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Research Papers

# A Two-Dimensional Computational Model of Lymph Transport Across Primary Lymphatic Valves

[+] Author and Article Information
Peter Galie

Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590

Robert L. Spilker

Department of Biomedical Engineering and Department of Mechanical, Aeronautical and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590spilker@rpi.edu

J Biomech Eng 131(11), 111004 (Oct 16, 2009) (9 pages) doi:10.1115/1.3212108 History: Received September 11, 2008; Revised May 22, 2009; Published October 16, 2009

## Abstract

This study utilizes a finite element model to characterize the transendothelial transport through overlapping endothelial cells in primary lymphatics during the uptake of interstitial fluid. The computational model is built upon the analytical model of these junctions created by Mendoza and Schmid-Schonbein (2003, “A Model for Mechanics of Primary Lymphatic Valves,” J. Biomed. Eng., 125, pp. 407–414). The goal of the present study is to investigate how adding more sophisticated and physiologically representative biomechanics affects the model’s prediction of fluid uptake. These changes include incorporating a porous domain to represent interstitial space, accounting for finite deformation of the deflecting endothelial cell, and utilizing an arbitrary Lagrangian–Eulerian algorithm to account for interacting and nonlinear mechanics of the junctions. First, the present model is compared with the analytical model in order to understand its effects on parameters such as cell deflection, pressure distribution, and velocity profile of the fluid entering the lumen. Without accounting for the porous nature of the interstitium, the computational model predicts greater cell deflection and consequently higher lymph velocities and flow rates than the analytical model. However, incorporating the porous domain attenuates the cell deflection and flow rate to values below that predicted by the analytical model for a given transmural pressure. Second, the present model incorporates recent experimental data for parameters such as lymph viscosity, transmural pressure measurements, and others to evaluate the ability of these junctions to act as unidirectional valves. The volume of flow through the valve is calculated to be $0.114 nL/μm$ per cycle for a transmural pressure varying between 8.0 mm Hg and −1.0 mm Hg at 0.4 Hz. Though experimental data for the absorption of lymph through these endothelial junctions are scarce, several measurements of lymph velocity and flow rates are cited to validate the present model.

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## Figures

Figure 1

1—Open endothelial valve, 2—collagen and elastin fibers, 3—anchoring filaments (not to scale)

Figure 2

Geometric representation of primary lymphatic valve—the coordinate axis designates the origin. The portions of the cell fixed to the extracellular matrix are distinguished from the part of the cell able to deform. The location of the transmural pressure boundary condition Δp=p1−p2 is indicated. xu and xov represent the portions of the endothelial cell that are free to deform, and overlapping, respectively. The thickness of the cell yt is 0.25 microns.

Figure 3

(a) Computed cell deflection without porous domain. 1—Lumen fluid flow, 2—fixed endothelial cell wall, 3—deforming endothelial cell, 4—porous extracellular flow. (b) Location of interface between porous extracellular and lumen fluid domains.

Figure 4

Comparison of the cell deflection—predicted by the analytical and computational models

Figure 5

Velocity profile at valve junction for a 10 dyne/cm2. Transendothelial pressure—the velocity is calculated at this junction, along the deflection of the cell. For this calculation, the cell is deflected about 0.25 microns at the junction (vx: velocity in the x direction, vy: velocity in the y direction, and total velocity is the resultant of these two components).

Figure 6

Pressure distribution comparison—this figure illustrates the effect of assuming a constant pressure for a portion of the deforming cell

Figure 7

Results of Transient Analysis, utilizing parameter values given in Table 2. (a)–(b) The results indicate that the overlapping cells do act as valves by preventing backflow when the lumen pressure exceeds the interstitial pressure. (c) Both the transmural pressure and afferent flow rate are plotted as a function of time (d) and as a pressure-flow relationship (the results are shown for a single cycle from t=0 to 2.5 s, although the computation was executed for three cycles to confirm the periodicity of the response).

Figure 8

Velocity profile at the valve junction for different values of porosity and permeability. The transmural pressure was set at 200 Pa (1.5 mm Hg) for these computations (elastic modulus=10,000 Pa).

Figure 9

Afferent flow rate as a function of transmural pressure for varying valve geometries

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