Temporal heterogeneity of the blood flow velocity at the middle cerebral artery in the normal human characterized by fractal analysis

Fractal Analysis of the Cerebrovascular System Pathophysiology

The cerebrovascular system is characterized by parameters such as arterial blood pressure (ABP), cerebral perfusion pressure (CPP), and cerebral blood flow velocity (CBFV). These are regulated by interconnected feedback loops resulting in a fluctuating and complex time course. They exhibit fractal characteristics such as (statistical) self-similarity and scale invariance which could be quantified by fractal measures. These include the coefficient of variation, the Hurst coefficient H, or the spectral exponent α in the time domain, as well as the spectral index ß in the frequency domain. Prior to quantification, the time series has to be classified as either stationary or nonstationary, which determines the appropriate fractal analysis and measure for a given signal class. CBFV was characterized as a nonstationary (fractal Brownian motion) signal with spectral index ß between 2.0 and 2.3. In the high-frequency range (>0.15 Hz), CBFV variability is mainly determined by the periodic ABP variability induced by heartbeat and respiration. However, most of the spectral power of CBFV is contained in the low-frequency range (<0.15 Hz), where cerebral autoregulation acts as a low-pass filter and where the fractal properties are found. Cerebral vasospasm, which is a complication of subarachnoid hemorrhage (SAH), is associated with an increase in ß denoting a less complex time course. A reduced fractal dimension of the retinal microvasculature has been observed in neurodegenerative disease and in stroke. According to the decomplexification theory of illness, such a diminished complexity could be explained by a restriction or even dropout of feedback loops caused by disease.

Variability Predictors of Vasospasm in Subarachnoid Hemorrhage: A Feasibility Study

BACKGROUND

Mean cerebral blood flow velocity (mean-CBFV) obtained from Transcranial Doppler (TCD) poorly predicts cerebral vasospasm in patients with aneurysmal subarachnoid hemorrhage (aSAH). Variability descriptors of mean-CBFV obtained during extended TCD recordings may improve this prediction. We assessed the feasibility of generating reliable linear and non-linear descriptors of mean-CBFV variability using extended recordings in aSAH patients and in healthy controls. We also explored which of those metrics might have the ability to discriminate between aSAH patients and healthy controls, and among patients who would go on to develop vasospasm and those who would not.

METHODS

Bilateral mean-CBFV, blood pressure, and heart rate were continuously recorded for 40 minutes in aSAH patients (n = 8) within the first 5 days after ictus, in age-matched healthy controls (n = 8) and in additional young controls (n = 8). We obtained linear [standard deviation, coefficient of variations, and the very-low (0.003-0.040 Hz), low (0.040-0.150 Hz), and high-frequency (0.15-0.4 Hz) power spectra] and non-linear (Fractality, deterministic Chaos analyses) variability metrics.

RESULTS

We successfully obtained TCD recordings from patients and healthy controls and calculated the desired metrics of mean-CBFV variability. Differences were appreciable between aSAH patients and healthy controls, as well as between aSAH patients who later developed vasospasm and those who did not.

CONCLUSIONS

A 40-minute TCD recording provides reliable variability metrics in aSAH patients and healthy controls. Future studies are required to determine if mean-CBFV variability metrics remain stable over time, and whether they may serve to identify patients who are at greatest risk of developing cerebral vasospasm after aSAH.

Complexity of cerebral blood flow velocity and arterial blood pressure in subarachnoid hemorrhage using time-frequency analysis

We investigated changes of time-frequency (TF) complexity, in terms of Rényi entropy and a measure of concentration, of middle cerebral blood flow velocity (CBFV) and arterial blood pressure in relation to the development of cerebral vasospasm in 15 patients after aneurysmal subarachnoid hemorrhage. Interhemispheric differences in the period of no vasospasm and vasospasm were also compared. Results show reduced complexity of TF representations of CBFV on the side of aneurysm before vasospasm was identified. This potentially can serve as an early-warning indicator of future derangement of cerebral circulation.

Fractal physiology and the fractional calculus: a perspective

This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. Not only are anatomical structures (Grizzi and Chiriva-Internati, 2005), such as the convoluted surface of the brain, the lining of the bowel, neural networks and placenta, fractal, but the output of dynamical physiologic networks are fractal as well (Bassingthwaighte et al., 1994). The time series for the inter-beat intervals of the heart, inter-breath intervals and inter-stride intervals have all been shown to be fractal and/or multifractal statistical phenomena. Consequently, the fractal dimension turns out to be a significantly better indicator of organismic functions in health and disease than the traditional average measures, such as heart rate, breathing rate, and stride rate. The observation that human physiology is primarily fractal was first made in the 1980s, based on the analysis of a limited number of datasets. We review some of these phenomena herein by applying an allometric aggregation approach to the processing of physiologic time series. This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data. Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation. A fractional operator (derivative or integral) acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a fractal statistical process. Control of physiologic complexity is one of the goals of medicine, in particular, understanding and controlling physiological networks in order to ensure their proper operation. We emphasize the difference between homeostatic and allometric control mechanisms. Homeostatic control has a negative feedback character, which is both local and rapid. Allometric control, on the other hand, is a relatively new concept that takes into account long-time memory, correlations that are inverse power law in time, as well as long-range interactions in complex phenomena as manifest by inverse power-law distributions in the network variable. We hypothesize that allometric control maintains the fractal character of erratic physiologic time series to enhance the robustness of physiological networks. Moreover, allometric control can often be described using the fractional calculus to capture the dynamics of complex physiologic networks.

Complexity of the human cerebral circulation

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.

Variability and fractal analysis of middle cerebral artery blood flow velocity and arterial blood pressure in subarachnoid hemorrhage

Higher biologic systems operate far from equilibrium resulting in order, complexity, fluctuation of inherent parameters, and dissipation of energy. According to the decomplexification theory, disease is characterized by a loss of system complexity. We analyzed such complexity in patients after subarachnoid hemorrhage (SAH), by applying the standard technique of variability analysis and the novel method of fractal analysis to middle cerebral artery blood flow velocity (FV) and arterial blood pressure (ABP). In 31 SAH -patients, FV (using transcranial Doppler sonography) and direct ABP were measured. The standard deviations (s.d.) and coefficients of variation (CV=relative s.d.) for FV and ABP time series of length 2(10) secs were calculated as measures of variability. The spectral index beta(low) and the Hurst coefficient H(bdSWV) were analyzed as fractal measures. Outcome was assessed 1 year after SAH according to the Glasgow Outcome Scale (GOS). Both FV (beta(low)=2.2+/-0.4, mean+/-s.d.) and ABP (beta(low)=2.3+/-0.4) were classified as nonstationary (fractal Brownian motion) signals. FV showed significantly (P<0.05) higher variability (CV=7.2+/-2.5%) and Hurst coefficient (H(bdSWV)=0.26+/-0.13) as compared with ABP (CV=5.5+/-2.7%, H(bdSWV)=0.19+/-0.11). Better outcome (GOS) correlated significantly (P<0.05) with higher s.d. of FV (Spearman's r(s)=0.51, r(s)(2)=0.26) and ABP (r(s)=0.57, r(s)(2)=0.32), as well as with a higher Hurst coefficient of ABP (r(s)=0.46, r(s)(2)=0.21). Cerebral vasospasm reduced CV of FV, but left H(bdSWV) unchanged. FV and ABP fluctuated markedly despite homeostatic control. A reduced variability of FV and ABP might indicate a loss of complexity and was associated with a less favorable outcome. Therefore, the decomplexification theory of illness may apply to SAH.

Independent component analysis of nondeterministic fMRI signal sources

Neuronal activation can be separated from other signal sources of functional magnetic resonance imaging (fMRI) data by using independent component analysis (ICA). Without deliberate neuronal activity of the brain cortex, the fMRI signal is a stochastic sum of various physiological and artifact related signal sources. The ability of spatial-domain ICA to separate spontaneous physiological signal sources was evaluated in 15 anesthetized children known to present prominent vasomotor fluctuations in the functional cortices. ICA separated multiple clustered signal sources in the primary sensory areas in all of the subjects. The spatial distribution and frequency spectra of the signal sources correspond to the known properties of 0.03-Hz very-low-frequency vasomotor waves in fMRI data. In addition, ICA was able to separate major artery and sagittal sinus related signal sources in each subject. The characteristics of the blood vessel related signal sources were different from the parenchyma sources. ICA analysis of fMRI can be used for both assessing the statistical independence of brain signals and segmenting nondeterministic signal sources for further analysis.

Fractal characterization of complexity in temporal physiological signals

This review first gives an overview on the concept of fractal geometry with definitions and explanations of the most fundamental properties of fractal structures and processes like self-similarity, power law scaling relationship, scale invariance, scaling range and fractal dimensions. Having laid down the grounds of the basics in terminology and mathematical formalism, the authors systematically introduce the concept and methods of monofractal time series analysis. They argue that fractal time series analysis cannot be done in a conscious, reliable manner without having a model capable of capturing the essential features of physiological signals with regard to their fractal analysis. They advocate the use of a simple, yet adequate, dichotomous model of fractional Gaussian noise (fGn) and fractional Brownian motion (fBm). They demonstrate the importance of incorporating a step of signal classification according to the fGn/fBm model prior to fractal analysis by showing that missing out on signal class can result in completely meaningless fractal estimates. Limitation and precision of various fractal tools are thoroughly described and discussed using results of numerical experiments on ideal monofractal signals. Steps of a reliable fractal analysis are explained. Finally, the main applications of fractal time series analysis in biomedical research are reviewed and critically evaluated.

Fractal dimensions of laser doppler flowmetry time series

Laser Doppler flowmetry (LDF) provides a non-invasive method of assessing cutaneous perfusion. As the microvasculature under the probe is not defined the measured flux cannot be given absolute units, but the technique has nevertheless proved valuable for assessing relative changes in perfusion in response to physiological stress. LDF signals normally show pronounced temporal variability, both as a consequence of the pulsatile nature of blood flow and local changes in dynamic vasomotor activity. The aim of the present study was to investigate the use of methods of nonlinear analysis in characterizing temporal fluctuations in LDF signals. Data were collected under standardised conditions from the forearm of 16 normal subjects at rest, during exercise and on recovery. Surrogate data was then generated from the original time series by phase randomization. Dispersional analysis demonstrated that the LDF data was fractal with two distinct scaling regions, thus allowing the calculation of a fractal dimension which decreased significantly from 1.23 +/- 0.09 to 1.04 +/- 0.02 during exercise. By contrast, dispersional analysis of the surrogate data showed no scaling region.

Is the sheet-flow design a 'frozen core' (a Bauplan) of the gas exchangers? Comparative functional morphology of the respiratory microvascular systems: illustration of the geometry and rationalization of the fractal properties

The sheet-flow design is ubiquitous in the respiratory microvascular systems of the modern gas exchangers. The blood percolates through a maze of narrow microvascular channels spreading out into a thin film, a "sheet". The design has been convergently conceived through remarkably different evolutionary strategies. Endothelial cells, e.g. connect parallel epithelial cells in the fish gills and reptilian lungs; epithelial cells divide the gill filaments in the crustacean gills, the amphibian lungs, and vascular channels on the lung of pneumonate gastropods; connective tissue elements weave between the blood capillaries of the mammalian lungs; and in birds, the blood capillaries attach directly and in some areas connect by short extensions of the epithelial cells. In the gills, skin, and most lungs, the blood in the capillary meshwork geometrically lies parallel to the respiratory surface. In the avian lung, where the blood capillaries anastomose intensely and interdigitate closely with the air capillaries, the blood occasions a 'volume' rather than a 'sheet.' The sheet-flow design and the intrinsic fractal properties of the respiratory microvascular systems have produced a highly tractable low-pressure low-resistance region that facilitates optimal perfusion. In complex animals, the sheet-flow design is a prescriptive evolutionary construction for efficient gas exchange by diffusion. The design facilitates the internal and external respiratory media to be exposed to each other over an extensive surface area across a thin tissue barrier. This comprehensive design is a classic paradigm of evolutionary convergence motivated by common enterprise to develop corresponding functionally efficient structures. With appropriate corrections for any relevant intertaxa differences, use of similar morphofunctional models in determining the diffusing capacities of various gas exchangers is warranted.

Spontaneous fluctuations in cerebral blood flow: insights from extended-duration recordings in humans

To determine the dependence of cerebral blood flow (CBF) on arterial pressure over prolonged time periods, we measured beat-to-beat changes in mean CBF velocity in the middle cerebral artery (transcranial Doppler) and mean arterial pressure (Finapres) continuously for 2 h in six healthy subjects (5 men and 1 woman, 18-40 yr old) during supine rest. Fluctuations in velocity and pressure were quantified by the range [(peak - trough)/mean] and coefficients of variation (SD/mean) in the time domain and by spectral analysis in the frequency domain. Mean velocity and pressure over the 2-h recordings were 60 +/- 7 cm/s and 83 +/- 8 mmHg, associated with ranges of 77 +/- 8 and 89 +/- 10% and coefficients of variation of 9.3 +/- 2.2 and 7.9 +/- 2.3%, respectively. Spectral power of the velocity and pressure was predominantly distributed in the frequency range of 0.00014-0.1 Hz and increased inversely with frequency, indicating characteristics of an inverse power law (1/f(alpha)). However, linear regression on a log-log scale revealed that the slope of spectral power of pressure and velocity was steeper in the high-frequency (0.02-0.5 Hz) than in the low-frequency range (0.002-0.02 Hz), suggesting different regulatory mechanisms in these two frequency ranges. Furthermore, the spectral slope of pressure was significantly steeper than that of velocity in the low-frequency range, consistent with the low transfer function gain and low coherence estimated at these frequencies. We conclude that 1) long-term fluctuations in CBF velocity are prominent and similar to those observed in arterial pressure, 2) spectral power of CBF velocity reveals characteristics of 1/f(alpha), and 3) cerebral attenuation of oscillations in CBF velocity in response to changes in pressure may be more effective at low than that at high frequencies, emphasizing the frequency dependence of cerebral autoregulation.

Nonlinear transcranial Doppler analysis demonstrates age-related changes of cerebral hemodynamics

We studied the age- and gender-related changes in complexity of arterial blood flow by analyzing the maximum velocity waveforms. Data were collected by insonation of the middle cerebral artery blood flow by a 2-MHz transcranial pulsed Doppler system. Following the paradigm of nonlinear dynamical systems or "chaos" theory, complexity is best evaluated by estimating the correlation dimension (D2) and the largest Lyapunov exponent (lambda 1). Forty healthy persons (male/female ratio: 1/1; mean age 48.6 y; range 19-86 y) were studied. No gender-related differences were observed. The age-effect showed a diminishing lambda 1 [df(1, 36) = 5.687; p < 0.022] and an increasing D2 at higher age [df(1, 36) = 4.997; p < 0.032]. The age-related decline of the lambda 1 implies a more prominent periodicity, explained by reduced fluctuations in R-R intervals and an altered gain of the baroreceptor reflex. The increased D2 might be related to more prominent vessel wall oscillations due to the increased vessel wall stiffness at higher age.

Shear stress in cerebral arteries supplying arteriovenous malformations

Arteries supplying cerebral arteriovenous malformations (AVMs) are known to dilate with time. These changes are reversible, and the feeders have been shown to slowly decrease in calibre after removal of the AMV. There is evidence that arteries alter their internal diameters in response to sustained changes of blood flow so that shear stress is kept constant. This implies that blood flow-induced shear stress might be the driving force for remodelling of the cerebral vascular network in the presence of an AVM, and for reversion of these changes after radical operation. The objective of this study is to examine the hypothesis that the shear stress in cerebral arteries supplying AMVs is of the same magnitude as in arteries supplying normal brain tissue in spite of larger blood flow rate. Fifteen patients with supratentorial cerebral AVMs admitted for endovascular treatment were examined with transcranial Doppler ultrasound in the distal Willisian vessels. Vessel calibres were measured in angiograms with magnification correction. Shear stress was estimated assuming a constant value for blood viscosity. Corresponding arteries in the cerebral hemisphere with AVM and in the contralateral one were compared in pairs. Thirty-four pairs of homonymous arteries were studied. The arteries on the AVM side presented larger calibres, higher axial blood flow velocities, lower pulsatility index and larger blood flow rates than the contralateral side. There was a clear positive correlation between blood flow velocities and vessel calibres. The estimates of shear stress did not differ significantly in corresponding arteries of both hemispheres (p = 0.18). The results indicate a precise adjustment of cerebral arterial calibre and blood flow-induced shear stress that presumably induces the progressive dilation of AVM feeders, and the slow regression of the vessel calibres to average dimensions after removal of the lesion. Each vessel seems to remodel itself in response to long-term changes in blood flow rate so that the vessel calibre is reshaped to maintain a constant level of wall shear stress.

Energetic and spatial constraints of arterial networks

The principle of minimum work (PMW) is a parametric optimization model for the growth and adaptation of arterial trees. A balance between energy dissipation due to frictional resistance of laminar flow (shear stress) and the minimum volume of the blood and vessel wall tissue is achieved when the vessel radii are adjusted to the cube root of the volumetric flow. The PMW is known to apply over several magnitudes of vessel calibers, and in many different organs, including the brain, in humans and in animals. Animal studies suggest that blood flow in arteries is approximately proportional to the cube of the vessel radius, and that arteries alter their caliber in response to sustained changes of blood flow according to PMW. Remodelling of the retinal arteriolar network to long-term changes in blood flow was observed in humans. Remodelling of whole arterial networks occurs in the form of increase or diminishing of vessel calibers. Shear stress induced endothelial mediation seems to be the regulating mechanism for the maintenance of this optimum blood flow/vessel diameter relation. Arterial trees are also expected to be nearly space filing. The vascular system is constructed in such a way that, while blood vessels occupy only a small percentage of the body volume leaving the bulk to tissue, they also crisscross organs so tightly that every point in the tissue lies on the boundary between an artery and a vein. This review describes how the energetic optimum principle for least energy cost for blood flow is also compatible with the spatial constraints of arterial networks according to concepts derived from fractal geometry.

Regulation of vascular tone

The intimal surface of the blood vessel in vivo is subject to shear stress resulting from blood flow, which in most of the circulation, at least at rest, is laminar. Turbulence can occur at bifurcations, especially those of the large arteries, and where vessels curve significantly. Shear stress is a frictional tangential force exerted at the fluid-intimal interface in the long axis of the vessel. It is now known that hemodynamic shear stress can influence a large variety of biological processes in endothelial cells, which vary from those with a short response time, just a few milliseconds, such as the opening of ion channels, to those that change over a period of minutes to several hours, for example, endocytosis and cytoskeleton rearrangement, and those features that alter much more slowly, such as cell shape and stiffness. In addition to these types of changes, there are suggestions that flow acting through shear stress may be responsible for several basic attributes of the vasculature, including the relative size and diameter of the components of a branching vascular system. In this symposium on the flow regulation of the blood vessel, the first presentation dealt with optimality principles that appear to govern the dimensions of the vasculature, in particular the geometry of the arterial branching and the role of shear stress. An optimally designed system is one that requires the least metabolic work to perform its function.(ABSTRACT TRUNCATED AT 250 WORDS)

Changes of blood flow velocity indicating mechanical compression of the vertebral arteries during rotation of the head in the normal human measured with transcranial Doppler sonography

The dynamical changes of blood flow velocity of the intracranial vertebral arteries (VA's) and proximal basilar artery (BA) provoked by rotation of the head in normal volunteers were measured using pulsed-wave transcranial Doppler sonography (TCD). In another group both VA's were examined simultaneously with 2-channel TCD. Blood flow velocities diminished compared to the neutral position in all vessels, independently of the side. Total obstruction of the flow was not observed. Our findings reveal a definitive decrease of blood flow velocity at the vertebrobasilar artery system provoked by rotation of the head in normal humans. This physiological phenomenon is suggested to have an impact on the cerebral blood flow in patients with impaired autoregulation of the cerebral vessels, low volume flow reserve in the contralateral VA or insufficient collateral channels because of normal anatomical variation, especially those patients under general anesthesia or comatose.