Mathematical modelling of blood flow through an overlapping arterial stenosis. (English) Zbl 0791.92009
Summary: Of concern in the paper is a theoretical study of blood flow in an arterial segment in the presence of a time-dependent overlapping stenosis using an appropriate mathematical model. A remarkably new shape of the stenosis in the realm of the formation of the arterial narrowing caused by atheroma is constructed mathematically. The artery is simulated as an elastic (moving wall) cylindrical tube containing a viscoelastic fluid representing blood. The unsteady flow mechanism of the present investigation is subjected to a pulsatile pressure gradient arising from the normal functioning of the heart.
The equations governing the motion of the system are sought in the Laplace transform space and their relevant solutions, supplemented by the suitable boundary conditions, are obtained numerically in the transformed domain through the use of an appropriate finite difference technique. Laplace inversion is also carried out by employing numerical techniques. A thorough quantitative analysis is performed at the end of the paper for the flow velocity, the flux, the resistive impedances, and the wall shear stresses together with their variations with the time, the pressure gradient, and the severity of the stenosis in order to illustrate the applicability of the present mathematical model under consideration.
The equations governing the motion of the system are sought in the Laplace transform space and their relevant solutions, supplemented by the suitable boundary conditions, are obtained numerically in the transformed domain through the use of an appropriate finite difference technique. Laplace inversion is also carried out by employing numerical techniques. A thorough quantitative analysis is performed at the end of the paper for the flow velocity, the flux, the resistive impedances, and the wall shear stresses together with their variations with the time, the pressure gradient, and the severity of the stenosis in order to illustrate the applicability of the present mathematical model under consideration.
MSC:
| 92C35 | Physiological flow |
| 76Z05 | Physiological flows |
| 92C30 | Physiology (general) |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
Keywords:
moving wall; blood flow; arterial segment; time-dependent overlapping stenosis; atheroma; viscoelastic fluid; unsteady flow; pulsatile pressure gradient; Laplace transform space; Laplace inversion; flow velocity; resistive impedances; wall shear stressesReferences:
| [1] | Young, D. F., Effect of a time-dependent stenosis on flow through a tube, J. Engng. Ind., Trans. ASME, 90, 248-254 (1968) |
| [2] | Forrester, J. H.; Young, D. F., Flow through a converging diverging tube and its implications in occlusive vascular disease, J. Biomech., 3, 297-316 (1970) |
| [3] | Lee, J. S.; Fung, Y. C., Flow in locally constricted tubes at low Reynolds numbers, J. Appl. Mech., 37, 9-16 (1970) · Zbl 0191.56105 |
| [4] | Young, D. F.; Tsai, F. Y., Flow characteristics in models of arterial stenosis—I. Steady flow, J. Biomech, 6, 395-411 (1973) |
| [5] | Misra, J. C.; Chakravarty, S., Flow in arteries in the presence of stenosis, J. Biomech., 19, 907-918 (1986) |
| [6] | Hershey, D.; Byrnes, R. E.; Deddens, R. L.; Rao, A. M., Blood rheology: Temperature dependence of the Power law model, A.I.Ch.E. Meeting (1964), Boston, MA |
| [7] | Huckaba, C. E.; Hahn, A. N., A generalised approach to the modelling of arterial blood flow, Bull. Math. Biophys., 30, 645-662 (1968) |
| [8] | Charm, S. E.; Kurland, G., Viscometry of human blood for shear rates \(0-100,000 sec^{−1}\), Nature, 206, 617-618 (1965) |
| [9] | Whitemore, R. L., Rheology of the Circulation (1968), Pergamon Press: Pergamon Press New York, NY |
| [10] | Han, C. D.; Barnett, B., Measurement of the rheological properties of biological fluids, (Gabelnick, H. L.; Litt, M., Rheology of Biological Systems (1973), Charles C. Thomas: Charles C. Thomas Illinois), 195 |
| [11] | Shukla, J. B.; Parihar, R. S.; Rao, B. R.P., Effects of stenosis on non-Newtonian flow of the blood in an artery, Bull. Math. Biol., 42, 283-294 (1980) · Zbl 0428.92009 |
| [12] | Chakravarty, S., Effects of stenosis on the flow behaviour of blood in an artery, Int. J. Engng. Sci., 25, 1003-1018 (1987) · Zbl 0618.76139 |
| [13] | Chakravarty, S.; Datta, A., Effects of stenosis on arterial rheology through a mathematical model, Math. Comp. Modelling, 12, 1601-1612 (1989) · Zbl 0703.76109 |
| [14] | Chakravarty, S.; Datta, A., Dynamic response of stenotic blood flow in vivo, Math. Comp. Modelling, 16, 3-20 (1992) · Zbl 0756.92012 |
| [15] | Lessner, A.; Zahavi, J.; Silberberg, A.; Frei, E. H.; Dreyfus, F., The viscoelastic properties of whole blood, (Hartert, H.; Copley, A. L., Theoretical and Clinical Hemorheology (1971), Springer-Verlag: Springer-Verlag New York, NY), 194-205 |
| [16] | Thurston, G. B., Frequency and shear rate dependence of viscoelasticity of human blood, Biorheology, 10, 375-381 (1973) |
| [17] | Burton, A. C., Introductory text, (Physiology and Biophysics of the Circulation (1966), Year Book Medical Publisher: Year Book Medical Publisher Chicago, IL) |
| [18] | von Rosenberg, D. U., Methods for the Numerical Solution of Partial Differential Equations (1969), Elsevier: Elsevier New York, NY · Zbl 0194.18904 |
| [19] | Krylov, V. I.; Skoblya, N. S., A Handbook of Methods of Approximate Fourier Transformation and Inversion of Laplace Transformation (1977), Mir Publisher: Mir Publisher Moscow · Zbl 0389.65051 |
| [20] | McDonald, D. A., Blood Flow in Arteries (1974), Edward |
| [21] | Milnor, W. R., Hemodynamics (1982), Williams and Williams: Williams and Williams Baltimore, MD · Zbl 0354.76079 |
| [22] | Thurston, G. B., Effects of viscoelasticity of blood on wave propagation in the circulation, J. Biomech., 9, 13-20 (1976) |
| [23] | Srivastava, L. M., Flow of couple stress fluid through stenotic blood vessels, J. Biomech., 18, 479-485 (1985) |
| [24] | Fry, D. L., Acute vascular endothelial changes associated with increased blood velocity gradient, Circulation Res., 22, 165-197 (1968) |
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