Fig 1.
Schematic of our multi-scale multi-physics model of ventilation-perfusion matching.
Block (A) illustrates the whole-lobe vascular network model. Black lines represent blood vessels, and colored regions represent discrete zones of perfusion. The network geometry is agnostically generated by a space-filling algorithm inspired by Wang et al. [33]. Block (B) shows how the mechanics of each vessel segment is represented as an equivalent circuit. Intravascular pressure (Pv) and flow into the vessel (qin) are the state variables; inlet pressure (Pin) and flow out of the vessel (qout) are the initial conditions at boundaries for a given vessel segment; outlet pressure (Pout) is an algebraic constraint; time-varying alveolar pressure (Palv) is a dynamic pressure source; and hydraulic resistance (R), inertance (L), compliance (C), and vessel wall resistance (RD) are anatomical parameters calculated from the geometry of the vessel segment (length and radius) and can be modulated by vasoregulation (purple boxes). Block (C) depicts a representative gas exchange unit. Gases flowing through the capillary tube are exchanged with an alveolar compartment. Block (D) portrays the oxygen-sensitive vasoregulatory mechanism hypoxic pulmonary vasoconstriction (HPV). Hypoxia in the alveolar space induces conducted vasoconstriction. Conducted vascular responses are spatially propagated in a upstream through the endothelium—these responses modulate the values of the anatomical parameters (R, C, and RD) in the arterial network.
Table 1.
Model variables, fixed parameters, and adjustable parameters.
Values are omitted if they are time varying or defined for each vessel segment. For the adjustable parameters estimated by the BELUGA Genetic Algorithm, values are reported for each network (A,B,C).
Fig 2.
Example of angiogenesis-inspired network generation algorithm.
Blue points define a mesh that the algorithm seeks to cover with a network. The black and red dots are the nodes generated by each iteration of the algorithm; red nodes denote network terminals; black lines are connections between nodes. The black scale bar is 1000 μm. The first two nodes (1 and 2) are manually specified by the user. The algorithm determines the mesh points furthest from the network. For nodes 3-5, the southwest corner is the furthest mesh point. For node 6, the southeast corner is furthest and then for node 7 the southwest corner is furthest. For node 8, the southeast corner is the furthest point and is connected to closest node of the network—node 6. The northern most mesh point is furthest from both node 7 and 8. Node 9 is connected to node 2 since it is the closest node in the existing network. To implement the user needs to supply a domain of points to cover, the “grow distance” between nodes, and the number of nodes to generate.
Fig 3.
Morphometrically realistic pulmonary vascular networks.
(A,B,C) Synthetic population of rat lungs. Blue dots are the finite-element mesh points made by Polymesher [34], the cyan circles denote the root node for each network, the black lines represent the network topology, and the red squares denote the network terminals. The black scale bar is 3000 μm. All networks were made with 1,000 nodes and Δg set to 50 μm. Note that the radius of each vessel is not plotted to size. The total area spanned by each finite element mesh is the same. (D, E) Comparison of our network geometries to measured diameters and lengths as a function of diameter-defined Strahler orders from silicone-elastometer casts of rat pulmonary arterial trees in Jiang et al. [35].
Fig 4.
Whole-organ pressure-flow relationships.
Discrete data points are taken from isolated-perfused-ventilated lung experiments performed in Molthen et al. [50] and Chlopicki et al. [20]. Simulations are run under Zone 3 conditions (venous outlet pressure (7 mmHg) > alveolar pressure (6 mmHg)). The model simulation curves that are near the 95% N2 (Chlopicki) data point were run with T* = 0.8, and the curves near the Papaverine (Molthen) data points were run with T* = 0.5.
Fig 5.
Fits from BELUGA Genetic Algorithm.
(A) Oxygen partial pressure in the pulmonary vein. The block circle and error bars denote data from Cheng et al., the blue circle denotes network A, the red square denotes network B, and the green diamond denotes network C. The plotted point is the best fit individual from BELUGA (parameters reported in Table 2). (B) The same data and results from (A) but expressed in terms of oxyhemoglobin saturation.
Table 2.
Results from BELUGA Genetic Algorithm.
The top block contains oxygen tensions from Cheng et al., and most fit individuals for each network from BELUGA optimization. The middle block contains the value of our objective function for each network from the most fit individuals. The bottom block contains the best fit parameter values for each network.
Table 3.
Standard Deviation (SD) and Correlations between parameters estimated by BELUGA Genetic Algorithm.
Table 4.
Percent change in oxygen tension of the primary vein given a ±10% perturbation in nominal parameter value. *—The sensitivity index is less than 0.01.
Fig 6.
Effect of HPV and uniform vasoconstriction on Network A regional perfusion, V/Q matching, and oxygen flux.
(A-C) Leftmost column of networks are without regulation from HPV, middle column of networks are with regulation from HPV, and the rightmost column of networks are with uniform vasoconstriction, where T* = 0.8 is used for the global vascular tone. The black scale bar is 1000 μm. (A) Perfusion in the capillary compartments; (B) Ventilation-perfusion (V/Q) ratios in the capillary compartments; (C) Oxygen flux in the capillary compartments. (D-G) Blood flow, RBC transit times, V/Q ratios, and oxygen flux expressed as probability densities—calculated with a kernel density estimator. (D) Distribution of flow; (E) RBC transit times normalized to mean transit time; (F) Distribution of V/Q ratios; (G) Distribution of oxygen flux.
Fig 7.
Effect of HPV and uniform vasoconstriction on Network B regional perfusion, V/Q matching, and oxygen flux.
(A-C) Leftmost column of networks are without regulation from HPV, middle column of networks are with regulation from HPV, and the rightmost column of networks are with uniform vasoconstriction, where T* = 0.8 is used for the global vascular tone. The black scale bar is 1000 μm. (A) Perfusion in the capillary compartments; (B) Ventilation-perfusion (V/Q) ratios in the capillary compartments; (C) Oxygen flux in the capillary compartments. (D-G) Blood flow, RBC transit times, V/Q ratios, and oxygen flux expressed as probability densities—calculated with a kernel density estimator. (D) Distribution of flow; (E) RBC transit times normalized to mean transit time; (F) Distribution of V/Q ratios; (G) Distribution of oxygen flux.
Fig 8.
Effect of HPV and uniform vasoconstriction on Network C regional perfusion, V/Q matching, and oxygen flux.
(A-C) Leftmost column of networks are without regulation from HPV, middle column of networks are with regulation from HPV, and the rightmost column of networks are with uniform vasoconstriction, where T* = 0.8 is used for the global vascular tone. The black scale bar is 1000 μm. (A) Perfusion in the capillary compartments; (B) Ventilation-perfusion (V/Q) ratios in the capillary compartments; (C) Oxygen flux in the capillary compartments. (D-G) Blood flow, RBC transit times, V/Q ratios, and oxygen flux expressed as probability densities—calculated with a kernel density estimator. (D) Distribution of flow; (E) RBC transit times normalized to mean transit time; (F) Distribution of V/Q ratios; (G) Distribution of oxygen flux.
Table 5.
Effect of HPV on regional flow, V/Q matching, and oxygenation.
Model simulations were performed with uniform vasoconstriction (UVC), and with and without regulation by HPV. All simulations used a fixed 25 mmHg arterial-venous pressure drop. The top block summarizes how HPV and uniform vasoconstriction (UVC) affects the total flow through the organ, and the variability of blood flow, V/Q ratio by the Coefficient of Variation (CV = μ/σ). The bottom block summarizes how HPV impacts the oxygenation in terms of the total alveolar-capillary oxygen mass flux, oxygen mass flux CV, oxygen mass flux normalized to total oxygen flux, venous oxygen tension, and venous oxyhemoglobin saturation.
Fig 9.
Effect of the fraction of inspired oxygen on HPV-mediated redistributions in hemodynamics, V/Q matching, and oxygen flux.
(A) Distributions of capillary blood flow; (B) Distributions of V/Q ratios; (C) Distributions of oxygen flux; (D) RBC transit time distributions; (E) Coefficient of variation of blood flow, V/Q ratio, and oxygen flux plotted as a function Fi,O2.
Fig 10.
Effect of airway occlusions on HPV-mediated redistributions in hemodynamics, V/Q matching, and oxygen flux.
(A-C) Leftmost column of networks are with 0% of the alveoli occluded, middle column of networks are with 17% of the alveoli occluded, and the rightmost column of networks are with 28% of the alveoli occluded. The first airway occlusion is at the most southwestern perfusion zone, and progressive airway occlusions are made following the structure of the vascular network. The black scale bar is 1000 μm. (A) Perfusion in the capillary compartment; (B) Ventilation-Perfusion (V/Q) ratios in the capillary compartment; (C) Oxygen flux in the capillary compartments; (D) End-venous oxygen tension as a function of the fraction of occluded alveoli; (E); Pearson correlation between ventilation and perfusion as a function of the fraction of occluded alveoli; (F) Coefficient of Variation for blood flow (circles), V/Q ratios (x’s), and oxygen flux (squares); (G) Distribution of blood flow; (H) Distribution of V/Q ratios; (G) Distribution of oxygen flux.