Heart and Circulatory Physiology

Chasing the wave. Unfashionable but important new concepts in arterial wave travel

Robert A. Bleasdale, Kim H. Parker, Christopher J. H. Jones


A continuing focus on subcellular processes could result in neglect of the framework through which these small processes are integrated. This focus could be particularly problematic in the cardiovascular system in which a patient's clinical condition depends on the function of the whole system, specifically the pumping of blood and its distribution. These are currently less fashionable areas of research. However, the ability to characterize the intact system, for example during and after pharmacological or genetic perturbation, must be retained to realize the potential of molecular biology. Paradoxically, and unusually in medicine, the integrating processes within the circulation may be best understood by adopting a reductionist approach. Such an approach involves breaking down blood flow into its constituent elements. In this short editorial, we outline recent work on wavefronts, mathematically derived elemental forces in arterial flow that appear to provide a new perspective on the integrated system for clinicians and basic scientists alike. Although clinical applications remain to be elucidated, we speculate on likely areas of interest.

The problem for intact system research is that elements are interdependent and are not easily held constant. There are many examples, e.g., cardiac output depends on the heart and the arteries but they are linked through afterload, coronary flow depends on aortic pressure and ventricular contraction and they interact, etc. In a system of tubes, the challenge frequently is how to resolve the directional basis of a perturbation, does it originate upstream or downstream, or both? How can interaction between constituent parts be considered as an interchange of energy? In the circulation, how do arteries serve as cables or transmission lines? It has long been recognized that the answers may be found in the mechanics of arterial wave travel (41), an area that has been mathematically redefined during the last few years (26). A wave-intensity analysis of forward and backward wave travel has the potential to quantify the amplitude and direction of energy transfer in the systemic (11, 12), pulmonary (9), or coronary arterial systems (32). This theoretical approach also has practical implications, because there is increasing evidence that upstream-downstream cardiovascular interactions have real prognostic (1) and therapeutic significance (24) for cardiology patients in the clinic. The disadvantageous development of left ventricular hypertrophy in hypertension and the advantageous use of vasodilators in heart failure serve as testimony to the importance of ventriculoarterial interaction.


In the circulation, wave travel results from the exchange of energy between the kinetic energy of the flowing blood and the potential energy of the elastic vascular wall (4). For the case of the aorta, a pressure increase associated with left ventricular ejection leads to and is buffered by aortic expansion. Elastic recoil accounts for a limited pressure increase, however, and thus blood flows by expansion and contraction along the vessel wall. In situations associated with arterial stiffening, such as atherosclerotic disease, hypertension, or aging, the systolic pressure change increases as vessel expansion decreases. The pulse wave travels as this interchange of energy spreads through the system. The speed of wave travel increases as the wall becomes stiffer, or less distensible, so that the state of a vessel can be described functionally by the pulse-wave velocity (PWV) along it. As for other variables associated with the elasticity of the arterial wall, PWV is pressure dependent, varying nonlinearly with pressure as distension shifts the load between different wall components (7). Medial smooth muscle is viscoelastic and contributes variably depending on its state of activation and the pressure range. Vasorelaxation tends to reduce PWV (27) and vice versa.

Since the pioneering work of Taylor and MacDonald in the 1950s using Fourier transform methods, it has been a valuable approach to think of arterial waves as combinations of sinusoidal wave trains of different amplitudes and frequencies (3, 20). This may seem natural, because everything that we see and hear reaches us as sinusoidal wave trains. The problems with this form of analysis are that it assumes linearity and periodicity, and it has been difficult for clinicians to understand instinctively. There are other forms of waves that could be used to describe the observed arterial waves. One choice, particularly useful in gas dynamics, is to consider the propagation of discrete wavefronts, elemental waves that cause a step change in pressure and velocity as they pass. One distinct and practical advantage of these wavefronts is that they can be located in time and space unlike sinusoidal waveforms that have only a phase and a frequency. Furthermore, if we consider infinitesimal, or transient, wavefronts that either increase the pressure (compression waves) or decrease it (expansion waves), it is possible to build up any observed waveform from a time-dependent succession of wavefronts. These infinitesimal wavefronts are convected with the blood velocity and travel forward and backward in the artery with a wave speed usually referred to as the PWV.

A measured pressure or flow waveform generally results from coincident successions of forward and backward wavefronts. In the aorta, forward wavefronts arise in the heart, and backward wavefronts arise from reflection in the systemic periphery. In a pulmonary vein, backward wavefronts originate in the left atrium (31). In a coronary artery, forward wavefronts originate in the left ventricular cavity and backward wavefronts originate in the small vessels within the relaxing or contracting myocardium (32). The physiologically important point is that the wave travel in any given direction carries information about its initiating event.

Pressure and flow change together in forward wavefronts but in opposite ways in backward, reflected wavefronts (Fig.1). Thus reflected compression wavefronts arriving in the aorta increase pressure but decelerate flow. Conversely, in a coronary artery, a backward expansion wave caused by myocardial relaxation decreases pressure but accelerates the flow along the vessel (Fig. 2). The nature of a wave reflection will depend on the nature of the reflecting site; a positive pressure change will reflect as a positive pressure change from a closed end but becomes a negative pressure change when reflected from an opening or expansion. This gives rise to four possibilities: forward and backward compression and expansion wavefronts (Fig. 1). Most reflecting sites in the normal systemic arterial system are closed end in type, but this appears not to be the case for the pulmonary circulation (9), in which the daughter vessels have greater area than the main vessel and is probably not the case for an aneurysmal aorta. The different relations between pressure and flow in forward and backward wavefronts allow their mathematical separation so that upstream and downstream information can be distinguished from a single set of measurements at an accessible site remote from where the waves were initiated (12).

Fig. 1.

A sketch representation of the impact on the pressure waveform and the flow (velocity) waveform, measured at any specific site in the vascular system, by the passage of each of the four types of wavefronts (forward traveling compression or expansion waves or backward traveling compression or expansion waves).

Fig. 2.

A diagrammatic representation of an epicardial coronary artery and a section of myocardium. Solid arrows represent myocardial fiber lengthening in early diastole. Interrupted arrow represents the resultant backward traveling expansion wave, which decreases pressure but accelerates coronary flow.


Impedance Analysis

For the last half century, arterial waves have been envisioned almost exclusively in terms of sinusoidal wave trains. The well-known theorem of Fourier tells us that any time series, arterial pressure and flow, for example, can be represented exactly as the superposition of harmonic waves with the appropriate frequencies and magnitudes and phase shifts. This Fourier analysis has been the basis of the vast majority of studies on arterial haemodynamics, but the standard textbooks tend not to mention that it can be useful to analyze waves in other ways (20, 21).

Fourier analysis has contributed much to our understanding of arterial hemodynamics, but it is not necessarily the most convenient mode of analysis. The principle drawback is that it is a frequency-domain analysis in which it is fundamentally impossible to relate events occurring at specific times in the cardiac cycle to particular features of the frequency spectrum; each Fourier component is determined by its fit over a periodic cardiac waveform. Thus, for example, an intervention that affects only systole will affect the Fourier components of all frequencies to some extent. Second, Fourier analysis is essentially a steady-state analysis that cannot be used to study transient or nonperiodic events. Practically, this means that an irregular rhythm such as atrial fibrillation can only be approximated and that short-lived changes following intervention cannot be properly assessed. Finally, in its most common form, impedance analysis, Fourier analysis makes the generally unwarranted assumption that pressure and flow are linearly related through an Ohms law for each frequency component: pressure = impedance × flow.

Generally, clinical cardiologists find it difficult to follow the meaning of results explained in terms of phase and frequency and would prefer a more intuitive, time-dependent approach.

Wave-Intensity Analysis

There is another way to analyze waves that might lead to more readily understandable information. This alternative approach has its origins in gas dynamics and takes infinitesimal discrete wavefronts as its basic elements. The basic one-dimensional equations describing the conservation of mass and momentum in an elastic tube can be solved by using the method of characteristics (26, 30). The solution process is rather complicated, but the results are surprisingly simple. Any disturbance applied to the tube will propagate downstream and upstream with velocity, U ± c, whereU is the velocity of the fluid and c is the wave speed determined by D, the distensibility of the tube, and ρ is the density of the fluid, c = (1/ρ · D)1/2. In a nonlinear analysis, wave speed is allowed to be a function of pressure and therefore to change during the cardiac cycle. If the disturbance is a transient wavefront, then there is a simple relationship between the change in pressure across the wavefront (dP) and the change in velocity (dU): − dP = ± ρ · c · dU, where the + sign refers to the forward direction and the − sign is to the backward direction. For each direction, there are two types of wavefronts: compression wavefronts that increase pressure (generated by “blowing”) and expansion wavefronts that decrease pressure (generated by “sucking”). These wavefronts are the elemental waves of this approach to wave mechanics, and any complex wave can be synthesized from a succession of compression and expansion wavefronts. The simplest way to imagine this synthesis of a complex wave is to consider measuring the wave at short, discrete time intervals as in analog-to-digital sampling. The change in pressure between successive sampling times (dP) is the size of the pressure wavefront at each time of sampling. When all of these waves are added together, the original wave is generated.

The four possibilities of forward and backward compression and expansion waves, each potentially occurring simultaneously in the artery, mean that the interpretation of measured waveforms can be challenging. Fortunately, this task is made simpler by the mathematical observation that the wave intensity, the product of the change in pressure, and the change in velocity is positive for forward waves and negative for backward waves. Furthermore, the wave intensity calculated from a measured pressure and velocity at any time is equal to the algebraic sum of the wave intensities of the forward and backward waves intersecting at the site and time of measurement. Because of the importance of the wave intensity, this approach to the study of arterial hemodynamics is now generally known as wave-intensity analysis.

This approach has several physiological advantages. It is carried out in the time domain instead of the frequency domain, and therefore temporal events in the cardiac cycle can be interpreted directly in terms of wavefronts arriving at specific times. This allows the identification and quantification of upstream and downstream events. For example, waves arriving at the ascending aorta before and after the closure of the aortic valve can be identified and their impact on (or isolation from) the ventricle can be assessed. In addition, the method does not assume linearity between pressure and velocity and easily accommodates the main nonlinearities in the arteries, the change in wave speed with pressure, and the essential nonlinearity of the convective derivative in the conservation equations (11). The detailed mechanics have been published elsewhere and do not require repetition herein.

The most direct application of this analysis is the calculation of the wave intensity throughout the cardiac cycle from simultaneous measurements of pressure (or diameter) and velocity (or flow). The measurement of the pressure waveform generally requires an invasive approach, but this can be substituted by a wall displacement waveform measured noninvasively. The sign and magnitude of the measured wave intensity indicate the net effect of the forward and backward waves. Wave intensity is positive if the forward wave is larger than the backward wave and negative if the backward wave is larger than the forward wave. Zero wave intensity indicates either an absence of wave travel or that the forward and backward waves are of similar magnitude at the time of measurement. This simple calculation can tell us much about the amplitude and timing of reflected waves.

It is also possible to use wave-intensity analysis to separate the measured pressure and velocity waveforms into their forward and backward components, if one assumes that the forward and backward waves are linearly additive. This separation is formally equivalent to the separation based on the measurement of characteristic impedance proposed by Van den Bos et al. (34). This method depends on the knowledge of the local pulse-wave velocity. The basic wave equations allow the wave speed to be calculated as a function of pressure from pressure and velocity waveforms (pressure-velocity loop) measured during the short period in early systole when forward wave travel is predominant (13) (Fig.3). These separated wave intensities give an even more detailed description of the pattern of forward and backward waves, which determines the pressure and flow that can be measured in the artery.

Fig. 3.

A pressure-velocity loop. During the short period in early systole, between the vertical lines, when only forward waves are present, pressure and velocity are linearly related. The wave speed (c) can be derived from the slope of this relation if blood density (ρ) is known.


Because waves are blood-borne mediators of energy transfer, a wave pattern will inform about all the mechanical factors that influence the pressure or flow. One example in disease is in hypertension, in which reflected waves assume central importance. In the elderly or other individuals with stiff or diseased large arteries, wave reflections are increased and arrive early due to an increased PWV (22). Early and accentuated reflections in the aorta account primarily for late systolic pressure augmentation and thus the absolute systolic pressure level (14, 23). Reflected waves also account for isolated systolic hypertension (24). Early arriving wave reflections are the major determinant of the systolic-diastolic pressure difference, and recent evidence points to pulse pressure as a more powerful determinant of adverse events and mortality than either systolic or diastolic pressures alone (16). Systolic ventricular loading by early and accentuated wave reflections can result in left ventricular hypertrophy and left ventricular failure (5). Reversal of systolic hypertension with reduction of pulse pressure and ventricular hypertrophy brings outcome benefit and points to large conduit arteries as a target for therapeutic intervention. Evidence is already accumulating that different blood pressure-lowering agents exert differential effects on PWV and central arterial wave reflections. Pannier et al. (25) showed that, despite causing similar decreases in systolic and diastolic blood pressures, an angiotensin-converting enzyme inhibitor reduced aortic wave reflections more than a β-adrenergic antagonist, but only the latter reduced central aortic stiffness. Because the measurable contribution of wave reflections to hypertension varies between patients, these findings offer the opportunity for closer matching of therapy to phenotype in the future. Care must be taken in the clinic environment, however, because the increasing amplitude of reflections in the periphery results in peripheral pressures that are higher than central pressures, and these may be affected differently by vasoactive interventions (35). These differences have led some to advocate a peripheral measurement with computation of central pressure using a transfer function (38).

Conduit artery stiffening resulting in increased PWV has thus become a powerful indicator of adverse prognosis. PWV generally increases with neurohumorally mediated vasoconstriction and decreases with endothelium-dependent dilatation (27). PWV is independently associated with cardiovascular and all-cause mortality in hypertensives (15). PWV has long been known to be increased in patients with Type I diabetes (28). More recently, PWV has been shown to be increased in patients with Type II diabetes and their first-degree relatives (10). In these patients, arterial compliance decreases as fasting glucose increases and accordingly may be seen before any changes in mean blood pressure (19). In patients with diabetes, PWV correlates positively with intima-media thickness (33), and the elastic behavior of arteries varies with blood glucose control (19). Sustained hyperglycemia results in the irreversible glycation of proteins, including collagen and other matrix and plasma proteins, to form advanced glycation end products (AGEs) (2). Elastin and collagen may be glycated with a corresponding increase in vascular stiffness in vitro (39). Glycation and intimal and medial cross-links account for the increased vascular stiffness seen in patients with diabetes (29). AGE cross-link breakers reverse the diabetes-induced increase in arterial stiffness in vitro (40). They also increase arterial compliance and tend to decrease PWV in elderly nondiabetic patients. The pressure-velocity loop approach to PWV measurement developed from wave-intensity analysis is a relatively easy way of assessing PWV in clinical studies. Furthermore, this single-site approach to local wave speed measurement provides greater precision than the time delay method applied to measurements at different sites, which usually results in a PWV averaged over a segment of artery containing widely differing wall characteristics (13). Because local wave speed reflects local wall elastic properties, it has potential utility for the longitudinal assessment of therapy directed at large artery stiffness and pulse pressure augmentation in patients with systemic hypertension and perhaps chronic heart failure.

Not all systems are as simple as the systemic arteries, in which backward traveling waves result from the passive reflection of actively generated forward waves. In more complex systems, waves can be generated from either direction. Wave intensity has been shown to reveal the net direction of wave travel at any instant in any structure in which one-dimensional blood flow can be assumed. One such complex system is the coronary circulation, in which flow is determined by waves generated actively by both upstream and downstream events (32). Wave-intensity analysis has been applied to measurements derived from the canine coronary circulation at rest and during atrial pacing. During isovolumic contraction, negative values of wave intensity in coronary arteries indicate backward compression waves generated by the compression of the intramyocardial vessels that increase coronary pressure but decrease coronary flow. During early systole, increasing aortic pressure initiates forward waves, indicated by positive values of wave intensity that increase both coronary pressure and flow, despite continuing high left ventricular pressure. Similarly, the relaxation of the myocardium during diastole generates a backward expansion wave that initiates the diastolic coronary flow. These new concepts, yet to be measured in the cardiac catheter laboratory, could be a possible means of assessing the functional significance of coronary stenoses and their response to intervention.

A different complex system is the pulmonary venous circulation where there has been debate about the origin of changes in pulmonary venous flow; does systolic forward flow result from left atrial “suction” or transmission of the right ventricular pressure pulse through the pulmonary vascular bed? An analysis of recent measurements shows that the wave intensity is negative in early systole, indicating a dominant backward traveling wave that decreases pressure and increases forward flow velocity (31). The backward traveling wave in this case results from atrial events. Thus backward traveling waves impose opposite effects on pressure in the coronary arterial and pulmonary venous circulations. In a pulmonary vein, a backward traveling expansion wave manifests “suction” from within the left atrium, caused by atrial relaxation and the downward apical shift of the atrioventricular ring caused by left ventricular contraction. The next event in the pulmonary venous cardiac cycle is a forward traveling wave in later systole, which originates in the right ventricle. Finally, during early diastole, pulmonary forward flow into the atrium is motivated by a further backward traveling expansion wave, this time caused by the reduction of left atrial pressure that follows mitral valve opening.

The wave intensity approach has also been adapted for the study of diastolic left ventricular filling by assuming one-dimensional blood flow between the left ventricle and atrium during the filling interval (17). As for the pulmonary veins, suction is apparent as a backward traveling expansion wave, in this case generated by left ventricular relaxation and falling left ventricular pressure. Every detail of the diastolic mitral inflow trace is explained in terms of forward or backward waves generated by the pressure changes occurring within either the left ventricular or left atrial cavities. Wave-intensity analysis has also allowed the calculation of left ventricular wave speed, a measure of myocardial compressibility that varies throughout the cardiac cycle, in animal experiments (36). The clinical assessment of diastolic filling, in particular, may be advanced by echocardiographic assessment of intraventricular wave speed and the direction of wave travel.

Finally, PWV can be measured noninvasively in rats and mice, and variations with vasoactive medications have been confirmed (8). Apolipoprotein E-knockout mice demonstrate reduced endothelium-dependent dilatation and increased atherosclerosis and, as expected, increased aortic PWV (6), despite unchanged blood pressure. This change is probably caused by decreased endothelial NO-dependent dilatation and by fragmentation of the elastic arterial lamina (37). This study advances our understanding of wave travel in transgenic mice by detailed consideration of changes in aortic velocity and acceleration waveforms. Increased PWV coincides with an early systolic inflection of the blood velocity signal in the upper descending aorta. This is consistent with a reflection from the carotid bifurcation traveling in the direction of aortic blood flow. The acceleration caused by the reflection exhibits a time delay consistent with the distance to the reflecting site and the wave speed. The authors speculate whether the same waveform may be seen in patients with severe atherosclerosis. This finding of increased peripheral wave reflection suggests widespread abnormalities of conduit vessels and smaller peripheral branching arteries in this animal model. A transgenic mouse model of Marfan syndrome with underexpression of fibrillin exhibits increased thoracoabdominal PWV (18). The wave mechanics of small animals can therefore change independently of blood pressure and may be used to monitor genetic and therapeutic manipulation.


Hemodynamic studies are unfashionable in this era of molecular and genetic biology. However, we will increasingly need to understand how the elemental building blocks of life are incorporated into complex systems in intact organs and organisms. In the intact circulation, traveling wavefronts represent elemental units of energy transmitted within and between blood vessels and the heart. Energy transfer within traveling wavefronts represents a powerful integrating, or “messaging,” mechanism operating through blood flow. Wave patterns, analyzed mathematically, inform about the upstream and downstream events that influence the flow, and the speed of wave travel informs about the mechanical state of the blood vessel in the locality. The measurement of pulse-wave velocity has physiological and potentially clinical significance in animals and humans and can be determined noninvasively and used to monitor treatment. Wave-intensity analysis, a relatively new approach, can explain the forces underlying blood flow in such complex systems as the coronary and pulmonary circulations and the diastolic heart. Despite its ease of measurement and interpretation, it has not yet been used extensively in the clinical environment. Indeed, the work referred to in this editorial has been preliminary.

Further study will be needed before we can “catch the wave” and realize its full potential on the catwalk of contemporary cardiovascular medicine and physiology. However, most fashions turn full circle in the fullness of time; this editorial shows how arterial hemodynamics has recently been redesigned. It may yet have its day.


The contributions of Dr. A. Khir and Dr. A. Zambinini to the figures are greatly appreciated.


  • C. J. H. Jones and R. A. Bleasdale are funded by the British Heart Foundation.

  • Address for reprint requests and other correspondence: C. J. H. Jones, Wales Heart Research Institute, Heath Park, Cardiff CF14 4XN, UK (E-mail:jonescj4{at}cf.ac.uk).

  • 10.1152/ajpheart.00070.2003