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Research Papers

A Mathematical Model for Understanding Fluid Flow Through Engineered Tissues Containing Microvessels

[+] Author and Article Information
Kristen T. Morin, Michelle S. Lenz, Caroline A. Labat

Department of Biomedical Engineering,
University of Minnesota,
Minneapolis 55455, MN

Robert T. Tranquillo

Department of Biomedical Engineering,
University of Minnesota,
Minneapolis 55455, MN;
Department of Chemical Engineering and
Materials Science,
University of Minnesota,
Minneapolis 55455, MN
e-mail: tranquillo@umn.edu

1Corresponding author.

Manuscript received July 24, 2014; final manuscript received November 8, 2014; published online February 24, 2015. Assoc. Editor: Ram Devireddy.

J Biomech Eng 137(5), 051003 (May 01, 2015) (11 pages) Paper No: BIO-14-1340; doi: 10.1115/1.4029236 History: Received July 24, 2014; Revised November 08, 2014; Online February 24, 2015

Knowledge is limited about fluid flow in tissues containing engineered microvessels, which can be substantially different in topology than native capillary networks. A need exists for a computational model that allows for flow through tissues dense in nonpercolating and possibly nonperfusable microvessels to be efficiently evaluated. A finite difference (FD) model based on Poiseuille flow through a distribution of straight tubes acting as point sources and sinks, and Darcy flow through the interstitium, was developed to describe fluid flow through a tissue containing engineered microvessels. Accuracy of the FD model was assessed by comparison to a finite element (FE) model for the case of a single tube. Because the case of interest is a tissue with microvessels aligned with the flow, accuracy was also assessed in depth for a corresponding 2D FD model. The potential utility of the 2D FD model was then explored by correlating metrics of flow through the model tissue to microvessel morphometric properties. The results indicate that the model can predict the density of perfused microvessels based on parameters that can be easily measured.

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Figures

Grahic Jump Location
Fig. 1

Diagram showing case of interest: Uniaxially aligned microvessels confined to the plane. Only two planes are drawn for clarity.

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Fig. 2

Comparison of 3D FD and FE models for a single microvessel. Velocity vector plot (a) and pressure map (b) for the 3D FD model, and velocity vector plot (c) and pressure map (d) for the FE model. In (b) and (d), the pressure units are mm Hg. Values plotted are for the plane containing the microvessel. Parameter values used are from Tables 1 and 2. Tissue thickness = 200 μm.

Grahic Jump Location
Fig. 3

2D FD model results for a single microvessel. (a) Velocity vector plot of the outlet of the microvessel for standard parameter values (Table 2). The thick line represents the microvessel. The arrows in the bulk represent interstitial velocities, whereas the arrow at the end of the microvessel represents the velocity through the microvessel. (b) Pressure map of the whole tissue. (c)–(h) Velocity through the microvessel and the pressure at the tissue inlet plotted against the following parameters: (c) x-velocity at the tissue inlet, (d) ECM permeability, (e) fluid viscosity, (f) microvessel radius, (g) microvessel angle relative to the x-direction, and (h) ECM permeability anisotropy. For each plot, all other input parameters were kept at their standard values.

Grahic Jump Location
Fig. 4

FE model results for a single microvessel. (a) Velocity vector plot of the region near the microvessel outlet for standard parameter values (Table 1, tissue thickness = 200 μm). The arrows indicating the velocity through the microvessel were removed to more clearly show the interstitial velocities. (b) Pressure map of the whole tissue. (c)–(i) Velocity through the microvessel and the pressure at the tissue inlet plotted against the following parameters: (c) x-velocity at the tissue inlet, (d) ECM permeability, (e) fluid viscosity, (f) microvessel radius, and (g) microvessel angle relative to the x-direction. For each plot, all other input parameters were kept at their standard values.

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Fig. 7

Evaluation of the FD model predictive capability. Example pressure plots for tissues with microvessel densities near 25 (a) or 550 (b) microvessels/mm2. The black lines represent the location of microvessels. The color scale is the same for both plots. Pressure at the tissue inlet and the effective hydraulic permeability are plotted for varying levels of the following parameters: (c) percentage of bifurcations, (d) microvessel anisotropy index, (e) microvessel length, (f) ECM anisotropy, and (g) (perfused) lumen density. (f) and (g) The data were fit with power equations, which are shown on the plots.

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Fig. 8

Schematic of a node (i) at which a microvessel begins. The microvessel is indicated by the solid straight line, and the arrow overlapping the microvessel indicates the fluid velocity in the microvessel (vt). An interstitial velocity (vi) was also defined at node i and is indicated by the second arrow. θ represents the angle between the microvessel and the x-direction.

Grahic Jump Location
Fig. 6

Comparison between the 2D FD and FE models for two microvessels. (a) Velocity vector plot showing the FD model results for a separation distance of 5 μm. The thick lines represent the microvessels. Arrows in the bulk represent interstitial velocities and the arrow at the tip of the microvessel represents velocities through the microvessels. (b) Velocity vector plot results from the FE model for a separation distance of 5 μm. No velocity vectors for flow within the microvessels are shown so that the interstitial velocities can be seen. The white rectangles represent the microvessels. (c) The percent error in the downstream microvessel velocity and pressure at the tissue inlet between relative to the FE model for a variety of separation distances between the two microvessels.

Grahic Jump Location
Fig. 5

Dependence of 2D FD model accuracy on microvessel radius for a single microvessel. The percent error in the microvessel fluid velocity relative to the FE model for varying microvessel radius is plotted.

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