Vascular Wall Shear Stress: Basic Principles and Methods

Hellenic J Cardiol 46: 9-15, 2005 Special Article Vascular Wall Shear Stress: Basic Principles and Methods

THEODOROS G. PAPAIOANNOU, CHRISTODOULOS STEFANADIS 1st Department of Cardiology, Unit of Biomedical Engineering, Hippokration General Hospital, Athens Medical School, Greece

Key words: he investigation of various me- where n is the order of the tensor. Thus Shear stress, shear chanical forces and the tissue de- stress, which is a 2nd. order tensor, consists rate, endothelium, 2 viscosity. T formations they induce have been of 3 = 9 components (terms); six tangential extensively studied for over three centu- (parallel) and three normal ones. As an ex- ries. The basic principles concerning the ample, pressure or temperature is a gradient relationship between mechanical stress quantity with zero order and one compo- and strain of various materials were demon- nent, while force or velocity is a vector quan- strated by Robert Hooke (1635-1703) and tity of one order and three components. proved to be essential for the improvement Strain is defined as the mechanical de- of bio-materials and the understanding of formation of a physical body under the pathophysiological mechanisms in the vas- action of applied forces. Depending on cular bed. Subsequently, the ground-break- the direction of the applied force there is Manuscript received: ing studies of Jean Poiseuille (1799-1869) a different kind of deformation. Shear July 14, 2004; led to the development of mercury mano- strain is defined as the deformation of an Accepted: November 2, 2004. meters for blood pressure measurement object in which parallel planes remain par- and resulted in theories describing the flow allel but are shifted in a direction parallel inside a cylindrical conduit, widely known to themselves, as opposed to normal stress as Poiseuille’s law. or force, which when applied to an object Address: The present work aims to describe induces normal (direct) deformation (Fig- Theodoros G. Papaioannou and provide an interpretation of basic me- ure 1). For a one-dimensional object, de- chanical and haemodynamic phenomena formation is either lengthening or short- 13 Iak. Patatsou St., related to forces applied to arterial walls ening. For a two-dimensional object, there N. Kipseli, 113 63 Athens, Greece and especially shear stresses. are a total of four strain components: nor- e-mail: mal strain along the x or y axis, and shear [email protected] strain causing distortion of the shape along Basic principles and definitions the x or y axis. In physics, stress is the internal distribu- The haemodynamic conditions inside tion of forces within a body that balance blood vessels lead to the development of and react to the external loads applied to superficial stresses near the vessel walls, it. Stress is a 2nd. order tensor. A tensor is which can be divided into two categories: a mathematical quantity that is defined by a) circumferential stress due to pulse pres- the order and the number of its compo- sure variation inside the vessel; b) shear nents. In Euclidian space (three dimen- stress due to blood flow. It should be noted sions) the components of a tensor are 3n, that intravascular hydrostatic pressure

(Hellenic Journal of Cardiology) HJC ñ 9 T.G. Papaioannou, C. Stefanadis

Shear rate The notions of shear rate and fluid viscosity should be first clearly apprehended, since they are crucial for the assessment and development of shear stress. Shear rate is defined as the rate at which adjacent layers of fluid move with respect to each other, usually expressed as reciprocal seconds. The size of the shear rate gives an indication of the shape of the velocity profile for a given situation. The determination of shear stresses on a surface is based on the fundamental assumption of fluid me- Figure 1. Deformations induced by normal and shear forces. chanics, according to which the velocity of fluid upon the surface is zero (no-slip condition).1 This leads to the establishment of velocity gradient. Thus, as the produces not only circumferential but also normal fluid particles “travel” parallel to the wall, their veloc- tensions. The direction of the shear stress vector is ity increases from zero at the wall to a maximum val- determined by the direction of the blood flow velocity ue at some distance from the wall (Figure 2). vector very close to the vessel wall. Shear stress is ap- If we consider a blood vessel as a straight, cylin- plied by the blood against the vessel wall. On the oth- drical tube with rigid walls then the velocity gradient er hand, the force applied to the blood by the wall is ÁØ (shear rate) will be given by the relation: considered as friction and has a direction opposite to the blood flow. The tensions acting against the vessel du ÁØ = –– (1) wall are likely to be determined by blood flow condi- dr tions. Shear stresses during turbulent flow, regions of where u is the fluid velocity and r the radius of the flow recirculation or flow separation, develop more tube. Figure 2 illustrates two cases with different ve- complicated local characteristics. locity profiles corresponding to flows with different Normal stresses due to blood pressure are trans- velocity gradients. Assuming that the blood is an ideal ferred to all vessel wall layers (intima, media and ad- Newtonian fluid with constant viscosity, the flow is ventitia). On the other hand, shear stress is applied steady and laminar and the vessel is straight, cylindri- mainly to the inner layer of the arterial wall in con- cal and inelastic, Poiseuille’s law could be applied to tact with the blood, the vascular endothelium. The determine shear rate as follows: normal stresses applied (directly), as well as shear stresses (indirectly), regulate blood vessel diameter 8 Ø u depending on vascular wall elasticity (distensibility) ÁØ = ––––– (2a) d and endothelial function (endothelial induced va- or Q sodilatation), respectively. Arterial distensibility is ÁØ =32 ––––– (2b)  d3 one of the most important mechanical properties of Ø the arterial walls, and deserves a more extensive where u is the average velocity, Q the mean volumet- analysis, which, however, is beyond the scope of the ric flow and d the vessel diameter. Under these condi- present review. tions a parabolic velocity profile could be assumed.

Figure 2. Velocity profiles in cylindrical tubes with different shear rates.

10 ñ HJC (Hellenic Journal of Cardiology) Shear Stress, Basic Principles and Methods

where Q is the mean volumetric flow rate, u the mean velocity, and d the vessel diameter. It should be emphasised that the Haagen-Pois- seuille equation is applied in vivo only after the fol- lowing assumptions have been made: ñ The blood is considered as a Newtonian fluid. ñ The vessel cross sectional area is cylindrical. ñ The vessel is straight with inelastic walls. ñ The blood flow is steady and laminar. The Haagen-Poisseuille equation indicates that shear stress is directly proportional to blood flow rate and inversely proportional to vessel diameter.

Figure 3. Shear stress-rate relationship for Newtonian (n=1) and non-Newtonian (n >1 or n <1) fluids. Blood viscosity Viscosity is an internal property of a fluid that offers resis- tance to flow. It is measured in centipoise (cP). Viscosity Determination of shear stress is a characteristic property of fluids and is a measure of Shear rate and viscosity are directly related to the the combined effects of adhesion and cohesion. Viscosity properties of the fluid. Non-Newtonian fluids are of fluids increases as temperature decreases. Blood vis- governed by a non-linear relationship between shear cosity (non-Newtonian fluid) depends on shear rate, stress and shear rate (Figure 3). In this case (non- which is determined by blood platelets, red cells, etc. Newtonian fluid), the slope of the shear stress-rate Lipowsky et al investigated the changes of blood curve, which is equal to fluid viscosity, depends on viscosity in relation to changes in: a) shear rate for dif- pressure, temperature and shear rate. In contrast, the ferent levels of haematocrit and b) haematocrit for dif- viscosity of Newtonian fluids is independent from ferent shear rates.2,3 It was shown that blood viscosity is shear rate, as indicated by the linear relation between slightly affected by shear rate changes at low levels of shear stress and shear rate (Figure 3). haematocrit. In contrast, as haematocrit increases, the For non-Newtonian fluids shear stress (Ù) is de- effect of shear rate changes on blood viscosity becomes -1 fined by the Ostwald de Waele equation: greater. Additionally, at low shear rates (~54 sec ) haematocrit variations induce greater changes in blood Ù=ÎØ ÁØ n (3) viscosity. Consequently, in arteries where high shear Ø -1 -1 where Á is the shear rate (sec ), n and k are indices of rates (~300 sec ) are present (e.g. the aorta) blood vis- fluid internal properties, i.e. adhesion, cohesion.1 cosity is approximately 3.5 cP (for 45% haematocrit), -1 For Newtonian fluids flowing upon a planar sur- while in vessels with lower shear rates (~5 sec ) such as face, shear stress is determined according to New- the veins, blood viscosity is markedly higher (~10 cP). ton’s law by the equation: Therefore, for large arteries the assumption of 3.5 cP blood viscosity is commonly made. Blood viscosity mea- du Ù= Ì Ø ––– (4) surement is required for the accurate calculation of dy shear stress in veins or microcirculation. Viscosity is where Ì is the kinetic viscosity, u the fluid velocity, y measured with various types of viscometer at a specified the distance from the surface and du/dy the velocity constant temperature, typically 25ÆC. gradient, namely the shear rate ÁØ (sec-1). For blood flow within a vessel (inelastic, cylindri- Relation of vessel diameter with haematocrit cal and straight), shear stress is defined by the Haa- gen-Poisseuille equation: It has to be emphasised that the dependence of blood Q viscosity on haematocrit is more pronounced in the Ù=32 Ø ÌØ ––––– (5a) Ø d3 microcirculation than in larger vessels, due to haema- or tocrit variations observed in small vessels (lumen di- u ameter <100 Ìm). The significant change of haemat- Ù =8 Ø ÌØ –––– (5b) d ocrit in relation to vessel diameter is known as the

(Hellenic Journal of Cardiology) HJC ñ 11 T.G. Papaioannou, C. Stefanadis

Fahraeus-Lindquist phenomenon.4 This phenomenon used instead. This value is based upon the assumption is associated with the tendency of red blood cells to that blood acts like a Newtonian fluid in large vessels. travel closer to the centre of the vessels. Thus, the Magnetic resonance imaging (MRI), by applying greater the decrease in vessel lumen, the smaller the various flow quantification methods (phase contrast, number of red blood cells that pass through, resulting phase velocity mapping, Fourier velocity mapping, in a decrease in blood viscosity. etc.), has been used for shear rate evaluation, with rather direct estimation of the flow velocity distribution in the cross-section of a vessel.12 Methods of shear stress measurement at vascular walls Phase contrast MRI is somewhat superior to other Shear stress on arterial walls has been estimated in nu- imaging techniques, providing better image resolution merical models, in vitro, in animals and in humans. In and contrast, while it allows measurement of blood ve- many cases approximations and assumptions are neces- locity at various points within a given cross-sectional sary in order to calculate shear stress; they may thus vessel area. Every pixel of the recorded image corre- sometimes diverge from the real flow phenomenon. sponds to a specific value of velocity. The determina- Mathematical models often require a specific de- tion of velocity gradient is made using various methods. scription of geometrical characteristics and the flow A simple method is to consider a linear velocity change conditions of the vessel studied. The distribution of between two adjacent points on the vessel’s diameter. shear stresses in a vessel area can be calculated by re- In this case shear rate is determined as follows: solving the mathematical equations describing the 5,6 ¢u flow velocity in the vessel. ÁØ =–– (6) In vitro models have been used previously to in- ¢r vestigate shear stresses applied to endothelial cells under various flow conditions: It has been found that shear rates calculated by eq. ñ Steady, laminar flow using parallel planar or coni- 6 are overestimated by approximately 10-45%.13 Non- cal moving surfaces.7 linear models describing the velocity profile more real- ñ Unsteady, pulsatile, laminar flow by using peri- istically have also been used,14 without, however, in- staltic pumps which allow the study of time varying creasing the accuracy of shear rate measurement, espe- shear stresses.8 cially in regions very close to the arterial wall. More so- ñ Oscillatory, turbulent flow.9 phisticated models have also been proposed for the es- In vitro estimation of shear stresses is achieved by timation of shear rate regardless of a vessel’s geometric determination of flow velocity profiles (e.g. by flow vi- details or diameter velocity distribution.15 sualisation techniques). Advanced experimental tech- One of the major shortcomings of imaging techniques also exist that allow the 3-D analysis of shear niques for the accurate estimation of wall shear stresses stresses and deformation of endothelial cells.10,11 is the inability of accurate arterial wall tracking. This, In vivo estimation of shear stress often requires however, is expected to be overcome by imaging sys- fewer assumptions and approximations than do the tems with more advanced spatial and temporal resolu- numerical or in vitro models. Detailed geometric tion. characteristics and specific flow conditions are partic- To date, in clinical practice there is no single com- ularly variable under in vivo conditions. Shear stress monly used technique of shear stress determination, measurement requires only the estimation of shear something that is also evident with respect to the mea- rate values and blood viscosity for a specific area of a surement of other physical variables such as the arterial vessel (considering the blood as a Newtonian fluid). mechanical properties. This renders it difficult to per- Estimation of local velocity gradient along a cross form a comparison and interpretation of data between sectional area of a vessel can be realised by various various clinical studies. techniques. Assessment of shear stress with Doppler echocar- Shear stress and vascular endothelium diography can be performed using equation 5, by calcu- lating the mean or maximum blood flow velocity within Endothelium responds to shear stress through various a vessel of known diameter, considering blood as a pathophysiological mechanisms depending on the kind Newtonian fluid. In most cases where measurement of and the magnitude of shear stresses. More specifically, blood viscosity is not feasible, the value 3.5 cP can be the exposure of vascular endothelium to shear forces in

12 ñ HJC (Hellenic Journal of Cardiology) Shear Stress, Basic Principles and Methods

Figure 4. Shear stress values in various vessels according to data reported by Lipowsky et al.2,3

the normal value range stimulates endothelial cells to sion of leukocytes, smooth muscle proliferation and release agents with direct or indirect antithrombotic endothelial apoptosis. properties, such as prostacyclin,16 nitric oxide (NO)17, The arterial and venous vascular tree is exposed to calcium18, thrombomodulin19, etc. Further insights con- different levels of shear stress, a fact that is determined cerning the interaction of shear stress and endothelium by flow velocity characteristics. Shear stress at arterial can be found elsewhere8,20,21. walls ranges between approximately 10 and 70 How the mechanical forces are detected and are dynes/cm2, whereas the corresponding normal values transformed into biological signals that stimulate in- for veins are considerably lower, from 1 to 6 dynes/cm2. travascular processes remains unresolved. The possi- Higher shear stress values have been observed in the ble existence of so-called “mechanoreceptors” has vasculature, especially in regions with an anatomy/ provoked a number of research groups to localise those geometry promoting turbulent flow or increased flow receptors which “translate” mechanical forces into bi- velocity (e.g. arteries with strong curvature such as the ological signals. Two models of mechanotransduction aortic arch, bifurcations, anastomoses, etc). In those have been demonstrated so far: cases, shear stress shows increased values, while the di- a) The localised model, where the mechano-re- rection may also change due to retrograde flow, which ceptor, like other receptors, is considered to be lo- depends on the haemodynamic conditions. cated in cellular membrane. Possible channels also Indicative values of shear stress and shear rate for located in membrane, such as those of potassium, several vessels are outlined in figure 4, based on pre- sodium and calcium, respond to shear stress alter- viously published data.22,23 The estimation of these ation.9 values requires the mean volumetric flow and velocity b) The decentralised model, where shear stress of blood in the specific vessel, the vessel diameter and forces acting on a cell’s surface are transmitted through the blood viscosity, which in this case was considered a cytoskeleton, allowing the activation of several to be equal to 3.5 cP. mechano-receptors within the cell. Integrines connect- Under normal shear conditions, endothelial as ed with cytoskeleton have been related to the afore- well as smooth muscle cells have a rather low rate of mentioned mechanism of mechanoreception. proliferation. Changes in shear stress magnitude acti- vate cellular proliferation mechanisms as well as vascular remodelling processes. More specifically, a high Shear stress in clinical practice grade of shear stress increases wall thickness and ex- Under normal conditions, shear stresses maintain pands the vessel’s diameter,24 so that shear stress val- their direction and their magnitude within a range of ues return to their normal values. In contrast, low shear values that impedes atherogenesis, thrombosis, adhe- stress induces a reduction in vessel diameter and leads

(Hellenic Journal of Cardiology) HJC ñ 13 T.G. Papaioannou, C. Stefanadis to intima-media hyperplasia. Consequently, shear References stresses stimulate vasoregulatory mechanisms which, 1. Papaioannou A: Fluid Mechanics, Vol. I, II. Athens 1998. together with alterations of arterial diameter, attempt 2. Lipowsky HH, Usami S, Chien S: In vivo measurements of to maintain a mean shear stress level of approximate- hematocrit and apparent viscosity in the microvasculature of ly 15 dynes/cm2.25,26 cat mesentery. Microvasc Res 1980; 19: 297-319. The presence of low shear stresses is frequently 3. Lipowsky HH, Kovalcheck S, Zweifach BW: The distribution of blood rheological parameters in the microcirculation of cat accompanied by unstable flow conditions (e.g. turbu- mesentery. Circ Res 1978; 43: 738-749. lence flow, regions of blood recirculation, “stagnant” 4. Fahraeus R, Lindqvist T: the viscosity of blood in narrow cap- blood areas). Such conditions are often observed in illary tubes. Am J Physiol 1931; 96: 562-568. the aorta or at arterial bifurcations.27-29 5. Perktold K, Thurner E, Kenner T: Flow and stress characteristics in rigid walled and compliant carotid artery bifurcation models. Med Biol Eng Comput 1994; 32: 19-26. Atherogenesis 6. Milner JS, Moore JA, Rutt BK, Steinman DA: Hemodynam- ics of human carotid artery bifurcations: Computational stud- Shear stress with a low mean or maximum value and ies with models reconstructed from magnetic resonance varying direction (oscillating shear stress) has been as- imaging of normal subject. J Vasc Surg 1998; 28: 143-156. 7. Dewey CF, Bussolari SR, Gimbrone MA, Davies PF: The dy- sociated with development of atherosclerotic impair- namic response of vascular endothelial cells to fluid shear ment in arteries.27-29 stress. J Biomech Eng 1981; 103: 177-185. 8. Ballerman BJ, Dardik A, Eng E, Liu A: Shear stress and the endothelium. Kidney Int (suppl) 1998; 67: S100-S108. Grafts - anastomoses 9. Paszkowiak JJ, Dardik A: Arterial wall shear stress: Observa- tions from bench to bedside. Vasc Endovasc Surg 2003; 37: In regions of anastomoses, thrombosis or intimal hy- 47-57. perplasia has been reported, mainly due to the pres- 10. Sakurai A, Nakano A, Yamaguchi T, et al: A computational ence of low shear stresses.30 fluid mechanical study of flow over cultured endothelial cells. Adv Bioeng 1991; 20: 299-302. 11. Barbee KA, Davies PF, Lal R: Shear stress-induced reorgani- Aneurysms zation of the surface topography of living endothelial cells imaged by atomic force microscopy. Circ Res 1994; 74: 163- Shear stresses play an important role in the pathogene- 171. sis of aneurysms. Vessel wall remodelling as a result of 12. Hoeks, Samijo S, Brands P, Reneman R: Noninvasive determination of shear-rate distribution across the arterial lumen. shear stress alteration is accompanied by synthesis and Hypertension 1995; 26: 26-33. secretion of NO, growth factors and metalloproteins, 13. Lou Z, Yang WJ, Stein PD: Errors in the estimate of arterial which contribute to aneurysm pathogenesis. Shear wall shear rates that result from curve fitting of velocity pro- stress has been also associated with the rupture of files. J Biomech Eng 1993; 26: 383-390. 31 14. Fatemi RS, Rittgers SE: Derivation of shear rates from near- aneurysms. wall LDA measurements under steady and pulsatile flow conditions. J Biomech Eng 1994; 116: 347-368. 15. Cheng C, Parker D, Taylor C: Lagrangian interpolation func- In-stent restenosis tions with cine phase-contrast magnetic resonance imaging. Ann Biomed Eng 2002; 30: 1020-1032. Low shear stress has also been linked to in-stent re- 16. Frangos JA, Eskin SG, McIntire LV, Ives CL: Flow effects of 32 stenosis. prostacyclin production by cultured human endothelial cells. Science 1985; 227(4693): 1477-1479. 17. Rubanyi GM, Romero JC, Vanhoutte PM: Flow-induced re- Conclusions lease of endothelium-derived relaxing factor. Am J Physiol 1986; 250 (6pt2): H1145-H1149. The pivotal role of shear stresses in vascular endothelial 18. Helmlinger G, Berk BC, Nerem RM: Calcium responses of function and in various pathological conditions renders endothelial cell monolayers subjected to pulsatile and steady shear stress investigation a matter of particular impor- laminar flow differ. Am J Physiol 1995; 269 (2Pt1): C367- tance at the molecular and clinical levels. Shear stress C375. 19. Malek AM, Jackman R, Rosenberg RD, Izumo S: Endothe- assessment in clinical practice may serve in the investi- lial expression of thrombomodulin is reversibly regulated by gation of many pathological conditions and mecha- fluid shear stress. Circ Res 1994; 74: 852-860. nisms affecting the cardiovascular system and the vas- 20. Malek AM, Alper SL, Izumo S: Hemodynamic shear stress cular endothelium. Understanding and development of and its role in atherosclerosis. JAMA 1999; 282: 2035-2042. 21. Resnick N, Yahav H, Shay-Salit A, et al: Fluid shear stress the existing techniques and methods for shear stress and the vascular endothelium: for better and for worse. Progr measurement may contribute greatly towards this goal. Biophys & Mol Biol 2003; 81: 177-199.

14 ñ HJC (Hellenic Journal of Cardiology) Shear Stress, Basic Principles and Methods

22. McDonald DA: Blood flow in arteries. Baltimore. The Wil- correlation between plaque location and low oscillating shear liams and Wilkins Co, 1974; pp. 356-378. stress. Arteriosclerosis 1985; 5: 293-302. 23. Whitmore RL: Rheology of the circulation. New York: Perg- 29. Zarins CK, Giddens DP, Bharadvaj BK, Sottiurai VS, Mabon amon Press, 1968. RF, Glagov S: Carotid bifurcation atherosclerosis: Quantita- 24. Irace C, Gnasso A, Cirillo F, et al: Arterial remodeling of the tive correlation of plaque localization with flow velocity pro- common carotid artery after aortic valve replacement in pa- files and wall shear stress. Circ Res 1983; 53: 502-514. tients with aortic stenosis. Stroke 2002; 33: 2446-2450. 30. Haruguchi H, Teraoka S: Intimal hyperplasia and hemody- 25. Glagov S, Zarins C, Giddens DP, Ku DN: Hemodynamics and namic factors in arterial bypass and arteriovenous grafts: a re- atherosclerosis - Insights and perspectives gained from studies view. J Artif Organs 2003; 6: 227-235. of human arteries. Arch Pathol Lab Med 1988; 112: 1018-1031. 31. Raghavan ML, Vorp DA, Federle MP, Makaroun MS, Web- 26. Zarins CK, Weisenberg E, Kolettis G, Stankunavicius R, Glagov ster MW: Wall stress distribution on three-dimensionally re- S: Differential enlargement of artery segments in response to en- constructed models of human abdominal aortic aneurysm. J larging atherosclerotic plaques. J Vasc Surg 1988; 7: 386-394. Vasc Surg 2000; 31:760-9. 27. Cybulsky MI, Gimbrone MA: Endothelial expression of a 32. Stone PH, Coskun AU, Kinlay S, Clark ME, Sonka M, Wahle mononuclear leukocyte adhesion molecule during atherogen- A, et al: Effect of endothelial shear stress on the progression esis. Science 1991; 251:788-791. of coronary artery disease, vascular remodeling and in-stent 28. Ku DN, Giddens DP, Zarins CK, Glagov S: Pulsatile flow restenosis in humans: In vivo 6-month follow-up study. Circu- and atherosclerosis in the human carotid bifurcation: Positive lation 2003; 108: 438-444.

(Hellenic Journal of Cardiology) HJC ñ 15