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256 Reverse Engineering – Recent Advances and Applications 14 Will-be-set-by-IN-TECH a 2 /mΛeff �� n =20 a/L =1/3, m =2 n =2 �� L/Λ Fig. 6. Scaled currents as given by the square-channel model for trees of differing depth, n. Here, lengths are scaled in units of channel length, L = 3a. Denoting the branch-wise Möbius matrix of gi/q2 by Gi, we construct � q Gi = −i/2φ0 tanh [ � p/ √ q �i φ0] q2−imφ2 0 tanh [�p/ √ q �i φ0] q2−i/2 � . (36) mφ0 The effective exploration length, Λ eff/L0 = D eff/L0W eff, can be calculated according to n ∏ Gi i=0 � � D/L0 = W � Deff/L0 W eff � . (37) In the special case of a “symmetric” tree, i.e. p = q = 1, the fully fractal Möbius matrix, Eqn. 37, reduces to � φ0 tanh φ0 Gi = mφ2 0 tanh φ0 � . (38) mφ0 A fully analytical formula can be derived from this simplification by diagonalizing this 2 × 2 matrix, but this procedure and its implications will be presented elsewhere (Mayo et al., 2011). Note that Eqns. 33 to 35 of the Cayley tree model, and Eqns. 27 and 28 of the square-channel model are predictions for the same quantity: the total current leaving the tree through its “reactive” side-walls. While the square-channel model is restricted to describe branches of equal width and length, the Cayley the model carries no such limitation. This scaling is quantified by inclusion of the trees’ two fractal dimensions, Dtree and Dcanopy, implied by the length and width ratios p and q, respectively. Although the Cayley tree model is more general than the square-channel model in this respect, there are firm physiological data demonstrating the length and width of alveolar ducts remain constant with increasing generation in the human lung, i.e. p = q = 1 (Weibel, 1984), allowing for a direct comparison between the Cayley tree and square-channel models under application of this data to these models. The implications of the fracticality of the Cayley tree model will be discussed elsewhere.
Reverse-Engineering the Robustness of Mammalian Lungs Reverse-Engineering the Robustness of Mammalian Lungs 15 πr 2 0 /mΛeff �� r 0/L0 =1/3 n =20 √ π, m =2 n =2 �� L0/Λ Fig. 7. Scaled currents given by the Cayley tree model for trees of varying depth, n. Here, r0 and L0 are the radius and length of the tree’s entrance branch, respectively. 5. Reverse-engineering the robustness of mammalian lungs 5.1 Plateaus in the oxygen current associated with “robustness” to changing parameters Despite differences in their geometrical treatment of the exchange surface, each of the models presented here (Rope-walk, square-channel, and Cayley tree models) predict the existence of a broad region in the exploration length, defined by ∂I/∂W = 0, wherein the current entering the tree remains unchanged with respect to the surface’s permeability. Figure 6 illustrates this current, scaled in units of Dc0 (fixed diffusivity), in the square-channel model. Here, L is the length of each channel, expressed in units of lattice constant a, and trees of differing depth, ranging from n = 2ton = 20, in steps of two, are compared against one another. For smaller trees, the plateau is shorter, but the plateau is the largest for the larger trees (n or m large). Figure 7 illustrates the scaled current computed for the Cayley tree model. Plateaus similar to the square-channel model are presented, with the diameter of the source chosen by equating trunk cross-sectional areas with the square-channel model. Finally, figure 8 depicts currents computed using the Rope-Walk Algorithm, wherein the plateau widths are more clearly defined. These plateaus describe a reduction in the active zones that is exactly matched by an increase in the permeability, or vice versa, keeping the current constant. Operation of mammalian lungs near the onset of these plateaus allows for the maximum current to be supplied to the blood by the lung while maintaining “robustness” necessary to mitigate any reduction in permeability that might occur from, for example, disease. 5.2 Screening regimes induced by diffusion of respiratory gases These models each predict the existence of various screening regimes, which limit molecular access to the exchange area for different values of the surfaces’ permeability. In the regime Λ = 0(W = ∞), termed complete screening above, molecules do not explore further than the 257
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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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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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