M.H. TawhaiBioengineering Institute, The University of Auckland, New Zealand.
Abstract
Detailed and accurate representation of geometry in computational models ofanatomical structures is essential to deliver meaningful results from numerical sim-ulations. Advances in imaging technologies – specifically CT (Computed Tomog-raphy) for the lung – means that high-resolution data is now readily available uponwhich the structure of anatomical computational models can be based. Three majorconsiderations are that 1) model geometry relates directly to individual data, 2) themodels are integrative, and 3) the models do not become too large. In this chapterwe present methods for constructing anatomically based finite-element models ofinter-related pulmonary structures. Segmented human lung-lobe data is fit to high-order elements, and we demonstrate how this model is customized to an individual.By employing high-order elements for the lung-volume mesh, detailed surfacegeometry is represented using relatively few elements. Finite-element models ofthe conducting airways are constructed within the lung-lobe model, using a com-bination of segmented data and a bifurcating-distributive algorithm to generate anairway-consistent mesh. We show how the lung parenchyma can be modeled asa space-filling three-dimensional Voronoi mesh, with a generated geometry con-sistent with the measured alveolated airway structure. The pulmonary capillariesare generated over the alveolar model, as a two-dimensional Voronoi mesh. Thecapillary model covers the surface of adjacent alveoli in a continuous network, witha single capillary plexus between the adjacent alveoli. All of the models presentedhere have been compared extensively with morphometric data, to verify that theirgeometry is representative of the real pulmonary structure. The models are designedto be integrative: they relate multiple structural systems within the same individual,and their use as computational meshes allows spatial distribution of, for example,air pressure or material properties within the lung.
Advances in medical-imaging technology and computing power provide both high-quality data for defining the geometry of pulmonary structures and the means toderive anatomically detailed models. This ‘anatomically based modeling’approachenables the derivation of geometric models that are specific to an individual, ratherthan seeking to be representative of the mean population geometry. By developingmodels that are patient specific, functional predictions can be made for the individ-ual by solution of model systems within the customized geometry. For example,predictions of three-dimensional flow distribution through numerical solution ofthe Navier–Stokes equations in models of the conducting airways have been shownto be very sensitive to the model branching pattern and the surface geometry [1].Airway models that are both detailed and subject specific are therefore importantfor accurate numerical predictions and for understanding the influence of inter-subject variability on flow, or on predicted particle transport and deposition. Themarked variability that exists in whole-lung geometry between individuals may alsoinfluence predictions of soft-tissue deformation, and hence the flow distribution towhich it is coupled. An anatomically accurate structure is also important for directcomparison of simulation predictions with regional functional measurements frommedical imaging.
Finite-element analysis solves model equations over a complex domain by divid-ing the domain into small ‘elements’ [2]. The discretized domain is referred toas a computational mesh. Because the lungs are comprised of several coupledsubsystems – including the airways, the pulmonary circulation, the lymphaticsystem, and the parenchymal tissue in which the airway and vascular trees areembedded – computational meshes of the lung must vary in structure over a rangeof scales of interest. The model structure is also dependent on the problem towhich it is applied: fluid-flow simulation requires three-dimensional meshes [1],whereas inert-gas mixing [3–5] or airway thermodynamics [6] can be modeledsatisfactorily using one-dimensional trees with diameter information. By devel-oping computational meshes of the interdependent structures for a specific indi-vidual’s lung geometry, the resulting meshes of the airways, circulation, andtissue are consistent and have anatomically based spatial relationships. This makesthe process of coupling functional models within the systems far more straight-forward than for models that have been developed independently from differentdata sets.
Creation of a computational mesh of an organ requires a data set that describeseither the representative or individual organ structure. For a muscular organ suchas the heart it is possible to extract data from post-mortem specimens to develop ageneric model [7, 8], or to use MR (Magnetic Resonance) imaging to customize thegeneric model to an individual [9]. However, these techniques are not suitable forthe lung: the delicate structure of the parenchyma precludes ex vivo measurementsunless the lung is air dried or similarly preserved, but this alters the lung geometry;and MR imaging does not currently provide adequate anatomical detail for theair-filled lung tissue. The Visible Man project [10] provides a data set from which
Anatomically Based Modeling of Pulmonary Structure 137
it is possible to develop a generic model, however, because the specimen is postmortem the lung geometry is quite deformed.
CT imaging provides a high level of anatomical detail for the lung. Currenttechnology provides enough resolution to enable segmentation of the human lungfissures (boundaries between the lobes), a limited number of major airways, anda large number of major blood vessels [11, 12]. A Lung Atlas is currently beingcollated by an interdisciplinary team of researchers, based upon measures fromhigh-resolution CT imaging [13]. The Lung Atlas aims to describe the structuraland functional properties of the healthy male and female lung and its normal rangeof variation over three decades of human life. CT data from the Lung Atlas projectwas used for development of the geometric models that are described in this chapter.
In Section 2 a geometry-fitting technique is described and applied to fit a high-order volume mesh to the human lobes, then a technique for customizing a genericmesh to low-resolution data is outlined. In Section 3 the same fitting technique isused to fit a high-order surface mesh to CT-segmented major airways. Section 3also describes the application of a bifurcating-distributive algorithm to generate amesh that is representative of the branching structure of the nonsegmented con-ducting airways. In Section 4 a technique is described for modeling the pulmonaryparenchymal tissue as a space-filling network of polyhedral cells, and in Section 5a technique is presented to generate a segmented capillary network over modelalveoli.
2 Finite-element models of the human lung lobes
The human lungs are typically divided into five lobes: two lobes in the left lung andthree lobes in the right lung. The lobes wrap around the heart and fill the pleuralcavity. The lobes are wrapped in the visceral membrane, with a small amount ofpleural liquid between them. Because the lung tissue is highly deformable, thepleural surface bears ‘imprints’ from the ribs (Fig. 1a) that change in location asthe lungs slide over the ribs during ventilation.
Simulation of soft-tissue deformation for the whole lung requires developmentof independent meshes of the lobes, so that the lobes can move freely over eachother. Fitting to geometry data that defines the fissures produces meshes of the lobesthat sit close to one another along their boundaries. That is, the definition of a singlefissure provides data to each lobe at the common interface.
2.1 High-order elements
In Fig. 1a the surface geometry of the right-upper and middle lobes from a high-resolution CT scan of a human male at 80% TLC (total lung capacity) is shownrendered in 3D from masked lobe images. A coarse linear mesh is shown in thesecond panel of the figure. To represent the lobe geometry using only a linear meshwould require a vast number of elements, which quickly becomes impractical whenapplying the mesh in numerical simulations. Therefore to maintain a small mesh
Figure 1: A single mesh fit to both the right-upper and right-middle lobes. (a) Ren-dered surfaces from masked CT images, and data-points generated overthe surface; (b) initial linear mesh of both lobes; (c) fitted cubic Hermitemesh showing surface curvature.
size high-order elements can be used to represent the surface curvature far moreefficiently.
High-order elements use polynomials to introduce boundary curvature. Thismeans that they provide a closer fit to geometry data with fewer elements than thelinear mesh, however, they do not guarantee derivative continuity across elementboundaries. One element type that exploits both the higher accuracy of polynomialsand ensures derivative continuity is the cubic Hermite. The formulation of this ele-ment type is detailed in Bradley et al [14]. The meshes described in the current andfollowing section use tri-cubic Hermite elements for the lobes (volume meshes)and bi-cubic Hermite elements for the airway surfaces (surface mesh).
2.2 Geometry-fitting
The geometry-fitting technique solves for nodal positions and first-derivative valuesto minimize the distance between data-points that represent the surface geometry ofan object and a finite-element mesh surface [8]. This technique has previously beenused to fit models of the heart [8], lungs [15], diaphragm [16], and musculoskeletalsystem [15].
Geometry-fitting can be applied to surface or volume meshes, but in each case itis only the nodes on the mesh surface that form part of the solution in the methoddescribed below. That is, the location of nodes internal to the mesh do not changeduring the solution. Data-points that are external to the mesh at the start of the fittingprocess are projected down to the mesh surface, and data-points that are internalare projected outwards.
Anatomically Based Modeling of Pulmonary Structure 139
(a) (b)
(c) (d)
Figure 2: The geometry-fitting technique. (a) initial linear mesh (3 nodes and 2 lineelements) and scatter of data-points; (b) orthogonal projection of data-points on to mesh surface; (c) intermediate solution for nodal positionsand derivatives, and re-projection of data-points; (d) final mesh solution.
The geometry-fitting technique illustrated in Fig. 2 can be summarized in thefollowing steps:
1. Data-points are gathered to describe the geometry of the object’s surface(Fig. 2a). This may necessitate manual image digitization, or image-thresholding techniques to identify the boundary between the structure of inter-est and the surrounding tissue. Data-points in Fig. 1a were calculated as arandom distribution scattered over the rendered 3D surfaces.
2. An initial linear mesh is positioned close to the data-points (Fig. 2a). An appro-priate location and structure for the initial mesh is critical for convergence ofthe fitting solution to a well-constructed final mesh.
3. Each data-point is projected orthogonally onto the closest surface element of theinitial mesh (Fig. 2b). The orthogonal projection defines the shortest distancefrom the point to the surface. The sum of all data-point to surface distances iscalculated as the current error in the mesh.
4. Each data-point projection, and therefore the summed error, can be written interms of nodal coordinates and derivatives. This expression is the objectivefunction that the geometry-fitting solution seeks to minimize. For each ele-ment, a linear system of equations is generated by differentiating the objectivefunction with respect to each element parameter, and equating the resultingexpression to zero.
5. The system of equations is solved for new nodal coordinates and derivatives,to minimize the objective function and excessive element curvature (Fig. 2c).
6. Steps 3–5 are repeated until the solution converges and the error is small.That is because the mesh location and surface curvature changes during thesolution the data-points must be reprojected onto the current mesh surface andan updated objective function formulated for solution.
The volume mesh in Fig. 1c results from applying the geometry-fitting techniqueto the data in Fig. 1a and initial volume mesh in Fig. 1b.
To create lobe models specific to an individual, the geometry-fitting technique canbe reapplied to a new dense data set. For the well-defined CT image boundariesfrom the Lung Atlas, compilation of a high-resolution data set and geometry-fittingfor an individual is straightforward. Although this approach produces a modelwith generally small error, an alternative technique – the affine transformation ofa generic mesh – is more appropriate for sparse or low-resolution data. An affinetransformation differs from a rigid-body transformation in that as well as includingtranslation and rotation, it also includes stretching and shearing.
Because CT is a high-dose imaging modality the number of scans a patient issubjected to must be limited, hence scanning the entire lung is not always possible.MRI is low dose, but this method generally produces poor anatomical detail for thelung. Sparse data sets from CT and low-resolution data sets from MRI can be usedto define target locations for affine transformation of a generic mesh.
An affine transformation procedure, also called ‘host-mesh customization’ hasbeen detailed by Fernandez et al [15] for both surface and volume meshes. Themethod is illustrated in Fig. 3 for a volume mesh, and can be summarized as follows:
1. A generic mesh – with structure representative of the body of interest (the left-upper lobe is shown in Fig. 3a) – is embedded within a host mesh. The host meshis comprised of one or more volume elements with any specified interpolationfunction. The example in Fig. 3 uses a single high-order element. For greateraccuracy several elements could be used, but the number of landmark and targetpoints required (step 3) would need to increase accordingly.
2. The location of each generic mesh node is calculated with respect to the hostmesh. That is, the host element that contains each node is determined and thenode’s local (ξ) coordinates with respect to the host element are calculated.
3. Pairs of landmark and target points on the generic mesh and data set, respec-tively, are identified (Fig. 3b). Target and landmark points are locations thatare clearly identifiable on both the image-based data set and the generic mesh.For example the carina, apex or base of each lung, or bifurcation points ofairways that are visible inside the lung and have been constructed as part of thegeneric mesh.
4. An affine transformation of the host mesh is calculated to minimize the sum ofthe distances between the pairs of target and landmark points (Fig. 3c).
5. The location of the generic mesh is updated by calculating the global positionof each node using the ξ coordinates calculated in step 2 and the new globallocations of the host nodes and elements (Fig. 3d).
The error in an affine transformation of a generic mesh is generally larger thanfor a full fit of a dense data set, but for sparse or low-resolution data there is littleadvantage in applying the geometry-fitting technique.
The affine transformation can also be used to position and arrange the initiallinear mesh before the projection of data-points in the geometry-fitting procedure.The initial mesh structure is very important for geometry-fitting: if a portion of the
Anatomically Based Modeling of Pulmonary Structure 141
(a) (b)
(c) (d)
Figure 3: Host-mesh customization technique. (a) An initial generic mesh of theleft-upper lobe surrounded by the host mesh (cube); (b) host meshenclosing landmark points (small crosses) and target locations (blackpoints); (c) transformation of the host mesh; (d) transformation of thegeneric mesh.
initial mesh is positioned far from the data-points, or the initial element sizes arevery different, this will persist throughout the fit and the final mesh will have anunsatisfactory structure. By using the affine transformation to position the meshfirst, the initial mesh can be forced to sit within the entire data set, and to haveelements of similar size.
3 Finite-element models of the conducting airways
The conducting airways extend from the trachea to the terminal bronchioles. Begin-ning with the trachea, each generation bifurcates into two (or occasionally more)child branches of generally unequal diameter, length, and branching angle [17–19].
Each terminal bronchiole (the last purely conductive portion of each airway path)supplies a single pulmonary acinus, which is the basic ‘unit’ of gas exchange[20]. The terminal bronchioles are positioned on average at generation 16 [19],but because of branching asymmetry the terminal bronchiole generation rangesfrom approximately generation 8 to generation 25 [21].
Anatomical studies of airway-branching pattern and dimensions have given riseto two widely used models of the human conducting airways: Weibel’s symmetricmodel A [19], and Horsfield’s asymmetric � model [21]. Both of these models arestill in widespread use in computational studies because they summarize detailedanatomical data into idealized constructs that are relatively easy to implement.
In Weibel’s symmetric model A the mean length and diameter are presented foran average number of generations from the trachea to the terminal bronchioles,with each generation (N ) having 2N branches (where N = 0 for the trachea). TheWeibel symmetric model has been so widely used because it enables the vast con-ducting airway system to be represented by only 16 branches (one per generation),hence making it feasible to solve equations within a very small anatomically baseddescription of the airway system. With current computer technology we are notlimited to solution in such a small system, however, the symmetric model is stillpreferable for some simulation studies. For example, when investigating inert-gasmixing in the respiratory airways, Paiva and Engel [4] and Dutrieue et al [3] usedthe symmetric model to introduce an average ‘delay’ in transit of the inspired gasfrom the mouth to the terminal bronchiole. When it is sufficient to treat the con-ducting airways as an ‘averaged’ structure, it is preferable to use the symmetricmodel than the more accurate but computationally expensive asymmetric models.
The Horsfield � model is based upon the ordering system proposed by Horsfieldand Cumming [17]. In Horsfield ordering the terminal airways are order 1 andeach parent branch is one order higher than the child branch of highest order. TheHorsfield � model represents airway-branching asymmetry by �, the difference inorder of the two child branches. Horsfield et al [21] presented models with constant� through the majority of the conducting airway tree (the terminal airways requirea slightly different pattern), and models with different � values in each lobe orbronchopulmonary segment. Because each terminal branch in the � model hasa unique connectivity path, the model size cannot be reduced to the extent ofthe symmetric model. The Horsfield � model also does not include the spatialinformation that is necessary for some studies.
Recent computational studies have highlighted the need for accurate anatomi-cally based airway models with spatially distributed branches [6, 22, 23]. Tawhaiand Hunter [22] investigated the influence of conducting airway asymmetry onpredictions of inert-gas mixing using an anatomically based model. They con-cluded that while the branching asymmetry did influence the inert-gas dynamics, amore important factor was an accurate description of the flow distribution. How-ever, to accurately predict the distribution of inspired gas requires coupling of theanatomically based model to deformation of the lung during the breathing cycle. Toinvestigate the influence of heterogeneous airway constriction on global measuresof lung function, Tgavalekos et al [23] applied constrictions in an anatomically
Anatomically Based Modeling of Pulmonary Structure 143
based airway model, with the location of the constricted airways determined fromPET (Positron Emission Tomography) imaging. This type of simulation study canonly be performed in a model that has both accurate branching structure and spatialdistribution of the airways to link the applied model airway constrictions to theregional PET measures. Tawhai and Hunter [6] have demonstrated that the asym-metry of the branching pattern in the human airways has a significant influence onthe rate of heating and humidification in different airway paths when gas is inspiredat less than core conditions. Because the location of the airways partly determinesthe flow through them, the local thermodynamic processes are influenced by therate of air flow. Therefore this type of study also requires the use of accurate airwaymodels with spatial information.
3.1 Modeling the central conducting airways
Advanced segmentation algorithms can uniquely identify human airways down togeneration 7, and in some airway paths down to generation 9 [11]. Beyond thislevel it becomes increasingly difficult to identify the individual airways as thediameters decrease. Therefore to develop models from this limited data set weuse a combination of surface-mesh fitting to segmented larger airway data, and anairway-generation algorithm to create centerlines of branches down to the terminalbronchioles.
The geometry-fitting technique detailed in Section 2 for the lobes can also beapplied to fitting finite element meshes to segmented airway data. Fig. 4 comparestwo models of several major airways (the first 3 generations): in Fig. 4a the airwaysare modeled as straight elements, starting and ending at the airway branch points,
(a) (b)
Figure 4: Models of the trachea and several major bronchi (to generation 3).(a) 1D centerline model with average diameter values for each airway(7 × 1D elements); (b) surface mesh fit to CT data (112 × 2D elements).
and with associated mean diameters; in Fig. 4b CT data has been fit to a surfacemesh (note that the technique can also be used to fit a volume mesh for the airways,but in the current example only a surface mesh is fit). The second mesh has amore accurate representation of the local surface geometry, whereas the first meshassumes cylindrical airways with constant cross-sectional area along each branch.
The first mesh has the form of the ‘traditional’ airway model. Its structure canbe described simply by the location of the branch points, and the diameter ofeach airway. While the model is small and relatively simple to use, it does notincorporate any change in cross section down the airway, and has discontinuitiesin cross sections and overlap of the airways at the bifurcations. The second meshrequires storing more information, but it provides a more accurate description ofthe change in geometry along each airway.
The surface mesh can be used in a similar way to the simpler structure in finite-element simulation studies by integrating the governing equations over the airwaycross section. The local area is then used in the formulation of the solution system.
3.2 Modeling the conducting airway tree
To model the airway tree from the last segmented airways to the terminal bronchi-oles, a bifurcating-distributive algorithm can be used to fill the available volumewith an asymmetric tree [24]. The algorithm relates the model airways directly tothe fit lobe geometry described in Section 2: the airways are generated from the lastCT-based airways into each of the fit lobar volume meshes so that the airway lengths,orientation, and branching pattern depend directly on the different lobe shapes.
The algorithm generates linear centerlines – no curvature is included. The struc-ture is therefore similar to that shown in Fig. 4a, with straight airways and constantcircular cross sections. The algorithm produces only bifurcations: each division isfrom a single parent into two child branches. Each airway in the unrefined gener-ated tree is a single 1D finite element, and each bifurcation point is the location ofa node.
The airway algorithm is illustrated in a simple 2D geometry in Fig. 5. Figure5a shows an initial rectangular host mesh and several airway generations. Thealgorithm proceeds as follows:
1. The host mesh is filled with a uniformly spaced grid of seed points (Fig. 5a).The density of the seed points is equal to the volume of the whole lung dividedby the expected number of acini (=the expected number of terminal branches,approximately 30,000 for the human lung). The seed points can also be ran-domly distributed, but this has not been shown to produce a significantly dif-ferent tree structure, and the points are more time consuming to generate.
2. The seed points are grouped with the closest segmented branch end point bycalculating the minimum distance from each point to a branch end (Fig. 5b).
3. For each initial group of seed points and parent branch a plane is calculatedthat contains both the parent branch and the center of mass of the seed points.This plane is used to split each set of seed points into two new sets of generally
Anatomically Based Modeling of Pulmonary Structure 145
(a) (b) (c)
(d) (e) (f)
Figure 5: The airway-generation algorithm illustrated in 2D. (a) regular seed pointsinside the rectangular host area; (b) seed points grouped with the closestairway ending; (c) centers of mass of seed point sets are calculated, andused to split each set of points in two; (d) centers of mass of new sets arecalculated; (e) new airways generated from end of parent airway towardscenter of mass; (f) the resulting area-filling tree.
unequal size (Fig. 5c). The asymmetry in the seed point split is influenced bythe shape of the host volume and introduces asymmetry in the branching patternand branch geometry.
4. The center of mass is calculated for each of the new seed point sets by averagingthe xyz-coordinates of the points (Fig. 5d).
5. For each set of seed points, a new branch is created that starts at the end of theparent branch, and points towards the center of mass. The branch terminates40% of the distance from the parent to the center of mass (Fig. 5e).
6. Each generated branch and seed point set is inspected to determine whetherthe generated airway is terminal. An airway is terminal if only one seed pointremains in the set, or if the length is less than an anatomically defined minimumvalue. For terminal airways with more than one seed point, all but the seedpoint closest to the branch end are distributed to the other seed point sets bycalculating the closest current branch ending. This ensures that no ‘gaps’appearin the generated tree.
7. Each set of seed points (including any points that have been reassigned to theset from the terminal airways) is split using the plane containing the newlygenerated branch and its parent branch.
8. Steps 4–7 are repeated until all of the paths have terminated.
The airway diameters can be included using a variety of assumptions. Diameterscan be assigned based on generation, Horsfield order, Strahler order [25], or length-to-diameter ratio. The mesh in Fig. 4a has diameters calculated from the cross-sectional areas of the masked CT airways.
The conducting-airway mesh generated into the CT-based lobe geometrydescribed in Section 2 is shown in Fig. 6. The method started with the locationof airways defined down to generations 6–9. Airways were generated into eachlobe separately. Figure 6a views the model from the front, and Fig. 6b from theleft-hand side. The different grayscale of each lobe highlights the left-upper andleft-lower lobes in Fig. 6b.
The geometry of the models produced using the generation algorithm has beencompared extensively with published morphometric values [17–19, 21, 25] and hasbeen shown to have anatomically consistent numbers of airways in each generationand Horsfield order, accurate length and branching ratios, and consistent branchingangles and planes of branching [24]. A full description of the development of themethod, sensitivity to model parameters, and comparison of the model airway treeswith anatomical measurements is given in Tawhai [26].
Because the generated airway mesh is always fully contained by the host volume,the algorithm can be applied in any shaped host lung and is not constrained byassumptions about the lobe geometry. This makes the method simple to applyin different human subjects, and in volume models of different animal lungs. Theinclusion of additional CT-based airway generations to increase the model accuracyis straightforward using this method: additional airways are simply incorporated inthe initial airway-model description, and therefore automatically form part of themesh from which the algorithm starts branching.
(a) (b)
Figure 6: Airway model generated from 6–9 generations of CT-segmented airways,into a CT-defined volume mesh of the human lung lobes. The model iscomprised of approximately 60,000 individual airways (a) from front,(b) from left.
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4 Modeling the respiratory airways as a volume-filling mesh
Each terminal conducting airway (the terminal or transitional bronchiole) supplies asingle pulmonary acinus. The airspaces in the acinus have walls that contain, or arecomprised entirely of, alveoli. The alveoli are densely packed, mechanically con-nected, and have multifaceted shapes [20]. The alveoli and alveolar ducts thereforeform a tethered sponge-like structure.
For computational studies the respiratory airspaces have been modeled as a 1Dmultibranching tree for simulating inert-gas mixing [3, 5], a 2D reconstruction fromscanning electron micrographs of the parenchyma [27], a 3D polyhedral networkfor mechanics studies [28], and as a 3D space-filling base structure for modelingthe pulmonary microcirculation [29]. In this section we describe the Delaunay–Voronoi technique employed by Burrowes et al [29] to construct a space-fillingalveolar mesh.
4.1 3D Voronoi meshing technique
Burrowes et al [29] modeled the alveolar geometry using a Voronoi meshing tech-nique, which creates a volume-filling polyhedral mesh of Voronoi ‘cells’. One of thetraditional applications of this technique is for constructing computational meshesfor fluid-flow simulation [30, 31]. Here the technique is used to fill a host volumewith space-filling model alveoli (one Voronoi cell = one alveolus) and central ductspaces to produce a 3D mesh with anatomically consistent alveolar dimensions andsurface-to-volume ratio. The meshing procedure is illustrated for a simple 2D hostmesh in Fig. 7.
1. Seed points are generated within a host volume and around the host boundary(Fig. 7a). In the method used by Burrowes et al [29], the host boundary isdefined by creating sets of seed points orthogonal to the bounding surface,such that a sphere passing through the points has its center located exactly onthe boundary [30, 31]. That is, an area bounded by a set of lines is defined bypairs of orthogonal points along the lines and two external and one internalpoint at any convex corner (one point orthogonal to each joining line). Anda volume bounded by planes is defined by pairs of orthogonal points on thebounding surfaces, and sets of three external and one internal point at anyconvex corner (where three planes meet).
2. The seed points are triangulated (Fig. 7b). This is a ‘Delaunay triangulation’,which means that the circle passing through three points defining any tri-angle (or sphere passing through four points defining any tetrahedral ele-ment in 3D) does not contain any other point. Note that the host boundarypasses through the triangulation (tetrahedralization) of the orthogonal boundarypoints.
3. The circumcenters of the triangles or tetrahedra are calculated. The circum-centers are the center points of the circle or sphere. The Voronoi ‘dual’ ofthe Delaunay triangulation has vertices at the circumcenters of neighboring
Figure 7: Space-filling Voronoi meshing technique for modeling the alveolar (res-piratory) tissue, illustrated in 2D. (a) Host area filled with regular grid ofseed points; (b) Delaunay triangulation of seed points; (c) circumcentersof adjacent triangles are joined; (d) Voronoi cells comprised of joinedcircumcenters; (e) faces removed to form ‘ducts’ and to aerate the entirestructure.
triangles or tetrahedra (Figs. 7c and d). The circumcenters of the boundarytriangles or tetrahedra lie on the boundary lines or surfaces.
4. Specific Voronoi cells are allocated as airway duct cells. This step starts at onecell close to the boundary, and with a specified initial direction. Every faceadjacent to the first duct cell is removed. The algorithm steps to the adjacentcell that is closest to the initial direction and removes all of its faces. Forwardstepping continues until 3–4 ‘rings’ of alveoli have been formed, at whichpoint the duct bifurcates. Two new directions are calculated, and the forwardstepping, face removal, and bifurcation continues as above until all of the cellshave been classified as either duct or alveolar cells (Fig. 7e).
5. Each cell face is converted into triangular finite elements. A node is placedat the center of the face and at each face vertex. The final structure forms avolume-filling honeycomb-like mesh.
The Delaunay–Voronoi meshing technique provides a relatively simple way ofcreating alveolated airway structures within general host-volume shapes. The finite-element meshes produced by the method are consistent with the alveolar meshdeveloped for mechanical analysis by Denny and Schroter [28]. Several ductsisolated from a larger 3D mesh are shown in Fig. 8a, and a single alveolar sac
Anatomically Based Modeling of Pulmonary Structure 149
(a) (b)
Figure 8: Model alveolated ducts isolated from an alveolar mesh generated bythe 3D Delaunay–Voronoi meshing technique. (a) system of isolatedducts; (b) a single alveolar duct.
is isolated in Fig. 8b. The triangular elements are shown dividing each cell face.The mesh is continuous, space-filling, and has an alveolar surface to volume ratiowithin 2.7% of the published value [19].
5 Modeling the microcirculation as a segmented network
The pulmonary capillaries form a dense sheet-like meshwork composed of shortinterconnected capillary segments. The capillaries are wrapped over the alve-oli, with only a single sheet of capillaries between adjacent alveoli on the samealveolar duct.
The classic model of the pulmonary microcirculation treats the capillary bed asa ‘sheet’ of blood bounded by two compliant layers of endothelium [32–34]. Thesheet-flow model of alveolar blood flow simplifies the complex capillary geometryinto a model that is appropriate for many computational studies of the pulmonarymicrocirculation. However, the sheet-flow model cannot be used for investigat-ing individual cell transit, segment blocking by neutrophils, or identification ofpreferential pathways for cell transit [35].
Segmented models of the pulmonary microcirculation have been used by Huanget al [35], Dhadwal et al [36], and Burrowes et al [29] to investigate cellular transit.Dhadwal et al [36] modeled the capillaries on a single alveolar septum as a 6 × 6square grid of interconnected segments. Huang et al [35] extended this model togenerate a random orientation of the capillary segments, and connected six of theseptae at opposite corners for simulation of red blood cell and neutrophil transit.Blood exiting a septal face was constrained to enter the adjacent face at the cornerjunction. Burrowes et al [29] developed an anatomically based capillary model withmultiple inlet and outlet vessels, for solution of the Huang et al [35] blood-flow
model in different vertical locations in the lung. The model developed by Burroweset al [29] is detailed in the following section.
5.1 2D Voronoi meshing technique
Burrowes et al [29] used a 2D Delaunay–Voronoi meshing method to generatea segmented capillary mesh that wraps over the surface of model alveoli, with asingle model capillary sheet between the adjacent alveoli. The method generatesa capillary mesh to cover the 2D alveolar septae of the volume-filling alveolarmodel described in Section 4. The meshing technique can be applied to not onlythe fairly regular alveolar structure shown in Section 4, but also to any irregularfinite-element representation of the alveoli.
The meshing technique as described below is illustrated over the surface of asimplified single alveolus in Fig. 9:
(a)
(c) (d)
(b)
Figure 9: Voronoi-mesh construction over a portion of the surface of a unit sphere,for modeling the pulmonary capillary bed. (a) Regular grid of seed pointsgenerated on the surface of a unit sphere, with a simplified mesh of a singlealveolus placed at the center; (b) Delaunay triangulation of initial points;(c) circumcenters of triangles joined to form Voronoi cells; (d) Voronoicells projected onto alveolar mesh, with each cell edge representing asingle capillary segment.
Anatomically Based Modeling of Pulmonary Structure 151
1. Regularly spaced points are generated over the surface of a portion of a unitsphere (Fig. 9a). The perimeter of the spherical portion is defined by pairs ofpoints that are orthogonal to the base and lie on the sphere surface.
2. The points are triangulated (Delaunay triangulation) over the 2D sphere surface(Fig. 9b).
3. The circumcenters of the triangles are calculated, and neighboring circum-centers are joined to form Voronoi cells (Fig. 9c). The cells are 2D polygonsin this method, in contrast to the 3D polyhedra generated by the method inSection 4.
4. The circumcenters are projected onto the surface of a single alveolus (Fig. 9d).This is a normal projection from the curved sphere surface to the flat alveolarfaces.
5. Each Voronoi cell edge represents a single capillary segment. A finite-elementmesh of the capillary network is constructed by transforming the circumcentercoordinates into nodal positions, and each Voronoi cell edge into a 1D ele-ment. The ‘rings’ of capillaries that are formed typically have either six or fivesegments.
6. For the next alveolus, any Delaunay points that are on a shared face (betweenadjacent alveoli) are replaced with the points from the neighboring alveolusthat lie on the common face. The triangulation produces an identical networkon the shared face that is continuous with the networks on each of the adjoiningfaces in each direction. This ensures that the entire generated capillary networkis continuous over the alveoli that surround a common duct, with only a singlecapillary layer between the adjacent alveoli.
The capillary mesh generated over the entire alveolar sac model in Fig. 8bis shown in Fig. 10. The mesh has approximately 1,000 capillary segments per
Figure 10: Capillary model generated over the surface of an alveolar sac. Fivevenules are shown on the external surface of the alveoli. Two of the foursupplying arterioles are visible at the bottom of the figure.
alveolus, and 85 segments on average that a red blood cell would pass through as ittraverses from a supplying arteriole to a draining venule. These values are consis-tent with published measurements [19, 37, 38]. Capillary diameters were includedusing published values from the literature [35].
6 Summary
In this chapter we have described geometric models of pulmonary structures thatare inter-related through development from the same imaging set (lung, lobes,airways), or through generation within the geometry of another model (capillarynetwork). That is, the models have spatial relationships that can be exploited whensimulating pulmonary behavior. But to be integrative the models also need to relateto one another functionally.
Because the large-scale models presented here were derived from a single lunggeometry, spatial positioning of their embedded structures allows accurate distribu-tion of material properties, or interpolation of parameters from one model subsystemfor inclusion in another. For example, the soft-tissue deformation of the lobes canbe coupled to the embedded conducting airway model to drive air flow throughthe tree. Conversely, the airways can be treated as ‘fibers’ within the parenchyma,which will influence the solution of tissue deformation.
Because the human lungs display marked variability in shape, volume, and airwayand vessel orientation, it is important to be able to produce individualized computa-tional meshes. The meshing techniques described in this chapter – geometry-fitting,affine transformation, Voronoi meshing, and algorithms to generate host-shape-dependent airway models – allow us to create computational models that relateanatomical information for an individual to function that can be predicted throughnumerical simulations.
Acknowledgements
This work was supported by The Royal Society of New Zealand Marsden Grant01-UOA-070 MIS.
The author would like to acknowledge the contributions of Professor P.J. Hunter,Ms K.S. Burrowes, Dr P. Mithraratne, and Mr J. Fernandez at the University ofAuckland to the modeling work described in this chapter. The author also thanksDr E.A. Hoffman and collaborators on the Lung Atlas project for access to, andassistance with, CT images.
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