Mathematical modeling of pulse wave propagation along human aorta
Abstract
Background: Physical characteristics of pulse waves, which are generated by the heart contractions and propagated along the arteries, are used in medicine for diagnostics of the blood circulation system and blood supply to the organs and tissues. At the sites with significant wave reflections the high local pressure oscillations appear that may lead to damage of the endothelium, development of atherosclerotic plaques and aortic aneurysm. Therefore, elaboration of a detailed biophysical model of the individual aorta based on tomography and determination of the dangerous sites with high wave reflections are important for medical diagnostics.
Objectives: The aim of the work is to study the regularities of the pulse wave propagation and reflection along the aorta and to propose new methods for early diagnosis of disorders in the blood circulation system.
Materials and methods: The measurement data on diameters and lengths of segments of aorta and its branches conducted on 5 corpses have been used. Calculations of the wave conduction and reflection coefficients are based on the linear theory of pulse waves developed by J. Lighthill.
Results: It is shown that from the biophysical point of view, the aorta is an optimal waveguide, which provides almost zero local reflections of the pulse waves. Most of the branches possess negative reflection, which accelerates the blood flow and decreases the load on the heart due to the suction effect. The calculated values of the branching coefficients and pulse waves speeds correspond to the data of the previous experimental measurements. It is shown that most of the branches have an optimal Murray coefficient close to one. It implies, aorta also provides the optimal volumetric blood flow over the period of cardiac contraction with minimal energy expenses.
Conclusions: Human aorta and its branches possess optimal biophysical properties, which ensure the blood flow with minimal energy consumption. Aorta as an optimal waveguide provides pulse wave propagation with almost without reflection. The proposed method of estimation of the biophysical properties of aorta as a waveguide can be useful for medical diagnostics, allowing early identification of the regions which are dangerous in terms of the progressive development of vascular pathologies in the individual geometry of the patient's vasculature.
Downloads
References
Nichols, W., & O’Rourke, M. (2005). McDonald’s blood flow in arteries. Theoretical, Experimental and Clinical Principles. Oxford: Hodder Arnold Oxford University Press.
Grotenhuis, H. B., Westenberg, J. M., Steendijk, P., et al. (2009). Validation and Reproducibility of Aortic Pulse Wave Velocity as Assessed with Velocity-Encoded MRI. J. Magn. Res. Imag., 30(3), 521–526.
Latham, R. D., Westerhof, N., Sipkema, P., et al. (1985). Regional Wave Travel and Reflections Along the Human Aorta: A Study with Six Simultaneous Micromanometric Pressures. Circul., 72(6), 1257-1269.
Rogers, W. J., Hu, Y.-L., Coast, D., et al. (2001). Age-Associated Changes in Regional Aortic Pulse Wave Velocity. J. Amer. College Cardiol., 38(4), 1123–1129.
O'Rourke, M. F., Blazek, J. V., Morreels, Ch. L., & Krovetz, L. J. (1968). Pressure Wave Transmission along the Human Aorta. Circul. Res., 23(4), 567-579.
Cruickshank, K., Riste, L., Anderson, S. G., et al. (2002). Aortic Pulse-Wave Velocity and its Relationship to Mortality in Diabetes and Glucose Intolerance. Circul., 106(16), 2085-2090.
Kizilova, N. N. (2006). Novye Napravleniia i Perspektivy Teorii Pulsovykh Voln v Arteriiakh. In: Sovremennye Problemy Biomekhaniki. Moscow: Moskow University Press. 11, 44-63. (in Russian)
Caro, C. G., Fitz-Gerald, J. M., & Schroter, R. C. (1971). Atheroma and Arterial Wall Shear: Observations, Correlation and Proposal of a Shear Dependent Mass Transfer Mechanism for Atherogenesis. Proc. Royal Soc.
London, Ser.B., 177(1046), 109-159.
Khir, A. W., O’Brien, A., Gibbs, J. S. R., & Parker, K. H. (2001). Determination of Wave Speed and Wave Separation in the Arteries. J. Biomech., 34(9), 1145–1155
Savitzky, A, & Golem, M. (1964). Smoothing and Differentiation of Data by Simplified Least Squares Procedures. Analyt. Chemistry, 36(8), 1627–1639.
Michoux, N., Joannides, R., Gouesbet, G., et al. (1999). Physical Determinism in Human Arterial Dynamics. Eur. Phys. J., 8(3), 265-268.
Sugawara, M., Uchida, K., Kondoh, Y., et al. (1997). Aortic Blood Momentum - the More the Better for the Ejecting Heart in vivo? Cardiovasc. Res., 33(2), 433-446.
Ovadia-Blechman, Z., Einav, S., Zaretsky, U. et al. (2002). The Area of the Pressure – Flow Loop for Assessment of Arterial Stenosis: A New Index. Technol. Health Care, 10(1), 39–56.
Quick, Ch. M., Mohiuddin, M. W., Laine, G. A., & Noordergraaf, A. (2005). The Arterial System Pressure–Volume Loop. Physiol. Meas., 26(1), 29–35.
Milnor, W. R. (1989). Hemodynamics. Baltimore: Williams &Wilkins.
Westerhof, N., Sipkema, P., Bos, C. G. V., & Elzinga, G. (1972). Forward and Backward Waves in the Arterial System. Cardiovasc. Res., 6(6), 648-656.
Parker, K. H., & Jones, J. H. (1990). Forward and Backward Running Waves in Arteries: Analysis Using the Method of Characteristics. ASME J. Mech. Eng., 112(3), 322-326.
Sun, Y.- H., Anderson, T. J., Parker, K. H., & Tyberg, J. V. (2000). Wave-Intensity Analysis: a New Approach to Coronary Hemodynamics. J. Appl. Physiol., 89(4), 1636–1644.
Li, Y., Parker, K. H., & Khir, A. W. (2016). Using Wave Intensity Analysis to Determine Local Reflection Coefficient in Flexible Tubes. J. Biomech., 49(13), 2709–2717.
Flaws, B. (1995). The Secret of Chinese Pulse Diagnosis, 2nd Ed. Boulder: Blue Poppy Press.
Flaws, B., & Sionneau, Ph. (2005). The Treatment of Modern Western Medical Diseases with Chinese Medicine. Boulder: Blue Poppy Press.
Kizilova, N. (2003). Рulse Wave Reflections in Branching Arterial Networks and Pulse Diagnosis Methods. J. Chinese Inst. Engin., 26(6), 869–880.
Kizilova, N. (2013). Blood Flow in Arteries: Regular and Chaotic Dynamics. In: Dynamical Systems. Applications / ed. Awrejcewicz, J., Kazmierczak, M., Olejnik, P., & Mrozowski, K. Lodz: Politechnical University Press.
Lighthill, M. J. (1978). Waves in Fluids. Cambridge: Cambridge University Press.
Westerhof, N., Bosman, F., de Vries, C. J., & Noordegraaf, A. (1969). Analog Studies of the Human Systemic Arterial Tree. J. Biomech., 2(1), 121–143.
Zujkov, A., & Volgina, L. (2014). Hydraulics. Vol. 2. Pressurized and Open Flows. Hudraulics of Constructions. Моscow: МGSU. (in Russian)
Hollander, E. H., Wang, J. J., Dobson, G. M., Parker, K. H., & Tyberg, J. V. (2001). Negative Wave Reflections in Pulmonary Arteries. Am. J. Physiol., 281(2), 895-902.
Kizilova, N. N. (2007). Моdeling of interorgan arteriel beds. II. Propagation of Pressure Waves. Biophysics, 52(1), 131-136. (in Russian)
Zenin, О. К., Кizilova, N. N., & Philippova, E. N. (2007). Investigation of Regularities of Construction of Human Coronary Vasculatures. Biophysics, 52(5), 924-930.
Zenin, O. K., Gusak, V. K., Kiriakulov, G. S. et al. (2006). Arterialnaia Sistema Cheloveka v Cyfrah i Formulah. Donetsk: «Donbass». (in Russian)
Alastruey, J., Khir, A. W., Matthys, K. S., et al. (2011). Pulse Wave Propagation in a Model Human Arterial Network: Assessment of 1-D Visco-Elastic Simulations Against in vitro Measurements. J. Biomech., 44(12), 2250–2258.
Folkov, B., & Nil, E. (1976). Circulation. Мoscow: Mir.
Olufsen, M. S. (1999). Structured Tree Outflow Condition for Blood Flow in Larger Systemic Arteries. Am. J. Physiol., 276(1), 257-268.
Levtov, V. A., Regirer, S. A., & Shadrina, N. Kh. (1982). Blood Rheology. Moscow: Medicine. (in Russian)
Karamanoglou, M., O'Rourke, M. F., Avolio, A. P., et al. (1993). An Analysis of the Relationship Between Central Aortic and Peripheral Upper Limb Pressure Waves in Man. Europ. Heart J., 14(2), 160–167.
Gao, M., Rose, W. C., Fetics, B., et al. (2016). A Simple Adaptive Transfer Function for Deriving the Central Blood Pressure Waveform from a Radial Blood Pressure Waveform. Sci. Rep., 14(6), 33230.
Mitchell, G.F., Hwang, Sh.-J., Larson, M.G., et al. (2016). Transfer Function-Derived Central Pressure and Cardiovascular Disease Events: the Framingham Heart Study. J. Hypert., 34(8), 1528–1534.
Sharman, J.E., Lim, R., Qasem, A.M., et al. (2006). Validation of a Generalized Transfer Function to Noninvasively Derive Central Blood Pressure During Exercise. Hypertension, 47(6), 1203-1208.
Shoshenko, K. A. (1982). Architectonics of Blood Vasculature. Novosibirsk: Nauka. (in Russian)
Zamir, M., & Bigelow, D.C. (1984). Cost of Depature from Optimality in Arterial Branching. J. Theor. Biol., 109(3), 401-409.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
