The Elastic Properties of Arteries in Relation to the Physiological Functions of the Arterial System

G .-\ ST RO EN TEROLOGY

Vol. 52, No.2, Part 2 Printed in U.S.A .

Copy right © 1967 by The Willia ms & Wilkins Co.

THE ELASTIC PROPERTIES OF ARTERIES IN RELATION TO THE PHYSIOLOGICAL FUNCTIONS OF THE ARTERIAL SYSTEM MICHAEL

G.

TAYLOR, M.D., PH.D.

Department of Physiology, University of Sydney, Sydney, Australia

In this communication it is not proposed to devote much time to the detailed structure of the arterial wall, since it can safely be assumed that this is familial' to most people. It is intended, however, to put forward some general observations on the properties of the system which appeal' to be of interest and importance in understanding the physiological functions of arteries. The approach adopted will be to treat the arterial system as a problem in design, beginning with an outline of its fun ctions and then attempting to determine wha t elastic properties would give optimal performance of these functions. Although this will be an exercise in teleology, it is hoped that it will permit one to obtain a more or less coherent view of the remarkable properties which the arteries do, in fact, possess. For the sake of simplicity, we shall define the arteries as those vessels which connect the heart with the small, muscular "resistance" vessels which finally control the flow of blood to an organ or region. Thi s effectively restricts the discussion to vessels of diameter greater than about 14 mm. We shall assume three things about the circulation: (a) the animal requires a certain volume of blood to be supplied to its tissues per unit of time, and because of t he resistance to flow through the peripheral vascular bed, this flow requires a cert ain driving pressure; (b) the action of the heart is intermittent, so that the flow into the arteries is pulsatile; (c) changing levels of activity of the whole or parts of the animal will involve changes in total blood flow requirement, probably accompanied The support of a Grant-in-Aid from the National Heart Foundation of Australia is gratefully acknowledged. 358

by changes in the pattern of the heart's action. In the light of these postulates, we may therefore define certain "tasks" of the arterial system: (1) it should provide a pathway of low resistance from the heart to the periphery, so that the total driving pressure is minimized; (2) it should possess a low input impedance for pulsatile flow to minimize the additional work which the heart must do, because its output is intermittent ; (3) its characteristics should be such that if optimal function is achieved under (1) and (2) ,it is maintained over a range of values of cardiac output and/ or heart rate. The usual method of approaching this kind of problem in design is to set up a cost function for the system and choose its parameters in such a way that the cost is minimized. At first sight, it might seem that the appropriate cost function here would be a n expression for cardiac work, which could be brought to a minimum by an appropriate choice of elastic and other properties of the arteries. It soon appears, however, that there are other costs to the animal which are of great importance, but which are very difficult to state quantitatively. These costs are related to two properties of the system, namely: (a) the total volume of blood in the arteries, in relation to both the whole animal and the volume of other parts of the vascular systern ; (b) the total distensibility of the arterial system. Thus, if we seek to minimize the pressure drop along the arteries by making their radii very large, then this obviously involves an increase in arterial blood volume. While it is not easy to evaluate the "cost" of this, one can see that if the volume became very large, then the animal would be under a number of disadvantages.

F eb1'Uary 1967

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ARTERIAL SYSTEM AND ELASTIC PROPERTIES

Two which come readily to mind are the expenditure of energy in simply carrying about an increased mass of blood (increased body weight) and, more remotely, the metabolic requirements of the increased mass of bone marrow needed to maintain the large red cell population. No doubt one could think of other similar disadvantages which would set an upper limit to the volume of the arterial system, but they will all be difficult to specify quantitatively. With regard to the distensibility of the system, one can see that if this is large, then the pulsatile pressures will be small, and this will tend to reduce the work of the heart. There is, however, a physiological cost which the animal must pay for a very distensible arterial system whenever there is a change in mean arterial blood pressure. The first component of this cost is the shifting of blood into or out of the venous side of the circulation. If there were large changes in venous volume accompanying changes in mean arterial pressure, special compensatory mechanisms would be required. The second component of cost is related to the speed with which changes in blood pressure can be brought about. If the arterial reservoir were very distensible, then an increase in blood pressure could only be slowly produced for a given card,i ac output, since the system would take a long time to inflate. The matter of the "response time" of the arterial reservoir is, of course, intimately bound up with the response times of the various other parts of the cardiovascular system and cannot readily be considered in isolation. However, for the present purposes we may observe that the broad aspects of the animal's way of life and modes of activity must be taken into account. An animal whose successful existence depends upon rapid changes in activity and rapid changes in mean arterial blood pressure would be severely penalized by an arterial system which was very distensible and had a response time that was long in relation to the time scale of the changes in activity. The preceding discussion may be summarized by saying that although one may consider the optimal design of the

arterial system to be one which provides minimum load on the heart, this must be achieved under conditions which keep the total arterial volume and the total arterial distensibility as small as possible. Resistance to Steady Flow

Although we are principally concerned with the elastic properties of the arteries, it is necessary to deal briefly with the question of steady flow, since we have stated that one of the "tasks" of the arterial system is to provide a low resistance pathway from the heart to the periphery. In an interesting paper, Cohn l has discussed the manner in which a system of bifurcating vessels could be designed in order to supply a certain volume of blood per unit of time to the capillary vessels of a tissue. He treated the tissue as divisible into cubes of the minimum size compatible with the diffusion of materials from the supplying capillary, and he obtained equations which prescribed the relations between the radius of any vessel and its order of branching, so that the pressure drop along the system was a minimum. His solution was obtained under the condition that the wall material was of fixed amount; this may appear somewhat strange, but it arose from considerations of the elastic properties of the wall and the relation between wall thickness and vessel radius. It is of particular interest, in terms of our present discussion, that the equations remain the same if one makes no assumptions about the wall but uses, instead, the condition that the total internal volume of the system remains fixed. The form of the solution is independent of the actual volume, but in any particular case it is ultimately determined by the dimensions of the terminal vessels. Cohn found that the minimum resistance to flow was obtained when the radius of a vessel after bifurcation (ri+d was related to the radius before bifurcation (r.) , by the following expression. Ti+l =

rJ(2)1/3

=

0.794 r,

(1)

In terms of the areas of the system, this leads to the result that the total area after a bifurcation has increased by a factor of

360

TAYLOR

1.26, which is of the order said to exist in t he arterial system. Input Impedance to Pulsatile Flow

While it is customary to deal with steady pressure-flow relations in terms of resistance, one may also consider pul~atile pressure-flow relations in terms of Impedance; t hat is, if one knows the input impedance of a system such as the arterial tree, then, given an oscillatory input of flow, one may calculate the associated oscillatory pressure. The impedance, being a complex quantity. determines not only the amplitude relationship between pressure and flow but also their relative phase, or timing. Input impedance is defined for oscillatory sinusoidal flow and pressure, but it is still an extremely useful quantity, since it can be used in calculations involving pulsatile signals, if these are expressed as t heir Fourier series. In what follows we shall, t herefore, consider the relations ?etween the elastic properties of the artenes and the input impedance of the system. We are concerned with the relation between pressure and flow at the origin of the arterial system, since it is the pressure here which will determine the load on the heart, and this is what we are trying ~o minimize. If the flow into the system IS intern1ittent (one of our primary assumptions), then the pressure at the origin will be pulsatile and will depend upon the input impedance. If we express the flow pattern (F) as the sum of a constant term and a Fourier series of oscillatory terms, where f is the heart frequency, F+

F(t) =

L

F(j )

(2)

then the pressure (P) which will be generated can be written P (t)

=

F·R

+L

F(j)·Zo(j)

(3)

where Zo (f) is the input impedance and R is the resistance to steady flow. The

external work (W) of the heart, averaged over one cycle, can be written TV =

T1

jT o

P(t) ·F (t) ·dt

(4)

Vol. 52, No.2, Par t 2

and, substituting the previous expressions and integrating, we have W

+ 7'2 L 1F(j) 1 .1 Zo(j) 1 cos if>(j) (steady) + (pulsatile)

= F2·R =

2

(5)

We havethus expressed the total external work of the heart as a steady component and a pulsatile component; it should be noted that the expression uses the quantity, (j), the phase of the impedance when written in complex form. Since we have assumed that both the mean flow and the peripheral resistance are given quantities, and since we have shown that by a suitable choice of radius relationships we may minimize the pressure drop along the arteries, it now only remains to consider ways in which the pulsatile component can be minimized. The pulsatile component represents the extra work which the heart must do because its ejection is intermittent; the steady component is irreducible. It may be pointed out, however, that in fact the mammalian arterial system is so constructed that the pulsatile component amounts to only about 10% of the total; the remarkable combination of properties of the arteries which bring this about is the main topic of this discussion. We have already considered the situation which would arise if the arteries were very distensible. It is clear that, although this would provide a very low impedance for pulsatile flow and thus minimize the work of the heart, the physiological disadvantages would be great. What, therefore, would be the result of choosing an arterial system with characteristics at the opposite extreme; that is, an entirely rigid system of tubes? The relations for steady flow would be unchanged, but if the arteries were rigid, the input impedance would be made up of the resistance to steady flow itself and a term involving the mass of blood in the arteries which has to be accelerated at each beat. If the form of cardiac ejection were to be the same as is normally found (which is, of course, highly unlikely under these extreme conditions), then the pressure pulsations which such a system would generate would be 10 to 20 times greater

ARTERIAL SYSTEM AND ELASTIC PROPERTIES

February 1967

than those whi ch actually occur; furthermore, the flow of blood in the peripheral vessels would be intermittent instead of continuous. We may thus exclude both the very distensible and the very rigid arterial systems. It will be helpful at this stage to consider the relations between the elastic properties of a tu be (expressed as its wave velocity, c) and the input impedance. If we take the very simple case of a single tube made of ideally elastic material filled with an inviscid fluid, then provided that there are no reflected oscillations present, the pressure at the origin is related to the linear velocity of flow by the so-called water-hammer formula, P

= p·i! ·C, (Zo =

p'c)

(6)

where v is the linear velocity of flow and p is the fluid density. If there were no reflected waves in the arterial system, then the solution to our problem would be easy; all that would be required would be to choose a value of the wave velocity which provided a suitable compromise between the load on the heart and the total distensibility of the system. However, because the arteries terminate in vessels of high resistance, then we Ulust take account of possible reflections from these endings. If reflections are present, then the input impedance will not be constant but will be frequency-dependent, passing through maxima and minima as the system passes through its resonances. The imput impedance will be a minimum whenever the system is oscillating at a frequency such that there is a node of pressure at the origin (i.e., its length is an odd multiple of a quarter wave length and a maximum whenever there is an antinode of pressure there) . We have taken as one of our design criteria that not only should the input impedance be low, but that also it should be stable over a range of frequencies. We can meet this latter condition and eliminate reflections if we choose the elastic properties, i.e., the wave velocity, so that the terminations are "matched." However, the terminal resist ance is so high that, to

361

match it, the wave velocity would also have to be high, and a rough calculation can show that it would probably have to be at least 10 times higher than is actually found in animals. When we recall that the wave velocity is related to the elastic properties approximately by the expression, C

=

(Eh)lf 2 2Rp

(7)

where E is the Young's modulus of the wall material and hj 2R the wall thickness-diameter ratio, then we see that if the dimensions of the arteries remained unchanged, the elastic modulus would have to be more than 100 times its normal value. In a system without reflections, we would certainly have stable values of the input impedance and would satisfy the condition that the over-all distensibility of the system should be small, but that the cost in terms of cardiac work should be great. The pulsatile terms in equation 5 would now amount to about 50% of the cardiac work; the total work would be about double the previous value. It seems, therefore, that we should seek to achieve stability of the input impedance by means other than matching the terminal impedance. Before continuing with the question of stabilizing the impendance, let us turn to the question of making it small. This could be done if, for a given operating frequen cy, the wave velocity were chosen so that the system would be resonating over a quarter wave length with a node of pressure at the origin. This arrangement seems at first sight to be very attractive, but it cannot be accepted without some qualification. In the first place, although the input impedance may b e aminimum for some particular frequency, this will not be simultaneously true for all the harmonic components of the flow pUlse. Even though the magnitudes of the flow components decrease quite quickly with increasing order of the harmonic, so that, because of the quadratic terms in equation 5, we have only to consider the first f ew terms of the series, we must still take account of the fact that, if we place the first harmonic at a minimum of the impedance, then the second must lie at a max-

362

,TAYLOR

imum. Even so, this might be tolerable were it not for the condition that the optimum performance should be stable over a range of frequencies. If we were to depend for optimum performance upon the presence of some sharply defined resonant frequencies, then a shift in operating frequency could compromise its efficiency. At this stage, then, it appears that we are in something of an impasse; we can only achieve stability of performance by eliminating the influence of reflections, and we can only achieve low input impedance by taking advantage of their presence. Fortunately, there is a way out which involves not the choice of a single wave velocity for the whole system but the specification of a distribution of velocities along it. Further, the problem is made somewhat easier for us by the architectural arrangement of the arteries with their multiple branching. It can be shown that, in a system with many terminations scattered at different distances from the origin, if the frequency is high enough, the reflections returning to the origin may be sufficiently out of phase with each other to cancel. This can happen when the frequency is greater than that required to place the "average reflecting site" at a quarter wave length from the origin; when this is so, then the different path lengths from the origin represent appreciable fractions of a wave length, and the phases of the reflected components may lie randomly between 0 and 2 71'. If the reflected components can thus be made to cancel, then they can no longer influence the input impedance, so that it will become stable and no longer vary with frequency. For a system of given dimensions, the lower the wave velocity, the lower will be the limiting frequency at which this occurs. We see, therefore, that the very form of the arterial tree can make an important contribution to its function, and if we choose a wave velocity such that for the important harmonics of the flow pulse the impedance is small, at or near a resonant minimum, then the scattering of the arterial terminations will depress the influence of reflections and leave the input impedance rela-

Vol. 52, No.2, Part 2

tively stable for the higher frequencies. This property of the system will allow us to take advantage of a resonance due to reflections, without subsequently meeting increased impedance if the operating frequency is increased. We may achieve still further optimization of the system, if we take account of the special properties of a nonuniformly elastic tube. It can be shown that, if the wave velocity in an elastic tube increases along its length, then this has the effect of "uncoupling" the termination, so that reflections have far less influence upon the input impedance than they do in a tube in which the wave velocity is everywhere the same. Provided, once more, that the operating frequency is above a lower limit, the input impedance will become relatively stable and, furthermore, will be determined only by the local properties of the tube at its origin. Our best choice, therefore, is an arterial system in which the region near the heart is more distensible than the peripheral extensions, with a gradual transition between the two. If, in addition, we select the wave velocities, so that for the frequencies of the largest harmonic components of the flow pulse we make the input impedance a minimum by establishing quarter wave length resonance, then the effect of both elastic nonuniformity and interference of reflections from the scattered terminations will combine to maintain the input impedance at a low and stable value. It should be noted that this arrangement is also an excellent compromise as far as the. over-all distensibility of the system is concerned. The heart "sees" the low impedance provided by the distensible region near the origin, but the compliance of the whole system is kept small by the fact that the more distant branches have a high wave velocity and are, therefore, less distensible. Conclusion

From a consideration of the physiological role of the arterial system, we have thus arrived at a specification of its elastic and other properties which will permit the cir-

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363

DISCUSSION

culation of the blood by the heart with a minimum expenditure of energy. In conclusion, we may note that the mammalian arterial system does meet these specifications: (1) the elastic properties are nonuniformly distributed, so that the central vessels are more distensible (c = approximately 4 m per sec) than the peripheral ones (c = approximately 10 m per sec); (2) the relations between the dimensions of the animal and the wave velocity and heart rate are such that there is a broad resonant minimum of the input impedance over the range of frequencies for the major components of the input flow pulse; (3) the spatial distribution of the terminations of the major arteries is such as to contribute to the stabilization of the input impedance; (4) the total cross-sectional area progressively increases as it branches, in a manner which minimizes the steady pressure drop along it.

REFERENCES 1. Cohn, D. L. 1955. Optimal systems: II. The vascular system. Bull. Math. Biophys. 17: 219-227. The aim of this presentation has been to give a very general outline of the problem; the reader is referred to the following for detailed references to technical literature: Attinger, E. O. [ed.]. 1964. Pulsatile blood flow. McGraw-Hill Book Company, Inc., New York. Burton, A. C. 1954. Relation of structure to function of the tissues of the walls of bloodvessels. Physiol. Rev. 34: 619-642. McDonald, D. A. 1960. Blood flow in arteries. Edward Arnold and Company, London. Rudinger, G. 1966. Review of current mathematical methods for the analysis of blood flow. American Society of Mechanical Engineers. Biomedical Fluid Mechanics Symposium. Taylor, M. G. 1966. An introduction to some recent developments in hemodynamics. Aust. Ann. Med. 15: 71-86.

DISCUSSION OF "THE ELASTIC PROPERTIES OF ARTERIES IN RELATION TO THE PHYSIOLOGICAL FUNCTIONS OF THE ARTERIAL SYSTEM"

Alan C. Burton, Ph.D. Dr. Taylor's paper is, as we would expect from hini, a highly sophisticated exercise in "design" of the circulation and adds more illustrations of the remarkable way in which the design of living systems so often turns out to be optimal, and probably much better, than nonbiologist engineers would have been capable of inventing. It supports the view that I have long held that the growing interest of engineers and physicists in the function of living things, called by some "bioengineering," is likely to be more fruitful in providing new ideas for inanimate engineering than for biologists. (For example, if aeronautical engineers were willing to study the locomotion of the squid, they might achieve vertical take off jet aircraft much sooner.) Dr. Burton is from the Department of Biophysics, University of Western Ontario Medical School, London, Ontario, Canada.

As Dr. Taylor explains, optimal design involves "minimizing" some function with respect to some specific "cost" or a compromise in approximately minimizing more than one type of cost. He chose the cost of the external work of the heart as a major item to be minimized. However, in the total load of the heart, on which the O2 consumption which must be supplied by the coronary circulation to the heart muscle depends, the external work is, perhaps astonishingly, a rather minor factor though of some importance. Far more important is what is called the "tension-time integral" of the contraction of the muscle, which is proportional to the systolic pressure that has to be produced in the ventricles before and during ejection, and the duration of the contractions. This item is usually more than 80% (perhaps 90%) of the load on the heart. Experimen-