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248 Reverse Engineering – Recent Advances and Applications 6 Will-be-set-by-IN-TECH consider an elementary airway unit to be a “1/8 subacinus” (Weibel, 1984) that begins with branching generation 18, instead of an entire lung acinus that includes respiratory bronchioles, and supports a network of alveolar ducts spanning 5 generations. Successive subacinii corresponding to the other exercise states are termed 1/16- (moderate exercise), 1/32- (heavy exercise), and 1/64-subacinii (maximum exercise). 3.2 Equations for oxygen transport throughout and across the acinar tree While strenuous exercise pushes the diffusion source deeper into the lung, as evidenced by Eqn. 3 and Fig. 3, the evolution of the oxygen concentration difference between the subacini entrance and the blood side of the membranes, c (x), obeys the stationary diffusion equation (Felici et al., 2003): ∇ 2 c (x) = 0, with x ∈ diffusion space. (4) The concentration across the tree’s entrance (the diffusional “source”) gives the first boundary condition, c (x) = ca, with x ∈ source. (5) The oxygen flux entering the gas-exchange (the diffusional “receptor”) surface is matched with the flux moving through it (Hou, 2005): ∇c (x) · n (x) = c (x) − cbβa/β b , with x ∈ boundary, (6) Λ wherein the βa and β b are the solubility of oxygen in air and blood, respectively. The parameter Λ = Da/W is the ratio of the diffusivity of oxygen in air, Da, to the surface permeability, W, has units of length, and is the only length scale for the boundary-valued problem defined by Eqns. 4 to 6. Moreover, it has an elegant physical interpretation: it measures the length along the surface a diffusing molecule visits before it is absorbed by the surface, and for this reason is termed the exploration length (Pfeifer & Sapoval, 1995). 3.3 The effective surface area: area of the active zones involved in the oxygen transport Although the diffusion-reaction problem defined by Eqns. 4 to 6 give concentrations across the tree’s exchange surface, we are instead interested in the total current of molecules supplied to the blood by the whole of the airway tree. Computing this current for a single subacinus, Ig, from Eqns. 4 to 6 can be carried out by integrating the concentrations over its entire surface, Sg (Hou et al., 2010): � Ig = W (c (x) − cbβa/β b) dS x∈Sg (7) Manipulating this equation leads to an expression for the current in terms of a constant concentration difference: Ig = WSeff,g (ca − cbβa/β b) , wherein Seff,g is a function of the exploration length (Hou, 2005): (8) � Seff,g (Λ) = c (x; Λ) − cbβa/β b dS. (9) x∈Sg ca − c bβa/β b
Reverse-Engineering the Robustness of Mammalian Lungs Reverse-Engineering the Robustness of Mammalian Lungs 7 This quantity, termed the effective surface area (Felici et al., 2003; Felici et al., 2004; Grebenkov et al., 2005; Hou, 2005; Hou et al., 2010), measures the “active” portion of the tree’s surface, and is generally less than its total, i.e. S eff,g ≤ Sg. The difference in these areas is unused in any molecular transport, and is “inactive,” or screened from molecular access to it. This screening can be conceptualized in terms of the exploration length (see below). For example, Λ = 0 (i.e. permeability is infinite) describes a situation in which no molecules explore areas of the tree beyond the entrance, being immediately absorbed by the surface on their first contact with it. This regime is termed complete screening. In contrast, Λ = ∞ (i.e. permeability is zero) implies that oxygen molecule hits the surface many times and eventually visits the entire surface, marking the regime of no screening. These facts provide a powerful conceptual picture of the lung’s operational principles: to increase the current under constant conditions of membrane permeability and concentration difference, the effective surface area must increase proportionally. So, for breathing/exercise states of increasing exercise intensity, ever larger fractions of the lung’s exchange area must be recruited for the molecular transport across it. 3.4 Total current into the tree The current crossing the lung’s exchange areas is the sum of currents contributed from each individual branch of the subacinar tree, given by Eqn. 8: Ng I = ∑ Ig,i, (10) i=1 wherein Ng gives the number of gas exchangers in the lung in the current exercise state (explained below). Taking an average of both sides of Eqn. 10 over all such gas-exchangers gives I = Ng Ig, wherein Ig is the average current supplied by a single subacinus. The total current can then be written as I = NgWS eff,g (ca − c b/βa/β b) . (11) This equation, Eqn. 11, can be rewritten in terms of the physiological variables of Eqn. 2: I = DmβmS eff (pa − p τ b) = WSeff (ca − cbβ b/βa) , (12) wherein the diffusing capacity of the lung has been replaced by the diffusion coefficient of oxygen through the surface of alveolar membranes, Dm, the thickness of the membrane-barrier, τ, the solubility of oxygen in the membranes, βm , and the effective surface area of the entire lung, Seff. The permeability can itself be expressed in terms of measurable quantities, and dependent upon the thickness of the membranes: W = Dmβm . (13) τ To good approximation, the thickness of the membranes is constant and the permeability is position-independent. 249
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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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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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