# American Institute of Mathematical Sciences

2012, 9(1): 61-74. doi: 10.3934/mbe.2012.9.61

## An evaluation of dynamic outlet boundary conditions in a 1D fluid dynamics model

 1 Department of Biomedical Engineering, North Carolina State University, Raleigh, NC 27695-7115, United States 2 Department of Biomedical Engineering, Engineering Building 3, RM 4204B, North Carolina State University, Raleigh, NC 27695-7115, United States

Received  February 2011 Revised  August 2011 Published  December 2011

When modeling the cardiovascular system, the use of boundary conditions that closely represent the interaction between the region of interest and the surrounding vessels and organs will result in more accurate predictions. An often overlooked feature of outlet boundary conditions is the dynamics associated with regulation of the distribution of pressure and flow. This study implements a dynamic impedance outlet boundary condition in a one-dimensional fluid dynamics model using the pulmonary vasculature and respiration (feedback mechanism) as an example of a dynamic system. The dynamic boundary condition was successfully implemented and the pressure and flow were predicted for an entire respiration cycle. The cardiac cycles at maximal expiration and inspiration were predicted with a root mean square error of $0.61$ and $0.59$ mm Hg, respectively.
Citation: Rachel Clipp, Brooke Steele. An evaluation of dynamic outlet boundary conditions in a 1D fluid dynamics model. Mathematical Biosciences & Engineering, 2012, 9 (1) : 61-74. doi: 10.3934/mbe.2012.9.61
##### References:
 [1] A. P. Avolio, Multi-branched model of the human arterial system, Medical and Biological Engineering and Computing, 18 (1980), 709-718. doi: 10.1007/BF02441895. [2] D. J. Brown, Input impedance and reflection coefficient in fractal-like models of asymmetrically branching compliant tubes, IEEE Transactions on Biomedical Engineering, 43 (1996), 715-722. doi: 10.1109/10.503179. [3] A. C. Burton and D. J. Patel, Effect on pulmonary vascular resistance of inflation of the rabbit lungs, Journal of Applied Physiology, 12 (1958), 239-246. [4] R. B. Clipp and B. N. Steele, Impedance boundary conditions for the pulmonary vasculature including the effects of geometry, compliance, and respiration, IEEE Transactions on Bio-Medical Engineering, 56 (2009), 862-870. doi: 10.1109/TBME.2008.2010133. [5] M. R. de Leval, G. Dubini, F. Migliavacca, H. Jalali, G. Camporini, A. Redington and R. Pietrabissa, Use of computational fluid dynamics in the design of surgical procedures: Application to the study of competitive flows in cavo-pulmonary connections, Journal of Thoracic and Cardiovascular Surgery, 111 (1996), 502-513. doi: 10.1016/S0022-5223(96)70302-1. [6] D. A. de Zèlicourt, K. Pekkan, J. Parks, K. Kanter, M. A. Fogel and A. P. Yoganathan, Flow study of an extracardiac connection with persistent left superior vena cava, The Journal of Thoracic and Cardiovascular Surgery, 131 (2006), 785-791. doi: 10.1016/j.jtcvs.2005.11.031. [7] B. Flemming, E. Seeliger, T. Wronski, K. Steer, N. Arenz and P. B. Persson, Oxygen and renal hemodynamics in the conscious rat, Journal of the American Society of Nephrology, 11 (2000), 18-24. [8] T. J. R. Hughes and J. Lubliner, On the one-dimensional theory of blood flow in the larger vessels, Mathematical Biosciences, 18 (1973), 161-170. doi: 10.1016/0025-5564(73)90027-8. [9] T. J. R. Hughes, "A Study of the One-Dimensional Theory of Arterial Pulse Propagation," Ph.D. Thesis, Report 74-13, Structural Engineering Laboratory, University of California, Berkeley, CA, 1974. [10] K. Lagana, R. Balossino, F. Migliavacca, G. Pennati, E. L. Bove, M. R. de Leval and G. Dubini, Multiscale modeling of the cardiovascular system: Application to the study of pulmonary and coronary perfusions in the univentricular circulation, Journal of Biomechanics, 5 (2005), 1129-1141. doi: 10.1016/j.jbiomech.2004.05.027. [11] H. J. Kim, I. E. Vignon-Clementel, J. S. Coogan, C. A. Figueroa, K. E. Jansen and C. A. Taylor, Patient-specific modeling of blood flow and pressure in human coronary arteries, Annals of Biomedical Engineering, 38 (2010), 3195-3209. doi: 10.1007/s10439-010-0083-6. [12] A. L. Marsden, I. E. Vignon-Clementel, F. P. Chan, J. A. Feinstein and C. A. Taylor, Effects of exercise and respiration on hemodynamic efficiency in CFD simulations of the total cavopulmonary connection, Annals of Biomedical Engineering, 35 (2007), 250-263. doi: 10.1007/s10439-006-9224-3. [13] A. L. Marsden, V. M. Reddy, S. C. Shadden, F. P. Chan, C. A. Taylor and J. A. Feinstein, A new multiparameter approach to computational simulation for fontan assessment and redesign, Congenital Heart Disease, 5 (2010), 104-117. doi: 10.1111/j.1747-0803.2010.00383.x. [14] H. Ohuch, Y. Arakaki, Y. Hiraumi, H. Tasato and T. Kamiya, Cardiorespiratory response during exercise in patients with cyanotic congenital heart disease with and without a fontan operation and in patients with congestive heart failure, International Journal of Cardiology, 66 (1998), 241-251. doi: 10.1016/S0167-5273(98)00249-6. [15] M. S. Olufsen, A structured tree outflow condition for blood flow in the larger systemic arteries, The American Journal of Physiology, 276 (1999), H257-H268. [16] M. S. Olufsen, C. S. Peskin, W. Y. Ki, E. M. Pedersen, A. Nadim and J. Larsen, Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions, Annals of Biomedical Engineering, 28 (2000), 1281-1299. doi: 10.1114/1.1326031. [17] K. Pekkan, D. A. de Zèlicourt, L. Ge, F. Sotiropoulos, D. Frakes, M. A. Fogel and A. P. Yoganathan, Physics-driven CFD modeling of complex anatomical cardiovascular flows-a TCPC case study, Annals of Biomedical Engineering, 33 (2005), 284-300. doi: 10.1007/s10439-005-1731-0. [18] D. J. Penny and A. N. Redington, Doppler echocardiographic evaluation of pulmonary blood flow after the Fontan operation: The role of the lungs, British Heart Journal, 66 (1991), 372-374. doi: 10.1136/hrt.66.5.372. [19] P. B. Persson, H. Ehmke, H. R. Kirchheim, B. Janssen, J. E. Baumann, A. Just and B. Nafz, Autoregulation and non-homeostatic behaviour of renal blood flow in conscious dogs, Journal of Physiology, 462 (1993), 261-273. [20] F. Petak, G. Albu, E. Lele, Z. Hantos, D. R. Morel, F. Fontao and W. Habre, Lung mechanical and vascular changes during positive- and negative-pressure lung inflations: Importance of reference pressures in the pulmonary vasculature, Journal of Applied Physiology, 106 (2009), 935-942. doi: 10.1152/japplphysiol.00831.2007. [21] A. N. Redington, D. Penny and E. A. Shinebourne, Pulmonary blood flow after total cavopulmonary shunt, Heart, 65 (1991), 213-217. doi: 10.1136/hrt.65.4.213. [22] D. D. Soerensen, K. Pekkan, D. A. de Zélicourt, S. Sharma, K. Kanter, M. A. Fogel and A. P. Yoganathan, Introduction of a new optimized total cavopulmonary connection, The Annals of Thoracic Surgery, 83 (2007), 2182-2190. doi: 10.1016/j.athoracsur.2006.12.079. [23] R. L. Spilker, J. A. Feinstein, D. W. Parker, V. M. Reddy and C. A. Taylor, Morphometry-based impedance boundary conditions for patient-specific modeling of blood flow in pulmonary arteries, Annals of Biomedical Engineering, 35 (2007), 546-559. doi: 10.1007/s10439-006-9240-3. [24] B. N. Steele, J. Wan, J. P. Ku, T. J. R. Hughes and C. A. Taylor, In vivo validation of a one-dimensional finite-element method for predicting blood flow in cardiovascular bypass grafts, IEEE Transactions on Biomedical Engineering, 50 (2003), 649-656. doi: 10.1109/TBME.2003.812201. [25] B. N. Steele, M. S. Olufsen and C. A. Taylor, Fractal network model for simulating abdominal and lower extremity blood flow during resting and exercise conditions, Computer Methods in Biomechanics and Biomedical Engineering, 10 (2007), 39-51. doi: 10.1080/10255840601068638. [26] B. N. Steele, D. Valdez-Jasso, M. Haider and M. S. Olufsen, Predicting arterial flow and pressure dynamics using a 1D fluid dynamics model with a viscoelastic wall, SIAM Journal of Applied Mathematics, 71 (2011), 1123-1143. doi: 10.1137/100810186. [27] C. A. Taylor, M. T. Draney, J. P. Ku, D. Parker, B. N. Steele, K. Wang and C. K. Zarins, Predictive medicine: Computational techniques in therapeutic decision-making, Computer Aided Surgery, 4 (1999), 231-247. doi: 10.3109/10929089909148176. [28] M. G. Taylor, The input impedance of an assembly of randomly branching elastic tubes, Biophysics Journal, 6 (1966), 29-51. doi: 10.1016/S0006-3495(66)86638-9. [29] W. B. Troutman, T. J. Barstow, A. J. Galindo and D. M. Cooper, Abnormal dynamic cardiorespiratory responses to exercise in pediatric patients after fontan procedure, Journal of the American College of Cardiology, 31 (1998), 668-673. doi: 10.1016/S0735-1097(97)00545-7. [30] I. E. Vignon-Clementel, C. A. Figueroa, K. E. Jansen and C. A. Taylor, Outflow boundary conditions for three-dimensional simulations of non-periodic blood flow and pressure fields in deformable arteries, Computer Methods in Applied Mechanics and Engineering, 195 (2006), 3776-3796. [31] I. E. Vignon and C. A. Taylor, Outflow boundary conditions for one-dimensional finite element modeling of blood flow and pressure waves in arteries. New computational methods for wave propagation, Wave Motion, 39 (2004), 361-374. doi: 10.1016/j.wavemoti.2003.12.009. [32] J. Wan, B. N. Steele, S. A. Spicer, S. Strohband, G. R. Feijoo, T. J. R. Hughes and C. A. Taylor, A one-dimensional finite element method for simulation-based medical planning for cardiovascular disease, Computer Methods in Biomechanics and Biomedical Engineering, 5 (2002), 195-206. doi: 10.1080/10255840290010670. [33] N. Westerhof, F. Bosman, C. J. De Vries and A. Noordergraaf, Analog studies of the human systemic arterial tree, Journal of Biomechanics, 2 (1969), 121-143. doi: 10.1016/0021-9290(69)90024-4. [34] J. L. Whittenberger, M. McGregor, E. Berglund and H. G. Borst, Influence of state of inflation of the lung on pulmonary vascular resistance, Journal of Applied Physiology, 15 (1960), 878-882. [35] J. R. Womersley, Oscillatory flow in arteries: The constrained elastic tube as a model of arterial flow and pulse transmission, Physics in Medicine and Biology, 2 (1957), 178-187. doi: 10.1088/0031-9155/2/2/305. [36] J. R. Womersley, Oscillatory motion of a viscous liquid in a thin-walled elastic tube. I. The linear approximation for long waves, The Philosophical Magazine (7), 46 (1955), 199-221.

show all references

##### References:
 [1] A. P. Avolio, Multi-branched model of the human arterial system, Medical and Biological Engineering and Computing, 18 (1980), 709-718. doi: 10.1007/BF02441895. [2] D. J. Brown, Input impedance and reflection coefficient in fractal-like models of asymmetrically branching compliant tubes, IEEE Transactions on Biomedical Engineering, 43 (1996), 715-722. doi: 10.1109/10.503179. [3] A. C. Burton and D. J. Patel, Effect on pulmonary vascular resistance of inflation of the rabbit lungs, Journal of Applied Physiology, 12 (1958), 239-246. [4] R. B. Clipp and B. N. Steele, Impedance boundary conditions for the pulmonary vasculature including the effects of geometry, compliance, and respiration, IEEE Transactions on Bio-Medical Engineering, 56 (2009), 862-870. doi: 10.1109/TBME.2008.2010133. [5] M. R. de Leval, G. Dubini, F. Migliavacca, H. Jalali, G. Camporini, A. Redington and R. Pietrabissa, Use of computational fluid dynamics in the design of surgical procedures: Application to the study of competitive flows in cavo-pulmonary connections, Journal of Thoracic and Cardiovascular Surgery, 111 (1996), 502-513. doi: 10.1016/S0022-5223(96)70302-1. [6] D. A. de Zèlicourt, K. Pekkan, J. Parks, K. Kanter, M. A. Fogel and A. P. Yoganathan, Flow study of an extracardiac connection with persistent left superior vena cava, The Journal of Thoracic and Cardiovascular Surgery, 131 (2006), 785-791. doi: 10.1016/j.jtcvs.2005.11.031. [7] B. Flemming, E. Seeliger, T. Wronski, K. Steer, N. Arenz and P. B. Persson, Oxygen and renal hemodynamics in the conscious rat, Journal of the American Society of Nephrology, 11 (2000), 18-24. [8] T. J. R. Hughes and J. Lubliner, On the one-dimensional theory of blood flow in the larger vessels, Mathematical Biosciences, 18 (1973), 161-170. doi: 10.1016/0025-5564(73)90027-8. [9] T. J. R. Hughes, "A Study of the One-Dimensional Theory of Arterial Pulse Propagation," Ph.D. Thesis, Report 74-13, Structural Engineering Laboratory, University of California, Berkeley, CA, 1974. [10] K. Lagana, R. Balossino, F. Migliavacca, G. Pennati, E. L. Bove, M. R. de Leval and G. Dubini, Multiscale modeling of the cardiovascular system: Application to the study of pulmonary and coronary perfusions in the univentricular circulation, Journal of Biomechanics, 5 (2005), 1129-1141. doi: 10.1016/j.jbiomech.2004.05.027. [11] H. J. Kim, I. E. Vignon-Clementel, J. S. Coogan, C. A. Figueroa, K. E. Jansen and C. A. Taylor, Patient-specific modeling of blood flow and pressure in human coronary arteries, Annals of Biomedical Engineering, 38 (2010), 3195-3209. doi: 10.1007/s10439-010-0083-6. [12] A. L. Marsden, I. E. Vignon-Clementel, F. P. Chan, J. A. Feinstein and C. A. Taylor, Effects of exercise and respiration on hemodynamic efficiency in CFD simulations of the total cavopulmonary connection, Annals of Biomedical Engineering, 35 (2007), 250-263. doi: 10.1007/s10439-006-9224-3. [13] A. L. Marsden, V. M. Reddy, S. C. Shadden, F. P. Chan, C. A. Taylor and J. A. Feinstein, A new multiparameter approach to computational simulation for fontan assessment and redesign, Congenital Heart Disease, 5 (2010), 104-117. doi: 10.1111/j.1747-0803.2010.00383.x. [14] H. Ohuch, Y. Arakaki, Y. Hiraumi, H. Tasato and T. Kamiya, Cardiorespiratory response during exercise in patients with cyanotic congenital heart disease with and without a fontan operation and in patients with congestive heart failure, International Journal of Cardiology, 66 (1998), 241-251. doi: 10.1016/S0167-5273(98)00249-6. [15] M. S. Olufsen, A structured tree outflow condition for blood flow in the larger systemic arteries, The American Journal of Physiology, 276 (1999), H257-H268. [16] M. S. Olufsen, C. S. Peskin, W. Y. Ki, E. M. Pedersen, A. Nadim and J. Larsen, Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions, Annals of Biomedical Engineering, 28 (2000), 1281-1299. doi: 10.1114/1.1326031. [17] K. Pekkan, D. A. de Zèlicourt, L. Ge, F. Sotiropoulos, D. Frakes, M. A. Fogel and A. P. Yoganathan, Physics-driven CFD modeling of complex anatomical cardiovascular flows-a TCPC case study, Annals of Biomedical Engineering, 33 (2005), 284-300. doi: 10.1007/s10439-005-1731-0. [18] D. J. Penny and A. N. Redington, Doppler echocardiographic evaluation of pulmonary blood flow after the Fontan operation: The role of the lungs, British Heart Journal, 66 (1991), 372-374. doi: 10.1136/hrt.66.5.372. [19] P. B. Persson, H. Ehmke, H. R. Kirchheim, B. Janssen, J. E. Baumann, A. Just and B. Nafz, Autoregulation and non-homeostatic behaviour of renal blood flow in conscious dogs, Journal of Physiology, 462 (1993), 261-273. [20] F. Petak, G. Albu, E. Lele, Z. Hantos, D. R. Morel, F. Fontao and W. Habre, Lung mechanical and vascular changes during positive- and negative-pressure lung inflations: Importance of reference pressures in the pulmonary vasculature, Journal of Applied Physiology, 106 (2009), 935-942. doi: 10.1152/japplphysiol.00831.2007. [21] A. N. Redington, D. Penny and E. A. Shinebourne, Pulmonary blood flow after total cavopulmonary shunt, Heart, 65 (1991), 213-217. doi: 10.1136/hrt.65.4.213. [22] D. D. Soerensen, K. Pekkan, D. A. de Zélicourt, S. Sharma, K. Kanter, M. A. Fogel and A. P. Yoganathan, Introduction of a new optimized total cavopulmonary connection, The Annals of Thoracic Surgery, 83 (2007), 2182-2190. doi: 10.1016/j.athoracsur.2006.12.079. [23] R. L. Spilker, J. A. Feinstein, D. W. Parker, V. M. Reddy and C. A. Taylor, Morphometry-based impedance boundary conditions for patient-specific modeling of blood flow in pulmonary arteries, Annals of Biomedical Engineering, 35 (2007), 546-559. doi: 10.1007/s10439-006-9240-3. [24] B. N. Steele, J. Wan, J. P. Ku, T. J. R. Hughes and C. A. Taylor, In vivo validation of a one-dimensional finite-element method for predicting blood flow in cardiovascular bypass grafts, IEEE Transactions on Biomedical Engineering, 50 (2003), 649-656. doi: 10.1109/TBME.2003.812201. [25] B. N. Steele, M. S. Olufsen and C. A. Taylor, Fractal network model for simulating abdominal and lower extremity blood flow during resting and exercise conditions, Computer Methods in Biomechanics and Biomedical Engineering, 10 (2007), 39-51. doi: 10.1080/10255840601068638. [26] B. N. Steele, D. Valdez-Jasso, M. Haider and M. S. Olufsen, Predicting arterial flow and pressure dynamics using a 1D fluid dynamics model with a viscoelastic wall, SIAM Journal of Applied Mathematics, 71 (2011), 1123-1143. doi: 10.1137/100810186. [27] C. A. Taylor, M. T. Draney, J. P. Ku, D. Parker, B. N. Steele, K. Wang and C. K. Zarins, Predictive medicine: Computational techniques in therapeutic decision-making, Computer Aided Surgery, 4 (1999), 231-247. doi: 10.3109/10929089909148176. [28] M. G. Taylor, The input impedance of an assembly of randomly branching elastic tubes, Biophysics Journal, 6 (1966), 29-51. doi: 10.1016/S0006-3495(66)86638-9. [29] W. B. Troutman, T. J. Barstow, A. J. Galindo and D. M. Cooper, Abnormal dynamic cardiorespiratory responses to exercise in pediatric patients after fontan procedure, Journal of the American College of Cardiology, 31 (1998), 668-673. doi: 10.1016/S0735-1097(97)00545-7. [30] I. E. Vignon-Clementel, C. A. Figueroa, K. E. Jansen and C. A. Taylor, Outflow boundary conditions for three-dimensional simulations of non-periodic blood flow and pressure fields in deformable arteries, Computer Methods in Applied Mechanics and Engineering, 195 (2006), 3776-3796. [31] I. E. Vignon and C. A. Taylor, Outflow boundary conditions for one-dimensional finite element modeling of blood flow and pressure waves in arteries. New computational methods for wave propagation, Wave Motion, 39 (2004), 361-374. doi: 10.1016/j.wavemoti.2003.12.009. [32] J. Wan, B. N. Steele, S. A. Spicer, S. Strohband, G. R. Feijoo, T. J. R. Hughes and C. A. Taylor, A one-dimensional finite element method for simulation-based medical planning for cardiovascular disease, Computer Methods in Biomechanics and Biomedical Engineering, 5 (2002), 195-206. doi: 10.1080/10255840290010670. [33] N. Westerhof, F. Bosman, C. J. De Vries and A. Noordergraaf, Analog studies of the human systemic arterial tree, Journal of Biomechanics, 2 (1969), 121-143. doi: 10.1016/0021-9290(69)90024-4. [34] J. L. Whittenberger, M. McGregor, E. Berglund and H. G. Borst, Influence of state of inflation of the lung on pulmonary vascular resistance, Journal of Applied Physiology, 15 (1960), 878-882. [35] J. R. Womersley, Oscillatory flow in arteries: The constrained elastic tube as a model of arterial flow and pulse transmission, Physics in Medicine and Biology, 2 (1957), 178-187. doi: 10.1088/0031-9155/2/2/305. [36] J. R. Womersley, Oscillatory motion of a viscous liquid in a thin-walled elastic tube. I. The linear approximation for long waves, The Philosophical Magazine (7), 46 (1955), 199-221.
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