Mathematical Models of Real Geometrical Factors in Restricted Blood Vessels for the Analysis of CAD (Coronary Artery Diseases) Using Legendre, Boubaker and Bessel Polynomials
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Most cardiovascular emergencies are directly caused by coronary artery disease. Coronary arteries can become clogged or occluded, leading to damage to the heart muscle supplied by the artery. Modem cardiovascular medicine can certainly be improved by meticulous analysis of geometrical factors closely associated with the degenerative disease that results in narrowing of the coronary arteries. There are, however, inherent difficulties in developing this type of mathematical models to completely describe the real or ideal geometries that are very critical in plaque formation and thickening of the vessel wall. Neither the mathematical models of the blood vessels with arthrosclerosis generated by the heart and blood flow or the NMR/MRI data to construct them are available. In this study, a mathematical formulation for the geometrical factors that are very critical for the understanding of coronary artery disease is presented. Based on the Bloch NMR flow equations, we derive analytical expressions to describe in detail the NMR transverse magnetizations and signals as a function of some NMR flow and geometrical parameters which are invaluable for the analysis of blood flow in restricted blood vessels. The procedure would apply to the situations in which the geometry of the fatty deposits, (plague) on the interior walls of the coronary arteries is spherical. The boundary conditions are introduced based on Bessel, Boubaker and Legendre polynomials.
KeywordsBloch NMR flow equations Atherosclerosis Coronary artery disease Bessel polynomials Boubaker polynomials Legendre polynomials
The authors appreciate encouragement provided by the STEP B research programme through NUC, Nigeria and collaboration with Professor Sreenivasan, ICTP, Trieste, Italy and Professor P. Fantazzini, University of Bologna, Italy through the ICTP STEP programme.
- 1.U.S. Department of Health and Human services, Nat. Inst. of Health, Diseases and Conditions Index, Nov. 2007.Google Scholar
- 2.Dada, M., Awojoyogbe, O. B., Moses, O. F., Ojambati, O. S., De, D.K. and Boubaker, K., A mathematical analysis of Stenosis Geometry, NMR magnetizations and signals based on the Bloch NMR flow equations, Bessel and Boubaker polynomial expansions. J. Biol. Phys. Chem. 9, 24DA09A, 2009. (in press).Google Scholar
- 4.Tuzcu, E. M., Kapadia, S. R., Tutar, E., et al., High prevalence of coronary atherosclerosis in asymptomatic teenagers and young adults: Evidence from intravascular ultrasound. Circulation 103:2705–2710, 2001.Google Scholar
- 5.Topol, E. J., and Nissen, S. E., Our preoccupation with coronary luminology.The dissociation between clinical and angiographic findings inischemic heart disease. Circulation 92:2333–2242, 1995.Google Scholar
- 7.Falk, E., Shah, P. K., and Fuster, V., Coronary plaque disruption. Circulation 92:657–671, 1995.Google Scholar
- 10.Losordo, D. W., Rosenfield, K., and Kaufman, J., Focal compensatory enlargement of human arteries in response to progressive atherosclerosis: In vivo docum. using intravascular ultrasound. Circulation 89:2570–2577, 1994.Google Scholar
- 16.Schoenhagen, P., Ziada, K., Kapadia, S. R., et al., Extent and direction of arterial remodeling in stable versus unstable coronary syndromes: An intravascular ultrasound study. Circulation 101:598–603, 2000.Google Scholar
- 17.Burke, A. P., Kolodgie, F. D., Farb, A., et al., Healed plaque ruptures and sudden coronary death: Evidence that subclinical rupture has a role in plaque progression. Circulation 103:934–940, 2001.Google Scholar
- 23.Awojoyogbe, O. B., and Boubarker, K., A solution to Bloch NMR flow equations for the analysis of hemodynamic functions of blood flow system using m-Boubaker polynomials, Curr. Appl. Phys, (2008), doi: 10.1016/j.cap.2008.01.019.
- 25.Awojoyogbe, O.B., Faromika, O.P., Moses, O. F., Dada, M., Fuwape, I. A., Boubaker, K., Mathematical model of the Bloch NMR flow equations for the fnalysis of fluid flow in restricted geometries using Boubaker expansion scheme. Applied Physics Ms. Ref. No.: CAP-D-09-00222 June 2009 (Available Online from www.sciencedirect.com).
- 26.Awojoyogbe, O. B., Dada, M., Faromika, O. P., Moses, O. F., and Fuwape, I. A., Polynomial solutions of Bloch NMR flow equations for classical and quantum mechanical analysis of fluid flow in porous media. Open Magn. Reson. J. 2:46–56, 2009.Google Scholar