## Compliance

### Compliance Defined

The above animation suggests a left ventricle as it deforms over the cardiac cycle.  However the video is here only to motivate folks to look further.  I've encountered cardiologists who think about compliance solely as a diastolic property of the ventricle(s).  I'm NOT going to use the term that way so heads up!  Compliance is a concept that applies to any vessel.   If I say "vessel" you may be thinking specifically of a tubular structure that conveys blood or lymph.  However this term can be applied to any container including the chambers of the heart and many objects that have nothing to do with cardiology or physiology.  We'll jump right into it:

$$\Large C(p) = \frac{d}{dp} V(p) \equiv \frac{dV}{dp}$$

With volume expressed as a function of distending pressure ($$V(p)$$), compliance here is defined as the slope of this function, i.e. the derivative of volume with respect to pressure. Application of the term is not limited to cardiac chambers. It's not limited to diastole.  Any vessel  for which we can define a volume and a distending pressure can have a compliance associated with it.  We don't even have to be able to seal the vessel; e.g. the term also applies to a blood vessel open at both ends. In the simplest situation, the compliance of a vessel might be a single value (over a reasonably confined range of distending pressure), i.e. the slope $$dV/dp$$ is reasonably described as a constant.  However linearity this is not likely to be the case for any vessels in the mammalian circulation; all we have to do is distend these vessels a little further and we start to see that the compliance is a function – a curve that varies with the distending pressure. I've included this possibility explicitly by writing $$C(p)$$ with the $$p$$ in the parentheses.

### The Electrical Analog

The electrical analog for compliance is the capacitor.  This is a gadget that stores electrical charge.   The schematic symbol for the capacitor is:

This conjures the impression of 2 plates in close proximity which is one means by which a capacitor can be constructed. For practical purposes however, the "plates" are typically made of a flexible conducting foil separated by an insulating ("dielectric") material so that the whole thing can be rolled up and compacted.  The characteristic equation for a capacitor is:

$$\Large v(t) = \frac{1}{C} \int i(t) dt$$

or alternatively:

$$\Large C \frac{dv}{dt} = i(t)$$

$$v(t)$$ is the time-varying voltage across the plates of the capacitor, $$i(t)$$ is the time varying electrical current and we see that it is being summed in the upper equation, integrated over the period of time that current is flowing.   Electrical current has physical units of charge (e.g. Coulomb) per unit of time.  Summing (integrating) the current over time amounts to an electrical charge.  Capacitance is measured in units of electrical charge per unit of voltage; a "larger" capacitor holds more charge without generating much voltage.  A coulomb per volt has been designated a Farad.   For a capacitor with a low level of capacitance (small number of Farads), little charge is stored per unit of voltage. Another way of saying this is that a small amount of stored charge results in a large voltage difference if the capacitor is "small".   In the electrical world, you want the value of the capacitance to be a constant and linear.  You probably would end up paying extra to make sure that it's exactly what's stated on the label and that it's not going to change with voltage, temperature, etc.  That's what makes the behavior predictable by the above equations.

The hydraulic analog of voltage is pressure and current equates to (fluid) flow rate.

$$\Large p(t) = \frac{1}{C} \int q(t) dt$$

The integral on the left is not the electrical charge in this scenario but rather the volume of the vessel.  $$C$$ now stands for the linear compliance and I'll shamelessly represent a compliance with the symbol for an electrical capacitor throughout the website.  If we actually perform the integration on the right we get:

$$\Large p(t) = \frac{1}{C} (V(t)-V_0 )$$

$$p(t)$$ Is the distending pressure of the vessel ( internal pressure minus external ), $$V_0$$ is an integration constant – in this case the volume of the vessel when the distending pressure is 0.  The equation could be rearranged showing again that $$C$$ has physical units of volume/pressure.  We see again that a vessel with a "large" compliance does not change distending pressure much when it's volume is changed.  In terms of the flow rate:

$$\Large C \frac{dp(t)}{dt} = q(t)$$

As was done for the inertance in the last article, the Fourier transform of the characteristic equation yields:

$$\Large P(j\omega) = \frac{1}{j\omega C} Q(j\omega)$$

$$\Large Z = \frac{P(j\omega)}{Q(j\omega)} = \frac{1}{j\omega C}$$

$$Z$$ is the symbol for impedance as before.  The next figure illustrates a sinusoidal (filling) flow plotted against the distending pressure of a compliance.  The red shaded area of the flow curve represents a segment of time when flow into the compliance is positive.  During that time, distending pressure increases from a minimum (negative in this case) to a maximum when filling flow returns to 0.0.  Similarly a negative flow subtracts volume from the compliance and the distending pressure decreases throughout that portion of the cycle.  Don't forget that the distending pressure could be displaced in the positive OR negative direction depending on the original volume of the vessel.

The next figure is to clarify how the inflow and distending pressure curves are 90 degrees out of phase.  In the above, this is depicted mathematically by having $$j = \sqrt{-1}$$ in the denominator of the impedance expression, $$Z = 1/(j\omega C)$$. Notice how the peaks and troughs of the pressure curve coincides with the 0 - crossing points of the flow tracing (and vice verse).

The phase of a pure compliance is - 90 degrees meaning that the distending pressure lags the (in)flow.  This is obvious enough from looking at tracings (flow peak occurs before the pressure peak), but also easy to remember when you realize that the flow has to fill the vessel (first) to increase its volume (and distending pressure).

The next figure helps solidify the concept: the distending pressure is plotted against the (normalized) vessel area (cylindrical tube).  As we're illustrating a pure compliance, the 2 curves simply track each other.

### Frequency Dependence

For a linear compliance, a sinusoidal input (distending pressure, flow, or volume) results in a sinusoidal output of the same frequency (and no other frequencies.)  However the impedance modulus of a compliance decreases with increasing oscillation frequency, i.e. the impedance magnitude is proportional to $$1/\omega$$ as shown above.  In the next figure, a compliant vessel is subjected to an oscillatory distending pressure of increasing frequency. We see that at low frequency there is very little flow into and out of the vessel compared to higher rates of oscillatory distending pressure.

An equivalent statement is shown in the next figure where the magnitude of the oscillatory flow rate is held constant but frequency is increased.  We see that the distending pressure decreases with oscillation frequency in accordance with the characteristic equation shown above.

Now don't lose track of the simplicity of these diagrams.  The distending pressure of a linear compliance increases linearly with volume.  That's all there is to this really.  If I had put the volume plot next to the pressure plot, you would have seen that these 2 track each other exactly (for a pure linear compliance) In accordance with $$p(t) = \frac{1}{C} (V(t)-V_0 )$$. The frequency dependence is due simply to the fact that increasing rates of flow oscillation result in smaller and smaller volume oscillation.  As you look at the flow tracing in the most recent figure above, remember that the volume tracing would simply be the area under this curve. So higher flow oscillation frequency results in a shorter time interval for volume change.

### Compliances in Series and Parallel Arrangement

Like resistance and inertance, compliances can be placed in a physical arrangement that corresponds to either series or parallel arrangement.  We need to be able to figure out how a system will behave depending on both the quantitative ( magnitude ) of compliances but also their physical arrangement.  Here are 2 compliances (capacitors) placed in parallel with each other.

As you might imagine, this thing is more compliant than than either one alone.  To determine the quantitative value of the "equivalent compliance", we just have to be brave and follow the rules.  The distending pressure for both compliances is the same.  Here are the rules where $$p$$ is the distending pressure, $$V_{10}$$ and $$V_{20}$$ are the unstressed volumes of the two compliances(at distending pressure):

$$\Large V_1 = V_{10} + C_1 p$$

$$\Large V_2 = V_{20} + C_2 p$$

The total volume of the compliances as arranged:

$$\Large V_1+V_2 = V_{10}+V_{20} + (C_1+C_2) p$$

And the equivalent compliance:

$$\Large V_1+V_2 = V_{eq} = V_{10}+V_{20} + C_{eq} p$$

$$\Large C_{eq} = C_1+C_2$$

That's the direct "derivation" of how the compliances add in parallel.  However, now I'm going to introduce the trick that works for impedances in general.  It's pretty simple – we've already done it the hard way.  $$Z$$ stands for impedance and $$Z_{eq}$$ is an "equivalent" impedance.  Impedances added in series looks like this:

$$\Large Z_{eq} = Z_1+Z_2$$

Impedances added in parallel looks like this:

$$\Large \frac{1}{Z_{eq}} = \frac{1}{Z_1}+\frac{1}{Z_2} \rightarrow Z_{eq} = \frac{Z_1 Z_2}{Z_1 + Z_2}$$

That's all there is to it!  Now if we are talking about 2 compliances ( capacitors ) $$Z_1 = 1/(j\omega C_1)$$ and $$Z_2 = 1/(j\omega C_2)$$ added in parallel, we have:

$$\Large Z_{eq} = \frac{Z_1 Z_2}{Z_1 + Z_2} = \frac{\frac{1}{j\omega C_1} \frac{1}{j\omega C_2}}{\frac{1}{j\omega C_1} + \frac{1}{j\omega C_2}}= \frac{1}{j\omega (C_1+C_2)}$$

So the equivalent compliance $$C_{eq}$$ is just $$C_1+C_2$$ as we've already seen.  For the same 2 added in series:

$$\Large Z_{eq} = Z_1+Z_2 = \frac{1}{j\omega C_1} + \frac{1}{j\omega C_2} = \frac{C_1+C_2}{j\omega (C_1 C_2)}$$

$$\Large C_{eq} = \frac{C_1 C_2}{C_1+C_2}$$

### Compliance Distributes Volume

There's that old saying that flow takes the path of least resistance.  We saw that this isn't analytically true, but that when 2 resistances are in parallel the fraction of (average) flow through $$R_1$$ is $$R_2/(R_1+R_2)$$ and the fraction through $$R_2$$ is $$R_1/(R_1+R_2)$$.  There is an analogous rule for compliance that is one of the most important truisms in cardiology (though not adequately appreciated in my view): Volume is distributed to the (path, bed, chamber, vessel) of greatest complance.  Suppose we start running flow into the arrangement here:

With this parallel arrangement, the distending pressure for the 2 compliances remains the same, e.g. $$p(t)$$.

$$\Large p(t) = \frac{1}{C_1} (V_1(t)-V_{10} )$$

$$\Large p(t) = \frac{1}{C_2} (V_2(t)-V_{20} )$$

If the pressure changes,  the amount of volume ($$\Delta V$$) into or out of each compliance is determined from :

$$\Large \Delta p(t) = \frac{\Delta V_1(t)}{C_1} \rightarrow \Delta V_1(t) = C_1 \Delta p(t)$$

$$\Large \Delta p(t) = \frac{\Delta V_2(t)}{C_2} \rightarrow \Delta V_2(t) = C_2 \Delta p(t)$$

$$\Large \frac{\Delta V_1}{\Delta V_1+\Delta V_2} = \frac{\Delta p(t) C_1}{\Delta p(t) C_1+\Delta p(t)C_2} = \frac{C_1}{C_1+C_2}$$

$$\Large \frac{\Delta V_2}{\Delta V_1+\Delta V_2} = \frac{\Delta p(t) C_2}{\Delta p(t) C_1+\Delta p(t)C_2} = \frac{C_2}{C_1+C_2}$$

On the far right hand side of these last 2 equations we have the fraction of the volume change distributed into/out of each compliance.   We saw previously that when the distending pressure changes for linear compliances in parallel, the total change in volume is $$\Delta p/(C_1+C_2)$$ (compliances in parallel).  The last 2 equations show that the greater fraction of that volume ends up where the compliance is greater.

### Compliance in the Clinic

I'm going to skip the tracks slightly now.  This whole section of the website is about linear hemodynamics; the circulation is notorious for nonlinear compliance.   However I want to add some clinical/physiological relevance to the article so we'll apply the above principles to important situations. Compliance is immensely important in the everyday function of the cardiovascular system. These are things that every cardiologist likely knows (and aspiring cardiologists should), but saying them a little differently might turn out to be useful.  Since compliances of the circulation are both time-varying and nonlinear, we get to abandon the equations for a while and just talk qualitatively.  Mathematical models can incorporate all of these features and I hope I have time to develop a cardiovascular playground for the exploration of these topics.

#### Venous Return

Somewhere along the line the phrase was adopted that "venous contraction increases venous return to the heart".  I would argue that this is a weak statement that misrepresents the physical actuality.  To understand my beef, you must first select the physical units that will represent "venous return". If you think it's flow rate (e.g. mL/sec), I disagree with you.  Venous contraction doesn't increase flow rate through the heart appreciably – it's the Frank-Starling mechanism that alters the flow rate.  The (effective) cardiac output is equal to (effective) flow rate into the heart except for rather small transient imbalances that end up changing the volume of the heart.   I agree that venous contraction does increase venous return and the physical units of venous return arevolume.  Venous contraction decreases the compliance of the circulation and blood volume is consequently distributedoutof the veins and into the heart in accordance with the concepts noted above.   How much volume?  This is just the difference in end-diastolic volume of the heart between the 2 vasomotive states ( after minus before ).  Notice I didn't even say which heart, left or right. Venous constriction increases venous return to both.

Below is a basic schematic of the above situation with $$C_V$$ representing ( in a general sense) the compliance of the veins and $$C_H$$ the compliance of the heart.  These are 2 compliances in parallel, subjected to the same filling pressure ( approximately ).  Note: the gadget at the bottom of the figure is the schematic representation of a "ground" or fixed pressure ( voltage ).  Think of this as set to the pleural pressure for our purposes.  According to the development above, the compliance of the veins has to be of similar magnitude as the heart to be able to affect filling pressure and cardiac volume significantly.

#### Atrial Septal Defect

I've heard this one discussed it quite a few clinical conferences and it makes a good board question (!)  Suppose you have a patient with a good sized ASD - one large enough so that we might consider the filling pressure of the two ventricles to be the same.  ( Note: the pathophysiology doesn't really depend on this last stipulation to any great degree.)  What determines whether blood flow across the ASD is L to R OR R to L?

The principal answer is the relative compliances of the two ventricles of course.  Blood volume entering the "common atrium" is distributed according to the principles detailed above.  The garden-variety ASD is L to R because  the right ventricle is more compliant.  (Sure, there's some other pathology in effect if the right ventricle has become less compliant than the left, and you could argue pulmonic stenosis or pulmonary hypertension as the "cause" of a R to L shunt.)  Now as I keep saying, the circulation is nonlinear and the diastolic pressure-volume relationship of each ventricle is not a straight line.  This leads to a situation where one of the ventricles could be more compliant at low filling pressures with a reversal of that situation at higher pressures.  That would be one scenario by which a bidirectional shunt would be promoted.

#### Treatment of Congestive Heart Failure

There are several potential therapeutic strategies for the treatment of cardiogenic pulmonary edema; one of them is to alter venous compliance.  This is obviously what venous vasodilators do and this causes a redistribution of blood volume in accordance with the volume distribution principle.  In the figure below, $$C_S$$ is the compliance of the systemic circulation, $$C_P$$ the pulmonary.  While there's a lot of other stuff not in the diagram (so direct application is a bit of a stretch), this is the essence of how we can affect the pulmonary venous pressure by giving a drug that only affects systemic veins.

And of course this is one of the mechanistic means for the development of non-cardiogenic pulmonary edema also.  Intense venous constriction decreases compliance of those vessels and redistributes volume into the pulmonary circulation, thereby increasing pulmonary vascular distending pressure with edema as the result.

#### Diastology

The next figure could be said to encapsulate LV diastology where the capacitors shown represent the compliances of the pulmonary veins, left atrium, and left ventricle respectively.  We'd have to incorporate the facts of nonlinear and time-varying compliance into the model to reproduce the clinical observations (see below), but the essence is there.  Given that there are volumes (no pun intended) in the literature on this topic, I'll only call attention to the left atrium for the moment.  When the LA contracts, it changes (decreases) its compliance - the relationship between distending pressure and volume. (Again I'm using the term here in a mechanical sense which may not be the same as the clinical usage.)

Why does blood volume end up in the ventricle when there are no valves to enforce the flow direction?  Well of course the normal LV compliance (in diastole) is greater than that of the pulmonary veins, so volume is preferentially delivered there.  Restrictive physiology (incompliant LV) leads to a diminishing mitral A inflow and an increasing dramatic venous wave.

### Time Varying Compliance and Elastance

Up until now, we've been talking about compliance as a derivative of volume, $$V(p)$$, with respect to pressure  In cardiovascular physiology and cardiology texts, you'll see terms like "end systolic pressure-volume relationship" abbreviated as ESPVR (and EDPVR to represent the corresponding function in diastole).  We tend not to write things this way in the engineering literature where we are going to write equations that will subsequently be manipulated mathematically.  Furthermore I'm going to switch it all around now, so that we can start thinking about how this is applied to the heart, and talk about a different function – $$p(V)$$.  All that's been done here is to switch the axes of the volume function so that now pressure occupies the $$y$$-axis, volume the $$x$$-axis.  The slope of this function is called the elastance.  (Note: for some reason this terminology conjures inappropriate thought processes for many folks.  Increasing elastance DOES NOT MEAN  that the vessel is becoming more "elastic" or that it is more easily distended.  Elastance is the reciprocal of compliance.  If compliance increases, elastance decreasesand vice versa.  A vessel with greater elastance is stiffer;  a small change in volume will result in large change in pressure.

$$\Large E(p) = \frac{1}{C(p)} = \frac{d}{dV} p(V) = \frac{dp}{dV}$$

As before, we have no guarantee that the elastance is a constant for a particular vessel and it's been written here as $$E(p)$$ to indicate that it most likely depends on the particular value of distending pressure.  Just as we expected compliance might change as a function of volume, elastance will in general be a function of pressure – another way of saying that the slope of the pressure-volume function might not be a constant.  Furthermore biological vessels such as the heart change their compliance ( and elastance ) as a function of time.  This turns out to be a pretty good way to think about the function of the heart overall.

$$\Large p(t) = E(t)[V(t)-V_0]$$

This is the famous elastance model of left ventricular function of Suga and Sagawa

As a starting approximation, the cardiac chambers circulate blood by cycling chamber compliance ( or elastance ).  The ventricles must be appropriately compliant during diastole, incompliant ( high elastance ) at end systole, and cycle between these two states in a "timely" manner ( this last clause equates to the rates of relaxation and contraction ).  Blood volume redistributes in to the ventricles when they are compliant, and redistributes out of them when they are incompliant. The cardiac valves enforce the unidirectional flow when function is optimal.

In the clinical and cardiovascular physiology literature, the slope of the function $$p(V)$$ for the left ventricle at end systole and specific physiological state is considered to be a (relatively) constant (not a function of volume) and has been designated $$E_{max}$$.   This may be a reasonable approximation for many clinical and physiological purposes.  However there is no physical or physiological law to dictate this outcome and it serves my purpose is better not to impose any such restrictions on the concepts of time varying compliance.  Hence a somewhat more general approach is to write the cardiac chamber pressure mathematically as:

$$\Large p = p(V,A) = p(V(t),A(t))$$

In this notation, it's obvious that the distending pressure changes depending on the volume within the vessel, but also depends on a state of muscular activation $$A(t)$$. Hence distending pressure would change function of time i.e. due to activation and deactivation of the muscle, even if a volume remains constant.  Well-developed mathematical constructs allow us to write an expression for the changing pressure due to these 2 effects:

$$\Large \frac{d}{dt} p(V(t),A(t)) = \frac{dp}{dV} \frac{dV}{dt} + \frac{dp}{dA} \frac{dA}{dt}$$

$$p(V,A)$$ is the engine of the heart – it's what's most commonly used write software that does a reasonable job of modeling cardiac function. $$p(V,A)$$ may look rather obtuse, but it's also the information everyone in cardiology would like to be able to determine from their patients ( preferably noninvasively using a Star Trek "tricorder").  Mostly what we do on a daily basis is assess some basic aspects of structure and deformation.  At maximal activation, we have $$p(V,A_{max})$$ which is the $$E_{max}$$ "line"; it's essentially the pressure-volume relationship at end systole.  $$p(V,A_{min})$$ (zero activation) is the diastolic pressure-volume relationship. The function also describes the relaxation rate.  Many of the indices we using cardiology have to do with trying to catch a glimpse of this function.  Develop a widget to determine this function quickly and noninvasively – collect your Nobel Prize and retire.

Even so, this isn't the whole story.  It's long been known that contracting myocardial fibers