Research Papers

Effect of Blood Flow on Near-the-Wall Mass Transport of Drugs and Other Bioactive Agents: A Simple Formula to Estimate Boundary Layer Concentrations

[+] Author and Article Information
Sandra Rugonyi

Biomedical Engineering Department, Oregon Health & Science University, 3303 SW Bond Avenue, Portland, OR 97239rugonyis@ohsu.edu

J Biomech Eng 130(2), 021010 (Mar 28, 2008) (7 pages) doi:10.1115/1.2899571 History: Received October 26, 2006; Revised June 25, 2007; Published March 28, 2008

Transport of bioactive agents through the blood is essential for cardiovascular regulatory processes and drug delivery. Bioactive agents and other solutes infused into the blood through the wall of a blood vessel or released into the blood from an area in the vessel wall spread downstream of the infusion/release region and form a thin boundary layer in which solute concentration is higher than in the rest of the blood. Bioactive agents distributed along the vessel wall affect endothelial cells and regulate biological processes, such as thrombus formation, atherogenesis, and vascular remodeling. To calculate the concentration of solutes in the boundary layer, researchers have generally used numerical simulations. However, to investigate the effect of blood flow, infusion rate, and vessel geometry on the concentration of different solutes, many simulations are needed, leading to a time-consuming effort. In this paper, a relatively simple formula to quantify concentrations in a tube downstream of an infusion/release region is presented. Given known blood-flow rates, tube radius, solute diffusivity, and the length of the infusion region, this formula can be used to quickly estimate solute concentrations when infusion rates are known or to estimate infusion rates when solute concentrations at a point downstream of the infusion region are known. The developed formula is based on boundary layer theory and physical principles. The formula is an approximate solution of the advection-diffusion equations in the boundary layer region when solute concentration is small (dilute solution), infusion rate is modeled as a mass flux, and there is no transport of solute through the wall or chemical reactions downstream of the infusion region. Wall concentrations calculated using the formula developed in this paper were compared to the results from finite element models. Agreement between the results was within 10%. The developed formula could be used in experimental procedures to evaluate drug efficacy, in the design of drug-eluting stents, and to calculate rates of release of bioactive substances at active surfaces using downstream concentration measurements. In addition to being simple and fast to use, the formula gives accurate quantifications of concentrations and infusion rates under steady-state and oscillatory flow conditions, and therefore can be used to estimate boundary layer concentrations under physiological conditions.

Copyright © 2008 by American Society of Mechanical Engineers
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Figure 1

Mass transfer problem considered. (a) Sketch of circular tube considered. (b) Sketch of solute concentration at the tube wall. (c) Sketch of the thickness of the concentration boundary layer, δ, along the tube wall. In this paper, δ⪡R was assumed.

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Figure 2

Solute concentration along the wall in the infusion region (Region I in Fig. 1). The analytical solution was calculated using Eq. 6, and parameters used in the calculations are listed in Table 1.

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Figure 3

Comparison between solutions obtained with FEM and the developed formula in Region II. (a) Normalized concentrations across the boundary layer region. The approximate profile was calculated using Eqs. 10,11, at x=50mm. (b) Solute concentration along the wall in Region II. The approximate solution (formula) was calculated using Eqs. 13,7. Parameters used are listed in Table 1.

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Figure 4

Plots of (B∕dx+1)2∕3 versus 1∕Cw in Region II (see Eq. 13), obtained from results of FEM simulations for different values of D. The diffusion coefficients employed in the simulations are as follows (8): thrombin (Th), D=10−11m2∕s (this value represents only the order of magnitude of D for thrombin); heparin (Hep), D=10−10m2∕s; NO, D=10−9m2∕s. Other parameters used are listed in Table 1.

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Figure 5

Wall concentration of a solute infused through three wall regions of equal length (d=1mm) separated by regions (of length 1mm) with no transfer through the wall. Concentrations obtained by adding contributions from single infusion regions, calculated using Eqs. 6,13, were compared to concentrations obtained with FEM. Other parameters employed in the calculations are shown in Table 1.

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Figure 6

Dynamic variation of wall concentration during oscillatory flow (obtained with FEM) at two different wall locations. Flow rate oscillated sinusoidally between 100ml∕min and 300ml∕min. Diamonds, end of infusion region (x=0mm); squares, x=50mm. Note that the concentration scales at x=0 and x=50mm are different, and that the amplitudes of oscillation are much smaller than those of a quasisteady oscillatory flow (see also Table 2). Other parameters employed in the calculations are shown in Table 1.




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