Mathematical Model of Blood Flow in an Anatomically Detailed Arterial Network of the Arm
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 4, p. 961-985

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.

Classification:  76Z05,  92C10,  92C30
Keywords: hemodynamics, anatomical model, vascular territories, numerical simulation
     author = {Watanabe, Sansuke M. and Blanco, Pablo J. and Feij\'oo, Ra\'ul A.},
     title = {Mathematical Model of Blood Flow in an Anatomically Detailed Arterial Network of the Arm},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {4},
     year = {2013},
     pages = {961-985},
     doi = {10.1051/m2an/2012053},
     zbl = {06198326},
     mrnumber = {3082285},
     language = {en},
     url = {}
Watanabe, Sansuke M.; Blanco, Pablo J.; Feijóo, Raúl A. Mathematical Model of Blood Flow in an Anatomically Detailed Arterial Network of the Arm. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 4, pp. 961-985. doi : 10.1051/m2an/2012053.

[1] J. Alastruey, A.W. Khir, K.S. Matthys, P. Segers, S.J. Sherwin, P.R. Verdonck, K.H. Parker and J. Peiró, Pulse wave propagation in a model human arterial network: Assessment of 1-D visco-elastic simulations against in vitro measurements. J. Biomech. 44 (2011) 2250-8.

[2] J. Alastruey, K.H. Parker, J. Peiró, S.M. Byrd and S.J. Sherwin, Modelling the circle of Willis to assess the effects of anatomical variations and occlusions on cerebral flows. J. Biomech. 40 (2007) 1794-805.

[3] B.H. Amundsen, U. Wisloff, J. Helgerud, J. Hoff and S.A. Slordahl, Ultrasound recorded axillary artery blood flow during elbow-flexion exercise. Medicine and science in sports and exercise 34 (2002) 1288-93.

[4] A.P. Avolio, Multi-branched model of the human arterial system. Medical Biolog. Engrg. Comp. 18 (1980) 709-18.

[5] O. Bilge, Y. Pinar, M.A. Ozer and F. Gövsa, A morphometric study on the superficial palmar arch of the hand. Surg. Radiol. Anat.: SRA 28 (2006) 343-50.

[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] P.J. Blanco, R.A. Feijóo and S.A. Urquiza, A unified variational approach for coupling 3D-1D models and its blood flow applications. Comput. Methods Appl. Mech. Engrg. 196 (2007) 4391-4410. | MR 2348785 | Zbl 1173.76430

[8] P.J. Blanco, J.S. Leiva, R.A. Feijóo and G.S. Buscaglia, Black-box decomposition approach for computational hemodynamics: One-dimensional models. Comput. Methods Appl. Mech. Engrg. 200 138 (2011) 9-1405. | MR 2774753 | Zbl 1228.76203

[9] P.J. Blanco, M.R. Pivello, S.A. Urquiza and R.A. Feijóo, On the potentialities of 3D-1D coupled models in hemodynamics simulations. J. Biomech. 42 (2009) 919-930.

[10] P.J. Blanco, P.R. Trenhago, L.G. Fernandes and R.A. Feijóo, On the integration of the baroreflex control mechanism in a heterogeneous model of the cardiovascular system. Int. J. Numer. Methods Biomed. Engng. 28 (2012) 412-433. | MR 2911312

[11] P.J. Blanco, S.A. Urquiza and R.A. Feijóo, Assessing the influence of heart rate in local hemodynamics through coupled 3D-1D-0D models. Int. J. Numer. Methods Biomed. Engng. 26 (2010) 890-903. | Zbl 1193.92027

[12] P.J. Blanco, S.M. Watanabe and R.A. Feijóo, Identification of vascular territory resistances in one-dimensional hemodynamics simulations. J. Biomech. 45 (2012) 2066-2073.

[13] V. Casoli, E. Kostopoulos, P. Pélissier, P. Caix, D. Martin and J. Baudet, The middle collateral artery: anatomic basis for the extreme lateral arm flap. Surg. Radiol. Anat.: SRA 26 (2004) 172-7.

[14] S. Chen, D. Xu, M. Tang, H. Ding, W. Sheng and T. Peng, Measurement and analysis of the perforator arteries in upper extremity for the flap design. Surg. Radiol. Anat.: SRA 31 687-93 (2009).

[15] H. Claassen, O. Schmitt, D. Werner, W. Schareck, J.C. Kröger and A. Wree, Superficial arm arteries revisited: Brother and sister with absent radial pulse. Annals of anatomy = Anatomischer Anzeiger: official organ of the Anatomische Gesellschaft 192 (2010) 151-5.

[16] W. Dauber, Pocket Atlas of Human Anatomy by Feneis. Thieme, 5th edition (2007).

[17] T.A. Denton, L. Trento, M. Cohen, R.M. Kass, C. Blanche, S. Raissi, W. Cheng, G.P. Fontana and A. Trento, Radial artery harvesting for coronary bypass operations: neurologic complications and their potential mechanisms. The Journal of thoracic and cardiovascular surgery 121 (2001) 951-6.

[18] F. Duparc, J.M. Muller and P. Fréger, Arterial blood supply of the proximal humeral epiphysis. Surg. Radiol. Anat.: SRA 23 (2001) 185-90.

[19] V.P.S. Fazan, C.T. Borges, J.H. Da Silva, A.G. Caetano and O.A.R. Filho, Superficial palmar arch: an arterial diameter study. J. Anat. 204 (2004) 307-11.

[20] L. Formaggia, D. Lamponi and A. Quarteroni, One-dimensional models for blood flow in arteries. J. Engrg Math. 47 (2003) 251-276. | MR 2038983 | Zbl 1070.76059

[21] L. Formaggia, A. Moura and F. Nobile, On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations. ESAIM: M2AN 41 (2007) 743-769. | Numdam | MR 2362913 | Zbl 1139.92009

[22] L. Formaggia, F. Nobile, A. Quarteroni and A. Veneziani, Computing and Visualization in Science Regular article Multiscale modelling of the circulatory system: a preliminary analysis. 83 (1999) 75-83. | Zbl 1067.76624

[23] M. Haerle, F. Tonagel and H.E. Schaller, Collateral arterial pathways in the forearm. Surg. Radiol. Anat.: SRA 26 (2004) 208-11.

[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] T.J.R. Hughes and J. Lubliner, On the one-dimensional theory of blood flow in the larger vessels. Math. Biosci. 18 (1973) 161-170. | Zbl 0262.92004

[26] H.J. Kim, I.E. Vignon-Clementel, C.A. Figueroa, J.F. Ladisa, K.E. Jansen, J.A. Feinstein and C.A. Taylor, On coupling a lumped parameter heart model and a three-dimensional finite element aorta model 37 (2009) 2153-2169.

[27] K. Knobloch, S. Tomaszek, A. Lichtenberg, M. Karck and A. Haverich, Long-term palmar microcirculation after radial artery harvesting: an observational study. The Annal. Thorac. Surg. 81 (2006) 1700-7.

[28] C.A.D. Leguy, E.M.H. Bosboom, A.S.Z. Belloum, A.P.G. Hoeks and F.N. Van De Vosse, Global sensitivity analysis of a wave propagation model for arm arteries. Medical Engrg. Phys. 33 (2011) 1008-16.

[29] J.S. Leiva, P.J. Blanco and G.C. Buscaglia, Partitioned analysis for dimensionally-heterogeneous hydraulic networks. SIAM Multiscale Model. Simulat. 9 (2011) 872-903. | MR 2818423 | Zbl 1300.76011

[30] R.C. Mahabir, J.S. Williamson, N.J. Carr and D.J. Courtemanche, Vascular Resistance in Human Muscle Flaps. Annal. Plast. Surg. 47 (2001) 148-152.

[31] K.L. Moore, A.F. Dalley and A.M.R. Agur, Clinically Oriented Anatomy. Wolters Kluwer, 6th edition (2010).

[32] S.F. Morris, M. Tang, K. Almutari, C. Geddes and D. Yang, The anatomic basis of perforator flaps. Clinics in plastic surgery 37 (2010) 553-70.

[33] F.H. Netter, Atlas of Human Anatomy. Elsevier, 5th edition (2011).

[34] M.S. Olufsen, Structured tree outflow condition for blood flow in larger systemic arteries. Amer. J. Phys. 276 (1999) H257-68.

[35] S. Omokawa, Y. Tanaka, J. Ryu and V.L. Kish, The anatomical basis for reverse first to fifth dorsal metacarpal arterial flaps. J. Hand Surgery (Edinburgh, Scotland) 30 (2005) 40-4.

[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] C.D. Prevel, H.S. Matloub, Z. Ye, J.R. Sanger and N.J. Yousif, The extrinsic blood supply of the ulnar nerve at the elbow: an anatomic study. The J. Hand Surgery 18 (1993) 433-8.

[38] P. Reymond, F. Merenda, F. Perren, D. Rüfenacht and N. Stergiopulos, Validation of a one-dimensional model of the systemic arterial tree. Amer. J. Physiol. Heart circulatory Physiol. 297 (2009) H208-22.

[39] A.G. Royse, G.S. Chang, D.M. Nicholas and C.F. Royse, No late ulnar artery atheroma after radial artery harvest for coronary artery bypass surgery. The Annal. Thoracic Surgery 85 (2008) 891-4.

[40] M. Sauerbier and F. Unglaub, Perforator flaps in the upper extremity. Clinics in plastic Surg. 37 (2010) 667-76.

[41] I. Schafhalterzoppoth and A. Gray, The Musculocutaneous Nerve: Ultrasound Appearance for Peripheral Nerve Block. Regional Anesthesia and Pain Medicine 30 (2005) 385-390.

[42] T.F. Sherman, On Connecting Large Vessels to Small The Meaning of Murray's Law. J. General Physiol. (1981).

[43] N. Stergiopulos, D.F. Young and T.R. Rogge, Computer simulation of arterial flow with applications to arterial and aortic stenoses. J. Biomech. 25 (1992) 1477-1488.

[44] G. Taylor, The angiosomes of the body and their supply to perforator flaps. Clin. Plast. Surg. 30 (2003) 331-342.

[45] S. Trager, M. Pignataro, J. Anderson and J.M. Kleinert, Color flow Doppler: imaging the upper extremity. J. Hand Surg. 18 (1993) 621-5.

[46] S.A. Urquiza, P.J. Blanco, M.J. Vénere and R.A. Feijóo, Multidimensional modelling for the carotid artery blood flow. Comput. Methods Appl. Mech. Engrg. 195 (2006) 4002-4017. | MR 2229832 | Zbl 1178.76395

[47] G. Walther, S. Nottin, M. Dauzat and P. Obert, Femoral and axillary ultrasound blood flow during exercise: a methodological study. Med. Sci. Sports Exerc. 38 (2006) 1353.

[48] J.J. Wang and K.H. Parker, Wave propagation in a model of the arterial circulation. J. Biomech. 37 (2004) 457-70.

[49] G. Wavreille, J. Bricout, S. Mouliade, S. Lemoine, G. Prodhomme, P. Khanchandani, C. Chantelot and C. Fontaine, Anatomical bases of the free posterior brachial fascial flap. Surg. Radiol. Anat.: SRA 32 (2010) 393-9.

[50] G. Wavreille, C. Dos Remedios, C. Chantelot, M. Limousin and C. Fontaine, Anatomic bases of vascularized elbow joint harvesting to achieve vascularized allograft. Surg. Radiol. Anat.: SRA 28 (2006) 498-510.

[51] F.H. Zhang, S.G. Topp, W.J. Zhang, H.P. Zheng and F. Zhang, Anatomic study of distally based pedicle compound flaps with nutrient vessels of the cutaneous nerves and superficial veins of the forearm. Microsurgery 26 (2006) 373-385.