Bulletin of Mathematical Biology

, Volume 76, Issue 6, pp 1306–1332 | Cite as

Stabilizing Control for a Pulsatile Cardiovascular Mathematical Model

  • Aurelio A. de los ReyesV
  • Eunok JungEmail author
  • Franz Kappel
Original Article


In this paper, we develop a pulsatile model for the cardiovascular system which describes the reaction of this system to a submaximal constant workload imposed on a person at a bicycle ergometer test after a period of rest. Furthermore, the model should allow to use measurements for the pulsatile pressure in fingertips which provide information on the diastolic and the systolic pressure for parameter estimation. Based on the assumption that the baroreceptor loop is the essential control loop in this case, we design a stabilizing feedback control for the pulsatile model which is obtained by solving a linear-quadratic regulator problem for the linearization of a non-pulsatile counterpart of the pulsatile model. We also investigate the behavior of the model with respect to changes in the weight of the term in the cost functional for the linear-quadratic regulator problem which penalizes the deviation of the momentary pressure in the aorta from the pressure at the stationary situation which should be obtained.


Cardiovascular model Baroreceptor loop Linear-quadratic regulator problem Stabilizing control 

Mathematics Subject Classification

92C30 49J15 49K15 



Jung’s work was supported by Konkuk University Research Grant in 2012. A. de los Reyes was supported by an ASEA-UNINET PhD-Technology Grant (administered by the Austrian Academic Exchange Service (OeAD)), by the University of the Philippines and by Konkuk University.


  1. Arechavaleta G, Laumond JP, Hicheur H, Berthoz A (2008) An optimality principle governing human walking. IEEE Trans Robot 24(1):5–14CrossRefGoogle Scholar
  2. Batzel JJ, Kappel F, Timischl-Teschl S (2005) A cardiovascular-respiratory control system model including state delay with application to congestive heart failure in humans. J Math Biol 50:293–335MathSciNetCrossRefzbMATHGoogle Scholar
  3. Batzel JJ, Kappel F, Schneditz D, Tran HT (2007) Cardiovascular and respiratory systems: modeling, analysis and control, frontiers in applied mathematics, vol 34. SIAM, PhiladelphiaCrossRefGoogle Scholar
  4. Bazett HC (1920) An analysis of the time-relations of electrocardiograms. Heart 7:353–370Google Scholar
  5. Bowditch HP (1871) Über die Eigenthümlichkeiten der Reizbarkeit, welche die Muskelfasern des Herzens zeigen. Berichte der Königlichen Sächsischen Gesellschaft der Wissenschaften, Mathematisch-Physische Classe, Sitzung am 12 Dezember 1871 23:651–689Google Scholar
  6. Danielsen M (1998) Modeling of feedback mechanisms which control the heart function in a view to an implementation in cardiovascular models. PhD thesis, Roskilde UniversityGoogle Scholar
  7. Danielsen M, Ottesen JT (1997) A dynamical approach to the baroreceptor regulation of the cardiovascular system. In: Proceeding of the 5th International Symposium, Symbiosis 1997, pp 25–29Google Scholar
  8. Danielsen M, Ottesen JT (2001) Describing the pumping heart as a pressure source. J Theor Biol 212(1):71–81CrossRefGoogle Scholar
  9. de los Reyes VAA (2010) A mathematical model for the cardiovascular system with a measurable pulsatile pressure output. PhD thesis, University of Graz, Institute for Mathematics and Scientific ComputingGoogle Scholar
  10. de los Reyes VAA, Kappel F (2010a) A mathematical cardiovascular model with pulsatile and non-pulsatile components. Tech. Rep. 011, Spezialforschungsbereich F32, University of GrazGoogle Scholar
  11. de los Reyes VAA, Kappel F (2010b) Modeling pulsatility in the human cardiovascular system. Mathematica Balkanica (New Series) 24(3—4):229–242. In: Proceedings of the “SEE Young Researchers Workshop”, MASSEE International Congress on Mathematics - MICOM 2009, September 16–20, 2009, Ohrid, MacedoniaGoogle Scholar
  12. Doubek E (1978) Least energy regulation of the arterial system. Bull Math Biol 40:79–93CrossRefGoogle Scholar
  13. Ellwein LM, Tran HT, Zapata C, Novak V, Olufsen MS (2008) Sensitivity analysis and model assessment: mathematical models for arterial blood flow and blood pressure. Cardiovasc Eng 8:94–108CrossRefGoogle Scholar
  14. Fink M, Batzel JJ, Kappel F (2004) An optimal control approach to modeling the cardiovascular-respiratory system: an application to orthostatic stress. Cardiovasc Eng 4(1):27–38CrossRefGoogle Scholar
  15. Fister KR, McCarthy CM (2003) Optimal control of a chemotaxis system. Q Appl Math 61(2):193–211MathSciNetzbMATHGoogle Scholar
  16. Gompertz B (1825) On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philos Trans R Soc London 115:513–583CrossRefGoogle Scholar
  17. Grodins FS (1959) Integrative cardiovascular physiology: a mathematical synthesis of cardiac and blood vessel hemodynamics. Q Rev Biol 34(2):93–116CrossRefGoogle Scholar
  18. Grodins FS (1963) Control theory and biological systems. Columbia University Press, New YorkGoogle Scholar
  19. Gusev S, Johansson S, Kågström B, Shiriaev A, Varga A (2009) A numerical evaluation of solvers for the periodic Riccati differential equation. Tech. Rep. Report / UMINF 09.03, Institutionen för Datavetenskap, Umeå UniversitetGoogle Scholar
  20. Guyton A, Hall J (2006) Textbook of medical physiology, 11th edn. Elsevier, AmsterdamGoogle Scholar
  21. Heldt T, Shim EB, Kamm RD, Mark RG (2002) Computational modeling of cardiovascular response to orthostatic stress. J Appl Physiol 92:1239–1254CrossRefGoogle Scholar
  22. Huntsman LL, Noordergraaf A, Attinger EO (1978) Metabolic autoregulation of blood flow in skeletal muscle: a model. In: Baan J, Noordergraaf A, Raines J (eds) Cardiovascular system dynamics. MIT Press, Cambridge, pp 400–414Google Scholar
  23. Janssen PML (2010) Myocardial contraction-relaxation coupling (54th Bowditch Lecture). Am J Physiol Heart Circ Physiol 299(6):H1741–H1749. doi: 10.1152/ajpheart.00759.2010 CrossRefGoogle Scholar
  24. Johansson S, Kågström B, Shiriaev A, Varga A (2007) Comparing one-shot and multishot methods for solving periodic riccati differential equations. In: Proceedings of the 3rd IFAC Workshop on periodic control systems, PSYCO07, St. Petersburg, RussiaGoogle Scholar
  25. Kappel F (2012) Modeling the dynamics of the cardiovascular-respiratory system (CVRS) in humans, a survey. Math Model Nat Pheno 7:65–77MathSciNetCrossRefzbMATHGoogle Scholar
  26. Kappel F, Peer RO (1993) A mathematical model for fundamental regulation processes in the cardiovascular system. J Math Biol 31:611–631MathSciNetCrossRefzbMATHGoogle Scholar
  27. Kappel F, Fink M, Batzel JJ (2007) Aspects of control of the cardiovascular-respiratory system during orthostatic stress induced by lower body negative pressure. Math Biosci 206(2):273–308MathSciNetCrossRefzbMATHGoogle Scholar
  28. Kenner T, Pfeiffer KP (1980) Studies on the optimal matching between heart and arterial system. In: Baan J, Arntzenius AC, Yellin EL (eds) Cardiac dynamics, developments in cardiovascular medicine, vol 2. Martinus Nijhoff Publishers bv, The Hague, pp 261–270Google Scholar
  29. Klabunde RE (2011) Cardiovascular physiology concepts, 2nd edn. Lippincott Williams & Wilkins, PhiladelphiaGoogle Scholar
  30. Křivan V (1996) Optimal foraging and predator-prey dynamics. Theor Popul Biol 49(3):265–290CrossRefzbMATHGoogle Scholar
  31. Kwakernaak H, Sivan R (1972) Linear optimal control systems. Wiley-Interscience, HobokenzbMATHGoogle Scholar
  32. Lebiedz D, Maurer H (2004) External optimal control of self-organisation dynamics in a chemotaxis reaction diffusion system. Syst Biol 1(2):222–229CrossRefGoogle Scholar
  33. Lenhart S, Workman JT (2007) Optimal control applied to biological models. Chapman and Hall/CRC, LondonzbMATHGoogle Scholar
  34. Noldus EJ (1976) Optimal control aspects of left ventricular ejection dynamics. J Theor Biol 63(2):275–309CrossRefGoogle Scholar
  35. Noordergraaf A (1969) Hemodynamics in biological engineering. McGraw-Hill, New YorkGoogle Scholar
  36. Olufsen MS, Ottesen JT (2013) A practical approach to parameter estimation applied to model predicting heart rate regulation. J Math Biol 67(1):39–68. doi: 10.1007/s00285-012-0535-8 MathSciNetCrossRefzbMATHGoogle Scholar
  37. Olufsen MS, Tran HT, Ottesen JT (2004) Modeling cerebral blood flow control during posture change from sitting to standing. Cardiovasc Eng 4:47–58CrossRefGoogle Scholar
  38. Olufsen MS, Ottesen JT, Tran HT, Ellwein LM, Lipsitz LA, Novak V (2005) Blood pressure and blood flow variation during postural change from sitting to standing: model development and validation. J Appl Physiol 99(4):1523–1537CrossRefGoogle Scholar
  39. Olufsen MS, Tran HT, Ottesen JT, Experiences Research, for Undergraduates Program, Lipsitz LA, Novak V, (2006) Modeling baroreflex regulation of heart rate during orthostatic stress. Am J Physiol Regul Integr Comp Physiol 291(5):R1355b–R1368Google Scholar
  40. Olufsen MS, Alston AV, Tran HT, Ottesen JT, Novak V (2008) Modeling heart rate regulation—Part I: sit-to-stand versus head-up tilt. Cardiovasc Eng 8(2):73–87CrossRefGoogle Scholar
  41. Ono K, Uozumi T, Yoshimoto C, Kenner T (1982) The optimal cardiovascular regulation of the arterial blood pressure. In: Kenner T, Busse R, Hinghofer-Szalkay H (eds) Cardiovascular system dynamics: models and measurements. Plenum Press, New York, pp 119–139CrossRefGoogle Scholar
  42. Ottesen JT (1997) Modelling of the baroreflex-feedback mechanism with time-delay. J Math Biol 36:41–63MathSciNetCrossRefzbMATHGoogle Scholar
  43. Ottesen JT, Danielsen M (2003) Modeling ventricular contraction with heart rate changes. J Theor Biol 222(3):337–346MathSciNetCrossRefGoogle Scholar
  44. Ottesen JT, Olufsen MS (2011) Functionality of the baroreceptor nerves in heart rate regulation. Comput Methods Programs Biomed 101(2):208–219CrossRefGoogle Scholar
  45. Ottesen JT, Olufsen MS, Larsen JK (eds) (2004) Applied mathematical models in human physiology. SIAM Monographs on Mathematical Modeling and Computation, SIAM, PhiladelphiaGoogle Scholar
  46. Palladino JL, Noordergraaf A (2002) A paradigm for quantifying ventricular contraction. Cell Mol Biol Lett 7(2):331–335Google Scholar
  47. Peskin CS (1981) Mathematical aspects of physiology. In: Hoppenstaedt F (ed) Mathematical aspects of physiology. Lectures in Applied Mathematics, vol 19. American Mathematical Society, Providence, pp 69–93Google Scholar
  48. Pfeiffer K, Kenner T (1981) On the optimal strategy of cardiac ejection. In: Kenner T, Busse R, Hinghofer-Szalkay H (eds) Cardiovascular system dynamics: models and measurements. Plenum Press, New York, pp 133–136Google Scholar
  49. Pope SR, Ellwein LM, Zapata C, Novak V, Kelley CT, Olufsen MS (2009) Estimation and identification of parameters in a lumped cerebrovascular model. Math Biosci Eng 6(1):93–115MathSciNetCrossRefzbMATHGoogle Scholar
  50. Rhoades RA, Tanner GA (eds) (2003) Medical physiology, 2nd edn. Medical Physiology (Rhoades) Series, Lippincott Willliams & Wilkins, PhiladelphiaGoogle Scholar
  51. Russell DL (1979) Mathematics of finite-dimensional control systems: theory and design. Marcel Dekker, New YorkzbMATHGoogle Scholar
  52. Sontag ED (1998) Mathematical control theory: deterministic finite dimensional systems. Textbooks in Applied Mathematics, vol 6, 2nd edn. Springer, New YorkCrossRefGoogle Scholar
  53. Sunagawa K, Sagawa K (1982) Models of ventricular contraction based on time-varying elastance. Crit Rev Biomed Eng 7(3):193–228Google Scholar
  54. Swan G (1984) Applications of optimal control theory in biomedicine. Marcel Dekker, New YorkzbMATHGoogle Scholar
  55. Timischl S (1998) A global model of the cardiovascular and respiratory system. PhD thesis, University of Graz, Institute for Mathematics and Scientific ComputingGoogle Scholar
  56. Todorov E, Jordan MI (2002) Optimal feedback control as a theory of motor coordination. Nat Neurosci 5(11):1226–1235CrossRefGoogle Scholar
  57. Todorov E, Li W (2003) Optimal control methods suitable for biomechanical systems. In: Proceedings of the 25th Annual International Conference of the IEEE Engineering in Biology and Medicine Society, vol 2, September 2003 Cancun, Mexico pp 1758–1761Google Scholar
  58. Ursino M (1998) Interaction between carotid baroregulation and the pulsating heart: a mathematical model. Am J Physiol 275(5):H1733–H1747Google Scholar
  59. Ursino M (1999) A mathematical model of the carotid baroregulation in pulsating conditions. IEEE Trans Biomed Eng 46(4):382–392CrossRefGoogle Scholar
  60. Ursino M, Fiorenzi A, Belardinelli E (1996) The role of pressure pulsatility in the carotid baroreflex control: a computer simulation study. Comput Biol Med 26(4):297–314CrossRefGoogle Scholar
  61. Westerhof N, Stergiopulos N, Noble MIM (2005) Snapshots of hemodynamics, basic science for the cardiologist, vol 18. Springer, New YorkGoogle Scholar
  62. Zabczyk J (2007) Mathematical control theory: an introduction. Modern Birkhäuser Classics, Birkhäuser, BaselGoogle Scholar

Copyright information

© Society for Mathematical Biology 2014

Authors and Affiliations

  • Aurelio A. de los ReyesV
    • 1
    • 2
    • 3
  • Eunok Jung
    • 1
    Email author
  • Franz Kappel
    • 4
  1. 1.Department of MathematicsKonkuk UniversitySeoulRepublic of Korea
  2. 2.Institute of MathematicsC.P. Garcia St., U.P. CampusQuezon CityPhilippines
  3. 3.Renal Research InstituteNew YorkUSA
  4. 4.Institute for Mathematics and Scientific ComputingUniversity of GrazGrazAustria

Personalised recommendations