Journal of Medical Systems

, Volume 35, Issue 6, pp 1513–1520 | Cite as

Mathematical Models of Real Geometrical Factors in Restricted Blood Vessels for the Analysis of CAD (Coronary Artery Diseases) Using Legendre, Boubaker and Bessel Polynomials

  • O. B. Awojoyogbe
  • O. P. Faromika
  • M. Dada
  • Karem Boubaker
  • O. S. Ojambati
Original Paper


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.


Bloch 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. 1.
    U.S. Department of Health and Human services, Nat. Inst. of Health, Diseases and Conditions Index, Nov. 2007.Google Scholar
  2. 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
  3. 3.
    Strong, J. P., Malcom, G. T., McMahan, C. A., et al., Prevalence and extent of atherosclerosis in adolescents and young adults: Implications for prevention from pathobiological determinants of atherosclerosis in youth study. JAMA 281:727–735, 1999.CrossRefGoogle Scholar
  4. 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. 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
  6. 6.
    Little, W. C., Constantinescu, M., and Applegate, R. J., Can coronary angiography predict the site of a subsequent infarction in patients with mild-to-moderate coronary artery disease? Circulation 78:1157–1166, 1988.CrossRefGoogle Scholar
  7. 7.
    Falk, E., Shah, P. K., and Fuster, V., Coronary plaque disruption. Circulation 92:657–671, 1995.Google Scholar
  8. 8.
    Burke, A. P., Farb, A., Malcom, G. T., et al., Coronary risk factors and plaque morphology in men with coronary disease who died suddenly. N Engl J Med 336:1276–1282, 1997.CrossRefGoogle Scholar
  9. 9.
    Glagov, S., Weisenberg, E., Zarins, C., et al., Compensatory enlargement of human atherosclerotic coronary arteries. N Engl J Med 316:1371–1375, 1987.CrossRefGoogle Scholar
  10. 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
  11. 11.
    Mintz, G. S., Painter, J. A., Pichard, A. D., et al., Atherosclerosis in angiographically “normal” coronary artery reference segments: An intravascular ultrasound study with clinical correlations. J Am Coll Cardiol 25:1479–1485, 1995.CrossRefGoogle Scholar
  12. 12.
    Schoenhagen, P., Ziada, K. M., Vince, D. G., Nissen, S. E., and Tuzcu, E. M., Arterial remodeling and coronary artery disease. The concept of “dilated” versus “obstructive” coronary atherosclerosis. J. Am. Coll. Cardiol. 38:297–306, 2001.CrossRefGoogle Scholar
  13. 13.
    Smits, P. C., Pasterkamp, G., de Jaegere, P. P., de Feyter, P. J., and Borst, C., Angioscopic complex lesions are predominantly compensatory enlarged: An angioscopy and intracoronary ultrasound study. Cardiovasc. Res. 41:458–464, 1999.CrossRefGoogle Scholar
  14. 14.
    Pasterkamp, G., Schoneveld, A. H., van der Wal, A. C., et al., Relation of arterial geometry to luminal narrowing and histologic markers for plaque vulnerability: The remodeling paradox. J Am Coll Cardiol 32:655–662, 1998.CrossRefGoogle Scholar
  15. 15.
    Nakamura, M., Nishikawa, H., Mukai, S., et al., Impact of coronary artery remodeling on clinical presentation of coronary artery disease: An intravascular ultrasound study. J Am Coll Cardiol 37:63–69, 2001.CrossRefGoogle Scholar
  16. 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. 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
  18. 18.
    Kim, W. Y., Danias, P. G., Stuber, M., et al., Coronary magnetic resonance angiography for the detection of coronary stenoses. N Engl J Med 345:1863–1869, 2001.CrossRefGoogle Scholar
  19. 19.
    Buonocore, M. H., RF Pulse design using the inverse scattering transform. Magn. Reson. Med. 29:470–477, 1993.CrossRefGoogle Scholar
  20. 20.
    Awojoyogbe, O. B., A mathematical model of Bloch NMR equations for quantitative analysis of blood flow in blood vessels of changing cross-section II. Physica A 323:534–550, 2003.CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Awojoyogbe, O. B., Analytical solution of the time dependent Bloch NMR equations: A translational mechanical approach. Physica A 339:437–460, 2004.CrossRefGoogle Scholar
  22. 22.
    Awojoyogbe, O. B., A mathematical model of Bloch NMR equations for quantitative analysis of blood flow in blood vessels of changing cross-section I. Physica A 303(1–2):163–175, 2002.CrossRefMATHMathSciNetGoogle Scholar
  23. 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.
  24. 24.
    Awojoyogbe, O. B., A quantum mechanical model of the Bloch NMR flow equations for electron dynamics in fluids at the molecular level. Phys. Scr. 75:788–794, 2007.CrossRefMATHGoogle Scholar
  25. 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
  26. 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
  27. 27.
    Dada, M., Awojoyogbe, O. B., Rozibaeva, N., and Mahmoud, K. B. B., Establishment of a Chebyshev-dependent inhomogeneous second order differential equation for the applied physics-related Boubaker-Turki polynomials. Appl. Appl. Math 3(2):329–336, 2008.MATHMathSciNetGoogle Scholar
  28. 28.
    Fridjine, S., and Amlouk, M., A new parameter: An Abacus for optimizing PV-T hybrid solar device functional materials using the Boubaker polynomials expansion scheme. Mod. Phys. Lett. B 23:2179, 2009.CrossRefMATHGoogle Scholar
  29. 29.
    Tabatabaei, S., Zhao, T., Awojoyogbe, O., and Moses, F., Cut-off cooling velocity profiling inside a keyhole model using the Boubaker polynomials expansion scheme. Heat Mass Transf. 45:1247, 2009.CrossRefGoogle Scholar
  30. 30.
    Belhadj, A., Onyango, O., and Rozibaeva, N., Boubaker polynomials expansion scheme-related heat transfer investigation inside keyhole model. J. Thermophys. Heat Transf. 23:639, 2009.CrossRefGoogle Scholar
  31. 31.
    Zhao, T. G., Wang, Y. X., and Ben Mahmoud, K. B., Int. J. Math. Comp. 1:13, 2008.MathSciNetGoogle Scholar
  32. 32.
    Ghrib, T., Boubaker, K., and Bouhafs, M., Mod. Phys. Lett. B 22:2907, 2008.CrossRefGoogle Scholar
  33. 33.
    Guezmir, N., Ben Nasrallah, T., Boubaker, K., Amlouk, M., and Belgacem, S., J. Alloys and Comp. 481:543, 2009.CrossRefGoogle Scholar
  34. 34.
    Oyodum, O. D., Awojoyogbe, O. B., Dada, M., and Magnuson, J., Europ. Phys. J.–App. Phys. 46(2):21201, 2009.CrossRefGoogle Scholar
  35. 35.
    Belhadj, A., Bessrour, J., Bouhafs, M., and Barrallier, L., Experimental and theoretical cooling velocity profile inside laser welded metals using keyhole approximation and Boubaker polynomials expansion. J. Therm. Anal. Calorim 97(3):911–915, 2009.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • O. B. Awojoyogbe
    • 1
  • O. P. Faromika
    • 3
  • M. Dada
    • 1
  • Karem Boubaker
    • 2
  • O. S. Ojambati
    • 1
  1. 1.Department of PhysicsFederal University of TechnologyMinnaNigeria
  2. 2.Dép. de Physique et ChimieÉcole Supérieure des Science et Techniques de TunisTunisTunisia
  3. 3.Department of PhysicsFederal University of TechnologyAkureNigeria

Personalised recommendations