Cambridge University Press
9780521700443 - Physics, Pharmacology and Physiology for Anaesthetists - Key concepts for the FRCA - by Matthew E. Cross MB ChB MRCP FRCA, Emma V. E. Plunkett MBBS MA MRCP FRCA and Tom E. Peck MBBS BSc FRCA
Excerpt

Introduction

This book is aimed primarily at providing a reference point for the common graphs, definitions and equations that are part of the FRCA syllabus. In certain situations, for example the viva sections of the examinations, a clear structure to your answer will help you to appear more confident and ordered in your response. To enable you to do this, you should have a list of rules to hand which you can apply to any situation.

Graphs

Any graph should be constructed in a logical fashion. Often it is the best-known curves that candidates draw most poorly in their rush to put the relationship down on paper. The oxyhaemoglobin dissociation curve is a good example. In the rush to prove what they know about the subject as a whole, candidates often supply a poorly thought out sigmoid-type curve that passes through none of the traditional reference points when considered in more detail. Such an approach will not impress the examiner, despite a sound knowledge of the topic as a whole. Remembering the following order may help you to get off to a better start.

Size

It is important to draw a large diagram to avoid getting it cluttered. There will always be plenty of paper supplied so don’t be afraid to use it all. It will make the examiner’s job that much easier as well as yours.

Axes

Draw straight, perpendicular axes and label them with the name of the variable and its units before doing anything else. If common values are known for the particular variable then mark on a sensible range, for example 0–300 mmHg for blood pressure. Remember that logarithmic scales do not extend to zero as zero is an impossible result of a logarithmic function. In addition, if there are important reference points they should be marked both on the axis and where two variables intersect on the plot area, for example 75% saturation corresponding to 5.3 kPa for the venous point on the oxyhaemoglobin dissociation curve. Do all of this before considering a curve and do not be afraid to talk out loud as you do so – it avoids uncomfortable silences, focuses your thoughts and shows logic.

Beginning of a curve

Consider where a curve actually starts on the graph you are drawing. Does it begin at the origin or does it cross the y axis at some other point? If so, is there a specific value at which it crosses the y axis and why is that the case? Some curves do not come into contact with either axis, for example exponentials and some physiological autoregulation curves. If this is the case, then you should demonstrate this fact and be ready to explain why it is so. Consider what happens to the slope of a curve at its extremes. It is not uncommon for a curve to flatten out at high or low values, and you should indicate this if it is the case.

Middle section

The middle section of a curve may cross some important points as previously marked on the graph. Make sure that the curve does, in fact, cross these points rather than just come close to them or you lose the purpose of marking them on in the first place. Always try to think what the relationship between the two variables is. Is it a straight line, an exponential or otherwise and is your curve representing this accurately?

End of a curve

If the end of a curve crosses one of the axes then draw this on as accurately as possible. If it does not reach an axis then say so and consider what the curve will look like at this extreme.

Other points

Avoid the temptation to overly annotate your graphs but do mark on any important points or regions, for example segments representing zero and first-order kinetics on the Michaelis–Menten graph.

Definitions

When giving a definition, the aim is to accurately describe the principle in question in as few a words as possible. The neatness with which your definition appears will affect how well considered your answer as a whole comes across. Definitions may or may not include units.

Definitions containing units

Always think about what units, if any, are associated with the item you are trying to describe. For example, you know that the units for clearance are ml.min⁻¹ and so your definition must include a statement about both volume (ml) and time (min). When you are clear about what you are describing, it should be presented as succinctly as possible in a format such as

‘x’ is the volume of plasma …
‘y’ is the pressure found when …
‘z’ is the time taken for …
Clearance (ml.min⁻¹) is the volume (ml) of plasma from which a drug is completely removed per unit time (min)
Pressure (N.m⁻²) describes the result of a force (N) being applied over a given area (m²).

You can always finish your definition by offering the units to the examiner if you are sure of them.

Definitions without units

If there are no units involved, think about what process you are being asked to define. It may be a ratio, an effect, a phenomenon, etc.

Reynold’s number is a dimensionless number …
The blood:gas partition coefficient is the ratio of …
The second gas effect is the phenomenon by which …

Conditions

Think about any conditions that must apply. Are the measurements taken at standard temperature and pressure (STP) or at the prevailing temperature and pressure?

The triple point of water is the temperature at which all three phases are in equilibrium at 611.73 Pa. It occurs at 0.01 °C.

There is no need to mention a condition if it does not affect the calculation. For example, there is no need to mention ambient pressure when defining saturated vapour pressure (SVP) as only temperature will alter the SVP of a volatile.

Those definitions with clearly associated units will need to be given in a clear and specific way; those without units can often be ‘padded’ a little if you are not entirely sure.

Equations

Most equations need only be learned well enough to understand the components which make up the formula such as in

V = IR

where V is voltage, I is current and R is resistance.

There are, however, some equations that deserve a greater understanding of their derivation. These include,

The Bohr equation
The Shunt equation
The Henderson–Hasselbach equation

These equations are fully derived in this book with step by step explanations of the mathematics involved. It is unlikely that the result of your examination will hinge on whether or not you can successfully derive these equations from first principles, but a knowledge of how to do it will make things clearer in your own mind.

If you are asked to derive an equation, remember four things.

1. Don’t panic!
2. Write the end equation down first so that the examiners know you know it.
3. State the first principles, for example the Bohr equation considers a single tidal exhalation comprising both dead space and alveolar gas.
4. Attempt to derive the equation.

If you find yourself going blank or taking a wrong turn midway through then do not be afraid to tell the examiners that you cannot remember and would they mind moving on. No one will mark you down for this as you have already supplied them with the equation and the viva will move on in a different direction.

Section 1 · Mathematical principles

Mathematical relationships

Mathematical relationships tend not to be tested as stand-alone topics but an understanding of them will enable you to answer other topics withmore authority.

Linear relationships

y = x

y = ax + b

Hyperbolic relationships (y = k/x)

Asymptote

A curve that continually approaches a given line but does not meet it at any distance.

Parabolic relationships (y =kx²)

Exponential relationships and logarithms

Exponential

A condition where the rate of change of a variable at any point in time is proportional to the value of the variable at that time.
or
A function whereby the x variable becomes the exponent of the equation y = e^x.

We are normally used to x being represented in equations as the base unit (i.e. y = x²). In the exponential function, it becomes the exponent (y = e^x), which conveys some very particular properties.

Euler’s number

Represents the numerical value 2.71828 and is the base of natural logarithms. Represented by the symbol ‘e’.

Logarithms

The power (x) to which a base must be raised in order to produce the number given as for the equation x = log_base(number).

The base can be any number, common numbers are 10, 2 and e (2.71828). Log₁₀(100) is, therefore, the power to which 10 must be raised to produce the number 100; for 10² = 100, therefore, the answer is x = 2. Log₁₀ is usually written as log whereas log_e is usually written ln.

Rules of logarithms

Multiplication becomes addition

log(xy) = log(x)+log(y)

Division becomes subtraction

log(x/y) = log(x) − log(y)

Reciprocal becomes negative

log(1/x) = −log(x)

Power becomes multiplication

log(xⁿ) = n.log(x)

Any log of its own base is one

log₁₀(10) = 1 and ln(e) = 1

Any log of 1 is zero because n⁰ always equals 1

log₁₀(1) = 0 and ln(1) = 0

Basic positive exponential (y = e^x)

Clinical tear away positive exponential (y = a.e^kt)

Basic negative exponential (y = a^−x)

Physiological negative exponential (y = a.e^−kt)

Physiological build-up negative exponential (y = a − b.e^−kt)

Half life

The time taken for the value of an exponential function to decrease by half is the half life and is represented by the symbol t_1/2
or
the time equivalent of 0.693τ            τ = time constant

An exponential process is said to be complete after five half lives. At this point, 96.875% of the process has occurred.

Graphical representation of half life

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