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246 Reverse Engineering – Recent Advances and Applications 4 Will-be-set-by-IN-TECH (a) (b) Fig. 2. (a) Silicon rubber case of a rabbit lung acinus (Sapoval et al., 2002); (b) illustration of a model mammalian lung acinus, wherein oxygen (red) concentration gradients are established by limited access to the acinar volume by by diffusion throughout the airways (Sapoval et al., 2002). 3. Mathematical modeling of mammalian lungs 3.1 Transitions between rest and exercise determine the size of the physiological diffusion space Oxygen uptake in mammalian lungs occurs across the alveolar ducts of the acinar airways–the elementary gas exchange unit of mammalian lungs–which host the alveoli (Weibel, 1984), as illustrated in Figs. 1 and 2. These are the airways in which oxygen is transported entirely by diffusion in the resting state, and there are ∼ 10 5 of them in human lungs under these conditions (Hou et al., 2010). The branching generation at which the tracheobronchial tree ends defines the entrance to the acinar airways, and is marked by the convection-diffusion transition in the resting state, which can be located through mass-balance considerations. The mass flow-rate through a duct of the ith generation of bronchial tree, ˙M = ρvi Ai (in units of mass per time), is written in terms of the cross-sectional area of the duct, Ai = πr2 i , the speed of the gases across this area, vi and the local oxygen density, ρ. No molecules of the gas is lost across a branching point, the point in which a parent duct extends into m-many daughter ones, because there are no alveoli lining them. Mass-balance in this branch gives ˙M i = m ˙M i+1, so that the transport speed can be written in terms of its speed at the trachea, v0: v0 vi = ∑ i m − 1 = j=0 mj/3 m (i+1)/3 v0. (3) Here, we have applied Murray’s law for the bronchial tree, ri = r0m−i/3 (Weibel, 1984), connecting radii of downstream branches, ri, to that of the trachea, r0. Note that m = 2 for mammalian lungs. The tree’s total cross-sectional area therefore increases exponentially with branching generation according to Eqn. 3, resulting in a drastic decrease of the oxygen velocity within the deeper airway branches as compared to the trachea.
Reverse-Engineering the Robustness of Mammalian Lungs Reverse-Engineering the Robustness of Mammalian Lungs 5 (a) Peclet Number �� vi/v0 =10 vi/v0 = 1000 �� Generation Fig. 3. (a) Branching hierarchy of mammalian lungs (Weibel, 1984); (b)Velocity of respiratory gases, such as molecular oxygen, decrease with increasing depth into the bronchial tree. The convection-diffusion transition occurs when these gas velocities fall below the local diffusion speed. The Peclet number gives the ratio of diffusive to convective flow rate, and the branching generation at which the Peclet number equals 1 gives the convection-diffusion transition. In the human lung, for example, this transition occurs at approximately the 18 th branching generation in the resting state (Hou 2005; Hou et al., 2010). The airway “unit” defined by the remaining five generations are termed a 1/8 subacinus (Haefeli-Bleuer and Weibel, 1988). For different exercise states, however, this transition occurs deeper than generation 18, because the inspiration speed at the trachea increases. This effect is illustrated in Fig. 3 by evaluating Eqn. 3 for varying speeds at the trachea in units of the diffusional speed of oxygen in air. In reality, transitions between exercise conditions is smooth; however, in light of Fig. 3, we refer herein to an "exercise state" in terms of how the exercise conditions affect the lung’s convection-diffusion transition. In this context, four distinct exercise states can be identified in humans (Sapoval et al., 2002; Felici et al., 2004; Grebenkov et al., 2005; Hou 2005; Hou et al., 2010), which serve as the exemplary mammalian lung system for remainder of this article. These regimes are rest (i = 18), moderate exercise (i = 19), heavy exercise (i = 20), and maximum exercise (i = 21). On average, the human lung terminates with a total depth of 23 branching generations, giving an approximately equal length to all downstream branching paths. In terms of subacinar trees, these exercise states can be associated to airways of depth n = 23 − 18 = 5 (rest), n = 23 − 19 = 4 (moderate exercise), n = 23 − 20 = 3 (heavy exercise), and n = 23 − 21 = 2 (maximum exercise) (Hou et al., 2010). The subacinar trees identified in terms of these four discrete “exercise states,” define the corresponding diffusion space relevant for each breathing regime. Because diffusion occurs at rest in the network of alveolar ducts, rather than the respiratory bronchioles, we (b) 247
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REVERSE ENGINEERING - RECENT ADVANC
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Contents Preface IX Part 1 Software
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Preface Introduction In the recent
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Preface XI and con’s related to t
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Preface XIII the step-by-step appli
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Part 1 Software Reverse Engineering
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4 Reverse Engineering - Recent Adva
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10 Reverse Engineering - Recent Adv
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58 2. Model driven architecture: An
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62 Fig. 1. Model, metamodels and tr
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100 Reverse Engineering - Recent Ad
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108 Table 1. Number of smoothers an
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1. Introduction 60 Surface Reconstr
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Surface Reconstruction from Unorgan
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1. Introduction A Systematic Approa
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A Systematic Approach for Geometric
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A Review on Shape Engineering and D
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Integrating Reverse Engineering and
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