Biomechanics Principles and Applications - Donald R. Peterson & Joseph D. Bronzino

FIGURE 11.8 Pressure–radius curves for a fully orthotropic vessel calculated with an exponential strain energy density function.

for the canine carotid artery, where eθ and ez are the strains in a reference configuration at in situ length and pressure. Why should this work? One answer appears to be related to residual stresses and strains.

When residual stresses are ignored, large-deformation analysis of thick-walled blood vessels predicts steep distributions in σθ and σz through the vessel wall, with the highest stresses at the interior. This prediction is considered significant because high tensions in the inner wall could inhibit vascularization and oxygen transport to vascular tissue.

When residual stresses are considered, the stress distributions flatten considerably and become almost uniform at in situ length and pressure. Figure 11.10 shows the radial stress distributions computed for a

FIGURE 11.9 Longitudinal distending force as a function of radius for the orthotropic vessel.

FIGURE 11.10 Stress distributions through the wall at various pressures for the orthotropic vessel.

vessel with β = 1 and β = 1.11. Takamizawa and Hayashi have even considered the case where the strain distribution is uniform in situ [9]. The physiologic implications are that vascular tissue is in a constant state of flux. New tissue is synthesized in a state of stress that allows it to redistribute the internal loads more uniformly. There probably is no stress-free reference state [7,8,17]. Continuous dissection of the tissue into smaller and smaller pieces would continue to relieve residual stresses and strains [10].

References

[1]Von Maltzahn, W.-W., Desdo, D., and Wiemier, W. 1981. Elastic properties of arteries: a nonlinear two-layer cylindrical model. J. Biomech. 4:389.

[2]Carew, T.E., Vaishnav, R.N., and Patel, D.J. 1968. Compressibility of the arterial walls. Circ. Res. 23:61.

[3]Bergel, D.H. 1961. The static elastic properties of the arterial wall. J. Physiol. 156:445.

[4]Fung, Y.C., Fronek, K., and Patitucci, P. 1979. Pseudoelasticity of arteries and the choice of its mathematical expression. Am. J. Physiol. 237:H620.

[5]Dobrin, P.B. 1978. Mechanical properties of arteries. Physiol. Rev. 58:397.

[6]Choung, C.J. and Fung, Y.C. 1986. On residual stresses in arteries. J. Biomed. Eng. 108:189.

[7]Fung, Y.C., Liu, S.Q., and Zhou, J.B. 1993. Remodeling of the constitutive equation while a blood vessel remodels itself under strain. J. Biomech. Eng. 115:453.

[8]Rachev, A., Greenwald, S., Kane, T., Moore, J., and Meister J.-J. 1994. Effects of age-related changes in the residual strains on the stress distribution in the arterial wall. In J. Vossoughi (Ed.), Proceedings of the Thirteenth Society of Biomedical Engineering Recent Developments, pp. 409–412, Washington, DC, University of District of Columbia.

[9]Takamizawa, K. and Hayashi, K. 1987. Strain energy density function and the uniform strain hypothesis for arterial mechanics. J. Biomech. 20:7.

[10]Vassoughi, J. 1992. Longitudinal residual strain in arteries. Proceedings of the 11th South Biomed Engrg Conf, Memphis, TN.

[11]Vaishnav, R.N. and Vassoughi, J. 1983. Estimation of residual stresses in aortic segments. In C.W. Hall

(Ed.), Biomedical Engineering II, Recent Developments, pp. 330–333, New York, Pergamon Press.

The venous system not only serves as a conduit for the return of blood from the capillaries to the heart but also provides a dynamic, variable blood storage compartment that influences cardiac output. The systemic (noncardiopulmonary) venous system contains more than 75% of the blood volume of the entire systemic circulation. Although the heart is the source of energy for propelling blood throughout the circulation, filling of the right heart before the subsequent beat is primarily passive. The subsequent amount of blood ejected is exquisitely sensitive to the transmural filling pressure (e.g., a change of right heart filling pressure of 1 cm water can cause the cardiac output to change by about 50%).

Because the blood vessels are elastic and have smooth muscle in their walls, contraction or relaxation of the smooth muscle can quickly redistribute blood between the periphery and the heart to influence cardiac filling and thus cardiac output. Even though the right ventricle is not essential for life, its functioning acts to reduce the central venous pressure to facilitate venous return [1]. It largely determines the magnitude of the cardiac output by influencing the degree of filling of the left heart. Dynamic changes in venous tone, by redistributing blood volume, can thus, at rest, change cardiac output over a range of more than ±20%. The dimensions of the vasculature influence both blood flow — by way of their resistive properties — and contained blood volume — by way of their capacitive properties. The arteries have about 10 times the resistance of the veins, and the veins are more than 10 times as compliant as the arteries.

The conduit characteristics of the venous system primarily depend on the anatomy of the system. Valves in the veins of the limbs are crucial for reducing the pressure in dependent parts of the body. Even small movements from skeletal muscle activity tend to compress the veins and move blood toward the heart. A competent valve then blocks back flow, thus relieving the pressure when the movement stops. Even a few steps can reduce the transmural venous pressure in the ankle from as much as 100 mmHg to about 20 mmHg. Without this mechanism, transcapillary movement of fluid into the extravascular spaces results in edema. Varicose (swollen) veins and peripheral pooling of blood can result from damage to the venous valves. During exercise, the rhythmic contraction of the skeletal muscles, in conjunction with venous valves, provides an important mechanism — the skeletal muscle pump — aiding the large increases

12-1

12.1.5 Capacity

Capacity refers to the amount of blood volume contained in the blood vessels at a specific distending pressure. It is the sum of the unstressed volume and the stressed volume, V = V0 + Vs.

12.1.6 Mean Filling Pressure

If the inflow and outflow of an organ are suddenly stopped, and blood volume is redistributed so that all pressures within the vasculature are the same, this pressure is the mean filling pressure [2]. This pressure can be measured for the systemic or pulmonary circuits or the body as a whole. The arterial pressure often does not equal the venous pressure as flow is reduced to zero, because blood must move from the distended arterial vessels to the venous beds during the measurement maneuver, and flow may stop before equilibrium occurs. This is because smooth-muscle activity in the arterial vessels, rheological properties of blood, or high interstitial pressures act to impede the flow. Thus corrections must often be made [2,3]. The experimentally measured mean filling pressure provides a good estimate of Pv (the pressure in the minute venules), for estimating venous stressed volume.

12.1.7 Venous Resistance

Venous resistance (R) refers to the hindrance to blood flow through the venous vasculature caused by friction of the moving blood along the venous vascular wall. By definition it is the ratio of the pressure gradient between the entrance of the venous circulation, namely the capillaries, and the venous outflow divided by the venous flow rate. Thus

where R is the venous resistance, Pc is the capillary pressure, Pra is the right atrial pressure, and F is the venous flow. As flow is decreased to zero, arterial closure may occur, leading to a positive perfusion pressure at zero flow. With partial collapse of veins, a Starling resistor like condition is present in which an increase in outlet pressure has no influence on flow until the outlet pressure is greater than the “waterfall” pressure.

12.1.8 Venous Inertance

Venous inertance (Iv) is the opposition to a change in flow rate related to the mass of the bolus of blood that is accelerated or decelerated. The inertance Iv for a cylindrical tube with constant cross-sectional area is Iv = Lρ/ A, where L is the length of the vessel, ρ is the density of the blood, and A is the cross-sectional area [4].

12.2 Methods to Measure Venous Characteristics

Our knowledge of the nature and role of the capacitance characteristics of the venous system has been limited by the difficulty of measuring the various variables needed to compute parameter values. State-of- the-art equipment is often needed because of the low pressures and many disturbing factors present. Many of the techniques that have been used to measure venous capacitance require numerous assumptions that may not be correct or are currently impossible to evaluate [3].

12.2.1 Resistance

For the estimate of vascular resistance, the upstream to outflow pressure gradient across the tissues must be estimated along with a measure of flow. Pressures in large vessels are measured with a catheter connected to a pressure transducer, which typically involves measurement of minute changes in resistance elements

attached to a stiff diaphragm that flexes proportionally to the pressure. For the veins in tissue, the upstream pressure, just downstream from the capillaries, is much more difficult to measure because of the minute size ( 15 μm) of the vessels. For this a servo-null micropipette technique may be used. A glass micropipette with a tip diameter of about 2 μm is filled with a 1 to 2 mol saline solution. When the pipette is inserted into a vein, the pressure tends to drive the lower conductance blood plasma into the pipette. The conductance is measured using an AC-driven bridge. A servosystem, driven by the imbalance signal, is used to develop a counter pressure to maintain the interface between the low-conductance filling solution and the plasma near the tip of the pipette. This counter pressure, which equals the intravascular pressure, is measured with a pressure transducer. Careful calibration is essential.

Another approach for estimating the upstream pressure in the veins is to measure the mean filling pressure of the organ (see above) and assume that this pressure is the upstream venous pressure. Because this venous pressure must be less than the capillary pressure and because most of the blood in an organ is in the small veins and venules, this assumption, though tenuous, is not unreasonable. To measure flow many approaches are available including electromagnetic, transit-time ultrasonic, or Doppler ultrasonic flowmeters. Usually the arterial inflow is measured with the assumption that the outflow is the same. Indicator dilution techniques are also used to estimate average flow. They are based on the principle that the reduction in concentration of infused indicator is inversely proportional to the rate of flow. Either a bolus injection or a continuous infusion may be used. Adequacy of mixing of indicator across the flow stream, lack of collateral flows, and adequately representative sampling must be considered [5].

12.2.2 Capacitance

For estimating the capacitance parameters of the veins, contained volume, rather than flow, and transmural pressure, rather than the longitudinal pressure gradient, must be measured. Pressures are measured as described above. For the desired pressure-volume relationship the total contained volume must be known.

Techniques used to measure total blood volume include indicator dilution. The ratio of the integral of indicator concentration time to that of concentration is used to compute the mean transit time (MIT) following the sudden injection of a bolus of indicator [3,5]. The active volume is the product of MTT and flow, with flow measured as outlined above. Scintigraphy provides an image of the distribution of radioactivity in tissues. A radioisotope, such as technicium 99 that is bound to red blood cells that in turn are contained within the vasculature, is injected and allowed to equilibrate. A camera, with many collimating channels sensitive to the emitted radiation, is placed over the tissue. The activity recorded is proportional to the volume of blood. Currently it is not possible to accurately calibrate the systems to provide measures of blood volume because of uncertain attenuation of radiation by the tissue and distance. Furthermore, delimiting a particular organ within the body and separating arterial and venous segments of the circulation are difficult.

12.2.3 Compliance

To estimate compliance, changes in volume are needed. This is generally easier than measuring the total blood volume. Using plethysmography, a rigid container is placed around the organ, and a servo system functions to change the fluid volume in the chamber to maintain the chamber pressure-constant. The consequent volume change is measured and assumed to be primarily venous, because most of the vascular volume is venous. With a tight system and careful technique, at the end of the experiment both inflow and outflow blood vessels can be occluded and then the contained blood washed out and measured to provide a measure of the total blood volume [6].

12.2.4 Gravimetric Techniques

Gravimetric techniques can be used to measure changes in blood volume. If the organ can be isolated and weighed continuously with the blood vessels intact, changes in volume can be measured in response to drugs or reflexes. With an important modification, this approach can be applied to an organ or the systemic