Biomechanics Principles and Applications - Schneck and Bronzino

202 Biomechanics: Principles and Applications

Kim, W.Y., Bisgaard, T., Nielsen, S.L., Poulsen, J.K., Pederson, E.M., Hasenkam, J.M., and Yoganathan, A.P. 1994. Two-dimensional mitral flow velocity profiles in pig models using epicardial Doppler echocardiography. J. Am. Coll. Cardiol. 24(2):532–545.

Kim, W.Y., Walker, P.G., Pederson, E.M., Poulsen, J.K., Oyre, S. Houlind, K.C., and Yoganathan, A.P. 1995. Left ventricular blood flow patterns in normal subjects: a quantitative analysis by threedimensional magnetic resonance velocity mapping. J. Am. Coll. Cardiol. 26(1):224–230.

Komoda, T., Hetzer, R., Uyama, C., Siniawski, H., Maeta, H., Rosendahl, P., and Ozaki, K. 1994. Mitral ammular function assessed by 3D imaging for mitral valve surgery. J. Heart Valve Dis. 3:483–490.

Kunzelman, K.S., Cochran, R.P., Murphee, S.S., Ring, W.S., Verrier, E.D., and Eberhart, R.C. 1993a. Differential collagen distribution in the mitral valve and its influence on biomechanical behaviour.

J. Heart Valve Dis. 2:236–244.

Kunzelman, K.S., Cochran, R.P., Chuong, C., Ring, W.S., Verner, E.D., and Eberhart, R.D. 1993b. Finite element analysis of the mitral valve. J. Heart Valve Dis. 2:326–340.

Kunzelman, K.S., Cochran, R.P., Verner, E.D., and Eberhart, R.D. 1994. Anatomic basis for mitral valve modeling. J. Heart Valve Dis. 3:491–496.

Kunzelman, K.S., Reimink, M.S., and Cochran, R.P. 1998. Flexible versus rigid ring annuloplasty for mitral valve annular dilatation: a finite element model. J. Heart Valve Dis. 7:108–116.

Leeson-Dietrich, J., Boughner, D., and Vesely, I. 1995. Porcine pulmonary and aortic valves: a comparison of their tensile viscoelastic properties at physiological strain rates. J. Heart Valve Dis. 4:88–94.

Levine, R.A., Trivizi, M.O., Harrigan, P., and Weyman, A.E. 1987. The relationship of mitral annular shape to the diagnosis of mitral valve prolapse. Circulation 75:756–767.

Lim, K.O. and Bouchner, D.P. 1975. Mechanical properties of human mitral valve chordae tendineae: variation with size and strain rate. Can. J. Physiol. Pharmacol. 53:330–339.

Lo, D. and Vesely, I. 1995. Biaxial strain analysis of the porcine aortic valve. Ann. Thorac. Surg. 60(suppl. II):S374–378.

Oh, J.K., Appleton, C.P., Hatle, L.K., Nishimura, R.A., Seward, J.B., and Tajik, A.J. 1997. The noninvasive assessment of left ventricular diastolic function with two-dimensional and Doppler echocardiography. J. Am. Soc. Echocardiol. 10:246–270.

Ormiston, J.A., Shah, P.M., Tei, C., and Wong, M. 1981. Size and motion of the mitral valve annulus in man: a two-dimensional echocardiographic method and findings in normal subjects. Circulation 64:113–120.

Pai, R.G., Tanimoto, M., Jintapakorn, W., Azevedo, J., Pandian, N.G., and Shah, P.M. 1995. Volumerendered three-dimensional dynamic anatomy of the mitral annulus using a transesophageal echocardiographic technique. J. Heart Valve Dis. 4:623–627.

Paulsen, P.K. and Hasenkam, J.M. 1983. Three-dimensional visualization of velocity profiles in the ascending aorta in dogs, measured with a hot film anemometer. J. Biomechanics 16:201–210.

Raganathan, N., Lam, J.H.C., Wigle, E.D., and Silver, M.D. 1970. Morphology of the human mitral valve: the valve leaflets. Circulation 41:459–467.

Reul, H. and Talukdar, N. 1979. Heart valve mechanics. In Quantitative Cardiovascular Studies: Clinical and Research Applications of Engineering Principles. Hwang, N.H.C., Gross, D.R. and Patel, D.J., Eds. University Park Press, Baltimore, MD, 527–564.

Roberts, W.C. 1983. Morphologic features of the normal and abnormal mitral valve. Am. J. Cardiol. 51:1005–1028.

Rossvoll, O., Samstad, S., Torp, H.G., Linker, D.T., Skjærpe, T., Angelsen, B.A.J., and Hatle, L. 1991. The velocity distribution in the aortic annulus in normal subjects: a quantitative analysis of twodimensional Doppler flow maps. J. Am. Soc. Echocardiogr. 4:367–378.

Sahasakul, Y., Edwards, W.D., Naessens, J.M., and Tajik, A.J. 1988. Age-related changes in aortic and mitral valve thickness: implications for two-dimensional echocardiography based on an autopsy study of 200 normal human hearts. Am. J. Cardiol. 62:424–430.

Salisbury, P.F., Cross, C.E., and Rieben, P.A. 1963. Chordae tendinea tension. Am. J. Physiol. 25:385–392.

Samstad, O., Gorp, H.G., Linker, D.T., Rossvoll, O., Skjaerpe, T., Johansen, E., Kristoffersen, K., Angelson, B.A.J., and Hatle, L. 1989. Cross-sectional early mitral flow velocity profiles from colour Doppler.

Br. Heart J. 62:177–184.

Sanfilippo, A.J., Harrigan, P., Popovic, A.D., Weyman, A.E., and Levine, R.A. 1992. Papillary muscle traction in mitral valve prolapse: quantitation by two-dimensional echocardiography. J. Am. Coll. Cardiol. 19:564–571.

Scott, M.J. and Vesely, I. 1996. Morphology of porcine aortic valve cusp elastin. J. Heart Valve Dis. 5:464–471. Silver, M.A. and Roberts, W.C. 1985. Detailed anatomy of the normally functioning aortic valve in hearts

of normal and increased weight. Am. J. Cardiol. 55:454–461.

Silver, M.D., Lam, J.H.C., Raganathan, N., and Wigle, E.D. 1971. Morphology of the human tricuspid valve. Circulation 43:333–348.

Silverman, M.E. and Hurst, J.W. 1968. The mitral complex: Interaction of the anatomy, physiology, and pathology of the mitral annulus, mitral valve leaflets, chordae tendineae and papillary muscles.

Am. Heart J. 76:399–418.

Simonson, J.S. and Schiller, N.B. 1989. Descent of the base of the left ventricle: an echocardiographic index of left ventricular function. J. Am. Soc. Echocardiogr. 2:25–35.

Sloth, E., Houlind, K.C., Oyre, S., Kim, Y.K., Pedersen, E.M., Jørgensen, H.S., and Hasenkam, J.M. 1994. Three-dimensional visualiztion of velocity profiles in the human main pulmonary artery using magnetic resonance phase velocity mapping. Am. Heart J. 128:1130–1138.

Sung, H.W. and Yoganathan, A.P. 1990. Axial flow velocity patterns in a normal human pulmonary artery model: pulsatile in vitro studies. J. Biomechanics 23(3):210–214.

Thubrikar, M. 1990. The Aortic Valve. CRC Press, Boca Raton, FL.

Thubrikar, M., Heckman, J.L., and Nolan, S.P. 1993. High speed cine-radiographic study of aortic valve leaflet motion. J. Heart Valve Dis. 2:653–661.

Tsakiris, A.G., von Bernuth, G., Rastelli, G.C., Bourgeois, M.J., Titus, J.L., and Wood, E.H. 1971. Size and motion of the mitral valve annulus in anesthetized intact dogs. J. Appl. Physiol. 30:611–618.

Vesely, I. 1998. The role of elastin in aortic valve mechanics. J. Biomech. 31:115–123.

Vesely, I. and Noseworthy, R. 1992. Micromechanics of the fibrosa and the ventricularis in aortic valve leaflets. J. Biomech. 25:101–113.

Vesely, I., Macris, N., Dunmore, P.J., and Boughner, D. 1994. The distribution and morphology of aortic valve cusp lipids. J. Heart Valve Dis. 3:451–456.

Vollebergh, F.E.M.G. and Becker, A.E. 1977. Minor congenital variations of cusp size in tricuspid aortic valves: possible link with isolated aortic stenosis. Br. Heart J. 39:106–111.

Walker, P.G., Cranney, G.B., Grimes, R.Y., Delatore, J., Rectenwald, J., Pohost, G.M., and Yoganathan, A.P. 1996. Three-dimensional reconstruction of the flow in a human left heart by magnetic resonance phase velocity encoding. Ann. Biomed. Eng. 24:139–147.

Westaby, S., Karp, R.B., Blackstone, E.H., and Bishop, S.P. 1984. Adult human valve dimensions and their surgical significance. Am. J. Cardiol. 53:552–556.

Weyman, A.E. 1994. Principles and Practices of Echocardiography. Lea & Febiger, Philadelphia, PA. Yellin, E.L., Peskin, C., Yoran, C., Koenigsberg, M., Matsumoto, M., Laniado, S., McQueen, D., Shore,

D., and Frater, R.W.M. 1981. Mechanisms of mitral valve motion during diastole. Am. J. Physiol. 241:H389–H400.

Source: Milnor, W.R. 1989. Hemodynamics, 2nd ed., p. 45. Baltimore, Williams & Wilkins. With permission.

some arteries. Smooth-muscle tone is also affected by circulating vasoconstrictors such as norepinephrine and vasodilators such as nitroprusside. Blood vessels, like other soft biological tissues, generally do not obey Hooke’s law, becoming stiffer as pressure is increased. They also exhibit viscoelastic characteristics such as hysteresis and creep. Fortunately, for many purposes a linear elastic model of blood vessel behavior provides adequate results.

13.2 Flow Characteristics

Blood is a complex substance containing water, inorganic ions, proteins, and cells. Approximately 50% is plasma, a nearly Newtonian fluid consisting of water, ions, and proteins. The balance contains erythrocytes (red blood cells), leukocytes (white blood cells), and platelets. Whereas the behavior of blood in vessels smaller than approximately 100 m exhibits significant non-Newtonian effects, flow in larger vessels can be described reasonably accurately using the Newtonian assumption. There is some evidence

aMean (systolic/diastolic)

bMean (peak)

cOne of the usually four terminal pulmonary veins

Source: Milnor, W.R. 1989. Hemodynamics, 2nd ed., p. 148. Baltimore, Williams & Wilkins. With permission.

suggesting that in blood analog fluids wall shear stress distributions may differ somewhat from Newtonian values [Liepsch et al., 1991].

Flow in the arterial circulation is predominantly laminar with the possible exception of the proximal aorta and main pulmonary artery. In steady flow, transition to turbulence occurs at Reynolds numbers

where r = vessel radius, V = velocity, v = kinematic viscosity = viscosity/density.

Flow in the major systemic and pulmonary arteries is highly pulsatile. Peak-to-mean flow amplitudes as high as 6 to 1 have been reported in both human and dog [Milnor, 1989, p. 149]. Womersley’s analysis of incompressible flow in rigid and elastic tubes [1957] showed that the importance of pulsatility in the

where ω = frequency.

This is usually referred to as the Womersley number (Nw) or α-parameter. Womersley’s original report is not readily available; however, Milnor provides a reasonably complete account [Milnor, 1989, pp. 106–121].

Mean and peak Reynolds numbers in human and dog are given in Table 13.2, which also includes mean, peak, and minimum velocities as well as the Womersley number. Mean Reynolds numbers in the entire systemic and pulmonary circulations are below 2300. Peak systolic Reynolds numbers exceed 2300 in the aorta and pulmonary artery, and some evidence of transition to turbulence has been reported. In dogs, distributed flow occurs at Reynolds numbers as low as 1000, with higher Womersley numbers increasing the transition Reynolds number [Nerem and Seed, 1972]. The values in Table 13.2 are typical for individuals at rest. During exercise, cardiac output and hence Reynolds numbers can increase severalfold. The Womersley number also affects the shape of the instantaneous velocity profiles as discussed below.

aAll data are apparent pressure wave velocities (although the average of higher frequency harmonics approximates the true velocity in many cases), from relatively young subjects with normal cardiovascular systems, at approximately normal distending pressures.

bRanges for each vessel and species taken from Table 9.1 of source.

Source: Milnor, W.R. 1989. Hemodynamics, 2nd ed., p. 235. Baltimore, Williams & Wilkins. With permission.

13.3Wave Propagation

The viscoelasticity of blood vessels affects the hemodynamics of arterial flow. The primary function of arterial elasticity is to store blood during systole so that forward flow continues when the aortic valve is closed. Elasticity also causes a finite wave propagation velocity, which is given approximately by the Moens–Korteweg relationship:

where E = wall elastic modulus, h = wall thickness, ρ = blood density, r = vessel radius.

Although Moens [1878] and Korteweg [1878] are credited with this formulation, Fung [1984, p. 107] has pointed out that the formula was first derived much earlier [Young, 1808]. Wave speeds in arterial blood vessels from several species are given in Table 13.3. In general, wave speeds increase toward the periphery as vessel radius decreases and are considerably lower in the main pulmonary artery than in the aorta owing primarily to the lower pressure and consequently lower elastic modulus.

Wave reflections occur at branches where there is not perfect impedance matching of parent and daughter vessels. The input impedance of a network of vessels is the ratio of pressure to flow. For rigid vessels with laminar flow and negligible inertial effects, the input impedance is simply the resistance and is independent of pressure and flow rate. For elastic vessels, the impedance is dependent on the frequency of the fluctuations in pressure and flow. The impedance can be described by a complex function expressing the amplitude ratio of pressure to flow oscillations and the phase difference between the peaks.

Zi (ω)= P((ω))

Q ω

where Zi is the complex impedance, Zi is the amplitude, and θi is the phase.

For an infinitely long straight tube with constant properties, input impedance will be independent of position in the tube and dependent only on vessel and fluid properties. The corresponding value of input impedance is called the characteristic impedance Zo given by:

where A = vessel cross-sectional area.

In general, the input impedance will vary from point to point in the network because of variations in vessel sizes and properties. If the network has the same impedance at each point (perfect impedance matching), there will be no wave reflections. Such a network will transmit energy most efficiently. The reflection coefficient R, defined as the ratio of reflected to incident wave amplitude, is related to the relative characteristic impedance of the vessels at a junction. For a parent tube with characteristic impedance Z0 branching into two daughter tubes with characteristic impedances Z1 and Z2, the reflection coefficient is given by:

and perfect impedance matching requires:

The arterial circulation exhibits partial impedance matching; however, wave reflections do occur. At each branch point, local reflection coefficients typically are less than 0.2. Nonetheless, global reflection coefficients, which account for all reflections distal to a given site, can be considerably higher [Milnor, 1989, p. 217].

In the absence of wave reflections, the input impedance is equal to the characteristic impedance. Womersley’s analysis predicts that impedance modulus will decrease monotonically with increasing frequency, whereas the phase angle is negative at low frequency and becomes progressively more positive with increasing frequency. Typical values calculated from Womersley’s analysis are shown in Fig. 13.1. In the actual circulation, wave reflections cause oscillations in the modulus and phase. Figure 13.2 shows input impedance measured in the ascending aorta of a human. Measurements of input resistance, characteristic impedance, and the frequency of the first minimum in the input impedance are summarized in Table 13.4.

FIGURE 13.1 Characteristic impedance calculated from Womersley’s analysis. The top panel contains the phase of the impedance and the bottom panel the modulus, both plotted as a function of the Womersley number Nw , which is proportional to frequency. The curves shown are for an unconstrained tube and include the effects of wall viscosity. The original figure has an error in the scale of the ordinate which has been corrected. Source: Milnor, W.R. 1989. Hemodynamics, 2nd ed., p. 172. Baltimore, Williams & Wilkins. With permission.

13.4 Velocity Profiles

Typical pressure and velocity fluctuations throughout the cardiac cycle in man are shown in Fig. 13.3. Although mean pressure decreases slightly toward the periphery due to viscous effects, peak pressure shows small increases in the distal aorta due to wave reflection and vessel taper. A rough estimate of mean pressure can be obtained as 1/3 of the sum of systolic pressure and twice the diastolic pressure. Velocity peaks during systole, with some backflow observed in the aorta early in diastole. Flow in the aorta is nearly zero through most of the diastole; however, more peripheral arteries such as the iliac and renal show forward flow throughout the cardiac cycle. This is a result of capacitive discharge of the central arteries as arterial pressure decreases.

FIGURE 13.2 Input impedance derived from the pressure and velocity data in the ascending aorta of Fig. 13.3. The top panel contains the modulus and the bottom panel the phase, both plotted as functions of frequency. The peripheral resistance (DC impedance) for this plot was 16470 dyne s/cm5. Source: Mills, C.J, Gabe, I.T, Gault, J.N. et al. 1970. Pressure-flow relationships and vascular impedance in man. Cardiovasc. Res. 4:405. With permission.

TABLE 13.4 Characteristic Arterial Impedances in Some Mammals:

Average (±SE)a,b

Abbreviations: Rin, input resistance (mean arterial pressure/flow) in dyn s/cm5; Zo, characteristic impedance, in dyn s/cm5, estimated by averaging high-frequency input impedances in aorta and pulmonary artery; value at 5Hz for other arteries, fmin, frequency of first minimum of Zi .

aValues estimated from published figures if averages were not reported.

bRanges for each species and vessel taken from values in Table 7.2 of source.

c103 dyn s/cm5.

Source: Milnor, W.R. 1989. Hemodynamics, 2nd ed., p. 183. Baltimore,

Williams & Wilkins. With permission.