A three-dimensional model of the human pulmonary acinus

How to build an epithelial tree

Nature has evolved a variety of mechanisms to build epithelial trees of diverse architectures within different organs and across species. Epithelial trees are elaborated through branch initiation and extension, and their morphogenesis ends with branch termination. Each of these steps of the branching process can be driven by the actions of epithelial cells themselves (epithelial-intrinsic mechanisms) or by the cells of their surrounding tissues (epithelial-extrinsic mechanisms). Here, we describe examples of how these mechanisms drive each stage of branching morphogenesis, drawing primarily from studies of the lung, kidney, salivary gland, mammary gland, and pancreas, all of which contain epithelial trees that form through collective cell behaviors. Much of our understanding of epithelial branching comes from experiments using mice, but we also include examples here from avian and reptilian models. Throughout, we highlight how distinct mechanisms are employed in different organs and species to build epithelial trees. We also highlight how similar morphogenetic motifs are used to carry out conserved developmental programs or repurposed to support novel ones. Understanding the unique strategies used by nature to build branched epithelia from across the tree of life can help to inspire creative solutions to problems in tissue engineering and regenerative medicine.

Mathematical modeling of pulmonary acinus structure: Verification of acinar shape effects on pathway structure using rat lungs

The pulmonary acinus is the gas exchange unit in the lung and has a very complex microstructure. The structure model is essential to understand the relationship between structural heterogeneity and mechanical phenomena at the acinus level with computational approaches. We propose an acinus structure model represented by a cluster of truncated octahedra in conical, double-conical, inverted conical, or chestnut-like conical confinement to accommodate recent experimental information of rodent acinar shapes. The basis of the model is the combined use of Voronoi and Delaunay tessellations and the optimization of the ductal tree assuming the number of alveoli and the mean path length as quantities related to gas exchange. Before applying the Voronoi tessellation, controlling the seed coordinates enables us to model acinus with arbitrary shapes. Depending on the acinar shape, the distribution of path length varies. The lengths are more widely spread for the cone acinus, with a bias toward higher values, while most of the lengths for the inverted cone acinus primarily take a similar value. Longer pathways have smaller tortuosity and more generations, and duct length per generation is almost constant irrespective of generation, which agrees well with available experimental data. The pathway structure of cone and chestnut-like cone acini is similar to the surface acini's features reported in experiments. According to space-filling requirements in the lung, other conical acini may also be acceptable. The mathematical acinus structure model with various conical shapes can be a platform for computational studies on regional differences in lung functions along the lung surface, underlying respiratory physiology and pathophysiology.

Count- and mass-based dosimetry of MDI spray droplets with polydisperse and monodisperse size distributions

Most previous numerical studies of inhalation drug delivery used monodisperse aerosols or quantified deposition as the ratio of deposited particle number over the total number of released particles (i.e., count-based). These practices are reasonable when the aerosols have a sufficiently narrow size range. However, spray droplets from metered-dose inhalers (MDIs) are often polydisperse with a wide size range, so using monodisperse aerosols and/or count-based deposition quantification may lead to significant errors. The objective of this study was to develop a mass-based dosimetry method and evaluate its performance in lung delivery in a mouth-lung (G9) geometry with an albuterol-CFC inhaler. The conventional practices (monodisperse and polydisperse-count-based) were also simulated for comparison purposes. The MDI actuation in the open space was studied using both high-speed imaging and LES-Lagrangian simulations. Experimentally measured spray velocities and size distribution were implemented in the computational model as boundary conditions. Good agreement was achieved between recorded and simulated spray plume evolution spatially and temporally. The polydisperse-mass-based predictions of MDI doses compared favorably with the measurements in all three regions considered (device, mouth-throat, and lung). Significant errors in MDI regional deposition were predicted using the monodisperse and count-based methods. The new polydisperse-mass-based method also predicted local deposition hot spots that were one order of magnitude higher in intensity than the two conventional methods. The results of this study highlighted that a presentative polydisperse size distribution and appropriate deposition quantification method should be applied to reliably predict the MDI drug delivery in the human respiratory tract.

Lower Inspiratory Breathing Depth Enhances Pulmonary Delivery Efficiency of ProAir Sprays

Effective pulmonary drug delivery using a metered-dose inhaler (MDI) requires a match between the MDI sprays, the patient’s breathing, and respiratory physiology. Different inhalers generate aerosols with distinct aerosol sizes and speeds, which require specific breathing coordination to achieve optimized delivery efficiency. Inability to perform the instructed breathing maneuver is one of the frequently reported issues during MDI applications; however, their effects on MDI dosimetry are unclear. The objective of this study is to systemically evaluate the effects of breathing depths on regional deposition in the respiratory tract using a ProAir-HFA inhaler. An integrated inhaler mouth-throat-lung geometry model was developed that extends to the ninth bifurcation (G9). Large-eddy simulation (LES) was used to compute the airflow dynamics due to concurrent inhalation and orifice flows. The discrete-phase Lagrangian model was used to track droplet motions. Experimental measurements of ProAir spray droplet sizes and speeds were used as initial and boundary conditions to develop the computational model for ProAir-pulmonary drug delivery. The time-varying spray plume from a ProAir-HFA inhaler into the open air was visualized using a high-speed imaging system and was further used to validate the computational model. The inhalation dosimetry of ProAir spray droplets in the respiratory tract was compared among five breathing depths on a regional, sub-regional, and local basis. The results show remarkable differences in airflow dynamics within the MDI mouthpiece and the droplet deposition distribution in the oral cavity. The inhalation depth had a positive relationship with the deposition in the mouth and a negative relationship with the deposition in the five lobes beyond G9 (small airways). The highest delivery efficiency to small airways was highest at 15 L/min and declined with an increasing inhalation depth. The drug loss inside the MDI was maximal at 45–60 L/min. Comparisons to previous experimental and numerical studies revealed a high dosimetry sensitivity to the inhaler type and patient breathing condition. Considering the appropriate inhalation waveform, spray actuation time, and spray properties (size and velocity) is essential to accurately predict inhalation dosimetry from MDIs. The results highlight the importance of personalized inhalation therapy to match the patient’s breathing patterns for optimal delivery efficiencies. Further complimentary in vitro or in vivo experiments are needed to validate the enhanced pulmonary delivery at 15 L/min.

Evaluation of Impulse Oscillometry in Respiratory Airway Casts with Varying Obstruction Phenotypes, Locations, and Complexities

The use of impulse oscillometry (IOS) for lung function testing does not need patient cooperation and has gained increasing popularity among both young and senior populations, as well as in patients with breathing difficulties. However, studies of the IOS sensitivity to regional lung obstructions are limited and have shown mixed results. The objective of this study was to evaluate the performance of an IOS system in 3D-printed lung models with structural abnormalities at different locations and with different severities. Lung trees of two complexity levels were tested, with one extending to the sixth generation (G6) and the other to G12. The IOS responses to varying glottal apertures, carina ridge tumors, and segmental bronchial constrictions were quantified in the G6 lung geometry. Both the G6 and G12 lung casts were prepared using high-resolution 3D printers. Overall, IOS detected the progressive airway obstructions considered in this study. The resonant frequency dropped with increasing obstructions for all three disease phenotypes in the G6 lung models. R20Hz increased with the increase in airway obstructions. Specifically, R20Hz in the airway model with varying glottal apertures agreed reasonably well with complementary measurements using TSI VelociCalc. In contrast to the high-resistance (R) sensitivity to the frequency in G6 lung models, R was nearly independent of frequency in G12 lung models. IOS R20Hz demonstrated adequate sensitivity to the structural remodeling in the central airways. However, the changes of R5Hz and X5Hz vs. airway obstructions were inconclusive in this study, possibly due to the rigid lung casts and the difference of a container–syringe system from human lungs.

Effect of MDI Actuation Timing on Inhalation Dosimetry in a Human Respiratory Tract Model

Accurate knowledge of the delivery of locally acting drug products, such as metered-dose inhaler (MDI) formulations, to large and small airways is essential to develop reliable in vitro/in vivo correlations (IVIVCs). However, challenges exist in modeling MDI delivery, due to the highly transient multiscale spray formation, the large variability in actuation–inhalation coordination, and the complex lung networks. The objective of this study was to develop/validate a computational MDI-releasing-delivery model and to evaluate the device actuation effects on the dose distribution with the newly developed model. An integrated MDI–mouth–lung (G9) geometry was developed. An albuterol MDI with the chlorofluorocarbon propellant was simulated with polydisperse aerosol size distribution measured by laser light scatter and aerosol discharge velocity derived from measurements taken while using a phase Doppler anemometry. The highly transient, multiscale airflow and droplet dynamics were simulated by using large eddy simulation (LES) and Lagrangian tracking with sufficiently fine computation mesh. A high-speed camera imaging of the MDI plume formation was conducted and compared with LES predictions. The aerosol discharge velocity at the MDI orifice was reversely determined to be 40 m/s based on the phase Doppler anemometry (PDA) measurements at two different locations from the mouthpiece. The LES-predicted instantaneous vortex structures and corresponding spray clouds resembled each other. There are three phases of the MDI plume evolution (discharging, dispersion, and dispensing), each with distinct features regardless of the actuation time. Good agreement was achieved between the predicted and measured doses in both the device, mouth–throat, and lung. Concerning the device–patient coordination, delayed MDI actuation increased drug deposition in the mouth and reduced drug delivery to the lung. Firing MDI before inhalation was found to increase drug loss in the device; however, it also reduced mouth–throat loss and increased lung doses in both the central and peripheral regions.

Septal destruction enhances chaotic mixing and increases cellular doses of nanoparticles in emphysematous acinus

One hallmark of emphysema is the breakdown of inter-alveolar septal walls in pulmonary acini. How the acinar dosimetry of environmental aerosols varies at different stages of emphysema remains unclear; this is specifically pertinent to users of tobacco products, which is the leading cause of emphysema. The objective of this study is to systematically assess the impacts of septal destruction on the behavior and fate of nanoparticles (1–800 nm) in a pyramid-shaped sub-acinar model consisting of 496 alveoli. Four diseased geometry variants were created by gradually removing the septal walls from the base model. Particle motions within the acinar region were tracked for particles raging 1–800 nm at four emphysema stages using a well-tested Lagrangian tracking model. Both spatial profile and temporal variation of particle deposition were predicted in healthy and diseased sub-acinar geometries on both a total and regional basis. Results show large differences in airflow and particle dynamics among different emphysema stages. Large differences in particle dynamics are also observed among different particle sizes, with one order of magnitude’s variation in the speeds of particles of 1, 10, and 200 nm. The destruction of septal walls also changed the deposition mechanisms, shifting from connective diffusion to chaotic mixing with emphysema progression. The sub-acinar dosimetry became less sensitive to particle size variation with more septal destructions. The lowest retention rate was found at 200–500 nm in the healthy sub-acinar geometry, but at 800 nm in all emphysematous models considered. The acinus-averaged dose for nanoparticles (1–800 nm) increases with aggravating septal destructions, indicating an even higher risk to the acinus at later emphysema stages.

SARS COV-2 virus-laden droplets coughed from deep lungs: Numerical quantification in a single-path whole respiratory tract geometry

When an infected person coughs, many virus-laden droplets will be exhaled out of the mouth. Droplets from deep lungs are especially infectious because the alveoli are the major sites of coronavirus replication. However, their exhalation fraction, size distribution, and exiting speeds are unclear. This study investigated the behavior and fate of respiratory droplets (0.1-4 μm) during coughs in a single-path respiratory tract model extending from terminal alveoli to mouth opening. An experimentally measured cough waveform was used to control the alveolar wall motions and the flow boundary conditions at lung branches from G2 to G18. The mouth opening was modeled after the image of a coughing subject captured using a high-speed camera. A well-tested k-ω turbulence model and Lagrangian particle tracking algorithm were applied to simulate cough flow evolutions and droplet dynamics under four cough depths, i.e., tidal volume ratio (TVR) = 0.13, 0.20. 0.32, and 0.42. The results show that 2-μm droplets have the highest exhalation fraction, regardless of cough depths. A nonlinear relationship exists between the droplet exhalation fraction and cough depth due to a complex deposition mechanism confounded by multiscale airway passages, multiregime flows, and drastic transient flow effects. The highest exhalation fraction is 1.6% at the normal cough depth (TVR = 0.32), with a mean exiting speed of 20 m/s. The finding that most exhaled droplets from deep lungs are 2 μm highlights the need for more effective facemasks in blocking 2-μm droplets and smaller both in infectious source control and self-protection from airborne virus-laden droplets.

Virtual clinical trials in medical imaging: a review

The accelerating complexity and variety of medical imaging devices and methods have outpaced the ability to evaluate and optimize their design and clinical use. This is a significant and increasing challenge for both scientific investigations and clinical applications. Evaluations would ideally be done using clinical imaging trials. These experiments, however, are often not practical due to ethical limitations, expense, time requirements, or lack of ground truth. Virtual clinical trials (VCTs) (also known as in silico imaging trials or virtual imaging trials) offer an alternative means to efficiently evaluate medical imaging technologies virtually. They do so by simulating the patients, imaging systems, and interpreters. The field of VCTs has been constantly advanced over the past decades in multiple areas. We summarize the major developments and current status of the field of VCTs in medical imaging. We review the core components of a VCT: computational phantoms, simulators of different imaging modalities, and interpretation models. We also highlight some of the applications of VCTs across various imaging modalities.

Interplay Between Optimal Ventilation and Gas Transport in a Model of the Human Lung

Ventilation is at the origin of a spending of energy coming from air circulation in the bronchial tree and from the mechanical resistance of the tissue to motion. Both amplitude and frequency of ventilation are submitted to a trade-off related to this energy, but they are also submitted to a constraint linked to the function of the lung: to transport enough oxygen and carbon dioxide in order to respect metabolism needs. We propose a model for oxygen and carbon dioxide transport in the lung that accounts for the core physical phenomena: lung's tree-like geometry, transport of gas by convection and diffusion, exchanges with blood and a sinusoidal ventilation. Then we optimize the power dissipated by the ventilation of our model relatively to ventilation amplitude and period under gas flow constraints. Our model is able to predict physiological ventilation properties and brings interesting insights on the robustness of different regimes. Hence, at rest, the power dissipated depends very little on the period near the optimal value. Whereas, at strong exercise any shift from the optimal has dramatic effect on the power. These results are fully coherent with the physiological observation and raises the question: how the control of ventilation could select for the optimal configuration? Also, interesting insights about pathologies affecting ventilation could be derived, and our model might give insights on therapeutical responses, with specific breathing strategies or for better driving mechanical ventilation.

A simplified parametrised model for lung microstructures capable of mimicking realistic geometrical and mechanical properties

The respiratory zone of mammalian lungs contains several millions of so-called alveoli. The geometrical and mechanical properties of this microstructure are crucial for respiration and influence the macroscopic behaviour of the entire organ in health and disease. Hence, if computational models are sought to gain more insight into lung behaviour, predict lung states in certain scenarios or suggest better treatment options in early stages of respiratory dysfunction, an adequate representation of this microstructure is essential. However, investigating the real alveolar architecture requires complex medical-imaging methods and would be computationally extremely expensive. Even worse, there is currently no way of obtaining the real patient-specific microstructure in vivo. Hence, we present a fast and easy to compute parametrised model of lung microstructures based on tetrakaidecahedra which can represent both geometrical and mechanical properties of the parenchyma. We show that gas transport pathways and stress and strain distributions are comparable to real alveolar microstructures and even capable of capturing variations present in biology. The created parametrised lung microstructure models can be utilized in finite element simulations to study, e.g., alveolar flow phenomena, particle deposition, or alveolar stresses and strains during mechanical ventilation. Due to the simpler geometry of the parametrised microgeometries compared to imaging-based microstructures, remarkable savings in CPU time can be achieved. We show that our model requires a minimum of 10% of the computational time for computing the same strain state in structural mechanics simulations compared to imaging-based alveolar microstructures.

Diffusion lung imaging with hyperpolarized gas MRI

Lung imaging using conventional 1 H MRI presents great challenges because of the low density of lung tissue, lung motion and very fast lung tissue transverse relaxation (typical T2 * is about 1-2 ms). MRI with hyperpolarized gases (3 He and 129 Xe) provides a valuable alternative because of the very strong signal originating from inhaled gas residing in the lung airspaces and relatively slow gas T2 * relaxation (typical T2 * is about 20-30 ms). However, in vivo human experiments should be performed very rapidly - usually during a single breath-hold. In this review, we describe the recent developments in diffusion lung MRI with hyperpolarized gases. We show that a combination of the results of modeling of gas diffusion in lung airspaces and diffusion measurements with variable diffusion-sensitizing gradients allows the extraction of quantitative information on the lung microstructure at the alveolar level. From an MRI scan of less than 15 s, this approach, called in vivo lung morphometry, allows the provision of quantitative values and spatial distributions of the same physiological parameters as measured by means of 'standard' invasive stereology (mean linear intercept, surface-to-volume ratio, density of alveoli, etc.). In addition, the approach makes it possible to evaluate some advanced Weibel parameters characterizing lung microstructure: average radii of alveolar sacs and ducts, as well as the depth of their alveolar sleeves. Such measurements, providing in vivo information on the integrity of pulmonary acinar airways and their changes in different diseases, are of great importance and interest to a broad range of physiologists and clinicians. We also discuss a new type of experiment based on the in vivo lung morphometry technique combined with quantitative computed tomography measurements, as well as with gradient echo MRI measurements of hyperpolarized gas transverse relaxation in the lung airspaces. Such experiments provide additional information on the blood vessel volume fraction, specific gas volume and length of the acinar airways, and allow the evaluation of lung parenchymal and non-parenchymal tissue. Copyright © 2015 John Wiley & Sons, Ltd.

Probing lung microstructure with hyperpolarized noble gas diffusion MRI: theoretical models and experimental results

The introduction of hyperpolarized gases ((3)He and (129)Xe) has opened the door to applications for which gaseous agents are uniquely suited-lung MRI. One of the pulmonary applications, diffusion MRI, relies on measuring Brownian motion of inhaled hyperpolarized gas atoms diffusing in lung airspaces. In this article we provide an overview of the theoretical ideas behind hyperpolarized gas diffusion MRI and the results obtained over the decade-long research. We describe a simple technique based on measuring gas apparent diffusion coefficient (ADC) and an advanced technique, in vivo lung morphometry, that quantifies lung microstructure both in terms of Weibel parameters (acinar airways radii and alveolar depth) and standard metrics (mean linear intercept, surface-to-volume ratio, and alveolar density) that are widely used by lung researchers but were previously available only from invasive lung biopsy. This technique has the ability to provide unique three-dimensional tomographic information on lung microstructure from a less than 15 s MRI scan with results that are in good agreement with direct histological measurements. These safe and sensitive diffusion measurements improve our understanding of lung structure and functioning in health and disease, providing a platform for monitoring the efficacy of therapeutic interventions in clinical trials.

Mathematical model of a heterogeneous pulmonary acinus structure

The pulmonary acinus is a gas exchange unit distal to the terminal bronchioles. A model of its structure is important for the computational investigation of mechanical phenomena at the acinus level. We propose a mathematical model of a heterogeneous acinus structure composed of alveoli of irregular sizes, shapes, and locations. The alveoli coalesce into an intricately branched ductal tree, which meets the space-filling requirement of the acinus structure. Our model uses Voronoi tessellation to generate an assemblage of the alveolar or ductal airspace, and Delaunay tessellation and simulated annealing for the ductal tree structure. The modeling condition is based on average acinar and alveolar volume characteristics from published experimental information. By applying this modeling technique to the acinus of healthy mature rats, we demonstrate that the proposed acinus structure model reproduces the available experimental information. In the model, the shape and size of alveoli and the length, generation, tortuosity, and branching angle of the ductal paths are distributed in several ranges. This approach provides a platform for investigating the heterogeneous nature of the acinus structure and its relationship with mechanical phenomena at the acinus level.

Transport of gases between the environment and alveoli--theoretical foundations

The transport of oxygen and carbon dioxide in the gas phase from the ambient environment to and from the alveolar gas/blood interface is accomplished through the tracheobronchial tree, and involves mechanisms of bulk or convective transport and diffusive net transport. The geometry of the airway tree and the fluid dynamics of these two transport processes combine in such a way that promotes a classical fractionation of ventilation into dead space and alveolar ventilation, respectively. This simple picture continues to capture much of the essence of gas phase transport. On the other hand, a more detailed look at the interaction of convection and diffusion leads to significant new issues, many of which remain open questions. These are associated with parallel and serial inhomogeneities especially within the distal acinar units, velocity profiles in distal airways and terminal spaces subject to moving boundary conditions, and the serial transport of respiratory gases within the complex acinar architecture. This article focuses specifically on the theoretical foundations of gas transport, addressing two broad areas. The first deals with the reasons why the classical picture of alveolar and dead space ventilation is so successful; the second examines the underlying assumptions within current approximations to convective and diffusive transport, and how they interact to effect net gas exchange.

The influence of lung airways branching structure and diffusion time on measurements and models of short-range 3He gas MR diffusion

Hyperpolarized (3)He diffusion experiments have been shown to be sensitive to changes in acinar structure due to emphysematous lung disease. Extracting quantitative information about lung microstructure from the diffusion signal is complicated due its dependence on a number of factors including diffusion time and the complex branching acinar geometry. A theoretical model (cylinder model) has been proposed as a means of estimating acinar airway dimensions from measured diffusivities. This model assumes that the effects of acinar branching geometry and finite airway length upon (3)He diffusion behaviour are negligible. In this work, we use finite element simulations of diffusion in a model of branching alveolar ducts to investigate in detail the effects of acinar branching structure and finite airway length on short-range (3)He diffusion measurements. The results show that branching effects have a significant influence upon (3)He diffusivity, even at short diffusion times. The expressions of the cylinder model theory do not account for significant dependences upon diffusion time, branching geometry and airway length, as a consequence of the oversimplified geometrical model used. The effect of diffusion time on (3)He ADC was also investigated through experiments with healthy human volunteers. The results demonstrate that the cylinder model can produce inaccurate estimates of the airway dimensions as a consequence of incompletely accounting for the diffusion-time dependence in the model equations and confirmed the predicted limitations of the cylinder model for reliable lung morphometry measurements. The results and models presented in this work may help in the development of a more realistic theoretical framework for 'in vivo lung morphometry' using (3)He diffusion MR.

Experimental investigation and numerical simulation of 3He gas diffusion in simple geometries: implications for analytical models of 3He MR lung morphometry

Models of lung acinar geometry have been proposed to analytically describe the diffusion of (3)He in the lung (as measured with pulsed gradient spin echo (PGSE) methods) as a possible means of characterizing lung microstructure from measurement of the (3)He ADC. In this work, major limitations in these analytical models are highlighted in simple diffusion weighted experiments with (3)He in cylindrical models of known geometry. The findings are substantiated with numerical simulations based on the same geometry using finite difference representation of the Bloch-Torrey equation. The validity of the existing "cylinder model" is discussed in terms of the physical diffusion regimes experienced and the basic reliance of the cylinder model and other ADC-based approaches on a Gaussian diffusion behaviour is highlighted. The results presented here demonstrate that physical assumptions of the cylinder model are not valid for large diffusion gradient strengths (above approximately 15 mT/m), which are commonly used for (3)He ADC measurements in human lungs.

Random walk simulation of the MRI apparent diffusion coefficient in a geometrical model of the acinar tree

Apparent diffusion coefficient (ADC) measurement in the lung using gas magnetic resonance imaging is a promising technique with potential for reflecting changes in lung microstructure. Despite some recent impressive human applications, full interpretation of ADC measures remains an elusive goal, due to a lack of detailed knowledge about the structure dependency of ADC. In an attempt to fill this gap we have performed random walk simulations in a three-dimensional geometrical model of the lung acinus, the distal alveolated sections of the lung tree accounting for approximately 90% of the total lung volume. Simulations were carried out adjusting model parameters after published morphological data for the rat peripheral airway system, which predict an ADC behavior as microstructure changes with lung inflation in partial agreement with measured ADCs at different airway pressures. The approach used to relate experimental ADCs to lung microstructural changes does not make any assumption about the cause of the changes, so it could be applied to other scenarios such as chronic obstructive pulmonary disease, lung development, etc. The work presented here predicts numerically for the first time ADC values measured in the lung from independent morphological measures of lung microstructure taken at different inflation stages during the breath cycle.

Relationship between structural changes and hyperpolarized gas magnetic resonance imaging in chronic obstructive pulmonary disease using computational simulations with realistic alveolar geometry

Both the development of accurate models of lung function and their quantitative validation can be significantly enhanced by the use of functional imaging techniques. The advent of hyperpolarized noble gas magnetic resonance imaging (MRI) technology has increased the amount of local, functional information we can obtain from the lung. In particular, application of (3)He to measure apparent diffusion coefficients has enabled some measure of lung microstructure and airspace size within the lung. Models mimicking image acquisition in hyperpolarized gas MRI can improve understanding of the relationship between image findings and lung structure, and can be used to improve the definition of imaging protocols. In this paper, we review the state of the art in hyperpolarized gas MRI modelling. We also present our own results, obtained using a Monte Carlo approach and a realistic alveolar sac geometry, which has previously been applied in functional lung studies. In this way, we demonstrate the potential for models combining lung function and image acquisition, which could provide valuable tools in both basic studies and clinical practice.

A model morphology of the pulmonary acinus

A variation of a mathematical model of the structure of a pulmonary ventilatory unit is used to generate its internal ductal tree and associated alveolar architecture. The model unit comprises a space-filling block of regular polyhedra; ducts and alveoli were formed by opening specific common faces between polyhedra. The model employs a physically reasonable optimization strategy of maximizing gas exchange while minimizing the mean transit time to ventilate the ventilatory unit (assumed to be proportional to the mean path length) in order to create the ductal tree. The sensitivity of the global architecture to the competitive optimization parameters used and the tree structure are compared with independently published measurements. The study concludes that it is possible to model the detailed architecture of a unit using a simple space-invariant uniform modular structure for both alveoli and ductal parts. The close similarity between model and experimental measurement strongly suggests that the optimization used to create the unit is a likely one from a functional biological standpoint. The insensitivity of the architecture to the competition between the optimization parameters supports the belief that it is not important to consider the detailed measured size distribution of alveoli when considering how the large structure of the ventilatory unit is built up.

Role of collateral paths in long-range diffusion in lungs

The long-range apparent diffusion coefficient (LRADC) of (3)He gas in lungs, measured over times of several seconds and distances of 1-3 cm, probes the connections between the airways. Previous work has shown the LRADC to be small in health and substantially elevated in emphysema, reflecting tissue destruction, which is known to create collateral pathways. To better understand what controls LRADC, we report computer simulations and measurements of (3)He gas diffusion in healthy lungs. The lung is generated with a random algorithm using well-defined rules, yielding a three-dimensional set of nodes or junctions, each connected by airways to one parent node and two daughters; airway dimensions are taken from published values. Spin magnetization in the simulated lung is modulated sinusoidally, and the diffusion equation is solved to 1,000 s. The modulated magnetization decays with a time constant corresponding to an LRADC of approximately 0.001 cm(2)/s, which is smaller by a factor of approximately 20 than the values in healthy lungs measured here and previously in vivo and in explanted lungs. It appears that collateral gas pathways, not present in the simulations, are functional in healthy lungs; they provide additional and more direct routes for long-range motion than the canonical airway tree. This is surprising, inasmuch as collateral ventilation is believed to be physiologically insignificant in healthy lungs. We discuss the effect on LRADC of small collateral connections through airway walls and rule out other possible mechanisms. The role of collateral paths is supported by measurements of smaller LRADC in pigs, where collateral ventilation is known to be smaller.

Quantitative analysis of the oxygen transfer in the human acinus

The gas transport in the acinus is limited at rest by the finite oxygen diffusivity. Although the simplest steady-state model of diffusion permits understanding a number of observations, it fails to quantitatively fit the value of the absolute flux of oxygen at rest and exercise. In this paper, we show that the binding of oxygen to hemoglobin modifies these values only slightly. In contrast, the dynamics of breathing have a strong effect and allow recovery to known experimental values for oxygen transfer.

Restricted diffusion in a model acinar labyrinth by NMR: theoretical and numerical results

A branched geometrical structure of the mammal lungs is known to be crucial for rapid access of oxygen to blood. But an important pulmonary disease like emphysema results in partial destruction of the alveolar tissue and enlargement of the distal airspaces, which may reduce the total oxygen transfer. This effect has been intensively studied during the last decade by MRI of hyperpolarized gases like helium-3. The relation between geometry and signal attenuation remained obscure due to a lack of realistic geometrical model of the acinar morphology. In this paper, we use Monte Carlo simulations of restricted diffusion in a realistic model acinus to compute the signal attenuation in a diffusion-weighted NMR experiment. We demonstrate that this technique should be sensitive to destruction of the branched structure: partial removal of the interalveolar tissue creates loops in the tree-like acinar architecture that enhance diffusive motion and the consequent signal attenuation. The role of the local geometry and related practical applications are discussed.

A 4-Dimensional Model of the Alveolar Structure

The alveolar structure, a space-filling branching duct system with alveolar openings, is one of the most complicated structures in the living body. Although its deformation during ventilation is the basic knowledge for lung physiology, there has been no consensus on it because of technical difficulties of dynamic 3-dimensional observation in vivo. It is known that the alveolar duct wall (primary septa) in the fetal lung is deformed so as to obtain the largest inner space and the widest surface area, and that the secondary septa grow just before birth and their free ridges form the alveolar entrance rings (mouths) containing abundant elastin fibers. We have constructed a 4-dimensional alveolar model according to this morphogenetic process, where the alveolar deformation is modeled by a combination of springs and hinges, corresponding to elastin fibers at alveolar mouths and junctions of alveolar septa, respectively. The model includes a hypothesis that alveolar mouths are closed at minimum volume and that closed alveoli are stabilized by the alveolar lining liquid film containing a surfactant. Morphometric characteristics of the model were consistent with previous reports. Furthermore, the model explained how the alveolar number and size could change during ventilation. Using in vivo microscopy, we validated our model by an analysis of the dynamic inflation and deflation of subpleural alveoli. Our model, including the alveolar mouth-closure hypothesis, can explain the origin of phase IV in a single breath nitrogen washout curve (closing volume) and mechanism of alveolar recruitment/derecruitment.

Design of peripheral airways for efficient gas exchange

Peripheral airways combine branched tubes for ventilation with the gas exchanging alveoli in the pulmonary acini, defined as the complex of airways supplied by one first order respiratory or transitional bronchiole. In this part, the replenishment of oxygen at the alveolar surface occurs by a combination of convective air flow with diffusion of oxygen in the air. The transition between convection and diffusion depends on the morphometric properties of the airways. The design of the peripheral airways in the acinus of the human lung is described quantitatively on the basis of measurements obtained on casts of the acinar airways. Comparable data for rat and rabbit are also discussed. On the basis of this morphometric information, a typical path model for human acinar airways is derived. These studies also form the basis for advanced modeling studies of gas exchange and ventilation. In particular the problems occurring because of diffusional screening and the design conditions for minimizing this effect are discussed.

Diffusional screening in real 3D human acini--a theoretical study

Gas exchange at the acinar level involves several physico-chemical phenomena within a complex geometry. A gas transport model, which takes into account both the diffusion into the acinus and the diffusion across the alveolar membrane, is used to understand gas mixing in realistic systems. It is first shown that the behaviour of the system, computed on model geometries in 3D, only depends on the topological structure of the acinus. Taking advantage of this property, a new efficient method based on random walks on a lattice is used to compute gas diffusion in structures taken from real morphological data. This approach shows that, at rest, the human acinus efficiency is only 30-40%. These results provide a new evidence of the existence of diffusional screening at the acinar level. This implies permanent spatial inhomogeneity of oxygen and carbon dioxide partial pressure. The notion of an "alveolar gas" is reinterpreted as a spatial average of the gas distribution. This model casts new light on the respiratory properties of other gas mixtures, such as helium-oxygen.

Diffusion-reaction in branched structures: theory and application to the lung acinus

An exact "branch by branch" calculation of the diffusional flux is proposed for partially absorbed random walks on arbitrary tree structures. In the particular case of symmetric trees, an explicit analytical expression is found which is valid whatever the size of the tree. Its application to the respiratory phenomena in pulmonary acini gives an analytical description of the crossover regime governing the human lung efficiency.

Renormalized random walk study of oxygen absorption in the human lung

The possibility to renormalize random walks is used to study numerically the oxygen diffusion and permeation in the acinus, the diffusion cell terminating the mammalian airway tree. This is done in a 3D tree structure which can be studied from its topology only. The method is applied to the human acinus real morphology as studied by Haefeli-Bleuer and Weibel in order to compute the respiratory efficiency of the human lung. It provides the first quantitative evidence of the role of diffusion screening in real 3D mammalian respiration. The net result of this study is that, at rest, the efficiency of the human acinus is only of order 33%. Application of these results to CO2 clearance provides for the first time a theoretical support to the empirical relation between the O2 and CO2 partial pressures in blood.

Diffusional screening in the human pulmonary acinus

In the mammalian lung acini, O(2) diffuses into quasi-static air toward the alveolar membrane, where the gas exchange with blood takes place. The O(2) flux is then influenced by the O(2) diffusivity, the membrane permeability, and the acinus geometric complexity. This phenomenon has been recently studied in an abstract geometric model of the acinus, the Hilbert acinus (Sapoval B, Filoche M, and Weibel ER, Proc Natl Acad Sci USA 99: 10411, 2002). This is extended here to a more realistic geometry originated from the morphological model of Kitaoka et al. (Kitaoka K, Tamura S, and Takaki R, J Appl Physiol 88: 2260-2268, 2000). Two-dimensional numerical simulations of the steady-state diffusion equation with mixed boundary conditions are used to quantify the process. The alveolar O(2) concentration, or partial pressure, and the O(2) flux are computed and show that diffusional screening exists at rest. These results confirm that smaller acini are more efficient, as suggested for the Hilbert acini.

Computational morphology of the lung and its virtual imaging

The author proposes an entirely new approach called 'virtual imaging' of an organ based on 'computational morphology'. Computational morphology describes mathematically design as principles of an organ structure to generate the organ model via computer, which can be called virtual organ. Virtual imaging simulates image data using the virtual organ. The virtual organ is divided into cubic voxels, and the CT value or other intensity value for each voxel is calculated according to the tissue properties within the voxel. The validity of the model is examined by comparing virtual images with clinical images. Computational image analysis methods can be developed based on validated models. In this paper, computational anatomy of the lung and its virtual X-ray imaging are introduced.

Smaller is better--but not too small: a physical scale for the design of the mammalian pulmonary acinus

The transfer of oxygen from air to blood in the lung involves three processes: ventilation through the airways, diffusion of oxygen in the air phase to the alveolar surface, and finally diffusion through tissue into the capillary blood. The latter two steps occur in the acinus, where the alveolar gas-exchange surface is arranged along the last few generations of airway branching. For the acinus to work efficiently, oxygen must reach the last branches of acinar airways, even though some of it is absorbed along the way. This "screening effect" is governed by the relative values of physical factors like diffusivity and permeability as well as size and design of the acinus. Physics predicts that efficient acini should be space-filling surfaces and should not be too large. It is shown that the mammalian acini fulfill these requirements, small mammals being more efficient than large ones both at rest and in exercise.