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 **P _{n}** can be thought of as the sum of a local reservoir pressure

**P _{n} = P_{res,n} + P_{ex,n}**

We then make the assumption discussed in the previous page that **P _{res}** is the same waveform throughout the arterial system, but it is delayed by the time that it takes for a wave to reach vessel

**P _{res,n}(t) = P_{res}(t - &tau_{n})**

We further assume that **P _{res}** 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

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 = P _{res} + P_{ex}**

*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 = W _{res} + W_{ex}**

where **W _{res} = &int P_{res}(t)Q(t)dt** and

*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.

*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 **W _{ex}** 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

First, we can show that if **W** is constant during systole, then **W _{ex} > 0** for all

Second, if **Q** takes the form of a half-sinusoid, we can also show that **W _{ex} > 0** for all

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 = T _{S}**, the time of the end of systole and the start of diastole. For this case

This led us to explore the possible values of **S** and **T/T _{S}**. (Note that we consider

Finally, we looked at the asymptotic expansion of the expression for **W _{ex}** when

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