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Research Papers

Blood Flow in Capillaries of the Human Lung

[+] Author and Article Information
Shimon Haber

Faculty of Mechanical Engineering,
Technion-Israel Institute of Technology,
Haifa 32000, Israel
e-mail: mersh01@tx.technion.ac.il and
shimonhaber@gmail.com

Alys Clark

e-mail: alys.clark@auckland.ac.nz

Merryn Tawhai

e-mail: m.tawhai@auckland.ac.nz
Auckland Bioengineering Institute,
The University of Auckland,
Auckland 1142, New Zealand

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the Journal of Biomechanical Engineering. Manuscript received September 21, 2012; final manuscript received May 12, 2013; accepted manuscript posted July 29, 2013; published online September 20, 2013. Assoc. Editor: Ender A. Finol.

J Biomech Eng 135(10), 101006 (Sep 20, 2013) (11 pages) Paper No: BIO-12-1427; doi: 10.1115/1.4025092 History: Received September 21, 2012; Revised May 12, 2013; Accepted July 29, 2013

A novel model for the blood system is postulated focusing on the flow rate and pressure distribution inside the arterioles and venules of the pulmonary acinus. Based upon physiological data it is devoid of any ad hoc constants. The model comprises nine generations of arterioles, venules, and capillaries in the acinus, the gas exchange unit of the lung. Blood is assumed incompressible and Newtonian and the blood vessels are assumed inextensible. Unlike previous models of the blood system, the venules and arterioles open up to the capillary network in numerous locations along each generation. The large number of interconnected capillaries is perceived as a porous medium in which the flow is macroscopically unidirectional from arterioles to venules openings. In addition, the large number of capillaries extending from each arteriole and venule allows introduction of a continuum theory and formulation of a novel system of ordinary, nonlinear differential equations which governs the blood flow and pressure fields along the arterioles, venules, and capillaries. The solution of the differential equations is semianalytical and requires the inversion of three diagonal, 9 × 9 matrices only. The results for the total flow rate of blood through the acinus are within the ballpark of physiological observations despite the simplifying assumptions used in our model. The results also manifest that the contribution of the nonlinear convection term of the Navier-Stokes equations has little effect (less than 2%) on the total blood flow entering/leaving the acinus despite the fact that the Reynolds number is not much smaller than unity at the proximal generations. The model makes it possible to examine some pathological cases. Here, centri-acinar and distal emphysema were investigated yielding a reduction in inlet blood flow rate.

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References

Figures

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Fig. 1

An illustration of the anatomy of the lung. Proximal to the transitional bronchiole are the conducting airways shown on the left-hand side (right lung) of the schematic with green, blue, and red airways indicating those which lie in the upper, lower, and middle lobes of the right lung, respectively. Each conducting airway has an accompanying artery (red) and vein (blue), shown on the right-hand side (left lung) of the schematic. Distal to the terminal bronchiole is the acinus, of which there are approximately 32,000 in the human lung, which comprises of alveolar ducts/sacs, arterioles, venules, and capillaries. The schematic of the acinus is adapted from Clark et al. [19].

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Fig. 2

(a) A model of the arteriole–venule–capillary blood vessel system in the lung. Unlike the geometric description of the pulmonary acinus given by Clark et al. [9] where a single capillary sheet connects arterioles and venules at each generation g, there are multiple capillary connections. A blow up of a particular generation is depicted in (b). (b) A diagrammatic representation of the arterioles, venules, and capillaries in a single acinar generation. Arrows show the direction of flow (Q) and important parameters are labeled (i = a for arteriole and v for venules). CV stands for control volume for which mass conservation and linear momentum balance was applied to obtain the ordinary differential equations governing the flow.

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Fig. 3

(a) The blood flow rate at the various generations not accounting for inertia effects (linear case Sec. 4). (b) The blood flow rate at the various generations accounting for inertia effects (Appendix A).

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Fig. 4

The flow rate through a single capillary down the artery/vein tree calculated for the case of negligible inertia

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Fig. 5

The pressure difference between the arterioles and venules decreases as one moves distally through the acinus—this results in the flow rate decreasing through the acinus as seen in Fig. 4

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Fig. 6

The dependence of the inlet artery flow rate upon the pressure difference Pin − Pout. (*) denotes an exact solution accounting for inertia, and (+) an approximate solution neglecting inertia.

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Fig. 7

Inlet blood flow in centri-acinar or distal emphysema. Centri-acinar emphysema is regarded as destruction of either 1, 1 + 2 or 1 + 2 + 3 capillary generations, while distal emphysema is perceived as destruction of capillary generations 9, 9 + 8 or 9 + 8 + 7. Two cases are considered: 99.5% or 50% of capillaries being destroyed. (Pin − Pout = 784 Pa and ε = 1.)

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Fig. 8

Pressure distribution along the arterioles and venules calculated at the bifurcation points in case of centri-acinar or distal emphysema [assuming a single capillary is not destroyed in generations 1, 2, and 3, or, in 1, 7, 8, and 9 (Pin − Pout = 784 Pa and ε = 1)]

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