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Research Papers

A One-Dimensional Mathematical Model for Studying the Pulsatile Flow in Microvascular Networks

[+] Author and Article Information
Qing Pan

Department of Biomedical Engineering,
Key Laboratory of Biomedical
Engineering of MOE,
Zhejiang University,
Hangzhou 310027, China
College of Information Engineering,
Zhejiang University of Technology,
Hangzhou 310023, China

Ruofan Wang

Department of Biomedical Engineering,
Key Laboratory of Biomedical
Engineering of MOE,
Zhejiang University,
Hangzhou 310027, China

Bettina Reglin

Department of Physiology and CCR,
Charité, Charitéplatz 1,
Berlin 10117, Germany

Jing Yan

Department of ICU,
Zhejiang Hospital,
Lingyin Road 12,
Hangzhou 310013, China

Axel R. Pries

Department of Physiology and CCR,
Charité, Charitéplatz 1,
Berlin 10117, Germany
Deutsches Herzzentrum Berlin,
Augustenburger Platz 1,
Berlin D-13353, Germany
e-mail: axel.pries@charite.de

Gangmin Ning

Department of Biomedical Engineering,
Key Laboratory of Biomedical
Engineering of MOE,
Zhejiang University,
Zheda Road 38,
Hangzhou 310027, China
e-mail: gmning@zju.edu.cn

lCorresponding authors.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received April 9, 2013; final manuscript received October 19, 2013; accepted manuscript posted October 31, 2013; published online December 4, 2013. Assoc. Editor: Dalin Tang.

J Biomech Eng 136(1), 011009 (Dec 04, 2013) (11 pages) Paper No: BIO-13-1179; doi: 10.1115/1.4025879 History: Received April 09, 2013; Revised October 19, 2013; Accepted October 31, 2013

Techniques that model microvascular hemodynamics have been developed for decades. While the physiological significance of pressure pulsatility is acknowledged, most of the microcirculatory models use steady flow approaches. To theoretically study the extent and transmission of pulsatility in microcirculation, dynamic models need to be developed. In this paper, we present a one-dimensional model to describe the dynamic behavior of microvascular blood flow. The model is applied to a microvascular network from a rat mesentery. Intravital microscopy was used to record the morphology and flow velocities in individual vessel segments, and boundaries are defined according to the experimental data. The system of governing equations constituting the model is solved numerically using the discontinuous Galerkin method. An implicit integration scheme is adopted to increase computing efficiency. The model allows the simulation of the dynamic properties of blood flow in microcirculatory networks, including the pressure pulsatility (quantified by a pulsatility index) and pulse wave velocity (PWV). From the main input arteriole to the main output venule, the pulsatility index decreases by 66.7%. PWV obtained along arterioles declines with decreasing diameters, with mean values of 77.16, 25.31, and 8.30 cm/s for diameters of 26.84, 17.46, and 13.33 μm, respectively. These results suggest that the 1D model developed is able to simulate the characteristics of pressure pulsatility and wave propagation in complex microvascular networks.

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Figures

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Fig. 1

Topology of the rat mesenteric vascular network. Arterioles, capillaries, and venules are colored red, yellow, and blue, respectively. The main feeding arteriole and the main draining venule are indicated by red and blue arrows, respectively. Secondary boundaries are indicated by small white directional arrows.

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Fig. 3

Distribution of the mean pressure and flow velocity against vessel diameter in arterioles and venules. Shown in the figure are mean values for the indicated diameter range with standard deviations.

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Fig. 4

Distribution maps of PIP (pulsatility index of pressure, (a)) and PTTP (pulse transit time of pressure, (b)). (a) The main feeding arteriole and the main draining venule are indicated by red and blue arrows, respectively. PIP decreases in the arteriolar tree and remains constant in the venular portion. The white arrows indicate secondary input boundaries, which are far from the feeding arteriole but exhibit high PIP values. They influence network regions limited to two to three generations of bifurcations. (b) PTTP continuously increases from the main feeding arteriole to the venular portion.

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Fig. 5

Four exemplary flow pathways from the main feeding arteriole to the main draining venule. The selected segments for showing PIP are indicated by A1–A4, C1, and V1–V4, respectively.

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Fig. 6

PIP on the four exemplary arteriovenous pathways shown in Fig. 5 (mean ± standard deviation). The PIP decreases from the main feeding arteriole to the postcapillary level and remains almost constant in the venular pathway.

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Fig. 7

The pressure waveforms of the selected segments on the arteriovenous pathway “A”. The waveforms are substantially damped from the main feeding arteriole to the capillary level and remain almost unchanged in the venular section.

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Fig. 2

Flow diagrams of the strategies of applying phase separation and the Fahraeus–Lindqvist effect. Strategy I: The phase separation effect is used in each simulation step. The viscosity is updated by the hematocrit derived by the phase separation effect and the varying diameter in each simulation step. The simulation moves to the next step until the pressure and velocity are converged in the inner iteration. Strategy II: The simulation begins with fixed discharge hematocrits. In each simulation step, the viscosity is updated by the fixed hematocrit and varying diameter. The gray block demonstrates the strategy finally adopted in this study, which uses fixed hematocrits calculated by a steady state model.

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Fig. 8

Normalized PIP on the exemplary pathway “A” with input heart rates of 75, 150, 225, and 300 bpm

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Fig. 9

Pulsatility damping on the arteriovenous pathway “A” under different settings for Young's modulus. E1: initial modulus setting. EA2: 200% arteriolar modulus. EA0.5: 50% arteriolar modulus. EC2: 200% capillary modulus. EC0.5: 50% capillary modulus. EV2: 200% venular modulus. EV0.5: 50% venular modulus.

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