In a previous study, we have demonstrated that turbulences in arteries less than 1.5 mm in diameter perfused ex vivo by an oscillated flow can be determined using nonlinear mathematical models. The hypothesis tested here was that nonlinear analyses enable the in vivo detection of turbulences in blood flow in vessels with dimensions affected by microsurgery. Twenty Wistar male rats were studied. After an intraperitoneal anesthesia (50 mg/kg sodium pentobarbital), each right carotid artery was dissected and a transit time flowmeter was used to measure blood flow. Arteries were studied in control conditions and graded stenoses, which were performed by a collar system and progressively increased from 0 to 95%. For each flow signal, time delay, false-nearest neighbors, correlation dimension and the largest Lyapunov exponent were computed to characterize the level of turbulence. The level of turbulence was highly correlated with the degree of stenosis induced (p < 0.001). The correlation dimensions of all the flow signals varied between 3 and 5, thus implying that more than three independent noninteger control variables were necessary to account for the complexity of rat carotid artery dynamics. Turbulence flow significantly increased in arteries with 40–95% stenosis. Nonlinear analysis may be useful to determine the turbulence level of an in vivo flow in arteries less than 1.5 mm in diameter and might be clinical useful for turbulence detection after microsurgery.

Keywords:
Artery, Chaos, Correlation dimension, Hemodynamics, In vivo study, Lyapunov exponent, Microsurgery, Nonlinear analysis, Stenosis, Turbulence
1.
Johnson PC, Dickson CS, Garrett KO, Sheppeck RA, Bentz ML: The effect of microvascular anastomosis configuration on initial platelet deposition. Plast Reconstr Surg 1993;91:522–527.
2.
Linden M, Sirsjo A, Lindbom L, Nilsson G, Gidlof A: Laser Doppler perfusion imaging of microvascular blood flow in rabbit tenuissimus muscle. Am J Physiol 1995;269:H1496–H1500.
3.
Moskovitz MJ, Zhang L, Baron DA, Kanner BJ, Tuchler RE, Siebert JW: Transient postoperative stenosis in small-vessel anastomoses. Ann Plast Surg 1995;34:309–317.
4.
Asada Y, Sumiyoshi A: Histopathology of thrombotic vascular diseases. Nippon Rinsho 1999;57:1555–1560.
5.
Cloutier G, Allard L, Durand LG: Changes in ultrasonic Doppler backscattered power downstream of concentric and eccentric stenoses under pulsatile flow. Ultrasound Med Biol 1995;21:59–70.
6.
Padet J: Fluides en écoulement. Paris, Masson, 1991, pp 99–279.
7.
Ruelle D, Takens F: On the nature of turbulence. Commun Math Phys 1971;20:167–192, 1971;23:343–344.
8.
Ruelle D: Strange Attractors. Math Intelligencer 1980;2:126–137.
9.
Kantz H, Schreiber T: Nonlinear Time Series Analysis. London, Cambridge University Press, 1997, pp 29–171. http://www.mpipks-dresden.mpg.de/~tisean.
10.
Manneville P: Structures dissipatives chaos et turbulences. Gif-sur-Yvettes, Aléa-Saclay, 1990, pp 1–361.
11.
Pedley TJ: Nonlinear pulse wave reflection at an arterial stenosis. J Biomech Eng 1983;105:353–359.
12.
Roffeh Y, Einav S, Liaw J, Whiting J, Keren G: Cepstrum analysis of reflected pressure waves in stenosed arteries. Med Biol Eng Comput 1996;34:175–180.
13.
Sipkema P, Westerhof N: Mechanics of a thin walled collapsible microtube. Ann Biomed Eng 1989;17:203–217.
14.
Jensen OE: Chaotic oscillations in a simple collapsible-tube model. J Biomech Eng 1992;114:55–59.
15.
Groisman A, Steinberg V: Elastic turbulence in a polymer solution flow. Nature 2000;405:53–55.
16.
Grassberger P, Procaccia I: Characterization of strange attractors. Phys Rev Lett 1983;50:346–349.
17.
Wolf A, Swift JB, Swinney HL, Vastano JA: Determining Lyapunov exponents from a time series. Physica D 1985;16:285–317.
18.
Glass L, Kaplan D: Time series analysis of complex dynamics in physiology and medicine. Med Prog Technol 1993;19:115–128.
19.
Kaplan D, Glass L: Understanding Nonlinear Dynamics. New York, Springer,1995, pp 279–338.
20.
Griffith TM: Temporal chaos in the microcirculation. Cardiovasc Res 1996;31:342–358.
21.
Griffith TM, Edwards DH: Modulation of chaotic pressure oscillations in isolated resistance arteries by EDRF. Eur Heart J 1993;14:60–67.
22.
Wagner CD, Persson PB: Nonlinear chaotic dynamics of arterial blood pressure and renal blood flow. Am J Physiol 1995;268:H621–H627.
23.
Wagner CD, Nafz B, Persson PB: Chaos in blood pressure control. Cardiovasc Res 1996;31:380–387.
24.
May P, Gerbault O, Arrouvel C, Revol M, Servant JM, Vicaut E: Nonlinear analysis of arterial oscillated flow in experimental stenosis and microsurgical anastomosis. J Surg Res 2001;99:53–60.
25.
Labadie RF, Antaki JF, Williams JL, Katyal S, Ligush J, Watkins SC, Pham SM, Borovetz HS: Pulsatile perfusion system for ex vivo investigation of biochemical pathways in intact vascular tissue. Am J Physiol 1996;270:H760–H768.
26.
D’Almeida MS, Gaudin C, Lebrec D: Validation of 1- and 2-mm transit-time ultrasound flow probes on mesenteric artery and aorta of rats. Am J Physiol 1995;268:H1368–H1372.
27.
Tuchler RE, Zhang L, Siebert JW, Shaw WW: Simultaneous comparison of pre- and post-microanastomotic hemodynamic profiles using a tandem Doppler probe. Microsurgery 1991;12:35–42.
28.
Welch WJ, Deng X, Snellen H, Wilcox CS: Validation of miniature ultrasonic transit-time flow probes for measurement of renal blood flow in rats. Am J Physiol 1995;268:F175–F178.
29.
Ifeachor EC, Jervis BW: Digital Signal Processing. A Practical Approach. London, Addison & Wesley, 1994, p 56.
30.
Takens F: Detecting Strange Attractors in Turbulence. Lecture Notes in Mathematics. New York, Springer, 1981, vol 898, pp 366–381.
31.
Kennel MB, Brown R, Abarbanel HDI: Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys Rev A 1992;45:3403.
32.
Abarbanel HDI: Analysis of Observed Chaotic Data. New York, Springer, 1996.
33.
Baker GL, Gollub JP, Blackburn JA: Inverting chaos: Extracting system parameters from experimental data. Chaos 1996;6:528–533.
34.
Theiler J, Eubank S, Longtin A, Galdrikian B, Farmer JD: Testing for nonlinearity in time series: The method of surrogate data. Physica D 1992;58:77–94.
35.
Parthimos D, Edwards DH, Griffith TM: Comparison of chaotic and sinusoidal vasomotion in the regulation of microvascular flow. Cardiovasc Res 1996;31:388–399.
36.
Griffith TM, Edwards DH: Integration of non-linear cellular mechanisms regulating microvascular perfusion. Proc Inst Mech Eng [H] 1999;213:369–383.
37.
Bräuer K, Hahn M: Nonlinear analysis of blood flux in human vessels. Phys Med Biol 1999;44:1719–1733.
38.
Garfinkel A, Spano ML, Ditto WL, Weiss JN: Controlling cardiac chaos. Science 1992;257:1230–1235.
39.
Persson PB, Wagner CD: General principles of chaotic dynamics. Cardiovasc Res 1996;31:332–341.
© 2002 S. Karger AG, Basel
2002
Copyright / Drug Dosage / Disclaimer
Copyright: All rights reserved. No part of this publication may be translated into other languages, reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, microcopying, or by any information storage and retrieval system, without permission in writing from the publisher.
Drug Dosage: The authors and the publisher have exerted every effort to ensure that drug selection and dosage set forth in this text are in accord with current recommendations and practice at the time of publication. However, in view of ongoing research, changes in government regulations, and the constant flow of information relating to drug therapy and drug reactions, the reader is urged to check the package insert for each drug for any changes in indications and dosage and for added warnings and precautions. This is particularly important when the recommended agent is a new and/or infrequently employed drug.
Disclaimer: The statements, opinions and data contained in this publication are solely those of the individual authors and contributors and not of the publishers and the editor(s). The appearance of advertisements or/and product references in the publication is not a warranty, endorsement, or approval of the products or services advertised or of their effectiveness, quality or safety. The publisher and the editor(s) disclaim responsibility for any injury to persons or property resulting from any ideas, methods, instructions or products referred to in the content or advertisements.
You do not currently have access to this content.