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254 Reverse Engineering – Recent Advances and Applications 12 Will-be-set-by-IN-TECH 4.2 Cayley tree model Instead of modeling an acinus as a hierarchical tree of square channels, here we consider these branches as cylinders; this symmetry allows for a continuous description of the concentrations along its branches (Mayo 2009; Mayo et al., 2011). This model is flexible enough to include scaling relationships between parent and daughter branches, giving the total current in terms of the tree’s fractal dimension. 4.2.1 Scaling relationships and fractal dimensions of the fractal Cayley tree To account for scaling between the tree’s branches, we adopt the following labeling scheme. The width and length of each branch is given by 2rei and Lei , respectively, wherein ei = (i, i + 1) label the edges of the Cayley graph in terms of its node, as illustrated by Fig. 5, and i = 0, 1, . . . , n, n + 1, with n being the depth of the tree. The single entrance branch, e0 = (0, 1), is termed the “trunk,” whereas the mn-many terminal branches, each labeled by en = (n, n + 1), are those composing its canopy, or “leaves.” We decompose the tree’s surface into two parts: i) the canopy, or the surface composed of the union of end-caps of the terminal branches, and ii) the tree’s cumulative surface area, or the sum of all surfaces minus the end cross-sections. The width and length of daughter branches can be expressed in terms of the those quantities for the parent branch through scaling relationships for their length, m (Lei )Dtree Dtree = Lei−1 and width, m (rei )Dcanopy Dcanopy = rei−1 (Mandelbrot, 1982). The ratios of these quantities across successive generations can be expressed in terms of the scaling exponents for the cumulative surface of the tree, Dtree , and its canopy, Dcanopy: p = m −1/Dtree , and q = m −1/Dcanopy . (29) For simplicity, we assume the length and width of branches in the same generation are of equal length and width; however, we allow branches to scale in width or length across generations: p = pei = Lei /Lei+1 and q = qei = rei /rei+1 for i = 0, 1, . . . , n − 1. These scaling exponents can be equated with the fractal dimension of the tree’s cumulative and canopy surface areas (Mandelbrot, 1982; Mayo, 2009). 4.2.2 Helmholtz approximation of the full diffusion-reaction problem in a single branch For cylindrical branches in which the ratio of radius to length, termed the aspect ratio, r/L, is “small,” the oxygen concentration is, to very good approximation, dependent only on the variable describing its axis, x (Mayo, 2009). Mass-balance of the oxygen flux in the cylinder gives a Helmholtz-style equation for the concentration in the tube (Mayo et al., 2011): d2 � �2 φ c (x) − c (x) = 0. (30) dx2 L wherein the Thiele modulus of the branch is given in terms of the exploration length: � 2 φ = L . (31) rΛ The tree is composed of three classes of branches, each defined by type of boundary condition given across the beginning and end of the tube. Giving an exit concentration (written in terms
Reverse-Engineering the Robustness of Mammalian Lungs Reverse-Engineering the Robustness of Mammalian Lungs 13 of the exit current) and an entrance current for a terminal branch, c (x = L) = Iext/Wπr 2 and d/dxc (x = 0) = −Ient/Dπr 2 , gives a solution to Eqn. 30 connecting the entrance and exit currents (Mayo, 2009): c (x) = Ient sinh [φ (1 − x/L)]+Iextφ cosh (φx/L) (πρ2 . (32) D/L) φ cosh φ Similar equations can be found for the trunk and “intermediate branches,” resulting in equations that link entrance and exit concentrations, or a mixture of concentrations and currents (Mayo et al., 2011). These equations are used in a renormalization calculation, similar to the branch-by-branch calculation of the square-channel model above, to compute the total current entering the tree by its root. 4.2.3 Total current into the tree Elsewhere we demonstrate how equations of the type described by Eqn. 32 can be employed for a branch-by-branch calculation of the total current, similar to that of the square-channel model described above, and written in terms of the scaling ratios of Eqn. 29 (Mayo et al., 2011). This calculation results in I = Dπr2 0 mq2 c0, (33) Λeff wherein c0 = cent,0 denotes the concentration across the entrance of the trunk, consistent with the terminology of Eqn. 28 and r0 = re0 is the radius of the trunk. So, the effective exploration length, Λeff is a p, q–dependent expression, Λeff L0 = g0 q 2 ◦ g 1 q2 ◦···◦ gn (Λ/L0) q2 , (34) wherein L0 = Le0 is the length of the trunk. Equation 34 is the n + 1–fold functional composition of a branch-wise attenuating function gi = gei . This function itself is given by the formula gi (Λ/L0) q2 (Λ/L0) q = −i/2φ0 + tanh [ � p/ √ q �i φ0] , (35) (Λ/L0) q 2−i mφ 2 0 tanh [� p/ √ q � i φ0]+q 2−i/2 mφ0 √ wherein φ0 = L0 2/r0Λ is the Thiele modulus of the entrance branch, but depends on the exploration length penetrating into the tree. 4.2.4 Attenuating function as a Möbius transformation Equation 35 is a Möbius transformation (Needham, 2007), and using this fact, Eqn. 34 can be expressed analytically. Möbius transformations, denoted by μ, are mappings that rotate, stretch, shrink, or invert curves on the complex plane, and take the following form (Needham, 2007): μ (z) = az + b/cz + d, wherein a, b, c, d, and z are complex numbers. Remarkably, functional compositions of μ can also be calculated by multiplying matrices derived from it, termed Möbius matrices, drastically reducing the complexity of the functional-composition problem. 255
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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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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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