Sep 132021
 

Jon-Emile S. Kenny MD [@heart_lung]

The relative simplicity of extending one’s finger onto the tissue overlaying a major artery belies the complicated forces generating an arterial pulse.  Descriptions and physiological elucidations have been put forth across the decades, yet even today the machinations relating cardiac contraction to a waxing and waning arterial wall are a bit obstruse.  While more detailed analyses are available for the keen reader [1-3], this brief approach will consider a simplified model first proposed well over a century ago.

To introduce the physiology at play, it might be best to employ a thought experiment describing what the pulse is not.  The pulse is not blood pressure, nor is the pulse blood flow.  This statement might be surprising given how the pulse is discussed clinically and how the pulse is exploited as a measure of the ‘circulation’ – as in ‘return of spontaneous circulation’ [ROSC].  However, the following model should dispel some of these half-truths.

A Thought Experiment

Consider a mechanical piston pump generating a constant pulsatile volume at a constant frequency.  This is analogous to a left ventricle with constant stroke volume and heart rate, respectively.  This pump is attached to a short steel tube with an internal diameter of, say, 6 millimetres. As the pump moves fluid through the tube, both pressure and flow are generated within; however, placing one’s finger over this tube would not yield a pulse.  If the volume-per-cycle of the pump were increased, both flow and pressure would rise in proportion; despite this augmentation of the ‘circulation’, there is no pulse.  Similarly, if the pump volume were turned down below baseline, both flow and pressure would fall in pulseless proportion.

So, we encounter a situation with 3 different flows and pressures all without a discernible, palpable pulse.  We, therefore, intuit that the model needs another biophysical property to introduce a pulse in the tube and this relates to elasticity; the tube needs some ‘give’ to it, some ‘compliance’ such that its radius [i.e., area] expands towards the finger and recoils away from it.  And with this the pulse comes alive – a transient change in vascular cross-sectional area or volume that we feel beneath the skin.

Two Elements

Given the thought experiment above, we arrive at the ‘two-element windkessel’ model of the circulation first proposed by Otto Frank in 1899 [4, 5]. While this model is a simplified depiction of vascular resistance, it also provides a starting point for the basics of pulsatile blood flow.

In Frank’s model, the first element is thought of in terms of classic, ‘Poiseuille’ resistance [which really applies to non-branching, non-pulsatile flow].  Nevertheless, the key determinant of resistance in this element is vessel radius.  To add this variable to the model above, consider a variable resistor at the outflow end of the short tube, whereby the radius of the terminal end of the tube is controlled by the experimenter.  In vivo, this is represented by sphincters at the pre-capillary arterioles – increasing or decreasing the cross-sectional area of the arterial tree under autonomic control.

The second element in Frank’s model is the compliance of the vessel, described by the change in volume relative to the change in pressure [5].  The compliance is related to the elastic modulus of the material – a very compliant structure has a large change in volume relative to changing pressure, while a poorly compliant vessel is the converse. In the thought experiment above, the steel tube has a very low compliance, approaching zero.  In vivo, compliance is made up mostly by the elasticity of the large arteries.

Thought Experiment: compliance

With the aforementioned experimental set-up, fluid volume per cycle, downstream resistance and the compliance may all be independently changed.  Keeping the piston pump moving at the same, baseline volume, with an open-ended [i.e., 6 mm] outflow, we change the tube from steel to rubber – greatly increasing compliance. Now, there is a pulse despite flow from the pump being the same as before.  Perhaps paradoxically, the pressure gradient down the tube is blunted with the evolution of the pulse.  That is, the pulse demands some fraction of volume shifting in a vector perpendicular to the vessel axis; volume ‘outwards’ against the wall, stenting it open.  As this perpendicular fraction of volume is transiently ‘removed’ from the volume vector down the vessel axis, the pressure gradient is diminished [see equation 1].

Equation 1: Equation for two-element model. Q(t) is flow, C is compliance, dP/dt is change in pressure with respect to time, ∆P is pressure gradient and R is resistance. For a given flow, Q(t), when the vessel is very stiff [i.e., C is zero], then ∆P is maximal. As the vessel gains compliance for the same Q(t), ∆P falls, necessarily.  See ref. [3]

Clinically, the above may be akin to comparing the radial pulse between a 19-year-old and a 91-year-old.  While both could have the same flow [and oxygen delivery if hemoglobin and saturation are equal], the 91-year-old would have a much-diminished pulse with a greater pressure gradient.

Thought Experiment: resistance

Now imagine the piston pump moving its baseline volume against an elastic tube with the open outflow [i.e., 6 mm].  Again, the pulse that we feel here is a rise and recoil of the tube [or vessel] volume.  As such, there is a transient imbalance between volume into the vessel and volume out of the vessel – this difference being the volume felt as the pulse.  Accordingly, if volume pushed into the vessel by the piston pump is reduced, but the downstream resistance greatly increased [i.e., shrinking the outflow radius from 6 mm to 2 mm], then the volume in relative to the volume out per unit time may stay somewhat constant and a near normal pulse felt in the face of falling flow.  Therefore, a healthy individual with an elastic artery and robust rise in downstream resistance could maintain a palpable pulse in the face of significantly impaired oxygen delivery.  This isn’t to say that the nature of the pulse would be identical nor would the heart rate or other physical examination findings [e.g., capillary refill].  This is meant only to illustrate that flow, pressure and the pulse are not interchangeable phenomena.  So, 1.] compliance allows for a transient imbalance between 2.] volume in and 3.] volume out of a vessel [3].  Accordingly, the physical manifestation of this transient change in vessel volume – the pulse – varies as a function of these 3 general elements.

Happy birthday Penny,

JE

Dr. Kenny is the cofounder and Chief Medical Officer of Flosonics Medical; he also the creator and author of a free hemodynamic curriculum at heart-lung.org.  Download his free textbook here.

References

  1. O'Rourke MF, Avolio AP: Pulsatile flow and pressure in human systemic arteries. Studies in man and in a multibranched model of the human systemic arterial tree. Circulation research 1980, 46(3):363-372.
  2. Attinger E: The physics of pulsatile blood flow with particular reference to small vessels. Investigative Ophthalmology & Visual Science 1965, 4(6):973-987.
  3. Taylor KJW, Burns PN, Wells PNT: Clinical Applications of Doppler Ultrasound. 2nd Ed. 1995, Ch. 3; pg. 45.
  4. Tedford RJ: Determinants of right ventricular afterload (2013 Grover Conference series). Pulmonary circulation 2014, 4(2):211-219.
  5. Stergiopulos N, Westerhof BE, Westerhof N: Total arterial inertance as the fourth element of the windkessel model. The American journal of physiology 1999, 276(1 Pt 2):H81-88.

 

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