Estimation of the arterial mechanical properties of the lower limb with a patient specific wave propagation model. Lenette Simons BMTE 08.

Transcription

1 Estimation of the arterial mechanical properties of the lower limb with a patient specific wave propagation model Lenette Simons BMTE 8.28

2 Estimation of the arterial mechanical properties of the lower limb with a patient specific wave propagation model Lenette Simons l.simons@student.tue.nl Report of an internship realized at the Pharmacology Department of Hôpital Européen Georges Pompidou, Paris, France, between September and December 27 Supervisors : Pharmacology Department, Hôpital Européen Georges Pompidou, Paris, France: Dr. Erwan Bozec Prof. dr. Pierre Boutouyrie, MD Eindhoven University of Technology, Eindhoven, the Netherlands: Ir. Carole Leguy Dr. Ir. Marielle Bosboom Prof. Dr. Ir. Frans N. Van de Vosse 2

3 Index 1 Introduction Ultrasound and applanation tonometry measurements Study group Measurement sites in the lower limb Protocol of measurements on lower limb Ultrasound measurement techniques Applanation tonometry measurement techniques Treatment of measurement data set Calculation of distensibility from measurement results Measurement results Blood volume flow Distension Pressure: applanation tonometry versus the rescaled distension waveform Diameter and IMT PWV and arterial distensibility determined from the measurements Model Wave propagation model Geometry of the model Prescribed BVF Parameters of the distal vascular bed Leakage of blood to surrounding tissue Estimation of initial arterial distensibility Reverse method Simulation results Model input Simulated BVF and pressure after use of the reverse method Frequency analysis of the simulation results PWV and arterial distensibility from the simulations Discussion Ultrasound and applanation tonometry measurements One-dimensional model for wave propagation Recommendations...32 References...33 Appendix...34 A...34 B...35 C...37 D

4 1 Introduction Arterial stiffening due to aging and artheriosclerosis plays an important role in the development of cardiovascular disease. An increased arterial stiffness is associated with an increased risk on cardiovascular disease. Therefore, arterial stiffening is used for clinical assessment of patients [Laurent et al., 26]. Arterial stiffening is defined by the mechanical properties of arteries. Most used indicators for arterial stiffness are pulse wave velocity (PWV) and arterial distensibility. PWV is defined as the propagation velocity of the pressure waveform generated by contraction of the heart. PWV increases with arterial stiffness. Another parameter often used is arterial distensibility. Distensibility is determined from the relative change in arterial cross-section driven by a change in pressure [Laurent et al., 26]. There are two methods to determine the mechanical properties of arteries non-invasively. The first method makes use of regional wave propagation phenomena. The pulse transit time method calculates PWV from the phase difference between the pressure waveform and distance between two locations along the arterial tree. Currently the measurement of PWV is accepted as the best method to assess arterial stiffness [Laurent et al., 26]. However, it is difficult to measure the distance between two measurement sites accurately. The distance between the measurement sites is not equal to the length of the artery between these sites, introducing a bias in the PWV estimation. In addition, limitations due to the sample frequency of the pressure measurements introduce large errors in the determination of small phase differences. Therefore, measurement of PWV is only accurate when measured over a long distance. The second method is based on local wave propagation phenomena. Arterial distensibility is assessed from pulse pressure and wall distension measurements. However, pulse pressure can only be measured non-invasively at few positions in the body. As a consequence, pulse pressure often can t be measured at the same site as the site for which arterial distensibility is determined. Because pulse pressure is not constant along the arterial tree, assumptions have to be made on the progress of mean and pulse pressure in the arterial tree. Therefore, a bias is introduced in the estimation of distensibility. In conclusion, when both methods are used to calculate Young s modulus from measurements, many assumptions are made. Therefore, these methods introduce a large bias in the estimation of the Young s modulus. Mechanical properties can also be determined with a wave propagation model. Wave propagation models have mainly been used to understand wave propagation phenomena. However, a model can also be used to estimate the mechanical properties. Distensibility, which is an input of the model, can be changed until a good fit is obtained between the simulated and the measured BVF and pressure waveforms. Main advantage of a model over direct measurement of mechanical properties could be the possibility to determine the mechanical properties of small arterial segments. However, in the clinical assessment of patients the use of models is not yet widespread [Laurent et al., 26]. Aim of this project is to study the feasibility of a one-dimensional model of the circulation developed by Bessems et al. [Bessems et al., 27] to determine the mechanical properties of arteries. The study focuses on the circulation in the main arteries of the lower limb. A one-dimensional wave propagation model was used to model pressure and BVF in the lower limb [Bessems et al., 27]. A reverse method was developed to determine 4

5 distensibility of the arteries. Simulation results were fit to measured BVF and pressure waveforms by changing the parameters that were used as input for the model. For these simulations, the initial input parameters were also determined from the measurements. In section 2 the measurement protocol, techniques and data treatment are discussed. The measurement results are presented in section 3. In section 4, the model, the imposed boundary conditions and the determination of initial input of the model are described. Also, the reverse method used to determine the mechanical properties from simulated BVF and pressure waveforms, is described. Measurement results are compared to the simulation results in section 5. Mechanical properties determined with the reverse method are also presented. This report ends with a discussion and a conclusion in section 6. 5

6 2 Ultrasound and applanation tonometry measurements BVF, vessel wall distension and pressure waveforms were determined in the arteries of the lower limb. Ultrasound and applanation tonometry measurements were performed on five positions in different arteries of the lower limb. In this section the measurement protocol, measurement techniques and treatment of the measurement data set are explained. 2.1 Study group The study group consists of six healthy subjects, three male and three female. The mean age of the subjects is 25 ± 5 years, the BMI of the subjects is 2 ± 2 kg/cm 2 and the length of the subjects is 174 ± 9 cm. Amongst the subjects is one smoker. 2.2 Measurement sites in the lower limb Measurements were performed at five positions in the main arteries of the lower limb. These arteries form an arterial tree with two bifurcations (Figure 2.1). In the groin starts the iliac artery (IA). The IA becomes the femoral artery below the groin and bifurcates into the superficial femoral artery () and common femoral artery (CFA). The common femoral artery becomes the popliteal artery (), which bifurcates in the knee into the anterior (ATA) and posterior tibial artery (). The ATA becomes the pedal artery () in the ankle. Figure 2.1: Anatomy of the lower limb [Marieb et al., 21] and a schematic overview of measurement sites 1 to 5 6

7 Ultrasound measurements couldn t be performed at every position of the artery and not on all arteries of the lower limb. The suitability of a measurement site depends mainly on the depth of the artery in the tissue. For example the CFA lies too deep below the skin to be measured with Doppler ultrasound. Due to large signal attenuation in the tissue between the CFA and the probe, the measured signal is very weak and contains a lot of noise. Finally, five measurement positions were chosen (Figure 2.1). The first measurement position was situated in the IA, in the groin. The second site is the, which is measured on the inner side of the upper leg. Then the was measured at position 3, at the inside of the knee. Finally and were measured, respectively on top and at the inside of the ankle (position 4 and 5). 2.3 Protocol of measurements on lower limb During the measurements in position 1, 2, 4 and 5, the subject was in supine position. For the measurements on the, the subject had to turn around. All measurements were performed on the subject s right leg. During the day on which the measurements were performed, the subject had to abstain from coffee to prevent influences on blood pressure. Preferably, all measurements were finished in one measurement session. For three subjects this was not possible. One session took approximately two hours. Before the measurements, electrodes were placed on the left ankle and both wrists for ECG measurements. The ECG signal was used as a trigger for both ultrasound and applanation tonometry measurements. A cuff was placed at the left upper arm for the pressure measurements. Centerline velocity and distension were then measured three times, subsequently for each measurement position. The exact location of each measurement site was marked on the leg. When all ultrasound measurements had been completed, applanation tonometry measurements were performed. Finally, the distance between the measurement site at the IA and the other measurement positions was measured with a measuring tape. This was used as an estimation of the length of the arteries. 2.4 Ultrasound measurement techniques Both arterial velocity and distension were measured at the previously mentioned measurement sites in the lower limb using duplex ultrasound (Scanner 35, Pie Medical, the Netherlands). Duplex ultrasound combines B-mode imaging and Pulsed Wave Doppler ultrasound. A linear probe (7.5 MHz) was placed on the skin. First, B-mode imaging was used to locate the artery at the measurement position and to define the region of interest for Doppler measurements. For the velocity measurements the probe was placed in a 6 o angle to the arterial wall. Velocity was measured along the arterial diameter, such that a velocity profile along the entire profile was obtained. The acquisition time of the measurement was set to two seconds. For the distension measurements the probe was placed perpendicular to the arterial wall to measure the velocity of the arterial wall. The acquisition time was set to four seconds. Both intima- media thickness (IMT) and diameter were determined off-line from distension measurements. 7

8 2.5 Applanation tonometry measurement techniques First, systolic (SBP) and diastolic blood pressure (DBP) were measured in the brachial artery of the left arm with a cuff. The pressure waveform was obtained with an applanation tonometer (Spyghmocor, AtCor Medical, Australia). To obtain the pressure waveform, the pressure sensor was placed above the artery where the strongest signal from the pulse is felt. The sensor has to press on the artery. Because the measured waveform is displayed, the measurement data could be captured as soon as a reproducible waveform was observed. The acquisition time of the measurement was set to ten seconds. Pressure waveforms couldn t be obtained at every position in the leg. The artery should not lie too deep below the skin and a bone is needed to support the artery [Van Bortel et al., 21]. After several measurements on a healthy subject, the measurement sites in the IA and were found suitable for applanation tonometry. 2.6 Treatment of measurement data set Criteria for the exclusion of measurements If an error has occurred in the measurement it was excluded from the measurement data set. Several errors can occur in the measurements. An error that occured both in pressure and in ultrasound measurements was the absence of an ECG trigger. The ECG was used to separate the heartbeats of a measurement. Another source of error was movement of the probe or the subject during a measurement, which caused a shift in measured centerline velocity or a loss of signal. Also, when the angle of the probe with the artery differed from the angle prescribed in the measurement protocol, this measurement was excluded as in this case the ultrasound signal is too weak. Finally, a different vessel than the intended artery could be measured, which was often a neighboring vein. Ultrasound measurements Distension waveforms and velocity waveforms across the lumen were determined from the Doppler measurements with the program Wall Track System 2 (Pie Medical, the Netherlands). To determine centerline velocity, distension, diameter and intima-media thickness (IMT), an offline radio frequency (RF) method was used [Kornet et al., 1999]. A file containing the radio frequency data (echo as a function of depth) results from the ultrasound measurements. To obtain the desired information, a region of interest was determined by manually detecting the arterial walls. The program uses the magnitude of the measured RF-signal in the selected area to calculate centerline velocity from the velocity measurements and distension (Figure 2.2), diameter and IMT from the distension measurements. Figure 2.2: Distension waveform determined with Wall Track System 2 Centerline velocity (Figure 2.3) was determined for every heart beat from the velocity measurements. 8

9 The BVF waveform was calculated using Womersley velocity profile [Womersley, 1955], centerline velocity and mean diameter of the artery. BVF was assumed a pulsatile flow into a straight and fixed wall tube. Figure 2.3: Centerline velocity determined with Wall Track System. For both distension and BVF waveforms some characteristics were calculated (Figure 2.4). The maximum of the waveform is defined as the value of the waveform in systole, t 1% is defined as the time at which the wave has reached 1% of its pulse magnitude. Rising time, t r, is defined as the time in which the pulse increases from 1% to 9% of its maximal value. Finally, t sys t dia is the time interval between diastole and systole. Mean and standard deviation of the parameters were calculated for all measurements at each measurement site. A mean waveform was obtained both for the distension and the BVF waveforms for each measurement site. This mean waveform was determined by averaging the distension and BVF waveforms of all measured heart beats. The length of the heartbeat of the mean waveform of a position is equal to the length of the shortest measured heart beat. Figure 2.4: Parameters of waveform Pressure measurements The program of the applanation tonometer automatically scales the mean waveform to SBP and DBP measured in the brachial artery. Because MBP and DBP change very little in the arterial system, the pressure waveforms were rescaled to MBP and DBP measured in the brachial artery [Van Bortel et al., 21]. The ECG signal was used as a trigger to separate the heart beats. The same parameters as for distension and BVF waveforms were used to quantify the pressure waveform. Mean and standard deviation of the waveform parameters at both measurement sites were calculated. A mean waveform was calculated with the same method used for the ultrasound measurements. 9

10 As mentioned before it wasn t possible to determine the pressure waveform at every position of the lower limb. To compare measured to simulated pressure waveforms, it was necessary to determine the pressure waveform at every measurement site. Therefore, the distension waveforms were also rescaled to MBP and DBP. The pressure waveforms determined with applanation tonometry measurements were compared to the rescaled distension waveforms. 2.7 Calculation of distensibility from measurement results Two methods were used to determine arterial distensibility from the measurement results. One method uses pulse pressure and arterial distension, the other uses PWV. Distensibility was assumed constant in each artery. Linear elastic behaviour of the arterial wall and a thin wall thickness assume no circumferential stress. Determinaton of arterial distensibility The first method determines distensibility from the pulse pressure, measured in the brachial artery, and systolic and diastolic arterial diameter. Distensibility is estimated from the relative change in arterial cross section driven by the change in pressure between systole and diastole (equation 2.1) [Laurent, et al., 27]. 1 A d d D = A p d ( p p ) 2 2 sys dia 2 dia sys dia (2.1) Where d sys and d dia are systolic and diastolic arterial diameter, p sys is SBP, p dia is DBP and A is the arterial cross-section during diastole. Determination of distensibility using distension waveform transit time The second method uses the transit time of the distension waveform at different measurement sites to calculate PWV, c (equation 2.2) [Laurent et al., 26]. Transit time, t, is defined as the phase difference of the 1% foot of the pressure waveforms measured at two measurement sites. The distance between two measurement sites is L. For the determination of the phase difference, the mean t 1% of all measured heart beats of a measurement position is used. L c = (2.2) t BVF is assumed to have a large Womersley number and thus to be inviscid, which holds for the large arteries of the lower limb. Moens-Korteweg wave speed (equation 2.3) is used to calculate distensibility from PWV in the artery between two measurement sites [Nichols et al., 1998]. 1 c = (2.3) ρd 1

11 3 Measurement results In this section the results from the ultrasound and applanation tonometry measurements are reviewed. These results consist of BVF and distension measurements for all measurement sites and pressure measurements with applanation tonometry of two sites. PWV and distensibility were calculated from the ultrasound measurements. 3.1 Blood volume flow At all measurement sites a BVF waveform could be determined with a physiological shape (Figure 3.1). For subjects 2 and 3 the number of BVF measurements at a position is less than three (Appendix A). The fraction of initial BVF measured in the groin through each artery differs for all subjects. BVF of subject 2 and 6 is smaller than in all other subjects. For subject 2, BVF through the is very small compared to BVF through the and the IA. q [ml/s] IA q [ml/s] IA (a) subject 1 (b) subject IA IA q [ml/s] q [ml/s] (c) subject 3 (d) subject 4 11

12 IA IA q [ml/s] 5 q [ml/s] (e) subject 5 (f) subject 6 Figure 3.1: BVF in all measurement sites for all subjects (a-f) For all subjects mean BVF decreases down the arterial tree. This is partly caused by the bifurcation of the arteries, which increases the total surface of the arteries. Also blood flows from the artery to the surrounding tissue. Subjects 2 and 4 have a negative phase shift of the waveform between measurements at the IA and and the IA and. Therefore, PWV couldn t be estimated between these measurement positions from the BVF waveforms. The negative phase shift is possibly caused by the averaging of BVF waveforms of different measurements at the same position. All other subjects have positive wave propagation down the arterial tree. All results of the BVF measurements can be found in Appendix B. From the mean parameters of BVF (Table 3.1) and the standard deviations can be concluded that systolic BVF, mean BVF and the BVF pulse between the subjects vary significantly. The same can be said for the time index of the systolic peak, rising time and duration of systole. Table 3.1: Mean BVF parameters (with standard deviation) IA q sys [ml/s] 26 ± 8 14 ± ± ± ±.9 q mean [ml/s] 2.8 ± ±.5.5 ±.24.5 ±.6.23 ±.18 q sys -q dia [ml/s] 26 ± ± ± ±.5.25 ±.1 t sys [ s].28 ±.2.29 ±.2.33 ±.7.33 ±.1.31 ±.1 t 1% [s].18 ±.3.19 ±.1.22 ±.4.25 ±.1.24 ±.2 t r [s].68 ±.2.72 ±.11.8 ± ± ±.11 t sys -t dia [s].18 ±.6.18 ±.7.19 ±.8.19 ±.7.17 ± Wall distension Wall distension measurements resulted in a physiological mean waveform at all measurement sites for all subjects (Figure 3.2). For subjects 3, 4 and 5 the number of measurements at a position is less than three (Appendix A). Distension of the and is small in all subjects and early wave reflections and measurement noise is large compared to systolic and mean distension. Therefore, the standard deviation of the parameters of the waveform is large for the measurements at these positions for every subject. The large standard deviation gives difficulties to estimate the 1% foot of the wave. This is an important factor in the accuracy of the estimation of PWV between the and and the and. 12

13 distension [mm] IA distension [mm] IA (a) subject 1 (b) subject distension [mm] IA distension [mm] IA (c) subject 3 (d) subject distension [mm] IA distension [mm] IA (e) subject 5 (f) subject 6 Figure 3.2: Distension in all measurement sites for all subjects (a-f) Mean and systolic distension decrease down the arterial tree for all subjects. Subjects 3 and 5 have a negative phase shift of the distension waveform between the and. All other subjects have positive wave propagation down the arterial tree. Subject 1 has a positive phase shift of the distension waveform between and, but this shift is very small, leading to a very high estimation of PWV. All parameters of the results of the distension measurements can be found in appendix C. 13

14 Like the mean parameters of the BVF waveforms, the mean parameters of the distension waveforms have a large standard deviation (Table 3.2), varying between 5% and 5% of the mean value. The phase shift of the waveform between the and, determined from the mean t 1%, is negative. Table 3.2: Mean parameters and standard deviation of distension waveform IA d sys [mm[.44 ±.8.22 ±.1.22 ±.7.84 ± ±.15 d mean [mm].15 ±.3.79 ± ± ±.15.5 ±.1 d sys -d dia [mm].51 ±.8.24 ±.9.25 ±.7.11 ±.8.43 ±.27 t 1% [s].17 ±.1.19 ±.1.23 ±.2.27 ±.1.22 ±.6 t r [s].82 ±.9.11 ±.1.1 ±.1.91 ±.1.13 ±.5 t sys -t dia [s].17 ±.1.19 ±.2.19 ±.4.17 ±.3.22 ±.5 t sys [s].31 ±.2.36 ±.3.38 ±.3.39 ±.1.38 ± Pressure: applanation tonometry measurements and the rescaled distension waveform Because it wasn t possible to measure the pressure waveform with applanation tonometry measurements at all measurement sites, distension measurements were rescaled with MBP and DBP. At the measurement sites in the IA and, the pressure waveform was measured with applanation tonometry to compare this waveform to the rescaled distension waveform. For subject 6 it was impossible to measure pressure waveform in the. Therefore, the waveform in the was used instead. The phase difference between the pressure waveform measured with applanation tonometry and the rescaled distension waveform could only be evaluated for subject 2, 5 and 6. In all other subjects, the pressure waveform was measured without an ECG signal. Therefore the phase shift of the pressure waveform between both measurement sites couldn t be determined. p [mmhg] IA, tonometry, tonometry IA, distension, distension p [mmhg] IA, tonometry, tonometry IA, distension, distension (a) subject 1 (b) subject

15 p [mmhg] IA, tonometry, tonometry IA, distension, distension (c) subject 3 (d) subject 4 p [mmhg] IA, tonometry, tonometry IA, distension, distension IA, tonometry, tonometry IA, distension, distension IA, tonometry, tonometry IA, distension, distension p [mmhg] 1 9 p [mmhg] (e) subject 5 (f) subject 6 Figure 3.3: Pressure waveform measured with applanation tonometry and pressure waveform obtained by rescaling the distension waveform in the IA and for all subjects (a-f). For neither of the subjects, a phase shift between the waveform measured with applanation tonometry and the distension waveform exists in the IA at the foot of the waveform (Figure 3.3). In the a small phase difference can be observed in subject 2 and 5. For subject 6 a phase difference is observed in the. For the applanation tonometry and the distension measurements brachial MBP and DBP were used to scale the waveform. This means that the MBP and DBP are assumed to be equal at all measurement sites. Differences in SBP and pulse pressure (PP) are caused by differences in the shape of the waveform. PP and SBP increase from the IA to the peripheries (Table 3.3). The results of subject 6 in the are not included in the results below. In the IA rising time is equal in both waveforms but in the a difference in rising time is observed between the pressure and the rescaled distension waveform. Table 3.3: Mean and standard deviation of the parameters of pressure waveforms determined with the tonometry measurements or rescaling of the distension waveform. IA (n=6) (n=5) Distension Tonometry Distension Tonometry SBP [mmhg] 13 ± 7 18 ± ± ± 18 PP [mmhg] 45 ± 1 5 ± 8 55 ± ± 18 t r [s].88 ±.9.88 ±.9.9 ±.1.72 ±.15 t sys -t dia [s].17 ±.1.15 ±.2.17 ±.3.13 ±.3 15

16 3.4 Diameter and IMT From the distension measurements, mean diameter and IMT were determined (Table 3.4 and 3.5). It was not possible to determine the IMT of the for subject 4. All subjects have only one good measurement of IMT for at least one measurement site. No standard deviation could be determined for these positions. Table 3.4: Mean diameter [mm] of arteries mean IA 7.1 ± ± ± ± ± ± ± ±.1 4. ± ± ± ± ± ± ± ± ± ± ± ± ± ±.2.9 ± ± ± ± ± ± ± ± ±.1 2. ± ± ± ±.5 Table 3.5: Mean IMT [mm] of arteries mean IA.52 ± ±.4.69 ± ±.3.38 ±.1.55 ± ± ±.25.5 ±.9.43 ± ±.1.41 ±.6.45 ±.7.49 ± ± ±.6.53 ±.9.47 ±.1.5 ± ± ±.1.4 ± ±.8.42 ± ±.2.44 ±.4 From the mean diameters can be concluded that diameter decreases along the arterial tree. For IMT no direct relation with measurement position is observed. The ratio IMT/diameter increases along the arterial tree. 3.5 PWV and arterial distensibility estimation from the measurements PWV in the arteries of the lower limb was determined with the pulse transit time method (Figure 3.4). The estimation of PWV between the measurement sites in the and was not possible for subject 3 and 5, because a negative phase shift was found. Because the phase shift is very small compared to the sample frequency of the measurement, large errors were introduced, when PWV is estimated over small distance with the pulse transit time method. Therefore, PWV was also estimated between the measurement sites in the IA and and between the IA and (Figure 3.4). 16

17 PWV [m/s] subject IA- IA- - - IA- IA- Figure 3.4: PWV [m/s] for all subjects between all measurement sites. PWV determined in the measurements was compared to the PWV determined from the simulations. Both local wave phenomena and PWV were used to estimate distensibility (Figure 3.5). After the use of the reverse method these values are compared to the values estimated from the simulations. For each artery, distensibility determined with the PWV method was calculated from the PWV between the measurement sites at the beginning and at the end of the artery. Both in the IA and the, distensibility was calculated from PWV in the IA and the. 3 3 D [MPa^-1] IA D [MPa^-1] IA PWV Dist ensibilit y PWV Dist ensibilit y (a) subject 1 (b) subject D [MPa^-1] IA D [MPa^-1] IA PWV Dist ensibilit y PWV Dist ensibilit y (c) subject 3 (d) subject 4 17

18 3 3 D [MPa^-1] IA D [MPa^-1] IA PWV Dist ensibilit y PWV Dist ensibilit y (e) subject 5 (f) subject 6 Figure 3.5: Estimated distensibility of all subjects a-f. Because PWV couldn t be determined between the measurement sites in the and for subject 3 and 5, distensibility couldn t be calculated for the for these subjects. The estimations for distensibility resulting from both methods are not in agreement. The standard deviation of the mean distensibility for each artery is smaller for the distensibility method. No relation is found between distensibility and the position of the measurement. 18

19 4 Model In this chapter, the theory behind the one dimensional model and the parameters that were used as (initial) input for the model are explained. Finally, the reverse method used to determine the mechanical properties of the arteries of the lower limb is explained. 4.1 Wave propagation model For the simulation of BVF and pressure in the lower limb a one dimensional wave propagation model developed by Bessems et al. was used [Bessems et al., 27]. In this study, the model was linear elastic. Thus, visco-elastic properties were ignored. The set of equations governing this model follow from mass conservation, momentum balance and a constitutive law for the material of the arterial wall (equation 4.1-3). A q + = ψ t z (4.1) 2π η t z A ρ z ρ ρ z 2 2 q q A p a q + δ1 + = Af z + τ w + 2 (4.2) p( z, t) =ɶ p( A( z, t); z, t) (4.3) Where Ψ is leakage, ρ is the density of blood, A is the arterial cross-section, q is BVF and γ, is a non-linear convection term. δ 1 is explained in equation 4.9 and τ w in equation 4.8. The velocity profile used in the model is based on a boundary layer theory. The velocity profile consists of an inviscid area in the core of the artery and a viscid boundary layer (Stokes layer) near the arterial wall (Figure 4.1). Figure 4.1: Approximate velocity profile function of the 1D model, with δ s, the boundary layer. The velocity profile is assumed axi-symmetric. In the inviscid core, velocity is constant in the radial direction and depends on time and position in axial direction (equation 4.4). v ( r, z, t) = v ( z, t) for r < a c (4.4) z c Where v c is the centerline velocity, z is the coordinate in axial direction and r, the coordinate in radial direction. In the boundary layer, the Navier-Stokes equation is solved for viscous flow (equation 4.5). 19

20 p 1 v = + η ( r z ) z r r r for a c < r < a (4.5) Where p, is pressure and η, is the fluid viscosity. The velocity in axial direction in the Stokes layer can be determined by integrating twice with respect to radius, r, and with the boundary conditions v z (a c ) = v c and v z (a)=. The velocity profile function (equation 4.4 and 4.5) is integrated with respect to arterial cross-section, A. This results in a relation between centerline and mean velocity (equation 4.6). v ζ 2 ln c 1 c = 1 ζ c + ( ζ c + 1) lnζ c 1 ζ c A 4η 2 q a p z (4.6) This equation is made dimensionless with ζ c, the dimensionless core thickness. The dimensionless core thickness is defined as the square of the dimensionless core diameter (a c 2 /a 2 ). Furthermore, a is arterial radius. q/a is equal to mean velocity. The equation for v c is substituted in the dimensionless equation for v z derived from equation 4.5. v z ˆ 2 lnζ q a ˆ 1 1 ( 1) ln ˆ p = ζ + ζ c + ζ 1 ζ A 4η 2 z c (4.7) Where ζ is the square of the dimensionless radius (r/a) and ˆ ζ = max[ ζ, ζ c ]. Wall shear stress, τ w, can be derived from velocity in axial direction (equation 4.8). vz 2η q a p τ w = η = + (1 ζ c ) r (1 ζ ) a A 4 z r= a c (4.8) The parameter δ 1 in equation 4.2 is a function which depends on the dimensionless core thickness ζ c (equation 4.9). 2 2 ζ (1 ln ζ c ) δ ( ζ ) = c 1 c 2 (1 ζ c ) (4.9) A spectral element method was used to solve the set of equations. The governing equations of the model were discretized with a Galerkin method. At each timestep of.1 s a Newton- Raphson iterative scheme handles the nonlinear terms. The pressure and BVF waveforms were obtained along the considered arterial segments. 4.2 Geometry of the model A geometry of the arterial tree of the lower limb was built from the ankle to the groin. It consists of the five main arteries (Figure 2.1) of the lower limb: the IA, the, the, the ATA and the. Compared to the anatomy of the leg s arterial tree, this geometry is simplified. Several arteries in the lower limb were not included in the geometry. BVF through these arteries was modeled as leakage of blood to the surrounding tissue. 2

21 The geometry was discretized in the spatial domain into 62 8 th order spectral line elements. For each element arterial radius, wall thickness, length and arterial distensibility were prescribed. To make the model patient specific these parameters were determined from ultrasound measurements. Arterial parameters determined from the ultrasound measurements were interpolated between the measurement sites. 4.3 Prescribed BVF The Fourier coefficients of the BVF in the groin were used as an input for the model. BVF was determined and transformed to the frequency domain using fast fourier transform. Only the first 32 harmonics were used to describe the shape of the waveform. 4.4 Parameters of the distal vascular bed The, and were closed with a three element Windkessel model (Figure 4.2) [Westerhof et al., 1969]. This end-segment consists of the characteristic impedance, Z, of the artery that is cut off, a resistance, R v, and a compliance, C v, of the vascular bed behind the end-segment. Figure 4.2: Electrical analogue of the three element Windkessel model Equation were used to make a first estimation of Z, C v and R v. This estimation was used to determine the input values for the model with a least square method. Z was estimated from the inertance and compliance of the artery that is cut off (equation 4.1). For high frequencies this gives a good match of the impedance between the endsegment and the cut off artery. It is assumed that the artery shows linear elastic behaviour. Z L ρ A Amax Amin =, with L = and C = = C A p p p sys dia (4.1) Where L, is inertia, ρ, the density of blood (1.5e3 kgm -3 ), A, is the lumen area determined with mean diameter, C, is the linearized compliance, A max, is the maximal and A min, is the minimal lumen area, p sys, is the SBP and p dia, is the DBP, measured at the brachial artery. Next the resistance of the vascular bed behind the end-segment was estimated from the peripheral resistance and impedance (equation 4.11). pmean Rv = Rp Z with Rp = (4.11) q mean 21

22 Where R p is the peripheral resistance, p mean is the mean pressure and q mean is mean BVF in the cut off artery. Compliance of the vascular bed distal to the cut off arterial branch was estimated. C v τ = (4.12) R v Where τ is the time constant (1.5 s) at which the compliance discharges, taken from literature. 4.5 Leakage of blood to surrounding tissue Leakage is defined as the amount of blood going from an artery included in the model to the surrounding tissue per unit length. Besides the entrance BVF in the leg and at the endsegments, BVF was also measured in the. Therefore, leakage could be calculated separately for the upper (equation 4.13) and the lower leg (equation 4.14). q q q Ψ up = l + l mean, IA mean, mean, (4.13) Where, Ψ up, is the leakage per unit length in the upper leg, q mean,ia, q mean,, and q mean,, are mean BVF at measurement sites in the IA, and, l and l are the lengths of the arteries between the measurement sites. q q q Ψ low = l + l mean, mean, mean, (4.14) Where, Ψ low, is the leakage per unit length in the lower leg, q mean,, q mean,, and q mean,, are mean BVF at measurement sites in the, and, l and l are the lengths of the arteries between the measurement sites. 4.6 Estimation of initial arterial distensibility The PWV and the distensibility method were both used to calculate arterial distensibility. For each subject was determined which method gives the simulated waveforms that are most similar to the measured waveforms. This method was used as the input for the one dimensional model and served as initial value for the reverse method. 4.7 Reverse method When all input parameters were determined from measured distension, BVF and pressure waveforms they were used as input for the model (Figure 4.3). Distensibility, diameter, arterial compliance, vascular tapering, leakage and resistance of the end impedances were determined from the measurements and used as first estimations in the model. Prescribed BVF, impedance of the cut off arteries and IMT were fixed parameters. A reverse method was used to determine the parameters that give the best fit between simulation and measurement results. To quantify the fit, a number of parameters of the waveform were used to fit the simulated to the measured waveform. These parameters are mean BVF and pressure and pulse BVF and pulse pressure. 22

23 Figure 4.3: Schematic overview of the input parameters at every measurement site. After the model had been built with all the input parameters, pressure and BVF were simulated in the entire geometry of the lower limb. Simulation results were compared to measured BVF and pressure (Figure 4.4). By changing model parameters simulation results were fit to the measured BVF and pressure. The model parameters were changed until the difference between the parameters of the simulated waveforms and the measured waveforms are less than ε, which is 1% of the value of the measured parameter. The parameters that were changed are diameter, vascular tapering and distensibility of the arteries, compliance and resistance of the end-segments and magnitude of the leakage of blood to surrounding tissue. Vascular tapering is defined as the relative change in diameter of an artery per unit length. (a) 23

24 (b) Figure 4.4: Schematic overview of the first (4.4a) and the second (4.4b) step of the reverse method The reverse method consists of two steps. First the simulated pulse BVF and pulse pressure were fit to the measurement results by changing distensibility, compliance of the endsegments, vascular tapering and arterial diameter (Table 4.1, Figure 4.4a)). These parameters influence the parameters mentioned above, but also influence mean pressure and mean BVF. All parameters have a similar effect on the parameters of the simulated waveform in the artery in which they are changed, but have a different effect on the arteries proximal, distal or parallel to this artery. Therefore, all parameters had to be changed. Table 4.1: Effect of increasing an parameter on BVF and pressure characteristics. q sys -q dia p sys -p dia D > < Tapering < > Diameter > < C v > < Secondly, mean pressure and BVF were fit by changing leakage and resistance of the endsegments (Table 4.2, Figure 4.4b). These parameters have little or no influence on the pulse BVF and pulse pressure. Table 4.2: Effect of increasing parameter on BVF and pressure characteristics. q mean p mean R v < > Ψ < < Besides that the BVF and pressure changes in the artery for which the parameters are changed, also the waveform parameters in the arteries distal, proximal or parallel to this artery change. All parameters were increased or decreased by multiplying the initial value with a factor. Tapering, leakage and diameter were changed to a maximum of 2% from the initial value. These parameters were changed in small steps of 5% of the initial value. For distensibility, compliance and peripheral resistance, no boundary was set, because these parameters couldn t be directly measured and couldn t be found in literature. These parameters were changed rapidly starting with 5% to see the effect of all changes clearly in the results. 24

25 5 Simulation results In this section, the initial input parameters of the model are discussed. Next, simulation results are compared to measurement results. The mechanical properties and PWV estimated with the reverse method are discussed. 5.1 Model input For all volunteers C v, R v and Z of the end-segments were determined from the pressure and ultrasound measurements (Appendix D). In five subjects, the simulation didn t converge with the estimated Z, which had to be decreased. Decreasing the impedance increases the magnitude of reflections in the simulation results. For each subject the leakage of blood to the surrounding tissue was determined and used as model input (Table 5.1). Diameter and wall-thickness were determined from the ultrasound measurements for the measurement sites and linearly interpolated between the measurement sites. Table 5.1: Measured leakage of blood to surrounding tissue [ml/(s cm)] Subject Ψ up Ψ low Initial distensibility The first simulation was run with a geometry, with an arterial distensibility determined with the methods mentioned in chapters 3 and 4. This resulted in two models of the arterial tree of the lower limb, which differ only in the distensibility for every artery. For the reverse method, the model, which gave the best initial simulation results, was used. Simulated BVF in the IA is exactly the same as measured BVF (Figure 5.1), because it was used as an input for the model. In the results of the initial simulation, simulated pulse BVF, pulse pressure, mean BVF and mean pressure differ from the measurements in all other positions mes sim flow [ml/s] Figure 5.1: Simulated (sim) and measured (mes) BVF trough the IA for subject 4. 25

26 5.2 Simulated BVF and pressure after use of the reverse method BVF After an initial model was chosen, the reverse method was used for a better fit between the measurement and simulation results. After use of the reverse method, mean BVF of the simulated waveform are in agreement with the measured waveform in all evaluated measurement positions. In 15 out of 24 evaluated measurement positions, pulse BVF of the simulated waveform is comparable to the measured waveform (Figure 5.2a,b,d). In this case, all evaluated parameters are comparable to measurement results. In 9 evaluated measurement positions, the shape of the simulated waveform differs from the measured waveform (Figure 5.2c) mes sim 8 6 mes sim flow [ml/s] 5 flow [ml/s] (a) (b).8.6 mes sim.25.2 mes sim flow [ml/s].4.2 flow [ml/s] (c) (d) Figure 5.2: Simulated (sim) and measured (mes) BVF for subject 4. The fit after the systolic peak and the phase difference between the simulated and measured waveform were not compared. Reflections occur at different moments in the waveform and have a different magnitude than the measurements (Figure 5.2a,d). In other waveforms a phase shift is observed between the measured and simulated BVF (Figure 5.2a,b,d). Pressure The fit between simulated and measured pressure was more difficult. Simulated pulse pressure increases stronger along the arterial tree than in the measurement results (Figure 5.3)..5 26

27 Also, wave reflections occur at different positions in the simulated waveform than in the measured waveform (Figure 5.3d,e). 11 mes sim 11 mes sim pressure [mmhg] pressure [mmhg] (a) IA (b) pressure [mmhg] mes sim pressure [mmhg] mes sim (c) (d) 15 1 mes sim pressure [mmhg] (e) Figure 5.3: Simulated (sim) and measured (mes) pressure in all measurement sites (a-e) for subject 4. After use of the reverse method, mean pressure of the simulated waveform is comparable to the measured waveform in all evaluated measurement positions. 27

28 5.3 Frequency analysis of the simulation results Power spectra were made of both measured and simulated BVF and pressure (Figure 5.4). Both pressure and BVF waveforms have peaks at the same frequencies of the spectrum. However, the difference in magnitude of these peaks leads to differences in the shape of the simulated waveform..8.7 mes sim 8 7 mes sim.6 6 Q [(ml/s) 2 ] P [mmhg 2 ] frequency [Hz] frequency [Hz] Figure 5.4: Power spectra of BVF and pressure waveforms in the of subject PWV and arterial distensibility from the simulations PWV was estimated with the pulse transit time method (Figure 5.4). Because large errors are introduced, when PWV is calculated over small distances, PWV was also estimated between the measurement sites in the IA en and the IA and. 4 PWV [m/s] IA- IA- - - IA- IA subject Figure 5.5: PWV of all subjects determined from simulation results Mean PWV is 4. ± 1.4 m/s between the measurement sites in the IA and, 6.6 ± 1.8 m/s between the IA and, 12.7 ± 11.9 m/s between the and and 15.8 ± 7.2 m/s between the and. Mean PWV is 7.8 ± 3.5 m/s between the measurement sites in the IA and and 9.2 ± 2.8 m/s between the IA and. PWV determined from the measured distension waveform compared to the PWV determined from the simulated pressure waveform. 28

29 For every subject, simulated PWV in the IA and is smaller than measured PWV. For subject 1 and 4, simulated PWV is smaller in every artery. For subject 2, 3, 5 and 6 simulated PWV is higher than measured PWV. For 4 positions, the difference between PWV estimated from simulations and from measurements is smaller than.5 m/s: for subject 3 between the IA and and for subject 5 between the IA and, the and and the IA and. The arterial distensibility was determined with the reverse method (Figure 5.5). This leads to a mean distensibility of 18 ± 6.2 MPa -1 for the IA, 14 ± 7.3 MPa -1 for the, 7. ± 2.2 MPa -1 for the, 11 ± 8.9 MPa -1 for the and 7.7 ± 6.2 MPa -1 for the D [MPa^-1] IA subject Figure 5.6: Distensibility determined from simulation results. No relation was found between the location at the arterial tree and distensibility. Distensibility of the is low for all subjects, but distensibility of all other arteries differs between subjects. 29

30 6 Discussion Aim of this study was to determine the mechanical properties of the arteries of the lower limb with a one dimensional wave propagation model [Bessems et al., 27]. First ultrasound measurements were performed and used to estimate the input parameters of the model. Secondly, a reverse method was used to fit simulated to measured BVF and pressure waveforms. With this reverse method, distensibility of the arteries of the lower limb was determined. 6.1 Ultrasound and applanation tonometry measurements Measurements were both time consuming and difficult and not all subjects were suitable for the measurements. Measuring BVF, IMT and wall distension in the, and of the lower limb was very difficult, because both wall distension and BVF are small in these arteries. Nevertheless, BVF and wall distension measurements were successfully performed on six healthy subjects on five measurement sites in the lower limb. Diameter and IMT were also estimated from the distension measurements. Diameter and IMT estimated from the ultrasound measurements were compared to values measured in previous research. The mean measured arterial diameter of all arteries differs 22 percent from the values found in literature for healthy subjects. The mean diameter of the [Kornet et al., 1999], the [Debasso et al., 24], the [Shau et al., 1999] and the [Black et al., 23] are lower than values found in literature. With respect to the mean IMT, the values found in literature in the [Kornet et al., 1999[ and the [Debasso et al., 24] are in good agreement. The mean IMT of the found in literature is 48% lower [Osika et al., 27] than the measured IMT. Unfortunately no values were found in literature for the mean diameter of the IA and for the mean IMT of the IA and the. Two methods were used to estimate distensibility non-invasively. The first method estimates distensibility from the relative change in arterial cross section driven by the change in pressure between systole and diastole. For this diameter, wall distension and pressure measurements were used. The estimation of distensibility is very sensitive to errors in the measured pressure and distension waveform. A bias in pressure and distension measurements causes large differences in the distensibility estimation. The second method estimates distensibility from PWV. PWV was estimated from the measured pressure waveforms with the pulse transit time method. Early wave reflections in the measurements complicate the determination of the foot of the wave and therefore introduce a bias in the estimation of PWV with the pulse transit time method. In the and the, IMT can t be determined for every subject or in only one measurement. Large differences are obtained between the estimations of distensibility for all subjects. Because the first patient specific estimation of distensibility is not reproducible, the mean distensibility of all subjects could be a better initial input parameter for the model than a patient specific estimation of distensibility. Furthermore, applanation tonometry measurements could only be performed at two positions on the lower limb. Therefore, the pressure waveforms measured with applanation tonometry were not used for the estimation of model parameters and for the reverse method. Distension waveforms are assumed to resemble the pressure waveform and are rescaled to DBP and MBP for all measurement positions. 3