Skip to main content Accessibility help

Mathematical Model of Blood Flow in an Anatomically Detailed Arterial Network of the Arm

  • Sansuke M. Watanabe (a1) (a2) (a3), Pablo J. Blanco (a2) (a3) and Raúl A. Feijóo (a2) (a3)


A distributed-parameter (one-dimensional) anatomically detailed model for the arterial network of the arm is developed in order to carry out hemodynamics simulations. This work focuses on the specific aspects related to the model set-up. In this regard, stringent anatomical and physiological considerations have been pursued in order to construct the arterial topology and to provide a systematic estimation of the involved parameters. The model comprises 108 arterial segments, with 64 main arteries and 44 perforator arteries, with lumen radii ranging from 0.24 cm – axillary artery- to 0.018 cm – perforator arteries. The modeling of blood flow in deformable vessels is governed by a well-known set of hyperbolic partial differential equations that accounts for mass and momentum conservation and a constitutive equation for the arterial wall. The variational formulation used to solve the problem and the related numerical approach are described. The model rendered consistent pressure and flow rate outputs when compared with patient records already published in the literature. In addition, an application to dimensionally-heterogeneous modeling is presented in which the developed arterial network is employed as an underlying model for a three-dimensional geometry of a branching point to be embedded in order to perform local analyses.



Hide All
[1] Alastruey, J., Khir, A.W., Matthys, K.S., Segers, P., Sherwin, S.J., Verdonck, P.R., Parker, K.H. and Peiró, J., Pulse wave propagation in a model human arterial network: Assessment of 1-D visco-elastic simulations against in vitro measurements. J. Biomech. 44 (2011) 22508.
[2] Alastruey, J., Parker, K.H., Peiró, J., Byrd, S.M. and Sherwin, S.J., Modelling the circle of Willis to assess the effects of anatomical variations and occlusions on cerebral flows. J. Biomech. 40 (2007) 1794805.
[3] Amundsen, B.H., Wisloff, U., Helgerud, J., Hoff, J. and Slordahl, S.A., Ultrasound recorded axillary artery blood flow during elbow-flexion exercise. Medicine and science in sports and exercise 34 (2002) 128893.
[4] Avolio, A.P., Multi-branched model of the human arterial system. Medical Biolog. Engrg. Comp. 18 (1980) 70918.
[5] Bilge, O., Pinar, Y., Ozer, M.A. and Gövsa, F., A morphometric study on the superficial palmar arch of the hand. Surg. Radiol. Anat.: SRA 28 (2006) 34350.
[6] P.J. Blanco and R.A. Feijóo, The role of the variational formulation in the dimensionally-heterogeneous modelling of the human cardiovascular system, in Modeling of Physiological Flows, edited by H. Ambrosi, A. Quarteroni and G. Rozza. Springer, Italy (2012) 251–288.
[7] Blanco, P.J., Feijóo, R.A. and Urquiza, S.A., A unified variational approach for coupling 3D–1D models and its blood flow applications. Comput. Methods Appl. Mech. Engrg. 196 (2007) 43914410.
[8] Blanco, P.J., Leiva, J.S., Feijóo, R.A. and Buscaglia, G.S., Black-box decomposition approach for computational hemodynamics: One-dimensional models. Comput. Methods Appl. Mech. Engrg. 200 138 (2011) 91405.
[9] Blanco, P.J., Pivello, M.R., Urquiza, S.A. and Feijóo, R.A., On the potentialities of 3D–1D coupled models in hemodynamics simulations. J. Biomech. 42 (2009) 919930.
[10] Blanco, P.J., Trenhago, P.R., Fernandes, L.G. and Feijóo, R.A., On the integration of the baroreflex control mechanism in a heterogeneous model of the cardiovascular system. Int. J. Numer. Methods Biomed. Engng. 28 (2012) 412433.
[11] Blanco, P.J., Urquiza, S.A. and Feijóo, R.A., Assessing the influence of heart rate in local hemodynamics through coupled 3D-1D-0D models. Int. J. Numer. Methods Biomed. Engng. 26 (2010) 890903.
[12] Blanco, P.J., Watanabe, S.M. and Feijóo, R.A., Identification of vascular territory resistances in one-dimensional hemodynamics simulations. J. Biomech. 45 (2012) 20662073.
[13] Casoli, V., Kostopoulos, E., Pélissier, P., Caix, P., Martin, D. and Baudet, J., The middle collateral artery: anatomic basis for the extreme lateral arm flap. Surg. Radiol. Anat.: SRA 26 (2004) 1727.
[14] Chen, S., Xu, D., Tang, M., Ding, H., Sheng, W. and Peng, T., Measurement and analysis of the perforator arteries in upper extremity for the flap design. Surg. Radiol. Anat.: SRA 31 68793 (2009).
[15] Claassen, H., Schmitt, O., Werner, D., Schareck, W., Kröger, J.C. and Wree, A., Superficial arm arteries revisited: Brother and sister with absent radial pulse. Annals of anatomy = Anatomischer Anzeiger: official organ of the Anatomische Gesellschaft 192 (2010) 1515.
[16] W. Dauber, Pocket Atlas of Human Anatomy by Feneis. Thieme, 5th edition (2007).
[17] Denton, T.A., Trento, L., Cohen, M., Kass, R.M., Blanche, C., Raissi, S., Cheng, W., Fontana, G.P. and Trento, A., Radial artery harvesting for coronary bypass operations: neurologic complications and their potential mechanisms. The Journal of thoracic and cardiovascular surgery 121 (2001) 9516.
[18] Duparc, F., Muller, J.M. and Fréger, P., Arterial blood supply of the proximal humeral epiphysis. Surg. Radiol. Anat.: SRA 23 (2001) 18590.
[19] Fazan, V.P.S., Borges, C.T., Da Silva, J.H., Caetano, A.G. and Filho, O.A.R., Superficial palmar arch: an arterial diameter study. J. Anat. 204 (2004) 30711.
[20] Formaggia, L., Lamponi, D. and Quarteroni, A., One-dimensional models for blood flow in arteries. J. Engrg Math. 47 (2003) 251276.
[21] Formaggia, L., Moura, A. and Nobile, F., On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations. ESAIM: M2AN 41 (2007) 743769.
[22] Formaggia, L., Nobile, F., Quarteroni, A. and Veneziani, A., Computing and Visualization in Science Regular article Multiscale modelling of the circulatory system: a preliminary analysis. 83 (1999) 7583.
[23] Haerle, M., Tonagel, F. and Schaller, H.E., Collateral arterial pathways in the forearm. Surg. Radiol. Anat.: SRA 26 (2004) 20811.
[24] K.R.S. Holzbaur, W.M. Murray, G.E. Gold and S.L. Delp, Upper limb muscle volumes in adult subjects. J. Biomech. (2007) 742–749.
[25] Hughes, T.J.R. and Lubliner, J., On the one-dimensional theory of blood flow in the larger vessels. Math. Biosci. 18 (1973) 161170.
[26] Kim, H.J., Vignon-Clementel, I.E., Figueroa, C.A., LaDisa, J.F., Jansen, K.E., Feinstein, J.A. and Taylor, C.A., On coupling a lumped parameter heart model and a three-dimensional finite element aorta model 37 (2009) 21532169.
[27] Knobloch, K., Tomaszek, S., Lichtenberg, A., Karck, M. and Haverich, A., Long-term palmar microcirculation after radial artery harvesting: an observational study. The Annal. Thorac. Surg. 81 (2006) 17007.
[28] Leguy, C.A.D., Bosboom, E.M.H., Belloum, A.S.Z., Hoeks, A.P.G. and van de Vosse, F.N., Global sensitivity analysis of a wave propagation model for arm arteries. Medical Engrg. Phys. 33 (2011) 100816.
[29] Leiva, J.S., Blanco, P.J. and Buscaglia, G.C., Partitioned analysis for dimensionally-heterogeneous hydraulic networks. SIAM Multiscale Model. Simulat. 9 (2011) 872903.
[30] Mahabir, R.C., Williamson, J.S., Carr, N.J. and Courtemanche, D.J., Vascular Resistance in Human Muscle Flaps. Annal. Plast. Surg. 47 (2001) 148152.
[31] K.L. Moore, A.F. Dalley and A.M.R. Agur, Clinically Oriented Anatomy. Wolters Kluwer, 6th edition (2010).
[32] Morris, S.F., Tang, M., Almutari, K., Geddes, C. and Yang, D., The anatomic basis of perforator flaps. Clinics in plastic surgery 37 (2010) 55370.
[33] F.H. Netter, Atlas of Human Anatomy. Elsevier, 5th edition (2011).
[34] Olufsen, M.S., Structured tree outflow condition for blood flow in larger systemic arteries. Amer. J. Phys. 276 (1999) H25768.
[35] Omokawa, S., Tanaka, Y., Ryu, J. and Kish, V.L., The anatomical basis for reverse first to fifth dorsal metacarpal arterial flaps. J. Hand Surgery (Edinburgh, Scotland) 30 (2005) 404.
[36] M.F. O’Rourke and W.W. Nichols, McDonald’s Blood Flow in Arteries - Theoretical, Experimental and Clinical Principles. Arnold, 4th edition (1998).
[37] Prevel, C.D., Matloub, H.S., Ye, Z., Sanger, J.R. and Yousif, N.J., The extrinsic blood supply of the ulnar nerve at the elbow: an anatomic study. The J. Hand Surgery 18 (1993) 4338.
[38] Reymond, P., Merenda, F., Perren, F., Rüfenacht, D. and Stergiopulos, N., Validation of a one-dimensional model of the systemic arterial tree. Amer. J. Physiol. Heart circulatory Physiol. 297 (2009) H20822.
[39] Royse, A.G., Chang, G.S., Nicholas, D.M. and Royse, C.F., No late ulnar artery atheroma after radial artery harvest for coronary artery bypass surgery. The Annal. Thoracic Surgery 85 (2008) 8914.
[40] Sauerbier, M. and Unglaub, F., Perforator flaps in the upper extremity. Clinics in plastic Surg. 37 (2010) 66776.
[41] Schafhalterzoppoth, I. and Gray, A., The Musculocutaneous Nerve: Ultrasound Appearance for Peripheral Nerve Block. Regional Anesthesia and Pain Medicine 30 (2005) 385390.
[42] T.F. Sherman, On Connecting Large Vessels to Small The Meaning of Murray’s Law. J. General Physiol. (1981).
[43] Stergiopulos, N., Young, D.F. and Rogge, T.R., Computer simulation of arterial flow with applications to arterial and aortic stenoses. J. Biomech. 25 (1992) 14771488.
[44] Taylor, G., The angiosomes of the body and their supply to perforator flaps. Clin. Plast. Surg. 30 (2003) 331342.
[45] Trager, S., Pignataro, M., Anderson, J. and Kleinert, J.M., Color flow Doppler: imaging the upper extremity. J. Hand Surg. 18 (1993) 6215.
[46] Urquiza, S.A., Blanco, P.J., Vénere, M.J. and Feijóo, R.A., Multidimensional modelling for the carotid artery blood flow. Comput. Methods Appl. Mech. Engrg. 195 (2006) 40024017.
[47] Walther, G., Nottin, S., Dauzat, M. and Obert, P., Femoral and axillary ultrasound blood flow during exercise: a methodological study. Med. Sci. Sports Exerc. 38 (2006) 1353.
[48] Wang, J.J. and Parker, K.H., Wave propagation in a model of the arterial circulation. J. Biomech. 37 (2004) 45770.
[49] Wavreille, G., Bricout, J., Mouliade, S., Lemoine, S., Prodhomme, G., Khanchandani, P., Chantelot, C. and Fontaine, C., Anatomical bases of the free posterior brachial fascial flap. Surg. Radiol. Anat.: SRA 32 (2010) 3939.
[50] Wavreille, G., Dos Remedios, C., Chantelot, C., Limousin, M. and Fontaine, C., Anatomic bases of vascularized elbow joint harvesting to achieve vascularized allograft. Surg. Radiol. Anat.: SRA 28 (2006) 498510.
[51] Zhang, F.H., Topp, S.G., Zhang, W.J., Zheng, H.P. and Zhang, F., Anatomic study of distally based pedicle compound flaps with nutrient vessels of the cutaneous nerves and superficial veins of the forearm. Microsurgery 26 (2006) 373385.


Mathematical Model of Blood Flow in an Anatomically Detailed Arterial Network of the Arm

  • Sansuke M. Watanabe (a1) (a2) (a3), Pablo J. Blanco (a2) (a3) and Raúl A. Feijóo (a2) (a3)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed