There is a wide variability in the response of different pa-tients to identical doses of the same drug. Drugs do not always produce the same effect in all subjects and may not even produce identical responses when given to the same patient on different occasions. Consequently, dose–
response curves obtained in man may only be directly applicable to individual subjects. Nevertheless, they are often used to illustrate the results obtained in a large sam-ple derived from a relatively homogeneous population of subjects. In these conditions, a sigmoid log dose–response curve can be used to express the results, although it of-ten disguises the high degree of interindividual variability (Fig. 5.1).
Variability in the response to drugs can be considered as part of the general phenomenon of inherent or intrin-sic biological variability. Both physiological responses and pharmacological phenomena are subject to considerable interindividual variability. Although this variability can be expressed by the enumeration of individual results, this is usually unnecessary and impractical, since a large num-ber of observations may be involved. More commonly, the distribution and variability of individual measure-ments (variables) is expressed in terms of two ‘descriptive statistics’:
These two statistics reflect
(1) The central tendency or midpoint of the data, ex-pressed by the
rMode (the most commonly occurring value)
rMedian (the central or middle value that divides the results into two equal groups)
rMean (the arithmetic average)
(2) The variability or scatter of individual values, ex-pressed by the
rFrequency (how often each observation occurs)
rInterquartile range (the middle 50% of values)
rStandard deviation (the root mean square deviate) The choice of the parameter or statistic that is used to express the central tendency and the scatter of a series of individual results depends on the ‘level of
measurement’ of the data. In general, there are three
‘levels of measurement’ of biological data:
r Nominal
r Ordinal
r Continuous
Nominal measurements
Nominal or categorical measurements depend on the clas-sification of data or subjects into groups that are solely de-pendent on their names or characteristics. Thus, patients can be classified as male or female, premedicated or un-premedicated, conscious or unconscious etc. In general, quantal (‘all-or-none’) measurements are also nominal.
All measurements within a group are equivalent to each other, and there are no quantitative differences between individual members.
The central tendency of nominal measurements is usu-ally expressed as the most frequently occurring value, or the mode; thus, in the numerical series, 1, 2, 3, 3, 5, 5, 5, 6, 7, 9, 9, the mode is 5. The distribution of nominal measurements is expressed by the frequency of their oc-currence (Table 5.1).
Nominal measurements can be considered as the lowest level of description of data and are only used when mea-surements are obtained by unsophisticated techniques.
Ordinal measurements
Ordinal measurements (ranked data) depend on the rank-ing of the underlyrank-ing data into a sequence or order that de-pends on their magnitude or relationship to each other. In-dividual data can usually be graded as lower (<) or greater (>) than other members of the series, although they cannot be given a precise quantitative value. At their best, ordinal or ranked data can be regarded as semi-quantitative. For instance, pain scores on a visual analogue scale (Fig. 5.2)
86 Principles and Practice of Pharmacology for Anaesthetists, Fifth Edition T.N. Calvey and N.E. Williams
© 2008 Norman Calvey and Norton Williams. ISBN: 978-1-405-15727-8
Variability in Drug Response 87
0 20 40 60 80 100
1 10 100
Variability in dose
Variability in response
Dose (mg)
Response (% of maximum)
Fig. 5.1 Log dose–response curve showing the variability in dosage required to produce the same response (↔↔), and the variability in response produced by a single dose () in a population of subjects.
No pain Worst pain
imaginable 0 1 2 3 4 5 6 7 8 9 10
Fig. 5.2 A commonly used visual analogue scale for pain. The scale is a horizontal line which is 10 cm long. The origin of the line represents a score of 0 (i.e., no pain), while the end of the line corresponds to a score of 10 (the worst pain imaginable).
The patient is asked to indicate the point on the line that corresponds best to their pain.
and Apgar scores are ordinal measurements, since they can be placed in an order in which their relationship to each other is defined (although their quantitative value is imprecise).
The central tendency of ordinal measurements is ex-pressed as the median, or the middle value in a range of numbers (Table 5.1). Thus, in 11 patients with post-operative pain scores of 1, 2, 3, 3, 5, 5, 5, 6, 7, 9, 9, the
Table 5.1 Descriptive statistics used to describe the central tendency and variability of nominal, ordinal and continuous data.
Descriptive statistic
Level of Central
measurement tendency Variability
Nominal Mode Frequency
Ordinal Median Interquartile range
Continuous Mean Standard deviation
median score is 5. The median is sometimes defined as the 50th percentile, since it divides the numerical obser-vations or results into two equal halves, i.e. 50% of the observations are below the median while 50% are above.
The distribution or variability of ordinal measurements can be expressed as the range (1–9), or more correctly as the interquartile range (3–7), which corresponds to the 25th–75th percentiles of the ranked values.
Pain scores are not accurate quantitative data, since their magnitude merely implies that a higher value represents more intense pain than a lower one. It is therefore incorrect to express pain scores as a mean and standard deviation (or standard error), since this implies that the scores have a defined numerical value. Ordinal measurements can be considered as an intermediate level of measurement for the expression and description of data.
Continuous measurements
Continuously variable measurements are quantitative data, and there is a precisely defined and measurable differ-ence between the individual observations or values. Mea-surements of drug concentrations and responses, heart rate and blood pressure are continuously variable data, since they have a defined numerical value, and equal dif-ferences between individual values are comparable to each other. Continuously variable measurements are the high-est level of description that can be used for biological and clinical data.
The central tendency of continuous measurements is expressed as the arithmetic mean or average (x):
mean (x)= x (the sum of their individual values) n (the number of results or observations) The scatter of the individual observations about the mean (i.e. their distribution or variability) is usually de-scribed by the standard deviation (Table 5.1).
Standard deviation
The standard deviation (‘root mean square deviate’) is a commonly used indicator of scatter that is used for con-tinuously variable data and is derived from the sum of the squares of the numerical differences (‘deviate’) between each individual value and the mean (i.e.[x1− x]2+ [x2− x]2+ [x3− x]2etc.). The ‘mean deviate’ from the mean is given by
mean deviate =
(x− x) n
However, the numerator of this equation ((x − x)) will clearly be zero when the (+) and (−) signs of individ-ual results are taken into account, since the mean is defined as the value at which the summated positive and negative deviations are equal. This problem can be eliminated by squaring the deviations about the mean before their sum-mation, and subsequently finding their mean value: this results in an expression for the mean square deviate (more commonly known as the variance):
variance =
(x− x)2 n
The original units for the deviation about the mean can be restored by taking the square root of the variance, re-sulting in an expression for the root mean square deviate or standard deviation:
standard deviation =
(x− x)2 n
This expression can be used to estimate the standard deviation when n is>30. However, the standard devia-tion in continuously variable data is more usually based on a relatively small sample (i.e.< 30 values), which is only a small part of a much larger, unsampled population of results. In these conditions, it can be shown that a slight modification (Bessel’s correction) gives a rather better es-timate of the population standard deviation, using the expression:
standard deviation =
(x− x)2 n− 1
In practice, it is unnecessary to calculate each deviation from the mean (i.e. [x1− x], [x2− x], [x3− x] etc.), and to square and summate them, since it can be shown that
(x − x)2= x2− (x)2/n. Consequently, the stan-dard deviation of samples when n< 30 can be defined as
standard deviation =
x2− x 2
/n n− 1
Most electronic calculators that are designed for sta-tistical use can automatically calculate the mean and the standard deviation, after the individual data values have been entered.
Variance
The variance (‘mean square deviate’) is the square of the standard deviation. It is defined (when n< 30) by the expression:
variance =
x2− x 2
/n n− 1
The denominator of the variance (n− 1) defines the number of ‘degrees of freedom’. It is one less than the number of observations or results, because only (n− 1) results are independent from each other, and the value of the nth result is determined by the values of the remainder.
Coefficient of variation
The variability or scatter of different observations can-not be compared by their standard deviations when the means are very dissimilar, or when they are calculated in different units. For example, in SI units, the weight of a group of 36 patients was 68.0 ± 20.4 kg (mean ± SD).
When expressed in Imperial units, their weight was 150
± 45 lb. The variability of the measurements must be identical, although their standard deviations are differ-ent. Clearly, the standard deviations of different groups cannot be compared unless their units of measurement are identical. This problem can be avoided by the use of the coefficient of variation, which expresses the standard deviation as a percentage of the mean:
coefficient of variation (%) =standard deviation
mean × 100
Since the standard deviation and the mean are measured in the same units, the coefficient of variation is independent of the units of measurement. In the above example, the coefficient of variation of body weight was 30%, whether measured in kg or lb.
Standard error of the mean
The mean (x) and standard deviation (SD) of a small sam-ple (e.g. n< 30) from a large population (e.g. n > 1000) are often used to provide an estimate of the population mean () and the population standard deviation ().
Clearly, different samples of the population may provide different estimates of and . The accuracy of the sam-ple mean (x) in comparison with the population mean () can be estimated by the standard error of the mean (SEM):
standard error of the mean = SD
√n.
Clearly, the larger the size of the sample, the more accurate is the sample mean as an estimate of the population mean, and the smaller is the calculated value for the SEM.
Variability in Drug Response 89
Confidence limits
The SEM can also be used to predict the range within which the population mean will lie.
When n> 30, there is a 95% chance that the true pop-ulation mean lies within 2 standard errors of the sample mean x. In the example previously discussed, a group of 36 patients had a body weight of 68.0 ± 20.4 kg (mean ± SD); consequently:
mean ± SEM = 68.0 ± 20.4
√36kg= 68.0 ± 3.4 kg There is a probability of 95% that the true mean of the population lies within the range: mean± 2 SEM (in the example above, this range is 61.2–74.8 kg). Approximately 95% of the sample means obtained by repeated sampling of the entire population will also lie within this range.
Thus, the range from [mean− (2 × SEM)] to [mean + (2× SEM)] represents the interval within which the true population mean is likely to lie, and is called the 95% confi-dence limits of the mean. By these methods, a small sample can be used to make inferences and predictions about the parent population from which the sample was obtained.
The SEM is sometimes incorrectly used to indicate the variability of individual values in the sample (possibly be-cause it is always less than the standard deviation from which it is derived). However, the variability or scatter of individual values in a population sample should always be described by the standard deviation.
Frequency distribution curves
The variability of individual observations in a popula-tion sample can also be expressed as a histogram or a frequency distribution curve. For example, the variabil-ity in the dosage of a drug required to produce a specific pharmacological effect in 100 subjects can be expressed in this way (Fig. 5.3). Each shaded rectangle represents the additional number of patients responding to each 10 mg increment in dosage (e.g. 50–60 mg, 60–70 mg, 70–
80 mg etc.). A frequency distribution curve can then be superimposed on the histogram and used to determine the probability of obtaining a response to a given dose in one subject. The data can also be plotted as a cumulative frequency distribution curve (Ogive) by calculating the proportion of patients that respond to a given dose, or to a dose below it (Fig. 5.4).
The normal distribution
The frequency distribution curve, shown in Fig. 5.3, is roughly symmetrical and bell-shaped and is an example
0 5 10 15 20 25 30
40 60 80 100 120 140 160 180 200
Dose (mg)
Number responding
Fig. 5.3 Histogram showing the variability in dosage required to produce a specific effect. Each rectangle represents the number of patients that respond to each 10 mg increment in dosage. A frequency distribution curve ( ) has been imposed on the histogram, showing that the data approximately corresponds to a normal or Gaussian distribution.
0 20 40 60 80 100 120 140 160 180 200
40 60 80 100 120 140 160 180 200
Dose (mg) Number responding
Fig. 5.4 A cumulative frequency distribution curve ( ) or Ogive representing the summated results of Fig. 5.3.
of a common type of frequency distribution known as a normal or Gaussian1distribution. In many instances, the central tendency and variability of biological or clinical data is consistent with a normal distribution. The central tendency of a normal distribution is equal to the mean value, while the variability is expressed by the standard deviation (Fig. 5.5). Clearly, the shape and form of the Gaussian curve are not critical but are determined by the variability of the underlying data in relation to the mean.
The curve is high and narrow when the coefficient of
1Johann Karl Friedrich Gauss (1777–1855).
0 1 2 3 4 5 6 1
2 3 4 5 6 7 8 9 10
Number of observations (%)
3 SD 2 SD 1 SD 1 SD 2 SD 3 SD
Mean Median Mode
Fig. 5.5 A symmetrical frequency distribution curve in a large population which is consistent with a normal distribution. The mean, the median and the mode are all identical, and the variability is expressed by the standard deviation (SD). The mean± 1 SD includes 67% of the results, the mean ± 2 SD includes 95% of the results and the mean± 3 SD includes 99%
of the results.
variation is low, but small and wide when the coefficient of variation is high.
In a normal distribution, the mean (average value), the median (the central value that divides the sample into two equally sized groups) and the mode (the most commonly occurring value) are identical (Fig. 5.5). In addition, 67%
of the values are within one standard deviation of the mean (i.e. mean± 1 SD); 95% of the values are within two standard deviations of the mean (i.e. mean± 2 SD);
and 99% of the values are within three standard deviations of the mean (i.e. mean± 3 SD). These statistics will also apply to a sample from a normally distributed population, as long as the sample size is sufficiently large (n > 30).
Skewed distributions
Biological or clinical observations may be positively or negatively skewed (Fig. 5.6). In a skewed distribution, the mean, median and mode are all different, and the me-dian may be a better indication of the central tendency of the results. Similarly, the standard deviation may not be a reliable estimate of the variability of the underlying population.
0 2 4 6 8 10 12 14 16 18 20
56 58 60 62 64 66 68 70 72 74 76 78 80
Number of men
Age on retirement (years) Median Mean Mode
Fig. 5.6 A positively skewed frequency distribution curve showing the retirement age of a male population. In a positively skewed distribution, the mean is usually greater than the median or the mode.
Positively skewed data can usually be manipulated to conform to a normal distribution by a logarithmic trans-formation. In these conditions, parametric statistical tests designed for normally distributed data can be used on the transformed observations.
Tests of statistical significance
In general, tests of statistical significance are of two main types:
rParametric tests
rNon-parametric tests
Parametric statistical tests
Most parametric statistical tests rely on the assumption that the variability of the two or more groups of results that are analysed are consistent with a normal distribution.
Many of these tests are quite ‘robust’ and can be used for continuously variable data whose distribution is not grossly abnormal. They should not be used with ordinal or ranked data. Parametric statistical tests usually depend on the comparison of the mean values of two or more samples in relation to their variability. The most commonly used parametric tests (Table 5.2) are
rStudent’s t test (for paired and unpaired data) rAnalysis of variance (ANOVA, for paired and unpaired data)
Variability in Drug Response 91
Table 5.2 Commonly used tests of statistical significance.
Parametric tests Non-parametric tests Student’s t test∗ Mann–Whitney test Analysis of variance (ANOVA)∗ Wilcoxon signed rank test
Wilcoxon ranked sum test Kruskal–Wallis test Friedman test
∗These tests can be used for both paired and unpaired data.
Non-parametric statistical tests
Non-parametric tests (rank tests) do not make any as-sumptions about the distribution of the data or the popu-lation from which it is derived. They are most commonly used for comparing ordinal or ranked data (e.g. Apgar scores, pain scores), in which individual results or obser-vations can be assigned a rank and arranged in order in relation to each other. Although they can also be used with parametric data, quantitative differences between their numerical values will be largely ignored or obscured.
Non-parametric tests of statistical significance include (Table 5.2):
r Mann–Whitney test
r Wilcoxon signed rank test
r Wilcoxon rank sum test
r Kruskal–Wallis test
r Friedman test
Hypothesis testing
Tests of statistical significance are widely used in hypothe-sis testing, in order to predict whether there are real differ-ences between different groups or treatments, or whether these differences could have occurred by chance.
In general, there are three main steps in hypothesis test-ing:
(1) Form a null hypothesis, which assumes that there are no real differences observed between the groups or treat-ments, i.e. any differences observed are due to random variability.
(2) Calculate the probability that any differences observed are due to chance by the use of an appropriate statistical test.
(3) If the probability is more than 5% (P > 0.05) accept the null hypothesis and conclude that any differences be-tween the groups are due to chance variations. On the other hand, if the probability is less than 5% (P < 0.05)
reject the null hypothesis and conclude that the differences between the groups are unlikely to be due to chance. In a small proportion of cases, the conclusions reached by hypothesis testing may be incorrect, resulting in Type I or Type II errors. In a Type I error, P is< 0.05, although there is no real difference between the populations or groups.
In a Type II error, P is> 0.05, despite the presence of a real difference between the groups.
Physiological and social factors that affect the response to drugs
A number of physiological and social factors may affect the response to drugs (Table 5.3), including:
rAge
rPregnancy
rTobacco
rAlcohol
Childhood Drug absorption
In the neonatal period, absorption of drugs is slower than in children or adults due to a longer gastric emptying time and an increase in intestinal transit time. Neverthe-less, oral drugs may be more extensively absorbed due to their greater contact time with the intestinal mucosa.
The gastric contents are less acidic, and some drugs such as benzylpenicillin and ampicillin will have greater overall oral absorption. Vasomotor instability observed in neona-tal life may result in the unreliable absorption from tissue sites after subcutaneous or intramuscular administration.
Table 5.3 The principal causes of variability in drug responses.
Physiological
and social Pathological conditions Other causes
Age Liver disease Idiosyncrasy
Pregnancy Renal disease Hypersensitivity Tobacco Respiratory diseases Supersensitivity Ethyl alcohol Cardiac diseases Tachyphylaxis
Neurological diseases Tolerance Endocrine diseases
Drug distribution
The distribution of drugs is influenced by several factors including tissue mass, fat content, blood flow, membrane permeability and protein binding. Total body water as a percentage of body weight falls from 87% in the preterm baby to 73% at 3 months, and subsequently decreases to 55% in adult patients. Consequently, doses of water-soluble drugs that are calculated by scaling down adult doses in proportion to body weight can result in lower tissue concentrations in infants and neonates. However, drug distribution is also affected by the lower body fat content and by the increased permeability of the blood–
brain barrier in the neonate, and lipid-soluble drugs may be relatively concentrated in the CNS. In addition, the decrease in plasma protein levels in the neonate leads to the increased availability of unbound drug, resulting in enhanced pharmacological activity and drug metabolism.
Any change in plasma pH during neonatal life may in-fluence the degree of drug ionization and thus affect the membrane permeability of both acidic and basic drugs.
Drug metabolism and renal elimination
The rate of drug metabolism depends on both the size of the liver and the activity of microsomal enzyme systems.
In neonatal life, hepatic enzyme activity, particularly glu-curonide conjugation, is initially immature and may not assume the adult pattern for several months. In older chil-dren, enzyme activity is similar to adults, although most drugs are metabolized at faster rates due to the relatively greater liver volume.
In the neonatal period, glomerular filtration rates are 20–40% of those in adults, and drugs removed from the body by this means (e.g. digoxin, aminoglycoside antibi-otics) are eliminated relatively slowly. Glomerular filtra-tion rates comparable to those in adults occur at about 4 months of age.
Practical implications
These physiological and kinetic differences have signifi-cant practical implications. In the neonate, dose regimes for water-soluble drugs should be related to surface area rather than body weight, in order to produce similar blood levels to those in adults. However, the increased volume of distribution and the decreased renal clearance results in a longer elimination half-life, and dose intervals should therefore be prolonged. In addition, the effects of drugs on the neonatal CNS may be enhanced after administration in labour (e.g. morphine, diazepam). This phenomenon may
be due to the increased fraction of non-protein-bound drug, delayed hepatic metabolism, the presence of active metabolites or greater permeability of the CNS due to im-maturity of the blood–brain barrier.
For many years, it was considered that neonates were relatively resistant to suxamethonium but highly sensitive to non-depolarizing agents, particularly during the first 10 days of life. More recent studies with atracurium in-fusions indicate that dose requirements in the neonate in proportion to body weight do not differ greatly from those at other ages. Any variations in dose requirement may be related to lower body temperatures in the newborn, and the subsequent effect on drug distribution.
In older children, protein binding, hepatic microsomal enzyme activity, renal function and the permeability of the blood–brain barrier are similar to adults, and so dif-ferences in drug disposition are less likely. Nevertheless, drug dosage is best expressed in terms of surface area due to the proportional increase in body water during child-hood. More frequent rates of administration, especially of less polar compounds, may be necessary due to the relative increase in liver blood flow. Differences in pharmacody-namic activity between children and adults may also be present, but are not easy to assess.
Maturity and old age
Elderly patients often respond differently to standard adult doses of drugs and are more likely to react adversely to drugs prescribed in hospital. Compliance with drug ther-apy may be unreliable due to failing memory, confusion and poor eyesight.
Pharmacokinetic changes
In contrast to the neonate, there is a reduction in the pro-portion of total body water and a relative increase in body fat. Although drug absorption is not appreciably modified, the volume of distribution of polar drugs is reduced and their plasma and tissue concentrations are effectively in-creased. Since plasma albumin levels tend to fall with age, the unbound fraction of certain drugs (e.g. phenytoin, tolbutamide) increases, thus enhancing their availability to cells and tissues.
At the age of 65, hepatic blood flow may be 45% less than normal values in younger adults, and experimental evidence suggests that the activity of microsomal enzyme systems declines to a similar extent. Consequently, the sys-temic bioavailability of drugs that are subject to low or high hepatic clearance is increased, with enhancement of their pharmacological effects. Similarly, glomerular filtration