A short review of some reference solutions for the magnetohydrodynamic flow of blood is proposed in this paper. We present in details the solutions of Hartmann (1937), of Vardanyan (1973) and of Sud et al . (1974). In each case, a comparison is provided with the corresponding solution for the flow without any external magnetic field, namely Poiseuille (plane or cylindrical) and Womersley. We also present a synopsis of some other solutions for people who would like to go further in this topic. The interest in MHD flow of blood may be motivated by many reasons, such as Magnetic Resonance Imaging (MRI), Pulse Wave Velocity measurement, magnetic drug targeting, tissue engineering, mechanotransduction studies, and blood pulse energy harvesting… These fundamental solutions should also be used as particular limiting cases to validate any proposed more elaborated solutions or to validate computer codes.
KeywordsMagnetohydrodynamic Flow of BloodMagnetic Field in Biomedical ApplicationsHartmannWomersley
Nijm, G., Swiryn, S., Larson, A. and Sahakian, A. (2008) Extraction of the Magnetohydrodynamic Blood Flow Potential from the Surface Electrocardiogram in Magnetic Resonance Imaging. Medical & Biological Engineering & Computing, 46, 729-733. https://doi.org/10.1007/s11517-008-0307-1
Kyriakou, A., Neufeld, E., Szczerba, D., Kainz, W., Luechinger, R., Kozerke, S., Mc Gregor, R. and Kuster N. (2012) Patient-Specific Simulations and Measurements of the Magneto-Hemodynamic Effect in Human Primary Vessels. Physiological Measurements, 33, 117-130. https://doi.org/10.1088/0967-3334/33/2/117
Krug, K., Rose, G., Clifford, G. and Oster J. (2013) ECG-Based Gating in Ultra High Field Cardiovascular Magnetic Resonance Using an Independent Component Analysis Approach. Journal of Cardiovascular Magnetic Resonance, 15, 104-116. https://doi.org/10.1186/1532-429X-15-104
Drochon, A. (2016) Sinuso?dal Flow of Blood in a Cylindrical Deformable Vessel Exposed to an External Magnetic Field. European Physical Journal—Applied Physics, 73, Article ID: 31101. https://doi.org/10.1051/epjap/2016150530
Laffon, E., Marthan, R., Montaudon, M., Latrabe, V., Laurent, F. and Ducassou, D. (2005) Feasibility of Aortic Pulse Pressure and Pressure Wave Velocity MRI Measurement in Young Adults. Journal of Magnetic Resonance Imaging, 21, 53-58. https://doi.org/10.1002/jmri.20227
Ibrahim, E.S., Johnson, K., Miller, A., Shaffer, J. and White, R. (2010) Measuring Aortic Pulse Wave Velocity using High-Field Cardiovascular Magnetic Resonance: Comparison of Techniques. Journal of Cardiovascular Magnetic Resonance, 12, 26-39. https://doi.org/10.1186/1532-429X-12-26
Nacev, A., Beni, C., Bruno, O. and Shapiro, B. (2010) Magnetic Nanoparticle Transport within Flowing Blood and into Surrounding Tissue. Nanomedicine, 5, 1459-1466. https://doi.org/10.2217/nnm.10.104
Ciofani, G., Riggio, C., Raffa, V., Menciassi, A. and Cuschieri, A. (2009) A Bi-Modal Approach against Cancer: Magnetic Alginate Nanoparticles for Combined Chemotherapy and Hyperthermia. Medical Hypotheses, 73, 80-82. https://doi.org/10.1016/j.mehy.2009.01.031
Morega, A., Dobre, A. and Morega, M. (2011) Magnetic Field-Flow Interactions in Drug Delivery through an Arterial System. Revue Roumaine des Sciences Techiques—Serie électrotechnique et énergétique, 56, 199-208.
Riegler, J., Lau, K., Garcia-Prieto, A., Price, A., Richards, T., Pankhurst, Q. and Lythgoe, M. (2011) Magnetic Cell Delivery for Peripheral Arterial Disease: A Theoretical Framework. Medical Physics, 38, 3932-3943. https://doi.org/10.1118/1.3593363
Sensenig, R., Sapir, Y., MacDonald, C., Cohen, S. and Polyak, B. (2012) Magnetic Nanoparticle-Based Approaches to Locally Target Therapy and Enhance Tissue Regeneration in Vivo. Nanomedecine, 7, 1425-1442. https://doi.org/10.2217/nnm.12.109
Perea, H., Aigner, J., Hopfner, U. and Wintermantel, E. (2006) Direct Magnetic Tubular Cell Seeding: A Novel Approach for Vascular Tissue Engineering. Cells Tissues Organs, 183, 156-165. https://doi.org/10.1159/000095989
Perea, H., Aigner, J., Heverhagen, J.T., Hopfner, U. and Wintermantel, E. (2007) Vascular Tissue Engineering with Magnetic Nanoparticles: Seeing Deeper. Journal of Tissue Engineering and Regenerative Medicine, 1, 318-321. https://doi.org/10.1002/term.32
Xu, H., Othman, S. and Magin, R. (2008) Monitoring Tissue Engineering Using Magnetic Resonance Imaging. Journal of Biosciences and Bioengineering, 106, 515-527. https://doi.org/10.1263/jbb.106.515
Poirier-Quinot, M., Frasca, G., Wilhelm, C., Luciani, N., Ginefri, J.C., Darrasse, L., Letourneur, D., Le Visage, C. and Gazeau, F. (2010) High Resolution 1.5 Tesla Magnetic Resonance Imaging for Tissue-Engineered Constructs: A Noninvasive Tool to Assess Three-Dimensional Scaffold Architecture and Cell Seeding. Tissue Engineering Part C Methods, 16, 185-200. https://doi.org/10.1089/ten.tec.2009.0015
Castro, E. and Mano, J. (2013) Magnetic Force-Based Tissue Engineering and Regenerative Medicine. Journal of Biomedical Nanotechnology, 9, 1129-1136. https://doi.org/10.1166/jbn.2013.1635
Pfenniger, A., Wickramarathna, L., Vogel, R. and Koch, V. (2013) Design and Realization of an Energy Harvester Using Pulsatile Arterial Pressure. Medical Engineering & Physics, 35, 1256-1265. https://doi.org/10.1016/j.medengphy.2013.01.001
Gregory, T.S., Cheng, R., Tang, G., Mao, L. and Tse, Z. (2016) The Magnetohydrodynamic Effect and Its Associated Material Designs for Biomedical Applications: A State-of-the-Art Review. Advanced Functional Materials, 26, 3942-3952. https://doi.org/10.1002/adfm.201504198
Hartman, J. (1937) Hg-Dynamics I: Theory of the Laminar Flow of an Electrically Conductive Liquid in a Homogeneous Magnetic Field. K?benhavn, Levin & Munksgaard, Ejnar Munksgaard, 15, 1-28.
Hartman, J. and Lazarus, F. (1937) Hg-Dynamics II: Experimental Investigations on the Flow of Mercury in a Homogeneous Magnetic Field. K?benhavn, Levin & Munksgaard, Ejnar Munksgaard, 15, 1-45.
Vardanyan, V. (1973) Effect of a Magnetic Field on Blood Flow. Biofizika, 18, 491-496.
Sud, V.K., Suri, P.K. and Mishra, R.K. (1974) Effect of Magnetic Field on Oscillating Blood Flow in Arteries. Studia Biophysica, 46, 163-172.
Womersley, J.R. (1955) Method for the Calculation of Velocity, Rate of Flow, and Viscous Drag in Arteries When the Pressure Gradient Is Known. The Journal of Physiology, 127, 553-563. https://doi.org/10.1113/jphysiol.1955.sp005276
Abi-Abdallah, D., Drochon, A., Robin, V. and Fokapu, O. (2009) Effects of Static Magnetic Field Exposure on Blood Flow. European Physical Journal—Applied Physics, 45, 11301-11317. https://doi.org/10.1051/epjap:2008193
Drochon, A., Robin, V., Fokapu, O. and Abi-Abdallah-Rodriguez, D. (2016) Stationary Flow of Blood in a Rigid Vessel in the Presence of an External Magnetic Field: Considerations about the Forces and Wall Shear Stresses. Applied Mathematics, 7, 130-136. https://doi.org/10.4236/am.2016.72012
Fung, Y.C. (1997) Biomechanics—Circulation. Springer-Verlag, New York.
Mc Donald, D.A. (1955) The Relation of Pulsatile Pressure to Flow in Arteries. The Journal of Physiology, 127, 533-552. https://doi.org/10.1113/jphysiol.1955.sp005275
Gold, R. (1962) Magnetohydrodynamic Pipe Flow. Journal of Fluid Mechanics, 13, 505-512. https://doi.org/10.1017/S0022112062000889
Abi-Abdallah, D., Drochon, A., Robin, V. and Fokapu, O. (2009) Pulsed Magnetohydrodynamic Blood Flow in a Rigid Vessel under Physiological Pressure Gradient. Computer Methods in Biomechanics and Biomedical Engineering, 12, 445-458. https://doi.org/10.1080/10255840802687384
Kinouchi, Y., Yamaguchi, H. and Tenforde, T.S. (1996) Theoretical Analysis of Magnetic Field Interactions with Aortic Blood Flow. Bioelectromagnetics, 17, 21-32. https://doi.org/10.1002/(SICI)1521-186X(1996)17:1 3.0.CO;2-8