Key points

• Non-linear systems are not amenable to investigation by reductionist methods.

• Complexity theory offers an alternative approach to quantifying the degree of physiological derangement in multi-system disorders such as sepsis.

• The normal, healthy human heart rate displays fractal variation which is lost in numerous disease states.

• Statistical techniques, such as approximate entropy, allow us to quantify the degree to which this variation is lost.

• Large, multi-faceted organizations such as the NHS frequently behave as complex systems and as such may benefit from alternative management strategies, informed by complexity theory.

Medicine, like many other scientific fields, is founded upon the classical Cartesian method of reductionism, where a problem is broken down into its smallest components, examined, and then the information gleaned used to draw conclusions about the nature of the larger reality. Fundamental to this approach is the requirement that the problem being examined is a linear system (Table 1). When this is the case, the reductionist approach is a great success and the clinician may rightly feel confident in predicting the outcome of an intervention. An example of this might be the response of blood glucose to a dose of exogenous insulin.2 Frustrations arise however when the problem we wish to examine is not a simple linear system but rather shows non-linear behaviour (Table 1). Our inability to predict the outcome in these situations is all too painfully familiar, yet it was at this problematic interface, between reductionism and real life, that the science of complexity theory was born.

Table 1

Features of linear and non-linear systems

Linear system Non-linear system
Output is proportional to input Output disproportionate to input
Output is reproducible over time for a given input Output for the same input value may not be constant over time,1 or be reproducible
Events occur sequentially Events occur both sequentially and simultaneously
Each variable within a linear system acts independently of another Each component of the system influences the other, i.e. shows interdependence
Example: Response of blood sugar to insulin Example: Human immune response to a pathogen
Linear system Non-linear system
Output is proportional to input Output disproportionate to input
Output is reproducible over time for a given input Output for the same input value may not be constant over time,1 or be reproducible
Events occur sequentially Events occur both sequentially and simultaneously
Each variable within a linear system acts independently of another Each component of the system influences the other, i.e. shows interdependence
Example: Response of blood sugar to insulin Example: Human immune response to a pathogen

Edward Lorenz was a meteorologist at the Massachusetts Institute of Technology who in 1961 was trying to develop a computer model to allow accurate long-range weather forecasting. While inputting the data to rerun a previous weather model, he abbreviated one number from 0.50612 to 0.506 to save time. When he returned later that day to examine the ‘weather’ pattern generated by the computer, to his surprise, it was markedly different from the previous model he was trying to recreate. Rather than ignore it, he had the presence of mind to publish his findings that non-linear systems appeared to display what he later coined ‘sensitive dependence on initial conditions’.1 This has since been more poetically paraphrased by the expression that if ‘butterfly flaps its wings in Brazil will it set off a tornado in Texas?’ This was the first step towards identifying the characteristics that all complex systems share (Table 2) regardless of whether the system in question is biological, meteorological, or social. Furthermore, a certain number of these characteristics may present us with novel opportunities to better understand and even predict the outcome in conditions characterized by non-linear behaviour.

Table 2

Common features of complex (non-linear) systems

Feature Definition
Sensitive dependence on initial conditions Tiny, even imperceptible, factors at the onset of the event can have a disproportionately large effect upon the ultimate endpoint
Mathematical simplicity (determinism) Paradoxically even the most complex systems are governed and can therefore be explained by relatively simple mathematical equations or ‘rules’
Strange attractors A mathematical concept whereby non-linear systems tend to settle at a finite number of non-repeating fixed points in phase space. It is the main difference between a difficult to predict complex system and a truly unpredictable random system. Also underlies the principle of ‘variability as health’ whereby a system will oscillate between several stable states over a period of time
Fractal variation Complex systems tend to display a striking degree of self-similarity at different scales of length. They can be geometrical (i.e. the human vascular tree) or temporal (i.e. heart rate variability)
Connectivity and synchronization Each component of a complex system responds to and influences each other. This can allow information to spread more rapidly through such a system, i.e. a murmuration of starlings or a nervous system
Emergent order This is the tendency of complex systems to adopt macroscopically ordered structures despite being composed microscopically of seemingly disordered components and is due to the properties described above. This is also the feature most likely to be missed if a reductionist approach is taken to investigate a complex system
Negative entropy Complex systems are highly ordered with low entropy and as such stand opposed to the second law of thermodynamics which states that the entropy in a system will trend towards maximum, i.e. a system is always trying to move from a highly ordered state to a less ordered state. The implication here is that a huge amount of energy is required to maintain a system in a state of low or negative entropy. Biologically speaking, health or life in general could be thought of as a state of negative entropy
Decomplexification This may be simply thought of as the biological manifestation of entropy in progress. It occurs when a biological system moves from a highly ordered state (health) towards a less ordered state (disease or ultimately death)
Feature Definition
Sensitive dependence on initial conditions Tiny, even imperceptible, factors at the onset of the event can have a disproportionately large effect upon the ultimate endpoint
Mathematical simplicity (determinism) Paradoxically even the most complex systems are governed and can therefore be explained by relatively simple mathematical equations or ‘rules’
Strange attractors A mathematical concept whereby non-linear systems tend to settle at a finite number of non-repeating fixed points in phase space. It is the main difference between a difficult to predict complex system and a truly unpredictable random system. Also underlies the principle of ‘variability as health’ whereby a system will oscillate between several stable states over a period of time
Fractal variation Complex systems tend to display a striking degree of self-similarity at different scales of length. They can be geometrical (i.e. the human vascular tree) or temporal (i.e. heart rate variability)
Connectivity and synchronization Each component of a complex system responds to and influences each other. This can allow information to spread more rapidly through such a system, i.e. a murmuration of starlings or a nervous system
Emergent order This is the tendency of complex systems to adopt macroscopically ordered structures despite being composed microscopically of seemingly disordered components and is due to the properties described above. This is also the feature most likely to be missed if a reductionist approach is taken to investigate a complex system
Negative entropy Complex systems are highly ordered with low entropy and as such stand opposed to the second law of thermodynamics which states that the entropy in a system will trend towards maximum, i.e. a system is always trying to move from a highly ordered state to a less ordered state. The implication here is that a huge amount of energy is required to maintain a system in a state of low or negative entropy. Biologically speaking, health or life in general could be thought of as a state of negative entropy
Decomplexification This may be simply thought of as the biological manifestation of entropy in progress. It occurs when a biological system moves from a highly ordered state (health) towards a less ordered state (disease or ultimately death)

## Examples of complex systems in clinical medicine

Examples of non-linear or complex systems abound in medicine and perhaps the most relevant to our speciality is that of the multiple organ dysfunction syndrome (MODS). This is a clinical syndrome precipitated by any number of insults (i.e. infection, trauma, burns), whose response is orchestrated by several components (immune, humoral, neurological, and inflammatory) and whose outcome remains difficult to predict at the outset. When viewed in this light, MODS does indeed display features consistent with a complex system in that we have sensitive dependence on initial conditions, interdependent components, and an outcome that is not necessarily proportional to the original insult. It has also been suggested that this may be an underlying reason why multiple trials of pharmacological agents targeting individual inflammatory mediators have failed to improve mortality3 as we are adopting a linear approach to solve a non-linear problem. However, it would be a mistake to view disease as a complex state while viewing normal homeostasis as an ordered linear state.

Much work has already been done to show that health itself (i.e. normal homeostasis) can also be thought of as a complex system characterized by a high degree of biological variability, negative entropy, and emergent order.4,5 It is this variability that provides us with the resilience to withstand physiological insults. The loss of biological variability, with an attendant increase in entropy, is characteristic of several disease states, including MODS and even normal ageing.6 Therefore, although MODS represents an example of a complex system, it may also be viewed as a higher entropy state than health and therefore as a step on the decomplexification pathway that may lead ultimately to death (Table 2). The challenge exists therefore to identify decomplexification before it becomes irreversible.

## Identifying loss of complexity

As already mentioned in Table 2, one hallmark of a ‘healthy’ complex system is the degree of connectivity between individual components. This connectivity manifests itself by encouraging various physiological parameters to oscillate between a number of steady states. The physiological parameter in question will then display variability which rather than being purely random will be seen to display self-similarity over time or in other words show temporal fractal variation. One such physiological parameter that displays this is the human heart rate. In a healthy individual, the inter-beat variation, as measured by the R–R interval, fluctuates in a stable manner that is hypothesized to be due to the influence of biological ‘strange attractors’ (Table 2).4 In plain English, this means that the heart receives information from other organs, including the sympathetic nervous system and various endocrine glands while also being affected by changes in breathing and posture; an ability to respond to these (and therefore show variability) is clearly a requirement for normal homeostasis.

Several studies have now demonstrated a correlation between loss of heart rate variability (HRV) and various disease states including congestive cardiac failure and MODS, the same pattern is also seen in normal human ageing3–7 (see Fig. 1 where HRV on the y-axis changes over time on the x-axis). Here, we can clearly see decomplexification in action; there is a marked reduction in the degree of fractal variation present which implies a loss of connectivity and an increase in the entropy of the system. HRV therefore provides us with a window into complex systems and may represent a novel means of monitoring and identifying critically ill patients; of particular note is the fact that these changes appear to precede the clinical manifestation of disease by several hours.

Fig 1

Examples of changes in heart rate variability with ageing and disease. The middle tracing is from a healthy young patient. The bottom tracing is from a healthy but aged patient. The top tracing was obtained from a critically ill patient. The image is reproduced with permission from Wolters Kluwer Health Inc.

Fig 1

Examples of changes in heart rate variability with ageing and disease. The middle tracing is from a healthy young patient. The bottom tracing is from a healthy but aged patient. The top tracing was obtained from a critically ill patient. The image is reproduced with permission from Wolters Kluwer Health Inc.

Various methods exist to analyse HRV (Table 3), all of which at present are experimental8 but of particular interest is the statistical technique of approximate entropy (ApEn). This tool allows us to measure the degree of variability within a data set obtained over a given period of time. A low ApEn correlates to a high rate of variability (i.e. normal homeostasis), while a high ApEn represents randomness and therefore decomplexification. Furthermore, a derivative of this method, Cross-ApEn, allows us to quantify the degree of connectivity between two separate data sets, potentially giving an even more accurate insight into the degree of connectivity within the system as a whole.4 A further advantage of the ApEn method is that it can be performed on a relatively small data set (100–900 data points).

Table 3

Techniques to quantify HRV

Technique Explanation
Time domain analysis Relatively simple technique whereby the standard deviation of a collection of R–R intervals is used to generate a measure of variability (as standard deviation is equal to the square root of variance from the mean).
Highly sensitive to artifact
Frequency domain analysis A time domain analysis can be converted into a frequency domain analysis (FDA) via a Fourier transform. FDA may then be used to identify the relative contributions of different systems to HRV (i.e. sympathetic, parasympathetic and humoral)
Subject to the same limitations as time domain analysis
Entropy analysis Statistical technique whereby the degree of connectivity between data sets is quantified
Principle advantage over other methods is its ability to be applied to shorter, ‘noisier’ data sets
Technique Explanation
Time domain analysis Relatively simple technique whereby the standard deviation of a collection of R–R intervals is used to generate a measure of variability (as standard deviation is equal to the square root of variance from the mean).
Highly sensitive to artifact
Frequency domain analysis A time domain analysis can be converted into a frequency domain analysis (FDA) via a Fourier transform. FDA may then be used to identify the relative contributions of different systems to HRV (i.e. sympathetic, parasympathetic and humoral)
Subject to the same limitations as time domain analysis
Entropy analysis Statistical technique whereby the degree of connectivity between data sets is quantified
Principle advantage over other methods is its ability to be applied to shorter, ‘noisier’ data sets

HRV monitoring has already demonstrated its utility as a means of predicting patients at risk of developing postoperative cardiac failure or increased mortality post-myocardial infarction;3 however, the challenge now remains to refine the technique so as to be amenable to real-time bedside monitoring. In this role, HRV analysis could potentially represent a highly individualized, patient-specific, technique to detect and monitor the progress of MODS, allowing clinicians to escalate and de-escalate treatment regimens with greater accuracy.

## Implications of complexity theory in healthcare management

A full description of the myriad ways in which complexity theory has influenced thinking on organizational behaviour is beyond the remit of this article; however, the reader could do a lot worse than start with an excellent series of articles published in 2001 in the British Medical Journal edited by Plsek and Greenhalgh.9,10 Just as reductionism has influenced generations of scientific investigation by encouraging participants to adopt the classical Newtonian viewpoint of the ‘clockwork universe’, so too have traditional management models tended to view organizations as machines, composed of multiple individual components each of which can be ‘fine-tuned’ separately to improve performance within the organization as a whole. In this model, effective leadership involves introducing best practice via a top-down approach and any resistance to change is frequently viewed as symptomatic of a poorly working machine. Likewise, any variation in practice from a predetermined norm can be eliminated via the imposition of protocols and guidelines which can be fitted into the machine rather like new spark plugs on a car. When a true consensus exists as to what constitutes best practice (i.e. antibiotics before knife to skin), then such an approach may be perfectly valid; however, more frequently genuine uncertainty may exist as to how to meet a particular challenge (i.e. how to improve operating theatre efficiency). It is in these situations that adopting an alternative viewpoint, informed by complexity theory, may allow more productive solutions to emerge.

By viewing an organization as a complex system, then a greater focus is placed upon the connections between individual components, how their interactions may lead to the emergence of novel, unpredictable outcomes, and the understanding that human behaviour shows ‘attractor patterns’10 whereby individuals will tend to default to a certain set of attitudes which can be misconstrued as resistance to change. In these situations, effective leadership would involve placing a greater emphasis upon finding out how different services interact and influence each other and engaging with staff to introduce new ‘attractors’ to influence behaviour and attitudes. More importantly is the understanding that meaningful change may be more likely to occur if it is allowed to emerge spontaneously from the interactions of the individual services involved; rather than imposed from the top down. Clearly, this has to be guided and this may achieved by the use of ‘minimum specifications’.10 This is a management strategy where the emphasis is placed upon ‘direction pointing’, setting ‘boundaries’ and ‘resources’, and then giving ‘permission’ for the system to generate its own solutions. In addition to this, there must be effective feedback mechanisms in place to allow solutions to be shared throughout the organization.

As an example, one might consider how the best way to reduce anaesthetic drug errors within a theatre complex might be achieved. An organization could issue top-down warnings to promote vigilance and create a sense of heightened awareness or it could engage with theatre teams to discover if there are any recurrent themes (or attractors) in these errors, suggest options for how to minimize these in the future (direction pointing) but also encourage staff to take ownership of the problem themselves to ideally find their own solutions (permission). Any solutions generated would then be shared with the organization as a whole.

## Conclusion

The science of complexity theory has already led to advancements in other fields as diverse as mathematics, economics, meteorology, and ecology, yet its potential utility for clinical medicine and healthcare organization remains to be fully explored. However, it represents an exciting avenue for future research in the hope of yielding novel solutions to age old problems.

None declared.

## MCQs

The associated MCQs (to support CME/CPD activity) can be accessed at https://access.oxfordjournals.org by subscribers to BJA Education.

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