KOZABUROHAYASHI
Okayama University of Science Okayama, Japan
INTRODUCTION
The elastic properties of the arterial wall are very impor- tant because they are closely related to arterial phy- siology and pathology, especially via effects on blood flow and arterial mass transport. Furthermore, stresses and strains in the arterial wall are prerequisite for the understanding of the pathophysiology and mechanics of the cardiovascular system. Stresses and strains cannot be analyzed without exact knowledge of the arterial elasticity.
STRUCTURE OF ARETRIAL WALL AND BASIC CHARACTERISTICS
Arteries become smaller in diameter with increasing dis- tance from the heart, depending on functional demands (1). In concert with this reduction in size, their structure, chemical composition, and wall thickness-inner diameter ratio gradually change in a way that leads to a progressive increase both in stiffness and in their ability to change their inner diameter in response to a variety of chemical and neurological control signals.
Arterial wall is inhomogeneous not only structurally, but also histologically. It is composed of three layers (intima, media, and adventitia), which are separated by elastic membranes. Because the media is much thicker than the other two layers and supports load induced by blood pressure, its mechanical properties represent the properties of arterial wall. The media is mainly composed of elastin, collagen, and cells (smooth muscle cell and fibroblast). Roughly speaking, elastin gives an artery its elasticity, while collagen resists tensile forces and gives the artery its burst strength. Smooth muscle cells contract or relax in response to mechanical, chemical, and the other stimuli, which alters the deformed configuration of arteries. The wall compositions vary at different locations depending on required functions. For example, collagen
and smooth muscle increase and elastin decreases at more distal sites in conduit arteries; the ratio of collagen to elastin increases in more distally located arteries. Collagen and elastin are essentially similar proteins, but collagen is very much stronger and stiffer than elastin. Therefore, the change of arterial diameter developed by blood pressure pulsation depends on the arterial site; it is larger in more proximal arteries.
Like most biological soft tissues, arteries undergo large deformation when they are subjected to physiological loading, and their force-deformation and stress–strain relations are nonlinear partly because of the above- mentioned inhomogeneous structure and partly because of the nonlinear characteristics of each component itself. Since collagen is a long-chained high polymer, it is intrin- sically anisotropic. Moreover, not only collagen and elastin fibers, but also cells, are oriented in tissues and organs in order so that their functions be most effective. Inevitably, the arterial wall is mechanically anisotropic like many other biological tissues. Biological soft tissues including arterial wall demonstrate opened hysteresis loops in their force–deformation and stress–strain curves, which means that those tissues are viscoelastic. In such materials, the stress state is not uniquely determined by current strain, but depends also on the history of deformation. When a viscoelastic tissue is elongated and maintained at some length, load does not stay at a specific level, but decreases rather rapidly at first and then gradually (relaxation). If some constant load is applied to the tissue, it is elongated with time rather rapidly at first and then gradually (creep). Viscoelastic materials generally show different stress– strain properties under different strain rates. It is true, and higher strain rates give higher stresses. However, such a strain rate effect is not so much in biological soft tissues like arteries, namely, their elastic properties are not more sensitive to strain rate. Therefore, it is not always neces- sary to consider viscoelasticity for arterial mechanics; it is very often enough to assume wall material to be elastic. Many biological soft tissues contain water of > 70%. There- fore, they hardly change their volume even if load is applied, and they are almost incompressible. The incom- pressibility assumption is very important in the formula- tion of constitutive laws of soft tissues, because it imposes a constraint on the strains and they are not independent.
MEASUREMENT OF ARTERIAL ELASTICITY In VitroTests
It is widely recognized that the mechanical properties of blood vessels do not change for up to 48 h if tissues are stored at 4 8C (1). One of the basic methods for the determination of the mechanical properties of biological tissues is uniaxial tensile testing on excised specimens. In this test, an increasing force is steadily applied to the longitudinal direction of a specimen, and the resulting specimen deformation is measured, which gives relations between stress (force divided by specimen cross-sectional area) and strain (specimen elongation divided by reference specimen length). Thisin vitro test is simple but, never- theless, provides us with basic and useful information on
the mechanical properties of tissues. Dumbbell-shape spe- cimens, helically stripped specimens, and ring specimens are commonly used for arterial walls.
Under in vivo conditions, arteries are tethered to or constrained by perivascular connective tissues and side branches, and pressurized by blood from inside. These forces develop multiaxial stresses in the wall. For the determination of the mechanical characteristics of arteries under multiaxial conditions, biaxial tensile tests on flat specimens are utilized to simultaneously apply forces in the circumferential and longitudinal directions; however, the effect of wall radial stress is ignored in this case.
Although stress–strain data obtained from the above- mentioned uniaxial and biaxial tests on flat, strip, and ring specimens are often used to represent the elastic properties of arterial walls, the data obtained from pressure–diameter tests on the tubular segments of blood vessels are more important and realistic. An example of the test devices is shown in Fig. 1 (2). A tubular specimen is mounted in the bath filled with Krebs–Ringer solution, which is kept at 37 8C and aerated with 95% O2and 5% CO2gas mixture.
Then, it is extend to thein vivo length to mimic the in vivo condition, because arteries inside the body are tethered to the surrounding tissues as mentioned above and, there- fore, they are extended in the axial direction. A diaphragm- type actuator, which is controlled with a sequencer, is incorporated in the device for the application of internal pressure to the specimen. The internal pressure or speci- men diameter can be controlled with the sequencer during pressure–diameter tests. The pressure is measured with a fluid-filled pressure transducer, while the outer diameter of the specimen is determined with a video dimension analyzer combined with a CCD camera. If the measure- ments of axial force are required in order to obtain pres- sure–diameter–axial force relations for the purpose of determining multiaxial constitutive laws, a load cell attached to one end of the vessel can be used.
In VivoMeasurements
It may be more realistic to obtain data fromin vivo experi- ments underin situ conditions rather than to get data from in vitro biomechanical tests. As a result of recent progress
in ultrasonic techniques, arterial diameter and even arter- ial wall thickness can be measured noninvasively with fairly good precision. These methods are being used not only forin vivo animal experiments, but also for clinical diagnosis of vascular diseases. It is true that the data obtained from these experiments and clinical cases are very useful, and provide important information concerning arterial mechanics. On the other hand, it is also true that many factors considerably affect the results obtained. These include physiological reactions to momentary changes in body and ambient conditions as well as the effects of anesthesia and respiration. In addition, since there has been some difficulty in applying the methods to small-diameter blood vessels, accurate measurements of vascular diameter and wall thickness with current techni- ques have been mostly limited to aortas and large arteries. Before noninvasive ultrasonic techniques were devel- oped, in vivo measurements of vascular diameter were invasively performed following surgical exposure of blood vessels, using strain gauge-mounted cantilevers, strain gauge-pasted calipers, and sonomicrometers. For example, a pair of miniature ultrasonic sensors may be used for the measurement of the outer diameter of a blood vessel (3). They are attached to the adventitial surface of a blood vessel so as to face each other across the vascular diameter. The diameter is determined from the transit time of the pulses between the two sensors. Similar sonomicrometers have been used for the measurement of arterial diameter not only in anesthetized, but also in conscious animals.
The noninvasive measurement of the elastic properties of arteries offers several significant advantages over inva- sive techniques. First, the nontraumatic character of the measurement guarantees a physiological state of the arter- ial wall, whereas such key functional elements of the wall as endothelium and smooth muscle might be affected in certain invasive measurement techniques. Second, it is of great clinical interest because it allows the monitoring of many outpatients and, therefore, it is well adapted for epidemiological or cross-sectional studies.
Noninvasive measurement of the arterial diameter can be done with ultrasonic echo-tracking techniques; recent improvements of the original technique have been pro- posed, which include digital tracking, prior inverse filter- ing, and coupling with B-mode imaging (1).
There exist no direct ways to measure pressure non- invasively in large central arteries, such as the aorta. Thus, regardless of the progress of ultrasonic and magnetic reso- nance imaging techniques which allow for the noninvasive measurement of vascular diameter, mechanical properties, such as compliance and elastic modulus cannot be derived from first principles. Therefore, primarily for clinical use, as an indirect, but noninvasive way of estimating the mechanical properties, the pulse wave velocity, c (see the next section), is often obtained from the measurements of pulsation at two distinct points along the vessel. One of the major drawbacks of this technique is low accuracy. The other one is that it yields a single value for the wave velocity. Because of the nonlinear elastic properties of the arterial wall, the pulse wave velocity sensitively changes depending on blood pressure. Therefore, the determination of a single value or a typical value of the arterial stiffness
86 ARTERIES, ELASTIC PROPERTIES OF
Actuator CCD camera Diaphragm Pressure transducer Load cell Specimen Valve 95%O2 + 5%CO2 Sequencer Video dimension analyzer Strain
amp. Strainamp.
Reservoir Recorder
Krebs-Ringer solution (37uC)
Figure 1. An in vitro experimental setup for the pressure– diameter–axial force test of a tubular arterial specimen. Internal pressure or outer diameter can be controlled with a feedback system (2).
estimated from the pulsation does not provide a full descrip- tion of the mechanical properties of the arterial wall.
MATHEMATICAL EXPRESSION OF ARTERIAL ELASTICITY Uniaxial Tensile Behavior
There are many tensile test data from arterial walls in humans and animals (4). Arterial walls exhibit nonlinear force-deformation or stress–strain behavior, having higher distensibility in the low force or stress range and losing it at higher force or stress. To represent strain in such biological soft tissues that deform largely and nonlinearly, we com- monly use extension ratio, l, which is defined by the ratio of the current length of a specimen (L) to its initial length (L0). If we plot a stress/extension ratio curve as the slope of
a stress/extension ratio curve versus stress, we can see that the relation is composed of one or two straight lines (1). Each line is described by
dT=dl¼ BT þ C (1)
whereT is Lagrangian stress defined by F/A0(F, force; A0,
cross-sectional area of an undeformed specimen), and B andC are constants. This is also expressed by
T¼ A½exp Bðl 1Þ 1 (2)
whereA is equal to C/B. This type of exponential formula- tion is applicable to the description of the elastic behavior of many other biological soft tissues (5).
Pressure–Diameter Relations
For practical purposes, it is convenient to use a single parameter that expresses the arterial elasticity under living conditions. In particular, for noninvasive diagnosis in clinical medicine, material characterization should be simple, yet quantitative. For this purpose, several parameters have been proposed and commonly utilized (1). These include pressure–strain elastic modulus, Ep
and vascular compliance,Cv. Pulse wave velocity,c, which
was mentioned above, is also used to express elastic properties of the arterial wall. These parameters are described by
Ep¼ DP=ðDDo=DoÞ (3)
Cv¼ ðDV=VÞ=DP (4)
and
c2¼ ðS=rÞðDP=DSÞ ¼ ðV=rÞðDP=DVÞ (5) whereDo,V, and S are the outer diameter, volume, and
luminal area of a blood vessel at pressureP, respectively, and DDo, DV, and DS are their increments for the pressure
increment, DP. The parameter r is the density of the blood. To calculate these parameters, we do not need to measure the wall thickness; for Ep and Cv, we need to
know only pressure–diameter and pressure–volume data, respectively, at a specific pressure level. However, we should remember that these parameters express the stiff- ness or distensibility of a blood vessel. Therefore, they are
structural parameters, and do not rigorously represent the inherent elastic properties of the wall material; in this sense, they are different from the elastic modulus which is explained below. In addition, these parameters are defined at specific pressures, and give different values at different pressure levels because the pressure–diameter relations of arteries are nonlinear.
To overcome this shortcoming, several functions have been proposed to mathematically describe pressure– diameter, pressure–volume, and pressure-luminal area data, and one or several parameters included in these equations have been used for the expression of the elastic charac- teristics of arteries. Among these functions, the following equation is one of the simplest and most reliable for the description of pressure-diameter relations of arteries in the physiological pressure range (6):
lnðP=PsÞ ¼ bðDo=Ds 1Þ (6) wherePsis a standard pressure andDsis the wall dia-
meter at pressure Ps. A physiologically normal blood
pressure like 100 mmHg (13.3 kPa) is recommended for the standard pressure,Ps. As an example, Fig. 2 shows
the pressure–diameter relationships of a human femoral artery under normal and active conditions of vascular smooth muscle and the relations between the logarithm of pressure ratio, P/Ps, and distension ratio, Do/Ds.
Figure 2a demonstrates nonlinear behavior of the artery under both conditions, while Fig. 2b shows the close fit of the data to Eq. 6 over a rather wide pressure range. The coefficient, b, called the stiffness parameter, represents the structural stiffness of a vascular wall; it does not depend upon pressure. This parameter has been used for the evaluation of the stiffness of arteries not only in basic investigations, but also in clinical studies (1).
As can be seen from Fig. 2a, under the normal condition, arteries greatly increase the diameter with pressure under a low pressure range, say < 60 mmHg (8 kPa), and then gradually lose the distensibility at higher pressures. When vascular smooth muscle cells are activated by stimuli, arteries are contracted and their diameter decreases in a
8.0 7.0 6.0 5.0 0 50 100 150 200 1.3 1.2 1.1 1.0 0.9 0.8 2.5 2.0 1.5 1.0 0.5 Pressure ratio P /Ps Distension ratio Do/Ds Internal pressure P (mmHg) External diamater Do (mm) Ps Active (Contracted) Normal ln(P/Ps)=β(Do/Ds–1) (β=3.2) (β=11.2) Active Normal β=Stiffness parameter Ds Ds (a) (b)
Figure 2. Pressure–diameter (a) and pressure ratio–distension ratio (b) relations of a human femoral artery under normal and active conditions (in vitro study) (1,7).
physiological pressure range and below the range [< 200 mmHg (26.6 kPa) in Fig. 2], and their pressure–diameter curves become greatly different from those observed under the normal condition.
To express the elastic properties of wall material, it is necessary to use a material parameter such as elastic modulus or Young’s modulus, which is the slope of a linear stress–strain relation. For arterial walls that have non- linear stress–strain relations, the following incremental elastic modulus has been often used for this purpose (8):
Huu¼ 2D2iDoðDP=DDoÞ=ðD2o D2iÞ þ 2PD2o=ðD2o D2iÞ (7) whereDiis the internal diameter of a vessel. This equation
was derived using the theory of small elastic deformation superposed on finite deformation in the case of a pressur- ized orthotropic cylindrical tube.
To calculate this modulus, it is necessary to know the thickness or internal diameter of a vessel. In in vitro experiments, we can calculate them fromDo, the internal
and external diameters under no-load conditions measured after pressure–diameter testing, thein vivo axial extension ratio, and assuming the incompressibility of wall material. Noninvasive measurement of wall thickness or internal diameter on intact vessels has been rather difficult com- pared with the measurement of external diameter; how- ever, it is now possible with high accuracy ultrasonic echo systems as mentioned above.
Constitutive Laws
Mathematical description of the mechanical behavior of a material in a general form is called a constitutive law or constitutive equation. We cannot perform any mechanical analyses without knowledge of constitutive laws of materi- als. Strain energy functions are commonly utilized for formulating constitutive laws of biological soft tissues that undergo large deformation (5). LetW be the strain energy per unit mass of a tissue, and r0be the density in the zero-
stress state. Then, r0W is the strain energy per unit volume
of the tissue in the zero-stress state, and this is called the strain energy density function. Because arterial tissue is considered as an elastic solid, a strain energy function exists, and the strain energy W is a function solely of the Green strains:
W ¼ WðEi jÞ (8)
where Eijare the components of the Green strain tensor
with respect to a local rectangular Cartesian coordinate system.
Under physiological conditions, arteries are subjected to axisymmetrical loads, and the axes of the principal stresses and strains coincide with the axes of mechanical ortho- tropy. Moreover, the condition of incompressibility is used to eliminate the radial strainErr, and therefore the strain
energy function becomes a function of the circumferential and axial strainsEuuandEzz. Then, the constitutive equa-
tions for arteries are
suu srr¼ ð1 þ 2EuuÞ@ðr0WÞ=@Euu (9)
and
szz srr¼ ð1 þ 2EzzÞ@ðr0WÞ=@Ezz (10) where suu, szz, and srr are Cauchy stresses in the
circumferential, axial, and radial directions, respectively. Thus, we need to know the details of the strain energy function to describe stress–strain relations.
Three major equations have so far been proposed for the strain energy function of arterial wall. Vaishnav et al. (9) advocated the following equation:
r0W¼ ðc=2Þexpðb1E2rrþ b2E2uuþ b3E2zz
þ 2b4ErrEuuþ 2b5EuuEzzþ 2b6EzzErrÞ ð11Þ whereEuuandEzzare Green strains in the circumferential
and axial directions, respectively, andA, B, and so on, are constants.
Chuong and Fung (10) proposed another form with an exponential function:
r0W ¼ ðc=2Þexpðb1E2rrþ b2E2uuþ b3E2zz
þ 2b4ErrEuuþ 2b5EuuEzzþ 2b6EzzErrÞ ð12Þ wherec, b1,b2, and so on, are material constants.
Later, Takamizawa and Hayashi (11) proposed a loga- rithmic form of the function described by
r0W¼ C lnð1 auuE2uu=2 azzE2zz=2 auzEuuEzzÞ (13) whereC, auu,azz, andauzcharacterize the elastic properties
of a material.
By using one of these strain energy equations or another type of equation forW in Eqs. 9 and 10, and applying the equations of equilibrium and boundary conditions, we determine the values of material constants. Although all of the proposed formulations describe quite well the elastic behavior of arterial walls, we prefer to reduce the number of constants included in the equations in order to handle