Assessment of distributed arterial network models

I Assessment of distributed arterial network models P. Segers 1 N. Stergiopuios 2 P. Verdonck 1 R. Verhoeven 1 1Hydraulics Laboratory, IBrrECH, University of Gent, Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium 2Biomedical Engineering Laboratory, EPFL, Swiss Federal Institute of Technology, Lausanne, PSE-Ecublens, 1015 Lausanne, Switzerland AbstractmThe aim of this study is to evaluate the relative importance of elastic nonlinearities, viscoelasticity and resistance vessel modelling on arterial pressure and flow wave contours computed with distributed arterial network models. The computational resu/ts of a non-linear (time-domain) and a linear (frequency-domain) mode were compared using the same geometrical configuration and identical upstream and downstream boundary conditions and mechanical properties. Pressures were computed at the ascending aorta, brachial and femoral artery. In spite of the identical problem definition, computational differences were found in input impedance modulus (max. 15--20%), systolic pressure (max. 5%) and pulse pressure (max. 10%). For the brachial artery, the ratio of pulse pressure to aortic pulse pressure was practically identical for both models (3%), whereas for the femoral artery higher values are found for the linear model (+10%). The aortic/brachial pressure transfer function indicates that pressure harmonic ampl/f/cation is somewhat higher in the linear model for frequencies lower than 6 Hz while the opposite is true for higher frequencies. These computational disparities were attributed to conceptual model differences, such as the treatment of geometric tapering, rather than to elastic or convective non-linearities. Compared to the effect of viscoelasticity, the discrepancy between the linear and non-linear model is of the same importance. At peripheral locations, the correct representation of terminal impedance outweights the computational differences between the linear and non-linear models. Keywords--Arterial network model, Linear, Non-linear, Viscoelasticity J Med. Biol. Eng. Comput., 1997, 35, 729-736 1 Introduction NETWORKMODELSof the arterial circulation range from early electrical analogue (WESTERHOF et aL, 1969) to computer models, eventually incorporating non-linearities and complex boundary conditions (SNYDERe t aL, 1968; SCHAm and ABBRECHT, 1972; WEMPLE and MOCKROS, 1972; AVOLIO, 1980; STERGIOPULOSet al., 1992; KRUS et al., 1991; SHENGet al., I995). The computational results of these models are similar: the intmt impedance of each of the models is equivalent to that measured in vivo (MILLS et al., 1970; MURGO et aL, 1980), and each model computes the propagation of pressure and flow waves along the arterial tree, mimicking observed wave phenomena such as the systolic presmare rise, diastolic pressure decay, wave attenuation and wave reflection. Differences exist in the way boundary conditions are treated, in the computational procedure and in the modelling of non-linearities. The upstream boundary condition can be a pressure or flow wave measured downstream of the aortic valve (WEMPLEand MOCKROS, 1972; AVOLIO, 1980; STERGIOPULOSet al., 1992; SHF-NGet al., 1995); the cardiovascular interaction may be modelled as well (SNYDERet al., 1968; Correspondenceshould be addressedto Dr Patrick Segers; emaiJ: Patrick.Segers@rug.ac.be First received 18 December 1995and in final form 21 March 1997 IFMBE:1997 Medical & Biological Engineering & Computing SCHAAF and ABBRECHT,1972; KRUS et al., 1991). Downstream, the arterioles and capillaries are modelled as simple resistances (SCHAAF and ABBRECH'r, 1972; AVOLIO, 1980; KRUS et al., 1991), or more complex impedance models (SNYDER et al., 1968; WEMPLE and MOCKROS, 1972; STERGtOVULOSet al., 1992; S~mN6 et aL, 1995). From a computational point of view, two approaches exist. First, the imegral form of the continuity equation and the momentum equations may be solved, complemented with an expression for the pressure-area relationship and an expression for the shear stress. The computation is performed in the time domain via an appropriate numerical technique (SNYDER et aL, 1968; SCHAAF and ABBRECHT, 1972; WEMPLE and MOCKROS, 1972; STERGIOPULOSet al., 1992; KRUS, et aL, 1991). This method enables the incorporation of a non-linear pressure-area relation and the non-linear convective terms of the Navier--Stokes equations. Secondly, based on the electrical analogy and the Womersley theory, it is possible to treat the arterial tree as a transmission line, consisting of line segments over which the propagation of individual pressure and flow harmonics can be computed (AVOLIO, 1980). Assuming linearity of the system, superposition of the different harmonics yields the propagated pressure and flow waves. The computation is performed in the frequency domain and enables a direct incorporation of viscoelasticity and a frequency-dependent shear function. Both approaches co-exist, but it is still not clear whether non-linearities are important in human arterial circulation. November 1997 729 Whereas the role of the convective acceleration terms is thought to be small (SCHAAP and ABBP.ECWr, 1972; WEMPLE and MOCKROS, 1972; REUDERINKet aL, 1989), the impact of the non-linear vessel pressure-diameter relation is expected to be more important. Nevertheless, based on experimental latex tube studies, it has been suggested that 'even in cases where non-linear effects are signifieamly present, it is better to use a linear 1D model that includes a satisfactory description of the fluid friction and the viscoelastic wall behaviour than a nonlinear 1D model in which these damping effects are strongly underestimated' (REuDERIrO: et al., 1989). Arterial network models axe becoming increasingly common for applications such as the validation of compliance estimation techniques (STERGIOI'ULOS et al., 1995), wave reflection analysis (KARAMANOGLUer aL, 1994) or to provide boundary conditions for more detailed numerical flow studies (REUDER1NK et aL, 1993), replacing the human body as the object of study. Thus it is important to know which physical properties and quantities are essential to guarantee physiologically relevant results. In this study we evaluate the importance of i) elastic non-linearities, ii) viscoelasticity and iii) boundary conditions. 2 Materials and methods We compare the computational results of a linear and a nonlinear (STERG1OPULOSet aL I992) arterial network model. The linear model is further used for the implementation of different viscoelastic models and for the evaluation of the impact of the terminal impedance model on pressure and flow wave morphology. 2.1 Non-linear arterial network model An extensive description of this model is found in (STER- GIOPULOSet al., 1992)..The model allows the computation of pressure and flow wave propagation over the arterial tree. The computations involve numerical integration of the continuity and momentum equations, together with an expression for the wall shear stress (YOUNG and TSAt, 1973) and the pressm-ediameter (or pressure-area) relation of the arteries. The computations are performed in the time domain. Arterial distensibility and compliance are non-linearly depeaxdent on pressure (LANGEWOUTERSet al., 1984); this non-linear relation is expressed (STERGIOPUIs et al., 1994) as: - - = ~ 3P ~P loo=~ar~ ( (1) a-t 1+ where A is the vessel cross-section, P is the pressure, a and b are constants (taken as 0.4 and 5, respectively), Pa = 2 0 m m Hg and Pb = 30trmaHg, and ('~'x~ \or/ is the 2.2 Linear arterial network model The computational procedure for calculating pressure and flow in the linear transmission line model used here is similar to (AVOLIO, 1980). The computations are carried out in the frequency domain. Pressure and flow are written as the sum of a steady component (Poiseuille flow), on which sinusoidal harmonics are superimposed. The computational procedure is based on the theory of oscillatory flow (individual harmonics) in uniform, longitudinally tethered elastic tubes (WOMERSLEY, t957). The tapering arteries are modelled as a sequence of cylindrical segments, with stepwise varying segment properties. Each segment is modelled as a transmission line element (Fig. 2) for which the following equations apply: 1110 m m H g reference area compliance at 100 ram Hg. The arterial network configuration of (STERGIOPULOSet al., 1992) consists of 55 arterial network segments (Fig. 1). More specifications about the dimensions and configuration of the arteries, the terminating 3-element 'windkessel' models (representing the smallsized arteries, arterioles and capillaries) and the pressure-area relation for the different vessels are found in (STERGIOPULO$ et aL, 1992). 730 Fig. 1 Geometrical configuration used for the comparison of the linear and non-linear models of the arterial circulation. The segments correspond to the non-linear model. The computational results are compared at segment 3 (oacending aorta, 4cm), 7 (brachial artery, 44cm) and 50 (femoral artery, 80 cm) 1 (2) where P is the pressure, Q the flow, 7 the complex wave number, L the vessel segment length, Z0 the characteristic impedance of the vessel segment and the indices are 1 for the Medical & Biological Engineering & Computing November 1997 tax ( 2.0 ": rdx ~ I. " L~ ! ~.a" q- ~ Fig. 2 Schematic representation of an infinitesimal transmission L6" L4 - - line model segment of length dx. r = resistance, e = compliance and 1= inertance (all properties per unit of length) 1,0 V 0 upstream and 2 for the downstream site. The characteristic impedance of the line segment is given as r•+-jcol zo=V | 1 I 2 | 3 I 4 I 5 I 6 I 7 I 8 I 9 I 10 frequency, Hz (3) 14 12 I with 9~ i '%e ........ AA Pet2 sin(e'10),l = ~ c o s ( e ' 1 0 ) , r ---- rcg4Mi ~ 7tK'MIO o d 2nR 3 c ----E(co)ph the transmission line parameters representing resistance, inertianee and compliance per unit of length dx. /z is the dynamic blood viscosity (4.5mPa's), p the blood density (I 050 kg m-~), R the vessel segment radius at mean pressure, h the thickness a~ the angular frequency, E(co)p the complex dynamic elasticity modulus (a function of co and mean pressure), j the complex constant ~ and M~0 and e'10 complex functions of ~ the Womersley parameter (r~,VOMERSLEY, 1957). With known segment properties, one can calculate the input impedance for each frequency and each location of the arterial network in a retrograde computation, starting from the downstream end of the peripheral segment where the impedance is known. In a forward computation, the propagation of an individual harmonic, starting from the upstream inlet boundary condition (either pressure or flow) is determined. Summation of all harmonics and the steady flow component leads to the reconstruction of the propagating pressure and flow wave. The non-linear relation between instantaneous pressure and elasticity modulus is difficult to implement since it is the propagation of individual harmonics that is studied and so the instantaneous pressure (sum of all harmonics) is unknown. Nevertheless, AC and DC computations are uncoupled, and since mean pressure is known at all locations, one can easily account for a non-linear relation between the elastic (frequency-dependent) properties and mean medal pressure. In analogy with STERO[OPULOSet al. (1995), this relation is expressed as eqn. 2: g(co)100ramus E(co)p = (4) b a-I where E(co)1O0m~ns is Young's modulus measured at the reference pressure of 100mmHg as a function of co; a and b are constants taken as 0.4 and 5, and Po = 20mm Hg and Pb = 30 mm Hg. The computational procedure thus starts with the determination of mean arterial pressure (0 I-Iz harmonic, independent of compliance) and the computation of Ep(co). Computation in the frequency domain allows the incorporation of viscoelasticity into the model (AVOLIO, 1980), which can be expressed as a complex, frequency-dependent elasticity modulus ~ERGEL, 1960) with a modulus and a phase angle (HARDUNG, 1953; BERGEL, 1960; LEAROYDand TAYLOR, 1966). Fig. 3 (upper panel) gives the ratio of Medical & Biological Engineering & Computing 10 ~A A~ e~ 6~ A 4-2-o 0 ~ I--i"q- 1 2 3 4 I' 5 lrequermy, I 6 Hz I 7 I 8 I 9 I 10 Fig. 3 Modulus (upper panel) and phase (lower panel) representation of theoretical viscoelastic models of ('[.ANGEWOUTERS e t al., 1985; WESTXRttOFet al., 1970; WESSSLINGet al.. 1973: HOLENSTEIN et al., 1980." th ao---thoracal aorta; abd ao: abdominal aorta; other art: other arteries; (o): old subjects; 9 Westerhof- no; C' Langewouters- th ao (o); A Wesselingth ao (o); - - Holenstein; 9 Westerhof- other art," 9 Langewouters- abd ao (o); 9 W~seting- abd ao (0) E(co)to0m~ ng/E(0)100 ~ rig for viscoelastic models proposed by different authors (WES'rEP,HOF and NOORDEGRAAF, 1970; WESSELLING et aL, 1973; HOLENSTEINet aL, 1980; LANGEWOUTERS et aL, 1985) where E(0)10omaas is the static Young's modulus at 100mmHg. In general, the modulus IE(co)t0omagl increases with frequency. For the aorta, Langewouters, Holeustein and Westerhof give a dynamic modulus 10 to 20% higher than its static value; Wesseling gives values 50 to 60% higher. The dynamic modulus increases for the distal arteries. Because of viscoelasticity, there is a phase lag between the pressure wave and the vessel wall movement, given by the complex E-modulus phase angle (Fig. 3, lower panel). Langewouters reports a negligible phase lag; the other authors report values between 5e (Westerhof; aorta) and 12~ other arteries). The viscoelastic models of Westerhof and Holenstein are incorporated in the linear network model and the impact on pressure wave propagation is studied. For the Westerhof model, there is a distinction between the aorta and the other arteries; the Holenstein model is applied on all vessels. 2.3 Geometrical data and boundary condition The geometrical data-set of STERGIOPULOSet al. (1992) was used for the comparative study (Fig. 1). The sum of the compliance of the vessel segments at 100mmHg is 1.23 mlmmHg -~. The distal impedances, modelled as 3element 'windkessels,' add a constant compliance of 0.24 mlmmHg -~, so that the total compliance at 100mmHg is N o v e m b e r 1997 731 1.47 ml mm Hg-~. The dam,set is determined for a 'reference person" of 75 kg and 1.75 m tall; the compliance normalised for body surface area is 0 ' 7 7 m l m m H g - ~ m -2. In the nonlinear model, geometrical tapering is automatically accounted for in the segmentation process, whilst for the linear model tapering is modelled as a sequence of cylindrical tubes. For the non-linear model, 55 arterial segments cover the arterial tree. For the linear model, a further segnaentation results in a total of 156 different segments. An artery in the linear model therefore consists of multiple segments. The upstream and downstream artery diameters are matched. The sum of the segment volumes equals the volume of the artery in the non-linear model. In the linear model, the wall thicknesses and Young's moduli are determined so that the (volume) compliance of corresponding arteries is matched as well. In the linear model, bifurcations are treated as parallel impedances, while for the non-linear approach continuity of pressure and flow is expressed. In both models, the resistance vessels are modelled as 3-element 'windkessels.' A physiological aortic flow profile (MILNOR, 1989), representing flow in a healthy subject at rest, is taken as the proximal boundary condition. The heart rate is 60 beats min -1, mean flow 88mls -~ (cardiac output=5.21min - I ) and peak flow 450mls -1. For the computation with the linear model, tiffs signal is decomposed into its Fourier-components and 10 harmonics are used. Table 1 Derived haemodynamic parameters at the ascending aorta and brachiaI andfemoral arteriesfor the linear and non-linear model using the same geometrical and mechanical configuration, and identical upstream and downstream boundary conditions SDP DBP PP PP*~n~ ascending aorta (4 era) linear non-linear t 11.8 114.0 69.9 69.1 41.9 44.9 -- brachial artery (44 era) linear non-linear 125.4 132.0 66. I 66.6 59.3 65.4 1.41 1.46 femoral artery (80crn) linear non-linear 143.9 141.6 61.1 61.3 82.8 80.4 1.98 1.79 SBP = systolic blood pressure (ramHg), DBP = diastolic blood pressure (ramHg), PP = pressure pulse (ramHg), PPamp= (PP at measured location)/(PP at ascending aorta) 160 - aorta (4 crrl) ~4o~ E femoral artery (80 cm) brlchiaJ artery (44 cm) 12(]- g 3 Results 3.1 Linear against non-linear modelling Pressure and flow arc computed at the ascending aorta, braehial and femoral artery (Table 1 and Fig. 4). These locations are situated 4, 44 and 80 cm downstream of the aortic valve, respectively. At the ascending aorta, mean pressure is almost identical (91.3mmHg in the non-linear (nl) and 90.SmmHg for the linear (l) ease), but higher differences occur in systolic (114.0 (nO against 111.8 (1)mmHg), diastolic (69.1 (nl) against 69.9 (l)mmHg) and hence pulse pressure (44.9 (hi) against 41.9 (t)mmHg). The input impedance differences between both models are within 20% (Fig. 5), but especially for the low frequencies the input impedance is higher for the non-linear model. At 1 Hz the nonlinear model input impedance modulus is 0.11 mm Hg m l - t s against 0.09mmHgml-~s) in the linear model. The characteristic impedance is slightly increased in the non-linear case (0.059 (nl) against 0.056 (1)mmHgml-ls). Fig. 6 gives the ratio of braehial/aortic pressure harmonics in the linear and non-linear cases. The harmonic ratio is similar for the two cases between I and 6 Hz, but at 2 and 3 Hz the ratio is about 2 0 % higher in the non-linear case. For the higher harmonics (> 6 Hz), damping is higher in the linearmodel. The values of systolic and diastolic pressure, pulse pressure and pulse pressui-e amplification are shown in Table I. In the nonlinear model, the higher systolic pressure and pulse pressure at the aseending aorta are maintained at the brachial artery. For the femoral artery, there is a reversal (Fig. 4 and Table 1). At the ascending aorta and brachial artery, pressure peaks at approximately the same time in both models; at the femoral artery peaking is earlier in the non-linear model. o.o o., oo , os ~rf~, s Fig. 4 Aortic, brachial and femoral pressure for the linear and nonlinear models. Input signal was a physiological flow wave ('MILNOR, 1989). ~ non-linear; . . . . . . linear most models, damping is considerable at the brachial and femoral artery. For the brachial artery, the pulse pressure amplification is reduced by 15%; for the femoral artery, the reduction is ",-23%. 3.3 Downstream boundary condition To check the effect of the downstream boundary conditions, we replaced the 3-clement 'windkessel' impedances by linear resistances. In the first computation, we neglected the compliance present in the terminal windkessel' sections, so that the actual compliance was only 1.23mlmmHg -1 at 100mmHg. For the second computation, we compensated the loss of distal compliance by adjusting the Young's modulus with a constant factor, distributing the 0.24mlmmHg -1 over the entire tree. In this way, total compliance is matched, but proximal compliance is increased, lowering wave velocity, the ascending aorta input impedance, the characteristic impedance and the pulse pressure. The computational results are shown in Fig. 8 and the derived haemodynamie parameters are surnmarised in Table 3. 4 Discussion 3.2 Viscoelastic#y Different viscoelastic models were implemented in the linear arterial network model using the geometrical configuration described earlier. The computational results are presented in Table 2 and Fig. 7 (Westerhof and Holenstein model). For 732 o.o 4.1 Linear against non-linear modelling The comparison of a linear and a non-linear arterial network model on a geometry with identical boundary conditions, total compliance and peripheral resistance, reveals differences in Medical & Biological Engineering & Computing November 1997 ~/r~'Hinaar ~scoe~.,.,~ty ~ r -1 i -o.1 0.1= 0.01 I 2 I 4 I 6 I 8 I 10 r 0 i 2 i ~, frequency, Hz i 6 "1 8 1 10 i 0 I 2 tre~lueney, Hz '1 4 i .... 6 i 8 ,0.01 10 frequency, Hz ,'-40 40- o O~ i +- ---80 1- I I I I I I I I I I I I I I I 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 frequency, Hz . Fig. 5 frequency, Hz 10 frequency, Hz Ascending aorta input impedance (upper panel, modulus; lower panel, phase angle) computed at 4 cm downstream of the aortic valve. Left: linear ( 0 ) against non-linear (A) model, middle: linear (0) against viscoelastic model (ix: Westerhof 0 : Holenstein); right: resistive (A) against 3-element 'windkessel' (0) termination 3.0 160- aoaa (4 era) brachial artery (44 cm) femoral artery (80 ern) 0.0 ' ' 0.4 ' ' 0 '.8 ' time, s 0.0 ' ' 0,4 ' ' 0i8 ' ~402.5- ". loo: 2.0- '~ 6o "- 1.5- 0.0 " ' 0.4 " " 0,8 ' _= 1.0 Fig. 7 0.5 84 0.0 0 | I I I l I I ! I I 1 2 3 4 5 6 7 8 9 1o Pressure at the ascending aorta, brachial and femoral artery incorporating different viscoelastz'c models in the linear transmission line model. The solid line represents no ~scoelasticity frequency, Hz Fig. 6 Pressure transfer function modulus for the brachial artery for both the linear and non-linear arterial network models. linear; . . . A . . . non-linear the computational results. These disparities may be due to elastic and/or convective non-linearities, or to conceptual differences arising from the treatment of geometric and elastic taper, bifurcations, etc. For the mamformation of the geometrical data set of 55 arteries or segments (STERGIOPULOSet aL, 1992) into a set for the linear model (156 segments), attention was paid to the following aspects: equal up- and downstream vessel radius and wall thickness, equal volume and equal compliance. E varies from 300kPa for the ascending aorta to 1000kPa for the peripheral arteries (values are for a reference pressure of 100 mmHg), with wave velocities ranging from 4 m s - l in the ascending aorta) to 13 ms -~ in the lower limb arteries. In the linear model, the 55 arteries are subdivided into smaller segments with the elastic modulus increasing with distance Medical & Biological Engineering & Computing from the heart. This can explain the observed higher wave velocity in the non-linear model at the femoral location (Fig. 4): total compliance of the femoral artery is equal, but not its distribution. Apparently the compliance of the proximal part of the femoral artery is lower for the non-linear model, with a consequently higher wave velocity and decreased pulse pressure amplification. In spite of identical downstream boundary conditions, total compliance and resistance, the input impedance is not identical. Especially at the lowest frequencies, the input impedance modulus is higher in the non-linear case (note that strictly speaking, the input impedance is a non-existing feature in the non-linear model). This difference may arise from the elastic nonlinearity: the most important harmonics load the vessel up to higher pressures, where compliance is lower. While we compared the model under control conditions, these aspects can be expected to be more pronounced under exercise conditions. Another aspect contributing to the input impedance difference is the treatment of geometric and elastic taper. Whereas for the non-linear model the tapering is November 1997 733 Table 2 Derived haemodynamlc 15arameters at the brachial and femoral arteries for different viscoelastic models. Computaaons were made with the linear model, and for the same geometrical setup SBP. DBP PP ascending aorta linear model Westerhof (1970) Wesseling (y) (1973) Wesseling (o) (1973) Langcwouters (1984) Holenstein (1980) 111,7 113.1 69.9 71.1 70,9 70.8 70.0 71.7 113.1 113.7 112.4 112.0 (4 PPm,, era) SBP 41.8 42.0 42.2 43.0 t25.4 121.6 122.9 124.5 125.8 122.6 g+slt~red C 3 el WK R 42.4 108.9 111.7 112.0 40.3 66.1 70.3 69.5 69.1 66.7 69.8 DBF PP PPa~, ascending aorta (4 era) 70.3 69.9 67.5 38.6 41.8 44.5 -- brachial artery (44 cm) braehial artery (44 crn) linear model Westerhof (1970) Wesseling 0') (1973) Wesseling (o) (1973) Langewouters (1984) Holenstein (1980) Table 3 Derived haemodynamic parameters at the brachial and femoral artery for a) resistance terminations with total:compliance compensation (R+altered C), b) 3.-element "windhessel termination' 3 el Wk and c) resistance termination without compensation f o r lost compliance (R) 59.3 51.4 53.4 55.4 59.1 52.8 1.42 1.22 1.26 1.29 1.39 1.31 femoral artery (80 cm) R+altered C 3 el Wk R 127.2 125.4 132.7 66.9 66.1 64.4 60.3 59.3 68.3 1.56 1.42 1.53 femoral artery (80cm) R+altered C 3 el Wk R 138. l 143.9 149.3 62.6 61.1 54.7 75.5 82.8 94.3 1.95 1.98 2.12 1.98 1.54 1.56 1.53 1.93 1.65 SBP =~ystolic blood pressure (ram Hg), DBP =diastolic blood pressure (mm Hg), PP = pressure pulse (mm Hg), PP,,~p = (PP at measured location)/(PP at ascending aorta) SBP = systolic blood pressure (ramHg), DBP = diastolic blood pressure (mm Hg), PP=pressure pulse (mm Hg), PPo,w=(PP at measured location)/(PP at ascending aorta), y =young, o =oM pressure is further transmitted towards the periphery. At the peripheral sites, the morphological agreement is better, although the absolute pressure differences are maintained. At the bracial artery, the pulse pressure is further amplified; at the femoral artery, there is less difference. Evaluating the ratio of pressure harmonics at the brachial artery and ascending aorta (transfer function; Fig. 6), it is noted that for frequencies below 6 Hz, the general pattern is similar. At 2 and 3 Hz the modulus is --'20~ higher in the nonlinear case. For both models a peak value of 2.5 is found at 6 Hz. For frequencies above 6 Hz there is better damping for the linear model: a sharp peak (2.8) is found at 8 Hz for the non-linear model. This phenomenon, that non-linear models tend to under-estimate damping at higher frequencies, agrees with the findings of REUDERINKet al. (1989). linear model Westerhof (1970) Wesseling (y) (1973) Wesseling (o) (1973) Langewouters (1984) Holenstein (1980) 143.9 133.3 133.5 134.1 143.9 134.7 aorta (4 cm) 160 ~ 1 61.1 68.5 67.6 68.6 62.3 68.3 ~."ter 82.8 64.8 65.9 65.6 81.7 66.5 cm) femoralmlery (80 cm) ~. 14o 12o i,~176 8O 6O I 0.0 ( 0.4 | I """1 0.8 I 0.0 ~ I th'~, ~g. 8 I 0.4 I 0.8 1 ~ 0-0 [ 1 0.4 I I ! 0.8 s Pressure at the ascending aorta, brachial and femoral artery for different models o f the resistance vessels: pure resistance, with compensation for the loss in compliance (0: 'R+altered C') ; pure resistance, without compensation (0: 'R "); 3-element "windkessel impedance" (---: '3-el WTC'J embedded in the equations, the converging vessel is replaced by a sequence of uniform segments in the linear case. The morphological pressure wave differences between both models are most pronounced at the ascending aorta, computed 4 em downstream from the a o ~ c valve (Fig. 4). It is likely that this is an effect of the specific geometry used for the computation (Fig. 1): the right common carotid and axiliary artery originate from the inominate artery, a short segment with large diameter and severe tapering (length 3.4 era, proximal radius 1.01cm, distal radius 0.62cm). In the linear model, the inominate artery has been replaced by three cylindrical segments, creating important local changes in characteristic impedance and thus reflection sites. Different systolic and diastolic pressure levels are observed at all locations (Table I). They are in part explained by the higher input impedance at 1 Hz in the non-linear model (Fig. 5). For the same input flow wave, the ascending aorta pulse pressure is higher in the non-linear model and this higher pulse 734 4.2 Viscoelasticity Viscoelasticity enhances peripheral pressure and flow wave damping (Table 2). Pulse pressure amplification is reduced from 1.42 to --- 1.25 for the brachial artery ( - 12%), and from 1.98 to 1.55 for the femoral artery ( - 2 2 % ) . Comparison with the data shown in Table 1 suggests that the impact of viscoelasticity on the pulse pressure propagation is more important than the impact of elastie non-linearities. The effect of viscoelasticity on pressure values and pressure wave morphology is most pronounced at the peripheral sites. However, its effect is also reflected in the input impedance pattern (Fig. 5). The dynamic elasticity modulus is higher than the static value, and it increases with frequency (Fig. 3, upper panel). Consequently, the input impedance modulus is higher in the viscoelastic case. The characteristic impedance is only slightly increased (+ 6%). We implemented different viscoelastic models, expressed as a complex frequency-dependent elasticity modulus, in the linear arterial network model. The disparity between most models is restricted, due to the fact that they are derived from the same data (HARDtr~G, 1953; BERGEL, 1960; LEAROYDand TAYLOR, 1966). The only significantly different model is proposed by LANGEWOtrI'ERSet aL (1985) for aged arteries, Medical & Biological Engin~ring & Computing November 1997 exhibiting an almost complete loss of viscoelasticity. Implementation of this model yields results that are very close to the linear case, with respect to both the magnitude and morphology of the pressure curves. Table 4 IL~lS-value (in mn, Hg) of computed pre.~v.rea Pc.~. at the ascending aorta (4r brachial (44cm) and#moral artery (80 cm) compared to the linear case Plin I~MS = V 4.3 Linear resistance against 3-element 'windkessel' Representation of the resistive vascular beds is found to be critical for the morphology and amplitude of the peripheral pressure. This feature is especially important if one intends to use arterial network models for the computation of transfer functions, wave reflection analysis (KARAMANOGLU et aL, 1994) or for the evaluation of other models or diagnostic techniques (STEROIOPULOSet al., 1995). Attention should be paid not to create reflection zones, which are not present in vivo. We modelled the capillary beds as a linear resistance and as a 3-element 'windkessel' model, including some distal compliance, and studied the effect on the pressure wave magnitude and contour. However, simple qualitative and quantitative analysis is difficult, because omitting distal compliance (for the resistive model) changes the total compliance of the configuration from 1.47 to 1.23 ml mm Hg -1. This has only a limited effect on the input impedance (Fig. 5, right panel) and hence proximal pressures (Fig. 8, ascending aorta), but increased systolic pressure and pulse pressure are found at peripheral sites. To cover this distal loss in compliance, we lowered the Emoduli throughout the arterial tree (by a factor of 0.84) in order to restore the total compliance to 1.47 ml mm H g - 1. This obviously influences the relative distribution of elasticity and results in a lower input impedance and lower proximal pressures. Nevertheless, distal compliance is still neglected and wave reflection intensity is definitely increased at the peripheral sites, influencing the peripheral pressure and ftow wave contour. With the higher proximal compliance, the effects are modest, but still clearly present. When a purely resistive peripheral model is assumed, the trough following the steep pressure decay (at the end of systole) is deeper. Peripheral systolic pressure and pulse pressure are higher, as well as pulse pressure amplification. The effects are most pronounced close to the terminal (reflection) point. The most distal reflection point is 35 cm away for the brachial artery and 65 cm away for the femoral artery; the impact of the terminal impedance model is therefore more clear at the brachial artery. Nevertheless, one should not conclude that the choice of the peripheral model is unimportant for the pressure computed in the central aorta. Abdominal capillary beds in the kidneys, liver and other abdominal organs capture a large portion of the stroke volume and have been designated an important role in aortic wave reflection (KARAMANOGLUet aL, 1994; LATHAMet aL, 1985). These beds are situated close to the aorta and may therefore play a very important role in the pattern of pressure and flow wave propagation. They should therefore be modelled accurately in order to match their in vivo input impedance spectra. The choice of terminal model should also be related to the order of branching included in the model. With a higher level of bifurcation the distal beds become isolated from the central aorta and their impact will be smaller, as waves reflected at these beds will not reach the aorta due to re-reflection at the bifurcations. 4.4 Elastic non-linearity against viscoelasticity and terminal impedance model In this study, we have evaluated the relative importance of elastic non-linearities compared to viscoelasticity and the Medical & Biological Engineering & Computing NL VE VE PM PM Westerhof Holenstein R+altered C R N brachial femoral aorta artery artery 1.89 1.91 1.49 1.81 1.61 2.92 3.34 2.36 3.70 4.04 8.82 7.13 5.00 6.71 4.50 NL = nonlinearity, FE = viscoelasticity, PM = peripheral model terminal impedance model used for the small arteries, arterioles and capillaries. Table 4 gives the mean deviation (RMS value) of pressures computed under different conditions with respect to reference pressures, computed with the linear arterial network model, no visco-elasticity and 3-element 'windkessel' terminations. It is shown that the RMS-value for the non-linear model is of the same order of magnitude as the values arising from viscoelasticity and the terminal impedance model. Linear and non-linear arterial network models do give different computational results, but the impact on pressure wave amplitude and morphology is less important than the impact of viscoelasticity or the choice of the distal terminating model. The similarity of the pressure transfer function (and thus individual harmonic treatment) in the linear and non-linear model suggests that neither convective acceleration, nor elastic nonlinearities are the most important determinants of the arterial pressure and flow. It is also shown that the harmonics > 6 Hz are more damped in the linear than in the non-linear model. A purely resistive terminal impedance model augments pressure wave reflection and consequently systolic pressure and pulse pressure amplification. This effect was less clear for the pressure computed at the femoral artery, where the distance to the distal reflection site is larger and wave damping along the transmission path is more important. Viscoelasticity enhances peripheral pressure wave damping, with lower systolic and higher diastolic pressures and reduced pulse pressure amplification. In the central aorta, the damping is overruled by the increased dynamic elasticity modulus. Further away from the heart, the net effect is wave damping, and the difference with the reference computations becomes significant. We believe that the combination of different modelling features (e.g. resistive distal model and viscoelasticity) may result in an adequate simulation of arterial pressure wave propagation- However, this does not necessarily imply that the model parameters correspond to actual biomechameal properties of the vessels and their boundary conditions. Aclmowledgments~This research is funded by a specialisation grant fi'om the Flemish Institute for the Promotion of Scientific-Technological Research in Industry (IWT 943065), and a concerted action programme of the University of Gent, supported by the Flemish government (GOA 95003). References AVOLIO,A. P. (1980): "Multi-branched model of the human arterial system,' Med. Biol. Eng. Con,put., 18, pp. 709--718 BERGEL, D. H. (1969): 'The viscoelastic properties of the arterial walls. PhD thesis, University of London November 1997 735 HAROD~O, V. 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(1980): 'Aortic input impedance in normal man: relationship to pressure wave forms,' Circulation, 62 (1), pp. 105-116 REUDERrNK, P. J., HOOGSTRATEN,H. W., SIPKEMA,P., HILLEN, B., WESTE,R.HOF, N. (1989): 'Linear and nonlinear one-dimensional models of pulse wave transmission at high Womersley numbers,' J. Biomech., 22, pp. 819--827 RELrDERn'~ P. l., VAN DE VOSSE, F. N., VAN STEENHOVEN,A. A., VAN DONGEN, M. E. H., JANSSEN,J. D. (1993): 'Incompressible low-speed-ratio in non-tmiform distensible tubes,' Int. s Num. Meth. Fluids, 16, pp. 597-612 736 SCttAAF,B. W.,I ABBRECHT, P. H. (1972): 'Digital computer simulatitm of human systemic artmal pulse wave. transmission: a nonlinear model,! s Biomecl~, 5, 345-364 SHENG, G., SARWAL,S. N., WATTS, K. C., MABLE, A. E. (1995)2 'Computational simulation of blood flow in human systemic circulation incorporating an external force field,' Med. Biol. Eng. Comput., 33, pp. 8--t7 SNYDER, M. F., RIDEOU'T,V. C., HILLEST, R. J. (1968): ' C o m e r modelling of the human systemic arterial tree," s Biomech~, t, pp. 341-353 STERGIOPULOS,N., YOUN~ D. F., ROUGE, T. R. (1992): 'Computer simulation of arterial flow with applications to a r t f u l and aortic menoses,' d. Biomech., 25 (12), pp. 1477-1488 STF.RGIOPULOS,N., MEtSTER, H., WES'rF_.~OF,N. (1995): 'Evaluation of methods for the estimation of total medal compliance,' Am. s Physiol., 268, lap. HI540-H1548 WEMPLE, R. R., MOCKROS, I. F., 'Pressure and flow in the systemic arterial system,' J. giomech., 5, pp. 629-64 l WESTERMOF,N., BOSMAN)E., DE VRIES, C. J., NOORDERGRAAF,A. (1969): 'Analog studies of the human systemic arterial tree,' I.9 Bioraech., 2, pp. 121-143 WESTERHOF, N., NOOP,DERGRAAF, A., (1970): 'Arterial viscoelasticity: a generalised model. Effect on impedance and wave travel in the systemic tree,' d. Biomech., 3, pp. 357-379 WESSEL1NG, K. H., WEBER, H., DE WIT, B. (1973): 'Estimated five component viscoteastic model parameters for human arterial wails,' d. Biomech., 6, pp. 13-24 WOMERSLEY,J. R. (1957): 'An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries.' Wright Air Development Centre, Technical Report WADC-TR 56-614 YOUNG, D. F., TSAt, F. (1973): 'Flow characteristics in models of arteriol stenoses-II. Unsteady flow,' J. Biomech., 6, 547-559 Author's biography Patrick Segers was born in Ninove, Belgium in 1968. He obtained his M.Sc. in Civil Engineering at the Faculty of Applied Sciences of the University of Gent, Belgium in 1991. In April 1997 he obtained his Ph.D. at the Hydraulics Laboratory, member of the Institute of Biomedical Technology (IBITECH) at the University of Gent. His research interests are in the cardiovascular domain: mathematical and physical models of the cardiovascular circulation, blood theology and compliance estimation techniques. Medical & Biological Engineering & Computing November 1997