## Abstract

The cerebral circulation shows both structural and functional complexity. For time scales of a few minutes or more, cerebral blood flow (CBF) and other cerebrovascular parameters can be shown to follow a random fractal point process. Some studies, but not all, have also concluded that CBF is non-stationary. System identification techniques have been able to explain a substantial fraction of the CBF variability by applying linear and nonlinear multivariate models with classical determinants of flow (arterial blood pressure, arterial CO_{2} and cerebrovascular resistance, CVR) as inputs. These findings raise the hypothesis that fractal behaviour is not inherent to CBF but might be simply transmitted from its determinants. If this is the case, future investigations could focus on the complexity of the residuals or the unexplained variance of CBF. In the low-frequency range (below 0.15 Hz), changes in CVR due to pressure and metabolic autoregulation represent an important contribution to CBF variability. A small body of work suggests that parameters describing cerebral autoregulation can also display complexity, presenting significant variability that might also be non-stationary. Fractal analysis, entropy and other nonlinear techniques have a role to play to shed light on the complexity of cerebral autoregulation.

## 1. Introduction

Several pathologies of significant incidence and considerable impact on quality of life, such as stroke, head trauma, carotid artery disease and subarachnoid haemorrhage, involve disturbances of cerebral blood flow (CBF) and its regulatory mechanisms. Understanding the determinants of human CBF variability in health and disease, however, is a major challenge, compounded by the difficulty of performing measurements within the skull. In the past, CBF was estimated with ^{133}Xe clearance or other indicator-dilution methods, which did not have the temporal resolution to allow a description of dynamic changes in CBF and other cerebrovascular parameters with each cardiac cycle (Panerai 1998). In the last 20 years, the introduction of transcranial Doppler (TCD) ultrasound, near-infrared spectroscopy (NIRS) and MRI has changed the landscape of research in this area. The ability to obtain reliable estimates of CBF with very high temporal resolution has shown that, in addition to the well-known structural complexity of the brain and its circulation, CBF and other cerebrovascular parameters are also regulated by extremely complex mechanisms that operate at molecular, cellular and supracellular levels (Baumbach & Heistad 1985; Faraci & Heistad 1998; Andresen *et al.* 2006; Haydon & Carmignoto 2006).

Hitherto, research into the dynamics and variability of CBF has been split between two different camps. On the one hand, different techniques of system identification, mathematical modelling or signal processing have been used to quantify the main determinants of CBF, such as cerebral perfusion pressure (CPP). On the other hand, CBF variability has been approached within the framework of ‘complexity theory’ by means of fractal analysis, chaos theory and nonlinear dynamics. Broadly, total blood flow in the brain should be a function of the perfusion pressure of the main supplying arteries divided by a lumped parameter that can be defined as *cerebrovascular resistance* (CVR). In most organs, the perfusion pressure would be the arteriovenous difference, but because intracranial pressure (ICP) is normally greater than cerebral venous pressure, CPP in the brain is usually expressed as the difference between arterial blood pressure (ABP) and ICP. A second important distinction between the brain and other organs is the presence of active autoregulatory mechanisms that tend to maintain CBF relatively constant, despite the changes in CPP (Paulson *et al.* 1990). The ability of cerebral arterioles to regulate their diameter to protect the brain against ischaemia, resulting from hypotension, or capillary damage and oedema, provoked by hypertension, can also regulate CBF to maintain a close match between oxygen supply and demand in the brain (Heistad & Kontos 1983). In the absence of these regulatory mechanisms controlling vasomotion, it would be reasonable to expect that CBF variability could be explained simply by changes in CPP and measurement noise. However, it is the presence of active autoregulation, adjusting CBF in response to external and internal changes in several parameters, such as ABP, ICP, arterial *p*CO_{2}, mental activation, posture and others, which makes the variability of CBF a formidable puzzle and has stimulated the use of alternative approaches, such as fractal analysis, to shed light on the complexity of the cerebral autoregulation.

Ideally, the information and knowledge generated by more classical approaches, such as system identification, would inform and stimulate research by those tackling CBF variability from the perspective of nonlinear dynamics, and vice versa. Unfortunately, this does not seem to be the case. For this reason, the main objective of this review is to attempt to establish a bridge between the two camps.

Section 2 provides a short introduction to the topic of CBF variability. Fractal analysis studies of the cerebral circulation are reviewed in §3 and compared with the more classical approaches of CBF deterministic modelling and system identification in §4. Section 5 highlights the contribution of blood flow regulatory mechanisms to CBF complexity and how different perspectives on the problem can be synergistically combined.

## 2. CBF variability

Apart from the earlier studies using electromagnetic flowmeters to measure internal carotid blood flow in humans, all current measurement techniques are non-invasive, and can only provide approximate estimates of CBF. As an example, TCD ultrasound records the blood velocity in large intracranial vessels, such as the middle cerebral artery (MCA), instead of absolute flow. CBF velocity (CBFV) can reflect changes in CBF with acceptable accuracy, as long as the diameter of the insonated artery remains constant. CBFV will be assumed to reflect the changes in CBF, but it is important to keep in mind that this cannot be generalized. Likewise, both NIRS and MRI are subject to assumptions that require frequent scrutiny. For simplicity, in what follows, no distinction will be made between CBF estimates obtained with these different modalities, except when strictly necessary for the interpretation of results.

Figure 1 illustrates how CBFV variability changes with the duration of the observation window. For each cardiac cycle, CBFV shows a recognizable pattern that, despite general similar features in each individual, can present small changes from beat to beat (figure 1*a*). At the next scale (figure 1*b*), the quasi-periodic characteristic of the signal is still present, but slower variability, showing both periodic and random components, can be envisaged. These periodic fluctuations tend to disappear for longer scales of observation though, and the overall impression is of random fluctuations around a given mean value (figure 1*c*). In adults, the mean CBFV is approximately 50–70 cm s^{−1} and tends to decrease with age (Adams *et al.* 1992). In general, women have higher velocities than men (Adams *et al.* 1992).

CBF variability can be described in the time or frequency domain. In the time domain, the simplest measure is the intra-subject standard deviation divided by the mean, which is the coefficient of variation (CV). As discussed later, this figure is not independent of the duration of the observation window. Broadly, CV of 5–10% has been reported for time intervals lower than 10 min (Rossitti & Volkmann 1995; Mitsis *et al.* 2006; Soehle *et al.* 2008). Spectral analysis of CBF time series will also depend on the observation window. In most cases, the influence of the cardiac cycle is removed by calculating the mean CBF for each cycle (figure 1*d*), followed by interpolation and resampling, at frequencies higher than 1 Hz (Panerai *et al.* 1998). The corresponding power spectral density (PSD; figure 2) tends to show a peak at approximately 0.05–0.15 Hz, followed by a rapid decline in power. The very reduced amount of power (or variability) for frequencies above 0.25 Hz (Reynolds *et al.* 1997; Zhang *et al.* 1998, 2000; Eames *et al.* 2004; Mitsis *et al.* 2004) has been largely ignored by many investigators when interpreting the results of CBF variability studies.

The term *complexity* has many different definitions in different fields of science. When applied to physiological signals, ‘complexity’ usually implies the lack of clear temporal patterns, such as well-defined oscillations. Consequently, we can say that the signals in figure 1*c*,*d* are ‘complex’, while those in figure 1*a*,*b* are not. The obvious reason why the signals in figure 1*a*,*b* are not complex is that we understand the main determinants (but certainly not all) of their variabilities at each scale of observation. In figure 1*a*, the pattern in each cardiac cycle is the result of a pulsatile ABP waveform. Small beat-to-beat differences might be the result of similar variability in ABP and possibly the influence of respiration as well. The influence of respiration, acting through ABP or by modulating cerebral venous outflow and ICP, is possibly the main determinant of the slower quasi-periodic variability observed in figure 1*b*. However, when we contemplate the variability in figure 1*c*,*d*, no clear interpretation springs to mind, and it is this complexity that calls for particular analytical techniques to explain the mechanisms behind this random-like variability. But is it truly random?

The answer to the above question can also be addressed either in the time or frequency domain. In the time domain, the autocorrelation function of a random beat-to-beat signal would tend to become zero within one or two cardiac cycles. If that does not happen, the time series cannot be taken as random, and further questions will then arise about the mechanisms underpinning its non-random behaviour. One possibility is that the data follow a random fractal point process (RFPP; West 1990), characterized by the presence of ‘self-similarity’. In other words, the variability patterns reflected in the data are the same, independently of the duration of the observation window. This approach is developed in §3.

## 3. Fractal analysis of the cerebral circulation

The property of self-similarity is more obvious in other fractal structures observed in nature (West 1990; Glenny *et al.* 1991), but in physiological signals such as CBF it can be identified by the within-window variability that follows a power-law relationship with the duration of the observation window:(3.1)where s.d. (*τ*) is the standard deviation of the data for window size *τ* and *τ*_{0} is the shortest window considered. *H* is the Hurst coefficient (Hurst 1951). For relatively long scales of observation, most physiological signals will be non-stationary. As discussed by Eke *et al.* (2000), a different expression would apply to estimate *H* for stationary data. It can be demonstrated that truly random signals have values of *H*=0.5, while values of *H* tending to 1 correspond to fairly smooth signals with long-lasting correlations. On the other hand, when *H* tends to −1, the temporal pattern of signals will show considerable ‘roughness’ and there will be significant anti-correlation. Fractals can also be described by the *fractal dimension D*_{f} (Glenny *et al.* 1991) that is related to the Hurst coefficient by the well-known expression *D*_{f}=2−*H*.

In the frequency domain, a truly random signal would present a PSD that will tend to be constant, i.e. independent of frequency (‘white noise’). On the other hand, it can be demonstrated that fractal data tend to follow a power law as a function of frequency (Kobayashi & Musha 1982; Schlesinger 1987),(3.2)where *|A*(*f*)|^{2} is the power at each frequency *f* and *β* is the *spectral index*. The importance of *β* is twofold: first, as acknowledged by many authors, *β* is the most reliable parameter to identify the presence of fractal behaviour; second, it can also be used to distinguish between stationary and non-stationary fractal processes. Accordingly, stationary fractal signals are expected to have −1<*β<*1, while non-stationarity moves *β* to the range 1<*β*<3. In practice, the separation between the two types of behaviour is not always so clear-cut and Eke *et al.* (2000) have proposed a very detailed procedure to aid in this classification. In summary, the calculation of *β* via PSD can help in the identification of processes that have fractal properties, and also to select the appropriate expressions to calculate the Hurst coefficient and/or the fractal dimension *D*_{f}. These two measures can then provide information about the characteristics of the long-term correlation or ‘memory’ in the mechanisms controlling CBF or other physiological variables.

Fractal behaviour is not the only type of process that can explain the complexity of physiological signals. Processes presenting the properties of *deterministic chaos* can also be perceived as random and fit a 1/*f ^{β}* power law in the frequency domain (West

*et al.*1999). Keunen

*et al.*(1994, 1996) concluded that CBFV followed a chaotic process, but the reliability of their arguments was challenged by West

*et al.*(1999). Other studies also failed to confirm Keunen

*et al.*'s assertion (West

*et al.*1999; Eke

*et al.*2005). For this reason, and owing to the limited literature on the studies of deterministic chaos applied to the cerebral circulation, this approach will not be discussed further.

Fractal analysis has been applied to different aspects of the circulatory system (Kobayashi & Musha 1982; Glenny *et al.* 1991), but studies of the cerebral circulation are relatively more recent. Table 1 presents the majority of studies in this area in chronological order. In general, it is not possible to compare the results between studies owing to considerable methodological diversity. Most studies performed measurements of CBFV with TCD ultrasound. Two studies were based on measurements of haemoglobin volume with NIRS (Eke & Herman 1999; Eke *et al.* 2005), and one study looked into the spatial heterogeneity and complexity of SPECT in subarachnoid haemorrhage patients (Mustonen *et al.* 2006). Eke *et al.* (2000) also reported on the estimates of the Hurst coefficient of cortical blood flow measurements with laser Doppler in rats. Moreover, not all studies included simultaneous estimates of *β*, *H* and *D*_{f} (table 1). One important methodological aspect in the calculation of *β* is the frequency range adopted for fitting the power-law relationship (equation (3.2)) to PSD estimates. As emphasized by Eke *et al.* (2000) and observed in figure 1, an appropriate scale of observation needs to be considered and this involves rejecting higher frequencies, including the cardiac cycle and possibly other dominant fluctuations such as respiration (Eke *et al.* 2000; Soehle *et al.* 2008). Interestingly enough, several independent studies found that a single value of *β* could not fit the PSD in the frequency region of interest. This is illustrated by figure 3. In neonates, Reynolds *et al.* (1997) found a ‘break-point’ frequency of 0.06 Hz with average values of *β*=0.670 for CBFV in the frequency range below 0.06 Hz, increasing to *β*=1.592 above this break-point frequency. Using much longer recordings in six healthy adults, Zhang *et al.* (2000) also observed a break-point frequency, in this case at 0.02 Hz, again with much lower values of *β* below the break point (0.34±0.09) than above it (2.31±0.08). Using even longer recordings and NIRS, instead of TCD, Eke *et al.* (2005) found a break point that was significantly influenced by ageing, but not by gender. In their case, mean values of *β* ranged from 1.14 from young males to 1.39 (post-menopausal females) below the break point. Above the break point, however, the values of *β* were not significantly different from zero. These results are in striking contrast to those of Reynolds *et al.* (1997) and Zhang *et al.* (2000). The two main possibilities to explain the differences in the position of the break point on the frequency axis and the values of *β* below and above it are the much longer duration of the recordings and data windows used by Eke *et al.* (2005) or the use of NIRS to measure total haemoglobin concentration (HbT), which would be more an expression of tissue volume perfusion, rather than blood flow velocity in large vessels. Both possibilities would explain the higher values of *β* below the break point in the case of Eke *et al.* (2005) compared with the other two studies that measured CBFV. With longer observation windows, it is more likely that CBF would manifest non-stationary behaviour thus pushing values of *β* from less than 1 to the 1<*β*<3 range. In addition, by measuring blood volume, which is the integral or cumulative sum of flow, non-stationary behaviour would be enhanced since the cumulative sum of a stationary signal is known to lead to non-stationarity (Eke *et al.* 2000, 2002). More work is obviously needed to shed light on these matters.

Other TCD studies that reported values of *β* have not specified the frequency range adopted for the linear regression between log-transformed PSD and log frequency (equation (3.2)), or the presence of a break-point frequency. Blaber *et al.* (1997) compared the values of *β* between 11 patients with autonomic nervous system failure and age- and sex-matched controls. At rest, there were no significant differences (table 1), but *β* was significantly reduced in patients during 60° tilt. Soehle *et al.* (2008) calculated the spectral index of CBFV signals in 31 patients with subarachnoid haemorrhage. A control group was not included, but the values of *β* reported are much higher than those obtained in healthy subjects as observed in other studies in table 1. Moreover, values of *β* were significantly increased during vasospasm when compared with the segments of data before the occurrence of vasospasm. However, values of *H*, estimated according to equation (3.1), did not show the same significant differences as *β* during vasospasm. Some of the studies that concentrated only on time-domain measures of fractality, such as *H* or *D*_{f}, can be criticized for not checking for the presence of non-stationarity as recommended by Eke *et al.* (2000). Rossitti & Stephensen (1994), Rossitti & Volkmann (1995) and West *et al.* (1999) estimated *H* and *D*_{f} using *dispersional analysis* (Glenny *et al.* 1991), which can lead to wrong values in the case of non-stationary processes. Indeed, as concluded by a series of different studies (Eke & Herman 1999; Eke *et al.* 2000; Soehle *et al.* 2008), CBFV (from TCD), laser Doppler (rat) or HbT (NIRS) signals were non-stationary and the resulting values of *H* were in very good agreement (table 1). On the other hand, from the values of *D*_{f} reported by Rossitti & Stephensen (1994), Rossitti & Volkmann (1995), West *et al.* (1999) and Mustonen *et al.* (2006), the corresponding values of *H* (=2−*D*_{f}) would be in the range *H*>0.80 and hence clearly distinct from what was reported when equation (3.1) was adopted, instead of dispersional analysis.

## 4. Disorganized versus organized complexity

According to the classical perspective of Weaver (1948), the fractal behaviour of CBF and other cerebrovascular parameters would represent *disorganized complexity* that could result from the interaction of a large number of independent entities (e.g. arterioles) acting independently or under heterogeneous control mechanisms. On the other hand, attempts to explain CBF variability using more classical deterministic models, independent of their dimensionality or structure (e.g. linear or nonlinear), could be seen as a problem in *organized complexity* (Weaver 1948). These considerations are pertinent because of the two disjoint camps that have dominated research into cerebrovascular dynamics. All the studies in table 1 have analysed CBF or related quantities without considering any of its possible determinants. The main assumption behind that approach is that any influences on CBF would automatically be incorporated in its variability patterns and hence would be reflected by the measures of complexity assessed, such as *β*, *H* or *D*_{f}. Unfortunately, the validity of this assumption has not been tested, and, above all, the sensitivity of different fractal parameters to reflect the influence of covariates has not been demonstrated.

For the camp adopting an *organized complexity* approach, the *coherence function* has been the traditional tool to identify the influence of other variables on CBF (Bendat & Piersol 1986). As a measure of the fraction of CBF power (or variability) that can be *linearly* explained by one or more other variables at each frequency, the main limitation of the coherence approach is the difficulty of quantifying the contribution of nonlinear influences, for example by the presence of deterministic chaos. For reasons that will be explained later, in practice, this limitation is not dominant. A second limitation of the classical coherence function, which is worth mentioning, is the impossibility of determining causality, although the calculation of *partial coherence* might help to clarify this matter in some cases. In practice, *a priori* knowledge about the physiology or the physics of the relationships involved can identify causality, for example, by assuming that ABP and arterial *p*CO_{2} influence CBF, but not vice versa. Nevertheless, there are areas where directionality might be less clear, for example when assessing the influence of ICP on CBF (Panerai *et al.* 2002).

A very large number of studies have assessed the contribution of ABP as a determinant of CBF (Giller 1990; Panerai 1998, 2004; Panerai *et al.* 1998, 2006; Zhang *et al.* 1998). Most of these studies reported coherence functions that were relatively high for frequencies above 0.15 Hz, but then dropped, sometimes to non-significant values, for frequencies below 0.15 Hz (figure 4). These two distinct frequency regions deserve to be considered separately. Starting with the higher frequency range (HF, *f*>0.15 Hz), it is not unusual to find squared coherence values, *γ*^{2}(*f*)>0.8, as shown in the example given in figure 4 (Zhang *et al.* 2000; Panerai *et al.* 2006). This indicates that in this frequency region, the largest fraction of CBF variability is explained by fluctuations in ABP. When the very small power of CBF (figure 2) is also considered, it becomes questionable whether the fractal hypothesis is really applicable to CBF in the frequency range above 0.15–0.20 Hz. This view is reinforced by some of the results presented in table 1. Reynolds *et al.* (1997), Zhang *et al.* (2000) and Eke *et al.* (2005) found break points for frequencies below 0.20 Hz. Eke *et al.* (2005) did not find significant values of *β* above the break-point frequency, while the other two studies obtained values of *β* that were much higher than the corresponding values below the break point. Reynolds *et al.* (1997) also questioned the validity of the fractal hypothesis for frequencies above the break point, owing to the presence of non-random residuals in the log–log linear regression. Figure 3 provides an example of this situation. However, Zhang *et al.* (2000) also applied the Durbin–Watson test and reported random residuals. Other studies have not reported the frequency band adopted for the estimation of *β* or the distribution of the linear regression residuals. From the above, it can be concluded that much more work needs to be done before it can be accepted that CBF behaves as a fractal process for frequencies above 0.15 Hz. Moreover, given the high fraction of CBF that is explained by ABP variability in the high-frequency range, it seems appropriate to suggest that future studies attempt to test the fractal hypothesis not to CBF as a whole, but only to the residual variability that is not explained by variables with significant values of *γ*^{2}(*f*) in this frequency range, such as ABP and possibly respiration as well (Eames *et al.* 2004). Studying the interaction between HR, ABP and stroke volume (SV), Mullen (1998) suggested something similar to explain the 1/*f* behaviour of the HR spectra, since this characteristic was not shown by the model coupling mechanisms that were identified but could be owing to the properties of random noise sources.

Moving on to the low-frequency region (LF, *f*<0.15 Hz), the interpretation of both organized and disorganized complexity approaches is much less straightforward. First, this is the frequency range where dynamic cerebral autoregulation is expected to be active (Aaslid *et al.* 1989; Panerai *et al.* 1998; Zhang *et al.* 1998), thus increasing the complexity of CBF variability. Second, it is well documented that *γ*^{2}(*f*) values for the ABP input tend to fall with decreasing frequencies (figure 4), often reaching values that suggest a non-significant linear contribution of ABP (Giller & Mueller 2003; Panerai 2004; Panerai *et al.* 2006). Third, as indicated in figure 2, this is the frequency region containing most of the CBF variability. Combined with the previous observation, it means that a substantial fraction of the CBF variability cannot be linearly explained by fluctuations in ABP. Finally, the results given in table 1. For the LF region, some investigators reported values of *β*<1 (Reynolds *et al.* 1997; Zhang *et al.* 2000), which would indicate CBF following a stationary process. On the other hand, Eke *et al.* (2005) obtained estimates of *β*>1, suggesting that HbT shows non-stationary behaviour. Even higher values of *β* were reported by others (Blaber *et al.* 1997; Soehle *et al.* 2008). Soehle *et al.* (2008) obtained estimates of *β* for *f*<0.125 Hz, but Blaber *et al.* (1997) did not specify the frequency range. The discrepancies between different studies might be due to the populations studied and variables recorded. Greater disturbances of CBF in the autonomic failure patients studied by Blaber *et al.* (1997) and the SAH patients studied by Soehle *et al.* (2008) might have led to non-stationary behaviour, when compared with the subjects analysed by Reynolds *et al.* (1997) and Zhang *et al.* (2000). Although Eke *et al.* (2005) studied healthy volunteers, NIRS recordings of HbT might be seen as reflecting blood volume, which would be expected to follow a non-stationary process when regarded as the cumulative sum of blood flow (Eke *et al.* 2000). The additional observations that *β* increased during vasospasm (Soehle *et al.* 2008) and with 60° tilt in autonomic failure patients (Blaber *et al.* 1997) add further encouragement to future studies to explore the possibility that CBF in the low-frequency range becomes more non-stationary during pathological conditions.

How has the organized complexity camp addressed the much more complex behaviour of CBF variability in the frequency region below 0.15 Hz? Both linear and nonlinear methods of system identification have been adopted in the attempt to understand the determinants of CBF in this frequency range (Panerai *et al.* 1998, 1999; Zhang *et al.* 1998; Panerai 2004; Mitsis *et al.* 2004, 2006). When taken in isolation, ABP has a limited contribution to explain CBF variability in the LF range, but different studies have shown that the multiple coherence of CBF [*γ*_{M}^{2}(*f*)] can rise significantly when other covariates are taken into account. The reason why the univariate coherence of CBF for the ABP input tends to be low in healthy subjects becomes apparent when Poiseuille's law is considered:(4.1)where *R*(*t*) is the CVR. The fact that *R*(*t*) is not constant, owing to vessel diameter changes resulting from CBF autoregulation, means that the above relationship is highly nonlinear, thus leading to low values of *γ*^{2}(*f*) in the LF range (Bendat & Piersol 1986). However, expressing *R*(*t*)=*R*_{0}*+*Δ*r*(*t*), and assuming that Δ*r*(*t*)≪*R*_{0}, allows equation (4.1) to be linearized as (Panerai *et al.* 2006)(4.2)where ΔCBF(*t*) and ΔABP(*t*) are fluctuations around mean values CBF_{0} and ABP_{0}. According to equation (4.2), for small fluctuations in CVR, it is possible to express a single-input nonlinear system as a two-input linear system. The main result in this case though is that the multiple coherence for CBF, corresponding to both inputs, increases to significant values, implying that a large fraction of CBF variability in the LF range can be explained by fluctuations in ABP and CVR (Panerai *et al.* 2006). It is important to note though that estimates of multiple coherence will only be meaningful if CVR is independent of CBF and ABP. This can be achieved with estimates of resistance-area product (Panerai *et al.* 2006). If CVR is derived simply by the ratio ABP/CBF, all estimates of multiple coherence will be necessarily equal to 1.

Arterial *p*CO_{2} (PaCO_{2}) is a strong determinant of CBF (Heistad & Kontos 1983), but only recently was it demonstrated that breath-by-breath measurements of end-tidal CO_{2} can help to explain the CBF variability in the LF range (Panerai *et al.* 2000; Edwards *et al.* 2004; Mitsis *et al.* 2004). Peng *et al.* (2008) have also shown significantly higher values of CBF multiple coherence for ABP and PaCO_{2} as simultaneous inputs. However, there were no further significant rises in *γ*_{M}^{2}(*f*) when PaO_{2} was also included (Peng *et al.* 2008). Unfortunately, these authors have not included the CVR term proposed by Panerai *et al.* (2006) (equation (4.2)). Had they done so, it is conceivable that the multiple coherence could have risen to values as high as 0.9, thus implying that the CBF variability in the LF range could be almost entirely explained by spontaneous fluctuations of ABP, PaCO_{2} and the corresponding adjustments in cerebral autoregulation manifested by changes in Δ*r*(*t*) (equation (4.2)). Other potential determinants of CBF, such as sympathetic activity, respiration and cardiac output, should also be investigated (van Lieshout & Secher 2008). For this purpose, previous multivariate approaches used to model peripheral oscillations, such as the interaction of ABP, HR, respiration and peripheral control (Porta *et al.* 2002), could be extended to include the cerebral circulation, aiming at an integrated and more comprehensive model of the main intervening variables.

If most of the variability of CBF can be predicted by other variables, it can be argued that the fractal behaviour observed in CBF recordings is not something inherent to this variable, or the result of obscure mechanisms that cannot be identified. Instead, it seems more likely that the fractal behaviour detected has its origins in the input variables that control CBF or in random noise sources (Mullen 1998). These cogitations suggest that more research is needed on the complexity of ABP, PaCO_{2} and other potential determinants of CBF. In particular, the complexity of cerebral autoregulation as manifested by changes in CVR deserves further attention.

## 5. Complexity of the cerebral autoregulation

When interpreting the results in table 1, most authors made explicit reference to the mechanism of cerebral autoregulation as central to explaining the fractal properties of CBF recordings. This association is justified by the high dimensionality of the cellular and subcellular mechanisms involved in pressure and metabolic autoregulation (Baumbach & Heistad 1985; Faraci & Heistad 1998; Andresen *et al.* 2006; Haydon & Carmignoto 2006) and also the analogy with blood flow regulation in the myocardium (Bassingthwaighte *et al.* 1989). The majority of these studies have also concluded that CBF is inherently a non-stationary process. If that is the case, it is highly likely that the source of non-stationarity lies with the mechanisms involved in cerebral autoregulation. However, no studies could be found in the literature analysing cerebral autoregulation from the perspective of *disorganized complexity*. For the *organized complexity* camp, though, there seems to be increasing interest in the non-stationary behaviour of dynamic cerebral autoregulation. The general approach has been the extraction of parameters that reflect autoregulatory performance for relatively short segments of data, followed by analysis of their longitudinal (temporal) variability or time–frequency properties. Two main parameters have been adopted to provide a unidimensional scale to represent autoregulatory efficiency. The first is the phase lead of CBF in relation to fluctuations in ABP (Birch *et al.* 1995; Diehl *et al.* 1995), and the second is the dynamic autoregulation index (ARI) proposed by Tiecks *et al.* (1995). The ARI ranges from 0 (lack of autoregulation) to 9 (best observed autoregulation) and is estimated by fitting a second-order linear differential equation to induced or spontaneous fluctuations in ABP and CBF (Tiecks *et al.* 1995; Panerai *et al.* 1998, 2001; Zhang *et al.* 1998). Both the ARI and the LF CBF–ABP phase difference have been shown to be sensitive to changes in PaCO_{2} and to reflect deterioration of dynamic cerebral autoregulation in several different cerebrovascular conditions (Panerai 2008). Representative CBFV–ABP gain and phase frequency responses are shown in figure 4, together with the CBFV step response and the best fit second-order template curve given by the Tiecks *et al.* (1995) model.

Despite concerns about the intrinsic measurement error and noise of the ARI index (Simpson *et al.* 2004), two separate studies have shown large fluctuations in this parameter with time (Panerai *et al.* 2003*b*, 2008). Temporal variability of the CBF–ABP phase difference was reported by Latka *et al.* (2005) who calculated the phase synchronization index by means of the complex Morlet wavelet transform. For scales corresponding to the LF range, the average synchronization index approached zero, thus suggesting that the phase showed a uniform distribution over time, rather than small fluctuations around more stable mean values. The same wavelet transform was adopted by Rowley *et al.* (2007) to study the stationarity of phase between ABP and oxygenated haemoglobin (O_{2}Hb) measured with NIRS. Although the authors described their results as being in general agreement with those of Latka *et al.* (2005), they did not present values of phase synchronization index in the LF range.

In addition to the studies based on the ARI or the LF phase difference between CBF and ABP, other approaches have also addressed the question of cerebral autoregulation non-stationarity. Evidence about temporal variability of cerebral autoregulation was obtained from analysis of the CBF response to spontaneous ABP transients (Panerai *et al.* 2003*a*) and from time–frequency analysis of CBFV and ABP signals (Giller & Mueller 2003). The latter study pointed out that the agreement between bifurcations and chirps in the ABP and CBF data was usually intermittent (Giller & Mueller 2003). Significant non-stationarity of cerebral autoregulation parameters was also reported by Mitsis *et al.* (2004) using a multivariate nonlinear model to study the interaction between ABP, PaCO_{2} and CBFV. The non-stationarity was more pronounced in the LF range and also greater for the nonlinear contributions of ABP and PaCO_{2} than for the first-order, linear terms (Mitsis *et al.* 2004).

Despite their methodological differences and the need for considerably more work, the preliminary results described above suggest that dynamic cerebral autoregulation parameters are not only variable in time, but also likely to be non-stationary. This hypothesis represents an opportunity for collaboration between investigators pursuing a deterministic approach, with those with expertise in techniques belonging to the disorganized complexity camp. Indeed, fractal analysis, entropy and other approaches can provide a different perspective and additional information about the nature and mechanisms which might be responsible for the non-stationary behaviour of dynamic cerebral autoregulation.

## 6. Conclusions

The complexity of the cerebral circulation has been studied from two quite different perspectives, but with a common focus on the variability of CBF as presently estimated by non-invasive techniques such as TCD ultrasound or NIRS. On the one hand, classical methods of system identification have sought to model the determinants of CBF and thus explain its variability with a deterministic approach. In the low-frequency range (below 0.15 Hz), which contains most of the CBF variability (or spectral power), and where CBF regulatory mechanisms are active, linear and nonlinear multivariate models have been able to explain a substantial fraction of the total CBF variability. Nevertheless, many questions remain unanswered. Little is known, for example, about the stochastic properties of the unexplained variance, whether it is truly random, or if it follows a fractal process, or, still, if it can be explained by other, as yet unidentified, physiological variables. Of practical importance, most of these studies have been performed on healthy subjects at rest, in the supine position. Very different results and conclusions might be obtained under more variable physiological conditions (posture, mental activation, exercise and sleep). Of particular interest, but still in its infancy, are observations indicating that even at rest there might be considerable variability in the mechanisms responsible for cerebral autoregulation, including non-stationary behaviour. Again, the possibility that this unexplained complexity might follow a fractal process deserves further investigation.

The alternative approach to complexity represented mainly by the fractal analysis perspective has been focused almost exclusively on a single variable, such as CBF or haemoglobin content, despite the possibility that the self-similarity reported is the consequence of fractal behaviour in one or more of its determinants, such as ABP, PaCO_{2} or changes in CVR. Most studies concluded that CBF (or HbT) follows a non-stationary process and this should send a strong message to those using classical methods to develop models of cerebral haemodynamics intended to describe physiological behaviour over relatively long periods of time.

This review has highlighted the chasm that is still observed between the two dominant perspectives mentioned earlier, and the need for more cross-disciplinary work to accelerate progress in understanding the complexity of the human cerebral circulation in health and disease.

## Acknowledgments

All human data shown in the figures have been collected following research protocols approved by the Leicestershire Ethics Research Committee.

## Footnotes

One contribution of 15 to a Theme Issue ‘Addressing the complexity of cardiovascular regulation’.

- © 2009 The Royal Society