Compliance

Abstract

When the transmural or distending pressure of an organ is increased, the volume increases. For biological organs the relation between transmural pressure and volume is, in general, not straight but convex to the volume axis, implying that, Compliance C = ΔV/ΔP, and Elastance, inverse of compliance, depend on pressure or volume. When organs of different size are to be compared we can normalize both C and E with respect to volume. These normalized descriptions are distensibility, and bulk modulus or volume elasticity, respectively. When cross-sectional area is measured, as is often done in blood vessels, we derive’area compliance’ and ‘area elastance’, where luminal area A replaces volume V. The rules for addition of compliances and elastances are discussed.

Compliance quantifies a pressure-volume relation. When the transmural or distending pressure is increased, the volume increases (left figure, as customary in arterial research). When volume is increased pressure increases (right figure, customary in cardiac research, and now gaining more attention in vascular research). For biological organs the relation is, in general, not straight but convex to the volume axis, implying that both slopes, Compliance C = ΔV/ΔP, and Elastance E = ΔP/ΔV, depend on pressure or volume. Elastance and compliance are the inverse of each other: E = 1/C. When organs of different size are to be compared we can normalize both E and C with respect to volume. These normalized descriptions are distensibility, C/V = (ΔV/ΔP)/V, and bulk modulus or volume elasticity, EV = VΔP/ΔV, respectively. When cross-sectional area is measured, as is often done in blood vessels, we derive’area compliance’ and ‘area elastance’, where luminal area A replaces volume V. When diameters or radii are measured the relations are called diameter and radius compliance and elastance, respectively. The rules for addition of compliances and elastances are discussed. The equivalent of the bulk modulus is the ‘pressure-strain’ or ‘Peterson’s modulus’, E _p = rΔP/Δr. Compliance and elastance are not material parameters, but structural parameters, they can be derived from the material properties, i.e., elastic modulus, and the geometry, e.g., diameter, wall thickness, etc.

1 Description

The advantage of the pressure-volume (or -diameter and -radius) relation (Figures in the Box) is that in vivo compliance can be determined. It is important to note that pressure-volume relations do not characterize the material alone but include the structure of the organ as a whole.

If the pressure-volume relations were straight and going through the origin, the slope, compliance or elastance would give the full characterization of the organ by a single quantity. However, in biology the pressure-volume relations are never straight. For a small change around a chosen working point, the curve is approximately straight, and the tangent of the pressure-volume curve can be used. We can determine compliance in the ‘working point’ as C = ΔV/ΔP. The elastance, the inverse of compliance is E = ΔP/ΔV. Of course, these local slopes depend on the pressure or volume chosen. Thus, when comparing compliance or elastance data one should report the chosen working point, i.e., the pressure at which compliance or elastance was determined. For instance, when the elastance of a heart in diastole is studied and appears increased, the increase can result from either a higher filling pressure but otherwise normal heart, or a normal filling pressure but a hypertrophied heart with thicker wall (Fig. 11.1).

Fig. 11.1

Examples of diastolic pressure-volume relations of a normal and hypertrophied heart. If only the similar elastance values are reported without further information on pressure or volume, it cannot be decided if the heart is normal and overfilled, or hypertrophied and under-filled, since both have the same E. At similar pressure the hypertrophied heart is stiffer (larger E). The full graph is required to give the complete information

The curvature of the pressure-volume relation is mainly the result of the fact that Youngs modulus increases with stretch, therefore C decreases and E increases, with volume.

1.1 Measurement of Elastance and Compliance

Elastance is customarily used for the heart, while compliance was mostly used to describe blood vessels, but the term stiffness is (meaning elastance) is now more and more used in vascular research as well.

Ventricular elastance is best determined by the measurement of pressure and volume. A number of noninvasive techniques are now available to determine ventricular volumes, such as Computed Tomography, Magnetic Resonance Imaging, Ultrasound Echo, while the Pressure-Volume catheter allows for determination of elastance in experimental situations [1]. Cardiac elastance determination requires volume and pressure measurements in systole and diastole, because of the varying properties of the cardiac muscle. Diameter as estimate of volume is inaccurate.

Compliance used in vascular studies can be based on cross-sectional area or on diameter (Fig. 11.2). Diameter changes can be measured by wall-tracking [2] and for large vessels like the aorta by MRI. From the local diameter the cross-sectional area is calculated assuming a circular cross-section. When area and pressure are related the term area compliance, C _A = ΔA/ΔP, is used to distinguish it from (volume) compliance. For instance, the systolic-diastolic differences in vessel area, ΔA, and pressure, ΔP, i.e., Pulse Pressure, when measured in vivo, can be used to obtain the area compliance. With modern echo-tracking techniques it is possible to determine vessel diameter noninvasively. In that case it is customary to report diameter compliance, C _D = ΔD/ΔP. The area compliance, C _A , and diameter compliance, C _D , are related by: C _A = π⋅D⋅C _D /2. The relations between the different expressions of compliance are given in Table 11.1.

Fig. 11.2

For geometrically simple shapes , like blood vessels, measurement of changes in internal radius or diameter is sufficient to obtain (radial or diameter) compliance. In complex geometries as that of the heart this cannot be done

Table 11.1 Summary of structural^a parameters of elasticity of blood vessels (length assumed constant)

1.2 Distensibility and Bulk Modulus

Compliance and elastance depend on the size of the organ under study. To compare properties of blood vessels, or hearts from different animal species we can normalize compliance and elastance with respect to the volume of the organ. We use C/V = (ΔV/ΔP)/V, called distensibility, and the inverse, E⋅V = (ΔP/ΔV)⋅V, called bulk modulus or volume elasticity . Area and diameter distensibilities are also used, area distensibility is (ΔA/A)/ΔP and diameter distensibility is 2⋅(ΔD/D)/ΔP.

1.3 The Pressure-Strain Elastic Modulus

Peterson [3] introduced the pressure-strain elastic modulus in blood vessel research. This measure of blood vessel elasticity requires the measurement of diameter and pressure only, and can be used to compare vessels of different size. The pressure-strain elastic modulus, or Peterson modulus [3], is defined as E _p = ΔP/(Δr _o     /r _o ), where usually external radius, r _o , instead of the internal radius r _i is used. The E _p compares to the bulk modulus.

1.4 The Stiffness Index β

Compliance, distensibility and Peterson’s modulus depend strongly on pressure. To describe the nonlinearity with a minimal number of parameters Hayashi et al. [4] introduced the stiffness index (or parameter) β, defined by the following relation

$$ \ln \left(P/{P}_o\right)=\beta \cdotp \left(D/{D}_0-1\right) $$

where P _o is a reference pressure (working point), typically mean pressure or 100 mmHg and D ₀ is the outer diameter at that pressure. Basic research and also several clinical studies have shown that the stiffness parameter β does not depend on pressure within the physiological pressure range. However, outside the physiological pressure range, and for pressures far from the reference pressure P _o , the β- stiffness index is not pressure-independent anymore. A recent study by Spronck et al. showed that a correction can reduce the pressure dependency [5].

1.5 Describing the Pressure-Area or Pressure-Diameter Relation of Blood Vessels

The pressure-area and pressure-diameter relations of blood vessels have been described in a number of ways. At a working pressure the slope of the relation gives compliance. However, description of the relation over a range of pressures and volumes gives more insight. Although these descriptions are phenomenological, we mention them here because of their general utility in arterial mechanics (Fig. 11.3).

Fig. 11.3

Mathematical models of pressure-diameter and pressure-area relations

The relation proposed by Langewouters et al. [6], describes the pressure-area relation over the widest range of pressures, i.e., from 0 to 200 mmHg. The relations of Fung [7] and Hayashi [4] can be applied over the physiological range of pressures. The D ₀ and P ₀ are reference values for the relations of Fung [7] and Hayashi [4]. In the relation by Langewouters A _m /2 and P ₀ designate the inflection point and P ₁ relates to the slope at the inflection point; A _m is the maximal, asymptotic, vessel area. The relations can also be presented in terms of volumes.

1.6 Addition of Compliances and Elastances

Let us consider the compliance of the entire aorta (Fig. 11.4), the individual compliances of three sections of the aorta are shown. In all sections the pressure is virtually equal. This implies that

$$ {\displaystyle \begin{array}{l}{C}_1+{C}_2+{C}_3=\varDelta {V}_1/\varDelta P+\varDelta {V}_2/\varDelta P+\varDelta {V}_3/\varDelta P\\ {}=\left(\varDelta {V}_1+\varDelta {V}_2+\varDelta {V}_3\right)/\varDelta P=\varDelta {V}_{total}/\varDelta P\end{array}} $$

or

$$ {C}_1+{C}_2+{C}_3={C}_{total} $$

Thus, simple addition of compliances is allowed and the total compliance is their sum and is therefore larger than the individual compliances. For vessels in parallel compliances may also be added. Thus for the entire systemic circulation the total arterial compliance is the sum of all local compliances.

Fig. 11.4

Addition of compliances. Note the choice of axes. The compliances of different sections can be added to obtain total aortic compliance. The whole graphs may also be summed, by adding volumes at similar pressures (dashed lines). The aa, da, and ab are ascending, descending, and abdominal aorta

If an organ with compliance C ₁ is enveloped with an organ with compliance C ₂ , the total volume change equals the individual volume changes (Fig. 11.5). In that case the pressures need to be added. The distending pressure of the inner organ is the luminal pressure minus the pressure in between the organs. The distending pressure of the outer organ is the pressure between the organs minus the pressure of the environment. Thus, the distending pressure acting on the two organs combined, i.e., the luminal pressure minus the external pressure is the sum of the distending pressures of the individual organs.

$$ 1/{C}_{total}=1/{C}_1+1/{C}_2 $$

In this situation addition of the elastances is easier. As an example, we use the heart with pericardium. When the transmural pressure over the ventricular wall is ΔP _v and over the pericardium is ΔP _pe , for the heart inside the pericardium the transmural pressure equals ΔP _total = ΔP _v + ΔP _pe . Therefore

$$ {E}_{total}=\varDelta {P}_{total}/\varDelta V=\left(\varDelta {P}_v+\varDelta {P}_{pe}\right)/\varDelta V=\varDelta {P}_v/\varDelta V+\varDelta {P}_{pe}/\varDelta V={E}_v+{E}_{pe} $$

with E _v , and E _pe the ventricular elastance and pericardial elastance, respectively. The implicit assumption is that the intra-pericardial pressure, ΔP _pe , is the same at all locations. There are indications that the actual situation is more complex.

Fig. 11.5

Addition of elastances. Note the choice of axes. Heart and pericardium have their individual elastances and the total elastance of the heart in pericardium can be obtained directly from addition of their individual elastances. Addition of the whole graphs is also allowed at similar volumes. Thus, for structures within each other the elastances can be added directly to obtain overall elastance. Transmural pressure over the ventricular wall is Δ P _v = P _ventr − P _pericard and transmural pressure of the pericardium is Δ P _pe = P _pericard − P _external

1.7 Relating Compliance to Youngs Modulus

The measurement of pressure-volume or pressure-radius relationships in arteries allows for derivation of compliance. However, as discussed in Chap. 9, estimation of Youngs modulus or incremental elastic modulus requires, in addition to radius and pressure, the measurement of wall thickness.

An accurate relation between the Youngs modulus and compliance, for a cylindrical vessel, is given by Love [8], and is used to model transverse impedance of an arterial segment (Appendix 3):

$$ {C}_A=3\pi \cdot {r_i}^2{\left({r}_i+h\right)}^2/{E}_{inc}\left(2{r}_i+h\right)h=3\pi \cdot {r_i}^2{\left(a+1\right)}^2/{E}_{inc}\left(2a+1\right),\mathrm{with}\ a={r}_i/h $$

For thin-walled vessels the equation reduces to:

$$ {C}_A\approx k\left(\pi \cdot {r_i}^3\right)/\left({E}_{inc}\cdot h\right),\mathrm{with}\ \mathrm{k}=1.5 $$

Sometimes k = 2 instead of 1.5 is used.

(Area) compliance, being a structural property, should be plotted against distending pressure. The incremental elastic modulus, being a material property, should be plotted against stress or strain. Plotting E _inc against pressure, as is often done, leads to misinterpretation of vessel properties.

An example where the structural aspect of compliance can be seen, is the comparison of the elastic properties of veins and arteries. The main reason why pressure-volume relations of veins differ from those of arteries is not the difference in wall material properties but their difference in wall thickness. More accurately stated, the ratio of wall thickness to radius is much smaller in veins than in arteries.

2 Physiological and Clinical Relevance

Compliance or elastance gives a quantitative measure of the mechanical and structural properties of an organ. Changes with disease and aging can be quantitatively investigated.

Arterial compliance decreases with age, and thus elastance (stiffness) increases with age and this is the main reason why arterial Pulse Pressure, Systolic minus Diastolic pressure, increases with age. The concomitant increase in systolic pressure is an extra load on the heart possibly leading to (concentric) hypertrophy. Concentric hypertrophy increases the elastance of the left ventricle in both diastole and systole. The increase in diastolic elastance results in decreased filling and filling volume can only return to near normal values with an increase in diastolic filling pressure (Fig. 11.6), which in turn may lead to increased pressure in the pulmonary veins and pulmonary edema.

Fig. 11.6

Loss of diastolic ventricular compliance, i.e., increased diastolic elastance means that distension (filling) of the left ventricle in diastole becomes more difficult. Even a higher filling pressure is not capable to reach sufficient end-diastolic volume. An increase in diastolic filling pressure implies an increase in the pulmonary venous pressure, leading to pulmonary edema

With the now available wall-track technique arterial diameters can be measured noninvasively in superficial arteries and if pressure is determined simultaneously as well, diameter compliance can be derived in large groups of patients [2]. However, we should realize that this is the local area compliance of a single, often peripheral, artery, such as the carotid or radial artery, and may not be a good measure of aortic compliance or total arterial compliance (see below and Chap. 25).

Compliance and elastance depend on volume and pressure. Comparison should thus be carried out at similar pressure. However, compliance and elastance, in contrast to the Youngs modulus, also depend on the size of the organ. Distensibility and volume elasticity account for vessel size and are often used for comparisons of groups.

2.1 Buffering Function of Arterial Compliance

Arterial compliance is the buffering element for pressure so that the oscillations in pressure during the cardiac cycle are limited (Chap. 25). The Pulse Pressure in the aorta, the difference between systolic and diastolic aortic pressure, is about 40 mmHg in the young healthy adult. It was shown by Randall et al. [9], in vivo, that an acute reduction of total arterial compliance results in a considerable increase in Pulse Pressure (Fig. 11.7). The effect of long term, i.e., ~60 days reduction of aortic compliance to 60%, increased systolic pressure by 31 mmHg and diastolic pressure by 10 mmHg without affecting Cardiac Output and peripheral resistance [10].

Fig. 11.7

Aortic pressure and flow in the intact dog during an acute decrease in aortic compliance, but unaltered peripheral resistance. The decreased compliance causes an increase in pulse pressure and decrease in flow. (Adapted from Ref. [9], by permission)

It is now accepted knowledge that increased Pulse Pressure is the strongest pressure-based indicator of cardiac mortality and morbidity [11, 12]. It has also been reported that cardiac elastance is affected when arterial compliance is decreased [13]. The scientific community is becoming more and more convinced that decreased compliance plays a major role in hypertension. In Chap. 25 it is shown that the change in compliance, with age, is considerable and contributes importantly to Pulse Pressure.