Diffusion in Replica Healthy and Emphysematous Alveolar Models Using Computational Fluid Dynamics

Deposition of nanosized particles in the pulmonary region has the potential of crossing the blood-gas barrier. Experimental in vivo studies have used micron-sized particles, and therefore nanoparticle deposition in the pulmonary region is not well understood. Furthermore, little attention has been paid to the emphysematous lungs, which have characteristics quite different from the healthy lung. Healthy and emphysematous replica acinus models were created from healthy and diseased human lung casts using threedimensional reconstruction. Particle concentration and deposition were determined by solving the convective-diffusion equation numerically for steady and unsteady cases. Results showed decreased deposition efficiencies for emphysema compared to healthy lungs, consistent with the literature and attributed to significant airway remodeling in the diseased lung. Particle diffusionwas found to be six times slower in emphysema compared to healthy model. The unsteady state simulation predicted deposition efficiencies of 96% in the healthy model for the 1 nm and 3 nm particles and 94% and 93% in the emphysema model for the 1 nm and 3 nm particles, respectively. Steady state was achieved in less than one second for bothmodels. Comparisons between steady and unsteady predictions indicate that a steady-state simulation is reasonable for predicting particle transport under similar conditions.


Introduction
Emphysema is a chronic obstructive pulmonary disease characterized by irreversible damage to alveolar sacs in the pulmonary region of the lung [1,2].Specifically, the septa separating the individual acini are destroyed so that the tiny acini merge together to create one large air sac.The effect of emphysema-related lung remodeling on airflow and particle deposition is currently unknown.Understanding how deposition changes in the pulmonary region due to emphysema will help to improve differential risk assessment for those suffering from the disease as well as increase the accuracy of delivering inhaled medication used to treat disease.
Studies that model pulmonary airflow and deposition, both experimentally and numerically, are summarized in Table 1, including model characteristics and deposition mechanisms addressed in each study.Deposition in the pulmonary region is governed primarily by diffusion and sedimentation.Whole lung models, such as MPPD, Trumpet, and NCRP [3][4][5], use analytical deposition equations derived by Ingham [6][7][8] for straight tube geometries, whereas alveolar models have been published that account for the presence of individual acinar geometry.Some of the first studies made use of axisymmetry to create simplistic models of individual acini, whereas more recent models have begun to move closer to what many may consider actual geometries.Kumar et al. [9] and Sznitman et al. [10] presented a branching bronchiole numerical model with multihedron acini, and Berg et al. [11] presented a replica alveolar sac experimental model made from scans of human lung casts.No study has presented numerical results for diffusional deposition in replica alveolar geometries.
Studies that measured pulmonary deposition in vivo are summarized in Table 2. To some extent, the in vivo studies validated whole lung model predictions (Figure 1); however there is limited data for submicron-sized particles.Even fewer studies have looked at how the differences in healthy and  [3] S t r a i g h t t u b e X X Whole lung models Yu and Diu (Trumpet) [5] S t r a i g h t t u b e X X Anjilvel and Asgharian (MPPD) [4] S t r a i g h tt u b e X X Sturm and Hofmann [12] R o u n d X X X Alveolar models Tsuda et al. [13] R o u n d X Darquenne [14] R o u n d X Haber et al. [15] R o u n d X X Numerical Sznitman et al. [16] R o u n d X Kumar et al. [9] O c t a h e d r o n X Sznitman et al. [10] 1 4 -H e d r o n X X Harding and Robinson [17] R o u n d X Karl et al. [18] S q u a r e Experimental Oakes et al. [19] R o u n d X X Berg et al. [11] R e p l i c a X Berg and Robinson [20] R e p l i c a X X Table 2: Experimental studies involving pulmonary deposition (NR: not reported).Data is combined and plotted in Figure 1.7.5 breaths/minute 9 5 and 7 Emmett et al. [26] 10 breaths/minute 12 3.5-10 Kim and Jaques [27] 15 breaths/minute 22 0.04-0.1 emphysematous lung geometries affect particle transport and deposition.Sturm and Hofmann [12] presented a numerical model for particle deposition by the combined mechanisms of sedimentation and diffusion using various models of emphysema, for 0.001 to 10 m particles in generations from the 12th to terminal.Particle deposition was shown to be higher in the healthy model compared to models representing various types of emphysema.Oakes et al. [19] and later Berg and Robinson [20] measured flow fields in idealized and replica models, respectively, by particle image velocimetry and found flow characteristics that would imply decreased deposition in emphysema compared to healthy.Kohlhäufl et al. [28] measured, on average, a greater bolus dispersion in 29 emphysema subjects compared to 79 healthy patients [29], which they attributed to airway blockage, unequal spread of the disease, and uneven ventilation.In summary, limited data are available to describe the nature of diffusional particle deposition in the alveolar region, relying only on the whole lung models.Limited experimental data is available to validate whole lung models for particles in the submicron region.In our previous work,  2. Kim and Jaques [27] data show the average for 22 subjects.

Study
we showed that in the alveolar region, diffusive motion likely dominates for submicron particles [11,17].However, alveolar models to date have focused on sedimentation and to our knowledge, no study on diffusional deposition has been presented.The limited studies to date indicate that deposition is decreased in emphysema; however, more work is needed to fully characterize and understand the effects of diseased lung remodeling.In the present study, pure diffusional transport is simulated numerically in alveolar replica models of healthy and emphysemic geometries.

Model Creation.
The healthy and emphysemic geometric models used in this study were previously used to measure  2. The lung models were reconstructed from the 2D images using 3D Doctor (Able Software, Lexington, MA, USA), applying a thresholding segmentation technique.Thresholding values between the limits of 45 and 255 provided the most optimal results, as it included the largest majority of the cast surface without including too much of the surrounding static.After examining the preliminary reconstructions (Figures 3(a) and 3(b)), smaller sections were chosen (Figures 3(c) and 3(d)) which reduced the total number of 2D images to 492 in the healthy model and 919 for the emphysematous.Only these regions were further processed and cleaned, while the other portions present in the image were deleted.Figures 3(e) and 3(f) show the original casts for comparison of the final reduced sections.This final reduction in the model size was done to reduce the number of 2D images that would require detailed image cleanup, to reduce the total number of meshed elements below the maximum supported by available RAM, and to reduce the overall computational time for simulation.

ISRN Biomedical Engineering
The final healthy model consisted of 368,238 nodes and 736,416 faces, while the emphysema final model had 1,589,498 nodes and 3,174,083 faces.
After reconstructions were done in 3D Doctor, object files, or OBJ files, were created for each of the models and exported to allow an easy transition into VP Sculpt (Colorado State University, Fort Collins, CO, USA) for surface cleanup to prepare the model for meshing.Each model was smoothed roughly fifty times to eliminate any outlying features, while maintaining the original number of faces and nodes that make up the model's surface.The models were then decimated, which is a process that combines similar faces to create larger faces, thereby reducing the model size without losing detail to the surface.It was necessary to decimate the models four times before the size was reduced enough for importing into SolidWorks (Solid Works Corp., Waltham, MA, USA) for model scaling.The healthy human model was reduced to 27,012 nodes and 54,032 faces after decimating, while the emphysemic human model was reduced to 23,497 nodes and 46,982 faces.
Finally, improper, null, and appended facets and unconnected vertices that may have resulted from either the reconstruction or smoothing and decimation processes were removed, and outer boundaries and holes were filled in order to produce a complete volume.The final step was to scale the models to accurately depict the original in vivo dimensions.SEM (scanning electron micrographs) and high-powered microscope images were analyzed using ImageJ to determine millimeter (mm) to pixel conversion factors.A scaling factor of 0.00361 pixels/mm was used for the healthy model, and a factor of 0.00306 pixels/mm was used for the emphysemic model.
The final healthy human model extends 2.11 mm in the -direction, 1.34 mm in the -direction, and 1.74 mm in the -direction with an inlet duct diameter of 0.41 mm.The total volume of the model was 1.16 mm 3 with a total surface area of 10.46 mm 2 .The emphysema human model had a total volume of 12.49 mm 3 and surface area of 33.56 mm 2 and is 10.8 times larger than the healthy model by volume.The global size of the model measures 3.74 mm, 3.18 mm, and 3.49 mm in the , , and  directions, respectively, with an inlet duct diameter of 0.81 mm.These dimensions are well within published values [28,[31][32][33][34].

Mesh Generation.
Meshing was done in Harpoon (Sharc Ltd., Manchester, UK), an automatic hex dominant volume meshing software.The models were exporting from VP Sculpt as STL files and imported into Harpoon as one complete surface, which was then broken into two faces, the inlet surface and acinus walls.The healthy model was meshed using a max volume size of 0.04 and a surface cell size of 0.010, which created 774,380 elements and 627,214 nodes (Figure 4(a)).The surface cell size was approximately 0.6% of the healthy model's , , and  dimensions, similar to the initial boundary layer height of 0.5% of the radial and axial dimensions of the tube used in the Chen validation [35].The max volume size is 2.4% on average of the model's dimensions, which also compares well with the internal mesh of 2.5% of the dimensions used in Chen's work.The emphysemic model was meshed using a base level of 0.025, a surface cell size of 0.01250, or 0.4% of the model's size, and max volume size of 0.02500, or 0.7% of the overall size, which created 947,199 elements and 842,689 nodes (Figure 4(b)).An example of the hex-meshing scheme is shown in a blown-up portion of the healthy model (Figure 4(c)).

Theory of Brownian Diffusion and Convective-Diffusion
Equation.Fick's law of diffusion was applied to solve for particle transport in the healthy and emphysemic models.Particle diffusion is the mass transfer of particles from a region of higher concentration to a region of lower concentration, as a result of Brownian motion or random particle motion due to the random relentless bombardment of gas molecules [36].Fick's first law of diffusion is given by where  is the flux vector (particles/m 2 s),  is the diffusion coefficient (m 2 /s), and  (, , , ) is the particle concentration (particles/m 3 ).If Fick's first law is applied to a finite volume element, along with the conservation of mass theorem, the particle concentration can be found by solving the convection-diffusion equation for zero mass generation [37] by where  is the time.Total particle diffusion through any given surface area,   area (particles/s), can be found by integrating  (, , , ) over the surface area,  area , using where  is the direction normal to the surface area.Equation ( 3) can be integrated over the inlet surface area to determine the rate of particle diffusion into the mode,   inlet , and integrated over the wall surface area,   wall , to determine diffusional deposition on the model wall.Total diffusion can be found by summing (3) over time, , so that deposition efficiency can be determined from (4)

Numerical Techniques.
The convective-diffusion equation was solved to obtain concentration as a function of position and time using the Fine Particle Model (FPM), which is a user defined subroutine (UDF) for the Computational Fluid Dynamics (CFD) software package, FLUENT (ANSYS, Inc., Canonsburg, PA, USA) [38].The FPM solves an equation known as the Moment Dynamics Equation,   which is simplified to (2) for monodisperse aerosols dominated by diffusion.The particle flux at the inlet,   inlet , and the wall,   wall , was extracted from the CFD/FPM analysis after each simulation time step.The deposition efficiencies were determined manually by applying (4) to the exported particle flux data.
A constant inlet concentration of 1e15 particles/m 3 was set at each of the model inlets, while the wall boundary conditions were set to a concentration of 0 particles/m 3 .Particle sizes of 1 nm and 3 nm diameter were analyzed for a period of 5 seconds.

Concentration Contours in Healthy and
Emphysema Models.Unsteady concentration contours in the healthy and emphysemic models were taken at an - plane located in the center of the inlet duct for 1 and 3 nm particles.Figure 5 shows the concentration contours for the 3 nm particles in the healthy and emphysematous models at three points in time during the simulation period.Similar contours were obtained for the 1 nm particle (not shown).For both particle sizes, a larger time was needed to penetrate the emphysemic model compared to the healthy mode, and to a lesser extent, larger times were needed for the emphysemic model to reach steady-state contours compared to the healthy model.
For both the 1 nm and 3 nm particles, the emphysemic model took 6 times longer to reach similar concentration profiles compared to the healthy model, and steady state was reached in half the time for the healthy compared to the emphysemic model.The associated times for the 3 nm results are provided in Figure 5.For the 1 nm particles (not shown), the emphysemic model took 0.006 sec to reach the same penetration depth shown in Figure 5 (middle row), while the healthy model took only 0.001 sec.Furthermore, for the 1 nm particle, steady state was reached in 0.08 sec for the healthy model compared to 0.17 sec for the emphysemic model.
As expected, the steady-state concentration contours were the same for both the 1 nm and 3 nm particles since at steady state, the diffusion coefficient no longer contributes to the solution of the concentration (2).Because steady state is reached relatively quickly, it is reasonable to consider the steady-state solution as a first approximation for deposition calculations that rely on these particle concentration contours.
Steady-state particle contours are shown in Figure 6 for both the healthy and emphysemic models.Each model was sectioned into six planes.The healthy model planes are equally spaced by 0.248 mm, while the emphysemic model planes are equally spaced by 0.499 mm.Since these are steady-state concentrations, they are independent of particle size.It is evident from these isometric images that the emphysemic model has a much lower concentration of inhaled particles distributed throughout the airway volume as compared to the healthy model.These differences in concentration gradient, particularly near the airway wall, will result in smaller particle deposition for emphysemic lungs compared to healthy lungs.

Diffusional Deposition in Healthy and
Emphysemic Models.Total particle deposition in the healthy and emphysemic models was obtained by summing the wall flux output over various time periods using (3).The deposition efficiency was calculated using (4).Figure 7 shows the percent deposition for 1 nm and 3 nm particles in the healthy and emphysemic models over a 2-second period.The deposition rate is larger for the healthy model compared to the emphysemic model for both particle sizes.Figure 7 also demonstrates that each model simulation quickly approached steady state, more quickly for the healthy compared to the emphysemic model and for the 1 nm compared to 3 nm particle size (Table 3).Assuming steady-state deposition over the entire breathing period provides the opportunity to simplify the numerical simulation.The error associated with the steady-state assumption is provided in Table 3 for a range of particle residence times, assuming that steady state occurs when the slope of the deposition efficiency curve becomes less than 0.1% per second (Figure 7).The 1 nm healthy simulation reached steady state at 0.08 sec.The worst case shown is for a 3 nm particle in the emphysemic model, where the steady simulation yields a 4.12% overestimation compared to the unsteady simulation for 0.49 second residence time.The error reduces for larger, more realistic residence times.Therefore, it appears that the steady-state assumption is reasonable.

Effect of Emphysema on Particle Deposition.
The simulated deposition efficiencies for the emphysema model are less than the healthy model.Decreased deposition in emphysema can be attributed to alveolar remodeling due to the destruction of the septa walls.The large, nonbranching volume of the diseased sac results in larger diffusion distances and smaller particle concentration gradients in the emphysemic compared to the healthy lung, as shown in the simulation results.It takes longer for the particles to reach an alveolar surface in emphysema and longer to develop a steady-state particle concentration gradient.Once the particle gradient reaches steady state, the magnitude of the gradient, which is the driving force behind diffusive particle transport and deposition, is much smaller in emphysema compared to the healthy lung.

Comparison of Numerical and Experimental Studies.
Simulations presented in this study are consistent with the numerical predictions of Sturm and Hoffmann [12], in which they reported a 7% decrease in pulmonary region deposition of emphysema compared to healthy and are consistent with decreased deposition in animal models with emphysema [39][40][41][42][43].
Experimental studies of particle deposition in the submicron range are not available for comparison of simulation results of the present study.The smallest experimental data point is a mean deposition efficiency of 33% for 0.04 microns, reported by Kim and Jaques [27] for healthy humans (Figure 1).The experimental trend in the range from 0.1 to 0.4 microns is reported to be an increase in deposition efficiency for decreasing particle size.

Conclusions
Two in vivo replica alveolar models, healthy and emphysematous, were created using lung casting and three-dimensional reconstruction techniques.The resulting geometries of the healthy and emphysemic models were consistent with the literature in terms of general features, sizes, and shapes of the alveoli.The healthy and emphysemic models were simulated to determine deposition of 1 nm and 3 nm particles by pure diffusion, for the same inhaled particle concentration.The study predicts very high and localized deposition in the alveolar sacs, with a decreased efficiency for emphysemic compared to the healthy lung, which can be attributed to geometric remodeling as a result of alveolar wall destruction.Results agree qualitatively with in vivo experimental published data reporting decreased deposition in emphysema compared to healthy lungs.Additional studies are needed, both experimental and numerical, to better understand nanoparticle deposition in the pulmonary region of the lung.

Figure 1 :
Figure 1: Whole lung model pulmonary deposition predictions compared to pulmonary deposition measurement studies listed in Table2.Kim and Jaques[27] data show the average for 22 subjects.

Figure 2 :
Figure 2: Examples of microCT image slices shown in ImageJ from (a) the healthy lung cast and the (b) the emphysema lung cast.The white line in the images represents 1 mm.

Figure 3 :
Figure 3: Original (a) healthy and (b) emphysemic casts.The red circles represent the sections chosen for final models, while the black lines demonstrate model scales.Whole model reconstructions of (c) healthy and (d) emphysemic models and the final chosen sections for the (e) healthy and (f) emphysemic models.

Figure 4 :
Figure 4: Meshed (a) healthy and (b) emphysemic models and (c) an example of the the hex meshing scheme.

Figure 5 :Figure 6 :Figure 7 :
Figure 5: Unsteady concentration contours (particles/mm 3 ) for 3 nm particles, for the times shown.Bottom row contours are at steady state for each model.Left and right panels are for the healthy and emphysemic models, respectively.The plane shown is located at the center of each model's inlet.

Table 1 :
Summary of whole lung models and alveolar models found in the literature.

Table 3 :
Steady-state (SS) times, deposition efficiencies, and percent error associated with the steady-state approximation for various residence times.
[31]32]elds by particle image velocimetry[20].Healthy and emphysematous human cadaver lungs were cast using the procedure of Phalen et al.[30].Microscope pictures were taken of each of the casted pieces to determine which sections would be used for final reconstruction.Healthy humans typically have alveolar mouth diameters of approximately 230 to 336 m[31,32]and represent defined bulb structures.Our healthy human cast was within this range, measuring between 243 and 378 microns.The literature cites emphysematous diameters ranging from 430 to 830 m[31]while our model's average effective airway diameter measures 1556 microns, indicating that our model was obtained from a severely diseased lung.Chosen sections were cut from the larger cast and scanned using microscale computed tomography (microCT) (Micro Photonics, Allentown, PA, USA).The healthy lung cast, measuring roughly 6 mm in size, was scanned at 3.58 m/pixel and created 1,241 two-dimensional (2D) images, each containing a 2000 × 2000 pixel image with a file size of about 4000 KB.The emphysematous lung cast, measuring about 12 mm in size, was scanned at 3.35 m/pixel, which produced 3,550 2D images, each a 4000 × 4000 pixel image approximately 15,300 KB in size.A representative example of a 2D image from the microCT scan is shown in Figure