Combined Transfer Function Analysis and Modelling of Cerebral Autoregulation

  • S. J. Payne
  • L. Tarassenko


The clinical importance of cerebral autoregulation has resulted in a significant body of literature that attempts both to model the underlying physiological processes and to estimate the mathematical relationships between clinically measurable variables, the most common of which are Arterial Blood Pressure and Cerebral Blood Flow Velocity. These approaches have, however, rarely been used together to interpret clinical data. A simple model of cerebral autoregulation is thus proposed here, based on a flow dependent feedback mechanism with gain and time constant that adjusts arterial compliance. Analysis of this model shows that it closely approximates a second order system for typical values of physiological parameters. The model parameters can be optimally estimated from available experimental data for the Impulse Response (IR), yielding physiologically reasonable values, although there is one free parameter that must be fixed. The effects of changes in feedback gain and time constant are found to be significant on the predicted IR and can thus be estimated robustly from experimental data. The effects of elevated baseline Intracranial Pressure (ICP) are found to be exactly equivalent to a reduced feedback gain, although the solution is much less sensitive to the former effect. A transfer function approach can be used to estimate autoregulation status clinically using a physiologically-based model, thus providing greater insight into the processes that govern cerebral autoregulation.


Impulse response Frequency response 



Stephen Payne was funded by the UK Research Councils Inter-Disciplinary Research Consortium (IRC) ‘Medical Images and Signals.’ Thanks are due to the other members of the IRC for profitable discussions and to the two anonymous reviewers for helpful suggestions.


  1. 1.
    Blaber, A. P., R. L. Bondar, F. Stein, P. T. Dunphy, P. Moradshahi, M. S. Kassam, and R. Freeman. Transfer function analysis of cerebral autoregulation dynamics in autonomic failure patients. Stroke 28:1686–1692, 1997.Google Scholar
  2. 2.
    Carey, B. J., P. J. Eames, M. J. Blake, R. B. Panerai, and J. F. Potter. Dynamic cerebral autoregulation is unaffected by aging. Stroke 31:2895–2900, 2000.Google Scholar
  3. 3.
    Carey, B. J., B. N. Manktelow, R. B. Panerai, and J. F. Potter. Cerebral autoregulatory responses to head-up tilt in normal subjects and patients with recurrent vasovagal syncope. Circulation 104:898–902, 2001.CrossRefGoogle Scholar
  4. 4.
    Chiu, C.-C., and S.-J. Yeh. Assessment of cerebral autoregulation using time-domain cross-correlation analysis. Comp. Bio. Med. 31:471–480, 2001.CrossRefGoogle Scholar
  5. 5.
    Chon, K. H., Y.-M. Chen, N.-H. Holstein-Rathlou, and V. Z. Marmarelis. Nonlinear system analysis of renal autoregulation in normotensive and hypertensive rats. IEEE Trans. Biomed. Eng. 45:342–353, 1998.CrossRefGoogle Scholar
  6. 6.
    Czosnyka, M., S. Piechnik, H. K. Richards, P. Kirkpatrick, P. Smielewski, and J. D. Pickard. Contribution of mathematical modeling to the interpretation of bedside tests of cerebrovascular autoregulation. J. Neurol. Neurosurg. Psychiatry 63:721–731, 1997.CrossRefGoogle Scholar
  7. 7.
    Dorf, R. C., and R. H. Bishop. Modern Control Systems. Prentice-Hall, 2000.Google Scholar
  8. 8.
    Eames, P. J., M. J. Blake, S. L. Dawson, R. B. Panerai, and J. F. Potter. Dynamic cerebral autoregulation and beat to beat blood pressure control are impaired in acute ischaemic stroke. J. Neurol. Neurosurg. Psychiatry 72:467–473, 2002.Google Scholar
  9. 9.
    Giller, C. A., and M. Mueller. Linearity and non-linearity in cerebral haemodynamics. Med. Eng. Phys. 25:633–646, 2003.CrossRefGoogle Scholar
  10. 10.
    Kirkham, S. K., R. E. Craine, and A. A. Birch. A new mathematical model of dynamic cerebral autoregulation based on a flow dependent feedback mechanism. Physiol. Meas. 22:461–473, 2001.CrossRefGoogle Scholar
  11. 11.
    Lee, S. P., T. Q. Duong, G. Yang, C. Iadecola, and S. G. Kim. Relative changes of cerebral arterial and venous blood volumes during increased cerebral blood flow: Implications for BOLD fMRI. Magn. Reson. Med. 45:791–800, 2001.CrossRefGoogle Scholar
  12. 12.
    Liu, Y., A. A. Birch, and R. Allen. Dynamic cerebral autoregulation assessment using an ARX model:comparative study using step response and phase shift analysis. Med. Eng. Phys. 25:647–653, 2003.CrossRefGoogle Scholar
  13. 13.
    Mitsis, G. D., and V. Z. Marmarelis. Modeling of nonlinear physiological systems with fast and slow dynamics: I. Methodology. Ann. Biomed. Eng. 30:272–281, 2002a.CrossRefGoogle Scholar
  14. 14.
    Mitsis, G. D., R. Zhang, B. D. Levine, and V. Z. Marmarelis. Modeling of nonlinear physiological systems with fast and slow dynamics: II. Application to cerebral autoregulation. Ann. Biomed. Eng. 30:555–565, 2002b.CrossRefGoogle Scholar
  15. 15.
    Myers, C. W., M. A. Cohen, D. L. Eckberg, and J. A. Taylor. A model for the genesis of arterial pressure Mayer waves from heart rate and sympathetic activity. Auton. Neurosci. 91:62–75, 2001.CrossRefGoogle Scholar
  16. 16.
    Narayanan, K., J. J. Collins, J. Hamner, S. Mukai, and L. A. Lipsitz. Predicting cerebral blood flow response to orthostatic stress from resting dynamics: Effects of healthy aging. Am. J. Physiol. 281:R716–722, 2001.Google Scholar
  17. 17.
    Olufsen, M. S., A. Nadim, and L. A. Lipsitz. Dynamics of cerebral blood flow regulation explained using a lumped parameter model. Am. J. Physiol. 282:R611–622, 2002.Google Scholar
  18. 18.
    Panerai, R. B. System identification of human cerebral blood flow regulatory mechanisms. Cardiovascular Eng. 4:59–71, 2004.CrossRefGoogle Scholar
  19. 19.
    Panerai, R. B., S. L. Dawson, and J. F. Potter. Linear and nonlinear analysis of human dynamic cerebral autoregulation. Am. J. Physiol. 277:H1089–H1099, 1999.PubMedGoogle Scholar
  20. 20.
    Panerai, R. B., D. M. Simpson, S. T. Deverson, P. Mahony, P. Hayes, and D. H. Evans. Multivariate dynamic analysis of cerebral blood flow regulation in humans. IEEE Trans. Biomed. Eng. 47:419–423, 2000.CrossRefGoogle Scholar
  21. 21.
    Panerai, R. B., S. L. Dawson, P. J. Eames, and J. F. Potter. Cerebral blood flow velocity response to induced and spontaneous sudden changes in arterial blood pressure. Am. J. Physiol. 280:H2162–H2174, 2001.Google Scholar
  22. 22.
    Panerai, R. B., P. J. Eames, and J. F. Potter. Variability of time-domain indices of dynamic cerebral autoregulation. Physiol. Meas. 24:367–381, 2003.CrossRefGoogle Scholar
  23. 23.
    Reinhard, M., T. Muller, B. Guschlbauer, J. Timmer, and A. Hetzel. Transfer function analysis for clinical evaluation of dynamic cerebral autoregulation—a comparison between spontaneous and respiratory-induced oscillations. Physiol. Meas. 24:27–43, 2003.CrossRefGoogle Scholar
  24. 24.
    Schondorf, R., R. Stein, R. Roberts, J. Benoit, and W. Cupples. Dynamic cerebral autoregulation is preserved in neurally mediated syncope. J. Appl. Physiol. 91:2493–2502, 2001.Google Scholar
  25. 25.
    Tiecks, F. P., A. M. Lam, R. Aaslid, and D. W. Newell. Comparison of static and dynamic cerebral autoregulation measurements. Stroke 26:1014–1019, 1995.Google Scholar
  26. 26.
    Ursino, M., and M. Giulioni. Quantitative assessment of cerebral autoregulation from transcranial Doppler pulsatility: A computer simulation study. Med. Eng. Phys. 25:655–666, 2003.CrossRefGoogle Scholar
  27. 27.
    Ursino, M., and C. A. Lodi. A simple mathematical model of the interaction between intracranial pressure and cerebral hemodynamics. J. Appl. Physiol. 82:1256–1269, 1997.Google Scholar
  28. 28.
    Ursino, M., and C. A. Lodi. Interaction among autoregulation, CO2 reactivity, and intracranial pressure: a mathematical model. Am. J. Physiol. 274:H1715–H1728, 1998.PubMedGoogle Scholar
  29. 29.
    Ursino, M., A. Ter Minassian, C. A. Lodi, and L. Beydon. Cerebral hemodynamics during arterial and CO2 pressure changes: in vivo prediction by a mathematical model. Am. J. Physiol. 279:H2439–2455, 2000.Google Scholar
  30. 30.
    Vespa, P. What is the optimal threshold for cerebral perfusion pressure following traumatic brain injury? Neurosurg. Focus 15:Article 4, 2003.CrossRefGoogle Scholar
  31. 31.
    Zhang, R., J. H. Zuckerman, C. A. Giller, and B. D. Levine. Transfer function analysis of dynamic cerebral autoregulation in humans. Am. J. Physiol. 274:H233–H241, 1998.PubMedGoogle Scholar
  32. 32.
    Zhang, R., J. H. Zuckerman, and B. D. Levine. Spontaneous fluctuations in cerebral blood flow: insights from extended-duration recordings in humans. Am. J. Physiol. 278:H1848–H1855, 2000.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • S. J. Payne
    • 1
  • L. Tarassenko
    • 1
  1. 1.Department of Engineering ScienceUniversity of OxfordOxfordUK

Personalised recommendations