The Mathematical Relationship between Heart Rate, Cardiac Output and Pulse Pressure in the Human Systemic

Vasculature

by

E R Lowe

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Cardiac Output(litres/min)

Total Systemic Arterial Loop Resistance(mmHg per ml/min)

Introduction

It is a matter of observation, not conjecture, that any and all heart rate readings (HR) taken at random from an individual, regardless of the time of day or prevailing stress level, exhibit a simple mathematical relationship to each other, expressible in the form of an elementary algebraic equation involving two other parameters - the simultaneous indicated pulse pressure (Pp) and cardiac output (CO) at the time the pulse rate is taken. The equation states simply that pulse rate is proportional to the quotient of cardiac output divided by indicated pulse pressure.

As a consequence of this relationship, the behaviour of these three parameters, heart rate, the indicated pulse pressure and cardiac output, as well as several other variables of the systemic vasculature, can be shown to be precisely ordered, conforming to the basic laws of physics and natural science and neither as independent nor random in behaviour as at first appears. This paper shows that, based upon their mutual relationship, their instantaneous values may be calculated from observed values of the others and thereby their behaviour may be monitored non-invasively at all times.

Background

Superficially there appears to be no clear correlation between the rate at which the human heart beats and indicated systolic and diastolic pressures, as measured conventionally by auscultation or even more sophisticated and accurate methods including applanation tonometry at points over the brachial or radial arteries. High pressures are often associated with low pulse rates but just as often with high or ‘normal’ ones, and low pressures with high pulse rates or low ones, with any combination in between. Historically this has led to the assumption that no mathematical relationship exists between what is called ‘blood pressure’ and the heart rate, but this is not the case and the assumption is wholly incorrect.

Thesis

The relationship between true stroke volume and indicated pulse pressure is almost directly linear. For the present purpose of calculating the values of the parameters involved, any deviation from linearity is so small as to allow it to be ignored1 without significant loss of accuracy. From their mutual proportionality, therefore, the standard definition of arterial compliance2 at the proximal aorta is expressible as:

SV = C × Pp (equation 1)

where SV is stroke volume, C is the compliance of the walls of the ascending aorta, and Pp is the indicated pulse pressure. The product identity {C × Pp} can therefore be substituted for SV in the cardiac output equation (CO = HR × SV) and, by designating pulse rate as HR, cardiac output (CO) may be expressed as:

CO = HR × {C × Pp} (equation 2).

1 Levick R.J. An Introduction to Cardiovascular Physiology, 2nd Edition, Chapter 6.4, pp 73, Figure 6.4(c)

2 Morhrmann & Heller, Cardiovascular Physiology, 5th Edition, Chapter 6, pp. 112, para. 2

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From this, it is possible to compute the pulse rate as:

HR = (equation 3)

The ratio of any heart rate HR1 to any other, HR2, then clearly is:

HR1 : HR2 = :

where CO1, Pp1, CO2 and Pp2 are respectively the corresponding cardiac outputs and pulse pressures. The compliance coefficient C effectively cancels out, allowing the heart rate at any time to be calculated from any previous reading:

HR2 = HR1 × × (equation 4)

Stroke volume is measured nowadays routinely on a clinical basis using Doppler or other methods as a means of determining cardiac output (HR × SV), so if pulse rate and pulse pressure are both recorded simultaneously with CO, then C, the compliance modulus of the ascending aorta, may be obtained by dividing stroke volume by the indicated pulse pressure.

Under relaxed bodily conditions, the compliance of the walls of the aorta remains essentially unchanged over a wide range of internal arterial pressures even though other factors influence its value, so for working purposes C may be assumed to be a constant whose value is specific to that individual. Inserting the values for measured cardiac output (CO) and pulse pressure (Pp) into equation 3 shows that only one pulse rate satisfies it. All the elements of equation 3 being known, and the value of each term being determinable, allows the heart rate to be calculated and what follows is an actual clinical example of it in practice.

Calculation of Heart Rate (HR)

The stroke volume of a patient was measured using Doppler methods and recorded as 95mls and the simultaneous indicated measurements of radial arterial systolic and diastolic pressures were 180/98 mms.Hg. The simultaneous heart rate was monitored as 66 bpm.

Pulse pressure was therefore 82 mms.Hg, and by dividing the measured stroke volume by this value the aortal wall compliance C can be calculated as 1.1585 mls/mm.Hg.

Using equation 2, cardiac output can now be calculated as:

66 × {1.1585 × 82} = 6270

and by applying these values to equation 3, the pulse rate can be computed as:

= 66 bpm

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It can be seen that the pulse rate calculated using equation 3 is exactly equal to the measured rate.

The Common Relationship between all Random Heart Rate Measurements

In practice, HR is rarely constant and varies continuously but equation 3 makes clear that it is impossible for it to vary independently. All such HR variations must be offset by compensating changes in either (or, more usually, both) cardiac output or pulse pressure. This is because, rearranging equation 4, the generalized mathematical relationship between all HR, Pp, and CO readings is:

× × = 1

This is a permanent and universal mutual relationship and therefore none of the three parameters, HR, Pp, or CO can ever be independently variable, enabling the instantaneous pulse rate at any subsequent time to be calculated simply by using equation 4 and inserting the appropriate (simultaneous) pulse pressure and cardiac output values.

Calculation of Pulse Pressure

The ability to calculate and even predict pulse rate in this way may well appear superficially to be little more than an entertaining party-piece which can be verified by counting beats over any interval. However, many (and more significant) corollaries follow from it in relation to the other variables as shown below.

Using equation 3 as a basis, the instantaneous (indicated) value of Pp can also be calculated for any given values of cardiac output and pulse rate:

Pp =

So using the same values obtained from the patient above, the pulse pressure can be calculated as:

= 82 mms.Hg

This again demonstrates that the calculated value of a parameter (in this case pulse pressure) is exactly equal to the measured value.

Calculation of Cardiac Output

In precisely the same way as for pulse rate and pulse pressure, for any configuration of these parameters there can be only one value for cardiac output and because no further involvement of Doppler techniques is required its calculation is simpler still. From equation 2, and denoting the original measured cardiac output, heart rate, and pulse pressure as before as CO1 HR1, and Pp1 respectively and any subsequent cardiac

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output, pulse rate and pulse pressure as CO2, HR2 and Pp2, it is possible to calculate the value of the subsequent cardiac output as:

CO2 = CO1 × (equation 5)

As an example of this in practice, and again using the same original parameter values as before, let us assume that on a subsequent occasion the same patient’s heart rate HR was (say) 78 instead of 66 bpm, whilst the indicated pulse pressure Pp had dropped from 82 to (say) 50mms.Hg. Using equation 5, the patient’s cardiac output can be calculated as:

CO2 = = 4158 mls/min

The combined slight rise in pulse rate and fall in pulse pressure (and stroke volume) results from a reduction of roughly one-third of cardiac output.

Relationship between Stroke Volume and Pulse Rate

With regard to the mathematical relationship between pulse pressure and pulse rate however, this is not the end of the story. Whilst there is but one pulse pressure consistent with any pair of simultaneous cardiac output and pulse rate readings, its value is critically dependent upon yet another parameter, the ratio of stroke volume to

pulse rate, . This is explicit in equation 1 and manifests itself implicitly in

equation 2 as the indicated ratio . This makes clear that any one of an infinite

number of combinations of different but complementary pulse rates and stroke volumes can generate the same cardiac output.

For example, a cardiac output of 5000mls/minute can be achieved by the heart ejecting 50mls of blood one hundred times a minute, or 100mls fifty times a minute, or 72mls just over 69 times a minute, or any number of similar combinations. The range is limitless, as shown in the curve resulting from plotting stroke volume against pulse rate (see Figure 1).

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Figure 1

The significance of this ratio cannot be overstated, because ipso facto the proportionality existing between pulse pressure and stroke volume also causes pulse pressure to increase not linearly but parabolically with decreasing pulse rate. This means that, although in practice the ability to deliver the required cardiac output to meet the demands of the organs it serves at the prevailing heart rate is limited by the heart’s efficiency as a pump, the stroke volume necessary to achieve the cardiac output is theoretically unlimited. The lower HR is, the greater SV must be to generate the required cardiac output and, by extension, so must the pulse pressure rise to quite disproportionate levels.

As an example of this, and using the same original parameters as before, if the patient’s pulse rate for the 6270 mls/min cardiac output were not 66 but 60 bpm, then the pulse pressure would need to rise by 10%, from 82 to 90.2 mm/Hg, and if it were still lower, say 55 bpm, then the rise required would be from 82 to 98.4 bpm. (see Figure 2).

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Figure 2

Somewhat fortuitously, entirely due to the range scales involved, it is possible to combine (as shown in Figure 3) the two curves in Figures 1 and 2 to illustrate how the three Pp readings, when superimposed, almost precisely overlay the SV curve when plotted against pulse rate. The small discrepancy, of course, arises because in one curve cardiac output is 5 L/min and in the other it is 6.270 L/min.

Figure 3

The considerable rise in pulse pressure induced by artificially lowering the heart rate must by extension be reflected in excessive systolic pressure Ps, even at rest, but when the body is placed under stress and additional systemic arterial flow is

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demanded, then if the pulse rate is inhibited (typically by beta blocking) from rising properly to fulfil its natural role in delivering extra output, gross and disproportionally high systolic pressures clearly will result, posing a risk to the structural integrity of blood vessels at all levels on the arterial tree.

Mathematically, the processes above show that the three parameters CO, Pp and PR at all times have mutually interdependent values whose loci inhabit a 3-dimensional shell-surface, the shape of which is shown in Figure 4. All possible configurations of a patient’s CO, Pp, and PR lie on this curved plane surface and each triple configuration constitutes a single point on the surface shell. This graphically illustrates why, as pointed out in earlier paragraphs, there can be only one pulse rate for any specific value of CO and Pp, but more significantly, there is only one single value of CO for any specific combination of Pp and PR, and only one value of Pp corresponding to any pair of CO and PR values.

Figure 4 3

From a medical (as distinct from a mathematical) standpoint however, it is clear that the values of any two variables uniquely determine that of the third. Therefore, if the indicated level of any is perceived to be ‘too low’ or ‘too high’ (typical examples of which would be, for example, so-called ‘isolated hypertension’ indicated by an excessive pulse pressure, or tachycardia evinced by a pulse rate deemed excessive or through impaired cardiac output), then consideration of the area of the 3-D shell upon which the cardiovascular system is operating would appear to merit being the primary focus of investigation rather than artificial intervention simply to ‘correct’ the apparent abnormality.

An ‘abnormality’ in any of the three parameters cannot exist in isolation. An ‘abnormally high’ pulse rate, for instance, might well be the brain’s natural, optimum and perhaps only mechanism of achieving the correct cardiac output. Artificially lowering it (if the cause is an impaired stroke volume) can have only one outcome -

3 The topography of the surface of the 3-D shell is the same, irrespective of whether stroke volume or pulse pressure is used for

the third axis, due to their proportionality. As it is more relevant to the haemodynamics of the system, the former has been used in this instance.

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exacerbating the condition. Artificially lowering the pulse rate and thereby raising pulse pressure must, if cardiac output is maintained, inevitably raise systolic pressure unless diastolic pressure is reduced by a corresponding amount, simply because Ps is the sum of Pd and Pp.

The only circumstance under which artificially lowering pulse rate can result in a reduction in systolic pressure is where cardiac pump-action is impaired and the heart is incapable, because of its characteristic curve, of delivering the required flow of blood at the appropriate pressure. Cardiac output therefore falls, diastolic pressure (the product of the reduced cardiac output and total systemic loop resistance) falls as well, and if the fall is greater than the artificially induced rise in pulse pressure then systolic pressure will also fall. Lowering of systolic pressure has therefore been achieved under these conditions but at the expense of optimum flow.

Diastolic Pressure

Unlike pulse pressure, no simple mathematical relationship exists between heart rate and diastolic pressure. Nor is it possible to compute its behaviour over time in relation to other systemic parameters, because irrespective of wherever it is measured, its instantaneous value (whether indicated or true) at that point simply reflects the volume of blood at pressure at that instant in that section of the vascular loop. Thus the extent to which it is dilating and expanding the arterial walls, and thereby being pressurized by their reactive forces, determines the internal pressure.

In turn, because blood neither collects at any point within nor escapes from the elastic plenum comprising the systemic vasculature, the total pressurised contained volume at any moment depends firstly upon run-off, tending to deplete it, and secondly by the efficacy of its replenishment i.e., cardiac output, tending always to restore it. Thus, if run-off exceeds cardiac output, both true and indicated diastolic pressure fall. If cardiac output exceeds run-off, both true and indicated diastolic pressures rise, and if cardiac output equals run-off then equilibrium is established and diastolic pressures (both indicated and true) remain roughly constant.4

Systolic Pressure

The value of both true and indicated systolic pressure therefore is the sum of (a) the pulsatile component Pp whose value is highly sensitive to heart rate and quantifiable non-invasively and (b) a relatively steady non-pulsatile component upon which it is superimposed whose level is determined by a completely different mechanism wholly unrelated to the heart rate. This being the case, systolic pressure is not susceptible to mathematical analysis. In reality, it is little more than a by-product reflecting the behaviour of two widely differing physiological mechanisms.

Because the indicated pulse pressure of a young healthy adult at rest is much less than the diastolic pressure, systolic pressure, the sum of the two, by and large simply follows variations in diastolic pressure. Observed differences between peak systolic and diastolic values are due to the continuous but relatively minor ‘hunting’ of pulse pressure levels caused by heart rate variations as shown in Figure 2.

4 The qualification arises due to the transit time required for blood to traverse the loop. Pressures and flows in phase cause non-

linear localised gradients to develop, leading to minor non-linear gradients of diastolic pressure. 9

However, in the dynamic (as opposed to static) mode of operation, when the body is stressed (as opposed to static mode, when it is relaxed), indicated pulse pressure can easily equal and even surpass the diastolic pressure and the behaviour of systolic pressure becomes quite different. It rises, reflecting increased pulse pressure, while diastolic pressure frequently does not, being held steady (as far as possible) by vasodilatation and may be observed even to drop initially when run-off exceeds cardiac output.

The limits are set by (a) the heart’s performance as a pump and (b) the parabolic nature of the vasodilatation curve, which approaches maximum asymptotically when dilation of the blood vessels reaches its practical limit and the resistance to flow is the minimum attainable.

Mean Arterial Pressure

The so-called ‘Mean Arterial Pressure’ or MAP, commonly held to be the pressure impelling blood round the systemic vascular loop and computed by adding roughly one third of pulse pressure, or Pp/3, to the diastolic pressure, is considered to be the ‘weighted mean’ of the pulse pressure waveform and their sum is supposedly the equivalent of the analogous RMS value of a complex electrical waveform. This is used usually in relation to the Law of Flow, (sometimes called the Mean Pressure Equation) or Darcy’s Law, in which flow, pressure, and the resistance to flow conform to the equation:

MAP = CO × TPR

TPR being the effective resistance to flow presented by the systemic loop and taken to be mostly that of the peripheral arteries.

The concept of ‘Mean Arterial Pressure’ and all its derivatives such as ‘Central Aortic Mean Arterial Pressure’ are however entirely spurious, because the pulse pressure wave is a travelling longitudinal wave of pressure only, not flow, and in common with all such waves performs no work upon the medium through which it passes. Its velocity through the blood is roughly an order of magnitude greater than that of blood flow itself and, apart from losses, none of its energy is dissipated in the process.

Impelling blood requires the expenditure of energy and none being dissipated by the pulse pressure wave in its transit means that the pulse pressure wave plays no part in its translation down the vascular loop, leaving diastolic pressure as the only source of the forces driving blood round the systemic vascular loop.

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