The flow through a curved tube whose radius of curvature varies with time was studied in order to better understand flow patterns in coronary arteries. A computational flow model was constructed using commercially available software. The artery model featured a uniform circular cross section, and the curvature was assumed to be constant along the tube, and in one plane. The computational model was verified with the use of a dynamically similar in vitro apparatus. A steady uniform velocity was prescribed at the entrance at a Reynolds number of 300. Two sets of results were obtained: one in which the curvature was held constant at the mean, maximum and minimum radii of curvature (quasistatic), and another in which the curvature was varied sinusoidally in time at a frequency of I Hz (dynamic). The results of the dynamic analysis showed that the wall shear rates varied as much as 52% of the static mean wall shear rate within a region of 10 tube diameters from the inlet. The results of the dynamic analysis were within 6% of the quasistatic predictions. Realistic modeling of the deforming geometry is important in determining which locations in the coronary arteries are subjected to low and oscillating wall shear stresses, flow patterns that have been associated with atherogenesis.
The flow through a curved tube model of a coronary artery was investigated computationally to determine the importance of time-varying curvature on flow patterns that have been associated with the development of atherosclerosis. The entry to the tube was fixed while the radius of curvature varied sinusoidally in time at a frequency of 1 or 5 Hz. Angiographic data from other studies suggest that the radius of curvature waveform contains significant spectral content up to 6 Hz. The overall flow patterns were similar to those observed in stationary curved tubes; velocity profile skewed toward the outer wall, secondary flow patterns, etc. The effects of time-varying curvature on the changes in wall shear rate were expressed by normalizing the wall shear rate amplitude with the shear rate calculated at the static mean radius of curvature. It was found that the wall shear rate varied as much as 94 percent of the mean wall shear rate at the mid wall of curvature for a mean curvature ratio of 0.08 and a 50 percent change in radius of curvature. The effects of 5 Hz deformation were not well predicted by a quasi-static approach. The maximum values of the normalized inner wall shear rate amplitude were found to scale well with a dimensionless parameter equivalent to the product of the mean curvature ratio (delta), normalized change in radius of curvature (epsilon), and a Womersley parameter (alpha). This parameter was less successful at predicting the amplitudes elsewhere in the tube, thus additional studies are necessary. The mean wall shear rate was well predicted with a static geometry. These results indicate that dynamic curvature plays an important role in determining the inner wall shear rates in coronary arteries that are subjected to deformation levels of epsilon delta alpha > 0.05. The effects were not always predictable with a quasi-static approach. These results provide guidelines for constructing more realistic models of coronary artery flow for atherogenesis research.
The localized nature of atherosclerosis has led to extensive study of blood flow patterns and their possible involvement in atherogenesis. Vessel geometry has always been considered a primary factor in determining blood flow patterns. In the coronary arteries, the geometry varies dynamically due to myocardial contraction. The effects of physiologic axial (Moore et al., 1994) and lateral (Delfino et al., 1994) vessel movement on coronary blood flow patterns have been shown to be important in producing oscillations in wall shear stress. Those studies were limited to straight vessels that translated in one direction only. Previous studies of flow in curved tubes with time-varying curvature showed that large quasi-static wall shear rate amplitudes relative to the static case when the curvature change was 50% of the mean curvature (Santamarina et al., 1997). The largest variation in wall shear rate from the minimum curvature to the maximum curvature was 52%, and was found at the mid-wall location (halfway between the inner and outer wall of curvature along the circumference) for the highest mean curvature ratio studied (δ = 0.12). A comparison of the shear rate amplitudes found in a tube whose radius of curvature varies dynamically at 1 Hz to the variations noted in the quasi-static analysis revealed differences of less than 1%. Changes in the mean wall shear rate predicted with the dynamic analysis were less than 7%, relative to the wall shear rate at the static mean radius of curvature. It was concluded that, although the change in curvature creates a relatively large shear rate amplitude, the fact that the curvature varies dynamically at 1 Hz is not important in predicting wall shear rates.
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