Mathematical model representing blood flow in arteries John H. G e r r a r d

Linda A. T a y l o r

Department of the Mechanics of Fluids, University of Manchester, Manchester M13 9P/, England

mathematical model is described which can be used to calculate blood flow in a normal artery from pressures measured at two separated points. The equations of motion of fluid in an elastic tube are simplified but sufficient realism is retained for the application to arterial flow. A numerical solution to the equations averaged over each section of the tube is chosen and these equations are solved by the method of finite differences. A substitution is made for the hictional term which cann ot be expressed exactly in these n 9nlinear partial differentia I equations, The sensitivity of the results to changes in the friction term is demonstrated. A method is presented which bases the skin friction on a linear approximation, but within this/imitation uses a value which is correct in magnitude and phase. A correction is made for entrance-length effects. The wall properties are represented by a pressure-radius relationship developed from previous work by the authors (Taylor and Gerrard, 1976). The use of the mode/is illustrated by using experimental data quoted by Streeter et al. (1963). The solution compares favourably with the experimental results.

Abstract--A

Keywords--I-laemodynamics, Modelling, Pulsatile flow

1 Introduction PULSATILEblood flow in humans is at present difficult to measure without major surgery. It is therefore necessary to model the arterial system either theoretically or experimentally. When developing a theoretical model, one must simplify the equations of motion sufficiently to permit calculation of the required flow variables while at the same time maintaining the realism of the model. Various analytical and numerical approaches have been made using different simplifying assumptions. A thorough analytical investigation was made by WOMERSLEY (1955, 1957) of the linearised problem of a cylindrical vessel. The numerical technique can accommodate nonlinearity and has the advantage that solutions can be obtained as functions of time and of distance along the vessel. The same basic ldimensional equations have been used to calculate blood flow by STREETER et al. (1963), OLSEN and SHAPIRO (1967), HUNT (1969), ANLIKERet al. (1971) and WEMPLE and MOCKROS (1972). The technique employed by all these workers was the method of characteristics: LING and ATABEK (1972), RAINES et al. (1974) and the present authors have used the method of finite differences. The difference in the various approaches is in the manner in which account is taken of the motion of the wall and the skin friction between the wall and the fluid inside. The skin friction must be independently determined in these models because velocity variation across the artery is integrated out and thus the resultipg First received 20th September 1976 and in final form 4th April 1977

Medical & Biological Engineering & Computing

equations contain only the cross-sectional mean velocity. The use of the method of characteristics results in a restriction on the form of the skin, as pointed out by ANLIKER et al. (1971).It is possible to circumvent this restriction of the method of characteristics by employing an approximation and an iterative procedure as was shown by ROCKWELL et al. (1974) in their consideration of the effects of wall viscoelasticity. The formulation of LtNO and ATABEK (1972) does not demand an independent assessment of skin friction but does require values of pressure gradient as well as the pressures at two separated points. RAINES et al. (1974) employed a zero frequency form of the skin friction. TAYLOR and GERRARD (1977) have considered the pressure~adius relationship which takes account of the wall motion. The results of that work are applied in this paper. The main contribution of the present paper is in the manner in which the skin friction is calculated. In all cases the friction determination is approximate. We present an improvement on the previous work in the field and this is discussed separately below. The ultimate use of this model is to predict normal reference levels of blood flow in arteries for individual patients undergoing reconstructive arterial surgery of the superficial femoral artery. Currently the simplest, and what promises to be the most reliable, method of determining at operation whether the flow is sufficient, is to calculate the flow waveform from pressures measured at each end of the graft on the basis that the vessel in between is a normal artery. Comparison with the measured flow then indicates whether the graft is adequate.

November 1977

611

2 Assumptions and basic equations I n principle it would be possible to perform a complete numerical computation of the flow through an artery, but this has not yet been done. Computing time and storage would be very large and the problem is difficult. It is necessary to make calculations on a simplified model of the system. The basic assumptions are as follows: (a) Although there is some evidence (e.g. NEREM et aL 1972) that flow is disturbed in the decelerating phase there is no information of a general form on which the inclusion of such effects could be based. We therefore adopt the assumption that the flow is laminar; (/9) The flow is axisymmetric; (c) The fluid density is constant; (d) That the flow is Newtonian is justified in the larger arteries according to MCDONALD (1974); (e) That fluid properties are homogeneous and isotropic is related to assumption (d) and is also discussed by McDONALD (1974); (f) Pressure is assumed not to vary along the radius. This is reasonable since the wavelength of the oscillations is considerably greater than the vessel radius. The effect of radial dependence of the velocity is integrated out and as a consequence friction at the wall has to be independently introduced; (g) The material properties of the vessel wall are homogeneous and isotropic. The external tethering of the artery is described by PATEL and FRY (1966). That the vessel wall is incompressible is discussed by CAREW et al. (1968). The long-wavelength assumption results in the equations of motion of the wall simplifying to the static equations. TAYLORand GERRARD(1976) have.alreadyreported on the pressureradius relationship which is used. The excess pressure p is related to the radius R by p = (Eho/3Ro)(1 - Ro'*/R4)Oo

. . . .

(1)

where E = Young's modulus for the wall material, ho = wall thickness trader zero excess pressure, Ro = internal radius under zero excess pressure, 0o = (1 + ho/2Ro)/(1 +ho/Ro) 2 = thickness factor. The quantity Eho/Do is often written as pco 2, where p is the fluid density and Co is the pulse-wave speed for an inviscid fluid in a thin-walled tube. The wave velocity a for an inviscid-fluid-filled tube is given by a 2 = (R/2p)ep/#R

. . . . . . .

(2)

and for the pressure-radius relationship of eqn. 1 this becomes a 2 = 4Co20o/3--2pip = aoZ--2p/p

O)

There has been some expression of doubt about the use of wave-speed equations which indicate a decrease in speed with increasing pressure since, in arteries, the opposite is found experimentally. Fortunately the model is not sensitive to the changes in a 2 represented by the pressure term. The main 612

purpose of this paper is to present the method of skin-friction calculation and it is acknowledged that the use of a pressure-radius relationship which strictly only holds for rubber tubes could be improved uporl for arteries. A range of speeds ao have been used. The wave speed is set by the phase difference of the given pressures at the two ends of the vessel under consideration. For a given pressureradius relationship and a sinusoidal pulse, there is only one value of ao which produces the same phase difference between the velocities computed at the two ends. For nonsinusoidal arterial pulses, the value of ao should be a function of frequency. This is discussed further in Section 5. A derivation of the basic equations of motion of the fluid is to be found in STI~rETER et al. (1963). These are the equation of continuity ~A/~t+~( V A ) / g x = 0 . . . . . .

(4)

and the momentum equation, p~3V/~t + pe(3 V/c3x + (3p/r + 2T/R = 0

(5)

where x = distance along the tube, t = time, A = internal area of cross-section of the vessel, V = cross-sectional mean velocity, p = fluid density, p = pressure, R = internal radius of the vessel, and r = skin friction between the fluid and the wall. These equations become the wave equation with wave speed squared, a 2 = (R/2p)~3p/c3R when z = 0 and the amplitude is very small. Incorporating a, the continuity equation, eqn. 4, can be written V~p/~x + tgp/~t + pa2~V[~x + (2pVa2/Ro)~Ro/OX = 0 .

(6)

The last term in the equation arises from the taper of the vessel, Ro being the equilibrium internal iadius which is a ftmction of x. In some of the calculations a 2 was assumed to depend on x. The case in which ao 2 is replaced by a ' = ao2Ro(O)/Ro(x) has been investigated. When this variation is assumed, a 2 in the last term in eqn. 6 is replaced by a ' - 3p]2p. Eqns. 5 and 6 are solved together with eqn. 3. The value of the skin friction z has to be independently determined. 3 Inclusion of wall friction r The equations to be solved retain the nonlinearity of the problem. On the other hand, all the determinations of skin friction are based on linear models. The principal features of the skin friction which it is necessary to retain in the model are that it increases as the frequency of the oscillating flow increases and also that the phase of the skin friction leads that of the mean velocity by greater amounts as the frequency increases. That the friction term is a good representation is important, because the effect of altering r is greater than that of changing the pressure-radius relationship when the amplitude is

Medical & Biological Engineering & Computing

November 1977

not very large and greater than the effect of an incorrect value of the wave speed a. STREETER e t al. (1963), acknowledging that the skin friction is greater in oscillating than in steady flow, accounted for the increase by assuming a turbulent flow expression for r. RAINES e t al. (1974) assumed a steady laminar-flow friction, but increased the value of the coefficient of viscosity. The value of the friction factor or viscosity enters as an additional parameter which should be a function of frequency. In neither of these treatments is there a phase difference between z and flow. OLSEN and SHAaIRO (1967) and WEMVLEand MOCKROS (1972) have used the expressions for the wall shear, and hence r, derived from WOMERSLEY'S (1955) analysis of oscillating flow in a rigid cylindrical tube. OLSEN and SHAVmO (1967) consider only sinusoidal, singlefrequency waveforms. In their determination of r, the full expression is used but, by taking the real part of a complex solution at an early stage, the final result lacks a term in the rate of change of velocity--an omission which they acknowledge. The error thus produced may differ according to the waves studied. WEMPLE and MOCKROS assumed that the friction was as given by the extension of WOMERSLEY'S(1955) analysis at a single frequency equal to twice the pulse rate. This is a reasonable approximation, but is in error in both amplitude and phase. The approximation is necessary because the method of characteristics was used. In the present method the skin friction is determined from a synthesis of its Fourier components and so, as far as the linear approximation is adequate, we determine the correct value of the friction term. This involves using a method of successive approximation, since the Fourier analysis can only be made when the velocity waveform is known. A method is used, however, which ensures convergence with reasonable rapidity. The analytical values of r determined for each Fourier component are those relating to a rigid cylindrical tube of the local diameter. The analysis could be undertaken for a distensible tube but the effect on r is too small to warrant the added complication. In a rigid cylindrical tube the solution for axial velocity can be calculated as a function of pressure gradient. F r o m this the cross-sectional mean velocity can also be determined in terms of the pressure gradient. Elimination of the pressure gradient between the two results gives an expression for axial velocity in terms of the sectional mean velocity. This is shown in detail by UCH1DA (1956) and by GERRARD (1971). The resulting expression can be differentiated to obtain an expression for the skin friction (TAYLOR, 1972) m

r = 4 1 t V o / R o + Y~IuV~ r= l

o~

az

No

(72

cos ( r o g t - 2r + g - q~) .

(7)

Medical & Biological Engineering & Computing

where ot = Ro(og/v)*

/z = coefficient of viscosity and v = coefficient of kinematic viscosity. The cross-sectional mean velocity with fundamental angular frequency co is given by m

V = Vo + Y~ Vr cos (root - 2,)

. . . .

(8)

r=l

F o r the arterial pulse the number of harmonics m = 6 was found to be more than adequate. The quantities al, a2, 6 and ~bare the following functions of ~: 0"I

=

%/(DI 2 Jr D 2

2)

a2 = V'{(l - 2Dx/a) 2 + (2D2/~t) 2} tan 6 = (1--2D~/~)/(2Dz/ct) tan ~b = D 2 / D 1

D1 =

ber(~) b e i ' ( ~ ) - bei(~) ber'(~) ber2(~) + bei2(~)

D2 =

ber(a) ber'(=)+ bei(=) bei'(~) ber2(~ ) + bei2(~t)

where Dx and D2 involve the Kelvin functions and their derivatives. All the quantities in eqn. 7 except Vo, Vp and 2p are functions of the constant parameters of the problem and can be calculated once for the whole computation. To use eqn. 7 for r, the velocity must be known. To obtain the first approximation to the velocity, the value of r used in eqn. 5 was the laminar steady flow value in the vessel of equilibrium radius r

=

4#V/Ro

In this expression V is the velocity local in time and distance which was determined one time step previously. When the computational procedure furnishes the steadily oscillating velocity this is Fourier analysed to give Vo, 11, and 2r of eqn. 8. The corresponding value of z from eqn. 7 is now used as the skin friction and the process repeated until becomes constant. The steadily oscillating output with one set of values of z is within 1 ~ of the final value after two cycles but the calculation has been continued for five cycles between each F o u r i e r analysis. The number of repetitions of these five cycles which are required to produce the final waveform to a fraction of 1% is three. I n the development of the procedure 15 repetitions were used. The convergence of the procedure is discussed below. The skin friction has also been corrected for November 1977

613

entrance-length effects. This is necessary if the pressure measurements to be used in the calculation of velocity are taken in the inlet length following a bifurcation. The pressure drop Ap experienced in an entrance length is closely proportional to the square root of the axial distance. Following CAMPBELLand SLATTERY (1963) we write Ap 89 z

o~

xv . § D2 V

The friction term in eqn. 5 is therefore written 2~ --if- ( x d x ) * i f x < xr

where x~ is the entrance length. The entrance length is discussed by KASS~AN~OESand GERRARD (1975). I n the present case, because ~ is low, no correction is made for the reduction of entrance length in oscillating flow.

4 Solution of the equations by finite-difference methods The equations of motion, eqns. 5 and 6, were solved by the method of finite differences, after substitution of a 2, R and z from eqns. 3, 1 and 7, respectively. Central finite-difference forms of the gradients of quantities were used, resulting in the equations Vlk + l -- Vi~-- i 2dr

= _ [(p,+r _ p K l ) / p + VIK(VU+I_ V , r ) ] / 2 d x -

2r~K/pRi r

(9)

and ( p r + 1 - P K+i 1,,2d )/ t =

_[Vk(p,+Kl_pK+,) + (pao2 _ 2p~k). (Vi ~ - Vi K+,) + 2 V~ka2k/Ro(c3Ro/~X)]/2dx .

(10)

where the suffixes irefer to the number of the distance step along the vessel length and the superscripts K refer to the number of the time step. The pressures at the ends of the vessel are given functions of time. The time and distance steps dt and d x must be small compared with the period and the tube length, respectively; also, d x / d t must be larger than the pulse speed to ensure stability of the computation as well as to produce an accurate result. Typical values were dt = period/2000 and d x = l e n g t h / l O . For the initial conditions (of which the results are independent) zero flow and uniform pressure was assumed. Time was stepped in intervals dt through the cycle of the waveform. At each time step the 614

boundary conditions (the pressure at the ends) were assigned and the distance stepped in increments dx along the internal length of the tube, that is excluding the points at the ends. At each station the friction value is set, a new velocity calculated from eqn. 9 and a new pressure from eqn. 10. Before passing to the next time step, the velocity values at the ends are determined from the known end pressures by application of the continuity condition to the section between the end and the first internal point. The iteration required to obtain the correct value of the skin friction was described in Section 4. The program is sensitive to the precise value and phase of the friction term. This sensitivity is greatest where velocity changes sign, because here the direction of the sectional-mean velocity and the friction are the same. The computational procedure had a tendency to become unstable. Stability could be achieved by incorporating extra friction in phase with the flux. The stability was also sensitive to the precise values of the pressures used as end conditions. Initially the end pressures were inserted at 40 or 50 intervals in the cycle and linear interpolation used between these times. A considerably more stable situation followed the smoothing of the end pressures. This was achieved by a Fourier analysis of the end pressures, including six harmonics and performing a Fourier synthesis at each time step. The analysis had already been performed to determine the wave speed. This behaviour seems to be connected with the findings of AYL~KER et al. (1971) who found difficulty in the calculation of flux from end pressures. They concluded that either the mechanical description of the wall properties was inadequate or that the calculation w~s sensitive to small pressure changes and that the pressure may not be obtainable with sufficient accuracy. The in-phase extra friction was included in such a way that it had a diminishing effect as the calculation progressed. The mean velocity Vo in eqn. 7 was written as the difference between the velocity calculated minus the oscillating part determined from the Fourier analysis of the previous iteration. In this way the equation used for the friction automatically tended to eqn. 7 as the correct value of the friction was attained. This ensured stability of the convergence. The values of the disposable parameters are set by considerations of stability rather than accuracy. The inaccuracy of the initial data precludes the use of high accuracy in the calculations. The final computer programs continued for two cycles at each value of'r except in the last iteration where four cycles were followed. The number of iterations was five. In this condition the computing job used 117 seconds on the University of Manchester's 1906, A7600 machine. Jobs have been run with one cycle per iteration and three iterations in 56s, but could not be guaranteed to be stable even though the results were less than one part in 1000 different from those of the final programs referred to above.

Medical & Biological Engineering & Computing

November 1977

5 Results and conclusions

The data from a dog's femoral artery on which the model wastested were abstracted from the publication of STREETER et al. (1963). The relevant information including the wave speeds computed from the pressures measured at the ends of the segment is given in Table 1. Streeter calculated the flux from these end pressures. The calculations have been repeated using the present model and, as far as possible, the same data. The flux calculation was based on fixed diameters at the ends since flux was measured there with a cuff type of electromagnetic flowmeter which remains in contact with the arterial wall. It was assumed that the quoted diameters correspond to the minimum pressure at the measuring Table. 1 Wave speeds and other data used in calculating the flux from the measured pressures

Source : StreeteretaL (1963) Canine femoral artery: length 94 mm; end diameters 3"16 and 2"28 mm ; linear taper ; branches ligated

H armonic

Wave speed cm/s

1 2 3 4 5 6

987 779 583 492 676 288

Foot-tospeed cm/s

442

E L)

stations. A range of wave speed (a in eqns. 3 and 6) was tried. The values adopted were those for which the wave speeds of the first three harmonics of the flux most nearly equalled those of the pressures which are given in Table I. Values of a corresponding to the minimum of the pressure waveform will be quoted. For calculation with the skin friction equated to %, its zero frequency value, a =444 cm/s, was adopted and the corresponding wave speeds for the three harmonics were 1256, 782 and 418 cm/s. For the skin friction calculated from the present full linear treatment, a = 4 8 2 c m / s for which the corresponding wave speeds were 1434, 754 and 404 cm/s. Large discrepancies were found at the higher harmonics. As expected the wave speed producing the best agreement between the phase differences of the pressures and velocities corresponded closely with the foot-to-foot speed. Fig. 1 compares calculations and experiment. Besides Streeter's results, calculated values for three skin friction values are shown; those corresponding to zero frequency skin friction (to = 4 / t V / R ) and those corresponding to the method outlined above with and without the application of augmented friction in the entrance length. At the frequencies of this waveform, altering the friction is seen to affect the amplitude of the calculated flux and to slightly affect its phase. It must be stressed that mean flux cannot be determined accurately because this involves too fine an accuracy on the measurement of pressure. For this reason, in the medical application

2

X

9

E o

+|174 +~-+

-2

I

0.1

+ / i /

1

___

O. 2

[

0.3

0.4

Time seconds Fig. 1 Volume flux for one period of oscillation in a canine femoral artery Streeter et al. (1963) - - - -

experiment

. . . .

calculation

Medical & Biological Engineering & Computing

Present model 4- skin friction = zero frequency value, entrance-length correction included x full linear treatment of the skin friction, entrance-length correction included 9 . as last without entrance-length correction

November 1977

615

of this work, the correlation between the calculated and measured wave shapes is used as the criterion of a satisfactory result. The closest agreement with experiment is obtained with the m e t h o d employing the full friction but agreement with one set of measurements is not considered significant to the aim of this paper which presents a working m e t h o d of calculation of the skin friction by successive approximation. The advantages of this method of solution of the equations of m o t i o n are that (1) The boundary conditions depend upon pressure and not the pressure gradient.

the

(2) The nonlinear terms are included. (3) The frictional term is as near to the true pulsatile value as can be obtained and does not depend on arbitrary substitutions nor on measured volume flux. (4) A correction can be made for entrance-length effects. (5) Exactly the same m e t h o d can be applied at different sites in the arterial tree in animals and humans. Acknowledgment--We gratefully acknowledge the assistance we have received in this work from our colleagues D. Charlesworth and F. Cave in the Manchester University Department of Surgery.

References ANLIKER, M., ROCKWELL,R. L. and OGDEN, E. (1971) Nonlinear analysis of flow pulses and shock waves in arteries Z A M P 22, 217. CAMPBELL,W. D. and SLATTERY,J. C. (1963) Flow in the entrance of a tube. Trans. A S M E ser. D 85, 41. CAREW, T. E., VAISHNAV,R. N. and PATEL,D. J. (1968) Compressibility of the arterial wall. Circ. Res. 23, 61. GERRARD, J. H. (1971) An experimental investigation of pulsating turbulent water flow in a tube. J. Fluid Mech. 46, 43.

HUNT, W. A. (1969) Calculations of pulsatile flow across bifurcations in distensible tubes. Biophys. J. 9, 993. KASS~ANIDES,E. and GERRARD,J. H. (1975) The calculation of entrance length in physiological flow. Med. & Biol. Eng. 13, 558. LING, S. C. and ATAaEK, H. B. (1972) A nonlinear analysis ofpulsatile flow in arteries. J. FluidMech. 55, 493. McDONALD, D. A. (1974) Blood flow in arteries. Edward Arnold, 2nd edn. NEREM, R. M., SEED,W. A. and WOOD, N. B. (1972) An experimental study of the velocity distribution and transition to turbulence in the aorta. J. Fluid Mech. 52, 137. OLSEN, J. H. and SHAPIRO,A. H. (1967) Large amplitude unsteady flow in liquid-filled elastic tubes. Ibid. 29, 513. PATEL, D. J. and FRY, D. L. (1966) Longitudinal tethering of arteries in dogs. Circ. Res. 19, 1011. RAXNES,J. K., JAFFRIN,M. Y. and SHAPIRO,A. H. (1974) A computer simulation of arterial dynamics in the human leg. J. Biomech. 7, 77. ROCKWELL, R. L., ANLIKER, M. and ELSNER, J. (1974) Model studies of the pressure and flow pulses in a viscoelastic arterial conduit. J. Franklin Inst. 297, 405. STREETER,V. L., KE1TZER,W. F. and BOHR,D. F. (1963) Pulsatile pressure and flow through distensible vessels. Circ. Res. 13, 3. TAYLOR, L. A. (1972) Pulsatile flow through a tapered distensible tube. M.Sc. thesis. Manchester University. TAYLOR,L. A. and GERRARD,J. H. (1976) Pressure-radius relationship for elastic tubes and their application to arteries. Med. & Biol. Eng. & Comput. 15, 11. UCHIDA,S. (1956) The pulsating viscous flow superposed on the steady laminar motion of incompressible fluid in a circular pipe. Z A M P 7, 403. VAN DER WERFF, T. J. (1974) Significant parameters in arterial pressure and velocity development. J. Biomech. 7, 437. WEMPLE, R. R. and MOCKROS,L. F. (1972) Pressure and flow in the systemic arterial system. Ibid. 5, 629. WOMERSLEY,J. F. (1955) Oscillatory motion of a viscous liquid in a thin walled elastic tube. I. The linear approximation for long waves. Phil. Mag. 46, 199. WOMERSLEY, J. R. (1957) Oscillatory flow in arteries: the constrained elastic tube as a model of arterial flow and pulse transmission. Phys. Med. Biol. 2, 178.

Un mod61e math6matique de 1"6coulement du sang dans les art6res Sommaire---Est ddcrit un module mathrmatique pouvant 6tre utilis6 pour calculer la circulation sanguine dans une art~re normale ~t partir de pressions mesur6es en deux points diff&ents. Les 6quations 6tablissant le mouvement d'un fluide dans un tube 61astique sont simplifires mais suffisamment rralistes pour pouvoir 6tre appliqures ~tla circulation artrrielle. On choisit une solution numrrique pour les 6quations dont la moyenne a 6t6 6tablie pour chaque section du tube; ces 6quations sont r&olues par la mrthode des diffrrences finies. On choisit un terme de substitution pour la constante de frottement qui ne peut pas 6tre exprim6e avec exactitude par ces 6quations diffrrentielles partielles non linraires. On drmontre combien les rrsultats sont sensibles aux changements de la constante de frottement. On prrsente une mrthode qui 6tablit le frottement sur les parois grace b. une approximation linraire mais qui, dans le cadre des limites imposres, utilise nranmoins une valeur correcte en amplitude comme en phase. Les effects longueur-entr6e sont corrigrs. Les proprirt& des parois sont reprrsentres par un rapport pression/rayon dabor6 ~ partir d'un prr~dent ouvrage 6crit par les auteurs (TAYLOR et GERRARD, 1976). L'utilisation du modrle est illustrre par l'emploi de donnres exprrimentales cit&s par STREETERet al. (1963). La solution est parfaitement conforme aux r6sultats exp&imentaux. 616

Medical & Biological Engineering & Computing

November 1977

Ein mathematisches Modell zur Darstellung des Blutdurchflusses in den Arterien Zusammenfassung--Ein mathematisches Modell wird beschrieben, mit dessen Hilfe man den Btutdurchfluo in einer normalen Arterie aus dem an zwei verschiedenen Punkten gemessenen Druck berechnen kann. Die Gleichungen der Fltissigkeitsbewegung in einem elastischen Schlauch sind zwar vereinfacht, aber ausreichend der Wirklichkeit entsprechend, um auf die Arterien angewmdet werden zu k6nnen. Man entscheidet ftir eine numerische LOsung der Gleichungen im Durchschnitt der Schlauchabschnitre, und die Gleichungen werden durch die Methode der endlichen Differenzen gel6st. Der Reibungsausdruck, der in diesen nicht-linearen partiellen Differenzial-gleichungen nicht genau bestimmt werden kann, wJrd substituiert. Die Sensitivifiit der Ergebnisse gegeniiber ~nderungen des Reibungsausdruckes wird aufgezeigt. Eine Methode wird vorgeftihrt, die die Hautreibung auf einem Linearn~iherungswert begrtindet; innerhalb dieser Grenzen verwendet sie, was Gr6Be Phase betrifft, einenrichtigen Wert. Fiir Eintrittsl~ingeneffekte wird eine Eerichtigung vorgenommen. Die Eigenschaften der Arterienwand werden durch ein Druck/Radiusverh~iltnis dargestellt, das in frtkheren Artikeln yon den Verfassern entwickelt wurde (TAYLORund GEkR.ARD 1976). Die Verwen, dung dieses Modells wird anhand yon Versuchsdaten, die yon STREETERu . a . (1963) angefiihrt wurden, dargelegt. Die Losung laBt sich vorteilhaft mit Versuchsergebnissen vergleichen.

Medical & Biological Engineering & Computing

November 1977

617