PhD Course in Biomechanical Modelling Assignment: Pulse wave propagation in arteries

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1 PhD Course in Biomechanical Modelling Assignment: Pulse wave propagation in arteries Jonas Stålhand Division of Mechanics, Linköping University September The assignment Your first task is to derive a model which depends on the axial coordinate and time following the steps in Sect. 2. Your second task is to implement the model in Matlab (or a similar program) and use it to study the following questions: 1. What is the effect of hypertension on the pressure load experienced by the heart, i.e., the pressure at the aortic root? 2. How will standard drugs for hypertension affect the pressure at the aortic root? The input signals to the arterial tree is a measured mass flow curve which is found on the course homepage 1. Please note that this introduction is not complete, there are many open issues. You will, therefore, need to make assumptions based on your engineering experience and also look for information elsewhere. It is therefore important that you consider the validity of the assumption and its effect on the results. The assignment is to be reported as a submissible scientific paper. The page limit is 8 pages and the preferred font is Times 11pt, or similar. 1 data p10.txt 1

2 2 Theory The theory outlined in this paper concerns a model for the blood flow in systemic arteries. The governing equations are based on the continuity equation and Navier-Stokes equations. 2.1 Representing the geometry of an arterial segment Assume the vessel to be approximated by a thin-walled cylinder of length l, time dependent radius a = a(t), and wall thickness h. Further, consider the radial movement a to be a small oscillation around a neutral and stress free radius r 0, as shown in Fig. 1. It is natural to use cylindrical coordinates r a(t) r 0 θ z l Figure 1: The arterial segment is modelled as an elastic thin walled cylinder of length l. The wall oscillates around the neutral position r = r 0 and radius is given by a = a(t). for this type of problems and let us, therefore, introduce the velocity vector: v = v r e r + v θ e θ + v z e z, where e i (i = r, θ, z) denotes a base vector and the indices r, θ, and z, refer to the radial, tangential, and axial directions, respectively. 2.2 Modelling assumptions In all types of modelling it is necessary to introduce simplifying assumptions to have a solvable problem. Some assumptions are obvious while others are less evident. Assumptions in a particular research field are often a product of the development of the field itself - scientists usually start with a simple model and gradually refine it until its response fits that of the studied system. It is important to understand the assumption made when deriving a model because they affect the result as well as place restrictions the applicability of the model. For example, a linear model can describe the response of a nonlinear system quite well in a small region around a point but it fails 2

3 to describe the response in a larger region, or Bernoulli s equation may be acceptable for studying internal flow in a needle but cannot be applied to intravenous tubing. This must be kept in mind when using the model! The list below includes the assumptions you will make in this project. Some of them are questionable on mechanical or physiological grounds, but they a common in this research field since they usually give acceptable results. 1. An artery can be approximated by thin walled cylinder, i.e., (h/r 0 < 0.1) 2. An arterial segment s lengths is much greater than its radius (L/r 0 >> 1) 3. The radial (pressure) oscillations are small occurs around the neutral radius 4. Arterial soft tissue obeys Hooke s law 5. Blood flow is rotationally symmetric and non-rotating in arteries 6. There are no volumetric forces 7. Arterial wave speed is much higher than the free stream speed 8. No-slip conditions applies for the fluid at the arterial wall 9. Shear forces from blood on the arterial wall are negligible 10. Only first-order downstream pressure and flow reflections are considered Step 1: Are the assumptions listed above reasonable for the application considered here? Compare to Ch. 9 in Humphrey and Delange (2004). What are the differences and how will they affect the model? 2.3 Governing equations for blood Once the geometry and coordinated system are decided, it is time to set-up the governing equations for the blood. 3

4 Step 2: Set-up the governing equations for the blood flow using the velocity components v r, v θ, and v z, the pressure p, and the blood density ρ. Note that by using the assumptions listed above, the equation system can be significantly simplified. One of the equations will give a constraint on the pressure. What does it imply and what is the effect on the equation system in terms of the independent variables, i.e. r, θ, and z? See also Chs and 9.5 in Humphrey and Delange (2004) but note that their assumptions are different and it affects the result. The governing equations constitute a coupled system of nonlinear secondorder partial differential equations 2. Such systems can be very difficult to solve numerically and it is desirable to remove the nonlinearity if possible. A widely applied technique in these cases is to study the order of magnitude for each velocity term and neglect the nonlinear terms if they are much smaller than the other terms. Whether or not this is possible depends on the assumptions, but in this case, the following simplifications apply (Zamir, 2000) v r v r r, v z 2 v r z 2 2 v r r 2, v r z v r t, 2 v z z 2 v v z z z, v v z r r v z t 2 v z r 2. (1) Step 3: Use the simplifications in Eq. (1) to reduce the governing equations from Step 2. The result from Step 3 gives three equations: one continuity equation and two Navier-Stokes equations. Our objective is to derive a model which only depends on time and axial position, i.e., the t and z coordinates, and full details in the radial direction are not needed. We can, therefore, simply ignore the radial equation and treat the two remaining equations as given in mean cross sectional quantities. A simple trick to remove the radial derivatives from the two remaining equations is to average the flow over the cross section of the cylinder and introduce the flow rate q and average pressure p. 2 The nonlinearity arises because of the convective terms v r v r r, vz v r z, vr v z r, and v z v z z. 4

5 Step 4: Integrate the equations from Step 3 over the cylinder s cross section and introduce the flow rate and average pressure defined as q = v z da, p = 1 p da, A A A where A is the cross sectional area and da = rdθdr. The boundary conditions are given by the symmetry about the centre line v r (r = 0) = 0, the no-slip condition at the cylinder wall v r (r = a) = a/ t, and the shear stress at the wall for the Newtonian fluid τ w = µ v z (a)/ r. Further, by using that da = 2πada the deformed radius can be replaced by A. At this point, the system of equations should read: A t + q z = 0, q t + A (2) p ρ z = 0. These equations are often referred to as the transmission line equations. Note that there are three dependent variables (A, p, and q) and only two equations. To have a unique solution, one of the variables must be eliminated. This can be done by studying the arterial wall and the result is a coupled solid-fluid problem. 2.4 Governing equations for the coupled solid-fluid problem We all know that elastic tubes inflates when subjected to an internal pressure, e.g., a tyre or a garden hose, and the same is true for the arteries. A natural assumption based on this observation is to eliminate the cross sectional area in Eq. (2) by taking it to be a function of the pressure, i.e., A = A(p). Step 5: Derive an expression for the cross sectional area as a function of the pressure in the artery and use it to eliminate A in Eq. (2), see Chs. 2 and 3 in Humphrey and Delange (2004). In addition, rewrite the equations by introducing the wave speed given by Moens-Korteweg equation c 2 = Eh 2ρa. where E is Young s modulus for the arterial wall, see Duan and Zamir (1995). 5

6 2.5 Solving the coupled solid-fluid problem The two equations from Step 5 are coupled in terms of p and q, but it is straightforward to recast them into the two uncoupled wave equations 2 p t 2 = c2 2 p z 2, 2 (3) q t 2 = c2 2 q z 2. A consequence of this is that it suffices to solve one of the variables, say the p in Eq. (3) 1, since the other one has an identical solution apart from a constant. Without loss of generality, we can also confine ourselves to study the solution of one harmonic wave with the angular frequency ω. The reason will become clear later; at this point is enough to know that the total pressure wave is constructed by a superposition of the pressure solutions to different frequencies. Step 6: Suppose that the tube is fed with a oscillating pressure at the entrance (z = 0) given by the complex function p(0, t) = p 0 e iωt, where i is the imaginary number. Consider the downstream (forward) running wave only and show that the solution to Eq. (3) 1 reads p f (z, t) = p 0 e iω(t z/c), where p 0 is the amplitude. You may recall from math courses that the wave equation can be solved using separation of variables or d Alembert s method. 2.6 Reflections at the end of the cylinder The pressure wave at the entrance of the cylinder will propagate downstream. For realistic cylinders, the wave will sooner or later reach the end of the cylinder where a portion of the wave is reflected and moves upstream (backward) towards the entrance again. Equation (3) 1 is linear in p and the total pressure at any point is, therefore, the sum of the downstream and upstream running waves and we can write p(z, t) = p f (z, t) + p b (z, t), (4) where p b (z, t) is the upstream moving wave. Step 7: When stating equation (4), we have to assume something regarding the reflections. What? 6

7 Step 8: The upstream moving pressure wave p b is composed of two terms: an amplitude and a oscillation. Let the amplitude be given by the constant C 1 and set-up an expression for p b. Tips: the oscillation term is almost like the exponential function for p f, but not quite. Revisit the calculations in Step 6 and see if you can spot the difference. Step 9: Study the situation at the exit z = l. Let the reflection coefficient be defined by the quotient: R = p b(l, t) p f (l, t), substitute the result from Step 8 and calculate the total pressure p(z, t). Compare your result to Duan and Zamir (1995). 2.7 Linking pressure and flow The solution to the pressure and flow waves are the same apart from a constant, as mentioned earlier. This can be utilised to compute an important relation between the pressure and flow amplitudes referred to as the characteristic admittance. This property may be thought of as the cylinders ability to accept flow. Step 10: First, take q(z, t) to have the same functional form as p(z, t) but replace the amplitude p 0 by q 0 and the reflection coefficient R by C 2. Second, compute q 0 and C 2 by substituting p(z, t) and q(z, t) into either of the two equations in Step 5 and require of the result to hold for all z, t, and l. Note the sign for the constant C 2. What does it mean physically? Step 11: Show that q 0 = Y 0 p 0, Y 0 = A ρc (5) where Y 0 is the characteristic admittance. The relations in Step 11 will prove to be useful in the next section. 2.8 Effects of downstream reflections in the arterial tree So far we have only studied a single vessel, but the arterial system is made up of a network of vessels. Typically, two or more daughter vessels are branched from a parent vessel, which in turn is branched from its parent vessel and so on. The diameter of the vessel also becomes smaller towards 7

8 the periphery. For example, the abdominal aortic radius is about 1.4 cm while the femoral artery is only about 0.3 cm. As a consequence there are many reflection sites downstream which will impact on upstream conditions. Let us consider the pressure at the beginning of the thoracic artery (just after the aortic arch). The wavelength of the pressure wave can be computed by the formula λ = c/ν, where λ is the wavelength and ν is the frequency. For the systemic arteries, the wave speed is about 5 m/s and the upper limit for physiologically relevant frequencies is about 15 Hz (Fetics et al., 1999). Substituting these values into the formula gives λ = 0.3 m. The physical dimension of the arterial tree is on the order of meters and it is quite clear that the pressure is affected by downstream reflections. Further, longer vessels like the aorta or the femoral artery are typically tapered. Since the model assumes an arterial segments to have a constant neutral radius (r 0 ), such arteries must be divided into smaller segments with a stepwise reduction in neutral radius at which reflections will occur. The characteristic admittance in Step 11 is the ratio between the flow and pressure amplitudes at the entrance, q 0 and p 0, respectively. When downstream reflections are taken in to account, the admittance will change. To find out how it changes, consider the transition between a parent and a daughter vessel as in Fig Assume that the physical dimension of the transition is much smaller than the wave length so that it can be considered as discrete. Physically, we expect pressure to be continuous and mass to be R p p,f Y 0 p p,b p d,f Y e q p,f q p,b q d,f Y e,d Figure 2: Schematic figure of a transition between two tubes of different admittance. The subscripts p and d denote the parent and daughter tube, respectively, while the subscripts b and f indicates the backward and forward moving waves, respectively. conserved over the transition 3. If the transition has the coordinates x = l for the parent vessel and z = 0 for the daughter vessel, the conservation laws 3 These conservation laws are the mechanical equivalence to Kirchhoff s circuit laws. 8

9 can be written: { pp,f (l, t) + p p,b (l, t) = p d,f (0, t), q p,f (l, t) + q p,b (l, t) = q d,f (0, t), (6) where q f and q b are the flow wave equivalence to p f and p b, respectively, and the subscripts p and d denote parent and daughter vessels, respectively. Step 12: In analogy with Step 11, introduce the effective admittance defined by Y e = q(0, t) p(0, t). (7) Substitute the expressions for p f, p b, q f, and q b in eq. (6) and use (5) and (7) to show that R = Y 0 Y e,d Y 0 + Y e,d. Show that Y e can be expressed as Y e = Y e,d + iy 0 tan θ Y 0 + iy e,d tan θ Y 0, where θ = ωl/c. Can θ assume any value? Step 13: Generalise the results in Step 12 to branching of a parent vessel into two daughter vessels. Compare your results to Duan and Zamir (1995). 9

10 3 Computing pressure distribution in an arterial tree The arterial network is made up of several vessels which branches and tapers. To use the model derived in the previous section, we replace all vessels in the network by cylinders of different radii and lengths. In the previous section, we also discussed the implication of downstream reflections and saw that the effective admittance Y e depends on the effective admittance of the daughter vessels. Applying this argument recursively down the arterial tree, it is clear that the effective admittance of the current vessel depends on the complete downstream arterial tree. In addition, because of pressure continuity, the pressure at the end of one segment must equal the pressure at the entrance of its daughter segments. These observations tell us that we must first compute the reflection coefficient and the effective admittance at the most peripheral vessel and recursively work our way upwards along the arterial tree towards the heart. Only when the effective admittance and reflection coefficient for the first vessel is known, the pressure distribution can be computed, but this time starting at the heart and moving downstream along the arterial tree towards the periphery. From what is described above, a few important questions may be raised. First, how do we construct a realistic arterial tree, second, how do we keep track of which segments that are connected to each other, and, finally, how is the arterial tree truncated? Let us take a closer look at these questions. 3.1 The arterial tree There are several arterial trees suggested in the literature, see, for instance, Formaggia et al. (2006), Stergiopulos et al. (1992) and Westerhof et al. (1969). No matter which tree is used, it is essential to know the connectivity of the vessels, i.e., how they connect to each other. The connectivity should be transparent and at the same time allow for easy and efficient computer implementation. A good example of such a method is a scheme proposed in Duan and Zamir (1995) which can be implemented using matrices. The key idea is to let the arterial tree be described by a truncated binary tree and introduce a coordinate pair [k, j] for each segment, where k is the segment s generation in the tree and j denotes the sequential position within that generation, see Fig. 3. With this approach, a segment at position [k, j] will have its two daughter segments at positions [k + 1, 2j 1] and [k + 1, 2j], respectively. 10

11 [1, 1] [2, 2] [3, 4] [3, 3] [2, 1] [3, 2] [4, 2] position [3, 1] generation [4, 1] Figure 3: The labelling scheme for the arterial tree in Duan and Zamir (1995). Each segment is labelled by a coordinate pair [k, j] where k is the generation in the tree and j is the sequential position in that generation. Step 14: Using the k, j-labelling scheme above, set-up expressions for the effective admittance Y e, the parameter θ, wave speed c, reflection coefficient R, and the characteristic admittance Y 0. Further, set-up an expression for the change in pressure amplitudes in terms of R and θ, i.e., compute the transfer function given by the ratio T F (ω) = p 0[k, j] p 0 [k 1, s], (8) where s = j/2 if j is even and s = (j + 1)/2 if j is odd. Compare your results to Duan and Zamir (1995). Note that the transfer function derived in Step 14 is a complex function. This means that the pressure amplitudes p 0 [k, j] and p 0 [k 1, s] may be complex. Further, the transfer function may also be used to relate the pressures any two vessels in the arterial tree by recursive application. This observation allows for pressure estimation in central arteries like the abdominal aorta using non-invasive measurements in peripheral arteries such as the brachialis (Thore et al., 2008). A drawback with the connectivity scheme proposed by Duan and Zamir (1995) is that the binary tree quickly becomes very large; the number of daughter vessels double for each generation and the matrix size grows as 2 (k 1). Further, for a realistic arterial tree, many positions in the matrix will not correspond to a vessel because each parent vessel need not have 11

12 % Connectivity matrix file. % % Matrix rows correspond to the vessel number while % matrix columns are: generation k, daughter 1, and % daughter 2. Vessels where the arterial tree is % truncated are indicated by NAN for both daughters NAN NAN NAN Figure 4: An example of a connectivity matrix file. daughters at all junctions. For large arterial trees, it is sometimes more convenient to use a finite element inspired scheme where the connectivity is given as a list or matrix. An example of how such a matrix can be constructed is seen in Fig. 4. To find the daughters for vessel m, we simple enter the connectivity matrix at row m and read off the numbers in the second and third columns. Conversely, if we wish to find the parent to vessel n, we look for the row m which contains n in the second or third column. Vessels where the arterial tree is truncated and the daughter vessels are replaced by a boundary condition can be indicated by assigning NaN:s or negative numbers to the second and third columns Boundary conditions for truncated arterial trees After having addressed the questions about the tree and how to keep track of the connectivity, let us take a look at boundary conditions. The arterial system in mammals is connected to the venous system via the capillaries, and blood is continuously circulated within the system. From a computational point of view, it is both costly and inconvenient to model every vessel down to the capillary level (billions of vessels) plus the venous side. Instead, all studies truncate the arterial tree at some convenient level and model the response of the truncated part. There are three types of truncation models: first, a specified resistance at the boundary, second, awindkessel, and, finally, an asymmetric structured tree. The first type where the arterial tree is truncated by a simple resistance is the simplest and by far the most computationally efficient. A weakness of the method, however, is that it does not model the pulse wave propagation in 12

13 the truncated system. An effect of this is that the decrease in impedance 4 associated with higher angular frequencies cannot be seen. In the second group, the arterial tree is truncated by a Windkessel element. It consists of a resistance in parallel with a capacitance, in series with a resistance, see Fig. 5. The Windkessel s effective admittance is given by R 1 R2 C Figure 5: The three-element Windkessel consists of two resistances R 1 and R 2, and a capacitance C. Y e = 1 + iωcr 2 R 1 + R 2 + iωcr 1 R 2, (9) where R 1 and R 2 are the resistances, and C is the capacitance. By letting ω 0, we see from Eq. (9) that the effective admittance becomes the inverse of the total peripheral resistance R 1 +R 2 and the resistance boundary condition is recovered. The capacitance C is the total peripheral compliance 5 and it may be thought of as the elastic energy stored by the deformation of the arterial walls in the truncated part of the tree. In contrast to a pure reflection coefficient, the Windkessel is able to capture the change in the impedance with angular frequency. Note that other types of Windkessel models exist, e.g., a four-element Windkessels which predicts the total arterial compliance and characteristic impedance better than the standard Windkessel in Fig. 5 (Stergiopulos et al., 1999). Finally, the asymmetric structured tree proposed by Olufsen (1998) is, probably, the most accurate boundary condition. It is based on a binary tree where the daughter vessels radii are given as factors α and β of the parent radius. A segment in the tree is truncated by a pure resistance when the radius becomes smaller that a predefined value. By choosing α β, the tree will become asymmetric and, hence, the name. This boundary condition is capable of capturing the change in impedance with angular frequency. In addition, it is also able to capture the phase-lag between flow and pressure that 4 Impedance is a complex resistance given by Z = Y 1 e. 5 For an isolated arterial segment, the compliance is defined as C = V/ p, where V is the volume. The compliance is, therefore, a measure of the incremental volume change in the segment in response to an altered pressure. 13

14 arises after a reflection. The draw-back of the asymmetric structured tree is that it is computationally expensive. For a thorough discussion regarding boundary conditions, see Olufsen (1998) and references therein Boundary conditions at the tree inlet So far, we have considered how to compute the pressure (and flow) wave propagation in the arterial tree and how to substitute the arterioles and capillaries by a boundary condition. In order to solve the problem, we still need to specify the pressure at the inlet to the arterial tree, i.e., p 0 and ω. This need only be made for the inlet of the arterial tree since the entrance conditions in all other segments are given by recursive application of the transfer function T F computed in Step 14. Let us assume that the flow wave from the heart is given by the time discrete signal q[n] = p(nt ), where n = 0, ±1, ±2,... and T is the sample interval. Further, assume that the q[n] is periodic with the period N, such that q[n + N] = q[n]. The signal is, therefore, possible to write as sum of trigonometric functions, q[n] = 1 N N 1 m=0 Q[m]e inω 0mT, (10) where Q[m] are the Fourier coefficients and ω 0 = 2π/NT. These coefficients can be computed by taking Q[m] = N 1 n=0 q[n]e imω 0nT, (11) from which it is clear that Q[m] equals the contribution to the total signal from the frequency 2πm/NT = mω 0. The Fourier coefficients are easily computed using the discrete Fourier transform (DFT). In Matlab this is done using the command fft (and ifft to invert the discrete Fourier transform back to the time domain). The result is a vector where the component correspond to the frequencies 0, ω 0, 2ω 0,..., N/2 ω 0, (N/2 1) ω 0,..., (N 2)ω 0, (N 1)ω 0, where N/2 ω 0 is referred to as the Nyquist frequency. Because the input signal is real, the Fourier coefficients on either side of the Nyquist frequency are complex conjugated, i.e., the vector reads Q = ( a 0 + ib 0 a 1 + ib 1... a N/2 + ib N/2... a 1 ib 1 ), (12) for some constants a n and b n (n = 0, 1, 2,..., N/2). Note that a 0 and b 0 correspond to the steady-state component (zero frequency) of the signal. If Q in Eq. (12) is transformed back to the time domain using an inverted DFT, the result may include a small imaginary part. This part should obviously 14

15 be zero and is a result of round off errors. In Matlab, you can remove the imaginary part by using the command real(ifft(q)) where Q is the variable name for Q Decomposition of the flow wave The velocity wave data on the homepage is the total velocity wave and comprises both the downstream and upstream running waves. Obviously, only the downstream running part of the flow wave should be used as input to the arterial tree in the previous subsection since this part describes the flow wave from the heart; the upstream running part is the response from the arterial tree. The total velocity wave must, therefore, be decomposed into its downstream and upstream parts. This can be done using a method described in Feng and Khir (2010), but, first, we need to derive a slightly different equation for the wave speed. Step 15: Return to the derivation of the Moen-Korteweg equation in Step 5 and show that dp = ρc 2 da A. (13) The method in Feng and Khir (2010) assumes pressure and cross-sectional area increments to be linear summations of the downstream and upstream running waves, i.e., dp = dp f + dp b, da = da f + da b. (14) What makes this method attractive is that it replaces pressure with flow velocity which is much easier to measure using non-invasive techniques such as ultrasound or magnetic resonance imaging. The replacement is done by introducing the water hammer equation, dp f = ρc du f, dp b = ρc du b, (15) where is the mean flow velocity. Step 16: Assume the incremental flow speed is given by du = du f + du b and that the waves are unidirectional. Show that du f = 1 (du + c ) 2 A da, du b = 1 (du c ) 2 A da, (16) and da f = 1 2 (da + Ac ) du, da b = 1 (da Ac ) 2 du, (17) by using Eq. (13) to (15). 15

16 At time sample nt, the decomposed waves are obtained by summing the increments, i.e., and u f (nt ) = A f (nt ) = nt k=0 nt k=0 du f (k) + U, u b (nt ) = da f (k) + A, u b (nt ) = nt k=0 nt k=0 du b (k) + U, (18) da b (k) + A, (19) where U, U, A, and A are integration constants. Equation (18) 1 can now be used to compute the downstream running flow wave that should be input to the arterial tree. 16

17 References Duan B., Zamir M. (1995) Pressure peaking in pulsatile flow through arterial tree structures. Ann. Biomed. Engrg. 23: Feng J., Khir A.W. (2010) Determination of wave speed and wave separation in the arteries using diameter and velocity. J. Biomech. 43: Fetics B., Nevo E., Chen C.-H., Kass D.A. (1999) Parametric Model Derivation of Transfer Function for Noninvasive Estimation of Aortic Pressure by Radial Tonometry. IEEE Transaction on Biomedical Engineering 46: Formaggia L., Daniele L., Massimiliano T., Veneziani A. (2006) Numerical modeling of 1D arterial networks coupled with a lumped parameters description of the heart. Computer Methods in Biomechanics and Biomedical Engineering 9: Humphrey J.D., Delange S.L. (2004) An introduction to biomechanics. Solids and fluids, analysis and design. Springer Science+Bussines Media, New York Olufsen M.S. (1998) Modeling the arterial system with reference to and anesthesia simulator. Dissertation. Department of Mathematics, Roskilde, Denmark. Stergiopulos N., Young D.F., Rogge T.R. (1992) Computer simulations of arterial flow with application to arterial and aortic stenoses. J Biomechanics 25: Stergiopulos N., Westerhof B.E., Westerhof N. (1992) Total arterial inertance as the fourth element of the windkessel model. American Journal of Physiology: Heart and Circulatory Physiology 276: Thore C.-J., Stålhand J., Karlsson M. (2008) Toward a noninvasive subjectspecific estimation of abdominal aortic pressure. American Journal of Physiology. Heart and Circulatory Physiology 295: H Wang J.J, Parker K.H: (2004) Wave propagation in a model of the arterial circulation. Journal of Biomechanics 37: Westerhof N., Bosman F., De Vries C.J., Noordengraaf A. (1969) Analog studies of the huiman arterial tree. Journal of Biomechanics 2: Zamir M. (2000) The Physics of Pulsatile Flow. Springer-Verlag, New York 17