Using Mathematics and Engineering to Study Blood Flow in Constricted Vessels
In case you need more reminders, research continues to explore the darker side of your favourite double-bacon fast-food combo. As we know, the typical fatty, high-cholesterol North American diet can send fatty substances in our blood stream and ultimately leave fatty streaks lining the insides of our artery walls.
In medical terms, these abnormal points of narrowing or constriction, often caused by plaque buildup, are called stenosis. The sad truth is these constrictions are trouble spots where blood flow can be impeded, blood pressure sent spiraling upwards – and ultimately, they can lead to life-threatening events like heart attack or stroke.
While medical researchers understand much about vessel stenosis, there are still gaps in knowledge. For example, at Ryerson, Salahaldeen Rabba, who recently earned his MSc in Applied Mathematics while doing the research described, has sought to bring a deeper understanding of blood flow properties through a constriction, with the belief that knowledge here could improve early diagnosis, prevention and treatment of vascular disease.
In terms of methods, this is where math and geometry come in. Rabba, under the supervision of Katrin Rohlf, a Ryerson math professor, devoted his Master's studies to developing and testing a compelling mathematical model that, as he explains, “Analytically studies the flow characteristics of blood through an artery in the presence of stenosis or constriction.”
Describing his model, Rabba says, “An appropriate shape of stenoses of arterial narrowing caused is constructed mathematically. The artery is simulated as a cylindrical tube having a narrowed portion forming stenosis. The viscosity of blood is taken to be constant. The flow mechanism in the stenosed artery is subjected to a pulse-pressure-gradient.” Rabba says other factors were also taken into account in his model – including compressibility, slip and Reynolds number (an important dimensionless quantity in fluid mechanics).
Asked where his research furthers current knowledge, Rabba says, “Unlike previous studies, the blood was analyzed as a weakly compressible fluid. Wall slip was also accounted for while looking at the different constriction geometries.” More specifically, “We also implemented a Gaussian geometry and compared the results to existing results in the literature. We extended the comparison against two other geometries, namely, piece-wise Cosinusoidal and polynomial geometries.”
Ultimately, Rabba says, extensive quantitative analysis like this is useful, helping in estimating the significant effects of the severity of the stenoses, effects on blood pressure and density. “Investigation of the flow in a stenosed (constricted) geometry is of interest because of its significance in biomechanics, especially in human vascular diseases.”
Rabba is grateful to Dr. Rohlf for her continuous support during the rigorous research process and looks forward to seeing his research published this year in the International Journal of Applied Nonlinear Science (IJANS) in an article called, “Pressure curves for compressible flows with slip through asymmetric local constrictions.”
Rabba’s research was partially funded by an NSERC grant.