# Mathematics64

Mathematical relationships in anaesthesia and intensive care medicine
Ari Ercole, PhD FRCA
Clinical Lecturer
Division of Anaesthesia
University of Cambridge
Cambridge CB2 2QQ
UK
Paul Roe, FRCA⇓
+ Author Affiliations

Consultant Anaesthetist
Cambridge CB2 2QQ
UK
Tel: +44 1223 217434
Fax: +44 1223 217223
Key points
Understanding how physical, physiological, pharmacological, and chemical factors affect each other is central to the scientific basis of anaesthesia. Many anaesthetic decisions are based on relationships between clinical measurements and the underlying physiology. Mathematical functions allow relationships between varying factors to be expressed in a quantitative and precise way.

Sometimes these functions result from the underlying science. In other cases, mathematical description is a simplification which allows us to understand and make predictions even when, in reality, the behaviour is more complicated.

Mathematical functions can also be used to build conceptual models of processes. The predictions from such models can be useful in understanding biological behaviour.

Where an underlying behaviour is expected from scientific principles, functions can be used to transform experimental data into a form that is easier to interpret.

Properties such as the rate of change and area under the curve can be derived from functions and have a number or important applications.

The art of anaesthesia is underpinned by concepts from physics, chemistry, pharmacology, and physiology. These disciplines seek to explain the relationship between different factors, for example, arterial pressure and cardiac output, in a quantitative way. The real world is undeniably complex but there are many different kinds of simple mathematical relationship that occur again and again in science. In this article, we aim to describe some of these using examples from the scientific basis of anaesthesia.

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Introduction to functions

When thinking about functions, we are seeking to describe how various quantities might depend on each other. The dependence of one quantity (say ‘y’, the dependent variable) on another variable (‘x’, the independent variable) can be described mathematically as a function [Equation (1)]:
(1)
The function, ‘f()’, is a kind of ‘black box’ which tells us what y to expect for any given input, x. In some cases, it is easy to write down what ‘f()’ represents explicitly. Mathematics has some ‘ready-made’ functions (for example, linear, hyperbolic, exponential, logarithmic) that may be helpful and we will discuss some of the most important examples in this article.

In real situations, there may be more than one independent variable. For example, the pressure (P) of an ideal gas depends not only on its volume (V) but also on its absolute temperature (T). Such situations are difficult to visualize. One simplification that is sometimes helpful is to hold some of the variables constant. In this way, it may be possible to reduce the problem to a simple function. For instance, if we hold P constant, we find that V varies directly with T (Charles’ law, an example of a linear function). Alternatively holding T constant, we find that P varies inversely with V (Boyle’s law, an example of a hyperbolic function) (Fig. 1).

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Fig 1
Pressure/volume relationship for an ideal gas keeping temperature constant at different levels gives a family of simple curves (isotherms).

Real gases are not ideal but such simple mathematical descriptions may still be accurate enough for many purposes. Sometimes, however, it is impossible to write an equation for ‘f()’ easily. For example, neglecting dissolved oxygen, the arterial oxygen saturation, , is related to arterial oxygen partial pressure, , by the oxyhaemoglobin dissociation curve. Unfortunately, the familiar sigmoid shape of this curve does not correspond directly to any ‘ready-made’ mathematical functions and we must be satisfied with graphical representations and approximations of the curve shape, such as the Hill model.2

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Transformations and inverse functions

It is possible to take basic functions and change their size or position to make new, more useful functions. Such changes are called transformations and some important examples are illustrated in Figure 2. Multiplication of the original function by a constant, a, results in a scaling of the original curve (by a factor of a). Multiplication of the function by −1 causes a reflection in the horizontal axis. Adding a constant, b, leads to a vertical translation by a distance given by b. On the other hand, substituting −x for x reflects the curve in the vertical axis. A substitution of (x−c) for x (where c is a constant) results in a lateral translation.

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Fig 2
Effect of reflection, translation, and scaling on an arbitrary function y=f(x).

One final transformation shown in Figure 2 is a reflection of the original curve about a 45° line. The resulting function has a special name—the inverse function—and is denoted by f−1(). This function is special, in that it reverses the effect of the original function. So, for example, if y=f(x) then f−1(y)=x. One well-known example of an inverse function is the logarithm, which is the inverse of the exponential function. We will look at this in more detail later.

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Linear functions

Linear functions are perhaps the simplest and arguably the most useful mathematical relationships. A linear relationship between y and x can be written in the standard form:
(2)
In Equation (2), m is the gradient of the line and c represents the y-axis intercept.

To see some of the features of linear functions, consider the problem of calibrating blood oxygen partial pressure measurements. The polarographic Clark electrode produces an oxygen-dependent current and calibration is the process of relating this output of a transducer to what is to be measured. Helpfully, in the case of the Clark electrode, the measured current (which we shall call y) should vary approximately linearly1 with blood oxygen partial pressure (which we shall refer to as x). The exact relationship, however, depends on the temperature and exact physical and electrochemical properties of the polarographic electrode which may vary from electrode to electrode, and change with time (a phenomenon known as drift). Furthermore, the current may not, in practice, be exactly zero even when there is no oxygen present due to various effects. In principle, calibration is a simple process. By measuring y for several solutions of known x, it is possible to construct a graph such as that in Figure 3.

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Fig 3
Calibration plot for a polarographic oxygen electrode. Measured current is plotted for control solutions of known oxygen partial pressure (black dots). A best-fit straight (solid) line of the form y=mx+c represents the linear calibration and reveals an offset whose value equals c. The effect of changes in c is to translate the line up or down. The increase in current for a given increase in oxygen partial pressure is the sensor sensitivity and is equal to the gradient of the line which is m. Increasing m would steepen the curve as shown.

The best-fit straight line (solid line in Fig. 3) can be determined from the calibration and this gives the values of m and c for Equation (2) which fit the data best. As it happens, the line does not pass through the origin but crosses the y-axis at a value given by c. The value of c is called the intercept and in this case represents the ‘zero-offset’ of the electrode. The sensitivity of the electrode to oxygen determines the gradient of the calibration line which is given by m. The gradient is calculated as the increase in y for a given change in x. More sensitive electrodes give a greater change in current for a given change in oxygen partial pressure (dotted lines representing increasing values of m).

The gradient, m, and y-axis intercept, c, are all that is needed to describe any linear relationship. Increasing the value of m steepens the line (dotted lines in Fig. 3). Changing the value of c is an example of a mathematical translation—its effect is to shift the line vertically up or down. Numbers which completely describe a relationship are known as parameters. Linear functions only have two parameters, m and c. One important implication is that only two points are required, in principle, to completely determine a straight line. ‘Two-point calibrations’ exploit this and have the advantage of being fast. However, they may be error prone and a full calibration usually uses more points.

Linear functions are appealing because they are easy to understand and work with. Of course, truly straight lines do not generally exist in nature. Such deviations are known as non-linearity and generally make the mathematics more complicated. However, the linear approximation is often adequate even if only over a limited range of inputs (for example, the relationship between intracranial pressure and ). Because of this, relationships like y=mx+c are very common in practice.

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Hyperbolic functions

Often, two variables may be oppositely related: when one increases, the other is observed to decrease. This general behaviour is known as an inverse relationship (not to be confused with an inverse function which, as we have already described, is a specific function which reverses the effect of another).

One common kind of inverse relationship is the reciprocal (or hyperbolic) relationship:
(3)
This is an example of a curve known as a rectangular hyperbola. We have seen one example of a hyperbolic relationship already: the pressure (P) and volume (V) of a quantity of ideal gas vary reciprocally (if the temperature is constant). This is known as Boyle’s law and can be written as
(4)
The constant in Equation (4) depends on the absolute temperature (and number of gas particles). The curves of P against V are rectangular hyperbolae and a family of them for different temperatures are shown in Figure 1. Notice how the curves approach the axes but never cross them. Lines like these are known as asymptotes.

Other important examples of hyperbolic relationships include the relation between serum creatinine and glomerular filtration rate and between alveolar carbon dioxide (FACO2 and therefore end-tidal CO2 concentration) and alveolar ventilation (for constant and cardiac output)3.

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Exponential functions

Exponentials are functions where a number, b, (called the base) is raised to a power:
(5)
A family of exponentials with different bases are shown in Figure 4. All of these cross the y-axis at y=1. A particularly important base is the number e (∼2.718281…). This number has special properties in the mathematics describing change, and exponentials with base e are common in problems where the rate at which a quantity changes itself depends on that quantity (e.g. radioactive decay, drug kinetics, oscillation, and damping).

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Fig 4
Family of different exponential functions (examples of ‘tear away’ and ‘wash-out’ functions). Note that these curves all pass through the y-axis at 1 and that the x-axis is another example of an asymptote.

As we have already seen, substituting –x for x results in a reflection in the vertical axis and this is also shown in Figure 4. The result is known as a ‘decaying exponential’ and has some important features. As an example, consider the simple first-order elimination (or ‘wash-out’) of a drug from a single compartment. The concentration C (along the vertical axis) which starts at C0 at time zero varies inversely with time, t (along the horizontal axis), in this case according to the decaying exponential:4
(6)
where β (also represented by the symbol k in some texts) is a parameter called the ‘elimination rate constant’ and determines the rate of decay. Examples with two different values of β are plotted in Figure 5. In some situations, Equation (6) is instead written in terms of an alternative quantity τ (=1/β), known as the ‘time constant’, which is the time taken for the concentration to decrease to ∼37% (=1/e) of the starting value. The clearance, τ, β, and volume of distribution Vdistribution are related:
(7)
One unique property of the decaying exponential is that it decreases by a constant proportion each time period. This leads us to be able to identify a half-life, t1/2 which is the time taken for the concentration to halve. We will examine the relationship between β and t1/2 later.

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Fig 5
Decaying exponentials such as observed with first-order drug wash-out from a single compartment. Notice how the decrease in drug concentration is initially rapid, but this rate of change (given by the gradient of the straight-line tangents to the curve) decreases with time.

The wash-in behaviour of a continuous infusion is obtained by reflection and translation of the wash-out function (Fig. 6).

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Fig 6
Single compartment approach to steady-state concentration (Css) for a continuous infusion of drug (‘wash-in’, curve 3) derived from a simple exponential decay (curve 1).

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Logarithmic functions

If, in Equation (6), we knew C0 and measured C at different times, we could calculate the value of β. This would tell us about the ratio of clearance to volume of distribution [Equation (7)]. One difficulty with exponentials is that it is not immediately possible to rearrange this equation for β. What we need is the inverse of an exponential function.

We have already seen how to construct the inverse of any function graphically—it is a reflection in the 45° line y=x. This is depicted in Figure 7. This function is known as the logarithm (in this case using base b). This function is written as
(8a)
The special cases of log base 10 and log base e (also known as the natural logarithm) have short-hand versions:
(8b)
Mathematicians have devised methods for evaluating these functions to any degree of precision required. Furthermore, logarithms have a number of mathematical properties that turn out to be useful. In particular, the following rules are sometimes helpful:
(9a)
(9b)
(9c)
Logarithms allow us to invert expressions such as Equation (6). Taking natural logarithm on both sides and remembering the rule that log(AB)=log(A)+log(B), we get
(10)
From Equation (9c), ln(e−βt) is simply equal to (−βt). Therefore, Equation (10) can be simplified to
(11)
which is the equation for a straight line. We can see that if we were to plot ln(C) against time, this would be a straight line with gradient −β and vertical intercept equal to ln(C0) (Fig. 8A). This technique of transforming one relation into a straight line can be useful experimentally.

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Fig 7
The logarithm function. Because the logarithm is simply the exponential function reflected in the 45° line y=x, it crosses the horizontal axis at x=1, whereas the exponential crosses the vertical axis at y=1.

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Fig 8
Straight-line transformation of experimental data. (A) A log-transformation successfully transforms decaying exponential data into a straight line. (B) Bi-exponential data are not transformed into a single linear relationship by a log-transform but into two approximately linear relationships with different gradients and therefore rate constants.

Of course, such transformations rely on the underlying assumptions [in this case, Equation (6)] being correct. If they are not, the transformed data will deviate from a straight line. For example, if the drug is first rapidly redistributed, then this initial phase will have a different rate constant, α. In this case, the logarithmic transform does not give us the equation of a straight line. However, if α and β are very different, the transformation may still be useful, in that the initial and terminal phases may both appear linear even if the data do not fall on a straight line in its entirety (Fig. 8B).

Finally, one interesting special case of equation is when t=t1/2 and therefore, by definition, C=C0/2. If we substitute these into Equation (11) above and remember Equation (9b), we get

This can be rearranged to give
(12)
Equation (12) provides the link between rate constant and half-life, another example of a reciprocal relationship.

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When we looked at the linear relationship given by the equation y=mx+c, we saw that the slope of the line was given by m. This slope is known as the gradient and describes the rate of change of y with respect to x. For straight lines, this is a constant; it does not depend on x.

For a curve, we could calculate the rate of change by drawing straight-line tangents (Fig. 5) but would find that the gradient varies as we move along the horizontal axis.

For example, in the case of the wash-out curves in Figure 5, the gradient is initially high at time zero and decreases to approach (but never quite reach) zero for very large times. This is easy to understand if we assume that the rate at which a drug subject is eliminated is directly proportional to the plasma concentration, C (first-order kinetics). This is an example of a mathematical model:
(13)
where β is the rate constant we have seen before. As the concentration decreases, the rate of change of C must also decrease as the drug is eliminated less quickly. This leads to the familiar exponential shape of the wash-out curve, although we will not prove this here.

For some drugs (e.g. thiopental or alcohol), the elimination mechanism may become saturated. In this case, the drug is being eliminated according to zero-order kinetics, which is independent of C, that is
(14)
where γ is a constant. Since the rate of change is a constant then, in this case, the concentration must decrease linearly rather than exponentially.

Working out the rate of change of a curve by drawing straight-line tangents is tedious and inaccurate. There is a mathematical method for doing this called differentiation.

It is interesting to note that for curves with a rapid rate of change, the cumulative area under the curve (AUC) increases more quickly. Indeed, differentiation is the reverse process to integration—a mathematical way of finding the AUC. The details of how integration is performed are beyond the scope of this article. However, the AUC of certain relationships is important. Examples include blood temperature against time in a thermodilution experiment, where the AUC is reciprocally related to cardiac output; the AUC over a single pulse of the invasive arterial pressure/time curve is related, in a complex way, to the stroke volume.

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Conflict of interest

None declared.

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References

↵ Mapelson WW, Horton JN, Ng WS, Imrie DD. The response pattern of polarographic oxygen electrodes and its influence on linearity and hysteresis. Med Biol Eng 1970;8:585-93.
↵ Hill AV. The combinations of haemoglobin with oxygen and with carbon monoxide. I. J Physiol 1910;40:iv-vii.
↵ Hlastala MP, Berger AJ. Physiology of Respiration. 2nd Edn. Oxford: Oxford University Press; 2001.
↵ Peck TE, Hill S. Pharmacology for Anaesthesia and Intensive Care. 3rd Edn. Cambridge: Cambridge University Press; 2008.
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Contin Educ Anaesth Crit Care Pain (2011) 11 (2): 50-55.
doi: 10.1093/bjaceaccp/mkq059
First published online: January 21, 2011
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Introduction to functions
Transformations and inverse functions
Linear functions
Hyperbolic functions
Exponential functions
Logarithmic functions
Conflict of interest
References
Contin Educ Anaesth Crit Care Pain (2011) 11 (2): 50-55.
doi: 10.1093/bjaceaccp/mkq059
First published online: January 21, 2011
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Introduction to functions
Transformations and inverse functions
Linear functions
Hyperbolic functions
Exponential functions
Logarithmic functions
Conflict of interest
References
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