The reservoir pressure gives the minimum ventricular work

This statement relies upon some fairly complex mathematical analysis. In keeping with the spirit of accessibility of this web site, the mathematical arguments will be given separately in pdf pages [Reservoir and excess pressure at the aortic root] while the gist of the analysis without the mathematical proof will be given in the html pages.

The outline of the mathematical analysis is based on an averaged approximation of the detailed wave solution for the pressure throughout the arterial system. With this approximation, we can express the solution in terms of sums over the average conditions in each individual vessel rather than having to solve a highly complex set of equations describing the propagation of each wave in the system as we do in our detailed method of characteristics modelling of arterial pressure and flow.

We then assume that the measured local pressure Pn can be thought of as the sum of a local reservoir pressure Pres,n and a local excess pressure Pex,n.

Pn = Pres,n + Pex,n

We then make the assumption discussed in the previous page that Pres is the same waveform throughout the arterial system, but it is delayed by the time that it takes for a wave to reach vessel n from the aortic root.

Pres,n(t) = Pres(t - &taun)

We further assume that Pres satisfies the mass conservation equation applied to the whole arterial system. This is the same assumption that is made about the Windkessel pressure, but in that case the pressure is assumed to be constant throughout the arterial system. Allowing Pres to have a time delay between different vessels results in a more complex form of the global mass conservation equation. This equation is of the form known as a time invariant time-delay ordinary differential equation. There is no standard method for solving these equations, but there is a theorem proving that a solution exists and that it is unique. In any case, a detailed solution of the differential equation would entail detailed knowledge of the properties of all of the individual arteries - something that is impossible to know clinically. It is possible, however, to obtain an estimate of the solution and and we know that this approximate solution gives an approximation to the unique solution. The existence of a solution for Pres is enough for us to carry out the rest of the analysis.

By the separation of the measured pressure into a reservoir pressure and an excess pressure, we can apply this separation to the pressure at the root of the aorta n=0. In the following discussion, we will assume that we are talking about the aortic root and drop the subscript 0; in the detailed mathematical development we will use a slightly different notation that is easier to write and to read. Thus at the aortic root

P = Pres + Pex

Ventricular hydraulic work

The hydraulic work done by the ventricle as it pumps the volume flow rate Q(t) into the arteries at pressure P(t) (remember that we are dropping the subscript 0 for the moment), is given by the integral of the product of pressure and volume flow rate over the cardiac period T

W = ∫ P(t)Q(t)>dt

Using the separation of P, we can write the hydraulic work for a given Q as the sum of a reservoir work and an excess work.

W = Wres + Wex

where Wres = &int Pres(t)Q(t)dt and Wex = &int Pex(t)Q(t)dt. We now concentrate on the excess work Wex.

Minimum excess hydraulic work

The essence of the theoretical development is to find the excess pressure waveform that minimises the excess work. This is done using the calculus of variations which is an old method for finding a function which minimises some other function. The method is similar to the much more familiar algebraic minimisation where the goal is to find the value of a variable that minimises the function. However, the problem is much more complex because we are trying to find a function that minimises some property of another function, in this case the excess work done over a cardiac period.

The problem is further complicated since the excess pressure is not free, but must satisfy the mass conservation equation. Essentially this says that any changes in the excess pressure at the aortic root must be accompanied by corresponding changes elsewhere in the system. I.e. if we consider a new function of excess pressure which has a slightly higher pressure than another function of excess pressure, then the volume of blood in the aortic root is larger because of the compliance of the aortic root. This means that there must be corresponding changes where the excess pressure in other vessels is less, so that the volume of blood in the rest of the system is correspondingly lower in order to ensure that the total volume in the arteries at any time satisfies global mass conservation. After a lot of calculus and algebra, we find an expression P*ex that minimises the excess work done by the ventricle. P*ex is a function of the flow rate Q, the net compliance of the arterial system C and the net resistance to flow out of the terminal arteries R. These are the same parameters that determine the Windkessel solution for the mass conservation equation.

Minimum work

The next step in the argument is perhaps the most subtle. If the minimum excess work is positive for a given Q, R and C, then we argue that the reservoir work represents the minimum work that the ventricle must do under those conditions.

It should be noted at this point that we do not expect the excess work to be positive under all conditions. We know that it is possible to do work on the ventricle using ventricular assist devices and so the excess pressure generated in those cases will be negative. We thus turn to the problem of finding the conditions for which Wex is positive. This part of the analysis is most conveniently carried out using nondimensional variables. It is most convenient to nondimensionalise times with respect to the usual time constant of the arterial system τ = RC. In particular, we will define the nondimensional time of systole S = TS/RC. This seems to be the most critical factor in the analysis.

First, we can show that if W is constant during systole, then Wex > 0 for all S.

Second, if Q takes the form of a half-sinusoid, we can also show that Wex > 0 for all W. These two results led us to believe that Wex > 0 for any Q that was always positive. This turned out not to be the case.

Third, we can also solve the problem exactly for Q which is triangular during systole. I.e. rising linearly from zero at t = 0 to a maximum at some time during systole and then falling linearly to zero at t = TS, the time of the end of systole and the start of diastole. For this case Wex goes from positive when S is small to negative at some critical value S*. The critical value S* is a function of the ratio T/TS, the ratio of the cardiac period to the time of systole.

This led us to explore the possible values of S and T/TS. (Note that we consider T/TS instead of its inverse, the fraction of the cardiac period taken up by systole, for notational simplicity in the analysis.) Physiologically, it is improbable that T/TS < 2, since this would require the ventricle to fill more quickly than it takes to eject. Similarly, it is always observed that the time of systole TS is always much shorter that the arterial time constant &tau, which means that S < 1. Under normal conditions in the human, TS is of the order 0.3 s while &tau is of the order 1.5 s, so that S is of the order 0.2. Physically, if &tau was small than TS, then the blood pressure would fall to a very low level during diastole (remember that diastole must at least be of similar duration to systole). This would lead to a relatively small mean blood pressure, which is not what is observed physiologically.

Finally, we looked at the asymptotic expansion of the expression for Wex when S << 1, which we believe is the case for all physiological or clinical conditions. The results of this analysis show that to zeroth order, Wex > 0 as long as the stroke volume is positive. The first order term in S is quite complex, depending on integral of Q (the time variation of the volume of blood ejected), its average and first moment and the average of the square of the volume ejected. This result is the basis of our claim that for all physiologically and clinically relevant flow waveforms, the reservoir pressure represents the minimum work that the ventricle must do to generate the given flow waveform with the given arterial parameters R and C.

The corollary to this is that the excess pressure represents the excess work that the ventricle does over and above the minimum possible work.