A numerical study of gas transport in human lung...

A numerical study of gas transport in human lung models Ching-Long Lina and Eric A. Hoffmanb

aDepartment of Mechanical and Industrial Engineering and IIHR -Hydroscience & Engineering

The University of Iowa, Iowa City, Iowa 52242 bDepartments of Radiology and Biomedical Engineering

The University of Iowa College of Medicine, Iowa City, Iowa 52242

ABSTRACT

Stable Xenon (Xe) gas has been used as an imaging agent for decades in its radioactive form, is chemically inert, and has been used as a ventilation tracer in its non radioactive form during computerized tomography (CT) imaging. Magnetic resonance imaging (MRI) using hyperpolarized Helium (He) gas and Xe has also emerged as a powerful tool to study regional lung structure and function. However, the present state of knowledge regarding intra-bronchial Xe and He transport properties is incomplete. As the use of these gases rapidly advances, it has become critically important to understand the nature of their transport properties and to, in the process, better understand the role of gas density in general in determining regional distribution of respiratory gases. In this paper, we applied the custom developed characteristic-Galerkin finite element method, which solves the three-dimensional (3D) incompressible variable-density Navier-Stokes equations, to study the transport of Xe and He in the CT-based human lung geometries, especially emulating the washin and washout processes. The realistic lung geometries are segmented and reconstructed from CT images as part of an effort to build a normative atlas (NIH HL-064368) documenting airway geometry over 4 decades of age in healthy and disease-state adult humans. The simulation results show that the gas transport process depends on the gas density and the body posture. The implications of these results on the difference between washin and washout time constants are discussed.

Keywords: Computational fluid dynamics, gas transport, CT, MRI, and pulmonary airflow

1. INTRODUCTION

Xe-CT 1,2,3 is a method for the noninvasive measurement of regional lung ventilation, being determined from the washin and washout rates of stable Xe. Mathematical treatment of the washin and washout curve for the inert gas is based on the analysis developed by Kety4. Kety’s model assumes that ventilation is a continuous process whereby the inert gas has a solubility of zero, and the rates of buildup and clearance of the gas are identical. Thus, in previous studies of Xe-CT1,2, the time constants for Xe washin and washout were assumed to be the same, ignoring Xe solubility in blood and tissue as well as geometric issues. During a controlled washin/washout ventilation protocol, a time series of CT scans taken at a constant lung volume are used to generate a local exponential density curve for any specified region of interest (ROI) in the lungs. The density of Xe is measured in Hounsfield units (HU). The time constants of Xe washin and washout can be obtained by the least-squares curve fitting method. The time constant of this curve is equal to the inverse of the local ventilation per unit volume. By selecting different ROIs, a detailed map of pulmonary ventilation can be constructed to analyze their changes in response to various interventions and identify the areas with impaired ventilation. Therefore, it is of critical importance to measure accurately Xe washin/washout time constants that are determined by the transport of Xe in the complex lung geometry.

Chon et al.5,6 have conducted research to examine the time constant hypothesis of Kety by scanning anesthetized, supine sheep using multidetector-row computed tomography (MDCT). They found that washout takes a longer time than that of washin; that is, the time constant difference between washin and washout is always greater than zero. They also found spatial and temporal dependences of the time constant difference. For instance, washin and washout in both dependent and nondependent regions at the lung base and apex were larger when measured at end expiratory scans compared to end inspiratory scans. They also demonstrated that, with increasing tidal volume, the global time constant

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difference decreases, indicating that tidal volume has an influence on the rates of both retention and clearance of Xe. It is noteworthy that the anesthetized sheep in the experiments of Chon et al.5,6 were placed in the supine position. In fact it has been known that the distribution of lung ventilation can change dramatically in prone and supine body postures. For example, Marcucci et al.2 reported from the experiments of five mongrel dogs that ventilation in the supine position increases with dependent location, whereas no ventilation gradients were found anywhere in the prone position. These differences are occasionally utilized to improve oxygenation in prematurely born infants with respiratory distress and patients with acute lung injury by positioning them in the prone position7.

Although Xe-CT and He-MRI have been established for years, advances in Xe-CT and He-MRI technology have focused on increasing the speed and resolution of the scanner rather than understanding the transport of Xe and He. The transport of Xe and He is an important factor in that the physical properties of Xe and He differ from those of air, which might cause some unexpected flow phenomena and result in improper physiological and clinical interpretation of the CT images. Despite its importance, the present state of knowledge regarding the transport of Xe and He in the lungs is incomplete. Therefore the objective of the paper is to shed light on the transport of Xe and He in the CT-based lung model by using the computational fluid dynamics (CFD) technique. The paper is organized as follows. In section 2, the computational methodologies are described. In section 3, the reconstruction of the CT-based lung geometry and mesh is given. In section 4, the velocity waveform at the inlet and the boundary conditions at the exits are prescribed. The simulation results are presented in section 5. The conclusion is drawn in section 6.

2. COMPUTATIONAL METHODOLOGIES

The current CFD method solves the 3D incompressible Navier-Stokes equations with variable fluid properties for fluid motion in the finite element framework8. The method is further coupled with the level set method for modeling immiscible binary fluids and the scalar transport equation for modeling miscible binary fluids. The level set capability can handle immiscible binary fluid systems with high density ratio, such as air-water systems, and one potential application is simulation of air-liquid systems in the lungs. The capability of modeling miscible binary fluid flow permits study of transport of two gases in the airways as in this study. The implicit characteristic-Galerkin approximation together with the fractional four-step algorithm is employed to discretize the governing equations. The second-order accuracy of the method is confirmed by simulation of decay vortex. The coupled system of the Navier-Stokes and level set equations is validated by benchmark problems of solitary wave and broken dam. The simulation results are in excellent agreement with experimental data. Other benchmark tests8 further indicate that the current method is much more accurate than the 3D second-order finite volume method9.

3. CT-BASED LUNG GEOMETRY AND MESH

In this paper, we used already gathered image data sets acquired through an NIH Bioengineering Research Partnership Grant (HL-064368). These data are for both sheep and humans. The research group has a MDCT-based research suite (Siemens Sensation 64). Scanning protocols have been established whereby image data sets are gathered at multiple lung volumes both statically and dynamically. Our airway segmentation method is based upon a combination of 3D region growing and 2D mathematical morphology and the third generation of airway segmentation further incorporates fuzzy logic.10 We have demonstrated that our custom-developed segmentation software can successfully extract airway trees from most clinical quality standard-dose CT data. The surface geometry can be exported to an stereo-lithography (STL) file. The original lung surfaces in the STL format are rough due to data resolution. For example, figure 1(a) shows a snapshot of a 3D human lung geometry segmented from CT images. It is noteworthy that the concavity located near the end of the trachea and before the first bifurcation is to accommodate aortic arch and pulmonary artery. In order to create CFD mesh, the geometry must be further processed, such as smoothing and clipping as shown in Fig. 1(b). This geometry contains about 5-6 generations of branching. The cross section at the beginning of the trachea is referred to as the inlet of the system where the breathing waveform is imposed. The cross sections at the truncated small branches are referred to as the outlets. There are a total of 45 outlets in this specific lung model.

The cross-sectional area A at the inlet is 0.0002141 m2, corresponding to an effective diameter D of about 0.0165 m, where A= πD2/4. The sum of all cross-sectional areas at the outlets is 0.0001177 m2. Because the ratio of the outlet area to the inlet area is 0.55, it is expected that the speed at the exit is approximately twice that of the inlet. The commercial code Gambit is used to generate mesh as shown in Fig. 2, comprising 264,331 nodes and 1,170,624 tetrahedral elements. Although figure 2 shows that the cross-sectional areas at the outlets vary, the average diameter of the outlet is




0.001825 m. The diameters of the trachea and small airways of this specific human subject are roughly consistent with the dimensions of the Weibel A lung geometry (table 5.1 of Ref. 11).

Figure 1. (a) A pre-processed STL image of a human airway scanned at the total lung capacity, (b) a smoothed and clipped geometry.

Figure 2. A snapshot of the surface mesh.

Concavity

Inlet




4. INITIAL AND BOUNDARY CONDITIONS AND FLUID PROPERTIES

We consider Xe and He washin and washout processes in two body postures: upright and supine positions. In the latter position, the trachea has an inclination angle of 10 degree with respect to the horizontal axis. The fluid properties used in the simulation are listed in table 1 and the gravitational acceleration is set to 9.81 m/s2. Hereafter, Xe (He) denotes the Xe-O2 (He-O2) mixture as given in table 1. For the Xe (He) washin case, the initial condition is specified as that Xe (He) fills the space above y = 0.06 m (marked by the dashed line in Fig. 1(b)) and air fills the rest of the space. For Xe (He) washout case, the initial condition is specified as air fills the space above y = 0.06 m and Xe (He) fills the space below.

The speed waveform imposed at the inlet is shown in Fig. 3. The maximum speed during inspiration and expiration is about 2 m/s, being equivalent to a volumetric flow rate of 428 cm3/s or 25.7 L/min. Adopting air properties, the

maximum Reynolds number νuD=Re at the trachea, where u is the speed, D is the effective diameter of an airway

tube, and ν is the kinematic viscosity, is estimated to be 1,942, whereas the maximum Re at the exit based on average D and speed is about 390. Therefore, the flow is expected to be laminar. Pressure boundary condition is prescribed at the outlets and no-slip condition is imposed at the surface. It is noted that two types of boundary conditions have been examined on a Weibel lung model12. They are pressure boundary condition and velocity gradient-free boundary condition that enforces global mass conservation. Both boundary conditions yield almost the same results.

Table 1. Fluid properties of (1) air, (2) 33 % Xe-67 % O2 mixture, and (3) 80 % He-20 % O2 mixture. D12, binary diffusivity between air and Xe-O2; D13, binary diffusivity between air and He-O2.

Air Xe-O2 mixture He-O2 mixture

Density, kg/m3 1.2 2.7 0.34

Kinematic viscosity, m2/s 1.7×10-5 8.1×10-6 5.8×10-5

Diffusivity D12, m2/s 2×10-5 2×10-5

Diffusivity D13, m2/s 2×10-4 2×10-4

Figure 3. The breathing waveform imposed at the inlet.




5. RESULTS

5.1. Washin and Washout in the Upright Body Posture

We first consider Xe/air washin and washout in the upright body posture. To investigate interfacial phenomena of binary miscible fluids in the current setting, 3D isosurface defined by the average density of two gases is plotted in green in the following figures. For example, as Xe is inhaled into an air-saturated lung. The green-colored isosurface represents gas mixture with a density of 1.95 kg/m3 that demarcates two gases. If density increases toward the density of Xe 2.7 kg/m3, the color spectrum changes to red. On the other hand, if density reduces to the density of air 1.2 kg/m3, the color changes to blue. Namely, red denotes dense gas, while blue represents light gas in the current binary fluid system. In the following presentation, only one-sided color spectrum is displayed for clarity. For instance, figure 4 (left) exhibits Xe washin process at time 0.16 s. Thus, red in the figure represents Xe, green is the interface, and air is not displayed (thus only the gray-colored lung geometry is seen). The red lines at the inlet and outlets represent velocity vectors. In contrast, figure 4 (right) shows the Xe washout: air is inhaled into a Xe-saturated lung. Air is lighter than Xe, thus blue is shown in the upper part of the trachea.

At this very early stage of these processes (Fig. 4), washin and washout exhibit different characteristics. The front of the interface during washin seems to propagate faster than that of washout. This observation may be attributable to the gravity effect that denser gas is attracted toward the ground rapidly. It is noteworthy that depending on binary diffusivity, inlet condition and gravity effect, the interface between two gases may be diffused or distorted as inhaled gas is drawn into the lungs. As time progresses, figures 5 and 6 reveal completely different flow characteristics for washin and washout. During Xe washin (left figures), as the interfacial front reaches the first bifurcation, Xe is split into two fluid streams, running separately into left and right lungs and creating M-shaped (curved) interface. This is apparently due to inertia effect resulting partially from gravitational acceleration. On the contrary, during washout, the interface maintains a cone shape with a needle-like front, keeping the inhaled gas away from the airway wall. Comparison of figures 5 and 6 (left) indicates that the cone-shaped interface is elongated during washout because the velocity is greater away from the airway wall. This special interface deformation is attributed to stable stratification that could prohibit effective mixing in the conducting airways.

Figure 4. (Left) Xe washin, (right) He washin at time 0.16 s in the upright posture.




Figure 5. (Left) Xe washin, (right) Xe washout, at time 0.20 s in the upright posture.

Figure 6. (Left) Xe washin, (right) Xe washout, at time 0.21 s in the upright posture.




Figure 7. (Left) (a,c) Xe washin, (b,d) Xe washout; (right) (a,c) He washin, (b,d) He washout, at time 0.2 s in the upright posture. (c,d) Velocity vectors and contours of density in the plane perpendicular to the flow direction in (a,b).

Figures 7 (left) (a,b) exhibit the velocity vector (in black) in the planes cutting across the fluid stream during Xe washin and washout, respectively, at time 0.2 s. The inhaled dense gas (Xe) tends to create a velocity profile that is biased toward the inner region of the bifurcation due to inertia effect. The impact of the dense gas on the bifurcation region subsequently creates M-shaped Xe gas distribution and counter-rotating vortices as the velocity vectors show in Fig.7(c) (left). The flow characteristics is akin to the secondary flow observed in the air-only case. For Xe washout, the velocity profile is more symmetric with respect to the centerline of the airway tube. In addition, the velocity vector field displayed in Fig. 7(d) (left) does not show any counter-rotating vortices as in the washin process. Instead, the light gas, viz. air relative to Xe, is confined to the inner region of the flow as the density contours show.

We then repeat the simulations by replacing Xe with He to study He/air washin and washout. The density ratio of air over He is 3.53 compared to the density ratio 2.25 of Xe over Air. Nevertheless, the binary diffusivity of He and air is one order larger than that of Xe and air. As pointed out before, the interfacial dynamics depends upon several factors, including gravity, stable versus unstable interfaces and binary diffusivity and among others. The results for He/air washin and washout are displayed in Fig. 7 (right). Since He is lighter than air, He washin exhibits the same feature as Xe washout, and vice versa. However, the stably stratified interface formed during He washin is not as sharp as that of Xe washout. This is probably caused by the strong binary diffusion between He and air.

5.2. Washin and Washout in the Supine Body Posture

We next consider washin and washout in the supine body posture. It is assumed that in this position, the trachea has an inclination angle of 10 degree with the end of trachea in the lower elevation than its beginning. The same initial and boundary conditions as those in section 5.1 are specified. Figure 8 (left) shows a time sequence of Xe being inhaled into an air-saturated lung. The interfacial front exhibits a tongue-like shape protruding along the lower airway wall and running downstream in the gravity direction. This phenomenon bears a striking resemblance with the density current occurring in nature. During Xe washout, figure 8 (right) shows the opposite effect of gravity, namely light gas is floating above dense gas and has the tendency to fill the upper portion of the lung. Figures 9 (left) (a,c) display the close-up view of Xe washin that creates a big vortex when Xe strikes the bifurcation region and makes a turn of 45 degree. Figures 9 (left) (b,d) indicate that for Xe washout, the light gas (air) floats over Xe and creates a skewed velocity profile, resulting in counter-rotating vortices. Similarly, for the He-air system, the same features are observed in Figs. 9 (right) except that He washin resembles Xe washout and vice versa. Besides, the He-air interface is more diffused than the Xe-air case.




Figure 8. A time sequence of (Left) Xe washin, (right) Xe washout in the supine posture.

Figure 9. (Left) (a,c) Xe washin, (b,d) Xe washout; (right) (a,c) He washin, (b,d) He washout, at time 0.2 s in the supine posture. (c,d) Velocity vectors and contours of density in the plane perpendicular to the flow direction in (a,b).




6. CONCLUDING REMARKS

In this paper we aim to understand gas transport in the human lungs to advance Xe-CT and He-MRI technologies. To achieve this goal, we used the state-of-the-art CFD method together with advanced geometrical modeling and segmentation technologies to simulate washin and washout of Xe and He in a CT-based human lung geometry. It is found that the gas density difference and gravity play a significant role in these processes. During washin of dense gas into light resident gas, if dense gas is introduced into light gas from above, an unstable interface could be formed and an inertia effect manifests itself. The phenomenon is often known as Rayleigh-Taylor instability. Other factors could also affect the development of the instability, such as binary diffusivity and inlet and boundary conditions. If light gas is placed over dense gas, with increasing convection effect over diffusion owing to increasing inhaled gas volume a stably stratified cone-shaped interface is formed to keep the front region away from the airway wall. The correlation between these stability phenomena and the washin and washout time constant difference is currently under investigation through laboratory experiments.

ACKNOWLEDGMENTS

The authors would like to thank Dr. Merryn Tawhai, Dr. Geoffrey McLennan, Dr. Juerg Tschirren, Dr. Taehun Lee, Senthil Kabilan, Georgette Stern, and Deokiee Chon for assistance and discussion during the course of the study. The CFD code is developed in part during the financial support of the Office of Naval Research (ONR) through grants #N00014-01-1-0262 and #N00014-05-1-0295, and the Public Utility District No. 2 of Grant County, Washington. The authors also thank the National Center for Supercomputing Applications (NCSA) for allocating the computer time to perform the above simulation. The second author Dr. Hoffman acknowledges the financial support of the National Institutes of Health through NIH grant #HL-064368.

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