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A mathematical model of the relationship between cerebral blood volume and intracranial pressure changes: The generation of plateau waves

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Abstract

The relationship between intracranial pressure (ICP), cerebral blood volume (CBV), cerebrospinal fluid dynamics, and the action of cerebral blood-flow (CBF) regulatory mechanisms is examined in this work with the help of an original mathematical model. In building the model, particular emphasis is placed on reproducing the mechanical properties of proximal cerebral arteries and small pial arterioles, and their active regulatory response to perfusion pressure and cerebral blood flow changes.

The model allows experimental results on cerebral vessel dilatation and cerebral blood-flow regulation, following cerebral perfusion pressure decrease, to be satisfactorily reproduced. Moreover, the effect of cerebral blood volume changes—induced by autoregulatory adjustments — on the intracranial pressure time pattern can be examined at different levels of arterial hypotension.

The results obtained with normal parameter values demonstrate that, at the lower lumits of autoregulation, when dilatation of small arterioles becomes maximal, the increase in cerebral blood volume can cause a significant, transient increase in intracranial pressure. This antagonism between intracranial pressure and autoregulatory adjustments can lead to instability of the intracranial system in pathological conditions. In particular, analysis of the linearized system “in the small” demonstrates that an impairment in cerebrospinal fluid (CSF) reabsorption, a decrease in intracranial compliance and a high-regulatory capacity of the cerebrovascular bed are all conditions which can lead the system equilibrium to become unstable (i.e., the real part of at least one eigenvalue to turn out positive). Accordingly, mathematical simulation “in the large,” in the above-mentioned conditions, exhibits intracranial pressure periodic fluctuations which closely resemble, in amplitude, duration, frequency and shape, the well-known Lundberg A-waves (or plateau waves).

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Abbreviations

r j :

average inner radius

h j :

wall thickness

G j,R j :

hydraulic conductance and resistance, respectively

V j :

arterial blood volume

P j :

average intravascular pressure

T ej,T mj,T vj, andT j :

elastic, muscular, viscous, and total wall tension, respectively

σ ej , ε ej :

elastic wall stress and wall strain

r oj ,h oj :

unstressed inner radius and unstressed wall thickness

r mj :

value of inner radius at which smooth muscle exerts its optimal tension

T m0j :

optimal smooth muscle tension in absence of feedback regulatory actions

M j :

activation factor, synthetizing the effect of regulatory actions on smooth muscle tension

η j :

low frequency wall viscosity

G pv :

hydraulic conductance of proximal cerebral veins

G vs :

hydraulic conductance of the distal venous cerebrovascular bed (lateral lacunae and bridge veins)

G f :

hydraulic conductance to CSF formation

G o :

hydraulic conductance to CSF outflow

C ic :

intracranial tissue compliance

C vi :

cerebral vein compliance

C ve :

extracranial vein compliance

P a :

mean arterial pressure

P c :

mean cerebral capillary pressure

P v :

mean cerebral venous pressure

P ic :

mean intracranial pressure

P vs :

mean sinus venous pressure

P cv :

central venous pressure

K E :

intracranial elastance coefficient

1/P ref :

gain factor of the feedback pressure-dependent regulatory mechanism

1/q ref :

gain factor of the feedback flow-dependent regulatory mechanism

τ j :

time constant of the feedback regulatory mechanism

q :

cerebral blood flow

References

  1. Auer, L.M.; Ishiyama, N.; Hodde, K.C.; Kleinert, R.; Pucher, R. Effect of intracranial pressure on bridging veins in rats. J. Neurosurg. 6:263–268; 1987.

    Google Scholar 

  2. Auer, L.M.; Ishiyama, N.; Pucher, R. Cerebrovascular response to intracranial hypertension. Acta Neurochir. 84:124–128; 1987.

    Article  CAS  Google Scholar 

  3. Auer, L.M.; MacKenzie, E.T. Physiology of the cerebral venous system. In: Kapp, J.P.; Schmidek, H.H., eds. The cerebral venous system and its disorders. Orlando: Grune & Stratton; 1984:pp. 169–227.

    Google Scholar 

  4. Avezaat, C.J.J.; van Eijndhoven, J.H.M. The conflict between CSF pulse pressure and volume-pressure response during plateau waves. In: Ishii, S.; Nagai, H.; Brock, M., eds. Intracranial pressure V. Berlin-Heidelberg: Springer-Verlag; 1983:pp. 326–332.

    Google Scholar 

  5. Bergel, D.H. The dynamic elastic properties of the arterial wall. J. Physiol. 156:458–469; 1961.

    Google Scholar 

  6. Dillon, P.F.; Aksoy, M.O.; Driska, S.P.; Murphy, R.A. Myosin phosphorylation and the cross-bridge cycle in arterial smooth muscle. Science 211:495–497; 1981.

    CAS  PubMed  Google Scholar 

  7. Dobrin, P.B.; Rovick, A.A. Influence of vascular smooth muscle on contractile mechanics and elasticity of arteries. Am. J. Physiol. 217:1644–1652; 1969.

    CAS  PubMed  Google Scholar 

  8. van Eijndhoven, J.H.M.; Sliwka, S.; Avezaat, C.J.J. The constant pressure term (P o) of the volume-pressure relationship. Comparison between results of infusion test and pulse pressure analysis. In: Miller, J.D.; Teasdale, G.M.; Rowan, J.O.; Galbraith, S.L.; Mendelow, A.D., eds. Intracranial pressure VI. Berlin-Heidelberg: Springer-Verlag; 1986:pp. 48–53.

    Google Scholar 

  9. Gjerris, F.; Borgesen, S.E.; Hoppe, E.; Boesen, F.; Nordenbo, A.M. The conductance to outflow of CSF in adults with high pressure hydrocephalus. Acta Neurochir. 64:59–67; 1982.

    Article  CAS  Google Scholar 

  10. Gore, R.W.; Davis, M.J. Mechanics of smooth muscle in isolated single microvessels. Ann. Biomed. Eng. 12:511–520; 1984.

    Article  CAS  PubMed  Google Scholar 

  11. Grubb, R.L.; Raichle, M.E.; Higgins, C.S. Measurement of regional cerebral blood volume by emission tomography. Ann. Neurol. 4:322–328; 1978.

    Article  PubMed  Google Scholar 

  12. Hansen, K.; Gjerris, F.; Sorensen, P.S. Absence of hydrocephalus in spite of impaired cerebrospinal fluid absorption and severe intracranial hypertension. Acta Neurochir. 86:93–97; 1987.

    Article  CAS  Google Scholar 

  13. Harper, A.M.; Deshmukh, V.D.; Rowan, J.O.; Jennett, W.B. The influence of sympathetic nervous activity on cerebral blood flow. Arch. Neurol. 27:1–6; 1972.

    CAS  PubMed  Google Scholar 

  14. Hayashi, K.; Honda, H.; Nagasawa, S.; Okumura, A.; Moritake, K. Stiffness and elastic behavior of human intracranial and extracranial arteries. J. Biomech. 13:175–184; 1980.

    Article  CAS  PubMed  Google Scholar 

  15. Hayashi, M.; Kobayashi, H.; Handa, Y.; Kawano, H.; Ishii, H. Cerebrospinal fluid dynamics in patients with plateau waves. In: Miller, J.D.; Teasdale, G.M., Rowan, J.O.; Galbraith, S.L.; Mendelow, A.D., eds. Intracranial pressure VI. Berlin-Heidelberg: Springer-Verlag; 1986:pp. 305–309.

    Google Scholar 

  16. Heistad, D.D.; Marcus, M.L.; Abboud, F.m. Role of large arteries in regulation of cerebral blood flow in dogs. J. Clin. Invest. 62:761–768; 1978.

    CAS  PubMed  Google Scholar 

  17. Hoffman, O. Biomathematics of intracranial CSF and haemodynamics. Simulation and analysis with the aid of a mathematical model. Acta Neurochir. Suppl. 40:117–130; 1987.

    Google Scholar 

  18. Hogestatt, E.D.; Anderson, K.E.; Edvinsson, L. Mechanical properties of rat cerebral arteries as studied by a sensitive device for recording of mechanical activity in isolated small blood vessels. Acta Physiol. Scand. 117:49–61; 1983.

    CAS  PubMed  Google Scholar 

  19. Kety, S.S.; Schmidt, C.F. The effect of altered arterial tension of carbon dioxide and oxygen on cerebral blood flow and cerebral oxygen consumption of normal young man. J. Clin. Invest. 27:484–492; 1948.

    CAS  Google Scholar 

  20. Kontos, H.A.; Wei, E.P.; Navari, R.m.; Levasseur, J.E.; Rosenblum, W.I.; Patterson, J.L. Responses of cerebral arteries and arterioles to acute hypotension and hypertension. Am. J. Physiol. 234:H371-H383; 1978.

    CAS  PubMed  Google Scholar 

  21. Kosteljanetz, M. CSF dynamics in patients with subarachnoid and/or intraventricular hemorrhage. J. Neurosurg. 60:940–946; 1984.

    CAS  PubMed  Google Scholar 

  22. Lundberg, N. Continuous recording and control of ventricular fluid pressure in neurosurgical practice. Acta Psychiat. 36:1–193; 1960.

    Google Scholar 

  23. Lundberg, N.; Cronquist, S.; Kjallquist, A. Clinical investigation on interrelations between intracranial pressure and intracranial hemodynamics. Prog. Brain Res. 30:69–75; 1968.

    CAS  PubMed  Google Scholar 

  24. MacKenzie, E.T.; Farrar, J.K.; Fitch, W.; Graham, D.; Gregory, P.C.; Harper, A.M. Effects of hemorrhagic hypotension on the cerebral circulation: I cerebral blood flow and pial arteriolar caliber. Stroke 10:711–718; 1979.

    CAS  PubMed  Google Scholar 

  25. Marmarou, A.; Schulman, K.; La Morgese, J. Compartmental analysis of compliance and outflow resistance of the cerebrospinal fluid system. J. Neurosurg. 43:523–534; 1975.

    CAS  PubMed  Google Scholar 

  26. Matsuda, M.; Yoneda, S.; Handa, H.; Gotoh, H. Cerebral hemodynamics changes during plateau waves in brain tumor patients. J. Neurosurg. 50:483–488; 1979.

    CAS  PubMed  Google Scholar 

  27. Mchedlishvili, G. Arterial behavior and blood circulation in the brain. New York, London: Plenum Press; 1986.

    Google Scholar 

  28. Minorsky, N. Nonlinear oscillations. Princeton, NJ: Van Nostrand; 1962.

    Google Scholar 

  29. Moré, J.; Garbow, B.; Hillstrom, K. User guide for MINPACK-1. Argonne National Labs. report ANL-80-74, Argonne, IL, 1980.

  30. Murphy, R.A. Estimates of the dynamical mechanical properties of arterial smooth muscle cells in vascular strip. Abstract. Ann. Biom. Eng. 11:54; 1983.

    Google Scholar 

  31. Nakagawa, Y.; Tsuru, M.; Yada, K. Site and mechanism for compression of the venous system during experimental intracranial hypertension. J. Neurosurg. 41:427–434; 1974.

    CAS  PubMed  Google Scholar 

  32. Nordborg, C.; Fredriksson, K.; Johansson, B.B. The morphometry of consecutive segments in cerebral arteries of normotensive and spontaneously hypertensive rats. Stroke 16:313–319; 1985.

    CAS  PubMed  Google Scholar 

  33. Nyary, I.; Vajda, J. Relationship of cerebral blood volume changes and estimated intracranial compliance. In: Ishii, S.; Nagai, H.; Brock, M., eds. Intracranial pressure V. Berlin-Heidelberg: Springer-Verlag; 1983:pp. 316–319.

    Google Scholar 

  34. Nylin, G.; Heldlund, S.; Regnstrom, O. Studies of the cerebral circulation with labelled erythrocytes in healthy man. Circ. Res. 9:664–674; 1961.

    CAS  PubMed  Google Scholar 

  35. Poincaré, H. Les méthodes nouvelles de la mécanique céleste. Paris: Gauthier-Villars; 1892.

    Google Scholar 

  36. Powell, M.J.D. An efficient method for finding the minimum of a function of several variables without calculating derivatives. Computer J. 6:163; 1963.

    Google Scholar 

  37. Risberg, J.; Lundberg, N.; Ingvar, D.H. Regional cerebral blood volume during acute transient rises of the intracranial pressure (plateau waves). J. Neurosurg. 31:303–310; 1969.

    CAS  PubMed  Google Scholar 

  38. Rosner, M.J. Cerebral perfusion pressure: Link between intracranial pressure and systemic circulation. In: Wood, J.H., ed. Cerebral blood flow. New York: McGraw Hill; 1987:pp. 425–448.

    Google Scholar 

  39. Rosner, M.J.; Becker, D.P. Origin and evolution of plateau waves. Experimental observations and a theoretical model. J. Neurosurg. 60:312–324; 1984.

    CAS  PubMed  Google Scholar 

  40. Smith, B.T.; Boyle, J.M.; Dongarra, J.J.; Garbow, B.S.; Ikebe, Y.; Klema, V.C.; Moler, C.B. Matrix eigensystem routines, EISPACK guide. New York: Springer-Verlag; 1976.

    Google Scholar 

  41. Sorek, S.; Bear, J.; Karni, Z. Resistances and compliances of a compartmental model of the cerebrovascular system. Ann. Biom. Eng. 17:1–12; 1989.

    CAS  Google Scholar 

  42. Takemae, T.; Kosugi, Y.; Ikebe, J.; Kumagai, Y.; Matsuyama, K.; Saito, H. A simulation study of intracranial pressure increment using an electric circuit model of cerebral circulation. IEEE Trans. Biom. Eng. 34:958–962; 1987.

    CAS  Google Scholar 

  43. Ursino, M. A mathematical study of human intracranial hydrodynamics. Part 1: The cerebrospinal fluid pulse pressure. Ann. Biomed. Eng. 16:379–401; 1988.

    CAS  PubMed  Google Scholar 

  44. Ursino, M. A mathematical study of human intracranial hydrodynamics. Part 2: Simulation of clinical tests. Ann. Biomed. Eng. 16:403–416; 1988.

    CAS  PubMed  Google Scholar 

  45. Ursino, M.; Di Giammarco, P.; Belardinelli, E. A mathematical model of cerebral blood flow chemical regulation. Part I: Diffusion processes. IEEE Trans. Biom. Eng. 36:183–191; 1989.

    CAS  Google Scholar 

  46. Ursino, M.; Di Giammarco, P.; Belardinelli, E. A mathematical model of cerebral blood flow chemical regulation. Part II: Reactivity of cerebral vascular bed. IEEE Trans. Biom. Eng. 36:192–201; 1989.

    CAS  Google Scholar 

  47. Wagner, E.M.; Traystman, R.J. Cerebrovascular transmural pressure and autoregulation. Ann. Biomed. Eng. 13:311–320; 1985.

    CAS  PubMed  Google Scholar 

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  1. Department of Electronics, Computer Science, and Systems, University of Bologna, Viale Risorgimento 2, I-40136, Bologna, Italy

    Mauro Ursino & Patrizia Di Giammarco

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  1. Mauro Ursino
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  2. Patrizia Di Giammarco
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Ursino, M., Di Giammarco, P. A mathematical model of the relationship between cerebral blood volume and intracranial pressure changes: The generation of plateau waves. Ann Biomed Eng 19, 15–42 (1991). https://doi.org/10.1007/BF02368459

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  • DOI: https://doi.org/10.1007/BF02368459

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