistical Analysis.

All statistical analyses were performed with the R software, version 4.0.34. Data are presented as mean $\pm$ standard deviation and normality is assessed with the Shapiro-Wilk test. For independent samples, two-tailed t-tests were used to compare the mean for parametric data and the Mann–Whitney U tests for nonparametric data, respectively. A p-value less than 0.05 was considered statistically significant for all tests.

Results

Velocity and vorticity magnitudes for normal and rapid breathing are shown for patient-specific geometries. The impact of flow on wall weakening is addressed by evaluating TAWSS and OSI, respectively. Simulations in the straight tube (Re = 2122) served, as a baseline model for understanding flow dynamics in realistic airways. Out of the ten patient-specific cases, Fig. 2(a) shows the velocity magnitude variation on the sagittal plane subjected to Womersley and real inlets in cases (1, 2, and 3). Specifically, the tracheal curvature induces secondary recirculation zones as evident in the anterior-posterior view (Fig. 2(b)), leading to significant variations in the normalized cross-sectional area (CSA) and velocity, respectively. This variation is interesting from a fluid dynamics perspective, and, thus, case 3 is deliberately chosen for further evaluation. Furthermore, during the inhalation phase, the velocity magnitude distribution is revealed in the three-cycle instants in Fig. 2(c) for rapid breathing conditions. The peak phase (t* = 0.5) encounters strong recirculation zones along the length of the trachea, which otherwise is absent at t* = 0.17 and becomes weak for t* = 0.83, respectively. In turn, the results have been reported at the peak phase.

Fig. 2.

Rapid breathing—velocity distribution subject to Womersley and real inlets revealed on (a) the sagittal plane for three patients and (b) normalized CSA (CSA/Max CSA), local curvature (k = 1/r) where “r” is the local radius of curvature measured from the centerline at each cross section (CSA 1-9) along with the inlet and outlet, $v^{*}\hspace{0.17em}$ represents the normalized average velocity at each cross section (CSA 1-9) obtained at the peak phase (t* = 0.5), and (c) velocity contours in case 3 for time; t* = 0.17, 0.5, and 0.83, respectively

Normal and Rapid Breathing: Influence of Varying Inlet Profiles.

For patient-specific case 3, the superimposed inlet velocity profiles, i.e., parabolic, Womersley, flat, and real, generate a three-dimensional distribution of flow patterns within the trachea.

At the peak cycle phase, these are revealed on 2-D tracheal planes, i.e., cross-sectional and sagittal planes, and have been shown in Fig. 3. The cross-plane reveals velocity magnitude, while the sagittal plane contains both velocity and vorticity magnitudes, these are consequently disclosed for normal (Figs. 3(a)–3(c)) and rapid breathing (Figs. 3(d)–3(f)) rates. For each inlet condition, the velocity and vorticity are normalized by the inlet diameter and the maximum flow velocity in the fluid domain. In normal breathing, considering a parabolic profile, the magnitude of velocity contours for the five cross sections that include an inlet plane cross section, CS 1-3, and an outlet show the high-velocity region with magnitudes v/v _max as 0.85–0.97 that encircle the plane center. Moving along the trachea axis, low-velocity regions of 0.01–0.25 appear along the anterior wall and increase in size as we move toward the outlet. A similar distribution is seen for the Womersley inlet, however, the flat profile reveals profound differences showing larger spatial regions of high velocity along with diminished low-velocity regions. In terms of the spatial location of high- and low-velocity regions, although the real inlet is qualitatively similar to parabolic and Womersley profiles, it encounters a large spread of high-velocity regions. In contrast, for rapid breathing conditions, these enlarged regions are visible for the Womersley inlet, and instead, local peaks of high velocity are visible in a real inlet that arise due to inherent variations in the inflow which are rather uniform for normal breathing. Furthermore, the velocity contours for the parabolic and flat profile are similar to normal breathing, while the values diminish in a real profile, rather, small spatial high-velocity regions are observed along the inlet that weakens along the planar cross section, CS 2 & 3, respectively.

Fig. 3.

Normalized velocity and vorticity contours subjected to parabolic, Womersley, flat and real inlets are shown along the cross-sectional and sagittal planes for different breathing rates: (a)–(c) normal and (d)–(f) rapid, respectively. The anterior-posterior view and the sagittal plane are revealed as slices cut along these planes for reference. The normalized axial velocity plot indicates regions of negative velocity in the recirculation zone. Arrows indicate the flow direction.

Along the sagittal plane, under normal breathing large regions of low velocity recirculating fluid encircled in Fig. 3 occupy the downstream regions of the trachea and reside along the inner curvature. However, the flat profile confronts an entrance jet across the tracheal length surrounded by a small recirculation zone. However, the high-velocity core occupies a smaller region for the parabolic and Womersley, which for the real profile is slightly larger. For clarity, the sagittal planes superimposed with the axial velocity magnitudes reveal even a clear recirculation zone (Fig. 3). However, instead of axial velocities, the velocity magnitudes on the sagittal planes are revealed in Figs. 2 and 3, respectively since flow 3-dimensionality exists within the trachea and thus a significant contribution to flow velocities on this plane comes from the secondary flow established on the cross-planes. In contrast, rapid breathing reveals small zones of recirculating fluid, while the overall velocity contours remain similar. Recirculation zones are also visible in the vorticity fields in Figs. 3(c) and 3(f). For normal breathing, it is evident that vorticity magnitude plots of parabolic and Womersley profiles are close to the real profile. However, the flat profile has a distinctively different vorticity near the wall compared to other velocity profiles. In contrast, for rapid breathing, the vorticity contours for the Womersley profile are close to the real inlet.

Impact of Flow on the Tracheal Wall.

For all ten patient-specific cases, Fig. 4, shows a comparison of the normalized TAWSS that are averaged spatio-temporally over the inhalation cycle subject to varying velocity inlets. It is also evident that the parabolic and Womersley profiles are in close agreement with the real velocity inlet, while the flat profile predicts the highest TAWSS for both normal and rapid breathing in all patients. A further quantitative evaluation is obtained by computing the percentage difference between the idealized velocity inlets and the real profile for each patient, $\hspace{0.17em}{E}_{\mathrm{TAWSS}},\hspace{0.17em}$ is computed as

$$E_{\mathrm{TAWSS}}=\left(\frac{\left|{\tau }_{\mathrm{real}}^{\mathrm{*}}-{\tau }_{\mathrm{ideal}}^{\mathrm{*}}\right|}{{\tau }_{\mathrm{real}}^{\mathrm{*}}}\right)\times 100\hspace{0.17em}\mathrm{\%}$$ (6)

Fig. 4.

Bar graph for spatio-temporal averaged wall shear stress (TAWSS) in the trachea of ten patients and the straight pipe. Comparison of parabolic, Womersley, flat, and real velocity inlets for (a) normal and (b) rapid breathing conditions.

where ${\tau }_{\mathrm{real}}^{\mathrm{*}}\hspace{0.17em}$ is normalized TAWSS for real velocity profile and ${\tau }_{\mathrm{ideal}}^{\mathrm{*}}$ is normalized TAWSS for idealized velocity profile, that is, parabolic, Womersley, and flat, respectively.

For normal breathing the parabolic, Womersley, and flat profile against real inlet vary as 0.70–31.63%, 1.65–24.78%, and 24.85–66.90%, respectively. During rapid breathing, the range varies from 1.37–24.98%, 0.92–28.16%, and 11.09–88.31%, respectively.

However, significant variations were observed compared to the straight model, indicative of the role of geometrical contours within patient-specific geometry that cause changes in wall shear stresses, so a statistic of this cohort has been evaluated to study further.

In context to the Womersley number, the closest surrogate model for patient-specific simulation for both breathing conditions, the box and whisker plot shown in Fig. 5(a) reveals no statistically significant difference (p > 0.05). However, still $E_{\mathrm{TAWSS}}$ is seen as distinguished in each patient. For parabolic, Womersley, and flat profiles, in normal breathing, $E_{\mathrm{TAWSS}}$ variates by 8.65  $\pm$  8.83%, 8.48  $\pm$  6.51%, and 43.67  $\pm$  15.84%, while for rapid breathing, they take the values 8.71  $\pm$  7.40%, 5.33  $\pm$  7.79%, and 30.06  $\pm$  20.99%, respectively. A further observation of the average data shows that the absolute difference of ${\mathrm{E}}_{\mathrm{TAWSS}}$ between real versus parabolic and real versus Womersley for normal breathing is 0.17%, which is less than 3.38% for rapid breathing, as observed in Fig. 5(b), respectively.

Fig. 5.

(a) Box and whisker plots for TAWSS resulting from the parabolic, Womersley, flat, and real velocity profiles, (b) $E_{\mathrm{TAWSS}}$ evaluated and plotted as box and whisker plots for all patient cases for normal and rapid breathing conditions. The dots represent outliers, R—Real, P—Parabolic, W—Womersley, and F—Flat. Furthermore, * and ** represent statistically significant differences at the levels of p < 0.05 and p < 0.0001 levels, respectively.

Oscillatory shear index plots in Fig. 6 predict the lowest values for flat profiles for all the patient-specific cases. In addition, the parabolic and Womersley profiles give close predictions compared to the real input. In general, during rapid breathing, patients encounter heightened OSI values compared to normal breathing. The percentage difference of OSI for ideal velocity inlets against the real profile (E _OSI) is

$$E_{\mathrm{OSI}}=\left(\frac{\left|\mathrm{OS}{\mathrm{I}}_{\mathrm{real}}\hspace{0.17em}-\mathrm{OS}{\mathrm{I}}_{\mathrm{ideal}}\hspace{0.17em}\right|}{\mathrm{OS}{\mathrm{I}}_{\mathrm{real}}}\right)\times 100\hspace{0.17em}\mathrm{\%}$$ (7)

Fig. 6.

Semilog plots of OSI values for varying velocity inlets, i.e., parabolic, Womersley, flat and real, are shown as (a) bar graphs and (b) box and whisker plots for all patient cases for each normal and rapid breathing condition

where $\mathrm{OS}{\mathrm{I}}_{\mathrm{real}}$ and $\mathrm{OS}{\mathrm{I}}_{\mathrm{ideal}}$ are evaluated for real and idealized velocity profiles, respectively.

Compared to the real inlet, the idealized inlet velocity profiles affect the OSI values for normal breathing as 23.61  $\pm$  18.03%, 28.50  $\pm$  25.25%, and 42.87  $\pm$  21.50%, while for rapid breathing, these take the values as 25.43  $\pm$  13.71%, 18.40  $\pm$  19.38%, and 44.85  $\pm$  16.70%, respectively. Further observation reveals that the absolute difference in the average ${\mathrm{E}}_{\mathrm{OSI}}$ while comparing each parabolic and Womersley against real is 4.89% for normal breathing, which is less than 7.03% for rapid breathing, respectively.

Discussion

Airflow simulations reveal that for normal breathing, parabolic and Womersley profiles are close to real velocity inlet, however, the Womersley profile is a better substitute under rapid breathing. Nevertheless, the former underestimates velocity and vorticity magnitudes while the latter overestimates their magnitudes, but qualitatively they have a similar flow distribution. Therefore, for a proper understanding of airflow dynamics, a Womersley profile will serve as a surrogate model. In this study, the choice of using idealized profiles, reasons for inherent differences in magnitudes, variations in flow metrics when compared with real velocity inlet [20], and their biomedical relevance in terms of particle deposition within the trachea, are discussed henceforth.

Computed tomography (CT) offers high spatial resolution, providing detailed features of patient-specific tracheal geometry, but it cannot measure velocity data. Magnetic resonance imaging (MRI), however, can measure velocity profiles within arteries [26] but to this end, there has been no clinically established imaging modality for measuring the air flow velocities in human lungs and hence, it is difficult to identify the underlying flow metrics associated with disease progression. In turn, we combine computational fluid dynamics (CFD) approach and simultaneously employ CT scans to obtain air flow patterns within the human trachea. This essentially requires enforcing idealized profiles, i.e., flat, parabolic, or Womersley, which, in the event of limited clinical data, can act as a substitute for understanding flow dynamics in realistic airways. We then established our comparison relating idealized profiles across ten patient-specific models and found variations in geometry as a potent regulator to alterations in airflow dynamics. Contrary to this, for a particular patient, these variations diminished in different phases of inhalation.

Particle deposition is strongly influenced by the flow field within the trachea and can be qualitatively assessed through this study. This can further improve drug delivery for chronic airway diseases and help evaluate dosimetry for occupational hygiene and medical treatment [27]. The presence of curvature within the trachea results in a recirculation region, which induces turbulence and improves the transport and deposition of particles [28]. Low-velocity regions such as recirculation zones exhibit lower particle concentration while high-velocity regions are associated with denser particle concentration [29]. Further, the breathing rate is correlated to the extent to which the particles travel deeper into the lungs [28]. Thus, in normal breathing, the parabolic and Womersley profiles will encounter higher particle concentrations in the recirculation zone around the curvature. In contrast, for rapid breathing, low deposition rate due to a smaller recirculation zone is expected for the Womersley profile and, rather, particles may settle down in terminal airways. However, the flat profile will experience a small deposition rate, and instead, the particles are expected to move downstream as a result of the large jet inflow length over the trachea. Further, the particle residence time within these recirculation zones is expected to increase due to the establishment of a strong secondary flow [30] that forms as a result of a high-velocity zone seen on the sagittal plane within the trachea. Moreover, irregularities in the tracheal surface further create localized high and low vorticity regions within the recirculation zones, these tend to further influence particle transport as well.

Moreover, significant variations in the flow patterns for the flat profile are attributed to the absence of velocity gradients which instead are prevalent in parabolic and Womersley profiles, these, gradients, in turn, are directly proportional to WSS. In addition, it influences the bronchial epithelium, causing airflow inflammation, exacerbating epithelial cell damage, and promoting COPD [31–33]. A high WSS indicates an increased velocity gradient near the wall region where aerosol particles are expected to collide and deposit on the wall. In contrast, a low WSS indicates a smaller velocity gradient in which the aerosol particles remain suspended in the air, which can then be repelled or attracted toward the wall [34]. Furthermore, higher OSI values are found within recirculation zones [25] and it has the potential to indicate the exact location of particle deposition [34]. The $E_{\mathrm{TAWSS}}$ and $E_{\mathrm{OSI}}$ magnitudes are similar for parabolic and Womersley during normal breathing, while for rapid, they are small in Womersley as compared to parabolic and flat profiles. This is attributed to the higher Womersley number encountered in rapid breathing when compared to normal. It indicates that the flow is dominated by time-varying pressure resulting in a flat velocity profile. In contrast, at a low Womersley number, the flow is dominated by viscous forces, and a parabolic shape is expected [35]. Furthermore, the deviation of TAWSS and OSI values derived from idealized profiles close to the real inlets is further evident in previous research [18,19].

The factors contributing to these deviations involve, first, employing a perfect sinusoidal inflow rather than patient-specific breathing pulse, second, smoothening the actual tracheal geometry for meshing while removing the artifacts, i.e., noise and staircases in CT images. In general, smoothing requires a continual examination of the shape and curvature of the patient-specific geometry. With the unique nature of each patient, smoothening a large number of cases is challenging. Third, the rigid tracheal wall curtails flow variations arising from fluid-structure interactions (FSI). Finally, a small cohort of patients in a particular age group, along with limited breathing frequencies, is used for simulations. However, these errors can be minimized by incorporating a large cohort of patients of different age groups so that structural variations within the trachea [36] are considered, and a compliant tracheal wall is used for modeling and realistic breathing waveform is enforced. Additionally, careful segmentation and smoothening of geometry should be ensured for exploring tracheal airflow dynamics. These modifications will then ultimately improve the clinical diagnosis and are expected to improve patient outcomes.

Apart from this, the Womersley profile employed here is essentially the solution for a fully developed straight pipe that includes pulsatility effects. However, patient-specific models exhibit structural variations along the cross-sectional area due to curvature and tortuosity that affect flow pulsatility. In turn, these geometrical parameters modulate the flow features significantly more than the inherent pulsatility effects [37]. Therefore, future scope involves geometrically transforming the realistic geometries to an equivalent constant cross section cylinder [38] and establishing simulations with realistic boundary conditions. Moreover, the real velocity profile is obtained in benchtop models using the magnetic resonance velocimetry (MRV) technique [20], while the idealized inlets are constructed based on the same flow waveform, and the results are subsequently compared. Although the idealized profile is based on the same flow waveform, a patient-specific breathing velocity profile containing secondary flow structures at the tracheal inlet will likely underestimate the differences between the idealized velocity profile and the real profile, respectively.

In summary, the parabolic and Womersley profiles closely approximate the real velocity profile under normal conditions. Under rapid conditions, the Womersley inlet is in close agreement with the real velocity profile than the parabolic profile. These are expressed through flow patterns on the cross-plane and sagittal plane. The primary flow patterns tend to strengthen the secondary flow features and are evident through the appearance of a jet on the sagittal plane. Particle deposition and residence time are strongly correlated to airflow patterns that in turn are dependent on the varying nature of the superimposed velocity inlets. Breathing rates administer the rate of particle deposition deeper into the lungs or terminal airways. Furthermore, the near-wall transport of particles is also governed by the wall shear stress distribution as well. Deviations in flow metrics arise due to smoothening of patient-specific geometry, using a sinusoidal profile, and considering a small cohort of patients. Although these deviations remain inherent, qualitatively the flow patterns obtained while enforcing idealized profiles are similar to the real inlet. On the contrary, a flat profile offers maximum deviation for both the breathing frequencies considered, and hence, enforcing it at the model inlet will result in an inaccurate estimation of particle deposition rates. Compared to straight pipe simulations, the evaluated TAWSS is significantly influenced by patient-specific geometries, in turn making the inclusion of geometrical factors necessary for performing real-time simulations. Finally, to acquire realistic airflow patterns and the limited availability of experimental breathing velocity data, this study recommends a Womersley profile for rapid breathing and parabolic or Womersley for normal breathing conditions, respectively. Furthermore, if a tentative assessment of the airflow field is needed, the parabolic profile alone can be employed due to the mathematical complexity existing within the Womersley profile.

The present findings have significant potential for advancing our understanding of airflow dynamics in COPD patients with even realistic lower respiratory tract in terms of more generations. Specifically, the future scope of the study will involve a comparative analysis of airflow dynamics and particle deposition in healthy older adults against COPD subjects while including major bifurcations originating from the trachea and branching to smaller airways up to at least the 10th generation. These simulations, conducted on multiple COPD patients, will provide valuable insights into the relationship between anatomical variations in the airways and their impact on airflow patterns, pressure changes, and turbulence. These insights from airflow dynamics analysis ultimately will help to understand the anatomical features and their contribution to airflow limitation in COPD subjects. This will shed light on specific mechanism and predictive markers for disease development and progression that leads to impaired respiratory function and COPD. Finally, detailed airflow dynamics in COPD subjects will guide novel therapeutic strategies that relate to targeted drug delivery, bronchoscopic interventions, or surgical approaches. Addressing these future research directions will enhance the knowledge of pulmonary health and disease progression from a fluid dynamics perspective.

Funding Data

• COPD Foundation (Grant No. NCT00608764; Funder ID: 10.13039/100008184).

• National Institute of Biomedical Imaging and Bio-engineering of the National Institutes of Health (NIH) (Grant No. R21EB027891; Funder ID: 10.13039/100000070).

• National Heart, Lung, and Blood Institute (Grant Nos. U01 HL089897 and U01 HL089856; Funder ID: 10.13039/100000050).

Nomenclature

D =

inlet diameter

f =

breathing frequency

$k$ =

curvature

P =

pressure

Q =

breathing flow rate

Re =

Reynolds number

t =

time

u =

average velocity

Wo =

Womersley number

$\mu \hspace{0.17em}\hspace{0.17em}$ =

dynamic viscosity

$\nu \hspace{0.17em}\hspace{0.17em}$ =

kinematic viscosity

$\omega \hspace{0.17em}\hspace{0.17em}$ =

angular frequency

${\mathbf{\tau }}_w\hspace{0.17em}\hspace{0.17em}$ =

wall shear stress

$v^{*}\hspace{0.17em}\hspace{0.17em}$ =

normalized average velocity