A new Rayleigh-type instability is proposed to explain the `sausage-string'
pattern of alternating constrictions and dilatations formed in blood vessels
under influence of a vasoconstricting agent. Our theory involves the nonlinear
elasticity characteristics of the vessel wall, and provides predictions for the
conditions under which the cylindrical form of a blood vessel becomes unstable.Comment: 4 pages, 4 figures submitted to Physical Review Letter

A. E. Green

F. B. Byrom

F. Gustafsson

Finn Gustafsson

J. Giese

J. Plateau

L. J. Beilin

Lord Rayleigh

M. J. Davis

Morten Colding-Jørgensen

Niels-Henrik Holstein-Rathlou

P. Nelson

Preben Alstrøm

R. Bar-Ziv

R. E. Goldstein

R. Feldberg

R. S. Chadwick

R. W. Gore

Victor M. Eguíluz

Y. C. Fung

English

arXiv

Crossref

Physical Review Letters

Instability and “Sausage-String” Appearance in Blood Vessels during High Blood Pressure

PACS numbers: 87.19.Rr, 02.30.Jr, 47.20.DrA new Rayleigh-type instability is proposed to explain the “sausage-string” pattern of alternating
constrictions and dilatations formed in blood vessels under inﬂuence of a vasoconstricting agent.
Our theory involves the nonlinear elasticity characteristics of the vessel wall, and provides
predictions for the conditions under which the cylindrical form of a blood vessel becomes unstable.The present study was supported by grants from the
Danish Natural Science Research Council, the Danish Medical Research Council, the Novo-Nordisk Foundation,
and the Danish Heart Association.Peer reviewe

Alstrøm, P.

Eguíluz, Víctor M.

Holstein-Rathlou, Niels-Henrik

Digital.CSIC

VOLUME 82, NUMBER 9 PHY S I CA L REV I EW LE T T ER S 1 MARCH 1999Instability and “Sausage-String” Appearance in Blood Vessels during High Blood PressurePreben Alstrøm,1 Victor M. Eguı´luz,2 Morten Colding-Jørgensen,3Finn Gustafsson,4 and Niels-Henrik Holstein-Rathlou41CATS, The Niels Bohr Institute, DK-2100 Copenhagen, Denmark2Instituto Mediterráneo de Estudios Avanzados IMEDEA (CSIC-UIB), E-07071 Palma de Mallorca, Spain3Novo Nordisk, DK-2880 Bagsvaerd, Denmark4Department of Medical Physiology, The Panum Institute, DK-2200 Copenhagen, Denmark(Received 1 October 1998)A new Rayleigh-type instability is proposed to explain the “sausage-string” pattern of alternatingconstrictions and dilatations formed in blood vessels under influence of a vasoconstricting agent.Our theory involves the nonlinear elasticity characteristics of the vessel wall, and providespredictions for the conditions under which the cylindrical form of a blood vessel becomes unstable.[S0031-9007(99)08585-3]PACS numbers: 87.19.Rr, 02.30.Jr, 47.20.DrHigh blood pressure can experimentally be induced byintravenous infusion of a vasoconstricting agent such as an-giotensin II (which regulates the contraction of the smoothmuscle cells surrounding the blood vessel) [1–3]. Asthe infusion is continued, a substantial narrowing of thesmaller blood vessels is observed, and suddenly the nar-rowed vessels develop a peculiar pattern consisting ofalternating regions of constrictions and dilatations, giv-ing the vessels the appearance of sausages on a string(Fig. 1). The “sausage-string” pattern may cause severedamage to the blood vessels because plasma and macro-molecules are transported into the vessel wall in the dilatedregions. The sausage-string pattern has been observed insmall vessels from many organs, including the brain, thegut, and the kidney [4].Despite several decades of research, the mechanismcausing the sausage-string pattern has remained unknown[4]. It has been suggested that it represents a “blowout”of the vessel wall due to the high blood pressure [5], butthis seems unlikely for several reasons. The sausage-stringpattern occurs in the smaller vessels (small arteries andlarge arterioles), and here the pressure elevation is rela-tively small compared to that in the larger arteries. Second,the phenomenon is highly reproducible [2]. If the infusionof the vasoconstricting agent is stopped, the normal, uni-form “cylindrical” structure is restored. Restoring the in-fusion causes again an extreme, uniform vasoconstrictionfollowed by the reappearance of the sausage-string pattern.A third spectacular feature of the phenomenon is its pe-riodicity with constrictions and dilatations occurring in aregular and repetitive pattern.In this Letter, we present a simple anisotropic, elasticmodel of the vessel wall. We show that under certain hy-pertensive conditions an instability occurs which leads toa periodic pattern of constrictions and dilatations along thevessel. Our theory provides predictions for the conditionsunder which the cylindrical form of a blood vessel becomesunstable.0031-9007y99y82(9)y1995(4)$15.00To be specific, a cylindrical shaped blood vessel isunstable if a small axial symmetric perturbation of theinner radius, r ! r 1 uszd, grows (Fig. 2). To determinethe stability, we must therefore know the dynamic equationfor the perturbation usz, td. To this end, we invoke thecontinuity equation, ›tspr2d ­ 2›zJ, associating a localchange of the cross-sectional area at a downstream site zwith a fluid flux Jszd. The flux is related to the transmuralpressure P, by J ­ 2csrd›zP, where csrd is the vascularconductance [6]. From the continuity equation and theflux-pressure relation, the dynamic equation to lowestorder in the perturbation follows:›tu ­csrd2pr›2zP . (1)As a simple illustration, consider first a very thin vesselwall, for which the pressure is given by the Laplace formFIG. 1. In vivo micrograph of rat intestinal arterioles show-ing a typical “sausage-string” pattern following an acute in-crease in blood pressure induced by intravenous infusion ofangiotensin II. The neighboring vessels not showing constric-tions and dilatations are the corresponding venules. From [4]with permission.© 1999 The American Physical Society 1995VOLUME 82, NUMBER 9 PHY S I CA L REV I EW LE T T ER S 1 MARCH 1999Zu(z)rw(z)FIG. 2. A schematic picture of a blood vessel of inner radiusr undergoing a perturbation uszd. The wall thickness wszd islarger at smaller radii since the circumference is smaller.[7,8] P ­ sTyRd 1 sTzyRzd, where T and Tz are thetensions in the principal angular and vessel directions,and 1yR and 1yRz are the corresponding curvatures [9].Assume furthermore that the tensions are constant andidentical, Tz ­ T . Inserting the above expression for thepressure into Eq. (1) and retaining only first order terms inu, we get›tu ­ 2Tcsrd2pr3f›2zu 1 r2›4zug . (2)For a given periodic perturbation, u ­ ukstd cosskzd, wehave ukstd , uks0delk t , wherelk ­Tcsrd2pr3k2f1 2 r2k2g . (3)Thus, the vessel wall is unstable to modes with rk , 1.The dominant mode, where lk is maximal, is atk ­ 1yp2 r .The above instability is the well-known Rayleigh insta-bility [10,11]. The theory explains why a cylindrical col-umn of water with surface tension T is unstable at all radii.However, cylindrical structures may be stable due to a re-luctance against bending [12]. Still an instability may oc-cur if the tension T can be brought to exceed a criticalvalue of order kyr2, k being the bending modulus [12].This is demonstrated by the so-called “pearling” instabil-ity, recently observed by Bar-Ziv andMoses [13] in tubularlipid membranes.For blood vessels the width w of the vessel wall cannotbe neglected. Here, the vessel wall is viewed as a layeredstructure, where the only stresses are in the principalangular and vessel directions [8]. There are no shearstresses, and the stress in the direction normal to the layersis zero. With these assumptions, the Laplacian form [7,8]1996for the pressure is replaced by an integral [14,15],P ­Z r1wr"S1r˜f1 1 s›z r˜d2g1y22 Sz›2z r˜f1 1 s›z r˜d2g3y2#dr˜ , (4)where S and Sz are the stresses in the angular and vesseldirections. The stresses, defined as the forces per actualcross-sectional area, are related to the experimentally mea-sured idealized stresses s and sz , defined as the forces perrelaxed cross-sectional area, S ­ ggzs and Sz ­ ggzsz[15]. Here, g and gz are the normalized lengths [16] inthe angular and vessel direction. The length of a vesselremains almost constant during a contraction, and gz istherefore assumed to be constant, gz ­ g0. Correspond-ingly, the stress sz is also replaced by a constant s0. Wenote that the width w of the vessel wall changes when theinner radius r changes (Fig. 2). Assuming that the cross-sectional area of the vessel wall is constant, the (inner)radius dependence of w is given, when the inner radiusr and wall thickness v are known for the relaxed vessel(g ­ 1) [15,17].For small perturbations, the relevant expression for thepressure reduces toP ­ g0Z r1vrfs 2 s0r›2zrg fr˜2 2 r2 1 r2g21y2 dr˜ .(5)For the angular direction, the stress s depends on thenormalized length g ­ fr˜2 2 r2 1 r2g1y2yr˜. To firstorder in the perturbation usz, td, we findP ­ P0srd 1 Isrdu 2 I0srd›2zu , (6)whereP0srd ­ g0Z r1vrsfr˜2 2 r2 1 r2g21y2 dr˜ , (7)I0srd ­ g0s0rZ r1vrfr˜2 2 r2 1 r2g21y2 dr˜­ g0s0r logf1 1 sv 1 wdysr 1 rdg , (8)andIsrd ­ddrP0srd ­ g0Z r1vrr˜21ddg•sg‚›g›rdr˜ .(9)The partial derivatives of g with respect to r and r˜ arerelated, r˜21s›gy›rd ­ rsr2 2 r2d21s›gy›r˜d, and Isrdcan be expressed in terms of the normalized length g,Isrd ­g0grrs1 2 g2r d"ssgwdgw2ssgr dgr#, (10)VOLUME 82, NUMBER 9 PHY S I CA L REV I EW LE T T ER S 1 MARCH 1999where gr ­ ryr is the normalized inner radius, andgw ­ sr 1 wdysr 1 vd is the normalized outer radius.We note that Isrd is not singular at gr ­ 1, where alsogw ­ 1. Inserting Eq. (6) into Eq. (1) we get for a givenperiodic perturbation, u ­ ukstd cosskzd, that ukstd ,uks0delk t , wherelk ­csrd2prk2f2Isrd 2 I0srdk2g . (11)The value of I0srd is always positive. Thus, it is thesign of Isrd that determines the stability of the vesselwall. If Isrd is positive, the cylindrical shape is stablefor all modes. If Isrd is negative, the cylindrical shape isunstable to modes with k2 , jIjyI0.As seen from the expression for I , Eq. (10), the impor-tant quantity is syg. The point of instability is where sygcalculated at the inner radius equals the value of syg atthe outer radius. This can be illustrated geometrically bydrawing a line in the plot of s versus g (Fig. 3) from s0, 0dthrough sgr , ssgr dd. If the point sgw , ssgwdd lies abovethis line, Isrd is positive and the cylindrical form is stable.If, however, the point sgw , ssgwdd lies below this line, Isrdis negative, leading to an instability of the cylindrical form.Under normal physiological conditions, the angularstress s in blood vessels increases linearly to exponen-0.0 0.2 0.4 0.6 0.8 1.0 1.2 1. 404080120160200(kPa )sg10 0% 50% 0%0.20.1FIG. 3. A schematic plot of typical stress-strain relations forarterioles (adapted from [18]). The three heavy solid curvescorrespond to a vessel where the smooth muscle cells arenot activated (0%), where the smooth muscle cells are halfmaximally activated (50%), and where the smooth musclecells are maximally activated (100%). The thin solid linesshow how the points sgr , ssgr dd and sgw , ssgwdd [marked0.1] move with muscle cell activation for an arteriole withwall-to-lumen ratio vyr ­ 0.1. The point of instability(r ­ rc) for the cylindrical form of the blood vessel isillustrated geometrically by thin dashed lines from s0, 0dthrough sgr , ssgr dd. The instability point is where ssgwdygwequals ssgr dygr . The thick dashed line [marked 0.2] showshow the point sgw , ssgwdd moves with muscle cell activationfor an arteriole with vyr ­ 0.2, keeping the same curve forsgr , ssgr dd.tially with the normalized length [8,18] (Fig. 3), and thevalue of Isrd is therefore positive (Fig. 4). This ensuresthat the blood vessel keeps its cylindrical shape—nobending arguments are needed to explain the stability ofthe cylindrical shape of a blood vessel. However, whenacute hypertension is induced by infusion of a strongvasoconstricting agent like angiotensin II, there will bea substantial reduction of the inner radius in small arter-ies and large arterioles due to contraction of the smoothmuscle cells. The operating point for the vessel will nowbe on a less steep part of the s-g curve (Fig. 3), and,when the inner radius is reduced below a certain value rcat which ssgwdygw ­ ssgr dygr , the value of Isrd be-comes negative (Fig. 4). This will result in an instabilityof the cylindrical form, giving rise to the sausage-stringpattern. The dominant (fastest growing) mode is given byk ­ fjIjy2I0g1y2, which will correspond to “sausages” oflength , ­ 2pp2I0yjIj. Insertion of typical values [19]for the various parameters of the model yields , ø 2pr;hence, the length of the sausages will be 5–10 times theradius of the relaxed vessel. The theory therefore pre-dicts that the sausages will have an elongated shape witha length that decreases as the vessel radius gets smaller.This is in good agreement with experimental observations[2] (Fig. 1).A way to view the basic phenomenon underlying the in-stability is to note that, when Isrd becomes negative, thepressure at slightly larger radii is smaller than at slightlysmaller radii. Accordingly, the resulting flow J will be di-rected from small-radii regions to large-radii regions, caus-ing the small radii to become even smaller, and the largeradii to become larger. This continues until the pressurestabilizes at a value which is the same for both the large0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-50-250255075100I(r)(kPa)rr/0. 20.1rFIG. 4. A plot of the stability measure Isrd at large musclecell activation for two different wall-to-lumen ratios, vyr ­0.1 and vyr ­ 0.2. The cylindrical form of a blood vesselbecomes unstable when I becomes negative. An almost linearstress-strain relation in a region above rc gives rise to a decayof I. Above g ­ 1, where the stress increases exponentially,also Isrd increases exponentially.1997VOLUME 82, NUMBER 9 PHY S I CA L REV I EW LE T T ER S 1 MARCH 1999radius rmax and the small radius rmin. The stabilization isonly possible because the pressure for radii above the insta-bility, r . rc, again increases with r . The theory allowsan estimate of the radius in the dilated regions, rmax. As-suming that rminyr is small (close to zero), we can estimatethe final value of rmax by the condition Psrmaxd ­ Ps0d.Interestingly, the almost linear stress function in the regionabove rc (Fig. 3) gives rise to a decay of Isrd in the sameregion (Fig. 4). As a consequence rmax can become quitelarge. However, close to g ­ 1, the stress increases ex-ponentially due to the elastic properties of the vessel wall[8], and the value of Isrd will increase rapidly. This willeffectively prevent rmax from attaining a value substan-tially larger than the relaxed vessel radius, r. However,rmax may be larger than the working radius of the vesselunder normal physiological conditions, because the normalworking radius is smaller than the relaxed vessel radius [8].This may explain why previous articles have suggested thatthe dilated regions represented a blowout due to mechani-cal failure of the vessel wall [5].The sausage-string pattern following infusion of an-giotensin II have been found to occur predominantly insmall arteries and large arterioles [2]. The present analy-sis predicts that large vessels will be stable. Their operat-ing points are on the steep portion of the s-g curve due totheir high pressure. As seen from Fig. 3, the contractionis here limited, thus preventing the larger vessels from re-ducing their (inner) radius below the critical value rc. Asarterial vessels get smaller, the wall-to-lumen ratio vyrincreases [8]. From the expression for Isrd, Eq. (10), wefind that rc as well as rmax decreases with an increasingwall-to-lumen ratio. Hence, the sausage-string instabilityis less likely to appear in blood vessels with large wall-to-lumen ratios. It seems that the transmural pressure andthe contractile potential sets an upper limit, and the wall-to-lumen ratio a lower limit for vessels that will undergothe sausage-string instability in response to an acute in-crease in blood pressure.In summary, we have demonstrated that, during severevasoconstriction, the normal cylindrical shape of a bloodvessel may become unstable, and as a result the vesselexhibit a periodic pattern of constrictions and dilatations.The sausage-string pattern is not caused by a mechanicalfailure of the vessel wall due to the high blood pressure,but is the expression of an instability. The instability isrelated to the Rayleigh instability, and to the pearling in-stability seen in tubular lipid membranes. The mecha-nism behind the instability, however, is novel, involvingthe nonlinear elasticity characteristics of the vessel wall.The developed theory explains many of the key featuresobserved experimentally, e.g., the predominant occurrencein small arteries and large arterioles, and most likely inthose with small wall-to-lumen ratios.The present study was supported by grants from theDanish Natural Science Research Council, the Danish1998Medical Research Council, the Novo-Nordisk Foundation,and the Danish Heart Association.[1] F. B. Byrom, Lancet 2, 201 (1954); F. B. Byrom, Prog.Cardiovasc. Dis. 1, 31 (1974).[2] J. Giese, Acta Pathol. Microbiol. Scand. 62, 497 (1964).[3] J. Giese, The Pathogenesis of Hypertensive VascularDisease (Munksgaard, Copenhagen, 1966).[4] F. Gustafsson, Blood Pressure 6, 71 (1997).[5] L. J. Beilin and F. S. Goldby, Clin. Sci. Mol. Med. 52, 111(1977).[6] In the Hagen-Poiseuille approximation, the fluid conduc-tance is csrd ­ pr4y8h, where h is the dynamic viscos-ity of the fluid (blood), but the specific form of csrd is notcrucial for our purpose. Moreover, we neglect the pres-sure drop along the vessel, noting that this is much smallerthan the transmural pressure.[7] A. E. Green and J. E. Adkins, Large Elastic Deformations(Clarendon Press, Oxford, 1960).[8] Y. C. Fung, Biomechanics. Mechanical Properties ofLiving Tissues (Springer-Verlag, New York, 1990), 2nded.; Y. C. Fung, Biomechanics. Motion, Flow, Stress, andGrowth (Springer-Verlag, New York, 1990).[9] The principal curvatures 1yR and 1yRz are given by thefollowing two relations:1R­1rf1 1 s›zrd2g1y2, and 1Rz­2›2zrf1 1 s›zrd2g3y2.[10] J. Plateau, Statique Experimentale et Theorique des Liq-uides Soumis aux Seules Forces Moleculaires (Gautier-Villars, Paris, 1873).[11] Lord Rayleigh, Philos. Mag. 34, 145 (1892).[12] P. Nelson and T. Powers, Phys. Rev. Lett. 74, 3384(1995); R. E. Goldstein, P. Nelson, T. Powers, andU. Seifert, J. Phys. II (France) 6, 767 (1996).[13] R. Bar-Ziv and E. Moses, Phys. Rev. Lett. 73, 1392(1994).[14] R. S. Chadwick, Biophys. J. 39, 279 (1982).[15] R. Feldberg, M. Colding-Jørgensen, and N.-H. Holstein-Rathlou, Am. J. Phys. 269, F581 (1995).[16] The normalized length g at a given radius r˜ (r , r˜ ,r 1 w) is equal to the ratio between the circumference2p r˜ of the vessel strip situated at the radius r˜ and thecorresponding circumference 2pr˜ (r , r˜ , r 1 v) ofthe same vessel strip in the relaxed vessel, where thesmooth muscle cells are not activated and the transmuralpressure is zero. The corresponding strain is by definitionequal to g 2 1.[17] We have psr 1 wd2 2 pr2 ­ psr 1 vd2 2 pr2.This equation suggests a useful change of variable, fromr˜ to r˜, where r˜2 2 r2 ­ r˜2 2 r2. When r˜ variesbetween values r and r 1 w, which under perturbationschanges along the z axis, r˜ varies between the fixed innerand outer radii r and r 1 v of the relaxed vessel. Thenormalized length g at a radius r˜ is g ­ r˜yr˜.[18] R.W. Gore, Circ. Res. 34, 581 (1974); M. J. Davis andR.W. Gore, Am. J. Phys. 256, H630 (1989).[19] For vyr ­ 0.1, and s0 ­ 100 kPa, we have jIjr ø2I0yr ø 10 kPa.

Instability and `Sausage-String' Appearance in Blood Vessels during High
  Blood Pressure

Instability and `Sausage-String' Appearance in Blood Vessels during High Blood Pressure

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