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 lay- ers 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).