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Departments of 1Mechanical and Aeronautical Engineering, 2Civil and Environmental Engineering, 3Land, Air and Water Resources, University of California, Davis; and 4Department of Community and Environmental Medicine, University of California, Irvine, California
Submitted 12 June 2006 ; accepted in final form 3 September 2006
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ABSTRACT |
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 TOP ABSTRACT RESULTSDISCUSSIONAPPENDIX AGRANTSREFERENCES |
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airway development; promoter signaling
There is a large and growing body of evidence that suggests that the long period of human lung development is susceptible to disruption by environmental factors. This disruption has been linked to a number of negative health effects including the development of chronic airflow obstruction (5), impaired lung function growth in children (11–13, 33, 34), asthma (23), cell injury and death and small airway remodeling (38–40), and even cardiovascular causes of death (17). It seems that children are particularly susceptible to these adverse health effects; their potential for a healthy life is lowered by exposure to current ambient pollutant concentrations in certain areas. That exposure to airborne pollutants can have deleterious effects on one's health is intuitive and has been accepted for some time; however, there is currently no accurate model of lung growth as it occurs biologically. Here lung airway morphogenesis was simulated in a simplified diffusing environment in company with simple rules that govern local branching morphogenesis.
To date, researchers have focused on data-driven recreations of the fully grown lung or pulmonary acinus, producing models that are based on either fluid flow parameters or geometric relationships derived from measurements made from casts or other imaging technology. These models have important implications in the genesis (particle deposition, gas mixing, etc.) and treatment (drug delivery, imaging, etc.) of pathological conditions of the lung.
One approach to the development of a model of the human airways has been to digitally reconstruct the airways from detailed imaging data (32, 45). The potential of reconstruction-based models to produce realistic simulations of the fluid flow characteristics of conducting airways is considerable, but these models rely on expensive and time-consuming imaging procedures to recreate the architecture of a single lung.
Another approach in the modeling of lungs has been to specify certain rules that then govern the construction of fully grown airway models. The deterministic models of Kitoaka and coworkers (24, 25) have been based on relationships between the diameter of individual segments and the amount of flow though them. In Ref. 25, they propose an algorithm based on nine rules that specify the relationships between parent and daughter airways (i.e., airway diameters, lengths, and angles). The volume that is supplied by each parent branch determines the characteristics of its daughters. The other main principle governing the behavior of this model is that the distribution of terminal bronchioles in the organ is homogeneous. In addition, some researchers investigated the design principles of the biological tree structure. For example West (49) derived allometric scaling relations in mammalian circulatory systems on the basis of the assumptions that energy dissipation is minimized and terminal tubes do not vary with body size.
Using a space-filling algorithm, Tawhai and coworkers (43) took a similar, although less complicated, approach to the construction of a lung from basic rules. The authors were able to produce a lung architecture that agrees with the data of Horsfield and Cumming (20). Interestingly, this model has now been combined with high-resolution imaging data in an attempt to produce anatomically based models that integrate thoracic structures other than the conducting airways into a computational model (42). As the authors point out, it is worth remembering that, although this algorithm generates a realistic lung from simple rules, "it does not attempt to mimic the actual growth process of the lung" (43). This is true of all previous lung models.
The model presented here takes an approach that is fundamentally different from its predecessors. All models of lung architecture to date have been constructed without consideration for the actual developmental mechanisms that produce the fully grown lung. Researchers have sought to model the data and not the process that produces the data. This is not to discount the substantial contribution that mathematical modeling has made to the scientific debate, but merely to emphasize that an improved understanding of the growth of the lung can only enhance the efficacy of knowledge concerning treatment and/or prevention of lung abnormalities as they occur in the fully grown lung or in the continually remodeling lungs of the fetus or child. The model presented here has been constructed with the goal of representing the fundamental mechanism of lung growth. As a primitive step, lung development was simulated in a simplified environment where a morphogen gradient exists. The model illustrates regulatory developmental mechanisms and their susceptibility to environmental disruption, and consequently may lead to a better understanding of potential preventive and curative measures.
A complete review of the morphogenesis of the lung is beyond the scope of this paper, the interested reader is directed to Refs. 3, 4, 46, and 47 for reviews of the subject. What follows is a brief synopsis of lung development, emphasizing prenatal development of the conducting airways (22, 27).
Development of the human lung has been divided into several stages: embryonic, pseudoglandular, canalicular, saccular. The lung bud appears on day 26 and soon divides and then elongates into two lung sacs. These grow dorsal on either side of the esophagus. Meanwhile, upper, middle, and lower lobar buds have developed. On these appear the primary buds of the bronchopulmonary segments, and from these arise the future airways. The embryonic stage is complete at the 7th wk of gestation.
From approximately the 5th to the 17th wk of gestation, lung development is said to be in the pseudoglandular stage. The conducting airways will be formed by repeated dichotomous, asynchronous branching during this time. The process of reiterated bud outgrowth, elongation, and subdivision of terminal units (i.e., branching morphogenesis) underlies the development of the entire human bronchial tree; the dynamic and reciprocal interactions between the epithelium of the developing lung and the surrounding mesenchyme is vital in this process. The majority of branching morphogenesis, up to 24 orders of branching, occurs during the pseudoglandular stage.
Weeks 16 to 26 of gestation encompass the canalicular stage. During this time, the primordia of the respiratory portions of the lungs are formed. An extensive vascular system begins to proliferate through the developing lung, and rudimentary respiratory bronchioles begin to form. In addition, epithelial differentiation, formation of the air-blood barrier, and the appearance of surfactant are seen during this stage.
The saccular stage of lung development lasts from approximately week 24 to near term. The end result of the saccular stage is an overall expansion and maturation of the gas-exchange surface of the lung and a thinning of the interstitium. The lung's vascular structure also grows significantly during this time.
Modeling development of branching morphogenesis. Experiments have shown that morphogens such as fibroblast growth factors (FGF), retinoids, Sonic hedgehog (Shh), and transforming growth factors each play important roles in the branching of the mouse lung bud (4, 18, 19, 46).
FGF-7 appears to act as a long-range signal in developing lung and likely continues to be a proliferation and differentiation factor for distal lung throughout life. FFG-10 is expressed in a localized fashion in close association with distal epithelial tubules, and it appears to precede distal bud formation; signals are downregulated once the bud is formed (4). The regulation of FGF-10 appears to be controlled by Shh and bone morphogenetic protein 4 (BMP-4), and it has been suggested that bud formation during branching morphogenesis is controlled by expression and regulation of these morphogens locally near the epithelia (1, 26).
Shh knockout mice show that although Shh is essential for branching morphogenesis, it is not required for primary bud induction. At early stages of lung development, high levels of Shh and its receptor Ptc, but low levels of FGF-7 are found in the embryonic lung. Conversely, prior to birth, high levels of FGF-7 and low levels of Shh and Ptc are detected (4).
Because the coordinated interaction of the various known and unknown morphogenic growth factors is, at this point, not well understood, and because they serve a common purpose, we have chosen to lump the roles of various morphogens into several rules for branching and growth. In the present model, airways develop and grow according to the following basic rules.
Rule 1: airways grow until they reach a universal lumen length, at which point they bifurcate asymmetrically, producing major and minor airways that can continue the cycle. This assumption is plausible because observation of the highest generation number at various ages shows that generation number increases in a steady manner with postconceptional age (2). The rate of growth depends on morphogen concentration and whether it is the major or minor daughter. Major daughters and minor daughters, that is, the larger of the two daughters and the smaller, respectively, are modeled to have different initial bud sizes with a single variable growth rate controlling growth in length. The ratio of initial lengths between the major airway and minor airway is 1.8. After all growth has occurred, the ratio of major to minor length in the simulated adult lung is 
1.2. An average daughter-diameter ratio of 1.3 has been reported biologically (36), but we could not find a report on daughter-length ratio. The airways branch when they are 300 µm long because this value produced statistical features close to that of real lung airways. When a new bifurcation buds, the major and minor branches alternate down the tree; that is, if the main branch in generation i is on the left, the main branch in generation i+1 is on the right. This study did not include alveolarization.
Rule 2: airways continue to grow after bifurcation. The ratio of growth rates between the larger offspring and smaller offspring is 1.24. According to observed airway growth data, the diameter of terminal bronchioles increases steadily or linearly with time but the most proximal branches grow more quickly (15, 16, 35). Based on Phalen's airway data for the newborn and the adult, we modeled airways to grow linearly with time but with a different growth rate for each generation. Details on the growth algorithm used in the model are presented in APPENDIX A.
Rule 3: a rotation angle for successive branching planes of 79° is used (41) and the rotation angle alternates sign; that is, if the rotation angle between generation i and i+1 is 79°, then that angle between i+1 and i+2 is –79° and vice versa.
Rule 4: the structure of the upper airways (trachea, two main bronchi, left and right diaphragmatic bronchi, distal right main bronchus, left and right apical bronchi, and medial lobar bronchus) is prescribed embryonically and will be maintained throughout lung growth. The structure of five lobar airways is determined based on the five-lobe model (50) and lobar airways were set to be in the same plane.
Rule 5: development velocity, i.e., lengthening amount per a time interval, depends on which lobe the branch belongs. In the five-lobe model (50), the number of airways differs for each lobe, implying that development or branching rate will be different for each lobe, although the mechanism inducing this feature is not yet well understood. We used developmental velocity weightings of 1.2, 1.19, 1.05, and 1.0 for left inferior, right inferior, two superior, and right middle lobes, respectively, which roughly produced the number of terminal bronchioles per lobe observed in real lungs (50).
Although there are reports that length-to-diameter ratio tends to decrease for distal airways (36, 37), it is not well understood what biological mechanisms induce these anatomical trends. Recently Majumdar et al. (28) explored universal patterns of parent-to-daughter diameter ratio, but did not include any information about airway diameter growth. Because there were no literature reports on the prenatal variation of length-to-diameter ratio, we formulated this ratio for postnatal growth based only on the airway data of Phalen et al. (35) and Weibel (48), described in more detail in APPENDIX A. We used observed anatomical data on the diameter of airways of adult lung (48) to visualize the airway tree model, but this has no effect on the model such as whether the airways maintain a given length-to-diameter ratio or not during prenatal growth, but rather was done to produce a visually realistic lung growth.
Figure 1 summarizes lung growth model of current study.
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We have modeled this mesenchymal environment as a rectangular box with a size ratio of 50:50:26 (height:width:depth), which grows at the same rate as the trachea and contains four pairs of morphogen sources. These sources are grid points at which morphogens are produced throughout the course of lung development. Morphogen diffuses away from each source continuously through the course of the model. After a number of discrete time steps, diffusion away from these sources acts to create a strong gradient that then affects the growth of the lung. Airways at sites of high morphogen concentration develop normally, but airways with very little or no morphogen nearby cannot develop. This is directly analogous to morphogenic growth promotion whereby the concentrations or concentration gradients of certain molecules direct the growth and differentiation of the lung (4, 31, 46). Although in the current study the chest shape was modeled as a rectangular box, a realistic chest shape as well as lobe separation will no doubt improve the simulations.
Although there are several mechanisms for morphogen transport and complicated mechanisms will operate to control expression and regulation of morphogens, we modeled the morphogen gradient environment in a very simple manner. We used several point sources of morphogen and the simple diffusion of morphogen through the environment away from these sources. This was done because the main goal of the present study was to observe how the response of growing airway buds to morphogen gradient could affect the formation of global lung architecture.
The diffusion of morphogen in mesenchyme takes place according to the three dimensional diffusion equation, 
c/t = D
2c, where c is the morphogen concentration, which is simulated numerically by the finite difference method (6). The molecular radius of morphogens, such as FGF-2, is 1.5 nm, and aqueous diffusivity estimated using the Stokes-Einstein equation is 2.3x10–6 cm2/s (8). Detailed study of the effects of extracellular matrix structure (ECM) on the diffusion of morphogen has not been reported. Filion and Popel (9) assumed that most of the diffusive hindrance is due to the combined effect of the collagen molecules and glycosaminoglycan chains, and estimated diffusivity for FGF-2 in ECM of 2.11x10–7 cm2/s. Inasmuch as the ratios of molecular weights of various morphogens are 4; in terms of molecular diffusivity the ratios will be 2, so we selected a value of 2.0x10–7 cm2/s as a representative diffusivity. A uniform initial concentration of morphogen was used because the initial size of the lung bud is so small that diffusion uniformly distributes it. In addition, airways may also affect morphogen diffusion by physically blocking or inhibiting transport. This effect was not considered in the current study.
Experiments on the morphogenesis of mouse salivary epithelium in a combination of basement membrane-like substratum (Matrigel) and epidermal growth factor (EGF) illustrate the response of epithelium to morphogens (31). At 10 ng/ml of EGF, the epithelium, covered with Matrigel, underwent branching morphogenesis, whereas only short, straight protrusions were formed at 0.1 ng/ml. EGF had an inhibitory effect on the epithelial morphogenesis at concentrations over 10 ng/ml. Threshold concentrations will be different for different morphogens and morphogens have a variety of effects on morphogenesis (4, 46). We simplified the response of epithelium to signaling factors using the three rules below because insufficient information is available at this time to provide a more detailed mechanism.
Rule 1: if the branch tip concentration is smaller than 0.1 times the average concentration throughout the environment, the corresponding branch does not develop.
Rule 2: if the branch tip concentration is between 0.1 and 1.5 times the average, the branch develops at a normal rate, i.e., 15 µm lengthening per one-half day for a minor daughter.
Rule 3: if the branch tip concentration is larger than 1.5 times the average concentration, the corresponding branch develops 3.0 times faster than a normal branch.
Although the selection of these values is somewhat arbitrary, this model still contains the rough features of real morphogenesis, whereby epithelium does not respond until the concentration of morphogen reaches a threshold and responds to morphogen in a concentration-dependent manner. More realistic responses of epithelia to the environment could be incorporated into future work when the interplay between the many morphogenic factors is understood more clearly.
It is suspected that the morphogenesis of the developing lung requires that cell proliferation be tightly controlled locally within different regions of the tissue to generate normal tissue microarchitecture, but the mechanisms that establish this crucial spatial heterogeneity of growth remain poorly understood (29). Overall lung shape will be regulated by mechanical variables such as lung volume, fetal breathing movements, lung fluid, and the size of the intrathoracic space (30) as well as the heterogeneous distribution of morphogens in mesenchyme, which is likely to influence the direction of airway development. To model this process, we set airways to preferentially lengthen in the direction of the morphogen concentration gradient. Although the current study did not take into account physical factors, previous work (e.g., Ref. 14) on the simulation of angiogenesis and vascular remodeling may guide future modeling efforts.
Biologically, the simplest way of achieving termination of extracellular signaling is to discontinue the supply of the extracellular signaling molecule or ligand. However, this may not completely represent the actual situation, because, in some cases, particularly with morphogens, receptor signaling can continue for some hours after the extracellular ligand has disappeared. Several other ways of achieving termination are the downregulation of receptors, provision of antagonistic factors, and cells losing competence to respond to signals (10). Because these mechanisms are quite detailed and because they are not well understood in the case of the lung, we have chosen different criteria for termination of development. Branching in the lung model will cease when the total number of airways reaches 450,000, but growth will continue until the trachea reaches an observed length of 10 cm. Airways stop developing when they reach the mesenchyme boundary.
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RESULTS |
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TOPABSTRACTRESULTSDISCUSSIONAPPENDIX AGRANTSREFERENCES |
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To describe the many airways of the adult lung in an efficient manner, some method of classification is required. In Weibel's (48) well known and much used symmetrical model A, the airways are classified by generations, starting with the trachea as zero and increasing by 1 at each bifurcation in a centrifugal direction. In the method described by Horsfield et al. (21) and Engel and Paiva (7) terminal bronchioles are labeled order 1 and ordering proceeds centripetally, increasing by 1 at each bifurcation. When airways of different order meet, the numbering is continued from higher of the two airways (7, 21).
Using Horsfield's method, a higher order label at the trachea indicates larger asymmetry of lung architecture. In a perfectly symmetric branching structure, the trachea has order around 16 but in reality its order is 28. Therefore, it is essential to compare the modeled airway tree with Horsfield's data to assess its realism, although this comparison does not confirm the model's success.
Because the number of terminal bronchioles is estimated to be around 27,000 (total number from trachea to terminal bronchioles is 50,000), we treated the 50,000th generated branch in the simulation as the last terminal bronchiole (i.e., conducting airway). All further branches are considered to be part of the respiratory portions of the lung.
The airways of the lung are asymmetric at both a local scale (the two daughters of a given airway display characteristic asymmetry) and a global scale (there are different numbers of airways for each lobe; Refs. 35, 50). To illustrate various factors that could induce this asymmetry, we simulated lung development for three different conditions: the lung grows with observed patterns of local development but without morphogen sources and without different development velocities for each lobe (case A), the lung grows with local developmental patterns and different development velocities for each lobe but without morphogen sources (case B), and the lung grows with all these mechanisms (case C). We compare modeled average airway length and its standard deviation as a function of generation to Weibel's generational statistics and Horsfield's order statistics in Tables 1 and 2 and Figs. 2 and 3, respectively.
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Figure 3 clearly shows that the airway model based on case A did not agree well with Horsfield's lung structure data. Taking into account lobe dependence (case B) improves the model agreement somewhat, whereas the airway model that considers both lobe dependence and interaction with mesenchyme (case C) was very close to Horsfield's data (Fig. 3). In Fig. 3, curves shifted to the left from observation indicate that the modeled trachea's order is much smaller than that of real lung for a given total number of airways between trachea and terminal bronchioles. Table 3 compares the number of terminal bronchioles for each lobe for each simulation case to the data of Yeh and Schum (50).
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To observe the effect of source variation we simulated several cases with different positions and concentrations, and compared the values obtained with case C as described above. When the sources in left lower and right lower region were moved lower, i.e., to 0.92L, the number of terminal bronchioles was 2,982, 2,984, 1,800, 9,062, and 8,710 in right upper, left upper, right middle, right lower, and left lower lobe, respectively; Weibel's statistics changed slightly, but Horsfield's order at the trachea was 24. When the concentrations were increased equally by 55 per day at each source position, the number of terminal bronchioles was 3,577, 3,814, 1,645, 8,240, and 7,853 and Horsfield's order at the trachea was 24, and again Weibel's statistics were essentially unchanged. Slight changes of position and concentration, for example, moving all sources deeper by 5%, did not cause appreciable change in Weibel's and Horsfield's statistics but caused a slight change in the number of terminal bronchioles in each lobe.
A representative value of diffusivity, 2.0x10–7 cm2/s, was selected because complex signaling by various morphogens was simplified into that of a single morphogen. Although the combined effect of many morphogens was not simulated in the current study, we did observe the effect of changes in diffusivity on airway structure. When diffusivity was increased to 4.0x10–7 cm2/s, the number of terminal bronchioles was 2,933, 2,932, 1,841, 8,862, and 8,885 and Horsfield's order at the trachea was 24. When diffusivity was decreased to 0.5x10–7 cm2/s, number of terminal bronchioles was 3,729, 5,363, 2,795, 7,180, and 8,008 and Horsfield's order at the trachea was 26. When diffusivity was changed to 1.0x10–7 cm2/s, the change in statistical features was negligible but the number of terminal bronchioles at each lobe changed a little. For all cases, change in Weibel's statistics was negligible.
In case C, airways alter their direction by taking the weighted average of the current heading and the direction of highest promoter concentration (Fig. 4 illustrates the shape of the resulting airways). The numeric values presented for case C use a weighting of 1% per one-half day (the simulation time step) toward the morphogen gradient. Change of gradient weighting in the range between 0.5% and 2% results only in slight changes both in the overall shape of the lung (Fig. 5A) and statistical features, but the structure of the lung changed appreciably for higher weighting values. Three percent weighting results in airways whose growth is overly skewed toward promoter sources, producing an airway tree that is far from realistic (Fig. 5B) and a lung structure that is more symmetric, i.e., Horsfield order at the trachea became 25.
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DISCUSSION |
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TOPABSTRACTRESULTSDISCUSSIONAPPENDIX AGRANTSREFERENCES |
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In the present study, we assumed that the overall asymmetry of lung airways results from different developmental velocities for each branch depending on which lobe the airway belongs to and on the concentration of morphogens at the developing branch tip. The case A lung model clearly shows that lung development that takes place without mesenchymal interaction and lobe-dependent growth results in a lung architecture that is far from what is observed. Wiebel's model A does not give any information about heterogeneity of lung structure, but contains only average size information for each generation. This is why the case A lung model agrees well with Weibel's data although it is far from Horsfied's (Figs. 2 and 3). The fact that the overall lung structure produced in case A results from repeated iterations of small-scale asymmetric behavior (i.e., the asymmetric development of matching daughter airways) implies that, within the limits of the model explored in the current study, asymmetric branching alone cannot produce the lung's observed global scale asymmetry.
In case B, we used lengthening velocity weightings of 1.2, 1.18, 1.052, 1.052, and 1.0 for the left inferior, right inferior, left superior, right superior, and right middle lobes respectively, a combination of parameters roughly producing the observed lobar branch number differences (50). As can be seen in Figs. 2 and 3, the different developmental velocities used in each of the different lobes produce only a slight change in the statistical features of the airway tree, so that, in the end, the lung model is still far from the data of Horsfield (7, 20).
Another factor that inevitably contributes to the observed asymmetry of the lung is the fact that airways develop in response to signaling factors. At dilute morphogen concentrations, airways will not grow or grow slowly, whereas at high morphogen concentrations, airways will grow relatively quickly. In case C, we shaped morphogen gradients using four pairs of sources and simulated the response of developing airways to morphogen gradients (see Fig. 4). By adjusting source position and source amount, it is possible to produce a lung model that agrees well with the data of both Horsfield (7, 20) and Weibel (48). The presentation of this agreement is a first among the models of which we are aware.
It is suspected that during the development of the lung in vivo the expression of various morphogens is elaborately controlled spatially and temporarily, but the details of the mechanism, distribution, and interaction of these morphogens is not well enough understood to embody them in the present model. Although lung growth was simulated in a simplified manner so that the lung model lacked many of the mechanisms and physical conditions of a real lung, these simulations imply two important points. First that simple repetition of observed local branching patterns are insufficient to produce realistic global features of the human airway tree and, second, that morphogen gradients play a critical role in inducing not only local branching patterns but also the overall heterogeneous structure of the lung. Again, these results are true only within the limits of current study.
During lung development in vivo, membranes separate lobes so that morphogens in a lobe may not readily affect growth in other lobes. Therefore the condition that forces airways to develop with different velocities in each lobe might not be necessary for obtaining similar results with case C. This is to say that setting different intensities of morphogen sources at each lobe would have a similar effect. In the present study, which does not include lobe membrane modeling, we could not find a parameter set that could produce realistic lung structure without the condition of lobe-dependent developments. Future work using a more realistic pleural cavity model is necessary to test the role of lobe membranes during lung development.
Although lobar distribution of terminal bronchioles produced in case C was similar to the observed data (Table 3), this result is somewhat trivial because this case prescribed the development velocity of airways. The lobar distribution of terminal bronchioles for case C was slightly different from that of case B because morphogen gradients also affect development velocity in case C. The number of terminal bronchioles is unrealistically similar between each of the lobes in case A because this condition does not contain any mechanism to induce a difference in the number of terminal bronchioles between lobes.
Lung structure depends on source position and strength. For example, when sources in the lower region were moved lower to 0.92L, the number of terminal bronchioles in lower lobes decreased a little and lung structure became more symmetric, i.e., Horsfield order of the trachea was 24. When sources were moved to farther positions with respect to airways in lower lobes, airway lengthening velocity decreased, in turn reducing branching velocity of the airways in lower lobes. The reason why this position change produces more symmetric structure is unclear but one possible explanation is that source concentration in the upper region of the chest will be more uniform than in the lower region so more upper region airways lengthen with similar velocity. When all source strengths were identical, the number of terminal bronchioles in upper lobes increased appreciably and the structure became more symmetric. Because source strengths are equal, morphogen distributions will become more uniform, resulting in a more symmetric structure. The number of terminal bronchioles in upper lobes increased because the source strength in the upper region increased.
We also observed the effect of different diffusivities. When diffusivity was increased to 4.0 x 10–7 cm2/s, lung structure became more symmetric, because higher diffusivity makes the morphogen concentration more uniform, in turn making the structure more symmetric. When diffusivity was decreased to 0.5x10–7 cm2/s, the number of terminal bronchioles in upper lobes increased appreciably and structure became a little more symmetric. Overall distance between upper lobe airways and upper sources is smaller than that between lower lobe airways and lower sources. Therefore airways in upper lobes are less affected by lower diffusivity than airways in lower lobes and this may increase the number of terminal bronchioles in upper lobes. Increased numbers of terminal bronchioles in upper lobes will yield a more symmetric structure for the same reason that it did when lower sources were moved to a lower position.
Changes in airway structure by slight variation of source position, concentration, and diffusivity were negligible, indicating that very strict regulation of sources is not necessary to produce plausible airway structure at least in this simulation. Obviously, it is still unclear whether this is true for lung growth in vivo because in reality there are many morphogens whose concentrations and biological effects are interdependent.
It is suspected that overall lung structure may be regulated by mechanical variables such as the size of the intrathoracic space (29, 30), but it is unclear whether the morphogen gradient influences the direction of airway development on a local level. We simulated lung growth with several different gradient weighting values. The overall structure of the lung proved to be quite sensitive to the weight that was placed on the direction of the gradient (Fig. 5). That is, by what amount each individual branch was allowed to bend toward the direction of the gradient. We were able to produce a plausible airway tree by adjusting the gradient weighting value in an iterative manner, implying that global distribution of morphogens contributes to the global structure of the airway tree; that is, the global morphogen distribution produces a realistic lung as long as the growing airway is appropriately sensitive to the local gradient. The local concentration of morphogen acts to create a lung that is statistically realistic; the global position of sources produces the local gradients and therefore a lung that is visually realistic.
In addition to statistical comparison with the fully developed lung, we compared the model to data describing the human lung at various stages of development to ensure that our growth model reproduces the developmental patterns of the biological lung at various stages. In our model, lobar buds appeared at 38 days, at which point the trachea length was 3.12 mm. This agrees with observations from the embryonic period of lung growth (22). In case C, 51,197 airways were generated at 113 days, which is consistent with the observation that all axial generations of the bronchial tree (the nonrespiratory portions) are formed by around the 16th wk (2, 22). Our model (case C) also agreed well with observations describing airway size at birth (Fig. 7). At very early ages, the shape of airways in the model (Fig. 6) seems different from the biological case, i.e., dissected lung at 45 days shows that distal airway diameter is rather close to proximal airway diameter (22). This discrepancy is attributed to the fact that the current model does not include airway growth in diameter but uses the adult length-diameter ratio for visualization.
Figure 8 shows that diameter data from the current simulation agrees well with both Weibel and Horsfield's data (7, 20, 48). This result indicates that length-to-diameter ratio from generation-based data (35, 48) fits well with the data in Horsfield's order, but it still does not give any information about the underlying mechanisms that lead to ratio variation. Biological studies of the effect of morphogens on length and diameter growth will improve the current simulations.
The simulated structure was validated by comparing its statistical features with those obtained from observation (7, 20, 48, 50), but it still lacks evaluation of three-dimensional aspects. Although Kitaoka et al. (25) assumed that uniform distribution of terminal bronchioles in chest can be a measure of the success of an airway model and defined a parameter to test space filling capacity, the lung model of Yeh and Schum (50) shows that the density of terminal bronchioles is quite different for different lobes. This means that uniformity of terminal bronchioles is not necessarily a good measure of three-dimensional authenticity. Observations of the distribution of terminal bronchioles at each lobe will be necessary for validating the three-dimensional aspects of the airway model.
It is interesting that five rules representing observed local branching patterns combined with adjustment of only four pairs of morphogen sources can produce an airway tree model that has the rough overall shape of the real lung as well as realistic statistical features. That this model can be made to agree with much of the published data on the structure of the bronchial tree is not surprising or novel, this is true of the models of Tawhai and colleagues (42, 43) as well. What sets this model apart from others is its simulation of the lung development process based partly on biological mechanisms and partly on observed anatomical variation during development.
In the biological case, the sources of growth promotion are not fixed in position relative to their surroundings or to the developing lung. Cells either express growth promoting factors transiently and this pattern of transient expression is controlled spatially and temporally, or certain cells express promoting factors continuously and move from one area of the developing thorax to another in a controlled fashion. Whichever is the case, the model presented here does not attempt to represent either.
The criterion used by this model to determine whether or not a growing airway should branch is based simply on a predetermined branching length. This may be partially true in the biological case, but is most likely not the whole story. What mechanical, chemical, or genetic factors determine the pattern of growth in the developing lung is poorly understood at this point. What this model does demonstrate is that repeated iterations of local growth and branching algorithms that are theoretically and practically simple can produce a realistic lung. That is, by modeling the actual mechanisms by which a lung grows, we are able to produce said lung without forcing it into a prescribed set of data.
Even with the above mentioned shortcomings, the model described here represents a step toward a model that is both faithful to the key processes that produce a lung biologically and capable of producing a realistic final product. This has many and varied potential applications, particularly those associated with the genesis and progression of disease or abnormal development of lung structure.
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APPENDIX A |
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TOPABSTRACTRESULTSDISCUSSIONAPPENDIX AGRANTSREFERENCES |
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Phalen et al. (35) described airway growth using the linear regression of body length measurements, which presented airway length and diameter from birth to adult for each generation from the trachea to terminal bronchioles. On the basis of the airway size observations and the fact that axial generations of the bronchial tree are formed by around the 16th wk (2, 22), airway growth in length is modeled using the following equations.
Prenatal growth rate in length (mm/day)
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We also formulated length-to-diameter ratio during postnatal growth based on the data of Phalen et al. (35) for generations from trachea to 15th generation and Weibel's (48) for generations 16–23.
Length-to-diameter ratio during postnatal growth (unit of t is postnatal days)
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is equal to 0.0 from trachea to 10th generation and equal to 2.283x10–5 from 11th to 23rd generation. Inasmuch as there were no reports on the variation of length-to-diameter ratio during growth for generations over 15, we assumed that this ratio will change in the same manner as that of the 15th generation in the work of Phalen et al. (35). |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
* S. Tebockhorst and D. Y. Lee contributed equally to this work. 
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