Understanding Pulsatile Pressure and Flow

 

 

 

 

The video depicts pressure and flow waves ( single frequency sinusoid ) transmitted through a compliant tube.  For this example, fluid viscosity has been set to a minimum; there is no apparent diminution (attenuation) of the waves with the distance traveled.  DOWNLOAD a (much larger) avi.

We've already determined the fact that pressure and flow disturbances both travel as waves.  This may seem like a rather obvious result; I believe it may be less apparent that the flow wave also is a disturbance. The disturbances - the waves - travel MUCH faster than the actual flow through the conduit.  In fact this turns out to be a requirement for the simplified analysis we're up to.  Pressure and flow waves travel at the same velocity ( in this linear model anyway ).  They are proportional to each other at every location, within a given compliant conduit, through the characteristic impedance. The maximal pressure region of the wave coincides with the location of maximal flow rate ( more or less, and absolutely for this 0 viscosity example), seemingly pushing the flow wave through the tube.  After all that's what happens with the vascular resistance situation; a higher input pressure results in greater the flow through the tube ( or network ) resistance : \(p = q R\).  However that thought process is not applicable to this situation and is INCORRECT ( more or less ).

Think you've got the concepts?  Here is a brief exam:

1)     In the ascending aorta, you observe that pressure has increased ( over some unspecified time interval ).  Which of the following is most probably true:

      A.      Vascular resistance has increased.
      B.      Aortic flow has increased.
      C.      Aortic volume has increased.

 

2)     The figure below is a freeze-frame from the above video.  At what location (or locations) is the pressure force on the fluid at a maximum  (at this particular moment in time)?

 

  

The annotated version of the figure below suggests where in the tube the pressure forces are. Gray vertical lines help you line up the pressure and flow waves and show that they are in phase in this situation with peaks, troughs, and zero crossings perfectly lined up.  Pressure force is due to pressure gradient, a difference in pressure with distance.  The gray shaded band is one section where pressure is greater upstream (left) than downstream and corresponds to a zone where there is net pressure force to the right; the arrow at the baseline suggests the direction of force.  Since friction is negligible in the example, it also corresponds to a location where flow is accelerating to the right.  Since there's no friction in the example, ALL of the pressure force goes into accelerating the fluid in the tube (none to overcoming friction).  The blue band corresponds to a section in the tube where pressure force is acting to decelerate the flow ( accelerate to the left ).  The pressure gradient is the slope of the pressure function, the derivative of pressure with respect to the axial coordinate (\(dp/dz\)).  This slope is suggested at several locations. Note that locations of maximal ( and minimal ) pressure also coincide with zero pressure gradient – no pressure force acting on the fluid.  It's the locations where pressure crosses zero that pressure gradient is maximal.  The first  vertical gray line  on the left corresponds to a location where \(dp/dz\) is maximal and negative, i.e. pressure higher upstream than downstream.  By this convention, it's a negative pressure gradient that forces flow to accelerate in the downstream ( positive ) direction.

 IN THE FIGURES BELOW THE X-AXIS IS LOCATION, NOT TIME, UNLESS OTHERWISE DESIGNATED.  THIS CAN BE A POINT OF CONFUSION.

In the next figure, pressure has been plotted along with negative pressure gradient, so the lower plot on the figure is simply the negative derivative of the upper plot.  (Note: "negative" is omitted on the figure; it's too much verbiage for the figure.)  As suggested above, the locations of maximal and minimal pressure also coincide with zero pressure gradient ( the change in pressure with respect to distance ) and no force on the fluid whatsoever due to pressure.  The maximal pressure force on the fluid actually occurs at locations of zero pressure in the figure. This would not necessarily be true if there were more than 1 sinusoid or if there was a net flow. The pressure sinusoid is 90° (\(\pi/2\) radians) out of phase with the negative pressure gradient. Take a moment to consider how pressure force (negative pressure gradient) relates to the pressure wave itself.  


So that was the long-winded answer to question #2.  For question #1, we go back to the original figure with some alternate annotations.  The gray band also corresponds to a region where flow into the conduit at the upstream end exceeds flow out of the conduit downstream; even if the flow rate is actually negative, there can be net flow into the conduit. The gray band is a region where the volume of the tube is increasing. The blue band is the opposite ( net flow leaving the conduit ).  Just as was seen for pressure, locations of 0 flow ( gray vertical line on the far left for example ) correspond to locations of maximal flow gradient, \(dq/dz\).  The negative flow gradient (\(- dq/dz\)) is mathematically identical to the rate of change of the conduit's cross-sectional area, \(dA/dt\).

 

The black arrows suggest where inflow exceeds outflow (arrow pointing right, increasing volume) and the opposite (arrow pointing left decreasing volume).

Below the flow wave is recapitulated with the negative flow gradient (even though it's labeled simply flow gradient) at the lower half of the figure.  The negative flow gradient sinusoid is always exactly 90° out of phase with the flow ( a mathematical fact ).  

The next figure shows the area ratio, \((A-A_0)/A_0\) ( normalized cross-sectional conduit area) and also the negative flow gradient.  These also are exactly 90° out of phase with each other.  The negative flow gradient is the same thing as the rate of conduit area change (\(dA/dt\)).  At every location, the tube distends  NOT because there is flow traveling through it , but because the rate of flow into the segment exceeds the rate of flow out of the segment.  DON'T FORGET IT'S LOCATION ON THE X-AXIS, NOT TIME.  

In the next figure we're measuring area ratio and negative flow gradient at a specific location in the tube (they would all look the same in this simplified example) and displaying the result AGAINST TIME.  The gray time band indicates a section of the tracing where the negative flow gradient was positive. We see the cross-sectional area increasing throughout that section of the tracing.  The blue band highlights a section of time where cross-sectional area decreased; there was a net flow out of the tube.

The answer to question #1 is C: Aortic volume is increased.  We could have aortic pressure decrease with increased vascular resistance ( if cardiac output decreased ), OR with increased cardiac output ( if vascular resistance decreased ); so both A and B are incorrect. The essential fact is that aortic pressure increases when the vessel distends – when it's volume increases.   This occurs when flow into the aorta ( or any other conduit ) exceeds exiting flow. Increasing vascular resistance OR increasing cardiac output both increase mean aortic pressure BECAUSE  they result in increase aortic volume  ( other things being equal ).  Furthermore the so-called "phasic" ( time-varying ) changes in aortic pressure are due to the time varying change of cross-sectional area and volume -- because of a flow gradient.

 

 

The figure above suggests 4 types of situations for the tube.  In A and B, more flow is entering the blue highlighted section of the tube than is leaving as suggested by the arrows; the volume of the tube segment is increasing as is the pressure.  In C and D, more flow is leaving the tube than is entering and the volume of the tube segment is decreasing as is pressure. It's the flow gradient that exerts this effect - NOT THE FLOW per se; the direction of flow can be either "forward" (A & C) or "backward" (B & D).  However the physics also dictates that the peak of the flow wave corresponds (roughly) with peak of the pressure wave (depending on the characteristic impedance).  We get an erroneous impression that it's the flow distending the tube.  You'll REALLY need to be thinking this way when we get to wave reflections!

Now I'm playing slightly fast and loose on this point. Whenever the word pressure arises here, we are talking about the distending pressure – the pressure inside the conduit minus pressure outside.  Also in the simplified tracings for the model above, a purely compliant conduit wall was assumed. In that situation, the distending pressure and cross-sectional area are perfectly in phase with each other.  A viscoelastic vessel wall however results in some degree of phase lag between the distending pressure and the cross-sectional area.  We would see that the pressure starts to rise before the conduit starts to distend. Furthermore the material properties of blood vessels are very complicated; they don't necessarily even exhibit classic viscoelastic behavior.  In another section, we will explore some of the effects of viscoelasticity and how the tracings shown on this page will be altered as a result.


 While I'm at it, I would like to dispel a myth whether it exists or not.  I believe I've seen physiology texts with a sentence that goes something like this:

Ejection of blood into the aorta distends the vascular wall, thereby storing mechanical energy.  During diastole, the elastic recoil of the wall converts the pulsatile ( or oscillatory ) energy into forward flow.

Again I just made up this sentence, but nothing could be further from the truth.  If the circulatory system is linear, there can be no conversion of sinusoidal or pulsatile inputs into net forward flow.  We've already agreed that the circulation is linear for many purposes and this means there will be no interactions between inputs and outputs of differing frequency; an oscillating input CANNOT result in a net forward flow.  However there are also components of the circulation that are necessarily highly nonlinear.  It is SPECIFICALLY the valves (to a great extent) that cause conversion of pulsatile pressure and flow into net forward flow.  The valves exhibit extremely low impedance for flow in the forward direction, and ( hopefully ) extremely high impedance for the reverse; this represents highly nonlinear behavior and is obviously essential to adequate function of the cardiovascular system.  It's the valves that result in the conversion of oscillatory (pulsatile) flow into net forward flow.


  

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