This chapter is relevant to Section G2(ii)  of the 2017 CICM Primary Syllabus, which asks the exam candidate to "define the components and determinants of cardiac output". Specifically, afterload seems to be the favourite determinant, as it was the main subject of the majority of past paper questions:

In summary:

  • Afterload can be defined as the resistance to ventricular ejection - the "load" that the heart must eject blood against. It consists of two main sets of determinant factors:
    • Myocardial wall stress, which represents intracardiac factors
    • Input impedance, which represents extracardiac factors
  • Wall stress is described by the Law of Laplace ( P × r / T)
    and therefore depends on: 
    • P, the ventricular transmural pressure, which is the difference between the intrathoracic pressure and the ventricular cavity pressure.
      • Increased transmural pressure (negative intrathoracic pressure) increases afterload
      • Decreased transmural pressure (eg. positive pressure ventilation) decreases afterload
    • r, the radius of the ventricle
      • Increased LV diameter increases wall stress at any LV pressure
    • T,  the thickness of the ventricular wall
      • A thicker wall decreases wall stress by distributing it among a larger number of working sarcomeres
  • Input impedance describes ventricular cavity pressure during systole and receives contributions from:
    • Arterial compliance
      • Aortic compliance influences the resistance to early ventricular systole (a stiff aorta increases afterload)
      • Peripheral compliance influences the speed of reflected pulse pressure waves (stiff peripheral vessels increase afterload)
    • Inertia of the blood column
    • Ventricular outflow tract resistance (increases afterload in HOCM and AS)
    • Arterial resistance
      • Length of the arterial tree (the longer the vessels, the greater the resistance)
      • Blood viscosity (the higher the viscosity, the greater the resistance)  
      • Vessel radius (the smaller the radius, the greater the resistance)

From what we can establish, the college examiners wanted the following elements only:

  • A definition
  • Factors which affect afterload: valve resistance, vascular resistance, vascular impedance, blood viscosity, intrathoracic pressure, and the relationship of ventricular radius and volume
  • Determinants which are specific to the right and left ventricles
  • What happens when afterload is suddenly increased, including effects on "ventricular end-systolic pressure, ventricular end-diastolic pressure, left atrial pressure, cardiac output, myocardial oxygen demand and myocardial work, coronary blood flow and systemic blood pressure"

Thus, without repeating oneself, it will suffice to say that the contents of this grey box here is the only essential element of this chapter, and the rest is self-indulgent word salad. It was a deep rabbit hole, and it would be unnecessary to confess that a profound mastery of the subject did not result from tumbling into it, as that will be quite obvious from even a superficial reading of the text.

"Afterload" by Amanda Vest (2019) is probably the single best most balanced discussion of the issues, including an excellent bibliography. Moreover, it appears to be available for free. Two 2010 papers by Chirinos & Segers (Part 1 and Part 2) fill in what few blanks were left by Vest. For the revising exam candidate, to read all three would be to drown in excessive detail.  And if that sounds appealing rather than frightening, one could acquire a copy of Mcdonald's Blood Flow In Arteries (2011) which dedicates about 300 of its 743 pages to an indepth mathematical exploration of afterload. 

Definition of afterload

CICM can often be accused of being definition-averse, insofar as they might expect something to be defined by the trainees but refuse to define it themselves in their examiner comments. Not so with afterload, where in their words for Question 17 from the first paper of 2012 they basically gave us a definition to work with:

"Afterload is the resistance to ventricular ejection - the "load" that the heart must eject blood 
against and is related to ventricular wall stress (Law of Laplace, T=Pt.r/u)"

This, henceforth, is the Official Definition of Afterload, and it would be frivolously wasteful to mention any others. However, there are many others. Norton (2001) lists about thirty. The trainee has some flexibility here ("many definitions of afterload were accepted") and conceivably they could choose another valid one and still score some marks.  As is often the case, Part One offer an excellent non-canonical alternative:

"Afterload is the sum of forces, both elastic and kinetic, opposing ventricular ejection"

The major advantage, to borrow a turn of phrase from the authors, is that this definition does not try to borrow misapplied terminology from physics and engineering, as this "may be leapt on by the cruel examiner". It also borrows legitimacy by resembling published works (eg. Textbook of Pulmonary and Critical Care Medicine, p.75). 

When one digs a little deeper, all of these definitions tend to boil down to the same thing. Myocytes contact, and by contracting, they move things. And the resistance of that thing to being moved is the afterload. It is much easier to do this with isolated myofibres: you can just hang them from a hook and make them lift weights, those weights being your afterload in its purest sense. Here's an example of such an experiment from 1963 (Sonneblick & Downing), where a cat papillary muscle was forced to lift a load of around 1.6g:

afterload - weightlifting by a cat papillary muscle

Obviously, where one encounters trouble is where one tries to relate all this to something realistic and measurable in the whole intact heart. This is where the definitions of afterload tend to diverge into two streams, which are mutually complementary. One stream describes the weight being lifted by all those combined myofibres in terms of wall stress, the opposing force they need to overcome when they contract against the blood volume in the ventricular chamber.  That takes the discussion to a Laplacian place, a Laplace if you will. The other approach is to describe afterload in terms of hydraulic "load", or arterial impedance, which is the ratio of change in pressure to change in flow. This approach represents the "weight" being lifted in terms of the arterial blood pressure, blood viscosity, the weight of the blood being ejected, the elasticity of the arteries, and other plumbing-related concepts.  Let us explore these two concepts individually:

Laplacian definition of afterload: ventricular wall stress

In the isolated myofibre experiment,  the afterload is basically the weight this muscle fibre is expected to lift. Weight is described by the same parameters as force, and the muscle fibre has a crossectional area across which this force acts, which makes this "stress" - in physics, defined as unit force per cross-sectional area. The stress on the wall of a sphere can therefore be expressed as the output of a rearranged Young-Laplace-Gauss equation:

Law of Laplace as it applies to the concept of afterload

where

  • σ is the wall stress
  • P is the Laplace pressure, or the pressure difference between the inside and the outside of the sphere (i.e. the ventricular transmural pressure, which is the difference between the intrathoracic pressure and the ventricular cavity pressure)
  • r is the radius of the ventricle
  • T  is the thickness

Thus, wall stress is proportional to the ventricular pressure during ventricular ejection, multiplied by the radius of the wall, and divided by the wall thickness.  But obviously we're taking some liberties with the Law of Laplace (which, strictly speaking, applies only to spheres), and so most honest people use the ∝ ("proportional to") operator instead of "=" to describe the relationship between ventricular wall stress and the other variables.

Hydraulic definitions of afterload: arterial input impedance

For lots of reasons other than the non-spherical shape of a ventricle, the Laplace definition is by istelf inadequate to describe afterload. To quote Milnor (1975),

"Stress unquestionably exerts a crucial influence... but when the ventricle is considered as a functioning unit it would be desirable to have some rigorous way of describing the external conditions (the properties of the arterial system) that contribute to the stress"

The muscle is not contracting to squeeze some sort of stress ball here, it is trying to displace a viscous fluid into a stretchy elastic receptacle. The hydraulically focused definitions of afterload focus on physical properties of that fluid and of the circulatory system as the main forces opposing ventricular ejection, which are described in terms of arterial input impedance.

That makes this definition much more complicated than just talking about the wall stress of a sphere. Without going into elaborate mathematical detail, it will suffice to say that impedance is like resistance, but different. "Resistance", by its strictly physical definition, is by convention applied to describing flow in non-oscillatory systems, i.e. where flow is constant- and is therefore probably inappropriate for describing pulsating arteries. "Impedance" is more appropriate: it is the measure of the opposition to flow presented by a system, and is a term usually applied to oscillatory mechanical systems or alternating current.  "Input impedance" specifically is the relationship between pulsatile pressure and pulsatile flow in an artery, which need to be described in terms of sinusoidal waves. And those aren't simple sinusoidal functions -  they are a series of sine waves of different frequency and amplitude ("harmonics") which superimpose to produce the pulse pressure and flow waveforms (thus input impedance is identical to resistance at a frequency of 0 Hz,  i.e. at constant flow). Moreover, the impedance cannot be described in terms of an isolated static measurement, as it incorporates input from flow at previous instants (because moving blood has inertia, arteries have distensibility, and pulse waves travel and reflect over time). Lastly, apart from input impedance, there are also other forms of impedance: longitudinal impedance, terminal impedance, characteristic impedance, and transverse impedance. 

At this stage, even the most committed student would complain that this is perhaps more than is required to answer a CICM written paper question. That is of course correct.  Fortunately, a few comforting things can be said to soothe the inflamed brain of the reader:

  • The modulus of aortic input impedance is maximal at 0 Hz (i.e. constant flow), and all the other harmonic moduli contribute much less to the total impedance, which basically means that aortic input impedance is very close to aortic resistance, a much more familiar property. Resistance can then be described in very simple Poiseuillean terms.
  • The other major determinants of arterial impedance are inertia and arterial compliance or elastance, which are actually quite easy to dismantle into components.
    • Inertia is related to the mass of the blood and its velocity
    • Compliance and elastance are physical properties of the arterial walls which can be expressed as a relationship of pressure and diameter.
  • The pulsatile dynamic component of impedance (which includes all those reflected waves) can be described in terms of the difference between the contours of pressure waves and flow waves. That makes it easier to visually appreciate from recorded waveform (thanks to Nichols et al,  2001), which means that, for educational purposes at least, one can ignore the difficulties of integrating this into a mathematical model.

Thus, after taking a few dodgy cognitive shortcuts, the hydraulic interpretation of afterload can be described in terms of arterial resistance, arterial compliance, and blood inertia.

A combined definition of afterload

The two main components of afterload overlap in the territory of pressure, specifically the aortic pressure to which the ventricle is exposed when the aortic valve opens in systole. At this stage, ventricular chamber pressure and aortic pressure are approximately equal, and their pressure/time waveforms are essentially superimposable (well, allowing for some resistance from the ventricular outflow tract). This combined "aortoventricular" pressure is a major determinant of ventricular wall stress, and is in turn determined by the factors which govern aortic input impedance. Thus, the hydraulic definition of afterload can be turned into one of the components of the Laplace definition, provided we avert our gaze from certain elements which do not fit this narrative (eg. pulsatile flow). 

On top of that, it seems that wall stress (rather than input impedance) is the dominant afterload-determining factor, if you had to pick one. Covell et al (1980), by looking at original and historic data, concluded that

"...alterations in characteristic impedance are indeed reflected by alterations in ventricular performance, but these alterations are also reflected by alterations in ventricular wall stress that more adequately predict alterations in ventricular shortening associated with changes in load. Moreover, changes in input impedance do not, in themselves, appear to influence the force-velocity-length framework for examining ventricular function."

Following from this, the determinants of afterload, arranged into some hierarchical order, must be as follows:

  • Ventricular radius
    • End-diastolic volume
  • Ventricular wall thickness
  • Ventricular transmural pressure
    • Intrathoracic pressure
    • Ventricular cavity pressure
      • Ventricular outflow impedance 
      • Aortic input impedance
        • Arterial compliance
        • Influence of reflected pressure waves
        • Inertia of the blood column
        • Arterial resistance
          • Length of the arterial tree
          • Blood viscosity
          • Vessel radius

They will be discussed in that order, mainly because to leave arterial resistance to last seemed like a suitable way to emphasise its importance. Of all the aforementioned determinants, most survive only in the minds of physiology professors, trapped there by bushy grey eyebrows. Consider: ventricular radius and wall thickness do not vary excessively. Intrathoracic pressure is usually within a range of around 0-10 mm Hg, unless you are really trying to kill your patient with the ventilator. Arterial compliance varies, but usually only over decades, as does the influence of the reflected waves. The inertia of the blood column is going to be the same for most (mildly anaemic) ICU patients, as will the viscosity of the blood, and for most people the length of the arterial tree is fairly fixed. This basically leaves the arterial vessel radius as the main determinant of "real" afterload, which varies over clinically relevant timeframes with pathology and therapy.

Effect of ventricular radius and end-diastolic volume on afterload

End-diastolic volume is the volume you would afterload from, if "to afterload" were a verb. It represents the degree of ventricular sarcomere stretch just before the beginning of systole, which is basically the definition of preload. This volume has dimensions, one of which could be described as a "radius", even though the crossection is very irregular and certainly not circular. This "radius", in turn, can sort-of plug into the Laplace equation, if we ignore the fact that neither ventricle has anything like spherical geometry.

Those caveats politely ignored, we find that plugging a changing radius into the Laplace equation while keeping all the other variables the same basically gives a linear increase in wall tension. This means that the bigger the ventricle dilates, the larger the amount of wall stress it experiences in the course of generating the same systolic pressure. Is this for real? Turns out, yes.  Hayashida et al (1990) compared seventeen patients with dilated cardiomyopathy and compared them to eleven normal controls, using clever maths to calculate regional wall stress from ventriculography data. As you can see from the stolen images below, the patients with dilated ventricles had a markedly increased wall stress. In most parameters, it was at least two to three times greater than controls, even with the lower systolic LV chamber pressures being generated by the DCM group.

effect of increased LV radius on LV wall stress from Hayashida et al (1990)

Effect of ventricular wall thickness on afterload

When a change in the thickness of the LV wall is substituted into the Laplace equation, a linear decrease in wall tension is observed. This is a purely mathematical thing (the thickness happens to be the denominator of the equation). That said, the effect of increased thickness also makes logical sense because a) that's what a smart ventricle would do in reaction to increased wall stress, and b) because more sarcomeres pulling the same yoke means the wall stress is shared and each individual sarcomere ends up under less stress.

Experimental evidence for this does exist, at least in terms of modelling what happens to wall stress when the thickness changes. Sayasama et al (1976) controllably constricted the proximal aortas of dogs and measured LV performance immediately and after enough time has passed for LV hypertrophy to develop. Immediately as afterload increased, LV wall stress increased by 55%, but after about twenty days the wall thickened by 15% and the wall stress decreased down to 22%:

decreased LV wall stress due to increased LV wall thickness from Ross et al (1976)

Effect of intrathoracic pressure on ventricular transmural pressure

This is discussed in greater detail in the chapter on the effects of positive pressure ventilation on cardiovascular physiology. In summary:

  • Normally, negative intrathoracic pressure increases afterload during spontaneous breathing.
  • This is because transmural pressure is the difference between LV chamber pressure and pleural pressure; i.e. a negative pleural pressure actually increases that difference:

LV transmural pressure

  • Positive intrapleural pressure decreases afterload because of a decreased LV transmural pressure (it is subtracted from the intra-LV pressure)
  • As the result, LV wall stress decreases:

LV%20transmurral%20pressure%20change%20with%20positive%20pressure%20breath.jpg

  • In terms of afterload reduction, this is roughly equivalent to a drop in arterial systolic blood pressure of the same magnitude as the intrathoracic positive pressure increase (Luecke et al, 2005)

Ventricular cavity pressure and ventricular outflow impedance 

Anybody who asks for a discussion of ventricular cavity pressure and how it contrasts aortic pressure must surely be asking for a Wiggers diagram. The most prominent visually appealing part of that diagram is the top section where the ventricular cavity pressure triumphantly overcomes aortic pressure in systole.  Obviously, there are several barriers in the way of it doing so, and the first of these can be the heart itself. Specifically, the meaty outflow tract in HOCM or the crusty valve in aortic stenosis can give rise to an increased ventricular outflow impedance.

normal AS and HOCM ventricular-aoric pressure gradients

Again, to reduce this concept to an interaction of pressures is an oversimplification, but it certainly helps to explain it. In the diagram above, the normal waveforms from Curtiss et al (1975) on the left are compared to abnormal ones recorded by Geske et al (2012) on the right. The colourised region represents an overlap of the graphs where the left ventricle pressure exceeds the aortic; the difference in pressures is produced by the mechanical resistance to ventricular outflow.

Of course, this is by no means common. With a competent aortic valve and a normal LVOT, the normal source of input impedance is the arterial circulation. 

Effect of arterial compliance on afterload

Arterial compliance is a complicated topic. The compliance (the opposity of stiffness) is a property of large arterial vessels (mainly owed to their tunica media) which permits them to expand in systole, store elastic energy, and then return it in diastole as the aortic valve closes (thereby maintaining flow - the Windkessel effect). It is usualy measured or calculated as a change in volume or diameter per change in pressure, over a fixed vessel length.  According to London et al (2004), 60% of every stroke volume is normally stored in distended arterial capacitance vessels, and 10% of the cardiac workload is spent on distending them for this purpose (it is not wasted, as the kinetic energy is reclaimed when the vessels contract again). From this, it follows logically that cardiac workload should increase if the arteries become less compliant.

One can imagine a nice elastic artery expanding readily to accommodate the stroke volume, and then contracting elastically to help it along into the capillary circulation - a process which smoothes the pulsatile flow from the ventricle and transforms it into the stable constant flow required by tissues. Conversely, one can imagine a stiff artery doing the opposite of those things, i.e. obstinately resisting the flow of blood and storing none of the energy. Experimentally, this is difficult to demonstrate, because it is practically challenging to alter the compliance of blood vessels without altering their diameter or other characteristics. Moreover, in a real life model, arterial compliance cannot be measured reliably because the blood volume keeps escaping into the venous circulation. However, it is possible to summarise things crudely for exam purposes, borrowing from Chirinos (2012):

  • Normally, the systolic ejection of the left ventricle generates some flow early in systole because of increased chamber pressure
  • If the proximal aorta is stiff and noncompliant (i.e. minimal change in volume per unit pressure), this means, for any given force generated by the ventricle, the pressure will be higher and the flow will be lower.
  • Ergo, to achieve the same early systolic flow, the ventricle would need to generate a higher pressure if the proximal aorta is stiff.
  • This higher ventricular chamber pressure translates into greater wall stress, i.e. increased afterload.

You could probably represent that in the form of a diagram, at a risk of giving the appearance that it was generated using some real experimental data (it wasn't).

effects of decreased aortic compliance on ventricular systole

Effect of reflected pressure waves on afterload

Increased stiffness is not only a phenomenon which affects the aorta. Distal peripheral arteries can also become stiffened and contribute to afterload, but they do so by a different mechanism. Decreased peripheral arterial compliance causes an increase in the pulse wave velocity, which means that the reflected wave from the distal circulation arrives too early - during systole - and contributes to the afterload.

Let's chew this slowly. 

  • The pulse pressure wave propagates into the distal circulation at about 1m/sec
  • It then reflects from the arterioles and returns to the heart
  • The circulatory system is long, and so the pulse wave usually returns in diastole, where it complements diastolic blood pressure and helps fill the coronaries
  • Decreased compliance of the peripheral circulation increases the speed of the pulse wave propagation, a well-known phenomenon applicable to incompressible fluids in elastic tubes.
  • This means the pulse wave will return sooner if it is reflecting off stiff arteries
  • An early return of the pulse pressure wave (i.e. during systole) adds extra pressure to the aortic pressure
  • As this is systole, the aorta is continuous with the ventricular chamber, so this also increases ventricular chamber pressure, and therefore wall stress (afterload) increases.

To illustrate the effect of this on afterload, the author abused Nichols & Edwards (2001), whose excellent original diagram has undergone whatever the image equivalent of paraphrasing is: 

effect of reflected pressure waves on left ventricular afterload

Effect of blood mass and inertia on afterload

One does not normally think about this variable or the role it plays in determining cardiac workload, but it is clearly there in the background. Blood has mass and therefore inertia - i.e. it resists being moved, and once moving it resists being stopped. Several important points can be made regarding this determinant of afterload, without going into excessive detail. And if excessive detail is for some reason required, it can be found in Sugawara et al (1997), from where most of this information was derived.

In short, blood inertia influences afterload in the following ways:

  • It increases afterload in early systole. Inertia is a force which opposes the acceleration of blood flow, which is maximal during early systole. At the beginning of systole, during isovolumetric contraction, the stroke volume is at rest, and so is the mass of blood in the aorta. As pressure is applied, it accelerates and acquires momentum.
  • It decreases afterload in late systole. Once momentum has been imparted to it, the inertia of the blood column moving out of the ventricle enhances ventricular relaxation in late systole. If the pressure from the ventricular muscle were to suddenly drop to zero, blood would still keep moving until it is gradually stopped by frictional forces.
  • Its influence is increased with increased heart rate, which makes logical sense because to accelerate the same mass over a shorter systolic timeframe would require greater force and therefore would call for a higher ventricular chamber pressure.

Arterial resistance and afterload

As has already been mentioned elsewhere, the modulus of arterial impedance is maximal at a frequency of 0 Hz, i.e. where flow is constant. That is thought to be due to the fact that the smallest arterioles are responsible for a lot of the impedance, and by the time it reaches these small vessels, blood flow has probably had most of the pulsatility windkessled out of it. Where flow is constant, "resistance" is the term we use to describe the force acting in opposition to forward flow. Resistance to the flow of fluids through tubes is described by the Poiseuille equation:

poiseuille equation for afterload

where

  •  is resistance
  • η (eta) represents blood viscosity
  • l is the length of the vessel 
  • r is the vessel diameter

As will be discussed here, of all these parameters, the one which has the greatest clinical significance is the vessel radius, but for completeness let's discuss the others.

The length of the vessel is important: the longer the vessel, the greater the resistance. This is obviously somewhat difficult to discuss in the circulatory system, which is a tree of many branches; various scaling models would need to be applied in order for it to make sense (vis. Huo & Kassab, 2009). The ICU being a dark weird place, plausible scenarios where arterial length changes dramatically can be generated by a restless imagination, but these are thankfully quite rare (eg. massive systemic embolism, aortic crossclamp or REBOA). In these scenarios, effective vessel length decreases - but at the same time the compliance of the system and the total radius take a massive dive. In short, in virtually every practical situation, the resistance-improving effects of reducing the length of the vascular tree will be massively overshadowed by the other effects. Fortunately, under normal circumstances, the length of the arterial tree of critically ill patients does not tend to vary overmuch during their ICU stay, and so this parameter can be safely ignored as something stable and boring.

 The viscosity of the blood is a much more variable parameter. Blood is a non-Newtonian fluid, and its apparent viscosity depends on things like shear forces, haematocrit, plasma protein interactions, and the deformability of RBCs (particularly where it comes to the small peripheral vessels). In general, it exhibits "shear-thinning" behaviour, where its viscosity decreases markedly with increasing shear stress. Slow blood moving though large vessels has a viscosity of 100 cP, whereas blood under maximum shear stress has a viscosity of 4-5 cP (Baskurt & Meiselman, 2003; where cP is centipoise).  Viscosity is affected by haematocrit (Clivati et al, 1980) and the unnatural excess of anything unusual (eg. LDL, in the 2016 study by Pop et al). As one might expect, increasing viscosity has the effect of increasing afterload, but it is probably a relatively minor player, and is often not amenable to direct control by the intensivist.

Arterial vessel radius is the most important determinant of arterial resistance because resistance is inversely proportional to the fourth power of the vessel radius. Miniscule changes in vessel radius, therefore, have massive effects on total peripheral resistance. This is amplified by the fact that the total radius of these small vessels is truly vast.  Attinger (1965) estimated the circulatory system of the dog has 40×106 such branches, each with a radius of around 0.01mm. Imagine all of those vessels simultaneously constricting even slightly. 

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