Tri-layered constitutive modelling unveils functional differences between the pig ascending and lower thoracic aorta.

The arterial wall's tri-layered macroscopic and layer-specific microscopic structure determine its mechanical properties, which vary at different arterial locations. Combining layer-specific mechanical data and tri-layered modelling, this study aimed to characterise functional differences between the pig ascending (AA) and lower thoracic aorta (LTA). AA and LTA segments were obtained for n=9 pigs. For each location, circumferentially and axially oriented intact wall and isolated layer strips were tested uniaxially and the layer-specific mechanical response modelled using a hyperelastic strain energy function. Then, layer-specific constitutive relations and intact wall mechanical data were combined to develop a tri-layered model of an AA and LTA cylindrical vessel, accounting for the layer-specific residual stresses. AA and LTA behaviours were then characterised for in vivo pressure ranges while stretched axially to in vivo length. The media dominated the AA response, bearing>2/3 of the circumferential load both at physiological (100 mmHg) and hypertensive pressures (160 mmHg). The LTA media bore most of the circumferential load at physiological pressure only (57±7% at 100 mmHg), while adventitia and media load bearings were comparable at 160 mmHg. Furthermore, increased axial elongation affected the media/adventitia load-bearing only at the LTA. The pig AA and LTA presented strong functional differences, likely reflecting their different roles in the circulation. The media-dominated compliant and anisotropic AA stores large amounts of elastic energy in response to both circumferential and axial deformations, which maximises diastolic recoiling function. This function is reduced at the LTA, where the adventitia shields the artery against supra-physiological circumferential and axial loads.

The mechanisms of adaptation for muscle fascicle length changes with exercise: Implications for spastic muscle.

We are proposing optimal training conditions that can lead to an increase in the number of serial sarcomeres (SSN) and muscle fascicle length (FL) in spastic muscles. Therapeutic interventions for increasing FL in clinical populations with neurological origin, in whom relative shortness of muscle fascicles contributed to the presentation of symptoms such as spasticity, contracture, and limited functional abilities, do not generally meet these conditions, and therefore, result in less than satisfactory outcomes. Based on a review of literature, we argue that protocols of exercise interventions that led to sarcomerogenesis, and increases in SSN and FL in healthy animal and human models satisfied three criteria: 1) all involved eccentric exercise at appropriately high velocity; 2) resulted in positive strain of muscle fascicles; and 3) momentary deactivation in the stretched muscle. Accordingly, to increase FL in spastic muscles, new exercise protocols in which the three presumed criteria are satisfied, must be developed, and long-term muscle architectural and functional adaptations to such trainings must be examined.

Noninvasive assessment of the common carotid artery hemodynamics with increasing exercise work rate using wave intensity analysis.

Noninvasively determined local wave speed ( c) and wave intensity (WI) parameters provide insights into arterial stiffness and cardiac-vascular interactions in response to physiological perturbations. However, the effects of incremental exercise and subsequent recovery on c and WI have not been fully established. We examined the changes in c and WI parameters in the common carotid artery (CCA) during exercise and recovery in eight young, healthy male athletes. Ultrasound measurements of CCA diameter and blood flow velocity were acquired at rest, during five stages of incremental exercise (up to 70% maximum work rate), and throughout 1 h of recovery, and noninvasive WI analysis [diameter-velocity ( DU) approach] was performed. During exercise, c increased (+136%), showing increased stiffness with work rate. All peak and area of forward compression, backward compression, and forward expansion waves increased during exercise (+452%, +700%, and +900%, respectively). However, WI reflection indexes and CCA resistance did not significantly change from rest to exercise. Furthermore, wave speed and the magnitude of all waves returned to baseline within 5 min of recovery, suggesting that the effects of exercise in the investigated parameters of young, healthy individuals were transient. In conclusion, incremental exercise was associated with an increase in local CCA stiffness and increases in all wave parameters, indicative of enhanced ventricular contractility and improved late-systolic blood flow deceleration. NEW & NOTEWORTHY We examined hemodynamics of the common carotid artery using noninvasive application of wave intensity analysis during exercise and recovery. The hemodynamic adjustments to exercise were associated with increases in local common carotid artery stiffness and all waves' parameters, with the latter indicating enhanced ventricular contractility and improved late systolic blood flow deceleration.

Multi-scale interaction of particulate flow and the artery wall.

We discuss, from the perspective of basic science, the physical and biological processes which underlie atherosclerotic (plaque) initiation at the vascular endothelium, identifying the widely separated spatial and temporal scales which participate. We draw on current, related models of vessel wall evolution, paying particular attention to the role of particulate flow (blood is not a continuum fluid), and proceed to propose, then validate all the key components in a multiply-coupled, multi-scale modeling strategy (in qualitative terms only, note). Eventually, this strategy should lead to a quantitative, patient-specific understanding of the coupling between particulate flow and the endothelial state.

Determination of wave speed and wave separation in the arteries using diameter and velocity.

The determination of arterial wave speed and the separation of the forward and backward waves have been established using simultaneous measurements of pressure (P) and velocity (U). In this work, we present a novel algorithm for the determination of local wave speed and the separation of waves using the simultaneous measurements of diameter (D) and U. The theoretical basis of this work is the solution of the 1D equations of flow in elastic tubes. A relationship between D and U is derived, from which, local wave speed can be determined; C=+/-0.5(dU(+/-)/dlnD(+/-)). When only unidirectional waves are present, this relationship describes a linear relationship between lnD and U. Therefore, constructing a lnDU-loop should result in a straight line in the early part of the cycle when it is most probable that waves are running in the forward direction. Using this knowledge of wave speed, it is also possible to derive a set of equations to separate the forward and backward waves from the measured D and U waveforms. Once the forward and backward waveforms of D and U are established, we can calculate the energy carried by the forward and backward waves, in a similar way to that of wave intensity analysis. In this paper, we test the new algorithm in vitro and present results from data measured in the carotid artery of human and the ascending aorta of canine. We conclude that the new technique can be reproduced in vitro, and in different vessels of different species, in vivo. The new algorithm is easy to use to determine wave speed and separate D and U waveforms into their forward and backward directions. Using this technique has the merits of utilising noninvasive measurements, which would be useful in the clinical setting.

Resolving the time lag between pressure and flow for the determination of local wave speed in elastic tubes and arteries.

It is well established that wave speed can be determined using the initial linear part of the pressure-velocity loop (PU-loop). However, the frequency response of most flow measuring devices is usually slower than that of solid-state pressure transducers; making flow waveforms lagging in time behind pressure waveforms. If this lag, which is traditionally determined by eye, is not corrected prior to the analysis, the PU-loop method may provide inaccurate wave speeds. The main aim of this work is therefore to introduce an objective technique to establish the value of this lag. The new technique relies on the linearity between pressure and velocity in the absence of reflections, and determines the highest correlation factor between pressure and velocity in the range of minimum pressure to maximum velocity. We shifted the flow waveform backwards in time steps equal to the sampling interval, and the time shift associated with the highest correlation indicates the correct time lag of the flow waveform. We first tested the new technique in vitro using a uniform latex tube and compared the results to those established using the traditional by eye method, whilst varying the filter setting of the flowmeter. Then we applied the new technique to pressure and flow measured in the ascending aorta of anaesthetised open-chested dogs. We found the time lag between pressure and velocity calculated by the new technique in good agreement with that determined by eye in vitro and that increasing the filtering power generated greater delay between the measured pressure and flow. The results obtained in vivo using the new technique were also in good agreement with those determined by eye. We therefore conclude that the new technique provides a convenient and objective way of correcting the lag and can reliably align pressure and flow.

Modelling the interaction of haemodynamics and the artery wall: current status and future prospects.

Clinical research has historically focused on the two main strategies of in vivo and in vitro experimentation. The concept of applying scientific theory to direct clinical applications is relatively recent. In this paper we focus on the interaction of wall shear stress with the endothelium and discuss how 'state of the art' computer modelling techniques can provide valuable data to aid understanding. Such data may be used to inform experiment and further, may help identify the key features of this complex system. Current emphasis is on coupling haemodynamics with models of biological phenomena to test hypotheses or predict the likely outcome of a disease or an intervention. New technologies to enable the integration of models of different types, levels of complexity and scales, are being developed. As will be discussed, the ultimate goal is the translation of this technology to the clinical arena.

The compression and expansion waves of the forward and backward flows: an in-vitro arterial model.

Although the propagation of arterial waves of forward flows has been studied before, that of backward flows has not been thoroughly investigated. The aim of this research is to investigate the propagation of the compression and expansion waves of backward flows in terms of wave speed and dissipation, in flexible tubes. The aim is also to compare the propagation of these waves with those of forward flows. A piston pump generated a flow waveform in the shape of approximately half-sinusoid, in flexible tubes (12 mm and 16 mm diameter). The pump produced flow in either the forward or the backward direction by moving the piston forward, in a 'pushing action' or backward, in a 'pulling action', using a graphite brushes d.c. motor. Pressure and flow were measured at intervals of 5 cm along each tube and wave speed was determined using the PU-loop method. The simultaneous measurements of diameter were also taken at the same position of the pressure and flow in the 16 mm tube. Wave intensity analysis was used to determine the magnitude of the pressure and velocity waveforms and wave intensity in the forward and backward directions. Under the same initial experimental conditions, wave speed was higher during the pulling action (backward flow) than during the pushing action (forward flow). The amplitudes of pressure and velocity in the pulling action were significantly higher than those in the pushing action. The tube diameter was approximately 20 per cent smaller in the pulling action than in the pushing action in the 16 mm tube. The compression and expansion waves resulting from the pushing and pulling actions dissipated exponentially along the travelling distance, and their dissipation was greater in the smaller than in the larger tubes. Local wave speed in flexible tubes is flow direction- and wave nature-dependent and is greater with expansion than with compression waves. Wave dissipation has an inverse relationship with the vessel diameter, and dissipation of the expansion wave of the pulling action was greater than that of the pushing action.

Determination of wave intensity in flexible tubes using measured diameter and velocity.

Wave intensity (WI) is a hemodynamics index, which is the product of changes in pressure and velocity across the wave-front. Wave Intensity Analysis, which is a time domain technique allows for the separation of running waves into their forward and backward directions and traditionally uses the measured pressure and velocity waveforms. However, due to the possible difficulty in obtaining reliable pressure waveforms non-invasively, investigating the use of wall displacement instead of pressure signals in calculating WI may have clinical merits. In this paper, we developed an algorithm in which we use the measured diameter of flexible tube's wall and flow velocity to separate the velocity waveform into its forward and backward directions. The new algorithm is also used to separate wave intensity into its forward and backward directions. In vitro experiments were carried out in two sized flexible tubes, 12mm and 16mm in diameters, each is of 2 m in length. Pressure, velocity and diameter were taken at three measuring sites. A semi-sinusoidal wave was generated using a piston pump, which ejected 40cc water into each tube. The results show that separated wave intensity into the forward and backward directions of the new algorithm using the measured diameter and velocity are almost identical in shape to those traditionally using the measured pressure and velocity. We conclude that the new algorithm presented in this work, could have clinical advantages since the required information can be obtained non-invasively.

Simultaneous determination of wave speed and arrival time of reflected waves using the pressure-velocity loop.

In a previous paper we demonstrated that the linear portion of the pressure-velocity loop (PU-loop) corresponding to early systole could be used to calculate the local wave speed. In this paper we extend this work to show that determination of the time at which the PU-loop first deviates from linearity provides a convenient way to determine the arrival time of reflected waves (Tr). We also present a new technique using the PU-loop that allows for the determination of wave speed and Tr simultaneously. We measured pressure and flow in elastic tubes of different diameters, where a strong reflection site existed at known distances away form the measurement site. We also measured pressure and flow in the ascending aorta of 11 anaesthetised dogs where a strong reflection site was produced through total arterial occlusion at four different sites. Wave speed was determined from the initial slope of the PU-loop and Tr was determined using a new algorithm that detects the sampling point at which the initial linear part of the PU-loop deviates from linearity. The results of the new technique for detecting Tr were comparable to those determined using the foot-to-foot and wave intensity analysis methods. In elastic tubes Tr detected using the new algorithm was almost identical to that detected using wave intensity analysis and foot-to-foot methods with a maximum difference of 2%. Tr detected using the PU-loop in vivo highly correlated with that detected using wave intensity analysis (r (2) = 0.83, P < 0.001). We conclude that the new technique described in this paper offers a convenient and objective method for detecting Tr, and allows for the dynamic determination of wave speed and Tr, simultaneously.

Wave dissipation in flexible tubes in the time domain: in vitro model of arterial waves.

Earlier work of wave dissipation in flexible tubes and arteries has been carried out predominantly in the frequency domain and most of the studies used the measured pressure waveform for presenting the results. In this work we investigate the pattern of wave dissipation in the time domain using the separated forward and backward travelling waves in flexible tubes. We tested four sizes of latex tubes of 2m in length each, where a single semi-sinusoidal in shape, pressure wave, was produced at the inlet of each tube. Simultaneous measurements of pressure and flow waveforms were recorded every 5cm along the tubes and wave speed was determined using the pressure-velocity loop method (PU-loop). The measured data and wave speed were used to separate the pressure waveform and wave intensity, into their forward and backward directions, using wave intensity analysis (WIA). Also, the energy carried by the wave was calculated by integrating the relevant area under the wave intensity curve. The peak of the measured pressure waveform increased downstream, however, the peak of the separated forward pressure waveform decreased exponentially along the tube. Wave intensity and energy also dissipated exponentially along the travelling distance. The peaks of the separated pressure and wave intensity decreased in the forward in a similar exponential way to that in the backward direction in all four tube sizes. Also, the smaller the size of the tube the greater wave dissipation it caused. We conclude that wave separation is useful in studying wave dissipation in elastic tubes, and WIA provides a convenient method for determining the dissipation of the energy carried by the wave along the travelled distance. The separated pressure waveform, wave intensity and wave energy dissipate exponentially with the travelling distance, and wave dissipation varies conversely with the diameter of elastic tubes.

Wave-energy patterns in carotid, brachial, and radial arteries: a noninvasive approach using wave-intensity analysis.

The study of wave propagation at different points in the arterial circulation may provide useful information regarding ventriculoarterial interactions. We describe a number of hemodynamic parameters in the carotid, brachial, and radial arteries of normal subjects by using noninvasive techniques and wave-intensity analysis (WIA). Twenty-one normal adult subjects (14 men and 7 women, mean age 44 +/- 6 yr) underwent applanation tonometry and pulsed-wave Doppler studies of the right common carotid, brachial, and radial arteries. After ensemble averaging of the pressure and flow-velocity data, local hydraulic work was determined and a pressure-flow velocity loop was used to determine local wave speed. WIA was then applied to determine the magnitude, timings, and energies of individual waves. At all sites, forward-traveling (S) and backward-traveling (R) compression waves were observed in early systole. In mid- and late systole, forward-traveling expansion waves (X and D) were also seen. Wave speed was significantly higher in the brachial (6.97 +/- 0.58 m/s) and radial (6.78 +/- 0.62 m/s) arteries compared with the carotid artery (5.40 +/- 0.34 m/s; P < 0.05). S-wave energy was greatest in the brachial artery (993.5 +/- 87.8 mJ/m2), but R-wave energy was greatest in the radial artery (176.9 +/- 19.9 mJ/m2). X-wave energy was significantly higher in the brachial and radial arteries (176.4 +/- 32.7 and 163.2 +/- 30.5 mJ/m2, respectively) compared with the carotid artery (41.0 +/- 9.4 mJ/m2; P < 0.001). WIA illustrates important differences in wave patterns between peripheral arteries and may provide a method for understanding ventriculo-arterial interactions in the time domain.

Wave intensity in the ascending aorta: effects of arterial occlusion.

We examine the effects of arterial occlusion on the pressure, velocity and the reflected waves in the ascending aorta using wave intensity analysis. In 11 anaesthetised, open-chested dogs, snares were used to produce total arterial occlusion at 4 sites: the upper descending aorta at the level of the aortic valve (thoracic); the lower thoracic aorta at the level of the diaphragm (diaphragm); the abdominal aorta between the renal arteries (abdominal) and the left iliac artery, 2 cm downstream from the aorta iliac bifurcation (iliac). Pressure and flow in the ascending aorta were measured, and data were collected before and during the occlusion. During thoracic and diaphragm occlusions a significant increase in mean aortic pressure (46% and 23%) and in wave speed (25% and 10%) was observed, while mean flow rate decreased significantly (23% and 17%). Also, the reflected compression wave arrived significantly earlier (45% and 15%) and its peak intensity was significantly greater (257% and 125%), all compared with control. Aortic occlusion distal to the renal arteries, however, caused an indiscernible change in the pressure and velocity waveforms, and in the intensities and timing of the waves in the forward and backward directions. The measured pressure and velocity waveforms are the result of the interaction between the heart and the arterial system. The separated pressure, velocity and wave intensity are required to provide information about arterial hemodynamic such as the timing and magnitude of the forward and backward waves. The net wave intensity is simpler to calculate but provides information only about the predominant direction of the waves and can be misleading when forward and backward waves of comparable magnitudes are present simultaneously.

A new approach to investigate wave dissipation in viscoelastic tubes: application of Wave Intensity Analysis.

Wave dissipation in elastic and viscoelastic medium has been investigated extensively in the frequency domain. The aim of this study is to examine the pattern of wave dissipation in the time-domain using Wave Intensity Analysis. A single semi-sinusoidal pulse was generated in 8mm and 16mm diameter tubes; each is of 200cm in length. Pressure and flow measurements were taken at intervals of 5 cm along the tube. In order to examine the effect of the wall mechanical properties on wave dissipation, we also modified the wall of the 16mm tube; a thread of strong cotton was wound with a pitch of approximately 30° around the circumference of the tube in the longitudinal direction. The separated forward pressure, wave intensity and wave energy were calculated using Wave Intensity Analysis. The amplitudes of the forward pressure wave, wave intensity and wave energy dissipated exponentially with distance. In the 8mm diameter tube, the dissipation of forward pressure, wave intensity and wave energy were greater than those in 16mm tube. For the same sized of tube, there was no significant difference in the dissipation of forward pressure, wave intensity and wave energy between the modified and normal wall tubes. It is concluded that the size of tube has a significant effect on the wave dissipation but the mechanical properties of the wall do not have a discernable effect on wave dissipation.

Local and regional wave speed in the aorta: effects of arterial occlusion.

Arterial wave speed is widely used to determine arterial distensibility and has been utilised as a surrogate marker for vascular disease. A comparison between the results of the traditional foot-to-foot method for measuring wave speed to those of the pressure-velocity loop (PU-loop) method is one of the primary objectives of this paper. We also investigate the regional wave speed along the aorta, and the effect of arterial occlusion on the PU-loop measured in the ascending aorta. In 11 anaesthetised dogs, a total occlusion lasting 3 min was produced at four sites: upper thoracic, diaphragm, abdominal and left iliac artery. Pressure and flow in the ascending aorta and pressure proximal to the occlusion site were measured, and data were collected before, during the occlusion and after the occlusion had been removed. In control conditions, the wave speeds determined by the PU-loop in the aortic root were systematically lower than those measured by the foot-to-foot method. During thoracic and diaphragm occlusions, mean aortic pressure and wave speed increased significantly but returned to control values after each occlusion had been removed. The PU-loop is an objective and easy to use method for determining wave speed and can be advantageous for use in short arterial segments when local measurements of pressure and velocity are available.

Measurements of wave speed and reflected waves in elastic tubes and bifurcations.

Wave intensity analysis is a time domain method for studying waves in elastic tubes. Testing the ability of the method to extract information from complex pressure and velocity waveforms such as those generated by a wave passing through a mismatched elastic bifurcation is the primary aim of this research. The analysis provides a means for separating forward and backward waves, but the separation requires knowledge of the wave speed. The PU-loop method is a technique for determining the wave speed from measurements of pressure and velocity, and investigating the relative accuracy of this method is another aim of this research. We generated a single semi-sinusoidal wave in long elastic tubes and measured pressure and velocity at the inlet, and pressure at the exit of the tubes. In our experiments, the results of the PU-loop and the traditional foot-to-foot methods for determining the wave speed are comparable and the difference is on the order of 2.9+/-0.8%. A single semi-sinusoidal wave running through a mismatched elastic bifurcation generated complicated pressure and velocity waveforms. By using wave intensity analysis we have decomposed the complex waveforms into simple information of the times and magnitudes of waves passing by the observation site. We conclude that wave intensity analysis and the PU-loop method combined, provide a convenient, time-based technique for analysing waves in elastic tubes.

Arterial waves in humans during peripheral vascular surgery.

The purpose of this study was to investigate the effect of aortic clamping on arterial waves during peripheral vascular surgery. We measured pressure and velocity simultaneously in the ascending aorta, in ten patients (70+/-5 years) with aortic-iliac disease intra-operatively. Pressure was measured using a catheter tip manometer, and velocity was measured using Doppler ultrasound. Data were collected before aortic clamping, during aortic clamping and after unclamping. Hydraulic work in the aortic root was calculated from the measured data, the reflected waves were determined by wave-intensity analysis and wave speed was determined by the PU-loop (pressure-velocity-loop) method; a new technique based on the 'water-hammer' equation. The wave speed is approx. 32% (P<0.05) higher during clamping than before clamping. Although the peak intensity of the reflected wave does not alter with clamping, it arrives 30 ms (P<0.05) earlier and its duration is 25% (P<0.05) longer than before clamping. During clamping, left ventricule (LV) hydraulic systolic work and the energy carried by the reflected wave increased by 27% (P<0.05) and 20% (P<0.05) respectively, compared with before clamping. The higher wave speed during clamping explains the earlier arrival of the reflected waves suggesting an increase in the afterload, since the LV has to overcome earlier reflected compression waves. The longer duration of the reflected wave during clamping is associated with an increase in the total energy carried by the wave, which causes an increase in hydraulic work. Increased hydraulic work during clamping may increase LV oxygen consumption, provoke myocardial ischaemia and hence contribute to the intra-operative impairment of LV function known in patients with peripheral vascular disease.

Assessment of cardiac risk before peripheral vascular surgery: a comparison of myocardial perfusion imaging and long axis echocardiography at rest.

OBJECTIVE: To compare resting long axis echocardiography with adenosine thallium-201 emission tomography in detecting myocardial ischaemic abnormalities and surgical related risk in patients before peripheral vascular surgery. DESIGN: A prospective and blinded pre-operative examination of resting left ventricular minor and long axes and myocardial perfusion during adenosine vasodilation using thallium-201 emission tomography. SETTING: A tertiary referral centre for cardiac and vascular disease equipped with invasive, non-invasive and surgical facilities. SUBJECTS: 65 patients (40 male) with significant peripheral vascular disease, mean age 63+/-10 (S.D.) years, and 21 normal subjects of similar age. RESULTS: Thallium-201 myocardial perfusion tomography was abnormal in 50/65 patients; 27 had fixed, 23 reversible abnormalities (19 of whom had both). Long axis was considered abnormal if one or more of two systolic long axis disturbances, reduced extent of total excursion <1 cm at any of the three (left, septal and posterior left ventricular) sites or prolonged shortening >1 mm after A2, and two diastolic abnormalities, delayed onset of lengthening >80 ms after A2 or reduced peak lengthening velocity <4.5 cm/s, was present. Long axis score (maximum 12) was based on the presence or absence of these four disturbances at each of the three sites. Myocardial perfusion imaging with thallium-201 classified the patients into three different groups according to their liability to low, moderate or high surgical risk (summed stress perfusion score of 36). Thirteen of 50 patients were identified as subjects at high surgical risk, with a perfusion score of 22/36 and below. Twelve of these demonstrated significantly greater impairment of systolic and diastolic long axis function, compared to those at low surgical risk, with a total long axis echo score of 6/12 or more. Seventeen of 18 patients identified as being at low surgical risk, with a perfusion score of 32/36 and above, had total long axis score of less than 6/12. The remaining 19 moderate risk patients had a wide range of long axis scores. In the 65 patients studied there were two post-operative deaths, one post-discharge death due to cerebrovascular accident, and one due to renal failure. CONCLUSION: The combination of both systolic and diastolic long axis disturbances in patients with peripheral vascular disease can be used to predict the thallium assessment of surgical risk. Long axis echocardiography may thus have value as a screening test before non-cardiac surgery as well as providing a means of monitoring myocardial perfusion.

Determination of wave speed and wave separation in the arteries.

Considering waves in the arteries as infinitesimal wave fronts rather than sinusoidal wavetrains, the change in pressure across the wave front, dP, is related to the change in velocity, dU, that it induces by the "water hammer" equation, dP=+/-rhocdU, where rho is the density of blood and c is the local wave speed. When only unidirectional waves are present, this relationship corresponds to a straight line when P is plotted against U with slope rhoc. When both forward and backward waves are present, the PU-loop is no longer linear. Measurements in latex tubes and systemic and pulmonary arteries exhibit a linear range during early systole and this provides a way of determining the local wave speed from the slope of the linear portion of the loop. Once the wave speed is known, it is also possible to separate the measured P and U into their forward and backward components. In cases where reflected waves are prominent, this separation of waves can help clarify the pattern of waves in the arteries throughout the cardiac cycle.