# Noninvasive Evaluation of Left Ventricular Afterload

## Part 1: Pressure and Flow Measurements and Basic Principles of Wave Conduction and Reflection

## Jump to

- Article
- Abstract
- Noninvasive Pressure Measurements
- Noninvasive Flow Measurements
- Basic Concepts for Afterload Assessments Using Pressure and Flow Measurements
- Input Impedance
- Model 1: Pulsatile Pressure and Flow in a Single Elastic Tube of Infinite Length (Absence of Reflected Waves)
- Model 2: Pressure and Flow in a Single Elastic Tube of Finite Length in the Presence of Wave Reflections
- Forward and Reflected Waves in the Human Arterial Tree
- Perspectives
- Acknowledgments
- References

- Figures & Tables
- Supplemental Materials
- Info & Metrics
- eLetters

## Abstract

The mechanical load imposed by the systemic circulation to the left ventricle is an important determinant of normal and abnormal cardiovascular function. Left ventricular afterload is determined by complex time-varying phenomena, which affect pressure and flow patterns generated by the pumping ventricle and cannot be expressed as a single numeric measure or described in terms of pressure alone. Left ventricular afterload is best described in terms of pressure-flow relations. High-fidelity arterial applanation tonometry can be used to record time-resolved central pressure noninvasively, whereas contemporary noninvasive imaging techniques, such as Doppler echocardiography and phase-contrast MRI, allow for accurate assessments of aortic flow. Central pressure and flow can be analyzed using simplified biomechanical models to characterize various components of afterload, with great potential for mechanistic understanding of the role of central hemodynamics in cardiovascular disease and the effects of therapeutic interventions. In the first part of this tutorial, we review noninvasive techniques for central pressure and flow measurements and basic concepts of wave conduction and reflection as they relate to the interpretation of central pressure-flow relations. Conceptual descriptions of various models and methods are emphasized over mathematical ones. Our review is aimed at helping researchers and clinicians apply and interpret results obtained from analyses of left ventricular afterload in clinical and epidemiological settings.

The mechanical load imposed by the systemic circulation to the pumping ventricle is an important determinant of normal cardiovascular function and plays an important role in the pathophysiology of various cardiac conditions, including hypertensive heart disease, heart failure, postinfarction left ventricular (LV) remodeling, left-sided regurgitant valvular lesions, aortic stenosis, aortic coarctation, and congenital conditions with a systemic right ventricle (eg, congenitally corrected transposition of the great arteries).

LV afterload is the impedance (load) against which the LV must work to promote forward flow (eject blood). In the presence of a normal aortic valve, LV afterload is largely determined by the properties of the arterial tree (“arterial load”).^{1,2} Although arterial pressure is taken as a useful surrogate of LV afterload in clinical practice, in reality, complex time-varying phenomena determine LV afterload, which ultimately affect, in a reciprocal fashion, the pressure and flow generated by the pumping ventricle. Pressure is not only dependent on afterload but also strongly influenced by ventricular structure and function.^{3} Therefore, LV afterload cannot be expressed as a single number, nor can it be described in terms of pressure alone, but should be described in terms of pressure-flow relations, which allow the quantification of “steady” (nonpulsatile) load and various components of pulsatile load.^{1,4} Although pulsatile LV afterload is fairly complex, it can be quantified and summarized using relatively simple mechanical models of the systemic circulation. In the first part of this tutorial, we review noninvasive techniques for aortic pressure and flow measurements and describe basic models of pulsatile wave conduction and reflection. In the second part, we review analytic methods used to assess LV afterload using basic physiological principles. Although material properties of the arterial wall (arterial stiffness) are important in cardiovascular disease, we consider them only in the context of their functional consequences for the pumping heart. We refer the reader to excellent literature regarding arterial stiffness,^{5,6} which is not the focus of this tutorial. Similarly, it would be impossible to extensively reference landmark contributions from investigators in this area. We cite only general reviews and a limited number of original research publications. Extensive references to original research contributions can be found elsewhere.^{1,4}

## Noninvasive Pressure Measurements

Arterial applanation tonometry can be used to obtain high-fidelity pressure wave forms from the carotid artery, which is morphologically similar to (and, therefore, may be an acceptable noninvasive surrogate of) the aortic pressure wave form.^{7} Arterial tonometry is based on the principle that, when the artery is immobilized and the arterial wall is flattened against a pressure sensor, pressure within the lumen is directly transmitted to the sensor. The carotid waveform needs to be calibrated, preferably using mean and diastolic pressure, which can be assessed at the brachial artery.^{8} This approach is justified because mean and diastolic pressures exhibit little variation between the upper limb and the central arteries, in contrast to systolic pressure, which increases variably from the aorta to the brachial artery because of the phenomenon of pulse pressure amplification.^{1,8} With proper technique, high-fidelity recordings of carotid pressure are feasible in most situations, although great operator skill may be required in some cases, particularly among obese individuals or those with a small stroke volume (ie, heart failure patients). Even with proper technique, high-quality recordings may not be possible in some cases. Technical issues regarding carotid tonometry are discussed in the online Data Supplement, available at http://hyper.ahajournals.org.

For completeness, it is to be mentioned that radial arterial tonometry and application of a transfer function are often used to assess aortic pressure.^{1} Although this approach allows for a reliable assessment of central systolic pressure (provided that radial artery waveforms are adequately calibrated), it may be less suited when the central pressure waveform is required in all of its detail, as will be the case here.^{9} However, it should also be recognized that there are systematic differences between the carotid and aortic pressure wave forms, such as a greater early systolic carotid upstroke slope compared with the aortic wave form.^{10} Although these differences may affect the assessment of specific components of LV afterload (and should be kept in mind), detailed analyses of LV afterload using tonometry-derived carotid pressure closely correlate with those derived from invasive aortic measurements.^{11}

## Noninvasive Flow Measurements

The 2 most accurate noninvasive techniques for flow measurements in the proximal arterial system are pulsed-wave Doppler (PW-Doppler) echocardiography and cardiac phase-contrast MRI (PC-MRI).^{12} Flow measurements with PW-Doppler rely on the principle that the Doppler shift of reflected ultrasound waves induced by moving blood particles is proportional to the velocity of those particles.^{1} PW-Doppler is an inexpensive, safe and convenient method to measure flow velocities, and there is widespread expertise in its use (Table). Important technical details for measuring blood flow with PW-Doppler include adequate sample position, sample volume, gain settings, and Doppler beam orientation. Although proximal aortic flow can be difficult to interrogate consistently and reproducibly, LV outflow tract (LVOT) velocities can be adequately interrogated with PW-Doppler in most subjects. Flow velocities must be converted to volume flow by multiplying velocity times LVOT cross-sectional area at the exact point of PW-Doppler interrogation.^{1,13,14} This method assumes a flat-flow profile across the LVOT and is highly dependent on the accuracy of cross-sectional area estimations, which in most studies have relied on LVOT anteroposterior diameter measurements from the parasternal long axis view, assuming a circular area (π*radius^{2}).^{13,14} This method has several limitations, including LVOT eccentricity (noncircularity); therefore, our preferred method is 3D echocardiography. Technical tips and considerations for measurements of LVOT velocity and area measurements with PW-Doppler and 3D echocardiography are described in the Data Supplement, figures, and movies available online at http://hyper.ahajournals.org.

PC-MRI relies on the fact that, when 2 opposing magnetic gradient pulses are applied to static nuclei aligned in a magnetic field, the effects of the pulses on nuclear spins cancel each other out, but if a particle moves in the time between the pulses, a phase shift of the nuclear spins within the moving particle is accumulated, which is proportional to the velocity of movement along the gradient’s direction. With PC-MRI, velocity maps along any given anatomic plane can be generated. When the gradient direction is applied perpendicular to the cross-sectional vessel plane (“through-plane” velocity encoding; Figure 1), the velocity distribution over the vessel cross-sectional area is measured, without assuming a flat-flow profile. Among other parameters, PC-MRI requires a user-defined velocity-encoding sensitivity, which should be as low as possible to adequately resolve flow velocities, yet higher than peak velocity in the region of interest to avoid aliasing. Although velocity-encoding sensitivity should be tailored to individual measurements, a good starting point for most proximal aortic flow measurements is 130 to 150 cm/s. PC-MRI data are acquired over several cardiac cycles, and consistent cardiac timing in each cycle is assumed. An important technical consideration is that flow measurements may be affected by phase-offset errors caused by in-homogeneities of the magnetic field environment (eddy currents and concomitant gradient effects). Even small background velocity offset errors can result in significant errors in volumetric flow, because summation of phase offset across the entire cross-sectional area of the aorta occurs (larger vessels being associated with greater phase-offset errors). The preferred method for phase-offset correction is based on data acquisition from a stationary phantom.^{15} If phantom data are not acquired, information from stationary tissue located in the imaging plane^{16} may be used to compensate for phase-offset. Other potential problems in PC-MRI that should be kept in mind include signal loss because of turbulent flow and pulsation artifacts. Finally, the anatomic plane should be prescribed exactly perpendicular to the flow direction, parameters should be set to obtain a temporal resolution that meets minimum requirements for planned pressure-flow analytic techniques, and the prescription of the region of interest (aortic lumen) during postprocessing should be as precise as possible. Magnitude images are particularly useful to identify the vessel lumen during systole (because of its high flow-related signal intensity). The precise delineation of the aortic lumen in diastole can be challenging, particularly when magnitude images are used in isolation; matched phase images should always be assessed simultaneously, because they can be very helpful in separating the diastolic lumen (in which low-velocity flow occurs) from the adjacent static tissue. Figure 1 shows an example of through-plane PC-MRI measurements of proximal aortic flow. Table 1 summarizes advantages and disadvantages of PW-Doppler and PC-MRI.

## Basic Concepts for Afterload Assessments Using Pressure and Flow Measurements

The terminology “resistance” (R) originates from electric circuit theory and applies to signals that do not vary in time (such as direct current). Ohm’s law is well known, with resistance being the ratio of the potential difference (voltage) over a conductor and the electric current. To describe resistance to electric current that fluctuates over time (eg, alternating current), the term impedance (Z) is used. Impedance is a more general formulation than resistance (which it actually includes) but varies with the frequency of fluctuations in the electric signal (R is the value of Z at zero-frequency). These terms have been “borrowed” to describe hemodynamic phenomena.^{1,4} Analogous to the dampening of electric flow by electric devices, R and Z in hemodynamics refer to dampening of blood flow by blood vessels, expressing the relation between pressure (voltage) and flow (current). By convention, the term “resistance” is typically used to describe nonoscillatory opposition to flow, whereas the term “impedance” is used for opposition to fluctuating (pulsatile) flow.

Assuming for the sake of simplicity that the downstream pressure at the end of a vessel of finite length (uniform in geometry and mechanical properties) is 0, resistance to flow imposed by the vessel equals the ratio between mean pressure at its upstream end and mean flow through the vessel (Pm/Qm) and is, according to Poiseuille’s law, directly proportional to vessel length and inversely proportional to the fourth power of vessel radius. Impedance to pulsatile flow imparted by a given vessel segment is defined, in analogy, as the ratio of pulsatile change in pressure/pulsatile change in flow in that particular vessel. In the absence of reflected waves in the vessel, this property is called characteristic impedance (Zc). It can be approximated as ρ*PWV/A, where ρ is blood density, pulse wave velocity (PWV) is the propagation speed of the pulse through that vessel, and A is vessel cross-sectional area (which, assuming a circular cross-section, is proportional to radius to the power of 2).^{1} PWV (a commonly used index of segmental stiffness), is directly related to the square root of wall elastic modulus and inversely proportional to the square root of vessel radius (ie, radius to the power of 0.5).^{4,5,17} Therefore, Zc depends on the stiffness of the vessel but is also highly dependent on vessel radius, being inversely proportional to its power of 2.5.^{17}

## Input Impedance

The resistance and impedance to flow imposed by a single vessel of finite length or by a vessel segment must be clearly distinguished from the summed resistance/impedance of an entire vascular bed. Because the arterial system is composed of a network of nonuniform, branching vessels with different geometries and wall properties that interact with each other, it is impossible to define the impedance of the arterial system based on single vessel properties. Indeed, the LV only senses the “summed” mechanical load imposed by all of the vessels downstream of the LVOT. The complex pattern of summed impedance imposed by a vascular bed downstream of a particular point (and which can be fully assessed by measuring time-varying flow and pressure at that particular point) is called input impedance^{1,4,13,14} (note the difference from characteristic impedance, Zc). Therefore, by analyzing proximal aortic pressure and flow, the impedance of the entire arterial tree is obtained, which is effectively what the heart “sees.” Aortic input impedance is, therefore, not a measure of aortic properties but rather reflects the load imparted by the proximal aorta and all of the arterial segments distal to it, including the effects of wave reflections.

Analyses of aortic input impedance are often done in the frequency domain. Any signal with periodicity, as can be reasonably assumed for arterial pressure and flow waves in steady-state (stable) conditions, can be decomposed into its steady (nonpulsatile or zero-frequency) component and its harmonic components, which, by definition, have a frequency that is a multiple of heart rate (fundamental frequency). The mathematical technique used for harmonic decomposition of pressure and flow signals is the Fourier transform.^{1,13,14} Each harmonic component is a pure sinusoidal wave with 3 basic properties: modulus (amplitude), period (which determines its temporal frequency, *f*), and phase angle (which is a numeric expression of its position in time relative to the beginning of each fundamental period, or cardiac cycle; Figure 2). When steady and harmonic components of a waveform are added arithmetically, the original waveform is obtained. The number of harmonic components within a waveform that can be discerned depends on the temporal resolution of the acquired signal and equals half the number of time points available within one fundamental period. Most of the energy and relevant details contained in human pressure and flow wave forms are found in the first 10 harmonics, which, therefore, can be obtained if ≥20 measurement points across the cardiac cycle are available.^{1} This is an important consideration, because it determines the minimal effective temporal resolution required for flow measurements, which influences prescribed PC-MRI acquisition parameters. As will be discussed in more detail later in this tutorial, once the harmonic components of pressure and flow are known, modulus of input impedance is calculated at each frequency as the ratio of pressure modulus/flow modulus, whereas phase angle of input impedance is calculated as pressure phase angle minus flow phase angle (Pφ−Qφ).

A more intuitive approach to pressure-flow analyses relies on assessing relationships between pressure and flow waves when these are analyzed “directly” in the time domain rather than decomposed into their harmonics.^{1,17} Before further discussing frequency- or time-domain analyses of pressure-flow relations, it is useful to consider extremely simple models of blood flow and pressure in elastic tubes.

## Model 1: Pulsatile Pressure and Flow in a Single Elastic Tube of Infinite Length (Absence of Reflected Waves)

The effects of intermittent flow injection into a single elastic tube of infinite length illustrate pressure-flow relations in the absence of reflected waves. Under such conditions, pulsatile energy imparted from one end of the tube promotes forward flow and increases pressure within the tube. Measured flow and pressure signals demonstrate exactly the same shape (Figure 3). The amount of pressure rise (ΔP) versus flow rise (ΔQ; vertical scale in pressure and flow curves in Figure 3) for any given amount of imparted energy is determined by the Zc of the tube (therefore, Zc=ΔP/ΔQ). The plot of instantaneous pressure versus instantaneous flow data points corresponds to a straight line. The slope of this line is identical to the ratio of ΔP/ΔQ and, therefore, represents Zc (Figure 3, bottom left panel); in reality, pressure-flow relations in human vessels are not perfectly linear, but these concepts do apply during early phases of the cardiac cycle. In addition, because pressure and flow wave forms are identical, the amplitude (modulus) of any harmonic relative to the parent wave is identical for pressure and flow, and, therefore, when analyzed in the frequency domain, modulus of input impedance is the same at all of the nonzero frequencies (ie, the ratio between the amplitude of pressure harmonics over the amplitude of the respective flow harmonics is the same for every harmonic pair). This constant impedance modulus is also identical to Zc (Figure 3, bottom middle panel). If Zc is known, pressure and flow can be scaled in the vertical axis and their wave forms superposed (Figure 3, bottom right panel).

## Model 2: Pressure and Flow in a Single Elastic Tube of Finite Length in the Presence of Wave Reflections

If we consider an alternative model in which a pulse of flow is injected into an elastic tube of finite length with a “reflector” of the closed-end type at its end (the “impedance” terminating the tube is higher than its characteristic impedance), the energy imparted to the tube promotes forward flow and increases pressure within the lumen. Flow and pressure signals measured during this initial period of time demonstrate exactly the same shape, like in the previous model. Assuming that the fluid within the tube is incompressible (as is the case with blood within vessels), the energy wave imparted to the tube is transmitted along its wall at a finite speed (PWV) until it encounters the reflector on the other side. The energy wave is then reflected and transmitted “backward,” promoting flow away from the reflector (backward flow) and an increase in pressure in the lumen. Assuming that no energy loss occurs along the tube and wave reflection is complete (100% of the energy is reflected), the reflected pressure signal is an exact copy of the forward pressure, whereas the flow signal is an exact “negative” version of the original flow (at the site of the reflection, the net pressure doubles, whereas net flow is 0). Therefore, the key difference between the forward and backward waves is that the forward wave manifests as increased pressure and flow, whereas the reflected wave manifests as increased pressure and inverse (or decreased forward) flow, the magnitude of pressure increase and backward flow induced by wave reflections being a function of the initial forward energy and the proportion of energy being reflected.

With progressive decreases in tube length or progressive increases in its PWV, progressively greater superimposition of forward and reflected pressure and flow waves occurs. Eventually, forward and backward components merge into a single wave from which forward versus backward signals may be difficult to discern based on pressure alone or flow alone. However, because, in the absence of wave reflections, pressure and flow wave forms should be identical (as in model 1), differences in the shape of the pressure versus the flow wave forms (which are induced by wave reflections) can be quantified as long as the 2 wave forms can be related quantitatively to each other (ie, properly “scaled” in the vertical axis). This scaling can be achieved if the parameter that describes the quantitative relationship between pulsatile pressure and flow in the absence of wave reflections is known. This parameter is Zc, which can be measured from early pressure and flow data points in the proximal tube, before the reflected wave returns, using the ratio of early ΔP/ΔQ or the slope of the early pressure-flow relation. This is the basis for wave separation analysis,^{13,18} which is used to quantify reflected waves in the arterial circulation in the time domain, as discussed in part 2 of this tutorial.

The description above is based on analysis of physics of wave reflection in the time domain, but a similar reasoning can be applied for the frequency domain.^{1,14} When considering the wave as a composite of sine waves with different frequencies, manifestations of the reflected wave will depend on the frequency of the sine wave considered. If a given harmonic of the incident wave has a wavelength equal to twice the length of the tube (ie, reflection site located at wavelength), the distance traveled by the wave before returning to the point of origin will correspond exactly to a full wavelength of that particular harmonic. Therefore, round-trip transit time corresponds exactly to the harmonic period (Figure 4A), and the reflected harmonic is delayed a full period relative to its forward counterpart. This causes forward and backward harmonics to be in phase. Two sine waves that are in phase, when added, will produce a sine wave of greater amplitude or modulus (constructive interference, Figure 4A, left). On the other hand, when a sine wave is subtracted from a wave that is in phase and has the same frequency, it will cancel it out (destructive interference, Figure 4A, right). Because reflected energy (for a reflection of the closed end type) increases pressure and decreases (“inverts”) flow, the former occurs for the pressure harmonic, whereas the latter occurs for the flow harmonic. Therefore, modulus of input impedance (pressure modulus/flow modulus) at that particular frequency will be high. On the other hand, for a harmonic of wavelength equal to 4 times the distance of the tube (reflection site located at wavelength), round-trip distance corresponds to half a wavelength, round-trip transit time corresponds to half the harmonic period, and the reflected harmonic is therefore delayed half a period relative to the forward harmonic (ie, is 180° out of phase). The reflected pressure harmonic adds to the incident pressure harmonic and cancels it out, whereas the reflected flow harmonic subtracts from the incident flow harmonic, increasing its amplitude (Figure 4B). Therefore, modulus of input impedance at that frequency is minimized. These principles also apply when partial rather than full reflections occur (Figure 5). It follows that input impedance tends to the minima at one-quarter wavelength frequency (and its odd multiples, eg, 3/4 and 5/4) and to the maxima at (1/2) wavelength (2/4) and all of the other even multiples of quarter wavelength (4/4, 6/4, etc), by an amount proportional to the amount of reflected energy. Therefore, the reflected wave in the frequency domain manifests as oscillations of impedance modulus about a mean non-zero value (which is the Zc of the tube). In addition, as illustrated in Figure 5, for frequencies corresponding with 1/4 wavelength (1/4, 1/2, 3/4, etc) the total pressure and flow harmonics are in phase, so that impedance phase angle (Pφ−Qφ) is 0. For intermediate frequencies, phase angle varies between −90 (eg, 3/8) and +90° (eg, 5/8). Finally, according to these principles, one can calculate the distance to the reflection site when the lowest frequency corresponding with a phase angle of 0 and impedance minimum are known, as long as the PWV of the tube is also known.

## Forward and Reflected Waves in the Human Arterial Tree

At the beginning of each cardiac cycle, the heart generates a forward-traveling energy pulse that results in increased pressure and forward flow in the proximal aorta during early systole. If proximal aortic Zc is high because of a stiff wall, a small aortic diameter, or both, the amount of pressure increase is relatively large for any given early systolic flow.^{1,14,17} The energy wave generated by the LV (incident wave) is transmitted by conduit vessels and partially reflected at sites of impedance mismatch, such as points of branching or change in wall diameter or material properties along the arterial tree. Multiple small reflections are conducted back to the heart and merge into a “net” reflected wave, composed of the contributions of the scattered backward reflections. This reflected wave is often looked at as one single discrete wave, originating from an “effective” reflection site (which adheres to the conceptual view of the arterial system as a tube with a single reflection site as described above), but it is important to realize that the reflected wave is the resultant of scattered reflections, originating from distributed reflection sites. The time of arrival of the reflected wave to the proximal aorta depends on the location of reflection sites and on the PWV of conduit vessels, particularly the aorta, which, in turn, is influenced by wall stiffness as described above.^{1,14,17} The distance to the reflection sites is strongly dependent on body height. Finally, it should be noted that complex reflection sites may induce phase shifts between the incident and reflected wave that result in differences between the calculated effective reflection distance and the actual anatomic distance to these sites.^{19}

## Perspectives

Wave conduction and reflections are important determinants of LV afterload and central arterial hemodynamics. Analyses of central pressure-flow relations can be very informative regarding the relatively complex set of events related to wave reflections in the arterial tree, as will be discussed in part 2 of this tutorial.

## Acknowledgments

**Disclosures**

None.

- Received May 31, 2010.
- Revision received June 24, 2010.
- Accepted July 29, 2010.

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## This Issue

## Jump to

- Article
- Abstract
- Noninvasive Pressure Measurements
- Noninvasive Flow Measurements
- Basic Concepts for Afterload Assessments Using Pressure and Flow Measurements
- Input Impedance
- Model 1: Pulsatile Pressure and Flow in a Single Elastic Tube of Infinite Length (Absence of Reflected Waves)
- Model 2: Pressure and Flow in a Single Elastic Tube of Finite Length in the Presence of Wave Reflections
- Forward and Reflected Waves in the Human Arterial Tree
- Perspectives
- Acknowledgments
- References

- Figures & Tables
- Supplemental Materials
- Info & Metrics
- eLetters

## Article Tools

- Noninvasive Evaluation of Left Ventricular AfterloadJulio A. Chirinos and Patrick SegersHypertension. 2010;56:555-562, originally published September 15, 2010https://doi.org/10.1161/HYPERTENSIONAHA.110.157321
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