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Themed Issue Papers |
1 Bioengineering Institute, University of Auckland, Private Bag 92019, Auckland, New Zealand2 Departments of Radiology and Biomedical Engineering, University of Iowa, Iowa City, IA 52242, USA
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Abstract |
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(Received 5 October 2005;
accepted after revision 2 January 2006; first published online 11 January 2006)
Corresponding author M. H. Tawhai: Bioengineering Institute, University of Auckland, Private Bag 92019, Auckland, New Zealand. Email: m.tawhai{at}auckland.ac.nz
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Introduction |
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The transport systems can be considered to be ‘embedded’ within the pulmonary parenchyma, or suspended in a vast fibre network as described by Weibel (1984). The fibre system initiates at the hilum and extends through the lung to the visceral pleura. The ‘axial’ fibre system begins at the main stem bronchus and progresses with the airways to the terminal bronchioles. The ‘peripheral’ fibre system is related to the visceral pleura and enwraps the lobar units. The pulmonary venous network follows the peripheral fibre system (between airways and arteries), while the arterial system follows the airways and therefore the axial fibre system. The axial fibre system continues into the respiratory tissue, encircling the entrances to the alveoli, and extending a fine fibre network over the alveolar septal surfaces and weaving through the capillary meshwork.
An accurate and detailed representation of geometry in computational models of anatomical structures is essential to deliver meaningful results from numerical simulations. Because the lungs are comprised of several mechanically and functionally coupled subsystems, computational models of the lung must vary in structure over a range of scales of interest. The model structure is also dependent on the problem to which it is applied: fluid-flow simulation requires three-dimensional meshes that accurately represent the complex surface airway geometry (Lin & Hoffman, 2005), whereas inert-gas mixing (Verbanck & Paiva, 1990; Dutrieue et al. 2000) or airway thermodynamics (Tawhai & Hunter, 2004) can be modelled satisfactorily using conventional (in pulmonary physiology) one-dimensional trees with diameter information.
State-of-the-art medical imaging (specifically multidetector row computed tomography (MDCT) imaging for the air-filled lung tissue) provides high-resolution data from which subject-specific interdependent models of lung geometry can be constructed as computational domains through which a set of governing equations will be solved to simulate and understand function. By developing models on a subject-specific basis, the resulting computational meshes have spatially consistent relationships. The process of coupling functional models within the systems is therefore far more straightforward than for models developed independently from different data sets. A model with accurate spatial relationships between airway, vessel and the tissue to which they are tethered is a necessity for computational analysis of airway–vessel–tissue interactions such as coupling of ventilation distribution in ‘embedded’ airway models to the large deformation of the lung tissue (Tawhai et al. 2005, 2006).
Physically realistic model geometry
The geometry of the conducting airways.
Airway model geometries have typically been constructed using measurements from painstaking morphometric studies such as Weibel (1963), Horsfield & Cumming (1968) or Phalen et al. (1978). Weibel's symmetric model A (Weibel, 1963) and Horsfield's asymmetric 
model (Horsfield et al. 1971) are still widely used for computational studies because they summarize detailed anatomical data into idealized constructs that are relatively easy and computationally inexpensive to use.
In Weibel's symmetric model the mean length and diameter are presented for an average number of generations from the trachea to the terminal bronchioles, with each generation (N) having 2N branches (where N= 0 for the trachea). The symmetric model enables representation of the vast conducting airway system by only 16 branches (one per generation), hence making it feasible to solve equations within a very small anatomically based description of the airway system.
The Horsfield model is based upon the assumption that the airway tree bifurcates with a pattern of regular asymmetry defined by , the difference in (Horsfield) order of two child branches. Horsfield et al. (1971) presented models with constant through the majority of the conducting airway tree (the terminal airways require a slightly different pattern), and models with different values in each lobe or bronchopulmonary segment. The model introduces a pattern of asymmetric branching and dimension asymmetry, but it does not include the spatial distribution of airways that is necessary for some computational studies.
Later anatomical studies, such as Krause et al. (1995) and Phillips & Kaye (1997), further analysed the geometry of the airway tree. The common limitation of these and previous anatomical studies is that measurements were made on relatively few excised airway or vascular casts, where the geometry of the measured structure might be quite different to that of the in vivo lung. This illustrates a major difference between anatomical studies of the lung and other body organs: under normal conditions the lung is far from its zero stress state, and its large elastic recoil means that excised tissue collapses when exposed to atmospheric pressure. Whereas a great deal of structural information can be gained by examining an excised muscular organ, the lung must be inflated or casts made of the conducting systems before the organ is in a suitable state for measurements. The upright human lung is known to deform in response to gravity, with tissue density highest in the gravitationally dependent region (that is, near the base of the lung as it rests against the diaphragm). Therefore cast measurements provide a ‘snapshot’ of the airway or vascular tree geometry under a uniform filling pressure that is generally higher than the normal pressures in vivo.
Advances in imaging technologies have provided a means to measure the lung geometry in its functioning state. Computed tomography (CT) studies have been carried out to measure relatively few in vivo airways (Sauret et al. 2002; Tawhai et al. 2004), and more extensive measurements have been made of single airway casts (Schmidt et al. 2004). As part of the development of the Human Lung Atlas (Hoffman et al. 2004), Saba et al. (2005) have presented an extensive analysis of human airway geometry at two volumes (functional residual capacity, FRC, the volume of the lung at rest prior to inspiration; and total lung capacity, TLC, the maximum volume of the lung) for 41 subjects based on imaging from MDCT. The analysis has demonstrated consistency between the geometry of the Human Lung Atlas airway tree at TLC and previous airway cast measures, CT measures, and a previous MDCT study of a single human airway tree (Tawhai et al. 2004). The extensive MDCT measures have identified a large decrease in the ratio of child-to-parent branch length from TLC to FRC, providing evidence that the more distal airways undergo a systematically larger increase in length during inspiration than the more central airways.
One challenge is to incorporate the increasing amount of imaging measures, such as those from the Human Lung Atlas, into computational models that can now feasibly be based on imaging for an individual. Tawhai et al. (2004) demonstrated how this can be achieved by developing subject-specific models of the human and ovine lung and conducting airways. A volume-filling branching algorithm (Tawhai et al. 2000) was adapted to model the full extent of the conducting airway tree, such that the algorithm utilized the maximum amount of information from imaging data (airway and lung or lobe geometry from the Human Lung Atlas). The method fills an MDCT-defined finite element mesh of the lobes or lung (Fernandez et al. 2004) with an anatomically consistent airway tree that incorporates any imaged airways (currently as high as 10 branch divisions in humans and 23 in sheep) and has a branching geometry dependent on the shape of the host mesh. This method was shown to generate airway trees consistent with both the bifurcating geometry of the human lung (Fig. 1) and the monopodial branching pattern of the ovine lung. Consistent with the aim of integrating the maximum amount of information from imaging, the approach automatically incorporates additional geometric features of the lung as they are segmented. The approach marries techniques that were developed independently for model derivation (Tawhai et al. 2000; Fernandez et al. 2004) and segmented image analysis (Tschirren 2003) to produce subject-specific models of the lung that are unique in their ability to relate imaged structure to predicted function through computational simulations.
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The geometry of the pulmonary arterial and venous trees. To facilitate computational studies, the complexity of the pulmonary arterial and venous trees has previously been reduced by representing the tree as symmetric (Parker et al. 1997), as a self-similar fractal tree (Glenny & Robertson, 1991; Krenz et al. 1992; Bennett et al. 1996), or by representing an average flow path via summary morphometric parameters (Dawson et al. 1999). These models were developed to investigate the effects of large-scale alterations of branching geometry on haemodynamics in the lung, and therefore represent only the average geometry of the branching structure. They were designed for effective use in specific functional investigations, and did not aim to reflect the complex branching geometry of the pulmonary vascular trees accurately. However, these studies did all illustrate the large dependence of flow distribution on network geometry.
To model the pulmonary vascular trees in detail, Burrowes et al. (2005a) used an approach similar to that taken by Tawhai et al. (2004) to model the airway tree. The geometry of the largest pulmonary arterial and venous blood vessels was defined from segmentation of the same human subject's contrast-enhanced CT imaging (Burrowes et al. 2005a), and the non-segmented conventional arteries that accompany the airways were modelled using the volume-filling algorithm advanced by Tawhai et al. (2004). ‘Conventional’ or ‘accompanying’ vessels refers to the well-documented pattern of each airway being matched by an artery that follows the same approximate path, to supply the same region of tissue. In addition to the conventional arteries, the lung is known to contain many more ‘supernumerary’ vessels that diverge from a parent artery at close to 90 deg and branch rapidly to supply the closest respiratory tissue (Elliot & Reid, 1965). Because supernumerary vessels are not visible on angiograms and owing to the presence of a sphincter at their entrance, they are hypothesized to be unperfused under normal conditions (Elliot & Reid, 1965; Shaw et al. 1999), so it is not likely that they are visible on standard MDCT image reconstructions. The combined imaging and volume-filling branching algorithm can therefore generate a representation of the conventional accompanying blood vessels within the geometry of the lobes, but additional steps must be included to model the extensive system of supernumerary vessels. Burrowes et al. (2005a) modelled the supernumerary arteries and veins by assuming that they would arise from parent branches with diameter less than 1.5 mm (Weibel, 1963) at approximately 90 deg, and branch rapidly until they supplied the nearest pulmonary acini. One criterion that guided the model derivation was that the resulting tree should have Strahler branching ratios close to 3.0 (Horsfield, 1978) and 3.3 (Horsfield & Gordon, 1981) for the arterial and venous trees, respectively. Vessel diameters were defined as 0.3 of the parent diameter, and lengths were calculated using anatomical length-to-diameter ratios (Horsfield, 1978; Horsfield & Gordon, 1981). The vascular models, including supernumerary vessels, are illustrated in Fig. 2.
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Burrowes et al. (2004) modelled a portion of the respiratory airways/alveolar tissue using a Voronoi meshing technique that creates a volume-filling polyhedral mesh of Voronoi ‘cells’ such that the resulting geometry is consistent with measured alveolated airway structure. In this approach (summarized in Fig. 3), a host volume is filled with space-filling model alveoli (one Voronoi cell = one alveolus) and central duct spaces to produce a 3-D mesh with anatomically consistent alveolar dimensions and surface-to-volume ratio. The technique exploits the geometric Delaunay–Voronoi relationship: a Delaunay triangulation has the property that for any set of triangulated points, the circle drawn through the triangle vertices (or sphere constructed through the tetrahedral vertices in 3-D) does not contain any other triangle vertex. A Voronoi tessellation is constructed by joining the circumcentres of adjacent Delaunay triangles (or tetrahedra).
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The pulmonary capillaries form a dense sheet-like meshwork composed of short interconnected capillary segments. The capillaries are wrapped over the alveoli, with only a single sheet of capillaries between adjacent alveoli on the same alveolar duct. Because of the density of the capillary network, the classic model of the pulmonary microcirculation treats the capillary bed as a ‘sheet’ of blood bounded by two compliant layers of endothelium (Fung & Sobin, 1969). The sheet-flow model of alveolar blood flow simplifies the complex capillary geometry into a model that is appropriate for many computational studies of microcirculatory flow. However, the sheet-flow model cannot be used for investigating individual cell transit, segment blocking by neutrophils, or identification of preferential pathways for cell transit (Huang et al. 2001).
Discrete segmented models of the pulmonary microcirculation have been used by Huang et al. (2001), Dhadwal et al. (1997) and Burrowes et al. (2004) to investigate cellular transit. Dhadwal et al. (1997) modelled the capillaries on a single alveolar septum as a 6 x 6 square grid of interconnected segments. Huang et al. (2001) extended this model to generate a random orientation of the capillary segments, and connected six of the septae at opposite corners for simulation of red blood cell and neutrophil transit.
Burrowes et al. (2004) used a 2-D Delaunay–Voronoi meshing method to generate a segmented capillary mesh that wraps over the surface of model alveoli, with a single model capillary sheet between the adjacent alveoli. The method generates a capillary mesh to cover the 2-D alveolar septae of the volume-filling alveolar model previously described. The meshing technique (summarized in Fig. 4) can be applied not only to the fairly regular alveolar structure, but also to any irregular finite element representation of the alveoli. The resulting mesh has approximately 1000 capillary segments per alveolus, and 85 segments on average that a red blood cell would pass through as it traverses from a supplying arteriole to a draining venule.
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Experimental studies. Early studies of perfusion distribution in the human lung used relatively low-resolution external measures (e.g. West & Dollery, 1960; West et al. 1964; Bryan et al. 1964; Anthonisen & Milic-Emili, 1966) averaged over large regions of the lung. These studies typically measured a vertical gradient of reducing blood flow from the least (gravitationally) dependent to the most dependent regions of the lung, with the conclusion that hydrostatic pressure differences were the main determinant of blood flow distribution; this led to the well known zonal model of pulmonary blood flow. One of the predictions of the zonal model is that reversal of posture should result in reversal of the flow gradient. Advances in experimental techniques (Glenny et al. 1991) have provided higher resolution data. Such studies have suggested that gravity is only a minor determinant of pulmonary blood flow. Using microsphere injection measurements in baboons, Glenny et al. (1999) have demonstrated a persistent blood flow gradient with respect to position in the lung, somewhat independent of body posture, suggesting that there are additional mechanisms responsible for maintaining the flow gradient. A small amount of flow reversal was observed on inversion of posture, but not a complete reversal of the flow gradient as would be predicted by the zonal flow model. Other experimental studies of perfusion heterogeneity in different postures have agreed that factors other than gravity are largely responsible for flow distribution. Musch et al. (2002) used positron emission tomography (PET) imaging to assess the distribution of ventilation and perfusion in prone versus supine humans. They demonstrated that both ventilation and perfusion gradients (favouring caudal lung regions) were maintained in both postures. They also found that flow heterogeneity, as measured by the coefficient of variation squared, was unaffected by posture. Jones et al. (2001) produced similar results using electron beam computed tomography scans on healthy humans, also comparing perfusion distribution in the prone versus supine positions. Gravity was estimated to contribute only 22–34% of perfusion heterogeneity in the supine posture and 27–41% when subjects were prone.
The more recent imaging and experimental studies therefore conclude that gravitational gradients in flow exist, but that they are overwhelmed by other mechanisms. The relative contributions of gravity, vascular branching, hypoxic vasoconstriction and other unknown factors to pulmonary perfusion are still unclear, and remain under debate (West, 2002).
Computational modelling studies. In addition to direct experimental and imaging measures, pulmonary perfusion has been investigated using a variety of computational models (Glenny et al. 1991; Krenz et al. 1992; Parker et al. 1997; Dawson et al. 1999). Burrowes et al. (2004, 2005b) and Burrowes & Tawhai (2005) used physically realistic models of the segmented alveolar capillary bed and pulmonary arterial tree to examine the influence of gravity, asymmetric network branching and posture on perfusion. In the microcirculatory model, blood transport equations were solved within a realistic geometry exposed to pressure boundary conditions that were consistent with the West zonal theory of flow; in the arterial model simplified versions of the Navier–Stokes equations were solved within an elastic vessel network subjected to gravitational gradients in pleural and blood pressure.
Burrowes et al. (2004, 2005b) and Burrowes & Tawhai (2005) examined blood transit through the capillary bed and flow through the arterial tree as two separate systems (note that venous flow was not presented in these studies). This was necessary because of computational constraints: explicitly modelling each vessel in the full arterial, venous and capillary network was not possible. The models were therefore partitioned at a physiologically sensible level, and pressure boundary conditions for each model were set in a way that attempted to compensate for the lack of the full blood flow circuit.
Microcirculatory flow. At the microcirculatory level blood must be considered as a two-phase non-Newtonian fluid. Particles suspended in blood, particularly the red blood cells (RBCs), strongly influence the apparent viscosity of the blood and therefore the blood flow through the segmented network. The apparent viscosity of blood varies with vessel diameter, haematocrit (volume fraction of RBCs in blood) and blood cell velocity. In blood vessels smaller than approximately 300 µm in diameter, RBCs are preferentially distributed near the vessel centre (Secomb, 1995; Pries et al. 1996). The creation of a layer of zero haematocrit near the vessel wall reduces the apparent viscosity of blood in proportion to the decreasing vessel diameter, termed the Fahraeus–Lindqvist effect (Fahraeus & Lindqvist, 1931). Because the RBCs travel in the centre of the vessel their velocity is higher than that of plasma near the vessel wall, hence the tube haematocrit (haematocrit in the capillary segment) is reduced relative to the discharge haematocrit (haematocrit of blood entering or leaving). This dynamic reduction is known as the Fahraeus effect (Fahraeus, 1929). In addition to these two effects, RBCs are distributed disproportionately at bifurcations (phase separation) proportional to the relative flow rates in child vessels (Pries et al. 1990). To incorporate these effects into a simulation of microcirculatory blood flow, an iterative procedure is therefore used: pressure and flow are solved throughout the network assuming a using a Poiseuille-based flow equation and assuming an initially uniform haematocrit; the haemodynamics of the system are updated using empirical equations that account for the Fahraeus, Fahraues–Lindqvist and phase separation effects; and a dimensional model (Huang et al. 2001) is solved to update the capillary diameters. Burrowes et al. (2004) implemented the blood flow model presented by Huang et al. (2001) in a capillary geometry over the surface of a single alveolar sac. By exposing the model to arterial, venous and pleural pressures that increased linearly with descent down the vertical lung, they were able to demonstrate preferential pathways for blood cell transit, more homogeneous flow in the most gravitationally dependent regions, and large unrecruited regions of the capillary bed even in the tissue region of highest flow. The pattern of flow through the model was consistent with West's zonal theory for zones 2 and 3 (flow increasing in the direction of gravity, alveolar pressure < arteriole pressure). Zone 1 (zero flow, alveolar pressure > arteriole pressure) was not modelled, and zone 4 (decreasing flow in gravitationally dependent region) was not apparent because West's hypothesis that flow reduction occurs due to compression of extra-alveolar vessels was not incorporated into the model. Transit times through the model were generally faster than those measured experimentally. This could be due to equal perfusion of the supplying arterioles in the model, whereas the arterioles that arise from supernumerary vessels would be likely to be unperfused.
Microcirculatory flow is difficult to quantify using imaging techniques; however, Won et al. (2003) have advanced a deconvolution–reconvolution method to determine regional pulmonary microvascular mean transit times (MTTs). The MTT measures from such an approach hold potential to provide a rich set of validation data for simulations of microcirculatory flow; conversely, the model flow predictions may provide guidance for parameterizing the analysis for calculation of MTT from the CT imaging.
Pulmonary arterial flow. To investigate the influence of normal arterial branching, gravity and posture, Burrowes et al. (2005b) and Burrowes & Tawhai (2005) solved a simplified version of the Navier–Stokes equations in elastic venous and arterial trees, subject to boundary conditions for pressures at the heart, pressures at the capillary bed, a gravitational acceleration vector and transpulmonary pressure. They demonstrated a persistent gradient of flow and flow heterogeneity in the absence of gravity (Fig. 5), and concluded that the long transit paths in the most apical and basal regions were the particular feature of the tree structure that caused a reduction of flow in these regions (Burrowes et al. 2005b). These predictions were consistent with measurements from high-resolution microsphere deposition studies (Glenny et al. 1999; Hlastala & Glenny, 1999), but by averaging the model results in thick ‘slices’ the prediction of an apparent gravitational flow gradient became more consistent with conclusions from the early, relatively low-resolution studies (West et al. 1964). That is, the mean pattern of flow in the upright lung model, with or without gravity, was similar to that measured by West et al. (1964) but attributed solely to gravity. Burrowes & Tawhai (2005) further extended the model to investigate incomplete reversal of flow upon reversal of body posture. Inversion of posture resulted in a clear effect on the gradient of pressure, and therefore radius, at all terminal vessels in the model. However, the effect on flow was less significant; while flow in the prone posture was consistently lower than flow supine, the flow at any terminal location was similar in either posture (Fig. 6). Because the arterial tree has regionally dependent path lengths (that is, longer branching paths to the most apical and basal regions) the resistance to flow in these model paths is similar regardless of posture, and therefore there is a persistent (non-gravitational) underlying pattern of flow in the model that is likely to exist in the real pulmonary arteries.
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The two main limitations of the perfusion analyses of Burrowes and coworkers are that the capillary and arterial models were not coupled, and that neither system was exposed to realistic geometry changes as a result of tissue deformation. Coupling the arterial (and venous) and capillary models to simulate perfusion of the full pulmonary circuit will provide a better prediction of the relative effects of vascular branching and gravity. Because the vascular trees are ‘embedded’ within the lung lobes, coupling the model of blood flow to soft tissue deformation of the lung tissue will provide: (1) a realistic change in geometry of the vascular models when exposed to gravity or when changing volume; and (2) realistic pressures acting on the vessels that result from expansion or recoil of the parenchymal tissue. Similar work to couple soft tissue mechanics and ventilation distribution has been presented by Tawhai et al. (2005, 2006).
The anatomically based models provide a uniquely controllable tool for probing the structure–function relationships that result in observed features of pulmonary perfusion. Current developments aim to link these computational predictions to functional measures by comparing subject-specific predictions of perfusion to MDCT measures of blood flow in the same subject. MDCT imaging can provide estimates of tissue density (to compare with the predicted soft tissue deformation) and of regional blood flow. One goal of this work is to understand the sensitivity of functional measures to changes in vascular geometry. This will provide a further set of functional parameters to populate the Human Lung Atlas.
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